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--- address: | University of Victoria\ ate@uvic.ca author: - Alejandro Erickson bibliography: - 'sos.bib' nocite: '[@wc]' title: Sums of squares and negative correlation for spanning forests of series parallel graphs ---

--- abstract: 'At mean-field level the $t$-$J$ model shows a phase diagram with close analogies to the phase diagram of hole doped cuprates. An order parameter associated with the flux or $d$ charge-density wave ($d$-CDW) phase competes and coexists with superconductivity at low doping showing characteristics identified with the observed pseudogap in underdoped cuprates. In addition, in the $d$-CDW state the Fermi surface is reconstructed toward pockets with low spectral weight in the outer part, resembling the arcs observed in angle-resolved photoemission spectroscopy experiments. However, the $d$-CDW requires broken translational symmetry, a fact that is not completely accepted. Including self-energy corrections beyond the mean, field we found that the self-energy can be written as two distinct contributions. One of these (called $\Sigma _{flux}$) dominates at low energy and originates from the scattering between carriers and $d$-CDW fluctuations in proximity to the $d$-CDW instability. The second contribution (called $\Sigma_{R\lambda}$) dominates at large energy and originates from the scattering between charge fluctuations under the constraint of non double occupancy. In this paper it is shown that $\Sigma_{flux}$ is responsible for the origin of low-energy features in the spectral function as a pseudogap and Fermi arcs. The obtained doping and temperature dependence of the pseudogap and Fermi arcs is similar to that observed in experiments. At low energy, $\Sigma_{R \lambda}$ gives an additional contribution to the closure of the pseudogap.' author: - 'Matías Bejas, Guillermo Buzon, Andrés Greco, and Adriana Foussats' title: 'Doping and temperature dependence of the pseudogap and Fermi arcs in cuprates from $d$-CDW short-range fluctuations in the context of the $t$-$J$ model ' --- Introduction ============ The origin of the pseudogap (PG) phase in cuprates is one of the most important and unresolved issues in solid-state physics.[@timusk99] Several experimental techniques are used for studying this phase, and its main characteristics remain unclear. For instance, in the superconducting state, some experiments are consistent with the existence of only one gap while others are in agreement with two order (competing) parameters. Two main scenarios, preformed pairs above $T_c$ and two competing gaps, dispute the explanation of the PG (see Refs. and ). Angle-resolved photoemission spectroscopy (ARPES) is presently a valuable tool for such research.[@damascelli03] Surprisingly, in underdoped cuprates, ARPES shows, in the normal state below a characteristic temperature $T^*$, Fermi arcs[@norman98; @kanigel06; @kanigel07; @shi; @terashima07; @hashimoto10; @yoshida09; @kondo07; @kondo09; @lee07; @tanaka06; @ma08] (FAs) (centered along the zone diagonal) instead of the full Fermi surface (FS) predicted by standard solid-state physics. Despite the general consensus about the existence of FAs, their main characteristics are controversial, even at an experimental level. For instance, while some experiments suggest that the arcs are disconnected,[@norman98; @kanigel06; @kanigel07] others claim that the arcs are associated with pockets.[@Nd; @meng; @jonson1; @jonson2; @jonson3] The number of theoretical studies about the PG and FAs is too large for a complete listing. Among them, early results where the PG was discussed in the framework of the Born approximation and the spin-polaron description for the $t$-$J$ model should be mentioned.[@sherman97] In addition, recent progress on dynamical cluster approximation (DCA) show the presence of a pseudogap [@huscroft01; @macridin06] and Fermi arcs[@parcollet04; @werner09; @lin10] in the two-dimensional Hubbard model. Recently, Norman [*et al.*]{} (Ref. ) have summarized some of the relevant models proposed for discussing ARPES experiments. These models are semiphenomenological or phenomenological, and the proposed Green function $G(k,\omega)$ has the following form: $$\begin{aligned} G^{-1}(k,\omega)=\omega-\epsilon_k + i \Gamma - \Sigma(k,\omega)\end{aligned}$$ where $\Gamma$ is a lifetime broadening, $\epsilon_k$ is the bare electronic dispersion, and $\Sigma(k,\omega)$ is the self-energy, which can be written as $$\begin{aligned} \Sigma(k,\omega) = \frac{\Delta^2_k}{\omega + \xi_k + i\Gamma}\end{aligned}$$ In Eq. (2) the phenomenological pseudogap $\Delta_k$ is assumed to be $d$ wave; $\Delta_k = \frac{\Delta}{2} (\cos k_x - \cos k_y)$. $\xi_k$ is model dependent; for instance: (a) $\xi_k = -\epsilon_{k+Q}$, where $Q=(\pi,\pi)$, in the $d$ charge-density wave ($d$-CDW) model,[@chakravarty01] (b) $\xi_k$ is the nearest-neighbor term of the tight binding dispersion in the model proposed by Yang, Rice, and Zhang (YRZ),[@yang06] and (c) $\xi_k = \epsilon_k$ in the $d$-wave preformed pairs model.[@norman07; @norman98p] Although these models represent different physical situations, the experimental distinction between them is a big challenge. In Ref. it was concluded that $d$-CDW and YRZ models lead to predictions that are not compatible with experiments. For instance, these models lead to FAs whose length is temperature independent and deviates from the underlying FS in contrast to the experiments. Finally, it was also concluded that experiments are better described in the framework of the $d$-wave pairs model. Since the early studies on high-Tc superconductivity, the $t$-$J$ model has been shown to be a basic model for describing the physics of cuprates.[@anderson87] This model, which can be considered a strong-coupling version of the Hubbard model,[@ZR; @Oles] contains (potentially) the main ingredients for describing cuprates, i.e., antiferromagnetism at zero doping, a metallic phase at finite doping, strong tendency to $d$-wave superconductivity and several candidates for the PG phase at low doping. Whether all these phases may be unambiguously associated to those known in cuprates is the big challenge for the $t$-$J$ model. The number of analytical and numerical techniques introduced for studying this model is too large to discuss here. One analytical approach for treating the model is the large-$N$ expansion where the two spin components are extended to $N$ and an expansion in powers of the small parameter $1/N$ is performed. The advantage of this approach is that the results are not perturbative in any model parameter and occur in strong coupling. For performing the large-$N$ expansion, treatments based on the slave boson [@wang92] and Hubbard operators[@zeyher2] were developed. However, evaluating fluctuations above mean field, as required for calculating dynamical self-energy effects, is not straightforward.[@wang92] On the basis of the path-integral representation for Hubbard operators[@foussats04] the large-$N$ approach to the $t$-$J$ model was implemented, yielding previous results[@wang92; @zeyher2] in leading order. At mean-field level ($N=\infty$) the well-known flux phase[@affleck88; @kotliar88; @morse91; @cappelluti99] (FP) instability at low doping was also reobtained.[@foussats04] In the FP a charge-density wave coexists with orbital currents in a staggered pattern and has the same momentum dependence of the superconducting state ($d$ wave), allowing the identification of the FP with the pseudogap. In addition, the FP scenario possesses the main properties to be identified with the phenomenological $d$-CDW proposal.[@chakravarty01] It is important to mention that the relevance of the FP for the physical case $N=2$, for instance, in the form of a phase with strong $d$-wave short-range order, is under dispute. While some exact diagonalization results[@leung00] show the presence of the $d$-CDW phase, DCA (Ref. ) and strong-coupling diagram technique[@sherman08] do not show the static long-range formation of the $d$-CDW state. In spite of this discussion it is important to note that the predicted mean-field phase diagram[@cappelluti99] has close similarities to the phase diagram of hole-doped cuprates where the FP competes and coexists with superconductivity. Importantly, the method developed in Ref. allows us to go beyond the mean field and to compute self-energy renormalizations. Here, following Refs. and , it will be shown that the doping and temperature dependence of the PG and FAs can be discussed after including self-energy effects in proximity to the FP instability, showing that $d$-CDW model is not inconsistent with the notion of arcs. This paper is organized as follows. In Sec. II we summarize the basic formalism. We show that the self-energy can be written in terms of two distinct contributions, $\Sigma_{flux}$ and $\Sigma_{R\lambda}$. The mean-field phase diagram is discussed together with the main characteristics of the self-energy. In Sec. III we describe the origin of the FAs and show that they are triggered by $\Sigma_{flux}$. Section III A discusses the topology of the FAs. Sections III B and III C discuss the temperature and doping dependence of the FAs, respectively. In Sec. III D we present the main characteristics of $\Sigma_{flux}$ at finite temperature. In Sec. IV we discuss the inclussion of $\Sigma_{R\lambda}$. Section V presents discussion and conclusions. Basic framework =============== The large-$N$ mean-field solution of the $t$-$J$ model yields a quasiparticle (QP) dispersion:[@foussats04] $$\begin{aligned} \epsilon_{k} &=& -2\left( t \frac{\delta}{2} + rJ \right) [cos(k_x)+cos(k_y)] \nonumber \\ & & -4 \: t'\: \frac{\delta}{2} \; cos(k_x) \: cos(k_y) - \mu,\end{aligned}$$ where $\delta$ is the doping away from half-filling. $t$, $t'$, and $J$ are hopping between nearest-neighbor, next-nearest-neighbor, and the nearest-neighbor Heisenberg coupling, respectively. The contribution $r$ to the mean-field band and the chemical potential $\mu$ must be obtained self-consistently from $$\begin{aligned} r=\frac{1}{N_s} \sum_{k} cos(k_x) n_F(\epsilon_{k})\end{aligned}$$ and $$\begin{aligned} (1-\delta)=\frac{1}{N_s} \sum_{k} n_F(\epsilon_{k}),\end{aligned}$$ where $n_F$ is the Fermi factor and $N_s$ the number of sites. Equations (3)–(5) define a homogeneous Fermi liquid (HFL) phase that, as discussed in Sec. I, is unstable against a flux phase or $d$-CDW state at low doping. Beyond the mean field the computation of fluctuations in $O(1/N)$ leads to the following expression for the self-energy:[@bejas06] $$\begin{aligned} \label{SigmaIm} {\mathrm{Im}}\Sigma(k,\omega) = -\frac{1}{N_{s}} \sum_{q,a,b} h_{a}(k,q,\omega-\epsilon_{{k-q}}) h_{b}(k,q,\omega-\epsilon_{{k-q}}) \nonumber \\ \times {\mathrm{Im}}[D_{ab}(q,\omega-\epsilon_{{k-q}})] [n_{F}( -\epsilon_{k-q}) +n_{B}(\omega-\epsilon_{{k-q}})] \nonumber \\\end{aligned}$$ where $n_B$ is the Bose factor and the six-component vector $h_{a} (k,q,\nu)$ is $$\begin{aligned} h_{a} (k,q,\nu) &=&\left\{ \frac{}{} \right. \frac{2\epsilon_{k-q}+\nu+2\mu}{2} + J r \left[ \cos\left(k_x-\frac{q_x}{2} \right) \cos \left( \frac{q_x}{2} \right) + \cos\left(k_y-\frac{q_y}{2} \right) \cos \left( \frac{q_y}{2} \right) \right] \; ; 1 \; ;\nonumber \\ && -J r \; \cos \left( k_{x}-\frac{q_{x}}{2} \right) \; ; -J r \; \cos \left( k_{y}-\frac{q_{y}}{2} \right) \; ; \; J r \; \sin \left( k_{x}-\frac{q_{x}}{2} \right) \; ; \; J r \; \sin \left( k_{y}-\frac{q_{y}}{2} \right) \left. \frac{}{} \right\}.\end{aligned}$$ The physical information contained in the vector $h_{a} (k,q,\nu)$ is as follows. The first component (called $\delta R$) is mainly dominated by the usual charge channel, the second component (called $\delta \lambda$) corresponds to the nondouble-occupancy constraint, and the last four components are driven by $J$. For the case $J=0$ the vector $h_a$ reduces to a two-component vector. In Eq. (6) $D_{ab}$ is a $6\times6$ matrix that contains contributions from the six different channels and their mixing. $$D^{-1}_{ab}(q,i\omega_{n})=[D^{(0)}_{ab}(q,i\omega_{n})]^{-1}- \Pi_{ab}(q,i\omega_{n})$$ where $$\label{eq:D0-1} D^{(0)}_{ab}(q,i\omega_{n}) = \left( \begin{array}{cccccc} \delta^2/2 (V-J/2) [\cos(q_x)+\cos(q_y)] & \delta/2 & 0 & 0 & 0 & 0 \\ \delta/2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & J\;r^2 & 0 & 0 & 0 \\ 0 & 0 & 0 & J\;r^2 & 0 & 0 \\ 0 & 0 & 0 & 0 & J\;r^2 & 0 \\ 0 & 0 & 0 & 0 & 0 & J\;r^2 \\ \end{array} \right)^{-1}$$ and $$\Pi_{ab}(q, i\omega_{n}) =- \frac{1}{N_s} \; \sum_{k}\;h_{a}(k,q,\epsilon_k-\epsilon_{k-q}) \;h_{b}(k,q,\epsilon_k-\epsilon_{k-q}) \; g(k,q,i\omega_{n}) - \delta_{a}^R\delta_{b}^R\;\frac{1}{N_s} \sum_{k} \frac{\epsilon_{k-q} - \epsilon_{k}}{2} \; n_{F}(\epsilon_{k}) \; ,$$ with $$\begin{aligned} \label{eq:gPi} g(k,q,i\omega_{n}) = \frac{[n_{F}(\epsilon_{k - q}) - n_{F}(\epsilon_{k})]} {i\omega_{n} + \epsilon_{k - q} - \epsilon_{k} } \; ,\end{aligned}$$ where $i\omega_n$ is the bosonic Matsubara frequency. Hereafter, $t'/t=-0.35$ and $J/t=0.3$, which are suitable parameters for cuprates, are used. The lattice constant $a$ of the square lattice and $t$ are considered to be a length unit and energy unit, respectively. In Eq.(\[eq:D0-1\]), $V$ is the nearest-neighbor Coulomb repulsion. The role of $V$ is to exclude phase separation. We choose $V=2J$. The instability of the mean-field solution occurs when $det[ D_{ab}(q,i\omega_n=0) ]=0$ (Ref. ). For the present parameters, at $T=0$, the instability takes place at $\delta=\delta_c\sim 0.23$ for $q\sim (\pi,\pi)$. It is important to note that $D_{ab}$ enters explicitly in the self-energy expression beyond the mean field \[Eq.(6)\]; thus, $\Sigma$ probes the proximity to the instability at low $\omega$ and for momenta $k-q$ near the FS. Since $D_{ab}$ contains contributions from six different channels and their mixing, it is important to isolate the most relevant channel dominating $\Sigma$ near the instability. The eigenvector with zero eigenvalue of $D_{ab}$ takes the form $\sim (0,0,0,0,-1,1)$ which is the eigenvector associated to the FP instability.[@foussats04] Projecting $\Sigma(k,\omega)$ on the FP eigenvector the following self-energy contribution is obtained[@greco08; @greco09] $$\begin{aligned} \label{eq:SigmaIm0flux} {\mathrm{Im}} \, \Sigma_{flux}(k,\omega) &=& -\frac{1}{N_{s}} \sum_{q} \gamma^2(q,k) {\mathrm{Im}} \chi_{flux}(q,\omega-\epsilon_{{k-q}}) \nonumber \\ &\times& \left[n_{F}(-\epsilon_{{k-q}}) + n_{B}(\omega-\epsilon_{{k-q}})\right]\end{aligned}$$ which shows the explicit contribution of the flux susceptibility $$\begin{aligned} \chi_{flux}(q,\omega)= [2J\;r^2-\Pi(q,\omega)]^{-1}\end{aligned}$$ where $\Pi(q,\omega)$ is an electronic polarizability $$\begin{aligned} \Pi(q, i\omega_n) = - \frac{1}{N_{s}}\; \sum_{k}\; \gamma^2(q,k) \frac{[n_{F}(\epsilon_{k + q}) - n_{F}(\epsilon_{k})]} {\epsilon_{k + q} - \epsilon_{k}-i \omega_n}\; \nonumber \\ \end{aligned}$$ calculated with a form factor $\gamma(q,k)=2 r [\sin(k_x-q_x/2)-\sin(k_y-q_y/2)]$. Since the instability takes place at $(\pi,\pi)$ the form factor $\gamma(q,k)$ transforms into $\sim [\cos(k_x)-\cos(k_y)]$, which indicates the $d$-wave character of the FP. Thus, the mode associated with the FP instability plays an important role in $\Sigma(k,\omega)$ at low doping near $\delta_c$. ![(Color online) Phase diagram of the $t$-$J$ model in the leading order of $1/N$ expansion where superconductivity was discarded. The instability (solid line), marked by $T_{FP}$, separates the homogeneous Fermi liquid state from the flux or $d$-CDW state and terminates at the quantum critical point at $\delta_{c} \sim 0.23$ at $T=0$. The inset shows the imaginary part of the flux susceptibility vs. $\omega$ for $\delta=0.10$ and for different temperatures \[$q=(\pi,\pi)$ is the momentum where the instability occurs\]. Approaching $T_{FP}$ from above the flux mode becomes better defined, accumulates weight, and softens. At $T=T_{FP}$ the flux mode reaches $\omega=0$, freezing the $d$-CDW phase. This flux mode contributes to the self-energy leading to a pseudogap and Fermi arcs features as discussed in text.](./figure1.eps "fig:"){width="8cm"} \[fig:intkTdp\] In Fig. 1, disregarding superconductivity, the solid line shows the temperature $T_{FP}$, which indicates the onset of FP instability, i.e., when the static ($i\omega_n=0$) flux susceptibility \[Eq.(13)\] diverges. At $T=0$ a phase transition occurs at the quantum critical point (QCP) placed at the critical doping $\delta_c$. At $T_{FP}$ a flux-mode \[${\rm Im} \chi_{flux}(q=(\pi,\pi),\omega)$\] reaches $\omega=0$, freezing the FP. In the inset in Fig. 1, we have plotted ${\rm Im} \chi_{flux}(q=(\pi,\pi),\omega)$ for $\delta=0.10$ for several temperatures approaching $T_{FP}/t \sim 0.018$, showing that when $T \rightarrow T_{FP}$, a low energy $d$-wave flux mode becomes soft and accumulates large spectral weight. We have used the small broadening $\eta/t=0.01$ in the analytic continuation ($i\omega_n \rightarrow \omega + i\eta$). Figure 2(a) shows $\rm-Im \, \Sigma_{flux}(k,\omega)$ at $T=0$ for several dopings at the antinodal Fermi wave vector $k_F^{AN}$. At large doping $\delta=0.40$, $\rm-Im \Sigma_{flux}$ is weak and behaves as $\sim \omega^2$ at low energy. Approaching $\delta_c$ ($\delta=0.26$ and $\delta=0.24$), $\rm-Im \Sigma_{flux}$ increases, and the behavior at low energy is nearly linear in $\omega$ and develops structures at low energy $\omega/t \sim 0.1-0.2$. Results for the nodal Fermi vector $k_F^N$ are not shown because they are nearly negligible due to the $d$-wave character of the flux instability. Inset (i) in Fig. 2(b) shows the QP weight $Z$ at $k_F^{AN}$ ($Z_{flux}^{AN}$) and at $k_F^N$ ($Z_{flux}^{N}$). While $Z_{flux}^{AN}$ is strongly doping dependent and tends to zero approaching $\delta_c$, $Z_{flux}^N \sim 1$ shows that $\Sigma_{flux}$ is also strongly anisotropic on the FS. $\Sigma_{flux}$ is written in terms of the flux or $d$-CDW susceptibility $\chi_{flux}(q,\omega)$, which shows explicitly the role played by the soft flux mode with momentum $(\pi,\pi)$ (see inset in Fig. 1). Therefore, near the antinode the QP on the FS is strongly distorted, leading to FA effects, as shown in Sec. III In addition, it is easy to check that the most important $J$ contribution to $\Sigma(k,\omega)$ enters only via $\Sigma_{flux}$. {width="8cm"} \[fig:imag\] In $\Sigma(k,\omega)$, there is another contribution that is nearly independent of $J$. This contribution belongs to the usual charge $\delta R$ and nondouble occupancy $\delta \lambda$ channels, (the first and second components of $h_a$ \[Eq.(7)\]) and can be written as[@greco08; @foussats08] $$\begin{aligned} \label{eq:SigmaIm0Rl} {\mathrm{Im}} \, \Sigma_{R \lambda}(k,\omega)= &-& \frac{1}{ N_{s}} \sum_{q} \left\{ \Omega^{2} \; {\mathrm{Im}} [D_{RR}(q,\omega-\epsilon_{{k-q}})] \right. \nonumber \\ &+& \;2\;\Omega \; {\mathrm{Im}}[D_{\lambda R}(q,\omega-\epsilon_{{k-q}})] \nonumber \\ &+& \left. {\mathrm{Im}} [D_{\lambda \lambda}(q,\omega-\epsilon_{{k-q}})] \right\} \nonumber \\ &\times& \left[ n_{F}(-\epsilon_{{k-q}}) + n_{B}(\omega-\epsilon_{{k-q}}) \right],\end{aligned}$$ where $\Omega=\frac{1}{2}(\epsilon_{{k-q}} + \omega + 2\mu)$. Figure 2(b) shows $\rm-Im \, \Sigma_{R \lambda}(k,\omega)$ at $T=0$ for $k_F^{AN}$ and for the same dopings as in Fig. 2(a). For $k_F^N$, $\rm Im \, \Sigma_{R \lambda}(k,\omega)$ (not shown) is nearly indistinguishable from results at $k_F^{AN}$, showing that $\Sigma_{R \lambda}$ is rather isotropic on the FS. In addition, $\Sigma_{R\lambda}$ behaves as $\sim \omega^2$ at low $\omega$, and contrary to $\Sigma_{flux}$, there is no energy scale at low energy. In inset (i) we show $Z_{R\lambda}^{AN}$ and $Z_{R\lambda}^{N}$. These results show that the doping dependence of $\Sigma_{R\lambda}$ is weaker than $\Sigma_{flux}$. Note that $Z_{R\lambda} \rightarrow 0$ when $\delta \rightarrow 0$. It is important to note that there are no structures in $\rm Im \, \Sigma_{R \lambda}(k,\omega)$ at low energy, and the main contribution appears at large energies of the order of $t$ \[see inset (ii)\]. Note also the strong asymmetry shown by $\Sigma_{R\lambda}$ that arises from nondouble-occupancy effects.[@foussats08] In summary, (a) $\Sigma_{flux}$ is strongly $J$ and doping dependent, is strongly anisotropic on the FS, and contributes at low energy, and (b) $\Sigma_{R \lambda}$ is nearly $J$ and doping independent, is strongly isotropic on the FS, and contributes at large energy. Thus, $\Sigma(k,\omega)$ can be written as the addition of two well-decoupled channels. $$\begin{aligned} \label{eq:SigmaIm0} {\rm Im} \, \Sigma(k,\omega) = {\rm Im} \, \Sigma_{R \lambda}(k,\omega) + {\rm Im} \, \Sigma_{flux}(k,\omega)\end{aligned}$$ Using the Kramers-Kronig relations, $\mathrm{Re} \Sigma(k,\omega)$ can be determined from $\mathrm{Im} \Sigma(k,\omega)$ and the spectral function $A(k,\omega)$, computed as usual: $$\begin{aligned} \label{A} A(k,\omega)= -\frac{1}{\pi}\frac{{\rm Im}\Sigma(k,\omega)} {[\omega- \epsilon_{k} - {\rm Re}\Sigma(k,\omega)]^2 + {\rm Im}\Sigma(k,\omega)^2} \nonumber \\\end{aligned}$$ Before concluding this section it is important to note that at mean-field level the $d$-CDW picture leads below $T_{FP}$, where the translational symmetry is broken, to four hole pockets with low spectral weight in the outer side resembling the FAs.[@chakravarty03] However, as discussed in Ref. , this picture has conflicting points when compared with some ARPES data. We will show in Sec.III that the inclusion of self-energy effects in proximity to the FP instability provides a possible scenario for describing several ARPES features. Therefore, although at mean-field level the instability to the static $d$-CDW occurs below $T_{FP}$, the PG and FA formation do not require the long-range $d$-CDW state, but they do require the enhancement of fluctuations due to proximity effects. Thus, we are always located in a homogeneous state with the presence of $d$-CDW fluctuations. $\Sigma_{flux}$ and Fermi arcs ============================== Topology of Fermi arcs ---------------------- Since $\Sigma_{flux}$ dominates at low energy we study here the spectral functions calculated with this contribution. In Sec. IV we show that the inclusion of $\Sigma_{R\lambda}$ does not change the main conclusion obtained in this section. {width="8cm"} \[fig:int\] Figure 3(a) shows for $\delta=0.10$ and $T/t=0.025$ (above but close to $T_{FP}$) the spectral function intensity at zero energy vs $k_x,k_y$. A well-defined FA is obtained. Similar to the experiment[@norman07] \[Fig. 3(b)\], the end of the arc does not turn away from the underlying FS, and there is no strong suppression of the intensity at the hot spots. In Fig. 4(a) the intensity of the spectral function on the FS is plotted as a function of the FS angle $\phi$ \[defined in Fig. 3(b)\] from the antinode ($\phi=0\circ$) to the node ($\phi=45\circ$). As in the experiment[@norman07] \[Fig. 4(b)\], the intensity monotonically decreases approaching the antinode but remains finite. ![(Color online) (a) Intensity of the spectral function at the FS vs the Fermi surface angle $\phi$ from the antinode ($\phi=0\circ$) to the node ($\phi=45\circ$) for $\delta=0.10$ and $T/t=0.025$. (b) The same as (a) but taken from the experimental results of Ref. for comparison. []{data-label="fig:angInt"}](./figure4.eps){width="8cm"} In Fig. 5(a) energy distribution curves (EDC) on the underlying FS are plotted. In agreement with the experiment[@norman07] \[Fig. 5(b)\], near the node, there are well-defined QP peaks; approaching the antinode, the spectral functions lose intensity at $\omega=0$, become broad, and develop a PG-like feature. The presence of a PG-like feature near the antinode means that the arc plotted in Fig. 3(a) is not simply related to a decrease in the intensity from the node to the antinode but that the FS near the antinode is gapped. Note that the present PG-like feature is not related to a true gap as in other models. It is developed dynamically in proximity to the $d$-CDW instability in the presence of strong and short-range $d$-CDW fluctuations. In summary, the effects described in Figs. 3–5 arise from self-energy effects due to the coupling between QPs and the soft flux mode (see inset in Fig. 1) in proximity to the FP instability (solid line in Fig. 1). Since the flux mode occurs mainly with momentum $(\pi,\pi)$, the QP near the antinode is distorted, leading to FAs. Note also that since $d$-CDW fluctuations are of short-range character, in the present picture, the translational symmetry is not broken. {width="8cm"} \[fig:EDC\] Temperature dependence of the Fermi arcs ---------------------------------------- The temperature dependence of the length of the FAs is puzzling. In spite of different views and interpretations most reports agree on the fact that the observed FAs depend on temperature. While there are reports that claim that the length of FAs collapse to one isolated point[@kanigel06; @kanigel07] (nodal metal) at $T=0$, others suggest a less-strong temperature dependence.[@jonson3; @storey08] In models discussed in Ref. the temperature dependence of the length of the arcs arises after assuming a temperature dependence for $\Delta_k$ or for the lifetime broadening $\Gamma$. In this subsection it is shown that the temperature dependence of the FAs emerges, in the framework of the present approach, without adjustable parameters, showing that the temperature dependence of the length of the arcs is entangled to their origin. Figure 6 shows the plot, for $\delta=0.10$, of FA for different temperatures. Clearly, the length of the arcs decreases when temperature decreases. We note that the temperature dependence of the arcs seems to be weaker than in some experiments[@kanigel06] but closer to others[@jonson3; @storey08] (this point is further discussed in Sec. IV). Beyond a quantitative comparison, the because no phenomenological parameter is assumed to be temperature dependent in the present model, the results can be considered satisfactory. Figure 7 plots the spectral function intensity on the FS for several temperatures normalized to the intensity at $k_F^N$. Consistent with the picture in Fig. 6, with decreasing temperature, the intensity is more concentrated around the node. {width="8cm"} \[fig:int21\] ![(Color online) (a) Intensity of the spectral function (normalized to the intensity at the node ) at the FS vs. $\phi$ for $\delta=0.10$ and for several temperatures. When the temperature decreases toward $T_{FP}$, the intensity is more and more concentrated around the node. []{data-label="fig:angIntNorm"}](./figure7.eps){width="7.5cm"} ![(Color online) EDC at $k_F^{AN}$ for $\delta=0.10$ and for the same temperatures as in Fig. 7. As in experiments when temperature increases, the PG feature fades out. Although the leading edge of the pseudogap partially closes, a filling is also observed, as in the experiments. []{data-label="fig:AkkT"}](./figure8.eps){width="7.5cm"} Finally, Fig. 8 shows EDC at $k_F^{AN}$ for the same temperatures as in Fig. 7. Although the PG feature partially closes[@hashimoto10] with increasing temperature, a filling is also observed.[@norman98; @kanigel06; @kanigel07] This feature is in agreement with experiments and in contrast to results from mean-field calculations where only a closure is expected. It is worth mentioning that while the arcs discussed here are dynamically generated, they necessarily occur at finite temperature. The present approach should be distinguished from other ones[@kampf; @lorenzana; @choi] where a phenomenological fitted susceptibility without explicit temperature dependence is proposed. Doping dependence of the Fermi arcs ----------------------------------- It is well known that with increasing doping, the PG feature closes[@kim98; @kaminski05] and, simultaneously, the length of the arcs increases.[@kanigel07] For describing this behavior, models discussed in Ref. need to assume a phenomenological doping dependence for the PG. In this subsection we will show that arcs whose length increases with increasing doping can be naturally discussed in the present context. In Fig. 9 the FA is shown for several dopings and for a fixed temperature $T/t=0.025$. With increasing doping, the length of the arcs increases, in agreement with experiments. Figure 10 shows EDC at $k_F^{AN}$ for several dopings. When doping increases, the PG-like feature closes and, simultaneously, the intensity increases at $\omega=0$. ![(Color online) Fermi arc for $T/t=0.025$ and for several dopings: (a) $\delta=0.05$, (b) $\delta=0.10$, and (c) $\delta=0.15$. As in experiments when doping increases toward overdoped, the length of the arc increases. []{data-label="fig:intdp"}](./figure9.eps){width="8cm"} ![(Color online) EDC at $k_F^{AN}$ for $T/t=0.025$ and for the same dopings as in Fig. 9. Like in experiments when doping increases, the PG feature washes out in a way consistent with the increment of the length of the arc reported in Fig. 9. []{data-label="fig:Akdp"}](./figure10.eps){width="8cm"} In summary, in Secs. III B and III C it is shown that with increasing doping and temperature the PG-like feature and the FA fade out like in the experiments. The origin for this behavior is easy to understand: By increasing doping and temperature we leave out the instability line $T_{FP}$. Then, the flux mode is less efficient, self-energy effects become weaker, and the long FS is smoothly recovered. It is important to note that from our approach $T^*$ must be distinguished from a true phase transition. Here at $T^* > T_{FP}$, where the PG features vanish, there is not a phase transition but a smooth crossover.[@tallon01] Finally, note that if $t=0.4 eV$, the energy scale for the pseudogap and temperature is of the order of the experiment. Main characteristics of $\Sigma_{flux}$ --------------------------------------- For a complete discussion about the origin of the arcs we have investigated the main characteristics of $\Sigma_{flux}$. Figure 11 shows $\rm-Im \Sigma_{flux}$ at $k_F^N$ and $k_F^{AN}$ for $T/t=0.025$ and $\delta=0.10$ \[Fig. 11(a)\], $\delta=0.25$ \[Fig. 11(b)\], and $\delta=0.40$ \[Fig. 11(c)\]. At $k_F^N$, $\rm-Im \Sigma_{flux}$ is smaller than for $k_F^{AN}$, leading to a well-defined and nearly no renormalized QP peak in the nodal direction \[Fig. 5(a)\]. However, the behavior at $k_F^{AN}$ is very different, especially at low doping. Instead of a minimum at $\omega = 0$, $\rm-Im \Sigma_{flux}$ shows a maximum clearly observed for $\delta=0.10$ \[Fig. 11(a)\]. This behavior, which is in contrast to the expected results from the usual many-body physics,[@katanin03; @dellanna06] is the main reason for the PG and FA formation. With increasing doping, the maximum at $\omega \sim 0$ washes out, and for large doping, $\rm-Im \Sigma_{flux}$ develops the expected minimum at $\omega = 0$. \[See results for $\delta=0.40$ in Fig. 11(c)\]. {width="8cm"} \[fig:int100\] Inclusion of $\Sigma_{R\lambda}$ ================================ {width="8.5cm"} \[fig:llena\] It was shown (Fig. 2) that the energy scale in $\Sigma_{R\lambda}$ is much larger ($\sim t$) than the energy scale in $\Sigma_{flux}$. Although this fact implies (as shown in Sec. III) that $\Sigma_{flux}$ is the relevant contribution for triggering the low-energy PG features, in this section, for completeness, we discuss the role of $\Sigma_{R\lambda}$ in the spectral functions. It was discussed in Sec. IIIB that the PG closes and fills smoothly with increasing temperature (see Fig. 8). In this section we show that the only role of including $\Sigma_{R \lambda}$ is to improve the vanishing of the PG.[@add] Figure 12(a) shows EDC for $\delta=0.10$ for several temperatures at $k_F^{AN}$. This figure shows that with increasing temperature the PG fills and a peak at $\omega=0$ emerges at $T/t \sim 0.035$. Note that in Fig. 8, where only $\Sigma_{flux}$ was considered, even at the high temperature $T/t =0.1$ the maximum of $A(k,\omega)$ is not yet fully formed at $\omega=0$. In Fig. 12(c) we have reproduced, for comparison, the experimental results,[@kanigel06] showing qualitative agreement between theory and experiment. In Fig. 12(b) we plot, for $T/t=0.035$, $A(k,\omega)$ for $k_F^{AN}$ (solid line) and $k_F^N$ (dashed line). Althought the entire FS is ungapped at this temperature, the QP are better defined near the node, as in the experiment.[@kim98] {width="8cm"} \[fig:intRL\] Figure 13 shows the spectral function intensity at $\omega=0$ vs $k_x$,$k_y$ for $\delta=0.10$ and for the same temperatures as in Fig. 6. At low temperatures a FA is obtained, and its length increases with increasing $T$. Note that while the FS is expected to be gapped near the antinode for $T/t=0.025$ (dot-dashed line in Fig. 12), for $T/t=0.050$ the full FS is ungapped. In summary, $\Sigma_{R\lambda}$ does not modify the main conclusion obtained in Sec. III. We have shown that its inclusion enhances the pseudogap closing and filling, and contributes to a faster reconstruction of the entire FS with increasing temperature. Discussion and conclusion ========================= The large-$N$ approach in the $t$-$J$ model leads, beyond the mean-field level, to two distinct dynamical self-energy contributions, namely, $\Sigma_{R\lambda}$ and $\Sigma_{flux}$. In this paper we have analyzed the role of these contributions in ARPES. The main characteristics of $\Sigma_{flux}$ are the following. $\Sigma_{flux}$ is strongly anisotropic on the FS, strongly doping dependent, and dominated by $J$ (if $J=0$ $\Sigma_{flux}$ is negligible), and it contributes at low energy. Thus, $\Sigma_{flux}$ is the relevant contribution for describing the Fermi arcs and pseudogap features. The fact that $\Sigma_{flux}$ is mainly dominated by $J$ may be understood as follows. At mean-field level the $t$-$J$ model shows (and only for finite $J$) the flux or $d$-CDW phase below a temperature $T_{FP}$. $T_{FP}$ decreases with increasing doping, approaching the QCP at $\delta_c$ and $T=0$. In the proximity of $T_{FP}$, $d$-CDW fluctuations enter $\Sigma_{flux}$ \[Eq.(12)\]. Since $d$-CDW fluctuations favor scattering between electrons with momentum transfer $q \sim (\pi,\pi)$, the FS near the antinode is gaped, leading to Fermi arcs being dynamically generated. With increasing doping and temperature beyond $\delta_c$ and $T_{FP}$, respectively, $d$-CDW fluctuations become weak, and the Fermi arcs and the pseudogap wash out, in agreement with experiments. It is important to note that the present picture does not require any phenomenological parametrization for the pseudogap or the lifetime broadening and their temperature and doping dependence. It is only necessary to be in the proximity of the $d$-CDW instability or in a situation with strong short-range fluctuations. In other words, under the present approach Fermi arcs originate dynamically due to the interaction between carriers and short-range and short-living $d$-CDW fluctuations, implying that long-range order is not broken. The present picture has similarities with some phenomenological approaches[@choi] where the pseudogap and Fermi arcs are described in a scenario where fermions interact with bosonic fluctuations of some special order. Importantly, our description offers a microscopic derivation from the $t$-$J$ model, and, as a corollary, the fluctuating spectrum is obtained with no assumptions of any fitted phenomenological parameter, such as coupling, correlation length, or bosonic frequency. Note that near the flux instability the flux mode (inset in Fig. 1) is overdamped and intrinsically temperature dependent and can not be easily considered as an Einstein mode as in other approaches.[@kampf; @lorenzana; @choi] A recent ARPES experiment[@hashimoto10] suggests a similar scenario to that presented here, i.e., density wave fluctuations without long-range order. As in that experiment, in our theory, the existence (and persistence with decreasing temperature) of broad and gapped spectral features near the antinode means that we are not sitting in a phase with long-range order. It is worth mentioning that under the present approach, below the mean-field temperature $T_{FP}$ the long-range $d$-CDW order occurs; that is, a true gap is formed, and sharp spectral features are expected with the corresponding reconstruction of the FS in the form of pockets.[@greco09] From an experimental point of view the existence of long-range order in underdoped cuprates is controversial and is tied to the following facts. (a) Some ARPES experiments show well-defined spectral peaks near the antinode in the superconducting state, while others show broad structures (see Ref. and references therein). (b) While some experiments support the existence of a second order parameter, distinct from but coexisting (and competing) with superconductivity,[@terashima07; @hashimoto10; @yoshida09; @kondo07; @kondo09; @lee07; @tanaka06; @ma08] others claim to observe only one gap feature.[@norman98; @kanigel06; @kanigel07; @shi] (c) While quantum oscillations[@QOs] and some ARPES experiments show a reconstruction of the FS in the form of pockets,[@Nd; @meng; @jonson1; @jonson2; @jonson3] other reports show only arcs.[@kanigel06; @kanigel07; @norman07] Although it is not our aim to solve these puzzles (which requires more theoretical and experimental work), we have shown that several aspects related to the Fermi arc phenomenology can be explained by $d$-CDW proximity effects, showing that this picture is not necessarily inconsistent with the notion of arcs. The characteristics of $\Sigma_{R\lambda}$ are very different from those of $\Sigma_{flux}$. $\Sigma_{R\lambda}$ is dominated by the usual charge channel and (nearly) independent of $J$. Thus, this is the relevant contribution for the $J=0$ case. In addition, it is strongly asymmetric in $\omega$ around the FS due to nondouble-occupancy effects, rather isotropic on the FS, and rather constant as a function of doping (for low to intermediate doping).[@foussats08] Finally, it contributes at large energy of the order of $t$. Although $\Sigma_{R\lambda}$ is not responsible for the pseudogap and Fermi arc formation, it gives an additional contribution to the temperature vanishing of the pseudogap. $\Sigma_{R\lambda}$ and $\Sigma_{flux}$ may also play a role in describing other experiments in cuprates. (a) Since $\Sigma_{R\lambda}$ shows high- energy contributions, it leads, in the spectral functions, to incoherent structures at high binding energy, which offers a possible explanation[@foussats08; @greco07] for the high-energy features or waterfall effects observed in cuprates.[@xie07; @meevasana07; @graf07; @zhang08] Other theoretical[@chinos; @zemljic] and experimental[@zhang08] reports show a similar conclusion. (b) The existence of two self-energy contributions is also consistent with recent angle-dependent magnetoresistance (ADMR) experiments.[@abdel06; @abdel07; @french09] These experiments show two different inelastic scattering rates with similar characteristics to the self-energy behavior discussed here, i.e., a strongly-doping-dependent and anisotropic scattering rate on the FS and another one that is weakly doping dependent and isotropic on the FS. Recently, ADMR experiments were discussed in the context of the present approach.[@buzon10] Here we want to comment on the recent progress on DCA. As discussed in Sec. I DCA shows the presence of a pseudogap[@huscroft01; @macridin06] and Fermi arcs.[@parcollet04; @werner09; @lin10] We wish to mention here the similarities between our results and those in DCA. For instance, the pole feature at $\omega\sim 0$ and near the antinode that occurs in $Im \Sigma$ (Fig. 11), which diminishes with increasing temperature and doping, is in remarkable agreement with similar results discussed in Ref. . This behavior for the self-energy leads also to a similar doping and temperature dependence for the PG and FAs. Note that in Ref. the temperature filling of the PG as discussed in the present paperwas also obtained. We note again that our results do not require the static long-range order of the $d$-CDW. What is needed is the enhancement of the $d$-CDW susceptibility due to fluctuations, as can be seen in the inset of Fig. 1. Interestingly, although the static $d$-CDW state was not found in Ref. , an enhancement of the $d$-CDW susceptibility was obtained. Finally, it is worth mentioning that although the origin of the PG and FAs is presumably of antiferromagnetic nature,[@huscroft01; @macridin06] a recent report[@lin10] is not conclusive about this affirmation. It is the aim of the present paper to show that $d$-CDW fluctuations may contribute to the PG and FAs formation. Although of one could certainly suppose that the large-$N$ is a particular approximation and some results may depend on its details, we think that our theory contains features that can be expected, qualitatively, in cuprates and in the $t$-$J$ model. Since the low-energy pseudogap feature increases with decreasing doping, it is reasonable to think that the pseudogap is associated with the same energy scale as the antiferromagnetism, i.e., $J$. This fact is contained in $\Sigma_{flux}$. On the other hand, there is a larger energy scale, the hopping $t$, which, together with nondouble-occupancy effects, enters through $\Sigma_{R\lambda}$. Finally, it is worth mentioning that besides $d$-CDW, other instabilities like stripes,[@stripes] antiferromagnetism,[@AF] and Pomeranchuk[@pomeranchuk] have been proposed to exist at low doping in cuprates. Thus, it is important to perform similar calculations for those cases and compare different predictions. [**Acknowledgments**]{} The authors thank H. Parent for suggestions on the presentation of the paper. A.G. thanks R.-H. He, W. Metzner, A. Muramatsu, Y. Yamase, and R. Zeyher for valuable discussions and the Max Planck Institute (Stuttgart) and the University of Stuttgart for hospitality. 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--- abstract: 'We present a [*nonequilibrium nonperturbative*]{} field theory for the Kondo effect in strongly interacting quantum dots at finite temperatures. Unifying the slave-boson representation with the Keldysh field integral an effective Keldysh action is derived and explored in the vicinity of the zero slave-bosonic field configuration. The theory properly reflects the essential features of the Kondo physics and at the same time significantly simplifies a field-theoretic treatment of the phenomenon, avoiding complicated saddle point analysis or $1/N$ expansions, used so far. Importantly, our theory admits a [*closed analytical*]{} solution which explains the mechanism of the Kondo effect in terms of an interplay between the real and imaginary parts of the slave-bosonic self-energy. It thus provides a convenient nonperturbative building block, playing the role of a “free propagator”, for more advanced theories. We finally demonstrate that already this simplest possible field theory is able to correctly reproduce experimental data on the Kondo peak observed in the differential conductance, correctly predicts the Kondo temperature and, within its applicability range, has the same universal temperature dependence of the conductance as the one obtained in numerical renormalization group calculations.' author: - Sergey Smirnov - Milena Grifoni title: 'Slave-boson Keldysh field theory for the Kondo effect in quantum dots' --- Introduction {#intro} ============ Experimentally discovered almost eighty years ago [@de_Haas_1934] and qualitatively explained thirty years after [@Kondo_1964], the Kondo effect, a minimum in the temperature dependence of the resistance in magnetic alloys, was later anew brought into the world and garbed with physics of quantum dots (QD) [@Ralph_1994; @Goldhaber-Gordon_1998]. That time this complicated phenomenon was supplemented by physics of nonequilibrium due to the dot coupling to contacts biased by an external voltage. In this setup the Kondo effect, a nonperturbative phenomenon induced by both the electron-electron interactions and the QD-contacts coupling, appears as a sharp many-particle resonance in the tunneling density of states (TDOS) at the Fermi energy. An immediate consequence of this resonance is the zero-bias maximum observed experimentally in the QD differential conductance at low temperatures. Theoretical predictions [@Glazman_1988; @Meir_1993; @Wingreen_1994] of this behavior made before the actual experiments use the single-impurity Anderson model (SIAM) [@Anderson_1961] with the on-dot interaction $U$. Those early works utilized either a quasi-particle transformation [@Glazman_1988] to analytically predict resonant transmission through QDs for arbitrary $U$ or the noncrossing approximation (NCA), equations of motion and perturbation theory [@Meir_1993; @Wingreen_1994; @Hettler_1995; @Sivan_1996; @Entin_2005] to numerically and analytically describe several important stationary and nonstationary features of the Kondo effect in strongly interacting QDs modeled by the infinite-$U$ SIAM, in particular, in a slave-boson representation [@Coleman_1984; @Coleman_1987; @Hewson_1997]. Later, as an alternative to the slave-boson representation, diagrammatic expansions within the reduced density matrix formalism [@Koenig_1998] were applied to SIAM for the infinite-$U$ case. Extracting a certain infinite subset of diagrams, an analytical nonperturbative expression for the TDOS was obtained at finite temperatures. Here the absence of the double occupancy played a crucial role in identifying the relevant diagrams. ![\[figure\_1\] (Color online) Classes of the slave-bosonic theories for the Kondo effect in QDs. Their range of quantitative reliability with respect to the Kondo temperature $T_\text{K}$ ($T_\text{K}/\Gamma\thicksim\exp[-2\pi(\mu_0-\epsilon_\text{d})/\Gamma]$, where $\Gamma$ is the total QD-contacts coupling strength, which is twice that of Ref. , $\mu_0$ the QD chemical potential, $\epsilon_\text{d}$ the single-particle energy level of the QD) is given by their horizontal location. The vertical location of a class shows its type, numerical, analytical or both. As to our field theory, what the blue class shows is what one generally expects from nonperturbative field theories in the vicinity of the zero slave-bosonic field configuration.](figure_1){width="7.6"} For small, intermediate and large $U$ the Kondo problem in QDs was widely explored numerically [@Gezzi_2007; @Anders_2008; @Heidrich_2009; @Eckel_2010; @Muehlbacher_2011]. In the present paper we develop an analytical nonequilibrium real-time field theory of the Kondo effect in strongly interacting QDs at finite temperatures using the infinite-$U$ SIAM. The field-theoretic approach is based on the slave-boson representation and the Keldysh field integral [@Kamenev_1999; @Altland_2010]. Importantly, our theory is nonperturbative in both the electron-electron interaction and QD-contacts coupling. The necessity in such a theory is obvious from its advantages. Firstly, it is a field theory and, thus, it has a clear systematic generalization from the present relatively simple basic model to more involved setups such as the ones with ferromagnetic contacts, superconducting contacts, finite $U$ systems, etc. This is especially important since there is a limited number of analytical theories nonperturbative in both the electron-electron interaction and QD-contacts coupling and which at the same time have a straightforward generalization scheme. Secondly, it is an analytical theory and, thus, it will help to reveal a relevant physical picture behind new and more complicated physical systems as, [*e.g.*]{}, those mentioned above. This is definitely an advantage over numerical methods such as, [*e.g.*]{}, NCA, which provide good quantitative description but hide the essence of the physics, giving only an indirect access to it. In Fig. \[figure\_1\] we show a comparative layout of the slave-bosonic theories for the Kondo effect in QDs. A basic nonperturbative analytical slave-bosonic field theory of the Kondo effect in QDs within the range from below $T_\text{K}$ to higher temperatures, [*i.e.*]{}, within the most relevant experimental temperature range (blue (uppermost) class in Fig. \[figure\_1\]), is formulated in this work. So far, using the slave-boson method in the context of QDs, the results were obtained nonperturbatively numerically [@Meir_1993; @Wingreen_1994; @Hettler_1995; @Aguado_2000] below and/or above $T_\text{K}$, perturbatively semi-analytically [@Sivan_1996] above $T_\text{K}$ and nonperturbatively analytically [@Ratiani_2009] below $T_\text{K}$. The slave-boson approach excludes the double occupancy of a QD restricting its Hilbert space to zero and single occupancy. This restriction was taken into account exactly in the numerical solutions [@Meir_1993; @Wingreen_1994; @Hettler_1995] using an additional integral removing the constraint [@Bickers_1987]. However, since the mean field [@Aguado_2000] and 1/$N$ expansion [@Ratiani_2009] theories, using the same integral trick, impose the constraint only approximately, they fail at higher temperatures [@Hewson_1997]. Here our goal is the development of an analytical ground for the slave-bosonic nonperturbative field theories which could be placed within the blue (uppermost) class in Fig. \[figure\_1\]. To avoid problems with high temperatures we do not use the integral trick to take into account the restriction of the Hilbert space but instead use an alternative method based on taking a certain limit with respect to a real parameter [@Bickers_1987]. This alternative method has not been applied so far in conjunction with the Keldysh field integral. We demonstrate that the corresponding limit can be exactly performed analytically after the Keldysh field integral for the TDOS has been analytically calculated using a certain approximation. Specifically, the approximation concerns the effective Keldysh action obtained after integrating out all electronic degrees of freedom. This action being a nonlinear functional of the slave-bosonic field is expanded up to second order in this field. Note, that this does not imply any perturbation since the action itself is the argument of an exponent. According to the general concept of the condensed matter field theory [@Altland_2010] the physics of our model is the physics in the vicinity of the zero slave-bosonic field configuration. We demonstrate that this physics contains the Kondo effect in QDs at finite temperatures. This scenario complements [@Altland_2010] the saddle point analysis [@Ratiani_2009] valid deep below the mean field theory slave-bosonic phase transition, i.e., at low temperatures when the slave-bosonic field in the mean field theory is condensed. Surprisingly, our simple or “bare” theory, which might play a role of a nonperturbative “free propagator” for more advanced nonperturbative theories of the blue class in Fig. \[figure\_1\], already provides a good description of the Kondo peak observed [@Ralph_1994] in the differential conductance at temperatures close to $T_\text{K}$. The paper is organized as follows. In Section \[of\] we formulate the problem on the operator level. Then in Section \[fts\] we translate this formulation into a field-theoretic framework using the Keldysh field integral and derive an analytic expression for the TDOS. Finally, we discuss our results and make conclusions in Sections \[dr\] and \[concl\], respectively. Operator formulation: Hamiltonian and observables {#of} ================================================= We start with the infinite-$U$ Anderson Hamiltonian in a slave-boson representation. As is well known [@Coleman_1984], in the case when $U=\infty$, the Anderson Hamiltonian, $\hat{H}_\text{d}=\sum_\sigma\epsilon_\text{d}\hat{n}_{\text{d},\sigma}+U\hat{n}_{\text{d},\uparrow}\hat{n}_{\text{d},\downarrow}$, where $\hat{n}_{\text{d},\sigma}=d^\dagger_\sigma d_\sigma$, $\sigma=\uparrow,\downarrow$, takes the form $\hat{H}_\text{d}=\sum_\sigma\epsilon_\text{d}f^\dagger_\sigma f_\sigma$, where the new fermionic operators are related to the original ones as $d_\sigma=f_\sigma b^\dagger$, $d^\dagger_\sigma=f^\dagger_\sigma b$, and $b$, $b^\dagger$ are the annihilation and creation operators of a slave-boson. Using these new fermionic and slave-bosonic operators the tunneling Hamiltonian, describing the coupling of the infinite-$U$ Anderson QD to contacts, can be written as $$\hat{H}_\text{T}=\sum_{a\sigma}\bigl(T_{a\sigma}c^\dagger_af_\sigma b^\dagger+T^*_{a\sigma}f^\dagger_\sigma c_a b\bigl), \label{tun_Ham}$$ where $c_a$, $c_a^\dagger$ are the annihilation and creation operators of the contacts fermions, $a$ includes the contacts complete set of quantum numbers and the contacts labels, left (L) or right (R), and $T_{a\sigma}$ are the tunneling matrix elements. The contacts are described by the Hamiltonian $\hat{H}_\text{C}=\sum_a\epsilon_ac^\dagger_ac_a$ and are assumed to be in equilibrium with the chemical potentials $\mu_\text{L,R}=\mu_0-eV_\text{L,R}$, with $V\equiv V_\text{L}-V_\text{R}$ being an external voltage. The total Hamiltonian is $\hat{H}=\hat{H}_\text{d}+\hat{H}_\text{T}+\hat{H}_\text{C}$. The restriction of the QD Hilbert space to zero and single occupancy requires the total number of the new fermions and slave-bosons, $\hat{Q}\equiv b^\dagger b+\sum_\sigma f^\dagger_\sigma f_\sigma$, to be equal to one, $\hat{Q}=\hat{I}$. This restriction must be taken into account in a QD observable $\langle\hat{O}\rangle$. There are two ways of doing that [@Bickers_1987]. In the QD context only the first one, [*i.e.*]{}, the integral way was used so far [@Meir_1993; @Wingreen_1994; @Hettler_1995; @Aguado_2000; @Ratiani_2009]. Here we employ the second way from Ref. , $$\langle\hat{O}\rangle(t)= \frac{\underset{\mu\rightarrow\infty}{\text{lim}} e^{\beta\mu}\text{Tr}[\hat{U}_{-\infty,t}\hat{O}\hat{U}_{t,-\infty}\hat{\rho}_0e^{-\beta\mu\hat{Q}}]} {\underset{\mu\rightarrow\infty}{\text{lim}}e^{\beta\mu}\text{Tr}[\hat{\rho}_0\hat{Q}e^{-\beta\mu\hat{Q}}]}, \label{observ_hsr}$$ where $\hat{U}_{t,t'}$, is the evolution operator with respect to the Hamiltonian $\hat{H}$, $\hat{\rho}_0=\exp[-\beta(\hat{H}_\text{d}-\mu_0\hat{N}_\text{d})]\otimes\exp[-\beta(\hat{H}_\text{C}-\sum_x\mu_x\hat{N}_x)]$, ($x=\text{L},\text{R}$) is the initial statistical operator with $\hat{N}_\text{d}$ and $\hat{N}_x$ being the number operators of the QD and contacts and $\beta$ is the inverse temperature. Eq. (\[observ\_hsr\]) may be rewritten in two equivalent forms, $$\langle\hat{O}\rangle(t)= \frac{\underset{\mu\rightarrow\infty}{\text{lim}} e^{\beta\mu}\text{Tr}[\hat{U}_{-\infty,\infty}\hat{U}_{\infty,t}\hat{O}\hat{U}_{t,-\infty}\hat{\rho}_0e^{-\beta\mu\hat{Q}}]} {\underset{\mu\rightarrow\infty}{\text{lim}}e^{\beta\mu}\text{Tr}[\hat{\rho}_0\hat{Q}e^{-\beta\mu\hat{Q}}]}, \label{observ_hsr_fb}$$ and $$\langle\hat{O}\rangle(t)= \frac{\underset{\mu\rightarrow\infty}{\text{lim}} e^{\beta\mu}\text{Tr}[\hat{U}_{-\infty,t}\hat{O}\hat{U}_{t,\infty}\hat{U}_{\infty,-\infty}\hat{\rho}_0e^{-\beta\mu\hat{Q}}]} {\underset{\mu\rightarrow\infty}{\text{lim}}e^{\beta\mu}\text{Tr}[\hat{\rho}_0\hat{Q}e^{-\beta\mu\hat{Q}}]}. \label{observ_hsr_bb}$$ These two forms have different interpretation: in Eq. (\[observ\_hsr\_fb\]) the observable is taken within the evolution from $-\infty$ to $\infty$ while in Eq. (\[observ\_hsr\_bb\]) it is taken within the evolution from $\infty$ to $-\infty$. To develop a Keldysh field theory one first equivalently rewrites Eqs. (\[observ\_hsr\_fb\]) and (\[observ\_hsr\_bb\]) on the Keldysh contour $C_\text{K}$. To this end one gives the creation and annihilation operators a formal temporal argument to allow the time-ordering operator to appropriately interlace operators. Afterwards one may take the half-sum of the two equivalent expressions to get the following symmetric form: $$\begin{split} \langle\hat{O}\rangle(t)=\frac{1}{\mathcal{N}_0}\underset{\mu\rightarrow\infty}{\text{lim}} &\frac{e^{\beta\mu}}{\text{Tr}[\hat{\rho}'_0(\mu)]} \text{Tr}[T_{C_\text{K}}e^{-\frac{i}{\hbar}\int_{C_\text{K}}d\tau\hat{H}'(\tau)}\times\\ &\times\frac{\hat{O}(t_+)+\hat{O}(t_-)}{2}\hat{\rho}'_0(\mu)], \end{split} \label{observ_hsr_1}$$ where $t_+$ and $t_-$ are the projections of $t$ onto the forward and backward branches of $C_\text{K}$, $$\begin{split} &\hat{\rho}'_0(\mu)=\hat{\rho}_0\exp(-\beta\mu\hat{Q}),\quad\hat{H}'=\hat{H}+\mu\hat{Q},\\ &\frac{1}{\mathcal{N}_0}\equiv\frac{\text{Tr}(\hat{\rho}_\text{C})}{\underset{\mu\rightarrow\infty}{\text{lim}}\{\exp(\beta\mu)\text{Tr}[\hat{Q}\hat{\rho}'_0(\mu)]\}} \end{split} \label{so_ham_pr}$$ and $\hat{\rho}_\text{C}$ is the statistical operator of the contacts. The expression under the limit in Eq. (\[observ\_hsr\_1\]) can be written as the Keldysh field integral [@Kamenev_1999; @Altland_2010] for a fixed value of $\mu$. The basic steps in the construction of the Keldysh field integral are identical to the ones presented in Ref. . The details specific to our application of this field integral are given in the next section. We would like to emphasize the absence in Eq. (\[observ\_hsr\_1\]) of any prefactor depending on the QD-contacts coupling. This is a great advantage of the Keldysh field theory over non-field-theoretic approaches [@Wingreen_1994] and imaginary-time field theories [@Bickers_1987]. Indeed, in our approach there is no need for an independent calculation of such prefactors. This greatly simplifies the analytical exact projection onto the physical subspace. This fact was not realized in the first attempt [@Ratiani_2009] to combine the Keldysh field theory and slave-boson approach and, as a result, this attempt was reduced to a nonequilibrium analog of equilibrium imaginary-time field theories with only an approximate projection onto the physical subspace. Field-theoretic solution: Keldysh field integral {#fts} ================================================ We obtain the effective Keldysh field theory in a way similar to the one used in Refs. for Coulomb-blockaded QDs. Namely, we first integrate out the QD and contacts Grassmann fields. After this step the field-theoretic description is given in terms of the effective Keldysh action, $S_\text{eff}[\chi^\text{cl}(t),\chi^\text{q}(t)]\equiv S_\text{B}^{(0)}[\chi^\text{cl}(t),\chi^\text{q}(t)]+S_\text{tun}[\chi^\text{cl}(t),\chi^\text{q}(t)]$, where $\chi^{\text{cl},\text{q}}(t)$ are the classical and quantum components of the slave-bosonic complex field, which is just a bosonic coherent state [@Altland_2010], $S_\text{B}^{(0)}[\chi^\text{cl}(t),\chi^\text{q}(t)]$ is the free slave-bosonic action with the standard matrix form in the Keldysh space and $S_\text{tun}[\chi^\text{cl}(t),\chi^\text{q}(t)]$ is the slave-bosonic tunneling action, $$S_\text{tun}[\chi^\text{cl}(t),\chi^\text{q}(t)]=-i\hbar\,\text{tr}\ln\bigl[I+\mathcal{T}G^{(0)}\bigl], \label{sbta_ex}$$ where the trace and matrix product are taken with respect to the temporal arguments and both single-particle and Keldysh indices. In Eq. (\[sbta\_ex\]) the matrices $G^{(0)}$ and $\mathcal{T}$ have the block form in the dot-contacts space, $$G^{(0)}= \begin{pmatrix} G^{(0)}_\text{d}(\sigma t|\sigma't')&0\\ 0&G^{(0)}_\text{C}(at|a't') \end{pmatrix}, \label{G0_matr}$$ $$\mathcal{T}= \begin{pmatrix} 0&M_\text{T}^\dagger(\sigma t|a't')\\ M_\text{T}(at|\sigma't')&0 \end{pmatrix}, \label{T_matr}$$ where the blocks $G^{(0)}_\text{d,C}(\alpha t|\alpha't')$ are the standard fermionic Keldysh Green’s function matrices ($2\times 2$ matrices in the Keldysh space) of the free QD ($\alpha=\sigma$) and contacts ($\alpha=a$), $$G^{(0)}_\text{d,C}(\alpha t|\alpha't')\!=\! \begin{pmatrix} G^{(0)+}_\text{d,C}(\alpha t|\alpha't')&G^{(0)\text{K}}_\text{d,C}(\alpha t|\alpha't')\\ 0&G^{(0)-}_\text{d,C}(\alpha t|\alpha't') \end{pmatrix}, \label{f_KGf_matr}$$ with $G^{(0)+,-,\text{K}}_\text{d,C}(\alpha t|\alpha't')$ being the retarded, advanced and Keldysh components [@Altland_2010], respectively, and the block $M_\text{T}(at|\sigma t')$ is the tunneling matrix ($2\times 2$ matrix in the Keldysh space), $$M_\text{T}(at|\sigma t')=\frac{1}{\hbar}\delta(t-t')T_{a\sigma}\frac{1}{\sqrt{2}} \begin{pmatrix} \bar{\chi}^\text{cl}(t)&\bar{\chi}^\text{q}(t)\\ \bar{\chi}^\text{q}(t)&\bar{\chi}^\text{cl}(t) \end{pmatrix}. \label{tun_matr}$$ Note that the only formal difference of Eq. (\[tun\_matr\]) from the corresponding expression in Ref. is that here instead of the bosonic phase field we have the slave-bosonic field and that this field is not exponentiated. Our goal is to investigate a strongly interacting QD in the Kondo regime when the Kondo resonance is not strong yet. This is the case, [*e.g.*]{}, at temperatures above the Kondo temperature when the Kondo resonance is already present but not fully developed. These range of temperatures is most relevant for both experiments and practical applications in devices. As the formation of the Kondo resonance is related to the QD population oscillations induced by the QD-contacts coupling, the QD empty state will weakly fluctuate and its probability will be small in the regime which we are interested in. Since the QD empty state is described by the slave-bosonic complex field, those weak oscillations, induced by the QD-contacts tunneling coupling, can be obtained from the expansion of the tunneling action around the zero slave-bosonic field configuration. This expansion is valid for small amplitudes of the slave-bosonic complex field and thus for small probabilities of the QD empty state. We then expand the tunneling action (\[sbta\_ex\]) around the zero slave-bosonic field configuration, [*i.e.*]{}, we keep in this action only the first non-vanishing term which is quadratic in the slave-bosonic field. Additionally, we assume a Lorentzian contacts density of states, $\nu_\text{C}(\epsilon)=\nu_\text{C}D(\epsilon)$, $D(\epsilon)=W^2/(\epsilon^2+W^2)$, $a=\{x,k,\sigma\}$ and $T_{xk\sigma\sigma'}=\delta_{\sigma\sigma'}T$. The expression for the tunneling matrix elements, in particular, means that we, for simplicity, consider the case of a symmetric coupling to the contacts. The tunneling action then becomes $$\begin{split} &S_\text{tun}[\chi^\text{cl}(t),\chi^\text{q}(t)]=(g/\hbar)\!\!\int \!\!dt\!\!\int \!\!dt' \begin{pmatrix} \bar{\chi}^\text{cl}(t)&\bar{\chi}^\text{q}(t) \end{pmatrix}\times\\ &\times\begin{pmatrix} 0&\Sigma^-(t-t')\\ \Sigma^+(t-t')&\Sigma^\text{K}(t-t') \end{pmatrix} \begin{pmatrix} \chi^\text{cl}(t')\\ \chi^\text{q}(t') \end{pmatrix}. \end{split} \label{sbta}$$ In Eq. (\[sbta\]) $g\equiv 2\pi^2\nu_\text{C}|T|^2$ and the retarded, advanced and Keldysh slave-bosonic self-energies are $$\begin{split} &\Sigma^\pm(t-t')\equiv(i/2)\sum_x[g^\text{K}_x(t'-t)g^\pm_\text{d}(t-t')+\\ &+g^\mp(t'-t)g^\text{K}_\text{d}(t-t')],\\ &\Sigma^\text{K}(t-t')\equiv(i/2)\sum_x\{g^\text{K}_x(t'-t)g^\text{K}_\text{d}(t-t')-\\ &\!\!\!-[g^+_\text{d}(t-t')-g^-_\text{d}(t-t')][g^+(t'-t)-g^-(t'-t)]\}, \end{split} \label{sb_se_def}$$ where the retarded, advanced and Keldysh components of the Green’s function matrix are $$\begin{split} &g^\pm(\omega)\equiv\int\frac{d\epsilon}{2\pi}D(\epsilon)\frac{\hbar}{\hbar\omega-\epsilon\pm i0},\\ &g^\pm_\text{d}(\omega)\equiv\frac{\hbar}{\pi(\hbar\omega-\epsilon_\text{d}-\mu\pm i0)}, \end{split} \label{gf_ra}$$ $$\begin{split} &g^\text{K}_{x}(\omega)\equiv[g^+(\omega)-g^-(\omega)]\tanh\biggl[\frac{\beta(\hbar\omega-\mu_x)}{2}\biggl],\\ &g^\text{K}_\text{d}(\omega)\equiv[g^+_\text{d}(\omega)-g^-_\text{d}(\omega)][1-2n_\text{d}(\hbar\omega)], \end{split} \label{gf_raK}$$ with $n_\text{d}(\epsilon)$ being the QD distribution of the new fermions. The QD TDOS is defined through the imaginary part of the QD retarded Green’s function, $\nu_\sigma(\epsilon)\equiv -(1/\hbar\pi)\text{Im}[G_{\text{d}\,\sigma\sigma}^+(\epsilon)]$. The Keldysh field integral expression for $\nu_\sigma(\epsilon)$ is $$\begin{split} &\nu_\sigma(\epsilon)=-\frac{1}{2\pi i\hbar}\frac{1}{\mathcal{N}_0}\underset{\mu\rightarrow\infty}{\text{lim}}e^{\beta\mu} \int_{-\infty}^\infty dte^{\frac{i}{\hbar}\epsilon t}\times\\ &\!\!\!\times\!\int\mathcal{D}[\chi^\text{cl}(t),\chi^\text{q}(t)] e^{\frac{i}{\hbar}S_\text{eff}[\chi^\text{cl}(t),\chi^\text{q}(t)]}\times\\ &\!\!\!\times\![\bar{\chi}_-(t)\chi_+(0)G^>_\text{d}(\sigma t|\sigma 0)-\bar{\chi}_+(t)\chi_-(0)G^<_\text{d}(\sigma t|\sigma 0)], \end{split} \label{tdos_kfi}$$ where $\chi_\pm(t)$ is the slave-bosonic field on the forward and backward branches of $C_\text{K}$, $$\begin{split} &iG^{>,<}_\text{d}(\sigma t|\sigma 0)=\\ &=\biggl[\frac{1}{2}\pm\frac{1}{2}-n_\text{d}(\epsilon_\text{d}+\mu)\biggl]\exp\biggl[-\frac{i}{\hbar}(\epsilon_\text{d}+\mu)t\biggl], \end{split} \label{gv_lg}$$ As in Refs. the distribution $n_\text{d}(\epsilon)$ is the double step. In the present context this is due to the noninteracting nature of the new fermions. Since the effective Keldysh action $S_\text{eff}[\chi^\text{cl}(t),\chi^\text{q}(t)]$ is quadratic, the functional integral in Eq. (\[tdos\_kfi\]) may be performed exactly for any real $\mu>0$. The limit $\mu\rightarrow\infty$ is readily taken afterwards. The final nonperturbative result for the QD TDOS is $$\nu_\sigma(\epsilon)=\frac{\mathcal{P}(\epsilon)} {[\epsilon_\text{d}-\epsilon+(g/\hbar)\Sigma_\text{R}(\epsilon)]^2+[(g/\hbar)\Sigma_\text{I}(\epsilon)]^2}, \label{tdos_f}$$ where $\Sigma_\text{R}(\epsilon)$, $\Sigma_\text{I}(\epsilon)$ are the real and imaginary parts of the retarded slave-bosonic self-energy for which we find the following analytical expressions, $$\begin{split} &\Sigma_\text{R}(\epsilon)=\hbar[D(\epsilon)/\pi]\{\epsilon/2W+\\ &\!\!\!+\!(1/2\pi)\text{Re}\!\!\sum_x[(1\!+\!i\epsilon/W)\psi[1/2\!-\!i\beta(iW\!\!+\!\mu_x)/2\pi]-\\ &-\psi[1/2-i\beta(\mu_x-\epsilon)/2\pi]]\},\\ &\Sigma_\text{I}(\epsilon)=\hbar[D(\epsilon)/2\pi]\sum_xn_x(\epsilon), \end{split} \label{rsbse_r_i}$$ where $\psi(x)$ is the digamma function and $n_x(\epsilon)$ are the contacts Fermi-Dirac distributions, and the numerator is $$\begin{split} &\mathcal{P}(\epsilon)=\frac{gD(\epsilon)}{2\pi^2}\frac{n_\text{L}(\epsilon_\text{d})+n_\text{R}(\epsilon_\text{d})- 2n_\text{L}(\epsilon_\text{d})n_\text{R}(\epsilon_\text{d})}{1-n_\text{L}(\epsilon_\text{d})n_\text{R}(\epsilon_\text{d})}. \end{split} \label{numer}$$ The imaginary part in Eq. (\[rsbse\_r\_i\]) has a particularly clear physical meaning. The slave-boson describes the empty state of the QD. From Eq. (\[rsbse\_r\_i\]) it follows that for energies above the chemical potentials $\mu_x$ the imaginary part of the slave-bosonic self-energy $\Sigma_\text{I}(\epsilon)\rightarrow 0$ which means that the life-time of the empty state goes to infinity, [*i.e.*]{}, the QD is empty. In contrast, below $\mu_x$ the imaginary part is finite leading to a finite life-time of the empty state, [*i.e.*]{}, the QD is filled. It is important to emphasize that the QD TDOS, Eq. (\[tdos\_f\]), is finite at any finite temperature but logarithmically diverges for $T=0$ at the chemical potentials. Thus, one expects that for very low temperatures the simple quadratic theory is not valid. This is in accordance with what one generally expects on purely theoretical grounds [@Altland_2010] from expansions around zero field configurations. Zero field expansions complement low temperature theories, being expansions around non-zero field configurations, like mean field theories [@Aguado_2000; @Ratiani_2009]. These non-zero field expansions fail to describe the high-temperature behavior of the Kondo effect in QDs as we have demonstrated in Fig. \[figure\_1\]. Therefore, one concludes that for a comprehensive description of the Kondo effect in the whole temperature range it is desirable to have in one’s disposal both types of field theories. Let us clarify the nature of the approximation used for the tunneling action, Eq. (\[sbta\]), and the applicability range of the result for the TDOS, Eq. (\[tdos\_f\]). Formally the small parameter of the expansion is the tunneling matrix element $T$. Hence, $\Gamma=2g/\pi$ should be small. However, the tunneling matrix elements always enter in product with the slave-bosonic fields. Thus our theory will be valid in physical situations where the large values of the slave-bosonic amplitude are not important, [*i.e.*]{}, when the probability of the QD empty state is not large. On one side this happens when $\mu_0-\epsilon_\text{d}$ is large and on the other side when the temperature is not too low and thus the Kondo resonance is not too strong so that the fluctuations of the QD empty state are not too large. Therefore, we assume that our theory should be applicable for $$\mu_0-\epsilon_\text{d}\gtrsim\Gamma\quad\text{and}\quad T\gtrsim T_\text{K}. \label{criteria}$$ One should keep in mind that this is a crude estimate and our theory may still work qualitatively and perhaps semiquantitatively outside those inequalities. As demonstrated in the next section, our simple quadratic approximation, accounting for the physics in the vicinity of the zero slave-bosonic field configuration, provides a reasonable description of the Kondo physics. It also gives a finite single-particle resonance. This situation is definitely better than the one taking place in perturbative approaches [@Sivan_1996] which have divergences at the single-particle resonance. Our theory being nonperturbative avoids this problem but requires the next order term, [*i.e.*]{}, the one which is quartic in the slave-bosonic field, in the expansion of the tunneling action (\[sbta\_ex\]) for a better quantitative description. This will be the subject of a more advanced theory. Here we only note that this advanced theory may be constructed in terms of the quadratic theory presented in this work. Namely, the present theory will play a role of a nonperturbative “free propagator” for the quartic theory. It is interesting to look at the result for the QD TDOS, Eq. (\[tdos\_f\]), from the physical point of view. Within the range of its applicability it suggests that the quasiparticle state in the QD represents a superposition of the bare electron state and the empty state of the QD coupled to contacts. Thus in this range the life-time of the QD quasiparticles can be estimated as the life-time of the QD empty state. This may be very attractive for the experiments measuring the quasiparticle life-times since observing QD states is easier than a direct observation of its quasiparticles. Finally, we would like to emphasize an additional fundamental advantage of our method: our field-theoretic approach is a truly field theory in contrast with mean field theories [@Aguado_2000; @Ratiani_2009]. The point is that here we work only with a physical field, being the slave-bosonic field, and not with artificial fields like the Lagrange multipliers fields used in mean field theories. Thus our theory has more transparent access to physics avoiding such artefacts of mean field theories as slave-bosonic condensation. ![\[figure\_2\] (Color online) The equilibrium and nonequilibrium TDOS in the Kondo regime. Here $kT=0.005\Gamma$, $\mu_0-\epsilon_\text{d}=1.876\Gamma$, $W=100\Gamma$. In equilibrium, $V=0$, there is a sharp many-particle peak at the Fermi energy. In nonequilibrium, $V\neq0$, the peak is reduced and split into two lower peaks (see inset). The total spectral weights of the equilibrium and nonequilibrium situations are almost the same differing by 1% which is due to the accuracy of the numerical integration involved in the total spectral weight.](figure_2){width="7.6"} Discussion of the results {#dr} ========================= In Fig. \[figure\_2\] the QD TDOS (\[tdos\_f\]) is shown. In addition to the single-particle resonance at the renormalized QD noninteracting energy level it reveals a many-particle peak at the Fermi energy, known as the Kondo resonance. The formation of the Kondo resonance is explained in Fig. \[figure\_3\] as an interplay between $\Sigma_\text{R}(\epsilon)$ and $\Sigma_\text{I}(\epsilon)$ from (\[rsbse\_r\_i\]). The distance between $\epsilon-\epsilon_\text{d}$ and $\Sigma_\text{R}(\epsilon)$ reaches a minimum where $\Sigma_\text{R}(\epsilon)$ has its maximum. This happens at the Fermi energy. At the same time $\Sigma_\text{I}(\epsilon)$ has a steep decrease at the Fermi energy. Therefore, the two terms in the denominator of Eq. (\[tdos\_f\]) become both minimal at the Fermi energy giving rise to the Kondo resonance. ![\[figure\_3\] (Color online) The mechanism of the Kondo resonance formation in the nonperturbative Keldysh field theory in the vicinity of the zero slave-bosonic field configuration. Here $V=0$, $kT=0.008\Gamma$, $\mu_0-\epsilon_\text{d}=1.95\Gamma$.](figure_3){width="7.6"} ![\[figure\_4\] (Color online) The Kondo peak in the differential conductance. Inset shows the temperature dependence of the differential conductance maximum at $V=0$. The solid line is the result of our nonperturbative Keldysh field theory. The circles show the experimental data of Ref. .](figure_4){width="7.6"} To verify our field-theoretic description of the Kondo physics we first compare it with experimental data. The presence of the Kondo resonance in the QD TDOS has an impact on the other QD observables. In particular, experiments [@Ralph_1994] show a peak in the differential conductance at $V=0$. Using the expression for the current (see Eq. (3) in Ref. ) through a QD together with Eqs. (\[tdos\_f\]) and (\[rsbse\_r\_i\]) we obtain this behavior of the differential conductance shown in Fig. \[figure\_4\]. To get Fig. \[figure\_4\] we have taken the values of the parameters, $\Gamma=2.6875$ meV, $W=5$ eV, $T=50$ mK, $\Gamma/[\pi(\mu_0-\epsilon_\text{d})]=0.1224$, close to the ones which were estimated in Ref. . ![\[figure\_5\] (Color online) Comparison of our field theory with the numerical renormalization group theory. The solid line is obtained from the numerical renormalization group calculations [@Costi_1994; @Goldhaber-Gordon_1998a; @Grobis_2008]. The circles show the differential conductance maximum obtained from our field theory for the values of the parameters used in Fig. \[figure\_4\].](figure_5){width="7.6"} ![\[figure\_6\] (Color online) Keldysh field-theoretic prediction of the Kondo temperature dependence on the QD single-particle energy level $(\mu_0-\epsilon_\text{d})/\Gamma$.](figure_6){width="7.6"} We would like further to compare our Keldysh field theory with existing theoretical approaches, in particular, with the numerical renormalization group theory from Ref. which was successfully employed to describe experiments [@Goldhaber-Gordon_1998a; @Grobis_2008] on the Kondo effect in QDs. As we have argued that our theory must be valid for temperatures $T\gtrsim T_\text{K}$, within this temperature range the differential conductance maximum in our theory must have the same temperature dependence as the one in Refs. (see, [*e.g.*]{}, Eq. (2) in Ref. , where we take $s=0.21$) for the case of a symmetric coupling. From this high-temperature comparison, for each value of $\mu_0-\epsilon_\text{d}$ used to calculate the differential conductance maximum in the Keldysh field theory, we can fix the value of $T_\text{K}$ to be used in the empirical form of Ref. . For example, for the parameters used in Fig. \[figure\_4\] we get the temperature dependence of the differential conductance maximum shown in Fig. \[figure\_5\]. The Kondo temperature $T_\text{K}=11$ mK agrees with the rough estimate $T_\text{K}<50$ mK given in Ref. , where a weak asymmetry in the capacitance of the QD to the left and right contacts was assumed. The same comparison between the Keldysh field theory and the numerical renormalization group theory can be done for any value of $\mu_0-\epsilon_\text{d}$ for which the Keldysh field theory is applicable. In this way we determine how in our Keldysh field theory $T_\text{K}$ depends on $\mu_0-\epsilon_\text{d}$. This dependence is shown in Fig. \[figure\_6\] and it is in full accordance with the standard expression [@Hewson_1997], $$\frac{T_\text{K}}{\Gamma}\thicksim\exp\biggl[-2\pi\frac{\mu_0-\epsilon_\text{d}}{\Gamma}\biggl], \label{TK}$$ where our definition of $\Gamma$ is twice that of Ref. (see also the caption of Fig. \[figure\_1\]). This proves that our theory for the Kondo effect in QDs correctly predicts the Kondo temperature. Moreover, our theory, within its applicability range, also predicts that the differential conductance maximum has a universal temperature dependence with the scaling given by the Kondo temperature, Eq. (\[TK\]). This universal temperature dependence is shown in Fig. \[figure\_7\] and additionally proves that our Keldysh field theory, within its applicability range, correctly describes the Kondo physics in QDs. As one can see from Fig. \[figure\_7\] the Keldysh field-theoretic description is quantitatively reliable for temperatures $T\geqslant 2T_\text{K}$, which perfectly agrees with our theoretical prediction made above in Section \[fts\]. ![\[figure\_7\] (Color online) Comparison of the universal temperature dependence of the differential conductance maximum in our field theory and in the numerical renormalization group theory. The solid line is the result of the numerical renormalization group calculations [@Costi_1994; @Goldhaber-Gordon_1998a; @Grobis_2008]. The circles show the result obtained from our field theory.](figure_7){width="7.6"} Finally, we would like to say a few words about the numerical consistency of our theory. To do this, we employ the sum rule given by Eq. (39) of Ref. . In NCA this sum rule is always satisfied within 0.5%. In our theory this depends on how well the applicability criteria, Eq. (\[criteria\]), of the Keldysh field theory are satisfied. For example, for the parameters presented in Fig. \[figure\_2\] the sum rule is satisfied within 15% while for $\mu_0-\epsilon_\text{d}=2.5\Gamma$ it is 8.5% and for $\mu_0-\epsilon_\text{d}=4.0\Gamma$ it is 4.4%. One should note that the sum rule is an integral estimate over the whole energy range. Thus, the error is gained over the whole range of energies. At the same time for a given energy the QD TDOS may have higher accuracy. Since the only approximation was the truncation of all the terms of higher orders than the terms quadratic in the slave-bosonic field, to improve the consistency of the method and extend its applicability criteria one should go beyond the quadratic approximation and this will be done in a subsequent study. Conclusion {#concl} ========== We have developed a basic slave-boson nonperturbative Keldysh field theory for the Kondo effect in quantum dots. The theory deals with the physics in the vicinity of the zero slave-bosonic field configuration where, as we have shown, the main fraction of the Kondo physics is located at experimentally relevant temperatures. The presented theory has a closed analytical solution for the quantum dot tunneling density of states and, despite being relatively simple, properly describes experimental data on the Kondo peak observed in the differential conductance, correctly predicts the Kondo temperature and, within its applicability range, has the same universal temperature dependence of the conductance as the one obtained in numerical renormalization group calculations. Therefore, it represents a convenient basis, as a free nonperturbative propagator, for more advanced theories which could extend the applicability of our approach to larger values of the slave-bosonic amplitude and, thus, to temperatures much lower than the Kondo temperature. Acknowledgments =============== The authors thank Alexander Altland and Dmitry Ryndyk for fruitful discussions. Support from the DFG under the program SFB 689 is acknowledged. [29]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , , , ****, (). , ****, (). , ****, (). , , , , , , ****, (). , ****, (). , , , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , , , ****, (). , ****, (). , ****, (). , ** (, ). , , , ****, (). , , , ****, (). , ****, (). , , , ****, (). , , , , , , ****, (). , , , ****, (). , ****, (). , ** (, ), ed. , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , , , , , , ****, (). , , , , , ****, ().

=-5mm [Generalization of the $N_pN_n$ Scheme and the Structure of the Valence Space]{} Y. M. Zhao$^{1}$, R. F. Casten$^{2}$, and A. Arima$^3$ $1$) Cyclotron Laboratory, the Institute of Physical and Chemical Research (RIKEN), Hirosawa 2-1, Wako-shi, Saitama, 351-0198 Japan $2$) A. W. Wright Nuclear Structure Laboratory, Yale University, New Haven, CT 06520 USA $3$)The House of Councilors, 2-1-1 Nagatacho, Chiyodaku, Tokyo 100-8962, Japan The $N_pN_n$ scheme, which has been extensively applied to even-even nuclei, is found to be a very good benchmark for odd-even, even-odd, and doubly-odd nuclei as well. There are no apparent shifts in the correlations for these four classes of nuclei. The compact correlations highlight the deviant behavior of the Z=78 nuclei, are used to deduce effective valence proton numbers near Z=64, and to study the evolution of the Z=64 subshell gap. PACS Number(s): 27.60+j, 27.70+q, 27.80+w, 27.80+b Many physical systems, including atoms, nuclei and metallic clusters, exhibit shell structure. Indeed, eigenvalues of the three-dimensional Schroedinger equation will tend to cluster in energy groupings (characterized by specific sets of principal and angular momentum quantum numbers) for any reasonable central potential. In the treatment of complex finite many-body systems, a common simplification is to invoke a “mean field” ansatz, replacing the sum of all the two-body interactions by a one-body potential. Generally, such a procedure is only an approximation and various residual interactions need to be incorporated. These will alter the predictions of the independent particle picture and may even lead to a breakdown of the shell structure, shell closures, and shell gaps. Nuclei provide an ideal venue to study shell structure and residual interactions since they are finite-body systems where the effective number of active bodies (the valence nucleons) is generally quite small (0-30, say) and where one can both count and change this number of bodies (the mass number) in a controlled way. Here, we wish to explore the evolution of collective behavior in nuclei and the associated evolution of shell structure using an empirical correlation scheme of collective observables that stresses the importance of the valence residual p-n interaction. The importance of the proton-neutron interaction in determining the evolution of nuclear structure was emphasized long-ago by de Shalit and Goldhaber [@Shalit], and Talmi [@Talmi]. Two decades ago, Federman and Pittel [@Pittel] emphasized that the driving mechanism in the development of nuclear deformation is the proton-neutron interaction between nucleons in spin-orbit partner orbits. If the proton-neutron interaction is a controlling factor in the determination of nuclear structure, a reasonable estimate of this interaction ought to be a useful systematizing parameter with which the evolution of structure could be correlated. In 1985 Casten described the $N_pN_n$ scheme for even-even nuclei [@Casten1], in which $E_{2_1^+}, E_{4_1^+}/E_{2_1^+}$, and $B(E2, 0_1^+\rightarrow 2_1^+)$ values were plotted against the product of valence proton number and valence neutron number, $N_pN_n$. The systematics for each observable is very smooth, and similar from region to region. It was found that the quantity $N_pN_n$ provides an excellent scaling factor that allows one to assess the rapidity of different transition regions and to predict the properties of new nuclei [@Casten3]. Moreover, the slopes of different observables plotted against $N_pN_n$ are related to the average interaction, per proton-neutron pair, in the highly overlapping orbits whose occupation induces structural change. However, most papers related to the $N_pN_n $ scheme have concentrated on the even-even case where there is a rich array of compiled nuclear data. It is therefore important to see whether the $N_pN_n$ scheme works, and how well it works, in odd-A and doubly odd nuclei. The $N_pN_n$ concept is more difficult to apply to odd-A and odd-odd cases because there can be a very strong interplay between collective and single particle excitations, and the low-lying excitation structures themselves are more complicated. Moreover, adjacent nuclei differ in ground state and low-lying $J^{\pi}$ values so it is sometimes not clear which data to use in a systematic comparision. Finally, observables related to odd-A nuclei and odd-odd nuclei are in general less well, and less systematically, known than those of even-even nuclei. The most extensive studies for odd-A nuclei to date have been for the A=80-100 region. In [@Bucu1] the $N_pN_n$ scheme was applied to both even-even and odd-A nuclei in the A$\sim$80 region; in [@Bucu2] a few odd-A nuclei with A$\sim$100 were considered; in [@Bucu3], it was shown that states based on different single-particle excitations behave differently with $N_pN_n$. However, there has not yet been any concerted effort towards a unified $N_pN_n$ treatment for even-even, odd-A and doubly odd nuclei over large mass regions. It is therefore the purpose of this Letter to show for the first time that the simple $N_pN_n$ scheme works equally well for large regions of medium-heavy nuclei for even-even, odd-A and the doubly-odd nuclei. This extension to the $N_pN_n$ scheme will significantly expand its usefulness for interpreting the sparse data soon-to-be-obtained on exotic nuclei far from stability. We will also use these results to extract effective valence proton numbers near Z=64 and N=83-91 in order to study the breakdown of the Z=64 shell gap in even, odd and odd-odd nuclei. We proceed by studying the deformation parameter $e_2$ against $N_pN_n$. The $e_2$ values are taken from the macroscopic-microscopic calculations of [@Moller] for nuclei with known ground and excited states. These deformations act as surrogates for directly measured observables, and therefore allow us to compare even and odd Z and N nuclei on the same footing. These calcualtions are highly refined, and widely used. For nuclei in or near the valley of stability, such as those considered here, they should provide an excellent guide to realistic deformations, although it would be useful to check them by experiment. Of course, far from stability, the importance of various residual interactions changes, as does the mean field itself, and hence care should be taken in extending these results to new regions. In any case, for known nuclei, we believe that the approximations used in [@Moller] are reasonably good individually, and fully adequate for a systematic study in large regions. Moreover, by using the deformation rather than excitation energies to gauge the structure, one avoids problems with comparing levels with different spins. In Fig. 1, we present the quadrupole deformation parameter in the Nilsson perturbed-spheroid parameterization, $e_2$, vs. $N_pN_n$ for the nuclei in four different regions ranging from Z=50 to 104, namely the 50$<$Z$\le$66, 82$<$N$\le$104 region, the 66$<$Z$<$82, 82$<$N$\le$104 region, the 66$<$Z$<$82, 104$<$N$<$126 region, and the 82$<$Z$\le$104, 126$<$N$<$155 region. The correlation between $e_2$ and $N_pN_n$ is extraordinarily compact not only for the even-even nuclei but also for the even-odd, odd-even and odd-odd cases as well (see solid symbols in Fig. 1a) and the full set of points in Figs. 1b, c, d). Moreover, the correlations are independent of the even-even, even-odd, odd-even or odd-odd nature of the nuclei considered. No discernible bias for these classes of nuclei is visible except for a slight difference between the points for even-proton number and odd-proton number for $N_pN_n$ values less than 50 in Fig. 1c). Among the correlations shown, Fig. 1a) shows a greater broadening near $N_pN_n$ $\sim$ 50-100 than the other regions. This is a region where there is a subshell at Z=64 which we did not take into account. That is, we used the proton magic numbers 50 and 82 for all nuclei. Below, we will examine the validity of these choices. To facilitate that discussion, Fig. 1a) uses open symbols for nuclei with N$\le91$ and $59\le$Z$\le$66. Another interesting point in Fig. 1a) is that there are a number of data points with $e_2$=0, which correspond to the N$=84$ isotones. These isotones are very soft, which means that the shallow part of the potential energy against the deformation parameter is wide. Hence there can be a large difference between the equilibrium deformation and the expectation value of the deformation. In b) and c) of Fig. 1, several data points clearly stand out to the upper left of the correlations. Nearly all have Z=78 (Pt) and lie in a complex region with large $\gamma$-softness, oblate shapes, prolate shapes, and transition regions between them. Nevertheless, other regions also show sharp shape changes but are not anomalous in the $N_pN_n$ plots. Therefore, it is worth further effort to understand the behavior of the Z=78 Pt region and whether these anomalous points reflect a different role for the p-n interaction in these nuclei or a shortcoming in the calculated deformations in [@Moller]. While the concept of the valence space is important in understanding the structure of nuclei, in many cases the conventional counting of valence protons and neutrons is inadequate. For example, near A=100 and 150, the Z=40 and 64 proton numbers take on magic character for certain neutron numbers but not for others [@Casten]. Likewise, the neutron number N=20 is no longer magic for the neutron rich nucleus $^{32}$Mg [@Motobayashi]. Indeed, it is expected that magicity may well be a fragile construct far from stability. This fragility is a result both of changes to the mean field and to the valence p-n residual interaction[@england; @Heyde]. Its effects might be expected to show up in the $N_pN_n$ scheme. Indeed, in even-even nuclei, effective $N_p$ values have been discussed for both the A=100 and 150 regions \[4,14-17\]. The present results give us the opportunity to probe this issue more deeply, by extracting effective $N_p$ values in the A=150 region from even, odd and odd-odd nuclei simultaneously and in a unified way. In Fig. 1a), the solid symbols are for the 59$\le$Z$\le$66 and N$\ge$92 nuclei, and all nuclei with 50$<$Z$\le$58. They form an extraordinarily compact trajectory, while the 59$\le$Z$\le$66 and N$\le$91 nuclei deviate strongly to the right. This arises because, for these latter nuclei, Z=64 acts as a magic or partially magic number whereas Fig. 1a) was constructed using Z=50 as magic. Hence these nuclei were plotted at inappropriately large $N_pN_n$ values. The opposite assumption, that Z=64 is magic for N$\le$91 is also too extreme. As shown in Fig. 2, this leads to an overshoot of these points to the left. Clearly, by assuming the validity of the compact correlation for nuclei not affected by a Z=64 gap, that is those marked by solid symbols in Fig. 1a), and shifting the “deviant" nuclei leftward to this correlation, we can extract the effective $N_p$ values for these nuclei and thereby assess the breakdown and dissolution of the Z=64 gap. Equivalently, one can shift the anomalous data points in Fig.2 to the right. The process is similar to that used in [@Casten1] for even-even nuclei but now is extended uniformly to all species. Fig. 3 illustrates how this approach works by looking at a subset of the points in Fig.2$--$those for even-odd nuclei. Here, the solid symbols are the nuclei unaffected by a Z=64 gap. The open symbols lie at various distances from the main correlation: consistently, the Z=64, 66 isotopes lie farthest, and the Z=62, and 60 isotopes occur successively closer. The amount of shifting required for each point is determined by fitting an exponential function to the normal (solid symbol) data in Fig.3, and such a fitting curve is used as a guide to deduce the appropriate $N_p$ value for that $e_2$. The resulting effective $N_p$ values for all the data of Fig. 1a) are summarized in Table 1 and shown in Fig. 4. They are given in the Table to the nearest odd(even) integers for odd(even)-Z nuclei. Note that in Table 1 we do not present effective valence proton numbers for the N=84 isotones since, as discussed above, these nuclei are soft and the equilibrium and mean deformations may differ considerably, and also the calculated deformations can be very sensitive to small perturbations. The results in Table 1 demonstrate a gradual breakdown of the Z=64 shell gap, which accelerates near N=90, and consistency regardless of whether the nuclei are even-even, odd-even, even-odd, or odd-odd. To summarize, the $N_pN_n$ scheme, which has been extensively studied for even-even nuclei, is found to be equally applicable to all species of medium-heavy nuclei: even-even, odd-even, even-odd, and odd-odd. The $N_pN_n$ correlations are not sensitive to the odd-even difference. This supports the idea that the proton-neutron interaction plays a similar role regardless of the even-odd character of the nuclei, and suggests that the average strength of the valence proton-neutron interaction is almost constant between even-even and their odd-A/odd-odd neighbors. The extremely compact $N_pN_n$ trajectories highlight a few deviant nuclei. Finally, effective valence proton numbers were extracted from these correlations and found to be also insensitive to the category of nucleus. This gives a deeper view of the breakdown of the Z=64 magicity near neutron number 90. The present work extends the realm of application of the $N_pN_n$ scheme to all types of nuclei. Given that compact correlation schemes, such as $N_pN_n$, magnify anomalous behavior (e.g., the Z=78 nuclei discussed above), and probe the valence space (i.e., the effective valence nucleon numbers), the present results and approach can provide a more general tool to disclose new and different types of shell structure or structural evolution (e.g., changes in shell structure and magicity) in exotic nuclei. [**ACKNOWLEDGEMENTS**]{} Discussions with Drs. N.V. Zamfir, P. Moeller, N. Yoshinaga, S. Yamaji, and S. G. Zhou are appreciated gratefully. One of the authors (YMZ) acknowledges the Science and Technology Agency of Japan (Contract: 297040) for supporting this project. Work supported in part by the U.S. DOE under grant number DE-FG02-91ER40609. [50]{} A. De Shalit and M. Goldhaber, Phys. Rev. [**92**]{}, 1211(1953). I. Talmi, Rev. Mod. Phys. [**34**]{}, 704(1962). P. Federman and S. Pittel, Phys. Lett. [**69B**]{}, 385(1977); [**77B**]{}, 29(1977); Phys. Rev. C[**20**]{}, 820(1979); P. Federman, S. Pittel, and R. Campas, Phys. Lett. [**82B**]{}, 9(1979). R. F. Casten, Phys. Rev. Lett. [**54**]{}, 1991(1985); Nucl. Phys. [**A443**]{}, 1(1985); Phys. Rev. Lett. [**58**]{}, 658(1987). For a recent review, see R. F. Casten and N. V. Zamfir, J. Phys. G [**22**]{}, 1521(1996). S.L. Tabor, Phys. Rev. C[**34**]{}, 311(1986). H.Dejbakhsh, Phys. Lett. B [**210**]{}, 50(1988). D. Bucurescu et al., Phys. Lett. B [**229**]{}, 321(1989). P. Moeller, J. R. Nix, W. D. Myers, and W. J. Swiatecki, Atomic Data Nucl. Data Tables [**59**]{}, 185(1995). R. F. Casten et al., Phys. Rev. Lett. 47, 1433(1981). T.Motobayashi et al., Phys. Lett. B[**346**]{}, 9(1995). J. Dobaczewski and W. Nazarewicz, Phil. Trans. R. Proc. Lond. [**A356**]{}, 2007(1998). K. Heyde et al., Phys. Lett. B[**155**]{}, 303(1985). Y. M. Zhao and Y. Chen, Phys. Rev. C[**52**]{}, 1453(1995); A. Wolf and R. F. Casten, Phys. Rev. C[**36**]{}, 851(1987). O. Scholten, Phys. Lett. [**127B**]{}, 144(1983). S. T. Hsieh et al., J. Phys. G[**12**]{}, L167(1986); D. S. Chuu et al., Nucl. Phys. [**A482**]{}, 679(1988); C. S. Han et al., Phys. Rev. C[**42**]{}, 280(1990). Captions: [FIG. 1. The deformation parameter $e_2$ vs. $N_pN_n$. a)   for nuclei with 50$<$Z$\le$66 and 82$<$N$\le$104. Open symbols for Z=59-66 and N$\le$91. Solid symbols for all other nuclei (i.e., 50$<$Z$\le$58 for all neutron numbers and 59$\le$Z$\le$66 for N$\ge$92.); b)    66$<$Z$<$82 and 82$<$N$\le$104; c)    66$<$Z$<$82 and 104$<$N$<$126; d)    82$<$Z$\le$104 and 126$<$N$<$155. Note the scale change in part d) to accomodate the larger $N_pN_n$ values in this mass region. ]{} [FIG. 2. Similar to Fig. 1a) except Z=64 is used as a magic number instead of 82 for N$\le$91. ]{} [FIG. 3. Extract from Fig. 2 for even-odd nuclei, where different symbols are used to denote nuclei with 60$\le$Z$\le$66 and N$\le$91.]{} [FIG. 4. Summary of the effective valence proton numbers obtained in this work. ]{} Z$/$N 83 85 86 87 88 89 90 91 92 ------- ---- ---- ---- ---- ---- ---- ---- ---- ---- 59 5 7 7 7 7 7 9 9 9 60 4 6 6 6 8 8 10 10 10 61 5 7 7 7 7 7 9 11 11 62 4 6 6 6 8 8 10 10 12 63 7 7 7 7 7 7 9 11 13 64 4 6 6 6 8 8 10 12 14 65 5 7 7 7 7 7 9 11 15 66 4 6 6 6 8 8 10 12 16 : Effective proton numbers for nuclei near the Z=64 subshell. \[two\]

--- abstract: 'By Jahnke-Peternell-Radloff and Takeuchi, almost Fano threefolds with del Pezzo fibrations were classified. Among them, there exists 10 classes such that the existence of members of these was not proved. In this paper, we construct such examples belonging to each of 10 classes.' address: | Graduate School of Mathematical Sciences\ The University of Tokyo\ 3-8-1 Komaba\ Meguro-ku, Tokyo 153-8914, Japan author: - Takeru Fukuoka title: On the existence of almost Fano threefolds with del Pezzo fibrations --- Introduction ============ Throughout this paper, we work over the field of complex numbers ${\mathbb{C}}$. Let $X$ be a non-singular projective three dimensional variety. If $-K_{X}$ is ample, $X$ is called a *Fano* threefold. As a weaker condition than Fano, we consider a non-singular projective threefold with a nef and big anti-canonical divisor. This variety is called a *weak Fano* threefold. If $-K_{X}$ is not ample but nef and big, we call $X$ an *almost Fano* threefold. The classification of Fano threefolds is very important and many people studied it. Roughly speaking, we may regard almost Fano threefolds as “degenerated” Fano threefolds. In particular, the classification of almost Fano threefolds is related to that of singular Fano threefolds. For example, some Gorenstein terminal Fano threefolds have almost Fano threefolds as their small resolutions. Therefore, the classification of almost Fano threefolds is also an important problem. To classify (almost) Fano threefolds, we are interested in possible tuples of values of some invariants. As examples of invariants, we consider the *Picard rank* $\rho(X):=\dim_{{\mathbb{Q}}} (\operatorname{NS}(X) \otimes_{{\mathbb{Z}}}{\mathbb{Q}})$, the *anti-canonical degree* $(-K_{X})^{3}$, the *Hodge number* $h^{1,2}(X):=\dim H^{2}(X,\Omega_{X})$, and types of contractions of extremal rays of $X$. For each fixed tuple of values, we call the set $$\{X \mid \text{ the tuple of invariants of } X \text{ is equal to the fixed one }\}$$ the *class* corresponding to the fixed tuple. To classify (almost) Fano threefolds, we need to reveal whether each class has a member or not. The classification of Fano threefolds is given by the papers : [@Isk77; @Isk78], [@Fuj80; @Fuj81; @Fuj84], [@Sho79a], [@Sho79b],[@Take89], [@Muk95], [@MM81; @MM86; @MM03], and so on. In particular, they classified the possible tuples of values of some invariants and proved that each class has a member. [@Fanobook] is a survey of the classification of Fano varieties including these results. The classification of almost Fano threefolds $X$ with the minimum Picard rank, that is, $\rho(X) = 2$ has been most intensively studied. Here we recall some known results. By virtue of Mori theory, $X$ has the contraction of the $K_{X}$-negative ray ${\varphi}\colon X \to W$ and the contraction of the $K_{X}$-trivial ray $\psi \colon X \to {\overline}{X} $. Then $\psi$ contracts a divisor or finitely many curves. In the latter case, let $\chi \colon X {\dashrightarrow}X^{+}$ denote the flop of $\psi$ [@Kol89]. Thus we have two cases as the following diagrams. $$\xymatrix{ \text{Case (A)}&&X \ar[rd]^{\psi\text{ : divisorial }} \ar[ld]_{{\varphi}}& \\ &W&&{\overline}{X}. }$$ $$\xymatrix{ \text{Case (B)}&&X \ar[rd]^{\psi} \ar[ld]_{{\varphi}} \ar@{-->}[rr]^{\chi \text{ : flop }}&&X^{+}\ar[rd]^{{\varphi}^{+}} \ar[ld]_{\psi^{+}}& \\ &W&&{\overline}{X}&&V. }$$ - Jahnke-Peternell-Radloff treated Case (A) in [@JPR05]. They narrowed down tuples of values of invariants. The existence of a member of each class was proved except for the two classes appearing in TABLE \[table:A\] below. - In [@JPR11] (resp. [@CM13]), Jahnke-Peternell-Radloff (resp. Cutrone-Marshburn) treated Case (B) when ${\varphi}$ or ${\varphi}^{+}$ is not a divisorial contraction (resp. ${\varphi}$ and ${\varphi}^{+}$ are divisorial contraction ). They narrowed down tuples of values of invariants. The existence of a member of each class was proved except for some classes. - In [@Take09], Takeuchi treated Case (B) when ${\varphi}$ is a del Pezzo fibration of degree $d \neq 6$. He classified tuples of values of invariants and prove the existence of members of each class. Takeuchi’s works and Jahnke-Peternell-Radloff’s works are independent each other. - In [@Vol01], Vologodsky treated Case (B) when ${\varphi}$ and ${\varphi}^{+}$ are del Pezzo fibrations. He classified the possible values of $(-K_{X})^{3}$. Unfortunately, due to mistakes in [@Vol01 Proposition 2.2], there are some missing values in his classification. [@Take09], [@JPR11] and this paper fill the whole missing values of anti-canonical degrees by constructing examples. In this paper, we mainly consider the case where ${\varphi}$ is a del Pezzo fibration. In this case, it is known that $W={\mathbb{P}}^{1}$. By summarizing the above known results, there exists 10 classes such that it is yet to be known whether these have members or not. Our main theorem is to show the existence of members of each class. \[Mainthm\] Each class belonging to the following TABLE \[table:A\] and \[table:B\] has a member. In particular, there exists examples of each almost Fano threefold with a del Pezzo fibration that appears in [@JPR05] and [@JPR11]. Name ${\varphi}$ : $dP_{d}$ Type of $\psi \colon X \to {\overline}{X}$ $(-K_{X})^{3}$ $D_{X}=\operatorname{Exc}(\psi)$ $h^{1,2}(X)$ $\exists$ ------- ------------------------ -------------------------------------------- ---------------- ---------------------------------- -------------- ---------------- (A-1) $6$ $(g,d)=(1,6)$ $12$ $(-K_{X})-F$ “2” $\S$ \[ss:A1\] (A-2) $5$ $(g,d)=(1,5)$ $10$ $(-K_{X})-2F$ “6” $\S$ \[ss:A2\] : [Case (B)]{} \[table:A\] Name ${\varphi}$ : $dP_{d}$ $V$ ${\varphi}^{+} \colon X^{+} \to V$ $(-K_{X})^{3}$ $D_{X}$ $h^{1,2}(X)$ $\exists$ ----------- ------------------------ -------------------- ------------------------------------ ---------------- ---------------- -------------- ------------------- -- (B-i-1) 6 $B(4)$ $(g,d)=(1,6)$ $8$ $3(-K_{X})-2F$ 3 $\S$ \[ss:Bi1\] (B-i-2) 6 $V(10)$ $(g,d)=(1,6)$ $6$ $2(-K_{X})-F$ 3 $\S$ \[ss:Bi2\] (B-i-3) 6 $V(9)$ $(g,d)=(1,6)$ $4$ $3(-K_{X})-F$ 4 $\S$ \[ss:Bi3\] (B-ii) 6 ${\mathbb{P}}^{2}$ $\deg \text{(disc.)}=4$ $14$ $(-K_{X})-F$ “2” $\S$ \[ss:Bii\] (B-iii-1) 6 ${\mathbb{P}}^{1}$ $dP_{6}$ $12$ $(-K_{X})-F$ “2” $\S$ \[ss:Biii1\] (B-iii-2) 6 ${\mathbb{P}}^{1}$ $dP_{6}$ $6$ $2(-K_{X})-F$ “4” $\S$ \[ss:Biii2\] (B-iii-3) 6 ${\mathbb{P}}^{1}$ $dP_{6}$ $4$ $3(-K_{X})-F$ “3” $\S$ \[ss:Biii3\] (B-iii-4) 6 ${\mathbb{P}}^{1}$ $dP_{6}$ $2$ $6(-K_{X})-F$ “5” $\S$ \[ss:Biii4\] : [Case (B)]{} \[table:B\] [Notation for TABLE \[table:A\].]{} - The second row from the left denotes the degree of the del Pezzo fibration ${\varphi}\colon X \to {\mathbb{P}}^{1}$. - The third row from the left denotes the types of $\psi$. “$(g,d)$” means that $\psi$ is the blowing-up along a non-singular curve $C$ of genus $g$ with $(-K_{{\overline}{X}}).C=d$. - The fifth row from the left denotes the description of $D_{X}:=\operatorname{Exc}(\psi)$ in $\operatorname{\mathrm{Pic}}(X)={\mathbb{Z}}\cdot [-K_{X}] \oplus {\mathbb{Z}}\cdot [F]$. Here, $F$ is a general ${\varphi}$-fiber. - The second rows from the right denote the Hodge numbers $h^{1,2}(X)$ of members of a class. “$n$” means that there exists at least one member $X$ of the class such that $h^{1,2}(X)=n$. - The rightmost rows denote subsections including proofs of the existence of a member. [Notation for TABLE \[table:B\].]{} - The second row from the left denotes the degree of the del Pezzo fibration ${\varphi}\colon X \to {\mathbb{P}}^{1}$. - The third row from the left denotes the types of $V$. $B(m)$ (resp. $V(g)$) denotes a del Pezzo (resp. Mukai) threefold of degree $m$ (resp. genus $g$). - The fourth row from the left denotes the types of ${\varphi}^{+}$. - “$(g,d)$” means that ${\varphi}^{+}$ is the blowing-up along a non-singular curve $C$ of genus $g$ with $-K_{V}.C=i_{V} \cdot d$. Here, $i_{V}$ denotes the Fano index of $V$. - “deg(disc.)=$4$” means that ${\varphi}^{+} \colon X^{+} \to V$ is a conic bundle and the degree of the discriminant divisor is 4. Note that we obtain $V={\mathbb{P}}^{2}$ in this case. - “$dP_{6}$” means that ${\varphi}^{+} \colon X^{+} \to V$ is a del Pezzo fibration of degree $6$. In this case, we obtain $V={\mathbb{P}}^{1}$. - The fifth row from the left denotes the description of the divisor $D_{X}:=\chi^{-1}_{\ast}D_{X^{+}}$ in $\operatorname{\mathrm{Pic}}(X)={\mathbb{Z}}\cdot [-K_{X}] \oplus {\mathbb{Z}}\cdot[F]$. Here, $F$ is a general ${\varphi}$-fiber and $D_{X^{+}}:=\operatorname{Exc}({\varphi}^{+})$ (resp. ${\varphi}^{+\ast}{\mathcal}O_{{\mathbb{P}}^{2}}(1)$, ${\varphi}^{+\ast}{\mathcal}O_{{\mathbb{P}}^{1}}(1)$) in Case (B-i) (resp. (B-ii),(B-iii)). - The second rows from the right denote the Hodge numbers $h^{1,2}(X)$ of members of a class. “$n$” means that there exists at least one member $X$ of the class such that $h^{1,2}(X)=n$. The Hodge numbers $h^{1,2}(X)$ of an arbitrary member of classes in Case (B-i) are given in $\S$ \[ss:hodge\]. - The rightmost rows denote subsections including proofs of the existence of a member. We explain a sketch of proof of Theorem \[Mainthm\]. In Case (B-i), the construction of a member of the class is reduced to the construction of a Fano threefold containing an elliptic curve of degree 6. In Case (B-i-1), we can construct a member by standard arguments. In Case (B-i-2) and (B-i-3), we use theory of Néron-Severi lattices of K3 surfaces. This strategy is the same one in [@CM13]. In Case (A-1), Case (B-ii) and Case (B-iii), the idea of our construction of a member of those classes based on an elementary birational transformation as follows. Let ${\mathbb{Q}}^{2} {\subset}{\mathbb{P}}^{3}$ be a smooth quadric surface and take general three points $p_{1},p_{2},p_{3}$ on ${\mathbb{Q}}^{2}$. Then the linear span of three points $p_{1},p_{2},p_{3}$ is a plane and the intersection of the plane and ${\mathbb{Q}}^{2}$ is a conic $C$. Let ${\sigma}\colon F \to {\mathbb{Q}}^{2}$ be the blowing-up of ${\mathbb{Q}}^{2}$ at $p_{1},p_{2},p_{3}$ and ${\widetilde}{C}$ the proper transform of $C$. Then ${\widetilde}{C}$ is $(-1)$-curve and hence we obtain the blowing down ${\tau}\colon F \to S$ of ${\widetilde}{C}$. Note that $S$ is a del Pezzo surface of degree 6. The following proposition is a relativization of this birational transformation ${\mathbb{Q}}^{2} \gets F \to S$. See Proposition \[prop:1\] for precise statement. Let $\pi \colon W \to {\mathbb{P}}^{1}$ be a quadric fibration and $B {\subset}W$ a smooth curve and ${\tau}\colon Z:=\operatorname{\mathrm{Bl}}_{B}W \to W$ the blowing-up along $B$. We assume the following condition for a pair $(\pi \colon W \to {\mathbb{P}}^{1},B)$. $$\left\{ \begin{array}{ll} \deg (\pi|_{B} \colon B \to {\mathbb{P}}^{1})=3 \text{ and } \\ -K_{Z} \text{ is $p$-nef and $p$-big with } p := \pi \circ {\tau}\colon Z \to {\mathbb{P}}^{1}. \end{array} \right.$$ Then there exists a birational map $\Phi \colon Z {\dashrightarrow}Y$ over ${\mathbb{P}}^{1}$ and a birational morphism $\mu \colon Y \to X$ over ${\mathbb{P}}^{1}$. Moreover, the following holds. - $\Phi$ is isomorphic in codimension 1. - $\mu$ is the blowing-up along a ${\varphi}$-section $C$. Here, ${\varphi}\colon X \to {\mathbb{P}}^{1}$ is a structure morphism onto ${\mathbb{P}}^{1}$. - ${\varphi}\colon X \to {\mathbb{P}}^{1}$ is a del Pezzo fibration of degree $6$. - $(-K_{X})^{3}=\frac{3(-K_{W})^{3}-16g_{B}-32}{4}$ and $-K_{X}.C=\frac{8 (-K_{W}).B-24g_{B}-(-K_{W})^{3}-32}{8}$. This proposition is a variant of Takeuchi’s 2-ray game [@Take89]. By using this proposition, the construction problem can be reduced to the construction of a pair $(\pi \colon W \to {\mathbb{P}}^{1},B)$ satisfying the condition above. As an example, assume that $W$ is the blowing-up of a quadric threefold ${\mathbb{Q}}^{3}$ along a conic ${\Gamma}$ (resp. the blowing-up of ${\mathbb{P}}^{3}$ along an elliptic curve ${\Gamma}$ of degree $4$) . Then the construction of $(\pi \colon W \to {\mathbb{P}}^{1},B)$ is reduced to the construction of a pair of curve $(B,{\Gamma})$ in ${\mathbb{Q}}^{3}$ (resp. ${\mathbb{P}}^{3}$), where ${\Gamma}$ is a conic (resp. elliptic curve of degree $4$) and $B$ satisfies $\deg B=\#(B \cap {\Gamma})+3$ (resp. $2\deg B=\#(B \cap {\Gamma})+3$) . We construct such pair by using a theory of Néron-Severi lattices of K3 surfaces except for Case (B-iii-1) and Case (A-1). Actually, our constructions of a member of each classes use the quadric fibrations above except for Case (B-iii-3). In Case (B-iii-3), we use a $(2,2)$ divisor in ${\mathbb{P}}^{3} \times {\mathbb{P}}^{1}$ for $W$. In Case (A-2), our construction of a member is similarly based on the birational transformation of surfaces as follows. We take general 5 points $p_{1},\ldots,p_{5}$ in ${\mathbb{P}}^{2}$ and consider the conic $C$ passing through $p_{1},\ldots,p_{5}$. Let ${\sigma}\colon F \to {\mathbb{P}}^{2}$ be the blowing-up at $p_{1},\ldots,p_{5}$ and ${\widetilde}{C}$ be the proper transform of $C$. Then ${\widetilde}{C}$ is a $(-1)$-curve and hence we obtain the blowing-down ${\tau}\colon F \to S$ of ${\widetilde}{C}$ from $F$ onto a del Pezzo surface $S$ of degree $5$. We prove a relativization of this birational transformation (see Proposition \[prop:2\]) and use it to construct an example belonging to Case (A-2). We basically adopt the terminology of [@HarBook] and [@KMBook]. - For a closed subvariety $Y {\subset}X$, $N_{Y}X$ denotes the normal sheaf. - For a birational morphism ${\varphi}\colon X \to Y$ with relative Picard rank one, $\operatorname{Exc}({\varphi})$ denotes the exceptional set of ${\varphi}$. If $\operatorname{\mathrm{codim}}_{X}\operatorname{Exc}({\varphi})=1$, we call ${\varphi}$ a *divisorial* contraction. If $\operatorname{\mathrm{codim}}_{X}\operatorname{Exc}({\varphi}) \geq 2$, we call ${\varphi}$ a *small* contraction. Assume that $X$ is non-singular projective variety. If ${\varphi}$ is a divisorial contraction and $K_{X} \sim_{{\varphi}} 0$, we call ${\varphi}$ a *crepant* contraction. If ${\varphi}$ is a small contraction and $K_{X} \sim_{{\varphi}} 0$, we call ${\varphi}$ a *flopping* contraction. - A *Mori fiber space* is a contraction ${\varphi}\colon X \to S$ of a $K_{X}$-negative ray with non-singular projective variety $X$ and normal projective variety $S$ such that $\dim S<\dim X$. - A *del Pezzo fibration* is a Mori fiber space ${\varphi}\colon X \to S$ with $\dim X=3$ and $\dim S=1$. The *degree* of del Pezzo fibration ${\varphi}\colon X \to S$ is a anti-canonical degree $(-K_{F})^{2}$ for a general ${\varphi}$-fiber $F$. A del Pezzo fibration of degree 8 is called a *quadric fibration*. - A *conic bundle* is a Mori fiber space ${\varphi}\colon X \to S$ with $\dim X=3$ and $\dim S=2$. The *discriminant divisor* ${\Delta}$ of conic bundle ${\varphi}\colon X \to S$ is a divisor of $S$ given by ${\Delta}=\{x \in S \mid {\varphi}^{-1}(x) \text{ is singular } \}$. Note that a conic bundle ${\varphi}\colon X \to S$ is always flat and hence $S$ is non-singular. - For a non-singular Fano variety $V$, $i_{V}:=\max\{ i \in {\mathbb{Z}}_{>0} \mid -K_{V}=i \cdot H \text{ for some Cartier divisor } H\}$ denotes the *Fano index* of $V$. - ${\mathbb{Q}}^{n}$ denotes a $n$-dimensional smooth quadric hypersurface. - $B(m)$ is a *del Pezzo threefold* of degree $m$, which means $V=B(m)$ is a Fano threefold with $i_{V}=2$ and $(-K_{V})^{3}=8m$. - $V(g)$ is a *Mukai threefold* of genus $g$, which means $V=V(g)$ is a Fano threefold with $i_{V}=1$ and $(-K_{V})^{3}=2g-2$. - For a locally free sheaf ${\mathcal}E$, we set ${\mathbb{P}}({\mathcal}E):=\operatorname{Proj}\operatorname{\mathrm{Sym}}^{\bullet}{\mathcal}E$ and ${\mathcal}O_{{\mathbb{P}}({\mathcal}E)}(1)$ as the tautological bundle of ${\mathbb{P}}({\mathcal}E)$. - The ${\mathbb{P}}^{1}$-bundle ${\mathbb{P}}_{{\mathbb{P}}^{1}}({\mathcal}O \oplus {\mathcal}O(n))$ over ${\mathbb{P}}^{1}$, a Hirzebruch surface, is denoted by ${\mathbb{F}}_{n}$. ${\mathcal}O_{{\mathbb{F}}_{n}}(1)$ denotes the tautological bundle of ${\mathbb{F}}_{n}={\mathbb{P}}_{{\mathbb{P}}^{1}}({\mathcal}O \oplus {\mathcal}O(n))$. - We say that ${\varphi}\colon X \to S$ is a *${\mathbb{P}}^{2}$-bundle* if ${\varphi}$ is a projection morphism of a projective space bundle ${\mathbb{P}}({\mathcal}E) \to S$ associated to some locally free sheaf ${\mathcal}E$ of rank 3 over $S$. - For a birational map $f \colon X {\dashrightarrow}Y$ and a closed subscheme $S {\subset}X$, $f_{\ast}S$ and $S_{Y}$ denotes the proper transform of $S$. I am deeply grateful to Professor Hiromichi Takagi, my supervisor, for his valuable comments, suggestions and encouragement. I am also grateful to Professor Kiyohiko Takeuchi who showed me his unpublished works about the classification of weak Fano threefolds with Picard rank two. This paper is based on my master thesis at the Graduate School of Mathematical Sciences, The University of Tokyo. Preliminaries ============= Divisors on K3 surfaces {#subsec:K3} ----------------------- In this subsection, we review some theory for K3 surfaces. First, we recall a result of a theory of Néron-Severi lattices of K3 surfaces. We also recall the fact that the fundamental domain, for the Picard-Lefschetz reflection on the positive cone of a K3 surface, is the closure of the Kähler cone. We collect these results as follows. \[thm:NSL\] The following hold. 1. Let $\rho$ be an integer with $1 \leq \rho \leq 10$ and ${\Lambda}$ an even lattice with signature $(1,\rho-1)$. Then there exists a projective K3 surface $S$ and an isometry $i \colon {\Lambda}\to \operatorname{\mathrm{Pic}}(S)$. Here, the lattice structure of $\operatorname{\mathrm{Pic}}(S)$ is given by whose intersection form. 2. Let $H$ be a line bundle of a K3 surface $S$. If $H^{2}>0$, then there exists an isometry $i \colon \operatorname{\mathrm{Pic}}(S) \to \operatorname{\mathrm{Pic}}(S)$ such that $i(H)$ is nef and big. We review some characterizations of very ampleness or base point freeness of nef line bundles of K3 surfaces. This subject is mainly treated by [@SD74]. \[thm:SDR\] Let $L$ be a nef line bundle on a K3 surface $S$. Then the following hold. 1. The following are equivalent. 1. $L$ is globally generated. 2. $|L|$ is fixed part free. 3. There is no line bundle $D \in \operatorname{\mathrm{Pic}}(S)$ such that $D^{2}=0$ and $L.D=1$. 2. Assume $L^{2} \geq 4$. Then the following are equivalent. 1. $L$ is very ample. 2. There is no line bundle $D \in \operatorname{\mathrm{Pic}}(S)$ satisfying one of the following. 1. $D^{2}=-2$ and $L.D=0$. 2. $D^{2}=0$ and $L.D=1$ or $2$. 3. $D^{2}=2$ and $L \equiv 2D$. 3. Assume $L$ is base point free. If $L^{2} > 0$, then there exists a non-singular curve $C$ of $|L|$. If $L^{2}=0$, then there exists an elliptic curve $C$ such that $nC \in |L|$ for some $n>0$. In particular, if $L^{2}=0$ and $L$ is a generator of the lattice $\operatorname{\mathrm{Pic}}(S)$, there exists an elliptic curve $C$ such that $C \in |L|$. We prepare two easy lemmas to prove Theorem \[Mainthm\]. \[lem:1\] Let $L$ be an effective divisor on a K3 surface $S$ and $N$ be the fixed part of $|L|$. Then every irreducible component of $N$ is a $(-2)$-curve. Moreover, if $L^{2} \geq 0$, then $L-N \neq 0$. Let $C$ be a irreducible component of $N$. Then the multiplication map $H^{0}(S,{\mathcal}O(L-C)) \otimes H^{0}(S,{\mathcal}O(C)) \to H^{0}(S,{\mathcal}O(L))$ is bijective and hence $h^{0}(S,{\mathcal}O_{S}(C))=1$. By the Serre duality and the Riemann-Roch theorem, we have $C^{2}<0$ and hence $C$ is $(-2)$-curve. If $L^{2} \geq 0$, we have $h^{0}(L) \geq 2$ by the Riemann-Roch theorem and hence $L \neq N$. \[lem:2\] Let $S$ be a K3 surface and $H$ a very ample divisor with $H^{2}=6$. Then $S$ is a complete intersection of a quadric hypersurface $Q$ and a cubic hypersurface in ${\mathbb{P}}^{4}$. Moreover, $Q$ is smooth if there exists no effective divisors $C_{1},C_{2}$ such that $H=C_{1}+C_{2}$ , $C_{1}^{2}=C_{2}^{2}=0$ and $H.C_{1}=H.C_{2}=3$. Fix a closed embedding $S {\hookrightarrow}{\mathbb{P}}^{4}$ given by $|H|$. The former statement is well-known. Note that $Q$ is smooth along $S$ since $S$ is a non-singular Cartier divisor of $Q$. Let $q$ be a quadratic form defining $Q$. Since $S {\hookrightarrow}{\mathbb{P}}^{4}$ is non-degenerate, we have $\operatorname{rk}q \geq 3$. If $\operatorname{rk}q=3$, then $Q$ is isomorphic to an weighted projective space ${\mathbb{P}}(1,1,2,2)$. But it contradicts that $Q$ is smooth along $S$. If $\operatorname{rk}q = 4$, then we have $Q=\{x_{0}x_{3}=x_{1}x_{2}\} {\subset}{\mathbb{P}}^{4}=\operatorname{Proj}{\mathbb{C}}[x_{0},\ldots,x_{4}]$. Set $P_{01}:=\{x_{0}=x_{1}=0\}$, $P_{02}:=\{x_{0}=x_{2}=0\}$, $H_{0}=\{x_{0}=0\}$ and $C_{i}:=P_{0i} \cap S$. Note that $C_{i}$ is a divisor of $S$ since $S$ does not pass the vertex of $Q$. Since $C_{i}$ is a cubic curve of $P_{0i}$, we have $C_{i}^{2}=0$, $H.C_{i}=3$ and $H=C_{1}+C_{2}$. It contradicts our assumption. At the end of this subsection, we summarize a part of Mukai’s theory about K3 surfaces [@Muk95]. Let $(S,H)$ be a polarized K3 surface, that is a pair of a K3 surface $S$ and an ample divisor $H$. We say that $(S,H)$ is *Brill-Noether general* if $H=L+N$ for $L,N \in \operatorname{\mathrm{Pic}}(S)-\{0\}$, then $h^{0}(H)>h^{0}(L)h^{0}(N)$. Let $(S,H)$ be a Brill-Noether general polarized K3 surface. Assume that $H$ is very ample, $H^{2}=2g-2$ and $g \in \{7,8,9,10\}$. Then there exists a non-singular Mukai threefold $V$ of genus $g$ such that $V$ has $S$ as an anti-canonical member. Computations of $h^{1,2}$ {#ss:hodge} ------------------------- We can compute the Hodge numbers $h^{1,2}(X)$ of the members belonging to Case (B-i) by using the following well-known lemmas. \[lem:3\] Let $X$ be a non-singular projective threefold and $\chi \colon X {\dashrightarrow}X^{+}$ be a flop. Then, for all $i,j \in {\mathbb{Z}}_{\geq 0}$, $h^{i,j}(X)=h^{i,j}(X^{+})$. \[lem:4\] Let $V$ be a smooth projective variety and $B {\subset}V$ a smooth closed subvariety with $\operatorname{\mathrm{codim}}_{V}(B) \geq 2$. Let ${\widetilde}{V} \to V$ be the blowing-up along $B$. Then we obtain that $$b_{i}({\widetilde}{V})=b_{i}(V)+\sum_{j=1}^{\operatorname{\mathrm{codim}}_{V}B-1} b_{i-2j}(B), \text{ where } b_{i}(X):=\dim H^{i}(X,{\mathbb{Q}}).$$ In particular, $h^{1,2}({\widetilde}{V})=h^{1,2}(V)+h^{0,1}(B)$ holds. The Hodge number $h^{1,2}$ of a Fano threefold with Picard rank 1 is well-known [@Fanobook]. \[lem:h12ofFano\] We have $h^{1,2}(B(4))=2$, $h^{1,2}(B(5))=0$, $h^{1,2}(V(9))=3$ and $h^{1,2}(V(10))=2$. By using Lemma \[lem:3\], \[lem:4\] and \[lem:h12ofFano\], we obtain the Hodge numbers $h^{1,2}(X)$ for $X$ appearing in Case (B-i) as in TABLE \[table:B\]. Constructions ============= Case (B-i-1) {#ss:Bi1} ------------ Let us construct an example belonging to Case (B-i-1). Let $C {\subset}{\mathbb{P}}^{5}$ be an elliptic curve of degree 6. Then there exists a linear subvariety ${\mathbb{P}}^{5} {\subset}{\mathbb{P}}^{7}$ such that $C={\mathbb{P}}^{5} \cap ({\mathbb{P}}^{1})^{3}$, where $({\mathbb{P}}^{1})^{3} {\hookrightarrow}{\mathbb{P}}^{7}$ is the Segre embedding. In particular, $C$ is defined by quadratic equations. Thus we can take general quadric hypersurfaces $Q_{1},Q_{2}$ containing $C$ such that $V:=Q_{1} \cap Q_{2}$ is a non-singular del Pezzo threefold of degree 4. Let ${\varphi}^{+} \colon X^{+}:=\operatorname{\mathrm{Bl}}_{C}V \to V$ be the blowing-up along $C$ and set $H:={\varphi}^{+\ast}{\mathcal}O_{{\mathbb{P}}^{5}}(1)|_{V}$. Since $C$ is defined by quadratic equations, we obtain that $-K_{X^{+}}=2H-D$ is nef. Also we obtain that $(-K_{X^{+}})^{3}=8$ by a straightforward calculation. In particular, $-K_{X^{+}}$ is nef and big. By the classification in [@MM81], $-K_{X^{+}}$ is not ample and thus there exists the contraction of the $K_{X^{+}}$-trivial ray $\psi \colon X^{+} \to {\overline}{X}$. $\psi^{+}$ is not a divisorial contraction by the classification in [@JPR05] and thus there exists the flop $\eta \colon X^{+} {\dashrightarrow}X$ of $\psi$ and the contraction of the $K_{X}$-negative ray ${\varphi}\colon X \to V$. By the classification in [@CM13] and [@JPR11], ${\varphi}$ is neither a divisorial contraction nor a conic bundle. Hence ${\varphi}$ is a del Pezzo fibration. By the classification in [@JPR11], ${\varphi}\colon X \to {\mathbb{P}}^{1}$ is an example of Case (B-i-1). ------------------------------------------------------------------------ Case (B-i-2) {#ss:Bi2} ------------ In order to construct an example belonging to Case (B-i-2), we consider the following lattice : $$\begin{pmatrix} &H&C\\ H&18&6\\ C&6&0 \end{pmatrix}.$$ This is an even lattice with signature $(1,1)$. By Theorem \[thm:NSL\], there exists K3 surface $S$ such that $\operatorname{\mathrm{Pic}}(S)={\mathbb{Z}}\cdot H \oplus {\mathbb{Z}}\cdot C$ is isometry to this lattice. Moreover, we may assume that $H$ is nef and big. For every $D=xH+yC \in \operatorname{\mathrm{Pic}}(S)={\mathbb{Z}}\cdot H \oplus {\mathbb{Z}}\cdot C$, we have $$D^{2}=18x^{2}+12xy \text{ and } H.D=18x+6y.$$ Note that $D^{2}$ and $H.D$ are multiple by $6$. In particular, $S$ has no $(-2)$-curve and hence every effective divisor on $S$ is base point free. Thus $H$ is very ample by Theorem \[thm:SDR\]. Since $C^{2}=0$ and $H.C=6$, $C$ is effective and hence has no base point. Therefore, we may assume that $C$ is non-singular. \[lem:BN1\] This polarized K3 surface $(S,H)$ is Brill-Noether general. Let $L$ and $N$ be non-zero effective divisors such that $H=L+N$ and set $L=xH+yC$ for some $x,y \in {\mathbb{Z}}$. Thus we have $N=(1-x)H-yC$. Since $L$ and $N$ are effective, we have $H.L>0$, $H.N>0$, $C.L \geq 0$, $C.N \geq 0$, $L^{2} \geq 0$ and $N^{2} \geq 0$. By this observation, we obtain the following: $$0 \leq x \leq 1, \quad 0 \leq 3x+y <3 \text{ and } 0 \leq 3x+2y \leq 3.$$ By a straightforward calculation, the pairs of integers satisfying this inequalities are $(x,y)=(0,1),(1,-1)$. In each case, $h^{0}(L)h^{0}(N)=10 < h^{0}(H)=11$. Therefore, there exists a Mukai threefold $V$ of genus $10$ having $S$ as an anti-canonical divisor. Let ${\varphi}^{+} \colon X^{+}:=\operatorname{\mathrm{Bl}}_{C}V \to V$ denotes the blowing-up of $V$ along $C$. Since $C {\subset}S$, we have $S \simeq ({\varphi}^{+})^{-1}_{\ast}S \in |-K_{X^{+}}|$. It is clear that $-K_{X^{+}}$ is nef if and only if $-K_{X^{+}}|_{S}$ is nef. For the line bundle $-K_{X^{+}}|_{S}=H-C$, we have $(H-C)^{2}=6$ and $H.(H-C)=12$ and hence $|H-C|$ is base point free. Hence $-K_{X^{+}}$ is nef and $(-K_{X^{+}})^{3}=6$. Thus $-K_{X^{+}}$ is nef and big. By applying similar arguments in $\S$ \[ss:Bi1\], we obtain the flop $\chi \colon X^{+} {\dashrightarrow}X$ and a del Pezzo fibration ${\varphi}\colon X \to {\mathbb{P}}^{1}$ of degree $6$ which belongs to Case (B-i-2). ------------------------------------------------------------------------ Case (B-i-3) {#ss:Bi3} ------------ In order to construct an example belonging to Case (B-i-3), we consider the following lattice : $$\begin{pmatrix} &H&C\\ H&16&6\\ C&6&0 \end{pmatrix}.$$ This is an even lattice with signature $(1,1)$. By Theorem \[thm:NSL\], there exists a K3 surface $S$ such that $\operatorname{\mathrm{Pic}}(S)={\mathbb{Z}}\cdot H \oplus {\mathbb{Z}}\cdot C$ is isometry to this lattice and $H$ is nef and big. For every $D=xH+yC \in \operatorname{\mathrm{Pic}}(S)$ with $x,y \in {\mathbb{Z}}$, we have $$D^{2}=16x^{2}+12xy \text{ and } H.D=16x+6y.$$ Note that $D^{2}$ is multiple by $4$. In particular, $S$ has no $(-2)$-curve and every effective divisor on $S$ is base point free. By using Theorem \[thm:SDR\], it is easy to see that $H$ is very ample. Since $C^{2}=0$ and $H.C=6$, $C$ is an effective divisor and hence it is base point free. Therefore, we may assume that $C$ is non-singular by Theorem \[thm:SDR\]. The following lemma can be proved similarly to Lemma \[lem:BN1\]. \[lem:BN2\] This polarized K3 surface $(S,H)$ is Brill-Noether general. Therefore, there exists a Mukai threefold $V$ of genus $9$ having $S$ as an anti-canonical divisor. Let ${\varphi}^{+} \colon \operatorname{\mathrm{Bl}}_{C}V=:X^{+} \to V$ denotes the blowing-up of $V$ along $C$. By a straightforward calculation, $(-K_{X^{+}})^{3}=4$ holds. By the similar argument in $\S$ \[ss:Bi2\], $-K_{X}$ is nef and big and we obtain the flop $\chi \colon X^{+} {\dashrightarrow}X$ and a del Pezzo fibration ${\varphi}\colon X \to {\mathbb{P}}^{1}$ of degree $6$ which belongs to Case (B-i-3). ------------------------------------------------------------------------ Quadric fibrations with tri-sections {#ss:dP6dP8} ------------------------------------ In this subsection, we prove Proposition \[prop:1\]. This proposition plays an important role on the constructions of the examples belonging to Case (A-1) , (B-ii) and (B-iii). In order to prove Proposition \[prop:1\], we prepare the following two lemmas. \[lem:5\] Let $Y$ be a smooth projective threefold and $S$ a smooth projective surface and $g \colon Y \to S$ and ${\varphi}\colon S \to {\mathbb{P}}^{1}$ morphisms. Assume $g$ is a contraction of $K_{Y}$-negative ray over ${\mathbb{P}}^{1}$. Set $\pi:={\varphi}\circ g$. $$\xymatrix{ Y\ar[r]^{g} \ar[rd]_{\pi} & S \ar[d]^{{\varphi}} \\ &{\mathbb{P}}^{1} }$$ If $\rho(Y)=3$ and $\pi^{\ast}{\mathcal}O(1).(-K_{Y})^{2}>0$, then $S \simeq {\mathbb{F}}_{n}$ and ${\varphi}\colon S={\mathbb{F}}_{n} \to {\mathbb{P}}^{1}$ is the ${\mathbb{P}}^{1}$-bundle structure. Let ${\Delta}{\subset}S$ be a discriminant divisor of the conic bundle $g \colon Y \to S$. Then $g_{\ast}(-K_{Y})^{2} \equiv -(4K_{S}+{\Delta})$ holds [@MM86 Corollary 4.6]. Let $F$ be a ${\varphi}$-fiber. By assumptions of this lemma, we have that $-F.(4K_{S}+{\Delta})=F.g_{\ast}(-K_{Y})^{2}=g^{\ast}F.(-K_{Y})^{2}=\pi^{\ast}{\mathcal}O(1).(-K_{Y})^{2}>0$. Hence we have $4K_{S}.F<-{\Delta}.F \leq 0$, which means $-K_{S}$ is ${\varphi}$-ample. Since $\rho(S)=2$, all fibers of ${\varphi}$ are isomorphic to ${\mathbb{P}}^{1}$. \[lem:6\] Let $f \colon X \to S$, $f' \colon X' \to S'$ be Mori fiber spaces and $\Phi \colon X {\dashrightarrow}X'$ an isomorphism in codimension 1. Assume there exists a rational map ${\varphi}\colon S {\dashrightarrow}S'$ such that the following diagram is commutative: $$\xymatrix{ X\ar[d]_{f} \ar@{-->}[r]^{\Phi} & X' \ar[d]^{f'} \\ S \ar@{-->}[r]_{{\varphi}} & S'. }$$ Then $\Phi$ and ${\varphi}$ are isomorphic. See [@Cor95 Proposition 3.5]. \[prop:1\] Let $W$ be a smooth projective threefold with $\rho(W)=2$, $\pi \colon W \to {\mathbb{P}}^{1}$ a quadric fibration, $B {\subset}W$ a smooth projective curve and ${\tau}\colon Z:=\operatorname{\mathrm{Bl}}_{B}W \to W$ the blowing-up along $B$. We assume the following condition $(\dag_{6})$ for the pair $(\pi \colon W \to {\mathbb{P}}^{1},B)$. $$(\dag_{6}) \cdots \left\{ \begin{array}{ll} \deg (\pi|_{B} \colon B \to {\mathbb{P}}^{1})=3 \text{ and }\\ -K_{Z} \text{ is $p$-nef and $p$-big with } p := \pi \circ {\tau}\colon Z \to {\mathbb{P}}^{1}. \end{array} \right.$$ Then the following hold. 1. The morphism from $Z$ to the relative anti-canonical model over ${\mathbb{P}}^{1}$ $$\Psi \colon Z \to {\overline}{Z}:=\operatorname{Proj}_{{\mathbb{P}}^{1}}\bigoplus_{n \geq 0} p_{\ast}{\mathcal}O_{Z}(-nK_{Z})$$ is a small contraction or an isomorphism. 2. Define a variety $Y$ and a birational map $\Phi \colon Z {\dashrightarrow}Y$ over ${\mathbb{P}}^{1}$ as follows: - If $-K_{Z}$ is $p$-ample, we set $Y:=Z$ and $\Phi:=\operatorname{id}_{Z} \colon Z \to Y$, or - if $-K_{Z}$ is not $p$-ample, let $\Phi \colon Z {\dashrightarrow}Y$ be the flop of $\Psi \colon Z \to {\overline}{Z}$. Let $q \colon Y \to {\mathbb{P}}^{1}$ be a structure morphism onto ${\mathbb{P}}^{1}$ and $\mu \colon Y \to X$ a $K_{Y}$-negative ray contraction over ${\mathbb{P}}^{1}$. If $-K_{Z}$ is $p$-ample, we choose the other ray which is different to the one corresponding to ${\tau}\colon Z \to W$. Let ${\varphi}\colon X \to {\mathbb{P}}^{1}$ denotes the structure morphism onto ${\mathbb{P}}^{1}$. Then $\mu$ is the blowing-up along a ${\varphi}$-section $C$ and ${\varphi}\colon X \to {\mathbb{P}}^{1}$ is a del Pezzo fibration of degree 6. $$\xymatrix{ &D \ar[ld]& \ar@{}[l]|{{\subset}} \ar[ld]_{\mu}Y=\operatorname{\mathrm{Bl}}_{C}X \ar[dd]^{q}\ar@{<--}[r]^{\Phi}& Z=\operatorname{\mathrm{Bl}}_{B}W \ar@{}[r]|{{\supset}} \ar[rd]^{{\tau}} \ar[dd]_{p}& E \ar[rd]& \\ C&\ar@{}[l]|{{\subset}} X \ar[rd]_{{\varphi}}& && W \ar@{}[r]|{{\supset}} \ar[ld]^{\pi} &B&(\bigstar_{6}) \\ &&{\mathbb{P}}^{1} \ar@{=}[r]&{\mathbb{P}}^{1}&& }$$ 3. Let $F_{Y}$ be a general $q$-fiber and we set $D:=\operatorname{Exc}(\mu)$, $E:=\operatorname{Exc}({\tau})$ and $E_{Y}:=\Phi_{\ast}E$. Then we obtain $$D \equiv \frac{1}{2}(-K_{Y})-\frac{1}{2}E_{Y}+zF_{Y}.$$ Moreover, the following equalities hold: $$\begin{aligned} (-K_{X})^{3}&=\frac{3(-K_{W})^{3}-16g_{B}-32}{4}, \\ -K_{X}.C&=\frac{8 (-K_{W}).B-24g_{B}-(-K_{W})^{3}-32}{8} \text{ and } \\ z&=\frac{4(-K_{W}).B-8g_{B}-(-K_{W})^{3}}{8}.\end{aligned}$$ <!-- --> 1. Assume $\Psi \colon Z \to {\overline}{Z}$ is a divisorial contraction. Set $D:=\operatorname{Exc}(\Psi)$. Let $F_{Z}$ be a general $p$-fiber and $F_{W}$ be a general $\pi$-fiber such that ${\tau}(F_{Z})=F_{W}$. Now ${\tau}|_{F_{Z}} \colon F_{Z} \to F_{W}\simeq {\mathbb{P}}^{1} \times {\mathbb{P}}^{1}$ is the blowing-up at reduced three points. Let us prove that $D|_{F_{Z}}$ is a disjoint union of $(-2)$-curves $l_{1},\ldots,l_{n}$ and $n \in \{1,2\}$. Let $e_{1},e_{2},e_{3}$ be exceptional curves of ${\tau}|_{F_{Z}}$ and set $f_{1}:=({\tau}|_{F_{Z}})^{\ast}{\mathcal}O_{{\mathbb{P}}^{1} \times {\mathbb{P}}^{1}}(1,0)$ and $f_{2}:=({\tau}|_{F_{Z}})^{\ast}{\mathcal}O_{{\mathbb{P}}^{1} \times {\mathbb{P}}^{1}}(0,1)$. For every irreducible curve $l {\subset}F_{Z}$, $l=a_{1}f_{1}+a_{2}f_{2}-\sum_{j=1}^{3} b_{j}e_{j}$ holds for some $a_{i},b_{j} \in {\mathbb{Z}}_{\geq 0}$. If $l$ is contracted by ${\tau}|_{F_{Z}}$, then $l$ is $(-2)$-curve and we have the following: $$0=2(a_{1}+a_{2})-\sum_{j=1}^{3}b_{j} \text{ and } -2=2a_{1}a_{2}-\sum_{j=1}^{3}b_{j}^{2}.$$ By the inequality $$2+2a_{1}a_{2}=\sum_{j=1}^{3}b_{j}^{2} \geq 3 \left( \frac{\sum_{j=1}^{3}b_{j}}{3} \right)^{2}=\frac{4}{3}(a_{1}+a_{2})^{2},$$ we have $(a_{1},a_{2})=(0,1)$ or $(1,0)$. Thus we have $(b_{1},b_{2},b_{3})=(1,0,0)$ or $(0,1,0)$ or $(0,0,1)$ and hence $l$ is the proper transform of a line passing through exactly two points of the three points. By our assumption, $-K_{F_{Z}}$ is nef and hence the three points are not collinear. Thus the number of lines passing through exactly two points of the three points is at most two. Moreover, if there exists two lines passing through two points of the three points, then the proper transforms of those do not meet since the two lines meet transversally at one point of the three point. Denote $D \equiv x(-K_{Z})+yE+zF_{Z}$ with $x,y,z \in {\mathbb{Q}}$. Then $D|_{F_{Z}} \equiv x(-K_{F_{Z}})+y(E|_{F_{Z}})$ and hence we have the following: $$\begin{aligned} 0&=(-K_{F_{Z}})(D|_{F_{Z}})=5x+3y \text{ and } \\ -2n&=(D|_{F_{Z}})^{2}=5x^{2}-3y^{2}+6xy. \end{aligned}$$ Thus we have $x=\pm \frac{\sqrt{15n}}{10}$. It contradicts $x \in {\mathbb{Q}}$ and $n \in \{1,2\}$. Therefore, $\Psi$ is an isomorphism or a small contraction. 2. For a general $p$-fiber $F_{Z}$ and the proper transform $F_{Y}:=\Phi_{\ast}F_{Z}$, the birational map $\Phi|_{F_{Z}} \colon F_{Z} {\dashrightarrow}F_{Y}$ is an isomorphism. We abbreviate $F_{Y},F_{Z}$ to $F$ for simplicity. Let us prove that $\mu \colon Y \to X$ is the blowing-up along a non-singular curve $C$. Since $\rho(Y)=3$, then $\dim X \geq 2$. If $\dim X=2$, $X$ is isomorphic to a Hirzebruch surface ${\mathbb{F}}_{n}$ by Lemma \[lem:5\]. Set $D=g^{\ast}{\mathcal}O_{{\mathbb{F}}_{n}}(1)$ and $D \equiv x(-K_{Z})+yE+zF_{Z}$ with $x,y,z \in {\mathbb{Q}}$. Then we have the following equations: $$\begin{aligned} 2&=(-K_{Y}).D.F=(-K_{F})D|_{F}=5x+3y \text{ and } \\ 0&=D^{2}F=(D|_{F})^{2}=5x^{2}-3y^{2}+6xy.\end{aligned}$$ Hence $x=\frac{4 \pm \sqrt{6}}{10}$, $ y=\mp \frac{1}{\sqrt{6}}$ and it contradicts $x,y \in {\mathbb{Q}}$. Hence $\mu$ is a divisorial contraction. Set $D=\operatorname{Exc}(\mu)$. If $\mu(D)$ is a point, $D$ does not meet the general fibers $F$ of $Y \to {\mathbb{P}}^{1}$. Thus we have the following: $$\begin{aligned} 0&=(-K_{Y}).D.F=(-K_{F})D|_{F}=5x+3y \text{ and } \\ 0&=D^{2}F=(D|_{F})^{2}=5x^{2}-3y^{2}+6xy.\end{aligned}$$ Hence we have $x=y=0$ and $D=zF$. In particular, $D$ or $-D$ is nef which is impossible. Therefore, $\dim \mu(D)=1$ and $\mu$ is the blowing-up along the smooth curve $C:=\mu(D)$. Let $F_{X}$ denotes a general ${\varphi}$-fiber. Set $D \equiv x(-K_{Z})+yE+zF$ with $x,y,z \in {\mathbb{Q}}$ and $m:=F_{X}.C \in {\mathbb{Z}}_{\geq 0}$. Since $(-K_{F_{Y}})^{2}=5$ and $(-K_{F_{X}})^{2} \leq 9$, we have $0 \leq m \leq 4$. By a calculation of some intersection numbers, we have the following: $$\begin{aligned} m&=(-K_{Y})DF=5x+3y, \\ -m&=D^{2}F=5x^{2}-3y^{2}+6xy \text{ and } \\ {\mathbb{Z}}&\ni 3x-3y=EFD. \end{aligned}$$ By these equality, we have the following: $$\begin{aligned} &x=\frac{4m \pm \sqrt{6m^{2}+30m}}{20}, y=\mp \sqrt{\frac{m^{2}+5m}{24}} \text{ and } \\ &3x-3y=\frac{3m \pm \sqrt{24m(m+5)}}{32} \in {\mathbb{Z}}.\end{aligned}$$ Then the possibilities of $(m,x,y)$ are as follows: $$(m,x,y)=(0,0,0),\left(1,\frac{1}{2},-\frac{1}{2} \right) , (3,0,1).$$ It is impossible that $x=y=0$ as we have seen. If $(x,y)=(0,1)$, then we have $D|_{F}=E|_{F}$ for a general $F$ and hence the birational map $\psi \colon W {\dashrightarrow}X$ is isomorphic in codimension 1. Since $F_{W}$ is birationally transformed into $F_{X}$ by the birational map $\psi \colon W {\dashrightarrow}X$, $\psi$ is an isomorphism over ${\mathbb{P}}^{1}$ by Lemma \[lem:6\]. Moreover, we have $\psi(C)=B$. In the case that $-K_{Z}$ is $p$-ample, it contradicts that the rays which corresponding ${\tau}$, $\mu$ are different. In the case that $-K_{Z}$ is not $p$-ample, it contradicts that $\Psi \colon Y {\dashrightarrow}Z$ is not an isomorphism over ${\mathbb{P}}^{1}$. Therefore, we obtain that $(m,x,y)=\left(1,\frac{1}{2},-\frac{1}{2} \right)$ and hence $(-K_{F_{X}})^{2}=6$. Therefore, ${\varphi}|_{C} \colon C \to {\mathbb{P}}^{1}$ is isomorphic and ${\varphi}\colon X \to {\mathbb{P}}^{1}$ is a del Pezzo fibration of degree $6$. 3. Since $x=\frac{1}{2}$, $y=-\frac{1}{2}$, $(-K_{Z})^{3}=(-K_{Y})^{3}$, $(-K_{Z})^{2}D=(-K_{Y})^{2}D_{Y}$ and $(-K_{Z})D^{2}=(-K_{Y})D_{Y}^{2}$, we obtain the following: $$\begin{aligned} &(-K_{X})^{3}-2(-K_{X}).C-2=(-K_{W})^{3}-2(-K_{W}).B+(2g_{B}-2), \\ &(-K_{X}).C+2=\frac{(-K_{W})^{3}-2(-K_{W}).B+(2g_{B}-2) }{2}-\frac{(-K_{W}).B+2-2g_{B}}{2}+5z \text{ and } \\ &-2=\frac{(-K_{W})^{3}-2(-K_{W}).B+(2g_{B}-2)}{4} +\frac{2g_{B}-2}{4}-\frac{(-K_{W}).B+2-2g_{B}}{2}+2z.\end{aligned}$$ The last equalities in Proposition \[prop:1\] are obtained by solving these equations. Case (B-ii) {#ss:Bii} ----------- In order to construct an example belonging to Case (B-ii), we consider the following lattice: $$\begin{pmatrix} & H & F & B \\ H & 6 & 4 & 6 \\ F & 4 & 0 & 3 \\ B & 6 & 3 & 2 \end{pmatrix}.$$ This is an even lattice with signature $(1,2)$. By virtue of Theorem \[thm:NSL\], there exists a K3 surface $S$ such that $\operatorname{\mathrm{Pic}}(S)={\mathbb{Z}}\cdot [H] \oplus {\mathbb{Z}}\cdot [F] \oplus {\mathbb{Z}}\cdot [B]$ is isometry to this lattice. Moreover, we can assume that $H$ is nef and big. Set $C_{2}=H-F$ and $C_{4}=H+F-B$. Then we have $(C_{i})^{2}=-2$ and $H.C_{i}=i$ for $i \in \{2,4\}$. Since $H$ is nef, $C_{2},C_{4}$ are $(-2)$-curves. Let $C$ be a divisor of $S$. Then there exists $x,y,z \in {\mathbb{Z}}$ such that $C=xH+yF+zB$ and we have $$\begin{aligned} C^{2}=6x^{2}+2z^{2}+8xy+12xz+6yz \text{ and } H.C=6x+4y+6z. \end{aligned}$$ By solving these equations for $x,y$, we have the following: $$\begin{aligned} &x=\frac{2(H.C)-9z \pm \sqrt{4(H.C)^{2}-24C^{2}-87z^{2}}}{12} \text{ and } \\ &y=\frac{-3z \mp \sqrt{4(H.C)^{2}-24C^{2}-87z^{2}}}{8}.\end{aligned}$$ Hence $x,y,z \in {\mathbb{Z}}$ and $H.C$ is even, we can prove that the following (i)-(iv) by a straightforward calculation. - It is impossible that $C^{2}=0$ and $H.C=1$. - It is impossible that $C^{2}=0$ and $H.C=2$. - It is impossible that $C^{2}=0$ and $H.C=3$. - If $C^{2}=-2$ and $H.C \leq 9$, then $C=C_{2}$ or $C_{4}$. \[lem:Bii\] 1. $H$ is very ample and $S$ is embedded to ${\mathbb{Q}}^{3}$ by $|H|$ as an anti-canonical member. 2. The linear systems $|F|$ and $|B|$ have non-singular members and the general members meet transversely in three point. 3. $2H+F-B$ is nef. <!-- --> 1. By (i),(ii),(iv) and Theorem \[thm:SDR\], $H$ is very ample. By (iii) and Lemma \[lem:2\], $S$ is embedded in ${\mathbb{Q}}^{3}$ as an anti-canonical divisor. 2. By Theorem \[thm:SDR\], it is enough to show that $|F|$ and $|B|$ are movable. Let us prove that $|F|$ is movable. Since $H.F=4$ and $F^{2}=0$, $F$ is effective. Let $M$ be the movable part of $|F|$ and $N$ the fixed part. Then we have $4=H.F>H.N$. By Lemma \[lem:1\] and (iv), we have $N=aC_{2}$ for some $a \in {\mathbb{Z}}_{\geq 0}$. Since $M$ is movable, we have $0 \leq M.F=(F-aC_{2}).F=-4a$ and hence $a=0$. Therefore, $F=M$ is movable. Let us prove that $|B|$ is movable. By the similar argument, there exists $a,b \in {\mathbb{Z}}_{\geq 0}$ such that $B=M+aC_{2}+bC_{4}$, where $M$ is the movable part of $|B|$. Since $M$ is movable, we have $0<M^{2}=2-2a^{2}-2b^{2}-6a-14b+6ab$ and it is easy to find that $(a,b)=(0,0)$. Hence $B=M$ is movable. 3. Since $(2H+F-B)^{2}=12$ and $H.(2H+F-B)=10$, we obtain that $2H+F-B$ is effective. Let $M$ be the movable part and $N$ the fixed part of $2H+F-B$. Since $10=H.(2H+F-B)>H.N$ and Lemma \[lem:1\] and (iv), there exists $a,b \in {\mathbb{Z}}_{\geq 0}$ such that $2H+F-B=M+aC_{2}+bC_{4}$. Since $(2H+F-B).C_{2}=5$ and $(2H+F-B).C_{4}=2$, $2H+F-B$ is nef. By Lemma \[lem:Bii\] (1), $S$ is a smooth anti-canonical member of ${\mathbb{Q}}^{3}$ and $C_{2} {\subset}S$ is a conic in ${\mathbb{Q}}^{3}$. Let ${\sigma}\colon W:=\operatorname{\mathrm{Bl}}_{C_{2}}{\mathbb{Q}}^{3} \to {\mathbb{Q}}^{3}$ denotes the blowing-up of ${\mathbb{Q}}^{3}$ along $C_{2}$. It is well-known that $W$ is a Fano threefold and there exists a quadric fibration $\pi \colon W \to {\mathbb{P}}^{1}$ that is given by the complete linear system $|{\sigma}^{\ast}{\mathcal}O_{{\mathbb{Q}}^{3}}(1)-\operatorname{Exc}({\sigma})|$ [@MM81 No.29 of Table 3]. Note that $S_{W} \simeq S$ and $B_{W} \simeq B$. In particular, we can treat divisors of $S_{W}$ as if those were divisors of $S$. Then we obtain $\pi^{\ast}{\mathcal}O_{{\mathbb{P}}^{1}}(1)|_{S_{W}}={\mathcal}O_{S}(F)$ for $F=H-C_{2}$ and thus $\deg(\pi|_{B_{W}} \colon B_{W} \to {\mathbb{P}}^{1})=3$. Let $Z:=\operatorname{\mathrm{Bl}}_{B_{W}}W$ denotes the blowing-up of $W$ along $B_{W}$. Note that $S_{Z} \simeq S$ is a member of $|-K_{Z}|$ and $(-K_{Z})|_{S_{Z}}={\mathcal}O_{S}(2H+F-B)$. The nefness of $-K_{Z}$ follows by Lemma \[lem:Bii\] (4). Moreover, we have $(-K_{Z})^{3}=12$ and hence $-K_{Z}$ is nef and big. By these arguments, $(\pi \colon W \to {\mathbb{P}}^{1},B)$ satisfies the condition $(\dag_{6})$ in Proposition \[prop:1\]. Therefore, we obtain a del Pezzo fibration ${\varphi}\colon X \to {\mathbb{P}}^{1}$ of degree 6 satisfying the following : $$(-K_{X})^{3}=14,\quad -K_{X}.C=0 \text{ and } D \equiv \frac{1}{2}(-K_{X})-\frac{1}{2}E-\frac{1}{2}F.$$ Since $Y$ is almost Fano and $-K_{X}.C=0$, $X$ is almost Fano. By TABLE \[table:A\] and TABLE \[table:B\], $X$ belongs to Case (B-ii). By Lemma \[lem:3\] and \[lem:4\], we have $h^{1,2}(X)=2$. ------------------------------------------------------------------------ Case (B-iii-1) {#ss:Biii1} -------------- Our construction of an example of Case (B-iii-1) does not need Theorem \[thm:NSL\]. Let $S_{0} {\subset}{\mathbb{P}}^{3}$ be a smooth cubic surface and ${\varepsilon}\colon S_{0} \to {\mathbb{P}}^{2}$ the blowing-up at 6 points in general position. Let $h$ (resp. $e_{1},\ldots,e_{6}$) denotes ${\sigma}^{\ast}{\mathcal}O_{{\mathbb{P}}^{2}}(1)$ (resp. ${\varepsilon}$-exceptional curves). Consider the following curves of $S_{0}$: $$B \in \left| 3h-\sum_{i=1}^{5} e_{i} \right| \text{ and } {\Gamma}\in \left| 3h-\sum_{i=2}^{6} e_{i} \right|.$$ Note that general $B$ and ${\Gamma}$ are degree 4 elliptic curves in ${\mathbb{P}}^{3}$ and meet transversely in 5 points. Let ${\sigma}\colon W:=\operatorname{\mathrm{Bl}}_{{\Gamma}}{\mathbb{P}}^{3} \to {\mathbb{P}}^{3}$ denotes the blowing-up of ${\mathbb{P}}^{3}$ along ${\Gamma}$ and $G$ denotes the ${\sigma}$-exceptional divisor. It is well-known that $W$ is a Fano threefold and a morphism $\pi \colon W \to {\mathbb{P}}^{1}$ that is given by the complete linear system $|{\sigma}^{\ast}{\mathcal}O_{{\mathbb{P}}^{3}}(2)-\operatorname{Exc}({\sigma})|$ is a quadric fibration [@MM81 No.25 of Table 3]. Let ${\tau}\colon Z:=\operatorname{\mathrm{Bl}}_{B_{W}}W \to W$ denotes the blowing-up of $W$ along $B_{W}$. Set $E:=\operatorname{Exc}({\tau})$ and $H:=f^{\ast}{\sigma}^{\ast}{\mathcal}O_{{\mathbb{P}}^{3}}(1)$. $-K_{Z}$ is nef and $(-K_{Z})^{3}=10$. $(-K_{Z})^{3}=10$ is followed by a straightforward calculation. Note that $-K_{Z}=4H-G_{Z}-E$ and $S_{0,Z}=3H-G_{Z}-E$. Since $-K_{Z}=H+S_{0,Z}$ and $H$ is nef, it is enough to show that $-K_{Z}|_{S_{0,Z}}$ is nef. Under an identification of $S_{0,Z} \simeq S_{0}$, we have $$\begin{aligned} \label{eq:Biii1} (-K_{Z})|_{S_{0,Z}}=4(-K_{S_{0}})-{\Gamma}-B=2(-K_{S_{0}})-(e_{1}+e_{6}). $$ This divisor is nef and we are done. By these arguments, $(\pi \colon W \to {\mathbb{P}}^{1},B)$ satisfies the condition $(\dag_{6})$ in Proposition \[prop:1\]. Hence we obtain the diagram $(\bigstar_{6})$. Because of $(-K_{W})^{3}=32$, $(-K_{W}).B=11$ and $g_{B}=1$, we obtain a del Pezzo fibration ${\varphi}\colon X \to {\mathbb{P}}^{1}$ of degree 6 satisfying the following: $$D \equiv \frac{1}{2}(-K_{Y})-\frac{1}{2}E_{Y}+\frac{1}{2}F_{Y},\quad (-K_{X})^{3}=12 \text{ and } -K_{X}.C=0$$ Since $Y$ is almost Fano and $-K_{X}.C=0$, $X$ is almost Fano. By TABLE \[table:A\] and TABLE \[table:B\], $X$ belongs to Case (A-1) or Case (B-iii-1). Let us prove that $X$ belongs to Case (B-iii-1). Now we obtain that $F_{W}=2H-G$ and $-K_{Z}=4H-G_{Z}-E$, where $F_{W}$ is a general fiber of $\pi \colon W \to {\mathbb{P}}^{1}$. Hence we obtain $$D_{Z} \equiv \frac{1}{2}(4H-G_{Z}-E)-\frac{1}{2}E+\frac{1}{2}(2H-G_{Z})=3H-G_{Z}-E =S_{0,Z}.$$ It means that $D=\operatorname{Exc}(\mu)=S_{0,Y}$. The $K_{X}$-trivial ray contraction of $X$ is a small contraction. Let ${\gamma}{\subset}X$ be a integral curve such that $K_{X}.{\gamma}=0$ and ${\gamma}\neq C$. Since $-K_{Y}$ is nef, we have $0 \geq K_{Y}.{\gamma}_{Y}=(g^{\ast}K_{X}+D).{\gamma}_{Y}=D.{\gamma}_{Y}$. Since ${\gamma}\neq C$, ${\gamma}_{Y}$ is not contained in $D$. Thus $-K_{Y}.{\gamma}_{Y}=D.{\gamma}_{Y}=0$ holds. Since ${\gamma}_{Y}$ is not contracted by $Y \to {\mathbb{P}}^{1}$, ${\gamma}_{Y}$ is not a flopped curve of $\Phi \colon Z {\dashrightarrow}Y$. In particular, we have $D_{Z}.{\gamma}_{Z} \geq 0$. On the other hand, we have $0=-K_{Z}.{\gamma}_{Z}=(H+D_{Z}).{\gamma}_{Z}$. Hence we have $H.{\gamma}_{Z}=D_{Z}.{\gamma}_{Z}=0$. Therefore, ${\gamma}_{Z}$ is a fiber of $G \to {\Gamma}$ or a fiber of $E \to B$. Since $-K_{Z}|_{E}$ is relatively ample for $E \to B$, ${\gamma}_{Z}$ is a fiber of $G \to {\Gamma}$. The fiber $l$ of $G \to {\Gamma}$ satisfying $-K_{Z}.l=0$ is the fiber of one of the 5 points ${\Gamma}\cap E$. Therefore, the $K_{X}$-trivial curves are exactly the proper transformations of these fibers and $C$. In particular, the number of the $K_{X}$-trivial curves is finite. Hence $X$ belongs to Case (B-iii-1). By using Lemma \[lem:3\] and \[lem:4\], we have $h^{1,2}(X)=3$. ------------------------------------------------------------------------ Case (A-1) {#ss:A1} ---------- The construction of an example belonging to Case (A-1) is obtained by the slightly modified manners in Case (B-iii-1). Let ${\Gamma}{\subset}S_{0}$, ${\sigma}\colon W \to {\mathbb{P}}^{3}$, $G=\operatorname{Exc}({\sigma})$ be as in $\S$ \[ss:Biii1\]. Recall that $W$ has a quadric fibration structure $\pi \colon W \to {\mathbb{P}}^{1}$ and a general $\pi$-fiber $F_{W}$ is linearly equivalent to ${\sigma}^{\ast}{\mathcal}O_{{\mathbb{P}}^{3}}(2)-G$. The ${\sigma}$-exceptional divisor $G={\mathbb{P}}({\mathcal}E)$ has a ${\mathbb{P}}^{1}$-bundle structure ${\sigma}|_{G} \colon G \to {\Gamma}$, where ${\mathcal}E=N_{{\Gamma}}W^{\vee}$. Let $h={\mathcal}O_{{\mathbb{P}}({\mathcal}E)}(1)$ be a tautological bundle. Since ${\Gamma}$ is an elliptic curve of degree 4, ${\Gamma}$ is a complete intersection of two quadrics. Hence ${\mathcal}E \simeq {\mathcal}L_{1} \oplus {\mathcal}L_{2}$ for some line bundles ${\mathcal}L_{1},{\mathcal}L_{2}$ of degree $-8$. For an arbitrary line bundle ${\mathcal}M$ of degree $11$ on ${\Gamma}$, ${\mathcal}L_{i} \otimes {\mathcal}M$ is very ample for $i \in \{1,2\}$ and hence ${\mathcal}E \otimes {\mathcal}M$ is globally generated. Therefore, the linear system $|h+({\sigma}|_{G})^{\ast}{\mathcal}M|$ on $G={\mathbb{P}}({\mathcal}E)$ has a non-singular member $B$. Since $B$ is $({\sigma}|_{G})$-section, $g_{B}=1$. Moreover, we have $${\sigma}^{\ast}{\mathcal}O_{{\mathbb{P}}^{3}}(1).B=4, \quad G.B=5,\quad F_{W}.B=3 \text{ and } (-K_{W}).B=11.$$ Let ${\tau}\colon Z:=\operatorname{\mathrm{Bl}}_{B}W \to W$ be the blowing-up of $W$ along $B$, $E$ the ${\tau}$-exceptional divisor. Set $H:={\tau}^{\ast}{\sigma}^{\ast}{\mathcal}O(1)$. Note that we have $$G_{Z}={\tau}^{\ast}G-E \text{ and } -K_{Z}=4H-{\tau}^{\ast}G-E=4H-G_{Z}-2E.$$ $-K_{Z}$ is nef and $(-K_{Z})^{3}=10$. By a straightforward calculation, it is easy to see that $(-K_{Z})^{3}=10$. Let us prove the nefness of $-K_{Z}$. Since $-K_{Z}=4H-G_{Z}-2E=2(2H-G_{Z}-E)+G_{Z}=2F_{Z}+G_{Z}$, it is enough to show the nefness of $-K_{Z}|_{G_{Z}}$. Since $G_{Z} \simeq G \simeq {\mathbb{P}}({\mathcal}E)$, we can describe as follows: $$H|_{G}=({\sigma}|_{G})^{\ast}{\mathcal}M_{4}, \quad G|_{G}=-h \text{ and } E|_{G_{Z}}=h+({\sigma}|_{G})^{\ast}{\mathcal}M_{11}.$$ Here, ${\mathcal}M_{i}$ denotes some line bundle on ${\Gamma}$ of degree $i$. In this notation, we have $(-K_{Z})|_{G_{Z}}= ({\sigma}|_{G})^{\ast}{\mathcal}M_{5}$, which complete the proof. Therefore, $(\pi \colon W \to {\mathbb{P}}^{1},B)$ satisfies the condition $(\dag_{6})$ in Proposition \[prop:1\] and hence we obtain the diagram $(\bigstar_{6})$ and a del Pezzo fibration ${\varphi}\colon X \to {\mathbb{P}}^{1}$ of degree 6 satisfying the following: $$D \equiv \frac{1}{2}(-K_{Y})-\frac{1}{2}E_{Y}+\frac{1}{2}F_{Y},\quad (-K_{X})^{3}=12 \text{ and } -K_{X}.C=0$$ Since $Y$ is almost Fano and $-K_{X}.C=0$, $X$ is almost Fano. Therefore, $X$ belongs to Case (A-1) or Case (B-iii-1). The $K_{X}$-trivial ray contraction of $X$ is crepant. For the general fiber $f$ of $G={\mathbb{P}}({\mathcal}E) \to {\Gamma}$, we have $-K_{Z}.f_{Z}=0$. Since $F_{Z}.f_{Z}={\tau}^{\ast}(2H-G).f_{Z}=1$, $f_{Z}$ is not contracted by $p \colon Z \to {\mathbb{P}}^{1}$. Since general $f_{Z}$ does not meet the flopping curves of $\Phi$, we have $-K_{Y}.f_{Y}=0$. Since $D$ is linearly equivalent to $3H-G-E$, $D.f_{Y}=0$ and hence $f_{Y}$ is not contracted by $\mu \colon Y \to X$. Therefore, $\mu(f_{Y})=f_{X}$ is a curve such that $-K_{X}.f_{X}=0$ and this argument tells us that $X$ has infinitely many $K_{X}$-trivial curves. Hence $X$ belongs to Case (A-1). By similarly arguments in $\S$ \[ss:Biii1\], we obtain $h^{1,2}(X)=2$. ------------------------------------------------------------------------ Case (B-iii-2) {#ss:Biii2} -------------- In order to construct an example belonging to Case (B-iii-2), we consider the following lattice: $$\begin{pmatrix} &H&{\Gamma}&B \\ H &6&2&9 \\ {\Gamma}&2&-2&6 \\ B&9&6&6 \end{pmatrix}.$$ This is an even lattice with signature $(1,2)$. By virtue of Theorem \[thm:NSL\], there exists a K3 surface $S$ such that $\operatorname{\mathrm{Pic}}(S)={\mathbb{Z}}\cdot [H] \oplus {\mathbb{Z}}\cdot [F] \oplus {\mathbb{Z}}\cdot [B]$ is isometry to this lattice. Moreover, we can assume that $H$ is nef and big. Set $C_{2}:={\Gamma}$ and $C_{5}:=3H-2{\Gamma}-B$. Note that $C_{i}$ is a $(-2)$-curve since $(C_{i})^{2}=-2$ and $H.C_{i}=i>0$. Let $C=xH+y{\Gamma}+zB$ be a divisor on $S$ with $x,y,z \in {\mathbb{Z}}$. Then we have $$C^{2}=6x^{2}-2y^{2}+6z^{2}+4xy+18xz+12yz \text{ and }H.C=6x+2y+9z.$$ By solving these equations for $x$ and $y$, we obtain the following: $$\begin{aligned} &x=\frac{4H.C-45z \pm \sqrt{4(H.C)^{2}-24C^{2}-99z^{2}}}{24} \text{ and }\\ &y=\frac{9z-\sqrt{4(H.C)^{2}-24C^{2}-99z^{2}}}{8}.\end{aligned}$$ Hence $x,y,z \in {\mathbb{Z}}$, we can prove that the following (i)-(iv) by a straightforward calculation. - It is impossible that $C^{2}=0$ and $H.C=1$. - It is impossible that $C^{2}=0$ and $H.C=2$. - It is impossible that $C^{2}=0$ and $H.C=3$. - If $C^{2}=-2$ and $H.C \leq 10$, then $C=C_{2}$ or $C_{5}$. \[lem:Biii2\] 1. $H$ is very ample and $S$ is embedded to ${\mathbb{Q}}^{3}$ by $|H|$ as an anti-canonical member. 2. The general member of $|B|$ is non-singular. 3. $3H-{\Gamma}-B$ is nef. <!-- --> 1. It can be proved similarly to Lemma \[lem:Bii\] (1). 2. By Theorem \[thm:SDR\], it is enough to show that $|B|$ is movable. Let $M$ be the movable part of $|B|$ and $N$ the fixed part. Then we have $9=H.B>H.N$. Due to Lemma \[lem:1\] and (iv), there exists $a,b \in {\mathbb{Z}}_{\geq 0}$ such that $B=M+aC_{2}+bC_{5}$. Thus we have $M^{2}=6-2a^{2}-2b^{2}-12a-18b+8ab \geq 0$. It is easy to find that $(a,b)=(0,0)$ is the only non-negative integer solution of this inequality. Therefore, $|B|=M$ is movable. 3. $3H-{\Gamma}-B=C_{2}+C_{5}$ is nef since $C_{2}.C_{5}=4$. By Lemma \[lem:Biii2\] (1), $S$ is embedded in ${\mathbb{Q}}^{3}$ as an anti-canonical member. Let ${\sigma}\colon W = \operatorname{\mathrm{Bl}}_{{\Gamma}}{\mathbb{Q}}^{3} \to {\mathbb{Q}}^{3}$ be the blowing-up of ${\mathbb{Q}}^{3}$ along ${\Gamma}$ and $G:=\operatorname{Exc}({\sigma})$ a ${\sigma}$-exceptional divisor. As we have seen, there exists a quadric fibration $\pi \colon W \to {\mathbb{P}}^{1}$ given by $F_{W}:={\sigma}^{\ast}{\mathcal}O_{{\mathbb{Q}}^{3}}(1)-G$. In what follows, we regard divisors of $S_{W}$ as those of $S$ because of $S_{W} \simeq S$. Then we have ${\mathcal}O_{S_{W}}(F_{W})=H-{\Gamma}$ and hence $(F_{W}.B)_{W}=((H-{\Gamma}).B)_{S}=3$. Let $Z:=\operatorname{\mathrm{Bl}}_{B_{W}}W$ denotes the blowing-up of $W$ along $B_{W}$. Then we have $(-K_{Z})^{3}=4$ by a straightforward calculation. By Lemma \[lem:Biii2\] (3), $-K_{Z}$ is nef. Therefore, the pair $(\pi \colon W \to {\mathbb{P}}^{1},B)$ satisfies the condition $(\dag_{6})$ in Proposition \[prop:1\]. Thus we obtain a del Pezzo fibration ${\varphi}\colon X \to {\mathbb{P}}^{1}$ of degree 6 satisfying the following: $$(-K_{X})^{3}=6,\quad -K_{X}.C=0 \text{ and } D \equiv \frac{1}{2}(-K_{Y})-\frac{1}{2}E_{Y}+\frac{3}{2}F_{Y}.$$ Since $Y$ is almost Fano and $-K_{X}.C=0$, $X$ is almost Fano. By Lemma \[lem:3\] and \[lem:4\], we have $h^{1,2}(X)=4$. Therefore, $X$ does not belong to Case (B-i-2). Thus $X$ belongs to Case (B-iii-2). ------------------------------------------------------------------------ Case (B-iii-3) {#ss:Biii3} -------------- In order to construct an example belonging to Case (B-iii-3), we consider the following lattice: $$\begin{pmatrix} &H_{{\alpha}}&H_{{\beta}}&F&B \\ H_{{\alpha}}&0&2&2&0 \\ H_{{\beta}}&2&0&2&3 \\ F&2&2&0&3 \\ B &0&3&3&-2 \end{pmatrix}.$$ This is an even lattice with signature $(1,3)$ and hence there exists a K3 surface $S$ such that $\operatorname{\mathrm{Pic}}(S)$ is isometry to this lattice by Theorem \[thm:NSL\]. Moreover, we may assume that $H:=H_{{\alpha}}+H_{{\beta}}$ is nef and big. Set $C_{1}:=2H_{{\alpha}}-B$, $C_{1}':=-H_{{\alpha}}+B$, $C_{3}:=B$ and $C_{3}':=3H_{{\alpha}}-B$. Then we have $C_{i},C_{i}'$ are $(-2)$-curves with $H.C_{i}=H.C_{i}'=i$ for $i \in \{1,3\}$. Let $C=xH_{{\alpha}}+yH_{{\beta}}+zF+wB$ be a divisor on $S$ with $x,y,z,w \in {\mathbb{Z}}$. Then we have $$C^{2}=-2w^{2}+4xy+4xz+4yz+6yw+6zw \text{ and } H.C=2x+2y+4z+3w.$$ By solving these for $x$ and $y$, we have the following: $$\begin{aligned} x&=\frac{(H.C)-4z-6w \pm \sqrt{(H.C)^{2}-4C^{2}-8w^{2}-16z^{2}}}{4} \text{ and } \\ y&=\frac{(H.C)-4z \mp \sqrt{(H.C)^{2}-4C^{2}-8w^{2}-16z^{2}}}{4}\end{aligned}$$ Hence the following (i) and (ii) holds by a straightforward calculation. - It is impossible that $C^{2}=0$ and $H.C=1$. - If $C^{2}=-2$ and $H.C \leq 4$, then $C=C_{1}$, $C_{1}'$, $C_{3}$ or $C_{3}'$. \[lem:Biii3\] 1. $H$ is base point free and ample but not very ample. 2. $F$ is base point free. 3. $H+F$ is very ample. 4. Let $f \colon S \to {\mathbb{P}}^{3}$ be a morphism given by $|H|$. Then the image $f(S)$ is non-singular quadric surface. 5. Let $p \colon S \to {\mathbb{P}}^{1}$ be a morphism given by $|F|$. Then the composition of $f \times p \colon S \to {\mathbb{P}}^{3} \times {\mathbb{P}}^{1}$ and the segre embedding ${\mathbb{P}}^{3} \times {\mathbb{P}}^{1} \to {\mathbb{P}}^{7}$ is exactly given by $|H+F|$. In particular, $f \times p \colon S \to {\mathbb{P}}^{3} \times {\mathbb{P}}^{1}$ is closed embedding. 6. There exists non-singular quadric $Q {\subset}{\mathbb{P}}^{3}$ such that $S {\subset}Q \times {\mathbb{P}}^{1} {\subset}{\mathbb{P}}^{3} \times {\mathbb{P}}^{1}$. 7. $2H-B$ is nef. <!-- --> 1. Since (i),(ii) and Theorem \[thm:SDR\], $H$ is base point free and ample. Since $H.H_{{\alpha}}=2$ and $H_{{\alpha}}^{2}=0$, $H$ is not very ample by Theorem \[thm:SDR\]. 2. It is enough to show that $|F|$ is movable. Let $M$ be the movable part of $|F|$ and $N$ the fixed part. Then we have $4=H.F>H.N$. By Lemma \[lem:1\] and (iv), there exists $a,b,c,d \in {\mathbb{Z}}_{\geq 0}$ such that $F=M+aC_{1}+bC_{1}'+cC_{3}+dC_{3}'$. Since $H.M > 0$ and $M^{2}>0$, we have $a=b=0$ and hence $F=M$ is movable. 3. By using Theorem \[thm:SDR\], the very ampleness of $H+F$ can be obtained by the same manner as we have seen. 4. $f \colon S \to {\mathbb{P}}^{3}$ is not closed embedding but finite morphism onto the image. Since $f$ is given by the complete linear system $|H|$, $Q:=f(S)$ is an irreducible and reduced quadric surface and $S \to Q$ is $2:1$. Assume $Q$ is a singular cone with a vertex $o \in Q$. Then a line $l$ of $Q$ is a non-Cartier Weil divisor and the closure ${\overline}{f^{-1}(l \setminus \{o\})}=D$ is a Cartier divisor of $S$. Since $f^{\ast}(2l)=H$, we have $H=2D$ and it contradicts that $H$ is a generator of $\operatorname{\mathrm{Pic}}(S)$. Thus $Q$ is non-singular. 5. It is clear. 6. It is clear from (4) and (5). 7. Let us prove the nefness of $2H-B$. Since $H.(2H-B)=5$ and $(2H-B)^{2}=2$, we have $2H-B$ is effective. Let $M$ be the movable part of $|2H-B|$ and $N$ the fixed part. Since $M \neq 0$ and $H$ is ample, we have $H.N < 5$. By Lemma \[lem:1\] and (iv), there exists $a,b,c,d \in {\mathbb{Z}}_{\geq 0}$ such that $F=M+aC_{1}+bC_{1}'+cC_{3}+dC_{3}'$. Since $(2H-B).C_{1}=0$, $(2H-B).C_{1}'=4$, $(2H-B).C_{3}=8$ and $(2H-B).C_{3}'=4$, we have that $2H-B$ is nef. Set $H_{\xi}={\mathcal}O_{{\mathbb{P}}^{3} \times {\mathbb{P}}^{1}}(1,0)$ and $H_{\eta}={\mathcal}O_{{\mathbb{P}}^{3} \times {\mathbb{P}}^{1}}(0,1)$. Then there exists a non-singular member $W \in |2H_{\xi} + 2H_{\eta}|$ which contains $S$. Consider the following exact sequence: $$0 \to {\mathcal}I_{S} \otimes {\mathcal}O(2H_{\xi}+2H_{\eta}) \to {\mathcal}O(2H_{\xi}+2H_{\eta}) \to {\mathcal}O(2H_{\xi}+2H_{\eta})|_{S} \to 0.$$ We have $h^{0}({\mathbb{P}}^{3} \times {\mathbb{P}}^{1},{\mathcal}O(2H_{\xi}+2H_{\eta}))=30$ and $h^{0}(S,{\mathcal}O(2H_{\xi}+2H_{\eta})|_{S})=26$ and hence there exists a member of $|2H_{\xi}+2H_{\eta}|$ which contains $S$. Set $V:=H^{0}({\mathbb{P}}^{3} \times {\mathbb{P}}^{1},{\mathcal}O(2H_{\xi}+2H_{\eta}) \otimes {\mathcal}I_{S})$ and let ${\Lambda}:=|V|$ be the linear system corresponding to $V$. Let us prove that $\operatorname{\mathrm{Bs}}{\Lambda}=S$. By Lemma \[lem:Biii3\] (6), there exists a non-singular quadric surface $Q {\subset}{\mathbb{P}}^{3}$ such that $S {\subset}Q \times {\mathbb{P}}^{1}$. Set $R:=Q \times {\mathbb{P}}^{1}$. Then $R$ is linear equivalent to $2H_{\xi}$ as a divisor of ${\mathbb{P}}^{3} \times {\mathbb{P}}^{1}$. Hence we have $R+2H_{\eta} \in {\Lambda}$ and $\operatorname{\mathrm{Bs}}{\Lambda}{\subset}R$. Thus $\operatorname{\mathrm{Bs}}{\Lambda}=\operatorname{\mathrm{Bs}}{\Lambda}\cap R=\operatorname{\mathrm{Bs}}{\Lambda}|_{R}$ holds. Let us fix an isomorphism $R \simeq {\mathbb{P}}^{1}_{{\alpha}} \times {\mathbb{P}}^{1}_{{\beta}} \times {\mathbb{P}}^{1}_{\eta}$ and define $H_{{\alpha}}:={\mathcal}O(1,0,0)$ and $H_{{\beta}}:={\mathcal}O(0,1,0)$. Note that $H_{\eta}|_{R}={\mathcal}O(0,0,1)$. Now ${\Lambda}|_{R}$ is the linear system corresponding to the linear space $V_{R}=H^{0}({\mathbb{P}}^{1}_{{\alpha}} \times {\mathbb{P}}^{1}_{{\beta}} \times {\mathbb{P}}^{1}_{{\gamma}} , {\mathcal}O(2H_{{\alpha}}+2H_{{\beta}}+2H_{\eta}) \otimes {\mathcal}I_{S/R})$. Since ${\mathcal}I_{S}={\mathcal}O(-2H_{{\alpha}}-2H_{{\beta}}-2H_{\eta})$, we have $\dim V_{R}=1$ and hence $|{\Lambda}|_{R}|=\{S\}$. Thus we have that $\operatorname{\mathrm{Bs}}{\Lambda}=S$. Therefore, a general member $W$ of ${\Lambda}$ have singularities only at $S$. Since ${\Lambda}$ is movable and $\dim {\Lambda}\geq 3$, a general member $W$ is irreducible and reduced by Bertini’s theorem. Moreover, $S=W \cap R$ is a non-singular Cartier divisor of $W$ and hence $W$ is non-singular along $S$. By this argument, general members of ${\Lambda}$ are non-singular. Let $W$ be a general member of ${\Lambda}$ and $\pi \colon W \to {\mathbb{P}}^{1}$ the restriction to W of the projection ${\mathbb{P}}^{3} \times {\mathbb{P}}^{1} \to {\mathbb{P}}^{1}$. Thus $\pi$ gives a quadric fibration structure and $\deg (\pi|_{B})=3$. Let $Z=\operatorname{\mathrm{Bl}}_{B_{W}}W$ denotes the blowing-up of $W$ along $B_{W}$. It is easy to see that $(-K_{Z})^{3}=2$. By Lemma \[lem:Biii3\] (7), $-K_{Z}$ is nef and big. Therefore, the pair $(\pi \colon W \to {\mathbb{P}}^{1},B)$ satisfies the condition $(\dag_{6})$ in Proposition \[prop:1\]. Thus we have a del Pezzo fibration of degree 6 ${\varphi}\colon X \to {\mathbb{P}}^{1}$ satisfying the following: $$(-K_{X})^{3}=4,\quad -K_{X}.C=0 \text{ and } D\equiv\frac{1}{2}(-K_{Y})-\frac{1}{2}E_{Y}+F.$$ Since $Y$ is almost Fano and $-K_{X}.C=0$, $X$ is almost Fano and belongs to Case (B-i-3) or Case (B-iii-3). $h^{1,2}(X)=3$． In the diagram $(\bigstar_{6})$ for this case, $Y \to X$ is the blowing-up along $B \simeq {\mathbb{P}}^{1}$ and $Y {\dashrightarrow}Z$ is flop and $Z \to W$ is the blowing-up along $C \simeq {\mathbb{P}}^{1}$. By Lemma \[lem:3\] and \[lem:4\], we have $h^{1,2}(X)=h^{1,2}(W)$. Note that $W$ is a non-singular member of $|2H_{\xi}+2H_{\eta}|$ in ${\mathbb{P}}^{3}_{\xi} \times {\mathbb{P}}^{1}_{\eta}$. Consider a $4 \times 4$ symmetric matrix $M(\eta_{0},\eta_{1})=(f_{ij}(\eta_{0},\eta_{1}))_{0 \leq i,j \leq 3}$ over ${\mathbb{C}}[\eta_{0},\eta_{1}]$ such that each $f_{ij}(\eta_{0},\eta_{1})$ is a homogenous polynomial of degree 2. Let $D=\left(\sum f_{ij}(\eta_{0},\eta_{1})\xi_{i}\xi_{j}=0\right)$ be a member of $|2H_{\xi}+2H_{\eta}|$. By taking general $M(\eta_{0},\eta_{1})$, we may assume that $D$ is non-singular and $\det M(\eta_{0},\eta_{1})$ has no multiple root. Thus the singular fibers of $D \to {\mathbb{P}}^{1}$ is determined by $\det M(\eta_{0},\eta_{1})=0$. In particular, the number of the singular fibers of $D \to {\mathbb{P}}^{1}$ is 8. Hence we have $$\operatorname{\chi_{\mathrm{Top}}}(D)=\operatorname{\chi_{\mathrm{Top}}}({\mathbb{P}}^{1} \times {\mathbb{P}}^{1}) \cdot \operatorname{\chi_{\mathrm{Top}}}({\mathbb{P}}^{1}) + 8 \left( \operatorname{\chi_{\mathrm{Top}}}({\mathbb{Q}}^{2}_{0})-\operatorname{\chi_{\mathrm{Top}}}({\mathbb{P}}^{1} \times {\mathbb{P}}^{1}) \right)=0,$$ where $\operatorname{\chi_{\mathrm{Top}}}$ denotes the topological Euler number and ${\mathbb{Q}}^{2}_{0}=\{xy-z^{2}=0\} {\subset}{\mathbb{P}}^{3}$ denotes the quadric cone. On the other hand, we have $b_{0}(D)=b_{6}(D)=1$, $b_{1}(D)=b_{5}(D)=0$ and $b_{2}(D)=b_{4}(D)=2$, where $b_{i}$ denotes the $i$-th Betti number. Thus we have $b_{3}(D)=6$. Due to Bertini’s theorem, $\{D \in |2H+2F| \mid D \text{ is non-singular }\}$ is an open set of $|2H+2F|$ and connected in the Euclidean topology. In particular, $D$ is deformation equivalent to $W$. Thus we have $b_{3}(D)=b_{3}(W)$. Since $h^{0,3}(W)=0$, we obtain $h^{1,2}(W)=3$. By this lemma, $X$ can not belong to Case (B-i-3). Therefore, $X$ belongs to Case (B-iii-3). ------------------------------------------------------------------------ Case (B-iii-4) {#ss:Biii4} -------------- In order to construct an example belonging to Case (B-iii-4), we consider the following lattice: $$\begin{pmatrix} &H&{\Gamma}&B \\ H&6&2&11 \\ {\Gamma}&2&-2&8 \\ B&11&8&8 \end{pmatrix}.$$ This is an even lattice with signature $(1,2)$. By virtue of Theorem \[thm:NSL\], there exists a K3 surface $S$ such that $\operatorname{\mathrm{Pic}}(S)$ is isometry to this lattice. Moreover, we may assume that $H$ is nef and big. Set $C_{2}:={\Gamma}$, $C_{5}:=3H-{\Gamma}-B$ and $C_{7}:=4H-3{\Gamma}-B$. Note that $C_{i}$ is $(-2)$-curve with $H.C_{i}=i$ for $i \in \{2,5,7\}$. Let $C=xH+y{\Gamma}+zB$ be a divisor on $S$ with $x,y,z \in {\mathbb{Z}}$. Then we have $$C^{2}=6x^{2}-2y^{2}+8z^{2}+4xy+22xz+16yz \text{ and } H.C=6x+2y+11z.$$ By solving these equations, we have the following: $$\begin{aligned} &x=\frac{4l-57z \pm \sqrt{4(H.C)^{2}-24C^{2}-123z^{2}}}{24} \text{ and } \\ &y=\frac{13z \mp \sqrt{4(H.C)^{2}-24C^{2}-123z^{2}}}{8}.\end{aligned}$$ By a straightforward calculation, we can prove that the following (i)-(iv). - It is impossible that $C^{2}=0$ and $H.C=1$. - It is impossible that $C^{2}=0$ and $H.C=2$. - It is impossible that $C^{2}=0$ and $H.C=3$. - If $C^{2}=-2$ and $H.C \leq 10$, then $C=C_{2}$, $C_{5}$ or $C_{7}$. 1. $H$ is very ample and $S$ is embedded in ${\mathbb{Q}}^{3}$ as an anti-canonical divisor. 2. There exists a non-singular member of $|B|$ meeting ${\Gamma}$ transversally. <!-- --> 1. It can be proved similarly to Lemma \[lem:Bii\] (1). 2. It is enough to show that $|B|$ is movable. Let $M$ denotes the movable part of $|B|$ and $N$ the fixed part. Then we have $11=H.B>H.N$ By Lemma \[lem:1\] and (iv), there exists $a,b,c \in {\mathbb{Z}}_{\geq 0}$ such that $B=M+a{\Gamma}+bC_{5}+cC_{7}$. Since $H$ is very ample and $M$ is movable, we have the following inequalities: $$\begin{aligned} &11=H.B=(H.M)+2a+5b+7c>2a+5b+7c \text{ and } \\ &M^{2}=8-(18a+36b+26c)+12ac+6bc \geq 0. \end{aligned}$$ It is easy to see that $a=b=c=0$ is the only non-negative integer solution of these inequalities. Hence $|B|=M$ is movable. Hence there exists an embedding $S {\hookrightarrow}{\mathbb{Q}}^{3}$. Let ${\sigma}\colon W:=\operatorname{\mathrm{Bl}}_{{\Gamma}}{\mathbb{Q}}^{3} \to {\mathbb{Q}}^{3}$ be the blowing-up of $W$ along ${\Gamma}$ and $G=\operatorname{Exc}({\sigma})$. As we have seen, there exists a quadric fibration $\pi \colon W \to {\mathbb{P}}^{1}$ given by $F_{W}:={\sigma}^{\ast}{\mathcal}O_{{\mathbb{Q}}^{3}}(1)-G$. Let us identify $S_{W},B_{W}$ with $S,B$ respectively and regard divisors of $S_{W}$ as those of $S$. Then ${\mathcal}O_{S_{W}}(F_{W})=H-{\Gamma}$ and hence $(F_{W}.B)_{W}=((H-{\Gamma}).B)_{S}=3$. Set ${\tau}\colon Z:=\operatorname{\mathrm{Bl}}_{B}W \to W$ and $p:=\pi \circ {\tau}\colon Z \to {\mathbb{P}}^{1}$. $-K_{Z}$ is $p$-nef and $p$-big. It is clear that $-K_{Z}$ is $p$-big. Let ${\gamma}{\subset}Z$ be a curve which is contained in some $p$-fiber. If $-K_{Z}.{\gamma}<0$, then $S_{Z} \simeq S$ contains ${\gamma}$. Since $-K_{Z}|_{S_{Z}}=3H-{\Gamma}-B=C_{5}$, we have ${\gamma}=C_{5}$. Since $F_{Z}|_{S_{Z}}=H-{\Gamma}$, we have $F_{Z}.{\gamma}=(H-{\Gamma}).C_{5}=5$, which contradicts that ${\gamma}$ is contracted by $p$. Hence $-K_{Z}$ is $p$-nef. Therefore, the pair $(\pi \colon W \to {\mathbb{P}}^{1},B)$ satisfies the condition $(\dag_{6})$. By Proposition \[prop:1\], we have the diagram $(\bigstar_{6})$. In particular, we obtain a del Pezzo fibration of degree 6 ${\varphi}\colon X \to {\mathbb{P}}^{1}$ satisfying the following: $$(-K_{X})^{3}=2,\quad -K_{X}.C=1 \text{ and } D \equiv \frac{1}{2}(-K_{Y})-\frac{1}{2}E_{Y}+\frac{5}{2}F.$$ \[lem:almFBiii4\] $-K_{X}$ is nef and big but not ample. By Mori-Mukai’s classification, $X$ is not Fano. Hence it is enough to show the nefness of $-K_{X}$. At first, we prove that $\Phi|_{S_{Z}} \colon S_{Z} {\dashrightarrow}S_{Y}$ is an isomorphism. Let ${\gamma}$ be an irreducible flopping curve of $\Phi$ and assume $S_{Z} \simeq S$ contains ${\gamma}$ as a divisor. Thus there exists $x,y,z \in {\mathbb{Z}}$ such that ${\gamma}=xH+y{\Gamma}+zB$ holds in $\operatorname{\mathrm{Pic}}(S)$. Note that $-K_{Z}|_{S_{Z}}=C_{5}$, $F_{Z}|_{S_{Z}}=H-{\Gamma}$ and ${\gamma}$ is $(-2)$-curve. Hence the following equations hold: $$(3H-{\Gamma}-B).{\gamma}=(H-{\Gamma}).{\gamma}=0 \text{ and }{\gamma}^{2}=-2.$$ But these equations have no integer solution for $x,y,z$. Hence $S_{Z}$ has no flopping curve of $\Phi$. Thus $\Phi|_{S_{Z}} \colon S_{Z} {\dashrightarrow}S_{Y}$ is an isomorphism. Let us prove the nefness of $-K_{X}$. Let ${\gamma}{\subset}X$ be a curve satisfing $(-K_{X}).{\gamma}<0$. Then we have ${\gamma}\neq C=\mu(D)$ and hence $0>\mu^{\ast}(-K_{X}).{\gamma}=(-K_{Y}+D).{\gamma}_{Y}$, where $D=\operatorname{Exc}(\mu)$. Since ${\gamma}\neq C$, $D$ does not contain ${\gamma}_{Y}$ and hence we have $D.{\gamma}_{Y} \geq 0$. Thus $-K_{Y}.{\gamma}_{Y} < 0$ which means $S_{Y}$ contains ${\gamma}_{Y}$. Since $-K_{Y}|_{S_{Y}}=3H-{\Gamma}-B=C_{5}$, we have ${\gamma}_{Y}=C_{5}$. Now, $D|_{S_{Y}}=4H-3{\Gamma}-B=C_{7}$ and hence we have $(-K_{Y}+D)|_{S_{Y}}=C_{5}+C_{7}$. Therefore, we have $(-K_{Y}+D).{\gamma}_{Y}=(C_{5}+C_{7}).C_{5}=-2+3=1$ and it is contradiction. Thus $-K_{X}$ is nef. By Lemma \[lem:almFBiii4\], ${\varphi}\colon X \to {\mathbb{P}}^{1}$ is an almost Fano del Pezzo fibration of degree $6$ with $(-K_{X})^{3}=2$. Hence $X$ belongs to Case (B-iii-4). By Lemma \[lem:3\] and \[lem:4\], we have $h^{1,2}(X)=5$. ------------------------------------------------------------------------ ${\mathbb{P}}^{2}$-bundles with quinque-sections ------------------------------------------------ In this subsection, we state and prove Proposition \[prop:2\], which is similar to Proposition \[prop:1\], to construct an example belonging to Case (A-2). \[prop:2\] Let $\pi \colon W \to {\mathbb{P}}^{1}$ be a ${\mathbb{P}}^{2}$-bundle and $B {\subset}W$ a smooth projective curve and ${\tau}\colon Z:=\operatorname{\mathrm{Bl}}_{B}W \to W$ the blowing-up along $B$. We assume the following condition $(\dag_{5})$ for a pair $(\pi \colon W \to {\mathbb{P}}^{1},B)$. $$(\dag_{5}) \cdots \left\{ \begin{array}{ll} \deg (\pi|_{B} \colon B \to {\mathbb{P}}^{1})=5 \text{ and } \\ -K_{Z} \text{ is $p$-nef and $p$-big with } p := \pi \circ {\tau}\colon Z \to {\mathbb{P}}^{1}. \end{array} \right.$$ Then the following holds. 1. The morphism from $Z$ to the relative anti-canonical model over ${\mathbb{P}}^{1}$ $$\Psi \colon Z \to {\overline}{Z}:=\operatorname{Proj}\bigoplus_{n \geq 0} p_{\ast}{\mathcal}O_{Z}(-nK_{Z})$$ is a small contraction or an isomorphism. 2. Define a variety $Y$ and a birational map $\Phi \colon Z {\dashrightarrow}Y$ as follows: - If $-K_{Z}$ is $p$-ample, set $Y:=Z$ and $\Phi:=\operatorname{id}_{Z} \colon Z \to Y$; or - if $-K_{Z}$ is not $p$-ample, let $\Phi \colon Z {\dashrightarrow}Y$ be the flop over ${\mathbb{P}}^{1}$ of $\Psi \colon Z \to {\overline}{Z}$. Let $q \colon Y \to {\mathbb{P}}^{1}$ be a structure morphism onto ${\mathbb{P}}^{1}$ and $\mu \colon Y \to X$ a $K_{Y}$-negative ray contraction over ${\mathbb{P}}^{1}$. If $-K_{Z}$ is $p$-ample, we choose the other ray which is different to the one corresponding to ${\tau}\colon Z \to W$. Let ${\varphi}\colon X \to {\mathbb{P}}^{1}$ denotes the structure morphism onto ${\mathbb{P}}^{1}$. Then $\mu$ is the blowing-up along a ${\varphi}$-section $C$ and ${\varphi}\colon X \to {\mathbb{P}}^{1}$ is a del Pezzo fibration of degree 5. $$\xymatrix{ &D \ar[ld]& \ar@{}[l]|{{\subset}} \ar[ld]_{\mu}Y=\operatorname{\mathrm{Bl}}_{C}X \ar[dd]^{q}\ar@{<--}[r]^{\Phi}& Z=\operatorname{\mathrm{Bl}}_{B}W \ar@{}[r]|{{\supset}} \ar[rd]^{{\tau}} \ar[dd]_{p}& E \ar[rd]& \\ C&\ar@{}[l]|{{\subset}} X \ar[rd]_{{\varphi}}& && W \ar@{}[r]|{{\supset}} \ar[ld]^{\pi} &B&(\bigstar_{5}) \\ &&{\mathbb{P}}^{1} \ar@{=}[r]&{\mathbb{P}}^{1}&& }$$ 3. Let $F_{Y}$ be a general $q$-fiber and we set $D:=\operatorname{Exc}(\mu)$, $E:=\operatorname{Exc}({\tau})$ and $E_{Y}:=\Phi_{\ast}E$. Then we obtain $$D \equiv \frac{2}{3}(-K_{Y})-\frac{1}{3}E_{Y}+zF_{Y}.$$ Moreover, the following equalities hold: $$\begin{aligned} (-K_{X})^{3}&=22-2g_{B}, \\ -K_{X}.C&=(-K_{W}).B-2g_{B}-16 \text{ and } \\ z&=\frac{2}{3}(-K_{W}).B-g_{B}-12.\end{aligned}$$ We can prove this proposition similarly to Proposition \[prop:2\]. 1. Let us assume that $\Psi \colon Z \to {\overline}{Z}$ is a divisorial contraction and set $D=\operatorname{Exc}(\Psi)$. Let $F_{Z}$ be a general $p$-fiber and $F_{W}$ a general $\pi$-fiber such that ${\sigma}(F_{Z})=F_{W}$. Now ${\sigma}|_{F_{Z}} \colon F_{Z} \to F_{W} \simeq {\mathbb{P}}^{2}$ is the blowing-up at reduced five points. Let us prove that $D|_{F_{Z}}$ is a disjoint union of $(-2)$-curves $l_{1},\ldots,l_{n}$ and $n \in \{1,2\}$. Let $e_{1},e_{2},e_{3},e_{4},e_{5}$ be exceptional curves of ${\tau}|_{F_{Z}}$ and set $h:=({\tau}|_{F_{Z}})^{\ast}{\mathcal}O_{{\mathbb{P}}^{2}}(1)$. For every irreducible curve $l {\subset}F_{Z}$, there exists $a,b_{j} \in {\mathbb{Z}}_{\geq 0}$ such that $l=ah-\sum_{j=1}^{5} b_{j}e_{j}$. If $l$ is contracted by ${\tau}|_{F_{Z}}$, then $l$ is $(-2)$-curve and we have the following: $$0=3a-\sum_{j=1}^{5}b_{j} \text{ and } -2=a^{2}-\sum_{j=1}^{5}b_{j}^{2}.$$ The following inequality $$a^{2}+2=\sum_{j=1}^{5}b_{j}^{2} \geq 5 \left( \frac{\sum_{j=1}^{5} b_{j}}{5} \right)=\frac{9}{5}a^{2}$$ implies that $a=1$. Thus three of the five numbers $b_{1},\ldots,b_{5}$ are 1 and the remaining two are 0. Hence $l$ is the proper transform of a line passing through exactly three points of the five points. By our assumption, $-K_{F_{Z}}$ is nef and hence any four points are not collinear. Therefore, the number of lines passing through exactly three points of the five points is at most two. Moreover, if there exists two lines passing through three points of the five points, then the proper transforms of those do not meet since the two lines meet transversally at one point of the five points. There exists $x,y,z \in {\mathbb{Q}}$ such that $D \equiv x(-K_{Z})+yE+zF_{Z}$. Then we have the following equations: $$\begin{aligned} 0&=(-K_{F_{Z}})(D|_{F_{Z}})=4x+5y \text{ and } \\ -2n&=(D|_{F_{Z}})^{2}=4x^{2}-5y^{2}+10xy. \end{aligned}$$ By these equations, we have $x=\pm \sqrt{\frac{5n}{18}}$. This contradicts $x \in {\mathbb{Q}}$ and $n \in \{1,2\}$. Therefore, $\Psi$ is an isomorphism or a small contraction. 2. Let us prove that $\mu \colon Y \to X$ is the blowing-up along a non-singular curve $C$. Since $\rho(Y)=3$, we have $\dim X \geq 2$. If $\dim X=2$, then the following equations holds by the same argument in the proof of Proposition \[prop:1\] : $$\begin{aligned} 2=4x+5y \text{ and } 0=4x^{2}-5y^{2}+10xy \text{ with } x,y\in {\mathbb{Q}}. \end{aligned}$$ Hence $x=\frac{3 \pm \sqrt{5}}{6}$, $y=\mp \frac{2}{3\sqrt{5}}$ and it contradicts $x,y \in {\mathbb{Q}}$. Hence $\mu$ is a divisorial contraction. By the similar arguments in the proof of Proposition \[prop:1\], we can prove that $\mu(D)$ is not a point. Hence we have that $\mu(D)=:C$ is a smooth curve, where $D=\operatorname{Exc}(\mu)$. Denote $D \equiv x(-K_{Z})+yE+zF_{Z}$ with $x,y,z \in {\mathbb{Q}}$ and $m:=F_{X}.C \in {\mathbb{Z}}_{\geq 0}$. Since $(-K_{F_{Y}})^{2}=4$ and $(-K_{F_{X}})^{2} \leq 9$, we have $0 \leq m \leq 5$. By a calculation of some intersection numbers, we have the following: $$\begin{aligned} m&=(-K_{Y})DF=4x+5y, \\ -m&=D^{2}F=4x^{2}-5y^{2}+10xy \text{ and } \\ {\mathbb{Z}}&\ni 5x-5y=EFD. \end{aligned}$$ Hence we have the following: $$\begin{aligned} &x=\frac{3m \pm \sqrt{5m^{2}+20m}}{12},y=\mp \sqrt{\frac{m^{2}+4m}{45}} \text{ and } \\ &5x-5y=\frac{5m \pm 3 \sqrt{5m^{2}+20m}}{4} \in {\mathbb{Z}}.\end{aligned}$$ By using these relations, the possibilities of $(m,x,y)$ are as follows: $$(m,x,y)=(0,0,0),\left(1,\frac{2}{3},-\frac{1}{3} \right), (5,0,1)$$ We can see that $(x,y) \neq (0,0),(0,1)$ by the same argument in the proof of Proposition \[prop:1\]. Hence we have that $m=1$, $x=\frac{2}{3}$, $y=-\frac{1}{3}$ and $(-K_{F_{X}})^{2}=5$. Thus $\mu$ is the blowing-up along a ${\varphi}$-section and ${\varphi}$ is a del Pezzo fibration of degree $5$. 3. Since $x=\frac{2}{3}$, $y=-\frac{1}{3}$, $(-K_{Z})^{3}=(-K_{Y})^{3}$, $(-K_{Z})^{2}D=(-K_{Y})^{2}D_{Y}$ and $(-K_{Z})D^{2}=(-K_{Y})D_{Y}^{2}$, we obtain the following equalities. $$\begin{aligned} &(-K_{X})^{3}-2(-K_{X}).C-2=(-K_{W})^{3}-2(-K_{W}).B+(2g_{B}-2), \\ &(-K_{X}).C+2=\frac{2\left((-K_{W})^{3}-2(-K_{W}).B+(2g_{B}-2) \right)}{3}-\frac{(-K_{W}).B+2-2g_{B}}{3}+4z \text{ and } \\ &-2=\frac{4\left( (-K_{W})^{3}-2(-K_{W}).B+(2g_{B}-2)\right)}{9}+\frac{2g_{B}-2}{9}-\frac{4\left( (-K_{W}).B+2-2g_{B}\right)}{9}+2z.\end{aligned}$$ By solving these equalities, we have the following. $$\begin{aligned} (-K_{X})^{3}&=\frac{5(-K_{W})^{3}-18g_{B}-72}{9}, \\ -K_{X}.C&=\frac{9 (-K_{W}).B-18g_{B}-2(-K_{W})^{3}-36}{9} \text{ and } \\ z&=\frac{6(-K_{W}).B-9g_{B}-2(-K_{W})^{3}}{9}.\end{aligned}$$ Note that $(-K_{W})^{3}=54$ since $W \to {\mathbb{P}}^{1}$ is ${\mathbb{P}}^{2}$-bundle. Thus we have the last statement in Proposition \[prop:2\]. Case (A-2) {#ss:A2} ---------- Let us construct an example belonging to Case (A-2). Let $C {\subset}{\mathbb{P}}^{2}$ be a smooth elliptic curve and set $S,W$ as follows: $$S:=C \times {\mathbb{P}}^{1} {\subset}W:={\mathbb{P}}^{2} \times {\mathbb{P}}^{1}.$$ Set $H:={\mathcal}O_{{\mathbb{P}}^{2} \times {\mathbb{P}}^{1}}(1,0)$ and $F:={\mathcal}O_{{\mathbb{P}}^{2} \times {\mathbb{P}}^{1}}(0,1)$. Note that $-K_{W}=3H+2F$. Let $\operatorname{pr}_{1} \colon S \to C$ is the projection of the first factor and $\operatorname{pr}_{2} \colon S \to {\mathbb{P}}^{1}$ is that of the second factor. Let $B \in |\operatorname{pr}_{1}^{\ast}{\mathcal}O_{C}( 5p ) \otimes \operatorname{pr}_{2}^{\ast}{\mathcal}O_{{\mathbb{P}}^{1}}(2)|$ be a smooth divisor of $S$, where $p$ is a point of $C$. Then we have $g_{B}=6$ by adjunction formula and we have $F.B=5$, $H.B=6$ and $-K_{W}.B=28$. Let ${\tau}\colon Z:=\operatorname{\mathrm{Bl}}_{B}W \to W$ be the blowing-up of $W={\mathbb{P}}^{2} \times {\mathbb{P}}^{1}$ along $B$ and $E$ the ${\tau}$-exceptional divisor. Note that $S_{Z}:={\tau}^{-1}_{\ast}S$ is linearly equivalent to ${\tau}^{\ast}(3H)-E$ and ${\tau}|_{S_{Z}} \colon S_{Z} \to S$ is isomorphic. $-K_{Z}$ is nef and big. It is easy to find that $(-K_{Z})^{3}=12$. Since $-K_{Z}={\tau}^{\ast}(3H+2F)-E=S_{Z}+2F_{Z}$, it is enough to show that $-K_{Z}|_{S_{Z}}$ is nef. Under the identifying $S_{Z}$ with $S$, we have $-K_{Z}|_{S_{Z}}=\operatorname{pr}_{1}^{\ast}{\mathcal}O_{C}( \text{4pts.} )$ and hence $-K_{Z}$ is nef. Therefore, the pair $(\pi \colon {\mathbb{P}}^{2} \times {\mathbb{P}}^{1} \to {\mathbb{P}}^{1},B)$ satisfying the condition $(\dag_{5})$ and we obtain the diagram $(\bigstar_{5})$ and a del Pezzo fibration ${\varphi}\colon X \to {\mathbb{P}}^{1}$ of degree 5 satisfying the following: $$\begin{aligned} (-K_{X})^{3}=10,\quad -K_{X}.C=0 \text{ and } D \equiv \frac{2}{3}(-K_{Y})-\frac{1}{3}E_{Y}+\frac{2}{3}F_{Y}.\end{aligned}$$ Since $-K_{Y}$ is nef and big and $-K_{X}.C=0$, $X$ is almost Fano. According to [@JPR05], [@JPR11] and [@Take09], there are two possibilities of a type of the contraction of the $K_{X}$-trivial ray: a divisorial type or a flopping type. The $K_{X}$-trivial elementary contraction of $X$ is a divisorial contraction. For $\Phi$-flopping curve $l {\subset}Z$, we have $0=-K_{Z}.l=(S_{Z}+2F_{Z}).l=S_{Z}.l$. Since every fiber of $p|_{S_{Z}} \colon S_{Z} \to {\mathbb{P}}^{1}$ is an elliptic curve, $S_{Z}$ has no $\Phi$-flopping curve. 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--- abstract: | We propose a superconducting phase qubit on the basis of the radio-frequency SQUID with the screening parameter value $\beta_L \equiv (2\pi/\Phi_0)LI_c \approx 1$, biased by a half flux quantum $\Phi_e=\Phi_0/2$. Significant anharmonicity ($> 30\%$) can be achieved in this system due to the interplay of the cosine Josephson potential and the parabolic magnetic-energy potential that ultimately leads to the quartic polynomial shape of the well. The two lowest eigenstates in this global minimum perfectly suit for the qubit which is insensitive to the charge variable, biased in the optimal point and allows an efficient dispersive readout. Moreover, the transition frequency in this qubit can be tuned within an appreciable range allowing variable qubit-qubit coupling. `ACS numbers: 85.25.Dq, 74.50.+r, 03.67.Lx, 03.67.`p author: - 'A. B. Zorin$^1$ and F. Chiarello$^2$' date: 'August 27, 2009' title: Superconducting phase qubit based on the Josephson oscillator with strong anharmonicity --- The superconducting qubits based on the Josephson tunnel junctions (see, e.g., the reviews in Refs. [@Makhlin; @Devoret2004]) have already demonstrated their great potential for the quantum computation [@WendinShumeiko]. The so-called phase qubits present the class of devices which are particularly suitable for integration with microwave on-chip transmission lines and resonators, i.e. the elements which significantly extend the scope of the quantum circuit designs [@Martinis2009]. These qubits are based on the energy quantization in the shallow wells of the inclined cosine Josephson potential [@Martinis]. This shape is ensured either by finite bias current $I_s$ with the value slightly below the critical current of the Josephson junction $I_c$ or a finite flux bias $\Phi_e$ applied to the qubit loop (in the case of a loop configuration of the circuit) [@Steffen2003]. In both cases, the energy potential can be approximated by the cubic parabola with a smooth energy barrier isolating the well from one side and allowing escape out of this well enabling a simple readout. The low depth of the cubic parabola well leads to anharmonicity, viz. successive reduction of the transition energies $\Delta E_n = (E_{n+1} - E_{n}),$ $n=0, 1, ...$, from bottom to top, necessary for the qubit operation within the basis states $|n=0\rangle$ and $|n=1\rangle$, excluding unwanted excitation of the higher energy states ($n>1$). Usually the phase qubit is designed such that for appropriate phase bias the cubic potential well includes three-four energy levels with anharmonicity of a few per cent [@Devoret2004; @Steffen2003]. This is achieved by adjusting the plasma frequency of the Josephson junction both by designing appropriate parameters of the junction and, possibly, by applying external capacitor shunting. The lowering of the energy barrier by applying the so-called measuring pulse, makes possible the reduction of the number of the levels to two ($n=0$ and 1), with notably different rates of escape to a running-phase state (in the case of current bias), or to the lower-energy state in the adjacent well (in the case of the loop configuration of the qubit). The large (but finite) difference of these tunneling rates sets the maximum theoretical value for the fidelity of such measurement to 96.6%. In the carefully designed and optimally biased qubit the best experimental fidelity values approach 90% [@Lucero]. The main disadvantage of such phase qubit is the necessity of resetting it after each measurement. ![(a) Electric diagram of the qubit coupled to a resonant circuit and (b) possible equivalent compound (two-junction SQUID) circuit of the Josephson element included into the qubit loop. Capacitance $C$ includes both the self-capacitance of the junction and the external capacitance. Due to inclusion in the resonant circuit of a Josephson junction JJ’, the resonator may operate in the nonlinear regime, enabling a bifurcation-based readout.[]{data-label="EqvSchm"}](fig1){width="3in"} In contrast to the charge [@Nakamura], charge-phase [@Vion; @Z-Phys-C], flux [@Chiorescu], transmon [@Koch] and recently proposed, the so-called fluxonium [@Manucharyan-arx09] qubits, the conventional phase qubits cannot inherently operate in an optimal point, i.e. in the symmetric working point insensitive in the first order to the noise that could give drastic improvement to the qubit performance [@Vion; @Z-JETP]. (The exceptions are the recently proposed three-junction interferometer circuit [@Chiarello] and the so-called camelback potential phase qubit based on the two-junction SQUID [@Hoskinson].) Moreover, the limited anharmonicity of the phase qubit makes the observable reactance impedance (i.e. the Josephson inductance) values in the ground and exited states hardly distinguishable. This poses serious problems for dispersive readout schemes, which proved advantageous where applicable [@Z-Phys-C; @Z-JETP; @Lupascu2004] and have allowed quantum nondemolition measurements as well as high fidelity measurements based on bifurcation amplifiers [@Siddiqi2004; @Lupascu2006]. In this paper we propose an improved phase qubit with significant anharmonicity, in which the manipulation and dispersive readout are both possible in a symmetry point. The circuit diagram of our qubit is shown in Fig.1a. It comprises the superconducting loop with geometrical inductance $L$ closed by a Josephson junction, generally shunted by an external capacitance, and rf-driven $L_RC_RG_R$ resonance circuit inductively coupled to the qubit loop. The peculiarity of this qubit is the unity value of the SQUID screening parameter $\beta_L \equiv (2\pi/\Phi_0)LI_c \approx 1$, where $\Phi_0=h/2e$ is the flux quantum. This can be achieved by an accurate design of the circuit including replacement of the single junction with a two-junction SQUID, allowing more precise adjustment of the resulting critical current $I_c$ (see Fig.1b). The potential energy of the stand-alone qubit biased by external magnetic flux $\Phi_e$ includes the magnetic and Josephson components and can be written as $$\label{U-phi} U(\phi, \phi_e) = E_L \left[0.5(\phi -\phi_e)^2-\beta_L (1+\cos \phi) \right],$$ where $E_L = (\Phi_0/2\pi)^2/L = E_J/\beta_L$ is characteristic magnetic energy associated with the loop inductance, the Josephson coupling energy $E_J=(\Phi_0/2\pi) I_c$, the phase variable $\phi$ and the phase bias $\phi_e = 2\pi\Phi_e/\Phi_0$. For small values of $\beta_L \ll 1$, the potential Eq.(\[U-phi\]) yields the almost parabolic shape of the global single well (first term in Eq.(\[U-phi\])), whereas for the values $\beta_L$ appreciably greater that 1, the series of wells are superimposed on the global parabola, so the bottom parts of these local minima can also be approximated by the quadratic parabolas. In the case of large density of the levels within these parabolas, the energy spectrum is also close to that of a harmonic oscillator. So, neither of these cases allows significant anharmonicity necessary for the efficient qubit operation. ![(Color online) Position of the lowest six levels (solid lines) in the potential Eq.(\[U-phi\]) for $\phi_e=\pi$ as a function of parameter $\beta_L$ for typical values of $L$ and $C$, yielding $E_J/E_c \sim E_L/E_c \approx 5.1 \times 10^4$. With an increase of $\beta_L$, the spectrum crosses over from that of the harmonic oscillator type (left inset) to the set of the doublets (right inset), corresponding to the weak coupling of the oscillator-type states in two separate wells. The spectrum in the central region $\beta_L \approx 1$ is strongly anharmonic. The dashed line shows the bottom energy of the potential $U(\phi,\phi_e=\pi)$, which in the case of $\beta_L > 1$ is equal to $-\Delta U\approx -1.5 E_L (\beta_L-1)^2/\beta_L$ (in other words, $\Delta U$ is the height of the energy barrier in the right inset) [@Chiarello2000; @Chiarello2007]. The dotted (zero-level) line indicates the energy in the symmetry point $\phi = 0$, i.e. at the bottom of the single well ($\beta_L\leq 1$) or at the top of the energy barrier ($\beta_L > 1$). The black dot shows the critical value $\beta^c_L$ at which the ground state energy level touches the top of the barrier separating the two wells. []{data-label="En_b"}](fig2){width="3.4in"} \[levels-beta\] The essentially different shape of the potential Eq.(\[U-phi\]) with $\phi_e=\pi$ is, however, achieved for $\beta_L \approx 1$, i.e. when the quadratic magnetic term is partially compensated by the quadratic term in the Josephson energy expansion near the bottom $(U=0)$ of the single well centered at $\varphi \equiv \phi -\pi = 0$, i.e. $$\label{U-phi-apprx} U(\varphi) \approx E_J \left[ -\frac{(\beta_L-1)}{2\beta_L}\varphi^2+ \frac{1}{24}\varphi^4+O(\varphi^6)\right].$$ In the ultimate case of $\beta_L=1$, Eq.(\[U-phi-apprx\]) provides the obtuse shape of the quartic parabola. Taking into account the finite kinetic energy of the system in corresponding Schrödinger equation, $\hat{Q}^2/2C = -4E_c \partial_{\phi\phi}$, where $\hat{Q}=-i(2e) \partial_{\phi}$ is the charge operator [@Anderson], one can perform quantization of the system. Application of the quasiclassical quantization rule of Bohr-Sommerfeld yields for this quartic oscillator the energy levels obeying the 4/3 power law [@Sanchez]: $$\label{B-S-apprx} E_n^{(\textrm{qc})} = \epsilon (n+1/2)^{4/3},$$ with prefactor $\epsilon$ which in terms of the parameters of our circuit is equal to $$\label{B-S-apprx2} \epsilon = 2^{-5/3} 3 \,[\pi/\textrm{K}(1/2)]^{4/3} (E_J E_c^2)^{1/3} \approx 1.9 (E_J E_c^2)^{1/3},$$ where $\textrm{K}(k)$ is the complete elliptic integral of the first kind. Thus, the energy spectrum in the quartic potential takes intermediate position between the equidistant spectrum of the harmonic oscillator $E_n \propto (n+1/2)$ and that of the rectangular well, $E_n \propto (n+1)^2$, having extremely high anharmonicity. Expressions (\[B-S-apprx\]) and (\[B-S-apprx2\]) are exact for the higher levels ($n\gg 1$) and large “mass” (capacitance $C$), ensuring the very large ratio of the Josephson energy $E_J$ to the charging energy $E_c=e^2/2C$. An estimate of the anharmonicity factor in this quasiclassical approximation can be immediately obtained from Eq.(\[B-S-apprx\]): $$\label{delta} \delta_{\textrm{qc}} =(\Delta E_1-\Delta E_0)/\Delta E_0 \approx 26\%.$$ The numerical solution of the corresponding Schrödinger equation with potential energy Eq.(1) yields in the limit $E_J/E_c \gg 1$ an even larger value of the anharmonicity factor, $\delta \approx 33\%$ (see the energy spectrum in Fig.2). These values substantially exceed the typical anharmonicity values of the conventional phase qubit, $|\delta_{\textrm{phase}}| \approx\,3\%$, for the number of levels inside the cubic-parabola well equal to four [@Devoret2004; @Steffen2003], and transmon-qubit, $|\delta_{\textrm{transmon}}| \approx (E_c/8E_J)^{1/2} \lesssim 5\%$ for optimum values $E_J/E_c \gtrsim 50$ [@Koch]. Moreover, in contrast to the negative values of $\delta$ in these examples, the series of the energy levels in the quartic potential has positive $\delta > 0$, i.e. corresponds to successively increasing level spacings $\Delta E_1 < \Delta E_2 < \Delta E_3 ...$ ![(Color online) (a) The qubit frequency as a function of parameter $\beta_L$ for fixed $L=50$pH and several values of capacitance $C=0.1$, 0.3, 1.0 and 3.0pF (from top to bottom), corresponding to the values of the ratio $E_L/E_c\approx 1.7\times 10^4, 5.1\times 10^4, 1.7\times 10^5$ and $5.1\times 10^5$. (b) Anharmonicity parameter $\delta$ as a function of parameter $\beta_L$ for the same as in (a) inductance $L$ and capacitance values (from top to bottom).[]{data-label="frequency-anharmonicity"}](fig3){width="3.2in"} Such a large, positive anharmonicity is a great advantage of the quartic potential qubit allowing manipulation within the two basis qubit states $|0\rangle$ and $|1\rangle$ not only when applying resonant microwave field, $\nu_{\mu\textrm{w}} \approx \nu_{10}$, but also when applying control microwave signals with large frequency detuning or using rather wide-spectrum rectangular-pulse control signals. The characteristic qubit frequency $\nu_{10}=\Delta E_0/h$ and the anharmonicity factor $\delta$ computed from the Schrödinger equation for the original potential Eq.(\[U-phi\]) in the range $0.9 \leq \beta_L\leq 1.02$ are shown in Fig.\[frequency-anharmonicity\]. One can see that the significant range in the tuning of the qubit frequency within the range of sufficiently large anharmonicity ($\sim 20-50\%$) is attained at a rather fine (typically $\pm1\textrm{-}2\%$) tuning of $\beta_L$ around the value $\beta_L=1$. Such tuning of $\beta_L$ is possible in the circuit having the compound configuration shown in Fig.1b. For values of $\beta_L>1$, the symmetric energy potential has two minima and a barrier between them. The position of the ground state level depends on $\beta_L$ and the ratio of the characteristic energies $E_J/E_c = \beta_L E_L/E_c$. The value of $\beta_L$ at which the ground state level touches the top of the barrier sets the upper limit $\beta^c_L$ for the quartic qubit (marked in Fig.\[levels-beta\] by solid dot). At $\beta_L >\beta^c_L$, the qubit energy dramatically decreases and the qubit states are nearly the symmetric and antisymmetric combinations of the states inside the two wells (see the right inset in Fig.2). Although the qubit with such parameters has very large anharmonicity and can be nicely controlled by dc flux pulses [@Chiarello2007; @Poletto], its readout can hardly be accomplished in a dispersive fashion. ![(Color online) The values of the Josephson inductance of the quartic potential qubit in the ground (solid lines) and excited (dashed lines) states calculated for the geometric inductance value $L=50$pH and the set of capacitances $C$, increasing from top to bottom for both groups of curves.[]{data-label="L_J"}](fig4){width="3.3in"} \[inductance-L01\] Another advantage of the phase qubit having the energy potential of the shape close to the quartic one is a strong dependence of its Josephson inductance $L_J(\Phi_e,n)$ on the quantum state $|n\rangle$. The observed value of the reverse inductance is related to the local curvature of the dependence of corresponding energy $E_n$ on flux $\Phi_e$ (see, e.g., Ref.[@Z-JETP]), $$\label{L_J} L_{J}^{-1}(\Phi_e,n) = \frac{2\pi}{\Phi_0}\langle n| \frac{\partial \hat{I}(\phi,\phi_e)}{\partial \phi_e} |n\rangle = \frac{\partial^2 E_n(\Phi_e)}{\partial \Phi_e^2},$$ where $\hat{I}$ is the operator of supercurrent circulating in the qubit loop. The dependence of the reverse inductance $L_{J}(\Phi_e=\Phi_0/2,n)$ calculated numerically in the two lowest quantum states ($n=0$ and 1) for $L=50$pH and the same set of capacitances $C$ as in Fig.\[frequency-anharmonicity\] is shown in Fig.\[inductance-L01\]. One can see that the ratio of the geometrical to Josephson inductances $L/L_J$ takes large and very different values that can be favorably used for the dispersive readout, ensuring a sufficiently large output signal. Note that for $\beta_L<1$, both inductances $L_J(n=0)$ and $L_J(n=1)$ are negative, whereas at $\beta_L>1$ the inductance $L_J(n=1)$ changes the sign to positive. The readout of this qubit is based on the measurements of the reactive part (inductance) of the loop impedance probed by a low-frequency ac signal, $f \ll \nu_{10}$ of sufficiently small amplitude [@Rifk] (see also Ref. [@Z-Phys-C]). This signal is supplied by a rf-driven oscillator (Fig.1a) as an alternating biasing flux, $\Phi_e = 0.5\Phi_0 + M I_L $ or $\phi_e = \pi + \delta\phi_e$, where $\delta\phi_e = a \cos(2\pi ft)$ with $a\ll1$. Here $M=\kappa (LL_R)^{1/2}$ is mutual inductance, $\kappa$, a dimensionless coupling coefficient and $I_L$, the ac current in inductance $L_R$. Coupling of the qubit to the resonance tank circuit causes renormalization of the circuit inductance (see, e.g. [@Z-Phys-C; @Z-JETP]), $$\label{frequency-shift} L_{R}^{(n)}= L_R(1-\kappa^2 L/L_J(n)),$$ and the resonance frequencies $\omega_n=[L_R^{(n)}C]^{-1/2}$, where $n = 0$ and 1. The relative difference of the resonance frequencies for the qubit in the excited and ground states is $$\label{frequency-shift} \frac{\delta\omega}{\omega_0}= \frac{\omega_1-\omega_0}{\omega_0} = \sqrt{ \frac{1-\kappa^2 L/L_J(0)}{1-\kappa^2 L/L_J(1)} } -1.$$ Figure \[f-shift\] shows this relative frequency shift versus parameter $\beta_L$. One can see that for the rather conservative value of dimensionless coupling $\kappa=0.05$, the relative frequency shift can achieve the easily measured values of about $10\%$. The efficiency of the dispersive readout can be improved in the non-linear regime with bifurcation [@Siddiqi2004]. With our device this regime can be achieved in the resonance circuit including, for example, a Josephson junction (marked in the diagram in Fig.1 by a dashed cross). Due to the high sensitivity of the amplitude (phase) bifurcation to the threshold determined by the effective resonance frequency of the circuit, one can expect a readout with high fidelity even at a rather weak coupling of the qubit and the resonator (compare with the readout of quantronium in Ref.[@Siddiqi2006]). Further improvement of the readout can be achieved in the QED-based circuit including this qubit [@Metcalfe]. The loop configuration and frequency detuning of the quartic qubits should allow their inductive coupling with variable strength keeping both qubits in optimal points. Variable coupling of the optimally biased qubit to a superconducting resonator is also possible. More sophisticated coupling of the pairs of quartic qubits can be accomplished, for example, using a Josephson-junction coupler in a fashion recently proposed by Harris et al. [@Harris2009]. ![(Color online) The resonance frequency shift in the circuit due to excitation of the qubit with the inductance value $L=50$pH and the set of capacitances $C$, decreasing from top to bottom. The dimensionless coupling coefficient $\kappa=0.05$.[]{data-label="dOmega"}](fig5){width="3.2in"} \[f-shift\] In conclusion, we have shown that the phase qubit of the rf-SQUID configuration with parameter $\beta_L \approx 1$ and flux bias $\Phi_e=\Phi_0/2$ has remarkable characteristics. Still, we expect that implementation of this qubit requires the solution of several experimental problems. For example, due to a high sensitivity of the qubit parameters to the magnitude of $\beta_L$, whose optimum values lie within a rather narrow range ($\pm 1\textrm{-}2\%$), particular precaution should be taken against fluctuations in the line controlling the effective Josephson coupling in the circuit (see Fig.1b), because otherwise it may cause significant dephasing of the qubit. Furthermore, flux bias $\Phi_e=\Phi_0/2$ should also be set as precisely as possible. Experimentally, it can be realized either by freezing the flux $\Phi_e=\Phi_0$ in the main loop having a symmetric gradiometer configuration [@Majer], or by including in the loop of a Josephson $\pi$-junction [@Ioffe99] with a sufficiently high critical current ensuring the steady phase shift of $\pi$. Of course, similar to properties of the conventional types of the phase qubits, the coherence characteristics of the quartic qubit will be strongly dependent on the material properties of the circuit. Minimizing the losses due to the qubit coupling to microscopic degrees of freedom (two-level systems) inside the dielectrics surrounding the superconducting circuit (the substrate, insulator inside the capacitor, the junction barriers, etc.) play crucial role for improving the qubit coherence [@Martinis2009]. Since the operation and tuning of the quartic qubit is possible without leaving the optimal point, one may expect a weaker coupling of the qubit to these microscopic two-level systems located inside dielectrics and, therefore, a better quantum coherence. Moreover, the zero persistent supercurrent circulating in the qubit loop at the optimum bias, $\phi_e =\pi$, may also reduce the effect of quasiparticle tunneling on the qubit coherence. 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--- abstract: 'Three quantitative measures of the spatiotemporal behavior of the coupled map lattices: reduced density matrix, reduced wave function, and an analog of particle number, have been introduced. They provide a quantitative meaning to the concept of coherence which in the context of complex systems have been used rather intuitively. Their behavior suggests that the logistic coupled-map lattices approach the states which resemble the condensed states of systems of Bose particles. In addition, pattern formation in two-dimensional coupled map lattices based on the logistic mapping has been investigated with respect to the non-linear parameter, the diffusion constant and initial as well as boundary conditions.' address: - 'Institute of Physics of the Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland; e-mail: mjanow@ifpan.edu.pl' - 'Faculty of Computer Science, Warsaw University of Life Sciences - SGGW, ul. Nowoursynowska 159, 02-786 Warsaw, Poland' author: - Maciej Janowicz - 'Arkadiusz Or[ł]{}owski' title: 'Coherence and pattern formation in coupled logistic-map lattices' --- [coupled map lattices ,Bose-Einstein condensation ,pattern formation ,classical field theory]{} Introduction ============ Coupled map lattices (CMLs) [@CF; @Ilachinski] have long become a useful tool to investigate spatiotemporal behavior of extended and possibly chaotic dynamical systems [@Kaneko1; @WK; @Kapral; @Kaneko2]. It is so even though the most standard CML, that based on the coupling of logistic maps, is physically not particularly appealing as it is fairly remote from any model of natural phenomena. Other, more complicated CMLs, have found interesting applications in physical modeling. One should mention here CMLs developed to describe the Rayleigh-Benard convection [@YK1], dynamics of boiling [@Yanagita; @GC], formation and dynamics of clouds [@YK2], crystal growth processes and hydrodynamics of two-dimensional flows [@Kaneko3]. The most important characteristic quantities employed to study various types of CMLs include co-moving Lyapunov spectra, mutual information flow, spatiotemporal power spectra, Kolmogorov-Sinai entropy density, pattern entropy [@Kaneko3]. More recently, the detrended fluctuation analysis, structure function analysis, local dimensions, embedding dimension and recurrence analysis have also been introduced for CMLs [@MFMF]. The purpose of this paper is to analyze the interesting features of the above-mentioned most standard coupled map lattices which resemble the characteristisc of the condensates of Bose particles as well as those associated with formation of patterns in two spatial dimensions. In particular, we investigate the formation of such patterns for relatively short times; their dependence on two parameters which define CML as well as the initial conditions is found numerically. Thus, the present work is very much in the spirit of classical papers [@Kapral; @Kaneko2; @Kaneko3]. We believe, however, that the subject is very far from being exhausted as it is quite easy to find interesting patterns not discovered in the above works. More importantly, we combine searching for interesting patterns with the introduction of three additional quantities with the help of which one can characterize the dynamics and statistical properties of CMLs. These are the reduced density matrix, the reduced wave function, and a quantity which is an analog of the number of particle. This is motivated, in part, by what we feel is the need to slightly deemphasize the connection of CMLs with finite-dimensional dynamical systems, and make their analysis similar to that of classical field theory, especially the Gross-Pitaevskii equation which is used in the physics of Bose-Einstein condensation [@DGPS; @Leggett]. Application of the classical field-theoretical methods in the physics of condensates have been described, e.g., in [@GGR1; @GGR2; @GGR3]. Many interesting patterns emerge in the system while it still exhibits a transient behavior as can be seen, e.g., in the plots of the “number of particles". We have not attempted here to reach the regime of stationary dynamics in each case. The problem for which times such a stationary regime becomes established is beyond the scope of this work. We are content with the transient regime as long as something interesting about the connection with condensates and about the patterns can be observed. Let us notice that remarkable results on the transient behavior of extended systems with chaotic behavior have been obtained, e.g., in [@Sinha; @Janosi]. In addition, we observe that the condensate-like behavior has been reported in other systems which are not connected with many Bose particles. Of particular interest are the developments in the theory of complex networks [@BB; @RB]. Here, however, we explore the condensate-like behavior in the coupled map lattices. The main body of this work is organized as follows. The mathematical model as well as the basic definitions of reduced density matrix and reduced wave function are introduced in Section 2. Section 3 provides a justification of our claim that the coupled map lattices based on logistic map exhibit properties which are analogous to those of the Bose-Einstein condensates (BEC). The description of numerical results concerning pattern formation are contained in Section 4, while Section 5 comprises a few concluding remarks. The model ========= Let us consider the classical field $\psi(x, y, t)$ defined on a two-dimensional spatial lattice. Its evolution in (dimensionless, discrete) time $t$ is given by the following equation: $$\begin{aligned} \psi(x,y,t+1) &=& (1 - 4 d) f(\psi(x,y,t)) + d \left[ f(\psi(x+1,y,t)) + f(\psi(x-1,y,t)) \right. \nonumber \\ & + & \left. f(\psi(x,y+1,t)) \right. + \left. f(\psi(x,y-1,t)) \right]\end{aligned}$$ where the function $f$ is given by: $$f(\psi) = c \psi (1 - \psi),$$ and the parameters $c$ and $d$ are constant. The set of values taken by $\psi$ is the interval $[0, 1]$. From the physical point of view the above diffusive model is rather bizarre, containing a field-dependent diffusion. There is no conserved quantity here which could play the role of energy or the number of excitations. In the following the coefficient $d$ will be called the “diffusion constant", and the coefficient $c$ - the “non-linear parameter". It is assumed that $\psi$ satisfies either the periodic boundary or Dirichlet (with $\psi = 0$) conditions on the borders of simulation box. The size of that box is $N \times N$. All our simulations have been performed with $N = 256$. Let ${\tilde \psi}$ be the two-dimensional discrete Fourier transform of $\psi$, $${\tilde \psi}(m, n) = \sum_{x = 0}^{N-1} \sum_{y=0}^{N-1} e^{2 \pi i m x/N} e^{2 \pi i n y/N} \psi(x, y),$$ Thus, ${\tilde \psi}$ may be interpreted as the momentum representation of the field $\psi$. Below we investigate the relation between a CML described by Eq.(1) and a Bose-Einstein condensate. Therefore, let us invoke the basic characteristics of the latter which are so important that they actually form a part of its modern definition. These are [@PO; @Yang]: 1. The presence of off-diagonal long-range order (ODLRO). 2. The presence of one eigenvalue of the one-particle reduced density matrix which is much larger than all other eigenvalues. Let us notice that the property 2. corresponds to the well-known intuitive definition of the Bose-Einstein condensate. Namely, taking into account that the one-particle reduced density matrix $\rho^{(1)}$ has the following decomposition in terms of eigenvalues $\lambda_j$ and eigenvectors $|\phi_j \rangle$: $$\rho^{(1)} = \sum_j \lambda_j |\phi_j \rangle \langle \phi_j|$$ we realize that if one of the eigenvalues is much larger than the rest, then the majority or at least a substantial fraction of particles is in the same single-particle quantum state. In addition, for an idealized system of Bose particles with periodic boundary conditions and without external potential, the following signature of condensation is also to be noticed: 3. The population of the zero-momentum mode is much larger than population of all other modes. The properties 1. and 2. acquire quantitative meaning only if the one-particle reduced density matrix is defined. However, as our model is purely classical, the definition of that density matrix is not self-evident. Here we make use of the classical-field approach to the theory of Bose-Einstein condensation [@GGR2; @KGR] and define the quantities: $${\bar \rho}(x, x^{\prime}) = \langle \sum_{y = 0}^{N - 1} \psi(x, y) \psi(x^{\prime}, y) \rangle_{t},$$ and $$\rho(x, x^{\prime}) = {\bar \rho}(x, x^{\prime}) / \sum_{x} {\bar \rho}(x, x).$$ We shall call the quantity $\rho(x, x^{\prime})$ the reduced density matrix of CML. The above definition in terms of an averaged quadratic form made of $\psi$ seems quite natural, especially because $\rho$ is a real symmetric, positive-definite matrix with the trace equal to $1$. The sharp brackets $\langle ... \rangle_{t}$ denote the time averaging: $$\langle (...) \rangle_{t} = \frac{1}{T_{s}} \sum_{t = T - T_{s}}^{T} (...),$$ where $T$ is the total simulation time and $T_{s}$ is the averaging time. In our numerical experiments $T$ has been equal to 3000, and $T_{s}$ has been chosen to be equal to 300. We can provide the quantitative meaning to the concept of off-diagonal long-range order (ODLRO) by saying that it is present in the system if $$\rho(x_{1} + x, x_{1} - x)$$ does not go to zero with increasing $x$ [@Yang]. If this is the case, the system possesses the basic property 1. of Bose-Einstein condensates. Let $W$ be the largest eigenvalue of $\rho$. We will say that CML is in a “condensed state” if $W$ is significantly larger that all other eigenvalues of $\rho$. If this is the case, the system possesses property 2. of the Bose-Einstein condensates. The corresponding eigenvector, $F(x)$, will be called the reduced wave function of the (condensed part of) coupled map lattice. One thing which still requires explanation is that the above definition of the reduced density matrix involves not only temporal, but also spatial averaging over $y$. This is performed just for technical convenience, namely, to avoid dealing with too large matrices. Strictly speaking, we are allowed to assess the presence or absence of ODLRO only in one ($x$) direction. But that direction is arbitrary, as we might equally well consider averaging over $x$ without any qualitative change in the results. In the classical field theory the quantity ${\tilde \psi}^{\star} {\tilde \psi}$ represents the particle density in momentum space; in the corresponding quantum theory, upon the raising of $\psi$, $\psi^{\star}$ to the status of operators, ${\tilde \psi}^{\star} {\tilde \psi}$ would be called the particle number operator. Analogously, we introduce the number $P$ which - just for the purpose of the present article - will be called the “particle number", and is defined as: $$P = \sum_{m = -N/2}^{N/2-1} \sum_{n=-N/2}^{N/2-1} |{\tilde \psi}(m,n)|^{2}.$$ All the above definitions are modeled after the corresponding definitions in the non-relativistic classical field theory. Similarity to Bose-condensed systems ==================================== We have performed our numerical experiment with six values of the non-linear parameter $c$ ($3.5 + 0.1 \cdot i$, $i = 0, 1,...,5$), twenty five values of the diffusion constant $d$ ($0.01 \cdot j$, $j = 1,2,..,25$), five different initial conditions, and two different boundary conditions. The boundary conditions have been chosen either as periodic or “Dirichlet" ones, the latter with $\psi = 0$ at all boundaries. To save some space, the tables below contain the results for $d$ being multiples of $0.05$, but the results for other $d$ do not differ qualitatively from those reported below. The following initial conditions have been investigated. The first - type (A) - initial conditions are such that $\psi(x, y, t)$ is “excited" only at a single point at $t = 0$: $\psi(N/2, N/2, 0) = 0.5$, and $\psi(x, y, 0)$ is equal to zero at all other $(x,y)$. Type (B) initial conditions are such that $\psi(x, y, t)$ has initially two non-vanishing values: $\psi(N/4, N/4, 0) = \psi(3 N/4, 3 N/4, 0) = 0.5$. By type (C) initial conditions we mean those with $\psi(x, y, 0)$ being a Gaussian function, $\psi(x, y, 0) = 0.5 \exp(-0.01 ((x - N/2)^2 + (y - N/2)^2))$. In type (D) initial conditions the Gaussian has been replaced with a sine function, $\psi(x, y, 0) = 0.5 \sin(10 x/(N-1)$, and type (E) are characterized by $\psi(x, y, 0)$ being equal to $0.5$ at all internal points except of one point - $(N/4, N/4)$ - where $\psi$ is equal to $0.50001$. Results for periodic boundary conditions ---------------------------------------- Tables 1-5 show the dependence of the largest eigenvalue of the time-averaged reduced density matrix on $c$ and $d$: $ d \backslash c$ 3.5 3.6 3.7 3.8 3.9 4.0 ------------------- ------- ------- ------- ------- ------- ------- 0.05 0.920 0.909 0.905 0.908 0.905 0.902 0.10 0.929 0.911 0.905 0.914 0.904 0.902 0.15 0.948 0.917 0.929 0.945 0.907 0.904 0.20 0.999 0.996 0.986 0.912 0.908 0.905 0.25 0.499 0.496 0.493 0.483 0.454 0.453 : Largest eigenvalue of the reduced density matrix. Periodic boundary conditions and type (A) initial conditions $ d \backslash c$ 3.5 3.6 3.7 3.8 3.9 4.0 ------------------- ------- ------- ------- ------- ------- ------- 0.05 0.921 0.909 0.906 0.909 0.905 0.902 0.10 0.940 0.918 0.902 0.923 0.904 0.902 0.15 0.945 0.910 0.914 0.954 0.907 0.904 0.20 0.912 0.911 0.898 0.899 0.908 0.905 0.25 0.457 0.457 0.459 0.481 0.455 0.453 : Largest eigenvalue of the reduced density matrix. Periodic boundary conditions and type (B) initial conditions $ d \backslash c$ 3.5 3.6 3.7 3.8 3.9 4.0 ------------------- ------- ------- ------- ------- ------- ------- 0.05 0.925 0.911 0.909 0.907 0.902 0.902 0.10 0.938 0.920 0.901 0.906 0.904 0.902 0.15 0.953 0.928 0.910 0.915 0.906 0.905 0.20 0.923 0.913 0.923 0.893 0.907 0.905 0.25 0.910 0.903 0.888 0.909 0.906 0.904 : Largest eigenvalue of the reduced density matrix. Periodic boundary conditions and type (C) initial conditions $ d \backslash c$ 3.5 3.6 3.7 3.8 3.9 4.0 ------------------- ------- ------- ------- ------- ------- ------- 0.05 0.927 0.913 0.901 0.899 0.893 0.858 0.10 0.934 0.924 0.911 0.904 0.896 0.887 0.15 0.940 0.934 0.919 0.918 0.901 0.882 0.20 0.940 0.932 0.925 0.924 0.902 0.881 0.25 0.934 0.931 0.925 0.921 0.919 0.880 : Largest eigenvalue of the reduced density matrix. Periodic boundary conditions and type (D) initial conditions $ d \backslash c$ 3.5 3.6 3.7 3.8 3.9 4.0 ------------------- ----- ------- ------- ------- ------- ------- 0.05 1.0 0.990 0.895 0.903 0.903 0.902 0.10 1.0 0.991 0.899 0.915 0.904 0.902 0.15 1.0 0.990 0.914 0.906 0.907 0.904 0.20 1.0 0.991 0.902 0.937 0.907 0.905 0.25 1.0 0.994 0.931 0.918 0.903 0.453 : Largest eigenvalue of the reduced density matrix. Periodic boundary conditions and type (E) initial conditions There are several interesting observations which can be made in connection with Tables 1-5. Firstly, with exception of the case $d = 0.25$ and arbitrary $c$ for type (A) initial conditions, the system exhibits one eigenvalue of the reduced density matrix which is much larger than all other eigenvalues for all other values of $c$ and $d$ and both types of initial conditions. This is one of the most important features of the Bose-condensed matter, as explained in Section 2. Our system clearly has the property 2. of BEC. Secondly, for the case $d = 0.25$ and types (A) and (B) initial conditions, the largest eigenvalue is slightly lower than $0.5$. We have checked that, for each $c$, there are [*two*]{} almost equal eigenvalues which are much larger than all other eigenvalues. The presence of two such eigenvalues of the reduced density matrix also has its analog in the physics of Bose-Einstein condensation; it is characteristic for the so-called quasi-condensates [@PSW1; @PSW2; @KGR]. Further, it seems there are certain regularities in the $c$ and $d$ dependence of the maximal eigenvalue. In most (but not all) cases, the value of $W$ appears to decrease with growing $c$ for given $d$. In all cases except of $d=0.25$, $W$ has had the largest value for $c$ equal to 3.5, that is, below the threshold of chaos for a single logistic map. The fact that the largest eigenvalue for type (E) initial conditions does not practically differ from $1$ for $c = 3.5$ and is still very close to $1$ for $c = 3.6$ can obviously be attributed to the fact that those values of the nonlinear parameters ar too “weak” to introduce sufficient variation into the system which is initially almost perfectly homogeneous. In Figure 1(a-b) we have displayed the spatial dependence of the quantity (“one-particle correlation function”) $\sigma(x) = \rho(N/2 + x, N/2 - x)$ for $x = 0,1,2,...,N/2-1$, $d = 0.20$, $c = 3.7$, periodic boundary conditions, and five types of initial conditions. {width="11.0cm" height="10.0cm"} While the values of the above “one-particle correlation function” for $x = 0$ and $x = N/2$ must be equal due to the boundary conditions, a strong decrease of $\sigma(x)$ for $x$ being far from $0$ or $N/2$ would have to take place if there were [*no*]{} long-range order. However, $\sigma(x)$ never falls below the $75\%$ of its value for $x = 0$. In addition, for types (A), (C) and (D) initial conditions the change of $\sigma$ with $x$ reduces itself to very small fluctuations. We can conclude that our system exhibits the property 1. of the Bose-Einstein condensates. Still, the considerable variation of $\sigma(x)$ wwith respect to initial conditions seems quite interesting and probably deserves some further investigation. The plots in Fig. 2(a-b) illustrate the dependence of eigenvectors $F(x)$ (“wave functions of the condensate”) corresponding to the largest eigenvalue $W$ on $x$ for all types of initial conditions. The most important feature of those plots is very weak dependence of $F(x)$ on $x$, with a single exception of type (D) initial conditions where the variation slightly exceeds $10\%$. This feature has also been observed for all other values of our parameters and for much longer times as well, regardless of the final state of the system. {width="11.0cm" height="10cm"} Thus, one may say that the correlation length is virtually infinite, which, again, is a characteristic feature of the strongly condensed physical systems. We note that this is true even for the $d = 0.25$ and type (A) initial conditions, where the system resembles quasi-condensates. In such a case the density fluctuations should not differ from those of the “true" condensates; the difference lies in the phase fluctuations. Elaboration of that interesting point is, however, beyond the scope of the present work. To make our case of pointing out the CML resemblance to Bose condensates even stronger, we have checked the behavior of the field $\psi$ in momentum space. In Figures 3(a-e) the plots of the moduli $|{\tilde \psi}|$ as functions of two components of their “momentum” argument are shown for periodic boundary conditions and five types of initial conditions. The function $|{\tilde \psi}(m, n)|$ is normalized in such a way that its maximal value is $1$. {width="11.0cm" height="14.0cm"} Almost all plots in Fig. 3 are qualitatively the same except, again, of that corresponding to the sinusoidal initial conditions. In addition, they are representative for the entire spectrum of values of $c$ and $d$, even for $d = 0.25$ with type (A) initial conditions (that is, for “quasi-condensates”). Strong peak at the zero momentum clearly dominates all the other maxima. The fact that the zeroth mode is the only one which is so strongly populated is yet another feature of Bose-condensed system of particles - our system exhibits the property 3. of condensates. However, the lateral amplitudes for the sinusoidal initial conditions are relatively high, as can be seen in Fig. 3(d). Although even for this case the dominance of the central mode is clear, it appears that it is weaker than in the case of other initial conditions. Let us finally consider the function $P(t)$, which is an analog of the particle number. In our system $P(t)$ is a genuine function of time, and there is no conservation law for it. The figure 4(a-e) contains several plots of time dependence of $P(t)$ for $d = 0.020$ and $c = 3.7$, periodic boundary conditions and five types of initial conditions. {width="11.0cm" height="15.0cm"} Let us first observe that the asymptotic dynamics of $P(t)$ for large $t$ which are very different for does indeed depend on initial conditions. The “bands" which are very characteristic for type (A) and (B) initial conditions practically disappear for type (C) and (E) initial conditions. The sinusoidal (type (D)) initial conditions are again quite special for two reasons. Not only is the change of $P(t)$ very erractic with no visible “bands", but its mean value is, in addition, almost one order of magnitude smaller than in the case of other initial conditions. So far, we cannot offer any explanation of this feature. Results for Dirichlet boundary conditions ----------------------------------------- Tables 6-10 show the dependence of the largest eigenvalue of the time-averaged reduced density matrix on $c$ and $d$: $ d \backslash c$ 3.5 3.6 3.7 3.8 3.9 4.0 ------------------- ------- ------- ------- ------- ------- ------- 0.05 0.920 0.909 0.904 0.908 0.905 0.902 0.10 0.922 0.916 0.913 0.911 0.903 0.901 0.15 0.940 0.944 0.929 0.933 0.905 0.903 0.20 0.997 0.997 0.986 0.909 0.906 0.905 0.25 0.499 497 0.493 0.483 0.453 0.453 : Largest eigenvalue of the reduced density matrix. Dirichlet boundary conidtions and type (A) initial conditions $ d \backslash c$ 3.5 3.6 3.7 3.8 3.9 4.0 ------------------- ------- ------- ------- ------- ------- ------- 0.05 0.920 0.909 0.906 0.910 0.905 0.902 0.10 0.944 0.915 0.903 0.910 0.904 0.901 0.15 0.950 0.973 0.899 0.920 0.905 0.904 0.20 0.953 0.956 0.958 0.916 0.906 0.905 0.25 0.486 0.489 0.483 0.483 0.453 0.453 : Largest eigenvalue of the reduced density matrix. Dirichlet boundary conidtions and type (B) initial conditions $ d \backslash c$ 3.5 3.6 3.7 3.8 3.9 4.0 ------------------- ------- ------- ------- ------- ------- ------- 0.05 0.925 0.912 0.909 0.907 0.902 0.902 0.10 0.937 0.920 0.902 0.897 0.904 0.901 0.15 0.948 0.928 0.907 0.903 0.905 0.904 0.20 0.923 0.913 0.923 0.906 0.906 0.905 0.25 0.911 0.904 0.887 0.894 0.906 0.904 : Largest eigenvalue of the reduced density matrix. Dirichlet boundary conidtions and type (C) initial conditions $ d \backslash c$ 3.5 3.6 3.7 3.8 3.9 4.0 ------------------- ------- ------- ------- ------- ------- ------- 0.05 0.913 0.912 0.900 0.898 0.896 0.902 0.10 0.936 0.926 0.916 0.897 0.904 0.901 0.15 0.942 0.936 0.925 0.911 0.905 0.904 0.20 0.942 0.934 0.929 0.895 0.906 0.905 0.25 0.934 0.931 0.964 0.936 0.905 0.904 : Largest eigenvalue of the reduced density matrix. Dirichlet boundary conidtions and type (D) initial conditions $ d \backslash c$ 3.5 3.6 3.7 3.8 3.9 4.0 ------------------- ----- ------- ------- ------- ------- ------- 0.05 1.0 0.993 0.922 0.926 0.913 0.902 0.10 1.0 0.992 0.922 0.900 0.903 0.901 0.15 1.0 0.992 0.929 0.906 0.905 0.904 0.20 1.0 0.992 0.922 0.913 0.906 0.905 0.25 1.0 0.991 0.922 0.904 0.906 0.904 : Largest eigenvalue of the reduced density matrix. Dirichlet boundary conidtions and type (E) initial conditions The most important conclusion one can draw from the Tables (1-5) is the same as in the case of periodic boundary conditions: there exist one dominant eigenvalues for almost values parameters except of the value $d=0.25$ where exactly two dominant eigenvalues are present. This feature suggests that the change of boundary conditions does not diminish the similarity of our system to the Bose-Einstein condensates, or quasi-condensates. Also, it seems that the general trend of change (namely, decrease) of the largest eigenvalue with growing $c$ for given $d$ is here observed, but again, the number of exceptions is considerable. In Figure 5 we have displayed the spatial dependence of the quantity (“one-particle correlation function”) $\sigma(x) = \rho(N/2 + x, N/2 - x)$ for $x = 0,1,2,...,N/2-1$, $d = 0.20$, $c = 3.7$, periodic boundary conditions, and five types of initial conditions. As in the case of periodic boundary conditions, the long-range order is very transparent. The function $\rho(N/2 - x, N/2 + x)$ approaches zero only for the values of its arguments approaching boundaries. {width="11.0cm" height="13.0cm"} Figure 6(a-c) contains the plots of the eigenvectors (“wave functions of the condensate”) corresponding to the largest eigenvalue $W$ for all types of initial conditions. {width="11.0cm" height="14cm"} All the above plots are very flat, except of the arguments close to the boundaries, and the values of the “wave functions” never never approach zero. The system appears to be globally correlated, even though it is still in the transient regime with only local synchronization. As in the previous Section, we have checked the behavior of the field $\psi$ in momentum space. In Figures 7(a-e) the plots of the moduli $|{\tilde \psi}|$ as functions of two components of their “momentum” argument are shown for two types of initial conditions. The function $|{\tilde \psi}(m, n)|$ is normalized in such a way that its maximal value is $1$. {width="11.0cm" height="15.0cm"} The dominance of a single $(m = 0, n = 0)$ mode is transparent, except that in the case the type (D) (sinusoidal) initial conditions the population of lateral modes is substantially bigger that in the other cases. Figure 8(a-e) contains the plots of time dependence of the variable $P$ for $d = 0.020$ and $c = 3.7$. {width="11.0cm" height="13.0cm"} It seems that the dynamics of the particle is quite sensitive to the boundary conditions for we can see here substantial deviation from the dynamics of $P$ in the case of periodic boundary conditions. In particular, in the time regime which is investigated here, the “bands" appear to be far better visible for the Dirichlet boundary conditions. What is more, the case of sinusoidal initial conditions is no longer much different from all the others, although the mean number of particles is still smallest in that case. Large-scale pattern formation ============================= We have observed the following general rules in the process of pattern formation in our system. Firstly, the patterns are incomparably better developed (much better visible) for any “structured" initial conditions (like those considered in this work) than in the case of random initial conditions. The initial inhomogeneities (or “seeds") serve the building of large structures much better than fully random conditions, which is fairly intuitive. The patterns are best developed for smaller values of the non-linear parameter and intermediate values of the diffusion constant. In Figs. 9-15 we show shaded-contour plots representing the values of the field $\psi(x,y)$ after 3000 time steps for periodic boundary conditions and types (A-C) and (E) initial conditions. There are no figures for type (D) (sinusoidal) boundary conditions because they are quite uninteresting, displaying merely the stripes corresponding to the sinusoidal initial “excitation". {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} Naturally, the large structures visible in Figs. 9-14 reflect, to some extent, the symmetry of the simulation box. More interesting observation is that the change from periodic ($c = 3.5$) to chaotic ($c = 3.6$) regime - as defined for individual maps - does not lead, in the case of very slow diffusion, to any spectacular change of the pattern. {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} The most characteristic feature of the fast-diffusion (i.e. large $d$) case is the disappearance of the large-scale structures even for any type of initial conditions. However, somewhat more pronounced grainy structures reappear for $c = 3.9$. {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} In Figs. 16-22 we show shaded-contour plots representing the values of the field $\psi(x,y)$ after 3000 time steps for Dirichlet boundary conditions and five types of initial conditions. {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} Naturally, the large structures visible in the figures reflect, to some extent, the symmetry of the simulation box. More interesting observation is that the change from periodic ($c = 3.5$) to chaotic ($c = 3.6$) regime - as defined for individual maps - does not lead, in the case of very slow diffusion, to any spectacular change of the pattern. {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} {width="4.5cm" height="4.5cm"} The most characteristic feature of the fast-diffusion (i.e. large $d$) case is again the disappearance of the large-scale structures for all types of initial conditions. This is also reflected by flat curves in the plots of reduced wave functions. However, somewhat more pronounced grainy structures again reappear for $c = 3.9$ and large $d$. Concluding remarks ================== Perhaps the most interesting of the various features of the considered system of coupled map lattices is that it appears to be “condensed” if the most standard measures of the classical field theory of Bose condensates are applied. That is, for a majority of parameter values we have observed that a gap between the largest eigenvalue of the reduced density matrix and the rest has been developed. Only for $d = 0.25$ we have observed not a single, but rather two eigenvalues which are much larger than all remaining ones. The latter fact might be of independent interest, as it seems to correspond with the so-called “quasi-condensates". Secondly, the prominent characteristic of the system is the presence of large-scale patterns for smaller values of the “diffusion constant" $d$, $d \leq 0.2$ and not too large values of the non-linear parameter, $c \leq 3.8$. Thirdly, a very strong dependence of both the presence and qualitative features of the patterns on the initial conditions is to be noticed. The latter fact should be a warning against restricting oneself to one type of initial conditions - namely, the purely random ones - which is very often met in the literature. The most interesting facts can be overlooked this way. Interestingly, the strong dependence of patterns on the initial conditions takes place even in the non-chaotic regime of the parameter $c$. Fourthly, for very slow diffusion ($d = 0.05$) we have found that the “number of particles” - defined in a natural way - is an approximate constant of motion for sufficiently large times (because the period-2 oscillations have very small amplitude). If the system exhibits period-2 or period-4 oscillations, the number of particle fluctuates around two (or four) values, as if there were two (four) different systems. We have, in addition, performed similar numerical experiments with another version of the logistic map, reaching similar conclusions. The same statement seems to be valid in the case of standard (rather than logistic) map employed as a basis for the coupled map lattice; however, we have only very preliminary results in that case. Finally, we would like to observe that the domination of zeroth mode in the momentum space suggests that a kind of Bogoliubov approximation could be applicable. This might lead to an efficient analytical approach to the dynamics of CML. We plan to develop further our attempt of using classical field-theoretical concepts in coupled map lattices. Work is in progress of their using in the case of three-dimensional CMLs based on logistic maps as well as other physically more appealing discrete systems. [**Acknowledgments**]{} It is a pleasure to thank Professor Mariusz Gajda and Dr. Emilia Witkowska for offering several helpful discussions [00]{} , edited by J.R. 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--- abstract: 'Suppose an interval is put on a horizontal line with random roughness. With probability one it is supported at two points, one from the left, and another from the right from its center. We compute probability distribution of support points provided the roughness is fine grained. We also solve an analogous problem where a circle is put on a rough plane. Some applications in static are given.' author: - 'D.Treschev' title: On the support of a body by a surface with random roughness --- Introduction ============ Motivations ----------- The Amonton-Coulomb law of friction (dry friction) says that if the motion of a body is a translation along a fixed plane, the friction force is up to a constant multiplier (the dry friction coefficient) equals total normal load. If the body slides along a plane with nonzero angular velocity, to obtain total friction force and total friction momentum, one has to integrate infinitesimal friction forces over the contact spot. This makes the problem of sliding of a body along a plane in the presence of dry friction non-trivial. There is a series of publications where dynamical problems of this kind are studied: [@Con; @Zh1; @ISC; @McM; @Far; @27_n]. A key role in these models is played by the hypothesis on the distribution of the normal load on the contact spot. All such hypotheses are essentially phenomenological although some quasistatic argument is usually attached. The uniform distribution [@ISC; @Far; @27_n] or rotational symmetric ones (for cylindrical bodies with rotational symmetric base) [@Kir1; @Kir2] are compatible with dynamics only for bodies of infinitesimal height. Dynamically compatible deformations of the above distributions are considered in [@Iva], see also [@GST; @BS; @EIT], where qualitative analysis of the motion is presented. Very careful experiments [@27_n], where a plastic disk slides along nylon, stretched over the surface of a flat table, essentially confirm (even quantitatively) theoretical predictions. Other experiments, where a rigid disk slides along a rigid surface [@Kir4; @GST] produce much more noisy data which correspond to the the above theoretical works only qualitatively. We believe that the main reason for such noisy and unstable data is that when both the disk and the support surface are sufficiently rigid, it is hard to expect that their surfaces are perfectly flat: very small deviations from ideal flatness can change unpredictably the distribution of the normal load and break any deterministic hypothesis on the distribution of a load over the contact spot. In this case one should use some probabilistic assumptions. For example, it is possible to consider a (perfectly) flat body on a rough surface with random roughness. Instead of a disk on a plane in this paper we consider two simpler problems: an interval on a rough line and a circle on a rough plane. We also consider some static problems which appear in this context. An interval on a line --------------------- Consider the points $$w_j = (x_j,0), \quad x_j = -1 + 2/N, \qquad j=1,\ldots,N$$ on the horizontal interval $$I = \{(x,z)\in{\mathbb{R}}^2 : x\in [-1,1], z=0\} .$$ Each point $w_j$ is supposed to be the lower end of a vertical interval whose length $\xi_j$ is uniformly distributed on $[0,1]$. We call any such vertical interval a tooth and the whole set of these intervals a random comb, see Fig. \[fig:comb\] An interval $J$, lying on this random comb and projecting exactly on $I$, with probability 1 is supported by two teeth with horizontal coordinates $$a_1 = x_{j_1}\in I_- = [-1,0], \quad a_2 = x_{j_1}\in I_+ = [0,1], \qquad 1\le j_1,j_2 \le N.$$ We say that in this case the event $S^a$ takes place. \[theo:int\] In the limit $N\to\infty$ the density $p : I_-\times I_+ \to {\mathbb{R}}_+$ of probability distribution of the random event $S^a$ is $$\label{p} p(a) = (a_2-a_1) \Big( \frac{4}{3(1 + a_2)^3} + \frac{4}{3(1 - a_1)^3} + \frac16 \Big).$$ Graph of the function $p$ is presented in Fig. \[fig:p\_density\]. We see that $p(0,0)=0$ and $p$ attains global maximum at the points $(-1,0)$ and (0,1). To get “mechanical” interpretations, suppose that the heavy interval $J$ is drawn along a rough line. Where it will be scratched more: near ends or in the middle? Density of probability distribution for the right support point $a_2$ is as follows: $$p_2 = \int_{-1}^0 p(a)\, da_1 = \frac{4}{3(1+a_2)^2} - \frac{2}{3(1+a_2)^3} + \frac{2a_2}{3} + \frac14.$$ We have: $p_2(1)/p_2(0) = 14/11$. Therefore endpoints of $J$ are support points 14/11 times more frequently than points near the center of $J$. However we should take into account that the rate of scratching depends also on the normal load. Hence we have to perform another calculation. Suppose that the rate of scratching is proportional to the normal load. If $J$ is supported at the points $a_1 < 0 < a_2$, the left and right tooth carries the weight $$l_1(a) = \frac{P a_2}{a_2 - a_1} \quad \mbox{and}\quad l_2(a) = \frac{P a_1}{a_1 - a_2}$$ respectively, where $P$ is the weight of $J$. Therefore the rate of scratching at the left support point is proportional to $$\mbox{\rm scr}_1(a_1) = \int_0^1 \frac{a_2}{a_2-a_1} p(a)\, da_2 = \frac{2}{3(1-a_1)^3} + \frac12.$$ Analogously $\mbox{\rm scr}_2(a_2) = \frac{2}{3(1+a_2)^3} + \frac12$. Since $\frac{\mbox{\rm scr}_2(0)}{\mbox{\rm scr}_2(1)} = \frac{10}{7}$, we see that the middle point will be scratched stronger than the end. Another application of (\[p\]) is as follows. Suppose that $J$ is a heavy beam of mass $M$ lying on an uneven surface. A man of mass $m$ walks along the beam. At some moment it may happen that under the weight of the man the beam will leave its initial equilibrium, starting to rotate on one of the support points. We compute the probability $$p_* = p_*(\mu), \qquad \mu = \frac{m}{m+M} \in [0,1]$$ of the random event that this does not happen. This event is equivalent to the following two inequalities: $$-a_1 > \mu, \quad a_2 > \mu.$$ Therefore $$p_* = \int_{-1}^\mu \int_\mu^1 (a_2 - a_1) \Big( \frac 4{3(1+a_2)^3} + \frac 4{3(1-a_1)^3} + \frac16 \Big)\, da_2 da_1 = \frac{(1-\mu)^2}{6}\, \Big( 6+\mu -\Big(\frac{2\mu}{1+\mu}\Big)^2\Big) .$$ In particular, if $m=M$, we have $p_* \approx 1/4$. Graph of the function $p_*(\mu)$ is presented in Fig. \[fig:\*\]. Circle on a plane ----------------- Consider the points $$w_j = w(\alpha_j) = (x_j,y_j,z_j) = (\cos\alpha_j,\sin\alpha_j,0), \quad \alpha_j = 2\pi j/N \qquad j = 1,\ldots,N$$ on the horizontal circle $${\mathbf{c}}= \{(x,y,z)\in{\mathbb{R}}^3 : x^2 + y^2 = 1, \; z=0 \}.$$ Each point $w_j$ is supposed to be the lower end of a vertical interval whose length $\xi(w_j)$ is uniformly distributed on ${\mathbf{c}}$. We call any such vertical interval a tooth and the whole set of these intervals a random circular comb, see Fig. \[fig:can\]. A thin hoop $J$, lying on this random comb, with probability 1 is supported by three teeth $$w_{n_i} = (\cos\alpha_{n_i},\sin\alpha_{n_i},0), \qquad 1\le n_i \le N, \quad i = 1,2,3.$$ We say that in this case the event $S^{\varphi}$ takes place $S^{\varphi}$ takes place, where $$\label{ph=n} {\varphi}= ({\varphi}_1,{\varphi}_2,{\varphi}_3), \quad {\varphi}_i = 2\pi n_i / N \bmod 2\pi, \qquad i = 1,2,3.$$ We are interested in probability distribution of the random event $S^{\varphi}$. We assume that orientation of the triangle ${\varphi}= ({\varphi}_1,{\varphi}_2,{\varphi}_3)$, ${\varphi}_i = \alpha_{n_i}$ is positive i.e., $${\varphi}_{i+1} - {\varphi}_i = {\vartheta}_{i-1}\bmod 2\pi, \quad \mbox{for some real } {\vartheta}_1,{\vartheta}_2,{\vartheta}_3 > 0, \quad {\vartheta}_1 + {\vartheta}_2 + {\vartheta}_3 = 2\pi,$$ where it is convenient to assume the subscript $i$ to lie in the cyclic group ${\mathbb{Z}}_3$. The mass center of $J$ should lie inside the triangle with vertices $w_{n_1},w_{n_2},w_{n_3}$ (otherwise $J$ can not be in equilibrium on the teeth $n_1,n_2,n_3$). This condition is equivalent to the inequalities $0 < {\vartheta}_i < \pi$. Moreover, the events $S^{({\varphi}_1,{\varphi}_2,{\varphi}_3)}$, $S^{({\varphi}_2,{\varphi}_3,{\varphi}_1)}$, and $S^{({\varphi}_3,{\varphi}_1,{\varphi}_2)}$ are the same. Therefore $${\varphi}\in \widehat{\mathcal{S}}= {\mathcal{S}}/{\mathbb{Z}}_3, \qquad {\mathcal{S}}= \{{\varphi}\in{\mathbb{T}}^3 : 0 < {\vartheta}_i({\varphi}) < \pi, \; i= 1,2,3 \},$$ where ${\mathbb{Z}}_3$ acts on ${\mathbb{T}}^3$ by cyclic permutations: $$({\varphi}_1,{\varphi}_2,{\varphi}_3) \mapsto ({\varphi}_2,{\varphi}_3,{\varphi}_1) \mapsto ({\varphi}_3,{\varphi}_1,{\varphi}_2).$$ In the limit $N\to\infty$ distribution of the random event $S^{\varphi}$ has density $p_{\mathcal{S}}: \widehat{\mathcal{S}}\to{\mathbb{R}}_+$. This density is invariant with respect to the action $R_\alpha$ of the circle ${\mathbb{T}}$: $$\label{action} \widehat{\mathcal{S}}\ni\hat{\varphi}\mapsto R_\alpha(\hat{\varphi}) = \hat{\varphi}+ \alpha\, {\mathbf{1}}, \qquad {\mathbf{1}}= (1,1,1)^T\in{\mathbb{R}}^3, \quad \alpha\in{\mathbb{T}}.$$ Therefore it is natural to consider this distribution on the quotient $$\label{triangle} \widehat{\mathcal{T}}= \widehat{\mathcal{S}}/ {\mathbb{T}}= {\mathcal{T}}/ {\mathbb{Z}}_3, \qquad {\mathcal{T}}= \{{\vartheta}= ({\vartheta}_1,{\vartheta}_2,{\vartheta}_3) : 0 < {\vartheta}_i < \pi, \; {\vartheta}_1 + {\vartheta}_2 + {\vartheta}_3 = 2\pi \} .$$ More precisely, let ${\operatorname{\mbox{pr}}}: \widehat{\mathcal{S}}\to\widehat{\mathcal{T}}$ be the natural projection. Then there exists a function $p_{\mathcal{T}}: \widehat{\mathcal{T}}\to{\mathbb{R}}_+$ such that $p_{\mathcal{T}}\circ{\operatorname{\mbox{pr}}}= p_{\mathcal{S}}$. The space $\widehat{\mathcal{T}}$ should be considered with the measure $\mu_{\mathcal{T}}$: $$\label{mu_T} d\mu_{\mathcal{T}}= \frac13 \big| d{\vartheta}_3\wedge d{\vartheta}_2 + d{\vartheta}_1\wedge d{\vartheta}_3 + d{\vartheta}_2\wedge d{\vartheta}_1 \big| .$$ Then $d\hat{\varphi}= d\hat{\varphi}_1 d\hat{\varphi}_2 d\hat{\varphi}_3$ is the pull-back of $d\mu_{\mathcal{T}}$: $\,{\operatorname{\mbox{pr}}}_* (d\mu_{\mathcal{T}}) = d\hat{\varphi}$. \[theo:circ\] The density $p_{\mathcal{T}}$ satisfies the equation $$p_{\mathcal{T}}(\hat{\vartheta}) = 2\pi \sin\frac{{\vartheta}_1}{2} \sin\frac{{\vartheta}_2}{2} \sin\frac{{\vartheta}_3}{2} \Big( \frac1{\pi^2} + \sum_{i=1}^3 f(\hat{\vartheta}_i) \Big),\quad f(\xi) = \int_0^{\xi/2} \! \frac{(\xi - 2{\varphi})\sin{\varphi}} {((\pi-{\varphi})\cos{\varphi}+ \sin{\varphi})^3} \, d{\varphi}.$$ Graph of the function $p_{\mathcal{T}}$ is presented in Fig. \[fig:comb\_circ\]. Here we take ${\vartheta}_1,{\vartheta}_2$ as coordinates on $\widehat{\mathcal{T}}$. Hence $\widehat{\mathcal{T}}$ can be regarded as the triangle $$\{({\vartheta}_1,{\vartheta}_2) : 0<{\vartheta}_1<\pi,\, 0<{\vartheta}_2<\pi,\, {\vartheta}_1 + {\vartheta}_2 < \pi\}$$ with identification $$\label{equivalence} ({\vartheta}_1,{\vartheta}_2) \sim ({\vartheta}_2,2\pi-{\vartheta}_1-{\vartheta}_2) \sim (2\pi-{\vartheta}_1-{\vartheta}_2,{\vartheta}_1).$$ We see that $p=0$ if one of the angles ${\vartheta}_i$ vanishes. Maximal value of $p$ is attained at points ${\vartheta}$ such that for some $i\in{\mathbb{Z}}_3$ $\; {\vartheta}_i = \pi$ and ${\vartheta}_{i\pm 1} = \pi/2$. As an illustration consider a man of mass $m$ going around the hoop of mass $M$. Let $$p_* = p_*(\mu), \qquad \cos\alpha = \mu = \frac{m}{m+M},\quad 0\le\alpha\le\pi/2$$ be the probability of the random event that the hoop stands motionless during all the walk. This event is equivalent to the 3 inequalities $$0 < {\vartheta}_i < 2\alpha, \qquad i = 1,2,3.$$ Hence $$p_* = \int_{D(\alpha)} p_{{\mathcal{T}}}({\vartheta})\, d{\vartheta}_1 d{\vartheta}_2, \qquad D(\alpha) = \big\{({\vartheta}_1,{\vartheta}_2) : {\vartheta}_1 < 2\alpha,\, {\vartheta}_1 < 2\alpha,\, 2\pi - 2\alpha < {\vartheta}_1 + {\vartheta}_2 \big\} / \sim ,$$ where $\sim$ is the equivalence relation (\[equivalence\]). Since $D(\alpha)$ is empty for $\alpha < \pi/3$, we only have to consider the case $\pi/3 < \alpha < \pi/2$. Graph of the function $\mu\mapsto p_*(\mu)$, $$p_* = \frac{2\pi}3 \int_{2\pi-4\alpha}^{2\alpha} d{\vartheta}_1 \int_{2\pi-2\alpha-{\vartheta}_1}^{2\alpha} \sin\frac{{\vartheta}_1}{2} \sin\frac{{\vartheta}_2}{2} \sin\frac{{\vartheta}_1+{\vartheta}_2}{2} \Big( \frac1{\pi^2} + f({\vartheta}_1) + f({\vartheta}_2) + f(2\pi - {\vartheta}_1 - {\vartheta}_2) \Big)\, d{\vartheta}_2$$ is presented in Fig. \[fig:man\_on\_circ\]. In particular, $p_* = 1/2$ for $\mu\approx 1/6$. Proof of Theorem \[theo:int\] ============================= Let $\Omega$ be the configuration space of the random comb: $\Omega = [0,1])^N$. We consider large integer $L$, and put $$n = (n_1,n_2), \quad 1\le n_1 \le N/2 < n_2 \le N.$$ Then we define two random events $\nu$ and $Q_n$, where by definition - $\nu = n$ iff $J$ is supported by the teeth $n_1$ and $n_2$, - $Q_n = (K_1,K_2)$ iff length of the tooth with number $n_i$ equals $$\xi_{n_i} \in \big(1 - (K_i-1) / (NL),1 - K_i / (NL)\big), \qquad i = 1,2.$$ For any $K\in \{1,\ldots,NL\}^2$ we have: $${\mathsf P}\{Q_n = K\} = (NL)^{-2}.$$ Therefore by the formula of total probability $${\mathsf P}\{\nu = n\} = \sum_K {\mathsf P}\{\nu = n | Q_n = K\}\, {\mathsf P}\{Q_n=K\} = \sum_K \frac{{\mathsf P}\{\nu = n | Q_n = K\}} {(NL)^2}.$$ In the limit $L\to\infty$ we obtain: $$\label{limit} {\mathsf P}\{\nu=n\} = \frac{1}{N^2} \int_0^N\!\!\!\int_0^N {\mathsf P}\{\nu=n | \xi_n = {\mathbf{1}}- A/N\} \, dA, \qquad \xi_n = (\xi_{n_1},\xi_{n_2}), \quad {\mathbf{1}}= (1,1)^T .$$ In the limit $N\to\infty$ we obtain densities of probability distributions $$\begin{aligned} & p : I_-\times I_+ \to {\mathbb{R}}_+, \quad p_{\nu|Q}:I_-\times I_+\times [0,N]^2 \to{\mathbb{R}}_+, & \\ &\displaystyle p(a) = \lim_{N\to\infty} \frac{N^2}{4}\,{\mathsf P}\{\nu=n\}, \quad p_{\nu|Q}(a,A) = \lim_{N\to\infty} {\mathsf P}\{\nu = n, \xi_n = 1-A/N\}, & \\ & a = (a_1,a_2) = \big(-1 + 2n_1/N,-1 + 2n_2/N\big). &\end{aligned}$$ Equation (\[limit\]) implies $$\label{p(a)} p(a) = \frac14 \int_0^N\!\!\int_0^N p_{\nu|Q}(a,A)\, dA .$$ Now we turn to computation of $p_{\nu|Q}$. The interval $J = J(a,A)$ is determined by the equation $$z = 1 - \frac{A_1a_2-A_2a_1}{N(a_2-a_1)} - \frac{A_2-A_1}{N(a_2-a_1)}x, \qquad x\in I.$$ We have to consider two cases. \(1) The interval $J$ does not intersect the line segment $I_+$ joining the points $(-1,1)$ and $(1,1)$. This happens provided $$|A_1-A_2| < A_1a_2 - A_2a_1.$$ \(2) $J\cap I_+ = (x_*,1)$. In this case $|A_1-A_2| \ge A_1a_2-A_2a_1$ and $$x_* = \frac{A_1a_2-A_2a_1}{A_1-A_2}.$$ In case (1) probability for the point $w_j=(x_j,0)$ to have the tooth (entirely) under $J$ is $$\label{under} z_j = 1 - \frac{A_1a_2-A_2a_1}{N(a_2-a_1)} - \frac{A_2-A_1}{N(a_2-a_1)} x_j < 1.$$ Therefore probability for the whole comb to be under $J$ is $$\begin{aligned} &\displaystyle p_{\nu|Q}^{(1)} = \prod_{j=1}^N \Big( 1 - \frac{A_1a_2-A_2a_1}{N(a_2-a_1)} - \frac{A_2-A_1}{N(a_2-a_1)} \Big(-1 + \frac{2j}{N}\Big) \Big) = e^{F_1}, & \\ &\displaystyle F_1 = \sum_{j=1}^N \log \Big( 1 - \frac{A_1a_2-A_2a_1}{N(a_2-a_1)} - \frac{A_2-A_1}{N(a_2-a_1)} \Big(-1 + \frac{2j}{N}\Big) \Big). &\end{aligned}$$ In the limit $N\to\infty$ we have: $F_1 = - \frac{A_1a_2-A_2a_1}{a_2-a_1}$. Hence $$p_{\nu|Q}^{(1)} = e^{- \frac{A_1a_2-A_2a_1}{a_2-a_1}} .$$ Consider case (2). For definiteness we assume that $A_1>A_2$ i.e., $x_*>0$. Then probability for any point $w_j$ to have the tooth under $J$ is determined by (\[under\]) if $x_j\in [-1,x_*]$ and equals 1 if $x_j\in [x_*,1]$. Probability for the whole comb to be under $J$ is $p^{(2)}_{\nu|Q} = e^{F_2}$, $$\begin{aligned} F_2 &=& \sum_{j\ge 1, -1+2j/N\le x_*} \log \Big( 1 - \frac{A_1a_2-A_2a_1}{N(a_2-a_1)} - \frac{A_2-A_1}{N(a_2-a_1)} \Big(-1 + \frac{2j}{N}\Big) \Big) \\ &=& -\frac12 \int_{-1}^{x_*} \Big( \frac{A_1a_2-A_2a_1}{a_2-a_1} + \frac{A_2-A_1}{a_2-a_1}x \Big)\, dx + O(1/N) .\end{aligned}$$ For $N\to\infty$ we obtain: $$p_{\nu|Q}^{(2)} = e^{- \frac{(A_1(a_2+1)-A_2(a_1+1))^2}{4(a_2-a_1)(A_1-A_2)}} \qquad \mbox{if $A_1>A_2$}.$$ The case $A_1<A_2$ can be obtained from this one by the exchange $A_1\leftrightarrow A_2$, $a_1\leftrightarrow -a_2$. Therefore $$p_{\nu|Q}^{(2)} = e^{- \frac{(A_2(-a_1+1)-A_1(-a_2+1))^2}{4(a_2-a_1)(A_2-A_1)}} \qquad \mbox{if $A_2>A_1$}.$$ Considering in (\[p(a)\]) the limit $N\to\infty$ we see that $$\begin{aligned} \label{pQQQ} &\displaystyle p(a)|_{N\to\infty} = Q_1 + Q_2^+ + Q_2^-, \qquad Q_1 = \frac14 \int_{D_1} p_{\nu|Q}^{(1)}\, dA, \quad Q_2^\pm = \frac14 \int_{D_2^\pm} p_{\nu|Q}^{(2)}\, dA ,& \\ \nonumber & D_1 = \{ A \in {\mathbb{R}}_+^2 : |A_1-A_2| < A_1a_2-A_2a_1 \}, & \\ \nonumber & D_2^\pm = \{ A \in {\mathbb{R}}_+^2 : |A_1-A_2| \ge A_1a_2-A_2a_1, \; \pm(A_1-A_2) > 0 \}. &\end{aligned}$$ Change of the variables $$a_2A_1 - a_1A_2 = r, \quad - A_1 + A_2 = q$$ transforms the integrals as follows: $$\begin{aligned} \label{Q1} Q_1 &=& \frac14 \int_{|q|<r} \frac{1}{a_2-a_1} e^{- \frac{r}{a_2-a_1}}\, drdq = \frac12 (a_2-a_1), \\ \nonumber Q_2^+ &=& \frac14 \int_{D^+} \frac{1}{a_2-a_1} e^{\frac{r^2}{4(a_2-a_1)q}} \, drdq, \qquad D^+ = \{ -(1+a_2)q < r < -2q \}.\end{aligned}$$ The quantity $Q_2^-(a)$ is obtained from $Q_2^+(a)$ by the exchange $a_1\leftrightarrow -a_2$. It is convenient to compute $Q_2^+$ in the variables $u = -r^2/q$, $v = -r/q$. Direct computation gives: $$\label{Q2} Q_2^+ = \frac{4(a_2-a_1)}{3} \Big(\frac{1}{(1+a_2)^3} - \frac18\Big),\quad Q_2^- = \frac{4(a_2-a_1)}{3} \Big(\frac{1}{(1-a_1)^3} - \frac18\Big) .$$ Now equation (\[p\]) follows from (\[pQQQ\]), (\[Q1\]), and (\[Q2\]). Proof of Theorem \[theo:circ\] ============================== Let $\Omega$ be the configuration space of the circular random comb: $\Omega = [0,1]^N$. The teeth that support $J$ are determined by equation (\[ph=n\]). We consider large integer $L$, and define two random events $\nu$ and $Q_n$, where by definition - $\nu = n$ iff $J$ is supported by the teeth $n = (n_1,n_2,n_3)$, - $Q_n = K = (K_1,K_2,K_3)$ iff length of the tooth with number $n_i$ equals $$\xi_{n_i} \in \big(1 - (K_i-1) / (NL),1 - K_i / (NL)\big), \qquad i = 1,2,3.$$ For any $K\in \{1,\ldots,NL\}^3$ we have: ${\mathsf P}\{Q_n = K\} = (NL)^{-3}$. Therefore by the formula of total probability $${\mathsf P}\{\nu = n\} = \sum_K {\mathsf P}\{\nu = n | Q_n = K\}\, {\mathsf P}\{Q_n=K\} = \sum_K \frac{{\mathsf P}\{\nu = n | Q_n = K\}} {(NL)^3}.$$ Putting ${\mathbf{1}}= (1,1,1)^T\in{\mathbb{R}}^3$, in the limit $L\to\infty$ we obtain: $$\label{limit_o} {\mathsf P}\{\nu=n\} = \frac{1}{N^3} \int_0^N\!\!\!\int_0^N\!\!\!\int_0^N {\mathsf P}\{\nu=n | \xi_n = {\mathbf{1}}- A/N\} \, dA, \qquad \xi_n = (\xi_{n_1},\xi_{n_2},\xi_{n_3}) .$$ In the limit $N\to\infty$ we obtain densities of probability distributions $$\begin{aligned} \nonumber & p_{\mathcal{S}}: \widehat{\mathcal{S}}\to {\mathbb{R}}_+, \quad \tilde p_{\nu|Q} : {\mathcal{S}}\times [0,N]^3 \to{\mathbb{R}}_+, & \\ \label{limlim} &\displaystyle p_{\mathcal{S}}(\hat{\varphi}) = \lim_{N\to\infty} \Big(\frac{N}{2\pi}\Big)^3\,{\mathsf P}\{\nu=n\}, \quad \tilde p_{\nu|Q}({\varphi}|A) = \lim_{N\to\infty} {\mathsf P}\{\nu = n | \xi_n = 1-A/N\}, &\end{aligned}$$ where $\hat{\varphi}\in\widehat{\mathcal{S}}$ and ${\varphi}\in{\mathcal{S}}$. Equations (\[limit\_o\])–(\[limlim\]) imply $$\label{p(a)o} p_{\mathcal{S}}(\hat{\varphi}) = \frac1{8\pi^3} \int_{{\mathbb{R}}^3_+} \tilde p_{\nu|Q}({\varphi}|A)\, dA , \qquad {\mathbb{R}}_+^3 = \{A = (A_1,A_2,A_3)\in{\mathbb{R}}^3 : A_i > 0,\; i=1,2,3\} ,$$ where $\hat{\varphi}$ is the image of ${\varphi}$ under the natural map ${\mathcal{S}}\to\widehat{\mathcal{S}}$. Both densities $p_{\mathcal{S}}$ and $\tilde p_{\nu|Q}$ are invariant with respect to the action $R_\alpha$ of the group ${\mathbb{T}}$, see (\[action\]). Hence we obtain the densities $p_{\mathcal{T}},p_{\nu|Q}$ on $\widehat{\mathcal{T}}= \widehat{\mathcal{S}}/{\mathbb{T}}$ and ${\mathcal{T}}\times [0,N]^3$ respectively: ${\mathcal{T}}= {\mathcal{S}}/ {\mathbb{T}}$, $$p_{\mathcal{T}}(\hat{\vartheta}) = 2\pi p_{\mathcal{S}}(\hat{\varphi}), \quad p_{\nu|Q}({\vartheta}|A) = \tilde p_{\nu|Q}({\varphi}|A),$$ where measures on $\hat{\mathcal{T}}$ and ${\mathcal{T}}$ are determined by (\[mu\_T\]). Then (\[p(a)o\]) implies $$\label{p(thet)} p_{\mathcal{T}}(\hat{\vartheta}) = \frac1{4\pi^2} \int_{{\mathbb{R}}_+^3} p_{\nu|Q}({\vartheta}|A)\, dA.$$ Now we turn to computation of $p_{\nu|Q}$. The plane passing through $J = J(a({\varphi}),A)$ is determined by the equation $$\begin{aligned} \nonumber &\displaystyle z = 1 - \frac{\sigma_0}{N} - \frac{\sigma_x}{N}x - \frac{\sigma_y}{N}y , & \\ \label{sigma0} &\displaystyle \sigma_0 = \frac{1}{\Delta} \left|\begin{array}{ccc} \cos{\varphi}_1 & \sin{\varphi}_1 & A_1 \\ \cos{\varphi}_2 & \sin{\varphi}_2 & A_2 \\ \cos{\varphi}_3 & \sin{\varphi}_3 & A_3 \end{array}\right| = \frac{A_1\sin{\vartheta}_1 + A_2\sin{\vartheta}_2 + A_3\sin{\vartheta}_3} {\Delta} > 0, & \\ \label{s+s+s} &\displaystyle \Delta = \left|\begin{array}{ccc} \cos{\varphi}_1 & \sin{\varphi}_1 & 1 \\ \cos{\varphi}_2 & \sin{\varphi}_2 & 1 \\ \cos{\varphi}_3 & \sin{\varphi}_3 & 1 \end{array}\right| = \sin{\vartheta}_1 + \sin{\vartheta}_2 + \sin{\vartheta}_3, & \\ \label{sigmaxy} &\displaystyle \sigma_x = \frac{1}{\Delta} \left|\begin{array}{ccc} A_1 & \sin{\varphi}_1 & 1 \\ A_2 & \sin{\varphi}_2 & 1 \\ A_3 & \sin{\varphi}_3 & 1 \end{array}\right|, \quad \sigma_y = \frac{1}{\Delta} \left|\begin{array}{ccc} \cos{\varphi}_1 & A_1 & 1 \\ \cos{\varphi}_2 & A_2 & 1 \\ \cos{\varphi}_3 & A_3 & 1 \end{array}\right| .\end{aligned}$$ We consider two cases. \(1) The disk $J$ does not intersect the disk $I_+$, obtained as a shift of the disk $I$ by the vector $(0,0,1)$. This happens provided $\sigma_x\cos{\varphi}+ \sigma_y\sin{\varphi}+ \sigma_0 > 0$ for all real ${\varphi}$ i.e., $$\sigma_x^2 + \sigma_y^2 < \sigma_0^2.$$ \(2) $J\cap I_+ \ne \emptyset$. In this case $J$ is below $I_+$ over the domain $$D_\sigma = \{x^2 + y^2 \le 1, \; \sigma_x x + \sigma_y y + \sigma_0 \ge 0 \}.$$ In case (1) probability for the point $w_j = w(\alpha_j)$ to have a tooth (entirely) under $J$ is $$1 - \frac{\sigma_0}{N} - \frac{\sigma_x}{N} \cos\alpha_j - \frac{\sigma_y}{N} \sin\alpha_j \le 1.$$ Therefore probability for the whole comb to be under $J$ equals $$\begin{aligned} & p_A^{(1)} = \prod_{j\ne n_1,n_2,n_3} \Big( 1 - \frac{\sigma_0}{N} - \frac{\sigma_x}{N} \cos\alpha_j - \frac{\sigma_y}{N} \sin\alpha_j \Big) = e^{F_1}, \qquad \alpha_j = \frac{2\pi j}{N}, \quad j = 1,\ldots,N, & \\ & F_1 = \sum_{j\ne n_1,n_2,n_3} \log\Big( 1 - \frac{\sigma_0}{N} - \frac{\sigma_x}{N} \cos\alpha_j - \frac{\sigma_y}{N} \sin\alpha_j \Big) = - \sigma_0 + O(1/N). &\end{aligned}$$ For $N\to\infty$ we obtain: $$p_A^{(1)} = e^{-\sigma_0} .$$ In case (2) the tooth is under $J$ with probability $$\begin{aligned} &\displaystyle 1 - \frac{\sigma_0 + \sigma_x\cos\alpha_j + \sigma_y\sin\alpha_j}{N} \quad \mbox{if } \alpha_j \in B^+, \quad \mbox{and $1$ if } \alpha_j \in B^-, & \\[1mm] & B^\pm = \big\{\alpha\in{\mathbb{T}}: \pm (\sigma_x\cos\alpha + \sigma_y\sin\alpha + \sigma_0) \ge 0 \big\}. &\end{aligned}$$ Therefore probability for the whole comb to be under $J$ equals $$\begin{aligned} & p_A^{(2)} = \prod_{\alpha_j\in B^+,\, j\ne n_1,n_2,n_3} \Big(1 - \frac{\sigma_0 + \sigma_x\cos\alpha_j + \sigma_y\sin\alpha_j}{N} \Big) = e^{F_2}, \qquad \alpha_j = \frac{2\pi j}{N}, \quad j = 1,\ldots,N, & \\ & F_2 = \sum_{\alpha_j\in B^+,\, j\ne n_1,n_2,n_3} \log \Big(1 - \frac{\sigma_0 + \sigma_x\cos\alpha_j + \sigma_y\sin\alpha_j}{N} \Big) = \sigma_0{\mathcal{A}}+ O(1/N), & \\ &\displaystyle {\mathcal{A}}= \frac{1}{2\pi\sigma_0} \int_{B^+} (\sigma_x\cos{\varphi}+ \sigma_y\sin{\varphi}+ \sigma_0)\, d{\varphi}. &\end{aligned}$$ \[prop:calA\] ${\mathcal{A}}= \frac1\pi ({\varphi}_\sigma - \tan{\varphi}_\sigma)$, where ${\varphi}_\sigma = \arccos\big(-\sigma_0 / \sqrt{\sigma_x^2 + \sigma_y^2}\big)$. In the limit $N\to\infty$ we obtain the probability $$p_A^{(2)} = e^{-\sigma_0{\mathcal{A}}} .$$ By (\[p(thet)\]) we have the equation $$\begin{aligned} \label{ppp} &\displaystyle p = p_1 + p_2, \qquad p_1 = \frac1{4\pi^2}\int_{B_1} p_A^{(1)}\, dA, \quad p_2 = \frac1{4\pi^2}\int_{B_2} p_A^{(2)}\, dA, & \\ \nonumber & B_1 = \big\{ A\in{\mathbb{R}}^3_+ : \sigma_x^2 + \sigma_y^2 < \sigma_0^2 \big\},\quad B_2 = \big\{ A\in{\mathbb{R}}^3_+ : \sigma_x^2 + \sigma_y^2 \ge \sigma_0^2 \big\} . &\end{aligned}$$ Computation of the integrals $p_1,p_2$ requires some preliminary work. First, we introduce new coordinates $$\tau_i = (1 - \cos{\vartheta}_i) A_i / \Delta, \qquad i=1,2,3$$ and put $${\mathbf{1}}= (1,1,1)^T \in{\mathbb{R}}^3, \quad c_i = \cot \frac{{\vartheta}_i}{2}, \qquad i=1,2,3 .$$ \[prop:iden\] For any ${\vartheta}\in{\mathcal{T}}$ $$\label{iden} c_1c_2 + c_2c_3 + c_3c_1 = 1, \quad \frac{\sum\sin{\vartheta}_i}{\prod\sin({\vartheta}_i/2)} = 4, \quad \frac{1}{\prod\sin({\vartheta}_i/2)} = \langle{\mathbf{1}},c\rangle - c_1c_2c_3.$$ Combining (\[sigma0\])–(\[sigmaxy\]) and (\[iden\]), we have: $$\begin{aligned} \label{Delta} \Delta &=& \frac{4}{\langle c,{\mathbf{1}}\rangle - c_1c_2c_3} , \quad \sigma_0 \;=\; \langle c,\tau\rangle, \\ \label{J} \sigma_x^2 + \sigma_y^2 - \sigma_0^2 &=& -\langle\tau, J\tau\rangle, \qquad {\mathbf{J}}= \left(\begin{array}{rrr} -1 & 1 & 1 \\ 1 & -1 & 1 \\ 1 & 1 & -1 \end{array}\right) = {\mathbf{1}}\otimes{\mathbf{1}}- 2 .\end{aligned}$$ Integrals (\[ppp\]) in the new coordinates take the form $$\begin{aligned} \nonumber &\displaystyle\!\!\!\!\!\! p_i = \frac{\Delta^3\hat p_i} {32\pi^2\prod\sin^2\frac{{\vartheta}_i}2} = \frac{2\hat p_i} {\pi^2(\langle c,{\mathbf{1}}\rangle - c_1c_2c_3)} = \frac{2}{\pi^2} \sin\frac{{\vartheta}_1}{2} \sin\frac{{\vartheta}_2}{2} \sin\frac{{\vartheta}_3}{2}, & \\ \label{hatp_i} &\displaystyle\!\!\!\!\!\! \hat p_1 = \int_{C_1} e^{-\langle c,\tau\rangle} \, d\tau, \quad \hat p_2 = \int_{C_2} e^{-\langle c,\tau\rangle\,{\mathcal{A}}} \, d\tau, &\\[1mm] \label{C2} & C_1 = \{ \tau\in{\mathbb{R}}^3 : \langle c,\tau \rangle > 0, \; \langle\tau,{\mathbf{J}}\tau\rangle > 0 \} , \quad C_2 = \{ \tau\in{\mathbb{R}}_+^3 : \langle\tau,{\mathbf{J}}\tau\rangle < 0 \} . &\end{aligned}$$ Convenient variables -------------------- To compute integrals (\[hatp\_i\]), it is convenient to introduce new variables. We put $$w = 1 - \frac{\langle \tau,{\mathbf{J}}\tau\rangle} {\langle c,\tau\rangle^2}, \qquad \lambda = \frac{|c\times{\mathbf{1}}|}{\sqrt 2} \equiv \sqrt{\langle c,{\mathbf{1}}\rangle^2 - 3}.$$ Equation (\[J\]) implies $$\label{sigsig} (\sigma_x^2 + \sigma_y^2) / \sigma^2 = w^2 .$$ In the domain $C_2$ (see (\[C2\])) we have: $w > 1$. The identity $$\lambda^2 w = \Big(\frac{\langle {\mathbf{1}},\tau\rangle} {\langle c,\tau\rangle} - \langle c,{\mathbf{1}}\rangle \Big)^2 + \frac{\langle c\times{\mathbf{1}},\tau\rangle^2} {\langle c,\tau\rangle^2}$$ suggests the following change of variables: $(\tau_1,\tau_2,\tau_3)\mapsto (u,w,\psi)$, $$\langle c,\tau\rangle = u, \quad \frac{1}{\lambda}\, \Big( \frac{\langle {\mathbf{1}},\tau\rangle} {\langle c,\tau\rangle} - \langle c,{\mathbf{1}}\rangle \Big) = \sqrt{w}\cos\psi , \quad \frac{\langle c\times{\mathbf{1}},\tau\rangle} {\lambda \langle c,\tau\rangle} = \sqrt{w}\sin\psi.$$ The Jacobian $\det\frac{\partial(u,w,\psi)}{\partial(\tau_1,\tau_2,\tau_3)}$ equals the product $$\det\frac{\partial(u,w,\psi)}{\partial(u,v_1,v_2)} \det\frac{\partial(u,v_1,v_2)}{\partial(\tau_1,\tau_2,\tau_3)}, \qquad v_1 = \sqrt{w}\cos\psi, \quad v_2 = \sqrt{w}\sin\psi.$$ These determinants equal 2 and $2u^{-2}$ respectively. Therefore $$\det\frac{\partial(u,w,\psi)}{\partial(\tau_1,\tau_2,\tau_3)} = \frac{4}{u^2}.$$ Assuming $i$ to be an element of the cyclic group ${\mathbb{Z}}_3$, we put $$a_i = \frac{c_i\langle c,{\mathbf{1}}\rangle - c^2} {(c_{i-1}+c_{i+1})\lambda}, \quad b_i = \frac{c_{i-1} - c_{i+1}}{(c_{i-1}+c_{i+1})\lambda}.$$ Direct computations show that $$a_i^2 + b_i^2 = 1, \quad a_{i+1} b_{i-1} - a_{i-1} b_{i+1} = \frac{2c_i}{c_i^2+1} = \sin{\vartheta}_i, \quad b_{i+1} b_{i-1} + a_{i+1} a_{i-1} = \frac{c_i^2-1}{c_i^2+1} = \cos{\vartheta}_i.$$ Therefore for some $\psi_1,\psi_2,\psi_3\in{\mathbb{T}}$ $$\label{ab} a_i = \sin\psi_i, \quad b_i = \cos\psi_i, \qquad \psi_{i+1} - \psi_{i-1} = {\vartheta}_i.$$ The integrals $\hat p_1$ and $\hat p_2$ --------------------------------------- By using the variables $(u,v,\psi)$ in (\[hatp\_i\]), we obtain: $$\label{intduint} \hat p_1 = \int_0^\infty du \int_{G_1} \frac{u^2}{4} e^{-u}\, dwd\psi, \quad \hat p_2 = \int_0^\infty du \int_{G_2} \frac{u^2}{4} e^{-u{\mathcal{A}}}\, dwd\psi,$$ where the domains $G_1,G_2$ are as follows: $$G_1 = \big\{ (w,\psi) : 0 < w < 1 \big\} , \quad G_2 = \Big\{ (w,\psi) : w > 1, \; \frac{\tau_i}{\langle c,\tau\rangle} > 0, \; i = 1,2,3 \Big\} .$$ \[prop:p1p2\] $$\label{p1p2} \hat p_1 = \pi , \quad \hat p_2 = \pi^3 \big( f({\vartheta}_1) + f({\vartheta}_2) + f({\vartheta}_3) \big) .$$ [*Proof of Proposition \[prop:p1p2\]*]{}. The first equation (\[p1p2\]) is obvious. To prove the second one, we note that $$\begin{aligned} \label{hatp2} &\displaystyle \hat p_2 = \frac12 \int_{w>1} dw \int_{\psi\in G(w)} {\mathcal{A}}^{-3} \, d\psi , &\\ \label{G} & G(w) = \{\psi\in{\mathbb{T}}: \sin(\psi+\psi_i) < 1/\sqrt{w}, \quad i\in{\mathbb{Z}}_3\}. &\end{aligned}$$ Indeed, by Proposition \[prop:calA\] and equation (\[sigsig\]) we have: ${\mathcal{A}}= {\mathcal{A}}(w)$. Therefore we can perform integration in (\[intduint\]) in the variable $u$ which implies (\[hatp2\]). To check that the domain $G$ is determined by (\[G\]), we define $$\begin{aligned} \nu &=& \langle {\mathbf{1}},\tau\rangle = u(\lambda\sqrt{w} \cos\psi + \langle c,{\mathbf{1}}\rangle), \\ \beta &=& \langle c\times{\mathbf{1}},\tau\rangle = u\lambda \sqrt{w} \sin\psi.\end{aligned}$$ Then $$\begin{aligned} \left( \begin{array}{c} u\\ \nu\\ \beta \end{array}\right) &=& \left( \begin{array}{ccc} c_1 & c_2 & c_3 \\ 1 & 1 & 1 \\ c_2-c_3 & c_3-c_1 & c_1 - c_2 \end{array}\right) \left( \begin{array}{c} \tau_1\\ \tau_2\\ \tau_3 \end{array}\right) , \\ \left( \begin{array}{c} \tau_1\\ \tau_2\\ \tau_3 \end{array}\right) &=& \frac{1}{\lambda^2} \left( \begin{array}{ccc} 3c_1 - \langle c,{\mathbf{1}}\rangle & c^2 - c_1\langle c,{\mathbf{1}}\rangle & c_2 - c_3 \\ 3c_2 - \langle c,{\mathbf{1}}\rangle & c^2 - c_2\langle c,{\mathbf{1}}\rangle & c_3 - c_1 \\ 3c_3 - \langle c,{\mathbf{1}}\rangle & c^2 - c_3\langle c,{\mathbf{1}}\rangle & c_1 - c_2 \end{array}\right) \left( \begin{array}{c} u\\ \nu\\ \beta \end{array}\right)\end{aligned}$$ Hence the inequalities $\tau_i>0$ take the form $$3c_i - \langle c,{\mathbf{1}}\rangle + (c^2 - c_i\langle c,{\mathbf{1}}\rangle) (\lambda\sqrt{w}\cos\psi + \langle c,{\mathbf{1}}\rangle) + (c_{i+1} - c_{i-1}) \lambda\sqrt{w} \sin\psi > 0.$$ After simple transformations we get: $a_i\cos\psi + b_i\sin\psi < 1/\sqrt{w}$ which implies (\[G\]). Equations (\[hatp2\])–(\[G\]) imply $$\hat p_2 = \frac12 \int_{w > 1} \frac{|G(w)|}{{\mathcal{A}}^3(w)}\, dw ,$$ where $|G(w)|$ is the measure of the set $G(w)$. The set ${\mathbb{T}}\setminus\{\pi/2-\psi_1,\pi/2-\psi_2,\pi/2-\psi_3\}$ has 3 connected components: $U_1,U_2$, and $U_3$, where the interval $U_i$ has endpoints $\pi/2 - \psi_{i-1}$ and $\pi/2 - \psi_{i+1}$. Hence $$\begin{aligned} & G(w) = G_1(w) + G_2(w) + G_3(w), \qquad G_i(w) = G\cap U_i, & \\ & \hat p_2 = \hat p_2^{(1)} + \hat p_2^{(2)} + \hat p_2^{(3)}, \qquad \hat p_2^{(i)} = \int_{w > 1} \frac{|G_i(w)|}{{\mathcal{A}}^3(w)} \, dw . &\end{aligned}$$ By using (\[ab\]) and (\[G\]), we get: $$|G_i(w)| = {\vartheta}_i - 2(\pi - {\varphi}_\sigma(w))$$ provided the right-hand side is non-negative. By using the change $w = 1 / \cos^2{\varphi}_\sigma$, ${\varphi}_\sigma\in (\pi - {\vartheta}_i/2,\pi)$ in the integral $$\hat p_2^{(i)} = \frac{\pi^3}{2} \int_{w>1} \frac{{\vartheta}_i - 2(\pi - {\varphi}_\sigma(w))} {({\varphi}_\sigma(w) - \tan{\varphi}_\sigma(w))^3} \, dw ,$$ we obtain the equation $$\hat p_2^{(i)} = \int_{\pi - {\vartheta}_i/2}^\pi \frac{\pi^3 ({\vartheta}_i - 2(\pi - {\varphi}_\sigma))\sin{\varphi}_\sigma} {(-{\varphi}_\sigma\cos{\varphi}_\sigma + \sin{\varphi}_\sigma)^3} \, d{\varphi}_\sigma = 2\pi^3 f({\vartheta}_i) .$$ Several proofs ============== [**Proof of Proposition \[prop:calA\]**]{}. We define $\sigma_*$ and ${\varphi}_*$ by the equations $$\sqrt{\sigma_x^2 + \sigma_y^2} = \sigma_*, \quad \cos{\varphi}_* = \sigma_x / \sigma_*, \quad \sin{\varphi}_* = \sigma_y / \sigma_*$$ Then ${\mathcal{A}}$ takes the form $$\begin{aligned} & {\mathcal{A}}= \frac{1}{2\pi} \int_{\hat B_+} \Big(1 + \frac{\sigma_*}{\sigma_0} \cos({\varphi}-{\varphi}_*) \Big)\,d{\varphi}, & \\ & \hat B_+ = \{{\varphi}\in{\mathbb{T}}: \sigma_0 + \sigma_*\cos({\varphi}-{\varphi}_*) \ge 0\} = \{{\varphi}\in{\mathbb{T}}: -{\varphi}_\sigma\le{\varphi}-{\varphi}_*\le{\varphi}_\sigma\}. &\end{aligned}$$ This implies the required assertion. [**Proof of Proposition \[prop:iden\]**]{}. The first identity (\[iden\]) follows from the equation $$c_3 = -\cot({\vartheta}_1/2 + {\vartheta}_2/2) = \frac{1 - c_1c_2}{c_1 + c_2}.$$ To prove the second one we note that $$\sin{\vartheta}_1 = -2\sin({\vartheta}_1/2)\cos({\vartheta}_2/2+{\vartheta}_3/2) = 2\prod\sin({\vartheta}_i/2) - 2\sin({\vartheta}_1/2)\cos({\vartheta}_2/2)\cos({\vartheta}_3/2).$$ Adding to this equation two analogous ones and dividing by $\prod\sin({\vartheta}_i/2)$, we get: $$\frac{\sum\sin{\vartheta}_i}{\prod\sin({\vartheta}_i/2)} = 6 - 2(c_2c_3 + c_3c_1 + c_1c_2) = 4.$$ Finally, adding up the equations $$\begin{aligned} \sin^2\frac{{\vartheta}_1}2 &=& \sin\frac{{\vartheta}_1}2 \sin\frac{{\vartheta}_2+{\vartheta}_3}2 \;=\; \sin\frac{{\vartheta}_1}2 \Big(\sin\frac{{\vartheta}_2}2 \cos\frac{{\vartheta}_3}2 + \cos\frac{{\vartheta}_2}2 \sin\frac{{\vartheta}_3}2 \Big), \\ \cos^2\frac{{\vartheta}_1}2 &=& -\cos\frac{{\vartheta}_1}2 \cos\frac{{\vartheta}_2+{\vartheta}_3}2 = \cos\frac{{\vartheta}_1}2 \Big(\sin\frac{{\vartheta}_2}2\sin\frac{{\vartheta}_3}2 - \cos\frac{{\vartheta}_2}2 \cos\frac{{\vartheta}_3}2 \Big).\end{aligned}$$ dividing by $\prod\sin({\vartheta}_i/2)$, we obtain the third identity (\[iden\]). Discussion ========== Our computation of probability distributions in Theorems \[theo:int\] and \[theo:circ\] are based on the assumptions that length of a tooth is uniformly distributed on $[0,1]$ and the teeth are situated on the base of the comb with a constant step. However we believe that the answers (i.e., formulas for densities of these distributions) are not sensitive to these details. For example, the answers should be the same if the teeth are randomly uniformly distributed on the base and/or lengths of the teeth are identical independently distributed random values with a continuous distribution density on $[0,b]$, $0<b<\infty$. It would be interesting to obtain a proof of this conjecture. We have already mentioned that it is interesting to consider analogous problems where base of a random comb is two-dimensional, for example, a disk. Also we would be happy to see dynamical applications of these problems. [20]{} T.Salnikova, D.Treschev, S.Galliamov. Motion of a free puck on a rough horizontal surface. Nonlinear Dynamics, 2012, V.8, N.1, P.83-101 (in Russian) Burlakov D., Seslavina A. Erdakova N., Ivanova T., Treschev D. Ivanov A.P. A dynamically consistent model of the contact stresses in the plane motion of a rigid body. J. Appl. Math. Mech. 2009 N.2, P. 134-144 Ishlinsky A.Yu., Sokolov B.N., Chernousko F.L. On motion of flat bodies with friction // Izvestiya of Russian Acad. Sci.: Rigid Body Mechanics. 1981. No. 4., P. 17-28. Kireenkov A.A. On the motion of a homogeneous rotating disk along a plane in the case of combined friction. J. Mechanics of Solids 2002, no.1, P. 47-53. Kireenkov A.A. A method for the calculation of the force and torque of friction in a combined model of dry friction for circular contact areas. J. Mechanics of Solids, 2003, no. 3 P. 39-43. Kireenkov A.A., Semendyaev S.V. and Filatov V.F. Experimental Study of Coupled Two-Dimensional Models of Sliding and Spinning Friction. J. Mechanics of Solids, 2010, no. 6, P. 921-930. Macmillan W.D., Dynamics of Rigid Bodies. McGraw-Hill, New York (1936). Contensou P. Couplage entre frottement de glissement et frottement de pivotement dans la teorie de la toupie. Kreiselsprobleme. Gyrodynamics. Symp. Celerina, 1962. Derlin etc.: Springer, 1963. Farkas Z., Bartels G., Unger T., Wolf D. Frictional coupling between sliding and spinning motion. Phys.Rev.Letters 2003, V.90, no. 24. 248302. Weidman P.D. Malhotra Ch.P. On the terminal motion of sliding spinning disks with uniform Coulomb friction // Physica D: Nonlinear Phenomena, 2007, vol. 233, pp. 1–13. 2007 Elsevier. Zhuravlev, V.F. On a model of dry friction in the problem of the rolling of rigid bodies. (Russian) Prikl. Mat. Mekh. 62 (1998), no. 5, 762–767; translation in J. Appl. Math. Mech. 62 (1998), no. 5, 705–710 (1999) 70F40 (70E18)

--- abstract: | The purpose of this paper is to survey some of the important results on Langlands program, global fields, $D$-shtukas and finite shtukas which have influenced the development of algebra and number theory. It is intended to be selective rather than exhaustive, as befits the occasion of the 80-th birthday of Yakovlev, 75-th birthday of Vostokov and 75-th birthday of Lurie.\ Under assumptions on ground fields results on Langlands program have been proved and discussed by Langlands, Jacquet, Shafarevich, Parshin, Drinfeld, Lafforgue and others. This communication is an introduction to the Langlands Program, global fields and to $D$-shtukas and finite shtukas (over algebraic curves) over function fields. At first recall that linear algebraic groups found important applications in the Langlands program. Namely, for a connected reductive group $ G $ over a global field $ K $, the Langlands correspondence relates automorphic forms on $ G $ and global Langlands parameters, i.e. conjugacy classes of homomorphisms from the Galois group $ {\mathcal Gal} ({\overline K} / K) $ to the dual Langlands group $ \hat G ({\overline {\mathbb Q}} _ p) $. In the case of fields of algebraic numbers, the application and development of elements of the Langlands program made it possible to strengthen the Wiles theorem on the Shimura-Taniyama-Weil hypothesis and to prove the Sato-Tate hypothesis. V. Drinfeld and L. Lafforgue have investigated the case of functional global fields of characteristic $ p> 0 $ ( V. Drinfeld for $ G = GL_2 $ and L. Lafforgue for $ G = GL_r, \; r $ is an arbitrary positive integer). They have proved in these cases the Langlands correspondence. Under the process of these investigations, V. Drinfeld introduced the concept of a $ F $-bundle, or shtuka, which was used by both authors in the proof for functional global fields of characteristic $ p> 0 $ of the studied cases of the existence of the Langlands correspondence. Along with the use of shtukas developed and used by V. Drinfeld and L. Lafforge, other constructions related to approaches to the Langlands program in the functional case were introduced. G. Anderson has introduced the concept of a $ t $-motive. U. Hartl, his colleagues and students have introduced and have explored the concepts of finite, local and global $ G $-shtukas. In this review article, we first present results on Langlands program and related representation over algebraic number fields. Then we briefly present approaches by U. Hartl, his colleagues and students to the study of $ D $ –shtukas and finite shtukas. These approaches and our discussion relate to the Langlands program as well as to the internal development of the theory of $ G $-shtukas. address: | Department of Electronics\ National Aviation University\ 1 Komarova Pr.\ Kiev\ 03680\ [Ukraine]{} author: - 'Nikolaj [Glazunov]{}' date: 'June 25, 2020' title: 'On Langlands program, global fields and shtukas' --- Introduction {#introduction .unnumbered} ============ This communication is an introduction to the Langlands Program and to ($D$-)shtukas and finite shtukas (over algebraic curves) over function fields. The Langlands correspondence over number fields in its full generality is facing with problems [@Langlands1979; @Lan1980; @JacLan1970; @Sha96; @Parshin; @Drinfeld1980; @Lafforgue2002]. So results from Galois theory, algebraic number theory and function fields can help understand it. Elements of algebraic number theory and field theory. ----------------------------------------------------- The questions what is a Galois group of a given algebraic closure of the number field or the local field, embedding problems of fields and extensions of class field theory belong to fundumental questions of Galois theory and class field theory. A.V. Yakovlev, S.V. Vostokov, B.B. Lur’e works spans many areas of Galois theory, fields theory and class field theory. The results obtained indicate that these questions connect with module theory, homological algebra and with other topics of algebra and number theory[@Sha; @Yak1967; @Yak; @Vos11; @Vos85; @Lur1964; @Lur1991]. The development and applications of these theories are discribed in papers by I.R. Shafarevich[@Sha96] and by F.N. Parshin [@Parshin] (and in references therein). For further details we refer the reader to papers themselves. By the lack of author‘s competence we discuss here very shortly only connection of local fields with formal modules. The Hensel-Shafarevich canonical basis in complete discrete valuation fields. ----------------------------------------------------------------------------- Vostokov has constructed a canonical Hensel-Shafarevich basis in ${\mathbb Z}_p-$module of principle units for complete discrete valuation field with an arbitrary residue field [@Vos11]. Vostokov and Klimovitski in paper [@VosKl] give construction of primary elements in formal modulei. Ikonnikova, Shaverdova [@IKSh] and Ikonnikova [@IK] use these results under construction, respectively, the Shafarevich basis in higher-dimensional local fields and under proving two theorems on the canonical basis in Lubin-Tate formal modules in the case of local field with perfect residue field and in the case of imperfect residue field. These canonical bases are obtained by applying a variant of the Artin-Hasse function. ${^L} G$ for reductive group $G$ -------------------------------- Here we follow to [@Langlands1979; @JacLan1970; @Arthur; @Borel; @Tate]. At first recall that linear algebraic groups found important applications in the Langlands program. Namely, for a connected reductive group $ G $ over a global field $ K $, the Langlands correspondence relates automorphic forms on $ G $ and global Langlands parameters, i.e. conjugacy classes of homomorphisms from the Galois group $ {\mathcal Gal} ({\overline K} / K) $ to the dual Langlands group $ \hat G ({\overline {\mathbb Q}} _ p) $. Let $\overline K$ be an algebraic closure of $ K$ and $ K_s$ be the separable closure of $ K $ in $\overline K$. Let $G$ be the connected reductive algebraic group over $\overline K$. The root datum of $G$ is a quadruple $(X^*(T), \Delta, X_*(T), \Delta^v)$ where $X^*$ is the lattice of characters of the maximal torus $T$, $ X_*$ is the dual lattice, given by the 1-parameter subgroups, $\Delta$ is the set of roots, $\Delta^v$ is the corresponding set of coroots. The dual Langlands group $ \hat G$ is a complex reductive group that has the dual root data: $(X_*(T), \Delta^v, X^*(T), \Delta )$. Here any maximal torus $ \hat T$ of $ \hat G$ is isomorphic to the complex dual torus $ X^*(T) \otimes {\mathbb C}^* = Hom(X_*(T),{\mathbb C}^*) $ of any maximal torus $T$ in $G$. Let $\Gamma_{\mathbb Q} = {\mathcal Gal} ({\overline Q} / Q) $. Given $G$, the Langlands ${L}$-group of $ G$ is defined as semidirect product $${^L} G = \hat G \rtimes \Gamma_{\mathbb Q}.$$ In the case of fields of algebraic numbers, the application and development of elements of the Langlands program made it possible to strengthen the Wiles theorem on the Shimura-Taniyama-Weil hypothesis and to prove the Sato-Tate hypothesis. Langlands reciprocity for $GL_n$ over non-archimedean local fields of characteristic zero is given by Harris-Taylor [@HarrisTaylor]. Langlands correspondence over functional global fields of characteristic $ p> 0 $ ---------------------------------------------------------------------------------- V. Drinfeld [@Drinfeld1980] and L. Lafforgue [@Lafforgue2002] have investigated the case of functional global fields of characteristic $ p> 0 $ ( V. Drinfeld for $ G = GL_2 $ and L. Lafforgue for $ G = GL_r, \; r $ is an arbitrary positive integer). They have proved in these cases the Langlands correspondence. In the process of these studies, V. Drinfeld introduced the concept of a $ F $ -bundle, or shtuka, which was used by both authors in the proof for functional global fields of characteristic $ p> 0 $ of the studied cases of the existence of the Langlands correspondence [@Drinfeld1987]. Along with the use of shtukas developed and used by V. Drinfeld and L. Lafforge, other constructions related to approaches to the Langlands program in the functional case were introduced. G. Anderson has introduced the concept of a $ t $-motive [@Anderson1986]. U. Hartl, his colleagues and postdoc students have introduced and have explored the concepts of finite, local and global $ G $ -shtukas[@Hartl2005; @HartlViehmann2011; @HartlRad2014; @Singh2012; @Rad2012; @Weiss2017]. In this review, we first present results on Langlands program and related representation over algebraic number fields. Then we briefly present approaches by U. Hartl, his colleagues and students to the study of $ G $ –shtukas. These approaches and our discussion relate to the Langlands program as well as to the internal development of the theory of $ G $-shtukas. Some results on commutative formal groups and commutative formal schemes can be found in [@Glazunov2015A; @Glazunov2015B; @Glazunov2015C] and in references therein. The content of the paper is as follows:\ Introduction.\ 1. Some results of the implementation of the Langlands program for fields of algebraic numbers and their localizations.\ 2. Elliptic modules and Drinfeld shtukas.\ 3. Finite $ G $ -shtukas.\ Some results on Langlands program over algebraic number fields and their localizations ====================================================================================== Langlands conjectured that some symmetric power $ L$-functions extend to an entire function and coincide with certain automorphic $L$-functions. Abelian extensions of number fields ------------------------------------- In the case of algebraic number fields Langlands conjecture (Langlands correspondence) is the global class field theory:\ Representations of the abelian Galois group $Gal(K^{ab}/K)$ = characters of the Galois group $Gal(K^{ab}/K) $ correspond to automorphic forms on $GL_1 $ that are characters of the class group of ideles. Galois group $ Gal(K^{ab}/K)$ is the profinite completion of the group ${\mathbb A}^* (K)/K^* $ where $ {\mathbb A}(K) $ denotes the adele ring of $ K$.\ If $K$ is the local field, then Galois group $ Gal(K^{ab}/K)$ is canonically isomorphic to the profinite completion of $K^*$. $l$ - adic representations and Tate modules -------------------------------------------- Let $K$ be a field and $\overline K $ its separate closure, $ E_n=\{P \in E(\overline K )| nP=0\}$ the group of points of elliptic curve $E(\overline K)$ order dividing n. When $char K $ does not divide $n$ then $E_n$ is a free ${\mathbb Z}/n{\mathbb Z}$ -module of rank $2$. Let $l$ be prime, $l \ne char K$. The projective limit $T_l (E)$ of the projective system of modules $E_{l^m}$ is free ${\mathbb Z}_l$-adic Tate module of rank $2$. Let $V_l (E)=T_l (E) \otimes_{{\mathbb Z}_l} {\mathbb Q}_l$. Galois group $Gal({\overline K}/K)$ acts on all $E_{l^m}$ , so there is the natural continuous representation ($l$-adic representation) $$\rho_{E,l} : Gal({\overline K}/K) \to Aut \; T_l (E) \subseteq Aut \; V_l (E).$$ $V_l (E)$ is the first homology group that is dual to the first cohomology group of $ l$-adic cohomology of elliptic curve $E$ and Frobenius $F$ acts on the homology and dually on cohomology. The characteristic polynomial $ P(T)$ of the Frobenius not depends on the prime number $ l$. Zeta functions and parabolic forms ---------------------------------- Let (in P. Deligne notations) $X$ be a scheme of finite type over ${\mathbb Z}$, $|X|$ the set of its closed points, and for each $x \in |X|$ let $N(x)$ be the number of points of the residue field $k(x)$ of $X$ at $x$. The Hasse-Weil zeta-function of $X$ is, by definition $$\zeta_X (s)= \prod_{ x \in|X|}(1 - N(x)^{-s} )^{-1} .$$ In the case when $X$ is defined over finite field ${\mathbb F}_q$, put $ q_x= N(x)$, $deg(x)=[k(x):{\mathbb F}_q ]$, so $q_x=q^{deg(x)}$ . Put $t=q^{-s}$. Then $$Z(X ,t) = \prod_{ x \in|X|}(1 - t^{deg(x)} )^{-1} .$$ The Hasse-Weil zeta function of $ E$ over ${\mathbb Q}$ (an extension of numerators of $\zeta_E (s)$ by points of bad reduction of $E$) is defined over all primes $p$: $$L(E({\mathbb Q}),s) = \prod_p (1 - a_pp^{-s} + \epsilon(p)p^{1 - 2s} )^{-1} ,$$ here $\epsilon(p) = 1$ if $E$ has good reduction at $ p$, and $\epsilon(p) = 0$ otherwise. Put $T = p^{-s}$. For points of good reduction we have $$P(T) = 1 - a_pT + pT^2 = (1- \alpha T)(1 - \beta T)$$. For symmetric power $L$-functions (functions $ L(s;E; Sym^n), \; n > 0;$ see below) we have to put $$P_p(T) = \prod_{i=0}^n (1- \alpha^i \beta^{n - i}T)$$ For $GL_2 ({\mathbb R})$, let $C$ be its center, $O(2)$ the orthogonal group. Upper half complex plane has the representation: ${\mathbb H}^2=GL_2 ({\mathbb R}) / O(2) C$. So it is the homogeneous space of the group $GL_2 ({\mathbb R})$. A cusp (parabolic) form of weight $k \ge 1$ and level $N \ge1$ is a holomorphic function $f$ on the upper half complex plane ${\mathbb H}^2$ such that\ a) For all matrices $$g = \left( \begin{array}{cc} a& b \\ c& d \end{array} \right), a,b,c,d \in {\mathbb Z}, a \equiv1(N),d\equiv1(N),c\equiv0(N)$$ and for all $z \in {\mathbb H}^2$ we have $$f(gz)=f((az+b)/(cz+d))=(cz+d)^k f(z)$$ (automorphic condition).\ b) $$|f(z)|^2 (Im z)^k$$ is bounded on ${\mathbb H}^2$ .\ Mellin transform $ L(f,s)$ of the parabolic form $ f$ coincides with Artin $ L$-series of the representation $\rho_f$. The space ${\mathcal M}_n(N)$ of cusp forms of weight $ k$ and level $N$ is a finite dimensional complex vector space. If $f \in {\mathcal M}_n (N)$, then it has expansion $$f(z)=\sum_{n=1}^{\infty} c_n (f)\exp(2\pi inz)$$ and $L$-function is defined by $$L(f,s)=\sum_{n=1}^{\infty} c_n (f) / n^s .$$ Modularity results ------------------ The compact Riemann surface $\Gamma\backslash{\mathbb H}^2$ is called the modular curve associated to the subgroup of finite index $\Gamma$ of $GL_2 ({\mathbb Z})$ and is denoted by $X(\Gamma)$. If the modular curve is elliptic it is called the elliptic modular curve.\ The modularity theorem states that any elliptic curve over ${\mathbb Q}$ can be obtained via a rational map with integer coefficients from the elliptic modular curve.\ By the Hasse-Weil conjecture (a cusp form of weight two and level $ N$ is an eigenform (an eigenfunction of all Hecke operators)). The conjecture follows from the modularity theorem.\ Recall the main (and more stronger than in Wiles [@Wiles] and in Wiles-Taylor [@TaylorWiles] papers) result by C. Breuil, B. Conrad, F. Diamond, R. Taylor [@BreuilConradDiamondTaylor]. ([**Taniyama-Shimura-Weil conjecture - Wiles Theorem.**]{}) For every elliptic curve $E$ over ${\mathbb Q}$ there exists $f$, a cusp form of weight 2 for a subgroup $\Gamma_0 (N)$, such that $ L(f,s)=L(E({\mathbb Q}),s)$. Here ${\Gamma_0(N)}$ is the modular group $${\Gamma_0(N)} = \left\{ \left( \begin{array}{cc} a& b\\ c& d \\ \end{array} \right), a,b,c,d \in {\mathbb Z}, c \equiv 0\pmod{N}, \det \left( \begin{array}{cc} a& b\\ c& d \\ \end{array} \right)= 1 \right \}.$$ Recall that for projective closure $\overline E$ of the elliptic curve $E$ we have $$\overline E({\mathbb F}_p) = 1 - a_p +p.$$ By H. Hasse $$a_p = 2\sqrt p \cos \varphi_p.$$ ([**Sato-Tate conjecture**]{}) Let $E$ be an elliptic curve without complex multiplication. Sato have computed and Tate gave theoretical evidence that angles $\varphi_p$ in the case are equidistributed in $[0,\pi]$ with the Sato-Tate density measure $\frac{2}{\pi} \sin^2 \varphi.$ We have two theorems from Serre [@Serre] which give the theoretical explanation in terms of Galois representations. Here we recall the corollery of the theorems. (Serre [@Serre]) The elements are equidistributed for the $v$ normalized Haar measure of $G$ if and only if $c = 0$ for every $X$ irreducible character of $G$, i. e. , if and only if the $L$-functions relative to the non trivial irreducible characters of $G$ are holomorphic and non zero at $s = 1$. The current state of Sato-Tate conjecture is now Clozel–Harris–Shepherd-Barron–Taylor Theorem [@Clozel; @HarrisTaylor; @HarrisShepherd-BarronTaylor]. (Clozel, Harris, Shepherd-Barron, Taylor). Suppose $E$ is an elliptic curve over ${\mathbb Q}$ with non-integral $j$ invariant. Then for all $n > 0; \; L(s;E; Sym^n)$ extends to a meromorphic function which is holomorphic and non-vanishing for $Re(s) \ge 1 + n/2$. These conditions and statements are sufficient to prove the Sato-Tate conjecture. Under the prove of the Sato-Tate conjecture the Taniyama-Shimura-Weil conjecture oriented methods of A. Wiles and R. Taylor are used. Recall also that the proof of Langlands reciprocity for $GL_n$ over non-archimedean local fields of characteristic zero is given by Harris-Taylor [@HarrisTaylor]. Elliptic modules and Drinfeld shtukas. ====================================== Let\ ${\overline{\mathbb F}}_q$ be the algebraic closure of ${\mathbb F}_q$,\ $\mathcal C$ be a smooth projective geometrically irreducible curve over ${\mathbb F}_q$,\ $K$ be the function field ${\mathbb F}_q (\mathcal C)$ of $\mathcal C$,\ $\nu$ be a close point of $\mathcal C$,\ $A$ be the ring of functions regular on $\mathcal C - \nu$,\ $K_{\nu}$ be the complation of $K$ at $\nu$ with valuation ring ${\mathcal O}_{\nu}$,\ ${\mathbb C}_{\nu}$ be the complation of the algebraic closure of $K_{\nu}$.\ At first recall some known facts about algebraic curves over finite fields. We will identify the set $|\mathcal C|$ of closed points of $\mathcal C$ with $\mathcal C({\overline{\mathbb F}}_q) = Hom_{{\mathbb F}_q} (Spec \; {\overline{\mathbb F}}_q, \mathcal C)$. Let $k(\nu)$ be the residue field of $\nu$. Then the degree of $\nu$ is equal of the number of elements $[k(\nu):{\mathbb F}_q]$. Below in this section we follow to [@Drinfeld1974; @Drinfeld1987; @DeHu]. Elliptic modules ---------------- Let $k$ be a field of characteristic $p > 0$ and let $R$ be a $k$-commutative ring with unit (there exists a morphism $k \to R$). The additive scheme ${\mathbb G}_a$ over $R$ is represented by the polynomial ring $R[X] $ with structural morphism $\alpha:R[X] \to R[X]\otimes_R R[X] $, given by $ \alpha(X) = X \otimes 1 + 1 \otimes X$. A morphism $\varphi: {\mathbb G}_a \to {\mathbb G}_a$ of additive schemes over $R$ is defined by an additive polynomial. If $\psi$ is another such morphism, then $\varphi\circ\psi = \varphi(\psi(T))$. So the set of (endo)morphisms of additive scheme has the structure of a ring. Let $a \in R[X], \; pa = 0$. Then the morphism $\varphi(T) = a T^{p^n}, \; (n \ge 0)$ is additive. Any additive morphism $\varphi(T)$ in characteristic $p$ has the form $\varphi(T) = a_0T + a_1T^p + \cdots + a_n T^{p^n}.$ Let $k$ be a field of characteristic $p > 0$. Put $\tau a = a^p\tau$. There is an isomorphism between $End_k({\mathbb G}_a)$ and the ring of noncommutative polynomials $k\{\tau\}$. For any $\varphi(T) = a_0T + a_1T^p + \cdots + a_n T^{p^n} \in End_k({\mathbb G}_a)$ and any $ \varphi(\tau) = a_0 + a_1\tau + \cdots + a_n\tau^n \in k\{\tau\}$ Lubin morphisms [@Lub] $c_0$ and $c$ are defined: $$c_0(\varphi(T)) = a_0, c(\varphi(\tau)) = a_0.$$ Respectively we define $$deg(\varphi(T)) = p^n, d(\varphi(\tau)) = n.$$ Any ring morphism $A \to End_k({\mathbb G}_a)$ is either injective or has image contained in the constants $k \subset k\{\tau\}$. Sketch of the proof. $k\{\tau\}$ is a domain. $End_k({\mathbb G}_a)$ is isomorphic to $k\{\tau\}$. $A$ is a ring with divisor theory ${\mathfrak D}$ and for any prime divisor ${\mathfrak p} \in {\mathfrak D}$ the residue ring $A/{\mathfrak p}$ is a field. From these statements the proposition follows.\ Assume now that $k$ is an $A$-algebra, i.e. there is a morphism $i: A \to k$ . An elliptic module over $k$ (of rank $r =2$) is an injective ring homomorphism $$\varphi: A \to End_k({\mathbb G}_a)$$ $$a \mapsto \varphi_a ,$$ such that for all $a \in A$ we have $$d(\varphi(\tau)) = 2\cdot deg(a),$$ $$c(\varphi(\tau)) = i(a).$$ Let $k = {\mathbb F}_q(T)$, $A = {\mathbb F}_q[{\mathbb P}^1 - \nu] = {\mathbb F}_q[T]$. Let $i(T) = T^2 +1$. In this case an elliptic module $\varphi$ is given by $$\varphi = T^2 +1 + c_1\cdot\tau + c_2\cdot\tau^2, c_1, c_2 \in k, \; c_2 \ne 0.$$ By the same way it is possible to define a Drinfeld module (over a field) for any natural $r$. Now consider the case of Drinfeld modules over a base scheme. Let $S$ be an $A$-scheme, $\mathcal L$ a line bundle over $S$, $i^*: S \to Spec \; A$ be an $A$ scheme morphism dual to the ring homomorphism $i:A \to {\mathcal O}_S$ (Drinfeld module over a base scheme) A Drinfeld module over $k$ of rank $r$ is an ring homomorphism $$\varphi: A \to End_S(\mathcal L)$$ $$a \mapsto \varphi_a ,$$ such that for all $a \in A$ we have\ 1) locally, as a polynomial in $\tau$, $\varphi_a$ has the degree $$d(\varphi(\tau)) = r\cdot deg(a),$$ 2) a unit as its leading coefficient $a_n$ and $$c(\varphi(\tau)) = i(a).$$ Drinfeld shtukas. ----------------- In notations of previous subsection let $x \in k$, $a \in A$, $\varphi_a(\tau)$ be a Drinfeld module of rank $r$. Put $L = k\{\tau\}$, $f(\tau) \in L$, $k[A] = k\otimes_{{\mathbb F}_q}A$, $deg_{\tau}f(\tau)$ the degree in $\tau$ of $f(\tau)$. Define the action of $k[A]$ on $L$ by the formula: $$x\otimes a\cdot f(\tau) = x\cdot f(\varphi_a(\tau) .$$ Then $L$ is a free $k[A]$-module of rank $r$. Let $E_s = \{f(\tau) \in L | deg_{\tau}f(\tau) \le s\}, \; E = \oplus_{s=0}^\infty E_s$, $ E[1] = \oplus_{s=0}^\infty E_{s+1}.$ $ E, E[1]$ are graded modules over the graded ring and give rise to locally free sheaves ${\mathcal F}$, ${\mathcal E}$ of rank $r$ over ${\mathcal C}$. Put ${\mathcal C}_S = C \times_{{\mathbb F}_q}S$, $\sigma_q = id_C \otimes Frob_{q,S}:{\mathcal C}_S \to {\mathcal C}_S $ A (right) $\mathcal D$-shtuka ($F$-sheaf [@Drinfeld1987]) of rank $r$ over an ${{\mathbb F}_q}$-scheme $S$ is a diagram $({\mathcal F} \stackrel{c_1}\to {\mathcal E} \stackrel{c_2}\gets ( id_C \otimes Frob_{q,S})^*{\mathcal F} )$, such that coker $c_1$ is supported on the graph ${\Gamma}_{\alpha}$ of a morphism ${\alpha}: S \to {\mathcal C}$ and it is a line bundle on support, coker $c_2$ is supported on the graph ${\Gamma}_{\beta}$ of a morphism ${\beta}: S \to {\mathcal C}$ and it is a line bundle on support. If ${\Gamma}_{\alpha} \cap {\Gamma}_{\beta} = \emptyset$ it is possible to give the next definition of $\mathcal D$-shtuka [@HartlRad2014; @HartlSingh]. A global shtuka of rank $r$ with two legs over an ${{\mathbb F}_q}$-scheme $S$ is a tuple $\underline{\mathcal N} = ({\mathcal N}, (c_1, c_2), {\tau}_{\mathcal N})$ consisting of 1) a locally free sheaf ${\mathcal N}$ of rank $r$ on ${\mathcal C}_S$; 2) ${{\mathbb F}_q}$-morphisms $c_i: S \to {\mathcal C} \; ( i=1,2)$, called the legs of $\underline{\mathcal N}$; 3) an isomorphism $\tau_N: \sigma_q^*{\mathcal N}|_{{\mathcal C}_S - ({\Gamma}_{c_1} \cup {\Gamma}_{c_2})}\simeq {\mathcal N}|_{{\mathcal C}_S - ({\Gamma}_{c_1} \cup {\Gamma}_{c_2})}$ outside the graphs ${\Gamma}_{c_i}$ of $c_i$, ${\Gamma}_{c_1} \cap {\Gamma}_{c_2} = \emptyset$. A global shtuka over $S$ is a $\mathcal D$-shtuka if $\tau_N$ satisfies $ \tau_N(\sigma_q^*{\mathcal N}) \subset {\mathcal N} $ on ${\mathcal C}_S - {\Gamma}_{c_2}$ with cokernel locally free of rank 1 as $\mathcal O_S$-module, and $\tau_{\mathcal N}^{-1}({\mathcal N}) \subset \sigma_q^*{\mathcal N}$ on ${\mathcal C}_S - {\Gamma}_{c_1}$ with cokernel locally free of rank 1 as $\mathcal O_S$-module. Finite $ G $-shtukas. ===================== We follow to [@Drinfeld1987; @Hartl2005; @HartlViehmann2011; @Singh2012; @HartlSingh]. We start with very short indication on the general framework of the section. In connection with Drinfeld‘s constructions of elliptic modules Anderson [@Anderson1986] has introduced abelian t-modules and the dual notion of t-motives. Beside with mentioned papers these are the descent theory by A. Grothendieck [@Grothendieck], cotangent complexes by Illusie [@Illusie], by S. Lichtenbaum and M. Schlessinger [@LichtenbaumSchlessinger], by Messing [@Messing] and by Abrashkin [@Abrashkin]. In this framework to any morphism $f: A \to B$ of commutative ring objects in a topos is associated a cotangent complex $ L_{(B/A)}$ and to any morphism of commutative ring objects in a topos of finite and locally free $Spec (A)$-group schemes $G$ is associated a cotangent complex $L_{(G/Spec (A))}$ as has presented in books by Illusie [@Illusie]. Finite shtukas and formal groups -------------------------------- Let $S$ be a scheme over $Spec \; {\mathbb F}_q$. A finite ${\mathbb F}_{q}$-shtuka over $S$ is a pair ${\underline M} = (M, F_M )$ consisting of a locally free ${\mathcal O}_S$-module $M$ on $S$ of finite rank and an ${\mathcal O}_S$-module homomorphism $F_M : \sigma_q^*M \to M$. Author[@Singh2012] investigates relation between finite shtukas and strict finite flat commutative group schemes and relation between divisible local Anderson modules and formal Lie groups. The cotangent complexes as in papers by S. Lichtenbaum and M. Schlessinger [@LichtenbaumSchlessinger], by W. Messing [@Messing], by V. Abrashkin [@Abrashkin] are defined and are proved that they are homotopically equivalent.\ Then the deformations of affine group schemes follow to the mentioned paper of Abrashkin are investigated and strict finite $\mathcal O$-module schemes are defined.\ Next step of the research is devoted to relation between finite shtukas by V. Drinfeld [@Drinfeld1987] and strict finite flat commutative group schemes.\ The comparison between cotangent complex and Frobenius map of finite ${\mathbb F}_p$-shtukas is given.\ Local shtukas and local Anderson modules ---------------------------------------- Recall some notions and notations. An ideal $I$ in a commutative ring $A$ is locally nilpotent at a prime ideal $\varrho$ if the localization $I_{\varrho}$ is a nilpotent ideal in $A_{\varrho}$. In the framework of smooth projective geometrically irreducible curves $\mathcal C$ over ${\mathbb F}_q$ let $Nilp_{{A}_{\nu}}$ denote the category of $A_{\nu}$-schemes on which the uniformizer $\xi$ of $A_{\nu}$ is locally nilpotent. Here $A_{\nu} \simeq {\mathbb F}_{\nu}[[\xi]]$ is the completion of the local ring $\mathcal O_{\mathcal C, \nu}$ at a closed point $\nu \in \mathcal C$.\ Let $Nilp_{{\mathbb F}_{q}[[\xi]]}$ be the category of ${\mathbb F}_{q}[[\xi]]$-schemes on which $\xi$ is locally nilpotent. Let $S \in Nilp_{{\mathbb F}_{q}[[\xi]]}$. Let $M$ be a sheaf of $\mathcal O_S[[z]]$-modules on $S$ and let $\sigma_q^*M = M \otimes_{\mathcal O_S[[z]],\sigma_q^*} O_S[[z]]$, $M[\frac{1}{z - \xi}] = M\otimes_{\mathcal O_S[[z]]} {\mathcal O_S[[z]]}[\frac{1}{z - \xi}]$ . A local shtuka of height $r$ over $S$ is a pair $M = (M, F_M )$ consisting of a locally free sheaf $M$ of ${\mathcal O}_S[[z]]$-modules of rank $r$, and an isomorphism $F_M: \sigma_q^*M[\frac{1}{z - \xi}] \simeq M[\frac{1}{z - \xi}]$. The next lemma is proved [@HartlSingh]. Let $R$ be an ${\mathbb F}_{q}[[\xi]]$-algebra in which $\xi$ is nilpotent. Then the sequence of $R[[z]]$-modules $$0 \to R[[z]] \to R[[z]] \to R \to 0$$ $$1 \mapsto z- \xi, z \mapsto \xi$$ is exact. In particular $R[[z]] \subset R[[z]][\frac{1}{z - \xi}]$. In the conditions of the lemma authors [@HartlSingh] give the next A $z$-divisible local Anderson module over $R$ is a sheaf of ${\mathbb F}_{q}[[z]]$-modules $G$ on the big $fppf$-site of $Spec \; R$ such that\ (a) $G$ is $z$-torsion, that is $G = \varinjlim G[z^n]$, where $G[z^n] = ker(z^n: G \to G)$,\ (b) $G$ is $z$-divisible, that is $z : G \to G$ is an epimorphism,\ (c) For every $n$ the ${\mathbb F}_{q}$-module $G[z^n]$ is representable by a finite locally free strict ${\mathbb F}_{q}$-module scheme over $R$ in the sense of Faltings ([@Faltings; @HartlSingh]), and\ (d) locally on $Spec \; R$ there exists an integer $ d \in {\mathbb Z}_{\ge 0}$, such that $ (z - \xi)^d = 0$ on $\omega_G$ where $\omega_G = \varprojlim \omega_{G[z^n]}$ and $\omega_{G[z^n]} = \varepsilon^* {\Omega^1}_{G[z^n]/{Spec \; R}}$ for the unit section $\varepsilon$ of $G[z^n]$ over $R$. $ z$-divisible local Anderson modules by Hartl [@Hartl2005] with improvements in [@HartlSingh] and local shtukas are investigated. The equivalence between the category of effective local shtukas over $S$ and the category of $z$-divisible local Anderson modules over $S$ is treated by the authors [@Singh2012; @HartlSingh]. The theorem about canonical ${\mathbb F}_p[[\xi]]$ -isomorphism of $z$-adic Tate-module of $z$ -divisible local Anderson module $G$ of rank $r$ over $S$ and Tate module of local shtuka over $S$ associated to $G$ is given. The main result of [@Singh2012] is the following (section 2.5) interesting result: it is possible to associate a formal Lie group to any $z$-divisible local Anderson module over $S$ in the case when $\xi$ is locally nilpotent on $S$. We note that related with [@Singh2012] and in some cases more general results have presented in the paper by U. Hartl, E. Viehmann [@HartlViehmann2011]. [40]{} Langlands R. P. On the notion of an automorphic representation, Automorphic Forms,Representations, and L-Functions, Proc. Symp. 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--- abstract: 'Estimating the values of unknown parameters from corrupted measured data faces a lot of challenges in ill-posed problems. In such problems, many fundamental estimation methods fail to provide a meaningful stabilized solution. In this work, we propose a new regularization approach and a new regularization parameter selection approach for linear least-squares discrete ill-posed problems. The proposed approach is based on enhancing the singular-value structure of the ill-posed model matrix to acquire a better solution. Unlike many other regularization algorithms that seek to minimize the estimated data error, the proposed approach is developed to minimize the mean-squared error of the estimator which is the objective in many typical estimation scenarios. The performance of the proposed approach is demonstrated by applying it to a large set of real-world discrete ill-posed problems. Simulation results demonstrate that the proposed approach outperforms a set of benchmark regularization methods in most cases. In addition, the approach also enjoys the lowest runtime and offers the highest level of robustness amongst all the tested benchmark regularization methods.' author: - 'Mohamed Suliman, , Tarig Ballal, , and Tareq Y. Al-Naffouri,  [^1]' bibliography: - 'References\_j14.bib' title: 'Perturbation-Based Regularization for Signal Estimation in Linear Discrete Ill-posed Problems' --- Linear estimation, ill-posed problems, linear least squares, regularization. Introduction {#sec:intro} ============ We consider the standard problem of recovering an unknown signal $\xv_{0} \in \mathbb{R}^{n}$ from a vector $\yv \in \mathbb{R}^{m}$ of $m$ noisy, linear observations given by $\yv = \Am \xv_{0} + \zv$. Here, $\Am \in \mathbb{R}^{m\times n}$ is a known linear measurement matrix, and, $\zv \in \mathbb{R}^{m\times 1}$ is the noise vector; the latter is assumed to be additive white Gaussian noise (AWGN) vector with unknown variance $\sigma_{\zv}^{2}$ that is independent of $\xv_{0}$. Such problem has been extensively studied over the years due to its obvious practical importance as well as its theoretical interest [@kailath2000linear; @poor2013introduction; @groetsch1993inverse]. It arises in many fields of science and engineering, e.g., communication, signal processing, computer vision, control theory, and economics. Over the past years, several mathematical tools have been developed for estimating the unknown vector $\xv_{0}$. The most prominent approach is the ordinary least-squares (OLS)[@kay2013fundamentals] that finds an estimate $\hat{\xv}_{\text{OLS}}$ of $\xv_{0}$ by minimizing the Euclidean norm of the residual error $$\label{eq:ls problem} \hat{\xv}_{\text{OLS}} = \argmin_{\xv} || \yv - \Am \xv||^{2}_{2}.$$ The behavior of the OLS has been extensively studied in the literature and it is now very well understood. In particular, if $\Am$ is a full column rank, (\[eq:ls problem\]) has a unique solution given by $$\begin{aligned} \label{eq:pure LS solution} {\hat{\xv}}_{\text{OLS}} &=& \left(\Am^{T} \Am\right)^{-1} \Am^{T} \yv =\Vm \Sigmam^{-1} \Um^{T} \yv, %{\dagger}\end{aligned}$$ where $\Am = \Um \Sigmam \Vm^{T} = \sum_{i=1}^{n} \sigma_{i} \uv_{i} \vv_{i}^{T}$ is the singular value decomposition (SVD) of $\Am$, $\uv_{i}$ and $\vv_{i}$ are the left and the right orthogonal singular vectors, while the singular values $\sigma_{i}$ are assumed to satisfy $\sigma_{1} \ge \sigma_{1} \geq \dotsi \geq \sigma_{n}$. A major difficulty associated with the OLS approach is in discrete ill-posed problems. A problem is considered to be well-posed if its solution always exists, unique, and depends continuously on the initial data. Ill-posed problems fail to satisfy at least one of these conditions [@fischler2014readings]. In such problems, the matrix $\Am$ is ill-conditioned and the computed LS solution in (\[eq:pure LS solution\]) is potentially very sensitive to perturbations in the data such as the noise $\zv$ [@kilmer2001choosing]. Discrete ill-posed problems are of great practical interest in the field of signal processing and computer vision [@piotrowski2008mv; @liu2008kernel; @bertero1988ill; @poggio1985computational]. They arise in a variety of applications such as computerized tomography [@natterer1986mathematics], astronomy [@craig1986inverse], image restoration and deblurring [@katsaggelos1991regularized; @hansen2006deblurring], edge detection [@torre1986edge], seismography [@scales1988robust], stereo matching [@scharstein2002taxonomy], and the computation of lightness and surface reconstruction [@blanz2004statistical]. Interestingly, in all these applications and even more, data are gathered by convolution of a noisy signal with a detector [@aster2005parameter; @hansen1993use]. A linear representation of such process is normally given by $$\label{eq:kernal equation} \int_{b_{1}}^{b_{2}} a\left(s,t\right) \xv_{0}\left(t\right) \text{dt} = \yv_{0}\left(s\right) + \zv\left(s\right) = \yv\left(s\right),$$ where $ \yv_{0}\left(s\right)$ is the true signal, while the kernal function $a\left(s,t\right)$ represents the response. It is shown in [@chen1993new] how a problem with a formulation similar to (\[eq:kernal equation\]) fails to satisfy the well-posed conditions introduced above. The discretized version of (\[eq:kernal equation\]) can be represented by $\yv = \Am \xv_{0} + \zv$. To solve ill-posed problems, regularization methods are commonly used. These methods are based on introducing an additional prior information in the problem. All regularization methods are used to generate a reasonable solution for the ill-posed problem by replacing the problem with a well-posed one whose solution is acceptable. This must be done after careful analysis to the ill-posed problem in terms of its physical plausibility and its mathematical properties. Several regularization approaches have been proposed throughout the years. Among them are the truncated SVD [@varah1983pitfalls], the maximum entropy principle [@smith2013maximum], the hybrid methods [@hanke1993regularization], the covariance shaping LS estimator [@eldar2003covariance], and the weighted LS [@eldar2007improvement]. The most common and widely used approach is the regularized M-estimator that obtains an estimate $\hat{\xv}$ of $\xv_{0}$ from $\yv$ by solving the convex problem $$\label{eq:m regularization} \hat{\xv} := \argmin_{\xv} \mathcal{L}\left(\yv - \Am \xv \right) + \gamma f\left(\xv\right),$$ where the loss function $\mathcal{L}:\mathbb{R}^{m} \to \mathbb{R} $ measures the fit of $\Am\xv$ to the observation vector $\yv$, the penalty function $f:\mathbb{R}^{m} \to \mathbb{R}$ establishes the structure of $\xv$, and $\gamma$ provides a balance between the two functions. Different choices of $\mathcal{L}$ and $f$ leads to different estimators. The most popular among them is the Tikhonov regularization which is given in its simplified form by [@tikhonov1977methods] $$\label{eq:tik-minimization} \hat{\xv}_{\text{RLS}} := \argmin_{\xv} \ ||\yv - \Am \xv ||_2^{2} + \gamma \ ||\xv ||_2^{2}.$$ The solution to (\[eq:tik-minimization\]) is given by the regularized least-square (RLS) estimator $$\label{eq:R-LS} \hat{\xv}_{\text{RLS}} = \left(\Am^{T}\Am+\gamma\Id_{n}\right)^{-1}\Am^{T}\yv,$$ where $\Id_{n}$ is ($n \times n $) identity matrix. In general, $\gamma$ is unknown and has to be chosen judiciously. On the other hand, several regularization parameter selection methods have been proposed to find the regularization parameter in regularization methods. These include the *generalized cross validation* (GCV) [@wahba1990spline], the *L-curve* [@hansen1992analysis; @hansen2007adaptive], and the *quasi-optimal* method [@morozov2012methods]. The GCV obtains the regularizer by minimizing the GCV function which suffers from the shortcoming that it may have a very flat minimum that makes it very challenging to be located numerically. The L-curve, on the other hand, is a graphical tool to obtain the regularization parameter which has a very high computational complexity. Finally, the quasi-optimal criterion chooses the regularization parameter without taking into account the noise level. In general, the performance of these methods varies significantly depending on the nature of the problem[^2]. Paper Contributions ------------------- 1. *New regularization approach*: We propose a new approach for linear discrete ill-posed problems that is based on adding an artificial perturbation matrix with a bounded norm to $\Am$. The objective of this artificial perturbation is to improve the challenging singular-value structure of $\Am$. This perturbation affects the fidelity of the model $\yv = \Am \xv_{0} + \zv$, and as a result, the equality relation becomes invalid. We show that using such modification provides a solution with better numerical stability. 2. *New regularization parameter selection method*: We develop a new regularization parameter selection approach that selects the regularizer in a way that minimizes the mean-squared error (MSE) between $\xv_{0}$ and its estimate $\hat{\xv}$, $\mathbb{E} \ ||\hat{\xv} - \xv_{0}||_{2}^{2} $. [^3] 3. *Generality*: A key feature of the approach is that it does not impose any prior assumptions on $\xv_{0}$. The vector $\xv_{0}$ can be deterministic or stochastic, and in the later case we do not assume any prior statistical knowledge about it. Moreover, we assume that the noise variance $\sigma_{\zv}^{2}$ is unknown. Finally, the approach can be applied to a large number of linear discrete ill-posed problems. Paper Organization ------------------ This paper is organized as follows. Section \[sec:COPRA\] presents the formulation of the problem and derive its solution. In Section \[sec:MSE\], we derive the artificial perturbation bound that minimizes the MSE. Further, we derive the proposed approach characteristic equation for obtaining the regularization parameter. Section \[sec:Properties\] studies the properties of the approach characteristic equation while Section \[sec:Results\] presents the performance of the proposed approach using simulations. Finally, we conclude our work in Section \[sec:conclusion\]. Notations --------- Matrices are given in boldface upper case letters (e.g., $\Xm$), column vectors are represented by boldface lower case letters (e.g., $\xv$), and $\left(.\right)^{T}$ stands for the transpose operator. Further, $\mathbb{E}\left(.\right)$, $\Id_{n}$, and $\bm{0}$ denote the expectation operator, the $\left(n \times n \right)$ identity matrix, and the zero matrix, respectively. Notation $||.||_{2}$ refers to the spectral norm for matrices and Euclidean norm for vectors. The operator $\text{diag}\left(.\right)$ returns a vector that contains the diagonal elements of a matrix, and a diagonal matrix if it operates on a vector where the diagonal entries of the matrix are the elements of the vector. Proposed Regularization Approach {#sec:COPRA} ================================ Background {#subsec:background} ---------- We consider the linear discrete ill-posed problems in the form $\yv = \Am \xv_{0} + \zv$ and we focus mainly on the case where $m\geq n$ without imposing any assumptions on $\xv_{0}$. The matrix $\Am$ in such problems is ill-conditioned that has a very fast singular values decay [@roy2001inverse]. A comparison between the singular values decay of the full rank matrices, the rank deficient matrices, and the ill-posed problems matrices is given in Fig. \[fig:sv decay\]. From Fig. \[fig:sv decay\], we observe that the singular values of the full column rank matrix are decaying constantly while in the rank definite matrix there is a jump (gap) between the nonzero and the zero singular values. Finally, the singular values of the ill-posed problem matrix are decaying very fast, without a gap, to a significantly small positive number. Problem Formulation {#subsec:analysis} ------------------- Let us start by considering the LS solution in (\[eq:pure LS solution\]). In many ill-posed problems, and due to the singular-value structure of $\Am$ and the interaction that it has with the noise, equation (\[eq:pure LS solution\]) is not capable of producing a sensible estimate of $\xv_{0}$. Herein, we propose adding an artificial perturbation $\Delta \Am \in \mathbb{R}^{m\times n}$ to $\Am$. We assume that this perturbation, which replaces $\Am$ by $\left(\Am+\Delta \Am\right)$, improves the challenging singular-value structure, and hence is capable of producing a better estimate of $\xv_{0}$. In other words, we assume that using $\left(\Am+\Delta \Am\right)$ in estimating $\xv_{0}$ from $\yv$ provides a better estimation result than using $\Am$. Finally, to provide a balance between improving the singular-value structure and maintaining the fidelity of the basic linear model, we add the constraint $||\Delta \Am||_2 \leq \delta$, $\delta \in \mathbb{R}^{+}$. Therefore, the basic linear model is modified to $$\label{eq:BDU equation} \yv \approx \left(\Am+\Delta \Am\right)\xv_{0} + \zv; \ ||\Delta \Am||_2 \leq \delta.$$ The question now is what is the best $\Delta \Am$ and the bound on this perturbation. It is clear that these values are important since they affect the model’s fidelity and dictate the quality of the estimator. This question is addressed further ahead in this section. For now, let us start by assuming that $\delta$ is known[^4]. Before proceeding, it worth mentioning that the model in (\[eq:BDU equation\]) has been considered for signal estimation in the presence of data errors but with strict equality (e.g., [@eldar2005robust; @el1997robust; @chandrasekaran1998parameter]). These methods assume that $\Am$ is not known perfectly due to some error contamination, and that a prior knowledge on the real error bound (which corresponds to $\delta$ in our case) is available. However, in our case the matrix $\Am$ is known perfectly, whereas $\delta$ is unknown. To obtain an estimate of $\xv_{0}$, we consider minimizing the worst-case residual function of the new perturbed model in (\[eq:BDU equation\]) which is given by $$\begin{aligned} \label{eq:worst-error} %%W\left(\Delta \Am, \hat{\xv}\right) = &\underset{\xv}{\operatorname{\min}} \ \underset{||\Delta \Am||_2 \leq \delta}{\operatorname{\max}}_{} \ Q\left(\xv,\Delta \Am\right) := \ ||\yv - \left(\Am + \Delta \Am\right) \xv ||_2. \end{aligned}$$ The unique minimzer $\hat{\xv}$ of the min-max constrained problem in (\[eq:worst-error\]) for fixed $\delta > 0$ is given by $$\label{eq:copra solution} \hat{\xv} = \left(\Am^{T}\Am+\rho\left(\delta,\hat{\xv}\right)\Id_{n}\right)^{-1}\Am^{T}\yv,$$ where $\rho\left(\delta,\hat{\xv}\right)$ is a regularization parameter that is related to the perturbation bound $\delta$ through $$\label{eq:sec1} \rho\left(\delta,\hat{\xv}\right) = \delta \ \frac{ || \yv- \Am \hat{\xv} ||_2 }{|| \hat{\xv} ||_2}.$$ By using Minkowski inequality [@maligranda1995simple], we find an upper bound for the cost function $Q\left(\xv,\Delta \Am\right)$ in (\[eq:worst-error\]) as $$\begin{aligned} \label{eq:bound minko} %\label{eq:Minkowski inequality} ||\yv - \left(\Am + \Delta \Am\right) \xv ||_2 & \leq ||\yv - \Am \xv ||_2 + || \Delta \Am \ \xv ||_{2} \nonumber\\ %\label{eq:Minkowski inequality2} &\leq ||\yv - \Am \xv ||_2 + || \Delta \Am ||_{2} || \xv ||_{2} \nonumber\\ %\label{eq:Minkowski inequality3} &\leq ||\yv - \Am \xv ||_2 + \delta \ || \xv ||_{2}.\end{aligned}$$ However, upon setting $\Delta \Am$ to be the following rank one matrix $$\label{eq:perturbtion upper bound} \Delta \Am = \frac{\left( \Am \xv -\yv \right) }{ ||\yv - \Am \xv ||_2} \frac{\xv^{T}}{|| \xv ||_{2}} \delta,$$ we can show that the bound in (\[eq:bound minko\]) is achievable by $$\begin{aligned} \label{eq:proof of upper bound} || \yv - \left(\Am + \Delta \Am\right) \xv ||_2 & = ||\left(\yv -\Am \xv \right) + \frac{\left(\yv - \Am \xv \right)}{||\yv - \Am \xv ||_2} \frac{\xv^{T} }{|| \xv ||_{2}} \xv \delta ||_2 \nonumber\\ & = ||\left(\yv -\Am \xv \right) + \frac{\left(\yv - \Am \xv \right)}{|| \yv- \Am \xv ||_2} || \xv ||_{2} \delta||_2. \end{aligned}$$ Since the two added vectors $\left(\yv -\Am \xv \right)$ and $\frac{\left(\yv - \Am \xv \right)}{|| \yv- \Am \xv ||_2} || \xv ||_{2} \delta $ in (\[eq:proof of upper bound\]) are positively linearly dependent (i.e., pointing in the same direction), we conclude that $$\begin{aligned} \label{eq:proof of upper bound2} ||\left( \yv - \Am \xv\right) + \frac{\left(\yv - \Am \xv \right)}{|| \yv - \Am \xv ||_2} || \xv ||_{2} \ \delta ||_2 = \underbrace{|| \yv - \Am \xv||_2 + \delta||\xv ||_2}_{W\left(\xv\right)}\end{aligned}$$ As a result, (\[eq:worst-error\]) can be expressed equivalently by $$\label{eq:costfunction} \underset{\xv}{\operatorname{\min}} \ \underset{||\Delta \Am||_2 \leq \delta}{\operatorname{\max}}_{} \ Q\left( \xv, \Delta \Am\right) \equiv \underset{\xv}{\operatorname{\min}} \ W\left(\xv\right).$$ Therefore, the solution of (\[eq:worst-error\]) depends only on $\delta$ and is agnostic to the structure of $\Delta \Am$[^5]. It is easy to check that the solution space for $W\left(\xv\right)$ is convex in $\xv$, and hence, any local minimum is also a global minimum. But at any local minimum, it either holds that the gradient of $W\left(\xv\right)$ is zero, or $W\left(\xv\right)$ is not differentiable. More precisely, $W\left(\xv\right)$ is not differentiable only at $\xv$ = 0 and when $\yv - \Am \xv= 0$. However, the former case is a trivial case that is not being considered in this paper, while the latter case is not possible by definition. The gradient of $W\left(\xv\right)$ can be obtained as $$\begin{aligned} \label{eq:gradientI} &\nabla_{\xv} W\left(\xv\right) = \frac{1}{|| \yv- \Am \xv ||_2} \Am^{T} \left(\Am \xv - \yv \right) + \frac{\delta \ \xv}{|| \xv ||_2} \nonumber\\ &= \frac{1}{|| \yv- \Am \xv ||_2} \left( \Am^{T} \Am \xv + \frac{\delta \ || \yv- \Am \xv ||_2\ \xv }{|| \xv ||_2} - \Am^{T}\yv \right).\end{aligned}$$ Solving for $\nabla_{\xv} W\left(\hat{\xv}\right) = 0$ upon defining $\rho\left(\delta,\hat{\xv}\right)$ as in (\[eq:sec1\]), we obtain (\[eq:copra solution\]). The regularization parameter $\rho$ in (\[eq:sec1\]) is a function of the unknown estimate $\hat{\xv}$, as well as the upper perturbation bound $\delta$ (we have dropped the dependence of $\rho$ on $\delta$ and $\hat{\xv}$ to simplify notation). In addition, it is clear from (\[eq:proof of upper bound2\]) that $\delta$ controls the weight given to the minimization of the side constraint relative to the minimization of the residual norm. We have assumed that $\delta$ is known to obtain the min-max optimization solution. However, this assumption is not valid in reality. Thus, is it impossible to obtain $\rho$ directly from (\[eq:sec1\]) given that both $\delta$ and $\hat{\xv}$ are unknowns. Now, it is obvious with (\[eq:copra solution\]) and (\[eq:sec1\]) in hand, we can eliminate the dependency of $\rho$ on $\hat{\xv}$. By substituting (\[eq:copra solution\]) in (\[eq:sec1\]) we obtain after some algebraic manipulations $$\begin{aligned} \label{eq:secular equation} &\delta^{2} \Big[\yv^{T}\yv - 2\yv^{T} \Am\left(\Am^{T}\Am+\rho\Id_{n}\right)^{-1}\Am^{T}\yv \nonumber\\ & + ||\Am\left(\Am^{T}\Am+\rho\Id_{n}\right)^{-1}\Am^{T}\yv||^{2} \Big] \nonumber\\ &= \rho^{2} \yv^{T}\Am\left(\Am^{T}\Am+\rho\Id_{n}\right)^{-2}\Am^{T}\yv.\end{aligned}$$ In this following subsection, we will utilize and simplify (\[eq:secular equation\]) using some manipulations to obtain $\delta$ that corresponds to an optimal choice of $\rho$ in ill-posed problems. Finding the Optimal Perturbation Bound {#choosing optimal bound} -------------------------------------- Let us denote the optimal choices of $\rho$ and $\delta$ by $\rho_{\text{o}}$ and $\delta_{\text{o}}$, respectively. To simplify (\[eq:secular equation\]), we substitute the SVD of $\Am$, then we solve for $\delta^{2}$, and finally we take the trace $\text{Tr}\left(.\right)$ of the two sides considering the evaluation point to be $\left(\delta_{\text{o}},\rho_{\text{o}}\right)$ to get $$\begin{aligned} \label{eq:secular equation trace} &\underbrace{\delta_{\text{o}}^2 \ \text{Tr}\left( \left(\Sigmam^2 + \rho_{\text{o}} \Id_{n} \right)^{-2} \Um^{T} \left(\yv \yv^{T}\right) \Um \right)}_{D\left(\rho_{\text{o}}\right)} \nonumber\\ &= \underbrace{\text{Tr}\left( \Sigmam^2 \left(\Sigmam^2 + \rho_{\text{o}} \Id_{n} \right)^{-2} \Um^{T} \left(\yv \yv^{T}\right) \Um \right)}_{N\left(\rho_{\text{o}}\right)}.\end{aligned}$$ In order to obtain a useful expression, let us think of $\delta_{\text{o}}$ as a single universal value that is computed over many realizations of the observation vector $\yv$. Based on this perception, $\yv\yv^{T}$ can be replaced by its expected value $\mathbb{E}(\yv \yv^{T})$. In other words, we are looking for an optimal choice of $\delta$, say $\delta_{\text{o}}$, that is optimal for all realizations of $\yv$. At this point, we assume that such value exists. Then this parameter $\delta_{\text{o}}$ is clearly deterministic. If we sum (\[eq:secular equation trace\]) for all realizations of $\yv$, and a fixed $\delta_{\text{o}}$, we end replacing $\yv\yv^{T}$ with $\mathbb{E}(\yv \yv^{T})$ which can be expressed using our basic linear model as $$\begin{aligned} \label{•eq:yy'} \mathbb{E} \left(\yv \yv^{T} \right) & = \Am \Rm_{\xv_{0}} \Am^{T} + \sigma_{\zv}^2 \Id_{m} \nonumber\\ &= \Um \Sigmam \Vm^{T}\Rm_{\xv_{0}} \Vm \Sigmam \Um^{T}+ \sigma_{\zv}^2 \Id_{m},\end{aligned}$$ where $\Rm_{\xv_{0}} \triangleq \mathbb{E}\left(\xv_{0} \xv_{0}^{T} \right)$ is the covariance matrix of $\xv_{0}$. For a deterministic $\xv_{0}$, $\Rm_{\xv_{0}} = \xv_{0}\xv_{0}^{T}$ is used for notational simplicity. Substituting (\[•eq:yy’\]) in both terms of (\[eq:secular equation trace\]) results $$\begin{aligned} \label{eq:N term} N\left(\rho_{\text{o}}\right)&= \text{Tr}\left( \Sigmam^2 \left(\Sigmam^2 + \rho_{\text{o}} \Id_{n} \right)^{-2}\Sigmam^{2} \Vm^{T} \Rm_{\xv_{0}} \Vm \right)\nonumber\\ &+\sigma_{\zv}^{2} \text{Tr}\left(\Sigmam^{2}\left(\Sigmam^{2}+\rho_{\text{o}}\Id_{n}\right)^{-2}\right),\end{aligned}$$ and $$\begin{aligned} \label{eq:D term} D\left(\rho_{\text{o}}\right)&= \delta_{\text{o}}^{2}\Big[\text{Tr}\left( \left(\Sigmam^2 + \rho_{\text{o}} \Id_{n} \right)^{-2}\Sigmam^{2} \Vm^{T} \Rm_{\xv_{0}} \Vm \right) \nonumber\\ &+\sigma_{\zv}^{2}\text{Tr}\left(\left(\Sigmam^{2}+\rho_{\text{o}}\Id_{n}\right)^{-2}\right)\Big].\end{aligned}$$ Considering the singular-value structure for the ill-posed problems, we can divide the singular values into two groups of *significant*, or relatively large, and *trivial*, or nearly zero singular value[^6]. As an example, we can see from Fig. \[fig:sv decay\] that the singular values of the ill-posed problem matrix are decaying very fast, making it possible to identify the two groups. Based on this, the matrix $\Sigmam$ can be divided into two diagonal sub-matrices, $\Sigmam_{n_{1}}$, which contains the first (significant) $n_{1}$ diagonal entries, and $\Sigmam_{n_{2}}$, which contains the last (trivial) $n_2 = n - n_1$ diagonal entries[^7]. As a result, $\Sigmam$ can be written as $$\label{eq:sigma parti} \Sigmam = \begin{bmatrix} \Sigmam_{n_{1}} & \bm{0} \\ \bm{0}& \Sigmam_{n_{2}} \end{bmatrix}.$$ Similarly, we can partition $\Vm$ as $\Vm = [\Vm_{n_{1}} \ \Vm_{n_{2}}]$ where $\Vm_{n_{1}}\in \mathbb{R}^{n\times n_{1}}$, and $\Vm_{n_{2}}\in \mathbb{R}^{n\times n_{2}}$. Now, we can write $N\left(\rho_{\text{o}}\right)$ in (\[eq:N term\]) in terms of the partitioned $\Sigmam$ and $\Vm$ as $$\begin{aligned} \label{eq:numer eqaution 1} N\left(\rho_{\text{o}}\right) &= \text{Tr} \left(\Sigmam_{n_{1}}^2 \left(\Sigmam_{n_{1}}^2 + \rho_{\text{o}} \Id_{n_{1}} \right)^{-2} \Sigmam_{n_{1}}^2 \Vm_{n_{1}}^{T} \Rm_{\xv_{0}}\Vm_{n_{1}}\right) \nonumber\\ &+ \text{Tr} \left(\Sigmam_{n_{2}}^2 \left(\Sigmam_{n_{2}}^2 + \rho_{\text{o}} \Id_{n_{2}} \right)^{-2} \Sigmam_{n_{2}}^2 \Vm_{n_{2}}^{T} \Rm_{\xv_{0}}\Vm_{n_{2}}\right)\nonumber\\ &+\sigma_{\zv}^2 \text{Tr}\left(\Sigmam_{n_{1}}^2 \left(\Sigmam_{n_{1}}^2 + \rho_{\text{o}} \Id_{n_{1}} \right)^{-2} \right)\nonumber\\ &+ \sigma_{\zv}^2 \text{Tr}\left(\Sigmam_{n_{2}}^2 \left(\Sigmam_{n_{2}}^2 + \rho_{\text{o}} \Id_{n_{2}} \right)^{-2} \right). \end{aligned}$$ Given that $\Sigmam_{n_{1}}$ contains the significant singular values and $\Sigmam_{n_{2}}$ contains the relatively small (nearly zero) singular values, we have $\lVert \Sigmam_{n_{2}} \rVert \approx 0$, and so we can approximate $N\left(\rho_{\text{o}}\right)$ as $$\begin{aligned} \label{eq:numer eqaution 2} N\left(\rho_{\text{o}}\right) &\approx \text{Tr} \left(\Sigmam_{n_{1}}^2 \left(\Sigmam_{n_{1}}^2 + \rho_{\text{o}} \Id_{n_{1}} \right)^{-2} \Sigmam_{n_{1}}^2 \Vm_{n_{1}}^{T} \Rm_{\xv_{0}}\Vm_{n_{1}}\right)\nonumber\\ &+ \sigma_{\zv}^2 \text{Tr}\left(\Sigmam_{n_{1}}^2 \left(\Sigmam_{n_{1}}^2 + \rho_{\text{o}} \Id_{n_{1}} \right)^{-2} \right) .\end{aligned}$$ Similarly, $D\left(\rho_{\text{o}}\right)$ in (\[eq:D term\]) can be approximated equivalently as $$\begin{aligned} \label{eq:denumer approximation} &D \left(\rho_{\text{o}}\right) \approx \sigma_{\zv}^2 \ \text{Tr}\left(\left(\Sigmam_{n_{1}}^2 + \rho_{\text{o}} \Id_{n_{1}} \right)^{-2} \right) + \frac{ n_{2} \sigma_{\zv}^2}{\rho_{\text{o}}^{2}} \nonumber\\ &+ \text{Tr} \left( \left(\Sigmam_{n_{1}}^{2} + \rho_{\text{o}} \Id_{n_{1}} \right)^{-2} \Sigmam_{n_{1}}^{2} \Vm_{n_{1}}^{T} \Rm_{\xv_{0}}\Vm_{n_{1}}\right).\end{aligned}$$ Substituting and in (\[eq:secular equation trace\]) and manipulating, we obtain $$\begin{aligned} \label{eq:eta min 2} \delta_{\text{o}}^2 &\approx \Big[\sigma_{\zv}^2 \text{Tr}\left(\Sigmam_{n_{1}}^2 \left(\Sigmam_{n_{1}}^2 + \rho_{\text{o}} \Id_{n_{1}} \right)^{-2} \right)\nonumber\\ &+ \text{Tr} \left(\Sigmam_{n_{1}}^2 \left(\Sigmam_{n_{1}}^2 + \rho_{\text{o}} \Id_{n_{1}} \right)^{-2} \Sigmam_{n_{1}}^2 \Vm_{n_{1}}^{T} \Rm_{\xv_{0}}\Vm_{n_{1}}\right)\Big] \Big/ \nonumber\\ &\Big[\sigma_{\zv}^2 \text{Tr}\left( \left(\Sigmam_{n_{1}}^2 + \rho_{\text{o}} \Id_{n_{1}} \right)^{-2} \right) + \frac{n_{2} \sigma_{\zv}^{2} }{\rho_{\text{o}}^{2}} \nonumber\\ &+ \text{Tr} \left( \left(\Sigmam_{n_{1}}^{2} + \rho_{\text{o}} \Id_{n_{1}} \right)^{-2} \Sigmam_{n_{1}}^{2} \Vm_{n_{1}}^{T} \Rm_{\xv_{0}}\Vm_{n_{1}}\right)\Big].\end{aligned}$$ The bound $\delta_{\text{o}}$ in (\[eq:eta min 2\]) is a function of the unknowns $\rho_{\text{o}}$, $\sigma_{\zv}^{2}$, and $\Rm_{\xv_{0}}$. In fact, estimating $\sigma_{\zv}^{2}$ and $\Rm_{\xv_{0}}$ without any prior knowledge is a very tedious process. The problem becomes worse when $\xv_{0}$ is deterministic. In such case, the exact value of $\xv_{0}$ is required to obtain $\Rm_{\xv_{0}} = \xv_{0}\xv_{0}^{T}$. In the following section, we will use the MSE as a criterion to eliminate the dependence of $\delta_{\text{o}}$ on these unknowns and a result to set the value of the perturbation bound. Minimizing the MSE for the solution of the proposed perturbation approach {#sec:MSE} ========================================================================= The MSE for an estimate $\hat{\xv}$ of $\xv_{0}$ is given by $$\label{eq:MSE} \text{MSE} =\mathbb{E}\big[ ||\hat{\xv} - \xv_{0}||^{2} \big] =\text{Tr}\left( \mathbb{E}\left( (\hat{\xv} - \xv_{0}) (\hat{\xv} - \xv_{0})^{T} \right) \right).$$ Since the solution of the proposed approach problem in (\[eq:worst-error\]) is given by (\[eq:copra solution\]), we can substitute for $\hat{\xv}$ from (\[eq:copra solution\]) in (\[eq:MSE\]) and then use the SVD of $\Am$ to obtain $$\begin{aligned} \label{eq:MSE2} \text{MSE}\left(\rho\right) &= \sigma_{\zv}^{2} \text{Tr}\left(\Sigmam^{2} \left(\Sigmam^{2} + \rho\Id_{n} \right)^{-2} \right) \nonumber \\ &+ \rho^{2} \text{Tr}\left( \left(\Sigmam^2 + \rho\Id_{n} \right)^{-2}\Vm^{T}\Rm_{\xv_{0}}\Vm \right).\end{aligned}$$ For $\sigma_{\zv}^{2} >0$, the approximate value for the optimal regularizer $\rho_{\text{o}}$ of (\[eq:MSE2\]) that approximately minimizes the MSE is given by $$\label{eq:gamma min approx} \rho_{\text{o}}\approx \frac{ \sigma_{\zv}^2}{\text{Tr}\left(\Rm_{\xv_{0}}\right) /n}.$$ We can easily prove that the function in (\[eq:MSE2\]) is convex in $\rho$, and hence its global minimizer (i.e., $\rho_{\text{o}}$) can be obtained by differentiating (\[eq:MSE2\]) with respect to $\rho$ and setting the result to zero, i.e., $$\begin{aligned} \label{eq:MSE'} &\nabla_{\rho} \ \text{MSE}\left(\rho\right) = -2\sigma_{\zv}^2 \text{Tr}\left(\Sigmam^2 \left(\Sigmam^2 + \rho\Id_{n} \right)^{-3} \right) \nonumber \\ &+ 2 \rho \underbrace{\text{Tr}\left( \Sigmam^{2} \left(\Sigmam^2 + \rho\Id_{n} \right)^{-3}\Vm^{T}\Rm_{\xv_{0}}\Vm \right)}_{S} = 0.\end{aligned}$$ Equation (\[eq:MSE’\]) dictates the relationship between the optimal regularization parameter and the problem parameters. By solving (\[eq:MSE’\]), we can obtain the optimal regularizer $\rho_{\text{o}}$. However, in the general case, and with lack of knowledge about $\Rm_{\xv_{0}}$, we cannot obtain a closed-form expression for $\rho_{\text{o}}$. As a result, we will seek to obtain a suboptimal regularizer in the MSE sense that minimizes (\[eq:MSE2\]) approximately. In what follows, we show how through some bounds and approximations, we can obtain this suboptimal regularizer. By using the trace inequalities in \[[@wang1986trace], eq.(5)\], we can bound the second term in (\[eq:MSE’\]) by $$\begin{aligned} \label{eq:inequality} &\lambda_{\text{min}}\left(\Rm_{\xv_{0}}\right)\text{Tr}\left(\Sigmam^2 \left(\Sigmam^2 + \rho\Id_{n} \right)^{-3} \right) \nonumber\\ & \leq S= \text{Tr}\left( \Sigmam^{2} \left(\Sigmam^2 + \rho\Id_{n} \right)^{-3}\Vm^{T}\Rm_{\xv_{0}}\Vm \right) \nonumber\\ & \leq \lambda_{\text{max}}\left(\Rm_{\xv_{0}}\right)\text{Tr}\left(\Sigmam^2 \left(\Sigmam^2 + \rho\Id_{n} \right)^{-3} \right),\end{aligned}$$ where $\lambda_{i}$ is the $i$’th eigenvalue of $\Rm_{\xv_{0}}$. Our main goal in this paper is to find a solution that is approximately feasible for all discrete ill-posed problems and also suboptimal in some sense. In other words, we would like to find a $\rho_{\text{o}}$, for all (or almost all) possible $\Am$, that minimizes the MSE approximately. To achieve this, we consider an *average* value of $S$ based on the inequalities in (\[eq:inequality\]) as our evaluation point, i.e., $$\begin{aligned} \label{eq:d} S \approx \text{Tr}\left(\Sigmam^2 \left(\Sigmam^2 + \rho\Id_{n} \right)^{-3} \right) \frac{\text{Tr}\left(\Rm_{\xv_{0}} \right)}{n}.\end{aligned}$$ Substituting (\[eq:d\]) in (\[eq:MSE’\]) yields[^8] $$\begin{aligned} \label{eq:MSE' approx} &\nabla_{\rho} \ \text{MSE}\left(\rho\right) \approx -2\sigma_{\zv}^2 \text{Tr}\left(\Sigmam^2 \left(\Sigmam^2 + \rho\Id_{n} \right)^{-3} \right) \nonumber \\ &+ 2 \rho \frac{\text{Tr}\left(\Rm_{\xv_{0}} \right)}{n}\text{Tr}\left(\Sigmam^2 \left(\Sigmam^2 + \rho\Id_{n} \right)^{-3} \right) = 0.\end{aligned}$$ Note that the same approximation can be applied from the beginning to the second term in (\[eq:MSE2\]) and the same result in (\[eq:MSE’ approx\]) will be obtained after taking the derivative of the new approximated MSE function. In Appendix \[Apen error\], we provide the error analysis for this approximation and show that it is bounded in very small feasible region.\ Equation (\[eq:MSE’ approx\]) can now be solved to obtain $\rho_{\text{o}}$ as in (\[eq:gamma min approx\]).[^9] $$\begin{aligned} \label{eq:IBPRfunction} G\left(\rho_{\text{o}}\right) &= \underbrace{\text{Tr}\left( \Sigmam^2 \left(\Sigmam^2 + \rho_{\text{o}} \Id_{n} \right)^{-2} \Um^{T}\yv\yv^{T}\Um \right)\text{Tr}\left( \left(\Sigmam_{n_{1}}^2 + \rho_{\text{o}} \Id_{n_{1}} \right)^{-2}\left(\beta\Sigmam_{n_{1}}^2 + \rho_{\text{o}} \Id_{n_{1}}\right) \right) + \frac{n_{2}}{\rho_{\text{o}}}\text{Tr}\left( \Sigmam^2 \left(\Sigmam^2 + \rho_{\text{o}} \Id_{n} \right)^{-2}\Um^{T}\yv\yv^{T}\Um \right)}_{G_{1}\left(\rho_{\text{o}}\right)}\nonumber\\ &- \underbrace{\text{Tr}\left(\left(\Sigmam^2 + \rho_{\text{o}} \Id_{n} \right)^{-2} \Um^{T}\yv\yv^{T}\Um\right)\text{Tr}\left( \Sigmam_{n_{1}}^2 \left(\Sigmam_{n_{1}}^2 + \rho_{\text{o}} \Id_{n_{1}} \right)^{-2} \left(\beta\Sigmam_{n_{1}}^2 + \rho_{\text{o}} \Id_{n_{1}}\right) \right)}_{ G_{2}\left(\rho_{\text{o}}\right)} = 0.\end{aligned}$$ The solution in (\[eq:gamma min approx\]) shows that there always exists a positive $\rho_{\text{o}}$, for $\sigma_{\zv}^{2} \neq 0$, which approximately minimizes the MSE in (\[eq:MSE2\]). The conclusion that the regularization parameter is generally dependent on the noise variance has been shown before in different contexts (see for example [@zachariah2015online; @hemmerle1975explicit]). For the special case where the entries of $\xv_{0}$ are independent and identically distributed (i.i.d.) with zero mean, we have $\Rm_{\xv_{0}} = \sigma_{\xv_{0}}^{2}\Id_{n}$. Since the linear minimum mean-squared error (LMMSE) estimator of $\xv_{0}$ in $\yv = \Am \xv_{0} + \zv$ is defined as [@kay2013fundamentals] $$\label{eq:lmmse2} \hat{\xv}_{\text{LMMSE}}= \left(\Am^{T} \Am + \sigma_{\zv}^{2} \Rm_{\xv_{0}}^{-1} \Id_{n}\right)^{-1} \Am^{T} \yv,$$ substituting $\Rm_{\xv_{0}} = \sigma_{\xv_{0}}^{2}\Id$ makes the LMMSE regularizer in (\[eq:lmmse2\]) equivalent to $\rho_{\text{o}}$ in (\[eq:gamma min approx\]) since $\rho_{\text{o}}= \frac{\sigma_{\zv}^{2}}{\text{Tr}\left(\Rm_{\xv_{0}}\right)/n} = \frac{\sigma_{\zv}^{2}}{\sigma_{\xv_{0}}^{2}}$. This shows that (\[eq:gamma min approx\]) is exact when the input is white, while for a general input $\xv_{0}$, the optimum matrix regularizer is given in (\[eq:lmmse2\]). In other words, the result in (\[eq:gamma min approx\]) provides an approximate optimum scalar regularizer for a general colored input. Note that since $ \sigma_{\zv}^2$ and $\Rm_{\xv_{0}}$ are unknowns, $\rho_{\text{o}}$ cannot be obtained directly from (\[eq:gamma min approx\]). We are now ready in the following subsection to use the result in (\[eq:gamma min approx\]) along with the perturbation bound expression in (\[eq:eta min 2\]) and some reasonable manipulations and approximations to eliminate the dependency of $\delta_{\text{o}}$ in (\[eq:eta min 2\]) on the unknowns $\sigma_{\zv}^{2}$, and $\Rm_{\xv_{0}}$. Then, we will select a pair of $\delta_{\text{o}}$ and $\rho_{\text{o}}$ from the space of all possible values of $\delta$ and $\rho$ that minimizes the MSE of the proposed estimator solution. Setting the Optimal Perturbation Bound that Minimizes the MSE ------------------------------------------------------------- We start by applying the same reasoning leading to (\[eq:d\]) for both the numerator and the denominator of (\[eq:eta min 2\]) and manipulate to obtain $$\begin{aligned} \label{eq:eta min 3} &{\text{Tr}\left( \Sigmam_{n_{1}}^2 \left(\Sigmam_{n_{1}}^2 + \rho_{\text{o}} \Id_{n_{1}} \right)^{-2} \left(\Sigmam_{n_{1}}^2 + \frac{n_1 \sigma_{\zv}^2}{\text{Tr}\left(\Rm_{\xv_{0}}\right)} \Id_{n_{1}} \right) \right)} \nonumber\\ &\approx \delta_{\text{o}}^2 \Bigg[ \text{Tr}\left(\left(\Sigmam_{n_{1}}^2 + \rho_{\text{o}} \Id_{n_{1}} \right)^{-2} \left(\Sigmam_{n_{1}}^2 + \frac{n_1 \sigma_{\zv}^2}{\text{Tr}\left(\Rm_{\xv_{0}}\right)} \Id_{n_{1}} \right) \right) \nonumber\\ & + \frac{n_2 n_1 \sigma_{\zv}^2 }{\rho_{\text{o}}^{2}{\text{Tr}\left(\Rm_{\xv_{0}}\right)}} \Bigg].\end{aligned}$$ In Section \[sec:Results\], we verify this approximation using simulations. Now, we will use the relationship of $\sigma_{\zv}^{2}$ and $\text{Tr}\left(\Rm_{\xv_{0}}\right)$ in (\[eq:gamma min approx\]) to the suboptimal regularizer $\frac{n_{1}}{n}\rho_{\text{o}} \approx \frac{n_{1} \sigma_{\zv}^{2}} {\text{Tr}(\Rm_{\xv_{0}})}$ to impose a constrain on (\[eq:eta min 3\]) that makes the selected perturbation bound minimizes the MSE and as a result to make (\[eq:eta min 3\]) an implicit equation in $\delta_{\text{o}}$ and $\rho_{\text{o}}$ only. By doing this, we obtain after some algebraic manipulations $$\begin{aligned} \label{eq:eta min final 1} \delta_{\text{o}}^2 \approx \frac {\text{Tr}\left( \Sigmam_{n_{1}}^{2} \left(\Sigmam_{n_{1}}^2 + \rho_{\text{o}} \Id_{n_{1}} \right)^{-2} \left(\frac{n}{n_1}\Sigmam_{n_{1}}^2 + \rho_{\text{o}} \Id_{n_{1}} \right) \right)} {\text{Tr}\left( \left(\Sigmam_{n_{1}}^2 + \rho_{\text{o}} \Id_{n_{1}} \right)^{-2} \left(\frac{n}{n_1}\Sigmam_{n_{1}}^2 + \rho_{\text{o}} \Id_{n_{1}} \right) \right) +\frac{n_{2}}{\rho_{\text{o}}}}.\end{aligned}$$ The expression in (\[eq:eta min final 1\]) reveals that any $\delta_{\text{o}}$ satisfying (\[eq:eta min final 1\]) minimizes the MSE approximately. Now, we have two equations (\[eq:secular equation\]) (evaluated at $\delta_{0}$ and $\rho_{0}$) and (\[eq:eta min final 1\]) in two unknowns $\delta_{0}$ and $\rho_{0}$. Solving these equations and then applying the SVD of $\Am$ to the result equation, result in the characteristic equation for the proposed constrained perturbation regularization approach (COPRA) in (\[eq:IBPRfunction\]), where $\beta = \frac{n}{n_1}$. The COPRA characteristic equation in (\[eq:IBPRfunction\]) is a function of the problem matrix $\Am$, the received signal $\yv$, and regularization parameter $\rho_{\text{o}}$ which is the only unknown in (\[eq:IBPRfunction\]. Solving for $G\left(\rho_{\text{o}}\right)= 0$ should lead to the regularization parameter $\rho_{\text{o}}$ that approximately minimizes the MSE of the estimator. Our main interest then is to find a positive root $\rho^{*}_{\text{o}} > 0$ for (\[eq:IBPRfunction\]). In the following section, we study the main properties of this equation and examine the existence and uniqueness of its positive root. Before that, it worth mentioning the following remark A special case of the proposed COPRA approach is when all the singular values are significant and so no truncation is required (full column rank matrix, see Fig. \[fig:sv decay\]). This is the case where $n_{1}=n$, $n_{2}=0$, and $\Sigmam_{n_{1}}=\Sigmam$. Substituting these values in (\[eq:IBPRfunction\]) we obtain $$\begin{aligned} \label{eq:IBPRfunction specail} &\bar{G}\left(\rho_{\text{o}}\right) = \nonumber\\ &\text{Tr}\left( \Sigmam^2 \left(\Sigmam^2 + \rho_{\text{o}} \Id_{n} \right)^{-2} \Um^{T}\yv\yv^{T}\Um \right)\text{Tr}\left( \left(\Sigmam^{2} + \rho_{\text{o}} \Id_{n} \right)^{-1} \right) \nonumber\\ &- \text{Tr}\left(\left(\Sigmam^{2} + \rho_{\text{o}} \Id_{n} \right)^{-2} \Um^{T}\yv\yv^{T}\Um\right)\text{Tr}\left( \Sigmam^{2} \left(\Sigmam^2 + \rho_{\text{o}} \Id_{n} \right)^{-1}\right)\nonumber\\ &= 0.\end{aligned}$$ Analysis of the function $G\left(\rho_{\text{o}}\right)$ {#sec:Properties} ======================================================== In this section, we provide a detailed analysis for the COPRA function $G\left(\rho_{\text{o}}\right)$ in (\[eq:IBPRfunction\]). We start by examining some main properties of $G\left(\rho_{\text{o}}\right)$ that are straightforward to proof. \[p4\] $G \left(\rho_{\text{o}}\right)$ is continuous over the interval $\left(0, +\infty\right)$. \[p44\] $G \left(\rho_{\text{o}}\right)$ has $n$ different discontinuities at $\rho_{\text{o}} = -\sigma_{i}^{2}, \forall i \in [1, n]$. However, these discontinuities are of no interest as far as COPRA is considered. \[p1\] $\lim_{\rho_{\text{o}} \to 0^{+}} G\left(\rho_{\text{o}}\right) = +\infty$. \[p2\] $\lim_{\rho_{\text{o}} \to 0^{-}} G\left(\rho_{\text{o}}\right) = -\infty$. \[p3\] $\lim_{\rho_{\text{o}} \to +\infty} G\left(\rho_{\text{o}}\right) = 0$. Property \[p1\] and \[p2\] show clearly that $G\left(\rho_{\text{o}}\right)$ has a discontinuity at $\rho_{\text{o}} = 0$. \[p5\] Each of the functions $G_{1}\left(\rho_{\text{o}}\right)$ and $G_{2}\left(\rho_{\text{o}}\right)$ in (\[eq:IBPRfunction\]) is completely monotonic in the interval $(0 ,+\infty)$. According to [@feller2008introduction; @widder2015laplace], a function $F\left(\rho_{\text{o}}\right)$ is completely monotonic if it satisfies $$\begin{aligned} \label{eq:completeMonotoneCond} \left(-1\right)^{n} F^{\left( n \right)}\left(\rho_{\text{o}}\right) \geq 0, \ 0 <\rho_{\text{o}} < \infty , \forall n\in \mathbb{N},\end{aligned}$$ where $F^{(n)}\left(\rho_{\text{o}}\right)$ is the $n$’th derivative of $F\left(\rho_{\text{o}}\right)$.\ By continuously differentiating $G_{1}\left(\rho_{\text{o}}\right)$ and $G_{2}\left(\rho_{\text{o}}\right)$, we can easily show that both functions satisfy the monotonic condition in (\[eq:completeMonotoneCond\]). \[th1\] The COPRA function $G\left(\rho_{\text{o}}\right)$ in (\[eq:IBPRfunction\]) has at most two roots in the interval $\left(0, +\infty \right).$ The proof of Theorem \[th1\] will be conducted in two steps. Firstly, it has been proved in [@kammler1976chebyshev; @kammler1979least] that any completely monotonic function can be approximated as a sum of exponential functions. That is, if $F\left(\rho_{\text{o}}\right)$ is a completely monotonic, it can be approximated as $$\begin{aligned} \label{eq:completeMonotoneApprox} F\left(\rho_{\text{o}}\right) \approx \sum_{i=1}^{l} l_i e^{-k_{i} \rho_{\text{o}}},\end{aligned}$$ where $l$ is the number of the terms in the sum and $l_{i}$ and $k_{i}$ are two constants. It is shown that there always exists a best uniform approximation to $F\left(\rho_{\text{o}}\right)$ where the error in this approximation gets smaller as we increase the number of the terms $l$. However, our main concern here is the relation given by (\[eq:completeMonotoneApprox\]) more than finding the best number of the terms or the unknown parameters $l_{i}$ and $k_{i}$. To conclude, both functions $G_{1}\left(\rho_{\text{o}}\right)$ and $G_{2}\left(\rho_{\text{o}}\right)$ in (\[eq:IBPRfunction\]) can be approximated by a sum of exponential functions. Secondly, it is shown in [@shestopaloff2008sums] that the sum of exponential functions has at most two intersections with the abscissa. Consequently, and since the relation in (\[eq:IBPRfunction\]) can be expressed as a sum of exponential functions, the function $G\left({\rho}_{\text{o}}\right)$ has at most two roots in the interval $\left(0,+\infty \right)$. \[th2\] There exists a sufficiently small positive value $\epsilon$, such that $\epsilon \to 0^{+}$ and $\epsilon \ll \sigma_{i}^{2}$, $ \forall i \in [1,n]$ where the value of the COPRA function $G\left(\rho_{\text{o}}\right)$ in (\[eq:IBPRfunction\]) is zero (i.e., $\epsilon$ is a positive root for (\[eq:IBPRfunction\])). However, we are not interested in this root in the proposed COPRA. The proof of Theorem \[th2\] is in Appendix \[Apen A\]. \[th3\] A sufficient condition for the function $G\left(\rho_{\text{o}}\right)$ to approach zero at $\rho_{\text{o}} = +\infty $ from a positive direction is given by $$\label{eq:solutioncondition} n \text{Tr}\left(\Sigmam^{2} \bv\bv^{T}\right) > \text{Tr}\left(\Sigmam_{n_{1}}^{2}\right) \text{Tr}\left(\bv\bv^{T}\right)$$ where $\bv = \Um^{T}\yv.$ Let $\bv = \Um^{T}\yv$ as in (\[eq:IBPRfunction\]). Given that $\Sigmam^{2}$ is a diagonal matrix, $ \Sigmam^{2}= \text{diag}\left(\sigma_{1}^{2}, \sigma_{2}^{2}, \dotsi,\sigma_{n}^{2}\right)$, and from the trace function property, we can replace $\bv \bv^{T} = \Um^{T}\yv\yv^{T}\Um$ in (\[eq:IBPRfunction\]) by a diagonal matrix $\bv\bv^{T}_{d}$ that contains $\bv\bv^{T}$ diagonal entries without affecting the result. By defining $\bv \bv^{T}_{d}= \text{diag}\left(b_{1}^{2}, b_{2}^{2}, \dotsi,b_{n}^{2}\right)$, we can write (\[eq:IBPRfunction\]) as $$\begin{aligned} \label{eq2:MSE B sum 1} &G\left(\rho_{\text{o}}\right) = \frac{\beta}{\rho_{\text{o}}^{4}}\sum_{j=1}^{n}\frac{\sigma_{j}^{2} b_{j}^{2}}{\left(\frac{\sigma_{j}^{2}}{\rho_{\text{o}}}+1\right)^{2}}\sum_{i=1}^{n_{1}}\frac{ \sigma_{i}^{2}}{\left(\frac{\sigma_{i}^{2}}{\rho_{\text{o}}}+1\right)^{2}} \nonumber\\ &+ \frac{1}{\rho_{\text{o}}^{3}}\sum_{j=1}^{n}\frac{\sigma_{j}^{2} b_{j}^{2}}{\left(\frac{\sigma_{j}^{2}}{\rho_{\text{o}}}+1\right)^{2}}\sum_{i=1}^{n_{1}}\frac{1}{\left(\frac{\sigma_{i}^{2}}{\rho_{\text{o}}}+1\right)^{2}} \nonumber\\ &- \frac{\beta}{\rho_{\text{o}}^{4}}\sum_{j=1}^{n}\frac{b_{j}^{2}}{\left(\frac{\sigma_{j}^{2}}{\rho_{\text{o}}}+1\right)^{2}}\sum_{i=1}^{n_{1}}\frac{\sigma_{i}^{4}}{\left(\frac{\sigma_{i}^{2}}{\rho_{\text{o}}}+1\right)^{2}}\nonumber\\ &- \frac{1}{\rho_{\text{o}}^{3}}\sum_{j=1}^{n}\frac{b_{j}^{2}}{\left(\frac{\sigma_{j}^{2}}{\rho_{\text{o}}}+1\right)^{2}}\sum_{i=1}^{n_{1}}\frac{\sigma_{i}^{2}}{\left(\frac{\sigma_{i}^{2}}{\rho_{\text{o}}}+1\right)^{2}} \nonumber\\ &+ \frac{n_{2}}{\rho_{\text{o}}^{3}}\sum_{j=1}^{n}\frac{\sigma_{j}^{2}b_{j}^{2}}{\left(\frac{\sigma_{j}^{2}}{\rho_{\text{o}}}+1\right)^{2}}.\end{aligned}$$ Then, we use some algebraic manipulations to obtain $$\begin{aligned} \label{eq2:MSE B sum 2} &G\left(\rho_{\text{o}}\right) = \frac{1}{\rho_{\text{o}}^{3}}\sum_{j=1}^{n}\frac{\sigma_{j}^{2} b_{j}^{2}}{\left(\frac{\sigma_{j}^{2}}{\rho_{\text{o}}}+1\right)^{2}} \Bigg[ \frac{\beta}{\rho_{\text{o}}}\sum_{i=1}^{n_{1}}\frac{\sigma_{i}^{2}}{\left(\frac{\sigma_{i}^{2}}{\rho_{\text{o}}}+1\right)^{2}} \nonumber\\ &+ \sum_{i=1}^{n_{1}}\frac{1}{\left(\frac{\sigma_{i}^{2}}{\rho_{\text{o}}}+1\right)^{2}} + n_{2} \Bigg]- \frac{1}{\rho_{\text{o}}^{3}}\sum_{j=1}^{n}\frac{b_{j}^{2}}{\left(\frac{\sigma_{j}^{2}}{\rho_{\text{o}}}+1\right)^{2}} \times \nonumber\\ &\Bigg[ \frac{\beta}{\rho_{\text{o}}}\sum_{i=1}^{n_{1}}\frac{\sigma_{i}^{4}}{\left(\frac{\sigma_{i}^{2}}{\rho_{\text{o}}}+1\right)^{2}} + \sum_{i=1}^{n_{1}}\frac{\sigma_{i}^{2}}{\left(\frac{\sigma_{i}^{2}}{\rho_{\text{o}}}+1\right)^{2}}\Bigg].\end{aligned}$$ Now, evaluating the limit of (\[eq2:MSE B sum 2\]) as $\rho_{\text{o}} \to +\infty$ we obtain $$\begin{aligned} \label{eq2:MSE B sum 3} \lim_{\rho_{\text{o}} \to +\infty} G\left(\rho_{\text{o}}\right) &= \left(\lim_{\rho_{\text{o}} \to +\infty} \frac{1}{\rho_{\text{o}}^{3}}\right) \nonumber\\ &\times \Bigg[\sum_{j=1}^{n}\sigma_{j}^{2} b_{j}^{2} \Big( \tau \beta\sum_{i=1}^{n_{1}}\sigma_{i}^{2} + \sum_{i=1}^{n_{1}}1 + n_{2} \Big)\nonumber\\ &- \sum_{j=1}^{n}b_{j}^{2} \Big( \tau \beta\sum_{i=1}^{n_{1}}\sigma_{i}^{4} +\sum_{i=1}^{n_{1}}\sigma_{i}^{2}\Big)\Bigg],\end{aligned}$$ where $\tau = \lim_{\rho_{\text{o}} \to +\infty} \frac{1}{\rho_{\text{o}}}$. The relation in (\[eq2:MSE B sum 3\]) can be simplified to $$\begin{aligned} \label{eq2:MSE B sum 4} &\lim_{\rho_{\text{o}} \to +\infty} G\left(\rho_{\text{o}}\right) = \left(\lim_{\rho_{\text{o}} \to +\infty} \frac{1}{\rho_{\text{o}}^{3}}\right) \nonumber\\ &\times \Bigg[\sum_{j=1}^{n}\sigma_{j}^{2} b_{j}^{2} \Big( \tau \beta\sum_{i=1}^{n_{1}}\sigma_{i}^{2} + n \Big)- \sum_{j=1}^{n}b_{j}^{2} \Big( \tau \beta\sum_{i=1}^{n_{1}}\sigma_{i}^{4} +\sum_{i=1}^{n_{1}}\sigma_{i}^{2}\Big)\Bigg]\end{aligned}$$ It is obvious that the limit in (\[eq2:MSE B sum 4\]) is zero. However, the direction where the limit approaches zero depends on the sign of the term between the square brackets. For $G\left(\rho_{\text{o}}\right)$ to approach zero from the positive direction, knowing that the terms that are independent of $\tau$ are the dominants, the following condition must hold: $$\label{eq2:MSE B sum 5} n \left(\sum_{j=1}^{n} \sigma_{j}^2 b_{j}^{2} \right) > \left(\sum_{i=1}^{n_{1}} \sigma_{i}^2 \right) \left(\sum_{j=1}^{n} b_{j}^{2}\right).$$ Which is the same as (\[eq:solutioncondition\]). \[theo2:th4\] If (\[eq:solutioncondition\]) is satisfied, then $G\left(\rho_{\text{o}}\right)$ has a unique positive root in the interval $\left(\epsilon, +\infty \right)$. According to Theorem \[th1\], the function $G\left(\rho_{\text{o}}\right)$ can have no root, one, or two roots. We have already proved in Theorem \[th2\] that there exists a significantly small positive root for the COPRA function at $\rho_{\text{o,1}} =\epsilon$ but we are not interested in this root. In other words, we would like to see if there exists a second root for $G\left(\rho_{\text{o}}\right)$ in the interval $\left(\epsilon, +\infty\right)$. From Property \[p1\] and Theorem \[th2\], we can conclude that the COPRA function has a positive value before $\epsilon$, then it switches to the negative region after that. The condition in (\[eq:solutioncondition\]) guarantees that $G\left(\rho_{\text{o}}\right)$ approaches zero from a positive direction as $\rho_{\text{o}}$ approaches $+\infty$. This means that $G\left(\rho_{\text{o}}\right)$ has an extremum in the interval $\left(\epsilon, +\infty\right)$, and this extremum is actually a minimum point. If the point of the extremum is considered to be $\rho_{\text{o,m}}$, then the function starts increasing for $\rho_{\text{o}} > \rho_{\text{o,m}}$ until it approaches the second zero crossing at $\rho_{\text{o},2}$. Since Theorem \[th1\] states clearly that we cannot have more than two roots, we conclude that when (\[eq:solutioncondition\]) holds, we have only one unique positive root over the interval $\left(\epsilon, +\infty\right)$. Finding the Root of $G\left(\rho_{\text{o}}\right)$ {#subsec:find root} --------------------------------------------------- To find the positive root of the COPRA function $G\left(\rho_{\text{o}}\right)$ in (\[eq:IBPRfunction\]), Newton’s method [@zarowski2004introduction] can be used. The function $G\left(\rho_{\text{o}}\right)$ is differentiable in the interval $\left(\epsilon, +\infty\right)$ and the expression of the first derivative $G^{'}\left(\rho_{\text{o}}\right)$ can be easily obtained. Newton’s method can then be applied in a straightforward manner to find this root. Starting from an initial value $\rho_{\text{o}}^{n=0} > \epsilon $ that is sufficiency small, the following iterations are performed: $$\label{eq:Newton} \rho_{\text{o}}^{n+1} = \rho_{\text{o}}^{n} - \frac{G\left(\rho_{\text{o}}^{n}\right)}{G^{'}\left(\rho_{\text{o}}^{n}\right)}.$$ The iterations stop when $|G\left(\rho_{\text{o}}^{n+1}\right)|< \bar{\xi}$, where $\bar{\xi}$ is a sufficiently small positive quantity. Convergence {#subsec:converge} ----------- When condition (\[eq:solutioncondition\]) is satisfied, the convergence of Newton’s method can be easily proved. As a result from Theorem \[th3\], the function $G\left(\rho_{\text{o}}\right)$ has always a negative value in the interval $\left(\epsilon, \rho_{o,2}\right)$. It is also clear that $G\left(\rho_{\text{o}}\right)$ is an increasing function in the interval $[\rho_{\text{o}}^{n= 0}, \rho_{\text{o},2}]$. Thus, starting from $\rho_{\text{o}}^{n= 0}$, (\[eq:Newton\]) will produce a consecutively increasing estimate for $\rho_{\text{o}}$. Convergence occurs when $G\left(\rho_{\text{o}}^{n}\right) \rightarrow 0$ and $\rho_{\text{o}}^{n+1}\rightarrow \rho_{\text{o}}^{n}$. When the condition in (\[eq:solutioncondition\]) is not satisfied, the regularization parameter should be set to $\epsilon$. COPRA Summary {#subsec2:I-BPR summery} ------------- The proposed COPRA discussed in the previous sections is summarized in Algorithm \[COPRA ALGORITHM\]. $\rho_{\text{o}} = \epsilon$. Define $\tilde{\xi}$ as the iterations stopping criterion. Set $\rho_{\text{o}}^{n =0}$ to a sufficiently small positive quantity. Find [$G\left(\rho_{\text{o}}^{n=0}\right)$]{} using (\[eq:IBPRfunction\]), and compute its derivative [$G^{'}\left(\rho_{\text{o}}^{n=0}\right)$.]{} Solve (\[eq:Newton\]) to get $\rho_{\text{o}}^{n+1}$. $\rho_{\text{o}}^{n} =\rho_{\text{o}}^{n+1}$. Find $\hat{\xv}$ using (\[eq:copra solution\]). \[COPRA ALGORITHM\] Numerical Results {#sec:Results} ================= In this section, we perform a comprehensive set of simulations to examine the performance of the proposed COPRA and compare it with benchmark regularization methods. Three different scenarios of simulation experiments are performed. Firstly, the proposed COPRA is applied to a set of nine real-world discrete ill-posed problems that are commonly used in testing the performance of regularization methods in discrete ill-posed problems. Secondly, COPRA is used to estimate the signal when $\Am$ is a random rank deficient matrix generated as $$\Am = \frac{1}{n}\Bm \Bm^{T},$$ where $\Bm \left(m \times n, m > n\right)$ is a random matrix with i.i.d. zero mean unit variance Gaussian random entries. This is a theoretical test example which is meant to illustrate the robustness of COPRA and to make sure that the obtained results are applicable for broad class of matrices. Finally, an image restoration in image tomography ill-posed problem is considered[^10]. Real-World Discrete Ill-posed Problems -------------------------------------- The Matlab regularization toolbox [@hansen1994regularization] is used to generate pairs of a matrix $\Am \in \mathbb{R}^{50 \times 50}$ and a signal $\xv_{0}$. The toolbox provides many real-world discrete ill-posed problems that can be used to test the performance of regularization methods. The problems are derived from discretization of Fredholm integral equation as in (\[eq:kernal equation\]) and they arise in many signal processing applications[^11]. *Experiment setup*: The performance of COPRA is compared with three benchmark regularization methods, the quasi-optimal, the GCV, the L-curve in addition to the LS. The performance is evaluated in terms of normalized MSE (NMSE); that is the MSE normalized by $\lVert \xv_{0} \lVert_{2}^{2}$. Noise is added to the vector $\Am \xv_{0}$ according to a certain signal-to-noise-ratio (SNR) defined as SNR $\triangleq \lVert \Am\xv_{0}\rVert_2^{2}/n \sigma_{\zv}^{2}$ to generate $\yv$. The performance is presented as the NMSE (in dB) (NMSE in dB = $10\log_{10}$ $\left( \text{NMSE}\right)$) versus SNR (in dB) and is evaluated over $10^{5}$ different noise realizations at each SNR value. Since some regularization methods provide a high unreliable NMSE results that hinder the good visualization of the NMSE, we set different *upper thresholds* for the vertical axis in the results sub-figures. Fig. \[fig:ill-posed performace\] shows the results for all the selected 9 problems. Each sub-figure quote the condition number (CN) of the problem’s matrix. The NMSE curve for some methods disappears in certain cases. This indicates extremely poor performance for these methods such that they are out of scale. For example, LS does not show up in all the tested scenarios, while the other benchmark methods disappear in quite a few cases. Generally speaking, It can be said that an estimator offering NMSE above 0 dB is not robust and is worthless. From Fig. \[fig:ill-posed performace\], it is clear that COPRA offers the highest level of robustness among all the methods as it is the only approach whose NMSE performance remains below 0 dB in almost all cases. Comparing the NMSE over the range of the SNR values in each problem, we find *on average* that COPRA exhibits the lowest NMSE amongst all the methods in 8 problems (the first 8 sub-figures). Considering all the problems, the closest contender to COPRA is the quasi method. However, this method and the remaining methods show lack of robustness in certain situations as evident by the extremely high NMSE. In Fig. \[fig:bounds\], we provide the NMSE for the approximation of the perturbation bound expression in (\[eq:eta min 2\]) by (\[eq:eta min 3\]) for a selected ill-posed problem matrices. The two expressions are evaluated at each SNR using the suboptimal regularizer in (\[eq:gamma min approx\]). The sub-figures show that the NMSE of the approximation is extremely small (below -20 dB in most cases) and that the error increases as the SNR increases. The increase of the approximation error with the SNR is discussed in Appendix \[Apen error\]. Rank Deficient Matrices ----------------------- In this scenario, a rank deficient random matrix $\Am$ is considered. This is the case where $\lVert \Sigmam_{n_{2}} \rVert = 0$ in (\[eq:sigma parti\]). This theoretical test is meant to illustrate the robustness of COPRA. *Experiment setup*: The matrix $\Am$ is generated as a random matrix that satisfies $\Am = \frac{1}{50}\Bm \Bm^{T}$, where $ \Bm \in \mathbb{R}^{50\times45}, B_{ij} \sim \mathcal{N}\left(0, 1\right)$. The elements of $\xv_{0}$ are chosen to be Gaussian i.i.d. with zero mean unit variance, and i.i.d. with uniform distribution in the interval $(0, 1)$. Results are obtained as an average over $10^{5}$ different realizations of $\Am$, $\xv_{0}$, and $\zv$. From Fig. \[fig:rank deficinet\](a), we observe that when the elements of $\xv_{0}$ are Gaussian i.i.d. COPRA outperforms all the benchmark regularization methods. In fact, COPRA is the only approach that provides a NMSE below 0 dB overall the SNR range while other algorithms are providing a very high NMSE. The same behavior can be observed when the elements of $\xv_{0}$ are uniformly distributed as Fig. \[fig:rank deficinet\](b) shows. Finally, the performance of the LS is above 250 dB for both cases. Image Restoration ----------------- The tomo example in Section \[sec:Results\]-A and Fig. \[fig:tomonmse\] discusses the NMSE of the tomography inverses problem solution. In this subsection, we present visual results for the restored images. *Experiment setup*: The elements of $\Am \xv_{0} $ are a representative of a line integrals along direct rays that penetrate a rectangular field. This field is discretized into $n^{2}$ cells, and each cell with its own intensity is stored as an element in the image matrix $\Mm$. Then, the columns of $\Mm$ are stacked into $\xv_{0}$. On the other hand, the entries of $\Am$ are generated as $$a_{ij} = \begin{cases} &l_{ij}, \ \ \text{pixel}_{j} \in \text{ray}_{i} \\ &0 \ \hspace{10pt} \text{else}, \end{cases} \nonumber$$ where $l_{ij}$ is the length of the $i$’th ray in pixel $j$. Finally, the rays are placed randomly. A noise with SNR equal to 30 dB is added to the image of size $16 \times 16$ and the performance is evaluated as an average over $10^{6}$ noise and $\Am$ realizations. In Fig. \[fig:image tomo results\], we present the original image, the received image, and the performance of the methods. Fig. \[fig:image tomo results\] demonstrates that COPRA outperforms all methods through providing a clear image that is very close to the original image. This also appear when we compare the peak signal-to-noise ratio (PSNR) of the algorithms as in Table. \[tab:psnr\] with COPRA having the largest PSNR among them. Moreover, algorithm such GCV provides an unreliable result, while L-curve and quasi fail to restore the internal parts clearly, especially those who have colors close to the image background color. Method COPRA L-curve GCV Quasi -------- ------------- --------- --------- --------- PSNR **29.8331** 13.6469 10.5410 15.9080 \[tab:psnr\] Average Runtime --------------- In Fig. \[fig:run time\], we plot the average runtime for each method against the SNR as calculated in the simulation. The figure is a good representation for the runtime of all the problems (no significant runtime variation between problems has been seen). The figure shows that COPRA is the fastest algorithm as it has the lowest runtime in compare to all benchmark methods. Conclusion {#sec:conclusion} ========== In this work, we developed a new approach to find the regularization parameter for linear discrete ill-posed problems. Due to the challenging singular-value structure for such problems, many regularization approaches fail to provide a good stabilize solution. In the proposed approach, the singular-value structure of the model matrix is modified by allowing an artificial perturbation into it. To maintain the fidelity of the model, an upper bound constraint on the perturbation is allowed. The proposed approach minimizes the worst-case residual error of the estimator and selects the perturbation bound in a way that approximately minimizes the MSE. As a result, the approach combines the simplicity of the least-squares criterion with the robustness of the MSE based estimation. The regularization parameter is obtained as a solution of a non-linear equation in one unknown variable. Simulation results demonstrate that the proposed approach outperforms a set of benchmark regularization methods. Error Analysis {#Apen error} ============== In this appendix, we analyze the error of the approximation that is made to obtain (\[eq:gamma min approx\]). To simpify the analysis, we will consider the case when the approximation is applied directly to the MSE function in (\[eq:MSE2\]). Let us start by defining $\Hm\triangleq {\Sigmam}^{2k} \left({\Sigmam}^2 + \rho \Id_n \right)^{-p}$, whose diagonal entries can be written as $$\label{eq:diagonal} h_{ii} = \frac{\sigma_i^{2k}}{\left(\sigma_i^2+\rho\right)^p} ; \ i=1,2,\cdots, n.$$ Note that in our case $k=0$ and $p=2$ for the diagonal matrix inside the trace function of the second term in (\[eq:MSE2\]). However, we will use these two variables to obtain the error expression for the general case, then we will substitute for $k$ and $p$. By using the inequalities in \[[@wang1986trace], eq.(5)\], we obtain $$\label{eq:eneq 1} \lambda_\text{min}(\Rm_{\xv_{0}}) \text{Tr}\left( \Hm \right) \leq \text{Tr}\left( \Hm \Vm^T\Rm_{\xv_{0}}\Vm \right) \leq \lambda_\text{max}(\Rm_{\xv_{0}}) \text{Tr}\left(\Hm \right).$$ Similarly, we can write $$\label{eq:eneq 2} \lambda_\text{min}\left(\Hm\right) \text{Tr}\left( \Rm_{\xv_{0}} \right) \leq \text{Tr}\left( \Hm \Vm^T\Rm_{\xv_{0}}\Vm \right) \leq \lambda_\text{max}\left(\Hm\right) \text{Tr}\left(\Rm_{\xv_{0}} \right).$$ Since $\Hm$ is diagonal, $\lambda_\text{min}\left(\Hm\right) = \text{min}\left(\text{diag}(\Hm)\right)$ and $\lambda_\text{max}\left(\Hm\right) = \text{max}\left(\text{diag}(\Hm)\right)$. Now, let us define the normalized error of the approximation as $$\label{eq:error def} \varepsilon = \frac{\text{Tr}\left( \Hm \Vm^T\Rm_{\xv_{0}}\Vm \right) - \frac{1}{n}\text{Tr}\left( \Hm \right) \text{Tr}\left( \Rm_{\xv_{0}} \right) } {\frac{1}{n}\text{Tr}\left( \Hm \right) \text{Tr}\left( \Rm_{\xv_{0}} \right)}.$$ Note that this is not the standard way of defining the normalized error. Typically, the error $\varepsilon$ is normalized by the true quantity, i.e., $\text{Tr}\left( \Hm \Vm^T\Rm_{\xv_{0}}\Vm \right)$. However, this way of defining the error is found to be more useful in carrying out the following error analysis. Based on (\[eq:error def\]), we see that $|\varepsilon| \geq 1$ indicates an inaccurate approximation. Although at the end it depends totally on the application, we will adopt $|\varepsilon| < 1$ as the reference for evaluating the accuracy of the approximation. In fact, we can observe from (\[eq:error def\]) that $|\varepsilon| = 1$ indicates that $\frac{1}{n}\text{Tr}\left( \Hm \right) \text{Tr}\left( \Rm_{\xv_{0}} \right) = 0.5 \ \text{Tr}\left( \Hm \Vm^T\Rm_{\xv_{0}}\Vm \right) $. To this end, we will derive two bounds based on and . Then, we will combine them to obtain the final error bound. *Absolute Error Bound Based on* : Subtracting $\frac{1}{n}\text{Tr}\left( \Hm \right) \text{Tr}\left( \Rm_{\xv_{0}} \right)$ from and dividing by the same quantity, we obtain $$\label{eq:eneq 1 bound} \frac{\lambda_\text{min}(\Rm_{\xv_{0}})}{\lambda_\text{avg}(\Rm_{\xv_{0}})}-1 \leq \varepsilon \leq \frac{\lambda_\text{max}(\Rm_{\xv_{0}})}{\lambda_\text{avg}(\Rm_{\xv_{0}})}-1,$$ where $\lambda_\text{avg}(\Rm_{\xv_{0}}) \triangleq \frac{1}{n} \text{Tr}\left( \Rm_{\xv_{0}}\right)$. Thus, $|\varepsilon|$ can be bounded by a positive quantity according to $$\label{eq:eneq 1 abs bound} |\varepsilon_{x}| \leq \mu_x = \max \left[ 1 - \frac{ \lambda_\text{min}(\Rm_{\xv_{0}}) }{\lambda_\text{avg}(\Rm_{\xv_{0}})}, \frac{ \lambda_\text{max}(\Rm_{\xv_{0}}) }{\lambda_\text{avg}(\Rm_{\xv_{0}})} -1 \right].$$ *Absolute Error Bound Based on* : Starting from , and by applying the same approach used to obtain , we derive the second bound as $$\label{eq:eneq 2 abs bound 0} |\varepsilon_{a}| \leq \mu_a = \max \left[ 1 - \frac{ \lambda_\text{min}(\Hm) }{\lambda_\text{avg}(\Hm)}, \frac{ \lambda_\text{max}(\Hm) }{\lambda_\text{avg}(\Hm)} -1 \right].$$ By using we can transform to $$\begin{aligned} \label{eq:eneq 2 abs bound} |\varepsilon_{a}| \leq \mu_a & = \max \left[1 - \frac{ \underset{i}{\min}\left[ \frac{\sigma_i^{2k}}{(\sigma_i^2 +\rho)^p}\right] } { \frac{1}{n}\sum_{i=1}^{n} \frac{\sigma_i^{2k}}{(\sigma_i^2 +\rho)^p} }, \frac{ \underset{i}{\max}\left[ \frac{\sigma_i^{2k}}{(\sigma_i^2 +\rho)^p}\right]} { \frac{1}{n}\sum_{i=1}^{n} \frac{\sigma_i^{2k}}{(\sigma_i^2 +\rho)^p} } -1 \right]\nonumber\\ & i=1, 2,\cdots, n. \end{aligned}$$ Note that the bound $\mu_x$ depends only on $\Rm_{\xv_{0}}$, while $\mu_a$ depends both the singular values of $\Am$ and the unknown regularizaer $\rho$. As in our case, $k=0$ and $p=2$, and therefore, (\[eq:eneq 2 abs bound\]) can be simplified to $$\begin{aligned} \label{eq:eneq 2 abs bound k0} |\varepsilon_{a}| \leq \mu_{a} & = \max \left[1 - \frac{ \frac{1}{(\sigma_1^2 +\rho)^2} } { \frac{1}{n}\sum_{i=1}^{n} \frac{1}{(\sigma_i^2 +\rho)^2} }, \frac{ \frac{1}{(\sigma_n^2 +\rho)^2}} { \frac{1}{n}\sum_{i=1}^{n} \frac{1}{(\sigma_i^2 +\rho)^2} } -1 \right] \end{aligned}$$ *Combined Bound*: By combining and , we obtain the final bound on the absolute error as $$\label{eq:eneq combined bound} |\varepsilon| \leq \mu = \min \left( \mu_x, \mu_a \right ).$$ From , , and , we notice that the bound is a minimum of two independent bounds. We will analyze each bound separately and then we will derive a conclusion concerning the overall error bound. Analysis of $\mu_x$ ------------------- When $\xv_{0}$ is deterministic, $\lambda_\text{min}(\Rm_{\xv_{0}}) = 0$, $\lambda_\text{max}(\Rm_{\xv_{0}}) = || \xv_{0} ||_{2}^{2}$ and $\lambda_\text{avg}(\Rm_{\xv_{0}}) = \frac{1}{n}|| \xv_{0} ||_{2}^{2}$. By substituting in , we obtain $$\label{eq:eneq 1 abs bound deterministic} \mu_x = \max \left[1, n-1\right] = n-1.$$ On the other hand, when $\xv_{0}$ is stochastic with i.i.d. elements, $\lambda_\text{min}(\Rm_{\xv_{0}})= \lambda_\text{avg}(\Rm_{\xv_{0}})= \lambda_\text{min}(\Rm_{\xv_{0}}) =\sigma_{\xv_{0}}^{2}$, and as a result $$\label{eq:eneq 1 abs bound iid} \mu_x = \max \left[0, 0\right] =0,$$ which means based on (\[eq:eneq combined bound\]) that the approximation is exact regardless of the contribution of the error from $\mu_a$. When $\xv_{0}$ deviate from being i.i.d., it will be very difficult to obtain a value for $\mu_x$. Therefore, and since no previous knowledge about $\xv_{0}$ is assumed in this paper, it seems that this bound is very loose for a general $\xv_{0}$ and we should rather rely on $\mu_a$ to tighten and evaluate the bound of the error as we will discuss. Thus, the modified bound, which can be larger than the actual bound, is given by $$\label{eq:eneq combined bound modified} |\varepsilon| \leq \mu = \mu_a.$$ Analysis of $\mu_a$ ------------------- By taking the derivative of each of the two terms inside (\[eq:eneq 2 abs bound k0\]) w.r.t. $\rho$, we can easily prove that the two functions are decreasing in $\rho$. This means that we can obtain the two extreme error bounds (the largest and the smallest possible value of the absolute error) by analyzing the two extreme SNR scenarios, i.e., the high SNR regime and the low SNR regime. ### Analysis for the low SNR regime In the extreme low SNR regime, we have $\rho\rightarrow \infty$, and therefore, we can obtain the minimum bound on the absolute error. Based on we can write $$\label{eq:D diag gammaInf} h_{ii} = \frac{1}{(\sigma_i^2 + \rho)^2}\rightarrow \frac{1}{\rho^2 } ; \ i=1,2,\cdots, n.$$ Consequently, will boil down after some manipulations to $$\begin{aligned} \label{eq:eneq 2 abs bound exp large gamma 2} |\varepsilon_{a}^{l}| \leq \mu_{a}^{l} & = \max \left[1 - \frac{ \frac{1}{\rho^{2}} } { \frac{1}{n}\sum_{i=1}^{n} \frac{1}{\rho^{2}} }, \frac{ \frac{1}{\rho^{2}}} { \frac{1}{n}\sum_{i=1}^{n} \frac{1}{\rho^{2}} } -1 \right]\nonumber\\ & = \max \left[1 - \frac{n} { \sum_{i=1}^{n}1 }, \frac{ n} { \sum_{i=1}^{n} 1 } -1 \right] = 0.\end{aligned}$$ The result in (\[eq:eneq 2 abs bound exp large gamma 2\]) indicates that as the SNR decreases, the approximation becomes more accurate. In the extreme low SNR regime (i.e., $\rho \to \infty$), the absolute error is zero and the approximated term is exactly equal to the original one. ### Analysis for the high SNR regime At the extremely high SNR, $\rho$ is sufficiently small and thus doing such analysis allows us to obtain the upper worst case possible value for $\varepsilon$. Since the main objective of this paper is to provide an estimator that minimizes the MSE, and based on the result proven in [@hoerl1970ridge], there always exists a positive regularizer $\rho > 0$, such that the regularized estimator offers lower MSE than the OLS estimator. This implies also for well-conditioned problems. However, if the condition number is too small, both the regularization parameter and the corresponding improvement in the MSE performance comparing to that of the OLS are too small. Therefore, we conclude that for the extreme high SNR, $\rho$ converges to a minimum value $\rho_{\text{min}}$. In what follow, we find a lower bound expression for $\rho_{\text{min}}$ and then examine the absolute error in this value. Starting from the definition of the SNR, we can write $$\label{eq:SNR} \text{SNR} = \frac{|| \Am \xv_{0} ||_{2}^{2}}{|| \zv ||_{2}^{2}}.$$ Applying the SVD of $\Am$ to (\[eq:SNR\]) then doing some algebraic manipulations, we obtain $$\label{eq:SNR2} \text{SNR} = \frac{\text{Tr}\left(\Vm \Sigmam^{2} \Vm^{T} \Rm_{\xv_{0}}\right)}{n \sigma_{\zv}^{2}},$$ where $\Rm_{\xv_{0}}= \xv_{0} \xv_{0}^{T}$. Now, using (\[eq:SNR2\]) with the suboptimal regularizer $\rho_{\text{o}}$ expression in (\[eq:gamma min approx\]), we can write $$\label{eq:regularizer based SNR} \rho_{\text{o}} = \frac{1}{\text{SNR}} \frac{\text{Tr}\left(\Vm \Sigmam^{2} \Vm^{T} \Rm_{\xv_{0}}\right)}{\text{Tr}\left(\Rm_{\xv_{0}}\right)}.$$ From (\[eq:regularizer based SNR\]) we deduce that for a given $\Am$ and $\xv_{0}$, the minimum achievable suboptimal regularizer $\rho_{\text{min}}$ dependents on the maximum SNR (i.e., $\text{SNR}_{\text{max}}$). That is $$\label{eq:regularizer based SNR2} \rho_{\text{min}} = \frac{1}{\text{SNR}_{\text{max}}} \frac{\text{Tr}\left(\Vm \Sigmam^{2} \Vm^{T} \Rm_{\xv_{0}}\right)}{\text{Tr}\left(\Rm_{\xv_{0}}\right)}.$$ Give the nature of the ill-posed problems and there singular values behavior as Fig \[fig:sv decay\] shows, we will partition $\Sigmam$ and $\Vm$ into to two sub-matrices (same as in Section \[choosing optimal bound\]) and then approximate (\[eq:regularizer based SNR2\]) as $$\label{eq:regularizer based SNR3} \rho_{\text{min}} \approx \frac{1}{\text{SNR}_{\text{max}}} \frac{\text{Tr}\left(\Vm_{n_{1}} \Sigmam_{n_{1}}^{2} \Vm_{n_{1}}^{T} \Rm_{\xv_{0}}\right)}{\text{Tr}\left(\Rm_{\xv_{0}}\right)},$$ where $\Sigmam_{n_{1}}^{2} = \text{diag}\left({\sigma}_{1}^{2}, \dots, {\sigma}_{n_{1}}^{2} \right)$. The value of $\rho_{\text{min}}$ in (\[eq:regularizer based SNR3\]) can be bounded by $$\label{eq:regularizer based SNR3 bounds} \frac{{\sigma}_{n_{1}}^{2}}{\text{SNR}_{\text{max}}} \frac{\text{Tr}\left( \Vm_{n_{1}}^{T} \Rm_{\xv_{0}} \Vm_{n_{1}}\right)}{\text{Tr}\left(\Rm_{\xv_{0}}\right)} \leq \rho_{\text{min}} \leq \frac{{\sigma}_{1}^{2}}{\text{SNR}_{\text{max}}} \frac{\text{Tr}\left( \Vm_{n_{1}}^{T} \Rm_{\xv_{0}} \Vm_{n_{1}}\right)}{\text{Tr}\left(\Rm_{\xv_{0}}\right)}$$ Since we are considering the worst case upper bound for the absolute error, and given that this absolute error increases as $\rho$ decreases, we will consider the lower bound of $\rho_{\text{min}}$ as in (\[eq:regularizer based SNR3 bounds\]). Moreover, and based on the unitary matrix property and the partitioning of $\Vm$, we can obtain a lower bound for the lower bound in (\[eq:regularizer based SNR3 bounds\]) as $$\label{eq:regularizer based SNR3 bounds 2} \rho_{\text{min}} \geq \frac{{\sigma}_{n_{1}}^{2}}{\text{SNR}_{\text{max}}} \frac{\text{Tr}\left( \Vm_{n_{1}}^{T} \Rm_{\xv_{0}} \Vm_{n_{1}}\right)}{\text{Tr}\left(\Rm_{\xv_{0}}\right)} \geq \frac{{\sigma}_{n_{1}}^{2}}{\text{SNR}_{\text{max}}} \frac{\text{Tr}\left(\Rm_{\xv_{0}} \right)}{\text{Tr}\left(\Rm_{\xv_{0}}\right)}.$$ Thus, a lower bound for $\rho_{\text{min}}$ (i.e., $\rho_{\text{min}}^{l}$) can be written as $$\label{eq:regularizer based SNR3 bounds 3} \rho_{\text{min}}^{l} = \frac{{\sigma}_{n_{1}}^{2}}{\text{SNR}_{\text{max}}}.$$ Now, we are ready to study the behavior of the error in the high SNR regime. When $\rho \to \rho_{\text{min}}^{l}$, (\[eq:eneq 2 abs bound k0\]) can be written as $$\begin{aligned} \label{eq:eneq 2 abs bound exp small gamma} |\varepsilon_{a}^{h}| \leq \max \left[1 - \frac{ \frac{1}{(\sigma_1^2 +\rho_{\text{min}}^{l})^2} } { \frac{1}{n}\sum_{i=0}^{n-1} \frac{1}{(\sigma_i^2 +\rho_{\text{min}}^{l})^2} }, \frac{ \frac{1}{(\sigma_{n}^2 +\rho_{\text{min}}^{l})^2}} { \frac{1}{n}\sum_{i=0}^{n-1} \frac{1}{(\sigma_i^2 +\rho_{\text{min}}^{l})^2} } -1 \right] \end{aligned}$$ To evaluate (\[eq:eneq 2 abs bound exp small gamma\]), we will relay on numerical results. Firstly, let us consider $\text{SNR}_{\text{max}}=40$ dB which is a realistic upper value in many signal processing and communication applications. Substituting this value in (\[eq:regularizer based SNR3 bounds 3\]) we find that $\rho_{\text{min}}^{l}= 0.018 {\sigma}_{n_{1}}^{2}$. Now, substituting the result in (\[eq:eneq 2 abs bound exp small gamma\]), then evaluating the expression for the 9 ill-posed problems in Section \[sec:Results\], we find that $|\varepsilon_{a}^{h}| \leq \mu_{a}^{h} \approx 1$. As this represents the worst case upper value for the absolute error, we conclude that $$\label{eq:fd} |\varepsilon_{a}^{h}| \leq \varrho; \ \ \ \varrho < 1.$$ Finally, based on (\[eq:eneq combined bound\]) and (\[eq:eneq combined bound modified\]), and by combining (\[eq:eneq 2 abs bound exp large gamma 2\]) and (\[eq:fd\]), we conclude that $$\label{eq:finall error} |\varepsilon|\in [0, \varrho]; \ \ \ \varrho <1.$$ The conclusion in (\[eq:finall error\]) indicates that $\frac{1}{n}\text{Tr}\left( \Hm \right) \text{Tr}\left( \Rm_{\xv_{0}} \right) = q \ \text{Tr}\left( \Hm \Vm^T\Rm_{\xv_{0}}\Vm \right) $ where $q \in (0.5, 1]$. Proof of theorem \[th2\] {#Apen A} ======================== We are interested in studying the behavior of $\lim_{\rho_{\text{o}} \to \epsilon} G\left(\rho_{\text{o}}\right)$, assuming that $ \epsilon $ is sufficiently small positive number (i.e., $ \epsilon \to 0^{+} $), and $ \epsilon \ll \sigma_{i}^{2}$, $\forall i \in [1,n]$. Starting from COPRA function in (\[eq:IBPRfunction\]), and by defining $\bv = \Um^{T}\yv$, we can write $$\begin{aligned} \label{eqC:MSE sum 1} &G\left(\rho_{\text{o}}\right) = \sum_{i=1}^{n}\frac{\sigma_{i}^{2} b_{i}^{2}}{\left(\sigma_{i}^{2}+\rho_{\text{o}}\right)^{2}}\sum_{j=1}^{n_{1}}\frac{\left(\beta \sigma_{j}^{2}+\rho_{\text{o}}\right)}{\left(\sigma_{j}^{2}+\rho_{\text{o}}\right)^{2}} \nonumber\\ &- \sum_{i=1}^{n}\frac{b_{i}^{2}}{\left(\sigma_{i}^{2}+\rho_{\text{o}}\right)^{2}}\sum_{j=1}^{n_{1}}\frac{\sigma_{j}^{2}\left(\beta\sigma_{j}^{2}+\rho_{\text{o}}\right)}{\left(\sigma_{j}^{2}+\rho_{\text{o}}\right)^{2}} + \frac{n_{2}}{\rho_{\text{o}}}\sum_{i=1}^{n}\frac{\sigma_{i}^{2}b_{i}^{2}}{\left(\sigma_{i}^{2}+\rho_{\text{o}}\right)^{2}}.\end{aligned}$$ Given how we choose $\epsilon$, Eq. (\[eqC:MSE sum 1\]) can be approximated as $$\begin{aligned} \label{eqC:MSE sum 2} G\left(\epsilon\right) &\approx \beta \sum_{i=1}^{n}\sigma_{i}^{-2} b_{i}^{2} \sum_{j=1}^{n_{1}} \sigma_{j}^{-2} - \beta n_{1} \sum_{i=1}^{n}\sigma^{-4}_{i} b_{i}^{2} +\frac{n_{2}}{\epsilon} \sum_{i=1}^{n}\sigma_{i}^{-2} b_{i}^{2}.\end{aligned}$$ Solving $G \left(\epsilon\right) = 0 $ from (\[eqC:MSE sum 2\]) leads to the following root $$\label{eqC:MSE sum 3} \epsilon = \frac{•n_{2} \sum_{i=1}^{n}\sigma_{i}^{-2} b_{i}^{2}}{\beta n_{1} \sum_{i=1}^{n}\sigma^{-4}_{i} b_{i}^{2} - \beta \sum_{i=1}^{n}\sigma_{i}^{-2} b_{i}^{2} \sum_{j=1}^{n_{1}} \sigma_{j}^{-2}}.$$ Now, we would like to know if the root define by (\[eqC:MSE sum 3\]) is positive. For (\[eqC:MSE sum 3\]) to be positive, the following relation should hold $$\label{eqC:MSE sum 4} n_{1} \sum_{i=1}^{n}\sigma_{i}^{-4} b_{i}^{2} \geq \sum_{i=1}^{n}\sigma_{i}^{-2} b_{i}^{2} \sum_{j=1}^{n_{1}} \sigma_{j}^{-2}.$$ Starting from the right hand side of (\[eqC:MSE sum 4\]), and given that $\sigma_{1} \ge \sigma_{2} \geq \dotsi \geq \sigma_{n}$, we can bound this term as $$\label{eqC:MSE sum 5} \sum_{i=1}^{n}\sigma_{i}^{-2} b_{i}^{2} \sum_{j=1}^{n_{1}} \sigma_{j}^{-2} \leq \sigma_{n_{1}}^{-2} \sum_{i=1}^{n}\sigma_{i}^{-2} b_{i}^{2} \sum_{j=1}^{n_{1}} 1 = n_{1} \sigma_{n_{1}}^{-2} \sum_{i=1}^{n}\sigma_{i}^{-2} b_{i}^{2}.$$ On the other hand, given how we choose $n_{1}$ and $n_{2}$, we have $$\label{eqC:MSE sum 6} \sum_{i={n_{1}+1}}^{n}\sigma_{i}^{-2} \geq \sum_{i=1}^{n_{1}}\sigma_{i}^{-2},$$ which can help us to bound the left hand side of (\[eqC:MSE sum 4\]) as $$\label{eqC:MSE sum 7} n_{1} \sum_{i=1}^{n}\sigma_{i}^{-4} b_{i}^{2} \geq n_{1} \sigma_{n_{1}}^{-2} \sum_{i=1}^{n}\sigma_{i}^{-2} b_{i}^{2}.$$ Now, from (\[eqC:MSE sum 5\]) and (\[eqC:MSE sum 7\]) we find that a lower bound for the left hand side of (\[eqC:MSE sum 4\]) is equal to the upper bound for its right hand side. Then we can conclude from these two relations that $$n_{1} \sum_{i=1}^{n}\sigma_{i}^{-4} b_{i}^{2} \geq \sum_{i=1}^{n}\sigma_{i}^{-2} b_{i}^{2} \sum_{j=1}^{n_{1}} \sigma_{j}^{-2}.$$ Thus, $\epsilon$ is a positive root for the COPRA function in (\[eq:IBPRfunction\]). Now, we would like to know if $\epsilon$ can be considered as a value for our regularization parameter $\rho_{\text{o}}$. A direct way to prove that can be noted from the fact that having $\epsilon \ll \sigma_{i}^{2}$   $\forall i \in [1,n]$ will not provide any source of regularization to the problem. Hence, the RLS in (\[eq:R-LS\]) converges to the LS in (\[eq:pure LS solution\]). As a remark, we can assume that the approximation in (\[eqC:MSE sum 2\]) is uniform such that it does not affect the position of the roots. Thus, we can claim that this root is not coming from the negative region of the axis. In fact, we can easily prove that (\[eqC:MSE sum 1\]) does not have a negative root that is close to zero. Thus, this root is not coming from the negative region as a result of the function approximation (i.e., perturbed root). [^1]: M. Suliman, T. Ballal, and T. Y. Al-Naffouri are with the Computer, Electrical, and Mathematical Sciences and Engineering (CEMSE) Division, King Abdullah University of Science and Technology (KAUST), Thuwal, Makkah Province, Saudi Arabia. e-mails:$\{$mohamed.suliman, tarig.ahmed, tareq.alnaffouri$\}$@kaust.edu.sa. [^2]: The work presented in this paper is an extended version of [@msulimanhybrid]. [^3]: Less work have been done in the literature to provide estimators that are based on the MSE as for example in [@eldar2005robust] where the authors derived an estimator for the linear model problem that is based on minimizing the *worst-case* MSE (as opposed to the actual MSE) while imposing a constraint on the unknown vector $\xv_{0}$. [^4]: We will use this assumption to obtain the proposed estimator solution as a function of $\delta$, then, we will address the problem of obtaining the value of $\delta$. [^5]: Interestingly, setting the norm of the penalty term in $W\left(\xv\right)$ to be of l1 norm (i.e., $||\xv ||_{1}$) leads to the square-root LASSO [@belloni2011square] which is used in sparse signal estimation. [^6]: This includes the special case when all the singular values are significant and so all are considered. [^7]: The splitting threshold is obtained as the mean of the eigenvalues multiplied by a certain constant $c$, where c $\in \left(0,1\right)$. [^8]: Another way to look at (\[eq:d\]) is that we can replace $\Vm^{T} \Rm_{\xv_{0}} \Vm $ inside the trace in (\[eq:MSE’\]) by a diagonal matrix $\Fm = \text{diag}\left(\text{diag} \left(\Vm^{T} \Rm_{\xv_{0}} \Vm \right)\right)$ without affecting the result. Then, we replace this positive definite matrix $\Fm$ by an identity matrix multiplied by scalar which is given by the average value of the diagonal entries of $\Fm$. [^9]: In fact, one can prove that when $\xv_{0}$ is i.i.d., (\[eq:MSE’ approx\]) and (\[eq:MSE’\]) are exactly equivalent to each other (see Appendix \[Apen error\]). [^10]: The MATLAB code of the COPRA is provided at http://faculty.kfupm.edu.sa/ee/naffouri/publications.html. [^11]: For more details about the test problems consult [@hansen1994regularization].

--- abstract: 'Baker’s method, relying on estimates on linear forms in logarithms of algebraic numbers, allows one to prove in several situations the effective finiteness of integral points on varieties. In this article, we give a generalisation of results of Levin regarding Baker’s method for varieties, and explain how, quite surprisingly, it mixes (under additional hypotheses) with Runge’s method to improve some known estimates in the case of curves by bypassing (or more generally reducing) the need for linear forms in $p$-adic logarithms.' author: - | Samuel Le Fourn[^1]\ University of Warwick [^2] title: 'Tubular approaches to Baker’s method for curves and varieties' --- Introduction ============ One of the main concerns of number theory is solving polynomial equations in integers, which amounts to determining the integral points on the variety defined by those equations. For a smooth projective curve over a number field, Siegel’s theorem says that there are generally only finitely many integral points on this curve, but this result is as of yet deeply ineffective in that it does not provide us with any way to actually determine this set of integral points. We focus here on Baker’s method (and to a lesser extent Runge’s method), which are both *effective*: when applicable, they give a bound on the height of the integral points considered. Our work is based on Bilu’s conceptual approach [@Bilu95] for curves, and its generalisation to higher-dimensional varieties by [@Levin14]. It is also heavily inspired (sometimes implicitly) by a previous article [@LeFourn3], dealing with Runge’s method. Before stating the main results, let us give some notations. $K$ is a fixed number field, $L$ is a finite extension of $K$ with set of places $M_L$ divided into its archimedean places $M_L^\infty$ and its finite places $M_L^f$, and $S$ a finite set of places of $L$ containing $M_L^{\infty}$ (the pair $(L,S)$ will be allowed to change). The set of $S$-integers of $L$ is denoted by ${{\mathcal O}}_{L,S}$ and the regulator of ${{\mathcal O}}_{L,S}^*$ by $R_S$. We also will denote by $P_S$ the largest norm of an ideal coming from a finite place of $S$ (equal to 1 if $S = M_L^\infty$). The notion of integral point on a variety will be precisely defined (model-theoretically) in paragraph \[subsecnotations\], but the result is compatible with all reasonable definitions, e.g. the one in ([@Vojtadiophapp], section I.4). The set $(X \backslash Y)({{\mathcal O}}_{L,S})$ will thus be the set of $S$-integral points of the variety $X \backslash Y$, where $Y$ is a closed subset of $X$. \[thmmainBaker\] Let $X$ be a smooth projective variety over $K$ and $D_1, \cdots, D_n$ be ample effective divisors on $X$, $D = \bigcup_{i=1}^n D_i$, and $h_D$ a choice of global logarithmic height relative to $D$. The number $m_{{\mathcal B}}$ (assumed to exist) is the smallest integer such that for any set $I \subset \{1, \cdots,n\}$ with $|I| = m_{{\mathcal B}}$, the intersection $T_I = \bigcap_{i \in I} \operatorname{Supp} (D_i) (\overline{K})$ is finite. For any point $P$ in the finite set $$T := \bigcup_{|I|=m_{{\mathcal B}}} T_I,$$ assume $(H)_P$ : “there exists $\phi_P \in K(X)$ whose support is included in ${\operatorname{Supp}}D \backslash P$ (and $\phi_P(P)=1$ for simplicity)”. Such a function will be fixed in the following. Let $Y$ be a closed subset of $X$. The number $m_Y$ (assumed to exist) is the smallest integer such that for any set $I \subset \{1, \cdots,n\}$ with $|I| > m_Y$, $\bigcap_{i \in I} \operatorname{Supp} (D_i) \subset Y$). Then, there exists an effectively computable constants $C>0$ and an explicit function $C_1(d,s)$ such that for any pair $(L,S)$ where $[L:{{\mathbb{Q}}}]=d$ and $s=|S|$ satisfying $$\label{eqtubBakerhyp} (m_{{\mathcal B}}-1) |M_L^{\infty}| + m_Y |S \backslash M_L^{\infty}| < n,$$ for every $Q \in (X \backslash D)({{\mathcal O}}_{L,S}) \cap (X \backslash Y) ({{\mathcal O}}_L)$, $$\label{eqboundBaker} h_D(Q) \leq C \cdot C_1(d,s) h_L R_S \log^* (h_L R_S),$$ where $\log^*(x) = \max(\log x,1)$, unless $$Q \in Z:= \bigcup_{P \in T} Z_{\phi_P}, \quad Z_{\phi_P} := \overline{\{ Q \in (X \backslash {\operatorname{Supp}}\phi_P)(\overline{K}), \, \phi_P(Q) = 1 \} },$$ this set being an effective strict closed subset of $X$ independent of $(L,S)$. In the case of curves, one obtains the following corollary. \[corBaker\] Let $C$ be a smooth projective curve over $K$ and $\phi \in K(C)$ nonconstant. Assume that every pole $P$ of $\phi$ satisfies $(H)_P$ (and to simplify, is defined over $K$). Then, there are a constant $C>0$ and an explicit function $C_1(d,s)$ such that for any pair $(L,S)$ satisfying $$\label{eqalmostunifconditioncurves} |S \backslash M_L^{\infty}| <n,$$ any point $Q \in C(L)$ such that $\phi(Q) \in {{\mathcal O}}_{L,S}$ satisfies $$h(\phi(Q)) \leq C \cdot C_1(d,s) h_L R_S\log^* (h_L R_S).$$ Let us make some comments about those results. - Under the assumption that $S$ does not have too many finite places (exactly translated by for curves), one obtains a bound on the height which only grows in $R_S^{1+\varepsilon}$. In particular, there is no linear dependence on $P_S$ (which would come from estimates of linear forms in $p$-adic logarithms in a straightforward application of Baker’s method), but rather in $\log P_S$, implicit in $R_S$, which might prove useful for some applications. - The set $Z$ can actually be made smaller: as done in [@Levin14], we can replace each $Z_{\phi_P}$ by the intersection of *all* $Z_{\phi_P}$ where $\phi_P$ runs through all functions satisfying the hypotheses of $(H)_P$. - The choice of subset $M_L^\infty$ of $S$ is quite arbitrary: replacing $M_L^\infty$ by a fixed $S' \supset M_L^{\infty}$ in each of its occurrences in Theorem \[thmmainBaker\] and Corollary \[corBaker\] (in particular the point $Q$ is now in $(X \backslash D)({{\mathcal O}}_{L,S}) \cap (X \backslash Y) ({{\mathcal O}}_{L,S'})$) and considering sets of places $S \supset S'$, we obtain estimates of the shape with an additional factor $P_{S'} \log^* P_{S'}$. With this consideration in mind, Theorem \[thmmainBaker\] applied for $Y=D$ and $S=S'$ retrieves Levin’s result ([@Levin14], Theorem 1), and for smaller $S'$ (hence more hypotheses) improves the quantitative estimates it implicitly gave. - For general $Y$, Theorem \[thmmainBaker\] improves qualitatively (when the hypotheses $(H)_P$ hold) a previous result based on Runge’s method ([@LeFourn3], Theorem 5.1 and Remark 5.2$(b)$), as condition is generally weaker than the tubular Runge condition defined there (the choice of set $(X \backslash Y) ({{\mathcal O}}_{L,S'})$ is inspired by the notion of tubular neighborhood defined in that article, see section 3 there). Indeed, the $m=m_\emptyset$ in the statement of ([@LeFourn3], Remark 5.2) is not $m_{{\mathcal B}}$, and in general one only knows that $m_{{\mathcal B}}\leq m+1$. If $m_{{\mathcal B}}=m+1$, the original Runge’s method can be applied as in [@Levin08] and gives better (and uniform) estimates than [@Levin14] (and the same holds for the tubular variants we propose), but if $m_{{\mathcal B}}\leq m$ (the most likely situation), the condition is indeed more easily satisfied than for Runge’s method. - The words “effectively computable” for $C$ deserve to be made more precise. One requires to know an embedding of $X$ in a projective space ${\mathbb{P}}^N_K$, explicit equations and formulas for $X$, the $D_i$, $D$, $h_D$, the points of $T$ and expressions of $\phi_P$ relative to this embedding. With this data, the effectivity boils down to an effective Nullstellensatz such as e.g. ([@MasserWustholz83], Theorem IV). Now, the functions $C_1(d,s)$ (as well as $R_S \log^*(R_S)$) are coming from the theory of linear forms in logarithms, and as such completely explicit. The addition of $h_L$ is a technicality due to the necessity of slightly increasing the ring of units ${{\mathcal O}}_{L,S}^*$ upon which to apply Baker’s estimates, and can probably often be removed in special cases. - Unless we are in the case of curves, $m_{{\mathcal B}}>1$ and then bounds $d$ and $s$ in terms of $n$, which allows us to replace $C_1(d,s)$ by an explicit functions $C_1(n)$. As an illustration of the effectivity of the method, we prove the following result on the $S$-unit equation: fix $L$ be a number field of degree $d$, $S \subset M_L^\infty$ a set of places of $L$ of cardinality $s$ and ${\alpha},{\beta}\in L^*$. We consider the $S$-unit equation $$\label{eqSunit} \alpha x + \beta y = 1, \quad x,y \in {{\mathcal O}}_{L,S}^*.$$ \[thmSunits\] Let $L,S,{\alpha},{\beta}$ as above. - If $S$ contains at most two finite places, all solutions of satisfy $$\max(h(x),h(y)) \leq c(d,s) R_S \log^*(R_S) H,$$ where $H = \max(h({\alpha}),h(\beta),1,\pi/d)$ and $c(d,s)$ is the constant defined as $c_{26}(s,d)$ in formula (30) of [@GyoryYu06]. - For any set of places $S$, all solutions of satisfy $$\max(h(x),h(y)) \leq c'(d,s) P'_S R_S (1+\log^*(R_S)/\log^* P'_S) H,$$ where $c'(d,s)=c_1(s,d)$ from Theorem 1 of [@GyoryYu06], and $P'_S$ the third largest value of the norms of ideals coming from finite places of $S$. This result provides an improvement on known bounds for solutions of the $S$-unit equations. More precisely, its dependence on $P'_S$ (instead of $P_S$ the largest norm of ideal coming from a place of $S$) becomes particularly interesting when there are at most two places of $S$ of large relative norm, and by construction it improves Theorem 1 of [@GyoryYu06]. One can also remark that such an estimate is likely to be close to optimality in terms of dependence on the primes in $S$, as replacing $R_S \log^*(R_S)$ by $o(R_S)$ in the first bound would imply that there are only finitely many Mersenne primes. On another hand, Theorem 4.1.7 of [@EvertseGyory15], based on slightly different Baker-type estimates, has a better dependence on $s$ and $d$. It is possible to combine the strategy of proof of the latter theorem with our own to obtain an improvement of both results, essentially replacing again $P_S$ by $P'_S$. This is achieved in a recent preprint of Györy [@Gyory19]. After proving Theorem \[thmmainBaker\] and Corollary \[corBaker\] in section 3 (section 2 gathering the necessary reminders and tools for the proof), we prove Theorem \[thmSunits\] in section 4. This application is heavily based on computations undertaken in [@GyoryYu06], hence we have chosen to refer to it whenever possible, and focus on pointing out where the improvements come from our approach. Acknowledgements {#acknowledgements .unnumbered} ================ I wish to thank Kálmán Györy for his insightful comments on this paper and his remarks regarding the existing results on $S$-unit equations and their applications. Reminders on Baker’s theory and local heights ============================================= For any place $w$ of $L$, the norm $|\cdot|_w$ associated to $w$ is normalised to extend the norm on ${{\mathbb{Q}}}$ defined by $v_0$ below $w$, where $|\cdot|_\infty$ is the usual norm on ${{\mathbb{Q}}}$ and for every prime $p$ and nonzero fraction $a/b$, $$\left| \frac{a}{b} \right|_p = p^{{\operatorname{ord}}_p b - {\operatorname{ord}}_p a}.$$ We also define $n_w = [L_w : {{\mathbb{Q}}}_{v_0}]$ the local degree of $L$ at $w$. In all discussions below, $X$ is a fixed projective smooth algebraic variety over the number field $K$ and closed subset of $X$ will mean a closed algebraic $K$-subvariety of $X$. Regarding the integrality, we choose a model-theoretic definition as follows. Assume ${{\mathcal X}}$ is a proper model of $X$ over ${{\mathcal O}}_K$, fixed until the end of this article. For every closed subset $Y$ of $X$, denote by ${{\mathcal Y}}$ the Zariski closure of $Y$ in ${{\mathcal X}}$. The set of integral points $(X \backslash Y) ({{\mathcal O}}_{L,S})$ will then implicitly denote the set of points $P \in X(L)$ whose reduction in ${{\mathcal X}}_v(\kappa(w))$ for a place $w$ of $M_L \backslash S$ above $v \in M_K$ (well-defined by the valuative criterion of properness) never belongs to ${{\mathcal Y}}$. MK-constants and bounded functions {#subsecnotations} ---------------------------------- The arguments below will be much simpler to present with the formalism of $M_K$-constants and $M_K$-functions briefly recalled here. 1. An *$M_K$-constant* is a family $(c_v)_{v \in M_K}$ of nonnegative real numbers, all but finitely many of them being zero. 2. An *$M_K$-function* $f$ (on $X$) is a function defined on a subset $E$ of $X({\overline{K}}) \times M_{{\overline{K}}}$ with real values (typically, a local height function as below). Equivalently, it is defined as a function on a subset of $\bigsqcup_{K \subset L \subset \overline{K}} X(L) \times M_L$, consistently in the sense that if $f$ is defined at $(P,w)$ with $P \in X(L)$ and $w \in M_{L}$, then it is defined at $(P,w')$ with $w'|w \in M_{L'}$ for any extension $L'$ of $L$, and $f(P,w) = f(P,w')$. 3. An $M_K$-function $f : E \rightarrow {{\mathbb{R}}}$ is *$M_K$-bounded* if there exists an $M_K$-constant $(c_v)_{v \in M_K}$ for which for all $(P,w) \in E$, $$|f(P,w)| \leq c_v \quad (w|v).$$ The notation $O_{M_K}(1)$ will be used for an $M_K$-bounded function depending on the context (in particular, its domain $E$ will often be implicit but obvious). 4. Two $M_K$-functions $f,g : E \rightarrow {{\mathbb{R}}}$ are *$M_K$-proportional* when there is an absolute constant $C>0$ and a $M_K$-constant $(c_v)_{v \in M_K}$ for which for all $(P,w) \in E$, $$\frac{1}{C} |f(P,w)| -c_v \leq |g(P,w)| \leq C |f(P,w)| + c_v \quad (w|v).$$ 5. Two functions $f,g$ defined on an open subset $O$ of $X({\overline{K}})$ (typically, global height functions) are *proportional* if there are absolute constants $C_1,C_2>0$ such that for every $P \in O$ : $$\frac{1}{C_1} f(P) - C_2 \leq g(P) \leq C_1 f(P) + C_2.$$ Local heights associated to closed subsets ------------------------------------------ We will now define explicitly local height functions relative to closed subsets of a projective variety $X$. 1. For any point $P \in {\mathbb{P}}^N(L)$, one denotes by $x_P=(x_{P,0}, \cdots, x_{P,n}) \in L^{n+1}$ a choice of coordinates representing $P$ and $\|x_P\|_w = \max_i |x_{P,i}|_w$. 2. For a polynomial $g \in L[X_0, \cdots, X_N]$ and $w \in M_L$, the norm $\|g\|_w$ is the maximum norm of its coefficients for $| \cdot|_w$. 3. Given a closed subset $Y$ of ${\mathbb{P}}^N_K$ and homogeneous polynomials $g_1, \cdots, g_m \in K[X_0, \cdots, X_N]$ generating the ideal of definition of $Y$, for any $w \in M_L$ and any $P \in ({\mathbb{P}}^N \backslash Y)(L)$, one defines explicitly a choice of local height of $P$ at $Y$ for $w$ by $$\label{eqdeflocheight} h_{Y,w} (P) := - \min_i \log \frac{|g_i(x_P)|_w}{\|g_i\|_w \| x_P\|_w^{\deg g_i}},$$ and the global height by $$h_{Y}(P):= \frac{1}{[L:{\mathbb{P}}]} \sum_{w \in M_L} n_w \cdot h_{Y,w}(P).$$ With this normalisation, for any $w \in M_L^f$ and $P \in ({\mathbb{P}}^N \backslash Y)(L)$, $h_{Y,w} (P) \geq 0$ and it is positive if and only if $P$ reduces in $Y$ modulo $w$. Let us now sum up the main properties of those functions that we will need. \[proplocalheights\] Let $X$ be a smooth projective variety over $K$, with an implicit embedding in a ${\mathbb{P}}^n_K$ and fixed choices of local heights as in for all closed subsets considered below. $(a)$ For any closed subsets $Y,Y'$ of $X$ the functions $h_{Y \cap Y',w}$ and $\min(h_{Y,w},h_{Y,w'})$ are $M_K$-proportional on $(X \backslash (Y \cup Y'))({\overline{K}}) \times M_{{\overline{K}}}$. $(b)$ For a disjoint union $Y \sqcup Y'$ of closed subsets of $X$, one has $$h_{Y,w}(P) + h_{Y',w}(P) = h_{Y \sqcup Y',w}(P) + O_{M_K}(1)$$ on $(X \backslash (Y \cup Y'))({\overline{K}}) \times M_{{\overline{K}}}$. $(c)$ For $Y \subset Y'$ closed subsets, one has $$h_{Y,w}(P) \leq h_{Y',w}(P) + O_{M_K}(1)$$ on $(X \backslash Y') ({\overline{K}}) \times M_{{\overline{K}}}$. $(d)$ If $\phi : X' \rightarrow X$ is a morphism of projective varieties, the functions $(P,w) \mapsto h_{Y,w}(\phi(P))$ (resp. $h_{\phi^{-1}(Y),w}(P)$) are $M_K$-proportional on $(X' \backslash \phi^{-1}((Y))({\overline{K}}) \times M_{{\overline{K}}}$. $(e)$ For any closed subset $Y$ of $X$, the function $(P,w) \mapsto h_{Y,w}(P)$ is $M_K$-bounded on the set of pairs satisfying $P \in (X \backslash Y) ({{\mathcal O}}_{L,w})$ (independently of the number field $L$). $(f)$ For any effective divisor $D$ on $X$ and any function $\phi \in K(X)$ with support of poles included in ${\operatorname{Supp}}D$, the function $(P,w) \mapsto |\phi(P)|_w$ is $M_K$-bounded on the set of pairs $(P,w) \in X(L) \times M_L^f$ satisfying $P \in (X \backslash D)({{\mathcal O}}_{L,w})$ (independently of the number field $L$). $(g)$ If $D$ and $D'$ are two ample divisors on $X$, for any two choices of global heights $h_D$ and $h_{D'}$, they are proportional on $(X \backslash {\operatorname{Supp}}(D \cup D')) ({\overline{K}})$. Furthermore, all the implied $M_K$-constants and constants are effective . This proposition is mostly a reformulation of results of [@Silverman87] already quoted in [@Levin14]. First, indeed defines local heights associated to closed subsets by ([@Silverman87], Proposition 2.4) so most of the proposition is contained in ([@Silverman87], Theorem 2.1). Let us point out the slight differences and explain how it is effective. In that article, local height functions are more precisely defined by their ideal sheaves, whereas we consider closed subsets hence reduced closed subschemes. Now, if two ideal sheaves ${{\mathcal I}}$ and ${{\mathcal I}}'$ have the same support, their local height functions are $M_K$-proportional. More concretely, let us fix $Y \subset Y'$ closed subsets of $X \subset {\mathbb{P}}^N_K$ and two systems of homogeneous generators $g_1, \cdots, g_m \in K[X_0, \cdots, X_N]$ and $h_1, \cdots, h_p$ of ideal sheaves with respective supports $Y$ and $Y'$ in ${\mathbb{P}}^N_K$. After multiplying by a suitable $n \geq 1$, one can assume all those polynomials’ coefficients belong to ${{\mathcal O}}_K$, and such an $n$ can be made effective in terms of the $\| g_i\|_v$ and $\|h_j\|_v$ for $v \in M_K^f$. Now, an effective Nullstellensatz (e.g. [@MasserWustholz83] applied to multiples of those generators), translated in the projective case, will give relations $$a g_i ^k = \sum_{j=1}^p f_{i,j} h_j$$ with $a \in {{\mathcal O}}_K$ nonzero, all the $f_{i,j}$ with coefficients in ${{\mathcal O}}_K$ and bounded $\|f_{i,j}\|_v$ and $|a|_v$ in terms of the norms of the polynomials for all $v \in M_K^\infty$. Furthermore, the power $k$ is effectively bounded in terms of $[K : {{\mathbb{Q}}}]$,$N$ and the degrees of the polynomials. This will clearly give an effective inequality $$h_{Y,w}(P) \leq k \cdot h_{Y',w}(P) + O_{M_K}(1).$$ for all $(P,w) \in (X \backslash Y')({\overline{K}}) \times M_{{\overline{K}}}$, and this argument works for parts $(a)$ to $(d)$ of the Proposition (the inequality in $(c)$ without a factor $k$ coming from the fact that we can extend generators of an ideal sheaf for $Y'$ to an ideal sheaf for $Y$). The only parts remaining to be proven are now $(e)$, $(f)$ and $(g)$. Part $(e)$, essentially saying that local height functions detect integral points up to some $M_K$-bounded error, is classical (see [@Vojtadiophapp], Proposition 1.4.7) and in fact automatic for the exact definition given in . Part $(f)$ then comes from $(e)$ and Lemma 11 of [@Levin14]. Finally, part $(g)$ is a classical result on heights (e.g. [@LangFundamentals], Chapter IV, Proposition 5.4), and there also, the constants implied can be made effective. Baker’s theory of linear forms in logarithms -------------------------------------------- Let us now give our second main tool: estimates from Baker’s theory in a special form sufficient for our purposes. For any place $w$ of $M_L$, $N(w)$ is defined to be 2 if $w$ is archimedean and the norm of the associated prime ideal otherwise. \[propBaker\] Define $\log^*(x) = \max(\log(x),1)$ for $x >0$. Let $d=[L: {{\mathbb{Q}}}]$ and $s = |S|$. Recall that $P_S$ is the largest norm of an ideal coming from a finite place of $S$ (equal to 1 if there is no such place). There is an effectively computable function $C(d,s)$ such that for any pair $(L,S)$, any $\alpha \in {{\mathcal O}}_{L,S}^* \backslash \{1\}$ and any $w \in M_L$, $$\label{eqBakertheory} \log |\alpha - 1|_w \geq - C(d,s) N(w) R_S \log^* h(\alpha).$$ In terms of local heights, one can choose the local height $h_{1,w}(\alpha)$ to be $\max(- \log |\alpha - 1|_w,0)$, which gives us $$h_{1,w}(\alpha) \leq C(d,s) N(w) R_S \log^* h(\alpha).$$ This result, although natural when one knows estimates for linear forms in logarithms, is not often presented in this form, so the following proof will explain how one can get to such an expression with known results. First, let us assume $s \geq 2$ (for $s=1$, the result is trivial). By Lemma 1 of [@BugeaudGyory96] ($\log h$ there is our logarithmic height here), one can choose a family of fundamental units $\varepsilon_1, \cdots, \varepsilon_{s-1}$ of ${{\mathcal O}}_{L,S}^*$ such that $$\prod_{i=1}^{s-1} h(\varepsilon_i) \leq c_1(s) R_S,$$ where $c_1 (s) = ((s-1)!)^2 /(2^{s-1} d^{s-2})$. Now, by Theorem 4.2.1 of [@EvertseGyory15] applied to $\Gamma = {{\mathcal O}}_{L,S}^*$ and this system of fundamental units, we obtain (taking into account our normalisation of $|\cdot|_w$) the bound $$\log |\alpha - 1|_w > - \frac{c_2(d,s)}{n_w} c_1(s) R_S \frac{N(w)}{\log N(w)} \log^* (N(w) h^*(\alpha)),$$ with $c_2(d,s) = c_8$ in the reference. We can put all effective constants in $C(d,s)$, and roughly bound $\frac{N(w)}{\log N(w)} \log^* (N(w) h^*(\alpha))$ by $2 N(w) \log ^* h({\alpha})$ to obtain the stated result. Proof of the main theorem ========================= We now have all the tools to prove the theorem. We keep the notations from its statement, and assume we have an embedding $X \subset {\mathbb{P}}^N_K$ from which all local heights considered below are defined. Recall that $s=|S|$, $d=[L:{{\mathbb{Q}}}]$ and $n_w$ is the local degree of $L$ at $w$. The constants $c_i$ below are absolute and can be made effective. First, let us notice that for every point $P \in T$, as the support of $\phi_P$ is in $D$, by Proposition \[proplocalheights\] $(f)$ applied to $\phi_P$ and $\phi_P^{-1}$, there is an absolute positive integer $m$ (independent on the choice of $(L,S)$ and $P \in T$) such that for every $Q \in (X \backslash D) ({{\mathcal O}}_{L,S})$, one has $m \phi_P(Q) \in {{\mathcal O}}_{L,S}$ and $m \phi_P(Q)^{-1} \in {{\mathcal O}}_{L,S}$. Defining $S_m$ the set of primes of $L$ dividing $m$, one thus has $\phi_P(Q) \in {{\mathcal O}}_{L,S \cup S_m}^*$ for all $Q \in (X \backslash D)({{\mathcal O}}_{L,S})$. By Proposition \[proplocalheights\] $(e)$, the map $(Q,w) \mapsto h_{D_i,w}(Q)$ is $M_K$-bounded on pairs $(Q,w)$ with $w \in M_L \backslash S$ and $Q \in (X \backslash D) ({{\mathcal O}}_{L,S})$, and $(Q,w) \mapsto h_{Y,w}(Q)$ is $M_K$-bounded on pairs $(Q,w)$ with $w \in M_L \backslash M_L^{\infty}$ and $Q \in (X \backslash Y) ({{\mathcal O}}_L)$. Let us assume now that $Q \in (X \backslash D)({{\mathcal O}}_{L,S}) \cap (X \backslash Y) ({{\mathcal O}}_L)$. The previous paragraphs imply that for every $i \in \{1, \cdots, n\}$ $$h_{D_i}(Q) = \frac{1}{[L:{{\mathbb{Q}}}]} \sum_{w \in S} n_w h_{D_i,w}(Q) + O(1)$$ where $O(1)$ is absolutely (and effectively bounded) on the set of such points $Q$ (even if $(L,S)$ is allowed to change). Thus, for all $i \in \{1, \cdots, n\}$, there is $w \in S$ such that $$h_{D_i,w} (Q) \geq \frac{n_w}{[L:{{\mathbb{Q}}}]} h_{D_i,w} (Q)\geq \frac{1}{s} h_{D_i} (Q) + O(1).$$ After choosing for every $i \in \{ 1, \cdots, n\}$ such a $w \in S$, we obtain a function $\{1, \cdots, n\} \rightarrow S$. Now, if the fiber above a place $w \in S \backslash M_L^{\infty}$ was a set $J$ with $|J| > m_Y$, by Proposition \[proplocalheights\] $(a)$, one would obtain an absolute effective (computable) upper bound on the minimum of such $h_{D_j,w}(Q)$, therefore on $h_{D_i}(Q)/s$ and $h_D(Q)/s$ by Proposition \[proplocalheights\] $(g)$. We can thus assume from now on that this is not the case. Therefore, the fibers of this function are of cardinality at most $m_Y$ above $S \backslash M_L^{\infty}$. Consequently, by hypothesis , one of the fibers above $M_L^\infty$, defined as $I$, has to be of cardinality at least $m_{{\mathcal B}}$ (if it’s more, we extract a subset of cardinality $m_{{\mathcal B}}$), which gives $w \in M_L$ such that $$\min_{i \in I} h_{D_i,w} (Q) \geq \frac{1}{s} \min_{i \in I} h_{D_i} (Q) \geq \frac{c_1}{s} h_D(Q) + O(1).$$ Now, by Proposition \[proplocalheights\] $(a)$ again and construction of $T$ (if $T=\emptyset$, we directly obtain an absolute $M_K$-constant bound), there exists $P \in T$ such that $$\label{eqheightscomparison} h_{P,w}(Q) \geq \frac{c_2}{s} h_D(Q) + O_{M_K}(1) \geq \frac{c_3}{s} h (\phi_P(Q)) + O_{M_K}(1),$$ using Proposition \[proplocalheights\] $(g)$, with absolute effective constants $c_1,c_2,c_3>0$. By Proposition \[proplocalheights\] $(d)$, as $\phi_P(P)=1$, if $\phi_P$ can be evaluated at $Q$ and $\phi_P(Q) \neq 1$, $$h_{1,w} (\phi_P(Q)) \geq \frac{c_4}{s} h(\phi_P(Q)),$$ for $c_4>0$ absolute effective, computable in terms of the initial data of embeddings and equations. Now, applying Proposition \[propBaker\] to ${{\mathcal O}}_{L,S \cup S_m}^*$ (here, $w$ is archimedean), this inequality implies the bound $$\label{eqintermformulaBaker} h(\phi_P(Q)) \leq 2 s \cdot C(d,s+|S_m|)/c_4 \cdot R_{S \cup S_m} \cdot \log^* h(\phi_P(Q))$$ By formula (1.8.3) of [@EvertseGyory15], one has $$\label{eqboundsregulator} R_{S \cup S_m} \leq h_L R_S \prod_{{{\mathfrak{P}}}\in S_n \backslash S} \log N({{\mathfrak{P}}})) \leq h_L R_S \prod_{p|m} e^{d/e \log p} \leq h_L R_S m^{d/e}$$ after optimising the products of logarithms. Combining and , we obtain an affine bound of the shape for $h(\phi_P(Q))$, hence on $h_{P,v} (Q)$ by Proposition \[proplocalheights\] $(d)$ applied the other way, which finally gives a bound on $h_D(Q)$ by (there is a constant term which we can absorb in the linear one as it is effectively boundable). To apply the same method with $S' \supset M_L^\infty$ instead of $M_L^\infty$, we apply exactly the same process under the condition $$(m_{{\mathcal B}}-1) |S'| + m_Y |S \backslash S'| <n,$$ to obtain for our point $Q$ a place $w \in S'$ and a point $P$ such that $h_{P,w}(Q) \geq c_3/s h(\phi_P(Q)) + O(1)$, and then follow the end of the proof with the general estimate of Proposition \[propBaker\] for places of $S'$ instead of the archimedean case (this is where the additional factor $P_{S'} \log^* P_{S'}$ comes from). Applications to the S-unit equation in the case of curves ========================================================= In this section, we realise our method in the practical case of the $S$-unit equation , to prove Theorem \[thmSunits\]. This problem is related to finding the integral points of $({\mathbb{P}}^1 \backslash \{0,1,\infty\})({{\mathcal O}}_{L,S})$ (up to taking into account the factors $\alpha,\beta$), this is the interpretation we will follow below to illustrate the main theorem. We follow closely the definitions and lemmas of [@GyoryYu06] (except their normalisations of norms), as our improvements intervene only at the beginning of the proof. As in that article, define $$d = [L:{{\mathbb{Q}}}], \quad H = \max(h(\alpha),h(\beta),1,\pi/d), \quad s=|S|.$$ For any $t \in L$, define $h_w (t) := h_{0,w} (t) = \log^+(1/|t|_w)$. For sake of symmetry of the exposition, we will do most computations with $ {\alpha}x $ and $ {\beta}y $, before coming back to $ H $. This means we deal with $h_w(P)$ for $$P \in E=\left\{ {\alpha}x,{\beta}y,\frac{{\beta}y}{{\alpha}x}\right\}.$$ \[lemSunits\] For any $x,y \in L$ with $\alpha x + \beta y = 1$: - For any place $w \in M_L$, at most one value of $h_w(P)$ for $P \in E$ can exceed ${\delta}_w \log 2$, where $ {\delta}_w =1 $ if $w$ is infinite, 0 otherwise. - The maximum module of the difference of logarithmic heights of any two of them amongst $h(x),h(y),h({\alpha}x), h(\beta y),h(\frac{{\beta}y}{{\alpha}x})$ is at most $3H$, and even $2H$ except for $|h(x)-h(y)|$. - If $x,y \in {{\mathcal O}}_{L,S}^*$ and $h=\max(h(x),h(y))$, we always have, for $P \in E$: $$\sum_{w \in S} \frac{n_w}{[L:{{\mathbb{Q}}}]}h_w(P) \geq h-3H.$$ The first item is the translation of the fact that is $z+z'=1$ one of $z,z'$ has to have norm at least 1 if the norm is ultrametric, and at least $1/2$ if it is archimedean. The second item uses that for any nonzero algebraic numbers $z,z'$, $h(zz') \leq h(z)+h(z')$ and $h(z+z') \leq h(z) + h(z') + \log 2$. For example, we obtain $|h(x) - h({\alpha}x)| \leq H$ and $|h({\alpha}x)-h({\beta}y)| \leq \log 2$, and by symmetric role this leads to all other bounds on difference of heights, as $\log 2 \leq H$. For the third item, there are six cases to deal with (depending on which of $h(x)$ and $h(y)$ is the maximum, and the values of $P$). If $P={\alpha}x$, we have $$\sum_{w \in S} \frac{n_w}{[L:{{\mathbb{Q}}}]}h_w(P) \geq \sum_{w \in S} \frac{n_w}{[L:{{\mathbb{Q}}}]} \log^+ |1/({\alpha}x)|_w \geq \sum_{w \in S} \frac{n_w}{[L:{{\mathbb{Q}}}]} (\log^+ |1/ x|_w-\log^+ |{\alpha}|_w) \geq h(x)-H$$ because $x \in {{\mathcal O}}_{L,S}^*$. In the same fashion, this term is also bounded from below by $h(P) - h({\alpha})$. By symmetric role, for $P=\beta y$, we obtain $$\sum_{w \in S} \frac{n_w}{[L:{{\mathbb{Q}}}]}h_w(P) \geq \max(h(y),h(P))-H.$$ Now, if $h=h(y)$ and $P={\alpha}x$, we use that $h({\alpha}x) \geq h({\beta}y) - \log 2$ to obtain the desired bound, and similarly for the symmetric case. There only remains the case $P=\frac{{\beta}y}{{\alpha}x}$. Using the fact that $1/P = 1 + 1/({\beta}y)$, we obtain $$\sum_{w \in S} \frac{n_w}{[L:{{\mathbb{Q}}}]}h_w(P) \geq h(P) - h(\beta) \geq h -3H$$ by the second item. First, notice that if $s \leq 2$, Lemma 4.1 alone gives immediately than there is $P \in E$ such that $h_w(P) \leq {\delta}_w \log 2$ for all $w \in S$, and elsewhere we have $h_w(P) = h_w({\alpha}),h_w({\beta})$ or $h({\beta}/{\alpha})$ depending on the value of $P$, because $x$ and $y$ are $S$-units. Consequently, $h(P) \leq 2H + \log 2$, hence $h \leq 4H + \log 2$ in this case. We can now assume that $s \geq 3$. For the first part of Theorem \[thmSunits\], the assumption amounts to saying that holds in this case. By Lemma \[lemSunits\], for any choice of $P \in E$, there is $w \in S$ such that $$\label{eqineqcleBaker} \frac{n_w}{[L:{{\mathbb{Q}}}]} h_{w}(P) \geq \frac{1}{|S|} (\max(h(x),h(y)) - 3 H),$$ We want to fall back on a case where for one of the three choices of $P$, one can impose that $w \in M_L^\infty$. If that is not possible, by pigeonhole principle and our hypothesis on $S$, there is a finite place $w$ and two points $P,Q \in \{E \}$ distinct with $$\frac{n_w}{[L:{{\mathbb{Q}}}]} \min(h_{w}(P),h_{w}(Q)) \geq \frac{1}{|S|}(\max(h(x),h(y)) -3 H).$$ By the same lemma, we get $\max(h(x),h(y)) \leq 3H$. This bound will be readily checked to be smaller than the other case. One can thus assume from now on that for some $w \in M_L^\infty$ and $P \in E $, holds. The only thing to do is then to get back to the situation of ([@GyoryYu06], page 24) in all three cases, after which we will obtain the exact same bounds. We fix a fundamental system $ \varepsilon_1, \cdots, \varepsilon_{s-1}$ of units of $ {{\mathcal O}}_{L,S}^* $ with the properties of ([@GyoryYu06], Lemma 2). $ \bullet $ Assume first $P={\alpha}x$, and write $$\label{eqfactory} y= \zeta \varepsilon_1 ^{b_1} \cdots \varepsilon_{s-1} ^{b_{s-1}},$$ with $\zeta$ a root of unity in $L$ and $b_i \in {{\mathbb{Z}}}$ for all $ i $. By the arguments of [@BugeaudGyory96] (p. 76), we obtain that $$B = \max(|b_1|, \cdots, |b_{s-1}|) \leq c_1(d,s) h(y)$$ with $$c_1(d,s) = \left\{ \begin{array}{lcr} ((s-1)!)^2/(2^{s-3} \log 2) & \textrm{if} & d=1 \\ ((s-1)!)^2/2^{s-2}) \log(3d)^3 & \textrm{if} & d \geq 2. \end{array} \right.$$ We set $\alpha_s = \zeta \beta$ and $b_s=1$ so that $$|{\alpha}x|_w = |1 - \varepsilon_1 ^{b_1} \cdots \varepsilon_{s-1} ^{b_{s-1}} {\alpha}_s^{b_s}|_w.$$ We set the $A_i$ and $A_s$ as in ([@GyoryYu06], equation (31)) and can make the same assumption (otherwise, we obtain a smaller bound). By ([@GyoryYu06], Proposition 4 and Lemma 5), we thus obtain $$\label{eqintermhw} h_w(P) = - \log |{\alpha}x|_w < c_2(d,s) c_3(d,s) R_S H \log \left( \frac{c_1(d,s) h(y)}{\sqrt{2}H}\right)$$ (as we always have $s \geq 3$ here), with $$c_2(d,s) = d^3 \log(ed) \min(1.451 (30\sqrt{2})^{s+4}(s+1)^{5.5},\pi 2^{6.5s+27}),$$ $$c_3(d,s) = e\sqrt{s-2} (((s-1)!)^2/(2^{s-2})) \pi^{s-2} \cdot \left\{ \begin{array}{lcr} 8.5 & \textrm{if} & d=1 \\ 29 d \log d & \textrm{if} & d \geq 2, \end{array} \right.$$ Let us now define $h=\max(h(x),h(y))$. We use inequality , and replace $h(y)$ by $h$ in the right-hand side of to obtain $$\label{eqfinaleBaker} \frac{h}{s} -H \leq \frac{h - 3H}{s} \leq \frac{n_w}{[L:{{\mathbb{Q}}}]} c_2(d,s) c_3(d,s) R_S H \log \left( \frac{c_1(d,s) h}{\sqrt{2}H}\right)$$ and these are equivalent to the two inequalities used page 24 to obtain the result. $ \bullet $ Assume $P={\beta}y$. We apply the same argument by symmetry, replacing $ {\alpha}$ by $ {\beta}$ and $ x $ by $ y $ everywhere, to finally obtain the same bound. $\bullet$ Assume $P=\frac{{\beta}y}{{\alpha}x}$. We thus write $$h_w(P) = - \log \left|\frac{{\beta}y}{{\alpha}x} \right|_w = - \log \left|1 - \frac{1}{{\alpha}x}\right|_w.$$ Write then $$\frac{1}{x} = \zeta \varepsilon_1 ^{b_1} \cdots \varepsilon_{s-1} ^{b_{s-1}}, \quad \alpha_s = \frac{{\zeta}}{{\alpha}}.$$ In the same fashion as above, we then get $ B \leq c_1(d,s) h(x)$, and the rest follows, with Lemma 4.1, in the exact same way until we obtain . For the second part of Theorem 4.1, we can play the same game, by defining $S'$ the set of places $S$ deprived of its two prime ideals with largest norm. The same elimination work as before will then give $w \in S'$ and $P \in E$ satisfying , and from there we can apply for $P = {\alpha}x$ the exact method of ([@GyoryYu06], page 25) with a prime ideal ${{\mathfrak{p}}}$ from $S'$. This finally leads to the same estimates with $P'_S$ instead of $P_S$, using again Lemma 4.1. [MW83]{} Y. Bugeaud and K. Győry. Bounds for the solutions of unit equations. , 74(1):67–80, 1996. Y. Bilu. Effective analysis of integral points on algebraic curves, 1995. J.-H. Evertse and K. Győry. , volume 146 of [*Cambridge Studies in Advanced Mathematics*]{}. Cambridge University Press, Cambridge, 2015. K. Győry and K. Yu. Bounds for the solutions of [$S$]{}-unit equations and decomposable form equations. , 123(1):9–41, 2006. K. Gy[ö]{}ry. Bounds for the solutions of [$S$]{}-unit equations and decomposable form equations [II]{}. arXiv:1901.11289. S. Lang. . Springer-Verlag, New York, 1983. A. Levin. Variations on a theme of [R]{}unge: effective determination of integral points on certain varieties. , pages 385–417, 2008. A. Levin. Linear forms in logarithms and integral points on higher-dimensional varieties. , 8:647–687, 2014. S. Le Fourn. A “tubular” variant of [R]{}unge’s method in all dimensions, with applications to integral points on [S]{}iegel modular varieties. . to appear. D. W. Masser and G. Wüstholz. Fields of large transcendence degree generated by values of elliptic functions. , 72(3):407–464, 1983. J. H. Silverman. Arithmetic distance functions and height functions in [D]{}iophantine geometry. , 279(2):193–216, 1987. P. Vojta. . Lecture Notes in Mathematics 1239. Springer-Verlag Berlin Heidelberg, 1987. [^1]: Supported by the European Union’s Horizon 2020 research and programme under the Marie Sklodowska-Curie grant agreement No 793646, titled LowDegModCurve. [^2]: samuel.le-fourn@warwick.ac.uk

--- author: - 'Kenji <span style="font-variant:small-caps;">Yonemitsu</span>$^{1,2}$[^1] and Keiichiro <span style="font-variant:small-caps;">Nasu</span>$^{3,4}$' title: Theory of Photoinduced Phase Transitions --- Introduction ============ Nonequilibrium phases can be generated from an equilibrium one by external stimulations. Photoirradiation is one of them and sometimes allows the emergence of a new phase, which cannot be reached by simply changing temperature or pressure, because the energy of a photon is much higher than thermal energies. Although these phases finally relax to the starting equilibrium phase, they are locally stable, separated from the initial one by a substantial free-energy barrier (except for the transition from a Mott insulator to a metal, as will be mentioned later). When the relaxation time is long enough, the properties of this nonequilibrium phase can be studied by laser spectroscopy techniques, which have recently made great progress. Thus, photoinduced phase transitions have been studied extensively, both experimentally and theoretically. [@Nasu_book97; @Nasu_book04; @Nasu_rpp04] Here, we review progress in theories for such phase transitions. Photoirradiation creates electrons and holes, or excitons if they are bound. In most of the cases, they are accompanied by local structural deformation. To develop it spatially, some cooperativity is needed. One would ask first what kind of interaction possesses such cooperativity. Section 2 introduces classical models, in which photoexcited states proliferate in some conditions. Ground and excited states are simply assigned to Ising variables, so that the electronic transitions are statistically described on the basis of diabatic potentials. Section 3 first introduces adiabatic pictures for electronic models, which are useful in discussing necessary conditions for the spontaneous growth of photoexcited states. Transition dynamics do not always proceed adiabatically. Next, this section introduces the electronic dynamics derived from the time-dependent Schrödinger equation and shows their relevance to experimental findings. Section 4 concludes and presents future issues. Statistic Theory of Photoinduced Phase Transitions ================================================== Linear chain system ------------------- In order to clarify how the local structural distortions lead to global ones, a theory based on stochastic processes was proposed at first. Hanamura and Nagaosa introduced a simple model: [@Hanamura_jpsj87] $$\begin{aligned} H =& \sum_l \mid e_l \rangle ( E_\mathrm{FC} - \sqrt{S} Q_l ) \langle e_l \mid + \frac12 \sum_l Q_l^2 \nonumber\\ & - \frac12 \sum_{ll'} K_{ll'} Q_l Q_{l'} \;, \label{eq:Hanamura}\end{aligned}$$ and $$\mid g_l \rangle \langle g_l \mid + \mid e_l \rangle \langle e_l \mid = 1 \;,$$ where $ E_\mathrm{FC} $ is the Franck-Condon excitation energy, $ Q_l $ the relevant displacement that dominantly interacts with an electron in the photoexcited $ \mid e_l \rangle $ state at the $ l $th molecule and is treated as a classical variable, and $ \sqrt{S} Q_l $ the Stokes shift in this state. The coupling strength between the displacements of the $ l $th and $ l' $th molecules is denoted by $ K_{ll'} $ with its diagonal element $ K_{ll} $ set at zero. Linear chain systems are first investigated in ref. . Suppose that electrons on $ m $ consecutive molecules ([*e.g.*]{}, on sites 1 to $ m $) are in the excited state, and others are in the ground state. Each molecule is located on the respective equilibrium position. This state is governed by the Hamiltonian $ H_\mathrm{g}^{(m)} $. Then, consider that an electron on a neighboring molecule ([*e.g.*]{}, on site 0) is excited. The corresponding molecule is now away from the equilibrium position. This state is governed by the Hamiltonian $ H_\mathrm{e}^{(m+1)} $. The difference between these Hamiltonians is given by $$H_\mathrm{e}^{(m+1)} = H_\mathrm{g}^{(m)} + E_\mathrm{FC} - \sqrt{S} Q_0 \;.$$ Then, an interaction mode $ q_0 $ (on site 0) is introduced, which is a linear combination of the displacements in the state with $ m $ excited molecules. The diabatic potentials for $ H_\mathrm{g}^{(m)} $ and $ H_\mathrm{e}^{(m+1)} $ are drawn as a function of $ q_0 $ (Fig. \[fig:two\_parabolas\]). ![Energy diagrams of 0th molecule, which is closest to cluster of $ m $ excited molecules, in ground and excited states. [@Hanamura_jpsj87; @Nagaosa_prb89][]{data-label="fig:two_parabolas"}](fig1_two_parabolas.eps){height="4cm"} They are displaced parabolas. The relative stability is determined by the minima of the two parabolas $ E_\mathrm{g} \equiv H_\mathrm{g}^{(m)}(q_0 = q_\mathrm{g}) $ and $ E_\mathrm{e} \equiv H_\mathrm{e}^{(m+1)}(q_0 = q_\mathrm{e}) $, where $ q_\mathrm{g} $ and $ q_\mathrm{e} $ give the respective minimum points, and by $ E^\mathrm{V}_\mathrm{g} \equiv H_\mathrm{g}^{(m)}(q_0 = q_\mathrm{e}) $ and $ E^\mathrm{V}_\mathrm{e} \equiv H_\mathrm{e}^{(m+1)}(q_0 = q_\mathrm{g}) $. If the inequality $ E_\mathrm{e} > E^\mathrm{V}_\mathrm{g} $ ($ E_\mathrm{g} > E^\mathrm{V}_\mathrm{e} $) is satisfied, the excited (ground) state at the 0th molecule is unstable and would decay into the ground (“excited”) state. Otherwise, the smaller one of $ E_\mathrm{g} $ and $ E_\mathrm{e} $ corresponds to the stable state, and the larger one to the metastable state. The analysis based on these relations reveals how a cluster of photoexcited molecules is stabilized by the electron-lattice interaction, whose dimensionless strength is the lattice relaxation energy $ S $ divided by $ E_\mathrm{FC} $, and by the coupling $ K $ among the displacements. A similar analysis also explains how a molecular excitation far from the cluster is attracted to the cluster. Mean-field theory ----------------- Nagaosa and Ogawa have extended this theory to general dimensions and considered the optical and thermal transitions. [@Nagaosa_prb89] When the relaxation of the interaction mode $ Q_l $ is rapid, the system is always in equilibrium with a given electronic state. These modes can be integrated out to reduce the original model (\[eq:Hanamura\]) to the Ising model, where the spin-up and spin-down states correspond to the excited and ground states: $$H = -h \sum_l \sigma_l - \frac12 \sum_{l\neq l'} J_{ll'} \sigma_l \sigma_{l'} \;,$$ where the field $ h $, the exchange $ J_{ll'} $, and the phonon dispersion $ \omega_k $ are given by $$h = - E_\mathrm{FC} + \frac{S}{2(1-K)} \;,$$ $$J_{ll'} = \frac{S}{N} \sum_{k} \frac{\cos[\mathbf{k}\cdot(\mathbf{R}_l-\mathbf{R}_{l'})]}{\omega_k^2} \;,$$ $$\omega_k^2 = 1 - \sum_{l'} K_{ll'} \exp[\mathrm{i} \mathbf{k}\cdot(\mathbf{R}_l-\mathbf{R}_{l'})] \;,$$ with $ \mathbf{R}_l $ being the lattice vector of the $ l $th molecule, and $ \omega_{k=0}^2 = 1 - K $. The intersite couplings are treated in the mean-field approximation. For instance, the thermal transition probabilities were assumed to be the product of the attempt frequency and the activation factor, $ P^\mathrm{th}_{{\rm e}\rightarrow{\rm g}} = \tau^{-1} \exp\left[-(E_\mathrm{c}-E_\mathrm{e})/(k_\mathrm{B}T)\right] $ and $ P^\mathrm{th}_{{\rm g}\rightarrow{\rm e}} = \tau^{-1} \exp\left[-(E_\mathrm{c}-E_\mathrm{g})/(k_\mathrm{B}T)\right] $, where $ E_\mathrm{c} $ is the energy of the intersection of the two parabolas. The stochastic dynamics of the system under and after the optical pumping are investigated on the basis of the rate equation for the density of the excited states $ n_\mathrm{e}(t) $, which is a function of time $ t $. When stable and metastable states are present, it is written as $$\frac{\mathrm{d}}{\mathrm{d}t} n_\mathrm{e}(t) = I(t) + P^\mathrm{th}_{{\rm g}\rightarrow{\rm e}} [ 1 - n_\mathrm{e}(t) ] - P^\mathrm{th}_{{\rm e}\rightarrow{\rm g}} n_\mathrm{e}(t) \;,$$ where $ I(t) $ is the optical pumping term, being positive to create the $ e $ state or negative to create the $ g $ state. The activation energies in the exponents of the thermal transition probabilities are quadratic functions of $ n_\mathrm{e}(t) $ in the mean-field approximation. The time evolution of $ n_\mathrm{e}(t) $ after $ n_\mathrm{e}(0) $=0 depends on the strength $ I_0 $ and duration $ t_0 $ of the optical pumping. When $ I_0 $ is less than a threshold value $ I_\mathrm{th} $, however long the optical pumping is applied, $ n_\mathrm{e}(t) $ does not exceed a certain value. After the pumping is switched off, the system always relaxes to the initial state $ n_\mathrm{e}(0) $=0. When $ I_0 $ is greater than $ I_\mathrm{th} $, on the other hand, $ n_\mathrm{e}(t) $ monotonically increases without saturation for $ 0 < t < t_0 $. After the pumping is switched off, $ n_\mathrm{e}(t) $ relaxes to the final value $ n_f $=0 for $ t_0 < t_\mathrm{cr} $ and to $ n_f $=1 for $ t_0 > t_\mathrm{cr} $. The critical value of the duration time $ t_\mathrm{cr} $, above which the system is switched from $ n_i $=0 to $ n_f $=1, is inversely proportional to $ I_0 $ for large $ I_0 $ and diverges as $ t_\mathrm{cr} \sim (I_0-I_\mathrm{th})^{-1/2} $ near the diverging point $ I_\mathrm{th} $. The relaxation times are also discussed. The theory is applied to photoinduced structural transformations in polydiacetylenes. Other theories -------------- The theories above are applied to systems with weakly interacting components to which thermal fluctuations are so dominant that the transition probability is a key quantity. Spin-crossover complexes are such examples, where spins indirectly interact with each other through couplings with lattice displacements. The Hanamura-Nagaosa theory is extended to systems with internal degrees of freedom, [*e.g.*]{}, spin-crossover complexes that are composed of dimers and show two-step transitions. [@Luty_jpsj04] In general, stochastic dynamics can be treated in classical models with master equations [@Romstedt_jpcs98; @Boukheddaden_epjb00; @Hoo_epjb00; @Nishino_prb01] and Monte Carlo simulations. [@Romstedt_jpcs98; @Nishino_prb01; @Kawamoto_apl02; @Sakai_jpsj02] Electronic Theories of Photoinduced Phase Transitions ===================================================== Domino effect ------------- Photoinduced dynamics are sometimes regarded as domino effects. This description is inappropriate for systems showing coherent dynamics, but gives a hint for the mechanisms of photoinduced phase transitions. Koshino and Ogawa theoretically show a domino-like structural transformation in a one-dimensional model composed of localized electrons and classical lattice displacements: [@Koshino_jpsj98] $$\begin{aligned} H = & \sum_j \left[ \left( \begin{array}{cc} \epsilon - 2 \gamma u_j & t_0 \\ t_0 & 2 u_j \end{array} \right) + u_j^2 + \left( \frac{\partial u_j}{\partial t} \right)^2 \right] \nonumber\\ & + \sum_{(i,j)} k_{ij} ( u_i - u_j )^2 \;,\end{aligned}$$ where $ u_j $ and $ \partial u_j/\partial t $ denote the $ j $th displacement and velocity, respectively, $ k_{ij} $ the elastic constants, $ \epsilon $ the energy difference between the two electronic states at $ u_j $=0, $ t_0 $ the overlap integral between them, and $ \gamma $ the electron-lattice coupling constant. After a photon is absorbed by an electron at a site, the lattice locally relaxes under friction to the minimum on the adiabatic potential. Then, after a photon is spontaneously emitted, three evolution patterns are shown to exist. When the elastic couplings $ k_{ij} $ are too weak, the local structural distortion remains locally. When they are too strong, the initial local distortion is pulled back by surrounding lattice displacements to the position before the absorption. Only when they are intermediately strong and short-ranged will the initial local distortion trigger a global structural transformation as the domino effect (Fig. \[fig:domino\]). ![(Color online) Schematic structural transformation. The standing and falling dominoes correspond to the displacements accompanied with stable and metastable localized electronic states.[]{data-label="fig:domino"}](fig2_domino.eps){height="1.5cm"} Adiabatic relaxation path ------------------------- Photoinduced phase transitions are often related to multistability, where different electronic phases are stable and metastable. Such stability may be manifested by the presence of a first-order phase transition induced by changing temperature or pressure. The metastable phase may be hidden in equilibrium so that it is manifested only by photoirradiation, which gives a much higher energy to the system than thermal energies. The photoinduced phase transition dynamics depend on the electronic state and especially on how the initial state is prepared. For this purpose, relevant itinerant-electron models are necessary. They have finite transfer integrals between neighboring molecular orbitals. They are off-diagonal elements of the model Hamiltonian, giving transition [*amplitudes*]{}. In contrast, the stochastic dynamics within classical statistical models are governed by transition [*probabilities*]{}, which are often given by Boltzmann factors satisfying detailed balance. Organic molecular tetrathiafulvalene-chloranil (TTF-CA) crystals are the most intensively studied. The donor TTF and acceptor CA molecules are alternately stacked along the most conducting axis. At low temperature or under high pressure with contraction, these molecules become ionic due to the long-range Coulomb interaction. [@Torrance_prl81] Otherwise, they are neutral due to the large difference between the ionization potential of the donor molecule and the electron affinity of the acceptor molecule. In the ionic phase at ambient pressure, these molecules are dimerized. Both ionic-to-neutral and neutral-to-ionic transitions are photoinduced, as shown by the optical reflectivity. [@Koshihara_jpcb99] It is found that photons with 2.3 eV corresponding to intramolecular excitations at TTF sites are much more efficient for generation of neutral domains in the ionic background than those with 0.6 eV corresponding to intrachain charge-transfer excitations. [@Suzuki_prb99] The latter clearly have a threshold in the photoexcitation density, below which a macroscopic neutral domain cannot be generated. Thus, the photoinduced dynamics are very sensitive to the initial condition of the electronic state, from which the lattice relaxation starts. These observations have motivated theoretical studies based on itinerant-electron models. Huai [*et al.*]{} have clarified the adiabatic relaxation path from the Franck-Condon-type photoexcited state above the ionic ground state to the formation of a large neutral domain, using a one-dimensional extended Peierls-Hubbard model with alternating potentials (and the interchain interaction mentioned later): [@Huai_jpsj00] $$\begin{aligned} H = & -t_0 \sum_{l,\sigma} \left( c^{\dagger}_{l,\sigma}c_{l+1,\sigma} + \mathrm{h.c.} \right) + \frac{\Delta}{2} \sum_{l} (-1)^{l} n_{l} \nonumber \\ & + U \sum_{l} n_{l,\uparrow} n_{l,\downarrow} + \sum _{l} V_{l}(q_l,q_{l+1}) \delta n_{l} \delta n_{l+1} \nonumber \\ & + \sum _{l} \frac{S_1}{2}( q_{l} - q_{l+1} )^{2} + \sum _{l} \frac{S_2}{4}( q_{l} - q_{l+1} )^{4} \;, \label{eq:Huai}\end{aligned}$$ where $ c^{\dagger}_{l,\sigma} $ ($ c_{l,\sigma} $) is the creation (annihilation) operator of an electron with spin $\sigma$ at site $l$, $ n_{l,\sigma} = c^{\dagger}_{l,\sigma} c_{l,\sigma} $, $ n_{l} = n_{l,\uparrow} + n_{l,\downarrow} $, $ \delta n_{l} = n_{l} - 2 $ for odd $l$, $ \delta n_{l} = n_{l} $ for even $l$, $ q_{l} $ is the dimensionless displacement of the $l$th molecule along the chain from its equidistant position. The distance between the $l$th and $(l+1)$th molecules is given by $ d_{l,l+1} = d_0 (1+q_{l+1}-q_l) $, where $d_0$ is the average intermolecular distance. The donor and acceptor molecules are located at odd and even sites, respectively. The nearest-neighbor repulsion strength is assumed to depend nonlinearly on the intermolecular distance as $$V_{l}(q_l,q_{l+1}) = V_0 + \beta_{1} ( q_{l} - q_{l+1} ) + \beta_{2} ( q_{l} - q_{l+1} )^{2} \;,$$ where $ V_0 $ is for the regular lattice, and $ \beta_{1} $ ($ \beta_{2} $) is the linear (quadratic) coefficient. The parameter $ t_0 $ denotes the transfer integral, $ \Delta $ the level difference between the neighboring orbitals in the neutral limit, and $ U $ the on-site repulsion strength. The elastic energy is expanded up to the fourth order: the parameters $ S_{1} $ and $ S_{2} $ are the linear and nonlinear elastic constants. In the neutral phase, the orbital of the donor molecule is almost doubly occupied, while that of the acceptor molecule is almost empty \[Fig. \[fig:NI\](a)\]. The total charge of the donor molecule at site $ l $ is $ +(2- n_l)= - \delta n_{l} $, while that of the acceptor molecule is $ -n_l = - \delta n_{l} $. In the ionic phase, both orbitals are almost singly occupied \[Fig. \[fig:NI\](b)\]. ![Schematic electronic and lattice structures in (a) neutral and (b) ionic phases.[]{data-label="fig:NI"}](fig3_NI.eps){height="4cm"} The values of the eight parameters are so determined that they reproduce the [*ab initio*]{} estimation of the transfer integral, the ionicity in the ionic phase, that in the neutral phase, the degree of dimerization in the ionic phase, the energies and the relative strength of the charge-transfer absorption peaks in the ionic phase, and the charge-transfer absorption energy in the neutral phase. The lattice relaxation path starting from the Franck-Condon state with a single charge-transfer exciton is assumed to be described by a domain, $$q_l = (-1)^l q_0 \left\{ 1 + \Delta q \left[ \tanh \left( \theta ( \mid l \mid - \frac{l_0}{2} ) \right) - 1 \right]\right\} \;,$$ where $ (-1)^l q_0 $ denotes the uniform dimerization in the ionic state, $ \Delta q $ the amplitude of a local distortion induced by an excited domain, $ \theta $ the width of the domain boundary, and $ l_0 $ the domain size (Fig. \[fig:N\_in\_I\]). ![Schematic excited domains in ionic background.[]{data-label="fig:N_in_I"}](fig4_N_in_I.eps){height="3.5cm"} An interchain interaction between displacements is introduced to confine the domain in the ionic background: $$H_\mathrm{inter}^\mathrm{ph} = \sum_l \sum_{i=1,2,3} K_i \left( q_l - (-1)^l q_0 \right)^{2i} \;,$$ where $ q_l $ denotes the displacement of a central chain surrounded by ionic environmental chains, and $ K_i $ the 2$ i $th coefficient with respect to the distortion in the central chain. The adiabatic energy surface of the ground state and that of the first excited state are drawn as a function of the amplitude $ \Delta q $ and the size $ l_0 $ of the domain with the width of the domain boundary $ \theta $ optimized. [@Huai_jpsj00] The energy surface of the ground state has a plateau corresponding to a neutral domain separated from the ground state by a low potential barrier. In the energy surface of the first excited state, a local minimum corresponding to the charge-transfer exciton appears around the Franck-Condon state. Another local minimum appears and corresponds to a neutral domain, which are separated from the Franck-Condon state by such a high potential barrier that the absorption of a single photon with 0.6 eV cannot overcome it. The domain is stable only when its size exceeds a critical value because the energy required for creation of the two domain boundaries makes a small domain unstable in the excited state. The spin- and charge-density distributions of the ground and excited states with a neutral domain are very similar: they are different only at the domain boundaries. Because a single charge-transfer exciton cannot trigger the formation of a macroscopic neutral domain, a large excess energy is needed at the very beginning of the relaxation to overcome the high barrier, for charge-transfer excitons to proliferate, and finally to form a macroscopic neutral domain. The process described above is only for the early nucleation stage of the photoinduced macroscopic transformation. Transition dynamics in TTF-CA ----------------------------- Along with the progress in experimental techniques, the pump-probe reflection spectroscopy acquires higher time resolution. Ultrafast optical switching from the ionic to the neutral state is observed. [@Iwai_prl02] Neutral strings are produced within 2 ps after a resonant excitation of the charge transfer band at 0.65 eV polarized along the stacking axis. Just below the transition temperature, a macroscopic conversion from the ionic to the neutral states is achieved within 20 ps. It is accompanied with coherent motion of the macroscopic neutral-ionic domain boundary with a much longer period than that of the optical lattice oscillation. Theoretically, real-time dynamics need to be calculated for further understanding. First, results are obtained by solving the time-dependent Schrödinger equation of an electron-lattice model without electron-electron interaction, where two charge-density-wave states with opposite polarizations are degenerate. [@Iwano_prb02] The lattice part is treated classically. The photoconversion is shown to be nonlinear. During the photoconversion process, macroscopic oscillations are observed, which are essentially different from the linear modes in equilibrium. Then, by adding the kinetic energy of the displacements, $$H_\mathrm{kin} = \frac12 m_{l} \dot{q}_{l}^{2} \;, \label{eq:kin}$$ with the $ l $th molecular mass $ m_l $, to the model (\[eq:Huai\]), Miyashita [*et al.*]{} have calculated real-time dynamics of ionicity and dimerization within the unrestricted Hartree-Fock approximation after the occupancy of orbitals in the ionic ground state is initially changed. [@Miyashita_jpsj03] As the number of orbitals whose occupancy is changed increases, i.e., with increasing photoexcitation density, more neutral domains are created. Above the threshold excitation density, the neutral phase is finally achieved. After the photoexcitation, ionic domains with wrong polarization generally appear, which reduce the correlation length of the staggered lattice displacement. As the degree of initial lattice disorder increases, more solitons appear between the ionic domains with opposite polarizations, which obstruct the growth of neutral domains and slow down the transition. From the Fourier transform of the ionicity as a function of time, three characteristic time scales are observed: rapid charge-transfer excitations corresponding to the peak energy of a charge-transfer exciton in the random-phase-approximation spectra, slow charge-transfer excitations strongly coupled with optical lattice oscillations, and much slower excitations due to the collective motion of neutral-ionic domain boundaries. The last corresponds to the experimentally observed, coherent motion of the macroscopic neutral-ionic domain boundary, [@Iwai_prl02] and is shown to be sensitively affected by collisions with the solitons. When the fraction of conversion from the ionic to the neutral phases is plotted as a function of the number of excited electrons, the curve evolves with time (Fig. \[fig:I\_to\_N\_fraction\]). ![Conversion from ionic to neutral phases, as a function of number of excited electrons in 100-site chain. [@Miyashita_jpsj03] The nonlinearity is strengthened with the passing of time after the photoexcitation, $ t $=1.5$/\omega$, 15$/\omega$, 100$/\omega$ and 250$/\omega$, where $ \omega $ denotes the bare phonon frequency. The initial lattice temperature is $ T $=$ 10^{-2}t_0 $.[]{data-label="fig:I_to_N_fraction"}](fig5_I_to_N_fraction.eps){height="5cm"} Immediately after the photoexcitation, it is close to a linear function. Above the threshold number, the fraction monotonically increases with time until the photoinduced phase transition is completed. Below the threshold, the fraction decreases and eventually becomes very small. This result is consistent with the experimental finding. [@Iwai_prl02] More recently, experimental results have been summarized covering both the ionic-to-neutral and the neutral-to-ionic transitions with different excitation energies, excitation densities and temperatures. [@Okamoto_prb04] The ionic-to-neutral transition induced by the resonant excitation of the charge-transfer band proceeds with (1) initial formation of a confined one-dimensional neutral domain, (2) multiplication of such domains to semi-macroscopic neutral states by 20 ps, and (3) evolution in the direction normal to the sample surface. The dynamics of the neutral-to-ionic transition that is induced by a similar excitation are clearly different from the above. Although one-dimensional ionic domains are initially produced by lights, they decay within 20 ps even if the excitation density is high. The initial conversion fraction is a linear function of the excitation density. Thus, we are motivated to incorporate a pulse of oscillating classical electric field, $$E(t) = E_\mathrm{ext} \sin \omega_\mathrm{ext} t \;,$$ with amplitude $ E_\mathrm{ext} $ and frequency $ \omega_\mathrm{ext} $ for $ 0 < t < 2 \pi N_\mathrm{ext} / \omega_\mathrm{ext} $ with integer $ N_\mathrm{ext} $, into the Peierls phase of the transfer integral. The time-dependent Schrödinger equation is solved again for the one-dimensional extended Peierls-Hubbard model with alternating potentials. Now, the frequency, the amplitude and the duration of the pulse can be explicitly and independently varied. The dynamics of the ionic-to-neutral transition are indeed qualitatively different from those of the neutral-to-ionic transition. Difference between I-to-N and N-to-I transitions ------------------------------------------------ When the dimerized ionic phase is photoexcited, the threshold behavior is observed again by plotting the final ionicity as a function of the increment of the total energy, i.e., as a function of the number of absorbed photons (Fig. \[fig:I\_to\_N\_threshold\]). [@Yonemitsu_jpsj04a] ![Final ionicity as a function of number of absorbed photons in 100-site chain, when electric field of strength $ eE_\mathrm{ext}d_0 $=5$ \omega $ and of frequency $ \omega_\mathrm{ext} $=28$ \omega $ below linear-absorption peak at 32$ \omega $ is applied to [*ionic*]{} phase. [@Yonemitsu_jpsj04a] $ \omega $ denotes the bare phonon frequency. The initial lattice temperature is $ T $=$ 10^{-3}t_0 $.[]{data-label="fig:I_to_N_threshold"}](fig6_I_to_N_threshold.eps){height="5cm"} Above the threshold photoexcitation density, the system finally enters a neutral state with equidistant molecules. This is the case even after the electric field is switched off. Once produced by the field, a neutral domain grows spontaneously. Its growth cannot be stopped by switching off the field. When compared with the threshold behavior derived from the rate equation, [@Nagaosa_prb89] we do not observe a quantity corresponding to the threshold intensity $ I_\mathrm{th} $. Even if the amplitude of the pulse is small (i.e., even if the pulse is weak), a transition seems finally achieved by setting the pulse duration long enough. The deterministic dynamics are more efficient than the stochastic ones because of the restricted energy dissipation. When compared with the domino effect, the motion of the neutral domain is similar in that its growth is spontaneous once triggered. However, it is quite different from the motion of dominoes under strong friction on the adiabatic potential, [@Koshino_jpsj98] which is far from being coherent. For a given amount of the increment of the total energy, the lattice dynamics relative to the charge dynamics in the ionic-to-neutral transition depend on the character of the pulse. When the pulse is strong and short, the charge transfer takes place on the same time scale with the disappearance of dimerization. When the pulse is weak and long, the dimerization-induced ferroelectric polarization is disordered first to restore the inversion symmetry, and then the charge transfer takes place to bring the system to a neutral state. The difference between the two time scales increases as the pulse is weakened. Between these time scales, a paraelectric ionic phase is realized. In the case of intramolecular excitations at 1.55 eV, such a new ionic phase with disordered polarizations is suggested to appear by time-resolved X-ray diffraction. [@Guerin_cp04] This similarity may be due to the fact that, when the pulse is weak, the supplied energy is not directly used to transfer charge density along the chain. It is furthermore shown theoretically that infrared light polarized along the chain can also induce the ionic-to-neutral transition if the excitation density exceeds a critical value, which is currently beyond the value experimentally achieved. When the neutral phase is photoexcited, the linear behavior is observed by plotting the final ionicity as a function of the increment of the total energy (Fig. \[fig:N\_to\_I\_linear\]). [@Yonemitsu_jpsj04b] ![Final ionicity as a function of number of absorbed photons in 100-site chain, when electric field of strength $ eE_\mathrm{ext}d_0 $=2$ \omega $ and of frequency $ \omega_\mathrm{ext} $=28$ \omega $ below linear-absorption peak at 30$ \omega $ is applied to [*neutral*]{} phase. [@Yonemitsu_jpsj04b] $ \omega $ denotes the bare phonon frequency. The initial lattice temperature is $ T $=$ 10^{-2}t_0 $.[]{data-label="fig:N_to_I_linear"}](fig7_N_to_I_linear.eps){height="5cm"} The neutral-to-ionic transition proceeds in an uncooperative manner. If the oscillating electric field is turned off before the transition is completed, the ionicity remains intermediate unless the energy dissipation is taken into account. The growth of a metastable domain is not spontaneous but forced by the external field. The final state is determined merely by how many photons are absorbed to increase the ionicity. This result is consistent with the experimental finding in the neutral-to-ionic transition induced by intrachain charge-transfer photoexcitations. [@Okamoto_prb04] By intramolecular excitations, on the other hand, a transition from the neutral phase to the ionic phase with three-dimensionally ordered ferroelectric polarizations is shown to be induced, as clearly demonstrated by X-ray diffraction. [@Collet_science03] The reason for this difference is unclear yet. The qualitative difference described above between the two transitions is not due to the different inversion symmetry between the dimerized ionic and the regular neutral states because the difference survives even if the electron-lattice coupling is turned off so that both ionic and neutral states have even parity with respect to the space inversion. The most plausible reason for the difference is the fact that the ionic state is a Mott insulator caused by the electron correlation, whereas the neutral state is a band insulator caused by the band structure. In a Mott insulator, all the electrons are so correlated that any one of them cannot easily make the first move. However, above the threshold excitation density, they cannot tolerate the increased total energy. Once some electrons move, they trigger the collective motion. One electron is transferred after another like dominoes. The collective dynamics in the ionic phase are due to the electron-electron interaction that realizes the Mott insulator phase. They cannot be described by stochastic approaches of refs.  and . In a band insulator, electrons move individually. The supplied energy is merely consumed to transfer electrons from the donor to acceptor molecules almost independently. They do not strongly influence the motion of other electrons. Thus far, we have focused on the dynamics in purely one-dimensional systems. Although the short-time behavior may not suffer from interchain interactions, they are important for the condition of coherence, which can be manifested by a double pulse. [@Yonemitsu_jpsj04c] The coherent motion of the macroscopic neutral-ionic domain boundary [@Iwai_prl02] is possible only when interchain interactions are strong enough. Although the effect of interchain elastic couplings is considered in the context of confinement of a metastable domain, [@Huai_jpsj00] interchain electron-electron interactions are much stronger. The intramolecular charge distribution in the TTF and CA molecules is calculated by the [*ab initio*]{} quantum chemical method. [@Kawamoto_prb01] The interchain electrostatic energies between neighboring molecules are smaller than but comparable to the intrachain ones. Because the molecules are tilted, the relative repulsion strengths are very different from the naive expectations based on the intermolecular distance measured in terms of the center of mass of each molecule. In the ionic phase, the interchain attractive interaction between the neighboring donor and acceptor molecules along the $ b' $ axis is larger than any interchain repulsive one between the donors or between the acceptors. The consequent electrostriction is theoretically shown [@Kishine_prb04] to be responsible for the pressure-temperature phase diagram of the TTF-CA complex containing the paraelectric ionic phase [@Lemee_prl97] and for the discontinuous contraction along the $ b $ axis. [@Luty02] Then, we add to the model (\[eq:Huai\]) with (\[eq:kin\]) interchain electron-electron interactions, [@Yonemitsu_jp_jltp_unpublished] $$H_\mathrm{inter}^\mathrm{el} = \sum_{l,j} \left( U_{\perp} \delta n_{l,j} \delta n_{l,j+1} + V_{\perp1} \delta n_{l,j+1} \delta n_{l+1,j} \right) \;,$$ where $ j $ is the chain index, and the donor-acceptor coupling strength $ V_{\perp1} $ is slightly larger than the donor-donor or acceptor-acceptor one $ U_{\perp} $. [@Kawamoto_prb01] When the ionic phase is photoexcited, the transition dynamics depend on the strengths of the interchain couplings. With weak interchain couplings, for instance, halves of the values for TTF-CA, the interchain correlation is very weak during the transition. A first neutral domain is easily created by a low density of photons. If the electric field is switched off immediately after this absorption, the residual chains remain ionic. By continuing the application of the electric field, next domains are created in the neighboring chains. Many photons are finally needed to complete the transition into the neutral phase. With strong interchain couplings comparable to those in TTF-CA, the interchain correlation is strong during the transition. To create a first neutral domain, more photons need to be absorbed. Once a domain is nucleated, nearby domains are almost simultaneously created in the neighboring chains to grow cooperatively. Their growth cannot be stopped even if the electric field is switched off immediately after the appearance of the first domain. As a consequence, the density of absorbed photons needed to complete the transition is lower than in the weak-coupling case. Because of the strong interchain correlation, the growth of metastable domains coherently proceeds, which is consistent with the experimentally observed, coherent motion of the macroscopic neutral-ionic domain boundary. [@Iwai_prl02] Photoinduced phases in MX and MMX chains ---------------------------------------- Besides the TTF-CA complex described above, many other molecular materials have been or are currently studied to show their own transition dynamics. Some of them show very asymmetric transitions between two different electronic phases. In a halogen-bridged nickel-chain compound, the ultrafast photoinduced Mott transition from a charge-transfer insulator to a metal is observed. [@Iwai_prl03] Photoinduced optical responses of one-dimensional half-filled Hubbard models with and without alternating potentials are studied by the exact diagonalization method to show a large shift of the spectral weight to low energies including the Drude component in the Mott insulator phase, which are similar to the response caused by chemical doping. [@Maeshima_jpsj05] In halogen-bridged binuclear platinum-chain compounds, R$_4$\[Pt$_2$(pop)$_4$I\]$ n $H$_2$O (pop=P$_2$O$_5$H$_2^{2-}$) with cation R, the transition from a charge density wave (CDW) to a charge-polarization (CP) state is photoinduced much more easily than its inverse transition. [@Matsuzaki_prl03] The electronic phases of these compounds are controlled by the distance between the neighboring binuclear units $ d_\mathrm{MXM} $ both when the counterion R is varied and when the applied pressure is varied. The nonmagnetic CDW phase is observed for small $ d_\mathrm{MXM} $, while the paramagnetic CP phase for large $ d_\mathrm{MXM} $. The charge density is disproportioned among binuclear units in the CDW phase, and within each binuclear unit in the CP phase (Fig. \[fig:CDW\_CP\]). In both phases, the iodine ion approaches the platinum ion with less electron density. ![Schematic electronic and lattice structures in MMX chains.[]{data-label="fig:CDW_CP"}](fig8_CDW_CP.eps){height="4.4cm"} The dependences of the electronic phase and its physical properties on $ d_\mathrm{MXM} $ are well explained by perturbation theories from the strong-coupling limit as well as the exact diagonalization method. [@Kuwabara_jmc01] The mechanisms of the photoinduced phase transitions are clarified by taking the charge transfer processes explicitly into account [@Yonemitsu_prb03] in a one-dimensional two-band three-quarter-filled Peierls-Hubbard model, $$\begin{aligned} H = & - t_{\rm MM} \sum_{l, \sigma} ( c^\dagger_{2l-1, \sigma} c_{2l, \sigma} + \mathrm{h.c.} ) \nonumber \\ & - t_{\rm MXM} \sum_{l, \sigma} ( c^\dagger_{2l, \sigma} c_{2l+1, \sigma} + \mathrm{h.c.} ) \nonumber \\ & + \beta \sum_l q_l ( n_{2l+1} - n_{2l} ) + U_{\rm M} \sum_l n_{l, \uparrow} n_{l, \downarrow} \nonumber \\ & + K_{\rm MX} \sum_l q_l^2 + (M/2) \sum_l \dot{q}_l^2 \;,\end{aligned}$$ where $ c^\dagger_{l, \sigma} $ creates an electron with spin $ \sigma $ at site $ l $, $ n_{l, \sigma} = c^\dagger_{l, \sigma} c_{l, \sigma} $, and $ n_l = \sum_\sigma n_{l, \sigma} $. The binuclear unit contains two M (M=Pt) sites, 2$ l-$1 and 2$ l $. The displacement of the X (X=I) ion between the two M sites 2$ l $ and 2$ l $+1, relative to that in the undistorted structure, is denoted by $ q_l $. The transfer integral within the unit is denoted by $ t_{\rm MM} $, while that between the neighboring units through the X $ p_z $ orbital by $ t_{\rm MXM} $. The energy level of the M $ d_{z^2} $ orbital depends on $ q $ with linear coefficient $ \beta $. The on-site repulsion strength is denoted by $ U_{\rm M} $, the elastic constant for the MX bond by $ K_{\rm MX} $, and the mass of the X ion by $ M $. With increasing pressure corresponding to decreasing $ d_\mathrm{MXM} $, $ \beta $ increases because it originates from the electrostatic energy between the positive M and the negative X ions. This is why the discontinuous transition from the CP to the CDW phases is induced by pressure. [@Matsuzaki_prl03] It is true that the dependence of the threshold excitation density in the photoinduced CDW-to-CP transition on the pressure, which controls the relative stability of these phases, can be explained qualitatively with their diabatic potentials, i.e., in a stochastic manner. However, the asymmetry in these transitions requires explanation based on the explicit charge transfer processes (Fig. \[fig:CDW\_to\_CP\]): in the CP state, low-energy charge-transfer processes occur only within a binuclear unit, and consequently they do not lead to the CDW state. ![(Color online) Schematic ground and low-energy photoexcited states in MMX chains. [@Yonemitsu_prb03] In the CDW phase, the charge is transferred to the neighboring unit to form a transient CP domain. In the CP phase, the charge is transferred within the unit to merely reverse the polarization.[]{data-label="fig:CDW_to_CP"}](fig9_CDW_to_CP.eps){height="5.5cm"} Even if the CP phase is irradiated by higher-energy photons, charge transfer among the binuclear units takes place incoherently. Since the interunit charge transfer is energetically unfavorable, the CDW domains never proliferate. Furthermore, the transition amplitude connecting the degenerate CDW phases with opposite polarizations is much smaller than those connecting the degenerate CP phases. The coherence recovery is achieved with much more difficulty in the CDW phase. Concluding Remarks ================== We have summarized the history of theories for photoinduced phase transitions, focusing on the development from stochastic to coherent transition dynamics. Both dynamics are characterized by cooperativity. Coherent dynamics are direct consequences of interactions among electrons and/or between electrons and lattice displacements. Photoinduced transition dynamics between different electronic phases of different materials will be studied further in appropriate models for understanding each mechanism and for exploring new possibilities. Current and future issues to be solved will include differences between photoinduced and thermally induced phases, origins of different nonlinear characters, coherence originating from phonons and/or that from electronic excitations, relative dynamics among spins, charge and lattice, relations to quantum phase transitions or quantum natures of phonons, and how to control the dynamics by tuning the laser pulse. Acknowledgment {#acknowledgment .unnumbered} ============== The authors are grateful to P. Huai, J. Kishine, M. Kuwabara, T. Luty, N. Maeshima, N. Miyashita and H. Zheng for theoretical collaboration, and especially H. Cailleau, S. Koshihara and H. 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--- abstract: 'Let $R$ be a commutative ring with identity. Let $\Gamma(R)$ be a graph with vertices as elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if $Ra+Rb=R$. In this paper we consider a subgraph $\Gamma_2(R)$ of $\Gamma(R)$ which consists of non-unit elements. We look at the connectedness and the diameter of this graph. We completely characterize the diameter of the graph $\Gamma_2(R)\setminus{\mbox{J}}(R)$. In addition, it is shown that for two finite semi-local rings $R$ and $S$, if $R$ is reduced, then $\Gamma(R)\cong\Gamma(S)$ if and only if $R\cong S$.' address: - | Hamid Reza Maimani\ Department of Mathematics, University of Tehran, Tehran, Iran\ and Institute for Theoretical Physics and Mathematics (IPM). - | Maryam Salimi\ Department of Mathematics, University of Tehran, Tehran, Iran. - | Asiyeh Sattari\ Department of Mathematics, University of Tehran, Tehran, Iran. - | Siamak Yassemi\ Department of Mathematics, University of Tehran, Tehran, Iran\ and Institute for Theoretical Physics and Mathematics (IPM). author: - Hamid Reza Maimani - Maryam Salimi - Asiyeh Sattari - Siamak Yassemi title: Comaximal graph of commutative rings --- [^1] [^2] [^3] Introduction ============ For the sake of completeness, first we state some definitions and notions used throughout to keep this paper as self contained as possible. We define a [*coloring*]{} of a graph $G$ to be an assignment of colors (elements of some set) to the vertices of $G$, one color to each vertex, so that adjacent vertices are assigned distinct colors. If $n$ colors are used, then the coloring is referred to as an [*$n$-coloring*]{}. If there exists an $n$-coloring of a graph $G$, then $G$ is called $n$-colorable. The minimum $n$ for which a graph $G$ is $n$-colorable is called the [*chromatic number*]{} of $G$, and is denoted by $\chi(G)$. For a graph $G$, the [*degree*]{} of a vertex $v$ in $G$ is the number of edges of $G$ incident with $v$. Recall that a graph is said to be [*connected*]{} if for each pair of distinct vertices $v$ and $w$, there is a finite sequence of distinct vertices $v=v_1,\cdots,v_n=w$ such that each pair $\{v_i,v_{i+1}\}$ is an edge. Such a sequence is said to be a path and the distance, ${\mbox{d}}(v,w)$, between connected vertices $v$ and $w$ is the length of the shortest path connecting them. The [*diameter*]{} of a connected graph is the supremum of the distances between vertices. The diameter is 0 if the graph consists of a single vertex and a connected graph with more than one vertex has diameter 1 if and only if it is complete; i.e., each pair of distinct vertices forms an edge. An [*$r$-partite*]{} graph is one whose vertex set can be partitioned into $r$ subsets so that no edge has both ends in any one subset. A [*complete $r$-partite*]{} graph is one in which each vertex is joined to every vertex that is not in the same subset. The [*complete bipartite*]{} (i.e., $2$-partite) graph with part sizes $m$ and $n$ is denoted by $K_{m,n}$. A graph in which each pair of distinct vertices is joined by an edge is called a [*complete*]{} graph. We use $K_n$ for the complete graph with $n$ vertices. A [*clique*]{} of a graph is its maximal complete subgraph and the number of vertices in the largest clique of graph G, denoted by ${\mbox{clique}\,}(G)$, is called the [*clique number*]{} of $G$. Obviously $\chi(G)\ge{\mbox{clique}\,}(G)$ for general graph $G$ (see [@CO page 289]). Let $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ be two graphs with disjoint vertices set $V_i$ and edges set $E_i$. The join of $G_1$ and $G_2$ is denoted by $G=G_1\vee G_2$ with vertices set $V_1\cup V_2$ and the set of edges is $E_1\cup E_2\cup\{xy|x\in V_1\,\, \mbox{and}\,\, y\in V_2\}$. From now on let $R$ be a commutative ring with identity. In [@B], Beck considered ${\Gamma}(R)$ as a graph with vertices as elements of $R$, where two different vertices $a$ and $b$ are adjacent if and only if $ab=0$. He studied finitely colorable rings with this graph structure and showed that $\chi({\Gamma}(R))=clique({\Gamma}(R))$ for certain classes of rings. in [@AN], Anderson and Naseer have made further study of finitely colorable rings and have given an example of a finite local ring with $5={\mbox{clique}\,}({\Gamma}(R))<\chi({\Gamma}(R))=6$. In [@SB], Sharma and Bhatwadekar define another graph on $R$, $\Gamma(R)$, with vertices as elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if $Ra+Rb=R$. They showed that $\chi(\Gamma(R))<\infty$ if and only if $R$ is a finite ring. In this case $\chi(\Gamma(R))={\mbox{clique}\,}(\Gamma(R))=t+\ell$, where $t$ and $\ell$, respectively, denote the number of maximal ideals of $R$ and the number of units of $R$. In this paper, we study further the graph structure defined by Sharma and Bhatwadekar. Let $\Gamma_1(R)$ be the subgraph of $\Gamma(R)$, generated by the units of $R$, and $\Gamma_2(R)$ be the subgraph of $\Gamma(R)$ generated by non-unit elements. In section 2, it is shown that the graph $\Gamma_2(R)\setminus{\mbox{J}}(R)$ is a complete bipartite if and only if the cardinal number of the set ${\mbox{Max}\,}(R)$ is equal 2 (see Theorem 2.2). Also we show that $R$ is a finite product of quasi-local rings if and only if $R$ is clean and ${\mbox{clique}\,}(\Gamma_2(R)\setminus{\mbox{J}}(R))<\infty$ (see Theorem 2.5). In section 3, the main result says that $\Gamma_2(R)\setminus{\mbox{J}}(R)$ is connected and ${\mbox{diam}\,}(\Gamma_2(R)\setminus{\mbox{J}}(R))\le 3$ (see Theorem 3.1). In addition, we completely characterize the diameter of the graph $\Gamma_2(R)\setminus{\mbox{J}}(R)$. In the final section, it is shown that for two finite semi-local rings $R$ and $S$, if $R$ is reduced, then $\Gamma(R)\cong\Gamma(S)$ if and only if $R\cong S$ (see Corollary 4.6). Bipartite graphs ================ Throughout this paper $R$ will be a commutative ring with identity, ${\mbox{U}}(R)$ its group of units, ${\mbox{J}}(R)$ its Jacobson radical, and ${\mbox{I}}(R)$ its set of idempotents. A ring $R$ is said to be quasi-local if it has a unique maximal ideal; if ${\mathfrak{m}}$ is the unique maximal ideal of $R$, we will often write $(R,{\mathfrak{m}})$. Let $\Gamma(R)$ be the graph represented by $R$ with definition of Sharma-Behatwadekar. Let $\Gamma_1(R)=<{\mbox{U}}(R)>$ and $\Gamma_2(R)=<R\setminus{\mbox{U}}(R)>$ be the subgraphs of $\Gamma(R)$. Then it is easy to see that $\Gamma(R)=\Gamma_1(R)\vee\Gamma_2(R)$. The following hold: - $\Gamma_1(R)$ is a complete graph. - $a\in{\mbox{J}}(R)$ if and only if $\deg_{\Gamma_2(R)}a=0$. Since (a) is clear we just prove (b). Suppose $a\in{\mbox{J}}(R)$. Then for any ${\mathfrak{m}}\in{\mbox{Max}\,}(R)$, $a\in{\mathfrak{m}}$. If $\deg_{\Gamma_2(R)}a\neq 0$, then there exists $b\in\Gamma_2(R)$ such that $Ra+Rb=R$. On the other hand there exists ${\mathfrak{n}}\in{\mbox{Max}\,}(R)$ with $b\in{\mathfrak{n}}$ and so $1\in{\mathfrak{n}}$ that is a contradiction. Conversely, assume that $\deg_{\Gamma_2(R)}a=0$. Assume contrary $a\notin{\mbox{J}}(R)$. Then there exists ${\mathfrak{m}}\in{\mbox{Max}\,}(R)$ such that $a\notin{\mathfrak{m}}$. Thus $Ra+{\mathfrak{m}}=R$. Therefore there exists $b\in{\mathfrak{m}}$ such that $Ra+Rb=R$. This contradicts our assumption. In the following we study the cases where $\Gamma_2(R)\setminus{\mbox{J}}(R)$ is complete bipartite graph and where this graph is $n$-partite. We know that each $x\in{\mbox{U}}(R)$ is adjacent to every vertex of $\Gamma(R)$ and it is shown that each $x\in{\mbox{J}}(R)$ is an isolated vertex of $\Gamma_2(R)$. Thus the main part of the graph $\Gamma(R)$ is the subgraph $\Gamma_2(R)\setminus{\mbox{J}}(R)$. For this reason the main aim of this paper is to study the structure of this subgraph. The following are equivalent: - $\Gamma_2(R)\setminus{\mbox{J}}(R)$ is a complete bipartite graph. - The cardinal number of the set ${\mbox{Max}\,}(R)$ is equal 2. (ii)$\Rightarrow$(i). Let ${\mbox{Max}\,}(R)=\{{\mathfrak{m}}_1,{\mathfrak{m}}_2\}$. Thus the vertices set of $\Gamma_2(R)\setminus{\mbox{J}}(R)$ is equal to the set $({\mathfrak{m}}_1\setminus{\mathfrak{m}}_2)\cup({\mathfrak{m}}_2\setminus{\mathfrak{m}}_1)$. Let $a\in{\mathfrak{m}}_1\setminus{\mathfrak{m}}_2$ and $b\in{\mathfrak{m}}_2\setminus{\mathfrak{m}}_1$. Thus $Ra+Rb\nsubseteq{\mathfrak{m}}_1\cup{\mathfrak{m}}_2$ and so $Ra+Rb=R$. (i)$\Rightarrow$(ii). Suppose $\Gamma_2(R)\setminus{\mbox{J}}(R)$ is a complete bipartite graph with two part $V_1$ and $V_2$. Set $M_1=V_1\cup{\mbox{J}}(R)$ and $M_2=V_2\cup{\mbox{J}}(R)$. We show that $M_1$ and $M_2$ are two maximal ideals of $R$ and ${\mbox{Max}\,}(R)=\{M_1,M_2\}$. Let $x,y\in M_1=V_1\cup{\mbox{J}}(R)$. Consider the following three cases: [*Case 1.*]{} Assume that $x,y\in{\mbox{J}}(R)$. Then $x-y\in{\mbox{J}}(R)$ and so $x-y\in M_1$. [*Case 2.*]{} Assume that $x\in{\mbox{J}}(R)$ and $y\in V_1$. Then $x-y\notin{\mbox{J}}(R)$. If $x-y\in{\mbox{U}}(R)$, then $Rx+Ry=R$ and so we obtain a contradiction. If $x-y\in M_2$, then $x-y\in V_2$ and so $R(x-y)+Ry=R$. Thus $Rx+Ry=R$ which is a contradiction. Therefore $x-y\in V_1\subseteq M_1$. [*Case 3.*]{} Assume that $x,y\in V_1$. If $x-y\in {\mbox{J}}(R)$ then there is nothing to prove. Therefore we assume $x-y\notin{\mbox{J}}(R)$. With the same proof as case 2, the assertion holds. Now suppose that $r\in R$ and $x\in M_1$. If $x\in{\mbox{J}}(R)$, then clearly $rx\in M_1$. Therefore suppose that $x\notin{\mbox{J}}(R)$. Also $rx$ is not unit. Suppose that $rx\in M_2$. Then $rx\in V_2$ and so $R(rx)+Rx=R$. Thus $x$ is a unit element of $R$ which is a contradiction. So $rx\in M_1$. To now we showed that $M_1$ is an ideal of $R$. By the structure of $\Gamma(R)$, for any $x\in R\setminus M_1$, we have $M_1+Rx=R$. This implies that $M_1$ is a maximal ideal. With the same argument $M_2$ is a maximal ideal of $R$. Now if $N\in{\mbox{Max}\,}(R)$ then $N\subseteq M_1\cup M_2$ and so $N=M_1$ or $N=M_2$.\ This finishes the proof. Let $n>1$. Then the following hold: - If $|{\mbox{Max}\,}(R)|=n<\infty$, then the graph $\Gamma_2(R)\setminus{\mbox{J}}(R)$ is $n$-partite. - If the graph $\Gamma_2(R)\setminus{\mbox{J}}(R)$ is $n$-partite, then $|{\mbox{Max}\,}(R)|\le n$. In this case if the graph $\Gamma_2(R)\setminus{\mbox{J}}(R)$ is not $(n-1)$-partite, then $|{\mbox{Max}\,}(R)|=n$. (a). Let ${\mbox{Max}\,}(R)=\{{\mathfrak{m}}_1,\cdots,{\mathfrak{m}}_n\}$ and set $V_1={\mathfrak{m}}_1\setminus{\mbox{J}}(R)$ and for each $i\ge 2$, $V_i={\mathfrak{m}}_i\setminus\cup_{t=1}^{t=i-1}{\mathfrak{m}}_t$. Using Prime Avoidence Theorem, $V_i\neq\varnothing$ for each $i$. It is easy to see that any two vertices belong to $V_i$ are not adjacent. (b). Let $V_1,\cdots,V_n$ be the $n$ parts of vertices of $\Gamma_2(R)\setminus{\mbox{J}}(R)$. Assume contrary $|{\mbox{Max}\,}(R)|>n$ and let ${\mathfrak{m}}_1,\cdots,{\mathfrak{m}}_{n+1}\in{\mbox{Max}\,}(R)$. For any $i$, choose $x_i\in{\mathfrak{m}}_i\setminus\cup_{j\neq i}{\mathfrak{m}}_j$. Then it is easy to see that $\{x_1,\cdots,x_{n+1}\}$ is a clique in $\Gamma_2(R)\setminus{\mbox{J}}(R)$. By the Pigeon Hole Principal, two of $x_i$’s should belong to one of $V_i$’s, that is a contradiction. Therefore $|{\mbox{Max}\,}(R)|\le n$. Now suppose that $\Gamma_2(R)\setminus{\mbox{J}}(R)$ is not $(n-1)$-partite and $|{\mbox{Max}\,}(R)|=m<n$. By (a) the graph will be $m$-partite and this is a contradiction. Let $R$ be a ring with $|{\mbox{Max}\,}(R)|\ge 2$. Then the following hold: If $\Gamma_2(R)\setminus{\mbox{J}}(R)$ is a complete $n$-partite graph, then $n=2$. If there exists a vertex of $\Gamma_2(R)\setminus{\mbox{J}}(R)$ which is adjacent to every other vertex then $R\cong\mathbb Z_2\times F$, where $F$ is a field. Let ${\mathfrak{m}}_1, {\mathfrak{m}}_2$ be two maximal ideals of $R$. Since the elements of ${\mathfrak{m}}_i\setminus{\mbox{J}}(R)$ are not adjacent, and at least one element of ${\mathfrak{m}}_1\setminus{\mbox{J}}(R)$ is adjacent to one element of ${\mathfrak{m}}_2\setminus{\mbox{J}}(R)$, so ${\mathfrak{m}}_1\setminus{\mbox{J}}(R)$ and ${\mathfrak{m}}_2\setminus{\mbox{J}}(R)$ are subsets of two distinct parts of $\Gamma_2(R)$. That means $({\mathfrak{m}}_1\setminus{\mbox{J}}(R))\cap({\mathfrak{m}}_2\setminus{\mbox{J}}(R))=\varnothing$. We claim that ${\mbox{J}}(R)={\mathfrak{m}}_1\cap{\mathfrak{m}}_2$. In other case, ${\mbox{J}}(R)\subsetneq({\mathfrak{m}}_1\cap{\mathfrak{m}}_2)$ and so there exists $x\in({\mathfrak{m}}_1\cap{\mathfrak{m}}_2)\setminus{\mbox{J}}(R)$. This elements belongs to ${\mathfrak{m}}_1\setminus{\mbox{J}}(R)$ and ${\mathfrak{m}}_2\setminus{\mbox{J}}(R)$, that is a contradiction. Therefore we obtain ${\mbox{J}}(R)={\mathfrak{m}}_1\cap{\mathfrak{m}}_2$ and so $|{\mbox{Max}\,}(R)|=2$. Now by theorem 2.2 we have $n=2$. (b). Let $x$ be a non-unit element of $R$ which is adjacent to every other vertex of $\Gamma_2(R)\setminus{\mbox{J}}(R)$. Since $x$ is comaximal with each nonunit outside the Jacobson radical, $x$ is idempotent, ${\mbox{J}}(R)=(0)$ and ${\mathfrak{m}}=\{0,x\}$ is a maximal ideal. Thus for each nonunit $s\in R\setminus{\mathfrak{m}}$, having $xR+sR=R$ implies $(1-x)sR=(1-x)R$ and this implies $(1-x)R=F$ is a field. Hence $R\cong{\Bbb Z}_2\times F$. A ring is said to be [*clean*]{} if each of its elements can be written as the sum of a unit and an idempotent cf. [@N] (see also [@AC]). For example, a quasi-local ring is clean. The following result gives an application of Sharma-Bhatwadegar graph to characterize clean rings. For the ring $R$, the following are equivalent: - $R$ is a finite product of quasi-local rings. - $R$ is clean and ${\mbox{clique}\,}(\Gamma_2(R)\setminus{\mbox{J}}(R))$ is finite. (a)$\Rightarrow$(b). Let $R=R_1\times\cdots\times R_n$ where each $R_i$ is quasi-local with unique maximal ideal ${\mathfrak{m}}_i$. Set $N_i=R_1\times\cdots\times R_{i-1}\times{\mathfrak{m}}_i\times R_{i+1}\times\cdots\times R_n$ for any $i=1,\cdots,n$. Then each $N_i$ belongs to ${\mbox{Max}\,}(R)$. For any $i$ choose $x_i\in N_i\setminus\cup_{\ell\neq i}N_{\ell}$. Then it is easy to see that $Rx_i+Rx_j=R$ for all $i\neq j$. In addition by using the Pigeon Hole Principal, there is no any $n+1$ family elements of $\cup_{i=1}^nN_i$ which pairwise adjacent. Thus ${\mbox{clique}\,}(\Gamma_2(R)\setminus{\mbox{J}}(R))=n<\infty$.\ On the other hand, each $R_i$ is clean and so by [@AC Proposition 2(3)], $R$ is clean. (b)$\Rightarrow$(a). Suppose that ${\mbox{clique}\,}(\Gamma_2(R)\setminus{\mbox{J}}(R))$ is finite. Assume contrary that ${\mbox{I}}(R)$ has infinitely many idempotent elements then by [@SB Lemma 2.1] there exists an infinite sequence $e_1,e_2,\cdots$ of non-trivial idempotents in $\Gamma_2(R)\setminus{\mbox{J}}(R)$ such that the set $S$ consisting of elements $e_i$ ($i\ge 1$) is an infinite clique. This is a contradiction. Diameter of the graph ===================== In this section we completely characterize the diameter of $\Gamma_2(R)\setminus{\mbox{J}}(R)$. The following result shows that $\Gamma_2(R)\setminus{\mbox{J}}(R)$ is a connected graph and its diameter is not greater than 3. The graph $\Gamma_2(R)\setminus{\mbox{J}}(R)$ is connected, and ${\mbox{diam}\,}(\Gamma_2(R)\setminus{\mbox{J}}(R))\le 3$. Let $a,b\in(R\setminus{\mbox{U}}(R))\setminus{\mbox{J}}(R)$. We consider two cases: [*Case 1*]{} Assume that $ab\notin{\mbox{J}}(R)$. There exists $x\in(R\setminus{\mbox{U}}(R))\setminus{\mbox{J}}(R)$ such that $Rab+Rx=R$. Thus $Ra+Rx=Rb+Rx=R$. So we have the path $a$—$x$—$b$, and so ${\mbox{d}}(a,b)\le 2$. [*Case 2*]{} Assume that $ab\in{\mbox{J}}(R)$. Set $S_a=\{{\mathfrak{m}}|{\mathfrak{m}}\in{\mbox{Max}\,}(R), a\in{\mathfrak{m}}\}$ and $S_b=\{{\mathfrak{m}}|{\mathfrak{m}}\in{\mbox{Max}\,}(R), b\in{\mathfrak{m}}\}$. Clearly, ${\mbox{Max}\,}(R)=S_a\cup S_b$. Now suppose that $x$ is adjacent to $a$ in $\Gamma_2(R)$. Then $x\notin{\mbox{J}}(R)$. If $a\in{\mathfrak{m}}$, then $x\notin{\mathfrak{m}}$ and so $x\in{\mathfrak{n}}\in{\mbox{Max}\,}(R)$, where ${\mathfrak{n}}\in S_b\setminus S_a$. Thus $bx\notin{\mbox{J}}(R)$. Therefore by Case 1, ${\mbox{d}}(b,x)\le 2$ and so ${\mbox{d}}(a,b)\le 3$. ${\mbox{diam}\,}(\Gamma_2(R)\setminus{\mbox{J}}(R))=1$ if and only if $R\cong\mathbb Z_2\times\mathbb Z_2$. If ${\mbox{diam}\,}(\Gamma_2(R)\setminus{\mbox{J}}(R))=1$, then $\Gamma_2(R)\setminus{\mbox{J}}(R)$ is complete graph. Thus there exists a vertex of $\Gamma_2(R)\setminus{\mbox{J}}(R)$ which is adjacent to every other vertex. Therefore $R\cong\mathbb Z_2\times F$, where $F$ is a field by Proposition 2.4(b). Since $\Gamma_2(R)\setminus{\mbox{J}}(R)$ is complete, we have that $F\cong\mathbb Z_2$. Thus $R\cong\mathbb Z_2\times\mathbb Z_2$. It is easy to see that for $R\cong\mathbb Z_2\times\mathbb Z_2$, ${\mbox{diam}\,}(\Gamma_2(R)\setminus{\mbox{J}}(R))=1$. Our next result characterizes the graphs where ${\mbox{diam}\,}(\Gamma_2(R)\setminus{\mbox{J}}(R))=2$. Assume that $R$ is not local. The diameter of the graph $\Gamma_2(R)\setminus{\mbox{J}}(R)$ is equal 2 if and only if one of the following holds: - ${\mbox{J}}(R)$ is a prime ideal. - $|{\mbox{Max}\,}(R)|=2$ and $R\ncong\mathbb Z_2\times\mathbb Z_2$. Note that if ${\mbox{J}}(R)$ is prime and $R$ is semi-local (i.e. has finite number of maximal ideals), then $R$ will be local. Let ${\mbox{J}}(R)$ be a prime ideal and $a,b\notin{\mbox{J}}(R)$. Then $ab\notin{\mbox{J}}(R)$, and so by the same argument as Theorem 3.1, there exists $x\in(R\setminus{\mbox{U}}(R))\setminus{\mbox{J}}(R)$, such that $a$—$x$—$b$ is a path. Thus ${\mbox{diam}\,}(\Gamma_2(R)\setminus{\mbox{J}}(R))\le 2$. If ${\mbox{diam}\,}(\Gamma_2(R)\setminus{\mbox{J}}(R))=1$, then by previous result $R\cong\mathbb Z_2\times\mathbb Z_2$. But ${\mbox{J}}(\mathbb Z_2\times\mathbb Z_2)$ is not a prime ideal. That is a contradiction. Now let $|{\mbox{Max}\,}(R)|=2$ and $R\ncong\mathbb Z_2\times\mathbb Z_2$, then by Theorem 2.2, $\Gamma_2(R)\setminus{\mbox{J}}(R)$ is a complete bipartite graph where at least one of the parts has at least two elements. Therefore ${\mbox{diam}\,}(\Gamma_2(R)\setminus{\mbox{J}}(R))=2$. Conversely, suppose that ${\mbox{diam}\,}(\Gamma_2(R)\setminus{\mbox{J}}(R))=2$ and ${\mbox{J}}(R)$ is not prime. let $a,b\notin{\mbox{J}}(R)$ but $ab\in{\mbox{J}}(R)$. We claim that $a$ and $b$ are adjacent. Otherwise, there exists $t$ in $\Gamma_2(R)$ such that $Ra+Rt=Rb+Rt=R$. Thus $Rab+Rt=R$ and so $ab\notin{\mbox{J}}(R)$ which is a contradiction. Therefore $Ra+Rb=R$ and so for some $p,q\in R$, $pa+qb=1$. Set $S=R/{\mbox{J}}(R)$ and $a_1=pa+{\mbox{J}}(R)$ and $b_1=qb+{\mbox{J}}(R)$. Then $a_1b_1=0$ and $a_1+b_1=1_S$. Therefore $a_1$ and $b_1$ are idempotent elements in $S$, and so $S=Sa_1\oplus Sb_1$. We will show that $Sa_1$ is a field. Let $0\neq x\in Sa_1$ and $0\neq y\in Sb_1$. Then there exists $\alpha, \beta$ such that $\alpha x+\beta y=1_S$ and so $\alpha(a_1+b_1)x+\beta(a_1+b_1)y=1_S$. Thus $(\alpha a_1)x+(\beta b_1)y=1_S$. On the other hand $a_1+b_1=1_S$ and so $(\alpha a_1)x=a_1$ and $(\beta b_1)y=b_1$. Therefore $x$ is a unit in $Sa_1$. Therefore $Sa_1$ and $Sb_1$ are fields and so $|{\mbox{Max}\,}(S)|=2$. therefore $|{\mbox{Max}\,}(R)|=2$. Let $R=\mathbb Z_n$ where $n=p_1^{\ell_1}\cdots p_r^{\ell_r}$. Assume $r\ge 3$. Let $x=p_1^{\ell_1}\cdots p_{r-1}^{\ell_{r-1}}$ and $y=p_2^{\ell_2}\cdots p_{r}^{\ell_{r}}$. Then $x$ and $y$ are not adjacent. Also if $x,y$ are adjacent $z$, then $(z,x)=(z,y)=1$, which is impossible. We have $Rx+Rp_r^{\ell_r}=R=Rp_r^{\ell_r}+Rp_1^{\ell_1}=Rp_1^{\ell_1}+Ry$. Hence there is path $x$—$p_r^{\ell_r}$—$p_1^{\ell_1}$—$y$. So ${\mbox{diam}\,}(\Gamma_2(\mathbb Z_n)\setminus{\mbox{J}}(\mathbb Z_n))=3$. Assume that $r=2$. In this case we have two maximal ideals $M_1=<p_1>$ and $M_2=<p_2>$. Then $\Gamma(R)$ is a complete bipartite graph and so ${\mbox{diam}\,}(\Gamma_2(\mathbb Z_n)\setminus{\mbox{J}}(\mathbb Z_n))=2$. Assume that $r=1$. Then $R$ is local and so $\Gamma_2(\mathbb Z_n)\setminus{\mbox{J}}(\mathbb Z_n)$ is empty graph. Let $R$ be an infinite PID. Then for any two non-unit elements $a,b$, there exists a prime element $p$ such that $p$ does not divide $a$ and $b$. Therefore $Ra+Rp=Rb+Rp=1$. So ${\mbox{d}}(a,b)\le 2$ and hence ${\mbox{diam}\,}(\Gamma_2(R)\setminus{\mbox{J}}(R))=2$. isomorphisms ============ Recall that two graphs $G$ and $H$ are isomorphic, denoted by $G\cong H$, if there is a bijection $\varphi : G\to H$ of vertices such that the vertices $x$ and $y$ are adjacent in $G$ if and only if $\varphi(x)$ and $\varphi(y)$ are adjacent in $H$. In this section, we consider the following question: If $R$ and $S$ are two rings with $\Gamma(R)\cong\Gamma(S)$, then do we have $R\cong S$? The following examples show that the above question is not valid in general. Let $R=\mathbb Z_4$ and $S=\mathbb Z_2[x]/(x^2)$. Then by simple computation we can see that $\Gamma(R)\cong\Gamma(S)(\cong K_2\vee\bar{K_2})$. But $\mathbb Z_4$ and $\mathbb Z_2[x]/(x^2)$ are not isomorphic. Let $R=\mathbb Z_8$ and $S=\mathbb Z_2[x]/(x^3)$. Then $\Gamma(R)\cong\Gamma(S) (\cong K_4\vee\bar{K_4})$. But $R$ and $S$ are not isomorphic. Let $R=\mathbb Z_2[x]/(x^3)$ and $S=\mathbb Z_2[x,y]/(x^2,y^2,xy)$. Then $\Gamma(R)\cong\Gamma(S) (\cong K_4\vee \bar{K_4})$). But $R$ and $S$ are not isomorphic. In the following theorem we give a partial answer to the above question. Let $\{(R_i,{\mathfrak{m}}_i)\}_{i=1}^m$ and $\{(S_j,{\mathfrak{n}}_j)\}_{j=1}^n$ be two finite families of finite quasi-local rings, and let $R=R_1\times\cdots\times R_m$ and $S=S_1\times\cdots\times S_n$. If $\Gamma(R)\cong\Gamma(S)$ then $m=n$ and there is a permutation $\sigma$ on the set $\{1,2,\ldots ,m\}$ such that $|R_i/{\mathfrak{m}}_i|=|S_{\sigma(i)}/{\mathfrak{n}}_{\sigma(i)}|$ for each $i=1,\cdots, m$, and hence $R_i/{\mathfrak{m}}_i\cong S_{\sigma(i)}/{\mathfrak{n}}_{\sigma(i)}$. In particular, if $\Gamma(R)\cong\Gamma(S)$ and each $R_i$ is a finite field, then each $S_j$ is also a finite field and $R_i\cong S_{\sigma(i)}$ for each $i=1,\cdots, m$, and thus $R\cong S$. First note that since $|{\mbox{Max}\,}(R)|=n$ and $|{\mbox{Max}\,}(S)|=m$, and $\Gamma(R)\cong\Gamma(S)$, we have that $m=n$. Set $M_i=R_1\times\cdots\times R_{i-1}\times{\mathfrak{m}}_i\times R_{i+1}\times\cdots\times R_m$ and $N_i=S_1\times\cdots\times S_{i-1}\times{\mathfrak{n}}_i\times S_{i+1}\times\cdots\times S_m$ for each $i=1,\cdots,m$. For any $i=1,\cdots ,m$ let $x_i\in M_i\setminus\cup_{j\neq i}M_j$. Clearly $\{x_1,\cdots, x_m\}$ is a clique in $\Gamma_2(R)$. Suppose that $\nu(x_i)$ is equal to the number of vertices of $\Gamma(R)$ which are not adjacent to $x_i$. Then $$\nu(x_i)=|R-\{y|y=x_i\,\,\mbox{or $y$ adjacent to $x_i$}\}|=|M_i|.$$ For each $i=1,\cdots ,m$, let $y_{\sigma(i)}\in N_{\sigma(i)}$ be the image of $x_i$ under the graph isomorphism. Then $\{y_1,\cdots,y_m\}$ is a clique in $\Gamma_2(S)$ and $y_{\sigma(i)}\in N_{\sigma(i)}\setminus\cup_{j\neq \sigma(i)}N_j$. It is easy to see that $\nu(y_{\sigma(i)})=|N_{\sigma(i)}|$. Thus $|M_i|=|N_{\sigma(i)}|$ and so $|R_i/{\mathfrak{m}}_i|=|S_{\sigma(i)}/{\mathfrak{n}}_{\sigma(i)}|$. Therefore $R/{\mathfrak{m}}_i\cong S_{\sigma(i)}/{\mathfrak{n}}_{\sigma(i)}$. In particular, if $\Gamma(R)\cong\Gamma(S)$ and each $R_i$ is a finite field. Thus ${\mbox{J}}(R)=(0)$ and so ${\mbox{J}}(S)=(0)$ and hence ${\mathfrak{n}}_i=(0)$ for each $i$. Therefore each $S_j$ is also a finite field and $R_i\cong S_{\sigma(i)}$ for each $i\in I$, and thus $R\cong S$. The following example shows that the condition “$R_i$ is a field” is necessary in Theorem 4.4. Let $R=\mathbb Z_2\times \mathbb Z_8$, and $S=\mathbb Z_4\times\mathbb Z_4$. Then ${\mbox{J}}(R)\cong\{0\}\times\mathbb Z_4$ and ${\mbox{J}}(S)\cong\mathbb Z_2\times\mathbb Z_2$. Also ${\mbox{U}}(R)=\{1\}\times\{1,3,5,7\}$ and ${\mbox{U}}(S)=\{1,3\}\times\{1,3\}$. Since $|{\mbox{Max}\,}(R)|=|{\mbox{Max}\,}(S)|=2$, then $\Gamma_2(R)\setminus{\mbox{J}}(R)\cong\Gamma_2(S)\setminus{\mbox{J}}(S)\cong K_{4,4}$. Therefore $\Gamma(R)\cong\Gamma(S)\cong(K_{4,4}\cup \bar{K_4})\vee K_4$. But it is clear that $R\ncong S$. Let $R$ and $S$ be two finite semi-local rings and let $R$ be reduced. Then $\Gamma(R)\cong\Gamma(S)$ if and only if $R\cong S$. It is clear that $R=F_1\times\cdots\times F_n$, where $F_i$ is a field for any $i=1,\cdots,n$. Now the assertion holds from Theorem 4.4. The following result shows that there exists a copy of $\Gamma(R/{\mbox{J}}(R))$ in the structure of $\Gamma(R)$. This result obtains that for two rings $R$ and $S$ if $\Gamma(R)\cong\Gamma(S)$, then $R/{\mbox{J}}(R)\cong S/{\mbox{J}}(S)$. The following hold: - If $a$ is adjacent to $b$ in $\Gamma(R)$, then every element of $a+{\mbox{J}}(R)$ is adjacent to $b+{\mbox{J}}(R)$. - The elements of $a+{\mbox{J}}(R)$ are adjacent if and only if $a$ is an unit. In this case, each element of $a+{\mbox{J}}(R)$ is unit too. - There exists a copy of $\Gamma(R/{\mbox{J}}(R))$ in the structure of $\Gamma(R)$. In particular, if $\Gamma(R)\cong\Gamma(S)$, then $R/{\mbox{J}}(R)\cong S/{\mbox{J}}(S).$ (a). Suppose that $Ra+Rb=R$. Let $x=a+r_1$ and $y=b+r_2$ for $r_1,r_2\in{\mbox{J}}(R)$. Then there exists elements $s,t\in R$ such that $sa+tb=1$. So $$\begin{array}{rl} sx+ty &\, =sa+tb+sr_1+tr_2\\ &\, = 1-(-sr_1-tr_2). \end{array}$$ Since $-sr_1-tr_2\in{\mbox{J}}(R)$, so $sx+ty$ is a unit and hence $Rx+Ry=R$. (b). Let $(a+r_1)$ be adjacent to $a+r_2$. Then $R(a+r_1)+R(a+r_2)=R$ and so there exist $s,t\in R$ such that $s(a+r_1)+t(a+r_2)=1$. This implies that $(s+t)a=1-(r_1s+r_2t)$ and so $(s+t)a$ is invertible. Therefore $a$ is invertible. (c). Choose a distinct representation $\{a_i\}$ from the cosets of $R/{\mbox{J}}(R)$. By parts (a) and (b), we have $<\{a_i\}>\cong\Gamma(R/{\mbox{J}}(R))$. Let $\varphi:\Gamma(R)\to\Gamma(S)$ be an isomorphism. Then $\varphi(<\{a_i\}>)=\{b_i\}$ and $<\{b_i\}>=S/{\mbox{J}}(S)$ and hence the assertion is easily obtained. Acknowledgments {#acknowledgments .unnumbered} =============== This paper was finalized when H.R. Maimani and S. Yassemi were visiting the Tata Institute of Fundamental Research (TIFR) under TWAS-UNESCO Associateship Scheme. It is a pleasure to thank both TWAS and TIFR for financial support and hospitality. The authors would like to thank S.M. Bhatwadekar for the stimulating discussions. The authors wish to thank an anonymous referee, whose comments have improved this paper. [10]{} D. D. Anderson and V. P. Camillo, *Commutative rings whose elements are a sum of a unit and idempotent*, Comm. Algebra **30** (2002), 3327–3336. D. D. Anderson and M. Naseer, *Beck’s coloring of a commutative ring*, J. Algebra **159** (1993), 500–514. I. Beck, *Coloring of Commutative Rings*, J. Algebra **116** (1988), no. 1, 208-226. G. Chartrand, O. R. Oellermann, [*Applied and Algorithmic Graph Theory*]{}, McGraw-Hill, Inc., New York, 1993. G. De Marco and A. Orsatti, *Commutative Rings in Which Every Prime Ideal is Contained in a Unique Maximal Ideal*, Proc. Amer. Math. Soc., **30** (1971), 459–466. W. K. Nicholson, *Lifting Idempotents and Exchange Rings*, Trans. Amer. Math. Soc., **229** (1977), 269–278. P. K. Sharma and S. M. Bhatwadekar, *A note on Graphical representation of rings*, J. Algebra **176** (1995), 124–127. [^1]: \* Corresponding author. Department of Mathematics, University of Tehran, P.O. Box 13145–448 Tehran, Iran [^2]: H. R. Maimani was supported in part by a grant from IPM No. 85050117 [^3]: S. Yassemi was supported was supported by a grant from IPM No. 85130214

--- abstract: 'The swept-field experiments on magnetic molecular solids such as 8 are studied using Monte Carlo simulations. A kinetic equation is developed to understand the phenomenon. It is found that the simulations provide a quantitatively accurate account of the experiments. The kinetic equation provides a similarly accurate account except at very low sweep velocities, where it fails modestly. This failure is due to the neglect of short-range correlations between the dipolar magnetic fields seen by the molecular spins. Both the simulations and the kinetic equation provide a good understanding of the distribution of these dipolar fields.' author: - Erik Lenferink - Avinash Vijayaraghavan - Anupam Garg title: 'Low-Temperature Magnetization Dynamics of Magnetic Molecular Solids in a Swept Field' --- Introduction {#intro} ============ As a prototype of magnetic molecular solids, consider the one generally known as 8. This material consists of molecules of \[Fe$_8$O$_2$(OH)$_{12}$(tacn)$_6$\]Br$_8$(H$_2$O)$_9$, in which the Fe(III) ions of one molecule are well separated from those of a neighboring molecule. At low temperatures, each molecule has a spin $S = 10$, a corresponding all-spin magnetic moment of $g\mu_B S$ with $g \simeq 2$, and an Ising-like anisotropy which translates into an energy barrier of $~22\,$K [@fe8props; @gat06]. Consider next the experiment of Wernsdorfer and Sessoli [@wer99], of which a partial and simplified description is as follows. At low temperatures ($T \ltwid 100\,$mK) they first saturate the magnetization of the sample by applying an external magnetic field $H_z$ along the Ising or $z$ axis, and also apply a magnetic field $H_x$ along the hard magnetic axis transverse to the easy axis. They then sweep $H_z$ so as to reverse the magnetization, and measure the rate at which the spins reverse. Astonishingly, this rate turns out to be an oscillatory function of $H_x$, even though the energy barrier and the angle between the two energy minimizing orientations of the spin are both monotonically decreasing functions of $H_x$. The interpretation of this phenomneon is that the sweeping of $H_z$ is inducing Landau-Zener-Stückelberg (LZS) transitions [@lan77; @kay84] between the two lowest states on opposite sides of the energy barrier, which take place at a rate proportional to $\Dta^2$, where $\Dta$ is the tunnel splitting between these states. The rate oscillates because $\Dta(H_x)$ oscillates with $H_x$, in accord with theory [@los92; @vdel92; @gar93]. 8 is just one of $\sim10^3$ magnetic molecular solids that are now known, and which have been the object of much study over the last fifteen years or so. Their main characteristics are that the spin of one molecule is large at low temperatures, and to a good first approximation, one may treat the spins on different molecules as non-interacting. (A comprehensive and authoritative review of the entire field is contained in Ref. [@gat06]. Shorter reviews may be found in Refs. [@can99; @vil00; @fri10].) These solids are often known as single-molecule-magnets (SMM’s), because many phenomena may be understood, at least qualitatively, in terms of the properties of the total ground state spin of a single molecule in a suitable crystal field, via an effective spin Hamiltonian that contains anisotropy terms reflecting the overall symmetry of the molecule and its local environment. The oscillatory tunnel splitting mentioned above is one such phenomenon. Another is that the hysteresis loops are sharply stepped, where the steps coincide with crossings of energy levels on opposite sides of the energy barrier [@fri96; @tho96]. This single molecule behavior has, unsurprisingly, led to suggestions and proposals for using these materials in devices [@bog07], but they are a new class of magnetic materials and worthy of study in their own right for the novel phenomena they display. Among the various experimental tools employed to study the low temperature magnetization dynamics of such solids, the swept-field or Landau-Zener-Stückelberg (LZS) protocol has proven to be one of the most fruitful. When the sweep is sufficiently rapid, the accompanying change in the magnetization can be interpreted in terms of LZS transitions as already mentioned, and thereby provides a measurement of the tunneling amplitude between energy levels on opposite sides of a barrier. Tunnel splittings measured by this technique are as low as $10^{-8}$K in temperature units. Such low splittings are beyond the reach of any other method. When the sweep rate is slow, on the other hand, the interpretation is not clear-cut. It is essential to consider the dipole-dipole interactions between different molecules, and the transition is influenced by the collective dynamics of all the spins. Indeed, in this case, the SMM designation falls short, and a more detailed analysis is called for. Collective, dipole-coupled dynamics of the spins are also seen in several magnetization-relaxation type experiments [@san97; @ohm98; @wer99b; @tho99; @wer00; @wer00b; @tup02]. Theoretical discussions and Monte-Carlo simulations of these experiments have been provided by [@gat06; @pro98ab; @cuc99; @fer0304; @vij12], and many aspects of the experiments are understood. The same is not true of the swept-field experiments, and we are unaware of any work that addresses the collective dynamics aspects. Theories that include the dipole-dipole interactions in a purely static [@kec07] or mean-fieldy [@avag09] way cannot explain all the experimental behavior, especially that at low sweep fields. It is important to have a more complete theory since experimentalists often interpret data in terms of the original LZS analysis [@ram08; @wer08; @ram08b; @wer08b]. In light of the fact that the spins in molecular magnets are subject to strong environmental influences, especially the dipolar couplings to other molecular spins, and that the LZS theory is based on fully quantum mechanically coherent time evolution, it is not at all clear that the LZS description is at all correct a priori. Indeed, as explained in Ref. [@avag09], it is a fortunate fact that it can be used even at high sweep velocities. Wernsdorfer has argued [@wer08], correctly in our view, that the LZS formula should not be used without first verifying that one is in the fast sweep limit, and that it can in general only yield a lower bound on the tunnel splitting. While the use of this formula may give one qualitative proof of the presence of a geometrical phase in the tunneling spectrum [@gar93], one is on shaky ground if one uses the extracted tunnel splittings to build detailed models of intramolecular magnetic interactions. It is clear that a recent debate about these matters [@ram08; @wer08; @ram08b; @wer08b; @ram09; @wer09] is caused by an incomplete understanding of how molecular magnets respond to a swept magnetic field. It is the purpose of this paper to provide a more complete analysis of the swept-field protocol including the collective behavior of all the spins. We study the coupled system of molecular spins by both Monte-Carlo simulations and by developing and solving a kinetic equation for the joint probability distribution of the spin (up or down) of a molecule and the local magnetic field seen by that spin. We compare the results of these two approaches with each other, and with experiments. We find that the Monte Carlo simulation agrees with the experimental results over almost the entire range of sweep velocities, giving us confidence that the physical picture underlying the simulations is correct. The kinetic equation agrees with the simulations up to moderately low sweep velocities, but fails at still slower velocities, although the failure is not as severe as for previous theoretical treatments [@avag09]. In all cases, we find that a model due to Kayanuma [@kay84] of a spin strongly coupled to a bath does a better job of explaining the physics than the LZS model, but it is never superior to the kinetic equations or the Monte Carlo simulations. The main advantage of the Kayanuma model is that it yields a simple formula for the magnetization reversal in terms of the tunneling matrix element, whereas we are unable to write a similar formula for the results of out kinetic equation or Monte Carlo simulations. We note here that a kinetic equation was written down in [@pro98ab], but this equation is formal in that the collision or interaction term between pairs of spins is written in terms of the two-site distribution of spin and local magnetic field. It is thus like the first of the equations of the BBGKY hierarchy, which are exact, but which cannot be solved without truncating the hierachy. The simplest truncation is at the level of the single-site distribution itself, and amounts to assuming that the two-site distribution factorizes. However, rather than obtain the kinetic equation in this formal way, we derive the equation by considering the various interaction processes which cause this distribution to change. This approach gives greater physical insight into the approximations made. The single-site approximation breaks down at ultra slow sweeps as we explain, leading to the divergence from the simulations mentioned above. Including two-site correlations would lead to an immensely more complex numerical problem, however, and experience with other problems suggests that if two-site correlations are indeed important, then we are faced with a humdinger, as three-site, four-site, and higher-order multi-site correlations are also likely to be important. The rest of the paper is organized as follows. In , we provide background information on 8, the LZS and Kayanuma models, and on a theoretical model for SMM’s that incorporates the influence of the environment [@avag09; @pro96]. We also describe how this model differs from our earlier rate equation approach [@vij12]. In , we discuss our Monte Carlo protocol, and the results of our simulations. The kinetic equation is discussed in . We first derive this equation, and then discuss how various partial sums of the collision term in this equation may be physically interpreted. We also discuss how we integrate this equation numerically, and the results of this integration. We summarize our conclusions briefly in . Background Information and Models {#background} ================================= Independent and Coherent Spin Approximation {#indy_spin} ------------------------------------------- Let us first assume (counterfactually) that the interactions between different molecular spins may be ignored. The dynamics of the spin of a single molecule would then be governed by an anisotropy Hamiltonian of the form = -k\_2 S\_z\^2 + \_ - g\_B , where $\bS = (S_x, S_y, S_z)$ is the total spin of the molecule, the first term in $\ham$ is the leading anisotropy, $g$ is a [*g*]{}-factor, $\mu_B$ is the Bohr magneton, $\bH$ is the external magnetic field, and $\ham_{\perp}$ is a term in the transverse spin components that is off-diagonal in the $S_z$ basis and reflects the intrinsic higher-order anisotropies of the molecule. For this reason it is time-reversal invariant, and so contains only even powers of the spin components. In 8, for example, \_ = (k\_1 - k\_2) S\_x\^2 - C(S\_+\^4 + S\_-\^4). \[Hperp\_fe8\] The various parameters in $\ham$ are well known for the most highly studied molecules. For 8, $S=10$, $g\simeq 2$, $k_1 \simeq 0.33\,$K, $k_2 \simeq 0.22\,$K, and $C \simeq 29\,\mu$K. In 12, by contrast, the anisotropy is tetragonal based on the symmetry of the molecule, but there are believed to be biaxial terms of the same type as in arising from chemical disorder, variable waters of crystallization, variant chemical species, etc., and there are also additional longitudinal terms such as $B S_z^4$. These details are largely immaterial for this paper, since we are interested in ultra-low temperatures only. We may therefore focus on the lowest two energy levels, $m = \pm S$, and presuppose that there is an amplitude per unit time, $-i\Dta/2\hbar$, to tunnel between these levels. We also assume that the effects of the transverse fields $H_x$ and $H_y$ have been incorporated in $\Dta$, and only the longitudinal field, $H_z$, is not. It is this field that is swept. Under these conditions, each molecular spin may be described as a two-level system governed by an effective single-spin Hamiltonian, \_[eff]{} = \(t) &\ & -(t)\ , \[ham2by2\] where $\eps(t)$ is the energy of the $m=S$ state relative to that of the $m=-S$ state, given by (t) = 2Sg \_B H\_z(t). \[eps\_vs\_H\] We shall refer to $\eps$ as the [*bias*]{} on the spin [@bias]. In writing , we have defined the zero of $H_z$ so that the levels $\pm S$ are degenerate at $H_z = 0$. In the laboratory, a nonzero offset field may be required to cancel demagnetizing fields and bring this degeneracy about; $H_z$ is supposed measured from this offset. We will interchangeably refer to the two states either as $m = \pm S$ states, or as pseudospin-1/2 states $\kup \equiv \ket{m= +S}$ and $\kdn \equiv \ket{m = -S}$ states, or as “up" and “down" states. We now suppose that the longitudinal field is swept at a steady rate so that (t) = t, and the spin is in the lower energy state, $\kup$ as $t \to -\infty$. Then, the probability that spin will flip into the $\kdn$ state as $t\to \infty$ is given by the classic LZS formula P\_[LZS]{} = 1 - (-\^2/2 || ). The limits of fast sweep, $\dot\eps \gg \Dta^2$, and slow sweep, $\dot\eps \ll \Dta^2$, are worth noting: P\_[LZS]{} , & \^2, 1, & \^2. The Kayanuma model {#kayanuma} ------------------ As mentioned in , the LZS model is inadequate to describe the actual experiments. An alternative single-spin model is that of Kayanuma [@kay84] wherein the bias field $\eps(t)$ has added to it a fluctuating part $\eta(t)$, which is taken as a Gaussian random process. In the limit where this process has very large amplitude and is delta-function correlated, corresponding to a very rapidly fluctuating bias, the probability that the spin will end up in the state $\kdn$ having started in $\kup$ is found to be P\_[K]{} = (1 - (-\^2/ || )). \[p\_kaya\] The fast and low sweep limits of the reversal probability are now given by P\_[K]{} , & \^2, , & \^2. This is identical to the LZS formula for fast sweeps, but very different for slow ones. Whereas the LZS process describes coherent adiabatic reversal of the spin, the Kayanuma process describes a spin which is able to make many transitions between the up and down states and is therefore completely randomized at $t = \infty$. We may think of the fluctuating field $\eta(t)$ as a qualitative way of describing the dipolar field of other spins, which are also undergoing transitions between up and down states. It was shown in Ref. [@avag09] that we also obtain Kayanuma’s answer (\[p\_kaya\]) if we assume that the external field is swept uniformly, and each spin flips between the up and down states independently of the others but with a bias-dependent rate. That is, if the probability for a spin to be in the up state is denoted $p_{\up}(t)$, we take \_ = (1 - 2p\_). \[rate\_ind\_eps\] follows if () d= \^2, \[int\_Gam\] and $\eps(t) = {\dot \eps}t$. Since this requirement on $\Gam(\eps)$ entails only the tunneling amplitude $\Dta$, the details of the physical decoherence mechanism that justifies writing down a rate equation in the first place are irrelevant. In Ref. [@avag09], these mechanisms involved the environments of nuclear spins and other molecular spins. The bias-dependent flip rate that we employ in this paper \[see \] is a special case of that in [@avag09], and satisfies . The Kayanuma model is attractive because it is simple and economical. It is a much better model for the reversal than the LZS process. It is plausible that for a large collection of molecular spins where each one may flip at different times in the sweep cycle because the offset field is not the same for all spins, the field seen by any one spin has a stochastic character. However, it is clearly an approximation to assume that the time-dependence of the bias on any one spin is uncorrelated with the configuration of the other spins. As shown in [@avag09], it is not even enough to take account of the spin configuration in a mean-fieldy way through the spatially averaged demagnetization field. Therefore, we must consider the interaction between the spins explicitly. Environmentally influenced interacting spins {#glauber_spins} -------------------------------------------- In reality, the spins are neither isolated nor noninteracting as noted before. As discussed in [@avag09; @pro96], there are two types of interactions to consider. First, each molecular spin is coupled to nuclear spins in its vicinity. These spins have the effect that the tunneling between the $m = \pm S$ states becomes fully incoherent for typical molecular magnets. The $\kup \tofro \kdn$ probability for the $i$th spin is given by p\_[[[flip]{}]{},i]{} = \_i dt, where \_i (\_i) = [ 4]{} [\^2 W]{} (-[\_i\^22W\^2]{} ). \[gamma1\] Here, $W \simeq 10 E_{dn}$ with $E_{dn}$ being the dipole-dipole interaction energy between the molecular spin and the nearby nuclear spins, and $\eps_i$ is the bias on site $i$ [@bias]. For 8, $E_{dn} \sim 1$ mK, and $\eps_i \sim 0.1$ K in temperature units (see below). Second, each molecular spin is coupled to all the other molecular spins via the dipole interaction. At first sight, this leads to a fully many-body quantum mechanical problem. Since the nuclear spins already render each molecular spin slow and incoherent, however, the field of one molecular spin on another may be taken as a c-number and quantum mechanical back action may be ignored. Let us define an Ising spin variable $\sig_i$ on every site $i$, such that $\sig_i = \pm 1$ corresponds to $m_i = \pm S$. The dipolar part of $\eps_i$ is then given by \_[i,[dip]{}]{} &=& \_[ji]{} K\_[ij]{} \_j, \[Ebias\]\ K\_[ij]{} &=& 2 [E\_[dm]{} a\^3 r\^3\_[ij]{}]{} (1-3[z\^2\_[ij]{} r\^2\_[ij]{}]{} ). \[Kij\] Here, $E_{dm}$ is the interaction energy scale between neighboring molecular spins, and $a$ is their separation. Further, $r_{ij}$ is the distance between spins $i$ and $j$, and $z_{ij}$ is the difference between their [*z*]{}-axis coordinates. We estimate the dipolar field in 8 as $\sim 100$ Oe, which implies the energy scale $\eps_i \sim E_{dm} \sim 0.1$ K quoted above. The net result of these two couplings is as follows. The ratio $\eps_i/W$ is large for most molecules most of the time, so these spins are essentially frozen. A spin is unfrozen only when the field it sees is essentially zero (more precisely of order $W$ or less). We refer to this range of bias values as the [*reversibility region*]{}. In magnetization relaxation experiments, the combination of a static and narrow reversibility region leads to very slow and non-exponential time decay, which many workers have investigated theoretically [@pro98ab; @cuc99; @fer0304; @vij12]. If the external field is swept, however, the odds of a spin being able to flip are greatly increased. Particularly if the sweep rate is low, the complete dynamics can be very rich and complex, as we now discuss. Let us focus on one spin as the bias is being increased. Since the dipole energy is so large compared to all other energies in the problem, the flip of even a far away spin can change the bias on the first or central spin from a big negative value (relative to $W$) to a big positive one. The central spin is then shunted past the $\eps = 0$ region and does not flip. This gives a qualitative explanation of why the net magnetization reversal can be less than that in Kayanuma’s model. However, the central spin can be returned to the region of negative bias if another spin not too far away were to flip subsequently. Thus, the actual amount of spin reversal is hard to calculate, especially analytically. We therefore turn to Monte Carlo simulations of the LZS protocol. Monte Carlo Simulations {#thefullmonty} ======================= The physical problem we have described in is characterized by four parameters: the molecular spin dipole-dipole energy $E_{dm}$, the reversible region width $W$, the external bias sweep rate $\dot\eps_a$, and the tunnel splitting $\Dta$. For purposes of analysis and Monte Carlo simulation, however, these parameters can be reduced to just two dimensionless variables: the scaled reversible region width $w$ and the scaled sweep rate $v$, given as w &=& ,\ v &=& . All bias energies are measured in units of $E_{dm}$. We use the symbol $\veps$ (note the slightly different font) for the dimensionless biases, both applied and dipolar. It is not always optimal to frame the discussion in terms of dimensionless variables, and we shall back and forth between dimensionless and dimensionful descriptions as needed. Simulation protocol {#how_we_simulate} ------------------- Our Monte Carlo simulation models a finite system of $N$ spins, labeled $\sig_i$, that can be in either the up or the down state ($\sig_i=\pm1$). The bias $\eps_i$ at site $i$ is the sum of the externally applied bias field $\eps_a(t)$ and the dipole field $\eps_{i,{\rm dip}}$: \_i(t) = \_a(t) + \_[i,[dip]{}]{}(t). We have explicitly shown that the dipole field is time dependent since the spin state of the system is time dependent. The spins are taken to lie on a cubic lattice with lattice constant $a$, and the sample is taken to be spherical so that the initial bias distribution is approximately a delta function centered at the value $\eps_a(0)$. For the majority of the simulations, the number of spins, $N$, is chosen to be 7[,]{}153, corresponding to 25 spins on the sphere’s diameter, and $\eps_a(t)$ is taken to vary linearly in time, \_a(t) = \_a t. Though the number $N$ is significantly less than that used for previous simulations of relaxation [@vij12], we found that the results converged even for $N \gtwid 1,419$ (diameter $\ge 15$ spins). In simulations of relaxation, large values of $N$ were necessary to ensure that the reversible region was never depleted entirely. In simulations of LZS sweeps, the reversible region moves across the bias distribution, so depletion is a much lesser concern. In each Monte Carlo timestep (which we arbitrarily denote by $dt$), the spin flip probabilities are calculated for every spin in the system from the rate function $\Gam(\eps)$. To save on computation time, for most of our simulations, we used a simplification of that is only nonzero for $|\eps| \leq W$: ’() = (W-||). \[gamma2\] (The constant multiplying $\Tta(W - |\eps|)$ is chosen so that the integrals of $\Gam(\eps)$ and $\Gam'(\eps)$ over all $\eps$ are identical, and holds with $\Gam'$.) In practice, this rate function gives very nearly the same results as and allows a simpler view of the spin flip process: the only spins that are able to flip are those strictly within the reversible region, a window of width $2W$ around zero. After the spins have flipped, the change in bias at every lattice site is calculated and the simulation moves to the next timestep. Over the course of one run of the simulation, the external bias is swept from $-25 E_{dm}$ to $25 E_{dm}$. If $d\eps_a$ is the change in the bias in one time step, the probability for a spin in the reversible region to flip in that timestep is p\_[flip]{} &=& dt\ &=&\ &=& d\_a. Naturally, care must be taken to ensure that $p_{\rm flip} \ll 1$. In practice, we required that it did not exceed $0.1$. We also want to ensure that we do not skip past the reversible region in just one time step, so $d\veps_a$ is also chosen not to exceed $0.1 w$. This becomes computationally intensive for slow sweeps, and for $v =0.1$, for example, we are only able to study values of $w$ exceeding $0.01$. The quantities that we record during the simulation are the magnetization per site, M = \_i \_i, and the bias field distributions, $f_{\pm}(\eps)$, defined so that $f_{\pm}(\eps) \, d\eps$ is the fraction of sites with spin $\sig = +1$ or $-1$ and a bias field in the range $\eps$ to $\eps + d\eps$. We are especially interested in the final magnetization, $M_f$. We have already touched on the feasibility of performing the simulations with spheres of as few as 15 spins on the diameter. However, to test the stability of the answers with system size, we have performed some simulations with as many as 55 spins on the diameter. Further, in some simulations, we have employed a triangular wave for $\eps(t)$ as in some of the experiments [@wer99; @wer00; @wer00b]. The excursion of the bias is again $\pm 25 E_{dm}$. In these cases, ${\dot\eps}$ refers to the value of $|d\eps/dt|$ on any leg of the wave. Finally, a few simulations are done using the full Gaussian rate function (\[gamma1\]). We shall mention these exceptional cases as they merit. Simulation Results ------------------ ![(Color online) Monte Carlo results for the magnetization of the system as a function of the externally applied bias as it is swept from negative to positive values for various values of $v$ and $w = 0.05$. Since $\eps_a$ varies linearly with time, this is essentially a plot of $M$ vs[.]{} $t$. Each curve is an average of 100 runs for a system of $N=7{,}153$ spins using the modified transition rate (\[gamma2\]). The triangles on the right edge of the plot show $M_{f,{\rm K}}$, the final magnetization for each value of $v$ as per Kayanuma’s model.[]{data-label="m-bias"}](m-e_t.pdf) In we show a plot of the time dependence of the magnetization for various values of $v$ and $w = 0.5$. Readers will note that irrespective of $v$, the major change in $M$ starts when $\veps_{\rm ext}$ reaches a value close to zero, and there is little change once $\veps_{\rm ext}$ exceeds $\sim 5$. Readers will also note that there appears to be a small drop in $M$ before the main one. The reason is that in the initial state, when all the spins are up, the bias field is zero at almost every site. However, there are always some spins with nonzero bias on the surface of the sample, though their fraction becomes smaller as $N$ increases. Thus the initial bias distribution is not quite a delta function centered at zero bias, and the surface spins start to flip before the external bias reaches $-w$, and this is responsible for the small dip. ![(Color online) Final Monte Carlo (MC) magnetization $M_f$ as a function of the scaled sweep rate $v$ for various values of $w$. Each data point is an average of 100 runs for a system of $N=7{,}153$ spins using the modified transition rate (\[gamma2\]). Also shown is for $M_{f,{\rm K}}$, the final magnetization in the Kayanuma model.[]{data-label="m_f-v"}](M_f-v.pdf) As can also be seen in , the final magnetization, $M_f$, agrees with the Kayanuma result only for large enough $v$ (the agreement with the LZS result is even poorer). To make this point clearer, in we show $M_f$ vs. the sweep rate $v$ for three different choices of $w$, along with the Kayanuma result M\_[f,[K]{}]{} = 1 - 2 P\_[K]{} = e\^[-\^2/|\_a|]{} = e\^[-/v]{}. \[m\_f\_k\] There is good agreement with the Kayanuma result for fast sweep rates, with $v \gtwid 10$. In fact for these velocities, the Kayanuma and LZS models both give good answers, but we do not compare the Monte Carlo answers with LZS because the disagreement for $v \ltwid 10$ is much worse. Below $v$ of about $10$, however, even the agreement with Kayanuma’s model starts to degrade, and we obtain a residual magnetization even for very slow sweep rates. The agreement is worse for smaller values of $w$, with a lower value of $w$ corresponding to a larger $M_f$. This is qualitatively understandable as larger $w$ allows for greater opportunities for relaxation. What is interesting is that for $v \gtwid 10$, there is virtually no $w$ dependence in $M_f$, and the divergence in $M_f$ starts to appear at about the same value of the scaled velocity, $v \sim 10$, as that where the disagreement with LZS and Kayanuma starts to show up. ![(Color online) The value of $\Dta$ that we would infer for 8 if we applied the LZS formula for $M_f$ to the Kayanuma model, our Monte Carlo (MC) simulations, and the experimental data from Wernsdorfer et al. [@wer99] (open diamonds). Since the actual dependence of $M_f$ on $v$ is incorrectly given by LZS, this leads to a sweep-rate dependent answer for the inferred $\Dta$. We deduce the conversion factor from $\dot\eps$ to $dH/dt$ by exploiting the fact that for high sweep rates, LZS and Kayanuma both give the correct answer for $\Dta$, viz. $1.12 \times 10^{-7}\,$K.[]{data-label="delta_inf"}](delta_inf-v.pdf) We now show what value of $\Dta$ we would infer if, as experimenters often do, we use the LZS formula for the final magnetization [@AM]. In other words, denoting this inferred value by $\Dta_{{\rm inf,LZS}}$, we use the formula \^2\_[[inf,LZS]{}]{} = - ( ). The results are shown in . We show what we get if we use the Kayanuma formula for $M_f$, the experimental data of Ref. [@wer99], and $M_f$ as given by our Monte Carlo simulations with $w = 0.01$ and $w=0.1$. From this plot, one cannot tell which value of $w$ is better, although as shows, $w = 0.01$ gives a better fit. Since we estimated $W \simeq 10\,$mK and $E_{dm} \simeq 100\,$mK, $w = 0.1$ certainly seems reasonable, and even $w = 0.01$ is not out of the question. This graph shows that the magnetization reversal is not a very sensitive function of $w$, and so is of limited value in trying to learn about nuclear spin decoherence. As can be seen, the disagreement between even the Kayanuma model and the experimental data is substantial (observe the logarithmic scale for $v$), while that between the data and our simulations is small. $\begin{array}{cc} \includegraphics[scale=.60]{bias-1.pdf} & \includegraphics[scale=.60]{bias-2.pdf} \\ \includegraphics[scale=.60]{bias-3.pdf} & \includegraphics[scale=.60]{bias-4.pdf} \end{array}$ We now attempt to understand the nonzero residual final magnetization. To this end, we consider the bias distributions $f_{\pm}(\eps)$ over the course of a field sweep. Initially, the bias distribution at up spin sites is sharply peaked at zero, with only spins on the surface of the sample having a significant bias. As the external bias $\eps_a$ approaches zero, spins begin to flip, causing the distribution to widen. Even though the dipolar interaction is long-ranged, when a spin flips from up to down, the resulting change in bias on the majority of the sample is small on the scale of $W$. However, spins within a couple of lattice spacings of the spin that flipped will experience a dramatic change in bias. Of the six nearest neighbors of a flipping spin, the four in the plane will experience a change in bias of $-4E_{dm}$, and the two along the axis will experience a change $8E_{dm}$. This can be seen in b as the bias distribution has two minor peaks spaced approximately $-4E_{dm}$ and $8E_{dm}$ from the central one. Suppose that we are at a point in the sweep where a particular test spin in the sample is seeing a net bias close to zero, and one of its nearby spins (not necessarily a nearest neighbor) flips. We refer to the second spin as the triggering spin. Let the first spin undergo a large negative change in bias as a result of this spin flip event. It will then be displaced in bias far from the reversible region, yet since the external bias is being swept from negative to positive values, it will reenter the reversible region at a later time provided that the dipolar contribution to the bias seen by it has not changed significantly. On the other hand, if the test spin undergoes a large positive change in bias, it will be displaced in bias far past the the reversible region. Unless the triggering spin flips yet again or another nearby spin flips so as to change the bias on the test spin by a large negative amount, it will continue to see a large positive bias, and remains in its original orientation, up. It will essentially have been shunted around the region of reversibility. These shunted spins give a net positive contribution to $M_f$, while unshunted spins will contribute an amount that is close to the Kayanuma result on average. ![(Color online) Final Monte Carlo magnetization as a function of the reversible region width $w$ for several small values of $v$, using the Gaussian rate function (\[gamma1\]). Each data point is an average of 100 runs for a system of N=7[,]{}153 spins.[]{data-label="m_f-w"}](M_f-w.pdf) We conclude this subsection by discussing two atypical simulations. First, to confirm that the dependence of $M_f$ on $w$ for small values of $v$ was not an artifact of using the modified spin flip rate (\[gamma2\]), we ran the simulation for fixed $v \le 10$ and variable $w$ using the original Gaussian rate function (\[gamma1\]). The results are plotted in and show that the $w$ dependence is weak, although it is more pronounced for smaller $v$ and $M_f$ tends to the Kayanuma result with increasing $w$. $\begin{array}{cc} \includegraphics[scale=.60]{M-N-p1.pdf} & \includegraphics[scale=.60]{M-N-1.pdf} \\ \includegraphics[scale=.60]{M-N-10.pdf} & \includegraphics[scale=.60]{M-N-100.pdf} \end{array}$ Second, we show the results of simulations in which $\eps(t)$ is a triangular wave. In we show the dependence of $M$ on the number of cycles of the wave for different values of $v$. Note that we show $M$ every half-cycle, where a cycle is a complete period of the wave. The relevant point here is that $M$ decreases essentially exponentially with the number of cycles, except for very small $v$, showing that it is the fraction $\Dta M/M_i$ that is the same each time the reversible region is traversed, where $M_i$ is the initial fraction before the traversal. This is true even when the agreement with the Kayanuma model is poor. Kinetic Equation {#kin_eqn} ================ In Ref. [@vij12], the results of Monte Carlo simulations of magnetization relaxation were analyzed in terms of coupled rate equations for three quantities: the magnetization $M$, the magnetization of the spins in the reversible bias region $M_r$, and the number of spins in the reversible region $N_r$. However, the equations did not form a closed system, and also involved a functional $\mathcal{F}$, given by \[f\_()\] = W\^2 \_ \_[| |&gt;W]{} d. The distribution $\sum_{\sig} f_{\sig} (\eps)$ was then handled via an approach called the Three Gaussian Approximation (TGA), wherein it was modeled as a sum of three Gaussians, centered at biases of $0$, corresponding to a spin with no neighbors flipped, and $-4E_{dm}$ and $8E_{dm}$, corresponding to spins with one nearest neighbor flipped. The widths of the Gaussians were assumed to be equal, and this common width and the heights of the three Gaussian were determined by matching the first three moments of $\sum_{\sig} f_{\sig}(\eps)$ with a model in which every spin was up or down with probabilities $(1 \pm M)/2$ independently of the others. The distribution determined in this way depends only on the system’s magnetization, and leads to a closed system of rate equations. These equations were then found to describe the short to moderate time behavior of the magnetization well. It is plain that this approach is inadequate for analyzing the swept-field problem. It is paramount to have a good approximation for the bias distribution near the reversible region to properly capture the probability of spin flips. For the relaxation problem, the reversible region is static and centered at zero bias, whereas in the case of swept field, the reversible region moves over the full range of the bias distribution. As a result, a larger number of spins are capable of flipping, and the bias distribution is altered over essentially it entire range. Thus, the TGA is fundamentally invalid and one cannot really identify just three peaks. The approximation may perhaps work for very fast sweeps since the fraction of spins that flip is then small, but even in this limited success it does not provide a physically satisfactory explanation. It is therefore necessary to analyze the time evolution of the full bias distribution $f_{\sig}(\eps)$. This is naturally done in terms of a kinetic equation. We develop this equation below in somewhat more generality than is necessary for the precise problem of this paper with a view to applying it to other settings in the future. The basic process responsible for changes in the distributions are exactly the same as in Ref. [@vij12], so our present discussion is more brief. Derivation of kinetic equation {#derive_ke} ------------------------------ Let us denote the rate at which a spin at bias $\eps$ flips from $\sig$ to $-\sig$ by (), where we write $\bar\sig$ for $-\sig$ in the suffixes. In general the rates $\gssb(\eps)$ and $\gsbs(\eps)$ need not be equal, although in this paper they are. One way in which $\fsi(\eps)$ can change is by what might be called direct or one-spin processes, wherein in a time $dt$, a spin flips at any given site $i$, but does not undergo any change in the bias seen by it. This gives a contribution . |\_[direct]{} = - () () + () (). \[df\_direct\] Next, we consider two-spin processes, where we focus on a spin at a particular site, $i$, and allow the bias seen by this spin to change by virtue of a spin-flip at another site $j$. We refer to the second spin as the [*triggering*]{} spin, and to the first as the [*central*]{} spin. The latter is taken to not flip, as the probability for a process where it also flips is of order $(dt)^2$ and therefore negligible. There are then two processes which cause $\fsi(\eps)$ to change: [rccccc]{} & [site ]{}i & [site ]{}j & & [site ]{}i & [ site ]{}j\ [I.]{} & , & ’, ’& & , ”& -’, ’\ [II.]{} & , ”& ’, ’& & , & -’, ’ Processes I and II can be thought of as loss and gain processes, respectively, since they lead to spins being knocked out of or into the bias region $(\eps, \eps + d\eps)$. Let us consider process I first. Since the spin at site $j$ flips from $\sig'$ to $-\sig'$, the change in this spin is $-2\sig'$. In order for the bias at site $i$ to change as shown, we must have ” = - 2 K\_[ij]{} ’, \[esp\_change\] i.e., K\_[ij]{} = (- ”) ’. If we let the final bias on site $i$ lie in the range ($\eps'', \eps'' + d\eps''$), the number of triggering spins, i.e., the number of spins that satisfy this condition is g((- ”)’ ) , where g(K) \_[j i]{} (K-K\_[ij]{}) \[def\_gk\] is the density of couplings, by which we mean that $g(K) dK$ is the number of sites which couple to the central site with couplings between $K$ and $K + dK$. The sum in is over an infinite lattice, and therefore independent of site $i$. The probability that a spin on a triggering site will indeed flip in time $dt$ is $\gspp\,dt$. To find the net loss in $\fsi(\eps)$, we must multiply the number of triggering sites with the probability of a flip at those sites and the fraction of sites $i$ and $j$ that have the stipulated spins and biases. We must then sum over all possible values of $\sig'$, $\eps'$, and $\eps''$. In this way we get . |\_[I]{} = - \_[’]{}d’ f\_[’]{}(’) (). \[df\_I\] The calculation for process II proceeds in identical fashion. This time we get . |\_[II]{} = \_[’]{}d’ f\_[’]{}(’) (”). \[df\_II\] Together, the last two equations describe what might be called the collision integral. Lastly, let us incorporate a swept or explicitly time dependent applied field. If there were no spin flip processes at all, then a site which had a bias $\eps$ at time $t$, would at time $t + dt$ have a bias (t+dt) = (t) + d\_a, where $d\eps_a$ is the change in the applied field in the interval $dt$, and we would have ((t), t) &=& ((t) + d\_a, t+dt )\ &=& ((t), t ) + d\_a + dt. Therefore, . |\_[sweep]{} = - . \[df\_sweep\] Adding Eqns. (\[df\_direct\]), (\[df\_I\]), (\[df\_II\]), and (\[df\_sweep\]), we get &=& - - () () + () ()\ && - \_[’]{} f\_[’]{}(’) .\ && \[df\_tot\] This is the kinetic equation we are seeking. We can rewrite this equation in various ways which help understand its structure, and also suggest algorithmic simplifications for numerical integration. So, we note that in the collision term, the integral over $\eps'$ can be factored out of the expression completely, reflecting the fact that the [*precise value of the bias*]{} at the triggering site is irrelevant to whether a flip on this site alters the bias on the central site by a given amount; what matters for that is [*where*]{} the triggering site is located relative to the central site, and the spin on the triggering site. We may therefore define a [*net rate per site*]{} for spin flip, averaged over the bias distribution, and over the entire sample, (t) = d f\_(, t) (). As indicated, this rate changes with time because the bias distribution changes. It is in fact a functional of the distribution. In terms of this rate, we may write the kinetic equation as &=& - - () () + () ()\ && - \_[’]{} d’ . \[df\_tot2\] We have also changed the dummy variable of integration from $\eps''$ to $\eps'$. An elementary but useful check on the kinetic equation is that the quantity \_d () should be time-independent, since it is just unity by normalization. Summing over $\sig$ and integrating over all $\eps$, we get &=& - - \_ d\ && - \_[, ’]{} dd’ .\ && \[d\_norm\] The sweep term integrates to zero directly, since $\fsi(\eps)$ must vanish for $\eps \to \pm\infty$. The direct terms also add to zero. To see that, we first note that a sum over $\sig$ is equivalent to a sum over $\bar\sig$. Recognizing this, and writing the second direct term as a sum over $\bar\sig$, and then interchanging the index labels $\sig$ and $\bar\sig$, the two direct terms are seen to cancel each other identically. To see that the two-spin terms also add to zero, we simply interchange the integration variables $\eps$ and $\eps'$ in the very last term. The two parts of the collision integral are then identical and they cancel each other. Hence, = 0, as desired. At this point it is useful to discuss the nature of $g(K)$, the density of dipole couplings. As shown in Ref. [@avag09], g(K) , K 0. \[gk\_asymptotic\] That is, for small $K$, $g(K)$ is approximately an even function, and diverges as $1/K^2$. The integrand of the $\eps'$ integral in therefore has a singularity at $\eps' = \eps$ of the form , which is equivalent to . This is an integrable singularity if we view the integral over $\eps'$ as a principal value. Hence, our kinetic equation is mathematically well posed. It is also clear that this singularity cannot have any physical consequences. It arises from flips of triggering spins which are very far away from the central spin, and these flips cannot affect the dynamical behavior of the central spin since they lead to miniscule changes in the bias seen by the latter. Fortunately, it is not necessary to take any special precautions about this singularity in the numerical integration, since it is automatically regulated by the finiteness of the sample. We can understand the last point further as follows. In the collision terms, the integrals over $\eps'$ are equivalent to a sum over sites. Consider, for example, the two-spin loss term (process I). We can write . |\_[I]{} = - () \_[’]{} \[df\_loss\_1\] Now, &=& \_[ji]{} d’ (’ - (- 2K\_[ij]{})’)\ &=& \_[ji]{} 1. \[df\_loss\_2\] In the same way, for the gain term (process II), we have g((’ - )’) (’) &=& \_[j i]{} ((’ - )’ - K\_[ij]{}) (’)\ &=& \_[j i]{} (+ 2 ’ K\_[ij]{}). Hence the combined contribution of these two processes can be written as . |\_[I + II]{} = \_[’]{} \_[ji]{} ( (+ 2’ K\_[ij]{}) - () ). Let us suppose that we cut off the sum so that site $j$ lies inside a large sphere (centered at site $i$) containing $N_K$ sites. For spins outside this sphere, $K_{ij}$ is very small in magnitude, and we may approximate \_[j &gt; N\_K]{} ( (+ 2’ K\_[ij]{}) - () ) = \_[j &gt; N\_k]{} 2 ’ K\_[ij]{}, which vanishes if we divide the sum into subsums carried out over sets of sites related by cubic symmetry. We also note that the form (\[gk\_asymptotic\]) for $g(K)$ is a poor estimate for near neighbor or short-distance couplings. It is these couplings which are in the end responsible for the peaked form of the bias distribution function, and must therefore be treated correctly taking the discontinuous delta-function nature into account. In other words, we must handle the part of the integral where $\eps' - \eps$ is large as a sum over neighboring sites, and a continuum form for $g(K)$ cannot be used. Numerical Integration --------------------- To integrate the kinetic equations, the bias distribution functions $f_+(\eps)$ and $f_-(\eps)$ were approximated by histograms with a scaled bias range $[-\veps_{\rm max},\veps_{\rm max}]$, divided into $N_b$ bins. As in the Monte Carlo simulations, $\veps_{\rm max}$ was set to 25. At first sight it appears that we should choose the bin width, $w_b = 2\veps_{\rm max}/N_b$, to be much less than $w$. Since realistic values of $w$ are quite small, however, this would require the number of bins, $N_b$ to be rather large. Now a larger number of bins leads to a smaller number of spins in each bin, and since we want the [*relative*]{} change in this number per time step to be small for accurate integration, it requires smaller integration timesteps. Hence too small a bin width is numerically expensive. In practice we find that changing $w_b$ only affects how very small changes in bias are handled, and it is perfectly acceptable to let the bin width equal the reversible region width $2w$, leading to $N_b$ in the range $10^3$–$10^4$. Smaller values of $w_b$ did not lead to appreciably different results. Since the kernel $K_{ij}$ is a property solely of the lattice, we determine it once and for all before doing any integration as soon as we have decided upon the bin width $w_b$. All values of $-K_{ij}$ are determined for a large sphere of $N_K$ spins with the site $i$ at the center. The quantity $-K_{ij}$ is half the amount by which the bias will change at the central site if a spin $\sig_j$ flips from up to down. We therefore compile a histogram from this data with a bin width matching that of the bias distribution functions, giving us $g\bigl((\eps-\eps')/2 \bigr) d\eps$. It should be noted that it does no good to bin $g(K)$ more finely. Let us denote this bin width by $w_g$. The value in a given bin of the $g(K)$ histogram essentially gives us the number of sites that can shift the bias by $2K$. The value of $g(K)$ in the next bin will gives us the sites that shift the bias by $2(K+w_g)$. If $2w_g$ is smaller than $w_b$, we are needlessly differentiating between sites that have the same effect as far the evolution of the histogram for $\fsi(\eps)$ is concerned. We therefore choose $2w_g$ equal to $w_b$. Combining for the contribution to $df/dt$ from process I (the loss term), we get . |\_[I]{} = - N\_K (T\_[[|1]{}1]{} + T\_[1[|1]{}]{}) () We can combine this with the term from process II to write . |\_[I + II]{} = \_[’]{} d’ (’), \[df\_coll\] where g\_R(K) = g(K) - N\_K (K) \[def\_greg\] is a regulated density of couplings. The form (\[df\_coll\]) once again shows that the infrared divergence mentioned in the previous subsection is a numerical nonissue. Once we have settled on a bin width $w_b/2$ for the histogram of $g(K)$, there is no point in increasing $N_K$ beyond a certain value. Beyond that value, we only change the number in the central bin around $K=0$. The delta-function term in goes into the same bin, so $g_R(K)$ does not change. Thus the numerics are essentially done for an infinite system, and the width of the bin serves as a cutoff that effects the principal value integral. It also pays to define the population [*transfer*]{} rate (, ’) = \_[’]{} . \[def\_Gam\_matrix\] When we multiply this quantity by the bin width $w_b$, the quantity $\tshf \gpepe w_b$ is the number of sites capable of triggering a shift in bias from $\eps'$ to $\eps$, and $\tspp$ is the spin-flip rate averaged over all sites, so $w_b \Gam(\eps, \eps')$ is the rate at which population is transferred from the bin containing the bias $\eps'$ to the bin containing $\eps$. It also gives the entire collision integral a nifty matrix multiplication form, . |\_[I + II]{} = d’ (,’) (’), \[df\_coll\_matrix\] and the complete kinetic equation can be written as = - - () () + () ()\ + d’ (,’) (’). \[df\_tot3\] The actual integration of the kinetic equations is simple once the histograms for $f_+$, $f_-$, and $g_R(K)$ are set up. In each timestep, there is some exchange of the populations in the central bins of the $f_+$ and $f_-$ histograms, corresponding to the reversible region, as dictated by . The reversible region is swept from $-\eps_{\rm max}$ to $\eps_{\rm max}$ for LZS runs, while it is allowed to remain static at the origin for relaxation runs. Next, the collision terms are evaluated for every bin in the two histograms, given by , and the integration moves to the next timestep. Kinetic Equation Results ------------------------ As a test of the kinetic equations and our numerical integration procedure, we first applied them to the problem of magnetic relaxation with zero external bias. The equations were integrated up to a time $10\tau$ where $\tau$ is the characteristic time for relaxation, given by = . ![(Color online) Comparison of Monte Carlo (MC) and kinetic equation (KE) results for magnetic relaxation with $w = 0.05$. MC data was obtained using $N=82{,}519$ and averaged over 20 runs. KE data was obtained using 2[,]{}001 bins and a scaled bias range $(-50,50)$.[]{data-label="demagke"}](m-t_relax.pdf) (Note that this time scale is very long in an absolute sense since it varies as $\Dta^{-2}$, and $\Dta$, being a tunnel splitting between deep levels on opposite sides of the anisotropy barrier, is very small.) The resulting demagnetization curve is plotted in Fig. \[demagke\] along with the results of the Monte Carlo simulation. As can be seen, it agrees extremely well with the Monte Carlo simulation for short times. However, after $t/\tau \gtwid 1$, the two curves begin to diverge, with the kinetic equations giving a higher value for the magnetization than the Monte Carlo simulation. This difference can be understood by comparing the bias distributions given by the two methods. These distributions are shown in . $\begin{array}{cc} \includegraphics[scale=.50]{bias-k-p1.pdf} & \includegraphics[scale=.50]{bias-mc-p1.pdf} \\ \includegraphics[scale=.50]{bias-k-1.pdf} & \includegraphics[scale=.50]{bias-mc-1.pdf} \end{array}$ For short times, $t \ltwid 0.1\tau$, the bias distributions match fairly well, with each peak in the Monte Carlo data also present in the solution to the kinetic equations. However, after $t \simeq \tau$, the bias distribution from the Monte Carlo is bounded by about $\pm 10 E_{dm}$, and still shows sharp side peaks, while that from the kinetic equations is significantly more spread out and lacking the peaks. This difference is due to the fact that the kinetic equations do not carry information about the orientations and biases of a spin’s neighbors while the Monte Carlo simulation does. More specifically, there is a short distance correlation between biases, as we now discuss. Consider a central spin and its 6 nearest neighbors all initially in the up state and with zero bias. If one of the neighboring spins flips, the bias on the central spin will change by an amount $8E_{dm}$ or $-4E_{dm}$, moving it out of the reversible region to one of the peaks that can be seen in Fig. \[bias\_relax\]b. Similarly, the bias at the 5 other neighboring sites will also change and these spins will also be removed from the reversible region. As the sample continues to relax, the bias for the central spin will not have a large probability to change significantly because of the fact that 5 of its nearest neighbors are displaced far from the reversible region. The sixth spin, the one that originally flipped, is still sitting in its original near-zero bias, and therefore has a higher probability to flip back, but this would cause the bias for the central spin to move back closer to the origin. Thus, by considering a spin’s neighbors, we can see that once a spin acquires a large bias, there is a low probability for the bias on that spin to change yet again by a large amount, and if such a change does occur it is more likely to make the bias regress back toward the mean. This explains why the bias distribution in the Monte Carlo simulation is more bounded and peaked. The kinetic equations do not keep track of these correlations, and thus yield too broad a distribution. We then applied the kinetic equations to the problem of a swept field. In this case, the equations are much more successful, and, as shown in , we find that the results are in good agreement with the Monte Carlo simulation for values of the scaled sweep rate, $v$, as low as 5, where the Kayanuma model does quite poorly. However, the limitations of the kinetic equations again show up for $v \ltwid 5$. ![(Color online) $M_f$ vs[.]{} $v$ as given by the Kayanuma model, our Monte Carlo data, and the kinetic equation. For the latter two we used a width parameter of $w = 0.05$ and for the kinetic equations, we used a bin width of $E_{dm}/30$.[]{data-label="m_f_ke"}](M_f-v-ke.pdf) Conclusions {#conc} =========== We have carried out Monte Carlo simulations of swept-field experiments on molecular magnetic solids based on the microscopic picture of spin reversal developed in Refs. [@avag09; @pro96]. We find that these simulations provide a very good picture of the time evolution of the entire system, and agree fairy well with experiments quantitatively. In order to understand the simulations, we have also developed a kinetic equation for the distribution of single-site spin and bias distribution. 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--- abstract: 'We demonstrate three-photon excitation in quantum dots with a mode-locked fiber laser operating in the telecommunications band. We compare spectra and intensity dependence of fluorescence from one- and three-photon excitation of commercially available 640 nm quantum dots, using a 372 nm diode laser for one-photon excitation and 116 fs pulses from a mode-locked fiber laser with a center wavelength of 1575 nm for three-photon excitation.' author: - 'M.J. Petrasiunas' - 'J.B.O. Wood' - 'D. Kielpinski' - 'E.W. Streed' title: 'Three-photon excitation of quantum dots with a telecom band ultrafast fiber laser' --- Introduction ============ Quantum dots are semiconductor nanocrystals, 1-20 nm in size [@Murray1993; @Pietryga2004], that possess a discrete energy level structure. The level structure is determined by the size, composition and shape of the quantum dot [@Empedocles1996; @Norris1996]. Particular focus has been placed on II-VI semiconductors, such as CdSe [@Murray1993]. By selecting for size, the fluorescence emission can be tailored to the application, which is attractive for use in biological labels [@BruchezJr.1998; @Alivisatos2004; @Michalet2005; @Resch-Genger2008]. In addition, the quantum dot surface can be chemically functionalized to bind to specific biomolecules. Compared to other fluorescent molecules, quantum dots possess high brightness, low bleaching rate and absorption over a wide wavelength range [@Michalet2005; @Resch-Genger2008]. Similar to the fluorescence from other labels, quantum dot fluorescence can be driven by both single- and multiphoton processes [@Zipfel2003; @Bentley2007; @Lad2007; @Chon2004; @He2007]. Fluorescent labels are conventionally excited by short wavelength light to produce fluorescence emissions. Ultraviolet (UV) light is often used with quantum dots due to their absorption that increases towards shorter wavelengths[@Resch-Genger2008]. However UV is not ideal for use in living cells, due to strong absorption and scattering of UV light by surrounding biological tissue. The use of shorter wavelengths leads to a higher potential for photochemical reactions and an increased possibility for the decomposition and subsequent release of component heavy metals, such as cadmium (Cd) [@Derfus2004; @Hardman2006]. Through excitation by multiphoton processes, it is possible to avoid the unwanted effects from exposure to shorter wavelengths in biological applications. Two-photon excitation of quantum dots is now routinely performed, often using Ti:sapphire lasers operating around 700-1000 nm [@He2007; @Dahan2001]. For imaging applications, it is often important for the excitation light to penetrate far into the sample, but the penetration depth is limited by Rayleigh scattering due to the effect on beam quality. Since Rayleigh scattering scales as $\lambda^{-4}$, long-wavelength multiphoton excitation seems promising for bioimaging applications. In a recent study, Horton et al. [@Horton2013] demonstrated in-vivo three-photon microscopy with an Er-doped fiber system and red fluorescent protein labels. They found that an optimum wavelength window due to scattering and absorption in tissue lies close to the telecom band. Quantum dots are attractive for use in such applications due to their large three-photon absorption cross-section [@Bentley2007; @Lad2007]. We demonstrate three-photon excitation and fluorescence in commercially available CdSe quantum dots using a custom built mode-locked Er-doped fiber laser (EDFL). Such lasers, and their associated optical components, are central to telecommunications and have therefore been the subject of extensive industrial research. These developments mean that they provide a stable, versatile and comparatively inexpensive method to generate ultrashort pulses for three-photon spectroscopy of quantum dots. We also excited single-photon fluorescence using a 372 nm diode laser to compare one- and three-photon fluorescence emission spectra, excitation intensity dependence and bleaching rate. Previous investigations have also demonstrated two- and three-photon excitation in quantum dots [@Chon2004; @He2007]. However, to our knowledge, multiphoton excitation and absorption have not yet been investigated in quantum dots for excitation wavelengths in the telecom band. The wavelength of expected fluorescence emissions from the sample is determined by the quantum dot energy level structure, which can be simplified to a ground state, fluorescent excited state and a continuum of absorption states [@Alivisatos1996; @Reimann2002]. Upon excitation to this continuum of states, a quantum dot undergoes a non-radiative decay to the excited state, followed by fluorescent decay back to the ground state[@Alivisatos1996]. In addition, intense excitation pulses will interact nonlinearly with the sample to generate harmonics of the excitation wavelength. In quantum dots, the third-harmonic generation (3HG) process is expected to be the most significant of these [@Dharmadhikari1999]. The process of second harmonic generation (SHG) is negligible in comparison to 3HG due to the centrosymmetric crystal structure of CdSe [@Banyai1993; @Feng2006]. As the product of a coherent process, the harmonics are emitted in line with the excitation beam. Method ====== ![\[fig1-setup\] Experimental Setup. Schematic for spectroscopy of quantum dots. (a) 372 nm light is focused onto the quantum dot sample and fluorescence is collected and coupled into a multimode fiber. Fluorescence is separated from residual excitation light using a Semrock FF511-Di01 dichroic mirror. (b) 1575 nm pulses are focused onto the quantum dot sample and fluorescence is collected off-axis to avoid excessive coupling of third harmonic (3HG) light. The majority of residual excitation and 3HG light continues to propagate on-axis past the slide and is not coupled into the multimode fiber (MMF). As some scattered light is still collected, KG3 coloured glass filters are used to block any remaining IR.](Fig1Setup.eps){width="7cm"} Figure \[fig1-setup\] shows the experimental setup used to probe the quantum dots. Colloidal Invitrogen Lumidot CdSe/ZnS 640 core-shell quantum dots, with a nominal fluorescence wavelength of 640 nm, were deposited on a sample slide. The toluene solvent was evaporated off, leaving a static quantum dot sample which was placed at the focus of the excitation beam. Collected fluorescence was coupled into a multimode fiber, where the spectrum was measured by a fiber-coupled spectrometer. For the three-photon spectroscopy of the quantum dot sample, a mode-locked EDFL was used to produce 1564 nm pulses at a repetition rate of 300 MHz. 116 fs pulses with a maximum average power of 186 mW were produced through Er-doped fiber amplification and soliton compression. The laser was focused to a spot with a $14.1\pm0.2\;\textrm{\ensuremath{\mu}m}$ $1/e^2$ radius, yielding an average intensity up to $48.8\pm0.5\;\textrm{kW/cm}^{2}$. The soliton self-frequency shift (SSFS) [@Mitschke1986; @Gordon1986] of the amplified pulses shifted the center wavelength to 1575 nm, increasing the spectral bandwidth to 85 nm. The pulse durations were measured by frequency resolved optical gating (FROG) using a MesaPhotonics FROGScan with an Ocean Optics HR2000+ spectrometer, and a wavelength range of 700-881 nm. The spectrum was measured using a HP 70950A optical spectrum analyser. The resulting fluorescence emitted from the quantum dots was collected off-axis with a 35 mm focal length lens to spatially separate it from the large amount of light produced by 3HG. However, some of the 3HG and excitation light was scattered by the sample and collected. The fiber-coupled fluorescence was detected by an Ocean Optics USB650 spectrometer. Three 2 mm thick KG3 coloured-glass filters were used to block any remaining IR, with a calculated attenuation of $-80$ dB, in order to prevent damage to the detector. The fluorescence detection spectrometer was used to capture all data for the fluorescence and 3HG spectra, and its dependence on the intensity of the excitation beam and exposure time. Measurements of the spectra were taken for different input intensities using a variable ND filter, and integration times from 5-30 s were used. The relative spectrum amplitudes were inferred from Gaussian fit profiles of the spectra. To check that the quantum dots behaved as expected under UV excitation, we applied up to 3.5 mW of 372 nm light from a fiber-coupled Nichia diode laser to the quantum dot sample. Basic optical isolation was employed, using a quarter waveplate and linear polariser, to ensure that the operation of the diode laser was not compromised by optical feedback from the single mode fiber. The UV light was focused to a spot with a $5.6\;\textrm{\ensuremath{\mu}m}$ $1/e^2$ radius at the quantum dot sample, with a maximum intensity of $3.54\;\textrm{kW/cm}^{2}$. The fluorescence excited by the UV laser was collected by an $\textrm{NA}=0.60$ aspheric lens with a 4.02 mm focal length. This was separated from the residual excitation light by a Semrock FF511-Di01 dichroic mirror, with an edge wavelength of 511 nm, and coupled into MMF. An Ocean Optics QE65000 spectrometer, with a 316-1021 nm wavelength range and 0.22 nm/pixel resolution, was used to measure the spectrum of the UV-induced fluorescence. Measurements were taken with varying input intensities and an integration time of 10 ms. Spectra ======= ![\[fig2-spectra\] Comparison of spectra from (a) $\lambda$=372 nm excitation and (b) $\lambda$=1575 nm excitation, shown with the energy level structures and excitation pathways. In (b), both the 640 nm fluorescence peak and a 540 nm 3HG peak are observed. (c) Spectrum of IR excitation beam on a linear scale, where the wavelength axis aligns at a ratio of 3:1 to that of (b), showing the correlation of the 3HG peak with the fundamental. (d) FROG trace of IR pulse, where the wavelength axis refers to the second harmonic of the excitation wavelength.](Fig2Spectra.eps){width="7cm"} Figure \[fig2-spectra\] shows the spectra from UV and IR excitation. From a Gaussian fit of the fluorescence spectra, the center wavelength from linear excitation is $632.6\pm0.1\;\textrm{nm}$, where it is $631.4\pm0.2\;\textrm{nm}$ in the spectrum produced by three-photon excitation. The FWHM linewidths are determined to be $47.8\pm0.1\;\textrm{nm}$ and $40.7\pm0.4\;\textrm{nm}$, respectively. The near-identical characteristics of the fluorescence spectra under UV and IR excitation indicate that the same excited energy band is reached in both cases. In the three-photon excitation spectrum, the secondary peak at $534.8\pm0.1\;\textrm{nm}$ corresponds with the third harmonic of the IR excitation pulse. The peak is visible in the off-axis configuration due to scattering from the quantum dot sample. As shown in Figure \[fig2-spectra\], the third harmonic is generated predominantly from the red-shifted soliton at the long-wavelength end of the excitation pulse spectrum. This part of the excitation spectrum, according to FROG trace measurements, corresponds to the most intense part of the excitation pulse. As a static sample of quantum dots were used for this experiment, the time dependent spectra were also observed in each case, using an integration time of 500 ms over 10 minutes, to compare bleaching rates of the linear and three-photon excitation processes. We observed that although a higher excitation intensity was used for the three-photon process, the emission signal of the quantum dots was bleached to 50% of the starting intensity after 566 s, compared to 103 s under linear excitation. However the bleaching of the sample under IR excitation was not fit well by the expected double-exponential profile, suggesting that it may not follow a conventional bleaching mechanism. Further study is needed to properly assess the meaning of these preliminary observations. Intensity dependence ==================== ![\[fig3-power\] Fluorescence rate dependence of one- and three-photon excitation. (a) Intensity dependent fluorescence measured under UV excitation shows a saturation intensity of $3.3\pm1.1\;\textrm{kW/cm}^{2}$ from the saturation fit, given by the dashed gray line. Excluding points above 50% of the saturation intensity, and fitting a power law $I^{n}$, gives $n=1.01\pm0.15$ (dashed blue line). (b) A power law fit of the intensity dependent fluorescence measured under three-photon excitation is shown, with an index $n=3.08\pm0.06$ (dashed red line).](Fig3IntSeries.eps){width="7.5cm"} To verify the photon orders of the two excitation processes and search for saturation of the excitation, we measured the dependence of the fluorescence on excitation intensity, as shown in Figure \[fig3-power\]. The fluorescence spectra for both the one- and three-photon measurements, taken at varying excitation intensities, were fit with Gaussian profiles. The amplitudes of the fit profiles were compared using power law fits, as shown in Figure \[fig3-power\], in order to determine a $F \propto I^{n}$ power law dependence between excitation intensity $I$ and fluorescence counts $F$, for each of the processes. First- and third-order dependencies are expected for the one- and three-photon processes, respectively. Saturation of the quantum dots is observed under UV excitation, with saturation intensity $I_{sat}=3.3\pm1.1\;\textrm{kW/cm}^{2}$, found using a fit of the form: $$F\propto\frac{s}{1+s} \qquad s=I/I_{sat}$$ By excluding data points greater than 50% of the saturation intensity found for the UV excitation data, the index of the UV intensity dependence was found to be $1.01\pm0.15$. The index for the IR induced fluorescence was $3.08\pm0.06$, confirming three-photon behaviour. Third harmonic intensity was also found to exhibit cubic behaviour. As such, the fluorescence under IR excitation could either occur via a direct three-photon mechanism or via absorption of a third harmonic photon. This experiment does not distinguish between the two mechanisms. Discussion ========== Our results show three-photon excitation and fluorescence in quantum dots, with the aid of ultrafast fiber lasers. With this comes the potential to improve the effectiveness of quantum dot based biological label schemes, using low-cost hardware. Telecom-band fiber lasers provide an economical source of laser hardware that has the potential to supplant more conventional laser hardware such as Ti:Sapphire in many applications - of which biomicroscopy applications are no exception. The experimental data for three-photon excitation shows that there are no significant changes or shifts in the fluorescence spectrum compared to the linear excitation case, and the third order dependence on excitation power shows that the quantum dot sample is indeed excited through a three-photon process. The process in the study by Horton et al. [@Horton2013] could thus be replicated using quantum dots as a more favourable label agent [@Michalet2005; @Resch-Genger2008; @Bentley2007; @Lad2007]. In future work, we aim to address the implementation of quantum dots in biological imaging within tissue, using multiphoton processes to maximize viewing depth. Further work could improve the design of the fiber laser system used for this process and use additional types of quantum dots - including more biocompatible materials such as InP and different fluorescence wavelengths. Optimising quantum dots for longer wavelength fluorescence will significantly reduce scattering of the fluorescence signal, once again increasing viewing depth. By opting for a lower repetition rate, thus increasing the pulse energy of the fiber laser, it will also be possible to increase the multiphoton fluorescence signal of the labels while reducing chances of cell damage due to high average power levels. Conclusion ========== We provide the first demonstration of three-photon excitation in quantum dots using a fiber laser operating in the telecommunications band. The experiment was conducted using a mode locked Er-doped fiber laser producing 116 fs pulses with a center wavelength of 1575 nm. We compare results from one- and three-photon excitation experiments to confirm that the three-photon spectrum is similar, and to demonstrate the third order intensity dependence of the process as an additional means of verification. We found that the fluorescence intensity scaled with input excitation power as expected for a three-photon process. Three-photon excitation in quantum dots using ultrafast fiber lasers provides an important step toward more effective biological imaging by minimizing scattering of the excitation light and increasing the possible penetration depth. D.K. was supported by an Australian Research Council Future Fellowship (FT110100513). We would like to acknowledge Prof. Jay L. Nadeau of McGill University for sparking our interest in the area. [24]{} ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****,  ()](http://pubs.acs.org/doi/abs/10.1021/ja00072a025) [****,  ()](\doibase 10.1021/ja047659f) [****,  ()](http://www.ncbi.nlm.nih.gov/pubmed/10062330) [****,  ()](http://www.ncbi.nlm.nih.gov/pubmed/9983472) [****,  ()](\doibase 10.1126/science.281.5385.2013) [****,  ()](\doibase 10.1038/nbt927) [****,  ()](\doibase 10.1126/science.1104274) [****,  ()](\doibase 10.1038/NMETH.1248) [****,  ()](\doibase 10.1038/nbt899) [****,  ()](\doibase 10.1117/1.2823156) [****,  ()](\doibase 10.1063/1.2714994) [****,  ()](\doibase 10.1063/1.1755420) [****, ()](http://www.ncbi.nlm.nih.gov/pubmed/19550551) [****,  ()](\doibase 10.1021/nl0347334) [****,  ()](\doibase 10.1289/ehp.8284) [****,  ()](http://www.ncbi.nlm.nih.gov/pubmed/18040463) [****,  ()](\doibase 10.1038/NPHOTON.2012.336) @noop [****,  ()]{} [****, ()](http://rmp.aps.org/abstract/RMP/v74/i4/p1283_1) [****,  ()](http://iopscience.iop.org/0953-8984/11/5/021) [**](http://books.google.com/books?hl=en&lr=&id=JdjVkrCsL8gC&pgis=1) (, ) p.  [****,  ()](\doibase 10.1016/j.physb.2006.03.012) [****,  ()](http://www.ncbi.nlm.nih.gov/pubmed/19738720) [****, ()](http://www.opticsinfobase.org/abstract.cfm?id=8765)

--- author: - | Naama Ben-David$^\ast$ Guy E. Blelloch$^\ast$ Jeremy T. Fineman$^\dag$\ Phillip B. Gibbons$^\ast$ Yan Gu$^\ast$ Charles McGuffey$^\ast$ Julian Shun$^\ddag$\ $^\ast$Carnegie Mellon University $^\dag$Georgetown University $^\ddag$UC Berkeley\ {nbendavi, guyb, gibbons, yan.gu, cmcguffe}@cs.cmu.edu\ jfineman@cs.georgetown.edu jshun@eecs.berkeley.edu bibliography: - '../../ref.bib' title: | Implicit Decomposition for\ Write-Efficient Connectivity Algorithms ---

--- abstract: 'Consider the Jacobian of a genus two curve defined over a finite field and with complex multiplication. In this paper we show that if the subgroup of the Jacobian is not cyclic, then the embedding degree of the Jacobian with respect to $\ell$ is one.' address: | Department of Mathematical Sciences\ University of Aarhus\ Ny Munkegade\ Building 1530\ DK-8000 Aarhus C author: - Christian Robenhagen Ravnshøj title: Embedding Degree of Hyperelliptic Curves with Complex Multiplication --- [^1] Introduction ============ In elliptic curve cryptography it is essential to know the number of points on the curve. Cryptographically we are interested in elliptic curves with large cyclic subgroups. Such elliptic curves can be constructed. The construction is based on the theory of complex multiplication, studied in detail by [@atkin-morain]. It is referred to as the *CM method*. [@koblitz89] suggested the use of hyperelliptic curves to provide larger group orders. Therefore constructions of hyperelliptic curves are interesting. The CM method for elliptic curves has been generalized to hyperelliptic curves of genus two by [@spallek], and efficient algorithms have been proposed by [@weng03] and [@gaudry]. Both algorithms take as input a primitive, quartic CM field $K$ (see section \[sec:CMfields\] for the definition of a CM field), and give as output a hyperelliptic genus two curve $C$ defined over a prime field ${\mathbb{F}}_p$. A prime number $p$ is chosen such that $p=x\overline x$ for a number $x\in{\mathfrak{O}_{K}}$, where ${\mathfrak{O}_{K}}$ is the ring of integers of $K$. We have $K={\mathbb{Q}}(\eta)$ and $K\cap{\mathbb{R}}={\mathbb{Q}}(\sqrt{D})$, where $\eta=i\sqrt{a+b\xi}$ and $$\xi=\begin{cases} \frac{1+\sqrt{D}}{2}, & \textrm{if $D\equiv 1\mod{4}$,} \\ \sqrt{D}, & \textrm{if $D\equiv 2,3\mod{4}$}. \end{cases}$$ In this paper, the following theorem is established. Let $C$ be a hyperelliptic curve of genus two defined over ${\mathbb{F}}_p$ with $\operatorname{End}(C)\simeq{\mathfrak{O}_{K}}$, where $K$ is a primitive, quartic CM field as defined in definition \[def:CMfieldPrimitive\]. Assume that the $p$-power Frobenius under this isomorphism is given by the number $\omega=c_1+c_2\xi+(c_3+c_4\xi)\eta$, where $\xi$ and $\eta$ are given as above and $c_i\in{\mathbb{Z}}$. Consider a prime number $\ell\mid|{\mathcal{J}_{C}}({\mathbb{F}}_p)|$ with $\ell\neq p$, $\ell\nmid D$ and $\ell\nmid c_2$. Assume that the $\ell$-Sylow subgroup of ${\mathcal{J}_{C}}({\mathbb{F}}_p)$ is not cyclic. Then $p\equiv 1\mod{\ell}$, i.e. the embedding degree of ${\mathcal{J}_{C}}({\mathbb{F}}_p)$ with respect to $\ell$ is one. Hyperelliptic curves ==================== A hyperelliptic curve is a smooth, projective curve $C\subseteq{\mathbb{P}}^n$ of genus at least two with a separable, degree two morphism $\phi:C\to{\mathbb{P}}^1$. Let $C$ be a hyperelliptic curve of genus two defined over a prime field ${\mathbb{F}}_p$ of characteristic $p>2$. By the Riemann-Roch theorem there exists an embedding $\psi:C\to{\mathbb{P}}^2$, mapping $C$ to a curve given by an equation of the form $$y^2=f(x),$$ where $f\in{\mathbb{F}}_p[x]$ is of degree six and have no multiple roots [see @cassels chapter 1]. The set of principal divisors $\mathcal{P}(C)$ on $C$ constitutes a subgroup of the degree 0 divisors $\operatorname{Div}_0(C)$. The Jacobian ${\mathcal{J}_{C}}$ of $C$ is defined as the quotient $${\mathcal{J}_{C}}=\operatorname{Div}_0(C)/\mathcal{P}(C).$$ Let $\ell\neq p$ be a prime number. The $\ell^n$-torsion subgroup ${\mathcal{J}_{C}}[\ell^n]<{\mathcal{J}_{C}}$ of elements of order dividing $\ell^n$ is then [@lang59 theorem 6, p. 109] $$\label{eq:J-struktur} {\mathcal{J}_{C}}[\ell^n]\simeq{\mathbb{Z}}/\ell^n{\mathbb{Z}}\times{\mathbb{Z}}/\ell^n{\mathbb{Z}}\times{\mathbb{Z}}/\ell^n{\mathbb{Z}}\times{\mathbb{Z}}/\ell^n{\mathbb{Z}},$$ i.e. ${\mathcal{J}_{C}}[\ell^n]$ is a ${\mathbb{Z}}/\ell^n{\mathbb{Z}}$-module of rank four. The order of $p$ modulo $\ell$ plays an important role in cryptography. Consider a prime number $\ell$ dividing the order of ${\mathcal{J}_{C}}({\mathbb{F}}_p)$, where $\ell$ is different from $p$. The embedding degree of ${\mathcal{J}_{C}}({\mathbb{F}}_p)$ with respect to $\ell$ is the least number $k$, such that $p^k\equiv 1\mod{\ell}$. An endomorphism $\varphi:{\mathcal{J}_{C}}\to{\mathcal{J}_{C}}$ induces a ${\mathbb{Z}}_\ell$-linear map $$\varphi_\ell:T_\ell({\mathcal{J}_{C}})\to T_\ell({\mathcal{J}_{C}})$$ on the $\ell$-adic Tate-module $T_\ell({\mathcal{J}_{C}})$ of ${\mathcal{J}_{C}}$ [@lang59 chapter VII, §1]. The map $\varphi_\ell$ is given by $\varphi$ as described in the following diagram: $$\xymatrix@C=40pt@R=40pt{ \dots \ar[r]^(0.4){[\ell]} & {\mathcal{J}_{C}}[\ell^{n+1}] \ar[r]^{[\ell]} \ar[d]^{\varphi} & {\mathcal{J}_{C}}[\ell^{n}] \ar[r]^{[\ell]} \ar[d]^{\varphi} & \dots \\ \dots \ar[r]^(0.4){[\ell]} & {\mathcal{J}_{C}}[\ell^{n+1}] \ar[r]^{[\ell]} & {\mathcal{J}_{C}}[\ell^{n}] \ar[r]^{[\ell]} & \dots \\ }$$ Here, the horizontal maps $[\ell]$ are the multiplication-by-$\ell$ map. Hence, $\varphi$ is represented by a matrix $M\in\operatorname{Mat}_{4\times 4}({\mathbb{Z}}/\ell{\mathbb{Z}})$ on ${\mathcal{J}_{C}}[\ell]$. Let $P(X)\in{\mathbb{Z}}[X]$ be the characteristic polynomial of $\varphi$ [see @lang59 pp. 109–110], and let $P_M(X)\in({\mathbb{Z}}/\ell{\mathbb{Z}})[X]$ be the characteristic polynomial of the restriction of $\varphi$ to ${\mathcal{J}_{C}}[\ell]$. Then [@lang59 theorem 3, p. 186] $$\label{eq:KarPolKongruens} P(X)\equiv P_M(X)\mod{\ell}.$$ Since $C$ is defined over ${\mathbb{F}}_p$, the mapping $(x,y)\mapsto (x^p,y^p)$ is an isogeny on $C$. This isogeny induces the $p$-power Frobenius endomorphism $\varphi$ on the Jacobian ${\mathcal{J}_{C}}$. The characteristic polynomial $P(X)$ of $\varphi$ is of degree four [@tate theorem 2, p. 140], and by the definition of $P(X)$ [see @lang59 pp. 109–110], $$|{\mathcal{J}_{C}}({\mathbb{F}}_p)|=P(1),$$ i.e. the number of ${\mathbb{F}}_p$-rational elements of the Jacobian is determined by $P(X)$. CM fields {#sec:CMfields} ========= An elliptic curve $E$ with ${\mathbb{Z}}\neq\operatorname{End}(E)$ is said to have *complex multiplication*. Let $K$ be an imaginary, quadratic number field with ring of integers ${\mathfrak{O}_{K}}$. $K$ is a *CM field*, and if , then $E$ is said to have *CM by ${\mathfrak{O}_{K}}$*. More generally a CM field is defined as follows. A number field $K$ is a CM field, if $K$ is a totally imaginary, quadratic extension of a totally real number field $K_0$. In this paper only CM fields of degree $[K:{\mathbb{Q}}]=4$ are considered. Such a field is called a *quartic* CM field. \[rem:quarticCM\] Consider a quartic CM field $K$. Let $K_0=K\cap{\mathbb{R}}$ be the real subfield of $K$. Then $K_0$ is a real, quadratic number field, $K_0={\mathbb{Q}}(\sqrt{D})$. By a basic result on quadratic number fields, the ring of integers of $K_0$ is given by ${\mathfrak{O}_{K_0}}={\mathbb{Z}}+\xi{\mathbb{Z}}$, where $$\xi=\begin{cases} \frac{1+\sqrt{D}}{2}, & \textrm{if $D\equiv 1\mod{4}$,} \\ \sqrt{D}, & \textrm{if $D\equiv 2,3\mod{4}$}. \end{cases}$$ Since $K$ is a totally imaginary, quadratic extension of $K_0$, a number $\eta\in K$ exists, such that $K=K_0(\eta)$, $\eta^2\in K_0$. The number $\eta$ is totally imaginary, and we may assume that $\eta=i\eta_0$, $\eta_0\in{\mathbb{R}}$. Furthermore we may assume that $-\eta^2\in{\mathfrak{O}_{K_0}}$; so $\eta=i\sqrt{a+b\xi}$, where $a,b\in{\mathbb{Z}}$. Let $C$ be a hyperelliptic curve of genus two. Then $C$ is said to have CM by ${\mathfrak{O}_{K}}$, if $\operatorname{End}(C)\simeq{\mathfrak{O}_{K}}$. The structure of $K$ determines whether $C$ is irreducible. More precisely, the following theorem holds. \[teo:reducibel\] Let $C$ be a hyperelliptic curve of genus two with $\operatorname{End}(C)\simeq{\mathfrak{O}_{K}}$, where $K$ is a quartic CM field. Then $C$ is reducible if, and only if, $K/{\mathbb{Q}}$ is Galois with Galois group $\operatorname{Gal}(K/{\mathbb{Q}})\simeq{\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/2{\mathbb{Z}}$. [@shi proposition 26, p. 61]. Theorem \[teo:reducibel\] motivates the following definition. \[def:CMfieldPrimitive\] A quartic CM field $K$ is called primitive if either $K/{\mathbb{Q}}$ is not Galois, or $K/{\mathbb{Q}}$ is Galois with cyclic Galois group. The CM method for constructing curves of genus two with prescribed endomorphism ring is described in detail by [@weng03] and [@gaudry]. In short, the CM method is based on the construction of the class polynomials of a primitive, quartic CM field $K$ with real subfield $K_0$ of class number $h(K_0)=1$. The prime number $p$ has to be chosen such that $p=x\overline x$ for a number $x\in{\mathfrak{O}_{K}}$. By [@weng03] we may assume that $x\in{\mathfrak{O}_{K_0}}+\eta{\mathfrak{O}_{K_0}}$. Properties of ${\mathcal{J}_{C}}({\mathbb{F}}_p)$ {#sec:properties} ================================================= Consider a primitive, quartic CM field $K$ with real subfield $K_0$ of class number $h(K_0)=1$, and let $p$ be an uneven prime number such that $p=x\overline x$ for a number $x\in{\mathfrak{O}_{K_0}}+\eta{\mathfrak{O}_{K_0}}$. The main result of this paper, given by the following theorem, concerns a curve of genus two with ${\mathfrak{O}_{K}}$ as endomorphism ring. \[teo:ed=1\] With the notation as in remark \[rem:quarticCM\], let $C$ be a hyperelliptic curve of genus two defined over ${\mathbb{F}}_p$ with $\operatorname{End}(C)\simeq{\mathfrak{O}_{K}}$. Assume that the $p$-power Frobenius under this isomorphism is given by the number $\omega=c_1+c_2\xi+(c_3+c_4\xi)\eta$, where $c_i\in{\mathbb{Z}}$. Consider a prime number $\ell\mid|{\mathcal{J}_{C}}({\mathbb{F}}_p)|$ with $\ell\neq p$, $\ell\nmid D$ and $\ell\nmid c_2$. Assume that the $\ell$-Sylow subgroup of ${\mathcal{J}_{C}}({\mathbb{F}}_p)$ is not cyclic. Then $p\equiv 1\mod{\ell}$, i.e. the embedding degree of ${\mathcal{J}_{C}}({\mathbb{F}}_p)$ with respect to $\ell$ is one. Consider a prime number $\ell\mid |{\mathcal{J}_{C}}({\mathbb{F}}_p)|$ with $\ell\nmid pc_2D$. If $\ell=2$, then obviously $p\equiv 1\mod{\ell}$. Hence we may assume that $\ell\neq 2$. Assume that the $\ell$-Sylow subgroup $S$ of ${\mathcal{J}_{C}}({\mathbb{F}}_p)$ is not cyclic. Then $S$ contains a subgroup $U\simeq({\mathbb{Z}}/\ell{\mathbb{Z}})^2$. So $$({\mathbb{Z}}/\ell{\mathbb{Z}})^2<{\mathcal{J}_{C}}({\mathbb{F}}_p)[\ell]<{\mathcal{J}_{C}}[\ell].$$ Let $\{e_1,e_2\}\subseteq{\mathcal{J}_{C}}({\mathbb{F}}_p)$ be a basis of $({\mathbb{Z}}/\ell{\mathbb{Z}})^2$. Expand by the isomorphism  this set to a basis $\{e_1,e_2,f_1,f_2\}$ of ${\mathcal{J}_{C}}[\ell]$. It then follows that $1$ is an eigenvalue of the Frobenius with eigenvectors $e_1$ and $e_2$, i.e. $1$ is an eigenvalue of multiplicity at least two. First we assume that $D\equiv 2,3\mod{\ell}$. Let $P(X)$ be the characteristic polynomial of the Frobenius. Since the conjugates of $\omega$ are given by $\omega_1=\omega$, $\omega_2=\overline{\omega}_1$, $\omega_3$ and $\omega_4=\overline{\omega}_3$, where $$\omega_3 = c_1-c_2\sqrt{D}+i(c_3-c_4\sqrt{D})\sqrt{a-b\sqrt{D}},$$ it follows that $$P(X) = \prod_{i=1}^4(X-\omega_i) = X^4-4c_1X^3+(2p+4(c_1^2-c_2^2D))X^2-4c_1pX+p^2.$$ Since $1$ is an eigenvalue of the Frobenius of multiplicity at least two, the characteristic polynomial $P(X)$ is divisible by $(X-1)^2$ modulo $\ell$. Now, $$P(X)=Q(X)\cdot(X-1)^2+R(X),$$ where $$\begin{aligned} R(X) = {} & 4(1-3c_1-(c_1-1)p+2(c_1^2-c_2^2D))X \\ & +p^2-2p-4(c_1^2-c_2^2D)+8c_1-3. \end{aligned}$$ Since $R(X)\equiv 0\mod{\ell}$, it follows that $$\label{eq:*} 1-3c_1-(c_1-1)p+2(c_1^2-c_2^2D) \equiv 0 \mod{\ell}.$$ Since $|{\mathcal{J}_{C}}({\mathbb{F}}_p)|=P(1)$, we know that $$\label{eq:***} (p+1)^2-4c_1(p+1)+4(c_1^2-c_2^2D)\equiv 0\mod{\ell}.$$ By equation  we see that $4(c_1^2-c_2^2D)\equiv 2(c_1-1)p-2+6c_1\mod{\ell}$. Substituting this into equation  we get $$(p+1)^2-4c_1(p+1)+2(c_1-1)p-2+6c_1\equiv 0\mod{\ell};$$ so either $p\equiv 1\mod{\ell}$ or $p\equiv 2c_1-1\mod{\ell}$. Assume $p\equiv 2c_1-1\mod{\ell}$. Then $$R(X)\equiv 4c_2^2D(-2X+1)\equiv 0\mod{\ell}.$$ Since $\ell\nmid 2c_2D$, this is a contradiction. So if $D\equiv 2,3\mod{4}$, then $p\equiv 1\mod{\ell}$. Now consider the case $D\equiv 1\mod{4}$. We now have $$\omega_3 = c_1+c_2\frac{1-\sqrt{D}}{2}+i\left(c_3+c_4\frac{1-\sqrt{D}}{2}\right)\sqrt{a+b\frac{1-\sqrt{D}}{2}},$$ and it follows that the characteristic polynomial of the Frobenius is given by $$P(X)=X^4-2cX^3+(2p+c^2-c_2^2d)X^2-2pcX+p^2,$$ where $c=2c_1+c_2$. We see that $P(X)=Q(X)(X-1)^2+R(X)$, where $$\begin{aligned} R(X) = {} & ((4-2c)p+2c^2-6c-2c_2^2D+4)X \\ & +p^2-2p-3+4c-c^2+c_2^2D. \end{aligned}$$ Since $R(X)\equiv 0\mod{\ell}$, it follows that $$\label{eq:*1} p^2-2p-3+4c-c^2+c_2^2D \equiv 0 \mod{\ell},$$ and since $|{\mathcal{J}_{C}}({\mathbb{F}}_p)|=P(1)$, we know that $$\label{eq:***1} (p+1)^2-2c(p+1)+c^2-c_2^2D \equiv 0 \mod{\ell}.$$ From equation  and it follows that $$p^2-cp+c-1 \equiv 0 \mod{\ell},$$ i.e. $p\equiv 1\mod{\ell}$ or $p\equiv c-1\mod{\ell}$. Assume $p\equiv c-1\mod{\ell}$. Then $$R(X)\equiv c_2^2D(-2X+1)\equiv 0\mod{\ell},$$ again a contradiction. So if $D\equiv 1\mod{4}$, then $p\equiv 1\mod{\ell}$. Consider the case $\ell\mid c_2$. Then the characteristic polynomial of the Frobenius modulo $\ell$ is given by $$P(X)\equiv (X^2-2c_1X+p)^2 \mod{\ell},$$ independently of the remainder of $D$ modulo $4$. Observe that $$X^2-2c_1X+p = (X+1-2c_1)(X-1)+p-2c_1+1.$$ Hence, $p\equiv 2c_1-1\mod{\ell}$, i.e. $$P(X)\equiv (X-1)^2(X-p)^2\mod{\ell}.$$ So the following theorem holds. \[teo:c2\] With the notation as in remark \[rem:quarticCM\], let $C$ be a hyperelliptic curve of genus two defined over ${\mathbb{F}}_p$ with $\operatorname{End}(C)\simeq{\mathfrak{O}_{K}}$. Assume that the $p$-power Frobenius under this isomorphism is given by the number $\omega=c_1+c_2\xi+(c_3+c_4\xi)\eta$, where $c_i\in{\mathbb{Z}}$. Consider a prime number $\ell\mid|{\mathcal{J}_{C}}({\mathbb{F}}_p)|$ with $\ell\neq p$, $\ell\mid c_2$. Assume that the $\ell$-Sylow subgroup of ${\mathcal{J}_{C}}({\mathbb{F}}_p)$ is not cyclic. Then either 1. ${\mathcal{J}_{C}}({\mathbb{F}}_p)[\ell]\simeq({\mathbb{Z}}/\ell{\mathbb{Z}})^2$, or 2. $p\equiv 1\mod{\ell}$ and ${\mathcal{J}_{C}}({\mathbb{F}}_p)[\ell]={\mathcal{J}_{C}}[\ell]$. If $p\not\equiv 1\mod{\ell}$, then $1$ is not an eigenvalue of the Frobenius of multiplicity three, i.e. ${\mathcal{J}_{C}}({\mathbb{F}}_p)[\ell]\simeq({\mathbb{Z}}/\ell{\mathbb{Z}})^2$. If $p\equiv 1\mod{\ell}$, then $1$ is an eigenvalue of the Frobenius of multiplicity four, i.e. ${\mathcal{J}_{C}}({\mathbb{F}}_p)[\ell]={\mathcal{J}_{C}}[\ell]$. Applications ============ Let $C$ be a hyperelliptic curve of genus two defined over ${\mathbb{F}}_p$ with $\operatorname{End}(C)\simeq{\mathfrak{O}_{K}}$. Write $$\label{eq:jac(Fp)} {\mathcal{J}_{C}}({\mathbb{F}}_p)\simeq{\mathbb{Z}}/n_1{\mathbb{Z}}\times{\mathbb{Z}}/n_2{\mathbb{Z}}\times{\mathbb{Z}}/n_3{\mathbb{Z}}\times{\mathbb{Z}}/n_4{\mathbb{Z}},$$ where $n_i\mid n_{i+1}$ and $n_2\mid p-1$ [see @hhec proposition 5.78, p. 111]. We recall the following result on the prime divisors of the number $n_2$. \[teo\] With the notion as above, let $\ell\mid n_2$ be an odd prime number. Then $\ell\leq Q$, where $$\begin{aligned} Q &= \max\{a,D,a^2-b^2D\}, \\ \intertext{if $D\equiv 2,3\mod{4}$, and} Q &= \max\{a,D,4a(a+b)-b^2(D-1),aD+2b(D-1)\}, \end{aligned}$$ if $D\equiv 1\mod{4}$. If $\ell>D$, then $c_1\equiv 1\mod{\ell}$ and $c_2\equiv 0\mod{\ell}$. [@me07a]. Let the Frobenius be given by the number $\omega=c_1+c_2\xi+(c_3+c_4\xi)\eta$, $c_i\in{\mathbb{Z}}$, and consider a prime number $\ell\mid|{\mathcal{J}_{C}}({\mathbb{F}}_p)|$, $\ell\neq p$. If $\ell\nmid c_2$ and $\ell>Q$, then the $\ell$-Sylow subgroup $S$ of ${\mathcal{J}_{C}}({\mathbb{F}}_p)$ is either of rank two and $p\equiv 1\mod{\ell}$, or $S$ is cyclic. By [@me07b], if $p\equiv 1\mod{\ell}$, then there exists an efficient, probabilistic algorithm to determine generators of the $\ell$-Sylow subgroup of ${\mathcal{J}_{C}}({\mathbb{F}}_p)$. Hence the following corollary holds. If $\ell\nmid D$ and $\ell\nmid c_2$, then there exists an efficient, probabilistic algorithm to determine generators of the $\ell$-Sylow subgroup $S$ of ${\mathcal{J}_{C}}({\mathbb{F}}_p)$. If $p\equiv 1\mod{\ell}$, then the corollary is given by [@me07b]. If $p\not\equiv 1\mod{\ell}$, then $S$ is cyclic by theorem \[teo:ed=1\]. Assume $|S|=\ell^n$. Then $S$ has $\ell^n-\ell^{n-1}$ elements of order $\ell^n$. Hence the probability that a random element $\sigma\in S$ generates $S$ is $1-\ell^{-1}$, and choosing random elements $\sigma\in S$ until an element of order $\ell^n$ is found will be an efficient, probabilistic algorithm to determine generators of $S$. Acknowledgement =============== I would like to thank my supervisor Johan P. Hansen for inspiration on theorem \[teo:c2\]. [99]{} <span style="font-variant:small-caps;">A.O.L. Atkin and F. Morain</span>. Elliptic curves and primality proving. *Math. Comp.*, vol. 61, pp. 29–68, 1993. <span style="font-variant:small-caps;">J.W.S. Cassels and E.V. Flynn</span>. *Prolegomena to a Middlebrow Arithmetic of Curves of Genus $2$*. London Mathematical Society Lecture Note Series. Cambridge University Press, 1996. <span style="font-variant:small-caps;">G. Frey and T. Lange</span>. Varieties over Special Fields. In H. Cohen and G. Frey, editors, *Handbook of Elliptic and Hyperelliptic Curve Cryptography*, pp. 87–113. Chapman & Hall/CRC, 2006. <span style="font-variant:small-caps;">P. Gaudry, T. Houtmann, D. Kohel, C. Ritzenthaler and A. Weng</span>. The $p$-adic CM-Method for Genus $2$. 2005. <http://arxiv.org>. <span style="font-variant:small-caps;">N. Koblitz</span>. Hyperelliptic cryptosystems. *J. Cryptology*, vol. 1, pp. 139–150, 1989. <span style="font-variant:small-caps;">S. Lang</span>. *Abelian Varieties*. Interscience, 1959. <span style="font-variant:small-caps;">C.R. Ravnshøj</span>. *Large Cyclic Subgroups of Jacobians of Hyperelliptic Curves*. 2007a. <http://arxiv.org>. <span style="font-variant:small-caps;">C.R. Ravnshøj</span>. *Generators of Jacobians of Hyperelliptic Curves*. 2007b. <http://arxiv.org>. <span style="font-variant:small-caps;">G. Shimura</span>. *Abelian Varieties with Complex Multiplication and Modular Functions*. Princeton University Press, 1998. <span style="font-variant:small-caps;">A.-M. Spallek</span>. *Kurven vom Geschlecht $2$ und ihre Anwendung in Public-Key-Kryptosystemen*. Ph.D. thesis, Institut für Experimentelle Mathematik, Universität GH Essen, 1994. <span style="font-variant:small-caps;">J. Tate</span>. Endomorphisms of abelian varieties over finite fields. *Invent. Math.*, vol. 2, pp. 134–144, 1966. <span style="font-variant:small-caps;">A. Weng</span>. Constructing hyperelliptic curves of genus $2$ suitable for cryptography. *Math. Comp.*, vol. 72, pp. 435–458, 2003. [^1]: Research supported in part by a PhD grant from CRYPTOMAThIC

--- abstract: 'In this paper, we present the first approximation algorithms for the problem of designing revenue optimal Bayesian incentive compatible auctions when there are multiple (heterogeneous) items and when bidders have arbitrary demand and budget constraints (and additive valuations). Our mechanisms are surprisingly simple: We show that a sequential all-pay mechanism is a $4$ approximation to the revenue of the optimal ex-interim truthful mechanism with a discrete type space for each bidder, where her valuations for different items can be correlated. We also show that a sequential posted price mechanism is a $O(1)$ approximation to the revenue of the optimal ex-post truthful mechanism when the type space of each bidder is a product distribution that satisfies the standard hazard rate condition. We further show a logarithmic approximation when the hazard rate condition is removed, and complete the picture by showing that achieving a sub-logarithmic approximation, even for regular distributions and one bidder, requires pricing bundles of items. Our results are based on formulating novel LP relaxations for these problems, and developing generic rounding schemes from first principles.' author: - 'Sayan Bhattacharya[^1]' - 'Gagan Goel[^2]' - 'Sreenivas Gollapudi[^3]' - 'Kamesh Munagala[^4]' title: Budget Constrained Auctions with Heterogeneous Items --- Introduction ============ In several scenarios, such as the Google TV ad auction [@nisan2] and the the FCC spectrum auctions [@pino], where auctions have been applied in the recent past, bidders are constrained by the amount of money they can spend. This leads to the study of auctions with budget-constrained bidders, which is the focus of this paper. The key difficulty with budgets is that the utility of a bidder is only quasi-linear as long as the price is below the budget constraint, but is $-\infty$ if the price exceeds the budget. As a consequence, well-known mechanisms such as the VCG mechanism are no longer directly applicable. Before proceeding further, we formally define our model. Our Model {#sec:model} --------- There are $m$ bidders and $n$ heterogeneous items. The [*type*]{} of a bidder is defined as the collection of her valuations for each item; we will assume throughout the paper that her valuations for different items are additive. Further, her type is private knowledge. We augment this basic model for bidder $i$ with two publicly known constraints: A [*demand constraint*]{} $n_i$ on the maximum number of items she is willing to buy, and a [*budget constraint*]{} $B_i$ on the maximum total price she can afford to pay. Her utility is defined as follows: Suppose she gets a subset $A$ of items where $|A| \leq n_i$, and pays a total price $P_i$. If $v_{ij}$ denotes her valuation for item $j$, then her utility is given by $\sum_{j \in A} v_{ij} - P_i$ if $P_i \leq B_i$, and $-\infty$ otherwise. The goal of the auctioneer is to design a mechanism that maximizes its revenue, which is defined as $\sum_i P_i$. We require that the mechanism be [*incentive compatible*]{} in that no bidder gains in utility by misreporting her valuations, and [*individually rational*]{}, meaning that a bidder gets nonnegative utility by reporting the truth. Some of our results generalize to the case where the budget constraints are private knowledge. There are two well-established ways of proceeding from here. In the adversarial setting, no assumptions are made on the valuations, while in the Bayesian setting, it is assumed the valuations are drawn from publicly known prior distributions. We take the latter Bayesian approach, which was pioneered by Myerson [@myerson]. As mentioned above, in this setting, the bidders’ private valuations are drawn from independent (but not necessarily identical) commonly known prior distributions. The auctioneer’s goal is to maximize her expected revenue, where the expectation is over the prior distributions, and possible randomization introduced by the mechanism itself. In the Bayesian setting, the notions of truthfulness and individual rationality can be of two kinds: Bayesian Incentive Compatible (BIC): : Here, each bidder’s expected utility is maximized by truth-telling, where the expectation is over the valuations of the other bidders (which are drawn according to their priors) and the randomness introduced by the mechanism. In other words, truth-telling is optimal only in expectation. Dominant Strategy Incentive Compatible (DISC): : In this setting, truth-telling is optimal for the bidder even if she knows the valuations of the other bidders and the random choices made by the mechanism. We note that traditionally, the terms BIC and DSIC only refer to randomness introduced by other bidders’ valuations via the prior distributions. Therefore, to be very precise, the space of mechanisms is two-dimensional, randomized versus universally truthful mechanisms on one dimension, and BIC versus DSIC mechanisms on the other. We collapse this space to randomized BIC versus universally truthful DSIC mechanisms, so as not to introduce additional terminology. Our Results {#sec:results} ----------- Our results depend on the structure of the type space of each bidder. In particular, the results depend on whether the distributions of the valuations of a bidder for different items are correlated or independent. We note that the correlations if any are [*only*]{} for valuations of one bidder for different items; the distributions of the valuations of [*different*]{} bidders are always assumed to be independent. All our results can be viewed as presenting simple characterizations of approximately optimal mechanisms in these contexts. We will first consider the situation when valuations of a bidder for different items are arbitrarily correlated, so that the type space of a bidder is simply specified as a poly-bounded discrete distribution. In Section \[sec:lot\], we present a simple [*all-pay*]{} mechanism: it charges each bidder a fixed price that depends only on its revealed type, while the allocation made to the bidder will be [*random*]{} depending on the types of other bidders. The resulting scheme is therefore only BIC; however, we show that it is a $4$ approximation to the revenue of the optimal BIC scheme (Theorem \[thm:allpay\]). An all-pay auction is unrealistic in several situations, since a bidder is forced to participate even if she obtains negative utility when the auction concludes. A natural question to ask is whether there exists a DSIC mechanism with good revenue properties. However, it is an easy exercise to show that the problem of designing an optimal DSIC mechanism for one unit-demand bidder ($n_i = 1$) whose type can take one of $n$ arbitrarily correlated possibilities with equal probability, reduces to the problem of unlimited supply [*envy-free pricing*]{} with $n$ bidders [@kempe]. For the envy free pricing problem, the best known is a logarithmic approximation, and there is strong evidence that a better approximation is not possible [@Briest]. Furthermore, in the correlated valuation setting, there are examples showing an unbounded gap between the revenues of the optimal randomized (BIC) and deterministic (DSIC) mechanisms [@shuchi2]. This significantly dims the possibility of a constant factor approximate DSIC mechanism with discrete correlated types. In view of the above negative result, we assume the valuations of a bidder for different items follow independent distributions. This [*product distribution*]{} assumption is also used by Chawla [*et. al.*]{} [@shuchi], who consider the special case of a single bidder with unit demand. In this special case, it is easy to see that any DSIC mechanism is a posted price mechanism, and they show a constant factor approximation to its revenue by an elegant connection to Myerson’s mechanism [@myerson]. Independently of our work, Chawla [*et al.*]{} [@shuchi1] extend their result to show a $O(1)$ approximation for multiple unit-demand bidders. However, this result also crucially require the unit-demand assumption. In Section \[sec:mon\], we show that for arbitrary demands and budgets, when the type of each bidder follows a product distribution where each dimension satisfies the standard monotone hazard rate (MHR), there is a constant factor approximation to the revenue of optimal BIC mechanism. Our mechanism is achieved by a DSIC mechanism with surprisingly simple structure: Consider the bidders sequentially in arbitrary order, and for each bidder, offer a subset of remaining items at a pre-computed price for each item; the bidder simply chooses the utility maximizing bundle of these items. This posted price mechanism shows a constant factor gap between the revenues of the optimal BIC and DSIC mechanisms (Theorem \[thm:optitem\]), which is in sharp contrast with the corresponding negative result [@shuchi2] when the type space of a bidder is correlated. The surprising aspect of our result is that since the optimal DSIC mechanism for general demands and budgets will price bundles of items, there is [*no a priori reason*]{} to expect that the more simplistic scheme of posting prices for each item would be a constant approximation. We show (Section \[sec:gen\]) that the MHR condition is indeed necessary for our item pricing result: If this condition is slightly relaxed to regular product distributions and one bidder, the optimal DSIC mechanism will price [*bundles of items*]{}, and will have a logarithmic gap against the revenue of the optimal item pricing scheme. We show that this gap is tight by showing the existence of a posted price mechanism achieving this approximation ratio (Theorem \[thm:gen\]). On a positive note, we conclude by proving that for regular distributions, if the space of feasible mechanisms is restricted to those that consider bidders in some adaptive order and post prices that may depend on the outcomes so far, there is a $O(1)$ approximation within this space, that considers bidders in an arbitrary but fixed order, and pre-computes the posted prices (Theorem \[thm:final\]). Related Work {#sec:work} ------------ The Bayesian setting is widely studied in the economics literature [@BenoitK; @pino; @Che1; @Che2; @laffont; @vincent; @vohra; @thas; @wilson]. In this setting, the optimal (either BIC or DSIC) mechanism can always be computed by encoding the incentive compatibility constraints in an integer program and maximizing expected revenue. However, the number of variables (and constraints) in this IP is exponential in the number of bidders, there being variables for the allocations and prices for each scenario of revealed types. Therefore, the key difficulty in the Bayesian mechanism design case is [*computational*]{}: [*Can the optimal (or approximately optimal) auction be efficiently computed and implemented?*]{} Much of the literature in economics considers the case where the auctioneer has one item (or multiple copies of the item). In the absence of budget constraints, Myerson [@myerson] presents the characterization of any BIC mechanism in terms of expected allocation made to a bidder: This allocation must be monotone in the revealed valuation of the bidder, and furthermore, the expected price is given by applying the VCG calculation to the expected allocation. This yields a linear-time computable optimal revenue-maximizing auction that is both BIC and DSIC. The key issue with budget constraints is that the allocations need to be thresholded in order for the prices to be below the budgets [@Che1; @laffont; @vohra]. However, even in this case, the optimal BIC auction follows from a polymatroid characterization that can be solved by the Ellipsoid algorithm and an all-pay condition [@Border; @vohra]. By [*all-pay*]{}, we mean that the bidder pays a fixed amount given his revealed type, regardless of the allocation made. This also yields a DSIC mechanism that is $O(1)$ approximation to optimal BIC revenue [@sayan], but the result holds only for homogeneous items. An alternative line of work deals with the [*adversarial setting*]{}, where no distributional assumption is made on the bidders’ private valuations. In this setting, the budget constrained auction problem is notorious mainly because standard auction concepts such as VCG, efficiency, and competitive equilibria do not directly carry over [@nisan2]. Most previous results deal with the case of multiple units of a homogeneous good. In this setting, based on the random partitioning framework of Goldberg [*et al.*]{} [@Goldberg04], Borgs [*et al.*]{} [@borgs] presented a truthful auction whose revenue is asymptotically within a constant factor of the optimal revenue (see also [@abrams]). If the focus is instead on optimizing social welfare, no non-trivial truthful mechanism can optimize social welfare [@borgs]. Therefore, the focus has been on weaker notions than efficiency, such as Pareto-optimality, where no pair of agents (including the auctioneer) can simultaneously improve their utilities by trading with each other. Dobzinski [*et al.*]{} [@nisan] present an ascending price auction based on the clinching auction of Ausubel [@ausubel], which they show to be the only Pareto-optimal auction in the public budget setting. This result was extended to the private budget setting by Bhattacharya [*et al.*]{} [@sayan]; see  [@ravi] for a related result. Finally, several researchers have considered the behavior specific types of auctions, for instance, auctions that are sequential by item and second price within each item [@BenoitK; @elkind], and ascending price auctions [@ausubel2; @pino]. The goal here is to analyze the improvement in revenue (or social welfare) by optimal sequencing, or to study incentive compatibility of commonly used ascending price mechanisms. However, analyzing the performance of sequential or ascending price auctions is difficult in general, and there is little known in terms of optimal mechanisms (or even approximately optimal mechanisms) in these settings. Our Techniques {#sec:techniques} -------------- If for every bidder, the valuations for different items are correlated (Section \[sec:lot\]), then the optimal BIC revenue can be bounded from above by a linear program ([LP1]{}) that requires the incentive compatibility, voluntary participation, supply and demand constraints to hold [*only in expectation*]{}. We construct a BIC all pay auction (Figure \[fig:allpay\]) that basically implements a rounding scheme on the optimal solution to [LP1]{}, loosing a constant factor in revenue (Theorem \[thm:allpay\]). This approach is based on the techniques used in [@sayan]. As mentioned before, Chawla [*et. al.*]{} [@shuchi] consider the Bayesian unit-demand pricing problem. There are $n$ heterogeneous items, a single bidder with unit demand, and her valuations ($v_j$ for item $j \in [1, \ldots , n]$) are drawn from independent distributions. They present an elegant pricing scheme that is a constant approximation to optimal revenue by upper bounding it using the revenue of Myerson’s auction in the following setting: There is a single item, $n$ bidders, and valuation of each bidder $j$ follows the same distribution as that of $v_j$. However, this technique cannot be applied if the unit demand assumption is removed. In contrast, our approach to the independent valuations setting (Section \[sec:seq\]) does [*not*]{} require the unit-demand assumption, and is based on a novel LP relaxation ([LPRev]{}) for the problem (Lemma \[lem:crux\]). Unlike the LP relaxation of Section \[sec:lot\] and perhaps surprisingly, [LPRev]{} does not encode any incentive compatibility constraints, and our DSIC mechanism (Figure \[fig:postprice\]), that competes against this LP, is in fact a constant approximation to optimal BIC revenue. One limitation of our approach is that we have to (necessarily) assume the bidders’ valuations satisfy montone hazard rate (MHR). In the process of proving our main result (Theorem \[thm:optitem\]), we describe a crucial property of MHR distributions (Lemma \[lem:hazard\]) that can be used to extend the type of results shown in [@HR]. For example, in multi-item settings with only demand constraints, [*posted price schemes generate revenue that is a constant factor of the optimal social welfare*]{}, assuming MHR distributions (Corollary \[cor1\]). The LP formulations also generalize the stochastic matching setting in Chen [*et al.*]{} [@chen]. Correlated Valuations: BIC Mechanisms {#sec:lot} ===================================== We first consider the problem of approximating the optimal Bayesian incentive compatible mechanism. There are $m$ bidders and $n$ indivisible items. The type $t$ of a bidder is a $n$-vector $\langle v_{1t}, v_{2t}, \ldots, v_{nt}\rangle$, which specifies her valuations for the items. For bidder $i$, we assume the types follow a publicly known discrete distribution with polynomial support, where for $t = t_1, t_2, \ldots, t_K$, we have $f_{i}(t) = \Pr[\mbox{Type } = t]$. The distributions for different bidders are independent. Note that the above model allows the valuations of a bidder for different items to possibly be correlated. The bidder can afford to pay at most a publicly known budget $B_i$, and is interested in at most $n_i \ge 1$ items. (Our scheme extends to private budgets using the techniques in [@sayan].) We will be interested in mechanisms that can be computed in time polynomial in the input size, [*i.e.*]{}, in $n, m$, and $K$. For reported types $\vec{t}$, the auctioneer computes allocation of items $S_i(\vec{t})$ for each bidder $i$. It also computes prices $P_i(\vec{t})$. The subsets $S_i(\vec{t})$ of items need to be disjoint, and $P_i(\vec{t}) \le B_i$. The utility of a bidder for obtaining subset $S$ at price $P$ when her type is $t$ is $\sum_{j \in S} v_{jt} - P$. A mechanism should be incentive compatible in the following sense: For any bidder $i$, her expected utility, where the expectation is over the types of other bidders, is maximized if she reveals her true type. The auctioneer is interested in designing an incentive compatible mechanism that maximizes revenue: ${\mathbf E}_{\vec{t}} \left[\sum_i P_i(\vec{t}) \right]$. We now show an all-pay auction mechanism that is a $4$-approximation to optimal revenue. Our LP relaxation and solution technique are inspired by the rounding scheme in [@sayan] for multi-unit auctions. [**Linear Programming Relaxation.** ]{} Throughout, we will denote bidders by $i$, items by $j$, and types by $t$. For any feasible mechanism, let $x_{ij}(t)$ denote the probability that bidder $i$ obtains item $j$ if her reported type is $t$. Let $P_{i}(t)$ denote the expected price paid by bidder $i$ when she bids type $t$. We have the following LP: $$\mbox{Maximize } \ \ \ \sum_{i,t} f_{i}(t) P_{i}(t) \qquad \mbox{({\sc LP1})}$$ $$\begin{array}{rcll} \sum_{i,t} f_{i}(t) x_{ij}(t) & \le & 1 & \forall j \\ \sum_{j} x_{ij}(t) & \le & n_i & \forall i, t \\ \sum_j v_{jt} x_{ij}(t) - P_{i}(t) & \ge & \sum_j v_{jt} x_{ij}(s) - P_{i}(s) & \forall i, t, s\\ \sum_j v_{jt} x_{ij}(t) - P_{i}(t) & \ge & 0 & \forall i, t\\ x_{ij}(t) & \in & [0, 1] & \forall i, j, t\\ P_{i}(t) & \in & [0,B_i] & \forall i, t \\ \end{array}$$ The optimal Bayesian incentive compatible mechanism is feasible for the above constraints: The first constraint simply encodes that in expectation, each item is assigned to at most one bidder; the second constraint encodes the demand. The third and fourth constraints encodes Bayesian incentive compatibility and individual rationality. Therefore, the [LP1]{} value is an upper bound on the expected revenue. \[thm:allpay\] The All Pay Auction (Figure \[fig:allpay\]) is Bayesian Incentive Compatible, and its revenue is a $4$ approximation to the optimal BIC mechanism for heterogeneous items. Note that $\sum_{i} X_{ij} \le \frac{1}{2}$ for all items $j$, since we scaled down the LP variables. Furthermore, $Z_{ij} = \prod_{i' < i} (1 - X_{ij}) \geq 1 - \sum_{i' < i} X_{ij} \ge \frac{1}{2}$. This implies $\frac{1/2}{Z_{ij}} \leq 1$. In Step 6, $J$ denotes the set of available items. When bidder $i$ is encountered in the predefined (but arbirtrary) ordering, the set $W_i$ is initialized to $\emptyset$. In each group ${\mathcal{G}}_{it_i k}$, a single item $j$ is selected with probability $\tilde{x}_{ij}(t_i)$. If $j \in J$, it is added to $W_i$ with probability $\frac{1/2}{Z_{ij}}$ (which is at most 1), and regardless of the outcome of this random event, $j$ is removed from $J$. In the discussion below, all expectations are with respect the bids revealed by bidders $i' < i$ and the random choices made in constructing $W_i$. Since the values revealed by bidders $i' < i$ are independent, item $j$ is available when bidder $i$ is considered with probability exactly $\prod_{i' < i} (1-X_{ij}) = Z_{ij}$. Therefore: $$\Pr[j \in W_i] = Z_{ij} \cdot \tilde{x}_{ij}(t_i) \cdot 1/(2Z_{ij}) = \tilde{x}_{ij}(t_i)/2 = x^*_{ij}(t_i)/4$$ By linearity of expectation, the expected welfare of items allocated to bidder $i$ is precisely $\frac{1}{4} \sum_j v_{jt_i} x^*_{ij}(t_i)$. Since bidder $i$ pays a fixed price $P^*_i(t_i)/4$, both the expected welfare and expected price are scaled down by a factor exactly $4$ relative to the LP values [*regardless of her revealed type*]{}. This preserves the incentive compatibility constraints in the LP, and makes the scheme be truthful and satisfy voluntary participation in expectation over the types of other bidders and the randomness introduced by the mechanism. The theorem follows. We note that by replacing the objective function with $\sum_{i,j,t} f_{i}(t) v_{jt} x_{ij}(t)$, the resulting scheme is a $4$ approximation to the optimal expected social welfare (or any linear combination of revenue and welfare). Independent Valuations: DSIC Mechanisms {#sec:seq} ======================================= The All Pay Auction (Figure \[fig:allpay\]) satisfies the incentive compatibility constraints only in expectation over the valuations of the other bidders and randomness introduced by the mechanism. In practice, it is more desirable to consider mechanisms that satisfy these properties ex-post. As mentioned before (Section \[sec:results\]), in general, it is likely to be hard to approximate such mechanisms, and this hardness comes from the type space of a bidder being correlated. However, we show that for the important special case of product distributions with the hazard rate condition, a constant factor approximation follows from sequential posted price mechanisms. Such a mechanism considers the bidders sequentially in arbitrary order, and for each bidder, posts a subset of remaining items at certain price for each item, and lets the bidder choose the utility maximizing bundle of these items. Such a scheme is ex-post incentive compatible by definition. Formally, there are $n$ distinct indivisible goods each of unit quantity, and $m$ bidders. Bidder $i$ has publicly known budget $B_i$ on the total price she is willing to pay. Her utility for paying more than this price is $-\infty$; as long as the budget constraint is satisfied, the utility for any item is quasi-linear, [*i.e.*]{}, the valuation obtained minus price, thresholded below at zero. The valuations for different items are additive, and she can buy more than one item, but at most some $n_i \ge 1$ items. Her valuation for item $j$ is a positive integer-valued random variable $v_{ij} \in [1,L_{ij}]$ following publicly known distributions. Though we assume the distributions are defined at positive integer values, this is w.l.o.g., since we can discretize continuous distributions in powers of $(1+{\epsilon})$ and apply the same arguments. This also holds when the values taken by the random variables are not polynomially bounded. [**Key Assumption.**]{} We will present ex-post incentive compatible mechanisms under the following key [*product distribution*]{} assumption: The valuations $v_{ij}$ for each $(i,j)$ follow independent distributions. Under the above assumption we consider two cases: (1) The random variables $v_{ij}$ satisfy the monotone hazard rate condition (Section \[sec:mon\]), and (2) The random variables $v_{ij}$ are arbitrary integer valued variables over the domain $[1,L]$ (Section \[sec:gen\]). In the former case, we show that sequential posted prices are a $O(1)$ approximation to the optimal BIC mechanism (Theorem \[thm:optitem\]), whereas in the latter case, they are a $O(\log L)$ approximation, and doing better necessarily requires pricing bundles of items rather than individual items (Theorem \[thm:gen\]). [**Proof Sketch of Theorem \[thm:optitem\].**]{} First, observe that in order to enforce individual rationality, whenever item $j$ is allocated to bidder $i$, the expected revenue generated across edge $(i,j)$ is at most ${\mathcal V}_{ij} = \min(v_{ij}, B_i)$. Let $x_{ij}(r)$ denote the probability that bidder $i$ gets item $j$ when ${\mathcal V}_{ij} = r$. If we require the demand, budget and supply constraints to hold [*only in expectation*]{}, the revenue of any mechanism can be upper bounded by a linear program with variables $x_{ij}(r)$ (Lemma \[lem:crux\]). Interestingly, this linear program ([LPRev]{}) does not encode incentive compatibility constraints. Thus, any DSIC scheme that is constant competitive against [LPRev]{} will also be a $O(1)$ approximation to optimal BIC revenue. Note that if $v_{ij}$ satisfies MHR, then ${\mathcal V}_{ij}$ also satisfies MHR (Claim \[claim1\]). Next, we make the crucial observation that if the prior on a bidder’s valuation is MHR, then with constant probability her virtual valuation is within a constant factor of her valuation (Lemma \[lem:hazard\]). Thus, replacing ${\mathcal V}_{ij}$ by the corresponding virtual valuation function $\varphi_{ij}$, we get a new linear program ([LP2]{}) that is still a good approximation to [LPRev]{} and hence to optimal revenue (Lemma \[lem:LP2\]). Using Myerson’s characterization of virtual valuations [@myerson], we show that the optimal solution to [LP2]{} has a nice structure (Lemma \[lem:round\]). Roughly speaking, it treats each edge $(i,j)$ as a separate bidder with valuation given by the random variable ${\mathcal V}_{ij}$. Finally, we employ an interesting rounding technique (Figure \[fig:postprice\]) on the optimal solution to [LP2]{}, and get a DSIC item pricing scheme that looses a constant factor in revenue. Since we need to apply Markov’s inequality during the rounding phase, we revisit the definition of ${\mathcal V}_{ij}$ and scale down the reported budget by a suitable factor (Definition \[def:scale\]). This implies [LPRev]{} is an upper bound on optimal revenue up to a constant factor; rest of the proof remains the same. One issue with auctions where bidders are budget and demand constrained is that given posted prices for the items, the bidder needs to solve a two-dimensional [Knapsack]{} problem to determine her optimal bundle, and this is [NP-Hard]{} in general. However, this aspect does not change our results: Our upper bound [LP2]{} is the best possible revenue regardless of what optimization the bidders perform, and furthermore, our analysis of the algorithm in Figure \[fig:postprice\] simply shows that with constant probability, the bidder solves a trivial knapsack instance where all items fit into the knapsack. Therefore, as long as the bidder uses any reasonable [Knapsack]{} heuristic, our results hold. Linear Programming Relaxation ----------------------------- We first present a simple bound on the revenue of any BIC mechanism. In the ensuing discussion, we will denote bidders by $i$ and items by $j$. \[def:scale\] Define ${\mathcal V}_{ij} = \min(v_{ij}, B_i/4)$ and let its density function be $f_{ij}$. $$\mbox{Maximize } \ \ \ \sum_{i,j} \sum_r r f_{ij}(r) x_{ij}(r) \qquad \mbox{({\sc LPRev})}$$ $$\begin{array}{rcllr} \sum_{j} \sum_r f_{ij}(r) x_{ij}(r) & \le & n_i & \forall i & \hfill (1)\\ \sum_{j} \sum_r r f_{ij}(r) x_{ij}(r) & \le & B_i & \forall i & \hfill (2) \\ \sum_{i} \sum_r f_{ij}(r) x_{ij}(r) & \le & 1 & \forall j & \hfill (3)\\ x_{ij}(r) & \in & [0,1] & \forall i, j, r & \hfill (4)\\ \end{array}$$ \[lem:crux\] The optimal value of the program ([LPRev]{}) is at least a factor $1/4$ of the revenue of the optimal BIC mechanism. Furthermore, in the LP solution, $x_{ij}(r)$ is a monotonically non-decreasing function of $r$. Without any loss of generality, we assume that whenever a bidder is allocated a bundle of items, the total price she has to pay is distributed amongst the individual items obtained. Since the mechanism satisfies the individual rationality condition in expectation, it is easy to ensure that the expected price on a single item is never greater than the valuation for the item or the overall budget. Let $x_{ij}(r)$ denote the expected amount of item $j$ bidder $i$ obtains when ${\mathcal V}_{ij} = r$, and let $p_{ij}(r)$ denote the expected price paid conditioned on obtaining the item. It is easy to see that $p_{ij}(r) \leq 4r$. The revenue of the optimal BIC mechanism can be relaxed as: $$\mbox{Maximize } \ \ \ \sum_{i,j} \sum_r p_{ij}(r) f_{ij}(r) x_{ij}(r)$$ $$\begin{array}{rcllr} \sum_{j} \sum_r f_{ij}(r) x_{ij}(r) & \le & n_i & \forall i & \hfill (1)\\ \sum_{j} \sum_r p_{ij}(r) f_{ij}(r) x_{ij}(r) & \le & B_i & \forall i & \hfill (2) \\ \sum_{i} \sum_r f_{ij}(r) x_{ij}(r) & \le & 1 & \forall j & \hfill (3)\\ x_{ij}(r) & \in & [0,1] & \forall i, j, r & \hfill (4)\\ p_{ij}(r) & \in & [0,4r] & \forall i, j, r & \hfill (5)\\ \end{array}$$ The above program is nonlinear. Scale $p_{ij}(r)$ down by a factor of $4$ so that $p_{ij}(r) \le r$. This preserves the constraints and loses a factor of $4$ in the objective. Now, for all $i, j$, if $p_{ij}(r) < r$, increase $p_{ij}(r)$ and decrease $x_{ij}(r)$ while preserving their product until $p_{ij}(r)$ becomes equal to $r$. This yields the constraints of [(LPRev)]{}, and preserves the objective; but now, the objective becomes $ \sum_{i,j} \sum_r r f_{ij}(r) x_{ij}(r)$. This shows [(LPRev]{}) is a $4$-approximation to the revenue of the optimal BIC mechanism. To show that the objective is maximized when the $x_{ij}(r)$ are monotonically non-decreasing in $r$, for any $(i,j)$, preserve $\sum_{r} r f_{ij}(r) x_{ij}(r)$ by increasing $x_{ij}(s_2)$ and decreasing $x_{ij}(s_1)$ for $s_1 < s_2$. In this process, $\sum_r f_{ij}(r) x_{ij}(r)$ must decrease, preserving all the constraints, which implies the monotonicity. We note that because of the presence of budget constraints, [(LPRev)]{} bounds only the expected revenue of the ex-post mechanism and [*not*]{} the expected social welfare, which can be larger by an unbounded amount. However, if the budget constraints are removed, the resulting LP also bounds the optimal social welfare. Monotone Hazard Rates {#sec:mon} --------------------- We will now present a constant factor approximation to the optimal BIC mechanism via sequential posted price schemes, assuming the random variables $v_{ij}$ satisfy the monotone hazard rate condition. The distribution of $v_{ij}$ satisfies the monotone hazard rate (MHR) condition if $\Pr[v_{ij} > r]/Pr[v_{ij} = r]$ is a non-increasing function of $r$. \[claim1\] If $v$ is a random variable satisfying the monotone hazard rate condition, then $\min(v,a)$ satisfies the MHR condition for any integer $a \ge 1$. It follows that the ${\mathcal V}_{ij}$ also satisfy the MHR condition. \[def:virt\] Let $G_{ij}(r) = \Pr[{\mathcal V}_{ij} > r]$. Define the [*virtual valuation*]{} as $\varphi_{ij}(r) = r - G_{ij}(r)/f_{ij}(r)$. This is said to be [*regular*]{} if it is a non-decreasing function of $r$. Clearly, MHR distributions are regular. We now present the crucial lemma for MHR distributions. \[lem:hazard\] Let $\phi(v)$ denote the virtual valuation function for a positive integer valued random variable $v \in [1,L]$ satisfying the MHR condition. Then, $\Pr[\phi(v) \ge \frac{v}{2}] \ge \frac{1}{e^2} $. Let $f(t) = \Pr[v = t]$ and $G(t) = \Pr[v > t]$. Let $h(t) = f(t)/G(t)$ denote the hazard rate at $t$. Since $f$ satisfies MHR, $\phi(t) < t/2$ iff $t \le k$ for some integer $1 \le k \le L$. Furthermore, $\phi(k) = k - 1/h(k) \le k/2$. This implies for all $t \le k$, $h(t) \le h(k) \le 2/k$. Therefore, $\Pr[\phi(t) \ge t/2] = \Pr[t > k]$. To bound the latter quantity, replace $f$ with a continuous-valued distribution $\hat{f}$ whose density in $[t,t+1)$ for any integer $t \in [1,L]$ is precisely $f(t)$. It is easy to see that this new density also satisfies the hazard rate condition. Let $\hat{G}(t) = 1 - \int_{q=1}^{t+1} \hat{f}(q) dq$, and let $\hat{h}(t) = \frac{\hat{f}(t)}{\hat{G}(t)}$. Thus, $$ \Pr[\phi(t) \ge t/2] = \Pr[t > k] = G(k) = \hat{G}(k) = e^{-\int_{t=1}^{k+1} \hat{h}(t) dt} \ge e^{-\int_{t=1}^{k+1} \frac{2}{k} dt} \ge e^{-2} $$ Let $v^*_{ij} = \mbox{argmin}_r \{\varphi_{ij}(r) \ge \frac{r}{2}\}$. Then, Lemma \[lem:hazard\] and Claim \[claim1\] imply $\Pr[{\mathcal V}_{ij} \ge v^*_{ij}] \ge e^{-2}$. ### Incorporating Virtual Valuations Now consider the following linear program obtained from ([LPRev]{}). $$\mbox{Maximize } \ \ \ \sum_{i,j} \sum_r f_{ij}(r) \varphi_{ij}(r) x_{ij}(r) \qquad \mbox{({\sc LP2})}$$ $$\begin{array}{rcllr} \sum_{j} \sum_r f_{ij}(r) x_{ij}(r) & \le & n_i & \forall i & \hfill (1)\\ \sum_{j} \sum_r f_{ij}(r) \varphi_{ij}(r) x_{ij}(r) & \le & B_i & \forall i & \hfill (2) \\ \sum_{i} \sum_r f_{ij}(r) x_{ij}(r) & \le & 1 & \forall j & \hfill (3)\\ x_{ij}(r) & \in & [0,1] & \forall i, j, r & \hfill (4) \end{array}$$ \[lem:LP2\] The value of [(LP2)]{} is at least $\frac{1}{2e^2}$ times the value of [(LPRev)]{}. In the optimal solution to [(LPRev)]{}, since $\Pr[{\mathcal V}_{ij} \ge v^*_{ij}] \ge e^{-2}$ by Lemma \[lem:hazard\] and since $x_{ij}(r)$ is monotone in $r$, setting $x_{ij}(r) = 0$ for $r < v^*_{ij}$ loses a factor of at most $e^2$ in the objective value and preserves feasibility. Now replacing $r$ by $\varphi_{ij}(r) \in [r/2,r]$ for $r \ge v^*_{ij}$ loses another factor of $2$ by Lemma \[lem:hazard\], and preserves feasibility. This shows a feasible solution to ([LP2]{}) of value at least $\frac{1}{2e^2}$ times that of [(LPRev)]{}. \[lem:round\] The optimal solution to [(LP2)]{} is as follows: For each $(i,j)$, we have a convex combination of two solutions. The first (resp. second) solution has variables $y_{ij}(r)$ (resp. $z_{ij}(r)$), and value $r^*_{ij} \le B_i/4$ (resp. $s^*_{ij} = r^*_{ij}+1 \le B_i/4$) so that if $r < r^*_{ij}$, then $y_{ij} = 0$, else $y_{ij} = 1$. Similarly, if $r < s^*_{ij}$, then $z_{ij} = 0$, else $z_{ij} = 1$. If the first solution has weight $p_{ij}$ in the convex combination, and the second $1-p_{ij}$, then: $$\label{eq1} \sum_r f_{ij}(r) \varphi_{ij}(r) x_{ij}(r) = p_{ij} r^*_{ij} \Pr[{\mathcal V}_{ij} \ge r^*_{ij}] + (1-p_{ij}) s^*_{ij} \Pr[{\mathcal V}_{ij} \ge s^*_{ij}] $$ For each $(i,j)$ if $x_{ij}(r_1), x_{ij}(r_2) \in (0,1)$ for $r_1 < r_2 \le B_i/4$, then $x_{ij}(r_1)$ can be increased and $x_{ij}(r_2)$ can be decreased to preserve $\sum_r \varphi_{ij}(r) f_{ij}(r) x_{ij}(r)$. Since $\varphi_{ij}(r)$ is a monotonically non-decreasing function of $r$, this implies $\sum_r f_{ij}(r) x_{ij}(r)$ must decrease in this process, preserving all constraints. When this process terminates, we must have $x_{ij}(r) = 0$ for all $r < r^*$ and $x_{ij}(r) = 1$ for all $r > r^*$ for some $r^*$. This implies the optimal solution to [(LP2)]{} can be written in the fashion implied by the lemma, with $r^*_{ij} = r^*$ and $p_{ij} = x_{ij}(r^*)$. By Myerson’s characterization of virtual valuations [@myerson], we have: $$\sum_r f_{ij}(r) \varphi_{ij}(r) x_{ij}(r) = \sum_r \left(f_{ij}(r) \left(r x_{ij}(r) - \sum_{s=1}^{r-1} x_{ij}(s) \right) \right)$$ Since $y_{ij}(r)$ is a $0/1$ step function, it is easy to see that $$\sum_r \left(f_{ij}(r) \left(r y_{ij}(r) - \sum_{s=1}^{r-1} y_{ij}(s) \right) \right) = r^*_{ij} \Pr[{\mathcal V}_{ij} \ge r^*_{ij}]$$ A similar equality holds for $z_{ij}$, and taking a convex combination completes the proof of the lemma. ### Posted Price Mechanism and Analysis {#sec:rounding} The posted price auction is described in Figure \[fig:postprice\]. Note that though the scheme is randomized, since it is posted price, it is universally and ex-post truthful and individually rational. \[thm:optitem\] The revenue of the posted price auction (Figure \[fig:postprice\]) is a $O(1)$ approximation to the optimal BIC mechanism, when the valuations of a bidder follow product distributions that satisfy the MHR condition. We show that for every pair $(i,j)$, the mechanism achieves a revenue that is within a constant factor of $ \sum_r f_{ij}(r) \varphi_{ij}(r) x_{ij}(r) $ in [(LP2)]{} for the pair. First note that if a price is posted according to the mechanism in Figure \[fig:postprice\] just along edge $(i,j)$, then by Equation (\[eq1\]), the expected revenue is simply: $$p_{ij} r^*_{ij} \Pr[{\mathcal V}_{ij} \ge r^*_{ij}] + (1-p_{ij}) s^*_{ij} \Pr[{\mathcal V}_{ij} \ge s^*_{ij}] = \sum_r f_{ij}(r) \varphi_{ij}(r) x_{ij}(r)$$ We will now show that even in the presence of other items and bidders, a constant factor of this revenue is extracted. For pair $(i,j)$, let $X_{ij}$ be the $0/1$ random variable denoting whether item $j$ is taken by bidder $i$, and let $P_{ij}$ denote the price at which it is taken (which is $0$ if the item is not taken by the bidder). These are random variables that depend on the prior distributions as well as the random choices made in choosing the structured solution and the mechanism. Since the probability that any $(i,j)$ is considered at all is $1/4$, the constraints of [(LP2)]{}, when scaled down by factor $4$, imply by linearity of expectation that for all $i,j$: $${\mathbf E}[\sum_{k \neq j}P_{ik}] \le B_i/4, \qquad {\mathbf E}[\sum_{k \neq j} X_{ik}] \le n_i/4, \qquad \mbox{and} \qquad {\mathbf E}[\sum_{m \neq i} X_{mj}] \le 1/4$$ Note that snce the valuation $v_{ij}$ is independent of other valuations, this implies the above statements hold regardless of $v_{ij}$. Now applying Markov’s inequality, we have for all $i,j$: $$\Pr[\sum_{k \neq j}P_{ik} \ge 3B_i/4] \le 1/3, \qquad \Pr[\sum_{k \neq j}X_{ik} \ge n_i ] \le 1/4, \qquad \mbox{and} \qquad \Pr[\sum_{m \neq i} X_{mj} \ge 1] \le 1/4$$ By union bounds, this implies that with probability at least $1/6$, we must have: $\sum_{k \neq j}P_{ik} < 3B_i/4$, $\sum_{k \neq j}X_{ik} < n_i$, and $\sum_{m \neq i} X_{mj} = 0$. In this event, item $j$ is offered to bidder $i$ w.p. $1/4$ (using one of two random choices of the posted price from Lemma \[lem:round\]). Furthermore, in this event, the bidder must take the item if $v_{ij} \ge \tilde{r}_{ij}$, since $\tilde{r}_{ij} \le B_i/4$ (so that the bidder has sufficient budget to purchase this item), and the bidder has not exhausted his demand $n_i$. Since the valuation $v_{ij}$ itself is independent of the event that the item $j$ is offered to bidder $i$, this implies that with probability at least $1/6 \times 1/4$, the posted price mechanism obtains revenue $ \sum_r f_{ij}(r) \varphi_{ij}(r) x_{ij}(r) $ along each $(i,j)$ (using the definition of the latter quantity from Equation (\[eq1\])). By linearity of expectation over all $(i,j)$, we have a $O(1)$ approximation. \[cor1\] Under the assumptions of Theorem \[thm:optitem\], if there are no budget constraints, then the revenue of the sequential posted price mechanism is a constant factor approximation to the optimal social welfare. If the Budget Constraint (2) is removed from [(LPRev)]{}, it is an upper bound on the optimal social welfare that can be obtained by any mechanism. The rest of the analysis remains the same. General Distributions {#sec:gen} --------------------- The key point shown by Theorem \[thm:optitem\] is that the optimal revenue of an ex-post truthful mechanism is approximated to a constant factor by a simple sequential posted price auction, where the prices are posted for each item and each bidder. This crucially used the monotone hazard rate condition. The natural question to ask is whether such a simple mechanism is a good approximation for more general classes of distributions. Note that even for one bidder, the optimal DSIC mechanism will in general price [*bundles*]{} of items. We show below that bundle pricing has a logarithmic advantage over item pricing even for regular distributions (see Definition \[def:virt\]), and that this gap is tight. \[thm:gen\] Suppose $v_{ij} \in [1,L]$, are independent for different $(i,j)$ but do not necessarily satisfy the monotone hazard rate condition. Then, there is a $\Theta(\log L)$ gap between the revenues of the optimal posted price scheme and the optimal DSIC mechanism. The lower bound holds for regular distributions and one bidder. To show the lower bound, consider the following scenario: There is only one bidder, $n$ items, and no budget or demand constraints. The valuations are $i.i.d.$ for each item $j$, and follow the common distribution with $G(r) = \Pr[v \ge r] = \frac{1}{r}$, for $r = 1,2, \ldots, n$, so that $n = L$. This distribution is regular, since $\varphi(r) = -1$ for $r < n$, and $\varphi(n) = n-1$. We have ${\mathbf E}[v_{1j}] = H_n$ for all item $j$, and by Chernoff bounds, $$\Pr[\sum_{j=1}^n v_{1j} \le (1-\epsilon) n H_n] \le e^{\frac{-n H_n {\epsilon}^2}{2n}} = o(1)$$ Therefore, a scheme that sets a price of $n H_n (1-{\epsilon})$ for the bundle of $n$ items sells the bundle with probability $1-o(1)$, so that the expected revenue is $\Omega(n \log n)$. Any posted price scheme can extract a revenue of $\max_i i \cdot G(i) = 1$ from each item, so that the expected revenue from $n$ items is at most $n$. This shows a gap of $\Omega(\log n) = \Omega(\log L)$. The upper bound follows from a standard scaling argument. We start with [(LPRev)]{} which upper bounds the optimal achievable revenue. For each $(i,j)$, group the $r$ values in powers of $2$ so that there are $O(\log L)$ groups. Let group ${\mathcal{G}}_k$ denote the interval $[2^k, 2^{k+1})$. The first step is to solve [(LPRev)]{}. Let its optimal value be $OPT$; note that this is a $4$ approximation to the optimal ex-post incentive compatible scheme. Now, for each $(i,j)$ perform the following rounding. Let $$\begin{split} & k_{ij} = \mbox{argmax}_k \sum_{r \in {\mathcal{G}}_k} r f_{ij}(r) x_{ij}(r) \\ & r^*_{ij} = 2^{k_{ij}} \\ & x^*_{ij} = \frac{\sum_{r \in {\mathcal{G}}_k} f_{ij}(r) x_{ij}(r)}{\sum_{r \in {\mathcal{G}}_k} f_{ij}(r)} \\ & Q_{ij} = \Pr[\tilde{V}_{ij} \ge r^*_{ij}] \end{split}$$ Since $x_{ij}(r)$ is monotone in $r$, it is easy to observe that: 1. $x^*_{ij} \le x_{ij}(r)$ for $r \in {\mathcal{G}}_k$ for $k > k^*_{ij}$. This implies $x^*_{ij} Q_{ij} \le \sum_r f_{ij}(r) x_{ij}(r)$. 2. $\sum_{r \in {\mathcal{G}}_{k_{ij}}} r f_{ij}(r) x_{ij}(r) \le 2 r^*_{ij} x^*_{ij} \Pr[r \in {\mathcal{G}}_{k_{ij}}]$. 3. $\max_k \sum_{r \in {\mathcal{G}}_k} r f_{ij}(r) x_{ij}(r) = \Omega(1/\log L) \sum_r r f_{ij}(r) x_{ij}(r) $. 4. Therefore, $r^*_{ij} x^*_{ij} Q_{ij} = \Omega(1/\log L) \sum_r r f_{ij}(r) x_{ij}(r) $. Furthermore, $r^*_{ij} x^*_{ij} Q_{ij} \le \sum_r r f_{ij}(r) x_{ij}(r) $. The above directly implies $\sum_{i,j} r^*_{ij} x^*_{ij} Q_{ij} = \Omega(OPT/\log L)$, and the following constraints: $$\begin{array}{rcll} \sum_{j} x^*_{ij} Q_{ij} & \le & n_i & \forall i \\ \sum_{j} r^*_{ij} x^*_{ij} Q_{ij} & \le & B_i & \forall i \\ \sum_{i} x^*_{ij} Q_{ij} & \le & 1 & \forall j \\ x^*_{ij} & \in & [0,1] & \forall i, j \\ r^*_{ij} & \in & [0,B_i/4] &\forall i,j \end{array}$$ The final mechanism chooses an arbitrary ordering of bidders. When bidder $i$ is encountered, let $S_i$ denote the subset of items that have survived so far. For each item $j \in S_i$, with probability $x^*_{ij}/4$, post price $r^*_{ij}$, and with the remaining probability skip this item. This process is done simultaneously for all items, and the resulting items and prices are simultaneously offered to the bidder. Note that though the scheme is randomized, since it is posted price, it is universally and ex-post truthful and individually rational. Using the same proof as Theorem \[thm:optitem\] now shows that this posted price scheme extracts revenue that is a constant factor of $\sum_{i,j} r^*_{ij} x^*_{ij} Q_{ij}$. This shows a $O(\log L)$ approximation. ### Adaptive Posted Price Schemes The above implies we cannot generalize the result in Section \[sec:mon\] even to the class of regular distributions (see Definition \[def:virt\]). This is because an $\omega(1)$ gap is introduced in going from [(LPRev)]{} to [(LP2)]{}. However, we can show that if the space of mechanisms is restricted, [(LP2)]{} itself is a good relaxation to the optimal revenue in this space. In particular, suppose we restrict the space of mechanisms to be those that are posted price, and sequential by bidder: Depending on the subset of items and bidders left, the mechanism adaptively chooses the next bidder and posts prices for a subset of the remaining items. The bidder being a utility maximizer, solves a knapsack problem to choose the optimal subset of items. Once this bidder is dealt with, the mechanism again adaptively chooses the next bidder depending on the acceptance strategy of this bidder. Assuming the valuations of a bidder follow a product distribution, and $v_{ij}$ follows a regular distribution, we will show a $O(1)$ approximation to the optimal mechanism. Since the optimization problem the bidders need to solve is a two-dimensional [Knapsack]{} problem which is [NP-Hard]{}, we we assume bidders are [*monotone optimizers*]{} in the following sense: For any edge $(i,j)$, if all valuations except $v_{ij}$ are fixed, the quantity of item $j$ taken by bidder $i$ is monotonically non-decreasing in $v_{ij}$. This is true if the bidder solves [Knapsack]{} optimally. As with the posted price scheme in the previous section, our algorithm itself will allow arbitrary [Knapsack]{} heuristics as long as a bidder chooses all items if she is not constrained by either demand or budget. [(LP2)]{} is a $4$-approximation to the revenue of the optimal adaptive posted price scheme assuming bidders are monotone optimizers. In the discussion below, recall the definition ${\mathcal V}_{ij} = \min(v_{ij}, B_i/4)$ from Definition \[def:scale\], and observe that its density function $f_{ij}$ satisfies the regularity condition. For any adaptive mechanism, let $x_{ij}(r)$ denote the expected allocation of item $j$ to bidder $i$ conditioned on ${\mathcal V}_{ij} = r$, and let $p_{ij}(r)$ denote the corresponding expected price paid scaled down by a factor of $4$. We will show that the LP relaxation that is a $4$-approximation to maximizing expected revenue is the following: $$\mbox{Maximize } \ \ \ \sum_{i,j} \sum_r f_{ij}(r) p_{ij}(r) \qquad \mbox{({\sc LPSeq})}$$ $$\begin{array}{rcllr} \sum_{j} \sum_r f_{ij}(r) x_{ij}(r) & \le & n_i & \forall i & \hfill (1)\\ \sum_{j} \sum_r f_{ij}(r) p_{ij}(r) & \le & B_i & \forall i & \hfill (2) \\ \sum_{i} \sum_r f_{ij}(r) x_{ij}(r) & \le & 1 & \forall j & \hfill (3)\\ r x_{ij}(r) - \sum_{s=1}^{r-1} x_{ij}(s) & \ge & p_{ij}(r) & \forall i, j, r & \hfill (4) \\ x_{ij}(r) & \ge & x_{ij}(s) & \forall i,j, r \ge s \ge 0 & \hfill (5)\\ p_{ij}(r) & \ge & p_{ij}(s) & \forall i,j, r \ge s \ge 0 & \hfill (6)\\ x_{ij}(r) & \in & [0,1] & \forall i, j, r & \hfill (7)\\ \end{array}$$ To see this, fix all valuations except $v_{ij}$. If the mechanism considers item $j$ for bidder $i$, then it posts a unique price $p^* \le B_i$ for $(i,j)$. Let $X_{ij}(r) \in [0,1]$ denote the expected quantity taken when ${\mathcal V}_{ij} = r$, and let $P_{ij}(r) = p^* X_{ij}(r)$ denote the expected price paid. Note that as long as $r < B_i / 4$, $v_{ij} = r$ and $X_{ij}(r)$ is either $0$ or $1$. However, at $r = B_i / 4$, $v_{ij}$ is not unique and $X_{ij}(r)$ can take a fractional value. Since the bidder is a utility maximizer, $X_{ij}(r)$ (and hence $P_{ij}(r)$) are monotone functions of $r$. Furthermore, it is easy to check that $r X_{ij}(r) - \sum_{s=1}^{r-1} X_{ij}(s) \ge p^* X_{ij}(r)/4 = P_{ij}(r)/4$. Defining $x_{ij}(r) = {\mathbf E}[X_{ij}(r)]$ and $p_{ij}(r) = {\mathbf E}[P_{ij}(r)]/4$, where the expectation is over the remaining valuations, we see that all constraints in the above LP hold. Consider the optimal solution to [(LPSeq)]{}, and focus on edge $(i,j)$. Let $g_{ij}(r) = r x_{ij}(r) - \sum_{s=1}^{r-1} x_{ij}(s) $. This is a monotone function of $r$, and so is $p_{ij}(r) \le g_{ij}(r)$. Consider the [*lexicographically maximal*]{} solution where it cannot be the case that $p_{ij}(r)$ can be increased and some $p_{ij}(s)$ for $s > r$ can be decreased preserving all constraints and the objective. Let $r_0$ be the largest integer such that for all $r < r_0$, $g_{ij}(r) = p_{ij}(r)$. It is easy to see that for all $r \ge r_0$, we have $p_{ij}(r) = p_{ij}(r_0)$, else the lexicographic maximality property is violated. Now, reduce $x_{ij}(r_0)$ until $p_{ij}(r_0) = g_{ij}(r_0)$. Observe that if $x_{ij}(r)$ is set to $x_{ij}(r_0)$ for all $r > r_0$, all constraints are preserved and the objective is unchanged. This also ensures that $p_{ij}(r) = g_{ij}(r)$ for all $r \ge 0$. This makes Constraints (4) and (6) irrelevant since the former is tight and implies the latter via Constraint (5). Thus, [(LPSeq)]{} reduces to: $$\mbox{Maximize } \ \ \ \sum_{i,j} \sum_r f_{ij}(r) \left(r x_{ij}(r) - \sum_{s=0}^{r-1} x_{ij}(s) \right)$$ $$\begin{array}{rcllr} \sum_{j} \sum_r f_{ij}(r) x_{ij}(r) & \le & n_i & \forall i\\ \sum_{j} \sum_r f_{ij}(r) \left(r x_{ij}(r) - \sum_{s=1}^{r-1} x_{ij}(s) \right) & \le & B_i & \forall i \\ \sum_{i} \sum_r f_{ij}(r) x_{ij}(r) & \le & 1 & \forall j\\ x_{ij}(r) & \in & [0,1] & \forall i, j, r\\ \end{array}$$ Now, by Myerson’s characterization [@myerson], we have: $$\sum_r \left(f_{ij}(r) \left(r x_{ij}(r) - \sum_{s=1}^{r-1} x_{ij}(s) \right) \right) = \sum_r f_{ij}(r) \varphi_{ij}(r) x_{ij}(r)$$ Plugging this formula into the objective and the second constraint completes the proof. The final mechanism and analysis are the same as in Section \[sec:mon\]: Solve [(LP2)]{}, decompose it into a convex combination of posted prices per edge, and sequentially post these prices for every bidder. To complete the analysis, note that Lemma \[lem:round\] only requires that the distribution be regular. This shows the following theorem: \[thm:final\] There is a $O(1)$ approximation to the revenue of the optimal adaptive posted price scheme, when the types follow product distributions, and for each $(i,j)$, the distribution of valuation is regular. [**Acknowledgments:**]{} We thank Shuchi Chawla, Vincent Conitzer, and Peng Shi for helpful discussions. [10]{} Z. Abrams. Revenue maximization when bidders have budgets. In [*SODA*]{}, pages 1074–1082, 2006. L. M. Ausubel. An efficient ascending-bid auction for multiple objects. , 94(5):1452 – 1475, 2004. L. M. Ausubel and P. Milgrom. Ascending auctions with package bidding. , 2002. J. Benoit and V. Krishna. Multiple-object auctions with budget constrained bidders. , 68:155 – 179, 2001. S. Bhattacharya, V. Conitzer, K. Munagala, and L. Xia. Incentive compatible budget elicitation in multi-unit auctions. In [*SODA*]{}, 2010. K. C. Border. Reduced form auctions revisited. , 31(1):167–181, 2007. C. Borgs, J. T. Chayes, N. Immorlica, M. Mahdian, and A. Saberi. Multi-unit auctions with budget-constrained bidders. In [*ACM Conference on Electronic Commerce*]{}, pages 44–51, 2005. P. Briest. Uniform budgets and the envy-free pricing problem. In [*ICALP (1)*]{}, pages 808–819, 2008. P. Briest, S. Chawla, R. D. Kleinberg, and S. M. Weinberg. Pricing randomized allocations. In [*SODA*]{}, 2010. S. Brusco and G. Lopomo. Simultaneous ascending bid auctions with privately known budget constraints. , 56(1):113–142, 2008. S. Chawla, J. D. Hartline, and R. D. Kleinberg. Algorithmic pricing via virtual valuations. In [*ACM Conference on Electronic Commerce*]{}, pages 243–251, 2007. S. Chawla, J. D. Hartline, D. Malec, and B. Sivan. Sequential posted pricing and multi-parameter mechanism design. In [*STOC*]{}, 2010. Y. Che and J. Gale. Expected revenue of the all-pay auctions and first-price sealed bid auctions with budget constraints. , 50:373–380, 1996. Y. Che and J. Gale. The optimal mechanism for selling to a budget constrained buyer. , 92(2):198–233, 2000. N. Chen, N. Immorlica, A. R. Karlin, M. Mahdian, and A. Rudra. Approximating matches made in heaven. In [*ICALP (1)*]{}, pages 266–278, 2009. S. Dobzinski, R. Lavi, and N. Nisan. Multi-unit auctions with budget limits. In [*FOCS*]{}, pages 260–269, 2008. E. Elkind and S. S. Fatima. Maximizing revenue in sequential auctions. In [*WINE*]{}, pages 491–502, 2007. A. V. Goldberg, J. D. Hartline, and A. Wright. Competitive auctions and digital goods. In [*SODA*]{}, pages 735–744, 2001. V. Guruswami, J. D. Hartline, A. R. Karlin, D. Kempe, C. Kenyon, and F. McSherry. On profit-maximizing envy-free pricing. In [*SODA*]{}, pages 1164–1173, 2005. I. Hafalir, R. Ravi, and A. Sayedi. Sort-cut: A pareto optimal and semi-truthful mechanism for multi-unit auctions with budget-constrained bidders. , abs/0903.1450, 2009. J. D. Hartline and T. Roughgarden. Simple versus optimal mechanisms. In [*ACM Conference on Electronic Commerce*]{}, pages 225–234, 2009. J.-J. Laffont and J. Robert. Optimal auction with financially constrained buyers. , 52(2):181 – 186, 1996. A. M. Manellia and D. R. Vincent. Multidimensional mechanism design: Revenue maximization and the multiple-good monopoly. , 137(1):153–185, 2007. R. B. Myerson. Optimal auction design. , 6(1):58–73, 1981. N. Nisan. Google’s auction for tv ads. In [*ESA*]{}, page 553, 2009. M. Pai and R. Vohra. Optimal auctions with financially constrained bidders. , 2008. J. Thanassoulis. Haggling over substitutes. , 117(2):217–245, 2004. R. B. Wilson. . Oxford University Press, 1997. [^1]: Department of Computer Science, Duke University, Durham NC 27708. Email: [bsayan@cs.duke.edu]{} [^2]: College of Computing, Georgia Institute of Technology, Atlanta GA 30332. Email: [gagang@cc.gatech.edu]{}. [^3]: Microsoft Research Silicon Valley, Mountain View, CA 94043. Email: [sreenig@research.microsoft.com]{}. [^4]: Department of Computer Science, Duke University, Durham NC 27708. Supported by an Alfred P. Sloan Research Fellowship, and by NSF via CAREER award CCF-0745761 and grant CNS-0540347. Email: [kamesh@cs.duke.edu]{}

--- abstract: 'The high frequency peaked BL Lac object [PG 1553+113]{} underwent a flaring event in 2012. The High Energy Stereoscopic System (H.E.S.S.) observed this source for two consecutive nights at very high energies (VHE, $E>$100 GeV). The data show an increase of a factor of three of the flux with respect to archival measurements with the same instrument and hints of intra-night variability. The data set has been used to put constraints on possible Lorentz invariance violation (LIV), manifesting itself as an energy dependence of the velocity of light in vacuum, and to set limits on the energy scale at which Quantum Gravity effects causing LIV may arise. With a new method to combine H.E.S.S. and *Fermi* large area telescope data, the previously poorly known redshift of PG 1555+113 has been determined to be close to the value derived from optical measurements.' author: - 'D. A. Sanchez' - 'F. Brun' - 'C Couturier, A Jacholkowska, J.-P. Lenain' - 'On behalf of the [[*[*Fermi*]{}*]{}]{} and [<span style="font-variant:small-caps;">H.E.S.S.</span>]{} collaborations' title: 'Probe of Lorentz Invariance Violation effects and determination of the distance of [PG 1553+113]{}.' --- Introduction ============ [PG 1553+113]{} is a high frequency peaked BL Lac object located in the Serpens Caput constellation. The object has been detected in VHE by [<span style="font-variant:small-caps;">H.E.S.S.</span>]{} [@hess1553] in 2006 and in high energies (HE, 100 MeV$<$E$<$300 GeV) by [[*[*Fermi*]{}*]{}]{} [@fermi]. The $\gamma$-ray spectrum presents the largest HE-VHE spectral break measured to date [@fermi; @sanchez]. The source has an unknown redshift despite several attempts to measure it. The best estimate to-date, made by spectroscopy [@dan], is 0.43$<z<$0.58. In April 2012, the source underwent a flare reported by the MAGIC collaboration [@atel]. Subsequently, the source has been observed by the [<span style="font-variant:small-caps;">H.E.S.S.</span>]{} telescopes. The data are used in this work to constrain its redshift and to probe a possible Lorentz invariance violation. Data Analysis ============= H.E.S.S. data analysis ---------------------- The [<span style="font-variant:small-caps;">H.E.S.S.</span>]{} telescopes have observed [PG 1553+113]{} in April 2012 for two nights. Data were analysed using the [Model]{} analysis [@model] with [Loose]{} cuts. The object has been detected with a significance of 21.5$\sigma$ in 3.5 hours of live time. The source spectrum is well fitted by a power-law model of the form: $$F(E) \propto E_*^{-(4.85\pm 0.25)}$$ where $E_*=E/(327\ {\rm GeV})$ and the flux is found to be 3.5 times higher than the measurements made in 2005-2006 [@hess1553]. Indications of intra-night variability have been found with the fit to a constant of the light-curve yielding a $\chi^2$ of $21.34$ for $7$ d.o.f. ($P_{\rm \chi^2}=3.3\times 10^{-3}$). Data taken in 2005-2006 were re-analysed and the spectrum is, in this case, well fitted by a log-parabola model: $$F(E) \propto E_*^{-(5.39\pm0.43) -(3.95\pm1.40)~log_{10}E_*}$$ where $E_*=E/(360\ {\rm GeV})$. Both spectra (archival and flare) are presented on figure \[SED\]. [[[*[*Fermi*]{}*]{}]{}-LAT]{} data analysis ------------------------------------------- The [[[*[*Fermi*]{}*]{}]{}-LAT]{} data, from 300 MeV to 300 GeV, have been analysed with the ScienceTools [V9R32P5]{} and instrumental response functions [P7REP\_SOURCE\_V15]{}. A region of interest of 15 degrees has been used and the sky model has been built using the third *Fermi* catalog [@3FGL]. Data contemporaneous to the [<span style="font-variant:small-caps;">H.E.S.S.</span>]{} exposures taken in 2012 are well fitted by a power-law of index $\Gamma= 1.72\pm 0.26$. The pre-flare data are defined by the data taken from August 8, 2008 up to March 1st 2012. The measured spectrum is described by a log-parabola model (Fig. \[SED\]). Variability has been probed before, during and after the flare using a bayesian blocks analysis [@bb]. No counterpart to the VHE flare was found in the HE light curve. ![Spectral energy distribution in $gamma$-ray measured with [[[*[*Fermi*]{}*]{}]{}-LAT]{} and H.E.S.S. during the flare (red) and in the pre-flare state (see text) in blue.[]{data-label="SED"}](SED.eps){width="85mm"} Determination of the redshift ============================= The extragalactic background light (EBL) is a field of infrared photons that interacts with the VHE [$\gamma$-rays]{} on their way to Earth. This absorption leaves a footprint on the source spectrum that is used in this work to constrain its distance using a new bayesian model. The Bayes theorem reads $P(\Theta|Y) \propto P(\Theta) P(Y|\Theta)$ where $Y$ stands for the data and $\Theta$ the parameters. The likelihood, $P(Y|\Theta)$, is minimized during the spectrum determination. The model used here is a power-law corrected for EBL absorption i.e $ \phi = N\times (E/E_0)^{-\Gamma}\times e^{-\tau(E,z)}.$ and then $\Theta$ is $N$, $\Gamma$ and $z$. To construct the prior $ P(\Theta)$, the following assumptions have been made: - EBL-corrected power-law cannot be harder than the [[*[*Fermi*]{}*]{}]{} measurement, the prior being then a truncated Gaussian of mean $\Gamma_{{{\sl {\it Fermi}}}}$ and width $\sigma$ that accounts for statistical and systematic uncertainties of both instruments. - Softening of this power-law, that arises from emission effects, is permitted with a constant probability. - It is also assumed that distant sources are harder to detect and that $P(z) \propto exp(-\tau(z))$. {width="135mm"} The EBL absorption is computed using the model of [@fra]. Marginalising over the parameters gives a redshift [$z=0.49\pm0.04$]{} (Fig. \[Prob\]) in agreement with other measurement [@dan] or constraints derived using GeV-TeV data [@sanchez]. Lorentz invariance violation (LIV) ================================== Tests for a possible LIV effect were performed by searching for a non-zero dispersion parameter $\tau_n (\sim \frac{\Delta t}{(\Delta E)^n})$ in the H.E.S.S. data of the flare. This is done by testing a correlation between arrival times of the photons and their energies. A maximum likelihood analysis based on [@liv1], has been modified to tackle the non-negligible background present in the data. For $n_{\rm ON}$ events recorded in the ON-source region with arrival times $t_i$ and energies $E_i$, the likelihood reads: $\mathit{L}(\tau_n) = \prod_{i = 1}^{n_{\rm ON}} P(E_i, t_i|\tau_n)$ with: -.3cm $$P(E_i, t_i|\tau_n) = w_s \cdot P_{\textrm{\tiny Sig}}(E_i, t_i|\tau_n) + (1-w_s) \cdot P_{\textrm{\tiny Bkg}}(E_i, t_i)$$ -.3cm The probability $P_{\textrm{\tiny Sig}}$ was mainly determined from a parametrization of the light curve at low energies parametrization while $P_{\textrm{\tiny Bkg}}$ was built assuming a constant background. The factor $w_s$ accounts for the relative weight of signal events with respect to background events. Constraints on $\tau_n$ led to lower limits on the Quantum Gravity energy scale $E_{\rm QG}$. The 95% 1-sided lower limits for the subluminal case are: $\textrm{E}_{\rm QG,1}>4.32\times 10^{17}$ GeV and $\textrm{E}_{\rm QG,2}>2.11\times 10^{10}$ GeV for linear and quadratic LIV effects, respectively. Figure 3 compares these results with other limits from less distant AGN flares. While the statistics is more limited here, the distance of the source makes the sensitivity to possible LIV effects comparable to previous results. {width="80mm"} {width="80mm"} \[liv\] Conclusions =========== The VHE emitter [PG 1553+113]{} underwent a flaring event in VHE with an increase of its flux by a factor of 3.5. No counterpart of this flare was found in the HE regime by [[*[*Fermi*]{}*]{}]{}. This data set has been used to constrain the redshift of the source to be [$z=0.49\pm0.04$]{} using a novel method based on $\gamma$-ray data. The flare is also used to put lower limits on the LIV effect with $\textrm{E}_{\rm QG,1}>4.32\times 10^{17}$ GeV and $\textrm{E}_{\rm QG,2}>2.11\times 10^{10}$ GeV for linear and quadratic effects. Acknowledgments {#Ack .unnumbered} =============== The support of the Namibian authorities and of the University of Namibia in facilitating the construction and operation of H.E.S.S. is gratefully acknowledged, as is the support by the German Ministry for Education and Research (BMBF), the Max Planck Society, the French Ministry for Research, the CNRS-IN2P3 and the Astroparticle Interdisciplinary Programme of the CNRS, the U.K. Science and Technology Facilities Council (STFC), the IPNP of the Charles University, the Polish Ministry of Science and Higher Education, the South African Department of Science and Technology and National Research Foundation, and by the University of Namibia. We appreciate the excellent work of the technical support staff in Berlin, Durham, Hamburg, Heidelberg, Palaiseau, Paris, Saclay, and in Namibia in the construction and operation of the equipment. The *Fermi*-LAT Collaboration acknowledges support for LAT development, operation and data analysis from NASA and DOE (United States), CEA/Irfu and IN2P3/CNRS (France), ASI and INFN (Italy), MEXT, KEK, and JAXA (Japan), and the K.A. Wallenberg Foundation, the Swedish Research Council and the National Space Board (Sweden). Science analysis support in the operations phase from INAF (Italy) and CNES (France) is also gratefully acknowledged. The work of DS has been supported by the Investissements d’avenir, Labex ENIGMASS. [9]{} Abdo, A. A., et al. 2009 ApJ, 707, 1310 Aharonian F. et al. 2008, A&A, 477, 481 Cortina, J. 2012, The Astronomer’s Telegram, 3977 Danforth, C. W. et al. 2010, The Astrophysical Journal, 720, 976 de Naurois, M. & Rolland, L. 2009, Astroparticle Physics, 32, 231 , A., [Rodighiero]{}, G., & [Vaccari]{}, M. 2008, A&A, 487, 837 Martinez, M., & Errando, M. 2009, Astroparticle Physics, 31, 226 , D. A., [Fegan]{}, S., & [Giebels]{}, B. 2013, A&A, 554, A75 , J. D. 1998, ApJ, 504, 405 The Fermi-LAT Collaboration 2015, submited to ApJ, arXiv:1501.02003

--- abstract: 'We test local and semi-local density functionals for the electronic exchange for a variety of systems including atoms, molecules, and atomic chains. In particular, we focus on a recent universal extension of the Becke-Johnson exchange potential \[Räsänen, E.; Pittalis, S.; Proetto, C. R. *J. Chem. Phys.* **2010**, *132*, 044112\]. It is shown that when this potential is used together with the Becke-Roussel approximation to the Slater potential \[Becke, A. D.; Roussel, M. R. *Phys. Rev. A* **1989**, *39*, 3761–3767\], a good overall agreement is obtained with experimental and numerically exact results for several systems, and with a moderate computational cost. Thus, this approximation is a very promising candidate in the quest for a simple and all-round semi-local potential.' author: - 'Micael J.T. Oliveira' - 'Esa R[ä]{}s[ä]{}nen' - Stefano Pittalis - 'Miguel A.L. Marques' bibliography: - 'rpp.bib' title: 'Toward an all-round semi-local potential for the electronic exchange' --- Introduction ============ Density-functional theory [@dg90; @barth04:9] (DFT) has become the standard tool both in quantum chemistry and in atomic, molecular, and solid-state physics. The practical applicability of DFT crucially depends on the approximation for the exchange-correlation (xc) energy functional. The “Jacob’s ladder” of functionals developed in the past few decades [@pk03] has posed the following well-known problem: by climbing successive rungs of the ladder one increases the accuracy of the functional, but one also increases substantially the computational burden of the method. Finding a balance between accuracy and efficiency, together with [*universality*]{} (which is the ideal ability to deal equally well with any kind of system), has remained a major challenge in DFT. As the simplest density functionals, occupying the first two rungs of Jacob’s ladder, the local density approximation (LDA) and generalized-gradient approximations (GGA) are numerically efficient and surprisingly accurate for many (strongly inhomogeneous) systems. However, both these families of functionals exhibit well-known failures in the calculation of, e.g., band gaps of semiconductors and insulators [@hpsm05:174101], the response to electric fields [@cpgbssrk98:10489], etc. The problems are particularly dramatic in systems where long-range interactions play a crucial role, i.e., elongated molecules and atomic chains [@kkp04:213002; @gggb02:9591; @mwy03:11001; @nf07:191106; @kak09:712; @rpcsv08:060502; @mwdrs09:300; @ggb02:6435]. The main origin for these errors is the wrong (exponential) asymptotic behavior and the lack of the derivative discontinuity in the xc potential. Climbing further the ladder, the optimized-effective-potential (OEP) method [@sh53:317; @ts76:36; @kk08:3] or its simplification within the Krieger-Li-Iafrate (KLI) approximation [@kli92:101] provide, in principle, an access to the [*exact*]{} exchange energy and potential within DFT. Thus, as long as the electronic correlation is not significant, the OEP and KLI are free from the failures mentioned above. However, as non-local orbital functionals they are computationally demanding and therefore usable only for systems containing a small number of particles. To bridge the gap between the GGA and OEP, meta-GGAs [@pkzb99:2544; @tpss03:146401] are appealing candidates. They supplement the GGA by further semi-local information through the kinetic-energy density and/or the Laplacian of the density, and, in some cases, also through the paramagnetic current density. Recently, Räsänen, Pittalis, and Proetto [@rpp10:044112] (RPP) developed a meta-GGA for the exchange part of the xc potential. The RPP potential introduces a number of important constraints and features (see below), and performs well for, e.g., non-Coulombic systems and atomic chains. It is based on the Becke-Johnson (BJ) potential [@bj06:221101] – a simple meta-GGA close to the OEP accuracy for atoms – but, in contrast to BJ, the RPP potential is fully gauge-invariant, exact for any one-particle system, and has the correct asymptotic behavior for any $N$-particle system. Also other modifications to the BJ potential have been suggested to improve the performance for atomic chains [@akk08:165106] and band gaps [@tb09:226401]. In fact, the latter modification [@tb09:226401] allows the calculation of band gaps of semiconductors and insulators with an error of the same order of GW calculations, but at a very small fraction of the GW computational time. In this paper we test the RPP potential [@rpp10:044112], used together with the Becke-Roussel (BR) approximation to the Slater potential [@br89:3761], for a large variety of systems. We compare this approximation to the BJ one, also complemented by the BR potential. In order to allow for a comparison to experimental reference data, we have added to the above exchange potentials the correlation within the LDA. We compare the results also against the LB94 potential of van Leeuwen and Baerends [@lb94:2421] (a GGA with correct asymptotic behavior including also correlation). Moreover, for completeness, we include results calculated with standard LDA and GGA functionals. As a reference we use experimental or high-quality ab initio data. In some cases the performance of the exchange potentials alone, i.e., without the addition of correlation, is compared to the exact-exchange OEP results. The combination of RPP and BR potentials is found to yield the best overall performance of the tested approximations, and thus it provides a promising step toward an all-round semi-local exchange potential in DFT. Theory ====== Exact exchange {#subsec:exx} -------------- In a majority of atomic, molecular, and solid-state systems, the electronic exchange gives, in absolute terms, a much larger contribution to (most) observables than the correlation. Therefore, in practical applications the exchange is the most important term to be approximated in the functional. The exact exchange energy in Hartree atomic units (a.u.) is written as $$E_\text{x}[\rho_{\sigma}] = -\frac{1}{2} \sum_{\sigma=\uparrow,\downarrow} \sum_{j,k=1}^{N_\sigma} {\int \! \! {\rm d}^3 r}\!\!{\int \! \! {\rm d}^3 r'} \frac{\varphi_{j \sigma}^*({{\bf r}})\varphi_{k \sigma}^*({{\bf r}}') \varphi_{j \sigma}({{\bf r}}')\varphi_{k \sigma}({{\bf r}})}{|{{\bf r}}- {{\bf r}}'|} \; , \label{Ex}$$ and its functional derivative gives the Kohn-Sham (KS) exchange potential as $v_{\text{x}\sigma}({{\bf r}})=\delta E_\text{x}/\delta\rho_\sigma({{\bf r}})$. These quantities can be rigorously calculated with the OEP method [@sh53:317; @ts76:36; @kk08:3] through an integral equation that has to be solved together with the KS equations. At this point, it is useful to write the (KS) exchange potential as a sum $$\begin{aligned} v_{\text{x} \sigma}({{\bf r}}) = & v^\text{SL}_{\text{x} \sigma}({{\bf r}}) + \Delta v^\text{OEP}_{\text{x} \sigma} ({{\bf r}}) \;\nonumber \\ = & v^\text{SL}_{\text{x} \sigma}({{\bf r}}) + \Delta v^\text{KLI}_{\text{x} \sigma}({{\bf r}}) + \Delta v^\text{OS}_{\text{x} \sigma}({{\bf r}})\;, \label{vx}\end{aligned}$$ where $$v^{\rm SL}_{\text{x}\sigma}({{\bf r}}) = -\sum_{j,k=1}^{N_\sigma} {\int \! \! {\rm d}^3 r'} \frac{\varphi_{j \sigma}^*({{\bf r}})\varphi_{k \sigma}^*({{\bf r}}') \varphi_{j \sigma}({{\bf r}}')\varphi_{k \sigma}({{\bf r}})}{\rho_\sigma({{\bf r}}) |{{\bf r}}- {{\bf r}}'|} \; , \label{slater}$$ is the Slater potential, i.e., the average of the Fock potential felt by the electrons, and $\Delta v^\text{OEP}_{\text{x} \sigma} ({{\bf r}})$ is the exact (OEP) contribution [@sh53:317; @ts76:36; @kk08:3], which can be decomposed into the Krieger-Li-Iafrate [@kli92:101] (KLI) part and the orbital shifts. Apart from, e.g., atomic chains [@kkp04:213002], the orbital shifts in a ground-state calculation are usually of minor importance and therefore neglected, leading to so-called KLI approximation. This relieves the computational burden of solving the integral equation, but the tedious integrals in the Slater potential are still to be calculated. Therefore, even within the KLI approximation the efficiency of an OEP calculation is far from that of semi-local functionals. Becke-Johnson potential ----------------------- The BJ potential [@bj06:221101] is a simple approximation to the OEP contribution in Eq. (\[vx\]), $$\Delta v^{\text{OEP}}_{\text{x} \sigma} ({{\bf r}})\approx \Delta v^{\text{BJ}}_{\text{x} \sigma} ({{\bf r}}) = C_{\Delta v} \sqrt{ \frac{ \tau_{\sigma}({{\bf r}}) }{ \rho_\sigma({{\bf r}}) }},$$ where $$\label{stau} \tau_\sigma({{\bf r}})=\sum_{j=1}^{N_\sigma} |\nabla\varphi_{j\sigma}({{\bf r}})|^2$$ is (twice) the spin-dependent kinetic-energy density, and $C_{\Delta v} = \sqrt{5/(12\pi^2)}$. The BJ potential is exact for the hydrogen atom and for the homogeneous electron gas, and, regarding quantum chemistry applications, it has several beneficial properties. First, it yields the atomic step structure in the exchange potential (which was the main motivation for the approximation) very accurately [@bj06:221101]. Secondly, it has the derivative discontinuity for fractional particle numbers [@akk08:165106]. To improve on the numerical efficiency of this potential, one often replaces also the Slater potential $v^{\rm SL}_{\text{x}\sigma}({{\bf r}})$ by the Becke-Roussel potential [@br89:3761]. This is again a meta-GGA potential, written in terms of $\nabla^2 \rho_\sigma$ and of $\tau_\sigma$, that reproduces to a very high precision the Slater potential for atoms. Universal extension to Becke-Johnson ------------------------------------ The main limitations of the BJ potential are the facts that it is not gauge-invariant and that it is not exact for [*all*]{} one-particle systems. Both limitations were recently removed in the extension by RPP [@rpp10:044112], that proposed the form $$\Delta v^\text{OEP}_{\text{x} \sigma} ({{\bf r}})\approx \Delta v^\text{RPP}_{\text{x} \sigma} ({{\bf r}}) = C_{\Delta v} \sqrt{\frac{ D_{\sigma}({{\bf r}}) }{ \rho_\sigma({{\bf r}}) }},$$ where $$\label{D} D_{\sigma}({{\bf r}})= \tau_{\sigma}({{\bf r}})-\frac{1}{4}\frac{\left[ \nabla \rho_\sigma({{\bf r}}) \right]^2}{\rho_\sigma({{\bf r}})}-\frac{{{\bf j}}^2_{p \sigma}({{\bf r}})}{\rho_\sigma ({{\bf r}})},$$ describes the local curvature of the exchange (Fermi) hole [@dobson93:8870]. This quantity has already been useful in the derivation of several functionals [@br89:3761; @becke88:1053; @becke96:995; @prhg07:235314; @prpg09:085316; @rpp10:195103] and is the key ingredient of the electron-localization function [@bh90:5397; @bmg05:010501; @rcg08:115108], a standard tool used to analyze bonding in electronic systems. Finally, the spin-dependent paramagnetic current density is defined as $$\label{current} {{\bf j}}_{p \sigma}({{\bf r}})=\frac{1}{2i}\sum_{j=1}^{N_\sigma} \left\{ \varphi^*_{j \sigma}({{\bf r}}) \left[\nabla \varphi_{j \sigma}({{\bf r}})\right] - \left[\nabla \varphi^*_{j \sigma}({{\bf r}})\right] \varphi_{j \sigma}({{\bf r}}) \right\}.$$ The RPP approximation is gauge-invariant and it is exact for [*all*]{} one-particle systems. Furthermore, it has a correct asymptotic limit for finite $N$-electron systems (except on nodal surfaces of the energetically highest-occupied orbitals [@sg02:033003; @kp03:035103]). The universality of the approach, whose principles has also been shown to work in two dimensions [@prp10:115108], is reflected into a resulting potential that can be applied reasonably well to any kind of system. For example, the RPP potential has been seen to reproduce well the KLI potential in hydrogen chains in electric fields and in Hooke’s atoms subject to magnetic fields [@rpp10:044112]. In this respect, another recent extension of the BJ potential can be viewed as more restrictive [@kak09:712]. The present study aims at further evaluating the capability of this approximation for atoms, small molecules, and atomic chains. Numerical procedure =================== The evaluation of the Slater part in the BJ [@bj06:221101] and RPP [@rpp10:044112] potentials is computationally more demanding than the evaluation of the correction terms $\Delta v^{\text{BJ}}_{\text{x} \sigma}$ and $\Delta v^{\text{RPP}}_{\text{x} \sigma}$. Nevertheless, as already pointed out by Becke and Johnson [@bj06:221101], it is possible to approximate the Slater part by using the semi-local Becke-Roussel (BR) exchange-energy functional [@br89:3761]. In this way, the cost of evaluating the full BJ and RPP potentials becomes similar to the one of a usual LDA or GGA. To avoid any ambiguity, we will hereafter denote the BJ and RPP potentials, where the Slater part was replaced by the BR potential, as BJBR and RPPBR, respectively When using experimental results as a reference, it is necessary to add a correlation contribution to the BJBR and RPPBR potentials for a proper comparison. We use the correlation in the LDA level within the Perdew-Wang [@pw92:13244] (PW) form. The results are compared also to the standard LDA – with the PW parametrization for the correlation part, the GGA of Perdew, Burke, and Ernzerhof [@pbe96:3865] (PBE), and the GGA of Leeuwen and Baerends [@lb94:2421] (LB94) – again using the PW parametrization for the LDA part of the potential. In all cases we have applied the potentials self-consistently in the KS-DFT framework. Although in the case of PBE the correlation functional used is not the same as in the other cases, we expect this fact to result in negligible differences in the quantities and systems studied in this work. In the case of atoms and hydrogen chains, calculations are also performed using [*exchange-only*]{} potentials. Results are then compared with exact-exchange OEP data available in the literature. Besides the BJBR and RPPBR potentials, we also performed these calculations using the exchange part of the LDA (xLDA) and of the PBE (xPBE). It is important to bear in mind that BJ, RPP, and LB94 are such approximations to the exchange (or xc) potential that are not functional derivatives of corresponding exchange (or xc) [ *energies*]{} [@gs09:044107]. Here we focus on fairly standard quantities that may be accessed without the computation of total energies. These quantities includes ionization potentials and electronic affinities of atoms, ionization potentials and dipole polarizabilities of small molecules, and longitudinal polarizabilities of hydrogen chains. We believe that these benchmarks provide us with a fairly complete view on the properties of different approximations considered in this work. All the single-atom calculations are performed with the [APE]{} code [@on08:524], while molecules and atomic chains are dealt with [octopus]{} [@mcbr03:60; @caoralmgr06:2465]. In the latter case, the electron-ion interaction is handled through norm-conserving pseudopotentials generated with [APE]{} for each functional and approximation studied in this work. Results ======= Atoms ----- First we consider single atoms and focus on the ionization energies and electron affinities (see \[table\_atoms1\]). There are several ways to estimate these quantities within DFT. The most direct one is to calculate the differences in total energy of both the neutral atom, and of its anion and cation, respectively. In this way, traditional LDA and GGA functionals usually yield quite good ionization potentials. Electron affinities are more complicated as often LDAs and GGAs fail to bind the extra electron. The other approach, the one used in this work, is to look at the KS eigenenergy of the highest occupied atomic orbital (HOMO), that should be equal to the negative of the ionization potential. The electron affinity is computed simply from the ionization potential of the respective anion. This method samples much better the quality of the potential, and it is particularly sensitive to the asymptotic description of the potential. As known from previous studies [@lb94:2421], the LDA and PBE perform poorly for the ionization potential: the mean absolute error (last row of \[table\_atoms1\]) is larger than 40% for this set of atoms. The result indicates the crucial role of the correct asymptotic behavior in the exchange potential. The decay of the xc potential is properly described by LB94 potential showing good performance. For the same reason, good results have been obtained also with KLI-CS – a combination of KLI [@kli92:101] for the exchange and the Colle-Salvetti [@cs75:329] functional for the correlation – as reported by Grabo and Gross [@gg95:141]. It seems that RPPBR is slightly more accurate than the original BJBR potential, also when considering the exchange potentials alone. When compared against exact-exchange OEP results [@ev93:2800], xLDA and xPBE perform poorly, while BJBR and RPPBR perform better, the later being now more accurate. Atom xLDA xPBE BJBR RPPBR OEP$^b$ LDA PBE LB94 KLI-CS$^c$ BJBR-PW RPPBR-PW Expt.$^d$ ---------------- ------- ------- ------- ------- --------- ------- ------- ------- ------------ --------- ---------- ----------- He 0.517 0.553 0.857 0.924 0.918 0.570 0.585 0.851 0.945 0.922 0.982 0.903 Li 0.100 0.109 0.254 0.183 0.196 0.116 0.111 0.193 0.200 0.276 0.201 0.198 Be 0.170 0.182 0.355 0.300 0.309 0.206 0.201 0.321 0.329 0.401 0.338 0.343 B 0.120 0.128 0.279 0.226 0.151 0.143 0.296 0.328 0.321 0.260 0.305 C 0.196 0.204 0.399 0.332 0.227 0.218 0.401 0.448 0.440 0.366 0.414 N 0.276 0.285 0.526 0.451 0.571 0.309 0.297 0.510 0.579 0.567 0.486 0.534 O 0.210 0.224 0.391 0.383 0.272 0.266 0.516 0.559 0.472 0.450 0.500 F 0.326 0.339 0.564 0.526 0.384 0.376 0.647 0.714 0.636 0.588 0.640 Ne 0.443 0.456 0.743 0.686 0.851 0.498 0.491 0.788 0.884 0.810 0.745 0.792 Na 0.097 0.103 0.247 0.178 0.182 0.113 0.106 0.205 0.189 0.270 0.197 0.189 Mg 0.142 0.149 0.313 0.252 0.253 0.175 0.168 0.291 0.273 0.357 0.287 0.281 Al 0.086 0.092 0.227 0.160 0.111 0.102 0.216 0.222 0.263 0.188 0.220 Si 0.144 0.150 0.320 0.237 0.170 0.160 0.290 0.306 0.356 0.267 0.300 P 0.203 0.210 0.416 0.324 0.392 0.231 0.219 0.369 0.399 0.453 0.355 0.385 S 0.174 0.182 0.349 0.305 0.229 0.219 0.410 0.404 0.420 0.362 0.381 Cl 0.254 0.262 0.469 0.400 0.305 0.295 0.491 0.506 0.533 0.453 0.477 Ar 0.334 0.343 0.592 0.506 0.591 0.382 0.373 0.577 0.619 0.652 0.557 0.579 $\Delta\,(\%)$ 43 41 13.8 8.5 41 42 3.7 5.7 14.4 7.4 : Ionization potentials from the highest occupied Kohn-Sham orbital (in a.u.)$^a$ \[table\_atoms1\] $^a$ The last row shows the mean absolute error in percentage with respect to exact-exchange and experimental results for exchange potentials and combined exchange and correlation potential, respectively. $^b$ From the work of Engel and Volko. [@ev93:2800] $^c$ From the work of Grabo and Gross. [@gg95:141] $^d$ Experimental results taken from Ratzig and Smirnov [@rs85]. As noted already by Becke and Johnson [@bj06:221101], the BJ exchange potential goes asymptotically to a finite (non-zero) constant. In principle, this constant only redefines the zero of orbital energy, and should have no implication in the quality of the results, but it has to be taken into account when computing the ionization potential. This can be done by subtracting the value of the constant, which can be obtained from the asymptotic expansion of the density and the kinetic energy density, to the value of the KS eigenenergy of the HOMO. A perfectly equivalent procedure is to shift the BJ exchange potential so that it goes asymptotically to zero. In the case spin-uncompensated atoms the constant depends on spin. Then it is possible to shift the spin-up and spin-down potentials by different amounts, provided that this does not imply a change in the occupancies of the orbitals. Atom LB94 KLI-CS$^b$ BJBR-PW RPPBR-PW Expt.$^c$ ---------------- ------- ------------ --------- ---------- ----------- Li 0.020 0.024 - 0.036 0.023 B 0.016 0.033 - - 0.010 C 0.049 0.083 - 0.032 0.046 O 0.077 0.110 - - 0.054 F 0.128 0.208 - 0.110 0.125 Na 0.023 0.022 0.012 0.036 0.020 Al 0.018 0.024 - - 0.016 Si 0.050 0.065 0.019 0.039 0.051 P 0.061 0.048 - 0.026 0.027 S 0.098 0.106 - 0.069 0.076 Cl 0.140 0.174 0.118 0.127 0.133 $\Delta\,(\%)$ 29 66 38$^d$ 28$^d$ : \[table\_atoms2\] Electron affinities calculated from the highest occupied Kohn-Sham orbital of the anion (in a.u.)$^a$ $^a$ The last row shows the mean absolute error in percentage. $^b$ From the work of Grabo and Gross. [@gg95:141] $^c$ Experimental results taken from Ratzig and Smirnov [@rs85]. $^d$ Mean error calculated for bound solutions only. The electron affinities for our set of atoms are given in \[table\_atoms2\]. As it is well known, the LDA or most GGAs do not give bound solutions for most negative ions, so we chose not to include them in the table. In most cases BJBR failed to give bound solutions for the anions, while for RPPBR this happened only in a few cases. Considering only the cases were RPPBR gave bound solutions, the deviation from the exact values was around 28%. It seems that LB94, having a similar overall accuracy, works better for small ions, whereas RPPBR increases its accuracy for larger systems. For example, for the last three atoms in \[table\_atoms2\] (P, S, Cl) RPPBR has an error of only a few percent. Interestingly, KLI-CS results deviate by more than 60% from the exact values. This might be due to the poor compatibility between the exact non-local exchange and the correlation part, when the asymptotic regime is strongly dominated by the ionic HOMO. Molecules --------- Molecule LDA PBE LB94 BJBR-PW RPPBR-PW Expt.$^b$ ---------------- ------- ------- ------- --------- ---------- ----------- CS$_2$ 6.93 6.81 11.54 13.08 10.76 10.07 H$_2$S 6.4 6.3 11.33 12.51 11.05 10.46 C$_2$H$_4$ 6.92 6.74 11.85 12.71 10.96 10.51 PH$_3$ 6.69 6.64 11.65 12.88 11.62 10.59 NH$_3$ 6.28 6.19 11.55 12.58 11.3 10.8 Cl$_2$ 7.47 7.36 12.3 14.03 11.86 11.48 C$_2$H$_6$ 8.13 8.15 12.94 15.04 13.33 12 SiH$_4$ 8.53 8.53 13.44 15.44 14.04 12.3 SO$_2$ 8.3 8.09 14.06 15.2 13.29 12.35 H$_2$O 7.38 7.23 13.2 14.08 12.66 12.62 HCl 8.14 8.04 13.29 14.81 12.83 12.74 N$_2$O 8.6 8.35 14.48 15.4 13.37 12.89 CH$_4$ 9.46 9.45 14.29 16.69 14.65 13.6 CO$_2$ 9.31 9.05 15.32 16.37 14.2 13.78 CO 9.16 9.09 14.49 16.46 14.47 14.01 H$_2$ 10.28 10.4 15.27 17.92 17.54 15.43 N$_2$ 10.39 10.24 16.94 18.18 16.09 15.58 F$_2$ 9.79 9.54 17.03 17.56 16.18 15.7 HF 9.85 9.65 16.44 17.3 15.69 16.03 $\Delta\,(\%)$ 35 36 8.0 19 5.7 : \[table\_molecules1\] Ionization potentials for molecules calculated from the highest occupied Kohn-Sham orbital (in eV)$^a$ $^a$ The last row shows the mean absolute error in percentage. $^b$ Experimental results taken from Gr[ü]{}ning et al. [@gggb02:9591]. Next we test the approximations for a large set of small molecules by computing ionization potentials and static (isotropic) dipole polarizabilities. The ionization potentials are obtained from the HOMO as in the previous section, while the polarizabilites are computed as a derivative of the dipole moment of the system with respect to the applied electric field. The ionization potentials are listed in \[table\_molecules1\]. Interestingly, RPPBR is significantly more accurate than BJBR and deviates less than 6% from the experimental values. LB94 performs also well with a mean absolute error of 8%. In contrast, the LDA and PBE fail in a similar fashion as in the atomic cases considered in the previous section. Molecule LDA PBE LB94 BJBR-PW RPPBR-PW Expt.$^b$ ---------------- ------- ------- ------- --------- ---------- ----------- CS$_2$ 56.50 56.45 51.72 55.44 55.29 55.28 H$_2$S 26.21 25.91 21.95 24.24 22.51 24.71 C$_2$H$_4$ 28.71 28.52 24.93 27.71 25.21 27.7 PH$_3$ 32.29 31.72 27.39 29.99 27.47 30.93 NH$_3$ 15.58 15.45 12.41 13.83 12.32 14.56 Cl$_2$ 32.33 32.21 30.92 31.41 32.39 30.35 C$_2$H$_6$ 30.17 29.73 27.41 28.23 26.52 29.61 SiH$_4$ 34.03 33.07 30.17 31.11 28.47 31.9 SO$_2$ 27.44 27.53 22.97 25.78 23.68 25.61 H$_2$O 10.74 10.73 8.28 9.49 8.53 9.64 HCl 18.61 18.47 15.85 17.18 16.21 17.39 N$_2$O 20.7 20.74 17.42 19.46 18.47 19.7 CH$_4$ 17.77 17.45 15.87 16.46 15.41 17.27 CO$_2$ 18.21 18.24 15.66 17.39 16.16 17.51 CO 13.91 13.87 11.6 13.13 12.29 13.08 H$_2$ 5.87 5.64 5.02 5.27 4.56 5.43 N$_2$ 12.64 12.63 10.79 11.9 11.4 11.74 F$_2$ 8.86 8.97 7.23 8.31 7.73 8.38 HF 6.23 6.27 4.8 5.52 4.89 5.6 $\Delta\,(\%)$ 6.1 5.3 9.8 2.0 8.9 : \[table\_molecules2\] Static (isotropic) dipole polarizabilities for molecules (in a.u.)$^a$ $^a$ The last row shows the mean absolute error in percentage. $^b$ Experimental results taken from Gr[ü]{}ning et al. [@gggb02:9591]. For static (isotropic) dipole polarizabilities (see \[table\_molecules2\]) the situation is different in the sense that the LDA and PBE perform rather well, which is surprising in view of the fact that the polarization is largely a non-local and collective effect. It is noteworthy, however, that the present test set does not include problematic elongated molecules or chains (see next section), for which going beyond LDA (and GGA) is essential [@kkp04:213002; @gggb02:9591; @mwy03:11001; @nf07:191106; @kak09:712; @rpcsv08:060502; @mwdrs09:300; @ggb02:6435]. For the present cases BJBR works remarkably well with a mean error of only 2%, whereas RPPBR and LB94 deviate almost 10% from the experiments. Nevertheless, no dramatic failures are obtained by using any of the tested approximations. Hydrogen chains --------------- Chain xLDA xPBE BJBR RPPBR OEP$^b$ LDA PBE LB94 BJBR-PW RPPBR-PW CCSD(T)$^c$ MP4$^c$ ---------------- ------- ------- ------- ------- --------- ------- ------- ------- --------- ---------- ------------- --------- H$_2$ 13.1 12.5 12.4 11.2 12.4 12.0 11.2 11.8 10.8 H$_4$ 39.6 37.2 36.3 33.3 32.2 37.7 36.1 35.5 34.9 32.4 29 29.5 H$_6$ 76,4 70.7 68.6 63.6 65.6 72.9 69.4 70.5 65.8 61.6 50.9 51.9 H$_8$ 120.6 110.2 106.0 99.0 84.2 115.2 108.8 112.9 101.6 95.8 74.4 76.2 H$_{10}$ 169.9 153.2 146.1 137.1 162.2 152.1 160.5 140.8 132.7 H$_{12}$ 222.4 199.2 188.4 177.2 138.1 212.2 197.8 211.6 182.1 171.1 124 127.3 H$_{14}$ 277.0 246.1 231.9 218.0 264.0 245.2 264.3 224.1 210.6 155 H$_{16}$ 333.0 294.1 277.5 259.5 317.2 293.4 318.6 267.3 250.5 H$_{18}$ 389.8 342.5 323.0 301.5 371.1 342.2 373.2 309.8 290.8 205.39 H$_{20}$ 447.3 391.4 367.2 343.6 425.4 391.4 425.0 353.9 331.3 $\Delta\,(\%)$ 40.6 28.9 24.0 15.4 56.2 46.5 53.8 36.2 27.7 : \[table\_chains\] Longitudinal polarizabilities of hydrogen chains (in a.u.)$^a$ $^a$ The last row shows the mean absolute error in percentage, calculated against OEP and MP4 (when available) for exchange only potentials and combined exchange and correlation potentials, respectively. $^b$ Results from the work of Kümmel et al. [@kkp04:213002] $^c$ The MP4 and CCSD(T) results have been taken from the work of Ruzsinszky et al. [@rpcsv08:060502] apart from the MP4 result for H$_{18}$ taken from Champagne et al. [@cmva95:178]. In \[table\_chains\] we show the polarizabilities calculated for hydrogen chains from H$_2$ up to H$_{20}$. As the reference results we use available data from the CCSD(T) (coupled-cluster with single and double and perturbative triple excitations) and MP4 (fourth-order M[ø]{}ller-Plesset perturbation theory) [@rpcsv08:060502]. This well-studied system has proved to be a remarkable challenge for DFT [@kkp04:213002; @rpcsv08:060502; @mwdrs09:300; @ggb02:6435; @akk08:165106]. For example, LDA severely overestimates the polarizability, as demonstrated also by our results in \[table\_chains\]. The error of PBE is slightly smaller. The failure of LDA and PBE to capture the electric response is believed to be due to the inherent self-interaction error [@rpcsv08:060502; @psb08:121204; @kmk08:133004]. We find that the mean error of LB94 is almost the same as the one of LDA, whereas for BJBR it is smaller. RPPBR has the best performance of all the tested potentials when compared to MP4, although it is still quite large (27.7%). Possible sources of error in RPPBR (and BJBR) results are the ultra-non-local effects in long chains, which might be beyond reach of any semi-local functionals without [*ad hoc*]{} modifications, and the using of LDA for the correlation part. This last point seems to be confirmed by the results obtained without adding a correlation part to the exchange potentials: when comparing the polarizabilities obtained from the exchange-only potentials against exact-exchange OEP results [@kkp04:213002], all the average errors are reduced, while the overall trend remains the same. Summary and outlook =================== In summary, we have tested recently constructed meta-generalized-gradient (meta-GGA) functionals for the exchange potential, in particular the potential of Räsänen, Pittalis, and Proetto (RPP) and that of Becke and Johnson (BJ), when complemented by the Becke-Roussel (BR) approximation to the Slater potential (denoted in total as RPPBR and BJBR, respectively), and by the correlation in the LDA level. These approximations were compared to the van Leeuwen and Baerends potential (LB94), a GGA that shares some properties with these new meta-GGAs, as well as to standard LDA and GGA functionals. As the reference data we used experimental results whenever available, numerically exact data, and, in the case of comparing the exchange-only results, the exact-exchange results obtained from the optimized-effective-potential method. Overall, the RPPBR potential fared best in the present testsuite consisting of ionization potentials and electronic affinities of atoms, ionization potentials and dipole polarizabilities of small molecules, and longitudinal polarizabilities of hydrogen chains. LB94 potential performed in an appealing fashion in several instances. The BJBR potential gave particularly good results for the calculation of static polarizabilities of small molecules. Desired future developments would include the development of correlation potentials compatible with the RPRBR potential. In conclusion, the RPPBR potential combines a proper theoretical foundation with very good results for a series of properties of atoms and molecules. Moreover, it is very light from the computational point of view, thus allowing an efficient calculation of large systems. Therefore, we believe that the RPPBR potential is an important step in the quest for a simple and all-round semi-local potential for applications of density-functional theory. Acknowledgements {#acknowledgements .unnumbered} ================ This work was supported by the Academy of Finland, and the EU’s Sixth Framework Programme through the ETSF e-I3. SP acknowledges support by DOE grant DE-FG02-05ER46203. MO thankfully acknowledges financial support from the Portuguese FCT (contract \#SFRH/BPD/44608/2008). MALM acknowledges partial funding from the French ANR (ANR-08-CEXC8-008-01), and from the program PIR Matériaux – MaProSu of CNRS. Part of the calculations were performed at the LCA of the University of Coimbra and at GENCI (project x2010096017).

--- address: | III. Physikalisches Institut, RWTH Aachen\ D-52056 Aachen, Germany author: - 'S. ROTH' title: STANDARD MODEL FITS --- Measurements at the Z Resonance =============================== The Standard Model (SM) is confirmed at the permille level using electroweak precision data. The analysis of electron-positron collisions at centre-of-mass energies around the Z resonance has delivered a wealth of precisely measured electroweak observables. One observable is the mass of the Z, which is derived mainly from the measurement of the total hadronic cross section as shown in Figure \[fig:mz-sef2\] (left). Today the Z mass is known with a precision of 23 ppm [@lep-z]: $$m_{\rm Z} = 91.1875 \pm 0.0021 \; {\rm GeV} \; .$$ Other observables are related to the coupling of the weak current to the fermions and can therefore be expressed in terms of the electroweak mixing angle. The different values of $\sin^2 \theta^{\rm lept}_{\rm eff}$ derived from the measurements of the lepton forward-backward asymmetry, $A^{\rm 0,l}_{\rm fb}$, the left-right asymmetry at SLD, $A_{\rm l}({\rm SLD})$, the $\tau$ polarization at LEP, $A_{\rm l}(P_\tau)$, and the forward-backward asymmetry of the b-quark and c-quark final state, $A^{\rm 0,b}_{\rm fb}$ and $A^{\rm 0,c}_{\rm fb}$ are compared in Figure \[fig:mz-sef2\] (right). A difference of 2.9 standard deviations is observed between the two most precise measurements, the left-right asymmetry and the b-quark forward-backward asymmetry. Combining all measurements results in an accuracy at the sub permille level [@winter04]: $$\sin^2 \theta^{\rm lept}_{\rm eff} = 0.23150 \pm 0.00016 \; .$$ Comparing measurement and theory prediction shows a preference for a light Higgs mass. ![Measurement of the total hadronic cross cross section on the Z resonance (left); measurements of the weak mixing angle from Z decays and comparison with the SM prediction (right).[]{data-label="fig:mz-sef2"}](xsh_all "fig:"){height="0.3\textheight"} ![Measurement of the total hadronic cross cross section on the Z resonance (left); measurements of the weak mixing angle from Z decays and comparison with the SM prediction (right).[]{data-label="fig:mz-sef2"}](w04_sef2_theta "fig:"){height="0.3\textheight"} Measurement of the W mass ========================= Within the SM the Z mass, $m_{\rm Z}$, the W mass, $m_{\rm W}$, and the weak mixing angle, $\sin^2 \theta_{\rm w}$, are related by $$\cos \theta_{\rm w} = \frac{m_{\rm W}}{m_{\rm Z}},$$ which is naturally predicted by the Higgs mechanism. To test this relation a precise measurement of the W mass is mandatory. Since 1996 the W mass is measured at LEP 2 studying the four-fermion production through ${\rm e}^+{\rm e^-} \to {\rm W}^+{\rm W}^- \to f \bar{f} f \bar{f}$. The W bosons are reconstructed from the measured momenta of the observed final state fermions. To extract the W mass the invariant mass spectrum is compared to the one obtained from Monte-Carlo events with complete detector simulation. In the four-jet final state momentum exchange between the decay products of two W bosons may occur due to colour reconnection or Bose-Einstein effects in the non-perturbative phase of the jet formation. Because of these additional uncertainties compared to the semileptonic channel the weight of the four jet channel in the LEP average is currently only about 10%. Further details on the measurement of the W mass at LEP are given elsewhere [@lep-wmass]. At hadron colliders leptonic W decays with electrons and muons are selected and the transverse mass is calculated. The transverse mass, i.e. the invariant mass of the transverse momentum of the charged lepton and the missing momentum vector in the plane transverse to the beam, is not affected by the unknown missing momentum along the beam axis. Recently the experiments CDF and D0 performed a precise measurement of the W mass using the Run I data set of the Tevatron collider [@tev-wmass]. The precision of the Tevatron W mass measurement is currently limited by data statistics. The uncertainty in the lepton energy scale gives the largest contribution to the systematic error. The results of the Tevatron and LEP experiments on the W mass are in good agreement as shown in Figure \[fig:mw-mt\] (left). All direct W mass measurements, of which most of them are still preliminary, result in a world average of: $$80.425 \pm 0.034 \; {\rm GeV} \; .$$ Another less precise indirect measurement of the W mass is coming from the measurement of neutrino nucleon scattering. Measurements of the NuTeV collaboration [@nutev] show a deviation from the world average of about three standard deviations. But recent theoretical studies [@nutev-theo] suspect that the uncertainties due to QCD corrections and due to electroweak radiative correction may be underestimated in this analysis. ![Measurement of the W mass at LEP and Tevatron (left); measurement of the Top mass at the Tevatron and comparison with the prediction from LEP data(right).[]{data-label="fig:mw-mt"}](w04_plot_mw "fig:"){height="0.25\textheight"} ![Measurement of the W mass at LEP and Tevatron (left); measurement of the Top mass at the Tevatron and comparison with the prediction from LEP data(right).[]{data-label="fig:mw-mt"}](w04_plot_mt "fig:"){height="0.25\textheight"} Other Observables {#other} ================= Electroweak radiative corrections have been calculated up to the two-loop level, but there accuracy is limited by the experimental uncertainties for the masses of the top quark and the unknown mass of the Higgs boson. A test of the quantum structure of the SM therefore requires a precise knowledge of the top quark mass, as the radiative corrections depent quadratically on this parameter. In the year 1995 the experiments at the Tevatron collider discovered in the mass range predicted by the electroweak measurements at LEP. They observe the top quark in the reaction ${\rm p} \bar{\rm p} \to {\rm t} \bar{\rm t} X \to {\rm b} \bar{\rm b} {\rm W}^+ {\rm W}^- X$ If the W boson decays into two quarks, the mass of the top quark can be reconstructed from the invariant mass of the b jet and the two jets coming from the W decay. The results based on data collected in Run I and partly in Run II have recently been combined [@tevewwg]: $$m_{\rm t} = 178.0 \pm 2.7({\rm stat}) \pm 3.3({\rm syst}) \; {\rm GeV} \; .$$ The new and improved preliminary Run-I based result of the D0 collaboration in the lepton-plus-jets channel is included in this value. A comparison between the different measurements of the top mass is shown in Figure \[fig:mw-mt\] (right). The calculation of the electromagnetic coupling constant $\alpha(m_{\rm Z})$ at the energy scale of the Z mass is dominated by the knowledge on the hadronic vacuum polarisation $\Delta \alpha_{\rm had}$. This part can not be calculated by pertubation theory, but has to be extracted using the total hadronic cross section in ${\rm e}^+ {\rm e}^-$ collisions. A very important part is coming from the region of the $\rho$ resonance. Precise measurements of the CMD-2 detector get now confirmed by new results from KLOE. The most recent compilation [@alpha] gives $$\Delta \alpha^{(5)}_{\rm had}({m_{\rm Z}}) = 0.02761 \pm 0.00036 \; .$$ Interpretation within the Standard Model and Higgs mass analysis ================================================================ The details of the combination of the electroweak precision data and the global SM fit is described elsewhere [@lepewwg]. The fit results in a rather poor $\chi^2$, but excluding the low $Q^2$ experiments gives a much better fit quality with a 27% probability. As stated above, the electroweak radiative corrections include a term proportional to the logarithm of the Higgs mass. Assuming the validity of the SM one can try to extract this term from a global fit of all electroweak precision measurement. This allows to predict the mass of the Higgs boson. In Figure \[fig:sm-fit\] the constraint from the electroweak precision measurements performed at LEP 1 and SLD in the $m_{\rm W}$ vs. $m_{\rm t}$ plane is shown together with the direct measurement of the W and the top mass. Additionally the SM prediction using $G_{\rm F}$ from muon decay is plotted for different Higgs masses. Both the indirect and the direct measurements prefer a low Higgs mass. The fit of SM modell prediction to all electroweak data with the Higgs mass as the only free parameter results in a $\chi^2$ curve as shown in Figure \[fig:sm-fit\]. It gives a central value of 117 GeV for the Higgs mass, which is consistent with the direct searches for the Higgs excluding masses below 114.4 GeV [@higgs]. At 95% C.L. an upper bound on the Higgs mass of 251 GeV is set. ![Comparison of direct mass measurements with prediction from the SM and from electroweak precision data (left); Prediction of the Higgs mass from global SM fit (right).[]{data-label="fig:sm-fit"}](w04_mt_mw_contours "fig:"){height="0.35\textheight"} ![Comparison of direct mass measurements with prediction from the SM and from electroweak precision data (left); Prediction of the Higgs mass from global SM fit (right).[]{data-label="fig:sm-fit"}](w04_blueband "fig:"){height="0.35\textheight"} [99]{} LEP Collaborations, CERN-EP/2000-153 and hep-ex/0101027 LEP Electroweak Working Group, [http://lepewwg.web.cern.ch/LEPEWWG/]{} LEP Collaborations, LEPEWWG/MASS/2003-01 Tevatron Collaborations, FERMILAB-FN-0716 G.P. Zeller, G.A. Miller and A.W. Thomas, hep-ex/0204007;\ K.P.O. Diener, S. Dittmaier and W. Hollik, LEP Collaborations, CERN-EP/2003-091 and hep-ex/0312023 Tevatron Collaborations, TEVEWWG/top 2004/01 and hep-ex/0404010 B. Pietrzyk and H. Burkhardt, LAPP-EXP 2003-07 LEP Collaborations,

--- abstract: 'Controlling the state of a Bose-Einstein condensate driven by a chirped frequency perturbation in a one-dimensional anharmonic trapping potential is discussed. By identifying four characteristic time scales in this chirped-driven problem, three dimensionless parameters $P_{1,2,3}$ are defined describing the driving strength, the anharmonicity of the trapping potential, and the strength of the particles interaction, respectively. As the driving frequency passes the linear resonance in the problem, and depending on the location in the $P_{1,2,3}$ parameter space, the system may exhibit two very different evolutions, i.e. the quantum energy ladder climbing (LC) and the classical autoresonance (AR). These regimes are analysed both in theory and simulations with the emphasis on the effect of the interaction parameter $P_{3}$. In particular, the transition thresholds on the driving parameter $P_{1}$ and their width in $P_{1}$ in both the AR and LC regimes are discussed. Different driving protocols are also illustrated, showing efficient control of excitation and de-excitation of the condensate.' author: - 'S.V. Batalov' - 'A.G. Shagalov' - 'L. Friedland' title: 'Autoresonant excitation of Bose-Einstein condensates' --- \[sec:1\] Introduction ====================== Unique properties of Bose-Einstein condensates (BEC) attracted enormous interest in the last decades as a very flexible framework for experimental and theoretical research in many-body physics and a promising basis for new technologies. Modern applications require understanding of nonlinear dynamics of the condensates. Nonlinear dynamics is especially interesting in anharmonic trapping potentials, when motion of the center of mass is coupled to internal degrees of freedom [@Dobson_1994], and may even become chaotic [@Ott_2003]. In this paper we take advantage of the anharmonic potential to excite a quasi-one-dimensional condensate from the ground state to a high energy level. The basic idea is to use a driving perturbation with a slowly varying frequency to transfer the population from the ground quantum state to the first excited state, then to the second, and so on. This dynamical process, when only two energy eigenstates are resonantly coupled at a time, is the ladder climbing (LC) regime. The classical counterpart of the ladder climbing is the autoresonance (AR), the phenomenon discovered by Veksler and McMillan in 1944 [@Veksler_1944] and referred to as the phase stability principle at the time. Nowadays, the AR has multitude of applications in such diverse areas as hydrodynamics [Friedland\_1999,Friedland\_2002,Assaf\_2005]{}, plasmas [Fajans\_1999,Lindberg\_2006,Yaakobi\_2008]{}, magnetism [Kalyakin\_2007,Batalov\_2013,Klughertz\_2015]{}, nonlinear optics [Barak\_2009]{}, molecular physics [@Marcus_2004], planetary dynamics [Friedland\_2001]{} etc. The AR in BECs was previously studied in [Nicolin\_2007]{} for the case of oscillating scattering length. The excitation of a BEC from the ground state to the first energy eigenstate using optimal control was investigated experimentally in Ref. [@Bucker_2013]. In this paper we consider Gross-Pitaevskii model [@GP] $$\begin{aligned} i\hbar \Psi _{t}& +\frac{\hbar ^{2}}{2m}\Psi _{xx}-\left( U+g|\Psi |^{2}\right) \Psi =0, \label{eq:GP} \\ U(x,t)& =m\omega _{0}^{2}\left( \frac{x^{2}}{2}-\beta \frac{x^{4}}{4}\right) +\varepsilon x\cos \varphi (t), \label{eq:U}\end{aligned}$$which describes a BEC in a trap with the anharmonic potential $U(x,t)$ perturbed by a small amplitude oscillating drive. Here $\beta >0$ is the anharmonicity parameter assumed to be small. The frequency of the drive $% \omega (t)=\dot{\varphi}(t)=\omega _{0}-\alpha t$ slowly decreases in time ($% \alpha >0$) and passes through the linear resonance frequency $\omega _{0}$ in the problem at $t=0$. We assume that the wave function is normalized to unity and, thus, parameter $g$ is proportional to the total number of particles in the condensate. Both the classical AR and the quantum LC in the linear limit ($g=0$) of Eq. , i.e., the quantum Duffing oscillator, were studied in Refs. [Marcus\_2004,Barth\_2011,Andresen\_2011,Shalibo\_2012]{}. In this work, we focus on the nonlinear effects due to the interaction of the particles in the condensate. As a first step, we adopt the notations used in Ref. [@Barth_2011] to allow comparison with the linear case. To this end, we classify different dynamical regimes of Eq. in terms of parameters $P_{1}$, $P_{2}$ used in Ref. [@Barth_2011] and introduce a new parameter $P_{3}$, characterizing the nonlinearity in the problem. These parameters are constructed using four characteristic time scales in the problem: the inverse Rabi frequency $T_{R}=\sqrt{2m\hbar \omega _{0}}/\varepsilon $, the frequency sweep time scale $T_{S}=\alpha ^{-1/2}$, the anharmonic time scale $T_{A}=3\hbar \beta /(4m\alpha )$ of the trapping potential, and the nonlinear time scale $T_{N}=g/(\hbar \alpha \ell )$, where $\ell =\sqrt{{\hbar }/{m\omega _{0}}}$ is the characteristic width of the harmonic oscillator. Time $T_{A}$ is the time of passage of the frequency $\omega (t)$ through the anharmonic frequency shift between the first two levels of the energy ladder. Similarly, $T_{N}$ is the time of passage through the nonlinear frequency shift. Then, our three dimensionless parameters are defined as $$\begin{split} P_{1}=\frac{T_{S}}{T_{R}}=\frac{\varepsilon }{\sqrt{2m\hbar \omega _{0}\alpha }}& ,\quad P_{2}=\frac{T_{A}}{T_{S}}=\frac{3\hbar \beta }{4m\sqrt{% \alpha }}, \\ P_{3}=\frac{T_{N}}{T_{S}}& =\frac{g}{\hbar \ell \sqrt{\alpha }} \end{split} \label{eq:par}$$and characterize the strength of the drive, the anharmonicity of the trapping potential and the nonlinearity of the condensate, respectively. We limit our discussion to the case of the positive nonlinearity (repulsion), $% P_{3}>0$. The scope of the paper is as follows. In Sec. \[sec:quantum\] we study the dynamics of our system in the energy basis of the quantum harmonic oscillator and find a domain in $P_{1,2,3}$ parameter space where the successive quantum energy ladder climbing process takes place. In Sec. [sec:classic]{}, we discuss the opposite limit of semiclassical dynamics. The details of our numerical simulations are presented in Sec. \[sec:numerics\] and the conclusions are summarized in Sec. \[sec:conclusions\]. \[sec:quantum\]Quantum LC regime ================================ Here we focus on the LC regime, where the quantum nature of the system is mostly pronounced. In this case, it is convenient to express Eq. in the energy basis of the linear harmonic oscillator $\Psi (x,t)=\sum_{n}c_{n}(t)\psi _{n}(x)$, yielding $$\begin{split} i\hbar \frac{dc_{n}}{dt}& =E_{n}c_{n}+\frac{\varepsilon \ell }{\sqrt{2}}% \left( \sqrt{n+1}\,c_{n+1}+\sqrt{n}\,c_{n-1}\right) \cos \varphi \\ & +g\sum_{klm}c_{k}c_{l}c_{m}^{\ast }\int_{-\infty }^{\infty }\psi _{n}\psi _{k}\psi _{l}\psi _{m}dx, \end{split} \label{eq:dcdt}$$where the approximate energy levels up to linear terms in $\beta $ are given by [@Landau1977quantum] $$E_{n}\approx \hbar \omega _{0}\left[ n+\frac{1}{2}-\frac{3\beta \hbar }{% 8m\omega _{0}}\left( n^{2}+n+\frac{1}{2}\right) \right] .$$Introducing new variables $B_{n}=c_{n}\,e^{i\left( \frac{E_{0}t}{\hbar }% +n\varphi \right) }$, we rewrite Eq.  in the form $$\begin{aligned} &i&\hbar \frac{dB_{n}}{dt}=\left[ n\hbar \alpha t-\frac{3\beta \hbar }{% 8m\omega _{0}}n(n+1)\right] B_{n} \label{eq:C1} \\ &&+\frac{\varepsilon \ell }{2\sqrt{2}}\left( \sqrt{n+1}\,B_{n+1}e^{-i\varphi }+\sqrt{n}\,B_{n-1}e^{i\varphi }\right) \left( e^{-i\varphi }+e^{i\varphi }\right) \notag \label{eq:C1} \\ &&+g\sum_{klm}B_{k}B_{l}B_{m}^{\ast }e^{i(n+m-k-l)\varphi }\int_{-\infty }^{\infty }\psi _{n}\psi _{k}\psi _{l}\psi _{m}dx \notag\end{aligned}$$ We also define the dimensionless time $\tau =\sqrt{\alpha }\,t$, coordinate $% \xi =x/\ell $, and basis functions $\chi _{n}(\xi )=\sqrt{\ell }\,\psi _{n}(x)$, and use the rotating wave approximation (preserve only stationary terms in the driving and nonlinear components in Eq. ) to get a dimensionless system $$\begin{aligned} i\frac{dB_{n}}{d\tau }& =\Gamma _{n}B_{n}+\frac{P_{1}}{2}\left( \sqrt{n+1}% \,B_{n+1}+\sqrt{n}\,B_{n-1}\right) \notag \\ & +P_{3}\sum_{kl}B_{k}B_{l}B_{l+k-n}^{\ast }\int_{-\infty }^{\infty }\chi _{n}\chi _{_{k}}\chi _{_{l}}\chi _{l+k-n}d\xi , \label{eq:dBdt}\end{aligned}$$where the frequencies $\Gamma _{n}$ are $\Gamma _{n}=n\tau -\frac{P_{2}}{2}% \,n(n+1)$. The resonant population transfer between levels $n-1$ and $n$ \[the Landau-Zener (LZ) transition [@Landau_1932]\] takes place when their time-dependent characteristic frequencies are matched: $\Gamma _{n-1}(\tau _{n})\approx \Gamma _{n}(\tau _{n})$, i.e. in the vicinity of $\tau _{n}=nP_{2}$. In terms of Eq. this corresponds to the resonance condition $E_{n}-E_{n-1}\approx \hbar \,\omega (\tau _{n})$. Note that the anharmonicity parameter determines the time interval between successive LZ transitions $\Delta \tau =\tau _{n}-\tau _{n-1}=P_{2}$. These transitions can be treated independently provided their duration is much shorter than $\Delta \tau $. As suggested in [@Barth_2011] for the linear case $(P_{3}=0)$, the well-separated LZ transitions are expected when $$P_{2}\gg 1+P_{1}, \label{eq:Barth}$$where the right hand side is a typical duration of a LZ transition in both adiabatic (slow passage through resonance) and fast transition limits [Vitanov\_1996]{}. This inequality defines the domain of essentially quantum dynamics in the parameter space $P_{1,2}$, if $P_{3}=0$. Later in this Section, we discuss how relation is modified in the case of the nonlinear LZ transitions ($P_{3}\neq 0$). Neglecting all states in , but those with amplitudes $% u=B_{n-1}$ and $v=B_{n}$, we obtain the nonlinear LZ-type equations describing the isolated transition between the two states: $$\begin{split} i\frac{du}{d\tau }& =\Gamma _{n-1}\,u+P_{3}\left( |u|^{2}I_{n-1}+2|v|^{2}J_{n}\right) u+\frac{P_{1}\sqrt{n}}{2}v, \\ i\frac{dv}{d\tau }& =\Gamma _{n}\,v+P_{3}\left( |v|^{2}I_{n}+2|u|^{2}J_{n}\right) v+\frac{P_{1}\sqrt{n}}{2}u, \end{split} \label{eq:NLZ}$$ where $J_{n}=\int |\chi _{n-1}|^{2}|\chi _{n}|^{2}d\xi $, $I_{n}=\int |\chi _{n}|^{4}d\xi $, The nonlinear LZ model attracted significant attention recently [Liu\_2002,Zhang\_2008,Trimborn\_2010]{}, especially in the context of BECs in optical lattices. Nonlinearity may change the dynamics of the system significantly compared to the linear case. One deviation from the linear case is the breakdown of adiabaticity due to the bifurcation of nonlinear stationary states [@Wu_2000; @Zobay_2000]. In this paper we focus on another property of the nonlinear LZ model, namely, the AR. Equations can be simplified using the conservation of the total probability [$K_{n}=|u|^{2}+|v|^{2}$]{}. Introducing the fractional population imbalance $S=({|v|^{2}-|u|^{2}})/{K_{n}}$ and the phase mismatch $% \delta =\arg \left( {v/u}\right) $, we obtain a set of real equations $$\begin{split} \frac{dS}{d\tau }& =-\mu _{n}\sqrt{1-S^{2}}\sin \delta , \\ \frac{d\delta }{d\tau }& =\frac{P_{3}K_{n}f_{n}}{2}S-(\tau -\tau _{\ast })+% \frac{P_{1}\sqrt{n}\,S}{\sqrt{1-S^{2}}}\cos \delta , \end{split} \label{eq:res}$$where $f_{n}=4J_{n}-I_{n-1}-I_{n}$ and $\tau _{\ast }=nP_{2}-P_{3}K_{n}(I_{n}-I_{n-1})/2$ . The AR in a similar system was studied in [@Barak_2009]. Its main characteristic is a continuing phase locking as the phase mismatch $\delta $ remains bounded due to a persistent self-adjustment of the nonlinear frequency of the driven system to slowly varying driving frequency, i.e., $P_{3}K_{n}f_{n}S/2\approx (\tau -\tau _{\ast })$. However, the slow passage through resonance does not guarantee the AR. Indeed, if one assumes that only state $u$ is populated initially, that is $S\rightarrow -1$, then the AR requires the driving parameter $P_{1% \text{ }}$to exceed a certain threshold [@Barak_2009] $$P_{1}>P_{1,cr}=\frac{0.82}{\sqrt{nP_{3}f_{n}K_{n}}}. \label{eq:P1cr}$$This condition shows that the AR is an essentially nonlinear phenomenon and disappears when $P_{3}\rightarrow 0$. A typical dynamics of the nonlinear LZ model is illustrated in Fig. [fig:add]{}. ![The phase mismatch $\protect\delta $ (solid lines) and amplitudes (1 – $|u|^{2}$ and 2 – $|v|^{2}$, dashed lines) versus slow time $\protect% \tau$ in the nonlinear Landau-Zener transition for $P_{2}=50$, $P_{3}=300$. Panel (a) below the AR threshold $P_{1}=0.14<P_{1,cr}=0.15$, panel (b) just above the threshold $P_{1}=0.16$, and panel (c) well above the threshold $% P_{1}=0.31$ when the AR phase-locking is observed; the parameters are $n=1$, $K_{1}=1$, and $f_{1}\approx 0.1$.[]{data-label="fig:add"}](fig0.eps) For a given nonlinearity $P_{3}$, the threshold condition (\[eq:P1cr\]) separates two different types of evolution of the system. If $P_{1}<P_{1,cr}$ (see Fig. \[fig:add\]$a$), the passage through the resonance at $\tau \approx \tau _{\ast }-P_{3}K_{n}f_{n}/2\approx 50$ yields a small excitation and a fast growth of the phase mismatch $\delta $ between the two states (see solid line in Fig. \[fig:add\]$a$). In contrast, we observe synchronization and nearly complete transition between the states when the coupling parameter exceeds the threshold, $P_{1}>P_{1,cr}$. Figures [fig:add]{}$b$ and \[fig:add\]$c$ illustrate this effect just beyond the threshold and far from the threshold, respectively. One can see that above the threshold the phase mismatch is bounded, exhibiting the phase-locking phenomenon characteristic of the AR. One can also see that the amplitudes vary significantly during the transition from $n-1$ to $n$ state. The phase-locking is destroyed only after the system almost completely transfers to the new state. As long as the phase-locking is sustained, the population imbalance increases on average as $S(\tau )\approx {2(\tau -\tau _{\ast })}/{% P_{3}f_{n}K_{n}}$ with some superimposed modulations. In particular, $% |u|^{2}\approx |v|^{2}$ at time $\tau \approx \tau _{\ast }\ $($\tau _{\ast }=65$ in our examples). Since the maximal change of the population imbalance is $\Delta S=2$, the duration of the complete autoresonant transition can be estimated as $$\tau _{AR}=P_{3}f_{n}K_{n}.$$ The most important characteristic of the LZ model is the transition probability $W_{n}={|v(\infty )|^{2}}/{K_{n}}$ for finding the system in the upper state if it was in the lower state initially. In the linear limit, this probability is given by the famous LZ formula $$W_{n}=1-e^{-\frac{n\pi P_{1}^{2}}{2}}. \label{eq:Wlinear}$$The numerical integration of the nonlinear LZ model gives the transition probability shown in Fig. \[fig:prob\]. The curve corresponding to $P_{3}=0$ coincides with the linear LZ result . The transition probability steepens as the nonlinearity $P_{3}$ increases and tends to a step-like function in the strongly nonlinear limit. The front of this step corresponds to the onset of the AR at $P_{1}\approx P_{1,cr}$. Once phase-locked, the system remains in this state until almost complete population inversion is achieved, as indicated by transition probability (the height of the step) close to unity in Fig. \[fig:prob\]. This means that the threshold $\eqref{eq:P1cr}$ obtained in Ref. [@Barak_2009] in a small-amplitude limit, i.e. assuming $S\approx -1+\delta S$, $\delta S\ll 1$, is applicable to the fully nonlinear equations as well. For interpreting numerical simulations covering both the linear and nonlinear LZ transitions we redefine the threshold $P_{1,cr}$ as the $P_{1}$ value corresponding to $50\%$ transition probability, i.e., $% W_{n}(P_{1,cr})=1/2$. Using this definition we numerically solve Eq. to find the threshold values $P_{1,cr}$ and compare these results with the theoretical prediction in Fig. [fig:threshold]{}. One can see that in the strongly nonlinear limit $% P_{3}f_{n}\gg 1$ the numerical results reproduce Eq. . On the other hand, for small nonlinearity the threshold approaches a constant $% P_{1,cr}\approx \sqrt{2\ln {2}/\pi n}$ corresponding to the linear LZ formula . ![The probability $W_{1}(P_{1})$ of the $0\rightarrow 1$ transition for different values of the nonlinearity parameter $P_{3}=0,50,300$ from numerical simulations of Eq. for $n=1$, $K_{1}=1$.[]{data-label="fig:prob"}](fig1.eps) ![The rescaled critical drive parameter versus rescaled nonlinearity parameter. Solid line – theory, Eq. ; circles – numerical simulations of Eq. .[]{data-label="fig:threshold"}](fig2.eps) The original system also allows excitation from the ground state to a $N$-th energy state via a sequence of $N$ independent LZ transitions with probability $$W=\prod_{n=1}^{N}W_{n}(P_{1},P_{3}). \label{eq:W}$$The equation $W(P_{1},P_{3})=1/2$ defines the threshold $P_{1,LC}(P_{3})$ for this LC process. As it was found numerically [@Barth_2011] in the linear $P_{3}=0$ case, the product quickly converges for $% N\geqslant 5$ and one finds $P_{1,LC}\approx 0.79$. On the other hand, in the strongly nonlinear limit one can approximate the transition probability by the Heaviside step function $W_{n}(P_{1},P_{3})\approx H(P_{1}-P_{1,cr}(P_{3},n))$, where $P_{1,cr}$ is given by Eq. . Since $P_{1,cr}$ decreases with the transition number $n$ and every transition leads to nearly complete population inversion, the product can be replaced by the single $n=1$ term $W\approx W_{1}$ and the capture into the LC regime occurs after the first transition. In this approximation the threshold is simplified $$P_{1,LC}\approx P_{1,cr}(P_{3},n=1)=\frac{0.82}{\sqrt{P_{3}f_{1}K_{1}}}.$$Note that the threshold width, defined as the inverse slope of the transition probability at $W=1/2$, equals $\Delta P_{1,LC}=0.66$ in the linear case [@Barth_2011] and tends to zero in the strongly nonlinear regime. Similarly to the linear case, we can assume that the successive transitions in the nonlinear regime will be well-separated provided the time between two successive transitions satisfies $\Delta \tau \gg \tau _{AR}$, i.e., $% P_{2}\gg f_{1}K_{1}P_{3}$. This estimate can be simplified because $f_{1}=% \sqrt{2/\pi }/8\approx 0.1$ and we can set $K_{1}=1$. Combining the above inequality with the linear result we find the condition for essentially quantum dynamics in the $P_{1,2,3}$ parameter space: $$P_{2}\gg 1+P_{1}+0.1P_{3}. \label{eq:qcl}$$This inequality together with the condition $P_{1}>P_{1,LC}$ defines the region of the parameter space, where efficient excitation of quantum states in the model via the autoresonant LC is achieved. \[sec:classic\]Semiclassical regime =================================== If the anharmonicity parameter $P_{2}$ of the trap decreases, the two-level approximation employed in the previous section breaks down as several levels can resonate with the drive simultaneously. In the limit $P_{2}\ll 1$, the number of coupled levels is so large that the dynamics becomes semiclassical. The linear $P_{3}=0$ case in this problem was already studied in Ref. [@Barth_2011]. It was shown that the autoresonant excitation of BEC oscillations is possible provided the drive strength $P_{1}$ exceeds the classical autoresonance threshold for the Duffing oscillator [Fajans\_2001]{} $$P_{1,AR}=\frac{0.82}{\sqrt{P_{2}}}. \label{eq:P1AR}$$In this regime, the center of mass of the condensate oscillates in the trap with an increasing amplitude. The frequency of these oscillations remains close to the driving frequency during the whole excitation process, despite the variation of the driving frequency. In this section we discuss the threshold value $P_{1,AR}$ in the nonlinear case $P_{3}>0$. Consider the Wigner representation of our problem [@hiley2004phase]: $$\frac{\partial f(x,p,t)}{\partial t}=\{H,f\}_{MB}, \label{eq:dfdt}$$where $f(x,p,t)$ is the Wigner function, $H$ is the Hamiltonian $$H=-\frac{\hbar ^{2}}{2m}\frac{\partial ^{2}}{\partial x^{2}}+U(x,t)+g|\Psi (x,t)|^{2}$$and $\{H,f\}_{MB}$ denotes the Moyal bracket. Since the Moyal bracket reduces to the Poisson bracket in the semiclassical limit $\hbar \rightarrow 0$: $\{H,f\}_{MB}\approx \{H,f\}+O(\hbar ^{2})$, equation reduces to the Liouville equation $$\frac{\partial f}{\partial t}+\frac{p}{m}\frac{\partial f}{\partial x}-\frac{% \partial }{\partial x}\left( U+g|\Psi |^{2}\right) \frac{\partial f}{% \partial p}\approx 0, \label{eq:dfdt_2}$$where in addition to the external potential $U(x,t),$ we have the self-potential $V=g|\Psi |^{2}$. This equation is reminiscent of the Vlasov equation for an ensemble of particles in the combined external and self-potentials. Note that the self-potential can be expressed via the Wigner function $$|\Psi (x,t)|^{2}=\int_{-\infty }^{\infty }f(x,p,t)dp,$$transforming into a closed integro-differential form. The characteristics (classical trajectories) for Eq.  are given by $$\frac{d^{2}x}{dt^{2}}+\frac{1}{m}\frac{\partial U}{\partial x}+\frac{g}{m}% \frac{\partial |\Psi |^{2}}{\partial x}=0. \label{eq:A}$$Suppose one starts in a localized state, so that the Wigner function has a local maximum at the phase-space point $[x_{0},p_{0}]$. In the semiclassical limit, the Wigner function is expected to continue having a local maximum near the phase-space point $[x(t),p(t)]$ moving along the classical trajectory starting at $[x_{0},p_{0}]$. Near this point,$$|\Psi (x+s,t)|^{2}\approx const-\kappa s^{2}/2$$and, thus, the term $(g/m)\partial |\Psi |^{2}/\partial x$ in Eq. vanishes along the trajectory of the maximum of $f$. Consequently, the nonlinearity (characterized by parameter $P_{3}$) in the semiclassical regime does not affect the evolution of the maximum. Then it also does not change the threshold of the AR and should not shift the transition probability versus $P_{1}$ in contrast to the quantum regime (see Fig. [fig:prob]{}). We confirm these conclusions in numerical simulations in the following section. \[sec:numerics\] Numerical simulations ====================================== In this section we present numerical simulations of the original Gross-Pitaevskii equation (\[eq:GP\]) in our driven problem. We rewrite this equation in a dimensionless form using the same $\xi =x/\ell $, but a different dimensionless time $T=\omega _{0}t$:$$\begin{aligned} i\Phi _{T}& +\frac{1}{2}\Phi _{\xi \xi }-\left( \tilde{U}+Q_{3}|\Phi |^{2}\right) \Phi =0, \label{eq:GPnd} \\ \tilde{U}(\xi ,T)& =\frac{\xi ^{2}}{2}-Q_{2}\frac{\xi ^{4}}{4}+Q_{1}\xi \cos \tilde{\varphi}(T), \notag\end{aligned}$$where $\Phi =\sqrt{\ell }\;\Psi $, $\tilde{\alpha}=\alpha /\omega _{0}^{2}$, $Q_{1}=\sqrt{2\tilde{\alpha}}\,P_{1}$, $Q_{2}=\frac{4}{3}\sqrt{\tilde{\alpha}% }\,P_{2}$, $Q_{3}=\sqrt{\tilde{\alpha}}\,P_{3}$ and $\tilde{\varphi}(T)=T-{% \tilde{\alpha}\,T^{2}}/2$. The simulations are based on the standard pseudo-spectral method [@canuto_spectral] with explicit 4-th order Runge-Kutta algorithm and adaptive step size control. The ground state of the harmonic oscillator was used as the initial condition: $$\Phi (\xi ,0)=\pi ^{-1/4}\;e^{-\xi ^{2}/2}. \label{incond}$$The state of the condensate was analyzed by calculating its energy $$E=\int_{-\infty }^{\infty }\left( \frac{1}{2}|\Phi _{\xi }|^{2}+\tilde{U}% |\Phi |^{2}+\frac{Q_{3}}{2}|\Phi |^{4}\right) d\xi ,$$the amplitudes $c_{n}(t)$ in the basis of Hermite functions, and the Wigner distribution. In order to study various regimes of excitation of a condensate, we performed a series of numerical simulations by varying parameters $P_{1}$ and $P_{2}$ in the linear ($P_{3}=0$) and nonlinear cases. The results of the simulations are presented in Fig. \[fig:regimes\]. The circles in the figure correspond to parameters yielding $50\%$ probability of capture into either the classical AR or the quantum LC regime. ![Different domains of phase-locking transition in the driven Gross-Pitaevskii equation in the linear ($P_{3}=0$, black open circles) and nonlinear ($P_{3}=100$, filled blue circles) cases. The symbols $\times $ and $\triangle $ show parameters used in subsequent figures \[fig:control\] and \[fig:control2\]. []{data-label="fig:regimes"}](fig3.eps) The classical autoresonance threshold is shown by the black long-dashed line. The two roughly horizontal lines $P_{2}=1+P_{1}+0.1\,P_{3}$ separate the regions of the quantum and semiclassical dynamics in the linear ($P_{3}=0$, red line) and nonlinear ($P_{3}=100$, blue line) regimes. The vertical lines show the theoretical LC transition thresholds $P_{1,LC}$ for the linear and nonlinear regimes. One can see that the classical AR threshold yields a good approximation for the transition boundary for $P_{2}$ versus $P_{1}$ in both the linear and the nonlinear cases. In the case $P_{3}=100$, the nonlinear ladder climbing transitions emerge at significantly smaller values of the driving parameter $P_{1}$ and larger anharmonicity $P_{2}$, compared to the linear case. This is in agreement with the shift of the threshold in the nonlinear model of the autoresonant LZ transitions discussed above (see Fig. 2). The important change due to the nonlinearity in the problem is the decrease of the width of the transition region. This effect is illustrated in Fig. \[fig:W\] for the case of the semiclassical autoresonant transition. The width of the transition for the nonlinear case decreases rapidly with the increase of the nonlinearity parameter $P_{3}$ and the transition probability assumes a nearly step-like shape for $P_{3}>15$. One also observes that the threshold location, where the probability crosses $50\%$, only slightly changes with the variation of $P_{3}$ as discussed at the end of Sec. II. ![The transition probability versus the driving parameter $P_{1}$ for $P_{2}=0.2$ and $P_{3}=0,5,10,15$ from numerical integration of Eq.  . []{data-label="fig:W"}](fig6.eps) The effect of the narrowing of the transition width in the semiclassical regime with the increase of the nonlinearity parameter can be associated with the improved stability of the autoresonant classical trajectories described by Eq. (\[eq:A\]) . Indeed, in dimensionless variables, Eq. ([eq:A]{}) for the characteristics of Eq. (\[eq:dfdt\_2\]) becomes $$\xi _{TT}+\xi -Q_{2}\xi ^{3}+Q_{3}\frac{\partial |\Phi (\xi ,T)|^{2}}{% \partial \xi }+Q_{1}\cos \widetilde{\varphi }(T)=0. \label{eq:char}$$Let the trajectory of the autoresonant maximum of the Wigner function be $% \xi _{0}(T)$. The dynamics of this trajectory is described by$$\xi _{0TT}+\xi _{0}-Q_{2}\xi _{0}^{3}+Q_{1}\cos \widetilde{\varphi }(T)=0 \label{artr}$$subject to $\xi _{0}=\xi _{0T}=0$ at large negative $T$ and is not affected by the nonlinearity, as described above. For studying the the evolution of a deviation $\eta =\xi -\xi _{0}(T)$ from $\xi _{0}(T)$, we linearize Eq. ([eq:char]{}) around $\xi _{0}$ and assume $|\Phi (\xi ,T)|^{2}\approx const-\kappa \eta ^{2}/2$ near the maximum, to get $$\eta _{TT}+[1-3Q_{2}\xi _{0}^{2}-\kappa Q_{3}]\eta =0. \label{eta}$$We analyze the solutions of system (\[artr\]), (\[eta\]) in the Appendix. Numerically, in the vicinity of the threshold for $P_{3}=0$, we observe the development of instability of $\eta $. By writing $\xi _{0}=a\cos\theta$, Eq. (\[eta\]) assumes the form of the Mathieu-type equation with slowly varying parameters$$\eta _{TT}+\left\{ 1-\frac{3Q_{2}a^{2}}{2}-\kappa Q_{3}-\frac{3}{2}% Q_{2}a^{2}\cos [2\theta (T)]\right\} \eta =0. \label{eq:paramet}$$We show in the Appendix that this equation predicts parametric-type instability for $P_{3}=0$ and we attribute the existence of the width in the transition to autoresonance as seen in Fig. \[fig:W\] to this effect. Note that the addition of the nonlinearity (the term $-\kappa Q_{3}$ in the last equation) shifts the eigenfrequency so that the parametric resonance can be avoided and characteristic trajectories with nearby initial conditions remain close to the classical autoresonant trajectory, narrowing the transition width as seen in Fig. \[fig:W\]. For the parameters in this figure, the system stabilizes at $P_{3}>0.3$ (see the Appendix). Nonetheless, the deviation from the autoresonant state is still large until $% P_{3}>10$, when the autoresonant transition width practically disappears. ![Numerical solutions of Eq.  in the LC regime for two scenarios. In the first scenario \[dashed red lines in panel (a)\], constant driving frequency chirp rate $\tilde{\protect\alpha}=10^{-6}$ and $P_{1}=0.5$, $P_{2}=30$, $P_{3}=100$ (as shown by cross symbol in Fig. [fig:regimes]{}) are used. In the second scenario \[solid line in panel (a)\] we set $\tilde{\protect\alpha}=0$ at $T=1.8\times 10^{5}$ and keep it constant until $T=2.2\times 10^{5}$. We restore the original value of $\tilde{\protect% \alpha}$, but with the opposite sign at $T=2.2\times 10^{5}$. The distribution of the populations of the energy levels and the Wigner quasi-probability distribution at $T=2\times 10^{5}$ for the second scenario are shown in panels (b) and (c), respectively. Here $T=\protect\omega _{0}t=% \protect\tau /\protect\sqrt{\tilde{\protect\alpha}}$.[]{data-label="fig:control"}](fig4a.eps "fig:") ![Numerical solutions of Eq.  in the LC regime for two scenarios. In the first scenario \[dashed red lines in panel (a)\], constant driving frequency chirp rate $\tilde{\protect\alpha}=10^{-6}$ and $P_{1}=0.5$, $P_{2}=30$, $P_{3}=100$ (as shown by cross symbol in Fig. [fig:regimes]{}) are used. In the second scenario \[solid line in panel (a)\] we set $\tilde{\protect\alpha}=0$ at $T=1.8\times 10^{5}$ and keep it constant until $T=2.2\times 10^{5}$. We restore the original value of $\tilde{\protect% \alpha}$, but with the opposite sign at $T=2.2\times 10^{5}$. The distribution of the populations of the energy levels and the Wigner quasi-probability distribution at $T=2\times 10^{5}$ for the second scenario are shown in panels (b) and (c), respectively. Here $T=\protect\omega _{0}t=% \protect\tau /\protect\sqrt{\tilde{\protect\alpha}}$.[]{data-label="fig:control"}](fig4b.eps "fig:") Our next numerical simulation shows that the system can be fully controlled by the AR in both the quantum and semiclassical regimes. Two protocols of such a control of the LC dynamics are shown in Fig. \[fig:control\]. Parameters $P_{i}$ are chosen so that both condition and $% P_{1}>P_{1,LC}$ are satisfied. Due to a relatively strong anharmonicity, the nonlinear LZ transitions are well separated in time. One can see that the energy in the system grows step-by-step from one energy level to another. In the first protocol, the linear frequency variation with a constant chirp rate $\alpha $ \[red dashed line in Fig. \[fig:control\](a)\] results in the excitation of the condensate to the sixth energy level. In the second protocol (solid line) we first excite the system to the fourth energy state by decreasing driving frequency until $T=1.8\times 10^{5}$, then keep the driving frequency constant for the time span of $\Delta T=4\times 10^{4}$, and finally return the system to the ground state by increasing the driving frequency back to its original value. One can see that the quantum state of can be efficiently controlled as long as the LC autoresonant conditions are met and the frequency and phase of the driving are continuous. The quantum state in the first protocol at $T=2\times 10^{5}$ is further illustrated in Figs. \[fig:control\] (b) and (c) showing the distribution of populations of different levels and the Wigner function. Small deviations of the solution from the fourth eigenfunction $\chi _{4}(\xi )$ of the linear harmonic oscillator are due to the anharmonicity and the nonlinearity in the problem. A similar control protocols for the semiclassical case are illustrated in Fig. \[fig:control2\]. One can see that resulting wave function has a Gaussian (Poissonian in the early stages of dynamics) population distribution, characteristic of coherent states. The Wigner function is positive everywhere (on the computational grid) and close to the $n$-squeezed coherent state. The second protocol (solid line in Fig. [fig:control2]{}) demonstrates that a more complex control scenarios are possible in the semiclassical case, as long as the driving parameter is within the region of autoresonant dynamics and the driving phase and frequency are continuous functions of time. ![The numerical solutions of Eq.  in the semiclassical regime for parameters $\tilde\protect\alpha=10^{-6}$, $P_1=1.6$, $P_2=0.3$, $% P_3=100$ (represented by the triangle in Fig. \[fig:regimes\]). In the second scenario (solid line) the frequency chirp is set to zero from $% T=1.8\times 10^{4}$, but restored to the original value with opposite sign at $T=3.8\times 10^{4}$. The distribution of populations of energy levels and the Wigner function at $T=2.8\times 10^4$ for the second scenario are shown in panels (b) and (c), respectively.[]{data-label="fig:control2"}](fig5a.eps "fig:") ![The numerical solutions of Eq.  in the semiclassical regime for parameters $\tilde\protect\alpha=10^{-6}$, $P_1=1.6$, $P_2=0.3$, $% P_3=100$ (represented by the triangle in Fig. \[fig:regimes\]). In the second scenario (solid line) the frequency chirp is set to zero from $% T=1.8\times 10^{4}$, but restored to the original value with opposite sign at $T=3.8\times 10^{4}$. The distribution of populations of energy levels and the Wigner function at $T=2.8\times 10^4$ for the second scenario are shown in panels (b) and (c), respectively.[]{data-label="fig:control2"}](fig5b.eps "fig:") \[sec:conclusions\]Conclusions ============================== In conclusion, we have studied the effect of the particles interaction on the excitation of Bose-Einstein condensate in a anharmonic trap under chirped-frequency perturbation. We have identified three dimensionless parameters $P_{1,2,3}$ \[see Eqs. (\[eq:par\])\] characterizing the driving strength, the anharmonicity and the strength of the interaction to show that there exist two very different regimes of excitation in this parameters space, i.e., the quantum-mechanical ladder climbing (LC) and the semiclassical autoresonance (AR). The transition boundary to the semiclassical AR in the $P_{1,2}$ parameter space is independent of the nonlinearity parameter $P_{3}$. In contrast, the LC transition boundary is significantly affected by the strength of the interaction of the particles, because the underlying nonlinear Landau-Zener (LZ) transitions behave differently than their linear counterpart. In the limit of strong interaction, the nonlinear LZ transition probability as a function of the driving strength parameter $P_{1}$ approaches the Heaviside step function due to the nonlinear phase-locking. We have also found that in both the quantum and the semiclassical regimes the width $\Delta P_{1}$ of the transition decreases as the strength of the interaction increases. In the quantum limit this effect is related to the autoresonance of the nonlinear LZ transitions, while in the semiclassical limit, the effect is due to the wave packet stability enhancement by avoiding parametric resonance between the center-of-mass motion and the internal dynamics of the condensate. Possible applications of the results of this paper may include a control of the quantum state of BECs and the implementation of precision detectors based on either the LC or the AR. Unlike the noninteracting case [Murch\_2010]{}, the resolution of such a detector is not limited by quantum fluctuations if the particles interaction is strong enough. ACKNOWLEDGEMENTS {#acknowledgements .unnumbered} ================ This work was supported by the Russian state assignment of FASO No.01201463332 and by the Israel Science Foundation Grant No. 30/14. APPENDIX: Stability of autoresonant trajectories. {#appendix-stability-of-autoresonant-trajectories. .unnumbered} ================================================= ![The energy $E=\frac{1}{2}\left\langle \protect\eta _{T}^{2}+\protect% \eta \right\rangle $ averaged over the integration time versus $P_{3}$ from the numerical solution of Eqs. , near the threshold of the autoresonance. The lines with squares and circles correspond to $\protect\alpha =10^{-4}$ and $\protect\alpha =10^{-6}$, respectively.[]{data-label="fig:appendix"}](fig7.eps) Here we discuss the Mathieu-type Eq. (\[eq:paramet\]) $$\eta _{TT}+[B-C\cos (2\theta )]\eta =0, \label{A1}$$where $B=1-\frac{3Q_{2}a^{2}}{2}-\kappa Q_{3}$, $C=\frac{3}{2}Q_{2}a^{2}$, describing the deviation $\eta $ of the trajectory from the autoresonant solution $\xi_{0}=a\;\cos \theta$ given by Eq. (\[artr\]). For sufficiently small amplitudes $a$ this solution is described by [Scholarpedia]{} $$\begin{aligned} \frac{{da}}{dT} &=&(Q_{1}/2)\sin \Phi , \label{A3} \\ \frac{{d\Phi }}{dT} &=&\alpha T-(3Q_{2}/8)a^{2}+(Q_{1}/2a)\cos \Phi , \label{A4}\end{aligned}$$where $\Phi =\theta -\widetilde{\varphi }+\pi $ is the phase mismatch, which starts at zero initially (at large negative $t$, where $a$ is also zero). We are interested in passage through resonance when parameter $Q_{1}$ is close to the autoresonance threshold $Q_{1th}$ (from either above or below). In both of these cases,  $\Phi $ slowly increases initially and then passes $% \pi /2$. Then when slightly above the threshold, $\Phi $ starts oscillating, but remains bounded ($|\Phi |<\pi $), while below the threshold $\Phi $ continues to increase reaching $\pi $ and the oscillator fully dephases from the drive. At this point we observe that there are three parameters ($\alpha ,Q_{1},Q_{2}$) in system (\[A3\]), (\[A4\]). However, by introducing the slow time $\tau ={\alpha ^{1/2}T}$ and rescaled amplitude $A=\sqrt{3/8}$ $% \alpha ^{-1/4}Q_{2}^{1/2}a$ we obtain a single parameter system $$\begin{aligned} \frac{{dA}}{d\tau } &=&\mu \sin \Phi , \label{A5} \\ \frac{{d\Phi }}{d\tau } &=&\tau -A^{2}+(\mu /A)\cos \Phi , \label{A6}\end{aligned}$$ where $$\mu =\sqrt{\frac{3}{32}}Q_{1}Q_{2}^{1/2}{\alpha ^{-3/4}.} \label{ttt}$$The autoresonant threshold in this system is $\mu _{th}=0.41$ [Scholarpedia]{}, which upon the return to the original parameters $P_{1,2}$ yields (\[eq:P1AR\]). Note that at the threshold ($\mu =\mu _{th}$) the rescaled problem has no free parameters. Next we discuss Eq. (\[A1\]). Here, using (\[A4\]), we have $$\frac{{d\theta }}{dT}=\frac{{d}\widetilde{\varphi }}{{dT}}+\frac{d\Phi }{dT}% =1-\alpha T+\frac{d\Phi }{dT}=1-\frac{3}{8}Q_{2}a^{2}+\frac{Q_{1}}{2a}\cos \Phi \label{A7}$$and assume that $S=-\frac{3}{8}Q_{2}a^{2}+\frac{Q_{1}}{2a}\cos \Phi $ is small compared to unity in the region of dephasing. This suggests to transform from $T$ to $\theta $ in (\[A1\]) and approximate the problem by the Mathieu equation with slow coefficients $$\frac{{d^{2}\eta }}{{d\theta ^{2}}}+[B^{\prime }-C^{\prime }\cos (2\theta )]\eta \approx 0 \label{A8}$$ where to lowest order in small parameters $$\begin{aligned} B^{\prime } &=&\frac{B}{(1+S)^{2}}\approx 1-\frac{3}{4}Q_{2}a^{2}-\frac{Q_{1}% }{a}\cos \Phi -\kappa Q_{3} \label{A9} \\ C^{\prime } &=&\frac{C}{(1+S)^{2}}\approx C=\frac{3}{2}Q_{2}a^{2} \label{A10}\end{aligned}$$From the theory of the Mathieu equation (with fixed parameters, which we assume locally in our case), the stability condition of the solution of ([A8]{}) is [@Hayashi] $$B^{\prime }<1-C^{\prime }/2 \label{A12}$$or$$\kappa Q_{3}>\max \left( -\frac{Q_{1}}{a}\cos \Phi \right) . \label{A13}$$Finally, in the last equation we use the rescaled amplitude $A$ instead of $% a $ to get the dimensionless condition $$\kappa P_{3}>P_{1}P_{2}^{1/2}\max \left( -\frac{\cos \Phi }{A}\right) _{\mu =\mu _{th}} \label{A11}$$ where the rescaled parameters are $P_{1}=Q_{1}/(2{\alpha )^{1/2}}$, $% P_{2}=(3/4)Q_{2}/{\alpha ^{1/2}}$, $P_{3}=Q_{3}/{\alpha ^{1/2}}$ and the value of $\max \left( -\frac{\cos \Phi }{A}\right) _{\mu =\mu _{th}}=0.37$ is found numerically by solving (\[A5\]), (\[A6\]). Thus, for $\kappa=0$, the solution remains stable only if $\Phi <\pi /2$ and, since we approach this value in the vicinity of the threshold as mentioned above, one encounters instability near the threshold, explaining the appearance of the width of the autoresonant transition as some initial conditions dephase. On the other hand, a sufficiently large $\kappa P_{3}$ stabilizes the solution and the width of the autoresonance threshold disappears. For the parameters of Fig. [fig:W]{} one finds that the solution is stable for $P_{3}>0.27$. Finally we check these conclusions by solving the set (\[artr\]) and (\[eta\]) numerically. 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--- abstract: 'The phylogenetic semigroup on a graph generalizes the Jukes-Cantor binary model on a tree. Minimal generating sets of phylogenetic semigroups have been described for trivalent trees by Buczy[ń]{}ska and Wi[ś]{}niewski, and for trivalent graphs with first Betti number 1 by Buczy[ń]{}ska. We characterize degree two minimal generators of the phylogenetic semigroup on any trivalent graph. Moreover, for any graph with first Betti number 1 and for any trivalent graph with first Betti number 2 we describe the minimal generating set of its phylogenetic semigroup.' author: - Kaie Kubjas bibliography: - 'low\_degree\_minimal\_generators\_of\_phylogenetic\_semigroups.bib' title: Low degree minimal generators of phylogenetic semigroups --- Introduction ============ Let $G$ be a graph. The phylogenetic semigroup on $G$ is a set of labelings of edges of $G$ by non-negative integers fulfilling some additional conditions. This set has naturally the structure of a semigroup by edge-wise addition. The phylogenetic semigroup on a trivalent graph was defined by Buczy[ń]{}ska [@Buczynska12] as a generalization of the affine semigroup of the Jukes-Cantor binary model on a trivalent tree. Jukes-Cantor binary model is the simplest group-based model with the underlying group $\mathbb{Z}_2$. In [@BBKM11], Buczy[ń]{}ska, Buczy[ń]{}ski, Micha[ł]{}ek and the author further generalized the definition of the phylogenetic semigroup to arbitrary graphs. This definition agrees with Buczy[ń]{}ska’s definition for trivalent graphs. Besides phylogenetic algebraic geometry, phylogenetic semigroups appear in several other contexts. In [@JW92], Jeffrey and Weitsman quantized the moduli space of flat $\textrm{SU}(2)$ connections on a two-dimensional surface of genus $g$ using a real polarization. The dimension of the quantization is counted by integral fibers of the polarization, which are in one-to-one correspondence with the labelings of a trivalent graph $G$ with first Betti number $g$ that satisfy the quantum Clebsch-Gordan conditions. These labelings are exactly the elements of the phylogenetic semigroup on $G$. Moreover, the number of labelings that satisfy the quantum Clebsch-Gordan conditions matches the Verlinde formula for the $\textrm{SU}(2)$ Wess-Zumino-Witten model in the quantum field theory [@Verlinde88]. In more recent work, Sturmfels and Xu [@SX10] showed that the projective coordinate ring of the Jukes-Cantor binary model is a sagbi degeneration of the Cox ring of the blow-up of $\mathbb{P}^{n+3}$ at $n$ general points. Manon generalized their construction showing that the algebra of ${\textrm{SL}}_2(\mathbb{C})$ conformal blocks for a stable curve of genus $g$ with $n$ marked points flatly degenerates to the semigroup algebra of the phylogenetic semigroup on a graph with first Betti number $g$ with $n$ leaves [@Manon09]. Low degree minimal generators of phylogenetic semigroups have been previously studied for trees and graphs with first Betti number 1. Phylogenetic semigroups on trees are generated by degree one labelings, known as networks [@BW07; @DBM12]. Buczy[ń]{}ska studied minimal generators of phylogenetic semigroups on trivalent graphs with first Betti number 1. She proved that any minimal generator of the phylogenetic semigroup on a trivalent graph with first Betti number 1 has degree at most two, and explicitly described minimal generating sets [@Buczynska12]. We extend this result from trivalent graphs to general graphs, i.e. we describe the minimal generating set of the phylogenetic semigroup on any graph with first Betti number $g\leq 1$. Moreover, we characterize degree two minimal generators on trivalent graphs with first Betti number $g>1$. We also specify the bound on the maximal degree of the minimal generating set for graphs with first Betti number 2. By [@BBKM11], the maximal degree of the minimal generating set of the phylogenetic semigroup on a graph with first Betti number 2 is at most three. We explicitly characterize when the maximal degree three is attained, and when the maximal degree is equal to two or one. If the degree three is attained, we describe the degree three minimal generators. Finally, we list maximal degrees of minimal generating sets of phylogenetic semigroups on some graphs with first Betti number 3, 4 or 5. We speculate that the maximal degree depends on the separateness of the cycles of the graph. Having low maximal degree is especially interesting from the perspective of SL$_2(\mathbb{C})$ conformal block algebras as this ensures low maximal degree for the minimal generators of these algebras, see [@Manon_bounds_on_generators]. In Section \[section:phylogenetic\_semigroups\], we introduce basics about phylogenetic semigroups on graphs. In Section \[section:first\_betti\_number\_one\], we give a shortened proof of Buczy[ń]{}ska’s theorem about minimal generators of phylogenetic semigroups on trivalent graphs with first Betti number 1, and we generalize the statement to general graphs with first Betti number 1. In Section \[section:degree\_two\_generators\], we characterize degree two minimal generators on an arbitrary trivalent graph. In Section \[section:first\_betti\_number\_two\], we study the explicit maximal degree of the minimal generating set of the phylogenetic semigroup on a graph with first Betti number 2. In Section \[section:degree\_three\_generators\], we describe minimal generating sets of phylogenetic semigroups on trivalent graphs with first Betti number 2. In the last section, we list examples of these maximal degrees for graphs with first Betti numbers 3,4 and 5. Phylogenetic semigroups {#section:phylogenetic_semigroups} ======================= In this section, we define phylogenetic semigroups on graphs as in [@BBKM11] and recall some basic properties about these semigroups. Let $G$ be a graph. A *path* in $G$ is a sequence of unrepeated edges which connect a sequence of vertices. Moreover, we require the first and the last vertex to be either both leaves or equal. In the latter case, a path is called a *cycle*. A *network* is a disjoint union of paths. A *cycle edge* is an edge on a cycle of $G$. A *cycle leg* is an edge incident to a cycle edge, but is not a cycle edge. We denote the *disjoint sum* of graphs $G_1$ and $G_2$ by $G_1\sqcup G_2$. We denote by $G^e$ the graph obtained from $G$ by cutting an internal edge $e$. More specifically, *cutting an internal edge* $e$ means replacing $e$ by two leaf edges $e_1$ and $e_2$ where $\partial_1(e_1)=\partial_1(e)$ and $\partial_1(e_2)=\partial_2(e)$. Here $\partial_1(e), \partial_2(e)$ denote endpoints of an edge $e$. Let $T$ be a tree with the set of edges $E$ and the set of inner vertices $I$. Define lattices $$L_T=\{x\in \mathbb{Z}^E:\sum _{v\in e}x_e \in 2\mathbb{Z} \textrm{ for every } v\in I\}$$ and $$L_T^{gr}=L_T\oplus \mathbb{Z}$$ together with the degree map $$\textrm{deg}:L_T^{gr}=L_T\oplus \mathbb{Z}\rightarrow \mathbb{Z}$$ given by the projection on the last summand. The lattice polytope associated with the Jukes-Cantor binary model on $T$ is $$P_T=\textrm{conv}\{x\in L_T:x_e\in\{0,1\} \textrm{ for every } e\in E\}.$$ The phylogenetic semigroup $\tau(T)$ on $T$ is $$\tau(T)=\textrm{cone}(P_T\times \{1\})\cap L_T^{gr}.$$ \[defin\_graph\_path\_network\] *First Betti number* of a graph is the minimal number of cuts that would make the graph into a tree. Given a graph $G$, we associate a tree $T$ with a set of distinguished pairs of leaves to $G$ and define the phylogenetic semigroup on $G$ using the phylogenetic semigroup on $T$. We construct the tree $T$ inductively on first Betti number $g$ of $G$. If $g=0$, then $G$ itself is the associated tree. If $g>0$, then we replace a cycle edge $\overline{e}$ by two leaf edges $e'$ and $e''$ where $\partial_1(e')=\partial_1(\overline{e})$ and $\partial_1(e'')=\partial_2(\overline{e})$. This replacement gives a graph with first Betti number $g-1$ and a distinguished pair of leaves $(e',e'')$. Doing this procedure $g$ times gives a tree $T$ and $g$ distinguished pairs of leaves. Although the tree $T$ and the set of distinguished pairs of leaves are in general not unique, the phylogenetic semigroup on $G$ does not depend on the choices we make. Let $G$ be a graph. Let $T$ be the associated tree with a set of distinguished pairs of leaves $\{(e_i',e_i'')\}$. We define the *phylogenetic semigroup on $G$* as $$\tau(G)=\tau(T)\cap \bigcap_{i}(x_{e_i'}=x_{e_i''}).$$ In other words, $\tau(G)$ consists of those labelings of $\tau(T)$ where the label on $e_i'$ is identical to the one on $e_i''$, and thus the labeling of $T$ gives a labeling of $G$. Similarly, define the lattice $$L_G^{gr}=L_T^{gr}\cap \bigcap_{i}(x_{e_i'}=x_{e_i''})$$ together with the degree map induced by the degree map of $L_T^{gr}$. The phylogenetic semigroup $\tau(G)$ has a unique minimal generating set. We call the elements of the minimal generating set minimal generators, or sometimes also indecomposable elements of $\tau(G)$. As we often deal with trivalent graphs, we introduce notation specific to these graphs. Let $G$ be a trivalent graph and $v$ be an inner vertex of $G$. Let $\{e_1, e_2, e_3\}$ be the edges of the tripod and $i_v$ a map that is locally an embedding of the tripod into $G$ and sends the central vertex of the tripod to $v$. For $\omega \in L_G^{gr}$ denote $$\begin{array}{lcr} a_v(\omega):=\omega_{i_v(e_1)}, & b_v(\omega):=\omega_{i_v(e_2)}, & c_v(\omega):=\omega_{i_v(e_3)}. \\ \end{array}$$ In other words, $a_v, b_v, c_v$ measure the coefficients of $\omega$ at the edges incident to $v$. The *degree* of $\omega\in L_G^{gr}$ *at an inner vertex* $v\in I$ is $$\deg_v(\omega):=\frac{1}{2}\bigl(a_v(\omega)+b_v(\omega)+c_v(\omega)\bigr).$$ \[def\_cone\_of\_G\] For a trivalent graph $G$ the *phylogenetic semigroup $\tau(G)$ on $G$* is the set of elements $\omega$ satisfying the following conditions 1. \[item\_parity\_condition\] parity condition: $\omega \in L_G^{gr}$, 2. \[item\_non-negativity\_condition\] non-negativity condition: $\omega_e \ge 0$ for any $e \in E$, 3. \[item\_triangle\_inequalities\] triangle inequalities: $ |a_v(\omega)-b_v(\omega)| \leq c_v(\omega) \leq a_v(\omega)+ b_v(\omega)$, for each inner vertex $v\in I$, 4. \[item\_degree\_inequality\] degree inequalities: $\deg(\omega) \geq \deg_v(\omega)$ for any $v\in I$. Let $G$ be a graph and $\overline{e}$ an inner edge of $G$. Let $e'$ and $e''$ be new leaf edges obtained by cutting $G$ at $\overline{e}$, as illustrated in Figure \[figure:cutting\_an\_edge\]. Then $\omega \in \tau(G)$ gives an element $\overline{\omega} \in \tau(G^{\overline{e}})$: $$\overline{\omega}_e=\left\{ \begin{array}{ll} \omega _e & \textrm{if } e\notin \{e',e''\},\\ \omega _{\overline{e}} & \textrm{if } e\in \{e',e''\}. \end{array}\right.$$ On the contrary, given $\overline{\omega} \in \tau (G^{\overline{e}})$, it gives an element $\omega \in \tau(G)$ if and only if $\overline{\omega}_{e'}=\overline{\omega}_{e''}$: $$\omega_e=\left\{ \begin{array}{ll} \overline{\omega} _e & \textrm{if } e\neq \overline{e},\\ \overline{\omega} _{e'} & \textrm{if } e= \overline{e}. \end{array}\right.$$ In [@Buczynska12], a polygon graph $G$ was defined as a graph with $2k$ edges, $k$ of which form the only cycle of $G$ and the remaining $k$ edges are cycle legs. The use of polygon graphs simplifies the study of phylogenetic semigroups on trivalent graphs with first Betti number 1. We generalize this definition to be able to simplify the study of phylogenetic semigroups on any graph. A graph $G$ with first Betti number $g\geq 1$ is called a *multiple polygon graph* if for no edge $e$ we can write $G^e=G'\sqcup G''$ with $G'$ or $G''$ a tree with more than one edge, see Figure \[figure:graphs\_with\_first\_betti\_number\_2\_cases\] for examples. A multiple polygon graph is a *polygon graph* if it has first Betti number 1. Given a graph $G$ with first Betti number $g\geq 1$, there exist non-cycle inner edges $e_1,\ldots ,e_k$ of $G$ such that $G^{e_1,\ldots ,e_k}=G_0\sqcup G_1 \sqcup \ldots \sqcup G_{k}$ where $G_0$ is a multiple polygon graph and $G_1,\ldots ,G_k$ are trees. Choose all non-cycle edges $e$ such that we can write $G^e=G'\sqcup G''$ with $G''$ a tree with more than one edge and $e$ maximal with this property, i.e. there is an edge $\overline{e}$ incident to $e$ such that we cannot write $G^{\overline{e}}={G}'\sqcup {G}''$ with $G'$ or $G''$ a tree with more than one edge. \[lemma:graph\_union\_tree\] Let $G$ be a graph and $\omega \in \tau(G)$. Let $e$ be a non-cycle inner edge such that $G^e=G'\sqcup G''$ with $G''$ a tree. Then any decomposition of $\omega|_{G'}\in \tau(G')$ lifts to a decomposition of $\omega\in \tau(G)$. This lemma is stated for trivalent graphs in [@Buczynska12 Lemma 2.31]. Since $\tau(T)$ is normal for any tree $T$ [@DBM12 Proposition 18], then the proof works for the general case exactly the same way as it does for the trivalent case. \[lemma:polygon\_graph\] Let $G$ be a graph and $\omega \in \tau(G)$. Let $e_1,\ldots ,e_k$ be non-cycle inner edges such that $G^{e_1,\ldots ,e_k}=G_0\sqcup G_1 \sqcup \ldots \sqcup G_{k}$ where $G_0$ is a multiple polygon graph and $G_1,\ldots ,G_k$ are trees. Then any decomposition of $\omega|_{G_0}\in \tau(G_0)$ lifts to a decomposition of $\omega\in \tau(G)$. We can use Lemma \[lemma:graph\_union\_tree\] iteratively. Graphs with First Betti Number 1 {#section:first_betti_number_one} ================================ In this section, we study minimal generating sets of phylogenetic semigroups on graphs with first Betti number 1. Buczy[ń]{}ska did this for trivalent graphs [@Buczynska12]. We give a shortened proof of her result, and as a corollary describe the minimal generating set of the phylogenetic semigroup on any graph with first Betti number 1. Let $G$ be a graph. Networks can be seen as degree one elements of $\tau(G)$. We define $\omega$ corresponding to a network $\Gamma$ in the following way: $$\begin{aligned} \omega _e=1 & & \quad \textrm{if } e \textrm{ belongs to } \Gamma,\\ \omega _e=0 & & \quad \textrm{otherwise}. \end{aligned}$$ It follows from the definition of a network that the parity condition is fulfilled for $\omega $ at every inner vertex of $G$. Hence $\omega \in \tau(G)$. We will often use the notion network for the corresponding labeling $\omega \in \tau(G)$. It has been shown for various classes of graphs that networks are in one-to-one correspondence with degree one elements of a phylogenetic semigroup [@BW07 Lemma 2.3],  [@Buczynska12 Lemma 2.26]. For an arbitrary tree this was stated in [@BBKM11 Section 2], but no proof was given. We did not find proofs for arbitrary trees or graphs in the literature and therefore will present them here. \[lemma:networks\_on\_trees\] Let $T$ be a tree. There is one-to-one correspondence between networks and degree one elements of $\tau(T)$. We will prove the lemma by induction on the number of inner vertices of $T$. Base case: The statement of the lemma clearly holds for claw trees. Induction step: Let $T$ be a tree with $n>1$ inner vertices and $\omega \in \tau(T)$ a degree one labeling. If $T$ has more than one connected component, then by induction $\omega$ restricted to any connected component is a disjoint union of paths. Hence $\omega$ is a disjoint union of paths. If $T$ has one connected component, let $e$ be an inner edge, $e_1,e_2$ new leaf edges obtained by cutting $T$ at $e$ and write $T^e=T_1\sqcup T_2$. Then $\omega $ restricted to either tree is a disjoint union of paths. If $\omega_e=0$, then $\omega$ is the disjoint union of exactly the same paths. If $\omega _e=1$, then the path of $T_1$ containing $e_1$ and the path of $T_2$ containing $e_2$ are combined to one path of $T$ containing $e$. Hence $\omega $ is a disjoint union of paths. \[lemma:networks\_on\_graphs\] Let $G$ be a graph. There is one-to-one correspondence between networks and degree one elements of $\tau(G)$. We will prove the lemma by induction on first Betti number $g$ of $G$. Base case: The statement of the lemma holds for trees by Lemma \[lemma:networks\_on\_trees\]. Induction step: Let $G$ be a graph with first Betti number $g>1$ and $\omega \in \tau(G)$ a degree one labeling. Let $e$ be a cycle edge of $G$ and $e_1,e_2$ new leaf edges obtained by cutting $G$ at $e$. The graph $G^e$ has first Betti number $g-1$. Then $\omega $ gives $\overline{\omega}\in G^e$ that is a disjoint union of paths containing both $e_1,e_2$ or neither of them. If $\omega_e=0$, then $\omega$ is the disjoint union of exactly the same paths. If $\omega _e=1$, then there are two possibilities. Either there is a path in $\overline{\omega}$ with first edge $e_1$ and last edge $e_2$ which lifts to a cycle in $\omega$. Or there is a path in $\overline{\omega}$ with first edge $e'$ and last edge $e_1$, and another path in $\overline{\omega}$ with first edge $e_2$ and last edge $e''$, where $e',e''$ are leaf edges. These paths in $\overline{\omega}$ lift to a single path in $\omega$ with first edge $e$ and last edge $e'$ in $G$. Let $G$ be a graph. All networks are included in the minimal generating set of $\tau(G)$. For any graded affine semigroup $\mathbb{N}\mathcal{A}$ all degree one minimal generators are included in the minimal generating set of $\mathbb{N}\mathcal{A}$. \[thm:buczynska\] Let $G$ be a trivalent graph with first Betti number 1 and $\omega \in \tau(G)$. Then $\omega$ is a minimal generator of $\tau(G)$ if and only if it satisfies one of the following conditions: - $\omega$ is a network, or - $\omega$ has degree two, and satisfies the following three conditions - $\omega _e=1$, for all cycle edges $e$, - $\omega _e=2$, for an odd number of cycle legs, - $\omega _e=0$, for the remaining cycle legs. We give a shortened proof of this theorem. The following lemma will be an important part of it. \[lemma:decomposition\_0\_or\_d\] Let $G$ be a graph with first Betti number 1. Let $\omega \in \tau(G)$ be of degree $d$. If there is a cycle edge $e$ with $\omega _e=0$ or $\omega _e=d$, then $\omega $ decomposes as a sum of degree one elements. Let $e$ be a cycle edge and $e_1,e_2$ new leaf edges obtained by cutting $G$ at $e$. Notice that $G^e$ is a tree. Then $\omega$ gives $\overline{\omega} \in \tau (G^e)$ that decomposes into degree one elements $\overline{\omega} = \overline{\omega _1}+\ldots +\overline{\omega _d}$. Since $(\overline{\omega _i})_{e_1}=(\overline{\omega _i})_{e_2}$ for all $i$, the decomposition $\overline{\omega} = \overline{\omega _1}+\ldots +\overline{\omega _d}$ gives a decomposition $\omega = \omega _1+\ldots +\omega _d$ of $\omega\in \tau(G)$. By Corollary \[lemma:polygon\_graph\] we can assume that $G$ is a trivalent polygon graph. First we prove that any minimal generator of $\tau(G)$ has degree at most two. Let $\omega \in \tau(G)$ be of degree $d$. Let $e$ be a cycle edge and $e_1,e_2$ new leaf edges obtained by cutting $G$ at $e$. Then $\omega$ gives $\overline{\omega} \in \tau (G^e)$ that decomposes as a sum of degree one elements $\overline{\omega} = \overline{\omega _1}+\ldots +\overline{\omega _d}$. If $(\overline{\omega _i})_{e_1}=(\overline{\omega _i})_{e_2}$ then $\overline{\omega _i}$ gives an element $\omega_i\in \tau (G)$. Otherwise there exists $j$ such that $(\overline{\omega _i})_{e_1}=(\overline{\omega _j})_{e_2}$ and $(\overline{\omega _j})_{e_1}=(\overline{\omega _i})_{e_2}$, because $\overline{\omega}_{e_1}=\overline{\omega}_{e_2}$. Thus $\overline{\omega _i}+\overline{\omega _j}$ gives a degree two element $\omega_i+\omega_j \in \tau (G)$. Degree one elements of $\tau (G)$ are networks by Corollary \[lemma:networks\_on\_graphs\]. By Lemma \[lemma:decomposition\_0\_or\_d\], all degree two indecomposable elements $\omega $ have $\omega _e=1$ on all cycle edges $e$. Since $G$ is a trivalent graph, we have $\omega_e\in \{0,2\}$ for all cycle legs because of the parity condition. Assume $\omega _e=2$ for an even number of cycle legs $e_1,\ldots e_{2k}$ in clockwise order. Denote by $P_i$ the path starting at $e_i$ and ending at $e_{i+1}$ (at $e_0$ for $i=2k$). Then $\omega$ decomposes as the sum of networks $P_1\cup P_3 \cup \ldots \cup P_{2k-1}$ and $P_2 \cup P_4 \cup \ldots \cup P_{2k}$. Hence for $\omega$ indecomposable $\omega _e=2$ for an odd number of cycle legs. Conversely, assume that $\omega \in \tau(G)$ has degree two and fulfills $(i),(ii),(iii)$. Suppose $\omega=\omega _1 + \omega _2$, where $\omega _1,\omega _2$ are networks. For all cycle legs $e$ with $\omega_e=2$ we have $(\omega _i)_e=1$, since $(\omega _i)_e\leq 1$ for all edges $e$. Hence $(\omega _i)_e=1$ for odd number of leaves of $G$. But this is contradiction to the fact that $\omega _i$ is a network. We know from [@Buczynska12; @BBKM11] that a minimal generator of the phylogenetic semigroup on a graph with first Betti number 1 has degree at most two. We showed this above to give a simple and self-containing proof. \[cor:first\_betti\_number\_1\] Let $G$ be a graph with first Betti number 1 and $\omega \in \tau(G)$. Then $\omega$ is a minimal generator of $\tau(G)$ if and only if it satisfies one of the following conditions: - $\omega$ is a network, or - $\omega$ has degree two, and satisfies the following three conditions - $\omega_e=1$ for all cycle edges $e$, - $\omega _e=2$, for an odd number of cycle legs, - $\omega _e=0$, for the remaining cycle legs. Let $G'$ be a trivalent graph constructed from $G$ in the following way: Replace all vertices $v$ with valency higher than three by two new vertices $v'$ and $v''$ together with a new edge between them, let two edges incident to $v$ be incident to $v'$ and the rest of the edges incident to $v$ be incident to $v''$. Moreover, if $v$ is on the cycle, let one cycle edge incident to $v$ be incident to $v'$ and let the other cycle edge incident to $v$ be incident to $v''$. This assures that we do not add any cycle legs. After a finite number of replacements we get a trivalent graph $G'$. As in [@BBKM11 Lemma 4.1], $\tau(G)$ is the coordinate projection of $\tau(G')$ that forgets coordinates corresponding to new edges. In particular, if $\omega'\in\tau(G')$ is decomposable, then its projection in $\tau(G)$ is also decomposable. By [@BBKM11], any minimal generator of $\tau(G)$ has degree at most two. Degree one elements are networks. We are left with describing degree two indecomposable elements of $\tau(G)$. A degree two indecomposable element $\omega \in \tau(G)$ is the coordinate projection of a degree two indecomposable element of $\tau(G')$. Since all cycle legs of $G'$ are also cycle legs of $G$, then by Theorem \[thm:buczynska\] the conditions $(i),(ii),(iii)$ are fulfilled for $\omega$. Conversely, assume that the conditions $(i),(ii),(iii)$ are fulfilled. Suppose $\omega=\omega _1 + \omega _2$, where $\omega _1,\omega _2$ are networks. For all cycle legs $e$ with $\omega_e=2$ we have $(\omega _i)_e=1$, since $(\omega _i)_e\leq 1$ for all edges $e$. Hence $(\omega _i)_e=1$ for odd number of leaves of $G'$. But this is a contradiction to the fact that $\omega _i$ is a network. Degree Two Minimal Generators {#section:degree_two_generators} ============================= In this section, we describe degree two indecomposable labelings for any trivalent graph $G$. \[cor:degree\_2\_generators\] Let $G$ be any graph and $\omega \in \tau(G)$ a degree two labeling. If there exists a cycle $G'$ of $G$ such that $\omega_e=1$ for all cycle edges $e\in G'$, $\omega_e=2$ for an odd number cycle legs $e$ of $G'$ and $\omega_e=0$ for the remaining cycle legs $e\in G'$, then the labeling $\omega$ is indecomposable. If $\omega$ decomposes, then a decomposition of $\omega$ restricts to a decomposition of $\omega |_{G'}\in \tau(G')$, where $G'$ is a cycle together with its cycle legs. Thus the statement follows from Corollary \[cor:first\_betti\_number\_1\]. \[prop:degree\_2\_generators\] Let $G$ be a trivalent graph and $\omega \in \tau(G)$ a degree two labeling. The labeling $\omega$ is indecomposable if and only if there exists a cycle $G'$ of $G$ together with its cycle legs such that $\omega |_{G'}\in \tau(G')$ is indecomposable. One direction follows from Lemma \[cor:degree\_2\_generators\]. We show by induction on first Betti number of $G$ that if $\omega \in \tau(G)$ is a degree two indecomposable labeling then there exists a cycle $G'$ together with its cycle legs such that $\omega|_{G'}$ is a degree two indecomposable labeling. Base case: If first Betti number of $G$ is 1, then the statement follows from Theorem \[thm:buczynska\]. Induction step: Assume that first Betti number of $G$ is $g>1$. If more than one connected component of $G$ contains a cycle, then there exists a connected component $C$ of $G$ containing a cycle such that $\omega |_{C}\in \tau(C)$ is an indecomposable element of degree two. Since first Betti number of $C$ is less than $g$, we know by induction that there exists a cycle $G'$ of $C$ together with its cycle legs such that $\omega |_{G'}\in \tau(G')$ is an indecomposable element of degree two. Otherwise all cycles of $G$ live in the same connected component of $G$. If $\omega_e=1$ for all cycle edges $e$, then by the parity condition $\omega_e\in\{0,2\}$ for all cycle legs $e$. In particular, none of the cycle legs is simultaneously a cycle edge and there exists a cycle leg $e$ that separates some cycles of $G$. Let $e_1,e_2$ be the new leaf edges obtained by cutting $G$ at $e$ and write $G^e=G_1\sqcup G_2$. Then $\omega$ gives $\omega_1 \in \tau(G_1)$ and $\omega_2 \in \tau(G_2)$ with at least one of them indecomposable, otherwise one could lift these decompositions to a decomposition of $\omega$. By induction, for $i$ with $\omega_i$ indecomposable there exists a cycle $G'$ of $G_i$ together with its cycle legs such that $\omega_i |_{G'}\in \tau(G')$ is indecomposable. Thus $\omega|_{G'} \in \tau(G')$ is indecomposable. If there exists a cycle edge $e$ with $\omega_e\in \{0,2\}$, then let $e_1$ and $e_2$ be new leaf edges obtained by cutting $G$ at $e$. The labeling $\omega\in \tau(G)$ gives a labeling $\overline{\omega}\in \tau(G^e)$ that is indecomposable. Otherwise one could lift a decomposition $\overline{\omega}=\overline{\omega}_1+\overline{\omega}_2$ to a decomposition $\omega=\omega_1+\omega_2$, because $(\overline{\omega_i})_{e_1}=(\overline{\omega_i})_{e_2}$. The graph $G^e$ has first Betti number less than $g$. By induction, there exists a cycle $G'$ of $G^e$ together with cycle legs such that $\overline{\omega}|_{G'}\in \tau(G')$ is indecomposable. Thus $\omega|_{G'} \in \tau(G')$ is indecomposable. Graphs with First Betti Number 2 {#section:first_betti_number_two} ================================ We know from [@BBKM11] that any minimal generator of the phylogenetic semigroup on a graph with first Betti number 2 has degree at most three. In this section, we will explicitly describe which phylogenetic semigroups have which maximal degrees of minimal generating sets for graphs with first Betti number 2. We will see that there are graphs with maximal degrees of minimal generators equal to one, two and three. Our analysis is based on five different cases depending on the structure of the graph - whether the cycles live in different components of the graph, share at least one edge, share exactly a single vertex, there is a single edge connecting the cycles, or the cycles are more than one edge apart from each other, see Figure \[figure:graphs\_with\_first\_betti\_number\_2\_cases\] for latter four cases. Assume a graph has a degree two vertex $v$. Denote the edges incident to $v$ by $e_1$ and $e_2$. By the definition of the phylogenetic semigroup on a graph, we have $\omega _{e_1}=\omega _{e_2}$ for $\omega \in \tau(G)$. Hence elements of $\tau(G)$ are in one-to-one correspondence with elements of $\tau(G')$, where $G'$ is obtained from $G$ by replacing $e_1$ and $e_2$ by a single edge. To simplify future analysis, from now on we will assume that graphs posses no degree two vertices. \[thm:graphs\_with\_first\_betti\_number\_2\] Let $G$ be a graph with first Betti number 2. The maximal degree of a minimal generator of $\tau(G)$ is - one if and only if $G$ does not contain any cycle legs that are not cycle edges; - two if and only if $G$ the cycles of $G$ live in different connected components, or $G$ contains at least one cycle leg that is not a cycle edge, all cycles of $G$ live in the same connected component and are not separated by an inner vertex; - three if and only if the minimal cycles of $G$ live in the same connected component and are separated by at least one inner vertex; We will study these different cases in Lemmas \[lemma:graph1\]–\[lemma:graph5\]. \[lemma:graph1\] Let $G$ be a graph with first Betti number 2 that does not contain any cycle legs that are not cycle edges. The maximal degree of a minimal generator of $\tau(G)$ is one. The cycles of $G$ live in the same connected component. Otherwise $G$ would have a degree two vertex. If the connected component of $G$ containing the cycles has one vertex, then it is isomorphic to the right graph in Figure \[figure:graph\_with\_three\_edges\]. If the connected component of $G$ containing the cycles has two vertices, then it is isomorphic to the left graph in Figure \[figure:graph\_with\_three\_edges\]. The connected component of $G$ containing the cycles cannot have three or more vertices, because every vertex must belong to at least two cycles. By computations with `Normaliz` [@normaliz], the phylogenetic semigroup of the left graph in Figure \[figure:graph\_with\_three\_edges\] is $$\mathbb{N} \{(1,0,0,0),(1,1,1,0),(1,1,0,1),(1,0,1,1)\},$$ where the first coordinate corresponds to the degree and the other three coordinates correspond to edges of $G$ in any fixed order. By simple observation, the phylogenetic semigroup of the right graph in Figure \[figure:graph\_with\_three\_edges\] is $$\mathbb{N} \{(1,0,0),(1,1,0),(1,0,1)\},$$ where the first coordinate corresponds to the degree and the other two coordinates correspond to edges of $G$ in any fixed order. \[lemma:graph2\] Let $G$ be a graph with first Betti number 2 and cycles living in different connected components. The maximal degree of a minimal generator of $\tau(G)$ is two. Define $\omega \in \tau(G)$ of degree two as follows: $\omega _e=1$ for all cycle edges $e$ of a cycle $G'$ of $G$, $\omega _e=2$ for one cycle leg of $G'$, and $\omega _e=0$ for all other cycle legs of $G'$. Extend this partial labeling of $G$ in any feasible way to a degree two labeling of $G$. By Lemma \[cor:degree\_2\_generators\], $\omega$ is indecomposable. Hence the maximal degree of a minimal generator of $\tau(G)$ is at least two. On the other hand, we show that every element $\omega \in \tau(G)$ can be decomposed as a sum of degree one and degree two elements. By Corollary \[cor:first\_betti\_number\_1\], $\omega$ restricted to each connected component decomposes as a sum of degree one and degree two elements. These decompositions can be combined to a decomposition of $\omega\in \tau(G)$ as a sum of degree one and degree two elements. Hence the maximal degree of a minimal generator of $\tau(G)$ is exactly 2. Let $G$ be a trivalent graph and $v$ be an inner vertex of $G$. Every element $\omega\in\tau(G)$ decomposes locally in a unique way into paths around an inner vertex $v$. This means that there exist non-negative integers $x_v(\omega)$, $y_v(\omega)$, $z_v(\omega)$ such that $$\begin{aligned} a_v(\omega)=y_v(\omega)+z_v(\omega),\\ b_v(\omega)=x_v(\omega)+z_v(\omega),\\ c_v(\omega)=x_v(\omega)+y_v(\omega), \end{aligned}$$ and $x_v(\omega)+y_v(\omega)+z_v(\omega)\leq \textrm{deg}(\omega)$, see also Figure \[fig\_building\_graph\_2\]. Let $T$ be a trivalent tree and $\omega_1,\omega _2\in \tau(T)$ networks. Let $v$ be an inner vertex of $T$. Then either $a_v(\omega_1)=a_v(\omega_2)$, $b_v(\omega_1)=b_v(\omega_2)$ or $c_v(\omega_1)=c_v(\omega_2)$, since $a_v(\omega_i)+b_v(\omega_i)+c_v(\omega_i)\in\{0,2\}$ for $i=1,2$. We denote this edge by $e$. By exchanging values of $\omega_1$ and $\omega_2$ on all edges of $T$ that are on the same side with $e$ from $v$, we get $\omega_1',\omega_2'\in \tau(T)$ such that $\omega_1+\omega_2=\omega_1'+\omega_2'$. We call this operation *branch swapping*. \[lemma:graph3\] Let $G$ be a graph with first Betti number 2 containing at least one cycle leg that is not a cycle edge and two cycles sharing at least one edge. The maximal degree of a minimal generator in $\tau(G)$ is two. By Corollary \[lemma:polygon\_graph\], we can assume that $G$ is a multiple polygon graph. There is at least one cycle leg $e'$ of $G$ that is not a cycle edge for any of the cycles of $G$. Assume that $e'$ is a cycle leg of a cycle $G'$. Define $\omega $ of degree two as follows: $\omega _e=1$ for all cycle edges $e$ of $G'$, $\omega _{e'}=2$, and $\omega _e=0$ for all other edges $e$ of $G$. By Lemma \[cor:degree\_2\_generators\], the labeling $\omega \in \tau(G)$ is indecomposable. Hence the maximal degree of a minimal generator is at least two. On the other hand, we show that every element $\omega \in \tau(G)$ can be decomposed as a sum of degree one and degree two elements. If $G$ is not trivalent, then by [@BBKM11 Lemma 4.1] we can construct a trivalent graph $G'$ with first Betti number 2 such that the maximal degree of the minimal generating set of $\tau(G)$ is less or equal than the one of $\tau(G')$. Moreover, two cycles of $G'$ share an edge. Hence we can assume that $G$ is a trivalent graph. If there is a cycle edge $e$ of $G$ with $\omega_e\in\{0,\textrm{deg}(\omega)\}$, we construct the graph $G^e$ with first Betti number 1 by cutting $G$ at $e$. Denote the new leaf edges by $e_1$ and $e_2$. The labeling $\omega$ gives a labeling $\overline{\omega}$ of $G^e$. By Theorem \[thm:buczynska\], the labeling $\overline{\omega}$ can be decomposed as a sum of degree one and two labelings $$\overline{\omega} =\sum _{i=1}^{\deg(\omega)}\overline{\omega _i},$$ where $$(\overline{\omega_i})_{e_1}=(\overline{\omega_i})_{e_2}=\left\{ \begin{array}{ll} 0 & \textrm{if } \omega_e=0\\ \textrm{deg}(\overline{\omega_i}) & \textrm{if } \omega_e=\textrm{deg}(\omega) \end{array} \right.$$ Hence the decomposition of $\overline{\omega}$ gives a decomposition of $\omega$ with all labelings having degree one or two. From now on we assume that there is no cycle edge $e$ of $G$ with $\omega_e\in\{0,\textrm{deg}(\omega)\}$. There are exactly two vertices of $G$ incident to three cycle edges. We denote them by $u$ and $v$. We construct a tree $G'$ from $G$ by replacing the vertex $u$ with three new vertices $u_1,u_2$ and $u_3$ as in Figure \[figure:replace\_vertex\_with\_three\_vertices\]. The labeling $\omega$ gives a labeling $\omega'$ of $G'$. Abusing the notation slightly, we denote by $a_u(\omega'),b_u(\omega'),c_u(\omega')$ the coordinates of $\omega'$ corresponding to leaf edges with endpoints $u_1,u_2,u_3$, respectively. The labeling $\omega'$ can be decomposed as a sum of degree one labelings $$\omega' =\sum _{i=1}^{\deg(\omega)}\omega' _i.$$ From this we want to construct a decomposition of $\omega\in\tau(G)$. To lift an element of $\tau(G')$ to an element of $\tau(G)$, the parity and the degree condition have to be satisfied at leaf edges with endpoints $u_1,u_2,u_3$. This is not true for all $\omega' _i$. We need to combine and alter these elements. We will use local paths to assure the parity and degree conditions are satisfied. We will construct the decomposition of $\omega \in \tau(G)$ iteratively. In each step we construct a degree one or two element $\omega^*$ and then take $\omega:=\omega-\omega^*$. Case 1. $\textrm{deg}_u(\omega)=\textrm{deg}(\omega)$. Note that $x_u(\omega),y_u(\omega),z_u(\omega)\geq 1$, otherwise there would be a cycle edge $e$ of $G$ with $\omega_e=\textrm{deg}(\omega)$. - If there is $\omega'_i$ with exactly two of $a_u(\omega'_i),b_u(\omega'_i),c_u(\omega'_i)$ equal to 1, then $\omega'_i$ can be lifted to a degree one labeling of $G$. - Otherwise if there is $\omega'_i$ with exactly one of $a_u(\omega'_i),b_u(\omega'_i),c_u(\omega'_i)$ equal to 1, then there is $\omega'_j$ with all of $a_u(\omega'_j),b_u(\omega'_j),c_u(\omega'_j)$ equal to 1. Then $\omega'_i+\omega'_j$ can be lifted to a degree two labeling of $G$. - Otherwise there has to be $\omega'_i$ with all of $a_u(\omega'_i),b_u(\omega'_i),c_u(\omega'_i)$ equal to 0. Then there is $\omega'_j$ with all of $a_u(\omega'_j),b_u(\omega'_j),c_u(\omega'_j)$ equal to 1. After branch swapping of $\omega'_i$ and $\omega'_j$ at $v$, we get a labeling with exactly two values corresponding to $a_u,b_u,c_u$ equal to 1. It can be lifted to a degree on labeling of $G$. Case 2. $\textrm{deg}_u(\omega)<\textrm{deg}(\omega)$. - If there exists $\omega' _i$ with $a_u(\omega'_i)=b_u(\omega'_i)=c_u(\omega'_i)=0$, then $\omega' _i$ lifts to a labeling of $\tau(G)$. Otherwise consider two subcases: Case 2.1. $x_u(\omega),y_u(\omega),z_u(\omega)\geq 1$. - If there is $\omega'_i$ with exactly two of $a_u(\omega'_i),b_u(\omega'_i),c_u(\omega'_i)$ equal to 1, then $\omega'_i$ can be lifted to a degree one labeling of $G$. - Otherwise if there is $\omega'_i$ with all of $a_u(\omega'_i),b_u(\omega'_i),c_u(\omega'_i)$ equal to 1, then there is $\omega' _j$ with exactly one of $a_u(\omega'_j),b_u(\omega'_j),c_u(\omega'_j)$ equal to 1. Then $\omega'_i+\omega'_j$ can be lifted to a degree two labeling of $G$. - Otherwise all $\omega'_i$ have exactly one of $a_u(\omega'_i),b_u(\omega'_i),c_u(\omega'_i)$ equal to 1. Since $x_u(\omega)\geq 1$ there is $\omega' _i$ with $a_u(\omega' _i)=c_u(\omega' _i)=0$ and $b_u(\omega' _i)=1$ and $\omega' _j$ with $a_u(\omega' _j)=b_u(\omega' _j)=0$ and $c_u(\omega' _j)=1$. Then $\omega' _i+\omega' _j$ can be lifted to a degree two labeling for $G$. Case 2.2. Exactly two of $x_u(\omega),y_u(\omega),z_u(\omega)\geq 1$. It is not possible to have only one $x_u(\omega),y_u(\omega),z_u(\omega)\geq 1$, because we assumed $\omega_e> 0$ for every cycle edge $e$. We assume that $x_u(\omega),y_u(\omega)\geq 1$, the other two cases are analogous. - If there is $\omega'_i$ with exactly $b_u(\omega'_i),c_u(\omega'_i)$ or $a_u(\omega'_i),c_u(\omega'_i)$ equal to 1, then $\omega'_i$ can be lifted to a degree one labeling of $G$. - Otherwise if there is $\omega'_i$ with exactly $a_u(\omega'_i),b_u(\omega'_i)$ equal to 1, there is $\omega' _j$ with exactly $c_u(\omega'_i)$ equal to 1, since $c_u(\omega)>a_u(\omega)$ and $c_u(\omega)>b_u(\omega)$. After branch swapping $\omega' _i$ and $\omega' _j$ at $v$, we either get a labeling with all values corresponding to $a_u,b_u,c_u$ equal to 0 or a labeling with values corresponding to $b_u,c_u$ equal to 1 or a labeling with values corresponding to $a_u,c_u$ equal to 1. They all can be lifted to a degree one labeling of $G$. - Otherwise if there is $\omega'_i$ with all of $a_u(\omega'_i),b_u(\omega'_i),c_u(\omega'_i)$ equal to 1, there is $\omega' _j$ with exactly $c_u(\omega'_j)$ equal to 1, since $c_u(\omega)>a_u(\omega)$ and $c_u(\omega)>b_u(\omega)$. Then $\omega_i+\omega_j$ can be lifted to a degree two labeling of $G$. - Otherwise all $\omega'_i$ have exactly one of $a_u(\omega'_i),b_u(\omega'_i),c_u(\omega'_i)$ equal to 1. Since $x_u(\omega)\geq 1$ there is $\omega' _i$ with $a_u(\omega' _i)=c_u(\omega' _i)=0$ and $b_u(\omega' _i)=1$, and $\omega' _j$ with $a_u(\omega' _j)=b_u(\omega' _j)=0$ and $c_u(\omega' _j)=1$. Then $\omega' _i+\omega' _j$ can be lifted to a degree two labeling for $G$. At each step a degree one or two element is constructed. This assures that the iterative process comes to an end, because the degree of $\omega$ decreases. \[lemma:graph3b\] Let $G$ be a graph with first Betti number 2 containing at least one cycle leg that is not a cycle edge and two cycles sharing exactly one vertex. The maximal degree of a minimal generator in $\tau(G)$ is two. By Corollary \[lemma:polygon\_graph\], we can assume that $G$ is a multiple polygon graph. There is at least one cycle leg $e'$ of $G$ that is not a cycle edge for any of the cycles of $G$. Assume that $e'$ is a cycle leg of a cycle $G'$. Define $\omega $ of degree two as follows: $\omega _e=1$ for all cycle edges $e$ of $G'$, $\omega _{e'}=2$ and $\omega _e=0$ for all other edges $e$ of $G$. By Lemma \[cor:degree\_2\_generators\], the labeling $\omega \in \tau(G)$ is indecomposable. Hence the maximal degree of a minimal generator is at least two. On the other hand, we show that every element $\omega \in \tau(G)$ can be decomposed as a sum of degree one and degree two elements. We construct a trivalent graph $G'$ from $G$ as in [@BBKM11 Lemma 4.1] such the that the maximal degree of the minimal generating set of $\tau(G)$ is less or equal than the one of $\tau(G')$. In particular, first we decrease the valency at the vertex $v$ that is on both cycles. We replace it by vertices $v',v''$ and an edge $e$ between them such that $e$ belongs to both cycles. We repeat replacing vertices until there are only trivalent vertices left. The graph $G'$ has two cycles that share at least one edge, thus we can apply Lemma \[lemma:graph3\]. \[lemma:graph4\] Let $G$ be a graph with first Betti number 2 where the two cycles are separated by a single edge $e$. The maximal degree of a minimal generator of $\tau(G)$ is two. Define $\omega \in \tau(G)$ of degree two as follows: $\omega _e=1$ for all cycle edges $e$, $\omega _e=2$ for the single edge separating cycles, and $\omega _e=0$ for all other edges. By Lemma \[cor:degree\_2\_generators\], $\omega$ is indecomposable. Hence the maximal degree of a minimal generator of $\tau(G)$ is at least two. On the other hand, we show that every element $\omega \in \tau(G)$ can be decomposed as a sum of degree one and degree two elements. If $G$ is not trivalent, then by [@BBKM11 Lemma 4.1] we can construct a trivalent graph $G'$ with first Betti number 2 such that the maximal degree of the minimal generating set of $\tau(G)$ is less or equal than the maximal degree of the minimal generating set of $\tau(G')$. Moreover, we may assume that every time we replace a vertex $v$ on a cycle by vertices $v',v''$ and an edge between them, then $v',v''$ belong to the same cycle. This assures that the two cycles of $G'$ are separated by a singe edge. Hence we can assume that $G$ is a trivalent graph. Let $e_1,e_2$ be new leaf edges obtained by cutting $G$ at $e$ and write $G^e=G_1\sqcup G_2$. The labeling $\omega$ gives labelings $\omega_1$ of $G_1$ and $\omega_2$ of $G_2$. By Corollary \[cor:first\_betti\_number\_1\], we can decompose $\omega_1$ and $\omega_2$ as a sum of degree one and degree two elements. Because all degree two labelings in these decompositions have values 0 or 2 corresponding to the edges $e_1$ and $e_2$, we can combine decompositions of $\omega_1$ and $\omega_2$ to get a decomposition of $\omega$ that consists of degree one and two elements. Hence the maximal degree of a minimal generator of $\tau(G)$ is exactly two. \[lemma:graph5\] Let $G$ be a graph with first Betti number 2 where the two cycles are separated by at least one inner vertex. The maximal degree of a minimal generator of $\tau(G)$ is three. By Corollary \[lemma:polygon\_graph\], we can assume that $G$ is a multiple polygon graph. We need to specify a degree three indecomposable element $\omega \in \tau(G)$. Fix an inner vertex $v$ on the path between the two cycles of $G$ and an edge $e^*$ incident to $v$ that is not on the path between the two cycles. Define $\omega _e=2$ for all cycle edges $e$ and all edges $e$ on the path between the cycles of $G$, $\omega _{e^*}=2$, and $\omega _e=0$ for all other edges $e$. We will show that $\omega $ is indecomposable as a degree three labeling. By contradiction, assume $\omega=\omega_1+\omega _2$, where deg$(\omega_1)=1$ and deg$(\omega_2)=2$. We must have $(\omega_2)_e=1$ for all cycle edges of $G$ and $(\omega_2)_e=2$ for both cycle legs $e$ that lie on the path between the two cycles. Hence also $(\omega _2)_e=2$ for all edges $e$ that lie on the path between the two cycles. Thus $(\omega _1)_{e^*}=2$ . This leads to a contradiction, because deg$(\omega _1)=1$. Hence $\omega$ is a degree three indecomposable element in $\tau(G)$. Theorem follows from Lemmas \[lemma:graph1\]–\[lemma:graph5\]. Degree Three Minimal Generators {#section:degree_three_generators} =============================== In this section, we describe degree three minimal generators of phylogenetic semigroups on trivalent graphs with first Betti number 2 having at least one inner vertex between two cycles. Together with Lemma \[lemma:networks\_on\_graphs\], Proposition \[prop:degree\_2\_generators\] and Theorem \[thm:graphs\_with\_first\_betti\_number\_2\] this completely characterizes minimal generating sets of phylogenetic semigroups on trivalent graphs with first Betti number 2. Let $G$ be a trivalent graph with first Betti number 1 and $\omega$ a degree two indecomposable labeling. Label the cycle legs of $G$ where $\omega$ has value two by $e_0,\ldots ,e_{2k}$ in clockwise order. Slightly abusing notation, we write $e_{i+j}$ for $e_{i+j \mod 2k}$ where $i+j>2k$. Label by $P_{e',e''}$ the path starting at a cycle leg $e'$ and going in the clockwise direction until reaching a cycle leg $e''$ . Write $P_i$ for $P_{e_i,e_{i+1}}$. We say a cycle leg $e$ is between cycle legs $e_i$ and $e_j$, when $e$ is between cycle legs $e_i$ and $e_j$ in clockwise direction. \[lemma:decompositions\_of\_degree\_three\_labelings\] Let $G$ be a trivalent polygon graph and $\omega \in \tau(G)$ a degree three labeling. Then $\omega $ cannot be decomposed as a sum of degree one labelings if and only if $\omega =\omega _1+\omega_2$ such that - $\deg(\omega_1)=1$ and $\deg(\omega_2)=2$, - $\omega_2$ is indecomposable with value two on cycle legs $e_0,\ldots ,e_{2k}$ and value zero on all other cycle legs, - $\omega_1$ is $P_0\cup P_2 \cup \ldots \cup P_{2k-2}$, $P_0 \cup P_2 \cup \ldots P_{2k-4} \cup P_{e_{2k-2},e_{2k}}$, $P_{0}\cup P_2 \cup \ldots P_{2k-2} \cup P_{e_{2k},e'}$ where $e'$ is a cycle leg between $e_{2k}$ and $e_{0}$ or also the cycle path if $k=0$. A degree three labeling $\omega$ can be always decomposed as $\omega=\omega_1+\omega_2$ with $\deg(\omega_1)=1$ and $\deg(\omega_2)=2$. We show that unless $\omega_1,\omega_2$ are as in the statement of the lemma, we can alter $\omega_1,\omega_2$ to get $\omega'_1,\omega'_2$ such that $\omega_1+\omega_2=\omega'_1+\omega'_2$ and $\omega'_2$ decomposes as a sum of two degree one labelings. If there is $P_i$ such that it does not intersect the network $\omega_1$, then the union of $\omega_1$ and $P_i$ is a network and the complement of $P_i$ in $\omega_2$ decomposes as the sum of $P_{i+1}\cup P_{i+3} \cup \ldots \cup P_{i-2}$ and $P_{i+2}\cup P_{i+4} \cup \ldots \cup P_{i-1}$. We will assume from now on that every $P_i$ intersects $\omega_1$. If there exist $e_i$ and $e_j$ such that neither of them is incident to a path in $\omega_1$, then either $e_j=e_{i+2l}$ or $e_i=e_{j+2l}$ for some $1\leq l \leq k$. In the first case, let $\Gamma$ be the union of paths in $\omega_1$ from $e_i$ to $e_j$ and define $\omega'_1=\omega_1 \backslash \Gamma \cup P_{i} \cup P_{i+2} \cup \ldots \cup P_{j-2}$ and $\omega'_2=\omega_2 \backslash (P_{i} \cup P_{i+2} \cup \ldots \cup P_{j-2}) \cup \Gamma$. Then $\omega'_2$ decomposes as the sum $\Gamma\cup P_{j} \cup P_{j+2} \cup \ldots \cup P_{i-1}$ and $P_{j+1}\cup P_{j+3}\cup \ldots \cup P_{i-2}\cup P_{i+1} \cup P_{i+3} \cup \ldots \cup P_{j-1}$. In the second case, the same discussion applies for $i$ and $j$ exchanged. We will assume from now on that there is at most one $e_i$ that is not incident to a path in $\omega_1$. If $\omega _1$ corresponds to the cycle path $P_{\textrm{cycle}}$ and $k\geq 1$, then $\omega$ decomposes as the sum of $P_{e_2,e_1}$ and $P_{e_0,e_2}\cup P_3 \cup P_5 \cup \ldots \cup P_{2k-1}$ and $P_{e_1,e_3}\cup P_4 \cup P_6 \cup \ldots \cup P_{2k}$. Here we use that $P_{\textrm{cycle}}\cup P_0 \cup P_1 \cup P_2 = P_{e_0,e_2} \cup P_{e_1,e_3} \cup P_{e_2,e_1}$. It there is a path $P_{e',e''}$ in $\omega_1$ such that not both $e',e''$ belong to $\{e_0,\ldots ,e_{2k}\}$, then we consider five different cases: - If there is a path $P_i$ such that $e',e''$ are both between $e_i$ and $e_{i+1}$, define $\omega_1'=\omega_1\backslash P_{e',e''}$ and $\omega_2''=\omega_2\cup P_{e',e''}$. Since $P_{e',e''}\cup P_i=P_{e_i,e''}\cup P_{e',e_{i+1}}$, the labeling $\omega'_2$ decomposes as the sum of $P_{e_i,e''}\cup P_{i+1}\cup P_{i+3}\cup \ldots \cup P_{i-2}$ and $P_{e',e_{i+1}}\cup P_{i+2}\cup P_{i+4}\cup \ldots \cup P_{i-1}$. - If there is a path $P_{i}$ such that $e'$ is before $e_i$ and $e''$ is between $e_{i}$ and $e_{i+1}$, define $\omega'_1=\omega_1\backslash P_{e',e''} \cup P_{e',e_i}$ and $\omega'_2=\omega_2 \backslash (P_{i-1}\cup P_i) \cup P_{e_{i-1},e_{i+1}} \cup P_{e_i,e''}$. Then $\omega_1+\omega_2=\omega'_1+\omega'_2$, since $P_{e',e''}\cup P_{i-1}\cup P_i=P_{e',e_i}\cup P_{e_{i-1},e_{i+1}} \cup P_{e_i,e''}$. Then $\omega'_2$ decomposes as the sum of $P_{e_{i-1},e_{i+1}}\cup P_{i+2},\ldots \cup P_{{i-3}}$ and $P_{e_i,e''}\cup P_{i+1} \cup \ldots \cup P_{i-2}$. - If there are paths $P_{e_i,e'}$ and $P_{e'',e_{j}}$ in $\omega_1$ such that $e'$ is between $e_i$ and $e_{i+1}$, and $e''$ is between $e_{j-1}$ and $e_{j}$ and $e_j=e_{i+2l+1}$ for some $0\leq l\leq k-1$, let $\Gamma$ be the union of paths between $e_i$ and $e_j$. Define $\omega'_1=\omega_1\backslash \Gamma \cup P_i \cup P_{i+2}\cup \ldots \cup P_{j-1}$ and $\omega'_2=\omega_2 \backslash (P_i \cup P_{i+2}\cup \ldots \cup P_{j-1}) \cup \Gamma$. Then $\omega'_2$ decomposes as the sum of $P_{e_i,e'}\cup P_{i+1} \cup P_{i+3} \cup \ldots \cup P_{i-2}$ and $\Gamma \backslash P_{e_i,e'} \cup P_{j+1} \cup P_{j+3} \cup \ldots \cup P_{i-1}$. - If there are paths $P_{e_i,e'}$ and $P_{e'',e_{j}}$ in $\omega_1$ such that the edge $e'$ is between $e_i$ and $e_{i+1}$, the edge $e''$ is between $e_{j-1}$ and $e_{j}$ and $e_j=e_{i+2l}$ for some $1\leq l\leq k$, let $\Gamma$ be the union of paths between $e_j$ and $e_i$ together with $P_{e_i,e'}$ and $P_{e'',e_{j}}$. Define $\omega'_1=\omega_1\backslash \Gamma \cup P_j \cup P_{j+2} \cup \ldots \cup P_{i-1}$ and $\omega'_2=\omega_2\backslash (P_j \cup P_{j+2} \cup \ldots \cup P_{i-1}) \cup \Gamma$. Then $\omega'_2$ decomposes as the sum of $\Gamma \backslash P_{e_i,e'} \cup P_i \cup P_{i+2} \cup \ldots \cup P_{j-2}$ and $P_{e_i,e'}\cup P_{i+1} \cup P_{i+3} \cup \ldots \cup P_{i-2}$. - If there are paths $P_{e_i,e'}$ and $P_{e_j,e''}$ in $\omega _1$ such that the edge $e'$ is between $e_i$ and $e_{i+1}$, the edge $e''$ is between $e_j$ and $e_{j+1}$ and $e_j=e_{i+2l}$ for some $1\leq l\leq k$, let $\Gamma$ be the union of paths in $\omega_1$ between $e_i$ and $e''$ without $P_{e_i,e'}$. Define $\omega'_1=\omega_1 \backslash \Gamma \cup P_{i+1} \cup P_{i+3} \cup \ldots \cup P_{j-1}$ and $\omega'_2=\omega_2\backslash (P_{i+1} \cup P_{i+3} \cup \ldots \cup P_{j-1}) \cup \Gamma$. Then $\omega'_2$ decomposes as the sum of $P_{e_j,e''}\cup P_{j+1} \cup P_{j+3} \cup \ldots \cup P_{j-2}$ and $P_{j} \cup P_{j+2} \cup \cdots \cup P_{i-1} \cup \Gamma \backslash P_{e_j,e''}$. If $e_j=e_{i+2l+1}$ for some $0\leq l\leq k-1$ then the same discussion works for $i$ and $j$ exchanged. If none of the five if-conditions holds, then the unique path of the form $P_{e',e''}$ in $\omega_1$ such that not both $e',e''$ belong to $\{e_0,\ldots ,e_{2k}\}$ must be $P_{e_i,e'}$ or $P_{e'',e_{i+1}}$ with $e,'e''$ between $e_i$ and $e_{i+1}$. If there is a path in $\omega_1$ of the form $P_{e_i,e_j}$, then we consider three different cases: - If there exists $P_{e_i,e_j}$ with $e_j=e_{i+2l+1}$ for $1\leq l \leq k-1$, then define $\omega'_1=\omega_1\backslash P_{e_i,e_j} \cup P_i \cup P_{i+2} \cup \ldots \cup P_{j-1}$ and $\omega'_2=\omega_2\backslash (P_i \cup P_{i+2} \cup \ldots \cup P_{j-1}) \cup P_{e_i,e_j}$. Since $P_{e_i,e_j}\cup P_{j-2}=P_{e_i,e_{j-1}}\cup P_{e_{j-2},e_j}$, then $\omega'_2$ decomposes as the sum of $P_{e_i,e_{j-1}}\cup P_j \cup P_{j+2} \cup \ldots \cup P_{i-2}$ and $P_{i+1} \cup P_{i+3} \cup \ldots \cup P_{j-4}\cup P_{e_{j-2},e_j}\cup P_{j+1} \cup P_{j+3} \cup \ldots \cup P_{i-1}$. - If there exists $P_{e_i,e_j}$ with $e_j=e_{i+2l}$ for $2\leq l \leq k$, assume that $j=0$. Define $\omega'_1=\omega_1 \backslash P_{e_i,e_j} \cup P_{e_i,e_{j-2}} \cup P_{j-1}$ and $\omega'_2=\omega_2 \backslash (P_{j-3} \cup P_{j-2} \cup P_{j-1}) \cup P_{j-3,j-1} \cup P_{j-2,j}$. Then $\omega_1+\omega_2=\omega'_1+\omega'_2$ since $P_{e_i,e_j}\cup P_{j-3}\cup P_{j-2} \cup P_{j-1}=P_{e_i,e_{j-2}} \cup P_{j-3,j-1} \cup P_{j-2,j}\cup P_{j-1}$. Then $\omega'_2$ decomposes as the sum of $P_0 \cup P_2 \cup \ldots \cup P_{j-3,j-1}$ and $P_1 \cup P_3 \cup \ldots \cup P_{j-2,j}$. - If there exist $P_{e_{i_1},e_{i_2}},P_{e_{j_1},e_{j_2}}$ such that $e_{i_2}=e_{i_1+2l},e_{j_2}=e_{j_1+2m}$ for $1\leq l,m \leq k$ and $e_{j_2}=e_{i_1+2n+1}$ for some $0\leq n\leq k-1$, assume that $P_{j_1,j_2}$ is the next path with such property after $P_{i_1,i_2}$ in clockwise direction. Denote all paths between $e_{i_1}$ and $e_{j_2}$ in $\omega_1$ by $\Gamma$. Define $\omega'_1=\omega_1 \backslash \Gamma \cup P_{i_1} \cup P_{i_1+2} \cup \ldots \cup P_{j_2-1}$ and $\omega'_2=\omega_2 \backslash (P_{i_1} \cup P_{i_1+2} \cup \ldots \cup P_{j_2-1}) \cup \Gamma$. Then $\omega'_2$ decomposes as the sum of $\Gamma \backslash \ P_{e_{j_1},e_{j_2}} \cup P_{j_1} \cup P_{j_1+2} \cup \cdots \cup P_{i_1-2}$ and $P_{i_1+1} \cup P_{i_1+3} \cup \ldots \cup P_{j_1-2} \cup P_{e_{j_1},e_{j_2}} \cup P_{j_2+1} \cup \ldots \cup P_{i_1-1}$. If $e_{j_2}=e_{i_1+2n}$ for some $1\leq n\leq k-1$ then the same discussion works for $i$’s and $j$’s exchanged. If none of the three if-conditions holds, then only paths in $\omega_1$ of the form $P_{e_i,e_j}$ can be $P_i$ and at most one $P_{e_j,e_{j+2}}$. Finally we have to show that $\omega_1$ cannot simultaneously contain paths $P_{e_i,e'}$ and $P_{e_j,e_{j+2}}$ where $i,j\in\{0,\ldots ,2k\}$ and $e'$ is between $e_i$ and $e_{i+1}$. We consider two different cases: $e_j=e_{i+2l}$ for $1\leq l \leq k-1$ and $e_j=e_{i+2l+1}$ for $0\leq l \leq k-1$. If we have $P_{e',e_i}$ instead of $P_{e_i,e'}$ then we can apply the same discussion in the counterclockwise direction. In the first case there must be $e_t$ with $t=i+2l+1$ between $e'$ and $e_j$ that is not incident to any of the paths in $\omega_1$. Otherwise paths between $e'$ and $e_j$ in $\omega_1$ would be $P_{i+1},P_{i+3},\ldots,P_{j-1}$, which is not possible, since $P_{e_j,e_{j+2}}$ is in $\omega _1$. Let $\Gamma$ be the union of paths in $\omega_1$ between $e_i$ and $e_t$. Define $\omega'_1=\omega_1 \backslash \Gamma \cup P_{i} \cup P_{i+2} \cup \ldots \cup P_{t-1}$ and $\omega'_2=\omega_2 \backslash (P_{i} \cup P_{i+2} \cup \ldots \cup P_{t-1}) \cup \Gamma$. Then $\omega'_2$ decomposes as the sum of $\Gamma \cup P_t \cup P_{t+2} \cup \ldots P_{i-2}$ and $P_{t+1}\cup P_{t+3} \cup \ldots P_{t-2}$. In the second case let $\Gamma$ be the union of paths in $\omega_1$ from $e_i$ to $e_{j+2}$. Define $\omega'_1=\omega_1 \backslash \Gamma \cup P_i \cup P_{i+2} \cup \ldots \cup P_{j+1}$ and $\omega'_2=\omega_2 \backslash (P_i \cup P_{i+2} \cup \ldots \cup P_{j+1}) \cup \Gamma$. Then $\omega'_2$ decomposes as the sum of $P_{e_i,e'}\cup P_{i+1} \cup P_{i+3} \cup \ldots \cup P_{i-2}$ and $\Gamma \backslash P_{e_i,e'} \cup P_{j+3} \cup P_{j+5} \cup \ldots \cup P_{i-1}$. If none of the previous is true for $\omega$, then $\omega$ is as in the statement of the lemma. We will show that the only decomposition of $\omega$ is $\omega=\omega_1+\omega_2$. This implies that we cannot decompose $\omega$ as a sum of degree one labelings as $\omega_2$ is indecomposable. In all four cases, we have $\omega_e\geq 2$ for a cycle leg $e$ if and only if $e\in \{e_0,\ldots ,e_{2k}\}$. Moreover, $\omega_{\overline{e}}=2$ holds for at most one $\overline{e}\in \{e_0,\ldots ,e_{2k}\}$. We construct a decomposition $\omega=\omega'_1+\omega'_2$ with $\deg(\omega'_1)=1$ and $\deg(\omega'_2)=2$. If $\omega_1=P_{0}\cup P_2 \cup \ldots P_{2k-2} \cup P_{e_{2k},e'}$, then $(\omega'_2)_e=2$ for all $e\in \{e_0,\ldots ,e_{2k}\}$ and $(\omega'_2)_e=0$ for all other cycle legs. Indeed, there is only one cycle leg $e'$ left with value one, but since the sum of values on all leaf edges must be even we have $(\omega'_2)_{e'}=0$. For the other three cases $(\omega'_2)_e=2$ for all $e\in \{e_0,\ldots ,e_{2k}\}\backslash{\overline{e}}$, since $\omega_e=3$ for $e\in \{e_0,\ldots ,e_{2k}\}\backslash{\overline{e}}$. Thus also $(\omega'_2)_{\overline{e}}=2$, since the sum of values on all leaf edges must be even. It follows that $(\omega'_2)_e=1$ for all cycle edges, hence $\omega'_i=\omega_i$ for $i\in\{1,2\}$. \[cor:unique\_decomposition\] Let $G$ be a polygon graph and $\omega \in \tau(G)$ a degree three labeling. Then $\omega $ cannot be decomposed as a sum of degree one labelings if and only if $\omega$ can be decomposed uniquely as $\omega =\omega _1+\omega_2$ with $\deg(\omega_1)=1$ and $\deg(\omega_2)=2$. Let $G$ be a trivalent graph with first Betti number 1 and $\omega \in \tau(G)$ a degree three labeling. Then $\omega $ cannot be decomposed as a sum of degree one labelings if and only if $\omega =\omega _1+\omega_2$ such that - $\deg(\omega_1)=1$ and $\deg(\omega_2)=2$, - $\omega_2$ is indecomposable with value two on cycle legs $e_0,\ldots ,e_{2k}$ and value zero on all other cycle legs, - $\omega_1$ restricted to the unique cycle with its cycle legs is $P_0\cup P_2 \cup \ldots \cup P_{2k-2}$, $P_0 \cup P_2 \cup \ldots P_{2k-4} \cup P_{e_{2k-2},e_{2k}}$, $P_{0}\cup P_2 \cup \ldots P_{2k-4} \cup P_{e_{2k-2},e'}$ where $e'$ is a cycle leg between $e_{2k-2}$ and $e_{2k-1}$ or also the cycle path if $k=0$. The statement follows directly from Lemma \[lemma:decompositions\_of\_degree\_three\_labelings\] and \[lemma:polygon\_graph\]. \[lemma:degree\_three\_generators\] Let $G$ be a trivalent graph with first Betti number 2 where the two cycles are separated by at least one inner vertex and $\omega \in \tau(G)$ a degree three labeling. Then $\omega $ is indecomposable if and only if the following conditions are fulfilled: - $\omega$ restricted to any cycle with its cycle legs does not decompose as a sum of degree one labelings, - $\omega$ restricted to an edge on the shortest path between two cycles has value one or two, - $\omega$ restricted to exactly one edge incident to an edge on the shortest path between two cycles that is not a cycle edge or an edge on the shortest path has value one or two, and has value zero or three on all other such edges. By Lemma \[lemma:polygon\_graph\], we can assume that $G$ is a multiple polygon graph. Depict the edges on the shortest path between the two cycles horizontally and edges incident to them vertically below them as in Figure \[figure:graphs\_with\_first\_betti\_number\_2\_cases\] $(c)$. Assume $\omega $ restricted to a cycle $G_1$ together with its cycle legs decomposes as a sum of degree one elements. Let $e$ be a cycle leg of $G_1$ on the shortest path between $G_1$ and the other cycle. Write $G^e=G_1\sqcup G_2$. Then $\omega $ decomposes on $G_2$, and this decomposition can be extended to $G$. Assume there is an edge $\overline{e}$ on the shortest path between two cycles of $G$ such that $\omega _{\overline{e}}\in\{0,3\}$. Let $e',e''$ be new leaf edges obtained by cutting $G$ at $e$ and write $G^{\overline{e}}=G'\sqcup G''$. Then $\omega |_{G'}$ and $\omega |_{G''}$ can be decomposed as $$\omega |_{G'}=\omega'_1+\omega'_2 \textrm{ and } \omega |_{G''}=\omega''_1+\omega''_2$$ with $\deg(\omega'_1)=\deg(\omega''_1)=1$ and $\deg(\omega'_2)=\deg(\omega''_2)=2$. Furthermore, $(\omega'_i)_{e'}=(\omega''_i)_{e''}$ for $i=1,2$ and hence they can be combined to a decomposition of $\omega$. Assume now that the conditions $(i),(ii)$ are fulfilled. The labeling $\omega $ can be decomposed if and only if it can be decomposed as $$\omega =\omega_1+\omega_2$$ with $\deg(\omega_1)=1$ and $\deg(\omega_2)=2$. There is a unique way of defining $\omega_1$ and $\omega_2$ on cycles and cycle legs by Corollary \[cor:unique\_decomposition\]. We try to construct a decomposition of $\omega$ on all other edges step-by-step going from left to right such that the decomposition is compatible with the decomposition on the cycle legs on the shortest path between the two cycles, and study when there exists no such decomposition. Let $e$ be the leftmost edge of the shortest path between two cycles where $\omega_1$ and $\omega_2$ are defined and let the vertex $v$ be the right endpoint of $e$. We want to define $b_v(\omega_i)$ and $c_v(\omega_i)$ given $a_v(\omega_i)$ for $i=1,2$. All possible local decompositions of $\omega$ at an inner vertex between the two cycles (assuming that horizontal edges have values one or two) are presented in Figures \[figure:degree\_three\_local\_decompositions\] and \[figure:degree\_three\_local\_decompositions\_with\_options\]. In Figure \[figure:degree\_three\_local\_decompositions\] the value of $\omega$ at the vertical edge is zero or three. In Figure \[figure:degree\_three\_local\_decompositions\_with\_options\] the value of $\omega$ at the vertical edge is one or two. Given a local decomposition at $v$ as in Figure \[figure:degree\_three\_local\_decompositions\] and $a_v(\omega_i)$, then there is a unique way of defining $b_v(\omega_i)$ and $c_v(\omega_i)$. In particular, if $a_v(\omega_2)\in\{0,2\}$ then $b_v(\omega_2)\in\{0,2\}$. If $a_v(\omega_2)=1$ then $b_v(\omega_2)=1$. Given a local decomposition at $v$ as in Figure \[figure:degree\_three\_local\_decompositions\_with\_options\] and $a_v(\omega_i)$, then there might be a unique way of defining $b_v(\omega_i)$ and $c_v(\omega_i)$ or not depending on the value of $a_v(\omega_2)$. If $a_v(\omega_2)\in\{0,2\}$ then $b_v(\omega _2)=1$. If $a_v(\omega_2)=1$ then one can define either $b_v(\omega _2)\in \{0,2\}$ or $b_v(\omega _2)=1$. Let $e$ be a cycle leg that is on the path between two cycles. If $\omega _{e}=2$, then $(\omega_1)_{e}=0$ and $(\omega_2)_{e}=2$, because a degree two indecomposable element on a cycle can have only values zero and two on cycle legs by Theorem \[thm:buczynska\]. If $\omega _{e}=1$, then $(\omega_{e})_1=1$ and $(\omega_{e})_2=0$ for the same reasons. Denote by $e_r$ the cycle leg of the right cycle that are on the path between two cycles. If the horizontal path contains labelings only as in Figure \[figure:degree\_three\_local\_decompositions\], then $b_v(\omega_2)\in\{0,2\}$ for every vertex $v$ on the horizontal path. In particular, $(\omega_2)_{e_r}\in\{0,2\}$, hence there exists a decomposition of $\omega$. If at more than one vertex the local decomposition is as in Figure \[figure:degree\_three\_local\_decompositions\_with\_options\], denote the first such vertex by $v'$ and the last one by $v''$. For all the vertices $v$ left from $v'$ the value $b_v(\omega)\in\{0,2\}$ is uniquely defined. For $v'$ we have $b_{v'}(\omega)=1$. For all the vertices $v$ between $v'$ and $v''$ we can define $b_v(\omega)=1$: If the local decomposition at $v$ is as in Figure \[figure:degree\_three\_local\_decompositions\_with\_options\] then we have this choice by the discussion below. If the local decomposition at $v$ is as in Figure \[figure:degree\_three\_local\_decompositions\] then $b_v(\omega_2)=1$ since $a_v(\omega_2)=1$ again by the discussion above. For $v=v''$ define $b_v(\omega_2)\in\{0,2\}$. At all vertices $v$ to the right of $v''$, we have local decompositions as in Figure \[figure:degree\_three\_local\_decompositions\], therefore $b_v(\omega_2)\in\{0,2\}$. In particular, $(\omega_2)_{e_r}\in\{0,2\}$ and the decomposition of $\omega$ on the horizontal path is compatible with the decompositions of $\omega$ on both cycles. Hence $\omega $ is decomposable. On the other hand, if at one vertex $v'$ the local decomposition is as in Figure \[figure:degree\_three\_local\_decompositions\_with\_options\] and at all other vertices the local decomposition is as in Figure \[figure:degree\_three\_local\_decompositions\], then $b_v(\omega_2)\in\{0,2\}$ for all vertices $v$ left from $v'$ and $b_v(\omega_2)=1$ for all vertices $v$ to the right of $v'$ including $v'$ itself. In particular, $(\omega_2)_{e_r}=1$ which is not compatible with the values of $\omega_2$ on the right cycle. Since all steps have been uniquely determined, then $\omega$ cannot be decomposed. This completes the proof. Examples {#section:examples_of_maximal_degrees} ======== In this section, we will list some examples of graphs with first Betti number 3, 4 and 5 together with the maximal degree of the minimal generating set of their phylogenetic semigroup.[^1] Maximal degrees have been computed with `Normaliz` [@normaliz]. We will also show that for any natural number $g$ there exists a graph $G$ with first Betti number $g$ such that the maximal degree of a minimal generator of $\tau(G)$ is one. We note that the maximal degree tends to depend on the “separateness” of the cycles, exactly as for graphs with first Betti number 2. Let $G$ be the graph with first Betti number $g$ that has two vertices and $g+1$ edges between the two vertices, as illustrated in Figure \[figure:two\_vertices\_graph\]. Then $\tau(G)$ is generated in degree one. By cutting all edges of $G$, we get two claw trees $T',T''$ with $g+1$ leaves. Let $\omega\in \tau(G)$ be a degree $d$ labeling. Then $\omega$ gives $\omega'\in T'$ and $\omega'' \in T''$ with $\omega'=\omega''$ that we can decompose as a sum of $d$ degree one labelings exactly the same way on both trees. Gluing the decompositions of $\omega'$ and $\omega''$ gives a decomposition of $\omega$ as a sum of degree one labelings. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Jaroslaw Buczy[ń]{}ski, Christopher Manon and Henning Meyer for their helpful comments. [^1]: We thank Christopher Manon for introducing us the trivalent graph with first Betti number 4 and maximal degree one, see Figure \[figure:first\_betti\_number\_four\].

--- abstract: | We give a proof of the Breuil-Schneider conjecture in a large number of cases, which [*[complement]{}*]{} the indecomposable case, which we dealt with earlier in \[Sor\]. In some sense, only the Steinberg representation lies at the intersection of the two approaches. In this paper, we view the conjecture from a broader global perspective. If $U_{/F}$ is any definite unitary group, which is an inner form of ${\text{GL}}(n)$ over ${\mathcal{K}}$, we point out how the eigenvariety ${\Bbb{X}}(K^p)$ parametrizes a global $p$-adic Langlands correspondence between certain $n$-dimensional $p$-adic semisimple representations $\rho$ of ${\text{Gal}}(\bar{{\Bbb{Q}}}|{\mathcal{K}})$ (or what amounts to the same, pseudo-representations) and certain Banach-Hecke modules $\mathcal{B}$ with an admissible unitary action of $U(F\otimes {\Bbb{Q}}_p)$, when $p$ splits. We express the locally regular-algebraic vectors of $\mathcal{B}$ in terms of the Breuil-Schneider representation of $\rho$. Upon completion, this produces a [*[candidate]{}*]{} for the $p$-adic local Langlands correspondence in this context. As an application, we give a weak form of local-global compatibility in the crystalline case, showing that the Banach space representations $B_{\xi,\zeta}$ of Schneider-Teitelbaum \[ScTe\] fit the picture as predicted. There is a compatible global mod $p$ (semisimple) Langlands correspondence parametrized by ${\Bbb{X}}(K^p)$. We introduce a natural notion of refined Serre weights, and link them to the existence of crystalline lifts of prescribed Hodge type and Frobenius eigenvalues. At the end, we give a rough candidate for a local mod $p$ correspondence, formulate a local-global compatibility conjecture, and explain how it implies the conjectural Ihara lemma in \[CHT\]. [^1] [^2] author: - 'Claus M. Sorensen' date: Preprint title: '[**[Eigenvarieties and invariant norms: Towards $p$-adic Langlands for $U(n)$]{}**]{}' --- Introduction ============ Let ${\mathcal{K}}$ be a number field. The Fontaine-Mazur conjecture \[FM\] predicts a characterization of all (irreducible) Galois representations $\rho:\Gamma_{{\mathcal{K}}}={\text{Gal}}(\bar{{\Bbb{Q}}}|{\mathcal{K}})\rightarrow {\text{GL}}_n({\Bbb{Q}}_p)$ occurring naturally. That is, in the etale cohomology $H^{\bullet}(X,{\Bbb{Q}}_p)$ of some smooth projective variety $X_{/{\mathcal{K}}}$. It is a major result (due to Tsuji and others) that every such $\rho$ is [*[geometric]{}*]{}, which means it is unramified at all but finitely many places, and potentially semistable at all places above $p$. Fontaine and Mazur assert the converse, that every geometric $\rho$ occurs in cohomology (up to a Tate twist). The potentially semistable representations are now more or less completely understood, by work of Colmez and Fontaine \[CM\]. They are given by admissibly filtered $(\phi,N)$-modules (with Galois action), which are objects of a more concrete combinatorial nature. The $p$-adic Langlands program, still in its initial stages, attempts to link $p$-adic Hodge theory with non-archimedean functional analysis. Locally, if $K$ is a fixed finite extension of ${\Bbb{Q}}_p$, and $L|{\Bbb{Q}}_p$ is another sufficiently large finite extension (the coefficient field), one hopes to pair certain Galois representations $\rho:\Gamma_K \rightarrow {\text{GL}}_n(L)$ with certain Banach $L$-spaces with a unitary admissible ${\text{GL}}_n(K)$-action. This is now well-understood for ${\text{GL}}_2({\Bbb{Q}}_p)$ thanks to recent work of Berger, Breuil, Colmez, Paskunas, and others. See \[Bg\] for a nice survey. The goal of this paper is to shed some light on a [*[global]{}*]{} analogue, for any $n$, and any CM field ${\mathcal{K}}$. To give the flavor, if ${\mathcal{K}}|{\Bbb{Q}}$ is a quadratic imaginary field, in which $p$ splits, we will set up a bijection between certain Galois representations $\rho:\Gamma_{{\mathcal{K}}}\rightarrow {\text{GL}}_n(L)$ (actually, pseudo-representations) and certain Banach-Hecke modules with a unitary admissible ${\text{GL}}_n({\Bbb{Q}}_p)$-action. This is most likely folklore. More importantly, we relate the algebraic vectors to the $p$-adic Hodge theory on the Galois side. The word [*[certain]{}*]{} here has a precise meaning. It means those representations which [*[come from an eigenvariety]{}*]{}, of some fixed tame level $K^p$. We will be precise below. We model the discussion on the ${\text{GL}}(2)$-case: With any continuous Galois representation $\rho:\Gamma_{{\Bbb{Q}}_p}\rightarrow {\text{GL}}_2(L)$, the $p$-adic Langlands correspondence associates a unitary Banach $L$-space representation $B(\rho)$ of ${\text{GL}}_2({\Bbb{Q}}_p)$. Moreover, $\rho$ is de Rham with distinct Hodge-Tate weights precisely when $B(\rho)^{alg}\neq 0$. If so, let us say the weights are $\{0,1-k\}$ with $k \geq 2$ (with the convention that the cyclotomic character has weight $-1$), then the algebraic vectors are given by $$B(\rho)^{alg}=\text{Sym}^{k-2}(L^2)\otimes \pi(\rho)$$ for a smooth generic representation $\pi(\rho)$, possibly reducible, obtained by a slight modification of the classical local Langlands correspondence. [*[The Breuil-Schneider conjecture]{}*]{}. The local $p$-adic Langlands program is somewhat vague, and a precise conjectural framework is still developing, beyond the case of ${\text{GL}}_2({\Bbb{Q}}_p)$, where pretty much everything is known. However, there is a weak (but precise) almost “skeletal” version formulated in \[BrSc\], which we now recall. We keep our finite extension $K|{\Bbb{Q}}_p$, and a finite Galois extension thereof $K'|K$. Pick a third field of coefficients $L\subset \bar{{\Bbb{Q}}}_p$, finite over ${\Bbb{Q}}_p$, but large enough so that it contains the Galois closures of $K$ and $K_0'$ (the maximal unramified subfield of $K'$). The roles of these fields are the following. We consider potentially semistable representations $\rho:\Gamma_K \rightarrow {\text{GL}}_n(L)$, which becomes semistable when restricted to $\Gamma_{K'}$. As mentioned above, such $\rho$ correspond to $(\phi,N)\times {\text{Gal}}(K'|K)$-modules $D$, with an admissible filtration. This makes use of Fontaine’s period ring $B_{st}$, $$D=(B_{st}\otimes_{{\Bbb{Q}}_p}\rho)^{\Gamma_{K'}}.$$ This is a finite free $K_0'\otimes_{{\Bbb{Q}}_p} L$-module, of rank $n$, with a semilinear Frobenius $\phi$, a (nilpotent) monodromy operator $N$ such that $N\phi=p\phi N$, a commuting action of ${\text{Gal}}(K'|K)$, and an admissible Galois-stable filtration on $D_{K'}$. Note $$K'\otimes_{{\Bbb{Q}}_p}L\simeq {\prod}_{\tau\in {\text{Hom}}(K,L)}K'\otimes_{K,\tau}L.$$ Accordingly, $D_{K'}\simeq \prod_{\tau}D_{K',\tau}$, and each $K'\otimes_{K,\tau}L$-module $D_{K',\tau}$ is filtered. - [*[Hodge-Tate numbers]{}*]{}. For each $\tau:K \hookrightarrow L$, we let $i_{1,\tau}\leq \cdots\leq i_{n,\tau}$ denote the jumps in the Hodge filtration (listed with multiplicity). That is, $$\text{gr}^i(D_{K',\tau})\neq 0 \Leftrightarrow i\in \{i_{1,\tau},\ldots, i_{n,\tau}\}.$$ We will denote this multiset of integers by $HT_{\tau}(\rho)=\{i_{j,\tau}:j=1,\ldots,n\}$. - [*[Weil-Deligne representation]{}*]{}. If we forget about the filtration, the resulting $(\phi,N)\times {\text{Gal}}(K'|K)$-module corresponds to a Weil-Deligne representation, once we fix an embedding $K_0'\hookrightarrow L$. This goes as follows, see Proposition 4.1 in \[BrSc\]. The underlying $n$-dimensional $L$-vector space is $$D_L=D\otimes_{K_0'\otimes_{{\Bbb{Q}}_p} L}L,$$ with the induced $N$ coming from $B_{st}$, and with $r:W_K \rightarrow {\text{GL}}(D_L)$ defined by $r(w)=\phi^{-d(w)}\circ \bar{w}$. Here $\bar{w}$ denotes the image of $w$ in ${\text{Gal}}(K'|K)$, and $d(w)$ gives the power of arithmetic Frobenius which $w$ induces. The ensuing Weil-Deligne representation becomes unramified upon restriction to $W_{K'}$. We will denote it by $WD(\rho)=(r,N,D_L)$ throughout the text. The Breuil-Schneider conjecture asks for a characterization of the data arising in this fashion, assuming all Hodge-Tate numbers are distinct. To state it, start with abstract data. Firstly, for each embedding $\tau:K \hookrightarrow L$, say we are given $n$ distinct integers $HT_{\tau}=\{i_{1,\tau}<\cdots<i_{n,\tau}\}$. Secondly, say we are given some $n$-dimensional Weil-Deligne representation $WD$, with coefficients in $L$, which become unramified on $W_{K'}$. With these data, below we will associate a locally algebraic representation $BS$ of ${\text{GL}}_n(K)$, with coefficients in $L$. The algebraic part is defined in terms of the $HT_{\tau}$, the smooth part in terms of $WD$. Our data should come from a $\rho$ precisely when $BS$ has a ${\text{GL}}_n(K)$-stable $\mathcal{O}_L$-lattice; the unit ball of an invariant norm. The following is Conjecture 4.3 in \[BrSc\], also announced as Conjecture 4.1 in Breuil’s 2010 ICM address \[Bre\]: [**[The Breuil-Schneider conjecture]{}**]{}. ** The following are equivalent: - The data $HT_{\tau}$ and $WD^{F-ss}$ arise from a potentially semistable $\rho$. - $BS$ admits a norm $\|\cdot\|$, invariant under the action of ${\text{GL}}_n(K)$. Before we recall the status of the conjecture, we return to the definition of $BS$. - [*[Algebraic part]{}*]{}. Introduce $b_{j,\tau}=-i_{n+1-j,\tau}-(j-1)$. That is, write the $i_{j,\tau}$ in the opposite order, change signs, and subtract $(0,1,\ldots,n-1)$. We let $\xi_{\tau}$ be the irreducible algebraic $L$-representation of ${\text{GL}}_n$, of highest weight $$b_{1,\tau}\leq b_{2,\tau}\leq \cdots \leq b_{n,\tau}$$ relative to the [*[lower]{}*]{} triangular Borel. Their tensor product $\xi=\otimes_{\tau}\xi_{\tau}$, with $\tau$ running over ${\text{Hom}}(K,L)$, is then an irreducible algebraic representation of ${\text{GL}}_n(K\otimes_{{\Bbb{Q}}_p}L)$ over $L$. We will view $\xi$ as a representation of ${\text{GL}}_n(K)$. - [*[Smooth part]{}*]{}. By the classical local Langlands correspondence \[HT\], the Frobenius-semisimplification $WD^{F-ss}\simeq \text{rec}_n(\pi^{\circ})$, for some irreducible admissible smooth representation $\pi^{\circ}$ of ${\text{GL}}_n(K)$, defined over $\bar{{\Bbb{Q}}}_p$. Here $\text{rec}_n$ is normalized as in \[HT\]. To define it over $\bar{{\Bbb{Q}}}_p$, we need to fix a square-root $q^{\frac{1}{2}}$, where $q=\#{\Bbb{F}}_K$. By the Langlands classification, one has $$\text{Ind}_P(Q(\Delta_1)\otimes \cdots \otimes Q(\Delta_r))\overset{!}{\twoheadrightarrow} \pi^{\circ},$$ a unique irreducible quotient, where the $Q(\Delta_i)$ are generalized Steinberg representation built from the $\Delta_i$, which are segments of supercuspidals, suitably ordered. The smooth part of $BS$ is now defined to be $$\pi=\text{Ind}_P(Q(\Delta_1)\otimes \cdots \otimes Q(\Delta_r))\otimes |\det|^{\frac{1-n}{2}},$$ or rather its model over $L$, which is independent of the choice of $q^{\frac{1}{2}}$. Note that $\pi\simeq \pi^{\circ}\otimes |\det|^{(1-n)/2}$ if and only if $\pi^{\circ}$ is generic (that is, has a Whittaker model). For that reason, the association $WD \mapsto \pi$ is often called the [*[generic]{}*]{} local Langlands correspondence. We let $BS=\xi\otimes_L \pi$, following \[BrSc\] (although they do not use the notation $BS$). In fact, we will find it more convenient to work with a different normalization. In the above construction there is a choice of a [*[sign]{}*]{}; essentially reflected in whether one twists by $|\det|^{(1-n)/2}$ or its inverse. The latter is more commonly used in the references we rely on. The resulting representation is just a twist of $BS$ by a harmless explicit $p$-adically unitary continuous character. Namely, $$\text{$\widetilde{BS}=BS \otimes_L \mu^{n-1}$, ${\hspace{6pt}}$ $\mu(g)=N_{K|{\Bbb{Q}}_p}(\det g)^{\times}$,}$$ where $a^{\times}=a|a|_p=BS(\chi_{cyc})(a)\in {\Bbb{Z}}_p^{\times}$ denotes the unit factor of an $a \in {\Bbb{Q}}_p^*$. Of course, $\widetilde{BS}$ has an invariant norm if and only if $BS$ does, so it makes no real difference. It reflects a Tate twist: $\widetilde{BS}(\rho)$ is nothing but $BS(\rho\otimes \chi_{cyc}^{n-1})$. The implication (2) $\Rightarrow$ (1) in the conjecture is in fact completely known. After many cases were worked out in \[ScTe\] and \[BrSc\], the general case was settled by Y. Hu in his thesis \[Hu\]. In fact Hu proves that (1) is equivalent to the [*[Emerton condition]{}*]{}, which is a purely group-theoretical condition: $$\text{(3) $J_P(BS)^{Z_M^+=\chi}\neq 0 \Longrightarrow \forall z \in Z_M^+: |\delta_P^{-1}(z)\chi(z)|_p\leq 1$.}$$ Here $J_P$ is Emerton’s generalization of the Jacquet functor, introduced and studied in \[Em1\] and \[Em2\]. The heart of Hu’s proof is to translate (3) into finitely many inequalities relating the Hodge polygon to the Newton polygon. In the vein \[FoRa\], he is then able to show the existence of an admissible filtration compatible with the given data. The implication (2) $\Rightarrow$ (3) is relatively easy. What remains, is to produce an invariant norm on $BS(\rho)$, for any potentially semistable $\rho$ (with distinct Hodge-Tate weights). One of the main motivations for writing this paper, was to make progress in this direction, (1) $\Rightarrow$ (2). We proved this in \[Sor\] when $WD(\rho)$ is indecomposable (in other words, $\pi^{\circ}=Q(\Delta)$ is generalized Steinberg). Here the Emerton condition boils down to just integrality of the central character, and in fact the resulting conjecture was stated explicitly as 5.5 in \[BrSc\]. The key point of \[Sor\] was to make use of the fact that $Q(\Delta)$ is a discrete series representation, and therefore admits a [*[pseudo-coeffcient]{}*]{}. Inserting this as a test-function in the trace formula for a certain definite unitary group, one can pass to a global setup (a la Grunwald-Wang). Finally, the desired norm was found by relating classical $p$-adic algebraic modular forms to completed cohomology $\tilde{H}^0$, as introduced in great generality in \[Emer\]. This whole argument is purely group-theoretical, and in fact carries over to any connected reductive group over ${\Bbb{Q}}_p$, exploiting a compact form (using a Galois cohomological computation of Borel and Harder, which shows the existence of locally prescribed forms). We should point out that the supercuspidal case is much easier. In this case there are several ways to produce a norm (compact induction, for example). One of the outcomes of this paper, is a [*[complement]{}*]{} to the main result of \[Sor\]. The idea of relating algebraic modular forms to $\tilde{H}^0$, already present in \[Emer\], can be pushed further, now that local-global compatibility at $p=\ell$ is available in the “book project” context. This was proved recently by Barnet-Lamb, Gee, Geraghty, and Taylor in the so-called Shin-regular case \[BGGT\], and this regularity hypothesis was then shown to be unnecessary by Caraiani, as part of her 2012 Harvard Ph.D. thesis \[Car\]. This results in the following somewhat vague Theorem A, which we will make more precise in Theorem B below. [**[Theorem A]{}**]{}. [*[The Breuil-Schneider conjecture holds for potentially semistable $\rho$, which come from a regular, classical, irreducible point on a unitary eigenvariety. ]{}*]{} [*[Eigenvarieties]{}*]{}. We will combine the approaches of \[Chen\] and \[Emer\]. Thus let ${\mathcal{K}}$ be a CM field, with maximal totally real subfield $F$. Let $D$ be a central simple ${\mathcal{K}}$-algebra of $\dim_{{\mathcal{K}}}(D)=n^2$, equipped with an anti-involution $\star$ of the second kind (that is, $\star|_{{\mathcal{K}}}$ is conjugation). We introduce the unitary $F$-group $U=U(D,\star)$, an outer form of ${\text{GL}}(n)$, which becomes the inner form $D^{\times}$ over ${\mathcal{K}}$. It will be convenient to also introduce $G=\text{Res}_{F|{\Bbb{Q}}}(U)$. We will always assume $U$ is totally definite. In other words, that $G({\Bbb{R}})$ is a [*[compact]{}*]{} Lie group, therefore a product of copies of $U(n)$ (hence also connected). We will fix a prime number $p$ such that every place $v|p$ of $F$ splits in ${\mathcal{K}}$, and such that $D_w^{\times}\simeq {\text{GL}}_n({\mathcal{K}}_w)$ for every $w|v$. To keep track of various identifications, it is customary to [*[choose]{}*]{} a place $\tilde{v}$ of ${\mathcal{K}}$ above every $v|p$. Once and for all, also fix an isomorphism $\iota: {\Bbb{C}}\overset{\sim}{\longrightarrow} \bar{{\Bbb{Q}}}_p$. This gives rise to an identification $${\text{Hom}}(F,{\Bbb{R}})={\text{Hom}}(F,{\Bbb{C}})\simeq {\text{Hom}}(F,\bar{{\Bbb{Q}}}_p)=\sqcup_{v|p}{\text{Hom}}(F_v,\bar{{\Bbb{Q}}}_p),$$ and similarly for ${\text{Hom}}({\mathcal{K}},{\Bbb{C}})$. By assumption $F_v \simeq {\mathcal{K}}_w$ for $w|v$, so the choices $\{\tilde{v}\}$ just amount to fixing a CM-type $\Phi$, which is ordinary for $\iota$, in the sense of \[Katz\]. This will ensure that the various identifications we make are compatible. The eigenvariety for $G$ depends on the choice of a tame level $K^p \subset G({\Bbb{A}}_f^p)$. It is a reduced rigid analytic space ${\Bbb{X}}_{/E}$, where we take $E$ to be the Galois closure of $F$ in $\bar{{\Bbb{Q}}}_p$, with additional structure: $$\text{$\chi:{\Bbb{X}}\rightarrow \hat{T}$, ${\hspace{6pt}}$ $\lambda: \mathcal{H}(K^p)^{\text{sph}}\rightarrow \mathcal{O}({\Bbb{X}})$.}$$ Here $\hat{T}_{/E}$ is weight space, parametrizing locally analytic characters of $T({\Bbb{Q}}_p)$, and $\mathcal{H}(K^p)^{\text{sph}}$ is the spherical central subalgebra of the Hecke $E$-algebra $\mathcal{H}(K^p)$. Moreover, there is a Zariski-dense subset $X_{cl}\subset {\Bbb{X}}(\bar{{\Bbb{Q}}}_p)$ such that the evaluation $$\text{${\Bbb{X}}(\bar{{\Bbb{Q}}}_p)\longrightarrow (\hat{T}\times \text{Spec}\mathcal{H}(K^p)^{\text{sph}})(\bar{{\Bbb{Q}}}_p)$, ${\hspace{6pt}}$ $x \mapsto (\chi_x,\lambda_x)$,}$$ identifies $X_{cl}$ with the set of [*[classical]{}*]{} points. Roughly this means that, first of all $\chi_x=\psi_x\theta_x$ is locally algebraic, and there exists an automorphic representation $\pi$ of weight $\psi_x$ such that $\pi_p\hookrightarrow \text{Ind}_{B}^{G}(\theta_x)$, and $\mathcal{H}(K^p)^{\text{sph}}$ acts on $\pi_f^{K^p}\neq 0$ by the character $\lambda_x$. (The condition that $\pi_p$ embeds in a principal series is the analogue of the “finite slope” requirement showing up in the classical works of Coleman, Mazur and others. The choice of a $\theta_x$ is called a refinement of $\pi$.) It is of utmost importance to us that the eigenvariety carries a family of Galois representations. To be more precise, if we let $\Sigma=\Sigma(K^p)$ be the set of ramified places, there is a unique continuous $n$-dimensional pseudo-representation $$\mathcal{T}: \Gamma_{{\mathcal{K}},\Sigma}\rightarrow \mathcal{O}({\Bbb{X}})^{\leq 1}$$ associated with $\lambda: \mathcal{H}(K^p)^{\text{sph}}\rightarrow \mathcal{O}({\Bbb{X}})$, in the sense that for all places $w \notin \Sigma$, $$\mathcal{T}(\text{Frob}_w)=\lambda(b_{w|v}(h_w)).$$ Here $h_w$ is the element of the spherical Hecke algebra for ${\text{GL}}_n({\mathcal{K}}_w)$, which acts via the sum of the (integral) Satake parameters on spherical vectors, and $b_{w|v}$ is the standard base change homomorphism between the pertaining spherical Hecke algebras, see \[Min\]. In particular, by Procesi and Taylor, for each $x \in {\Bbb{X}}(\bar{{\Bbb{Q}}}_p)$ there is a unique semisimple Galois representation $$\text{$\rho_x:\Gamma_{{\mathcal{K}},\Sigma}\rightarrow {\text{GL}}_n(\bar{{\Bbb{Q}}}_p)$, ${\hspace{6pt}}$ $\mathcal{T}_x={\text{tr}}(\rho_x)$.}$$ In fact, the way $\mathcal{T}$ is constructed, is by first defining $\rho_x$ for [*[regular]{}*]{} classical points $x\in X_{reg}$, by which we mean the dominant character $\psi_x$ is given by a strictly decreasing sequence of integers (at some place). Thanks to \[Whi\], this guarantees that $\pi$ has a base change to ${\text{GL}}_n({\Bbb{A}}_{{\mathcal{K}}})$ of the form $\Pi=\boxplus \Pi_i$, where the $\Pi_i$ are cohomological [*[cuspidal]{}*]{} (as opposed to just discrete) automorphic representations to which one can attach Galois representations. Now, $X_{reg}$ can be shown to be Zariski-dense, and a formal argument in \[Che\] interpolates the pseudo-characters ${\text{tr}}(\rho_x)$ for $x\in X_{reg}$ by a unique $\mathcal{T}$, which one can then specialize at [*[any]{}*]{} point $x \in {\Bbb{X}}(\bar{{\Bbb{Q}}}_p)$. Now we can clarify the statement in Theorem A: If $x \in X_{reg}$ is a classical point such that $\rho_x$ is irreducible (globally, as a representation of $\Gamma_{{\mathcal{K}}}$), and $w|p$ is a place of ${\mathcal{K}}$, then $\rho_x|_{\Gamma_{{\mathcal{K}}_w}}$ is potentially semistable, [*[and]{}*]{} its locally algebraic representation $BS(\rho_x|_{\Gamma_{{\mathcal{K}}_w}})$ admits a ${\text{GL}}_n({\mathcal{K}}_w)$-invariant norm $\|\cdot\|$. [*[A global $p$-adic Langlands correspondence]{}*]{}. The actual construction of an invariant norm $\|\cdot\|$ is more interesting than its mere existence. It comes out of a much more precise result, which we now describe. Fix a finite extension $L|E$. At each point $x \in {\Bbb{X}}(L)$, we have pseudo-representation $\mathcal{T}_x:\Gamma_{{\mathcal{K}},\Sigma} \rightarrow L$ (the trace of an actual representation $\rho_x$, which may or may not be defined over $L$). On the other hand, with $x \in {\Bbb{X}}(L)$ we associate the Banach $L$-space $$\mathcal{B}_x=(L\otimes_E \tilde{H}^0(K^p))^{\frak{h}=\lambda_x}$$ where $\frak{h}=\mathcal{H}(K^p)^{\text{sph}}$ is shorthand notation. This space is really very concrete. The completed cohomology $\tilde{H}^0(K^p)$, defined in much greater generality by Emerton, is here nothing but the space of all [*[continuous]{}*]{} functions $$\text{$f: Y(K^p)\rightarrow E$, ${\hspace{6pt}}$ $Y(K^p)=\underset{K_p}{\varprojlim} Y(K_pK^P)$, ${\hspace{6pt}}$ $Y(K)=G({\Bbb{Q}})\backslash G({\Bbb{A}}_f)/K$,}$$ with supremum norm. The superscript $\frak{h}=\lambda_x$ means we take the eigenspace for the character $\lambda_x: \frak{h}\rightarrow L$ (not the generalized eigenspace). Note that $\mathcal{B}_x$ is much more than just a Banach $L$-space: For one thing, it is a Banach module for the Banach-Hecke algebra $\hat{\mathcal{H}}(K^p)$ (see \[ScTe\] for a detailed discussion of these). For another thing, there is a natural $\hat{\mathcal{H}}(K^p)$-linear action of $G({\Bbb{Q}}_p)$ by [*[isometries]{}*]{} of $\mathcal{B}_x$, which is admissible (meaning that its reduction $\bar{\mathcal{B}}_x$ is a smooth admissible representation of $G({\Bbb{Q}}_p)$ over ${\Bbb{F}}_L$, in the usual sense). Now, for $x,x'\in {\Bbb{X}}(L)$, $$\mathcal{T}_x=\mathcal{T}_{x'} \Leftrightarrow \lambda_x=\lambda_{x'}\Leftrightarrow \mathcal{B}_x=\mathcal{B}_{x'},$$ since each $b_{w|v}$ is onto, see Corollary 4.2 in \[Min\] (a fact also used on p. 10 of \[CHL\]). In other words, the set of all pairs $(\mathcal{T}_x,\mathcal{B}_x)$, with $x \in {\Bbb{X}}(L)$, is the graph of a bijection between the images of the two projections: $$\left\{ \begin{matrix} \text{$n$-dimensional pseudo-representations} \\ \text{$\mathcal{T}: \Gamma_{{\mathcal{K}},\Sigma}\rightarrow L$ coming from ${\Bbb{X}}(L)$} \end{matrix} \right\} \longleftrightarrow$$ $$\left\{ \begin{matrix} \text{Banach $\hat{\mathcal{H}}_L(K^p)$-modules $\mathcal{B}$ with admissible} \\ \text{unitary $G({\Bbb{Q}}_p)$-action, coming from ${\Bbb{X}}(L)$} \end{matrix} \right\}.$$ Here $\mathcal{T}\leftrightarrow \mathcal{B}$ means there is a point $x \in {\Bbb{X}}(L)$ such that $\mathcal{T}=\mathcal{T}_x$ and $\mathcal{B}=\mathcal{B}_x$. (We say that a pseudo-character $\mathcal{T}: \Gamma_{{\mathcal{K}},\Sigma}\rightarrow L$ comes from ${\Bbb{X}}(L)$ if it is of the form $\mathcal{T}_x$ for a point $x\in {\Bbb{X}}(L)$. Similarly for Banach modules.) For ease of exposition, let us assume we have [*[split ramification]{}*]{}. In other words, $S(K^p)\subset \text{Spl}_{{\mathcal{K}}|F}$. Then local base change is defined everywhere, and there is a unique automorphic representation $\pi_x$ associated with a point $x\in X_{cl}$ such that $\rho_x$ is irreducible (indeed its global base change is cuspidal and determined almost everywhere). It is expected (and perhaps known?) that $m(\pi_x)=1$. Our main result in this paper is the following, proved in Section 6. [**[Theorem B]{}**]{}. ** Assume $S(K^p)\subset \text{Spl}_{{\mathcal{K}}|F}$. For each point $x \in X_{reg}\cap X_{irr}$, defined over $L$, such that $m(\pi_x)=1$, there is a unique (up to topological equivalence) Banach space $B(\rho_x)$ over $L$ with an unitary $G({\Bbb{Q}}_p)$-action such that - $B(\rho_x)^{ralg}\simeq \widetilde{BS}(\rho_x):=\bigotimes_{v|p}\widetilde{BS}(\rho_x|_{\Gamma_{{\mathcal{K}}_{\tilde{v}}}})$ is . - There is a $G({\Bbb{Q}}_p)\times \hat{\mathcal{H}}(K^p)$-equivariant topological isomorphism, $$B(\rho_x)\otimes (\otimes_{v\nmid p}\pi_{x,v}^{K_v}) \overset{\sim}{\longrightarrow} \overline{\mathcal{B}_x^{ralg}}.$$ (Here $\overline{\mathcal{B}_x^{ralg}}$ denotes the closure of the regular-algebraic vectors in $\mathcal{B}_x$.) - If $\rho_x$ is crystalline above $p$, there is a continuous $G({\Bbb{Q}}_p)$-map, with dense image, $$B_{\xi_x,\zeta_x} \longrightarrow B(\rho_x),$$ which restricts to an isomorphism $H_{\xi_x,\zeta_x} \overset{\sim}{\longrightarrow} B(\rho_x)^{ralg}$. (Here $H_{\xi,\zeta}$ and $B_{\xi,\zeta}$ are the spaces introduced by Schneider and Teitelbaum \[ScTe\], and we take $\xi_x$ of highest weight $\psi_x$, and $\zeta_x$ to be the eigensystem of $\theta_x$.) [*[A local $p$-adic Langlands correspondence?]{}*]{} The ${\text{GL}}_2({\Bbb{Q}}_p)$-case suggests there should be a local “correspondence” for any finite extension $K|{\Bbb{Q}}_p$. Being cautious, there should at least be [*[map]{}*]{} $\rho \mapsto \frak{B}(\rho)$ (defined for possibly non-semisimple $\rho$. In other words, for representations, as opposed to just pseudo-representations) $$\left\{ \begin{matrix} \text{continuous representations $\rho: \Gamma_K \rightarrow {\text{GL}}_n(L)$} \end{matrix} \right\} \overset{?}{\longrightarrow}$$ $$\left\{ \begin{matrix} \text{Banach $L$-spaces $\mathfrak{B}$ endowed with} \\ \text{admissible unitary ${\text{GL}}_n(K)$-action} \end{matrix} \right\},$$ which should map irreducible $\rho$ to topologically irreducible $\frak{B}(\rho)$, and one should be able to recover $\rho$ from $\frak{B}(\rho)$. If so, we believe in the provisional equality $$B(\rho_x)\overset{?}{=} \hat{\otimes}_{v|p}\mathfrak{B}(\rho_x|_{\Gamma_{{\mathcal{K}}_{\tilde{v}}}}),$$ for $x$ as in Theorem B, at least when the restrictions $\rho_x|_{\Gamma_{{\mathcal{K}}_{\tilde{v}}}}$ are irreducible. Of course, the flaring question here is whether $B(\rho_x)$ only depends on these restrictions at $p$ (and whether it factors as a tensor product). This is the crux of the matter, and seems hard. We have nothing to offer in this direction here. For definite unitary groups $U(2)$ over ${\Bbb{Q}}$, the above question is well-posed: The local $p$-adic Langlands correspondence $\frak{B}$ has been constructed for ${\text{GL}}_2({\Bbb{Q}}_p)$ by Berger, Breuil, Colmez, Kisin, and others, by ingenious use of $(\phi,\Gamma)$-modules; a truly monumental result! In the crystalline case, $\frak{B}$ is (typically) given by the universal modules $B_{\xi,\zeta}$. Thus, at least in this case, Theorem B gives some sort of weak local-global compatibility, in the vein of \[Eme\]. In joint work with Przemyslaw Chojecki, we are hoping to extends this to the [*[non]{}*]{}-crystalline case, by employing the deformation theoretical techniques used in \[Eme\]. [*[A global mod $p$ Langlands correspondence]{}*]{}. In the last section, we introduce the mod $p$ correspondence, for any finite extension $L|E$, and any tame level $K^p$, $$\left\{ \begin{matrix} \text{$n$-dimensional pseudo-representations} \\ \text{$t: \Gamma_{{\mathcal{K}},\Sigma}\rightarrow {\Bbb{F}}_L$ coming from ${\Bbb{X}}(L)$} \end{matrix} \right\} \longleftrightarrow$$ $$\left\{ \begin{matrix} \text{$\mathcal{H}_{{\Bbb{F}}_L}(K^p)$-modules $b$ with admissible} \\ \text{$G({\Bbb{Q}}_p)$-action, coming from ${\Bbb{X}}(L)$} \end{matrix} \right\}.$$ Here, $$t_x={\text{tr}}\bar{\rho}_x^{ss} \longleftrightarrow b_x=H^0(K^p,{\Bbb{F}}_L)^{\frak{h}^{\circ}=\bar{\lambda}_x}$$ for any $x \in {\Bbb{X}}(L)$. If $\omega$ is an irreducible $G({\Bbb{F}}_p)$-representation over $\bar{{\Bbb{F}}}_p$, a Serre weight, ${\text{Hom}}_{G({\Bbb{Z}}_p)}(\omega,b_x)$ is naturally identified with the generalized eigenspace for $\bar{\lambda}_x$ in the mod $p$ modular forms $\mathcal{A}_{\omega}(G({\Bbb{Z}}_p)K^p,\bar{{\Bbb{F}}}_p)$. This latter space carries a natural action of $\mathcal{H}_{\omega}(G,K)$, the Hecke algebra at $p$, which is commutative \[Her\]. This triggers the following notion. We define a [*[refined]{}*]{} Serre weight as a pair $(\omega,\nu)$, where $\nu: \mathcal{H}_{\omega}(G,K) \rightarrow \bar{{\Bbb{F}}}_p$ is an algebra character. Let $$(\omega,\nu) \in \mathcal{W}_+(\bar{\rho}_x) \Longleftrightarrow \mathcal{A}_{\omega}(G({\Bbb{Z}}_p)K^p,\bar{{\Bbb{F}}}_p)^{\mathcal{H}_{\omega}(G,K)\otimes\frak{h}^{\circ}=\nu\otimes\bar{\lambda}_x} \neq 0.$$ Equivalently, $\mathcal{W}_+(\bar{\rho}_x)$ contains $(\omega,\nu)$ when there is a nonzero $G({\Bbb{Q}}_p)$-map, $$\pi(\omega,\nu):=\text{c-Ind}_{K}^G(\omega)\otimes_{\mathcal{H}_{\omega}(G,K),\nu}\bar{{\Bbb{F}}}_p \rightarrow b_x.$$ In Proposition 4 below, we aim at a more Galois-theoretical description of $\mathcal{W}_+(\bar{\rho}_x)$, inspired by \[Gee\]. Namely, given $(\omega,\nu)$ in there, we note that $\bar{\rho}_x$ has a crystalline lift with Hodge-Tate weights and Frobenius eigenvalues prescribed by $(\omega,\nu)$. We establish the converse when $\omega$ has a $p$-small highest weight. [*[A local mod p Langlands correspondence?]{}*]{} It is widely believed that there should be a local mod $p$ correspondence, which is compatible with $\frak{B}$ above. At the very least, there should be map $\bar{\rho}\mapsto b(\bar{\rho})$ (on equivalence classes) $$\left\{ \begin{matrix} \text{continuous $\bar{\rho}:\Gamma_K \rightarrow {\text{GL}}_n({\Bbb{F}}_L)$} \end{matrix} \right\} \overset{?}{\longrightarrow}$$ $$\left\{ \begin{matrix} \text{${\Bbb{F}}_L$-spaces $b$ endowed with} \\ \text{admissible ${\text{GL}}_n(K)$-action} \end{matrix} \right\},$$ which takes an irreducible $\bar{\rho}$ to an irreducible $b(\bar{\rho})$. One should be able to recover $\bar{\rho}$ from $b(\bar{\rho})$. Furthermore, for any lift $\rho: \Gamma_K \rightarrow {\text{GL}}_n(\mathcal{O}_L)$ of $\bar{\rho}$, there ought to be at least a canonical nonzero map (perhaps an embedding), $$\frak{B}(\rho\otimes L)^{\circ}\otimes {\Bbb{F}}_L \overset{?}{\longrightarrow} b(\bar{\rho}).$$ At the end, we define an admissible mod $p$ representation $b(\bar{\rho}_x)$ of $G({\Bbb{Q}}_p)$, when $\bar{\rho}_x$ is irreducible, which we like to think of as a candidate for $\otimes_{v|p}b(\bar{\rho}_x|_{\Gamma_{{\mathcal{K}}_{\tilde{v}}}})$, at least when these local restrictions above $p$ are irreducible. However, we cannot say much about it. Even showing that $b(\bar{\rho}_x)\neq 0$ appears to be a difficult problem, closely tied to the conjectural Ihara lemma for unitary groups (Conjecture B in \[CHT\]). We formulate a precise local-global compatibility conjecture for $b(\bar{\rho}_x)$, and explain how it [*[implies]{}*]{} Ihara’s lemma. This makes heavy use of \[EH\]. [*[Acknowledgements]{}*]{}. With great pleasure and admiration, I want to acknowledge the impact of the visions of Breuil and Emerton on this work. I would like to thank both of them heartfully for discretely pointing out some (very) embarrassing misconceptions of mine, the first time I spoke about it (in hindsight prematurely). Also, conversations and correspondence with Chojecki, Dospinescu, Helm, Herzig, Newton, Ramakrishnan, and Shin, have been a great help and a source of inspiration. I am grateful to Minguez for clarifying Corollary 4.2 of \[Min\] to me, and for pointing to the reference \[CHL\]. Thanks are due to Caraiani, for making a preliminary version of \[Car\] available to me before circulating it. Automorphic Galois representations ================================== We start out by summarizing what is currently known about attaching Galois representations to automorphic representations of definite unitary groups. Due to the work of many people, we now have an almost complete understanding of this, and below we merely navigate the existing literature. We claim no originality in this section. Our goal is simply to state the precise result. Particularly, we want to emphasize the local-global compatibility at $p=\ell$, recently proved in \[BGGT\] and \[Car\], which is fundamental for this paper. Definite unitary groups ----------------------- Throughout this article, we fix a totally real field $F$, and a CM extension ${\mathcal{K}}$. We let $c$ denote the non-trivial element of ${\text{Gal}}({\mathcal{K}}|F)$. The places of $F$ will usually be denoted by $v$, those of ${\mathcal{K}}$ by $w$. We are interested in outer forms $U$ of ${\text{GL}}(n)_{F}$, which become an inner form $D^{\times}$ over ${\mathcal{K}}$. Here $D$ is a central simple ${\mathcal{K}}$-algebra, of $\dim_{{\mathcal{K}}}(D)=n^2$. These forms are unitary groups $U=U(D,\star)$, where $\star$ is an anti-involution on $D$ of the second kind ($\star|_{{\mathcal{K}}}=c$). Thus, for any $F$-algebra $R$, $$U(R)=\{x \in (D\otimes_F R)^{\times}: xx^{\star}=1\}.$$ We will always assume from now on that $U(F\otimes_{{\Bbb{Q}}}{\Bbb{R}})$ is [*[compact]{}*]{}. Thus, by making a choice of a CM-type $\Phi$, the group may be identified with $U(n)^{{\text{Hom}}(F,{\Bbb{R}})}$ (up to conjugation). It will be convenient to work over the rationals, and introduce $G=\text{Res}_{F|{\Bbb{Q}}}(U)$. With the same $\Phi$ one identifies $G({\Bbb{C}})$ with ${\text{GL}}_n({\Bbb{C}})^{{\text{Hom}}(F,{\Bbb{R}})}$. Weights of automorphic representations -------------------------------------- Following standard notation in the subject, $({\Bbb{Z}}^n)_+^{{\text{Hom}}({\mathcal{K}},{\Bbb{C}})}$ will denote the set of tuples $a=(a_{\tau})_{\tau\in {\text{Hom}}({\mathcal{K}},{\Bbb{C}})}$, where each $a_{\tau}=(a_{\tau,j})$ itself is a decreasing tuple, $$a_{\tau}=(a_{\tau,1}\geq a_{\tau,2}\geq \cdots \geq a_{\tau,n}),$$ of integers. In the obvious way, we can identify $a_{\tau}$ with a dominant weight for ${\text{GL}}(n)$, relative to the upper triangular Borel. We say $a_{\tau}$ is [*[regular]{}*]{} if all the inequalities above are strict. We say $a$ is regular if $a_{\tau}$ is regular for [*[some]{}*]{} $\tau$. Now, let $\pi=\pi_{\infty}\otimes \pi_f$ be an automorphic representation of $U({\Bbb{A}}_F)$. We will define what it means for $\pi$ to have weight $a$: Every embedding $\tau: {\mathcal{K}}\hookrightarrow {\Bbb{C}}$ restricts to a $\sigma: F \hookrightarrow {\Bbb{R}}$, which corresponds to an infinite place $v=v(\sigma)$ of $F$. With this notation, $\tau$ identifies $U(F_v)\simeq U(n)$, under which $\pi_v$ should be equivalent to the contragredient $\breve{V}_{a_{\tau}}$, or rather its restriction. Here $V_{a_{\tau}}$ is the irreducible algebraic representation of ${\text{GL}}_n({\Bbb{C}})$ of highest weight $a_{\tau}$. [*[Remark]{}*]{}. We must have $V_{a_{\tau c}}=\breve{V}_{a_{\tau}}$. In other words, $a_{\tau c,j}=-a_{\tau,n+1-j}$. Associating Galois representations ---------------------------------- We have introduced enough notation, in order to formulate the following main result, the foundation for our work. As mentioned already, this is the culmination of collaborative efforts of a huge group of outstanding mathematicians, as will become clear below. Choose a prime $p$, and an isomorphism $\iota: {\Bbb{C}}\overset{\sim}{\longrightarrow} \bar{{\Bbb{Q}}}_p$. Let $\pi$ be an automorphic representation of $U({\Bbb{A}}_F)$ such that $\pi_{\infty}$ has [**[regular]{}**]{} weight $a$. Then there exists a unique continuous semisimple Galois representation $$\rho_{\pi,\iota}: \Gamma_{{\mathcal{K}}}={\text{Gal}}(\bar{{\Bbb{Q}}}|{\mathcal{K}})\rightarrow {\text{GL}}_n(\bar{{\Bbb{Q}}}_p)$$ such that the following properties are satisfied: - $\breve{\rho}_{\pi,\iota}\simeq \rho_{\pi,\iota}^c \otimes \epsilon_{cyc}^{n-1}$, - For [**[every]{}**]{} finite place $v$, and every $w|v$ (even those above $p$), $$WD(\rho_{\pi,\iota}|_{\Gamma_{{\mathcal{K}}_w}})^{F-ss}\simeq \iota \text{rec}(BC_{w|v}(\pi_v)\otimes |\det|_w^{(1-n)/2}),$$ whenever $BC_{w|v}(\pi_v)$ is defined: If $\pi_v$ is unramified or $v=ww^c$ splits. - $\rho_{\pi,\iota}|_{\Gamma_{{\mathcal{K}}_w}}$ is potentially semistable for all $w|p$, with Hodge-Tate numbers $$HT_{\iota\tau}(\rho_{\pi,\iota}|_{\Gamma_{{\mathcal{K}}_w}})=\{a_{\tau,j}+(n-j): j=1,\ldots,n\}$$ for every $\tau:{\mathcal{K}}\hookrightarrow {\Bbb{C}}$ such that $\iota\tau$ lies above $w$. A word about our normalization here; $\rho_{\pi,\iota}\otimes_{\iota\tau,{\mathcal{K}}_w}{\Bbb{C}}_{{\mathcal{K}}_w}(i)$ has no $\Gamma_{{\mathcal{K}}_w}$-invariants unless $i$ is of the above form, in which case they form a line. Thus $HT_{\iota\tau}(\epsilon_{cyc})=\{-1\}$. [*[Proof]{}*]{}. Ngo’s proof of the fundamental lemma makes functoriality available in a slew of cases. In particular, weak base change from any unitary group associated with ${\mathcal{K}}|F$ to ${\text{GL}}_n({\Bbb{A}}_{{\mathcal{K}}})$ has matured. Building on work of Clozel and Labesse, White was recently able to work out the cohomological case completely \[Whit\]. In our given setup, $\pi_v$ is automatically discrete series for all $v|\infty$, in which case Theorem 6.1 in \[Whit\], or rather the pertaining remarks 6.2 and 6.3, yields an automorphic representation $$\Pi=\Pi_1\boxplus \cdots \boxplus \Pi_t$$ of ${\text{GL}}_n({\Bbb{A}}_{{\mathcal{K}}})$, which is an isobaric sum of mutually non-isomorphic conjugate self-dual cuspidal automorphic representations $\Pi_i$ of some ${\text{GL}}_{n_i}({\Bbb{A}}_{{\mathcal{K}}})$ such that $$\Pi_w=\text{BC}_{w|v}(\pi_v)$$ for all $w|v$, where $v$ is split or archimedean, or $\pi_v$ is unramified. The [*[regularity]{}*]{} of $\pi_{\infty}$ ensures that the $\Pi_i$ are cuspidal (as opposed to just discrete), which in turn implies the previous equality at the archimedean places $w|v$. Let us spell it out in that case: Fix an embedding $\tau:{\mathcal{K}}\hookrightarrow {\Bbb{C}}$ inducing ${\mathcal{K}}_w \simeq {\Bbb{C}}$. Then, $$\phi_{\Pi_w}: {\Bbb{C}}^*=W_{{\Bbb{C}}}\simeq W_{{\mathcal{K}}_w}\rightarrow {\text{GL}}_n({\Bbb{C}})$$ maps $$z \mapsto \begin{pmatrix}(z/\bar{z})^{-h_1+\frac{n-1}{2}} & & \\ & \ddots & \\ & & (z/\bar{z})^{-h_n+\frac{n-1}{2}}\end{pmatrix},$$ for certain $h_j \in {\Bbb{Z}}$, which are given in terms of the weight by $h_j=a_{\tau,j}+(n-j)$. This last formula is worked out in \[BeCl\] for example. These $h_j$ are distinct, so each $\Pi_i\otimes|\det|_{{\mathcal{K}}}^{(n_i-n)/2}$ is regular algebraic, [*[essentially]{}*]{} conjugate self-dual, and cuspidal. By Theorem A of \[BGGT\], and the references therein, we can associate a Galois representation $r_{\Pi_i,\iota}$ to it, satisfying the analogous properties of (a) to (c). As a remark, in loc. cit. local-global compatibility [*[at]{}*]{} $p$ is proved assuming Shin-regularity, which is much weaker than regularity. In any case, the Shin-regularity assumption is currently being removed by Caraiani as part of her 2012 Harvard Ph.D thesis. It is then straightforward to check that the representation $$\rho_{\pi,\iota}=r_{\Pi_1,\iota}\oplus \cdots \oplus r_{\Pi_t,\iota}$$ has the desired properties. It is uniquely determined by (b) by Tchebotarev. $\square$ [*[Remark]{}*]{}. It appears within reach to extend the previous argument to the [*[irregular]{}*]{} case. By \[Whit\], one still has a weak base change $\boxplus_{i=1}^t\Pi_i$, but the $\Pi_i$ are only [*[discrete]{}*]{}, not cuspidal. By Shapiro’s lemma in $(\frak{g},K)$-cohomology, these $\Pi_i$ should still be cohomological (of Speh type). By the Moeglin-Waldspuger description of the discrete spectrum of ${\text{GL}}(n_i)$, one can in turn express each $\Pi_i$ as an isobaric sum of cusp forms, with which one can associate Galois representations. After having consulted several experts in the field, we are quite optimistic about this line of argument, and that the ubiquitous regularity assumption (appearing throughout this paper) can safely be dropped. However, we have not made any serious attempt to work out the details. We are hopeful that forthcoming joint work of Kaletha-Shin-White should provide the strengthenings of \[Whit\] needed. The Breuil-Schneider recipe --------------------------- As described in detail in the introduction, to a potentially semistable representation $\rho:\Gamma_{{\mathcal{K}}_w}\rightarrow {\text{GL}}_n(\bar{{\Bbb{Q}}}_p)$, with distinct Hodge-Tate numbers, Breuil and Schneider attach a locally algebraic representation $\text{BS}(\rho)$ of ${\text{GL}}_n({\mathcal{K}}_w)$ on a $\bar{{\Bbb{Q}}}_p$-vector space. The algebraic part is given by all the Hodge-Tate numbers $\text{HT}(\rho)$ (for varying embeddings ${\mathcal{K}}_w \hookrightarrow \bar{{\Bbb{Q}}}_p$), and the smooth part is given by the Weil-Deligne representation $\text{WD}(\rho)$, or rather its Frobenius-semisimplification, via the classical local Langlands correspondence, slightly modified in the non-generic case. They conjecture the mere existence of an invariant norm on $\text{BS}(\rho)$. In fact, Conjecture 4.3 in \[BrSc\] also predicts a converse, which has been proved in \[Hu\]. Our goal is to first prove the Breuil-Schneider conjecture for $\rho=\rho_{\pi,\iota}|_{\Gamma_{{\mathcal{K}}_w}}$, for any place $w$ of ${\mathcal{K}}$ above $p$. We will achieve this below. For now, we will compute $\text{BS}(\rho)$ explicitly in this situation. In fact, we prefer to use a slightly different normalization: There is a choice of a [*[sign]{}*]{} involved in the recipe on p. 16 in \[BrSc\]. Instead of twisting by $|\det|_w^{(1-n)/2}$, we prefer to twist by $|\det|_w^{(n-1)/2}$ to make it more compatible with the previous notation. Consequently, $\text{BS}(\rho)$ becomes twisted by an integral character. For $a \in {\Bbb{Q}}_p^*$ we let $a^{\times}=a|a|_p$ denote its unit part. We introduce $$\mu: {\text{GL}}_n({\mathcal{K}}_w)\overset{\det}{\longrightarrow} {\mathcal{K}}_w^*\overset{N_{{\mathcal{K}}_w|{\Bbb{Q}}_p}}{\longrightarrow}{\Bbb{Q}}_p^*\longrightarrow {\Bbb{Q}}_p^*/p^{{\Bbb{Z}}}\simeq {\Bbb{Z}}_p^{\times}.$$ That is, $\mu(g)=N_{{\mathcal{K}}_w|{\Bbb{Q}}_p}(\det g)^{\times}$. We will normalize $\text{BS}(\rho)$ as follows: $$\widetilde{\text{BS}}(\rho):=\text{BS}(\rho)\otimes_{\bar{{\Bbb{Q}}}_p}\mu^{n-1}.$$ (Of course, this has an invariant norm precisely when $\text{BS}(\rho)$ does.) $\widetilde{\text{BS}}(\rho)=\text{BS}(\rho(n-1))$. [*[Proof]{}*]{}. Note that the character $a \mapsto a^{\times}$ (which maps $p \mapsto 1$, and is the identity on ${\Bbb{Z}}_p^{\times}$) corresponds to the $p$-adic cyclotomic character $\chi_{cyc}: \Gamma_{{\Bbb{Q}}_p}\rightarrow {\Bbb{Z}}_p^{\times}$ via local class field theory ${\Bbb{Q}}_p^*\rightarrow\Gamma_{{\Bbb{Q}}_p}^{ab}$. For any $p$-adic field $K$, it follows that $BS(\chi_{cyc})$ is simply the character $a \mapsto N_{K|{\Bbb{Q}}_p}(a)^{\times}$. Consequently, $\widetilde{\text{BS}}(\rho)=\text{BS}(\rho\otimes \chi_{cyc}^{n-1})$. $\square$ We compute it in the automorphic case. Given the local-global compatibility results of \[BGGT\] (generalized in \[Car\]), this is basically just “bookkeeping”. Let $\pi$ be an automorphic representation of $U({\Bbb{A}}_F)$ of regular weight $a$. Assume $\rho_{\pi,\iota}$ is absolutely [**[irreducible]{}**]{} (as a representation of the full Galois group $\Gamma_{{\mathcal{K}}}$). Let $v|p$ be a place of $F$, either split in ${\mathcal{K}}$, or such that $\pi_v$ is unramified. Then, for any place $w|v$ of ${\mathcal{K}}$, we have $$\widetilde{\text{BS}}(\rho_{\pi,\iota}|_{\Gamma_{{\mathcal{K}}_w}})=(\otimes_{\sigma: {\mathcal{K}}_w \hookrightarrow \bar{{\Bbb{Q}}}_p} \breve{V}_{a_{\iota^{-1}\sigma}}) \otimes_{\bar{{\Bbb{Q}}}_p}(BC_{w|v}(\pi_v)\otimes_{{\Bbb{C}},\iota}\bar{{\Bbb{Q}}}_p).$$ (We abuse notation a bit and let $V_{a_{\tau}}$ denote the irreducible algebraic representation of ${\text{GL}}_n(\bar{{\Bbb{Q}}}_p)$ of highest weight $a_{\tau}$; as opposed to the [*[complex]{}*]{} representation from earlier chapters) [*[Proof]{}*]{}. What is denoted $\pi^{\text{unit}}$ in \[BrSc\] equals $BC_{w|v}(\pi_v)\otimes |\det|_w^{(1-n)/2}$ in our case (more precisely, $\otimes_{{\Bbb{C}},\iota}\bar{{\Bbb{Q}}}_p$). When it is generic, the smooth part of $\text{BS}(\rho)$ is $$\pi^{\text{unit}}\otimes_{\bar{{\Bbb{Q}}}_p} |\det|_w^{(1-n)/2}=(BC_{w|v}(\pi_v)\otimes |\det|_w^{1-n})\otimes_{{\Bbb{C}},\iota}\bar{{\Bbb{Q}}}_p.$$ In the non-generic case, $\pi^{\text{unit}}$ has to be replaced by a certain parabolically induced representation. However, if we assume $\rho_{\pi,\iota}$ is (globally) irreducible, we see that $\Pi=\text{BC}_{{\mathcal{K}}|F}(\pi)$ must be cuspidal, and in particular $\Pi_w$ is generic. The algebraic part of $\text{BS}(\rho)$ is constructed out of the Hodge-Tate numbers: What is denoted $i_{j,\sigma}$ in \[BrSc\], for an embedding $\sigma:{\mathcal{K}}_w \hookrightarrow \bar{{\Bbb{Q}}}_p$, equals $a_{\tau,n+1-j}+(j-1)$ in our notation, where $\sigma=\iota\tau$. In (8) on p. 17 of \[BrSc\], the numbers become $$b_{\tau,j}:=-i_{n+1-j,\sigma}-(j-1)=-a_{\tau,j}-(n-1).$$ Breuil-Schneider’s $\rho_{\sigma}$ is the irreducible algebraic representation of ${\text{GL}}_n(\bar{{\Bbb{Q}}}_p)$ of highest weight $b_{\tau,1}\leq \cdots\leq b_{\tau,n}$ relative to the [*[lower]{}*]{} triangular Borel. Relative to the upper triangular Borel, $\rho_{\sigma}$ has highest weight $b_{\tau,n}\geq \cdots\geq b_{\tau,1}$, so that $\rho_{\sigma}\simeq \breve{V}_{a_{\tau}}\otimes \det^{1-n}$ (more precisely, $\otimes_{{\Bbb{C}},\iota}\bar{{\Bbb{Q}}}_p$). Altogether, the algebraic part is $$\xi=\otimes_{\sigma} \rho_{\sigma}\simeq \otimes_{\tau|w} (\breve{V}_{a_{\tau}}\otimes {\det}^{1-n})$$ (the tensor product ranging over $\tau:{\mathcal{K}}\hookrightarrow {\Bbb{C}}$ such that $\iota\tau$ induces $w$). Here we abuse notation a bit, and use $V_{a_{\tau}}$ to denote the irreducible algebraic representation of ${\text{GL}}_n(\bar{{\Bbb{Q}}}_p)$ of highest weight $a_{\tau}$. As a representation of ${\text{GL}}_n({\mathcal{K}}_w)$, embedded diagonally in $\prod_{\sigma: {\mathcal{K}}_w \hookrightarrow \bar{{\Bbb{Q}}}_p}{\text{GL}}_n(\bar{{\Bbb{Q}}}_p)$, the algebraic part becomes $$\xi=(\otimes_{\sigma: {\mathcal{K}}_w \hookrightarrow \bar{{\Bbb{Q}}}_p} \breve{V}_{a_{\iota^{-1}\sigma}}) \otimes (N_{{\mathcal{K}}_w|{\Bbb{Q}}_p}\circ \det)^{1-n},$$ which yields the result. $\square$ Completed Cohomology ==================== In this section we will prove the Breuil-Schneider conjecture, 4.3 in \[BrSc\], for the potentially semistable representations $\rho=\rho_{\pi,\iota}|_{\Gamma_{{\mathcal{K}}_w}}$ above. This will make heavy use of ideas of Emerton introduced in \[Emer\]. The basic idea is to view $\widetilde{\text{BS}}(\rho)$ as a component of the $p$-adic automorphic representation $\tilde{\pi}=\tilde{\pi}_p\otimes \pi_f^p$ attached to $\pi$, which in turn embeds into the completed cohomology $\tilde{H}^0$ for $G$. The $p$-adic automorphic representation --------------------------------------- We keep our automorphic representation $\pi$ of $U({\Bbb{A}}_F)$ of regular weight $a$. Recall that we introduced the group $G=\text{Res}_{F|{\Bbb{Q}}}(U)$. Interchangeably, below we will view $\pi$ as an automorphic representation of $G({\Bbb{A}})$. We will follow p. 52 in \[Emer\] in attaching a $p$-adic automorphic representation to $\pi$. (The ${\Bbb{G}}$ there will be our $G$, and $F$ there will be ${\Bbb{Q}}$.) This can be done for $W$-allowable $\pi$, where $W$ is an irreducible algebraic representation of $G({\Bbb{C}})$, which in this case (where $G$ is compact at infinity) simply means $\pi_{\infty}\simeq W|_{G({\Bbb{R}})}$. See Definition 3.1.3 in \[Emer\]. To make this more explicit, in terms of the weight $a$, we need to make some identifications. Let us choose a CM-type $\Phi$. For each $\sigma:F \hookrightarrow {\Bbb{R}}$ we let $\tilde{\sigma}$ denote its lift in $\Phi$. Thus the two extensions to ${\mathcal{K}}$ are $\{\tilde{\sigma},\tilde{\sigma}^c\}$. Via the choice of $\Phi$, $$\text{$G({\Bbb{C}})\overset{\sim}{\longrightarrow}_{\Phi} {\text{GL}}_n({\Bbb{C}})^{{\text{Hom}}(F,{\Bbb{R}})}$, ${\hspace{6pt}}$ $G({\Bbb{R}})\overset{\sim}{\longrightarrow}_{\Phi} U(n)^{{\text{Hom}}(F,{\Bbb{R}})}$.}$$ We immediately infer that $W\simeq \otimes_{\sigma \in {\text{Hom}}(F,{\Bbb{R}})} \breve{V}_{a_{\tilde{\sigma}}}$ under these identifications. Via $\iota: {\Bbb{C}}\overset{\sim}{\longrightarrow} \bar{{\Bbb{Q}}}_p$ we identify $W$ with an algebraic representation of $G(\bar{{\Bbb{Q}}}_p)$. $$G(\bar{{\Bbb{Q}}}_p)\overset{\sim}{\longrightarrow}_{\Phi} \prod_{v|p}{\text{GL}}_n(\bar{{\Bbb{Q}}}_p)^{{\text{Hom}}(F_v,\bar{{\Bbb{Q}}}_p)}$$ allows us to factor our $p$-adic $W$ accordingly, as $W\simeq \otimes_{v|p}W_v$, where we let $$W_v=\otimes_{\sigma \in {\text{Hom}}(F,{\Bbb{R}}): \sigma|v}\breve{V}_{a_{\tilde{\sigma}}}.$$ In the same vein, $G({\Bbb{Q}}_p)=\prod_{v|p}U(F_v)$. To go any further, from this point on we assume every $v|p$ splits in ${\mathcal{K}}$, and that $D_w\simeq M_n({\mathcal{K}}_w)$ for each divisor $w|v$. Then $U(F_v)\overset{\sim}{\longrightarrow} {\text{GL}}_n({\mathcal{K}}_w)$, defined up to conjugation. If we assume (as we may) that our CM-type $\Phi$ is ordinary at $\iota$, in the sense of \[Katz\], $\Phi$ singles out a place $\tilde{v}$ of ${\mathcal{K}}$ above each $v|p$ of $F$. With this selection of places at hand, $$G({\Bbb{Q}}_p) \overset{\sim}{\longrightarrow} \prod_{v|p} {\text{GL}}_n({\mathcal{K}}_{\tilde{v}}).$$ Moreover, the inclusion into $G(\bar{{\Bbb{Q}}}_p)$ corresponds to the diagonal embeddings, $${\text{GL}}_n({\mathcal{K}}_{\tilde{v}})={\text{GL}}_n(F_v)\hookrightarrow {\text{GL}}_n(\bar{{\Bbb{Q}}}_p)^{{\text{Hom}}(F_v,\bar{{\Bbb{Q}}}_p)}.$$ The following is Definition 3.1.5 in \[Emer\], except that we are working with representations over $\bar{{\Bbb{Q}}}_p$ instead of descending to a finite extension of ${\Bbb{Q}}_p$. The classical $p$-adic automorphic representation of $G({\Bbb{A}}_f)$ over $\bar{{\Bbb{Q}}}_p$ attached to the $W$-allowable automorphic representation $\pi$ of $G({\Bbb{A}})$ is $$\text{$\tilde{\pi}:=\tilde{\pi}_p\otimes_{\bar{{\Bbb{Q}}}_p}\pi_f^p$, ${\hspace{6pt}}$ $\tilde{\pi}_p:=W \otimes_{\bar{{\Bbb{Q}}}_p}\pi_p$.}$$ Here $G({\Bbb{Q}}_p)$ acts diagonally on $W \otimes_{\bar{{\Bbb{Q}}}_p}\pi_p$, and $G({\Bbb{A}}_f^p)$ acts through the second factor $\pi_f^p$. (Abusing notation, we write $\pi_p$ instead of $\pi_p\otimes_{{\Bbb{C}},\iota}\bar{{\Bbb{Q}}}_p$ and so on.) At each $v|p$ we introduce $\tilde{\pi}_v=W_v \otimes_{\bar{{\Bbb{Q}}}_p}\text{BC}_{\tilde{v}|v}(\pi_v)$, a locally algebraic representation of ${\text{GL}}_n({\mathcal{K}}_{\tilde{v}})$, which depends on the choice of an ordinary CM-type $\Phi$. Moreover, $\tilde{\pi}_p\simeq \otimes_{v|p}\tilde{\pi}_v$ under the isomorphism $G({\Bbb{Q}}_p)\simeq \prod_{v|p}{\text{GL}}_n({\mathcal{K}}_{\tilde{v}})$. This leads to the main result of this section. Hypothesis: Every $v|p$ of $F$ splits in ${\mathcal{K}}$, and $D_w\simeq M_n({\mathcal{K}}_w)$ for all $w|v$. For each $v|p$ of $F$ pick a place $\tilde{v}|v$ of ${\mathcal{K}}$ (this amounts to choosing an $\iota$-ordinary CM-type). Let $\pi$ be an automorphic representation of $U({\Bbb{A}}_F)$ of regular weight, and assume $\rho_{\pi,\iota}$ is (globally) irreducible. Then, for all $v|p$ of $F$, $$\widetilde{\text{BS}}(\rho_{\pi,\iota}|_{\Gamma_{{\mathcal{K}}_{\tilde{v}}}})\simeq \tilde{\pi}_v,$$ which embeds into $\tilde{\pi}|_{{\text{GL}}_n({\mathcal{K}}_{\tilde{v}})}$ (where we restrict via $U(F_v)\overset{\sim}{\longrightarrow}{\text{GL}}_n({\mathcal{K}}_{\tilde{v}})$). [*[Proof]{}*]{}. This follows from the preceding discussion, combined with the computation of the Breuil-Schneider representation in Corollary 1 above. $\square$ Algebraic modular forms ----------------------- We will study the space of modular forms for $G$ of a given weight. To put things in a broader perspective, we will use the cohomological framework of \[Emer\], although we will only work with $H^0$, which is explicit, and of a combinatorial nature. In our situation, $G({\Bbb{R}})$ is compact and connected, so things simplify tremendously, and we only have cohomology in degree zero. Indeed, for every compact open subgroup $K\subset G({\Bbb{A}}_f)$, the corresponding arithmetic quotient is a [*[finite]{}*]{} set: $$Y(K)=G({\Bbb{Q}})\backslash G({\Bbb{A}}_f)/K.$$ An irreducible algebraic representation $W$ of $G({\Bbb{C}})$ defines a local system $\mathcal{V}_W$ on each $Y(K)$, and $H^0(Y(K),\mathcal{V}_{\breve{W}})$ is identified with the space of modular forms, of level $K$, and weight $W$. That is, all functions $f: G({\Bbb{A}}_f)\rightarrow \breve{W}$, which are $K$-invariant on the right, such that $f(\gamma g)=\gamma f(g)$ for all elements $\gamma \in G({\Bbb{Q}})$. $$H^0(\mathcal{V}_{\breve{W}}):=\underset{K}{\varinjlim} H^0(Y(K),\mathcal{V}_{\breve{W}})\simeq \oplus_{\pi: \pi_{\infty}\simeq W}m_G(\pi)\pi_f$$ is then a smooth admissible semisimple representation of $G({\Bbb{A}}_f)$, which we wish to suitably $p$-adically complete. Via our choice of $\iota: {\Bbb{C}}\overset{\sim}{\longrightarrow} \bar{{\Bbb{Q}}}_p$, we will view $W$ as a representation of $G(\bar{{\Bbb{Q}}}_p)$ and so on. Occasionally it will be convenient to work over a field $E\subset \bar{{\Bbb{Q}}}_p$, finite over ${\Bbb{Q}}_p$. It suffices to take $E$ large enough, so that it contains the image of every embedding $F \hookrightarrow \bar{{\Bbb{Q}}}_p$. In that case $G$ splits over $E$, and by highest weight theory $W$ may be defined over $E$. Thus, from now on, $H^0(\mathcal{V}_{\breve{W}})$ is an $E$-vector space with a smooth admissible $G({\Bbb{A}}_f)$-action. For each tame level $K^p\subset G({\Bbb{A}}_f^p)$, following \[Emer\], we introduce $$H^0(K^p,\mathcal{O}_E/\varpi_E^s):=\underset{K_p}{\varinjlim} H^0(Y(K_pK^p),\mathcal{O}_E/\varpi_E^s),$$ and $$\tilde{H}^0(K^p):=E \otimes_{\mathcal{O}_E}\underset{s}{\varprojlim} H^0(K^p,\mathcal{O}_E/\varpi_E^s).$$ The latter is an $E$-Banach space with a unitary $G({\Bbb{Q}}_p)$-action, commuting with the action of the Hecke algebra $\mathcal{H}(K^p)$ of compactly supported $K^p$-biinvariant $E$-valued functions on $G({\Bbb{A}}_f^p)$. In fact, it becomes a Banach module over the completion $\hat{\mathcal{H}}(K^p)$. Also, $$\tilde{H}^0:=\underset{K^p}{\varinjlim} \tilde{H}^0(K^p),$$ a locally convex $E$-vector space with an action of $G({\Bbb{A}}_f)$. In our simple setup, they can all be realized very explicitly. For example, $$\text{$\tilde{H}^0(K^p)=\{\text{continuous $Y(K^p)\overset{f}{\rightarrow} E$}\}$, ${\hspace{6pt}}$ $Y(K^p)=\underset{K_p}{\varprojlim} Y(K_pK^p)$,}$$ with the supremum-norm $\|\cdot\|$. The [*[key]{}*]{} ingredient we will use is the isomorphism: For any absolutely irreducible algebraic representation $W$, $$W \otimes_E H^0(\mathcal{V}_{\breve{W}}) \overset{\sim}{\longrightarrow} (\tilde{H}^0)_{\text{$W$-alg}}.$$ (When $W$ is only irreducible over $E$, tensor over ${\text{End}}_{\frak{g}}(W)$, where $\frak{g}=\text{Lie}G({\Bbb{Q}}_p)$.) [*[Proof]{}*]{}. This is Corollary 2.2.25 in \[Emer\] (also spelled out in \[Sor\] for $H^0$). Let us briefly sketch the main idea. For any tame level $K^p$, one shows that $$W \otimes_E H^0(K^p,\mathcal{V}_{\breve{W}})=\underset{K_p}{\varinjlim} W \otimes_E H^0(Y(K_pK^p),\mathcal{V}_{\breve{W}})) \overset{\sim}{\longrightarrow} \tilde{H}^0(K^p)_{\text{$W$-alg}}.$$ This goes as follows: $H^0(Y(K_pK^p),\mathcal{V}_{\breve{W}}))$ is a space of classical $p$-adic modular forms, and it is an easy exercise to identify it with ${\text{Hom}}_{K_p}(W, \tilde{H}^0(K^p))$. Now, $$W \otimes_E {\text{Hom}}_{K_p}(W, \tilde{H}^0(K^p))\overset{eval.}{\longrightarrow} \tilde{H}^0(K^p)$$ is injective since $W$ is absolutely irreducible, even when restricted to $K_p$ (which is Zariski dense). The image of this evaluation map is the $W$-isotypic subspace of $\tilde{H}^0(K^p)$. As $K_p$ varies, the maps are compatible, and produces a map out of the direct limit onto $\tilde{H}^0(K^p)_{\text{$W$-alg}}$ as desired. $\square$ [*[Remark]{}*]{}. For higher degree cohomology $H^i$ there is an analogous canonical $G({\Bbb{A}}_f)$-equivariant map, which occurs as the edge map of a certain spectral sequence, but it is not known to be injective for groups other than ${\text{GL}}(2)_{{\Bbb{Q}}}$ (and groups $G$ which are compact at infinity mod center). Injectivity is what makes the whole machinery of \[Emer\] work, see his Theorem 0.7 and Proposition 2.3.8 on p. 47, for example. In particular, it is available in our case, where $G({\Bbb{R}})$ is compact. In general, one would have to localize the spectral sequence at a “cohomologically” non-Eisenstein maximal ideal ${\frak{m}}$ (which means it does not contribute to mod $p$ cohomology outside the middle degree). This is expected to hold when the Galois representation $\bar{\rho}_{{\frak{m}}}$ is absolutely irreducible, but this is difficult to show. Partial results are now available for $U(2,1)$. From the previous discussion, we get decompositions of completed cohomology: - $\bar{{\Bbb{Q}}}_p\otimes_E (\tilde{H}^0)_{\text{$W$-alg}} \simeq \oplus_{\pi:\pi_{\infty}\simeq W} m_G(\pi) \tilde{\pi}$, - $\bar{{\Bbb{Q}}}_p\otimes_E \tilde{H}^0(K^p)_{\text{$W$-alg}} \simeq \oplus_{\pi:\pi_{\infty}\simeq W} m_G(\pi) (\tilde{\pi}_p\otimes_{\bar{{\Bbb{Q}}}_p} (\pi_f^p)^{K^p})$. Now, suppose $\frak{h}\subset \mathcal{H}(K^p)$ is a central subalgebra. It then acts on $(\pi_f^p)^{K^p}$ by a character $\lambda_{\pi}: \frak{h}\rightarrow \bar{{\Bbb{Q}}}_p$. Conversely, say we start out with $\lambda: \frak{h}\rightarrow \bar{{\Bbb{Q}}}_p$. Then, $$\bar{{\Bbb{Q}}}_p\otimes_E \tilde{H}^0(K^p)_{\text{$W$-alg}}^{\frak{h}=\lambda} \simeq \oplus_{\pi:\pi_{\infty}\simeq W, \lambda_{\pi}=\lambda} m_G(\pi) (\tilde{\pi}_p\otimes_{\bar{{\Bbb{Q}}}_p} (\pi_f^p)^{K^p}).$$ As always, we assume $W$ has [*[regular]{}*]{} weight, so we know how to attach Galois representations. If $\frak{h}$ contains the spherical part $\mathcal{H}(K^p)^{\text{sph}}$, all the $\pi$ contributing to the right-hand side have the same Galois representation $\rho_{\lambda}$, by Chebotarev, which we assume is [*[irreducible]{}*]{}. By Proposition 1, we may factor the above, $$\bar{{\Bbb{Q}}}_p\otimes_E \tilde{H}^0(K^p)_{\text{$W$-alg}}^{\frak{h}=\lambda} \simeq (\otimes_{v|p} \widetilde{BS}(\rho_{\lambda}|_{\Gamma_{{\mathcal{K}}_{\tilde{v}}}})) \otimes_{\bar{{\Bbb{Q}}}_p} (\oplus_{\pi:\pi_{\infty}\simeq W,\lambda_{\pi}=\lambda} m_G(\pi)\pi_f^p)^{K^p}.$$ This has the form of a $G({\Bbb{Q}}_p)\simeq \prod_{v|p}{\text{GL}}_n({\mathcal{K}}_{\tilde{v}})$-representation tensor an $\mathcal{H}(K^p)$-module. In particular, since $\tilde{H}^0(K^p)$ carries a $G({\Bbb{Q}}_p)$-invariant norm, we finally deduce the Breuil-Schneider conjecture for automorphic Galois representations: If $\pi$ is an automorphic representation of $U({\Bbb{A}}_F)$, of regular weight, such that $\rho_{\pi,\iota}$ is irreducible. Then $\text{BS}(\rho_{\pi,\iota}|_{\Gamma_{{\mathcal{K}}_w}})$ admits a ${\text{GL}}_n({\mathcal{K}}_w)$-invariant norm, for all places $w|p$ of ${\mathcal{K}}$. The discussion leading up to the Theorem strongly suggests a better formulation in terms of eigenvarieties. We will employ this machinery in the next Chapter. Eigenvarieties ============== Eigenvarieties are rigid analytic spaces interpolating Hecke eigensystems occurring in spaces of automorphic forms, of varying weight. Historically, the first example is the Coleman-Mazur eigencurve for ${\text{GL}}(2)_{{\Bbb{Q}}}$, revisited by Buzzard, Emerton, Urban, and others. There are different constructions for any reductive group $G$, which each have their drawbacks and limitations. When $G({\Bbb{R}})$ is compact, however, the theory is in good shape, and all constructions are compatible. Below we will combine the approach of \[Emer\] with that of \[Chen\] (for arbitrary totally real $F$) extending parts of \[BeCh\] (when $F={\Bbb{Q}}$). The classical points -------------------- By our standing hypotheses, $G_{{\Bbb{Q}}_p}\simeq \prod_{v|p}\text{Res}_{{\mathcal{K}}_{\tilde{v}}|{\Bbb{Q}}_p}{\text{GL}}(n)$ is quasi-split, and we pick the Borel pair $(B,T)$, defined over ${\Bbb{Q}}_p$, corresponding to the product of the upper triangular pairs in each ${\text{GL}}_n({\mathcal{K}}_{\tilde{v}})$. As in \[Emer\], let $\hat{T}$ denote the [*[weight]{}*]{} space. That is, the rigid analytic variety over $E$ (a subfield of $\bar{{\Bbb{Q}}}_p$, finite over ${\Bbb{Q}}_p$, but large enough so that $G$ splits over $E$) which parametrizes the locally analytic characters on $T({\Bbb{Q}}_p)$. In other words, $$\hat{T}(A)={\text{Hom}}_{la}(T({\Bbb{Q}}_p),A^{\times})$$ for any affinoid $E$-algebra $A$. It comes with a universal map $T({\Bbb{Q}}_p)\rightarrow \mathcal{O}(\hat{T})^{\times}$. The eigenvariety depends on the choice of [*[tame]{}*]{} level $K^p\subset G({\Bbb{A}}_f^p)$, which we will always assume is decomposable as $\prod_{v\nmid p}K_v$, where $K_v$ is a compact open subgroup of $U(F_v)$, which is hyperspecial for all but finitely many $v$. Say, for all $v \notin S(K^p)$. Correspondingly, the Hecke algebra factors as a tensor product, $$\mathcal{H}(K^p)=\otimes_{v\nmid p} \mathcal{H}(K_v)=\mathcal{H}(K^p)^{\text{ram}}\otimes_E \mathcal{H}(K^p)^{\text{sph}}.$$ Here $\mathcal{H}(K^p)^{\text{sph}}=\otimes_{v\notin S(K^p)} \mathcal{H}(K_v)$ sits as a central subalgebra of $\mathcal{H}(K^p)$, hence it acts by a character on $\pi_f^{K^p}$, for any automorphic $\pi$ with $K^p$-invariants. We now make precise which points we wish to interpolate by an eigenvariety. Let $E(0,K^p)_{cl}\subset (\hat{T}\times \text{Spec} \mathcal{H}(K^p)^{\text{sph}})(\bar{{\Bbb{Q}}}_p)$ be the subset of pairs $x=(\chi,\lambda)$ for which there exists an irreducible $G({\Bbb{A}}_f)$-subquotient $\pi_f$ of $\bar{{\Bbb{Q}}}_p\otimes_E H^0(\mathcal{V}_{\breve{W}})$, where $W$ is an irreducible algebraic representation of $G_E$, such that - $\chi=\psi\theta$, where $\psi$ is the highest weight of $W$ (relative to $B$), and $\theta$ is a smooth character of $T({\Bbb{Q}}_p)$ such that $\pi_p \hookrightarrow \text{Ind}_{B({\Bbb{Q}}_p)}^{G({\Bbb{Q}}_p)}(\theta)$, - $\pi_f^{K^p}\neq 0$, and $\mathcal{H}(K^p)^{\text{sph}}$ acts on it via $\lambda$. This is the definition, and notation, used on p. 5 in \[Emer\]. Eigenvariety conventions ------------------------ Emerton defines the degree zero cohomological eigenvariety of $G$, of tame level $K^p$, to be the rigid analytic closure of $E(0,K^p)_{cl}$ in $\hat{T}\times \text{Spec} \mathcal{H}(K^p)^{\text{sph}}$. By the uniqueness part of Theorem 1.6 in \[Chen\], it coincides with the eigenvariety defined there. We will intertwine the two points of view. Thus, with $E(0,K^p)_{cl}$ is associated a quadruple $({\Bbb{X}},\chi,\lambda,X_{cl})$, consisting of the following data: - ${\Bbb{X}}_{/E}$ is an equi-dimensional reduced rigid analytic variety, - $\chi: {\Bbb{X}}\rightarrow \hat{T}$ is a finite morphism (Theorem 0.7 (i) on p. 6 in \[Emer\]), - $\lambda: \mathcal{H}(K^p)^{\text{sph}} \rightarrow \mathcal{O}({\Bbb{X}})$ is an $E$-algebra homomorphism, - $X_{cl}\subset {\Bbb{X}}(\bar{{\Bbb{Q}}}_p)$ is a Zariski-dense subset, satisfying various properties (listed in Theorem 1.6 in \[Chen\], for example), the most important of which is the following: The canonical evaluation map, $$\text{${\Bbb{X}}(\bar{{\Bbb{Q}}}_p)\longrightarrow (\hat{T}\times \text{Spec} \mathcal{H}(K^p)^{\text{sph}})(\bar{{\Bbb{Q}}}_p)$, ${\hspace{6pt}}$ $x \mapsto (\chi_x,\lambda_x)$,}$$ induces a [*[bijection]{}*]{}, $$X_{cl} \overset{\sim}{\longrightarrow} E(0,K^p)_{cl}.$$ Moreover, there is a classicality criterion, analogous to Coleman’s “non-critical slope implies classical”, which we will not use directly (we will use that $X_{cl}$ is Zariski dense, though). More properties will be recalled below when needed, such as the connection with Emerton’s Jacquet functor. [*[Notation]{}*]{}. Following standard usage, by ${\Bbb{X}}(\bar{{\Bbb{Q}}}_p)$ we mean the union (or direct limit) of all ${\Bbb{X}}(L)={\text{Hom}}_{E}(\text{Sp}(L),{\Bbb{X}})$, where $L$ ranges over all finite extensions of $E$. [*[Remark]{}*]{}. There is recent paper \[Loef\], in which Loeffler spells out how Chenevier’s construction is related to Emerton’s (in the case where $G({\Bbb{R}})$ is compact). In addition, he introduces so-called [*[intermediate]{}*]{} eigenvarieties, where one replaces $B$ with an arbitrary parabolic subgroup (and drops the assumption that $G$ should be quasi-split at $p$). It would be interesting to adapt our arguments to that setting, and thereby make progress towards the Breuil-Schneider conjecture when $\pi_p$ does not embed in a principal series (induced from the Borel). This ought to put the results of this paper, and that of \[Sor\], under the same roof. We hope to return to this question in the near future. The Galois pseudo-character --------------------------- At each point $x \in {\Bbb{X}}(\bar{{\Bbb{Q}}}_p)$ we will assign a continuous semisimple Galois representation $\rho_x:\Gamma_{{\mathcal{K}}}\rightarrow {\text{GL}}_n(\bar{{\Bbb{Q}}}_p)$, which is unramified outside $\Sigma=\Sigma(K^p)$, the places of ${\mathcal{K}}$ above $S(K^p)$. This is first done at a dense set of classical points, then by a formal argument one interpolates ${\text{tr}}(\rho_x)$ by a pseudo-character. We refer to Chapter 1 of \[BeCh\] for an extensive elegant introduction to pseudo-representations, a notion going back to Wiles for ${\text{GL}}(2)$, and Taylor for ${\text{GL}}(n)$. Let $X_{reg}\subset X_{cl}$ be the subset of points $x$ such that $\chi_x=\psi_x\theta_x$, where $\psi_x=\otimes_{\sigma \in {\text{Hom}}(F,\bar{{\Bbb{Q}}}_p)} \psi_{x,\tilde{\sigma}}$ is a regular character of $T$. That is, some $\psi_{x,\tilde{\sigma}}$ is a regular dominant character of $T_{{\text{GL}}(n)}$ in the usual sense. This is a Zariski-dense subset of ${\Bbb{X}}(\bar{{\Bbb{Q}}}_p)$, see p. 18 in \[Chen\], and the references given there. Now let $x \in X_{reg}$, and look at the corresponding pair $(\chi_x=\psi_x\theta_x,\lambda_x)$. There exists an irreducible $G({\Bbb{A}}_f)$-summand $\pi_f$ in $\bar{{\Bbb{Q}}}_p\otimes_E H^0(\mathcal{V}_{\breve{W}_x})$, where $W_x$ has regular highest weight $\psi_x$, such that $\mathcal{H}(K^p)^{\text{sph}}$ acts on $\pi_f^{K^p}\neq 0$ via $\lambda_x$, and $\pi_p \hookrightarrow \text{Ind}_{B({\Bbb{Q}}_p)}^{G({\Bbb{Q}}_p)}(\theta_x)$. Thus $\iota^{-1}\pi_f$ is the finite part of an automorphic representation of $U({\Bbb{A}}_F)$ of regular weight $W_x$, unramified outside $S(K^p)$, to which we can associate a continuous semisimple Galois representation $$\rho_x: \Gamma_{{\mathcal{K}}}\rightarrow {\text{GL}}_n(\bar{{\Bbb{Q}}}_p)$$ with the following properties: - $\rho_x^{\vee}\simeq \rho_x^c \otimes \epsilon_{cyc}^{n-1}$. - For every finite place $v\nmid p$ of $F$, [**[outside]{}**]{} $S(K^p)$, and every $w|v$ of ${\mathcal{K}}$, the local representation $\rho_x|_{\Gamma_{{\mathcal{K}}_w}}$ is unramified, and satisfies the identity: $${\text{tr}}\rho_x(\text{Frob}_w)=\lambda_x(b_{w|v}(h_w)).$$ (Here $\text{Frob}_w$ is a geometric Frobenius, $h_w$ is the element of the spherical Hecke algebra for ${\text{GL}}_n({\mathcal{K}}_w)$ acting on an unramified $\Pi_w$ by $\sum \alpha_i$, where the $\alpha_i$ are the integral Satake parameters. Finally, the map $$b_{w|v}: \mathcal{H}({\text{GL}}_n({\mathcal{K}}_w),K_w)\rightarrow \mathcal{H}(U(F_v),K_v)$$ is the base change homomorphism between the spherical Hecke algebras. See \[Min\] for a careful useful discussion of this latter map.) - For every finite place $v|p$ of $F$, the local representation $\rho_x|_{\Gamma_{{\mathcal{K}}_{\tilde{v}}}}$ is potentially semistable. Furthermore, - The semisimplification of the attached Weil-Deligne representation is $$WD(\rho_x|_{\Gamma_{{\mathcal{K}}_{\tilde{v}}}})^{ss}\simeq \oplus_{i=1}^n (\theta_{x,\tilde{v}}^{(i)} \circ \text{Art}_{{\mathcal{K}}_{\tilde{v}}}^{-1}).$$ (Here $\theta_x=\otimes_{v|p}\theta_{x,\tilde{v}}$, where $\theta_{x,\tilde{v}}$ is a smooth character of the diagonal torus $T_{{\text{GL}}(n)}({\mathcal{K}}_{\tilde{v}})\simeq ({\mathcal{K}}_{\tilde{v}}^*)^n$, factored as a product $\theta_{x,\tilde{v}}^{(1)}\otimes \cdots \otimes \theta_{x,\tilde{v}}^{(n)}$.) - The Hodge-Tate numbers are, for any embedding $\tau: {\mathcal{K}}_{\tilde{v}}\hookrightarrow \bar{{\Bbb{Q}}}_p$, $$\text{HT}_{\tau}(\rho_x|_{\Gamma_{{\mathcal{K}}_{\tilde{v}}}})=\{a_{\tau,j}+(n-j): j=1,\ldots, n\},$$ where the tuple $(a_{\tau,j})$ corresponds to the dominant character $\psi_{x,v,\tau}$ of $T_{{\text{GL}}(n)}$. (Here we factor $\psi_x=\otimes_{v|p}\otimes_{\tau: {\mathcal{K}}_{\tilde{v}}\hookrightarrow \bar{{\Bbb{Q}}}_p} \psi_{x,v,\tau}$.) Observe that there may be many automorphic representations associated to a given point $x\in X_{cl}$, but they are all isomorphic outside $S(K^p)$ (and of the same weight). In particular, by (b) and Chebotarev, the Galois representation is independent of the choice of $\pi_f$, justifying the notation $\rho_x$. There exists a unique continuous $n$-dimensional pseudo-character $\mathcal{T}: \Gamma_{{\mathcal{K}},\Sigma}\rightarrow \mathcal{O}({\Bbb{X}})^{\leq 1}$ such that $\mathcal{T}(\text{Frob}_w)=\lambda(b_{w|v}(h_w))$ for all places $w \notin \Sigma$. [*[Proof]{}*]{}. We are in the situation of Proposition 7.1.1 in \[Che\]: ${\Bbb{X}}$ is reduced, $\mathcal{O}({\Bbb{X}})^{\leq 1}$ is a compact subring, and for all $x \in X_{reg}$, a Zariski-dense subset, we have a representation $\rho_x$ of $\Gamma_{{\mathcal{K}},\Sigma}$ such that ${\text{tr}}\rho_x(\text{Frob}_w)=\lambda(b_{w|v}(h_w))(x)$. $\square$ For every $x \in {\Bbb{X}}(\bar{{\Bbb{Q}}}_p)$, there is a unique continuous semisimple Galois representation $\rho_x:\Gamma_{{\mathcal{K}},\Sigma}\rightarrow {\text{GL}}_n(\bar{{\Bbb{Q}}}_p)$ such that ${\text{tr}}\rho_x(\text{Frob}_w)=\lambda_x(b_{w|v}(h_w))$ for all $w \notin \Sigma$. [*[Proof]{}*]{}. This follows from Theorem 1 of \[Tay\]. $\square$ In particular, this applies to the classical point $x \in X_{cl}$, [*[not]{}*]{} in $X_{reg}$. One of the goals of \[Chen\] was to extend properties (a)-(c) above to this setting. This was partially accomplished. See Theorem 3.3 and 3.5 of \[Chen\]. Banach space representations ============================ With each point $x \in {\Bbb{X}}(L)$, we have associated an $n$-dimensional continuous pseudo-character $\mathcal{T}_x: \Gamma_{{\mathcal{K}}}\rightarrow L$, unramified outside $\Sigma(K^p)$. Here we will associate a Banach $\hat{\mathcal{H}}_L(K^p)$-module $\mathcal{B}_x$, with an admissible unitary $G({\Bbb{Q}}_p)$-action, such that the pairs $(\mathcal{T}_x,\mathcal{B}_x)$ form the graph of a one-to-one correspondence. We explicitly compute the locally (regular) algebraic vectors in $\mathcal{B}_x$, for $x \in X_{reg}$ such that $\mathcal{T}_x$ is absolutely irreducible, in terms of the Breuil-Schneider representation attached to $\mathcal{T}_x$, or rather, its corresponding Galois representation $\rho_x$. As a result, we prove the Breuil-Schneider conjecture for such $\rho_x$. A global $p$-adic Langlands correspondence ------------------------------------------ With the eigenvariety language set up, we can reformulate our findings at the end of Chapter 2. We let $X_{irr}\subset {\Bbb{X}}(\bar{{\Bbb{Q}}}_p)$ be the points $x$ for which $\rho_x$ is irreducible. Let $x \in X_{reg}\cap X_{irr}$, corresponding to $(\psi_x\theta_x,\lambda_x)$. Let $W_x$ be the irreducible algebraic representation of $G_{E}$ of highest weight $\psi_x$. Then, $$\bar{{\Bbb{Q}}}_p\otimes_E \tilde{H}^0(K^p)_{\text{$W_x$-alg}}^{\frak{h}=\lambda_x}\simeq (\oplus_{v|p}\widetilde{BS}(\rho_x|_{\Gamma_{{\mathcal{K}}_{\tilde{v}}}}))\otimes_{\bar{{\Bbb{Q}}}_p} (\oplus_{\pi:\pi_{\infty}\simeq W_x,\lambda_{\pi}=\lambda_x} m_G(\pi)(\pi_f^p)^{K^p}),$$ where we write $\frak{h}=\mathcal{H}(K^p)^{\text{sph}}$ for simplicity. This formula suggests the following definition. At each point $x \in {\Bbb{X}}(\bar{{\Bbb{Q}}}_p)$, we introduce the eigenspace $$\mathcal{B}_x:=(\bar{{\Bbb{Q}}}_p\otimes_E\tilde{H}^0(K^p))^{\frak{h}=\lambda_x}.$$ This is a Banach $\hat{\mathcal{H}}(K^p)$-module with a (commuting) unitary $G({\Bbb{Q}}_p)$-action. We remind ourselves that $\mathcal{B}_x$ is nothing but the space of [*[continuous]{}*]{} $\lambda_x$-eigenforms $f:Y(K^p)\rightarrow \bar{{\Bbb{Q}}}_p$. This sets up a one-to-one correspondence $\rho_x \leftrightarrow \mathcal{B}_x$. That is, $$\rho_x=\rho_{x'} \Leftrightarrow \lambda_x=\lambda_{x'} \Leftrightarrow \mathcal{B}_x=\mathcal{B}_{x'}$$ for any two $x,x' \in {\Bbb{X}}(\bar{{\Bbb{Q}}}_p)$. Let us say that a Galois representation $\rho$ [*[comes from]{}*]{} ${\Bbb{X}}$ if $\rho\simeq \rho_x$ for some $x \in {\Bbb{X}}(\bar{{\Bbb{Q}}}_p)$. Similarly for Banach modules $\mathcal{B}\simeq \mathcal{B}_x$. This leads to the main result of this section, which in some sense is the genesis of what follows. The eigenvariety ${\Bbb{X}}$ mediates a one-to-one correspondence between: - The set of continuous semisimple Galois representations $\rho: \Gamma_{{\mathcal{K}}}\rightarrow {\text{GL}}_n(\bar{{\Bbb{Q}}}_p)$ coming from ${\Bbb{X}}$. (In particular, $\rho$ is unramified outside $\Sigma(K^p)$.) - The set of Banach $\hat{\mathcal{H}}(K^p)$-modules $\mathcal{B}$, with unitary $G({\Bbb{Q}}_p)$-action, from ${\Bbb{X}}$. We write $\rho \leftrightarrow \mathcal{B}$ when there is a point $x \in {\Bbb{X}}(\bar{{\Bbb{Q}}}_p)$ such that $\rho\simeq \rho_x$ and $\mathcal{B}\simeq \mathcal{B}_x$. - Let $x \in {\Bbb{X}}(\bar{{\Bbb{Q}}}_p)$. If there is a regular $W$ for which $\mathcal{B}_x^{\text{$W$-alg}}\neq 0$, then $\rho_x$ is potentially semistable at all places $w|p$ of ${\mathcal{K}}$. - Let $x \in X_{cl}$. Then $\mathcal{B}_x^{\text{$W_x$-alg}}\neq 0$, and $\mathcal{B}_x^{\text{$W$-alg}}=0$ for all regular $W\neq W_x$. - For $x \in X_{reg}\cap X_{irr}$, the locally regular algebraic vectors of $\mathcal{B}_x$ are $$\mathcal{B}_x^{ralg}=\mathcal{B}_x^{\text{$W_x$-alg}}\simeq (\otimes_{v|p}\widetilde{BS}(\rho_x|_{\Gamma_{{\mathcal{K}}_{\tilde{v}}}}))\otimes_{\bar{{\Bbb{Q}}}_p} (\oplus_{\pi:\pi_{\infty}\simeq W_x,\lambda_{\pi}=\lambda_x} m_G(\pi)(\pi_f^p)^{K^p}).$$ [*[Proof]{}*]{}. First, (1) follows from Proposition 2, which shows there is an automorphic $\pi$, with $\pi_{\infty}\simeq W$, such that $\frak{h}$ acts on $\pi_f^{K^p}$ by $\lambda_x$. Since $W$ is regular, we know how to associate a Galois representation $\rho_{\pi,\iota}$, with the usual local properties, which must be $\rho_x$ by Tchebotarev. For (2), we follow the same line of argument. Since $x \in X_{cl}$, there is an automorphic $\pi$ contributing to $\mathcal{B}_x^{\text{$W_x$-alg}}$. Moreover, if $\mathcal{B}_x^{\text{$W$-alg}}\neq 0$, there is an automorphic $\pi$, of regular weight $W$, for which $\rho_{\pi,\iota}\simeq \rho_x$. From $\rho_{\pi,\iota}$ we can recover $W$ through its Hodge-Tate numbers. Similarly for $\rho_x$, even if $x$ is not in $X_{reg}$ (this is shown in section 3.15 of \[Chen\], based on results of Sen, and Berger-Colmez). Therefore, $W=W_x$. $\square$ [*[Remark]{}*]{}. As remarked earlier, we are optimistic that one can remove the regularity hypotheses in the Theorem. Indeed it seems possible to attach Galois representations to automorphic $\pi$ of $U({\Bbb{A}}_F)$ of [*[irregular]{}*]{} weight. When $\pi_p$ is of finite slope (that is, embeds in a principal series), this can be done by means of eigenvarieties, as in \[Chen\]. In general, it seems likely that one can push the ideas from the proof of Theorem 1. By \[Whit\], there is always a base change $\boxplus_{i=1}^t \Pi_i$, where the $\Pi_i$ are [*[discrete]{}*]{} automorphic representations of ${\text{GL}}_{n_i}({\Bbb{A}}_{{\mathcal{K}}})$, which in turn (by the Moeglin-Waldspurger classification) are isobaric sums of cohomological, essentially conjugate self-dual, cusp forms; with which one can associate Galois representations. Local-global compatibility at $p$ follows from Caraiani’s Harvard thesis. Perhaps it would be stylistically more elegant to simply admit the association of Galois representations in what follows. Then everywhere one could replace regular-algebraic vectors with algebraic vectors. The following result of Emerton shows how to detect the weights in $\mathcal{B}_x$. It utilizes his extension of the Jacquet functor $J_B$ to the locally analytic setting. We apply it to the locally ${\Bbb{Q}}_p$-analytic vectors $\mathcal{B}_x^{an}$. $J_B(\mathcal{B}_x^{an})^{T({\Bbb{Q}}_p)=\chi_x}\neq 0$, for all $x \in {\Bbb{X}}(\bar{{\Bbb{Q}}}_p)$. [*[Proof]{}*]{}. Combine Propositions 2.3.3 (iii) and 2.3.8 in \[Emer\]. $\square$ Analytic variation of the Breuil-Schneider recipe ------------------------------------------------- By the very definition of $J_B$ (as an adjoint functor, see Theorem 0.3 of \[Em1\]), this is equivalent to having a nonzero continuous $B$-equivariant map $$\mathcal{C}_c^{sm}(N,\delta_B^{-1}\chi_x)\rightarrow \mathcal{B}_x^{an},$$ where $B=TN$. If there is a so-called [*[balanced]{}*]{} map $\chi_x \hookrightarrow J_B(\mathcal{B}_x^{an})$ (Definition 0.8 in \[Em2\]), it is expected to arise from a nonzero continuous $G$-map (by applying $J_B$ and composing with the adjunction map) $$I_{\bar{B}}^G(\delta_B^{-1}\chi_x)\rightarrow \mathcal{B}_x^{an},$$ where, loosely speaking, $I_{\bar{B}}^G$ is the closed subrepresentation of $\text{Ind}_{\bar{B}}^G$, which can be detected by its Jacquet module (Corollary 5.1.4 in \[Em2\]). This expectation is known in many cases, see Theorem 0.13 in \[Em2\]. Note that, for $x \in X_{irr}$, $$I_{\bar{B}}^G(\delta_B^{-1}\chi_x)\twoheadrightarrow \widetilde{BS}(\rho_x):=\otimes_{v|p}\widetilde{BS}(\rho_x|_{\Gamma_{{\mathcal{K}}_{\tilde{v}}}})$$ according to Remark 5.1.8 in \[Em2\] (an isomorphism when $\text{Ind}_B^G(\theta_x)$ is irreducible). What we wish to point out in this paragraph, is that $I_{\bar{B}}^G(\delta_B^{-1}\chi_x)$ [*[always]{}*]{} satisfies the Emerton condition (that is, (3) in the introduction), for every point $x$ on the eigenvariety ${\Bbb{X}}$, even though we do [*[not]{}*]{} know if it admits a $G$-invariant norm: $I_{\bar{B}}^G(\delta_B^{-1}\chi_x)$ satisfies the Emerton condition for all $x \in {\Bbb{X}}(\bar{{\Bbb{Q}}}_p)$. [*[Proof]{}*]{}. Since $\mathcal{B}_x^{an}$ [*[has]{}*]{} an invariant norm, it satisfies the Emerton condition. (See Lemma 4.4.2 in \[Em1\].) In particular, $|\delta_B^{-1}(z)\chi_x(z)|_p\leq 1$ for $z \in T^+$, since $\chi_x \hookrightarrow J_B(\mathcal{B}_x^{an})$. Therefore, the sup-norm $\|\cdot\|$on $\mathcal{C}_c^{sm}(N,\delta_B^{-1}\chi_x)$ is non-increasing under the action of the submonoid $NT^+$. By restriction to $N$, $$I_{\bar{B}}^G(\delta_B^{-1}\chi_x)^{\frak{n}}:=\cup_{N_0}I_{\bar{B}}^G(\delta_B^{-1}\chi_x)^{N_0}\hookrightarrow \mathcal{C}_c^{sm}(N,\delta_B^{-1}\chi_x).$$ Consequently $I_{\bar{B}}^G(\delta_B^{-1}\chi_x)^{\frak{n}}$ carries a norm such that $\|gx\|\leq \|x\|$ for all $g \in NT^+$. This is sufficient to run the argument proving Lemma 4.4.2 in \[Em1\]. $\square$ Weak local-global compatibility =============================== In this section we deduce from our previous results that $\widetilde{BS}(\rho_x)$ admits an invariant norm such that the completion satisfies (a strong version of) local-global compatibility. However, we cannot show that this completion $\widetilde{BS}(\rho_x)^{\wedge}$ only depends on the restrictions of $\rho_x$ to places above $p$. Ultimately, we will restrict ourselves to the [*[unramified]{}*]{} case, and prove a weak version of local-global compatibility (somewhat similar to part (1) of Theorem 1.2.1 in \[Eme\]). The $p$-adic local Langlands correspondence, still mysterious in higher rank, is replaced by the coarse version in \[ScTe\], which associates a huge Banach representation $B_{\xi,\zeta}$ with a pair $(\xi,\zeta)$ satisfying the Emerton condition (here $\xi$ is an irreducible algebraic representation, and $\zeta$ is a suitable Weyl-orbit in the dual torus). The philosophy propounded in \[ScTe\] and \[BrSc\] is that the completed quotients of $B_{\xi,\zeta}$ should somehow correspond to the crystalline representations of type $(\xi,\zeta)$. This is well-understood for ${\text{GL}}_2({\Bbb{Q}}_p)$, where the admissible filtration is usually unique, and $B_{\xi,\zeta}$ essentially [*[is]{}*]{} the local $p$-adic Langlands correspondence in the crystalline case. We provide evidence supporting this philosophy for $n>2$. Completions of the algebraic vectors ------------------------------------ [*[Split ramification and the automorphic representation $\pi_x$]{}*]{}. Throughout, we will make the assumption that we have [*[split ramification]{}*]{}. That is, $S(K^p)\subset \text{Spl}_{{\mathcal{K}}|F}$. This has the effect that local base change $\text{BC}_{w|v}$ is defined at [*[all]{}*]{} places $v$. We fix a point $x \in X_{reg}\cap X_{irr}$, as above. Under our ramification hypothesis, there is a [*[unique]{}*]{} automorphic $\pi$ contributing to the (regular) algebraic vectors $\mathcal{B}_x^{ralg}$ in Theorem 4, part (3). Indeed, any such $\pi$ has an irreducible Galois representation $\rho_{\pi,\iota}\simeq \rho_x$, and therefore $\text{BC}_{{\mathcal{K}}|F}(\pi)$ must be cuspidal, and it is uniquely determined at the infinite places, and away from $\Sigma(K^p)$. By strong multiplicity one for ${\text{GL}}_n$, the base change is unique. Locally, $\text{BC}_{w|v}$ is injective (see Corollary 4.2 in \[Min\]), and therefore $\pi$ is uniquely determined. We denote it $\pi_x=\otimes \pi_{x,v}$. Its local components $\pi_{x,v}$ are given by $$WD(\rho_x|_{\Gamma_{{\mathcal{K}}_w}})^{F-ss}\simeq rec(BC_{w|v}(\pi_{x,v})\otimes |\det|_w^{(1-n)/2}).$$ We think of $\{\pi_x\}$ as a family of automorphic representations interpolated by ${\Bbb{X}}$. In general (without split ramification) the $\pi_x$ will be $L$-packets, not singletons. With this notation, part (3) of Theorem 4 becomes: For all $x \in X_{reg}\cap X_{irr}$, $$\mathcal{B}_x^{ralg}\simeq \widetilde{BS}(\rho_x) \otimes (\otimes_{v\nmid p}\pi_{x,v}^{K_v})^{m(\pi_x)}.$$ Most likely, $m(\pi_x)=1$, and this may already be in the literature. However, we have not been able to find a suitable reference. Now, since $\otimes_{v\nmid p}\pi_{x,v}^{K_v}$ is a simple $\mathcal{H}(K^p)$-module, we may think of $\widetilde{BS}(\rho_x)^{m(\pi_x)}$ as its multiplicity space in $\mathcal{B}_x^{ralg}$, $$\widetilde{BS}(\rho_x)^{m(\pi_x)} \overset{\sim}{\longrightarrow} {\text{Hom}}_{\mathcal{H}(K^p)}(\otimes_{v\nmid p}\pi_{x,v}^{K_v}, \mathcal{B}_x^{ralg}),$$ as representations of $G({\Bbb{Q}}_p)$. We will view the right-hand-side as sitting inside a Banach space of continuous transformations. For that purpose, we first look at each local component $\pi_{x,v}$, where $v \nmid p$. When $v$ splits, it can be identified with a $p$-integral irreducible representation of ${\text{GL}}_n(F_v)$. By Theorem 1 in \[Vig\], it has a unique commensurability class of stable lattices. Correspondingly, $\pi_{x,v}$ has a unique equivalence class of ${\text{GL}}_n(F_v)$-invariant norms $\|\cdot\|_v$. (By Theorem 1 in \[Vign\], the completion $\hat{\pi}_{x,v}$ is a topologically irreducible unitary Banach space representation of ${\text{GL}}_n(F_v)$.) When $\pi_{x,v}$ is unramified, its Satake parameters are $p$-units, and one easily finds a stable lattice in a suitable unramified principal series, again resulting in a $U(F_v)$-invariant (supremum) norm $\|\cdot\|_v$, which we may normalize such that a given spherical vector has norm one. The tensor product norm (see Proposition 17.4 in \[Sc\]) on $\otimes_{v\nmid p}\pi_{x,v}$ is then invariant under $G({\Bbb{A}}_f^p)$. By restriction, the finite-dimensional space $\otimes_{v\nmid p}\pi_{x,v}^{K_v}$ inherits a norm, and becomes a Banach-module for $\hat{\mathcal{H}}(K^p)$. With this extra structure at hand, $${\text{Hom}}_{\mathcal{H}(K^p)}(\otimes_{v\nmid p}\pi_{x,v}^{K_v}, \mathcal{B}_x^{ralg})\hookrightarrow \mathcal{L}_{\hat{\mathcal{H}}(K^p)}(\otimes_{v\nmid p}\pi_{x,v}^{K_v},\mathcal{B}_x).$$ (Here $\mathcal{L}$ denotes the space of continuous linear transformations, equipped with the usual transformation norm, see Corollary 3.2 in \[Sc\].) We have to check that any $\mathcal{H}(K^p)$-equivariant map $\otimes_{v\nmid p}\pi_{x,v}^{K_v} \overset{\phi}{\rightarrow} \mathcal{B}_x$ is automatically continuous: If $\phi\neq 0$, it must be injective (by simplicity), and thus $\|\phi(\cdot)\|_{\mathcal{B}_x}$ defines a norm on $\otimes_{v\nmid p}\pi_{x,v}^{K_v}$. However, all norms on a finite-dimensional space are equivalent (4.13 in \[Sc\]), so that $\|\phi(u)\|_{\mathcal{B}_x}\leq C\|u\|$ for some constant $C>0$, and all $u$. Altogether, this embeds $\widetilde{BS}(\rho_x)$ into a Banach space (Proposition 3.3 in \[Sc\]): $$\widetilde{BS}(\rho_x)^{m(\pi_x)} \hookrightarrow \mathcal{L}_{\hat{\mathcal{H}}(K^p)}(\otimes_{v\nmid p}\pi_{x,v}^{K_v},\mathcal{B}_x).$$ If we restrict the tranformation norm to $\widetilde{BS}(\rho_x)^{m(\pi_x)}$, we arrive at: Let $x \in X_{reg}\cap X_{irr}$ be a point such that $m(\pi_x)=1$. Then there is a $G({\Bbb{Q}}_p)$-invariant norm $\|\cdot\|$ on $\widetilde{BS}(\rho_x)$ such that the corresponding completion $\widetilde{BS}(\rho_x)^{\wedge}$ satisfies the following: There is a topological isomorphism, $$\widetilde{BS}(\rho_x)^{\wedge}\otimes (\otimes_{v\nmid p}\pi_{x,v}^{K_v}) \overset{\sim}{\longrightarrow} \overline{\mathcal{B}_x^{ralg}},$$ where $\overline{\mathcal{B}_x^{ralg}}$ is the closure of the regular-algebraic vectors $\mathcal{B}_x^{ralg}$ in $\mathcal{B}_x$. Moreover, - $\widetilde{BS}(\rho_x)^{\wedge}$ is an unitary Banach space representation of $G({\Bbb{Q}}_p)$. - Its regular-algebraic vectors $\widetilde{BS}(\rho_x)$ form a subspace. [*[Proof]{}*]{}. We obtain $\|\cdot\|$ by restricting the transformation norm to $\widetilde{BS}(\rho_x)$. Thus (2) becomes an isometry, and extends uniquely to an isometry $$\widetilde{BS}(\rho_x)^{\wedge} \hookrightarrow \mathcal{L}_{\hat{\mathcal{H}}(K^p)}(\otimes_{v\nmid p}\pi_{x,v}^{K_v},\mathcal{B}_x).$$ To ease the notation, let us write $M=\otimes_{v\nmid p}\pi_{x,v}^{K_v}$ throughout this proof; a finite-dimensional simple $\mathcal{H}(K^p)$-module. We tensor the isometry by this $M$, $$j:\widetilde{BS}(\rho_x)^{\wedge}\otimes M \hookrightarrow \mathcal{L}_{\hat{\mathcal{H}}(K^p)}(M,\mathcal{B}_x)\otimes M\overset{\sim}{\longrightarrow} \mathcal{B}_x[M].$$ Here $\mathcal{B}_x[M]$ denotes the closure of the sum of all closed $\mathcal{H}(K^p)$-submodules of $\mathcal{B}_x$ isomorphic to $M$ (a topological direct sum of a subcollection, by Zorn). Note that ${\text{End}}_{\mathcal{H}(K^p)}(M)=\bar{{\Bbb{Q}}}_p$. Note also that the tensor products (equipped with their tensor product norms, as on p. 110 in \[Sc\]) are already complete, as $M$ is finite-dimensional. The above isomorphism with $\mathcal{B}_x[M]$ is a [*[topological]{}*]{} isomorphism by the open mapping theorem (8.6, p. 55 in \[Sc\]), but not necessarily isometric. Consequently, ${\text{im}}(j)\subset \mathcal{B}_x$ is a closed subspace, containing $\mathcal{B}_x^{ralg}$ by Theorem 4. In fact, ${\text{im}}(j)$ is the closure of $\mathcal{B}_x^{ralg}$ in $\mathcal{B}_x$, since $\widetilde{BS}(\rho_x)$ is dense in the completion $\widetilde{BS}(\rho_x)^{\wedge}$. Again invoke the open mapping theorem to see that $j$ is a topological isomorphism onto $\overline{\mathcal{B}_x^{ralg}}$. Admissibility of $\widetilde{BS}(\rho_x)^{\wedge}$ follows from admissibility of $\mathcal{B}_x$. $\square$ [*[Remark]{}*]{}. Equivalently, there is a $G({\Bbb{Q}}_p)$-equivariant topological isomorphism, $$\widetilde{BS}(\rho_x)^{\wedge} \overset{\sim}{\longrightarrow} \mathcal{L}_{\hat{\mathcal{H}}(K^p)}(\otimes_{v\nmid p}\pi_{x,v}^{K_v},\overline{\mathcal{B}_x^{ralg}}).$$ We like to think of this Banach space representation $\widetilde{BS}(\rho_x)^{\wedge}$ as a “rough” candidate for a $p$-adic local Langlands correspondence $\frak{B}(\rho_x)=\hat{\otimes}_{v|p} \frak{B}(\rho_x|_{\Gamma_{{\mathcal{K}}_{\tilde{v}}}})$, at least when the various restrictions $\rho_x|_{\Gamma_{{\mathcal{K}}_{\tilde{v}}}}$ are irreducible. Of course, to really justify this point of view, one would need to show that the completion $\widetilde{BS}(\rho_x)^{\wedge}$ only depends on the restrictions $\rho_x|_{\Gamma_{{\mathcal{K}}_{\tilde{v}}}}$ at $p$, and that it factors as a tensor product $\hat{\otimes}_{v|p}$ of appropriate completions $\widetilde{BS}(\rho_x|_{\Gamma_{{\mathcal{K}}_{\tilde{v}}}})^{\wedge}$. Both appear to be very difficult questions. Universal modules: The crystalline case --------------------------------------- We now specialize to the [*[crystalline]{}*]{} case, where we can relate $\widetilde{BS}(\rho_x)^{\wedge}$ to the Schneider-Teitelbaum universal modules $B_{\xi,\zeta}$, which are given by a purely local construction at $p$. They are expected to be quite large. However, for $n>2$ it is not even known that $B_{\xi,\zeta}\neq 0$ (Conjecture 6.1, p. 24 in \[BrSc\]). For $n=2$ this is a deep result of Berger and Breuil. We will prove non-vanishing when $(\xi,\zeta)$ “comes from an eigenvariety”. This will be a by-product of a stronger result. A classical point $x \in X_{cl}$ is called if $\rho_x$ is crystalline at all places above $p$. That is, ${\text{Hom}}_{G({\Bbb{Z}}_p)}(W_x,\mathcal{B}_x)\neq 0$. Equivalently, $\pi_{x,v}$ is unramified for all $v|p$. We denote the set of old points by $X_{old}$. Thus, from now on, we fix a point $x \in X_{reg}\cap X_{irr} \cap X_{old}$. By Proposition 1, $$\widetilde{BS}(\rho_x)=W_x \otimes \pi_{x,p}\overset{\sim}{\longrightarrow} W_x \otimes \text{Ind}_{B}^{G}(\theta_x),$$ where $\theta_x$ is unramified smooth. (Indeed, for any point $x$, $\pi_{x,p}$ embeds into the (non-normalized) principal series $\text{Ind}_B^G(\theta_x)$. Since $x$ is old, $\pi_{x,p}$ is unramified, and hence so is $\theta_x$. Furthermore, as $x \in X_{irr}$, the base change $\text{BC}_{{\mathcal{K}}|F}(\pi_x)$ is cuspidal, and therefore generic. In particular, $\pi_{x,p}$ must be generic. As is well-known, this implies that $\pi_{x,p}$ must be the full unramified principal series.) As in \[ScTe\] we express $\pi_{x,p}\simeq \text{Ind}_B^G(\theta_x)$ in terms of the [*[universal module]{}*]{}. This goes back to Borel and Matsumoto, and is defined as follows. For any algebra character $\zeta: \mathcal{H}(G,K) \rightarrow \bar{{\Bbb{Q}}}_p$ (where $K=G({\Bbb{Z}}_p)$ is hyperspecial when $p$ is assumed to be unramified in $F$) we introduce the smooth representation $$\mathcal{M}_{\zeta}=\text{c-Ind}_{K}^{G}(1)\otimes_{\mathcal{H}(G,K),\zeta} \bar{{\Bbb{Q}}}_p=\mathcal{C}_c(K\backslash G,\bar{{\Bbb{Q}}}_p)\otimes_{\mathcal{H}(G,K),\zeta} \bar{{\Bbb{Q}}}_p.$$ The pair $(\mathcal{M}_{\zeta},1_K)$ is a universal initial object in the category of pairs $(V,v)$, where $V$ is an unramified smooth representation of $G({\Bbb{Q}}_p)$, and $v \in V^K$ is a nonzero vector on which $\mathcal{H}(G,K)$ acts via $\zeta$. That is, there is a unique $G({\Bbb{Q}}_p)$-map $\mathcal{M}_{\zeta} \rightarrow V$ which maps $1_K\mapsto v$. The image of this map is the span of the orbit $Gv$ (since $\mathcal{M}_{\zeta}$ is generated by $1_K$). In what follows we will take $\zeta_x=\hat{\theta}_x$, the eigensystem of $\text{Ind}_B^G(\theta_x)^K$. The choice of a spherical vector yields $$\text{$\mathcal{M}_{\zeta_x}\rightarrow \text{Ind}_B^G(\theta_x)$, ${\hspace{6pt}}$ $\zeta_x=\hat{\theta}_x$.}$$ It is a general fact that the two representations have the same semi-simplification (see the Orsay Ph.D. thesis of X. Lazarus for a thorough discussion in greater generality). Under our assumptions, $\text{Ind}_B^G(\theta_x)$ is irreducible, and therefore the above must be an isomorphism. Consequently, we may identify $$\widetilde{BS}(\rho_x)\simeq W_x \otimes \mathcal{M}_{\zeta_x}\simeq \text{c-Ind}_{K}^{G}(\xi_x)\otimes_{\mathcal{H}_{\xi_x}(G,K),\zeta_x} \bar{{\Bbb{Q}}}_p=:H_{\xi_x,\zeta_x}.$$ Here we have changed notation $\xi_x:=W_x$ to aid comparison with \[ScTe\]. The algebra $\mathcal{H}_{\xi_x}(G,K)$ is by definition the $G$-endomorphisms of $\text{c-Ind}_{K}^{G}(\xi_x)$. Or, more concretely, compactly supported $K$-biequivariant functions $G \rightarrow {\text{End}}(\xi_x)$, with convolution. However, since $\xi$ is an irreducible representation of $G$ (viewed as a representation of $K$), as on p. 639 in \[ScTe\] one can identify the algebras $$\text{$\mathcal{H}(G,K)\overset{\sim}{\longrightarrow} \mathcal{H}_{\xi_x}(G,K)$, ${\hspace{6pt}}$ $h \mapsto (g \mapsto h(g)\xi_x(g))$.}$$ In the definition of $H_{\xi_x,\zeta_x}$ we view $\zeta_x$ as a character of $\mathcal{H}_{\xi_x}(G,K)$ via this isomorphism, as at the bottom of p. 670 in \[ScTe\], where $H_{\xi,\zeta}$ is defined. The representation $H_{\xi_x,\zeta_x}$ has a natural locally convex topology, being a quotient of $\text{c-Ind}_{K}^{G}(\xi_x)$, which has a supremum norm: Pick any norm $\|\cdot\|_{\xi_x}$ on $\xi_x$, which is invariant under (the compact group) $K$. They are all equivalent since $\xi_x$ is finite-dimensional (4.13 in \[Sc\]). As in Appendix A, for $f \in \text{c-Ind}_{K}^{G}(\xi_x)$, we let $$\|f\|_{\xi_x,\infty}={\sup}_{g \in G({\Bbb{Q}}_p)} \|f(g)\|_{\xi_x}<\infty$$ This defines an norm $\|\cdot\|_{\xi_x,\infty}$ on the compact induction, which is obviously invariant under $G({\Bbb{Q}}_p)$, and it induces a quotient [*[seminorm]{}*]{} on the representation $$H_{\xi_x,\zeta_x}=(\text{c-Ind}_{K}^{G}(\xi_x))/(\ker\zeta_x)(\text{c-Ind}_{K}^{G}(\xi_x)).$$ We will show below that in fact this is a [*[norm]{}*]{}, but this is far from clear a priori! Following \[ScTe\], on p. 671 where they define $B_{\xi,\zeta}$, we introduce the space $$B_{\xi_x,\zeta_x}:=\hat{H}_{\xi_x,\zeta_x}=(H_{\xi_x,\zeta_x}/\overline{\{0\}})^{\wedge}=\text{Hausdorff completion of $H_{\xi_x,\zeta_x}$}.$$ (We refer to 7.5 in \[Sc\] for a general discussion of Hausdorff completions.) We have defined a Banach space $B_{\xi_x,\zeta_x}$ with a unitary $G({\Bbb{Q}}_p)$-action However, it is not clear at all that it is [*[nonzero]{}*]{}. This is in fact a fundamental problem! Conjecture 6.1 on p. 24 in \[BrSc\] says that $B_{\xi,\zeta}\neq 0$ whenever the Emerton condition is satisfied (the converse is known). This follows from our methods when the pair $(\xi,\zeta)$ comes from an eigenvariety. That is, when it is of the form $(\xi_x,\zeta_x)$ for an old irreducible point $x$. What we prove is a strengthening: Let $x \in X_{reg}\cap X_{irr}\cap X_{old}$ be a classical point such that $m(\pi_x)=1$. Then $H_{\xi_x,\zeta_x}$ is Hausdorff, $B_{\xi_x,\zeta_x} \neq 0$ is its universal completion. Furthermore: - There is a continuous map, with dense image, $B_{\xi_x,\zeta_x} \rightarrow \widetilde{BS}(\rho_x)^{\wedge}$ (into the completion from Corollary 3) which restricts to an isomorphism $H_{\xi_x,\zeta_x}\overset{\sim}{\longrightarrow} \widetilde{BS}(\rho_x)$ onto the regular-algebraic vectors. - There is a $G({\Bbb{Q}}_p)\times \hat{\mathcal{H}}(K^p)$-equivariant continuous map $$B_{\xi_x,\zeta_x} \otimes (\otimes_{v\nmid p}\pi_{x,v}^{K_v}) \rightarrow \overline{\mathcal{B}_x^{ralg}}$$ with dense image. [*[Proof]{}*]{}. From (2) of the previous section, we have a $G({\Bbb{Q}}_p)$-embedding $$H_{\xi_x,\zeta_x}\simeq \widetilde{BS}(\rho_x) \hookrightarrow \mathcal{L}_{\hat{\mathcal{H}}(K^p)}(M,\mathcal{B}_x),$$ where we keep writing $M=\otimes_{v\nmid p}\pi_{x,v}^{K_v}$. We claim this map is automatically continuous, when we equip the $\mathcal{L}$-space with the transformation norm, and $H_{\xi_x,\zeta_x}$ with the quotient seminorm induced by $\|\cdot\|_{\xi_x,\infty}$. Since $H_{\xi_x,\zeta_x}$ gets the quotient topology, we just have to check continuity of the inflated map $$\text{c-Ind}_{K}^{G}(\xi_x) \twoheadrightarrow H_{\xi_x,\zeta_x} \hookrightarrow \mathcal{L}_{\hat{\mathcal{H}}(K^p)}(M,\mathcal{B}_x).$$ This is part of Lemma 3 in Appendix A (in which Frobenius reciprocity is made explicit). In particular, the seminorm on $H_{\xi_x,\zeta_x}$ is actually a norm (as the kernel of the above map is closed). Therefore, $H_{\xi_x,\zeta_x}$ is Hausdorff, and $B_{\xi_x,\zeta_x}$ is its universal completion. That is, there is an isometry with dense image, $$H_{\xi_x,\zeta_x} \hookrightarrow B_{\xi_x,\zeta_x}.$$ ($\Rightarrow B_{\xi_x,\zeta_x}$ is nonzero.) By continuity of the initial map, it has a unique extension $$B_{\xi_x,\zeta_x} \rightarrow \mathcal{L}_{\hat{\mathcal{H}}(K^p)}(M,\mathcal{B}_x),$$ which is continuous (but not necessarily injective) and maps into the completion $\widetilde{BS}(\rho_x)^{\wedge}$ from Corollary 3, with dense image (but not necessarily onto). $\square$ [*[Remark]{}*]{}. This fits perfectly with the picture suggested in the papers \[ScTe\] and \[BrSc\]. If there is a local $p$-adic Langlands correspondence $\rho \mapsto \frak{B}(\rho)$, these references speculate that $B_{\xi,\zeta}$ maps to each $\frak{B}(\rho)$, with dense image, for all crystalline representations $\rho$ of type $(\xi,\zeta)$. The previous Theorem provides strong evidence that $\widetilde{BS}(\rho_x)^{\wedge}$ is at least closely related to the (elusive) local $p$-adic Langlands correspondence $\frak{B}(\rho_x)=\hat{\otimes}_{v|p} \frak{B}(\rho_x|_{\Gamma_{{\mathcal{K}}_{\tilde{v}}}})$. On the density of the algebraic vectors ======================================= In general, it is [*[not]{}*]{} expected that $\mathcal{B}_x^{alg}$ is dense in $\mathcal{B}_x$. In this section, we will adapt (and elaborate on) an argument from sections 5.3 and 5.4 in \[Eme\], which shows that the algebraic vectors [*[are]{}*]{} dense, if we substitute $\mathcal{B}_x$ by a bigger space. Injectivity of certain modules ------------------------------ We fix a finite extension $L|{\Bbb{Q}}_p$, and we will write $\mathcal{O}=\mathcal{O}_L$ and $\varpi=\varpi_L$, and so on. We will look at locally constant functions $f:Y(K^p)\rightarrow A$, taking values in various finite $\mathcal{O}$-modules $A=\mathcal{O}/\varpi^s\mathcal{O}$, where $s$ is a positive integer. These functions form a (discrete) torsion $\mathcal{O}$-module, denoted $H^0(K^p,A)$, carrying a natural action of the Hecke algebra $\mathcal{H}_{\mathcal{O}}(K^p)$, and a commuting smooth $G({\Bbb{Q}}_p)$-action, which is [*[admissible]{}*]{} in the following sense: For every compact open subgroup of $G({\Bbb{Q}}_p)$, its invariants form a finite $\mathcal{O}$-module (torsion and finitely generated means finite cardinality, since $A$ is a finite ring). Suppose $K^p$ is sufficiently small (for example, it suffices that $K_v$ has no $p$-torsion for some $v\nmid p$). Then, for any compact open subgroup $K_p \subset G({\Bbb{Q}}_p)$, $$\text{$H^0(K^p,\mathcal{O}/\varpi^s\mathcal{O})$ is an {\bf{injective}} smooth $(\mathcal{O}/\varpi^s\mathcal{O})[K_p]$-module for all $s \geq 1$.}$$ Consequently, every direct summand of $H^0(K^p,\mathcal{O}/\varpi^s\mathcal{O})$ is an injective module. [*[Proof]{}*]{}. We have to show the exactness of the functor sending a module $M$ to $${\text{Hom}}_{\mathcal{O}[K_p]}(M,H^0(K^p,\mathcal{O}/\varpi^s\mathcal{O})).$$ Here $M$ is an $\mathcal{O}[K_p]$-module with $\varpi^sM=0$. Therefore, it has Pontryagin dual, $$M^{\vee}={\text{Hom}}_{\mathcal{O}}(M,L/\mathcal{O})={\text{Hom}}_{\mathcal{O}}(M,\varpi^{-s}\mathcal{O}/\mathcal{O})\simeq {\text{Hom}}_{\mathcal{O}/\varpi^s\mathcal{O}}(M,\mathcal{O}/\varpi^s\mathcal{O}).$$ (Here $M$ is smooth, so we equip it with the discrete topology.) The initial module above can then be identified with that consisting of all functions, $$\text{$f: Y(K^p) \rightarrow M^{\vee}$, ${\hspace{6pt}}$ $f(gk)=k^{-1}f(g)$,}$$ for $k \in K_p$. Choosing representatives $g_i \in G({\Bbb{A}}_f)$ for the finite set $Y(K_pK^p)$, and mapping $f$ to the tuple of all $f(g_i)$, then identifies the latter with the direct sum $\oplus_i (M^{\vee})^{\Gamma_i}$, where the $\Gamma_i$ are certain finite subgroups of $K_p$, having prime-to-$p$ order by assumption. This ensures that $(\cdot)^{\Gamma_i}$ is exact, by averaging. Also, taking the Pontryagin dual is exact ($L/\mathcal{O}$ is divisible). Finally, as is well known (and easy to check) every summand of an injective module is itself injective. $\square$ [*[Examples]{}*]{}. Let us first introduce certain finite type Hecke algebras. For each $K_p$, we let ${\Bbb{T}}(K_pK^p)$ denote the image of $\frak{h}^{\circ}=\mathcal{H}_{\mathcal{O}}(K^p)^{\text{sph}}$ in the endomorphism algebra ${\text{End}}_{\mathcal{O}}H^0(Y(K_pK^p),\mathcal{O})$. Thus ${\Bbb{T}}(K_pK^p)$ is finite free over (the PID) $\mathcal{O}$, and we endow it with the $\varpi$-adic topology. If we have a subgroup $K_p'\subset K_p$, there is a natural restriction map ${\Bbb{T}}(K_p'K^p)\rightarrow {\Bbb{T}}(K_pK^p)$, and we take the limit, $${\Bbb{T}}(K^p):=\underset{K_p}{\varprojlim} {\Bbb{T}}(K_pK^p)\subset {\text{End}}_{\mathcal{O}}\tilde{H}^0(K^p)^{\circ},$$ the closure of the image of $\mathcal{H}_{\mathcal{O}}(K^p)^{\text{sph}}$. This defines a reduced, commutative, complete, topological $\mathcal{O}$-algebra. Moreover, ${\Bbb{T}}(K^p)$ has only [*[finitely]{}*]{} many maximal ideals: They correspond to the maximal ideals of ${\Bbb{T}}(K^p)\otimes {\Bbb{F}}$, which is the image of $\frak{h}^{\circ}$ in ${\text{End}}_{{\Bbb{F}}}H^0(K^p,{\Bbb{F}})$. Hence, the maximal ideals are in bijection with the (Galois conjugacy classes of) eigensystems $\frak{h}^{\circ} \rightarrow {\Bbb{F}}$ which occur in $H^0(K^p,{\Bbb{F}})$. If $K_p$ is any pro-$p$ group, they all must occur in $H^0(Y(K_pK^p),{\Bbb{F}})$, which is finite-dimensional. Therefore, since $\mathcal{O}$ is complete, we have $${\Bbb{T}}(K^p) \overset{\sim}{\longrightarrow} \prod_{{\frak{m}}}{\Bbb{T}}(K^p)_{{\frak{m}}},$$ where the product extends over the finitely many maximal ideals ${\frak{m}}\subset {\Bbb{T}}(K^p)$, and ${\Bbb{T}}(K^p)_{{\frak{m}}}$ denotes the corresponding localization, a complete local $\mathcal{O}$-algebra. (We refer to Chapter 4 of \[DDT\] for a discussion of the commutative algebra needed.) We will use this product decomposition as follows: Obviously, $$\tilde{H}^0(K^p)^{\circ}/\varpi^s \tilde{H}^0(K^p)^{\circ}\simeq H^0(K^p,\mathcal{O}/\varpi^s\mathcal{O})$$ carries an action of ${\Bbb{T}}(K^p)$. This gives rise to a [*[direct sum]{}*]{} decomposition, $$H^0(K^p,\mathcal{O}/\varpi^s\mathcal{O}) \overset{\sim}{\longrightarrow} \bigoplus_{{\frak{m}}} H^0(K^p,\mathcal{O}/\varpi^s\mathcal{O})_{{\frak{m}}},$$ into localized smooth admissible $G({\Bbb{Q}}_p)$-submodules over $\mathcal{O}/\varpi^s\mathcal{O}$, $$H^0(K^p,\mathcal{O}/\varpi^s\mathcal{O})_{{\frak{m}}}:=H^0(K^p,\mathcal{O}/\varpi^s\mathcal{O})\otimes_{{\Bbb{T}}(K^p)}{\Bbb{T}}(K^p)_{{\frak{m}}},$$ which are then [*[injective]{}*]{} $(\mathcal{O}/\varpi^s\mathcal{O})[K_p]$-modules, for every compact open $K_p$. To connect this to the previous discussion, one could take the maximal ideal ${\frak{m}}_x=\ker(\bar{\lambda}_x)$ for a point $x \in {\Bbb{X}}(L)$. A priori this is a maximal ideal in $\frak{h}^{\circ}$, but it is the pull-back of an ideal ${\frak{m}}\subset {\Bbb{T}}(K^p)$ since $\bar{\lambda}_x$ occurs in tame level $K^p$. Projective modules over certain Iwasawa algebras ------------------------------------------------ To simplify notation, we write $A=\mathcal{O}/\varpi^s\mathcal{O}$ in this section, where $s>0$ is fixed for the moment. We will briefly recall known facts about the Pontryagin duality functor $M \mapsto M^{\vee}$, which sends a discrete $A[K_p]$-module $M$ to the compact $$M^{\vee}={\text{Hom}}_{\mathcal{O}}(M,L/\mathcal{O})\simeq {\text{Hom}}_A(M,A).$$ If $M$ is smooth, $M=\underset{H}{\varinjlim}M^H$, with $H$ running over normal open subgroups of $K_p$, and therefore its dual $M^{\vee}=\underset{H}{\varprojlim}(M^H)^{\vee}$ becomes a module for $$A[[K_p]]:=\underset{H}{\varprojlim} A[K_p/H],$$ the [*[Iwasawa algebra]{}*]{}. Conversely, if $X$ is an $A[[K_p]]$-module, $X/I_HX$ becomes a module for $A[K_p/H]$, where $I_H$ is the kernel of the natural projection $A[[K_p]]\twoheadrightarrow A[K_p/H]$. It follows that $X^{\vee}$ is again a smooth $A[K_p]$-module, since $$(X^{\vee})^H \simeq (X/I_HX)^{\vee}.$$ Thus, duality sets up a one-to-one correspondence $M \leftrightarrow X$ between smooth discrete $A[K_p]$-modules and compact $A[[K_p]]$-modules, which reverses arrows. Suppose $M$ is a smooth $A[K_p]$-module, with Pontryagin dual $M^{\vee}$. - $M$ is admissible $\Longleftrightarrow$ $M^{\vee}$ is finitely generated over $A[[K_p]]$. - $M$ is injective $\Longleftrightarrow$ $M^{\vee}$ is a projective $A[[K_p]]$-module. [*[Proof]{}*]{}. For part (i), if $X$ is finitely generated over $A[[K_p]]$, we deduce that $X/I_HX$ is finitely generated over $A[K_p/H]$, which is a ring of finite cardinality. Therefore its dual $M^H$ is (physically) finite. For the converse, suppose $M$ is admissible. Then, first of all, $M^{\vee}$ is profinite, so we may apply the “converse” (topological) Nakayama lemma discussed in depth in \[BH\] (specifically, their main Theorem in Chapter 3, section (1), and its Corollary): To verify that $M^{\vee}$ is finitely generated over the compact ring $A[[K_p]]$, it suffices to check that $X/I_H X$ is finitely generated over $A[K_p/H]$, for [*[some]{}*]{} $H$ such that $I_H^n \rightarrow 0$ as $n \rightarrow \infty$. This limit holds for any pro-$p$-group $H$, see Lemma 3.2 in \[ScT\], for example. Finiteness of $X/I_H X$, or rather its dual $M^H$, is admissibility. For part (ii), use that Pontryagin duality is exact (divisibility of $L/\mathcal{O}$). It follows that ${\text{Hom}}_{A[K_p]}(-,M)$ is exact if and only if ${\text{Hom}}_{A[[K_p]]}(M^{\vee},-)$ is exact. $\square$ From the last two lemmas, we immediately conclude the following: Suppose $K^p$ is sufficiently small. Then, for any compact open subgroup $K_p \subset G({\Bbb{Q}}_p)$, the dual $H^0(K^p,A)^{\vee}$ is a [**[projective]{}**]{} finitely generated module over $A[[K_p]]$ for all $s \geq 1$. The same is true for any direct summand, such as the localized module $H^0(K^p,A)_{{\frak{m}}}^{\vee}$ for any maximal ideal ${\frak{m}}$. For later use, we will record the following fact here. Often, the Iwasawa algebra $A[[K_p]]$ is viewed as a distribution algebra. Indeed, there is a natural pairing with the continuous (that is, locally constant) functions $\mathcal{C}(K_p,A)$. $A[[K_p]]\overset{\sim}{\longrightarrow}\mathcal{C}(K_p,A)^{\vee}$, as modules over $A[[K_p]]$. [*[Proof]{}*]{}. For any normal open subgroup $H$, there is a canonical integration pairing, $$\text{$\mathcal{C}(K_p/H,A)\times A[K_p/H]\rightarrow A$, ${\hspace{6pt}}$ $(f,\mu)\mapsto {\sum}_{k \in K_p/H}f(k)\mu(k)$,}$$ which is non-degenerate, and therefore defines an isomorphism $$\text{$A[K_p/H]\overset{\sim}{\longrightarrow}\mathcal{C}(K_p/H,A)^{\vee}$, ${\hspace{6pt}}$ $\mu \mapsto (-,\mu)$.}$$ This is easily checked to preserve the $A[K_p/H]$-module structures on both sides. Moreover, as $H$ varies, these isomorphisms are compatible with the transition maps. Passing to the projective limit $\underset{H}{\varprojlim}$ gives the lemma. $\square$ In other words, $\mathcal{C}(K_p,A)\leftrightarrow A[[K_p]]$ under the correspondence discussed above. Local Iwasawa algebras of pro-$p$-groups ---------------------------------------- A local ring is a (possibly non-commutative) ring $R$, whose Jacobson radical $J(R)$ is a two-sided maximal ideal ${\frak{m}}_R$. In other words, there is a unique maximal left ideal, and a unique maximal right ideal, and they coincide. Nakayama’s lemma even holds for non-commutative local rings, as is easily checked. In particular, a finitely generated projective $R$-module is free; a key fact we will make use of below, by taking $R$ to be the Iwasawa algebra of a pro-$p$-group, which turns out to be local. We first assemble the following well-known facts. Let $K_p$ be a pro-$p$-group, and let $A$ be any $p$-ring (that is, its cardinality is a finite power of $p$, such as for $A=\mathcal{O}/\varpi^s\mathcal{O}$). Then, - Let $M$ be a left $A[[K_p]]$-module, and $H \subset K_p$ an open normal subgroup. Then $M/I_HM$ has a nonzero $K_p$-invariant element if $M \neq I_HM$. - $K_p$ acts trivially on any simple left $A[[K_p]]$-module. - $I_{K_p}\subset J(A[[K_p]])$. - $A$ local $\Longrightarrow$ $A[[K_p]]$ local. (Furthermore, $J(A[[K_p]])={\frak{m}}_A+I_{K_p}$.) (The same is true when left modules are replaced by right modules.) [*[Proof]{}*]{}. This is all standard. We cannot resist to briefly outline the argument. For (1) it is clearly enough to show that a $p$-group $K_p$ fixes a nonzero element of any $A[K_p]$-module $M \neq 0$. This is basic group action theory; the fact that $A$ is a $p$-ring allows to count fixed points modulo $p$. For (2), if $M$ is a simple left $A[[K_p]]$-module, we must have $I_HM=M$ or $I_HM=0$ for all $H$. There must be some $H$ for which $I_HM\neq M$, since $M\neq 0$ is the inverse limit of all quotients $M/I_HM$. Now (1) shows that $M^{K_p}\neq 0$. By simplicity, $K_p$ acts trivially on $M$. For (3) just use that $I_{K_p}$ is generated by elements $k-1$ with $k \in K_p$. We see from (2) that $I_{K_p}$ acts trivially on any simple left $A[[K_p]]$-module, and therefore, by the very definition of the Jacobson radical, we have the inclusion as claimed. Now (4) is immediate from (3). Indeed any maximal left ideal of $A[[K_p]]$ must be the pull-back of ${\frak{m}}_A$ under the augmentation map. $\square$ We will apply this to $A[[K_p]]$, where $A=\mathcal{O}/\varpi^s\mathcal{O}$, in conjunction with Proposition 4. Suppose $K^p$ is sufficiently small, and let $K_p \subset G({\Bbb{Q}}_p)$ be an open pro-$p$-group. Then there exists an integer $r>0$ such that $$\tilde{H}^0(K^p)^{\circ}\simeq \mathcal{C}(K_p,\mathcal{O})^r$$ as $\mathcal{O}[K_p]$-modules. Moreover, for any maximal ideal ${\frak{m}}\subset {\Bbb{T}}(K^p)$, the localization $\tilde{H}^0(K^p)_{{\frak{m}}}^{\circ}$ sits as a topologically direct summand. [*[Proof]{}*]{}. Since $A[[K_p]]$ is local, Nakayama’s lemma (and Proposition 4) tells us that $H^0(K^p,A)^{\vee}$ is a [*[free]{}*]{} $A[[K_p]]$-module, of finite rank $r_s$, say. Taking the Pontryagin dual, then yields an isomorphism of smooth $A[K_p]$-modules, $$H^0(K^p,\mathcal{O}/\varpi^s\mathcal{O})\simeq \mathcal{C}(K_p,\mathcal{O}/\varpi^s\mathcal{O})^{r_s}.$$ Now, we claim that $r_s$ is in fact independent of $s>0$ (and we will just write $r$ instead of $r_s$). To see this, scale both sides of the isomorphism by $\varpi$, compare the corresponding quotients, take $H$-invariants for some $H$, and compare dimensions over ${\Bbb{F}}$. This shows that $r_s=r_1$. This allows us to take the inverse limit over $s$, to obtain an isomorphism of modules over $\mathcal{O}[K_p]$, $$\tilde{H}^0(K^p)^{\circ}=\underset{s}{\varprojlim} H^0(K^p,\mathcal{O}/\varpi^s\mathcal{O})\simeq \mathcal{C}(K_p,\mathcal{O})^r.$$ In other words, an isometry $\tilde{H}^0(K^p)\simeq \mathcal{C}(K_p,L)^r$ of Banach representations of $K_p$. Finally, we may localize at any maximal ideal ${\frak{m}}\subset {\Bbb{T}}(K^p)$ and realize $\tilde{H}^0(K^p)_{{\frak{m}}}^{\circ}$ as a (topologically) direct summand of $\mathcal{C}(K_p,\mathcal{O})^r$. $\square$ Mahler expansions and full level at $p$ --------------------------------------- Proposition 5 already shows that the algebraic vectors are dense in $\tilde{H}^0(K^p)$ (by employing Mahler expansions, as below). In fact, this is even true for the unit ball $\tilde{H}^0(K^p)^{\circ}$. However, we can be more precise, and prove density of the smaller set of $G({\Bbb{Z}}_p)$-locally algebraic vectors: Those $f \in \tilde{H}^0(K^p)$ such that $\langle G({\Bbb{Z}}_p)f \rangle$ is an algebraic representation of $G({\Bbb{Z}}_p)$. $\tilde{H}^0(K^p)^{G({\Bbb{Z}}_p)-alg}$ is dense in $\tilde{H}^0(K^p)$. (Same for $\tilde{H}^0(K^p)_{{\frak{m}}}$.) [*[Proof]{}*]{}. Pick an open normal pro-$p$-subgroup $K_p \subset G({\Bbb{Z}}_p)$. From Proposition 5, we have an isometry $\tilde{H}^0(K^p)\simeq \mathcal{C}(K_p,L)^r$ of Banach space representations of $K_p$. We take the topological dual space $\mathcal{L}(-,L)$ on both sides, and get $$\text{$\tilde{H}^0(K^p)^{\vee}\simeq L[[K_p]]^r$, ${\hspace{6pt}}$ $L[[K_p]]:=L \otimes_{\mathcal{O}}\mathcal{O}[[K_p]]$.}$$ Here $L[[K_p]]$ is identified with the distribution algebra $\mathcal{C}(K_p,L)^{\vee}$ (equipped with the bounded-weak topology) as in \[ScT\]. Thus, $\tilde{H}^0(K^p)^{\vee}$ is a free $L[[K_p]]$-module of rank $r$. It follows that $\tilde{H}^0(K^p)^{\vee}$ is projective over $L[[G({\Bbb{Z}}_p)]]$, as $${\text{Hom}}_{L[[G({\Bbb{Z}}_p)]]}(\tilde{H}^0(K^p)^{\vee},-)={\text{Hom}}_{L[[K_p]]}(\tilde{H}^0(K^p)^{\vee},-)^{G({\Bbb{Z}}_p)/K_p}$$ is exact: $\tilde{H}^0(K^p)^{\vee}$ is projective over $L[[K_p]]$, and taking invariants under the finite group $G({\Bbb{Z}}_p)/K_p$ is exact, by averaging (we are in characteristic zero). Being projective, $\tilde{H}^0(K^p)^{\vee}$ is a direct summand of a free module (of finite rank by finite generation). That is, there is an $s>0$, and a submodule $Z$, such that $$\tilde{H}^0(K^p)^{\vee}\oplus Z \simeq L[[G({\Bbb{Z}}_p)]]^s.$$ Again, undoing the dual, and invoking Corollary 2.2 and Theorem 3.5 in \[ScT\], $$\tilde{H}^0(K^p) \oplus Z^{\vee} \simeq \mathcal{C}(G({\Bbb{Z}}_p),L)^s.$$ Comparing the $G({\Bbb{Z}}_p)$-algebraic vectors on both sides, we see that it suffices to show they are dense in $\mathcal{C}(G({\Bbb{Z}}_p),L)$. Now, topologically, we identify $G({\Bbb{Z}}_p)\simeq \prod_{v|p}{\text{GL}}_n(\mathcal{O}_{\tilde{v}})$ with a closed-open subset of $\prod_{v|p}\mathcal{O}_{\tilde{v}}^{n^2}\simeq {\Bbb{Z}}_p^t$, where we have introduced $t=[F:{\Bbb{Q}}]n^2$. Any continuous function on $G({\Bbb{Z}}_p)$ therefore extends (non-uniquely) to a continuous function on ${\Bbb{Z}}_p^t$, which has a (multi-variable) Mahler power series expansion \[Mah\], which shows that the polynomials are dense in $\mathcal{C}({\Bbb{Z}}_p^t,L)$. Finally, observe that polynomials obviously restrict to $G({\Bbb{Z}}_p)$-algebraic functions in $\mathcal{C}(G({\Bbb{Z}}_p),L)$. At last, localize at ${\frak{m}}$. $\square$ Zariski density of crystalline points ------------------------------------- Following Emerton, in section 5.4 of \[Eme\], we deduce from the previous Proposition that “crystalline points are dense”. The submodule $\oplus_{\lambda \in C}\tilde{H}^0(K^p)^{alg}[\lambda]$ is dense in $\tilde{H}^0(K^p)$, where $C$ denotes the collection of Hecke eigensystems $\lambda: \mathcal{H}(K^p)^{sph} \rightarrow \bar{{\Bbb{Q}}}_p$ associated with an automorphic $\pi$, which is unramified at $p$ (and of tame level $K^p$). Thus, the set of points $\ker(\lambda)$, with $\lambda \in C$, are Zariski dense in $\text{Spec} {\Bbb{T}}(K^p)[\frac{1}{p}]$. [*[Proof]{}*]{}. First off, recall from section 3.2 that we have a decomposition, $$\tilde{H}^0(K^p)^{alg}=\oplus_W W \otimes H^0(K^p,\mathcal{V}_{\breve{W}})=\oplus_W \oplus_{\pi: \pi_{\infty}\simeq W}m_G(\pi)(W\otimes \pi_f^{K^p}).$$ In particular, $$\tilde{H}^0(K^p)^{G({\Bbb{Z}}_p)-alg}=\oplus_W \oplus_{\pi: \pi_{\infty}\simeq W}m_G(\pi)(W\otimes \pi_p^{G({\Bbb{Z}}_p)}\otimes (\pi_f^p)^{K^p}),$$ which is dense in $\tilde{H}^0(K^p)$. A fortiori, so is the $G({\Bbb{Q}}_p)$-submodule it generates, $$\langle \tilde{H}^0(K^p)^{G({\Bbb{Z}}_p)-alg}\rangle_{G({\Bbb{Q}}_p)}=\oplus_W \oplus_{\pi: \pi_{\infty}\simeq W, \pi_p^{G({\Bbb{Z}}_p)}\neq 0} m_G(\pi)(W\otimes \pi_f^{K^p}).$$ We decompose the latter into eigenspaces for the action $\mathcal{H}(K^p)^{sph}$. That is, as $$\langle \tilde{H}^0(K^p)^{G({\Bbb{Z}}_p)-alg}\rangle_{G({\Bbb{Q}}_p)}=\oplus_{\lambda}\tilde{H}^0(K^p)^{alg}[\lambda]$$ where $\lambda: \mathcal{H}(K^p)^{sph} \rightarrow \bar{{\Bbb{Q}}}_p$ runs over all eigensystems of the form $\lambda=\lambda_{\pi}$, for some automorphic $\pi$, of tamel level $K^p$, which is [*[unramified]{}*]{} at $p$ (and of some weight $W$). Thus, elements of $\cap_{\lambda\in C}\ker(\lambda)$ act trivially on $\tilde{H}^0(K^p)$. $\square$ Reduction mod $p$ and refined Serre weights =========================================== We fix a tame level $K^p$ and point out how the eigenvariety ${\Bbb{X}}$ defines a (semisimple) global mod $p$ Langlands correspondence $t_x \leftrightarrow b_x$, analogous to the $p$-adic case discussed above. There is a natural notion of [*[refined]{}*]{} Serre weights of $\bar{\rho}_x$, and we relate them to the mod $p$ representation theory of $b_x$, and to the existence of crystalline lifts from ${\Bbb{X}}$ of compatible type $(\xi,\zeta)$. Integral models at $p$ ---------------------- We keep the notation from previous chapters. Thus, recall that $G=\text{Res}_{F|{\Bbb{Q}}}(U)$, where $U=U(D,\star)$ is a unitary group in $n$ variables over $F$, which becomes $D^{\times}$ over ${\mathcal{K}}$. Also, we fix a prime $p$ which is [*[unramified]{}*]{} in $F$, and such that every $v|p$ of $F$ splits in ${\mathcal{K}}$. Moreover, at each $w|v$ of ${\mathcal{K}}$, we assume $D_w \simeq M_n({\mathcal{K}}_w)$. Once and for all, above each $v|p$, we fix a place $\tilde{v}$ of ${\mathcal{K}}$, and use it to identify $$G({\Bbb{Q}}_p)\simeq \prod_{v|p}{\text{GL}}_n({\mathcal{K}}_{\tilde{v}}) \hookrightarrow G(\bar{{\Bbb{Q}}}_p)\simeq \prod_{v|p}{\text{GL}}_n(\bar{{\Bbb{Q}}}_p)^{{\text{Hom}}({\mathcal{K}}_{\tilde{v}},\bar{{\Bbb{Q}}}_p)}.$$ By our assumptions, $G$ is unramified over ${\Bbb{Q}}_p$. Indeed it has a Borel pair $(B,T)$ defined over ${\Bbb{Q}}_p$, and $G$ splits over an unramified extension (the compositum of all ${\mathcal{K}}_{\tilde{v}}=F_v$). By Bruhat-Tits theory, there is a smooth affine integral model $G_{/{\Bbb{Z}}_p}$ with connected reductive special fiber. Hence $G({\Bbb{Z}}_p)$ is hyperspecial, and $$G({\Bbb{F}}_p)\simeq \prod_{v|p}{\text{GL}}_n({\Bbb{F}}_{\tilde{v}}) \hookrightarrow G(\bar{{\Bbb{F}}}_p)\simeq \prod_{v|p}{\text{GL}}_n(\bar{{\Bbb{F}}}_p)^{{\text{Hom}}({\Bbb{F}}_{\tilde{v}},\bar{{\Bbb{F}}}_p)}.$$ (Here we have tacitly used the natural bijection ${\text{Hom}}({\mathcal{K}}_{\tilde{v}},\bar{{\Bbb{Q}}}_p) \overset{\sim}{\longrightarrow} {\text{Hom}}({\Bbb{F}}_{\tilde{v}},\bar{{\Bbb{F}}}_p)$, using that $p$ is unramified.) Moreover, one can spread out the Borel pair over ${\Bbb{Z}}_p$, and reduce it mod $p$. In particular, this allows us to identify dominant weights in different characteristics, $$X^*(T_{\bar{{\Bbb{Q}}}_p})_+ \simeq X^*(T_{\bar{{\Bbb{Z}}}_p})_+ \simeq X^*(T_{\bar{{\Bbb{F}}}_p})_+,$$ with tuples $a=(a_{\tau})_{\tau \in {\text{Hom}}({\mathcal{K}},{\Bbb{C}})}$, where each $a_{\tau}=(a_{\tau,j})$ is a decreasing $n$-tuple of integers, such that the following polarization condition is satisfied, $a_{\tau c,j}=-a_{\tau,n+1-j}$. This last step requires the choice of an $\iota: {\Bbb{C}}\overset{\sim}{\longrightarrow}\bar{{\Bbb{Q}}}_p$. Refined Serre weights --------------------- A [*[Serre weight]{}*]{} is commonly defined as an irreducible representation $\omega$ of $G({\Bbb{F}}_p)$ with coefficients in $\bar{{\Bbb{F}}}_p$ (usually inflated to a representation of $G({\Bbb{Z}}_p)$ by composing with the reduction map). Since $G$ has a simply connected derived group, a result of Steinberg (extended to reductive groups by Herzig) shows that such $\omega$ are restrictions of algebraic representations $\omega$ of $G(\bar{{\Bbb{F}}}_p)$. For the restriction $\omega|_{G({\Bbb{F}}_p)}$ to remain irreducible, the highest weight of $\omega$ has to be $p$-[*[restricted]{}*]{}. In the notation introduced in the previous subsection, this means all gaps are in the range $a_{\tau,j}-a_{\tau,j+1}<p$ (and non-negative). In what follows, we will simply write $G=G({\Bbb{Q}}_p)$ and $K=G({\Bbb{Z}}_p)$ when there is no risk of confusion. For each Serre weight $\omega$, we introduce the $\omega$-spherical Hecke algebra: $$\mathcal{H}_{\omega}(G,K)={\text{End}}_{G}(\text{c-Ind}_K^G(\omega)).$$ By Frobenius reciprocity, this can be thought of more concretely as the convolution algebra of compactly supported $K$-biequivariant functions $G\rightarrow {\text{End}}(\omega)$. $\mathcal{H}_{\omega}(G,K)$ is a [*[commutative]{}*]{} noetherian $\bar{{\Bbb{F}}}_p$-algebra, according to Corollary 1.3 in \[Her\]. In fact, there is a mod $p$ analogue of the Satake isomorphism. A refined Serre weight is a pair $(\omega,\nu)$, consisting of a Serre weight $\omega$, together with an algebra homomorphism $ \nu: \mathcal{H}_{\omega}(G,K)\rightarrow \bar{{\Bbb{F}}}_p$. Each such pair $(\omega,\nu)$ defines an $\bar{{\Bbb{F}}}_p$-representation of $G$, as the quotient $$\pi(\omega,\nu)=\text{c-Ind}_K^G(\omega)\otimes_{ \mathcal{H}_{\omega}(G,K),\nu}\bar{{\Bbb{F}}}_p=(\text{c-Ind}_K^G(\omega))/(\ker\nu)(\text{c-Ind}_K^G(\omega)).$$ (Analogous to the universal modules $H_{\xi,\zeta}$ from 6.2.) If $b$ is an admissible mod $p$ representation of $G$, there is a natural action of the Hecke algebra $\mathcal{H}_{\omega}(G,K)$ on the finite-dimensional multiplicity space, $${\text{Hom}}_K(\omega,b)\overset{\sim}{\longrightarrow}{\text{Hom}}_G(\text{c-Ind}_K^G(\omega),b),$$ which therefore decomposes as a direct sum of generalized eigenspaces. The character $\nu: \mathcal{H}_{\omega}(G,K)\rightarrow \bar{{\Bbb{F}}}_p$ occurs as an eigensystem in ${\text{Hom}}_K(\omega,b)$ if and only if there is a nonzero $G$-equivariant map $\pi(\omega,\nu)\rightarrow b$. [*[Proof]{}*]{}. Immediate from the definitions. $\square$ Types compatible with a refined Serre weight -------------------------------------------- For the moment, we fix a refined Serre weight $(\omega,\nu)$ as above. We will define what it means for a pair $(\xi,\zeta)$ to be compatible with $(\omega,\nu)$. As in previous sections, $\xi$ denotes an irreducible algebraic $\bar{{\Bbb{Q}}}_p$-representation of $G(\bar{{\Bbb{Q}}}_p)$, and $$\zeta:\mathcal{H}_{\xi}(G,K)\simeq \mathcal{H}(G,K)\longrightarrow \bar{{\Bbb{Q}}}_p$$ is an algebra homomorphism. Eventually we will take $(\xi,\zeta)$ to be the [*[type]{}*]{} of a crystalline representation, in the sense used previously in this paper (that is, $\xi$ is defined from the Hodge-Tate weights, and $\zeta$ gives the eigensystem of the associated unramified representation). We say $(\xi,\zeta)$ is compatible with $(\omega,\nu)$ if there is a $G({\Bbb{Z}}_p)$-invariant norm $\|\cdot\|_{\xi}$ (equivalently, a stable lattice $\xi^{\circ}$, the unit ball) such that - $\omega \hookrightarrow \xi^{\circ}\otimes \bar{{\Bbb{F}}}_p$ (as a $G({\Bbb{Z}}_p)$-submodule), - $\zeta$ is $\bar{{\Bbb{Z}}}_p$-valued on $\mathcal{H}_{\xi}(G,K)^{\circ}$, and its reduction $\zeta\otimes 1\leftrightarrow \nu$ under $$\mathcal{H}_{\xi}(G,K)^{\circ}\otimes \bar{{\Bbb{F}}}_p \overset{\sim}{\longrightarrow} \mathcal{H}_{\omega}(G,K).$$ (This is the comparison isomorphism $\alpha$ from Proposition 2.10 in \[Her\].) A few words of elaboration: The first bullet is really just saying that $\omega$ and $\xi$ have the [*[same]{}*]{} highest weight $a$. Indeed, in the notation of \[Jan\], if $\omega=L(a)$, we may take $\xi=H^0(a)$, a dual Weyl module, which can be defined over $\bar{{\Bbb{Z}}}_p$. However, note that for $\omega \hookrightarrow \xi^{\circ}\otimes \bar{{\Bbb{F}}}_p$ to be an isomorphism, the weight has to be $p$-[*[small]{}*]{} (according to Corollary 5.6 on p. 221 in \[Jan\]), which means $$0 \leq a_{\tau,1}-a_{\tau,n}\leq p-(n-1),$$ for all $\tau$. (Indeed, the highest weight paired with any positive coroot $\alpha^{\vee}$ should be at most $p-\langle\varrho,\alpha^{\vee}\rangle$, where $\varrho=\frac{1}{2}\sum_{\alpha>0}\alpha$. In the ${\text{GL}}_n$-case this translates into the inequalities $a_{\tau,i}-a_{\tau,j}\leq p-(j-i)$ whenever $i<j$, the strongest of which is the bound on the total gap, $a_{\tau,1}-a_{\tau,n}\leq p-(n-1)$, which is $p$-smallness.) Note that $p$-smallness is stronger than $p$-restrictedness for $n>2$. The isomorphism in the second bullet is defined as follows: Take an $h \in \mathcal{H}_{\xi}(G,K)$, of sup-norm at most one. In particular, $h(g)$ maps $\xi^{\circ}$ to itself for any $g \in G$. What is shown in Proposition 2.1 in \[Her\] is that the reduction $\overline{h(g)}$ in fact preserves any submodule of $\xi^{\circ}\otimes \bar{{\Bbb{F}}}_p$, hence $\omega$. This yields $g \mapsto \overline{h(g)}|_{\omega}$. A global mod $p$ Langlands correspondence ----------------------------------------- There is a natural mod $p$ analogue of the $p$-adic correspondence preceding Theorem B, obtained as follows. Take a point $x \in {\Bbb{X}}(L)$, where $L|E$ is an arbitrary finite extension. On the one hand, it gives rise to a pseudo-representation $t_x={\text{tr}}\bar{\rho}_x^{ss}$ (by reducing $\mathcal{T}_x={\text{tr}}\rho_x$ mod ${\frak{p}}_L$). On the other, we may reduce the eigensystem $\lambda_x: \frak{h}^{\circ}\rightarrow \mathcal{O}_L$ modulo ${\frak{p}}_L$, and look at the [*[generalized]{}*]{} eigenspace $$\text{$b_x=H^0(K^p,{\Bbb{F}}_L)^{ \frak{h}^{\circ}=\bar{\lambda}_x}=H^0(K^p,{\Bbb{F}}_L)_{\frak{m}_x}$, ${\hspace{6pt}}$ $\frak{m}_x=\ker(\bar{\lambda}_x)$.}$$ Here $H^0(K^p,{\Bbb{F}}_L)$ is nothing but the space of smooth functions $Y(K^p)\rightarrow {\Bbb{F}}_L$. Clearly $b_x$ carries an $\mathcal{H}_{{\Bbb{F}}_L}(K^p)$-module structure, and a commuting [*[admissible]{}*]{} action of $G({\Bbb{Q}}_p)$. As in the $p$-adic case, the pairs $(t_x,b_x)$ form the graph of a bijection, $$\left\{ \begin{matrix} \text{$n$-dimensional pseudo-representations} \\ \text{$t: \Gamma_{{\mathcal{K}},\Sigma}\rightarrow {\Bbb{F}}_L$ coming from ${\Bbb{X}}(L)$} \end{matrix} \right\} \longleftrightarrow$$ $$\left\{ \begin{matrix} \text{$\mathcal{H}_{{\Bbb{F}}_L}(K^p)$-modules $b$ with admissible} \\ \text{$G({\Bbb{Q}}_p)$-action, coming from ${\Bbb{X}}(L)$} \end{matrix} \right\}.$$ Here $t \leftrightarrow b$ if there is a point $x \in {\Bbb{X}}(L)$ such that $t=t_x$ and $b=b_x$, in which case $$t_x(\text{Frob}_w)=\bar{\lambda}_x(b_{w|v}(h_w))$$ for all $w \notin \Sigma$. This is compatible with the $p$-adic correspondence in the following sense: For any $x' \in {\Bbb{X}}(L)$ with eigensystem $\lambda_{x'}\equiv \lambda_x$ (mod ${\frak{p}}_L$) we have $$\bar{\mathcal{B}}_{x'}=\mathcal{B}_{x'}^{\circ}\otimes {\Bbb{F}}_L \hookrightarrow b_x,$$ a $G({\Bbb{Q}}_p)\times \mathcal{H}_{{\Bbb{F}}_L}(K^p)$-equivariant embedding. Algebraic modular forms mod $p$ ------------------------------- We will only introduce mod $p$ modular forms of level $G({\Bbb{Z}}_p)K^p$, and weight $\omega$, $$\mathcal{A}_{\omega}(G({\Bbb{Z}}_p)K^p;\bar{{\Bbb{F}}}_p)={\text{Hom}}_{G({\Bbb{Z}}_p)}(\omega, H^0(K^p,\bar{{\Bbb{F}}}_p)).$$ More concretely, they are $K^p$-invariant functions $f: G({\Bbb{Q}})\backslash G({\Bbb{A}}_f)\rightarrow \omega^{\vee}$ with $$\text{$f(gk)=\omega^{\vee}(k)^{-1}f(g)$, ${\hspace{6pt}}$ $k \in G({\Bbb{Z}}_p)$.}$$ There is a natural action of the algebras $\mathcal{H}_{\omega}(G,K)$ and $\mathcal{H}_{{\Bbb{F}}_L}(K^p)$, and hence it decomposes as a direct sum of generalized eigenspaces for the action of the commutative algebra $\mathcal{H}_{\omega}(G,K)\otimes \mathcal{H}_{{\Bbb{F}}_L}(K^p)^{\text{sph}}$. Which eigensystems occur? We fix a point $x \in {\Bbb{X}}(L)$ such that $\bar{\rho}_x$ is absolutely [*[irreducible]{}*]{}, as a representation of $\Gamma_{{\mathcal{K}}}$. The Serre weights $\mathcal{W}(\bar{\rho}_x)$, relative to $K^p$, are those $\omega$ such that $\bar{\lambda}_x$ occurs in the mod $p$ modular forms. That is to say, such that $${\text{Hom}}_{G({\Bbb{Z}}_p)}(\omega,b_x)\neq 0.$$ In other words, $\text{soc}_{G({\Bbb{Z}}_p)}(b_x)$ is of the form $\oplus_{\omega \in \mathcal{W}(\bar{\rho}_x)}m_x(\omega)\omega$. This naturally leads up to a more refined notion, taking into account the Hecke-action at $p$. $\mathcal{W}_+(\bar{\rho}_x)$ is the set of refined Serre weights $(\omega,\nu)$ such that $\nu\otimes \bar{\lambda}_x$ occurs as an eigensystem in the space of mod $p$ modular forms $\mathcal{A}_{\omega}(G({\Bbb{Z}}_p)K^p;\bar{{\Bbb{F}}}_p)$. Equivalently, by Lemma 1, $$(\omega,\nu)\in \mathcal{W}_+(\bar{\rho}_x)\Longleftrightarrow{\text{Hom}}_{G({\Bbb{Q}}_p)}(\pi(\omega,\nu),b_x)\neq 0.$$ (Note that $\mathcal{W}_+(\bar{\rho}_x)$ may depend on the tamel level $K^p$, at least a priori.) Crystalline lifts of compatible type ------------------------------------ The next result is inspired by Gee’s approach to Serre weights via [*[local]{}*]{} crystalline lifts with prescribed Hodge-Tate weights. See \[Gee\], for example. We will instead consider [*[global]{}*]{} lifts, which show up on our fixed eigenvariety, and prescribe both the Hodge-Tate weights [*[and]{}*]{} the Frobenius eigenvalues. For technical reasons, we will assume from now on that our fixed tame level $K^p$ is sufficiently small (for precision, that some $K_v$ has no elements of order $p$). Let $(\omega,\nu)$ be an arbitrary refined Serre weight, and let $x$ be a point of $X_{reg}$ such that $\bar{\rho}_x$ is [*[irreducible]{}*]{}. Then the following holds. - If $(\omega,\nu)\in \mathcal{W}_+(\bar{\rho}_x)$, then $\bar{\rho}_x$ has a crystalline lift from $X_{reg}$, whose type is compatible with $(\omega,\nu)$. - The converse holds when $\omega$ is $p$-: If $\bar{\rho}_x$ has a crystalline lift from $X_{reg}$, whose type is compatible with $(\omega,\nu)$, then $(\omega,\nu)\in \mathcal{W}_+(\bar{\rho}_x)$. [*[Proof of (a)]{}*]{}: Say $\omega=L(a)$. Pick $\xi^{\circ}$ to be a $\bar{{\Bbb{Z}}}_p$-structure in the dual Weyl module $\xi=H^0(a)$. Then $\omega$ is the unique irreducible submodule of $\xi^{\circ}\otimes \bar{{\Bbb{F}}}_p$. This yields a natural reduction map, $$\mathcal{A}_{\xi}(G({\Bbb{Z}}_p)K^p;\bar{{\Bbb{Z}}}_p)\otimes \bar{{\Bbb{F}}}_p \rightarrow \mathcal{A}_{\omega}(G({\Bbb{Z}}_p)K^p;\bar{{\Bbb{F}}}_p),$$ where $$\mathcal{A}_{\xi}(G({\Bbb{Z}}_p)K^p;\bar{{\Bbb{Z}}}_p)={\text{Hom}}_{G({\Bbb{Z}}_p)}(\xi^{\circ}, H^0(K^p,\bar{{\Bbb{Z}}}_p))$$ is naturally a $\bar{{\Bbb{Z}}}_p$-structure in a space of classical $p$-adic modular forms. The reduction map is easily seen to be surjective if $K^p$ is sufficiently small (that is, if $p$ does not divide the orders of the arithmetic subgroups showing up). By assumption, the inflated eigensystem $\nu\otimes \bar{\lambda}_x$, $$\mathcal{H}_{\xi}(G,K)^{\circ}\otimes \frak{h}^{\circ} \longrightarrow \mathcal{H}_{\omega}(G,K)\otimes \mathcal{H}_{{\Bbb{F}}_L}(K^p)^{\text{sph}}\longrightarrow \bar{{\Bbb{F}}}_p,$$ occurs in $\mathcal{A}_{\omega}(G({\Bbb{Z}}_p)K^p;\bar{{\Bbb{F}}}_p)$. Localizing the reduction map at $\ker(\nu\otimes \bar{\lambda}_x)$, and invoking the Deligne-Serre lifting lemma, we infer that there is an eigensystem $$\Lambda: \mathcal{H}_{\xi}(G,K)^{\circ}\otimes \frak{h}^{\circ} \longrightarrow \bar{{\Bbb{Z}}}_p,$$ with $\bar{\Lambda}=\nu\otimes \bar{\lambda}_x$, which occurs in the space $\mathcal{A}_{\xi}(G({\Bbb{Z}}_p)K^p;\bar{{\Bbb{Q}}}_p)$ of classical $p$-adic modular forms. Via $\iota: {\Bbb{C}}\overset{\sim}{\longrightarrow}\bar{{\Bbb{Q}}}_p$ we thus find an automorphic representation $\pi$ of $G({\Bbb{A}})$ with $\pi_{\infty}=\xi$, such that $\frak{h}$ acts on $\pi_f^{K^p}\neq 0$ by the character $\Lambda^p\equiv \lambda_x$, and such that $\pi_p$ is unramified with eigensystem $\Lambda_p$ lifting $\nu$ (where we use the usual identification $\mathcal{H}(G,K)\simeq \mathcal{H}_{\xi}(G,K)$ to view $\Lambda_p$ as a character of the spherical Hecke algebra). Say $\pi_p\subset \text{Ind}(\theta)$, and $\psi$ is the highest weight of $\xi$. Then $(\psi\theta,\Lambda^p)$ belongs to $E(0,K^p)_{cl}$. Let $x' \in X_{cl}$ be the corresponding classical point of the eigenvariety. Now, the Galois representation $\rho_{x'}\simeq \rho_{\pi,\iota}$ (comes from $X_{cl}$ and) is crystalline at all places avove $p$; since $\pi_p$ is unramified. Moreover, $\bar{\rho}_{x'}\simeq \bar{\rho}_x$ since $\Lambda^p\equiv \lambda_x$ (both are irreducible). Finally, $\rho_{x'}$ has type $(\xi,\Lambda_p)$, which is compatible with $(\omega,\nu)$. Indeed $\omega \subset \xi^{\circ}\otimes \bar{{\Bbb{F}}}_p$, and obviously $\bar{\Lambda}_p\leftrightarrow \nu$. [*[Proof of (b)]{}*]{}: Now suppose there is an old point $x' \in X_{reg}$ such that $\bar{\rho}_{x'}\simeq \bar{\rho}_x$, and $\rho_{x'}$ has type $(\xi,\zeta)$ compatible with $(\omega,\nu)$. By Theorem 4, part (3), we know that $$\xi_{x'}\otimes \text{Ind}_B^G(\theta_{x'})=\widetilde{BS}(\rho_{x'})\hookrightarrow \mathcal{B}_{x'},$$ as a $G({\Bbb{Q}}_p)$-representation. (Recall that $\rho_{x'}$ is even residually irreducible, so $\pi_{x'}$ is generic at all split places). Here $\xi=\xi_{x'}$, and $\mathcal{H}(G,K)$ acts on the newvectors in $\text{Ind}_B^G(\theta_{x'})$ via $\zeta$. Choosing a newvector (unique up to a constant) then gives a $K$-map $\xi \hookrightarrow \mathcal{B}_{x'}$ (or, equivalently, a nonzero $G$-map $\text{c-Ind}_K^G(\xi)\rightarrow \mathcal{B}_{x'}$) on which $\mathcal{H}_{\xi}(G,K)\simeq \mathcal{H}(G,K)$ acts by $\zeta$. Let $\|\cdot\|_{\xi}$ be the norm on $\xi$ (and $\xi^{\circ}$ its unit ball), whose existence is guaranteed by Definition 9; the compatibility with $(\omega,\nu)$. Since $\xi$ is finite-dimensional, all norms are equivalent. So $\xi \hookrightarrow \mathcal{B}_{x'}$ is automatically continuous, and upon scaling, we may assume it preserves the unit balls. That is, $\xi^{\circ}\hookrightarrow \xi \cap \mathcal{B}_{x'}^{\circ}$, with cokernel killed by some nonzero constant in $\bar{{\Bbb{Z}}}_p$ (all lattices are commensurable). By the Brauer-Nesbitt principle, $$(\xi^{\circ}\otimes \bar{{\Bbb{F}}}_p)^{ss}\simeq ((\xi \cap \mathcal{B}_{x'}^{\circ}) \otimes \bar{{\Bbb{F}}}_p)^{ss}.$$ Moreover, the cokernel of the inclusion $\xi \cap \mathcal{B}_{x'}^{\circ}\hookrightarrow \mathcal{B}_{x'}^{\circ}$ is clearly torsion-free, so $$(\xi \cap \mathcal{B}_{x'}^{\circ}) \otimes \bar{{\Bbb{F}}}_p\hookrightarrow \mathcal{B}_{x'}^{\circ} \otimes \bar{{\Bbb{F}}}_p\hookrightarrow b_x,$$ since $\lambda_{x'}\equiv \lambda_{x}$. Now, since we assume $\omega$ is $p$-small, the source of this map is isomorphic to $\omega$, and the way it was constructed shows that $\mathcal{H}_{\omega}(G,K)$ acts on $\omega \hookrightarrow b_x$ by $\nu\leftrightarrow \zeta\otimes 1$. Ths results in a nonzero $G$-map $\pi(\omega,\nu)\rightarrow b_x$. $\square$ [*[Remark]{}*]{}. If we are not assuming $p$-smallness in (b), the above argument still shows that $\mathcal{W}(\bar{\rho}_x)$ contains [*[some]{}*]{} constituent of $\xi^{\circ}\otimes \bar{{\Bbb{F}}}_p$ (of highest weight at [*[most]{}*]{} that of $\omega$), but we cannot say much about the Hecke action at $p$. Local-global compatibility and Ihara’s lemma ============================================ We define a candidate $b(\bar{\rho}_x)$ for $\otimes_{v|p}b(\bar{\rho}_x|_{\Gamma_{{\mathcal{K}}_{\tilde{v}}}})$ by means of the local Langlands correspondence in characteristic $p$ (see Theorem 5.1.5 in \[EH\]). We offer some (precise) [*[speculations]{}*]{} towards local-global compatibility mod $p$, from which the conjectural Ihara lemma (Conjecture B in \[CHT\]) is deduced. Local Langlands in characteristic $p$ ------------------------------------- For finite extensions $K|{\Bbb{Q}}_{\ell}$, where $\ell \neq p$, Emerton and Helm have defined a map $$\left\{ \begin{matrix} \text{continuous $\bar{\rho}:\Gamma_K \rightarrow {\text{GL}}_n({\Bbb{F}}_L)$} \end{matrix} \right\} \longrightarrow$$ $$\left\{ \begin{matrix} \text{${\Bbb{F}}_L$-spaces $\bar{\pi}$ endowed with} \\ \text{admissible ${\text{GL}}_n(K)$-action} \end{matrix} \right\},$$ denoted $\bar{\rho}\mapsto \bar{\pi}(\bar{\rho})$. This $\bar{\pi}(\bar{\rho})$ is of finite length, and is characterized uniquely by the following three properties (according to Theorem 5.1.5 in \[EH\]): - $\bar{\pi}(\bar{\rho})$ is essentially AIG. - If $\rho: \Gamma_K \rightarrow {\text{GL}}_n(\mathcal{O}_{L'})$ is a continuous lift of $\bar{\rho}\otimes {\Bbb{F}}_{L'}$, and $\Lambda$ is a ${\text{GL}}_n(K)$-invariant lattice in $\pi(\rho)$ (the generic local Langlands correspondent) with $\Lambda \otimes {\Bbb{F}}_{L'}$ essentially AIG, then there is a ${\text{GL}}_n(K)$-embedding, $$\Lambda \otimes {\Bbb{F}}_{L'} \hookrightarrow \bar{\pi}(\bar{\rho})\otimes {\Bbb{F}}_{L'}.$$ - $\bar{\pi}(\bar{\rho})$ is minimal in the following sense: If $\bar{\pi}$ is any representation satisfying (a) and (b), then there is a ${\text{GL}}_n(K)$-embedding, $\bar{\pi}(\bar{\rho}) \hookrightarrow \bar{\pi}$. The correspondence has a number of additional nice properties, which we will not recall. See (4) through (8) on p. 44 in \[EH\]. AIG stands for absolutely irreducible and generic. Paraphrasing Definition 3.2.1 in \[EH\], a smooth representation $\bar{\pi}$ is [*[essentially AIG]{}*]{} if $\text{soc}(\bar{\pi})$ is AIG, $\bar{\pi}/\text{soc}(\bar{\pi})$ has no generic constituents, and $\bar{\pi}$ is the sum of its finite length submodules. The correspondence $\bar{\rho}\mapsto \bar{\pi}(\bar{\rho})$ plays a key role in (strong) local-global compatibility mod $p$ for ${\text{GL}}(2)$. See Theorem 1.2.6 in \[Eme\], which will serve as a guide in what follows. Conjectural local-global compatibility mod $p$ ---------------------------------------------- We now return to the situation from Chapter 6. Thus, we have a tame level $K^p$ with [*[split ramification]{}*]{}, and we let $x \in {\Bbb{X}}(L)$ be a regular (classical) point on the eigenvariety such that $\rho_x$ is [*[residually]{}*]{} irreducible, and $m(\pi_x)=1$. For each place $v \nmid p$ of $F$, we define a representation $\bar{\pi}_{x,v}$ of $U(F_v)$ over ${\Bbb{F}}_L$. In the unramified case, where $v \notin S(K^p)$, $\pi_{x,v}$ is a $p$-integral unramified principal series, and we define $\bar{\pi}_{x,v}$ to be its canonical reduction. In the ramified case, where $v \in S(K^p)$ necessarily splits in ${\mathcal{K}}$, $$\bar{\pi}_{x,v}\simeq \bar{\pi}(\bar{\rho}_x|_{\Gamma_{{\mathcal{K}}_w}}),$$ under $U(F_v)\simeq {\text{GL}}_n({\mathcal{K}}_w)$, where $w|v$ is any of the two places above. Only the $\mathcal{H}(K^p)$-module $m=\otimes_{v\nmid p}\bar{\pi}_{x,v}^{K_v}$ will occur below, on which $\mathcal{H}(K^p)^{\text{sph}}$ acts by $\bar{\lambda}_x$. Thus the precise definition of $\bar{\pi}_{x,v}$ at $v \notin S(K^p)$ is not relevant here. The ramified places $S(K^p)$ are the interesting ones, where we invoke \[EH\]. Guided by Theorem 1.2.6 in \[Eme\], and the remark after Corollary 3 above, we define our candidate representation $b(\bar{\rho}_x)$ as follows. $b(\bar{\rho}_x)={\text{Hom}}_{\mathcal{H}(K^p)}(\otimes_{v\nmid p}\bar{\pi}_{x,v}^{K_v},b_x)$. This is an admissible representation of $G({\Bbb{Q}}_p)$ over ${\Bbb{F}}_L$. However, it is not clear at all whether $b(\bar{\rho}_x)$ is [*[nonzero]{}*]{}! This is implied by local-global compatibility: The tautological map $$\Phi:b(\bar{\rho}_x) \otimes (\bigotimes_{v\nmid p}\bar{\pi}_{x,v}^{K_v}) \rightarrow b_x$$ is an isomorphism (preserving the action of $G({\Bbb{Q}}_p)$ and $\mathcal{H}(K^p)$). This may very well turn out to be too naive. For now, it serves as a guiding principle. In the $p$-adic case, we had to “replace” $\mathcal{B}_x$ with the closure of the regular-algebraic vectors $\overline{\mathcal{B}_x^{ralg}}$ to achieve local-global compatibility on the nose. Maybe one should look for an analogous (finite length?) submodule $b_x^{\#}\subset b_x$, containing all the reductions $\bar{\mathcal{B}}_{x'}\hookrightarrow b_x^{\#}$, and use [*[it]{}*]{} to define $b(\bar{\rho}_x)$ as above. Of course, $b(\bar{\rho}_x)$ appears to be nothing more than a toy construction unless one can show it only depends on the various restrictions $\bar{\rho}_x|_{\Gamma_{{\mathcal{K}}_{\tilde{v}}}}$ at $p$, and that it factors as a tensor product $\otimes_{v|p}$. This is clearly at the heart of the whole discussion! As in the $p$-adic case, we have made no progress towards this problem. [*[Remark]{}*]{}. Conjecture 1 holds if $S(K^p)=\varnothing$, in which case $b(\bar{\rho}_x)\simeq b_x$. Conjectural $p$-adic and mod $p$ compatibility ---------------------------------------------- We keep the assumptions and the notation of the previous section. Thus, we fix a point $x \in {\Bbb{X}}(L)$ such that $\bar{\rho}_x$ is absolutely irreducible (as a representation of $\Gamma_{{\mathcal{K}}}$). There should be a relation between $b(\bar{\rho}_x)$ and the reduction of $B(\rho_x)$, the $p$-adic candidate introduced in Theorem B of the introduction (and in the remark after Corollary 3 in section 6.1). We will make this expectation precise. First, we will specify a lattice in $M=\otimes_{v\nmid p}\pi_{x,v}^{K_v}$. Let $v\nmid p$ be a ramified (hence split) place of $F$. By Proposition 3.3.2 in \[EH\], since $\pi_{x,v}$ is generic, it admits a $U(F_v)$-invariant lattice $\pi_{x,v}^{\circ}$ such that its (naive) reduction $\pi_{x,v}^{\circ}\otimes {\Bbb{F}}_L$ is essentially AIG (which is unique up to homothety). By (2) of Theorem 5.1.5 in \[EH\], one can then find a $U(F_v)$-equivariant embedding $\pi_{x,v}^{\circ}\otimes {\Bbb{F}}_L\hookrightarrow \bar{\pi}_{x,v}$ (not necessarily onto, when the target is reducible). As lattice in $M$, we take $M^{\circ}=\otimes_{v\nmid p} (\pi_{x,v}^{\circ})^{K_v}$. There is a $G({\Bbb{Q}}_p)$-equivariant map $\Psi$ such that the diagram $$\xymatrix{ B(\rho_x)^{\circ}\otimes {\Bbb{F}}_L \ar[d] \ar[r]^{\Psi} &b(\bar{\rho}_x) \ar[d]\\ {\text{Hom}}_{\mathcal{H}(K^p)^{\circ}}(M^{\circ}\otimes {\Bbb{F}}_L,\mathcal{B}_x^{\circ}\otimes {\Bbb{F}}_L) \ar[r] & {\text{Hom}}_{\mathcal{H}(K^p)^{\circ}}(M^{\circ}\otimes {\Bbb{F}}_L,b_x)}$$ commutes. (Here the right vertical map is composition with $M^{\circ}\otimes {\Bbb{F}}_L \hookrightarrow m$.) Note that such a map $\Psi$ is necessarily nonzero ($\Rightarrow b(\bar{\rho}_x)$ is nonzero). Connections to Ihara’s lemma ---------------------------- A key step in Wiles’s proof of Fermat’s Last Theorem was to reduce the non-minimal case of modularity lifting to the minimal case, by level-raising. What makes this work is a lemma of Ihara. This approach was mimicked in \[CHT\] for $U(n)$, where they deduce modularity lifting $R={\Bbb{T}}$ from a conjectural analogue of Ihara’s lemma (Conjecture B in \[CHT\]), originating from Mann’s Harvard Ph.D. thesis, which is still open. Taylor has since adapted techniques of Kisin to bypass Ihara’s lemma and prove modularity lifting theorems of the form $R^{\text{red}}={\Bbb{T}}$. As mentioned in the introduction of \[CHT\], it would be interesting to get rid of the nilradical. For example, this would have applications to special values of $L$-functions. Let us recast Conjecture B of \[CHT\] in our setup: [**[Ihara’s Lemma]{}**]{}. [*[Let $u \in S(K^p)$ be a place of $F$ such that $U(F_u)\simeq {\text{GL}}_n(F_u)$, and $p$ is banal for this group. Let $x\in X_{cl}$ be a point such that $\bar{\rho}_x$ is irreducible. $$\mathcal{S}_x\overset{df}{=}\text{$\mathcal{C}^{\infty}(U(F)\backslash U({\Bbb{A}}_{F,f})/K^{p,u},\bar{{\Bbb{F}}}_p)_{{\frak{m}}_x}$, ${\hspace{6pt}}$ ${\frak{m}}_x\overset{df}{=}\ker(\bar{\lambda}_x)\subset \mathcal{H}(K^p)^{\text{sph}}$.}$$ Then every simple ${\text{GL}}_n(F_u)$-submodule of $\mathcal{S}_x$ is generic.]{}*]{} This has close ties to Conjecture 1 (local-global compatibility mod $p$). Conjecture 1 $\Longrightarrow$ Ihara’s lemma. [*[Proof]{}*]{}. Let $\tau \subset \mathcal{S}_x$ be a simple ${\text{GL}}_n(F_u)$-submodule. In $U(F_u)$ we pick a small enough compact open subgroup $K_u$ such that $\tau^{K_u}\neq 0$, and such that taking $K_u$-invariants defines an equivalence of categories between the category of smooth representations of $U(F_u)$, which are generated by their $K_u$-invariants, and the category of modules for the Hecke algebra $\mathcal{H}(U(F_u),K_u)$. This is possible since $p$ is assumed banal for $U(F_u)$, by a result of Vigneras. Consequently, with a possibly smaller $K^p$ (namely $K_uK^{p,u}$) we have $$\tau^{K_u}\subset \mathcal{S}_x^{K_u}=H^0(K^p,\bar{{\Bbb{F}}}_p)_{{\frak{m}}_x}=b_x.$$ Admitting Conjecture 1, we conclude that the simple Hecke module $\tau^{K_u}$ embeds into $\bar{\pi}_{x,u}^{K_u}$. By choice of $K_u$, this arises from an embedding $\tau \hookrightarrow \bar{\pi}_{x,u}$. Hence, $$\tau=\text{soc}(\bar{\pi}_{x,u}),$$ since the latter is AIG, by the desiderata in \[EH\]. In particular, $\tau$ is generic. $\square$ Admittedly, this just replaces Ihara’s lemma by a stronger conjecture. However, we feel that the latter is more conceptual, and more intuitive. [*[Remark]{}*]{}. Conversely, it looks like something along the lines of Ihara’s lemma is needed to even show that $b(\bar{\rho}_x)\neq 0$. Indeed, this would require producing maps $\bar{\pi}_{x,u}^{K_u}\rightarrow b_x$. This would make use of the minimality (c). However, to do that, one would have to verify that $\mathcal{S}_x$ is AIG, among other things, such as dealing with the lifts in (b) above. It would be interesting to see if the modularity lifting theorems in \[CHT\], which [*[result]{}*]{} from Ihara’s lemma, shed any light on this. Appendix: Frobenius reciprocity =============================== For the convenience of the reader, we briefly recall the explicit formulas giving Frobenius reciprocity, and as a consequence deduce that [*[continuity]{}*]{} is preserved. This was used in section 6 on weak local-global compatibility. We will work in the following setup: Let $G$ be a topological group, and let $K$ be a compact open subgroup. Let $\xi:K \rightarrow {\text{GL}}(V)$ be a continuous representation on a finite-dimensional $L$-vector space $V$, where $L/{\Bbb{Q}}_p$ is finite. The compact induction $\text{c-ind}_K^G\xi$ consists of all compactly supported functions $f:G \rightarrow V$ such that $f(kg)=\xi(k)f(g)$. Such $f$ are automatically continuous. [*[Example]{}*]{}. For any vector $v \in V$ define $\phi_v$ by letting $\phi_v(g)=\xi(g)v$ when $g \in K$, and $\phi_v(g)=0$ otherwise. Observe that $\phi_{\xi(k)v}=k\phi_v$. Consequently, these $\{\phi_v\}$ generate $\text{c-ind}_K^G\xi$ as a $G$-representation: Indeed, any $f$ as above can be written as a (finite) sum $f=\sum_{g \in K\backslash G}g^{-1}\phi_{f(g)}$. Let $\rho:G \rightarrow {\text{GL}}(W)$ be any (possibly infinite-dimensional) representation of $G$ on an $L$-vector space $W$. Then there is a natural bijection $${\text{Hom}}_K(\xi,\rho|_K)\overset{\sim}{\longrightarrow} {\text{Hom}}_G(\text{c-ind}_K^G\xi,\rho).$$ Choose a $K$-invariant norm $\|\cdot\|_{\xi}$ on $V$, and let $\|\cdot\|_{\xi,\infty}$ be the corresponding $G$-invariant supremum-norm on $\text{c-ind}_K^G\xi$. Moreover, suppose $W$ has a $G$-invariant norm $\|\cdot\|_{\rho}$. [**[Then]{}**]{} the bijection preserves the transformation norms on both sides. (In particular, if $\xi$ is irreducible, every transformation $\text{c-ind}_K^G\xi\rightarrow \rho$ is continuous.) [*[Proof]{}*]{}. The two adjunction maps $$\text{$\alpha: \xi \rightarrow \text{c-ind}_K^G(\xi)|_K$, ${\hspace{6pt}}$ $\beta: \text{c-ind}_K^G(\rho|_K)\rightarrow \rho$,}$$ are defined by: $\alpha(v)=\phi_v$ and $\beta(f)=\sum_{g \in K\backslash G} \rho(g)^{-1}f(g)$. Easily checked to be equivariant under $K$ and $G$ respectively. They define the bijection as follows. For $\ell \in {\text{Hom}}_K(\xi,\rho|_K)$, we first induce it to a map $i(\ell): \text{c-ind}_K^G(\xi) \rightarrow \text{c-ind}_K^G(\rho|_K)$, and then compose it with $\beta$. More explicitly, with $\ell$ we associate $$r(f)={\sum}_{g \in K \backslash G}\rho(g)^{-1}\ell(f(g)).$$ Conversely, starting with an $r \in {\text{Hom}}_G(\text{c-ind}_K^G\xi,\rho)$, we define an $\ell$ by first restricting $r$ to $K$, and then composing with $\alpha$. In other words, by the formula $$\ell(v)=r(\phi_v).$$ It is straightforward to check these maps are mutually inverse. Suppose $\ell \leftrightarrow r$. Then, first of all, if $r$ is bounded, with transformation norm $\|r\|$, it follows that $$\|\ell(v)\|_{\rho}\leq \|r\|\cdot \|\phi_v\|_{\xi,\infty}=\|r\|\cdot\|v\|_{\xi},$$ so that $\ell$ is bounded, with transformation norm $\|\ell\|\leq \|r\|$. Conversely, $$\|r(f)\|_{\rho}\leq {\max}_{g\in K\backslash G}\|\ell(f(g))\|_{\rho}\leq \|\ell\|\cdot \|f\|_{\xi,\infty},$$ and hence $\|r\|\leq \|\ell\|$. Altogether, this must be an equality. 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--- abstract: | We investigate by means of Dynamical Mean-Field Theory the crossover from BCS superconductivity to Bose-Einstein (BE) condensation of preformed pairs which occurs in the attractive Hubbard model by increasing the attraction strength. We follow the evolution of the two energy scales underlying the superconducting phenomenon, the gap $\Delta_0$ and the superfluid stiffness $D_S$, which controls the phase coherence. The BCS-BE crossover is clearly mirrored in a change in the hierarchy of these two scales, the smallest of the two controlling the critical temperature. In the whole intermediate-to-strong coupling region $T_c$ scales with $D_S$, while $T_C$ is proportional to $\Delta_0$ only in the BCS regime. This evolution as a function of the interaction qualitatively resembles what happens in the cuprates when the doping is decreased towards the Mott insulator. This continuous change reflects also in the energetic balance at the superconducting transition. While, as it is well known, superconductivity is stabilized by a potential energy gain in the BCS regime, the strong-coupling superconductivity is made stable by a reduction of kinetic energy. Interestingly the intermediate-coupling region, where the maximum $T_c$ is achieved, behaves similarly to the strong-coupling regime, and its gain in kinetic energy is the largest as a function of the coupling. Since the integral of the optical conductivity is proportional to the kinetic energy, the above finding implies that the attractive Hubbard model can account qualitatively for the anomalous behavior of optical spectra around $T_c$, where an increase of spectral weight is observed in under and optimally doped cuprates, while the overdoped samples have a more standard behavior. This qualitative agreement is lost in the normal phase, specifically at strong-coupling, calling for the inclusion of strong correlation effects in the theoretical description. author: - 'A. Toschi$^{1,2}$, M. Capone$^{1,3}$, and C. Castellani$^1$' title: 'Energetic balance of the superconducting transition across the BCS-Bose Einstein crossover in the attractive Hubbard model' --- Introduction ============ High-temperature superconductors (HTSC) have attracted an unprecedented interest in the solid-state community, yet the number of open issues largely exceeds the number of generally accepted points. Among the latter, we can count the crucial role of strong electron-electron repulsion. This immediately raises the question of the competition and coexistence between this repulsion and the attraction between quasiparticles responsible of the superconducting phenomenon. In this work, we make a step back, and we consider a simpler and more “phenomenological” question, that is the possibility to explain some features of the HTSC physics simply in term of a crossover between a conventional (BCS) superconductivity (characterizing the overdoped compounds) and a sort of strong-coupling Bose-Einstein (BE) superconductivity (in the underdoped region).[@micnasrev; @varie] A number of properties of the cuprates can in fact be qualitatively described in this framework. Notable examples are the intermediate values ($\sim 10\div 20$ Å) of the superconducting coherence length $\xi_0$ in optimally doped compounds[@pan; @iguchi], and the pseudogap phenomenology in the underdoped compounds[@timusk], which can be ascribed to a “preformed pairs” phase, in which the superconducting order parameter has a finite amplitude, but it lacks phase coherence. The strongest connection with the physics of a BCS-BE crossover is perhaps the evolution with doping of the relevant energy scales controlling the stability of the superconducting condensate, namely the gap $\Delta_0$, which is proportional to the binding energy of the Cooper pairs, and the superfluid stiffness $D_S$, which represents the energy cost for phase fluctuations. In the underdoped compound there is a clear experimental evidence[@uemura1; @ding] that $D_S < \Delta_0$ and $T_c$ is proportional to the “weak link of the chain” $D_S$ according to the so-called Uemura plot[@uemura], whereas for higher doping a more conventional direct proportionality of $T_c$ on $\Delta_0$ is recovered. However some obstacles arise trying to push further this line of thought. For instance, the unconventional $d-$wave symmetry of the order parameter, with the related presence of quasiparticle excitations down to zero energy, is known to strongly affect the low-temperature thermodynamic properties. In particular the nodal quasiparticles can dominate the low-temperature charge[@arun; @lara] and thermal transport even in the strong-coupling regime, invalidating a purely bosonic description. It is also intuitively hard to reconcile the BCS-BE crossover scenario with the relevance of strong correlation effects, which are naturally larger and larger when the doping is reduced and the Mott insulator is approached. Then, in the BCS-BE scenario, the attraction should be stronger in the same region where the repulsion is maximum. This puzzling fact can be understood by interpreting the BCS-BE physics as relevant only for [*low-energy*]{} quasiparticles, while the high-energy physics has to be dominated by Coulomb repulsion. An explicit realization of a similar scenario has been obtained in models with orbital degeneracy and specific kind of interactions[@capone1; @capone2], where the superconducting properties of the strongly renormalized quasiparticles share similarity with those of an effective attractive Hubbard model[@capone1]. Keeping in mind those limitations, we find it important to understand if, and to what extent, a simple attractive picture for quasiparticles, modelized by an attractive Hubbard model, is able to reproduce some of the properties of the cuprates. We will find that while some of the properties of the superconducting phase can be put in this framework, the normal state behavior necessarily requires the inclusion of strong repulsive correlations. The Hamiltonian of the attractive Hubbard model reads $$\begin{aligned} \label{hubbard} {\cal H} = &-&t \sum_{<ij>\sigma} c_{i\sigma}^{\dagger} c_{j\sigma} -U\sum_{i}\left ( n_{i\uparrow}-{1\over 2}\right ) \left ( n_{i\downarrow}-{1\over 2}\right ), \end{aligned}$$ where $<ij>$ indicates that the first sum is restricted to nearest neighbors sites only, $c_{i\sigma}^{\dagger}$ ($c_{i\sigma}$) creates (destroys) an electron with spin $\sigma$ on the site $i$ and $n_{i\sigma} = c_{i\sigma}^{\dagger}c_{i\sigma}$ is the number operator; $t$ is the hopping amplitude and $U$ is the Hubbard on-site attraction. As it can easily seen, this model reproduces in the extreme weak- and strong-coupling limit the BCS and BE regime respectively, and allows to move in the whole crossover region, simply by adjusting a single parameter, the ratio $U/t$. From the theoretical point of view, a stringent test of the relevance of the crossover in the attractive Hubbard model for the cuprates has been mainly limited by the lack of reliable non perturbative approaches able to follow the evolution from weak to strong coupling without a bias in some direction. In this work we use for this purpose the Dynamical Mean-Field Theory (DMFT)[@dmft], a non perturbative approach which neglects the spatial correlations beyond the mean field level, but fully retains the local quantum dynamics, and becomes exact in the infinite coordination limit. The local nature of the attraction and the non-perturbative nature of DMFT are expected to lead to reliable results and allow for a democratic treatment of the different regimes and of the whole BCS-BE crossover. The attractive Hubbard model has been already investigated with DMFT in the past, but the attention was focused mainly on the normal phase or on the determination of the critical temperature for the superconducting transition, with the recent exception of Ref. , where the BCS-BE crossover at $T=0$ is analyzed using iterated perturbation theory as an impurity solver, but the focus is mainly on spectral properties. More precisely, by explicitly avoiding superconducting solutions, a phase transition has been found both at finite[@keller; @paolo] and at zero temperature[@prl] between a metal and a “paired” phase, i.e., a collection of independent pairs without the superconducting phase coherence. This paired phase represent the ’negative-$U$’ counterpart of the paramagnetic Mott insulator found for the repulsive Hubbard model[@prl; @keller]. With the present work we complete the DMFT analysis of this model reported in the Refs. with a careful investigation of the properties of the superconducting phase, certainly stable at low temperatures. Although the onset of the superconductivity smoothes the abrupt changes observed in the low-temperature metastable normal phase, and the evolution of the superconducting phase as the coupling is increased is a smooth crossover, the way in which the energetics changes in this process is extremely interesting. We fully characterize the crossover from BCS to BE superconductivity, and we establish that the intermediate regime, where the maximum $T_c$ is obtained, shares the behavior of the strong-coupling, BE superconductivity. In particular, this means that the “optimal” superconductor is stabilized by a kinetic energy gain, as it is expected in the bosonic regime, as opposed to the standard potential energy stabilization characteristic of weak-coupling superconductivity. The paper is organized as follows: Sec. II briefly describes the DMFT for an s-wave superconductor, Sec. III is devoted to the evolution of the superfluid stiffness and the gap as a function of the interaction, Sec. IV presents the energetic balance at the superconducting transition, and the relation with optical measurements. Sec. V is dedicated to concluding remarks. method ====== In this section we will briefly introduce DMFT and some aspects related to the study of superconducting solutions. DMFT maps a quantum lattice model onto an impurity model whose hybridization function (usually called Weiss field in analogy with classical mean-field theories) is determined by means a self-consistent equation. The latter equation contains the information about the original lattice only through the non-interacting density of states. In our case (\[hubbard\]) is mapped onto an Anderson model with attractive coupling. In order to describe the superconducting phase, the bath presents superconducting terms leading to an anomalous Weiss field $$\begin{aligned} \label{aim} {\cal{ H}}_{AM} & = & \sum_{l,\sigma} \, \left[ \epsilon_l \, c^{\dag}_{l\sigma} \, c_{l\sigma} + V_l\, ( c^{\dag}_{l\sigma} d_{\sigma} + \mbox{h.c.}) \right. \nonumber \\ & + & \left. \Delta_l \, (c_{l\downarrow}^{\dag} c_{l\uparrow}^{\dag} + \mbox{h.c.}) \right] + {\cal{H}}_{loc} \end{aligned}$$ where $ {\cal{H}}_{loc} = -U\left ( n_{0\uparrow}-\frac{1}{2}\right ) \left ( n_{0\downarrow}-\frac{1}{2}\right ) - \mu n_0$ is the on-site term, and the chemical potential $\mu$ is adjusted to fix the particle density on the impurity site (in our calculations we always fix $n=0.75$ as a generic density out of half-filling, as done in Refs. ). From the impurity model we compute the normal and anomalous Green’s functions, $G(\tau)=-\langle T c_{\uparrow}(\tau) c^{\dag}_{\uparrow}(0)\rangle$ and $F(\tau)=- \langle T c_{\uparrow} (\tau) c_{\downarrow}(0)\rangle$, which are used to build the matrix $\hat{G}(i\omega_n)$ in Nambu-Gor’kov formalism. Analogously one can define a matrix of Weiss fields whose diagonal ${\cal G}^0(i\omega_n)^{-1}$ and off-diagonal ${\cal F}^0(i\omega_n)^{-1}$ elements are related to the parameters appearing in (\[aim\]) by $$\begin{aligned} \label{gf0} {\cal G}_0^{-1}(i\omega_n) & = & i\omega_n +\mu + \sum_{l=1}^{n_s} \, V_l^2 \, \frac{i\omega_n+\epsilon_l}{\omega_n^2+ \epsilon_l^2 +\Delta_l^2} \nonumber \\ {\cal F}_0^{-1}(i\omega_n) & = & + \sum_{l=1}^{n_s} \, V_l^2 \, \frac{\Delta_l}{\omega_n^2+ \epsilon_l^2 +\Delta_l^2}. \end{aligned}$$ The two above quantity also define the local self-energy matrix $$\hat\Sigma(i\omega_n) = \hat{\cal G}_0^{-1}(i \omega_n) - \hat{G}^{-1}(i\omega_n).$$ The self-consistency condition relates the Weiss field to the Green’s function. We work with an infinite-coordination Bethe lattice with semicircular density of states of half-bandwidth $D$, (i.e., $N(\epsilon)=2/(\pi D^2) \, \sqrt{D^2-\epsilon^2}$), for which the self-consistency reads $$\label{selfbethe} \hat{\cal G}_0^{-1}(i \omega_n)= i\omega_n \hat{\tau}^0 + \mu \hat{\tau}_3 -t^2 \hat{\tau}_3 \hat{G}(i\omega_n) \hat{\tau}_3$$ $\hat{\tau}_{i}$ being the Pauli matrices. It is straightforward to check that Eqs. (\[aim\])-(\[selfbethe\]) reduce automatically to their normal-state counterparts as soon $\Delta_l = 0$ for each $l$. The inclusion of local quantum fluctuations rules out the possibility to analytically solve Eqs. (\[aim\])-(\[selfbethe\]), and requires numerical (or approximate) solutions of the Anderson impurity model (\[aim\]). Here we adopt Exact Diagonalization (ED) as the impurity solver. Thus, we discretize the Anderson model, by truncating the sums in Eqs. (\[aim\]) and (\[gf0\]) to a small finite number of levels $N_s$. It has been shown that extremely small values of $N_s$ provide really good results for thermodynamic properties and reliable results for spectral functions.[@krauth] Here we use both the Lanczos algorithm (at zero temperature) and the finite temperature algorithm in its simplest version, which requires the full spectrum of the Hamiltonian matrix. To obtain the full spectrum of the Hamiltonian, needed to compute the finite temperature properties, we are forced to a rather small value of $n_s$, namely $5$, whereas in the zero temperature case the Lanczos algorithm allow us to deal with larger systems, up to $n_s \sim 10$ also in the superconducting phase, where the number of particles is not conserved, making the size of the Hilbert space larger. Hierarchy of energy scales ========================== As we anticipated in the introduction, a landmark of the HTSC phase-diagram is the crossing of the energy scales relevant for superconductivity: the superconducting gap $\Delta_0$ and the superfluid stiffness $D_S$, which represent respectively the energetic cost to form a Cooper pair and to break the phase coherence of the superconducting phase. DMFT allows for a straightforward calculation of the order parameter $\Delta_0$ at zero temperature, defined through the local anomalous Green’s function $$\Delta_0= -U \langle T c_{\uparrow} (0) c_{\downarrow}(0)\rangle= -U / \beta \sum_n F(i \omega_n).$$ We note that the values for ${\Delta}_0$ obtained with this standard “mean-field” definition coincides almost exactly with the anomalous part of the self-energy $\Sigma_{12}(i\omega_n)$ at large $\omega_n$. In our approach $\Sigma_{12}$ is frequency dependent, with a small reduction at small Matsubara frequencies with respect to the large $\omega_n$ value. In most cases we find that the above reduction is quite small and $\Sigma_{12} \simeq \Delta_0$. On the other hand, the superfluid stiffness $D_S$, is defined in terms of the static limit of the electromagnetic response function as $$D_S= D_{dia} - \chi_{jj}({\bf q} \rightarrow 0, \Omega=0) \label{ds}.$$ The diamagnetic term $D_{dia}$ is given by $D_{dia} = - \langle E_{kin} \rangle = - 2/\beta \sum_{\omega_n} \int d \epsilon \epsilon N(\epsilon) G(\epsilon,\omega_n) $[@notadiam]). The paramagnetic term, which measures the normal component $D_N$ is defined as the sum over all the directions[@notadim] of the transverse part of the paramagnetic kernel in the static limit (i.e., $\chi_{jj}({\bf q} \rightarrow 0, \Omega=0) = \sum_{\alpha=x,y,z,\ldots}\chi_{jj}^{\alpha\alpha}({\bf q} \rightarrow 0, \Omega=0) $ being $\chi_{jj}^{\alpha\alpha}({{\bf r}},\tau)= \langle T_\tau \vec{j}_\alpha({{\bf r}},\tau) j_\alpha(0,0) \rangle$). In principle the evaluation of $\chi_{jj}({\bf q} \rightarrow 0, \Omega=0)$ would require the calculation of the corresponding four-field correlation function, but a remarkable simplification occurs in the infinite coordination limit where the DMFT is exact, since all the vertex corrections to the e.m. kernel vanish[@dmft]. The evaluation of $D_S$ requires therefore only the dressing of the non-interacting Green’s function in the electromagnetic kernel with the self-energy. As a results, the DMFT expression for $\chi_{jj}$ reads[@dmft; @chatt] $$\begin{aligned} \chi_{jj} & = & -\frac{2}{\beta} \sum_{n} \int \, d\epsilon \, N(\epsilon) \, V(\epsilon) \\ &\times & \left[ G(\epsilon,\omega_n) G^{*}(\epsilon, \omega_n) + F(\epsilon,\omega_n)F(\epsilon,\omega_n) \right] \label{chijj} \end{aligned}$$ where $V(\epsilon)=(4t^2-\epsilon^2)/3$ is the square current vertex for the Bethe lattice[@chung; @chatt; @bluemer] and $G(\epsilon,\omega_n)$, $F(\epsilon, \omega_n)$ are the normal and anomalous lattice Green function respectively [@notadefds]. ![\[fig:sup\_dens\] (Color online) Temperature dependence of the superfluid stiffness $D_S$ and of the normal component $D_N$ at $U=2.0 D$ (just before the maximum $T_c$) compared with the diamagnetic term $-\langle E_{kin} \rangle$.](fig1.eps){width="8cm"} Thanks to the specific form of the current vertex, it is possible to write a more compact expression for $D_S$. More specifically by exploiting the relation $ -\epsilon N(\epsilon) = \partial_\epsilon [ V(\epsilon) N(\epsilon)]$ and then transferring the energy-derivative on the Green functions one finally gets $$D_S = \frac{4}{\beta} \sum_{n} \int \, d\epsilon \, N(\epsilon) V(\epsilon) \; F(\epsilon,\omega_n) F(\epsilon,\omega_n)$$ The above expression only contains the anomalous Green functions, and makes it apparent the vanishing of $D_S$ at $T=T_c$. Before comparing the evolution of $\Delta_0$ and $D_S$, we briefly discuss the temperature behavior of the superfluid and normal densities. In particular, the evaluation of $D_N$ and $D_S$ reveals that the $T=0$ value of $D_N$ is basically negligible, as predicted by BCS mean-field, not only in the weak-coupling regime, but also for sizable interaction, as shown in Fig. \[fig:sup\_dens\] for $U=2D$, where $D_N/D_S$ at $T=0$ turns out to be less than 0.01. This means that it is possible to identify $D_S(T=0)$ with $ - \langle E_{kin} \rangle $ in a wide range of couplings. It is clear that this results depends on the neglect of small-momentum collective modes, which can deplete the condensate[@lungo]. It is worth noting that the presence of a frequency-dependent self-energy and of an incoherent part of the Green’s function does not lead to a reduction to a depletion of $D_S$, differently from what happens, for instance, for impurity scattering. We now come to the evolution of the energy scales inherent to the superconducting phase as a function of the interaction. Our results are summarized by Fig. \[fig:cross\_dmft\], where the superconducting gap $\Delta_0$, the superfluid stiffness $D_S$ and the critical temperature $T_c$ are plotted as a function of the ratio $U/D$. $\Delta_0$ is smallest in the weak-coupling regime, and the superfluid stiffness becomes the lowest scale in strong-coupling. It is natural that the system becomes superconductor only when the pairs are formed and they have phase coherence, namely when the temperature is lower than both the gap scale and the superfluid stiffness scale. Therefore the critical temperature is substantially determined by the smaller scale at each interaction. Thus the critical temperature is proportional to $\Delta_0$ in the weak-coupling regime, as predicted by BCS mean-field, while in the strong-coupling limit we recover $T_c \simeq D^2/4U \simeq \langle E_{kin}\rangle /2 \sim D_S/2 $, as predicted by the mean-field solution in the limit of an hard core boson system with Heisenberg coupling $J=D^2/(d U)$[@micnasrev; @notadensity] ![\[fig:cross\_dmft\] (Color online) Superconducting energy scales (in unit of $D$) (${\Delta}_0$ and $D_S$ have been normalized to be directly comparable with $T_c$ in the asymptotic regimes). The two energies cross at intermediate coupling at $U \sim D$, well before the maximum $T_c$ dome ($T_c$ is taken from Ref. )](fig2.eps){width="8cm"} The crossing between the two energy scales, which is characteristic of the BCS-BE crossover occurs at intermediate coupling for $U$ slightly smaller than maximum $T_c$ value. Therefore only in the proximity of the weak coupling regime the smallest energy scale, which ultimately determines $T_c$, is the binding energy of the Cooper pairs, whereas already at the optimal $U$ (and of course in all the strong coupling regime) the weak link for the stability of the superconducting phase is the superfluid stiffness $D_S$ (hence, the onset of the superconductivity is controlled by the phase-coherence). In other words, the optimal superconductivity is achieved in a regime where the physics is already that of strong-coupling, with pair formation occurring at a temperature higher than $T_c$. We notice that the optimal interaction value is noticeably larger than the attraction needed to form a bound state in the low-density limit $U_b \sim D$[@randeria]. Further insight on the influence of the crossing of the energy scales on $T_c$ can be highlighted by plotting the critical temperature versus $D_S$. In this way a sort of “Uemura-plot” relation, like that characterizing the physics of the underdoped cuprates[@uemura], is found in the attractive Hubbard model for $U > 2.4 D$, where $T_c$ is a decreasing function of $U$ (see Fig. \[fig:uem\_dmft\]). This result supports the identification of the superconductivity with a condensation of preformed bosons already at a moderate values of the interaction. ![\[fig:uem\_dmft\] (Color online) Uemura plot in the attractive Hubbard model. The direct proportionality between $D_S$ and $T_c$ in the strong-coupling limit is evident on the left side of the figure, corresponding to $U > 2.4 D$. ](fig3.eps){width="8cm"} On the other hand, the attractive Hubbard model, at least in our DMFT framework, cannot reproduce the deviations from the Uemura plot recently observed in the cuprates[@uem2002; @panago]. More specifically an evident “re-entrance” of the curve $T_c(D_S)$ has been observed in many overdoped compounds, and its shape appears to be strongly material dependent. In our calculation at small $U$, where $D_S \simeq -\langle E_{kin} \rangle$, the superfluid stiffness is a monotonically decreasing function of the interaction, and does not show the experimentally observed re-entrance. In order to push a little bit further the comparison with the most recent experimental data, it is useful to recast our DMFT results for the “Uemura plot” rescaling $T_c$ with the superconducting gap $\Phi_0$ in the single particle spectrum, as done by Tallon [*et al*]{}[@panago], in order to single out as much as possible the nonstandard behavior of $T_c$. The experimental $T_c/\Phi_0$ is found sub-linear in $D_S$ in the underdoped regime, while it is slightly super-linear at higher doping (with a not universal tail in most overdoped compounds). We notice that, beyond the BCS regime $\Phi_0$ can be different from the order parameter $\Delta_0$ that we have introduced before. Here $\Phi_0$ is evaluated directly as the gap in the density of states. The difference between the two quantities varies up to 30. Our results are summarized in Fig. \[fig:panag\_dmft\], where one can note the presence of a wide region of a sub-linear behavior of $T_c/\Phi_0$ on the left-side of the figure, which corresponds to the intermediate-to-strong coupling regime in fair agreement with Ref. . Also a partial of a super-linear behavior appears on the right-hand side of our plot. The observed behavior of $T_c/\Phi_0$ is therefore captured by our simple BCS-BE crossover, without invoking more involved explanations[@panago]. ![\[fig:panag\_dmft\] (Color online) Rescaled “Uemura-plot" (following Ref. ) for the attractive Hubbard model. We have plotted the $T_c$ divided by the gap $\Phi_0$ as a function of the superfluid density. It is evident the sub-linear behavior of $T_c/\Phi_0$ in the intermediate-to-strong coupling regime, which mirror rather well the data of Ref. for several underdoped or optimally doped cuprates.](fig4.eps){width="8cm"} The evolution of energy scales we have discussed in this section is reflected in an increased role of phase fluctuations as the coupling grows. As we have mentioned in the introduction, those fluctuations may have in principle relevant effects on the superconducting phase, in which finite dimensionality effects beyond DMFT can be important, possibly destroying the superconducting ordering. In DMFT, indeed, phase fluctuations at large ${\bf q}$ are properly taken into account, while the small ${\bf q}$ collective modes are neglected. In Ref. it is argued that superconductivity is destroyed by phase fluctuations for $U$ larger than a critical value due to the small charge compressibility $\kappa = \partial n/ \partial \mu \propto 1/U$ derived within a phase-only effective theory based on the atomic limit, . Since $\kappa$ measures the inertia of the system against the dynamic phase fluctuations, a small value would permit large zero-point fluctuations, eventually destroying phase coherence. On the other hand, an alternative derivation of a phase-only action within one-loop expansion gives instead $\kappa \sim 2U/D^2$, therefore increasing with $U$[@lungo]. ![\[fig:chi\_dmft\] (Color online) Charge compressibility in the superconducting phase at $T << T_c$. The DMFT data are compared respectively with the strong-coupling behavior from the phase-only action of Ref. and with the noninteracting value.](fig5.eps){width="8cm"} Even if DMFT neglects finite-dimensionality effects and cannot therefore treat the physics of the Goldstone modes associated with the phase-fluctuations, we can use this approach to evaluate the coefficients appearing in the phase-only effective theories. The lack of bias towards weak or strong coupling of DMFT should allow us to discriminate between the two derivations. The results shown in Fig. \[fig:chi\_dmft\], clearly display that $\kappa$ grows linearly with $U$ at strong coupling in agreement with Ref. : the cost of dynamic phase fluctuation at ${\bf q}\simeq 0$ tends to increase in the large $U$ limit, leaving the superconducting phase stable, in contradiction with Ref. . A more complete description of the phase-fluctuations would require the calculation of all the phase-only theory coefficient, including the anharmonic ones. This issue is beyond the aim of the present work. Energetic Balance of the superconducting phase and optical sum rule =================================================================== The characterization of the superconducting phase can be made more concrete by studying the temperature behavior of the kinetic ($E_{kin}$) and the potential ($E_{pot}$) energies. From one side this analysis allows to evaluate separately all the energetic contributions which characterize the mechanism for the stabilization of the superconducting long range order across the BCS-BE crossover. On the other hand, since in a lattice system the frequency integral of the optical conductivity $\sigma(\omega)$ can be related to the kinetic energy of the carriers[@Vdm], our calculations allow for a direct comparison with the optical measurements on the cuprates. Kinetic and Potential Energy behavior ------------------------------------- The calculation of both potential and kinetic energies is quite straightforward in DMFT, since it only involves local quantities. This is evident for the potential energy, because by definition it is proportional to the local density of double occupancies $n_d = \langle \sum_i n_{i\uparrow}n_{i\downarrow}\rangle$ $$E_{pot}= -U n_d.$$ As far as $E_{kin}$ is concerned, exploiting the simplified form of the self consistency equation for the Bethe lattice we can derive the following expression $$\begin{aligned} E_{kin} & = & t^2 \, T \sum_{i \omega_n} [G(i \omega_n) \, G_(i \omega_n) + G^{*}(i \omega_n) \, G^{*}(i \omega_n)\nonumber \\ \label{ekin_loc_expl} & - & 2 F(i \omega_n) \,F(i \omega_n)],\end{aligned}$$ which shows that also the kinetic energy can be expressed only in terms of the local (normal and anomalous) Green function. ![\[fig:ekin\_dmft\] (Color online) Low-temperature behavior of the kinetic (upper panels) and the potential (lower panels) energies. The critical temperature is marked by arrows. Notice that the kinetic energy is normalized by the half-bandwidth, and the potential energy by the attraction strength $U$](fig6.eps){width="82mm" height="77mm"} In Fig. \[fig:ekin\_dmft\] we report the kinetic energy and potential energies for $U=0.8 D, 2.4D$ and $3.6D$, chosen as representative of the BCS, the intermediate and the BE regimes, as a function of the temperature. The onset of the superconductivity is always marked by an abrupt change in the concavity of $E_{kin}(T)$ e $E_{pot}(T)$. The energetic balance of superconductivity displays instead clear differences between the various regimes. At $U= 0.8 D$ the onset of superconductivity is accompanied by a slight loss in the kinetic energy ($\Delta E_{kin}= E_{kin}(0)-E_{kin}(T_c)= + 0.003 D$), and potential energy gain ($\Delta E_{pot}= E_{pot}(T=0)-E_{pot}(T_c) = -0.008 D$), as in standard BCS theory. On the other hand, both at $U= 2.4 D$ and $U = 6.4 D$ the superconducting phase is characterized by a lower value of $E_{kin}$ and a loss of potential energy is observed below $T_c$. Such changes in the energetic balance at the superconducting transition clearly highlight the different mechanisms stabilizing superconductivity in the two regimes. In the BCS limit superconductivity coincides with pair formation, which determines a gain in potential energy and a consequent loss in kinetic energy. In the opposite BE regime, the electrons are paired at a high temperature of order $U$, but a true long-range superconducting order can take place only when phase coherence establishes between pairs. Therefore $T_c$ is associated to a gain in kinetic energy, while a small fraction of the potential energy gained with pair formation is lost at $T_c$. More interestingly, our calculations indicate that already at intermediate $U$ the “strong-coupling” mechanism is taking place, and superconductivity is associated with a consistent kinetic-energy gain, such as the one observed in the optimally doped cuprates (see next sub-section). The kinetic energy gain turns out to be maximum at intermediate $U/D$, where the highest $T_c$ is achieved. The overall picture is drawn in Fig. \[fig:ecvsU\] where we have reported the variation of the kinetic $\Delta E_{kin}$ , the potential $\Delta E_{pot}$ and the total energy $\Delta E_{tot}$ between $T=T_c$ and $T=0$ for a number of interaction values. From these data one can easily see that [*(i)*]{} the typical BCS feature of an increasing $E_{kin}$ below $T_c$ is rapidly lost (approximatively for $U \sim D \sim U_b$), in agreement with the discussion of the previous section; [*(ii)*]{} an intermediate region exists, in which the onset of the SC phase is associated with a gain of both potential and kinetic energy; [*(iii)*]{} the BE regime is effectively reached for $U$ larger than $1.8 D$ (hence before reaching the maximal critical temperature), where superconductivity is stabilized by kinetic energy. It is also worth noting in Fig. \[fig:ecvsU\] that the variation of the total energy $\Delta E_{tot}$ has approximatively the same dome-shape behavior of the $T_c$ dependence on $U$. It is tempting then, to consider this value as a measure of the condensation energy $E_c=E_{tot}^N(T=0)-E_{tot}^S(T=0)$ of the superconducting phase. This identification works well only if one can assume a small variation of $E_{tot}^N(T)$ between $T=T_c$ and $T=0$, an assumption which holds both at small (due to the small value of $T_c$) and at intermediate-large $U$ (where both $E_{kin}$ and $E_{pot}$ are almost constant in temperature above $T_c$). However this is not the case for $U \sim 1 \div 2$ D, where a strong temperature dependence of $E_{kin}(T)$ (and of $E_{pot}(T)$) in the normal phase is found, because of the presence of a strongly renormalized quasiparticle excitation at the Fermi level[@paolo; @lavoroopt]. Within DMFT the situation is even more involved for $ U_{c1}\sim 2.2 D < U < 2.9 D \sim U_{c2}$ where a coexistence between metallic and insulating solution is found and the temperature dependence of $E_{kin}$ and $E_{pot}$ in the normal phase below $T_c$ is subject to extremely abrupt changes[@paolo; @prl]. However this is a peculiar feature of the DMFT treatment of the Hubbard model which could not be so relevant when comparing our result with the experimental findings, also because these features of the Hubbard physics occur at temperatures much smaller than $T_c$. ![\[fig:ecvsU\] (Color online) Kinetic, potential and total energy variation in the superconducting region for different values of the pairing interaction $U$](fig7.eps){width="80mm" height="57mm"} Comparison with the optical measurements ---------------------------------------- The integral over all the frequencies of the real part of the optical conductivity in a lattice model is related to the second order derivative of the free particle dispersion[@hirsh; @Vdm] according to the following equation, $$\begin{aligned} \label{sum_expl1} \int_{0}^{+\infty} {\sigma}_{xx}(\omega) \, d\omega & = & \frac{\pi e^2}{2 N V} \sum_{{\bf k},{\sigma}} \frac{\partial^2 {\epsilon}_{{\bf k}}}{\partial k_x^2} n_{{\bf k},{\sigma}} \\ \nonumber & = & - \frac{\pi e^2 a^2} {2 d N V} \sum_{{\bf k},{\sigma}} {\epsilon}_{{\bf k}} n_{{\bf k},{\sigma}}= - \frac{\pi e^2 a^2}{2 d V} \langle E_{kin} \rangle \label{sum_expl}\end{aligned}$$ and the effect of the interactions is hidden in the values of the momentum distribution function $n_{{\bf k},{\sigma}}$. Here $\langle E_{kin} \rangle=\frac{1}{N}\sum_{{\bf k},{\sigma}} {\epsilon}_{\bf k} n_{{\bf k},{\sigma}}$ , $a$ the lattice spacing, $d$ the dimension of the system considered, and $V$ the unit-cell volume. The second equality, which holds for nearest-neighbor hopping only, implies that the kinetic energy data in Figs. \[fig:ekin\_dmft\] and \[fig:ecvsU\] can be compared directly with the the optical spectral weight of the conduction band in the cuprates. The experimental results for the low-energy behavior of the spectral weight behavior in the HTS have been the object of a recent intense debate, and consensus has been reached about a few general points. In all the cuprates a sensible enhancement of low-energy spectral weight $W(T) = $ $ \int_0^{\Omega_c} $ $\, d\omega {\sigma}(\omega,T)$ ($\Omega_C$ being a frequency cut-off of the order of 1 eV, which is the plasma frequency for those materials), is observed when lowering the temperature[@Vdm; @Bont; @homes; @michele]. In the superconducting phase, underdoped (UD) and the optimally (Opt) doped samples behave differently from the overdoped (OD) ones. More precisely, in the UD-Opt compounds below $T_c$ an upward bump is observed with respect to the normal phase behavior, in contrast with the decreasing of $W(\Omega_c,T)$ observed in OD samples[@bont04]. The gain of kinetic energy per $Cu$ atom $\Delta E_{kin} = E_{kin}(T_c) - E_{kin}(0)$ in UD-Opt BSCCO can be estimated using (\[sum\_expl1\]) as $\Delta E_{kin} = - (0.5 \div 1)$ meV [@Vdm; @Bont; @bont04]. Quite noticeably, this value of $\Delta E_{kin}$ is much higher than the condensation energy for the same material[@hirsh], estimated as $E_c \sim 0.1$ meV on the basis of the specific heat measurements[@specheat]. According to our DMFT results, the bump of $W(\Omega_C,T)$ observed below $T_c$, can be qualitatively understood in the framework of the BCS-BE crossover. More specifically the hypothesis of an intermediate-to strong coupling description of the superconductivity in the UD-Opt region appears to fit well with the observed enhancement of the low-frequency spectral weight below $T_c$, whereas the disappearance of the upward bump in the OD cuprates is perfectly compatible with a weak-coupling superconductivity. It is interesting therefore to check to what extent is possible to push forward such an analogy with the properties of the real systems. We can try to establish a comparison with the experimental data by choosing the value of the semi-bandwidth $D$ of the attractive Hubbard model in order to reproduce the maximum value $T_c^{max}$ (a $T_c^{max}$ of $90-100$ K is obtained with $D \sim 1000 K \sim 100 meV$). The theoretical values for $\vert\Delta E_{kin}\vert$ and $\vert\Delta E_{tot}\vert$ (that we take as an estimate of $E_c$) are respectively $4 \div 8$ meV and $2 \div 4$ meV, larger by less than one order of magnitude than the experimental values in the underdoped HTSC. It is anyway remarkable that the DMFT of the attractive Hubbard model correctly predicts the qualitative trend of a condensation energy which is only a fraction of the kinetic energy gain. As soon as the temperature exceeds $T_c$ and we enter the normal phase, our calculation ceases to properly describe the experimental results. The $T^2$ behavior of $E_{kin}$ is in fact found only in the BCS region, as shown by the $U=0.8 D$ data of Fig. \[fig:ekin\_dmft\], and it completely disappears in the intermediate-to-strong coupling regime: no relevant variation of $E_{kin}(T)$ is found at $U=2.4$, $6.4 D$ in a wide temperature range above $T_c$. The inadequacy of the present approach to describe the experimental findings in the normal phase has different origins. A first reason is the freezing of spatial fluctuations characteristic of single-site DMFT. It is indeed reasonable to expect a role of short-range fluctuations at least at strong coupling. In this limit the attractive Hubbard model can be mapped in an effective “pseudospin” hamiltonian[@micnasrev], and the kinetic energy is given basically by nearest-neighbor correlations between the pseudospin operators, i.e., the local pairs and the empty sites. The temperature dependence of these correlations is poorly described by DMFT at large $U/D$. The relevance of this effect can tested using cluster extensions of DMFT[@jarrel; @gabi], where the short-range spatial correlations are included. There is however a more basic reason for the inadequacy of the attractive Hubbard model to capture the temperature behavior of the normal phase of the cuprates, which is the inability to properly describe the approach to the Mott insulator as the doping is reduced. The underdoped region is indeed almost universally believed to be dominated by the physics of a doped Mott insulator, with strongly renormalized quasiparticles. On the other hand, the strong-coupling region of our BCS-BE framework is certainly a “correlated” regime, with renormalized quasiparticles, but the interaction is simply the attractive one, which does not lead to the Mott insulating state. Indeed in a recent work[@lavoroopt] it has been explicitly shown that the physics of a doped Mott insulating system, with its associated band-narrowing, may represent the natural explanation for the strong temperature dependence of the spectral observed in the normal phase of the cuprates. Conclusions =========== In this paper we have performed a detailed investigation of the physics of superconducting phase in the attractive Hubbard model both at zero and at finite temperature. The approach used here, namely the Dynamical Mean Field theory, is completely non-perturbative allowing for a treatment of the superconducting phase properties, which is not tied in principle either to the weak-coupling (BCS) or to the strong-coupling (BE) regime. In particular we have investigated the behavior of the energy scales relevant for the superconductivity as a function of the pairing interaction $U$. We have found that the evolution of the superfluid density $D_S$ and the superconducting gap $\Delta_0$ displays a clear crossing in the intermediate coupling regime, which is reminiscent of what is found experimentally in the phase-diagram of HTSC for different doping levels. In the weak-coupling region the BCS picture works well and the superconductivity is controlled by the binding energy of the Cooper pairs $\Delta_0$; on the contrary when the pairing interaction is high the superconductivity becomes essentially a phenomenon of superfluidity of preformed local pairs. This is clearly witnessed by the direct proportionality found in this regime between $T_c$ and $D_S$, and showed in a sort of “Uemura-plot” for the attractive Hubbard model. We also consider explicitly the problem of superconductivity suppression due to phase fluctuations in the hydrodynamic regime, which is usually neglected in a DMFT framework. Our DMFT calculation of the compressibility indicates that, contrary to the claim of Ref. [@kopec], the phase-fluctuation effects do not destroy superconductivity, even for extremely large values of $U$. The analysis of the superconducting phase properties in the attractive Hubbard model has been then further enriched by considering the energetic stabilization of the superconductivity and the f sum-rule of the model. Our results clearly show a remarkable difference in the energetic balance responsible for the stabilization of superconducting order when moving along the BCS-BE crossover. While in the weak-coupling regime the onset of superconductivity is associated to a gain of potential energy and a slight loss in kinetic energy as it is expected in a BCS picture, already in the intermediate coupling at $U \simeq D$ and specifically in proximity of the maximum of the $T_c$ dome, the situation is completely reversed. Here, and for higher values of $U$, the stabilization of the superconductivity is due to a marked reduction of $E_{kin}$, while a smaller (but sizable) loss of potential energy is observed. Therefore the onset of superconductivity is clearly distinct from Cooper pair formation already for moderate values of the pairing interaction. The energetic balance at the superconducting transition finds a direct experimental counterpart in the optical measurements, since the integral of the optical conductivity is proportional to the kinetic energy itself. Our results can be rephrased in this terms, so that in weak-coupling a slight reduction of the optical sumrule is found in our calculation at the superconducting transition, while a relevant enhancement of the sumrule is observed for $T < T_c$ in the whole intermediate-to-strong coupling region. This scenario is actually realized in optical measurements in the HTSC cuprates as a function of doping, with overdoped samples behaving more or less as standard BCS superconductors, and underdoped one, characterized by a gain in kinetic energy[@Vdm; @bont04; @homes]. A more quantitative comparison with the experiments reveals that the variations of both kinetic energy and condensation energy between $T=0$ and $T_c$ are reasonably well reproduced by our model, which also correctly predicts that a sizable fraction of the kinetic energy gain is canceled by the potential energy loss, leading to a condensation energy significantly smaller than the kinetic energy gain, as it indeed happens[@hirsh] in the HTSC. It must be noticed that our calculation tends to overestimate both kinetic energy gain and condensation energy. However, the main limitation of the attractive Hubbard model in this context is the inability to capture the $T^2$ enhancement of the spectral weight in the normal phase above $T_c$. The main reason for this disagreement is likely the complete neglect of the strong repulsive interactions characterizing the underdoped cuprates close to the Mott state. ![\[fig:cross\_vsD\] (Color online) Same quantities of Fig. 2 plotted as a function of $D/U$. Both $D_S$ and $\Delta_0$ are normalized by $U$. Now the crossing between the superconducting gap $\Delta_0$ and the superfluid stiffness at $D \sim U$, occurs in the proximity of the maximum value of $T_c$.](fig8.eps){width="8cm"} An important outcome of our analysis is that the optimal critical temperature is obtained for an interaction strength well inside the BE region. This finding is in good agreement with previous calculations of the pseudogap temperature $T^*$ extracted from the spin susceptibility and specific heat, where the $T^*$ is significantly larger than $T_c$ at optimal interaction[@paolo], but it is in contrast with experiments in Bismuth and Yttrium based cuprates, where $T^*$ tends to vanish close to optimal doping. It is interesting to notice that this discrepancy seems to disappear if we fix the attraction strength $U$ and follow the BCS-BE crossover by varying the half-bandwidth $D$, and measuring energies in units of $U$. Such a rescaling is meant to roughly describe a situation in which the crossover from BCS to BE is due to a shrinking of the coherent band due to strong repulsive correlation effects which are stronger and stronger as the Mott insulator is approached, while the attraction, which we might think to arise from antiferromagnetic superexchange for the sake of definiteness, is basically unrenormalized in the same process. As shown in Fig. \[fig:cross\_vsD\], the maximum critical temperature now occurs for $U\simeq D$, at the boundary of the BCS region, where the pseudogap is small. However this different perspective, in which the optimal $T_c$ moves closer to the BCS region is in contradiction with the experimental evidence of a kinetic-energy driven superconductivity around optimal doping, which is qualitatively obtained in the BE regime. As a summary of our results, we can draw some final considerations on the the connection of the BCS-BE crossover scenario and the physics of the HTSC: several qualitative feature of the HTSC properties (as the crossing of the hierarchy of $D_S$ and ${\Delta}_0$, the Uemura plot ,…) can be actually captured within a purely attractive description. However, as a general trend, it provides too high estimation of the values of many thermodynamic quantities as the kinetic energy variation below $T_c$ and the condensation energy. At the same time the purely attractive description fails in reproducing the magnitude of the temperature dependence of the optical spectral weight above $T_c$, calling for the inclusion of strong correlation effect and suggesting that a closer agreements with the physics of the cuprates may be obtained by applying the attractive Hubbard description, or more generally the BCS-BE crossover picture, only to the narrow quasiparticle excitations characteristic of a doped Mott insulator. Models in which explicit attractive and repulsive interaction are present[@capone1; @capone2] are natural candidates to explore this scenario. 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--- abstract: 'A theory of dynamical control by modulation for optimal decoherence reduction is developed. It is based on the non-Markovian Euler-Lagrange equation for the energy-constrained field that minimizes the average dephasing rate of a qubit for any given dephasing spectrum.' author: - Goren Gordon - Gershon Kurizki - 'Daniel A. Lidar' title: Optimal Dynamical Decoherence Control of a Qubit --- *Introduction*.— Quantum information processing (QIP) harbors enormous unleashed potential in the form of efficient algorithms for classically intractable tasks and unconditionally secure cryptography [@Jaeger:book]. Perhaps the largest hurdle on the way to a realization of this potential is the problem of decoherence, which results when a quantum system, such as a quantum computer, interacts with an uncontrollable environment or bath [@Breuer:book]. Decoherence reduces the information processing capabilities of quantum computers to the point where they can be efficiently simulated on a classical computer. In spite of dramatic progress in the form of a theory of fault tolerant quantum error correction (QEC) [@Gaitan:book], finding methods for overcoming decoherence that are both efficient and practical remains an important challenge. An alternative to QEC that is substantially less resource-intensive is dynamical decoupling (DD) [@Viola:98Zanardi:98bViola:99Vitali:99; @KhodjastehLidar:04; @Uhrig:07]. In DD one applies a succession of short and strong pulses to the system, designed to stroboscopically decouple it from the environment. This can significantly slow down decoherence, though not halt it completely, since unlike QEC, DD does not contain an entropy removal mechanism. Similar in spirit to DD (in the sense of being feedback-free), but more general, is the method we term here dynamical control by modulation(DCM), wherein one may apply to the system a sequence of arbitrarily-shaped pulses whose duration may vary anywhere from the stroboscopic limit to that of continuous dynamical modulation [@Kofman:04; @Agarwal:99; @Alicki:01a; @Facchi:05; @Brion:05; @gor06a]. In the DCM approach, the decoherence rate is governed by a universal expression, in the form of an overlap between the bath-response and modulation spectra, subject to finite spectral bandwidth and amplitude constraints. Neither DD [@Viola:98Zanardi:98bViola:99Vitali:99; @KhodjastehLidar:04; @Uhrig:07] nor DCM [@Kofman:04; @gor06a] studies have so far gone beyond particular schemes for suppression of decoherence. What is lacking is a systematic theory for finding the *optimal modulation* for any given decoherence process. Here we apply variational principles to the DCM approach in order to address this problem. We derive an equation for the *optimal, energy-constrained control by modulation* (ODCM) that minimizes dephasing, for any given dephasing spectrum. We numerically solve this equation, and compare the optimal modulation to energy-constrained DD pulses. We show that ODCM always outperforms DD when subjected to the same energy constraint. We note that Ref. [@Uhrig:07] developed an optimal DD pulse sequence for the diagonal spin-boson model of pure dephasing, but without an energy constraint, i.e., assuming zero-width pulses. This was improved upon by perturbatively accounting for pulse widths in Ref. [@Karbach:08]. *Model*.— We consider a driven two-level system (qubit) with ground and excited states ${|g\rangle }$ and ${|e\rangle }$ separated by energy $\omega _{a}$ (we set $\hbar =1$), and Hamiltonian $$H(t)=\left( \omega _{a}+\delta _{r}(t)\right) {|e\rangle }{\langle e|} +\left( V(t){|g\rangle \langle e|}+h.c.\right) , \label{Hamiltonian}$$ where $V(t)=\Omega(t)e^{-i\omega _{a}t}+c.c.$ is a time-dependent resonant classical driving field with amplitude $\Omega (t)$, and $\delta _{r}(\omega )$ describes random, Gaussian distributed, zero-mean energy fluctuations. Let ${|\psi (t)\rangle }$ denote the solution of the time-dependent Schrödinger equation with the Hamiltonian $H(t)$, and let the density matrix $\rho (t)= \overline{{|\psi (t)\rangle \langle \psi (t)|}}$ denote the corresponding ensemble average over realizations of $\delta _{r}(t)$. We are interested in the average fidelity ${\langle F(t)\rangle }$, where ${\langle \cdots \rangle }$ is the average over all possible initial pure states of the fidelity $F(t)=|{\langle \psi (0)|\rho (t)|\psi (0)\rangle }|$. It can be shown that [@gor06a]: $$\begin{aligned} {\langle F(t)\rangle } &=&1-\alpha R(t)t \label{R-def} \\ R(t) &=&2\mathrm{Re}\left\langle \int_{0}^{t_{1}}dt_{2}\Phi (t_{1}-t_{2})\epsilon ^{\ast }(t_{1})\epsilon (t_{2})\right\rangle _{t}^{t_{1}} \\ \Phi (t) &=&\overline{\delta _{r}(t)\delta _{r}(0)}\quad \epsilon (t)=e^{-i\int_{0}^{t}dt_{1}\Omega (t_{1})} \label{eps-def}\end{aligned}$$ where $0<\alpha \lesssim 1$ is a known constant, $\langle \cdot \rangle _{t}^{t_{1}}\equiv \frac{1}{t}\int_{0}^{t}\cdot dt_{1}$ is the time-average, $R(t)$ is the *average modified dephasing rate*, $\Phi (t)$ is the second ensemble-average moment of the random (stationary non-Markov) noise, and $\epsilon (t)$ is the phase factor associated with the modulation. We impose the energy bound constraint $$\int_{0}^{T}dt\left\vert \Omega (t)\right\vert ^{2}=E \label{energy-const}$$ where $T$ is the total modulation time and $E$ is the energy constraint. As a boundary condition we require that the field is turned on, i.e. $\Omega (0)=0$. Although the analysis below is given in the time-domain, it is advantageous to analyze the problem in the frequency domain, in terms of the universal expressions [@Kofman:04; @gor06a]: $$\begin{aligned} R(t) &=&2\pi \int_{-\infty }^{\infty }d\omega G(\omega )F_{t}(\omega ) \label{R-omega} \\ G(\omega ) &=&(2\pi )^{-1}\int_{-\infty }^{\infty }dt\Phi (t)e^{i\omega t} \\ F_{t}(\omega ) &=&|\epsilon _{t}(\omega )|^{2}/t\quad \epsilon _{t}(\omega )= \frac{1}{\sqrt{2\pi }}\int_{0}^{t}dt_{1}\epsilon (t_{1})e^{i\omega t_{1}} \label{Ft}\end{aligned}$$ where $G(\omega )$ is the dephasing spectrum, $\epsilon _{t}(\omega )$ is the finite-time Fourier transform of the modulation function, and $ F_{t}(\omega )$ is the normalized spectral modulation intensity. The model we have just described applies to a qubit undergoing dephasing due to coupling to a finite-temperature bath of harmonic oscillators with energies $\hbar \omega _{\lambda }$. The qubit then has an average modified dephasing rate of the form given by Eqs. , where the dephasing spectrum is given by [@gor06a]: $$\begin{aligned} G(\omega ) &=&\left( n(\omega )+1\right) G_{0}(\omega )+n(-\omega )G_{0}(-\omega ) \\ G_{0}(\omega ) &=&\sum_{\lambda }|\kappa _{\lambda }|^{2}\delta (\omega -\omega _{\lambda })\end{aligned}$$ where $G_{0}(\omega )$ is the zero-temperature bath spectrum, $\kappa _{\lambda }$ is the off-diagonal coupling coefficient of the qubit to the bath oscillator $\lambda $, and $n(\omega )=\left( e^{\beta \omega }-1\right) ^{-1}$ is the average number of quanta in the oscillator (bath mode) with frequency $\omega $, with $\beta $ the inverse temperature. Since $R(t)$ is the overlap between the dephasing and modulation spectra, it can be reduced by choosing an appropriate modulation that reduces this overlap [@Kofman:04; @gor06a; @Alicki:01a]. We shall show that the optimal modulation reduces the spectral overlap of the dephasing and modulation spectra (Fig. \[Fig-2\]). However, since the energy constraint in the frequency domain is non-trivial we shall derive the equations for optimal modulation using the time domain. *Optimization*.— We wish to find the optimal modulation, i.e., time-dependent near-resonant field, that minimizes $R(t)$. Calculus of variation is an often-used technique of optimal control theory, e.g., [@PhysRevA.37.4950; @Palao:02]. We apply it to derive the Euler-Lagrange (EL) equations for the energy-constrained optimal modulation. The accumulated phase due to the modulation is $\phi (t)=\int_{0}^{t}d\tau \Omega (\tau )$. Let us write $\Phi (t)=\tilde{\Phi}(t)e^{i\Delta t}$, where $\tilde{\Phi} (t)$ and $\Delta $ are the amplitude and spectral center of the correlation function, respectively. Using Eqs.  and , we can then derive the EL equation for the optimal modulation (see Appendix \[appA\] and \[appB\]): $$\begin{aligned} \lambda \ddot{\phi}(t)&=&-Z[t,\phi (t)] \label{Z-def} \\ Z[t,\phi (t)] &=&\left\langle \tilde{\Phi}(\left\vert t-t_{1}\right\vert )\sin [\phi (t)-\phi (t_{1})+\Delta (t-t_{1})]\right\rangle _{T}^{t_{1}}, \notag\end{aligned}$$ where $\lambda $ is the Lagrange multiplier. The boundary conditions for the accumulated phase are $\phi (0)=\dot\phi (0)=0$, which results in a smooth solution and accounts for turning the control field on. Eliminating $\lambda $ we find that the optimal control field shape is the solution to the following equation (see Appendix \[appB\]): $$\ddot{\phi}(t)=\frac{-\sqrt{E}Z[t,\phi (t)]}{\sqrt{\int_{0}^{T}dt_{1}\left \vert \int_{0}^{t_{1}}dt_{2}Z[t_{2},\phi (t_{2})]\right\vert ^{2}}}. \label{phi-eq}$$ Equation (\[phi-eq\]) is the central result of this work. It furnishes the optimal time-dependent field amplitude, that maximizes the average fidelity ${\langle F(t)\rangle }$ at the final time $T$, via $\Omega (t)=\dot{\phi}(t)$. Although Eq. (\[phi-eq\]) is a complicated non-linear integro-differential equation, it is very useful indeed, as we show next. *Linearized EL equation*.— Assuming that we have a good initial guess $\phi _{0}(t)$ for the modulation, we can look for the optimal *deviation* $\nu (t)$ by writing $\phi (t)=\phi _{0}(t)+\nu (t)$, where $\nu (t)\ll 1$. Expanding Eq. (\[Z-def\]) in powers of $\nu (t)$ and retaining only the first order, the linearized EL equation becomes (see Appendix \[appC\]): $$\begin{aligned} &&\lambda \ddot{\nu}(t)+\left\langle Q(t,t_{1};\phi _{0}(t))\left( \nu (t)-\nu (t_{1})\right) \right\rangle _{T}^{t_{1}} =-C(t;\phi _{0},\lambda ) \notag \\ &&Q(t,t_{1};\phi _{0}(t)) = \tilde{\Phi}(\left\vert t-t_{1}\right\vert ) \times \notag \\ &&\qquad \cos \left( \phi _{0}(t)-\phi _{0}(t_{1})+\Delta (t-t_{1})\right) \notag \\ &&C(t;\phi _{0}(t),\lambda ) = \lambda \ddot\phi _{0}(t)+Z[t,\phi _{0}(t)]. \label{Q-line-def}\end{aligned}$$ This linearized EL equation is valid also in the case of short time optimal modulation, for which we simply set $\phi _{0}(t)=0$, subject to $\nu (t)\ll 1$ for $0\leq t\leq T\ll 1$. *Numerical analysis*.— Armed with the equations for the optimal modulation, we turn to solving them numerically for specific decoherence scenarios, defined by their dephasing spectra $G(\omega )$. We obtain the numerical solution to the integro-differential Eq.  via an iterative process, where we guess a probable solution that satisfies the boundary conditions and the constraint, use it in the RHS of Eq.  to compute the integral, and solve the resulting differential equation. The solution is then used in the RHS of Eq.  , and so on. For the examples presented below, we checked that several initial guesses converged to the same optimal modulation. Most importantly, we found that the optimal modulation is [*robust against random control field imperfections*]{}. This is due to the fact that the decoherence rate is determined by the accumulated phase and not the instantaneous modulation, Eq. . Specifically, we found that a $10\%$ zero-mean random pulse fluctuation results in less than a $1\%$ increase in the optimal dephasing rate. We compare the optimal dephasing rate to the one obtained by the popular periodic DD control (“bang bang”) procedure [@Viola:98Zanardi:98bViola:99Vitali:99], but to make the comparison meaningful we impose the same energy constraint. Finite-duration periodic DD against pure dephasing is the “bang bang” application of $n$ $\pi $-pulses and is given in our setting by $$\Omega (t)=\left\{ \begin{array}{ll} \pi /\nu & j\tau \leq t<j\tau +\nu \quad j=0\ldots n-1 \\ 0 & \mathrm{otherwise} \end{array} \right. \label{BB-def}$$ where $\nu <\tau $ is the width of each pulse and $\tau $ is the interval between pulses. The energy constraint $E$ and the total modulation duration $ T=n\tau +\nu $ are related via $n=\nu E/\pi ^{2}$. In the frequency domain, the spectral modulation intensity can be described by a series of peaks, where the two main peaks are at $\pm \pi /\tau $. Thus, the peaks are shifted in proportion to the energy invested in the modulation. However, DD is not an admissible solution to our EL equation due to its discontinuous derivative. In order to improve the comparison, we apply our linearized EL equation with the DD modulation as an initial guess, and obtain the optimal modulation in the vicinity of the DD control. *(a) Single-peak resonant dephasing spectrum*.– This simple dephasing spectrum describes a common scenario where $\Phi (t)=e^{-t/t_{c}}\gamma /t_{c}$, where $\gamma $ is the long-time dephasing rate \[$R(t\rightarrow\infty )=2\pi \gamma $\] and $t_{c}$ is the noise correlation time. Fig. \[Fig-1\](a) shows $R(T)$, normalized to the bare (unmodulated) dephasing rate, as a function of the energy constraint. As expected, the more energy is available for modulation, the lower is the dephasing rate. For low energies the optimal modulation significantly outperforms DD, while at higher energies this difference disappears. These results can be understood from Fig. \[Fig-2\](a), by noticing that the two central DD peaks have significant overlap with $G(\omega )$ at the low energy value shown. As $E$ is increased at fixed $T$ the DD peaks move farther apart, and have less overlap with $G(\omega )$, leading to improved performance. Applying the linearized EL equation with DD as initial guess yields only mild improvements (not shown). The explanation for the superior performance of the optimal modulation is also evident from Fig. \[Fig-2\](a): since higher frequencies have lower coupling strength in this case, the optimal control reshapes so as to maximize its weight in the high-frequency range, to the extent permitted by the energy constraint. The modulation can be well approximated by $\Omega(t)=a[1+e^{-t/T}(t/T-1)]$, where $a$ is determined by the energy constraint, which fits the inset in Fig. \[Fig-1\](a). ![(color online) Average modified final decoherence rate $R(T)$, normalized with respect to the unmodulated rate, as a function of energy constraint. DD - dash, magenta. Optimal modulation - solid, blue. Insets: optimal modulation $\Omega (t)$ for different energy constraints. (a) Single-peak resonant dephasing spectrum (inset: $E=20$). (b) Single-peak off-resonant spectrum (inset: $E=50$). (c) $1/f$ spectrum (inset: $E=30$). (d) Multi-peaked spectrum (inset: $E=30$). []{data-label="Fig-1"}](PRL_Fig1.eps){width="9cm"} ![(color online) Dephasing spectrum $G(\protect\omega )$ (solid, red), optimal (dot-dash, green) and DD (dash, magenta) modulation spectra $F_{T}(\protect\omega )$, in arbitrary units (a.u.). Same parameters as in the insets of Fig. \[Fig-1\]. []{data-label="Fig-2"}](PRL_Fig2.eps){width="9cm"} *(b) Single-peak off-resonant dephasing spectrum*.— This dephasing spectrum describes a variation on the aforementioned scenario, where the spectral peak is shifted \[$\Delta \neq 0$ in $\Phi (t)=\tilde{\Phi} (t)e^{i\Delta t}$\], e.g., coupling to a non-resonant bath. With no other constraints, the optimal modulation is trivially similar to the one of the resonant spectrum, with a shifted energy-constraint $E^{\mathrm{non-res}}=E^{\mathrm{res}}+\Delta $. However, by imposing a positivity constraint, $\dot\phi (t)\geq 0$ (positive field amplitude), one obtains non-trivial behavior of $R(T)$ as a function of the energy constraint – see Fig. \[Fig-1\](b). Here we used the linearized EL equation with the DD modulation (\[BB-def\]) as an initial guess. For both the DD and optimal modulations, we observe an initial *increase* in the dephasing rate as a function of energy, followed by a decrease. For DD, this can be interpreted as a manifestation of the initial anti-Zeno effect and the subsequent quantum Zeno effect [@Kofman:04; @Facchi:05]. Because of the positivity constraint, the optimal modulation does worse than the unmodulated case, for low enough energy. The DD modulation is optimal for small energy constraints, hence the decoherence rates of DD and our optimal solution coincide. This is because the DD peaks do not overlap the off-resonant spectral peak. However, as the positive-frequency main DD peak \[Fig. \[Fig-2\](b)\] nears the off-resonant spectral peak, with increased energy, the optimal modulation diverges from the DD modulation, and reshapes itself so as to couple to higher modes of the bath. In the time domain \[Fig. \[Fig-1\](b) inset\], this is seen as a smoothing of the abrupt DD modulation. At even higher energy constraints, there is once more no improvement by the optimal modulation over DD, yet there is an improvement over the unmodulated case. Over the entire range of $E$, the optimal modulation results in a much flatter $R(T)$ than DD, which is an indicator of its robustness. While DD is strongly influenced by the off-resonant peak, the optimal modulation exploits the energy available to find the minimal overlap, irrespective of dephasing spectrum. *(c) $1/f$ dephasing spectrum*.— The ubiquitous $1/f$ dephasing spectrum that describes a variety of experiments – e.g., charge noise in superconducting qubits [@Nakamura:02] – is given in our notation by $G(\omega )\propto 1/\omega $, with cutoffs $\omega _{\mathrm{min}}$ and $\omega _{\mathrm{max}}$. Fig. \[Fig-1\](c) shows that as expected, the more energy is available for modulation, the lower the dephasing rate. Since, as in case (a), higher frequencies now have lower coupling strength, the optimal control reshapesso as to have as high a weight in the high frequency range as the energy constraint allows \[Fig. \[Fig-2\](c)\]. This is expressed in the time-domain \[Fig. \[Fig-1\](c) inset\] as the initial increase in the modulation strength ($t<50$). The later decrease in modulation strength can be attributed to the lower cutoff, where the optimal modulation benefits from lower frequencies, i.e., lower modulation amplitudes. Upon comparing the $1/f$ case to the Lorentzian spectrum, Fig. \[Fig-1\](a), we observe a similar optimal initial chirped modulation in the time domain. Despite the differences in the long-time behavior (due to the lower cutoff in the $1/f$ case), these two examples allow us to generalize to any dephasing spectrum with a monotonically decreasing system-bath coupling strength as a function of frequency. The optimal modulation for such spectra will be an energy-constrained chirped modulation, with variations due to other spectral characteristics, e.g., cutoffs. *(d) Multi-peaked dephasing spectrum*.— This describes the most general scenario, where there can be several resonances and noise correlation times. Fig. \[Fig-1\](d) shows $R(T)$ as a function of the energy constraint. Once again, because DD does not account for the dephasing spectrum, its performance is much worse than the optimal modulation, whose reshaping results in monotonically improving performance: the peaks of the optimal modulation are predominantly anti-correlated with the peaks of $G(\omega)$. *Conclusions*.— We have found the optimal modulation for countering pure dephasing upon imposing an energy constraint on the DCM approach [@Kofman:04; @gor06a], by deriving and solving the Euler-Lagrange equation . This yields optimal reduction of the overlap of the dephasing and the modulation intensity spectra. We stress that our optimal control theory results are also applicable to scenarios other than pure dephasing, such as amplitude noise (relaxation), due to the universality of Eqs. (\[R-def\])-(\[Ft\]) [@Kofman:04; @gor06a]. The form of the energy constraint will then differ in detail from the pure dephasing case. However, our general conclusions about the optimal modulation to minimize spectral overlap, will remain valid. We expect that the optimal modulation technique will find useful applications in quantum information processing and quantum computation. The price is that one must acquire intimate knowledge of the noise spectrum, which is often neglected, as previous control techniques such as DD and QEC had no use for it. We have shown that this information can result in the maximization of fidelity, under operational constraints. [*Acknowledgments*]{}.— G.K. acknowledges the support of GIF and EC (MIDAS STREP, FET Open). 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General comments on deriving optimal functions {#appA} ============================================== For the optimal control of a functional $$\mathcal{F}(y,\dot y)=\int_{0}^{T}dtF(t,y,\dot y)$$with the constraint $$\mathcal{K}(y,\dot y)=\int_{0}^{T}dtK(t,y,\dot y)=E$$one follows the following procedure: \(i) Solve the Euler-Lagrange equation: $$\frac{\delta F}{\delta y}-\frac{\partial }{\partial t}\frac{\delta F}{\delta \do y}=-\lambda \left[ \frac{\delta K}{\delta y}-\frac{\partial }{% \partial t}\frac{\delta K}{\delta \dot y}\right] \label{eq-1}$$where $\delta F$ is the variation of $F$, $\lambda $ is the Lagrange multiplier, and the boundary conditions are $y(0)=y_{0}$ and $\dot y(0)=y_{1}$. \(ii) Insert the solution $\tilde{y}(t;\lambda )$ into the constraint: $$\mathcal{K}(\tilde{y}(t;\lambda ),\tilde{\dot y}(t;\lambda ))=E \label{eq-2}$$and obtain $\lambda =\lambda (E)$. \(iii) Eliminate $\lambda $ by inserting $\lambda (E)$ into $\tilde{y}% (t;\lambda )$ and obtain the optimal solution, $\tilde{y}(t;E)$, that minimizes the functional $\mathcal{F}$, under the constraint $\mathcal{K}=E$. Derivation of the Euler-Lagrange equation {#appB} ========================================= The average modified decoherence rate is given by: $$\begin{aligned} R(T) &=&\frac{2}{T}\int_{0}^{T}dt\int_{0}^{t}dt_{1}\tilde{\Phi}(t-t_{1})\cos [\phi (t)-\phi (t_{1})+\Delta (t-t_{1})] \\ &=&\frac{2}{T}\int_{0}^{T}dt\int_{0}^{T}dt_{1}\Theta (t-t_{1})\tilde{\Phi}% (t-t_{1})\cos [\phi (t)-\phi (t_{1})+\Delta (t-t_{1})]\end{aligned}$$where $\Theta (t)$ is the Heaviside step function. One arrives at the following variation of the average modified decoherence rate: $$\begin{aligned} \delta R(T) &=&\frac{2}{T}\int_{0}^{T}dt\int_{0}^{T}dt_{1} \nonumber \label{d-J-3} \\ &&\left[ -\Theta (t-t_{1})\tilde{\Phi}(t-t_{1})\sin [\phi (t)-\phi (t_{1})+\Delta (t-t_{1})]\delta \phi (t)\right. \nonumber \\ &&\left. +\Theta (t-t_{1})\tilde{\Phi}(t-t_{1})\sin [\phi (t)-\phi (t_{1})+\Delta (t-t_{1})]\delta \phi (t_{1})\right] \label{d-R-1} \\ &=&\frac{2}{T}\int_{0}^{T}dt\int_{0}^{T}dt_{1} \nonumber \\ &&\left[ -\Theta (t-t_{1})\tilde{\Phi}(t-t_{1})\sin [\phi (t)-\phi (t_{1})+\Delta (t-t_{1})]\delta \phi (t)\right. \nonumber \\ &&\left. -\Theta (t_{1}-t)\tilde{\Phi}(t_{1}-t)\sin [\phi (t)-\phi (t_{1})+\Delta (t-t_{1})]\delta \phi (t)\right] \label{d-R-2} \\ &=&\frac{2}{T}\int_{0}^{T}dt\int_{0}^{T}dt_{1}\left[ -\tilde{\Phi}% (\left\vert t-t_{1}\right\vert )\sin [\phi (t)-\phi (t_{1})+\Delta (t-t_{1})]\delta \phi (t)\right] \label{R-var}\end{aligned}$$where we have made a $t\leftrightarrow t_{1}$ substitution in the second integrand of Eq. (\[d-R-1\]), and notice that $\Theta (t)f(t)+\Theta (-t)f(-t)=f(\left\vert t\right\vert )$. One can easily see that defining the constraint functional as: $$\mathcal{K}(t,\phi (t),\dot \phi(t))=\int_{0}^{t}dt_{1}|\dot \phi(t_{1})|^{2}=E, \label{const-functional}$$with $K(t,\phi (t),\dot \phi(t))=|\dot \phi (t_{1})|^{2}$ results in the variation: $$\delta K=2\ddot\phi(t)\delta \dot\phi(t). \label{const-var}$$Combining Eqs. (\[R-var\]), (\[const-var\]) and (\[eq-1\]) results in the Euler-Lagrange equation:$$\lambda \ddot\phi(t)+Z[t,\phi (t)]=0 \label{EL}$$where $$Z[t,\phi (t)]=\frac{1}{T}\int_{0}^{T}dt_{1}\tilde{\Phi}(\left\vert t-t_{1}\right\vert )\sin [\phi (t)-\phi (t_{1})+\Delta (t-t_{1})]. %\label{Z-def}$$ Derivation of the linearized Euler-Lagrange equations {#appC} ===================================================== In some cases it is advantageous to linearize the EL equations with respect to the modulation. If one looks for the optimal deviation $\nu (t)$ from a given pulse shape, $\phi _{0}(t)$, then one can write $\phi (t)=\phi _{0}(t)+\nu (t)$, where $\nu (t)\ll 1$. Equation (\[EL\]) then becomes: (\_0(t)+(t))&=-& \_[0]{}\^[T]{}dt\_[1]{}(t-t\_[1]{})\ &=-&\_[0]{}\^[T]{}dt\_[1]{}(t-t\_[1]{}) +O(\^2(t)) where we approximated $\sin[\nu(t)-\nu(t_{1})]\approx \nu(t)-\nu(t_{1})+O(\nu^3(t))$ and $\cos[\nu(t)-\nu(t_{1})]\approx 1-\frac{1}{2}(\nu(t)-\nu(t_{1}))^2 +O(\nu^4(t))$. The linearized Euler-Lagrange equation becomes: $$\lambda \ddot\nu(t)+\frac{1}{T}\int_{0}^{T}dt_{1}Q(t,t_{1};\phi _{0}(t))\left( \nu (t)-\nu (t_{1})\right) =-C(t;\phi _{0},\lambda ) \label{EL-line-def}$$where $$\begin{aligned} Q(t,t_{1};\phi _{0}(t)) &=&\tilde{\Phi}(\left\vert t-t_{1}\right\vert )\cos \left( \phi _{0}(t)-\phi _{0}(t_{1})+\Delta (t-t_{1})\right) %\label{Q-line-def} \\ C(t;\phi _{0}(t),\lambda ) &=&\lambda \ddot\phi _{0}(t)+Z[t,\phi _{0}(t)].\end{aligned}$$

--- abstract: 'Crystalline Fe$_{3}$O$_{4}$/NiO bilayers were grown on MgO(001) substrates using reactive molecular beam epitaxy to investigate their structural properties and their morphology. The film thickness either of the Fe$_{3}$O$_{4}$ film or of the NiO film has been varied to shed light on the relaxation of the bilayer system. The surface properties as studied by x-ray photo electron spectroscopy and low energy electron diffraction show clear evidence of stoichiometric well-ordered film surfaces. Based on the kinematic approach x-ray diffraction experiments were completely analyzed. As a result the NiO films grow pseudomorphic in the investigated thickness range (up to 34nm) while the Fe$_{3}$O$_{4}$ films relax continuously up to the thickness of 50nm. Although all diffraction data show well developed Laue fringes pointing to oxide films of very homogeneous thickness, the Fe$_{3}$O$_{4}$-NiO interface roughens continuously up to 1nm root-mean-square roughness with increasing NiO film thickness while the Fe3O4 surface is very smooth independent on the Fe$_{3}$O$_{4}$ film thickness. Finally, the Fe$_{3}$O$_{4}$-NiO interface spacing is similar to the interlayer spacing of the oxide films while the NiO-MgO interface is expanded.' author: - 'T. Schemme' - 'O. Kuschel' - 'F. Bertram' - 'K. Kuepper' - 'J. Wollschläger' title: 'Structure and morphology of epitaxially grown Fe$_{3}$O$_{4}$/NiO bilayers on MgO(001)' --- Introduction ============ The modification of magnetic properties of ferro(i)magnetic films (F) by antiferromagnetic films (AF) is of huge physical and technological interest for instance for the development of magnetoresistive (MR) devices like magnetic tunnel junctions (MTJs)[@MTJ; @MTJsMoodera; @OxideSpintronics; @SpintronicsApps]. MTJs primarily consist of two ferro(i)magnetic conducting films, which are separated by a non-magnetic insulator. In case both ferro(i)magnetic films are comprised of the same material, it is essential to shift the coercive field of one of the films. The AF/F exchange coupling causing an exchange bias can be utilized to have two different switching fields. Thus, the magnetization of the films can be switched separately and an alternation of the alignment of the magnetization between parallel and antiparallel is possible.\ Magnetite (Fe$_{3}$O$_{4}$) is a promising material for such applications due its half-metallic[@halfmet] character and a high spin polarization at the Fermi level. The ferrimagnetic oxide has a high Curie temperature (860K) and a saturation moment of almost 4$\mu_{B}$. It crystallizes in an inverse spinel structure with a cubic lattice constant of 0.83964nm at 300K. Below the so called Verwey temperature T$_{V}$ = 120K occurs a phase transition, whereby magnetite adopts a monoclinic structure, becomes insulting and its susceptibility changes.\ NiO is an antiferromagnetic ionic insulator with a high thermal stability. It is inert against corrosion and has a N$\rm \acute{e}$el temperature of T$_{N}$ = 523K and therefore well suited as exchange bias material NiO crystallizes in a rock salt structure with a lattice constant of 0.41769nm.\ MgO is isostructural to NiO with a slightly larger lattice constant of 0.42117nm. Hence, the lattice mismatch between NiO and MgO is about 0.8% while the misfit between Fe$_{3}$O$_{4}$ and MgO is about 0.3% and MgO is apparently well suited as substrate for both.\ The AF/F exchange coupling was first reported in 1957 in Co/CoO systems[@FirstExchangeCoupling]. This pinning effect increases the coercive field of the F film and leads to a shift of the hysteresis loop by a bias field, the so called exchange bias. The pinning effect can be induced by cooling the AF/F bilayer from high temp through the N$\rm \acute{e}$el temperature of the AF under the application of a magnetic field.\ The exchange bias on Fe$_{3}$O$_{4}$ films has been investigated in prior studies, i.e. by using different substrates to change the growth direction of the AF and analyzing the influence of a compensated NiO(001) and a fully uncompensated NiO(111) interface on the exchange bias[@Fertbilayer]. The influence of the stacking order was also examined in that study.\ Models have been developed to describe the exchange bias, i.e a domain state model (DSM) based on Monte Carlo simulations[@DomainI], which shows that the magnitude of the exchange bias is dependent on the degree of dilution of the antiferromagnet. This domain state model has been verified by experiments[@DomainII].\ Shortly after the exchange coupling, the trainings effect was found,too. The trainings effect denotes the descending of the initial exchange bias to a smaller residual value during measuring the hysteresis loops. D. Paccard has reported about the training effect[@TrainingsEffect1] in AF/F systems.\ Since the film thickness, the crystal quality of the films and AF/F interface structure and roughness strongly influence the magnetic properties of the bilayers, we have investigated the thickness dependence of the strutural quality of Fe$_{3}$O$_{4}$/NiO-bilayers on the film thickness of each film. Both NiO and Fe$_{3}$O$_{4}$ were grown at 250$^{\circ}$ substrate temperature to avoid the unwanted interdiffusion of magnesium ions from the substrate into the bilayer[@PressureRate]. A substrate temperature of 250$^{\circ}$ is also considered as the lower limit for the growth of well-ordered magnetite films[@NiOMagnetit].\ The stoichiometry of the bilayers were analyzed $in-situ$ by x-ray photoelectron spectroscopy (XPS), while the structural properties were investigated by $in-situ$ low energy electron diffraction (LEED) and $ex-situ$ x-ray diffraction (XRD). The XRD data were evaluated using kinematic diffraction theory.\ EXPERIMENTAL SETUP AND SAMPLE PREPARATION ========================================= The sample preparation was carried out in multi chamber ultra high vacuum (UHV) systems with a preparation chamber (base pressure of 10$^{-8}$mbar) and an analysis chamber (base pressure of 10$^{-10}$mbar). The available $in-situ$ characterization methods are low energy electron diffraction (LEED) and x-ray photoelectron spectroscopy (XPS). The XPS system operates with an hemispherical analyzer and an Al K$_{\alpha}$ x-ray anode (1486.6 eV). To perform reactive molecular beam epitaxy (RMBE) the preparation chamber is equipped with electron beam evaporators for nickel and iron, a heatable manipulator and an oxygen source.\ Before film deposition the MgO substrates were annealed at 400$^\circ$C in a 10$^{-4}$mbar oxygen atmosphere to obtain well-ordered and clean surfaces as proved by LEED and XPS, respectively. After the preparation of the substrates NiO films were deposited using RMBE at 250$^\circ$C in a 10$^{-5}$mbar oxygen atmosphere. Afterwards, Fe$_{3}$O$_{4}$ films with constant thickness were grown on the NiO films also via RMBE at 250$^\circ$C in a 5$\times$10$^{-6}$ mbar oxygen atmosphere. The thicknesses of the different films were in situ controlled by a quartz crystal microbalance. Later ex situ XRR measurements were performed to prove the thickness of all films of the samples. Two different series of Fe$_{3}$O$_{4}$/NiO bilayers were grown on MgO(001) supports. In series one the NiO film thickness was modified, while the magnetite film thickness was kept constant. The bilayers of series two has a constant NiO film thickness, while the magnetite thickness was varied.\ After each annealing step and each film deposition the samples were transferred to the analysis chamber for $in-situ$ LEED and XPS measurements.\ After the sample preparation the samples were exposed to ambient conditions for diverse $ex-situ$ experiments. The film thickness and structure of the films was analyzed by XRR and XRD, respectively. These experiments were carried out at Deutsches Elektronen-Synchroton (DESY), Hamburg at PETRA III beamline P08. This is an undulator beamline with a high heat-load double-crystal monochromator and large-offset monochromator to separate the beamline from the adjacent beamline. At the endstation a Kohzu 4S+2D type diffractometer is installed[@OSeeckP08]. A Mythen array detector[@Detektor] was used due to its higher dynamic range and the capability of creating reciprocal space maps (RSM) within a shorter period of time compared to a point detector. Results ======= $In-Situ$ surface characterization by LEED and XPS -------------------------------------------------- LEED experiments were performed at an electron energy of 135eV. Fig. \[LEED\] shows exemplary for one sample the surface structure of an initial clean MgO substrate (a), with an additional 7nm NiO film (b) and the subsequently grown 56nm Fe$_{3}$O$_{4}$ film (c). All bilayers show in each deposition step the same diffraction pattern. Since MgO and NiO have the same rocksalt structure and similar lattice constants, the surface diffraction pattern is in either case a quadratic 1 $\times$ 1 structure. The diffraction pattern of the magnetite film shows more spots due to the almost doubled lattice constant. In addition, we always observe the typical ($\sqrt{2} \times \sqrt{2})$R45$^{\circ}$ superstructure. This superstructure is only visible in the case of magnetite[@Cham1], while maghemite has only an quadratic 1 $\times$ 1 surface structure. ![LEED pattern obtained at 135eV electron energy of a clean MgO(001) substrate (a), a NiO film on top of this substrate (b) and an Fe$_{3}$O$_{4}$ film on top of this NiO film. The small white square indicate the ($\sqrt{2} \times \sqrt{2})$R45$^{\circ}$ surface superstructure of the magnetite, while the bigger white squares indicate the 1 $\times$ 1 surface structures.[]{data-label="LEED"}](./LEED.eps) The XP spectra of the Fe 2p peak region (a) and the Ni 2p region (b) are presented exemplarily in Fig.\[XPSCombo\] for the same sample. The Ni 2p spectra consists of the Ni 2p$_{3/2}$ peak at 854.6eV and the Ni 2p$_{1/2}$ peak at 872.6eV with their corresponding satellites at about 7eV higher binding energy. The measured Ni 2p spectra agree well with Ni 2p spectra for NiO reported in literature[@NiOXPS]. The XP spectra of the Fe 2p region of all samples exhibit the same behavior. The Fe 2p$_{3/2}$ is located at binding energy of 710.6eV, while the Fe 2p$_{1/2}$ is situated at binding energy of 723.6eV. Magnetite contains Fe$^{2+}$-ions as well as Fe$^{3+}$-ions at the ratio of 1:2. At this ratio the charge transfer satellites characteristic for maghemite (718.8eV[@Yamashita; @Fuji]) or wuestite (714.7eV[@Yamashita; @Fuji]) are not visible separately since both satellites overlap forming a flat plateau between the Fe 2p$_{3/2}$ and Fe 2p$_{1/2}$ peak. Hence, this spectrum is typical for stoichiometric Fe$_{3}$O$_{4}$. Therefore, combined with the LEED experiments it is safe that crystalline and stoichiometric magnetite has been deposited on crystalline and stoichiometric NiO. ![XP spectra of the Fe 2p (a) and the Ni 2p (b) of a 50nm iron oxide and a 7nm nickel oxide film, respectively.[]{data-label="XPSCombo"}](./XPScombo.eps) Film structure characterization by XRD -------------------------------------- The structures of the entire oxide bilayers were analyzed by X-ray diffraction experiments. Hence, the 3D reciprocal space spanned by the MgO substrate is indexed lateral by the MgO(001) surface unit cell while the layer distance has been used for the direction perpendicular to the surface. Compared to the well-known cubic bulk unit cell this surface unit cell has half the size of the bulk unit cell of MgO in vertical direction due to the spacing between (001) crystal planes and is rotated by 45$^{\circ}$ in the (001) plane. Thus, compared to the cubic bulk lattice, we use the base vectors $\vec{a}_{1}\,=\,\frac{1}{2}(1,1,0)$, $\vec{a}_{2}\,=\,\frac{1}{2}(-1,1,0)$ and $\vec{a}_{3}\,=\,\frac{1}{2}(0,0,1)$ to describe the lattice. As a consequence the MgO(002)$_{B}$ rock salt bulk reflection is denoted by MgO(001)$_{S}$. Since the magnetite and the maghemite spinel structures have almost doubled bulk lattice constants compared to MgO, the (004)$_{B}$ spinel bulk reflection is very close to the MgO(001)$_{S}$ reflection NiO crystallizes like MgO in the rock salt structure with a slightly smaller lattice constant, so the NiO(002)$_{B}$ rock salt bulk reflection is also close to the MgO(001)$_{S}$ reflection.\ ![Reciprocal space map of a sample with a 24nm NiO and a 51nm Fe$_3$O$_4$ film (a) and of a sample with 29nm NiO and 10nm Fe$_3$O$_4$ film (b).[]{data-label="SRSM"}](./RSMZus.eps) A reciprocal space ($K$,$L$) mapping (RSM) of one sample of the first series with 50nm Fe$_{3}$O$_{4}$ on 24nm NiO is shown in Fig.\[SRSM\](a). For comparison another RSM of a sample of the second series with a 28nm NiO film and a 10nm Fe$_{3}$O$_{4}$ film is depicted in Fig.\[SRSM\](b). Both RSMs show a sharp MgO substrate peak at $L$=2 due to diffraction at the MgO substrate. The sample of the first series in Fig.\[SRSM\](a) shows broad Bragg peaks and corresponding Laue oscillations of the deposited NiO and Fe$_{3}$O$_{4}$ films due to their finite film thickness. As opposed to this the sample of the second series in Fig.\[SRSM\](b) exhibits only the Bragg peak of the NiO film along with distinct Laue oscillations. Here, the diffraction signal of the iron oxide causes only weak modulations of the diffraction signal of the NiO since the Fe$_{3}$O$_{4}$ film is very thin. ![XRD rod scans along the (00$L$) of the samples with constant Fe$_{3}$O$_{4}$ thickness and increasing NiO thickness. Circles show experimental data and solid lines calculations.[]{data-label="fig:XRD1"}](./XRDfit1.eps) The crystal truncation rod (CTR) scans along the (00$L$) direction for the series with the nearly constant magnetite film and variation of the NiO film thickness are depicted in Fig. \[fig:XRD1\]. All scans show a sharp MgO substrate peak at $L$ = 2 due to diffraction at the MgO substrate. The series with the constant magnetite film thickness in Fig. \[fig:XRD1\] exhibits in all scans broad Bragg peaks of the deposited Fe$_{3}$O$_{4}$ films due to their finite film thickness. In addition well-pronounced Laue oscillations with high periodicity can be observed, which indicate well ordered films with homogeneous thickness. The Bragg peaks of NiO are also visible for NiO thicknesses higher than 7 nm. The increasing film thickness of the NiO films can be easily seen by comparing the periodicity of the oscillations and the FWHM of the corresponding Bragg peaks (Fig. \[fig:XRD1\]). However, at film thicknesses smaller than 7nm no Bragg peaks and only Laue oscillations of the NiO modulating the diffraction signal of the magnetite can be observed. At a NiO film thickness of 24nm the Laue oscillations of the magnetite film becomes weaker and are nearly vanished at a NiO film thickness of 33 nm. This observation indicates an increasing roughness of the NiO/Fe$_{3}$O$_{4}$-interface. Moreover, here the Bragg peak of the magnetite film is weaker than the Bragg of the NiO film at 33nm although the Fe$_{3}$O$_{4}$ film has a film thickness of 43nm.\ ![XRD rod scans along the (00$L$) of the samples with constant NiO thickness and increasing Fe$_{3}$O$_{4}$ thickness. Circles show experimental data and solid lines calculations.[]{data-label="fig:XRD2"}](./XRDfit2.eps) The specular rod scans of the second series of bilayers with a constant NiO film thickness of approximately 30nm are shown in Fig. \[fig:XRD2\]. All films feature Laue fringes and broad Bragg peaks of the NiO film independent of the Fe$_{3}$O$_{4}$ film thickness. In this series the Laue fringes for thin magnetite films cause very small modulations of the diffraction signal of the NiO. However, the thin magnetite films in this series modulate the diffraction signal of the NiO film much weaker than the thin NiO films modulate the thick magnetite films in series one. Another feature is that the Bragg peaks and fringes of the thicker magnetite films can hardly be seen. Even the Bragg peak of a 60nm thick magnetite film is almost not visible. One exception is the sample with a 37nm magnetite film, where some weak Laue fringes and a small Bragg peak are observable. However, overall the diffraction signal of the magnetite film is very weak compared to a well-ordered magnetite film. This observation indicates that the quality of the magnetite depends on the NiO film thickness. While small NiO thicknesses barely influence the crystal quality of the magnetite films, a NiO film thickness of 24nm increases the roughness of the NiO/Fe$_{3}$O$_{4}$-interface .\ ![Schematic cross section through a sample to explain the parameters used in the layer model. The different colored circles represent the different ions in the corresponding crystal lattice.[]{data-label="ModellNiFeO"}](./Modell.eps) We have applied full kinematic diffraction theory for analysis of the diffraction data lines in Figs. \[fig:XRD1\] and \[fig:XRD2\] in order to determine the structural parameters and to understand deeper the observations described above. The applied approach for the calculations shown in Fig.\[ModellNiFeO\] consists of the MgO rock salt substrate, the on top grown NiO with rock salt structure and the subsequently deposited Fe$_{3}$O$_{4}$ with spinel structure. In this approach oxygen, Ni and iron ions were primarily arranged in their respective bulk structures and the diffracted intensity was calculated using their atomic form factors. In the calculation the unit cells of the respective films were homogeneously deformed perpendicular to the surface to obtain the vertical layer distance. Further parameters of the calculation are the surface roughness of the Fe$_{3}$O$_{4}$ film and interface roughnesses as well as the Debye-Waller factors. The vertical layer distances determined from the curve fitting calculations of the (00$L$)-rod plotted against the film thickness are shown in Fig.\[fig:Lattieserie1\] and Fig.\[fig:Lattieserie2\]. ![Vertical layer distance of magnetite and nickel oxide of the samples with constant Fe$_{3}$O$_{4}$ film thickness and increasing NiO film thickness dependent on film thickness.[]{data-label="fig:Lattieserie1"}](./LagenabstandS1.eps) The vertical layer distances $c$ for completely strained pseudomorphic Fe$_{3}$O$_{4}$ and NiO were calculated using $\frac{\Delta c}{c} = \frac{2\nu}{\nu - 1} \frac{\Delta a}{a}$[@NiOPoisson] and assuming a Poisson number of $\nu$=0.356 for magnetite[@Poisson] and $\nu$=0.21 for NiO[@NiOPoisson]. The areas between these calculated values (dotted lines) and their corresponding bulks values (solid lines) are marked in grey. The vertical layer distances of NiO ($c_{1,NiO}$ ) and Fe$_{3}$O$_{4}$ ($c_{1,Fe_{3}O_{4}}$) for the series with increasing NiO film thickness are shown in Fig.\[fig:Lattieserie1\]. The vertical layer distance $c_{1,NiO}$ of the NiO (red dots) is completely strained (0.2079nm) and exhibits no dependence on the NiO film thickness. Therefore, the NiO does not relax with increasing film thickness in the investigated range of film thicknesses. The vertical layer distances of magnetite is also strained and nearly constant at 0.2098nm and is consequently also independent of the NiO film thickness. ![Vertical layer distance of magnetite and nickel oxide of the samples with constant NiO film thickness and increasing Fe$_{3}$O$_{4}$ film thickness dependent on film thickness.[]{data-label="fig:Lattieserie2"}](./LagenabstandS2.eps) A similar observation can be made in Fig.\[fig:Lattieserie2\], where the vertical layer distances $c_{2,NiO}$ and $c_{2,Fe_{3}O_{4}}$ of the second series are plotted against the film thickness. In agreement with the first series of bilayers the vertical layer distances $c_{2,NiO}$ of the NiO films with constant film thickness are completely strained at approximately 0.2079nm. The vertical layer distance $c_{2,Fe_{3}O_{4}}$ of Fe$_{3}$O$_{4}$ is heavily strained and relaxes distinctly from 0.2080nm at 5nm to 0.2092nm at 18nm and then slowlier to 0.2099nm for 60nm Fe$_{3}$O$_{4}$ film thickness. Thus, magnetite reaches the vertical layer distance of bulk magnetite. However, we have to admit that the vertical layer distance of the magnetite films in this series were more difficult to determine than in the other series since the intensity of the corresponding Bragg peaks and fringes were quite weak. The weak Fe$_{3}$O$_{4}$ Bragg peaks in this series confirms the observation for the first series that the quality of the magnetite films depends strongly on the NiO film thickness. Beyond 24nm NiO film thickness the magnetite film is not well-ordered. ![Surface and interface roughness of the samples with constant Fe$_{3}$O$_{4}$ film thickness and increasing NiO film thickness dependent on film thickness. []{data-label="fig:FeNiOS1roughness"}](./FeNiOS1roughness.eps) This finding is strongly supported by our results for the NiO/Fe$_{3}$O$_{4}$ interface roughness. First, we have to emphasize tha the MgO/NiO interface is very smooth. It’s roughness is 0.2($\pm$ 0.1)nm The roughness of the NiO/Fe$_{3}$O$_{4}$ interface and the Fe$_{3}$O$_{4}$ surface is plotted against film thickness in Fig.\[fig:FeNiOS1roughness\]. It can be seen that with increasing NiO film thickness the roughness of the interface is also continuously increasing from 0.1nm to 1nm. The surface of the magnetite films, however, exhibits a small roughness of 0.1nm independent of the film thickness with the exception of the Fe$_{3}$O$_{4}$ film with 33nm NiO underneath. Here, the magnetite film has also a roughness of 1nm. ![Surface and interface roughness of the samples with constant NiO film thickness and increasing Fe$_{3}$O$_{4}$ film thickness dependent on film thickness. []{data-label="fig:FeNiOS2roughness"}](./FeNiOS2roughness.eps) The interface and surface roughness of the NiO and Fe$_{3}$O$_{4}$ films of the second series of samples are shown in Fig. \[fig:FeNiOS2interface\]. In agreement with the first series the roughness of the NiO/Fe$_{3}$O$_{4}$ interface is 0.80 ($\pm$ 0.15)nm for the 30$\pm 2$nm NiO films, while the surface roughness of the magnetite films is 0.15($\pm$0.05)nm with the exception of the magnetite film with 60nm film thickness. Here, interface roughness is 1nm. ![Interface spacing of the samples with constant Fe$_{3}$O$_{4}$ film thickness and increasing NiO film thickness dependent on film thickness. []{data-label="fig:FeNiOS1interface"}](./FeNiOS1interface.eps) The spacings for the MgO/NiO and NiO/Fe$_{3}$O$_{4}$ interfaces (cf. Fig.\[ModellNiFeO\]) are depicted dependent on film thickness in Fig. \[fig:FeNiOS1interface\] for the first series. The interface spacing MgO/NiO is slighty increasing with increasing film thickness from 0.2106nm to 0.2211nm, while interface spacing NiO/Fe$_{3}$O$_{4}$ is 0.202($\pm$0.008)nm and shows no dependence on film thickness. ![Interface spacing of the samples with constant NiO film thickness and increasing Fe$_{3}$O$_{4}$ film thickness dependent on film thickness. []{data-label="fig:FeNiOS2interface"}](./FeNiOS2interface.eps) The spacings of the MgO/NiO and NiO/Fe$_{3}$O$_{4}$ interfaces of the second series of samples are plotted versus film thickness in Fig. \[fig:FeNiOS2interface\]. Both spacings show no dependence on film thickness and are 0.206($\pm$ 0.005)nm for the NiO/Fe$_{3}$O$_{4}$ interface while the MgO/NiO interface spacing is 0.217($\pm$ 0.014)nm. discussion ========== After preparation of the nickel oxide and iron oxide films the photoelectron spectra of the Fe2p and Ni2p peak reveal that the surface near region of the films is Fe$_{3}$O$_{4}$ and NiO, respectively. Both NiO and Fe$_{3}$O$_{4}$ films show LEED diffraction patterns with the expected surface structures. NiO films exhibit a (1 $\times$ 1) surface structure, since they crystallize like MgO in rock salt structure. Fe$_{3}$O$_{4}$ films have the the typical ($\sqrt{2}\times\sqrt{2}$)R45$^{\circ}$ superstructure of the surface.\ The photoelectron spectra of the Fe2p and Ni2p peak reveal that the surface near region of the films is Fe$_{3}$O$_{4}$ and NiO, respectively. In summary, LEED and XPS prove that NiO and Fe$_{3}$O$_{4}$ films have the expected surface structure and surface near stoichiometry and there is no dependence on film thickness.\ XRD specular rod scans were carried out to investigate the whole structure of the samples. The scans of all films reveal that all NiO films are crystalline and well-ordered. In the first series where the initially NiO film thickness is 3nm only a weak broad Bragg peak can be observed, however, the strong Laue fringes of the NiO modulate the diffraction signal of the Fe$_{3}$O$_{4}$ film. A separate broad Bragg peak of the NiO film is observable at 14 nm and it gets more distinct with increasing NiO film thickness. All NiO films in both series are fully strained with a vertical layer distance of 0.2079 nm and do not relax with increasing film thickness. The interlayer spacing of NiO normal to the surface has become smaller to compensate the tensile strain due to the lattice matching of the film with the surface unit cell of the MgO substrate. This strain should decrease rapidly with the stable formation of dislocation above the critical thickness $h_{c}$. The critical thickness $h_{c}$ for the formation of misfit dislocations can be calculated using the formula[@FormelcritThick] $$\begin{aligned} \frac{h_{c}}{b}= \frac{1 - \nu \cdot \cos^{2}(\alpha) (ln(\frac{h_{c}}{b}) +1)}{2\pi f(1 + \nu) \cos(\lambda)} \, \label{Gleichung} ,\end{aligned}$$ where $b=\frac{a_{NiO}}{\sqrt{2}}$ is the magnitude of the Burgers vector, $f$ = 0.8% is the misfit of NiO, $\alpha = 90^{\circ}$ is the angle between the dislocation line and the Burgers vector, $\lambda = 45^{\circ}$ is the angle between the Burgers vector and the direction that is both normal to the dislocation line and that lies within the plane of the interface and $\nu = 0.21$ is the Poisson ratio[@NiOPoisson]. We obtain a critical film thickness of $h_{c}$ = 39nm, which means that the NiO film grown for this study are all below this critical film thickness and it is reasonable to observe no strain relaxation at the vertical layer distance. In addition, we like to emphasize that the relaxation process is very slow also for reasonable thicker NiO films[@NiOPoisson].\ The determined interface spacings $g$ between the MgO substrate and the NiO film and between the NiO and the Fe$_{3}$O$_{4}$ (cf. Fig. \[ModellNiFeO\]) show no interpretable dependence on the film thickness. Only the spacing of the MgO/NiO interface of the first film series increases slightly with increasing film thickness. While the Fe$_{3}$O$_{4}$/NiO interface thickness does not differ significantly from the interlayer spacings of these oxide films the NiO/MgO interface thickness is expanded compared to their interlayer spacings. This may be attributed to some covalent character of NiO - MgO interactions[@NiOMgOcovalent].\ The roughness of the NiO/Fe$_{3}$O$_{4}$ interface in series one is increasing with increasing NiO film thickness from 0.1nm to 1nm. In series two the NiO/Fe$_{3}$O$_{4}$ interface roughness is nearly constant at 0.80 ($\pm$ 0.15)nm. This is consistent with series one since the constant NiO film thickness in series two confirms approximately the NiO film thickness of the thickest NiO film in series one.\ Gatel et al. have grown among others Fe$_{3}$O$_{4}$/NiO bilayers on MgO(001) with a constant NiO film thickness of 66nm and different Fe$_{3}$O$_{4}$ film thicknesses ranging from 5 - 100nm. NiO and magnetite films have been grown using RF-sputtering at 700$^{\circ}$C and at 400 $^{\circ}$C substrate temperature, respectively. Gatel et al. state a NiO/Fe$_{3}$O$_{4}$ interface roughness of 0.7nm of their bilayers on MgO(001)[@Fertbilayer]. Thus, the obtained interface roughness is in good agreement with literature. Growth studies of single NiO films on MgO(001) at different preparation temperatures ranging vom 500$^{\circ}$C to 900$^{\circ}$C have shown that the surface roughness of NiO films becomes higher with increasing growth temperature from 0.2nm to 5.0nm [@CoNiObilayer]. In both studies the NiO films were grown using sputter deposition. Thus, lower growth temperatures reduce the interface roughness. In another study James et al. have grown NiO films with different thickness (20nm - 162nm) on MgO(001) using NO$_{2}$ assisted RMBE. They obtain an average NiO surface roughness of 0.35nm. The latter study confirms that our growth temperature should not be the reason for the rough NiO/Fe$_{3}$O$_{4}$ interface, since a growth temperature of 250$^{\circ}$ is sufficient to obtain smooth surfaces even at 162nm NiO film thickness. One possible explanation is that the interface roughness is increasing during the deposition of magnetite leading to a high NiO/Fe$_{3}$O$_{4}$ interface roughness since intermixing effects may play an important role at least at higher temperatures. However, regarding the development of the roughness in the first series, this should not be the case since the interface roughness is increasing with growing NiO film thickness.\ While the structural quality of the NiO films is constantly high in both series, the structural quality of the Fe$_{3}$O$_{4}$ film gets worse with increasing NiO film thickness. Until 24nm NiO film thickness the Fe$_{3}$O$_{4}$ exhibits an obvious Bragg peak with corresponding Laue fringes. Above this thickness only weak Fe$_{3}$O$_{4}$ Bragg peaks are visible and nearly no Laue fringes can be seen. This observation at the first series is confirmed by CTR analysis of the second series, where the film thickness of all NiO films is approximately 30nm. While at small Fe$_{3}$O$_{4}$ film thickness the Laue fringes of magnetite causes weak modulations of the NiO diffraction, no distinct Bragg peak corresponding to Fe$_{3}$O$_{4}$ appears between the NiO and MgO Bragg peaks with increasing Fe$_{3}$O$_{4}$ film thickness. The Laue fringes corresponding to magnetite do not reach a comparable strength to the Laue fringes of the first three samples of the first series. So we have to note that the NiO film thickness has an influence on the structural quality of the magnetite films. The reason for the bad structural ordering of the magnetite is obviously the high NiO/Fe$_{3}$O$_{4}$ interface roughness which is increasing with advancing NiO film thickness (Fig. \[fig:FeNiOS1roughness\]). Calculating the critical film thickness h$_{C}$ for the formation of misfit dislocations in magnetite using formula \[Gleichung\] we obtain h$_{C}$ = 105nm ($b = 0.2969\,nm$, $f = 0.3\,\%$, $\nu = 0.356$, $\lambda = 45^{\circ}$, $\alpha = 90^{\circ}$ ), which is approximately twice the film thickness of magnetite in this study. Since NiO grow pseudomorph on MgO adapting its lateral lattice constant, the misfit $f$ is respective to the MgO.\ In the first series of bilayers the vertical layer distance of magnetite features no dependence on NiO film thickness, which is reasonable since the magnetite film thickness of every bilayer is almost the same and the NiO films are pseudomorph to MgO. However, in the second series the vertical layer distance of Fe$_{3}$O$_{4}$ relaxes with increasing Fe$_{3}$O$_{4}$ film thickness until it reaches nearly the bulk value, although the magnetite have not reached the critical film thickness. Our previous studies on the growth of magnetite on MgO(001)[@BertiBernd; @Berti2heizen; @Berti1] confirm this observation. In these studies the strain of the magnetite films has relaxed with increasing film thickness and the critical film thickness is very small.\ Although the structural ordering of the magnetite gets worse, the roughness of the magnetite surface is relative low ranging from 0.1nm to 0.2nm. Thus, magnetite films seem to compensate the interface roughness. Only the bilayer with 33nm NiO film thickness in series one and the bilayer with 60nm magnetite film thickness in series two have a distinct higher roughness with 1nm. This can be attributed to the great progression of the structural degradation of the magnetite films. Thus, in order to get structural better magnetite films on thick NiO films, the preparation of the Fe$_{3}$O$_{4}$/NiO bilayers has to be improved. Therefore, the influence of the growth temperatures of NiO and Fe$_{3}$O$_{4}$ on the Fe$_{3}$O$_{4}$-NiO interface roughness have to be investigated. As mentiond above a growth temperature of 500$^{\circ}$C leads to a smooth NiO film surface. Withal you have to keep in mind that a too high growth temperature may lead to an intermixing of the NiO and Fe$_{3}$O$_{4}$ films, which will also affect the magnetic properties of the bilayers. conclusion ========== The detailed structural characterization of Fe$_{3}$O$_{4}$/NiO bilayers grown on MgO(001) substrates have shown that the quality of the NiO/Fe$_{3}$O$_{4}$ interface has a huge impact on the quality of the Fe$_{3}$O$_{4}$ films. While magnetite grows homogenously and smoothly on NiO films with up to 24nm thickness, the structural quality of the magnetite films gets distinctly worse with higher NiO film thickness. 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--- abstract: | Object classification with 3D data is an essential component of any scene understanding method. It has gained significant interest in a variety of communities, most notably in robotics and computer graphics. While the advent of deep learning has progressed the field of 3D object classification, most work using this data type are solely evaluated on CAD model datasets. Consequently, current work does not address the discrepancies existing between real and artificial data. In this work, we examine this gap in a robotic context by specifically addressing the problem of classification when transferring from artificial CAD models to real reconstructed objects. This is performed by training on ModelNet (CAD models) and evaluating on ScanNet (reconstructed objects). We show that standard methods do not perform well in this task. We thus introduce a method that carefully samples object parts that are reproducible under various transformations and hence robust. Using graph convolution to classify the composed graph of parts, our method significantly improves upon the baseline.\ – Code will be made publicly available on acceptance. – author: - 'Jean-Baptiste Weibel, Timothy Patten and Markus Vincze [^1] [^2]' bibliography: - 'IEEEexample.bib' title: '**Addressing the Sim2Real Gap in Robotic 3D Object Classification** ' --- Introduction ============ Whether to recommend the most suitable CAD model to a designer or to enable a service robot to decide where to place objects when tidying a room, 3D object classification is an essential task. Research in this area has greatly benefited from the wide availability of 3D CAD models as well as the accessibility of depth sensors, as this has established a large amount of data to apply geometric reasoning. Deep learning has profoundly transformed computer vision in recent years, and in particular, object classification has seen spectacular improvements. There has been steady interest for applying these methods for 3D data but introducing geometric reasoning in deep learning is not without its pitfalls. Typical deep learning approaches cannot handle rotated objects and real-world objects might be observed in arbitrary poses. Some methods use the statistical distribution of the data to transform the unknown object to a canonical pose for the deep network [@jaderberg_spatial_2015; @qi_pointnet:_2017]. However, inaccessible viewpoints, partial occlusions, supporting surfaces, and over- or under-segmentation observed in real-world data all contribute to modifying the statistical distribution of data that these methods expect, thus hindering their performance. Most deep networks also expect a fixed size input. This is achieved by rescaling, however, applying this to an occluded object can lead to a significant difference in the final fixed size representation. Consider how rescaling and centering a model airplane to the unit sphere and the same model with one wing missing would produce vastly different coordinates. The effect becomes prevalent when transferring from CAD models to real-world objects, as scale information is not available during training since most CAD models are scaleless. ![Creating reproducible object parts with similar representations on all sources of data enables better transfer from artificial to real objects.[]{data-label="fig:noise"}](res/overview.pdf) In this work, we develop a method for 3D object classification based on object parts that are reproducible under orientation or scale changes and can be defined for any level of occlusion, as shown in Figure \[fig:noise\]. It should be noted that we define parts as a *continuous subset* of the original object without any specific semantic meaning. Indeed, semantic parts such as a cup handle or a chair leg are also likely to be occluded and impossible to recover from the original objects, whereas our non-semantic parts can always be defined. Once parts are extracted, a rotation-invariant representation is computed through the use of a reproducible local reference frame. Finally, a graph-convolution based architecture is used to classify the graph of parts. In summary, our contributions are the introduction of: 1. a carefully designed angle-based sampling procedure that creates object parts reproducible under various rotation, scale and occlusion and, 2. a general graph-based learning architecture for classification that preserves the relevant properties of 3D object parts. These aspects allow us to achieve high performance when transferring from artificial to real-world data. In particular, our approach transfers from the ModelNet dataset [@MODELNET] to objects segmented from the reconstructed scenes in the ScanNet dataset [@dai2017scannet] better than previous methods. Our approach also significantly outperforms state-of-the-art methods when training and testing on ScanNet, which further demonstrates the value of careful part design and the inclusion of geometric priors. The remainder of the paper is organized as follows. Section \[sec:related\_work\] reviews the relevant literature. Section \[sec:method\] describes our approach for creating a graph of parts and subsequent learning for object classification. Section \[sec:experiments\] presents results of our method in comparison to existing methods, both on artificial data, real data, and when transferring from the former to the latter. Finally, Section \[sec:conclusion\] concludes and discusses future work. Related Work {#sec:related_work} ============ This section will first discuss approaches for geometric deep learning, and then approaches specifically designed for noisy data. Geometric Deep Learning ----------------------- Due to the way cameras perceive the world, 3D object classification is mostly concerned with surfaces (i.e. a 2D manifold in 3D space) rather than volumes. These surfaces are best represented with meshes, which is a specific type of graph. The branch of deep learning concerned with such data has recently received much attention from fields such as computer graphics as it focuses on triangle meshes. Graph convolutional models were originally designed for citation graphs and other graph-structured high-level data [@kipf2017semi], but more complex models have since developed for object meshes [@verma_feastnet:_2018; @SurfNet_2018_CVPR]. These models are typically developed for and tested on CAD models, which allows certain design choices such as using vertex coordinates. Unfortunately coordinates change significantly with rotation or rescaling and therefore these methods are not suitable for the unpredictability of real-world data. High performance is achieved by taking advantage of the progress made in 2D classification by using a collection of 2D views of the object. They either pool over views [@su15mvcnn; @10.1007/978-3-030-11015-4_49], apply more advanced schemes such as intelligently clustering before pooling [@wang2017dominant] or jointly learning the corresponding object pose [@kanezaki2018_rotationnet]. Beyond the power of such representations, they also take advantage of networks pre-trained on large scale 2D datasets. Once again, however, they have been designed and tested only on CAD models. Experiments suggest they heavily depend on the outline of the object in the 2D view, which is significantly affected by occlusions [@10.1007/978-3-030-11015-4_49]. Another option for 3D data is to apply 3D convolution on voxel grids [@MODELNET; @maturana2015voxnet]. 3D fixed grids are, by design, very sensitive to differences in object orientation and occlusion. It is possible to train with objects in a variety of poses [@SZB17a; @zeng20163dmatch], but this amounts to learning as many representations as orientations. Resultingly, the methods require a larger number of parameters to attain a given accuracy. Working with an additional dimension compared to images also leads to the parameter count increasing much faster. The trade-off between the coarseness of the grid and model complexity inherent to this representation led to the exploration of more powerful fixed 3D grid representations such as the signed distance function [@Park_2019_CVPR]. The limitation can also be counterbalanced by using multiple resolutions [@Riegler2017OctNet]. KD-tree-based models [@Klokov_2017_ICCV] push the multi-resolution idea further. Learning from this data structure achieves high accuracy but the KD-tree itself is sensitive to sensor noise and slight rotations, which makes it unsuitable for real sensor data. Depth sensors sample perceived surfaces and provide point cloud data that is used for direct learning. PointNet [@qi_pointnet:_2017] exploits this data type by learning from one point at a time before using a global max pooling layer to optimize globally over all the positions in the unit sphere, independently of the order of the points in the set. Research is still very active in this area with methods creating local kernels [@thomas2019KPConv; @Tat2018] or exploring novel classifiers [@DBLP:journals/corr/abs-1811-02191]. Robust 3D Classification ------------------------ The methods mentioned until now are evaluated using the coordinates of the models in the unit sphere. Objects extracted from a reconstructed scene, however, come in any orientation, which is often detrimental to the performance of methods that ignore the difference. Common approaches to this challenge are to train over various orientations as in [@maturana2015voxnet] or to use a spatial transformer layer [@jaderberg_spatial_2015] to learn an alignment of the objects as in [@qi_pointnet:_2017]. Another direction is to learn from rotation-invariant features as is explored in [@Chen_2019_CVPR; @8794432; @eppf18]. While a new representation is introduced in [@Chen_2019_CVPR], most work take inspiration from classical feature descriptors and explore the potential combination with deep networks. For example, [@eppf18] takes inspiration from the Point Pair Features (PPF) descriptor [@drost_model_2010] and use convolutional neural networks to learn a multi-dimensional histogram without losing the correlation between the features. The ESF descriptor [@wohlkinger_ensemble_2011] is another handcrafted descriptor and is designed specifically for classification. It concatenates histograms over those sampled features, which is given to an SVM to classify. [@8794432] combines some of the features from ESF with PPF to learn local structures that are later combined with a graph convolutional network. Learning from Object Parts for Robust 3D Object Classification {#sec:method} ==============================================================  Our method is developed for over- or under-segmented objects represented by manifold triangle meshes (a mesh is a manifold if each edge is connecting at most two triangles). The reason for using a mesh is that surfaces are preserved. Reconstruction methods generate that representation either directly [@Schreiberhuber2019] or by applying a post-processing step such as the marching cube algorithm [@InfiniTAM_ECCV_2016]. However, the method presented here could easily be adapted for dense point clouds by using nearest neighbor approaches to retrieve the neighborhood of each point. This section describes the proposed method. We first explain the object part sampling process and the part representation. We then outline the learning approach for the graph of object parts and the design of our graph convolution architecture. Creating object parts --------------------- ### Object parts sampling To transfer from artificial object models to real reconstructed data, object parts should be repeatable under varying orientation, occlusions, scale and point density. Scale-invariance forbids the use of Euclidean distance for sampling parts. To avoid sampling a part that would span through an object, and thus being significantly more sensitive to occlusions, parts are grown by following the surface of the object. The average angle between a triangle and its neighbors on a surface is used when deciding whether a neighboring triangle should be added to the part. Since reconstruction algorithms account for sensor noise and artificial data does not suffer from any random noise, high quality normals can be computed for both type of data. The angle between two neighboring normals is independent of scale and orientation of the object, and in a perfectly noiseless case, even independent of the surface sampling density. Object parts are then extracted by performing a Breadth-First Search (BFS) on the graph defined by the object mesh, or in other words, incrementally adding a one-ring neighborhood around the sampled part center. Due to the strong unpredictability of occlusions, part centers are randomly sampled. Centers are sampled so long as they do not belong to a previously sampled part. The search is stopped when the accumulated angle over the object part reaches a set threshold. The accumulated angle is computed from the average angle of each triangle, which is simply the average of the angle with each triangle neighbor. Reconstructed scenes do not provide perfectly smooth surfaces, therefore, we perform low-pass filtering on the normals defined by the triangles. Normals for all points are first computed by averaging the normals of each triangle they belong to. Then triangle normals are computed by averaging the normals of the three points. The resulting normals are smoother than the original mesh. ### Object part features The object part representation should maintain the properties of the sampling. In this work, we sample a fixed number of points from the object part to generate a fixed size representation from parts of varying sizes. Orientation-invariance is then achieved by defining a local reference frame (LRF). The center of the LRF is defined by the mean of a set of points and we propose two different orientations. The first design option is to perform Principal Component Analysis (PCA) with the set of points and use the eigenvectors as the LRF. The first and last eigenvectors (when ordered by decreasing eigenvalues) are kept and the direction of the last eigenvector is flipped in order to follow the average direction of the surface normals of the set of points. This guarantees a different LRF for concave and convex sets. The last vector is the cross-product of the first two vectors. This LRF provides a total orientation invariance and is referred to as PCA-LRF. The second option is to define the LRF based on the global vertical axis (Z-axis) and the component of the mean surface normal of the part that is orthogonal. This is no longer independent of the orientation of the object but only independent of the orientation of the object around the Z-axis. This local frame of reference is referred to as Z-LRF. Although it is only partially orientation-invariant, it offers a more informative representation. Since many objects have a small number of canonical poses (e.g. most bottles stand upright), it remains beneficial when tested on realistic data. All point sets are rescaled to the unit sphere to make the representation independent of the scale. In most experiments in Section \[sec:experiments\], we also add the average angle value as a feature to the point coordinates. When sampling the point, we use the average angle value of the triangle it belongs to. It slightly improves the accuracy without any extra computation overhead because it is computed during sampling. Finally, the graph is constructed by connecting parts that overlap. In other words, parts are connected if at least one triangle in each of the parts was sampled in the original object. Model Architecture ------------------ ### General architecture The architecture, as shown in Figure \[fig:system\], follows the PointNet model where each point extracted from a part is independently fed to the same convolutional neural network. In difference to the architecture of PointNet, the spatial transformer network is unnecessary as the points’ coordinates are already defined in a LRF. Instead, four 1D convolutional layers are used with a kernel size of one. A progressively increasing number of filters are then applied before a max-pooling layer to pool over each object part. Each of the layers includes a batch normalization step [@ioffe_batch_2015]. A weighted version of the maximum value (over the whole set) of a given filter is subtracted from each output as described in [@iclr_2017_dl_sets_pc]. ReLU is used as an activation function. The proposed model includes graph convolution layers inspired by the GCN model introduced in [@kipf2017semi]. This is a simplification of larger graph convolutional models because only first-degree neighbors are considered. As a result, the model cannot differentiate neighbors from each other. To address this, we introduce an attention model in our graph convolutions. ### Attention model The GCN model is made more powerful by introducing an attention mechanism. Instead of considering all neighbors as equal, the neighbors are weighted according to a criterion that is specific to the attention model chosen. Examples of attention models have been developed in [@verma_feastnet:_2018] and [@velickovic2018graph]. To further improve the representational power of a network, multiple attention heads are used for the same layer, and the attention heads output are concatenated at each layer. Introducing attention to the GCN model amounts to learning a valid coefficient to replace the normalization factor. We follow the model defined in [@velickovic2018graph] where the graph convolution layer becomes $$h_{v_i}^{l+1} = \sigma \left ( \sum_{j \in \mathcal{N}_i} \gamma_{ij} h_{v_j}^l W^l \right ),$$ where $h_{v_i}$ is the feature vector of the $i$-th vertex, $\sigma$ is the activation function, and $W$ is the parameter vector of the layer $l$. The coefficient $\gamma_{ij}$ is defined as $$\begin{split} \gamma_{ij} & = \text{softmax}(e_{ij}), \\ & = \frac{\text{exp}(e_{ij})}{\sum_{k \in \mathcal{N}_i} \text{exp}(e_{ik})}, \\ \end{split}$$ where $e_{ij} = \text{LeakyReLU}(a^T . [h_{v_i} W || h_{v_j} W]))$, $(\cdot)^T$ denotes the transpose operation, $\cdot||\cdot$ denotes the concatenation operation and $a$ is the vector of learned parameters for the attention. In order to respect the part connectivity, we add a bias matrix to the $e_{ij}$ term before applying the softmax in which disconnected nodes have a value of $-10^{9}$ and connected nodes have a value of $0$. ### Summarizing over object parts In this work, we are interested in predicting the object-level class. The object part representation described so far affords a number of different options for this task. The most straightforward option is to simply perform a max-pooling operation on the feature vectors of each part and then classifying the object (referred to as MaxPool). However, the classifier will be trained expecting all nodes and is therefore less likely to transfer well to real reconstructed data that have missing nodes. A second option is to predict one class per node and average all predictions into an object-level prediction (referred to as SingleNode). Both options are evaluated in Section \[sec:experiments\]. The single node prediction trained on artificial data still provides a representation that assumes perfect connectivity. We therefore propose one last option in which a proportion of nodes are randomly disconnected (except for self-connections). Experiments {#sec:experiments} =========== This section presents the experimental results. The first set of experiments compares our proposed approach to state-of-the-art methods for object classification on artificial data using the ModelNet dataset [@MODELNET]. The second set of experiments evaluates the transfer abilities from artificial data (ModelNet) to real-world data (objects extracted from the ScanNet dataset [@dai2017scannet]) in comparison to the baseline PointNet [@qi_pointnet:_2017]. We also evaluate in more depth the impact of the object part size and the connectivity of the object parts graph. Lastly, our method is evaluated against the PointNet architecture when training and testing on real-world data with objects extracted from the ScanNet dataset. Experimental setup ------------------ ### Implementation Our final model is described in Figure \[fig:system\]. We sample up to 32 parts per object and 250 points per part. The representation of each object part is fed through four 1D convolutional layers (kernel size one with filters 16, 16, 32 and 256) and max-pooling is applied over the whole set of points from the object part. The feature dimension is reduced with two convolutional layer (kernel size one) of size 256 and 128. The output is then passed to four graph convolutional layers. Each of these layers has eight attention heads with respectively 16, 16, 32 and 32 filters. The output of each attention head is concatenated at each layer before being fed to the next. Finally, the resulting features are max-pooled over all object parts and the result is passed to the classification layers that consists of two fully connected layers of size 128 and 256. When predicting over single nodes, the same classification layers are applied directly on each object part representation. ### Datasets Evaluation is performed on two datasets: The ModelNet dataset [@MODELNET] and the ScanNet dataset [@dai2017scannet] (1513 reconstructed rooms). We use both ModelNet40 (12311 CAD models split between 40 classes) and the ModelNet10 subset (4899 models in 10 classes). The second version of the annotation for ScanNet is used with the train/test/val split defined in the first version. Objects are extracted according to the annotation in the dataset. Afterwards the object classes are mapped to the ModelNet classes. Evaluation on artificial data ----------------------------- Table \[table:expMN40\] compares the performance of our method to state-of-the-art methods on the ModelNet [@MODELNET] dataset. It should be noted that the models of the dataset are non-manifold meshes. To apply our method, we project views of those objects and reconstruct them using a TSDF-based reconstruction. As a result, the object models differ slightly. For reference, we provide the accuracy of the PointNet architecture [@qi_pointnet:_2017] on both versions and observe a drop in accuracy of one percent for the object models created for our approach. **Method** **MN40** **Input** ------------------------------------------------- ---------- ------------- VoxNet [@maturana2015voxnet] 83.0 Voxel Grid KD-Networks [@Klokov_2017_ICCV] 91.8 KD-Tree MVCNN [@su15mvcnn] 90.1 Views MVCNN-New [@10.1007/978-3-030-11015-4_49] **95.0** Views 3DmFV-Net [@ben20183dmfv] 91.6 Point Cloud 3DCapsules [@DBLP:journals/corr/abs-1811-02191] 92.7 Point Cloud PointNet [@qi_pointnet:_2017] 89.2 Point Cloud **\***PointNet [@qi_pointnet:_2017] 88.1 Point Cloud **\***Ours (PCA-LRF) 86.9 Mesh **\***Ours (Z-LRF) 89.4 Mesh : Classification accuracy on the ModelNet40 dataset [@MODELNET] (**\*** indicates that the method was evaluated on the reconstructed models)[]{data-label="table:expMN40"} This experiment shows that despite being specifically designed with real-world constraints in mind, our method still shows competitive results on artificial data. The results presented in Table \[table:expMN40\] correspond to our method with a max pooling and trained with the average angle values as an included feature. Table \[table:expMN10design\] shows the results of the addition of the average angle value as an extra feature. We see that performance slightly improves in all conditions without adding any computation as it is already calculated during the sampling process. Furthermore, the SingleNode type of pooling (i.e. predicting a class for each object part and averaging the prediction over the object) gives similar results to MaxPool on artificial data. All future results include the average angle as a feature and use the Z-LRF. **Pooling** **LRF** **Avg ang.** **Acc.** ------------- --------- -------------- ---------- MaxPool PCA-LRF 85.6 MaxPool PCA-LRF 86.3 MaxPool Z-LRF 87.2 MaxPool Z-LRF 89.2 SingleNode Z-LRF 89.6 : Evaluation of design choices on ModelNet10[]{data-label="table:expMN10design"} ![Illustration of the noisy objects extracted from the ScanNet dataset. Left to right: monitor (very noisy surface), potted plant (strong occlusions and many disconnected parts), cup (small object) and table (object on top merged with it).[]{data-label="fig:objects"}](res/scannet_objects.png) Evaluation of the gap between real and artificial data ------------------------------------------------------ This section evaluates the gap when training on artificial data (objects from ModelNet) and testing on real-world data (segmented objects from ScanNet). This is a difficult task because of the domain shift as well as some important characteristics of the ScanNet dataset (see Figure \[fig:objects\]). ScanNet was first and foremost designed for semantic segmentation, which is more concerned with large-scale structures. Additionally, since it represents natural environments, the dataset is strongly imbalanced. Chair is a highly dominating class, which is seen by the performance of the “chair predictor” baseline (i.e. always predicting the chair class) in Table \[table:expSNToMN40\]. As such, class accuracy is a more relevant metric than overall accuracy. Moreover, most structures tend to be oversmoothed by the reconstruction algorithm. This is a side-effect of the reconstruction algorithm that tries to reconcile various noisy measurements. Subtle differences also exist between ScanNet classes and ModelNet classes (e.g. a pack of bottles in ScanNet is mapped to the bottle class, whereas that class only contains single standing bottle in ModelNet). Finally, the segmentation is often inaccurate for smaller objects. This is due to the fact that scenes are first oversegmented and then clusters are annotated. Therefore, many extracted objects still include points from the surrounding elements. All these factors contribute to making this a very challenging dataset for the task. As shown in Table  \[table:expSNToMN40\], the PointNet model fails to transfer to the real-world domain, scoring only 2.2% accuracy and 3.3% class accuracy. In comparison, our model achieves much higher accuracy and class accuracy with scores of 34.6% and 19.1% respectively. The result when always predicting the chair class not only demonstrates the imbalance of the dataset but also provides a general reference for evaluating performance. The reason to focus specifically on the PointNet architecture, besides its versatility and proven reliability in various contexts [@Aoki_2019_CVPR; @Ge_2018_CVPR], is that it is the closest architecture to our approach. Considering that our object part representation could easily be swapped out in our pipeline, a comparison to PointNet allows for a fairer comparison of our contribution. For our method, the best results are achieved by increasing the sampling threshold by a factor of two compared to training. Also, SingleNode pooling is used with a disconnection rate of 75% during training. In the next section, we evaluate those design choices in more detail. **Acc.** **Cls Acc.** ----------------- ---------- -------------- Chair Predictor **36.2** 2.5 PointNet 2.2 3.3 Ours 34.6 **19.1** : Evaluation when transferring from ModelNet40 to ScanNet[]{data-label="table:expSNToMN40"} ### Influence of the part size in the transfer One significant parameter in the transfer performance is the threshold set for the part sampling algorithm. ScanNet scenes tend to be oversmoothed, which affects the angle-based sampling procedure for objects that have large flat surfaces in artificial models. Also, scenes are reconstructed at a constant density, which means that smaller objects have a smaller density than bigger objects. Combined with the low level of noise, the right trade-off needs to be found for increasing the threshold used in training and testing because classes react differently. Figure \[fig:size\] shows the impact of the sampling threshold on the accuracy when using the mapping from ModelNet40 classes to the ScanNet objects. The best threshold typically increases with the increase of the average size of the class. For small objects, such as “bowl”, the best threshold is close to the training value. For medium-sized objects, such as “toilet”, the best threshold is between three and fours times the original value. For very large objects, such as “piano”, the best best threshold is up to eight times. Larger objects simply have more triangles in the mesh. The threshold is reached much faster due to the noise, therefore, they require a larger threshold in order to reproduce parts similar to those observed during training. ![Accuracy in percent for various multiplicative factors applied to the sampling threshold (ModelNet40 mapping is used, SingleNode model trained with 75% Disconnect).[]{data-label="fig:size"}](res/neigh_size.png) ### Influence of the connectivity and pooling Table \[table:expTransferDesign\] shows the impact of different pooling strategies on the transfer performance. It also shows the impact of randomly disconnecting some object parts in the object part graph during training in order to better approximate occluded objects. The experiments are performed using the mapping from ModelNet10 to ScanNet. The significant increase in accuracy is due to the fact that the ModelNet10 mapping contains most of the well-reconstructed objects because classes of ModelNet10 corresponds to bigger structures. Clearly, performance improves with disconnection considered. Applying 0% has low accuracy because it does not model the occlusion. At 100% the performance is also low because this over-estimates the level of occlusion. The best compromise is achieved with 75% for the ScanNet dataset. The disconnect experiments are not applied to the MaxPool setup, as it would still take into account each part when max-pooling, thus failing to simulate occlusions. **Pooling** **Disconnect (%)** **Acc.** **Cls Acc.** ----------------- -------------------- ---------- -------------- MaxPool - 44.1 42.35 SingleNode 0 50.1 38.3 SingleNode 50 62.2 39.0 SingleNode 75 **62.9** **43.2** SingleNode 100 54.5 33.2 Chair predictor - 56.0 10.0 : Evaluation of design choices for transferring to ScanNet (using the ModelNet10 mapping)[]{data-label="table:expTransferDesign"} Evaluation on real data ----------------------- The large size of ScanNet makes it feasible to train methods on real-world data instead of artificial data. This is helpful because it can be used to establish an upper bound for the transfer to this dataset. The results in Table \[table:expSNeval\] are significantly higher than when transferring, which implies the existence of occlusion consistency for a given class. We, however, conjecture that this only holds for larger structures but not for smaller objects, such as household items. The results additionally show that an advantageous side-effect of our design is the ability to better learn from noisy data. Our method achieves an accuracy of 85.2% and class accuracy of 62.8%. The best results are achieved using the Z-LRF frame of reference. Prediction is performed for each node and averaged, and training is performed with a 75% random chance of disconnecting two neighbors. In comparison, the PointNet architecture achieves much lower scores of 73.6% accuracy and 52.3% class accuracy, which is approximately 10% less than our method. We also performed experiments with PointNet trained on objects rescaled to the unit sphere to prevent the method from taking advantage of scale information. The significant decrease in this case suggests that PointNet finds it more difficult to establish consistent shapes. This observation supports our argument that even though objects have inconsistent shapes, they always have consistent parts. **Acc.** **Cls Acc.** -------------------------- ---------- -------------- Chair Predictor 36.2 2.5 PointNet 73.6 52.3 PointNet (rescaled obj.) 53.4 17.2 Ours (Z-LRF) **85.2** **62.8** : Evaluation when training and testing on ScanNet (using the ModelNet40 mapping)[]{data-label="table:expSNeval"} Conclusion {#sec:conclusion} ========== This paper addressed the important robotics task of object classification from 3D data. We present an approach that transfers to real reconstructed objects when trained on clean CAD models only. Results show that our approach significantly outperforms state-of-the-art methods while maintaining competitive performance when training and testing on CAD models. The performance increase in the transfer is achieved through sampling object parts that are reproducible under rotation, occlusion and scale in combination with a graph-based deep learning architecture. Learning with a rotation-invariant and scale-invariant representation of parts enables objects to be recognized with significant portions of missing data. The next step for this work is to create a large-scale dataset focused on objects rather than scenes to support further research in the task of transferring from widely available CAD models to noisy real-world data. This will be particularly important for the task of retrieving CAD models from candidate segments in noisy data. This will be highly relevant for service robotics systems that have to prepare for a variety of contexts by building a general world knowledge when deployed in user homes. [^1]: The research leading to these results has received funding from the Austrian Science Foundation (FWF) under grant agreement No. I3968-N30 HEAP and No. I3969-N30 InDex [^2]: All authors are with the Vision for Robotics Laboratory, Automation and Control Institute, TU Wien, 1040 Vienna, Austria. [{weibel, patten, vincze}@acin.tuwien.ac.at]{}

--- abstract: 'A graph drawn in the plane with straight-line edges is called a geometric graph. If no path of length at most $k$ in a geometric graph $G$ is self-intersecting we call $G$ $k$-locally plane. The main result of this paper is a construction of $k$-locally plane graphs with a super-linear number of edges. For the proof we develop randomized thinning procedures for edge-colored bipartite (abstract) graphs that can be applied to other problems as well.' author: - | Gábor Tardos[^1]\ Rényi Institute, Budapest, Hungary and\ School of Computing Science, Simon Fraser University, Burnaby, BC\ [tardos@renyi.hu]{} title: Construction of locally plane graphs with many edges --- Introduction ============ A [*geometric graph*]{} $G$ is a straight-line drawing of a simple, finite (abstract) graph $(V,E)$, i.e., we identify the vertices $x\in V$ with distinct points in the Euclidean plane, and we identify any edge $\{x,y\}\in E$ with the straight line segments $xy$ in the plane. We assume that the edge $xy$ does not pass through any vertex of $G$ besides $x$ and $y$. We call $(V,E)$ the abstract graph [*underlying*]{} $G$. We say that the edges $e_1,e_2\in E$ [*cross*]{} if the corresponding line segments cross each other, i.e., if they have a common interior point. We say that a subgraph of $G$ is [*self-intersecting*]{} if it contains a pair of crossing edges. Geometric graphs without crossing edges are plane drawings of planar graphs: they have at most $3n-6$ edges if $n\ge3$ is the number of vertices. Avital and Hanani [@AH], Erdős, and Perles initiated in the mid 1960s the systematic study of similar questions for more complicated [*forbidden configurations*]{}: Let $H$ be set of forbidden configurations (geometric subgraphs). What is the maximal number of edges of an $n$ vertex geometric graph not containing any configuration belonging to $H$? This problem can be regarded as a geometric version of the fundamental problem of extremal graph theory: What is the maximum number of edges that an abstract graph on $n$ vertices can have without containing subgraphs of a certain kind. Many questions of the above type on geometric graphs have been addressed in recent years. In a number of papers linear upper bounds have been established for the number of edges of a graph, under various forbidden configurations. They include the configurations of three pairwise crossing edges [@AAPPS], four pairwise crossing edges [@Ack], the configurations of an edge crossed by many edges [@PT], or even two large stars with all edges of one of them crossing all edges of the other [@TT]. For a constant number of 5 or more pairwise crossing edges Pavel Valtr has the best result [@Valtr]: a geometric graph on $n$ vertices avoiding this configuration has $O(n\log n)$ edges but no construction is known with a super linear number of edges. Adam Marcus and the present author [@MT] building on an earlier result of Pinchasi and Radoičić [@PR] prove an $O(n^{3/2}\log n)$ bound on the number of edges of an $n$ vertex geometric graph not containing self-intersecting cycles of length four. No construction is known beating the $O(n^{3/2})$ edges an abstract graph having no cycles of length four can have. For surveys on geometric graph theory, consult [@P1], [@P2] and [@PRT]. In this paper we consider forbidding self-intersecting paths. For $k\ge3$ we call a geometric graph [*$k$-locally plane*]{} if it has no self-intersecting subgraph (whose underlying abstract graph is) isomorphic to a path of length at most $k$. Pach et al. [@PPTT] consider $3$-locally plane graphs, i.e., the case of geometric graphs with no self-intersecting paths of length three. They prove matching lower and upper bounds of $\Theta(n\log n)$ on the maximal number of edges of a $3$-locally plane graph on $n$ vertices. We extend the the lower bound result of [@PPTT] by considering self-intersecting drawings of longer paths as forbidden configurations. Technically $k$-locally plane graphs are defined by forbidding self intersecting paths of length $k$ [*or shorter*]{}, but forbidding only self-intersecting paths of length [*exactly $k$*]{} would lead to almost the same extremal function. Indeed, one can delete at most $nk$ edges from any graph on $n$ vertices, such that all the non-zero degrees in the remaining graph are larger than $k$. This ensures that all shorter paths can be extended to a path of length $k$. It is possible, but not likely, that if one only forbids paths of length $k$ with [*the first and last edges crossing*]{} significantly higher number of edges is achievable. For even $k$ a geometric graph is $k$-locally plane if and only if the $k/2$-neighborhood of any vertex $x$ is intersection free. Note that this requirement is much stronger than the similar condition on abstract graphs, namely that the $k/2$ neighborhood of any point is planar. One can construct graphs with girth larger than $k$ and $\Omega(n^{k\over k-1})$ edges. In such a graph the $k/2$-neighborhood of any vertex is a tree, still by [@PPTT] the graph does not even have $3$-locally plane drawing. Extending the lower bound result in [@PPTT] we prove in Theorem \[15\] that for arbitrary fixed $k\ge3$ there exist $k$-locally plane graphs on $n$ vertices with $\Omega(n\log^{(\lfloor k/2\rfloor)}n)$ edges. Here $\log^{(t)}$ denotes $t$ times iterated logarithm and the hidden constant in $\Omega$ depends on $k$. Given two arbitrarily small disks in the plane we can even ensure that all edges of the constructed graph connect a vertex from the first disk with another vertex from the second. This ensures that all the edges of the constructed geometric graph are arbitrarily close to each other in length and direction. In the view of the author this makes the existence of a high average degree (or for that matter high minimum degree) 100-locally plane graphs even more surprising. As a simple corollary we can characterize the abstract graphs $H$ such that any geometric graph having no self-intersecting subgraph isomorphic to $H$ has a linear number of edges. These graphs $H$ are the forests with at least two nontrivial components. To see the linear bound for the number of edges of a geometric graph avoiding a self-intersecting copy of such a forest $H$ first delete a linear number of edges from an arbitrary geometric graph $G$ until all non-zero degrees of the remaining geometric graph $G'$ are at least $|V(H)|$. If $G'$ is crossing free the linear bound of the number of edges follows. If you find a pair of crossing edges in $G'$ they can be extended to a subgraph of $G'$ isomorphic to $H$. On the other hand, if $H$ contains a cycle, then even an abstract graph avoiding it can have a super-linear number of edges. If $H$ is a tree of diameter $k$, then a $k$-locally plane geometric graph has no self-intersecting copy of $H$. Notice that the extremal number of edges in this case (assuming $k>2$) is $O(n\log n)$ by [@PPTT], thus much smaller than the $\Omega(n^\alpha)$ edges ($\alpha>1$) for forbidden cycles. The main tool used in the proof of the above result is a randomized thinning procedure that takes a $d$ edge colored bipartite graph of average degree $\Theta(d)$ and returns a subgraph on the same vertex set with average degree $\Theta(\log d)$ that does not have a special type of colored path (walk) of length four. The procedure can be applied recursively to obtain a subgraph avoiding longer paths of certain types. We believe this thinning procedure to be of independent interest. In particular it can be used to obtain optimal 0-1 matrix constructions for certain avoided submatrix problems, see the exact statement in Section \[s4\] and the details in [@T]. In Section \[s2\] we define two thinning procedures for edge colored bipartite graphs and prove their main properties. This is the most technical part of the paper. While these procedures proved useful in other setting too (and the author finds the involved combinatorics appealing) this entire section can be skipped if one reads the definition of $k$-flat graphs (the two paragraphs before Lemma \[10\]) and is willing to accept Corollary \[corol\] at the end of the section (we also use the simple observation in Lemma \[11\]). In fact, in order to understand the main ideas behind the main result of this paper it is recommended to skip Section \[s2\] on the first reading and to go straight to Section \[s3\] where we use Corollary \[corol\] to construct locally plane graphs with many edges. In Section \[s4\] we comment on the optimality of the thinning procedures and have some concluding remarks. Thinning {#s2} ======== In this section we state and prove combinatorial statements about edge colored abstract graphs, i.e., we do not consider here geometric graphs at all. The connection to locally plane geometric graphs will be made clear in Section \[s3\]. A [*bipartite graph*]{} is a triple $G=(A,B,E)$ with disjoint vertex sets $A$ and $B$ (called [*sides*]{}) and edge set $E\subseteq A\times B$. In particular, all graphs considered in this paper are [*simple*]{}, i.e., they do not have multiple edges or loops. The edge connecting the vertices $x$ and $y$ of $G$ is denoted by $xy$ or $(x,y)$. The latter notation is only used if $x\in A$ and $y\in B$. By a [*$d$-edge coloring*]{} of a graph we mean a mapping $\chi:E\to\{1,2,\ldots d\}$ such that adjacent edges receive different colors. When we do not specify $d$ we call such coloring a [*proper edge coloring*]{} but we always assume that the “set of colors” are linearly ordered. The degree of any vertex in $G$ is at most the number $d$ of colors, and our results are interesting if the average degree is close to $d$. Unless stated otherwise the subgraphs of an edge colored graph are considered with the inherited edge coloring. Our goal is to obtain a subgraph of $G$ with as many edges as possible without containing a certain type of colored path or walk. Heavy paths ----------- A simple example of the above concept is the following. We call a path $P=v_0v_1v_2v_3$ of length $3$ [*heavy*]{} if $v_0\in B$ and the colors $c_1=\chi(v_0v_1)$, $c_2=\chi(v_1v_2)$, $c_3=\chi(v_2v_3)$, satisfy $c_2<c_1\le c_3$. The next lemma describes a thinning procedure that gets rid of heavy paths. Although we do not need this lemma in our construction, we present it as a simple analogue of our results for more complicated forbidden walks. \[0\] Let $G=(A,B,E)$ be a bipartite graph with a proper edge coloring $\chi:E\to\{1,2,\ldots d\}$. Then there exists a subgraph $G'=(A,B,E')$ of $G$ with $|E'|\ge|E|/(3\lceil\sqrt d\,\rceil)$ that does not contain heavy paths. The constant $3$ in the lemma could be replaced by the base of the natural logarithm. Notice that if $G$ had average degree $\Theta(d)$, then the average degree of $G'$ is $\Omega(\sqrt d)$. Let $t=\lceil\sqrt d\,\rceil$ and select a uniform random value $i_y\in\{1,2,\ldots,t\}$ independently for each vertex $y\in B$. We say that an edge $e\in E$ is of [*class*]{} $\lceil \chi(e)/t\rceil$. We call an edge $e=(x,y)\in E$ [*eligible*]{} if its class is $i_y$. Let the subgraph $G'=(A,B,E')$ consist of those eligible edges $e=(x,y)\in E$ for which there exists no other eligible edge $(x,y')\in E$ of the same class. Note that the words “class” and “eligible” will be used in a different meaning when defining the two thinning procedures in the next subsection. By the construction, all edges incident to a vertex $x\in A$ have different classes and all edges incident to a vertex $y\in B$ have the same class. Let $e_1$, $e_2$ and $e_3$ form a path in $G'$ starting in $B$. Then $e_2$ and $e_3$ are of the same class, while the class of $e_1$ is different. For the colors $c_i=\chi(e_i)$ this rules out the order $c_2<c_1\le c_3$. Thus, $G'$ does not contain a heavy path. Note that another order, $c_3\le c_1<c_2$ is also impossible. The number of edges in $G'$ depends on the random choices we made. Any edge $(x,y)\in E$ is eligible with probability $1/t$, and this is independent for all the edges incident to a vertex $x\in A$. As edges of a fixed color form a matching, there are at most $t$ edges of any given class incident to $x$. Thus, we have $$\P[(x,y)\in E']\ge\frac{(1-1/t)^{t-1}}t>\frac1{3t}.$$ The expected number of edges in $G'$ is $$\E[|E'|]\ge\frac{|E|}{3t}.$$ It is possible to choose the random variables $i_y$ so that the size of $E'$ is at least as large as its expected value. This proves the lemma. Fast and slow walks ------------------- Next we turn to more complicated forbidden subgraphs. For motivation we mention that self-crossing paths of length $4$ in the 3-locally plane graphs of [@PPTT] (considered with their natural edge coloring) are exactly the [*fast walks*]{} (to be defined below). For technical reasons, it will be more convenient to consider walks, i.e., to permit that a vertex is visited more than once, but we will not allow [*backtracking*]{}, i.e., turning back on the same edge immediately after it was traversed. Thus, for us a [*walk*]{} of length $k$ is a sequence $v_0,v_1,\ldots,v_k$ of vertices in the graph such that $v_{i-1}v_i$ is an edge for $1\le i\le k$ and $v_{i-2}\ne v_i$ for $2\le i\le k$. The [*$\chi$-coloring*]{} (or simply coloring) of this walk is the sequence $(\chi(v_0v_1),\chi(v_1v_2),\ldots,\chi(v_{k-1}v_k))$ of the colors of the edges of the walk. If $\chi$ is a proper edge coloring, then any two consecutive elements of the coloring sequence are different. We use $\log$ to denote the binary logarithm. We introduce the notation $P(a,b)$ for two non-equal strings $a,b\in\{0,1\}^t$ to denote the first position $i\in\{1,2,\ldots,t\}$, where $a$ and $b$ differ. We consider the set $\{0,1\}^t$ to be ordered lexicographically, i.e., for $a,b\in\{0,1\}^t$ we have $a<b$ if $a$ has $0$ in position $P(a,b)$ (and thus $b$ has $1$ there). The following trivial observation is used often in this paper. We state it here without a proof. \[1\] Let $t\ge1$ and let $a$, $b$ and $c$ be distinct binary strings of length $t$ with $P(a,b)<P(a,c)$. We have $P(b,c)=P(a,b)$. Furthermore $a>b$ implies $c>b$, and $a<b$ implies $c<b$. A walk of length $4$ with coloring $(c_1,c_2,c_3,c_4)$ is called a [*fast walk*]{} if $c_2<c_3<c_4\le c_1$. Note that a fast walk may start in either class $A$ or $B$. We call a walk of length $4$ a [*slow walk*]{} if it starts in the class $B$ and its coloring $(c_1,c_2,c_3,c_4)$ satisfies $c_2<c_3<c_4$ and $c_2<c_1\le c_4$. Note that either the color $c_1$ or $c_3$ can be larger in a slow walk, or they can be equal. The two [*thinning procedures*]{} below find a random subgraph of an edge-colored bipartite graph. One is designed to avoid slow walks, the other is designed to avoid fast walks. [**Lexicographic thinning**]{} Let $G=(A,B,E)$ be a bipartite graph and let $\chi:E\to\{1,\ldots,d\}$ be a proper edge coloring with $d\ge2$. [*Lexicographic thinning*]{} is a randomized procedure that produces a subset $E'\subseteq E$ of the edges and the corresponding subgraph $G'=(A,B,E')$ of $G$ as follows: Let $t=\left\lceil\log d\over2\right\rceil+1$. Let $H$ be the set of triplets $(a,i,z)$, where $a\in\{0,1\}^t$, $i\in\{2,3,\ldots,t\}$, $z\in\{1,2,3,\ldots2^i\}$, and $a$ has $0$ in position $i$. Straightforward calculation gives that $|H|=2^{2t}-2^{t+1}\ge2d$. We order $H$ lexicographically, i.e., $(a,i,z)<(b,j,s)$ if $a<b$, or $a=b$ and $i<j$, or $(a,i)=(b,j)$ and $z<s$. Consider the following random function $F:\{1,\ldots,d\}\to H$. We select uniformly at random the value $F(1)=(a,i,z)\in H$ with the property that the first bit of $a$ is $0$. We make $F(2)$ to be the next element of $H$ larger than $F(1)$, and in general $F(k)$ is the next element of $H$ larger than $F(k-1)$ for $2\le k\le d$. As $|H|\ge2d$ and $F(1)$ is chosen from the first half of $H$, this defines $F$. In what follows we simply identify the color $k$ with the element $F(k)\in H$ without any reference to the function $F$. We say that $(a,i,z)\in H$ and any edge with this color is of [*class*]{} $a$ and [*type*]{} $i$, while $z$ will play no role except in counting how many values it can take. We choose an independent uniform random value $a_x\in\{0,1\}^t$ for each vertex $x\in A\cup B$. Let $e=(x,y)\in E$ be an edge of class $a$ and type $i$. We say that $e$ is [*eligible*]{} if $a=a_y<a_x$ and $P(a,a_x)=i$. Let the subgraph $G'=(A,B,E')$ contain those edges $e\in E$ that are eligible but not adjacent to another eligible edge $e'$ of the same type as $e$. [**Reversed thinning**]{} Let $G=(A,B,E)$ be a bipartite graph and let $\chi:E\to\{1,\ldots,d\}$ be a proper edge coloring with $d\ge2$. [*Reversed thinning*]{} is a randomized procedure that produces a subset $E'\subseteq E$ of the edges and the corresponding subgraph $G'=(A,B,E')$ of $G$. Reversed thinning is almost identical to the lexicographic thinning, the only difference is in the ordering of the set $H$. We define $t$ and $H$ as in the case of lexicographic thinning. Recall that $H$ is the set of triplets $(a,i,z)$ where $a\in\{0,1\}^t$, $i\in\{2,3,\ldots,t\}$, $z\in\{1,2,\ldots2^i\}$, and $a$ has $0$ in position $i$. We still order $\{0,1\}^t$ lexicographically, but now we reverse the lexicographic order of $H$ in the middle term $i$. That is, we have $(a,i,z)<(b,j,s)$ if $a<b$, or $a=b$ and $i>j$ or $(a,i)=(b,j)$ and $z<s$. We define the function $F:\{1,\ldots,d\}\to H$, the [*types*]{} and [*classes*]{} of colors and edges, [*eligible*]{} edges and the subset $E'$ of edges the same way as for the lexicographic thinning, but using this modified ordering of $H$. Note that we associated a type in $\{2,\ldots,t\}$ and a class in $\{0,1\}^t$ to each edge in either procedure and they satisfy that - an edge with a smaller class has smaller color; - among edges of equal class an edge with smaller type has smaller color in the case of lexicographic thinning and it has larger color in the case of reversed thinning; - among the edges incident to a vertex at most $2^i$ have the same class $a$ and the same type $i$. Proving most of the properties of the thinning procedures this is all we need to know about how classes and types are associated to the edges and we could use a deterministic scheme for $F$. But for Lemma \[5\] we need that all types are well represented and the randomization in the identification function $F$ (as well as the dummy first bit of the class) is introduced to ensure this on the average. This randomization is not needed if one assumes all color classes have roughly the same size. The next lemmas state the basic properties of the thinning procedure. Lemma \[2\] lists common properties of the two procedures, while Lemmas \[3\] and \[4\] state the result of the thinning satisfies its “design criteria” avoiding slow or fast walks. Finally Lemma \[5\] shows that enough edges remain in the constructed subgraphs on average. Note that Lemmas \[3\] and \[4\] are special cases of the more complex Lemmas \[7\] and \[8\] proved independently. We state and prove the simple cases separately for clarity, but these proofs could be skipped. \[2\] Let $G=(A,B,E)$ be a bipartite graph with a proper edge coloring $\chi:E\to\{1,\ldots,d\}$. If $G'=(A,B,E')$ is the result of either the lexicographic or the reversed thinning then we have a) : Adjacent edges in $G'$ have distinct types. b) : If two edges of $G'$ meet in $B$, they have the same class. c) : Suppose two distinct edges $e$ and $e'$ of $G'$ meet in $A$. Let their classes and types be $a$, $a'$ and $i$, $i'$, respectively. If $i<i'$ then $a<a'$ and $P(a,a')=i$. d) : $G'$ has no heavy path. The definition of $E'$ immediately gives a). For b) note that all eligible edges incident to $y\in B$ have $a_y$ for class. For c) let $x\in A$ be the common vertex of the two edges and apply Lemma \[1\] for $a_x$, $a$, and $a'$. Finally d) follows since if a walk of $G'$ starts in $B$ then its coloring $(c_1,c_2,c_3)$ must satisfy that $c_1$ and $c_2$ have different class by c) but $c_2$ and $c_3$ have the same class by b), so $c_2<c_1\le c_3$ is impossible. \[3\] Let $G=(A,B,E)$ be a bipartite graph with a proper edge coloring. Lexicographic thinning produces a subgraph $G'$ with no slow walk. Suppose $v_0v_1v_2v_3v_4$ is a walk in $G'$ starting at $v_0\in B$ and let its coloring be $(c_1,c_2,c_3,c_4)$. Assume $c_1>c_2<c_3<c_4$. We show that $c_1>c_4$, so this walk is not slow. By Lemma \[2\]/a,b, as $c_2$ and $c_3$ are colors of edges incident to $v_2\in B$, they have the same class, but they have different types: $c_2=(a,i,z)$, $c_3=(a,j,s)$ with $i\ne j$. We use lexicographic ordering, so $c_2<c_3$ implies $i<j$. Both $c_1$ and $c_2$ are colors of edges incident to $v_1\in A$, so by Lemma \[2\]/c their classes are different. Since $c_1>c_2$ we have $b>a$ for the class $b$ of $c_1$. Still by Lemma \[2\]/c $P(a,b)=i$. Similarly, $c_3$ and $c_4$ are colors of distinct edges in $E'$ incident to $v_3\in A$, so they have different classes. As $c_3<c_4$ we have $c>a$ for the class $c$ of $c_4$. We have $P(a,c)=j$. By Lemma \[1\] we have $b>c$. This proves $c_1>c_4$ as claimed. \[4\] Let $G=(A,B,E)$ be a bipartite graph with a proper edge coloring. Reversed thinning produces a subgraph $G'$ with no fast walk. Suppose $v_0v_1v_2v_3v_4$ is a walk in $G'$ with coloring $(c_1,c_2,c_3,c_4)$. Assume $c_1>c_2<c_3<c_4$. We show that $c_1<c_4$, so this walk is not fast. First assume the walk starts at $v_0\in A$. As $G'$ does not contain a heavy path, $v_3v_2v_1v_0$ is not heavy, so $c_1<c_3$. This implies $c_1<c_4$ as claimed. Now assume $v_0\in B$. Just as in the proof of the previous lemma, $c_2$ and $c_3$ are colors of edges incident to $v_2\in B$, so they have the same class, but they have different types: $c_2=(a,i,z)$, $c_3=(a,j,s)$ with $i\ne j$. We use the reversed ordering, so $c_2<c_3$ implies $i>j$. Both $c_1$ and $c_2$ are colors of edges incident to $v_1\in A$, so their classes are different. Since $c_1>c_2$ we have $b>a$ for the class $b$ of $c_1$ and $P(a,b)=i$. Similarly, $c_3$ and $c_4$ are colors of distinct edges in $E'$ incident to $v_3\in A$, so they have different classes. As $c_3<c_4$ we have $c>a$ for the class $c$ of $c_4$ and $P(a,c)=j$. By Lemma \[1\] we have $c>b$. This proves $c_1<c_4$ as claimed. Below we estimate the number of edges in $E'$. Recall that both thinning procedures are randomized. We can show that the subgraphs they produce have a large expected number of edges. We did not make any effort to optimize for the constant in this lemma. \[5\] Let $G=(A,B,E)$ be a bipartite graph with a $d$ edge coloring. Let $G'=(A,B,E')$ be the result of either the lexicographic or the reversed thinning. We have $$\E[|E'|]\ge{t-1\over240d}|E|\ge{\log d\over480d}|E|.$$ We compute the probability for a fixed edge $e=(x,y)\in E$ to end up in $E'$. For this we break down the random process producing $E'$ into three phases. In the first phase we select $F$. With $F$ the color $\chi(e)$ is identified with an element of $H$, most importantly, the type of $e$ is fixed. In the second phase we select $a_x$ and $a_y$ uniformly at random. These choices determine if $e$ is eligible. If $e$ is not eligible then $e\notin E'$. So in the third phase we consider $F$, $a_x$, and $a_y$ fixed and assume $e$ is eligible. We select the random values $a_z$ for vertices $z\ne x,y$. This effects if other edges are eligible and if $e\in E'$. Here is the detailed calculation: Let $e\in E$ have the color $\chi(e)=k\in\{1,2,\ldots, d\}$. The choice of $F$ in the first phase determines $F(k)=(a,i,z)\in H$. By the construction of $F$, if we call $a'$ the last $t-1$ bits of $a$ then $(a',i,z)$ is uniformly distributed among all its possible values. In particular, the probability that $e$ becomes a type $i$ edge is exactly $$\P[e\hbox{ is of type }i]=\frac{2^{t+i-2}}{2^{2t-1}-2^t} =\frac{2^i}{2^{t+1}-4}.$$ For phase two we consider the function $F$ identifying colors with elements of $H$ fixed. Consider an edge $e=(x,y)$ of color $(a,i,z)\in H$. This edge is eligible if $a_y=a<a_x$ and $P(a,a_x)=i$. This determines $a_y$ and the first $i$ bits of $a_x$. Recall that by the definition of $H$ the string $a$ has $0$ in position $i$. Thus, the probability that the edge $e$ of type $i$ is eligible is exactly $2^{-t-i}$. Assume for the third phase that $e$ is eligible. Consider another edge $e'=(x,y')\in E$ with color $\chi(e')=(a',i,z')$ of type $i$. If $a$ and $a'$ do not agree in the first $i$ positions, then $e'$ is not eligible. If they agree in the first $i$ positions, then $e'$ is eligible if and only if $a_{y'}=a'$, so with probability $2^{-t}$. Let $k'_e$ be the number of edges $(x,y')$ of type $i$ with the first $i$ digits of their class agreeing with $a$, but not counting $e$ itself. We have $k'_e<2^t$. Consider now an edge $e''=(x'',y)\in E$ with color $(a'',i,z'')$ such that $e''\ne e$. As $e$ is eligible, $e''$ can only be eligible if $a''=a$. If $a''=a$ then $e''$ is eligible if and only if $a_x$ and $a_{x''}$ agree in the first $i$ digits. This happens with probability $2^{-i}$. For the number $k''_e$ of the edges $(x'',y)\ne e$ of type $i$ and class $a$ we have $k''_e<2^i$. In phase three the eligibility of all these edges $e'$ and $e''$ adjacent with $e$ are independent events. Still consider the function $F$ fixed. The total probability for an edge $e$ of type $i$ to be in $E'$ is $$\begin{array}{rcl}\P[e\in E'|F] &=&2^{-t-i}(1-2^{-t})^{k'_e}(1-2^{-i})^{k''_e}\\ &\ge&2^{-t-i}(1-2^{-t})^{2^t-1}(1-2^{-i})^{2^i-1}\\ &>&2^{-t-i}/7.5.\end{array}$$ The total probability of $e\in E'$ can be calculated from the distribution of its type and the above conditional probability depending on its type: $$\P[e\in E']>\sum_{i=2}^t\frac{2^i}{2^{t+1}-4}\cdot\frac{2^{-t-i}}{7.5} >\frac{t-1}{15\cdot2^{2t}}>\frac{t-1}{240d}.$$ The expected number of edges in $E'$ is then $$\E[|E'|]>\frac{t-1}{240d}|E|\ge\frac{\log d}{480d}|E|.$$ \[6\] Let $G=(A,B,E)$ be a bipartite graph with a proper edge coloring. There exists a subgraph $G'=(A,B,E')$ of $G$ without a slow walk and with $|E'|>\frac{\log d}{480d}\cdot|E|$. Similarly, there exists a subgraph $G''=(A,B,E'')$ of $G$ without a fast walk and with $|E''|>\frac{\log d}{480d}\cdot|E|$. By Lemmas \[3\] and \[4\] the results of the lexicographic and reversed thinnings avoid the slow and fast walks, respectively. There exists an instance of the random choices with the size of $E'$ being at least its expectation given in Lemma \[5\]. This proves the theorem. Longer forbidden walks ---------------------- Here we generalize the concept of fast and slow walks to longer walks. Consider a bipartite graph $G=(A,B,E)$ with a proper edge coloring. For $k\ge2$ we call a walk of length $2k$ in $G$ a [*$k$-fast walk*]{} if its coloring $(c_1,\ldots,c_{2k})$ satisfies $c_1>c_2>\ldots>c_k<c_{k+1}<c_{k+2}<\ldots<c_{2k}$ and $c_1\ge c_{2k}$. For $k\ge2$ we call a walk of length $2k$ in $G$ a [*$k$-slow walk*]{} if its coloring $(c_1,\ldots,c_{2k})$ satisfies the following: $c_{2j-1}>c_{2j}$ for $1\le j\le k/2$; $c_{2j}<c_{2j+1}$ for $1\le j<k/2$; $c_{2j-1}<c_{2j}$ for $k/2<j\le k$; $c_{2j}>c_{2j+1}$ for $k/2\le j<k$; and finally $c_1\ge c_{2k}$. If a $k$-slow walk starts in the vertex set $B$ we call it a $(k,B)$-slow walk, otherwise it is a $(k,A)$-slow walk. Notice that $2$-fast walks are the fast walks and $(2,B)$-slow walks are the slow walks with their orientation reversed. For the coloring $(c_1,\ldots,c_{2k})$ of a $k$-fast walk $c_j$ is in between $c_{j-1}$ and $c_{j+1}$ for all $1<j<2k$, $j\ne k$, while $c_k$ is the smallest color on this list. For the coloring $(c_1,\ldots,c_{2k})$ of a $k$-slow walk the situation is reversed: the only index $1<j<2k$ with $c_j$ being in between $c_{j-1}$ and $c_{j+1}$ is the index $j=k$. In order to apply the lexicographic and reversed thinning recursively we have to change the coloring of the subgraph. Let $G=(A,B,E)$ be a bipartite graph with a proper edge coloring given by $\chi:E\to\{1,\ldots,d\}$. Let $G'=(A,B,E')$ the result of the lexicographic or the reversed thinning of $G$. Recall that the edges in $E'$ have a type $2\le i\le t$ with $t=\lceil(\log d)/2\rceil+1$. The [*type edge coloring*]{} of $G'$ is the map $\chi':E'\to\{1,\ldots,t-1\}$ defined by $\chi'(e)=t+1-i$ for an edge $e\in E'$ of type $i$. By Lemma \[2\]/a $\chi'$ is a proper edge coloring of $G'$. \[7\] Let $G=(A,B,E)$ be a bipartite graph with proper edge coloring given by $\chi:E\to\{1,\ldots,d\}$. Let $G'=(A,B,E')$ the result of the lexicographic thinning of $G$. Let $\chi'$ be the type edge coloring of $G'$ and let $k\ge2$. If a subgraph $G''=(A,B,E'')$ of $G'$ with its edge coloring given by $\chi'$ has no $(k',A)$-slow walk for $2\le k'<k$ then $G''$ with its edge coloring given by $\chi$ has no $(k,B)$-slow walk. Notice that the $k=2$ case of this lemma gives a second proof of Lemma \[3\]. Let $W=v_0v_1\ldots v_{2k}$ be a walk in $G''$ starting at $v_0\in B$ and let its $\chi$-coloring be $(c_1,c_2,\ldots,c_{2k})$. Assume that $c_i>c_{i+1}$ or $c_i<c_{i+1}$ for $1\le i<2k$ as required in the definition of a $(k,B)$-slow walk. We need to show $c_1<c_{2k}$. We identify the colors of $\chi$ with the triplets $(a,i,z)\in H$ as in the definition of lexicographic thinning. We let $c_j=(a_j,i_j,z_j)$. The $\chi'$-coloring of $W$ is $(t+1-i_1,\ldots,t+1-i_{2k})$. We have $i_j\ne i_{j+1}$ for $1\le j<2k$. For $1\le j<k$ the colors $c_{2j}$ and $c_{2j+1}$ are colors of distinct edges incident to $v_{2j}\in B$, so by Lemma \[2\]/b their class is the same: $a_{2j}=a_{2j+1}$. We consider lexicographic thinning, so the order between $c_{2j}$ and $c_{2j+1}$ is the same as the order between their types: $i_{2j}$ and $i_{2j+1}$. For $1\le j<k/2$ we have $i_{2j}<i_{2j+1}$ but for $k/2\le j<k$ we have $i_{2j}>i_{2j+1}$. For $1\le j\le k$ the colors $c_{2j-1}$ and $c_{2j}$ are colors of edges incident to $v_{2j-1}\in A$. By Lemma \[2\]/c the classes of these colors do not agree, and the ordering between the classes, between the types, and between the colors themselves are the same. Thus, for $1\le j\le k/2$ we have $a_{2j-1}>a_{2j}$ and $i_{2j-1}>i_{2j}$. For $k/2<j\le k$ we have $a_{2j-1}<a_{2j}$ and $i_{2j-1}<i_{2j}$. Also by Lemma \[2\]/c for all $1\le j\le k$ we have $P(a_{2j-1},a_{2j})=\min(i_{2j-1},i_{2j})$. The sequence $a_1,a_2,\ldots,a_k$ is monotone decreasing and it changes only in every other step. The first positions of change between distinct consecutive elements are $i_2,i_4,\ldots,i_{2\lfloor k/2\rfloor}$. So we have $a_1>a_k$ and $P(a_1,a_k)=\min(S_1)$ for the set $S_1=\{i_2,i_4,\ldots,i_{2\lfloor k/2\rfloor}\}$. Similarly, $a_k,a_{k+1},\ldots,a_{2k}$ is monotone increasing and it changes only in every other step. The first positions of change between distinct consecutive elements are $i_{2\lfloor k/2\rfloor+1},\ldots,i_{2k-3},i_{2k-1}$. So we have $a_k<a_{2k}$ and $P(a_k,a_{2k})=\min(S_2)$ for the set $S_2=\{i_{2\lfloor k/2\rfloor+1},\ldots,i_{2k-3},i_{2k-1}\}$. Let us consider an arbitrary value $1\le j<k/2$ and let $2\le k'=k-2j+1<k$. Consider the $2k'$ long middle portion $W'$ of the walk $W$: let $W'=v_{2j-1}v_{2j}\ldots v_{2k-2j+1}$. This is a walk of length $2k'$ in $G''$ starting at $v_{2j-1}\in A$. By our assumption on $G''$ this is not a $(k',A)$-slow walk if considered with the coloring $\chi'$. But the $\chi'$-coloring of $W'$ is $(t+1-i_{2j},t+1-i_{2j+1},\ldots,t+1-i_{2k-2j+1})$ and the consecutive values in this list compare as required for a $(k',A)$-slow walk. Therefore, we must have $t+1-i_{2j}<t+1-i_{2k-2j+1}$. We have just proved $i_{2j}>i_{2k-2j+1}$ for $1\le j<k/2$. For even $k$ and $j=k/2$ the same formula compares the types of two consecutive edges of $W$ and we have already seen its validity in that case too. For every element of the set $S_1$ we have just found a smaller element of the set $S_2$. Therefore, $\min(S_1)>\min(S_2)$. Using that $a_{2k}>a_k$ and $P(a_1,a_k)=\min(S_1)>\min(S_2)=P(a_k,a_{2k})$ Lemma \[1\] gives $a_1<a_{2k}$. This implies $c_1<c_{2k}$ and finishes the proof of the lemma. \[8\] Let $G=(A,B,E)$ be a bipartite graph with proper edge coloring given by $\chi:E\to\{1,\ldots,d\}$. Let $G'=(A,B,E')$ the result of the reversed thinning of $G$. Let $\chi'$ be the type edge coloring of $G'$ and let $k\ge2$. If a subgraph $G''=(A,B,E'')$ of $G'$ with its edge coloring given by $\chi'$ has no $(k',A)$-slow walk for $2\le k'<k$ then $G''$ with its edge coloring given by $\chi$ has no $k$-fast walk. Notice that the $k=2$ case of this lemma gives a second proof of Lemma \[4\]. The proof of this lemma is very similar to that of Lemma \[7\]. Let $W=v_0v_1\ldots v_{2k}$ be a walk in $G''$ and let its $\chi$-coloring be $(c_1,c_2,\ldots,c_{2k})$. Assume that $c_1>c_2>\ldots>c_k<c_{k+1}<c_{k+2}<\ldots<c_{2k}$ as required in the definition of a $k$-fast walk. We need to show $c_1<c_{2k}$. Instead, we prove the slightly stronger statement that the class of $c_1$ is smaller than the class of $c_{2k}$. We first do that for walks starting in $B$: assume that $v_0\in B$. We identify the colors of $\chi$ with the triplets $(a,i,z)\in H$ as in the definition of reverse thinning. We let $c_j=(a_j,i_j,z_j)$. Note that the $\chi'$-coloring of $W$ is $(t+1-i_1,\ldots,t+1-i_{2k})$. We have $i_j\ne i_{j+1}$ for $1\le j<2k$. For $1\le j<k$ the colors $c_{2j}$ and $c_{2j+1}$ are colors of distinct edges incident to $v_{2j}\in B$, so by Lemma \[2\]/b their classes are the same: $a_{2j}=a_{2j+1}$. Thus the order between $c_{2j}$ and $c_{2j+1}$ is determined by the order between $i_{2j}$ and $i_{2j+1}$, but as we use the reversed ordering in $H$ the order between $c_{2j}$ and $c_{2j+1}$ is reversed compared to the order between $i_{2j}$ and $i_{2j+1}$. Specifically, for $1\le j<k/2$ we have $i_{2j}<i_{2j+1}$ and for $k/2\le j<k$ we have $i_{2j}>i_{2j+1}$. For $1\le j\le k$ the colors $c_{2j-1}$ and $c_{2j}$ are colors of edges incident to $v_{2j-1}\in A$. By Lemma \[2\]/c the classes of these colors do not agree, and the ordering between the classes, between the types, and between the colors themselves are the same. Thus, for $1\le j\le k/2$ we have $a_{2j-1}>a_{2j}$ and $i_{2j-1}>i_{2j}$. For $k/2<j\le k$ we have $a_{2j-1}<a_{2j}$ and $i_{2j-1}<i_{2j}$. Also by Lemma \[2\]/c for $1\le j\le k$ we have $P(a_{2j-1},a_{2j})=\min(i_{2j-1},i_{2j})$. At this point we have the same ordering of the classes and types of the coloring of $W$ as in the proof of Lemma \[7\]. We also have the same assumption that $G''$ with the edge coloring $\chi'$ has no $(k',A)$-slow walk for $2\le k'<k$. So we arrive to the same conclusion $a_1<a_{2k}$ with an identical proof. We finish the proof of the lemma by considering the alternative case when $W$ starts in $A$. Now $c_1$ and $c_2$ are colors of edges sharing a vertex $v_1\in B$, so by Lemma \[2\]/b their classes are equal. Similarly, the classes of $c_{2k-1}$ and $c_{2k}$ are equal, so it is enough to prove that the class of $c_2$ is smaller than the class of $c_{2k-1}$. For $k=2$ this follows directly from Lemma \[2\]/c. For $k>2$ the walk $W'=v_1v_2\ldots v_{2k-1}$ is exactly the type of walk we considered for $k_0=k-1$. As it starts in $B$ we have already proved that the class of its first edge is smaller than the class of its last edge. This finishes the proof of the case of a walk starting in $A$ and also the proof of Lemma \[8\]. Lemmas \[7\] and \[8\] set the stage to use the thinning procedures recursively to get subgraphs avoiding $(k,B)$-slow or $k$-fast walks. In a single application of either thinning procedure the number $d$ of colors in the original coloring is replaced by $t-1=\lceil\log d/2\rceil$ colors in the type coloring. Here $4(t-1)>\log(4d)$, so after $k$ recursive calls we still have more than $\log^{(k)}(4d)/4$ colors, where $\log^{(k)}$ stands for the $k$ times iterated log functions. (Of course, this only makes sense if $\log^{(k)}(4d)>2$. Otherwise we can get stuck, as neither thinning procedure is defined in the pathetic case of $d=1$ colors.) Making optimal random choices we may assume that each thinning procedure yields at least the expected number of edges. Thus, the ratio of the number of edges and the number of colors decreases by at most a factor of $240$ in each iteration. Clearly, the only interesting case is when the original average degree was $\Theta(d)$ in which case the average degree after $k$ iterations remains $\Theta(\log^{(k)}d)$. The constant of proportionality depends on $k$. \[9\] Let $G=(A,B,E)$ be a bipartite graph with a $d$ edge coloring and let $k\ge2$. There exists a subgraph $G'=(A,B,E')$ of $G$ without a $(k',B)$-slow walk for any $2\le k'\le k$ and with $|E'|>{\log^{(k-1)}d\over4\cdot240^{k-1}d}|E|$. Similarly, there exists a subgraph $G''=(A,B,E'')$ of $G$ without a $k'$-fast walk for any $2\le k'\le k$ and with $|E''|>{\log^{(k-1)}d\over4\cdot240^{k-1}d}|E|$. We apply the thinning procedures recursively. First we use lexicographic and reversed thinning to obtain subgraphs $G_1$ and $G_2$ of $G$, respectively. We make sure these graphs have at least as many edges as the expected number given in Lemma \[5\]. If $k=2$ we are done, $G'=G_1$ and $G''=G_2$ satisfy the conditions of the theorem. Otherwise we consider $G_1$ and $G_2$ with the type edge coloring. We find recursively their subgraphs $G'$ and $G''$, respectively, avoiding $(k',A)$-slow walks for $2\le k'\le k-1$. This can be done because the sides $A$ and $B$ play symmetric roles. Finally, we apply Lemmas \[7\] and Lemma \[8\] to see that the subgraphs $G'$ and $G''$, if considered with the original edge coloring of $G$, avoid all walks required in the theorem. The number of edges guaranteed in the subgraphs is calculated in the paragraph preceding the theorem and is at least the stated bound. $k$-flat graphs --------------- In this subsection we establish that removing a linear number of edges from a $k$-fast walk free graph the resulting graph has special structural properties. We note here that the recursive thinning construction that we used to arrive at $k$-fast walk free graphs results in a graph that itself is $k$-flat as defined below. We chose however to keep the inductive part of the proof simple and concentrated only on $(k,B)$-slow and $k$-fast walks. We derive the more complicated properties from these simpler ones. Note that in this subsection we do not use that our graphs are bipartite. Let $G$ be a graph and $\chi$ a proper edge coloring of $G$. We define the [*shaving*]{} of the graph $G$ to be the subgraph obtained from $G$ by deleting the edge with the largesr color incident to every (non-isolated) vertex. Clearly, we delete at most $n$ edges, where $n$ is the number of vertices in $G$. We define the [*$k$-shaving*]{} of $G$ to be the subgraph obtained from $G$ by repeating the shaving operation $k$ times. Clearly, we delete at most $kn$ edges for a $k$-shaving. Let $W$ be a walk of length $m$ in a properly edge colored graph $G$, and assume its coloring is $(c_1,\ldots,c_m)$. We define the [*height function*]{} $h_W$ from $\{1,\ldots,m\}$ to the integers recursively letting $h_W(1)=0$ and $$h_W(i+1)=\left\{\begin{array}{lrl}h_W(i)+1&\hbox{~~~if }&c_{i+1}>c_i\\ h_W(i)-1&\hbox{if }&c_{i+1}<c_i\end{array}\right.$$ for $1\le i<m$. Note $h_W(i)+i$ is always odd. This function considers how the colors of the edges in the walk change, in particular, how many times the next color is larger and how many times it is smaller than the previous color. We call a graph $G$ with proper edge coloring [*$k$-flat*]{} if the following is true for every walk $W$ in $G$. Let $m\ge2$ be the length of $W$, let $(c_1,\ldots,c_m)$ be the coloring of $W$ and assume that the height function satisfies $h_W(i)<0$ for $2\le i\le m$. If $m\le 2k+1$ or $h_W(i)\ge-k$ for all $i$ then $c_1>c_m$. \[10\] Let $G$ be properly edge colored graph. Let $k\ge1$ and assume $G$ has no $k'$-fast walk for $2\le k'\le k$. Then the $(k-1)$-shaving $G'$ of $G$ is $k$-flat. We prove the following slightly stronger statement by induction on $m$. Let $W=v_0\ldots v_m$ be a walk of length $m$ in $G$ with coloring $(c_1,\ldots,c_m)$. Let $1\le j\le m$ be the largest index such that $h_W(j)=1-j$. Assume the walk $v_jv_{j+1}\ldots v_m$ is in the $(k-1)$-shaving $G'$ of $G$. Also assume that $h_W(i)<0$ for $2\le i\le m$. If $m\le 2k+1$ or $h_W(i)\ge-k$ for all $i$ then we claim $c_1>c_m$. This statement is stronger than Lemma \[10\] since it allows for the initial decreasing segment of $W$ be outside $G'$. If $j=m$ the statement of the claim is obvious from the definition of the height function. This covers the $m=2$ and $m=3$ base cases. Let $m\ge4$ and assume the statement is true for walks of length $m-1$ and $m-2$. If $j>k+1$ we have $h_W(j)=1-j<-k$ so we must have $m\le2k+1$. Consider the walk $W'=v_1\ldots v_m$ of length $m-1$. We have $h_{W'}(i)=1-i<0$ for $2\le i<j$ and $h_{W'}(i)\le h_{W'}(j-1)+(i-(j-1))=3+i-2j<0$ for $j\le i\le m-1$. Thus the inductive hypothesis is applicable and we get $c_1>c_2>c_m$. Finally consider the $j\le k+1$ case. As the trivial $j=m$ case was already dealt with we also assume $j<m$. Clearly, $h_W(2)<0$ implies $j\ge2$. We chose a $w_0\ldots w_{j-2}$ walk in $G$ ending at $w_{j-2}=v_j$ and with coloring $(c_1',\ldots,c_{j-2}')$ satisfying $c_1'>c_2'>\ldots>c_{j-2}'>c_{j+1}$. This is possible since the edge $v_jv_{j+1}$ is in the $(k-1)$-shaving $G'$ of $G$, so we can find the edge $w_{j-3}w_{j-2}$ in the $(k-2)$-shaving of $G$, $w_{j-4}w_{j-3}$ in the $(k-3)$-shaving, and so on. We must have $c_1>c_1'$ as otherwise $w_0w_1\ldots w_{j-3}v_jv_{j-1}\ldots v_0$ is a $(j-1)$-fast walk and no such walk exists in $G$. Now consider the walk $W'=w_0w_1\ldots w_{j-3}v_jv_{j+1}\ldots v_m$. This is a walk of length $m-2$ and satisfies $h_{W'}(i)=1-i$ for $1\le i\le j-1$ and $h_{W'}(i)=h_W(i+2)$ for $j-1\le i\le m-2$. All requirements of the inductive hypothesis are satisfied, so we have $c_1'>c_m$. Thus $c_1>c_1'>c_m$ as claimed. \[corol\] Let $G=(A,B,E)$ be a bipartite graph with a $d$ edge coloring and let $k\ge2$. There exists a $k$-flat subgraph $G'=(A,B,E')$ of $G$ with $|E'|>{\log^{(k-1)}d\over4\cdot240^{k-1}d}|E|-(k-1)(|A|+|B|)$. Combine Theorem \[9\] nwith Lemma \[10\] and the fact that $(k-1)$-shaving keeps all but at at most $(k-1)(|A|+|B|)$ edges of $G$. The final lemma in this section is a simple but useful observation on $k$-flat graphs. It can also be stated for longer walks with height function bounded from below, but for simplicity we restrict attention to short walks. \[11\] Let $k\ge1$ and let $G$ be a properly edge colored $k$-flat graph. Let $W=v_0\ldots v_m$ be a walk in $G$ of length $m\le2k+1$ with coloring $(c_1,\ldots,c_m)$. If $c_1\ge c_i$ for all $1\le i\le m$ then $h_W(i)\le0$ for all $1\le i\le m$. We prove the contrapositive statement. Assume $h_W(i)>0$ for some $1\le i\le m$ and let $i_0$ be the smallest such index. Clearly, $i_0\ge2$, $h_W(i_0)=1$ and for the walk $W'=v_{i_0}v_{i_0-1}\ldots v_0$ we have $h_{W'}(i)=h_W(i_0-i+1)-1<0$ for $1<i\le i_0$. So by the definition of $k$-flatness we have $c_{i_0}>c_1$. Locally plane graphs {#s3} ==================== Locally plane graphs were introduced in the paper [@PPTT] (though the name appears first in this paper). That paper gives a simple construction for $3$-locally plane graphs. We recall (a simplified version of) the construction as it is our starting point. Construction of $3$-locally plane graphs in [@PPTT] --------------------------------------------------- Let $d\ge1$ and consider the orthogonal projection of (the edge graph of) the $d$ dimensional hypercube into the plane. A suitable projection of the “middle layer” of the hypercube provides the $3$-locally plane graph. Here is the construction in detail: Let $d\ge1$ be fixed and set $b=\lfloor d/2\rfloor$. The bit at position $i$ in $x\in\{0,1\}^d$ (the $i$th coordinate) is denoted by $x_i$ for $1\le i\le d$. We let $A=\{x\in\{0,1\}^d\mid\sum_{i=1}^dx_i=b\}$ and $B=\{x\in\{0,1\}^d\mid\sum_{i=1}^dx_i=b+1\}$. The abstract graph underlying the geometric graph to be constructed is $G_d=(A,B,E)$ with $(x,y)\in E$ if $x\in A$, $y\in B$ and $x$ differs from $y$ in a single position. This is the middle layer of the $d$ dimensional hypercube. We define the edge coloring $\chi:E\to\{1,\ldots,d\}$ that colors an edge $e=(x,y)\in E$ by the unique position $\chi(e)=i$ with $x_i\ne y_i$. Notice that this is a proper edge coloring. The number of vertices is $n=|A|+|B|={d\choose b}+{d\choose b+1}\le2^d$, the number of edges is $|E|={d\choose b}(d-b)>nd/4$. The average degree is greater than $d/2\ge\log n/2$. To make the abstract graph $G_d$ into a geometric graph we project the hypercube into the plane. We give two possible projections here. The first is more intuitive and it is closer to the actual construction in [@PPTT]. We let $a_i=(10^i,i\cdot10^i)$ for $1\le i\le d$ and use this vector as the projection of the edges of color $i$. We will use that among the vectors $a_i$ higher index means higher slope and much greater length. We identify the vertex $x\in A\cup B$ with the point $P_x=\sum_{i=1}^dx_ia_i$. The edges are represented by the straight line segment connecting their endpoints. We give the second construction to obtain a graph where all edges are very close in length and direction. Let $0<\epsilon<10^{-d}$ be arbitrary and consider the vectors $b_i=(1+10^i\epsilon,\epsilon^{d+1-i}(1+10^i\epsilon))$ and identify a point $x\in A\cup B$ with $Q_x=\sum_{i=1}^dx_ib_i$. It is easy to verify that we get a geometric graph in both cases (i.e., the vertices are mapped to distinct points and no edge passes through a vertex that is not its endpoint). Note that edges of color $i$ are all translates of the same vector $a_i$ or $b_i$. We do not introduce separate notations for the two geometric graphs constructed this way as they will only be treated separately in the proof of Lemma \[12\], where we refer to them as the first and the second realization of $G_d$. Self-intersecting paths in $G_d$ -------------------------------- In [@PPTT] a graph very similar to $G_d$ was shown to be $3$-locally plane. Here we do more, we analyze all self-intersecting paths of $G_d$ as follows. \[12\] Let $W$ be a walk in $G_d$ with coloring $(c_1,\ldots,c_m)$ satisfying $c_1\ge c_m$. Assume that $W$ and all its non-empty subwalks have a unique edge of maximal color. The first and last edges of $W$ cross in either geometric realization if and only if $m$ is even and there is an odd index $1<j<m$ satisfying $c_1>c_m>c_j\ge c_i$ for all $1<i<m$. Let $W=v_0\ldots v_m$. Note that the first and the last edges cross if and only if $v_0$ and $v_1$ are on different sides of the line $\ell$ through $v_{m-1}$ and $v_m$ and similarly $v_{m-1}$ and $v_m$ are on different sides of the line $\ell'$ through $v_0$ and $v_1$. To analyze such separations consider the projection $\pi_i$ to the $y$ axis parallel to edges of color $i$. Let us consider the first realization of $G_d$ with the vectors $a_i$. We have $\pi_i(x,y)=y-ix$ and the projection of the vector $a_j$ is of length $|i-j|10^j$. Thus higher colored edges map to longer intervals (except color $i$ itself). Under the projection $\pi_{c_1}$ the direction of the highest colored edge in the walks $v_1\ldots v_{m-1}$ (respectively, $v_1\ldots v_m$) determines which side $v_{m-1}$ (respectively $v_m$) lies of the line $\ell'$. Indeed this highest color cannot be $c_1$, so the projections of the other edges will be much shorter and by the unique maximal color property we see that the walk contains at most $2^{k-1}$ edges having the $k$th largest color, so these shorter projections cannot add up to be more than the projection of the largest edge. We can only have $v_{m-1}$ and $v_m$ lying on different sides of $\ell'$ if these edges of maximal color are distinct, thus we must have $c_m>c_i$ for all $1<i<m$. From $c_1\ge c_m$ and the unique maximal color property we have $c_1>c_m$. Taking $c_j$ to maximize $c_i$ for $1<i<m$ (this is unique again) we have $c_1>c_m>c_j\ge c_i$ for all $1<i<m$. It is left to prove that the first and last edges of $W$ cross if and only if $m$ is even and $j$ is odd. To prove this claim one has to use that $G_d$ is bipartite with vertex sets $A$ and $B$, and every edge of color $c$ is a translate of the vector $a_c$ with its head in $B$ and tail in $A$. Thus, the vector $v_{i-1}v_i$ is either $a_{c_i}$ or $-a_{c_i}$ depending on the parity of $i$. Which sides of $\ell'$ $v_{m-1}$ and $v_m$ lie is determined by the projections of the $j$th and last edge, so they are on opposite sides if $j$ and $m$ have different parities. Similarly, the sides of $\ell$ on which $v_0$ and $v_1$ lies is detemined by the $\pi_{c_m}$ projection of the first and $j$th edges, but as we have $c_1>c_m>c_j$ they are on different sides if $1$ and $j$ has the same parity. See Figure 1 for a rough depiction of all four cases. This finishes the proof of the claim and the part of the lemma regarding the first realization of $G_d$. The proof for the second realization of $G_d$ as a geometric graph (involving the vectors $b_i$) is slightly more complicated. We have $\pi_i(x,y)=y-\epsilon^{d+1-i}x$ and $\pi_i(b_j)=\epsilon^{d+1-j}(1+10^j\epsilon)-10^j\epsilon^{d+2-i}-\epsilon^{d+1-i}$. The $\epsilon^{d+1-i}$ terms alternate in sign in the projection on the edges and cancel completely for a walk of even length. For a walk of odd length a single such term remains but it is dominated by the other terms if the walk has an edge with color above $i$. If, however, no such edge exists, the remaining uncanceled $\epsilon^{d+1-i}$ term dominates the other terms in the projection. Thus, if the projection of the unique edge with the largest color of a walk has color $c>i$ or if $c<i$ but the walk has even length, then the sign of the $\pi_i$ projection of the edge with the largest color determines the sign of the projection of the entire walk. But in case $c<i$ and the walk has odd length, then the non-canceling $\epsilon^{d+1-i}$ term determines the sign. Let the edge $v_{j-1}v_j$ be the one with the unique largest color in the walk $v_1\ldots v_{m-1}$. A case analysis of the parities of $j$, $m$ and whether $c_j>c_1$ or $c_j>c_m$ hold show that the first and last edges of $W$ cross if and only if $c_m>c_j$, $j$ is odd and $m$ is even – as claimed in the lemma. We call $v_m\ldots v_0$ the [*reverse*]{} of the walk $W=v_0\ldots v_m$. \[13\] Let $k\ge1$ and let $G'$ be a $k$-flat subgraph of $G_d$. If the length of a walk $W$ in $G'$ does not exceed $2k+1$, then $W$ has unique edge of largest color. Let $W$ be a walk of length $3\le m\le2k+1$ in $G'$ with coloring $(c_1,\ldots,c_m)$ and assume the largest color is not unique. We may assume $c_1=c_m>c_i$ holds for all $1<i<m$, otherwise one can take a suitable subwalk of $W$. By Lemma \[11\] we have $h_W(m)\le0$. Consider $W'$ the reverse of the walk $W$. Clearly, $h_{W'}(m)=-h_W(m)$, but also by Lemma \[11\] we have $h_{W'}(m)\le0$. So we must have $h_W(m)=0$ and $m$ must be odd. The contradiction that proves the statement of the lemma comes from the simple observation that between two consecutive appearances of a color in any walk of $G_d$ there always are an even number of edges, so $m$ must be even. To see this recall that $G_d$ is bipartite with sides $A$ and $B$ where $A$ consists of the 0-1 sequences of length $d$ with $\lfloor d/2\rfloor$ ones, while the 0-1 sequences in $B$ contain one more ones. The $A$ end of an edge of color $c$ has $0$ at position $c$, while the $B$ end of this edge has $1$ there. Along edges of other colors bit $c$ does not change. Thus if a walk traverses an edge of color $c$ from $A$ to $B$ say, then along the walk bit $c$ remains $1$ until the next time the walk traverses an edge of color $c$ and this has to be from $B$ to $A$. We note that Lemma \[13\] immediately implies that the girth of a $k$-flat subgraph of $G_d$ is at least $2k+2$. This estimate can be improved by observing that any cycle in $G_d$ has an even number of edges of any color (one has to flip a bit even times to get beck to the original state). In particular any cycle has at least two occurrences of the largest color. These edges break the cycle into two paths sharing two edges. Both path has to be of length at least $2k+2$, the length of the cycle is at least $4k+2$. As every properly edge colored graph is $1$-flat the $k=1$ case of the next lemma establishes that $G_d$ is $3$-locally plane. \[14\] For any $k\ge1$ any $k$-flat subgraph of $G_d$ is $(2k+1)$-locally plane in either realization. Let $G'$ be a $k$-flat subgraph of $G_d$. We need to show that no walk (or path) $W$ of length $m\le2k+1$ is self-intersecting. It is clearly enough to show that the first and the last edges of $W$ do not cross and we may assume that the color of the first edge is not smaller than that of the last edge (otherwise simply consider the same walk reversed). By Lemma \[13\] $W$ and all its subwalks have a unique edge of maximal color, so Lemma \[12\] applies. It is enough to show that the coloring $(c_1,\ldots,c_m)$ of $W$ does not satisfy the conditions of Lemma \[12\]. Assume the contrary. So $m$ is even, and there is an odd index $j$ such that $c_1>c_m>c_j\ge c_i$ for all $1<i<m$. Consider the walk $W_1=v_mv_{m-1}\ldots v_{j-1}$ of length $m-j+1$. Its coloring is $(c_m,\ldots,c_j)$ and $c_m$ is its largest color. By Lemma \[11\] $h_{W_1}(m-j+1)\le0$. In fact, as $m-j+1$ is even $h_{W_1}(m-j+1)$ is odd, so we have $h_{W_1}(m-j+1)\le-1$. Consider the walk $W_2=v_jv_{j-1}\ldots v_1$ of length $j-1$ and its coloring $(c_j,\ldots,c_2)$. The largest color in $W_2$ is $c_j$, so we have $h_{W_2}(j-1)\le0$ by Lemma \[11\]. And again by parity considerations $h_{W_2}(j-1)\le-1$. It is easy to see that $h_W(m)=-1-h_{W_2}(j-1)-h_{W_1}(m-j+1)$. So we have $h_W(m)\ge-1+1+1=1$ contradicting Lemma \[11\]. The contradiction proves Lemma \[14\]. \[15\] For any fixed $k>0$ and large enough $n$ there exists a $(2k+1)$-locally plane graphs on $n$ vertices having at least $({\log^{(k)}n\over240^k}-k)n$ edges. Given two arbitrary disks in the plane, one can further assume that all edges of these graph connect a vertex inside one disk to one inside the other disk. Simply combine the results of Corollary \[corol\] and Lemma \[14\]. If $n$ is not the size of the vertex set of $G_d$ for any $d$, add isolated vertices to the largest $G_d$ with fewer than $n$ vertices. Use the second realization of $G_d$ with a small enough $\epsilon>0$ to obtain a geometric graph with all edges connecting two small disks and apply a homothety and a rotation to get to the desired disks. Discussion on optimality of thinning {#s4} ==================================== The maximum number of edges of a $3$-locally plane graph on $n$ vertices is $\Theta(n\log n)$ as proved in [@PPTT]. The lower bound is reproduced here by the $k=1$ case of Theorem \[15\], which is therefore tight. The upper bound of [@PPTT] extends to [*$x$-monotone topological graphs*]{}, i.e., when the edges are represented by curves with the property that every line parallel to the $y$ axis intersects an edge at most once. Without this artificial assumption on $x$-monotonicity only much weaker upper bounds are known. For higher values of $k$ we do not have tight results even if the edges are straight line segments as considered in this paper. While the number of edges in a $k$-locally plane graph constructed here deteriorates very rapidly with the increase of $k$, the upper bound hardly changes. In fact the only known upper bound better than $O(n\log n)$ is for $5$-locally plane graphs: they have $O(n\log n/\log\log n)$ edges as shown in [@PPTT]. Slightly better bounds are known for geometric (or $x$-monotone topological) graphs with the additional condition that a vertical line intersects every edge. If such a graph is $(2k)$-locally plane for $k\ge2$, then it has $O(n\log^{1/k}n)$ edges. The first realization of $G_d$ does not satisfy this condition, but the second one does. Still, the lower and upper bounds for this restricted problem are far apart: for $4$-locally plane graphs with a cutting line the upper bound on the number of edges is $O(n\sqrt{\log n})$ while the construction gives $\Omega(n\log\log n)$. Although we cannot establish that the locally plane graphs constructed are optimal we can prove that the thinning procedure we use is optimal within a constant factor. It follows that any $4$-locally plane subgraph of either realization of $G_d$ has $O(n\log\log n)$ edges. This optimality result below refers to a single step of the thinning procedure. It would be interesting to establish a strong upper bound on the number of edges of a $k$-flat graph for $k\ge3$. Let us mention here that the thinning procedures described in this paper found application in the extremal theory of $0$-$1$ matrices, see [@T], and there the result is shown to be optimal within a constant factor. Consider an $n$ by $n$  0-1 matrix that has no $2$ by $3$ submatrix of either of the following two forms: $$\left(\begin{array}{ccc}1&1&*\\1&*&1\end{array}\right),\hskip1cm \left(\begin{array}{ccc}1&*&1\\{*}&1&1\end{array}\right),$$ where the $*$ can represent any entry. The maximal number of $1$ entries in such a matrix is $\Theta(n\log\log n)$ as proved in [@T]. The construction proving the lower bound is based on lexicographic thinning. The following lemma shows that the number of edges in the subgraphs claimed in Lemma \[0\] and Theorem \[6\] are optimal in a very strong sense: no properly edge colored graph with significantly more edges than the ones guaranteed by the above results can avoid heavy paths (slow or fast walks, respectively). \[16\] Let $G=(A,B,E)$ be a bipartite graph with proper edge coloring given by $\chi:E\to\{1,\ldots,d\}$. a) : If $G$ does not have a heavy path then $|E|\le2\sqrt{d|A|\,|B|}\le(|A|+|B|)\sqrt d$. b) : If $G$ does not have a slow walk then $|E|\le(|A|+|B|)(\log d+2)$. c) : If $G$ does not have a fast walk then $|E|\le2(|A|+|B|)(\log d+2)$. For any vertex $z\in A\cup B$ denote by $m(z)=\max(\chi(e))$, where the maximum is for edges $e$ incident to $z$. For an edge $e=(x,y)\in E$ we define its [*weight*]{} to be $w(e)=m(x)-\chi(e)$, while its [*$B$-weight*]{} is $w_B(e)=m(y)-\chi(e)$. Clearly, both $w(e)$ and $w_B(e)$ are integers in $[0,d-1]$. To prove part a) of the lemma assume $G$ does not contain a heavy path. We set a threshold parameter $t=\lfloor\sqrt{d|A|/|B|}\rfloor$ and call an edge $e$ [*$B$-light*]{} if $w_B(e)<t$, otherwise $e$ is [*$B$-heavy*]{}. All the edges incident to a vertex $y\in B$ have different colors, thus they also have different $B$-weights, so at most $t$ of them can be $B$-light. The total number of $B$-light edges is at most $|B|t\le\sqrt{d|A|\,|B|}$. Now consider two edges $e_1=(x,y_1)$ and $e_2=(x,y_2)$ incident to vertex $x\in A$. Assume $\chi(e_2)\le \chi(e_1)$. If $w_B(e_2)>0$ we can extend the path formed by these two edges with the edge $e_3$ incident to $y_2$ having maximum color $\chi(e_3)=m(y_2)$. Clearly, $\chi(e_3)=\chi(e_2)+w_B(e_2)$ and the resulting path is heavy unless $\chi(e_2)>\chi(e_1)+w(e_2)$. Therefore, the number of $B$-heavy edges incident to $x$ is at most $d/(t+1)$. The total number of $B$-heavy edges is at most $|A|d/(t+1)\le\sqrt{d|A|\,|B|}$. For the total number of edges we have $|E|\le2\sqrt{d|A|\,|B|}\le(|A|+|B|)\sqrt d$. For parts b) and c) of the lemma consider an edge $e=(x,y)\in E$. If $w(e)=0$ we call the edge $e$ [*maximal*]{}. Clearly, there are at most $|A|$ maximal edges. If $e$ is not maximal we define $n(e)$ to be the “next larger colored edge at $x$”, i.e., $n(e)$ is the edge in $E$ having minimal color $\chi(n(e))$ among edges incident to $x$ and satisfying $\chi(n(e))>\chi(e)$. We define the [*gap*]{} of $e$ to be $g(e)=\chi(n(e))-\chi(e)$. Clearly, $0<g(e)\le w(e)$. We call the edge $e$ [*heavy*]{} if $w(e)>2g(e)$, otherwise $e$ is [*light*]{}. Recall, that for maximal edges $n(e)$ and $g(e)$ are not defined and maximal edges are neither light nor heavy. Let $e_1$ and $e_2$ be distinct edges in $E$ incident to a vertex $x\in A$. If $\chi(e_1)<\chi(e_2)$ then $w(e_1)\ge w(e_2)+g(e_1)$. If $e_1$ is light, $w(e_1)\ge2w(e_2)$ follows, therefore at most $\lceil\log d\rceil$ light edges are incident to $x\in A$. Thus the total number of light edges in $E$ is at most $\lceil\log d\rceil|A|$. For part b) of the lemma assume $G$ does not contain a slow walk. Let $e_2=(x_2,y)$ and $e_3=(x_3,y)$ be distinct non-maximal edges in $E$ and assume $\chi(e_2)<\chi(e_3)$. Let $e_1=n(e_2)$ and let $e_4$ be the maximal edge incident to $x_3$ in $G$. We have $\chi(e_1)=\chi(e_2)+g(e_2)$ and $\chi(e_4)=\chi(e_3)+w(e_3)$. The edges $e_1$, $e_2$, $e_3$, and $e_4$ cannot form a slow walk. As $\chi(e_1)>\chi(e_2)<\chi(e_3)<\chi(e_4)$ we must have $\chi(e_4)<\chi(e_1)$. This implies $g(e_2)>w(e_3)$, and if $e_2$ is heavy $w(e_2)>2w(e_3)$. Therefore, at most $\lceil\log d\rceil$ heavy edges can be incident to $y\in B$. The total number of heavy edges in $G$ is at most $(\lceil\log d\rceil|B|$. For the total number of edges we add the bound obtained for light, heavy, and maximal edges and get $|E|\le(|A|+|B|)\lceil\log d\rceil+|A|$. Finally for part c) of the lemma we assume $G$ does not contain a fast walk. Let $E_1$ consist of the edges $(x,y)\in E$ for which $m(x)\ge m(y)$. Assume without loss of generality that $|E_1|\ge|E|/2$. If this is not the case consider the same graph with its sides switched. We use here that the definition of a fast walk and the claimed bound on the number of edges are both symmetric in the color classes. As in the previous case consider two non-maximal edges $e_2=(x_2,y)$ and $e_3=(x_3,y)$ in $E_1$ with $\chi(e_2)<\chi(e_3)$. Let $e_1$ be the maximal edge incident to $x_2$ and let $e_4=n(e_3)$. As $\chi(e_2)< \chi(e_3)<\chi(e_4)$ but $G$ does not contain a fast walk we must have $\chi(e_1)<\chi(e_4)$. As $e_2\in E_1$ we must also have $\chi(e_1)=m(x_2)\ge m(y)\ge \chi(e_3)$. If $e_3$ is heavy we also have $$\begin{array}{rcl} \chi(e_3)+w(e_3)-m(y)&>&\chi(e_3)+2g(e_3)-m(y)\\ &\ge&2(\chi(e_3)+g(e_3)-m(y))\\ &=&2(\chi(e_4)-m(y))\\ &>&2(\chi(e_1)-m(y))\\ &=&2(\chi(e_2)+w(e_2)-m(y)). \end{array}$$ For all the heavy edges $e\in E_1$ incident to $y\in B$ the values $\chi(e)+w(e)-m(y)$ increase strictly more than by a factor of $2$. As these values are integers from $[0,d-1]$, there are at most $\lceil\log d\rceil$ heavy edges in $E_1$ incident to $y$. The total number of heavy edges in $E_1$ is at most $\lceil\log d\rceil|B|$. For the total number of edges in $E_1$ we sum our bound on heavy edges in $E_1$ and the bounds on the light and maximal edges in $E$. We obtain $|E_1|\le(|A|+|B|)\lceil\log d\rceil+|A|$. Finally we get $|E|\le2(|A|+|B|)\lceil\log d\rceil+2|A|$. [99]{} E. Ackerman, On the maximum number of edges in topological graphs with no four pairwise crossing edges, [*Discrete and Computational Geometry*]{} [**41**]{} (2009) (3), 365–375. P. 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Tóth, Relaxing planarity for topological graphs, in: [*Discrete and Computational Geometry*]{} (J. Akiyama, M. Kano, eds.), Lecture Notes in Computer Science [**2866**]{}, Springer-Verlag, Berlin, 2003, pp. 221–232. J. Pach. G. Tóth, Graph drawn with few crossings per edge, [*Combinatorica*]{} [**17**]{} (1997), 427–439. R. Pinchasi, R. Radoičić, On the number of edges in geometric graphs with no self-intersecting cycle of length $4$, in: [*Towards a Theory of Geometric Graphs*]{}, 233–243, Contemporary Mathematics 342, American Mathematical Society, Providence, RI, 2004. P. Valtr, Graph drawing with no $k$ pairwise crossing edges, in [*Graph Drawing*]{} (Rome, 1997), 205–218, Lecture Notes in Comput. Sci., 1353, Springer, Berlin, 1997. G. Tardos, On 0-1 matrices and small excluded submatrices, [ *Journal of Combinatorial Theory, Series A*]{} [**111**]{} (2005), 266–288. G. Tardos, G. Tóth, Crossing stars in topological graphs, [ *SIAM Journal on Discrete Mathematics*]{} [**21**]{} (2007), 737–749. [^1]: Reserach partially supported by NSERC grant 329527 and by OTKA grants T-046234, AT048826 and NK-62321.

--- abstract: 'Waveguide arrays offer enormous potential to design circuit elements essential to fabricate optical devices capable to processing information codified by light. In this work we study the existence and stability of localized beams in one dimensional photonic lattices composed by a Kerr type waveguide array. We analyzed the case where discrete translation symmetry is broken in as much as one of the waveguides lacks a nonlinear response. Specifically, we determined the space of parameters where a coherent and robust mobility across the lattice is achieved. Moreover, we calculate the reflection and transmission coefficients when localized beams interact directly with the impurity, finding that it behaves as a variable filter depending of system parameters. Our results would shed light on develop solutions to keep unaltered information during its transmission within future optical devices.' address: 'Programa de Física, Facultad de Ciencias Básicas, Universidad del Atlántico, Puerto Colombia 081007, Colombia.' author: - 'M A Sabogal, I C Parra, M Bandera, J Gallardo and Cristian Mejía-Cortés' title: 'Mobility of localized beams in non-homogeneous photonic lattices' --- Introduction ============ Localized modes have been studied extensively since middle of last century [@PhysRev.109.1492]. Many physical systems exhibit different phenomena which lead to formation of this kind of excitation. For example, the first system where light localization was predicted and observed was in optical fibers [@doi:10.1063/1.1654836]. Here, light pulses were able to travel long distances without distortion, due to a balance between nonlinear response of material and chromatic dispersion of light [@MO35250] . This kind of pulse received the name of [*optical soliton*]{} and since then they have been widely used in telecommunications [@Segovia_Cabrera_2015]. Optical periodic systems have attracted enormous attention during last three decades because they bear enormous potential in technological applications. Their underlying characteristics offer the possibility of manage the light behaviour either over long distances or short ones. For example, at big scale photonic crystal fibers, optical fibers with a micro structured cross sections [@Russell:06; @Russell358], can be employed in fiber-optics communications [@Roberts:05], but also they can be used as sensors with high resolution [@Cregan1537]. They also offer the possibility to manage light propagation in short scale. Logical operations similar to those involved with electron currents can be mimic in a completely optical “microprocessor” or photonic chip. It can be plausible due to the refractive index in theses systems possesses a periodical distribution, hence, there are forbidden regions for light propagation [@Joannopoulos]. Experimentally, photonic lattices has been implemented by creating waveguide arrays in several media. For example, by using a femto-second pulsed laser on an amorphous (non-crystalline) phase silicon glass, it can be possible to “write” waveguides by modifying the nominal refractive index around the area where it is has been focused [@Szameit_2010]. Photo-refractive crystals are systems where these waveguide arrays also can be written by and induction process, due to its refractive index changes by the light intensity variation, i.e., by a non-linear response of the electric field [@Armijo:14]. On the other hand, research on coherent transfer of light stays as a hot topic over the years due to its direct implications in design technological devices for controlled transport of information. A change in the periodic distribution of the refractive index, by introducing an impurity into the lattice, results in the scattering of transverse traveling waves. For example, when the impurity comes from localized solutions, the scattering of plane waves by them has opened the possibility to observe Fano resonances [@PhysRev.124.1866]. In a nonlinear optical context, it has been observed that scattering of solitons in waveguide arrays moving towards impurity potentials, has a complex phenomenology [@PhysRevLett.99.133901]. Moreover, by adjusting the strength of linear impurities a completely trapping regime can be tailored [@Morales-Molina:06]. In the present study we address the case of nonlinear photonic lattices with an embedded linear impurity, which eventually improves the manipulation of light beam across the lattice. We hope that our results may be interesting in the design of optical limiters, barriers and gates for future photonic chips. The paper is organized as follows: in Section \[model\], we introduce the model and develop the main formalism employed to identify nonlinear stationary solutions, as well as, a favorable domain in terms of system parameters to achieve coherent and robust mobility. In Section \[modes\], we report findings on the existence and stability of localized stationary solutions around the lattice impurity. Section \[mobility\] is devoted to estimate the optimal domain for coherent and robust mobility of nonlinear modes. The analysis on scattering problem between nonlinear modes and lattice impurity is presented in Section \[scattering\]. Finally, in Section \[conclusions\], we summarize and draw our main conclusions. Model ===== The Discrete Nonlinear Schrödinger Equation (DNLSE) represents one of the most important models in nonlinear physics. For example, in classical mechanics, this equation describes a particular model for a system of coupled anharmonic oscillators [@eilbeck2003discrete]. On the other hand, this model predict the existence of localized modes of excitation of Bose–Einstein condensates in periodic potentials such as those generated by counter-propagating laser beams in an optical lattice [@Franzosi_2011]. In nonlinear optics, this equation combines phenomena related to the dispersion and/or diffraction of electromagnetic waves with those generated by higher order electric polarization in periodical media [@khare2006discrete]. When impurities are introduce to the system the translational symmetry becomes broken, which leads to the formation of localized modes around the defects [@kevrekidis2009discrete]. For the case when the effect of impurity is the lack of nonlinear response in a specific waveguide, we can model the propagation, along the $\hat z$-axis, of the corresponding electric field amplitude present in the $n$-th guide, $E_ {n} (z)$, as a variant of a more general DNLSE $$i\frac{dE_{n}}{dz} + \zeta_{n+1}E_{n+1}+\zeta_{n-1}E_{n-1} + \gamma(1-\delta_{n,n_{i}})|E_{n}|^{2}E_{n} = 0, \label{eq1}$$ where $ i = \sqrt {-1} $ and $\delta_{n,n_{i}}$ is the Kronecker symbol. The nonlinear response of the media is represented by the parameter $\gamma$, which is proportional to the nonlinear refraction index of the medium. Here we assume that $\zeta_ {n}$, the coupling between waveguides, is the same for each $n$, i. e., $\zeta_ {n}= \zeta $. The Equation (\[eq1\]) has two conserved quantities, the generating function that corresponds to the Hamiltonian ($H$) $$H = -\sum_{n=1}^{N}\left(\zeta E_{n}(z)E^{\ast}_{n+1}(z)+\text{c.c.} + \frac{\gamma}{2}(1-\delta_{n,n_{i}})|E_{n}(z)|^{4}\right), \label{eq2}$$ where the symbol $^\ast$ and c.c. denote the complex conjugate, and the norm or optical power ($P$) in the system $$P = \sum_{n=1}^{N}|E_{n}(z)|^{2}. \label{eq3}$$ It is worth to mention here that these two conserved quantities will be monitored during all the calculations along this work, because they will serve to check the validity of our numerical findings. Families of nonlinear modes {#modes} =========================== We look for stationary solutions of Equation (\[eq1\]) in the form $ E_ {n} = \phi_ {n} \exp {(i \lambda z)} $, where the amplitudes $ \phi_ {n} $ are real quantities that satisfy the following system of nonlinear algebraic equations $$-\lambda\phi_{n} + \zeta(\phi_{n+1}+\phi_{n-1}) + \gamma(1-\delta_{n,n_{i}})\phi_n^3=0, \label{eq4}$$ being $\lambda $ the propagation constant of the stationary solutions. According on the sign of $\gamma$ the nonlinear effect of the system can be of the self-focusing type ($\gamma>0$) or self-defocusing type ($\gamma<0$). Throughout the rest of the paper, we assume to deal with self-focusing type media. We solve the model (\[eq4\]) by implementing a Newton-Raphson scheme. We start from a localized seed around the impurity and in a few number of iterations the algorithm converges to localized stationary solutions. We check the stability of localized solutions by performing the standard linear stability analysis. From now on we use solid (dashed) lines to denote families with stable (unstable) solutions. It is interesting to study the consequences that the value of the nonlinear constant of the impurity of the lattice entails, specifically, the type of solutions that exist around the defect and its mobility around it. Recently, it has been found that modes centered at site are the only stable family of solutions that exist around the impurity [@mejia2019nonlinear], however, we found that the stability region of these modes depends on the value of the nonlinearity of impurity. Figure \[fig1\](a) shows the families of solution in the space of ($P$,$\lambda$) for the even and odd modes far from the defect. Families for odd modes around the defect for ten values of $ \gamma_ {i} $ between $ 0 $ and $ 0.90 $ is displayed at Figure \[fig1\](b). It can be seem here that when nonlinearity for defect diminish there is a reduction in the region of existence and stability of this solutions. The inset in Figure \[fig1\](a) sketches the odd (c) and even (d) modes belonging to the families represented by solid and dashed curves, respectively. On the other hand, bottom inset at Figure \[fig1\](b) displays three modes around the impurity that belongs to the odd (e), even (f) and symmetrical (g) families of solution, when $\gamma=0$, for three different values of $\lambda$. Last two families are not illustrated in this work but they have been reported recently in reference [@mejia2019nonlinear]. ![(a) $P$ vs $\lambda$ diagram for odd and even families of solutions far away from the impurity. Bottom inset displays both solutions for $ P $ = 3.00 and $\lambda$ = $2.85$ (c) and $2.62$ (d), respectively. (b) $P$ vs $\lambda$ diagram for odd families around impurity for different values of $\gamma$. Bottom inset displays three different type solutions, near to impurity for $ P $ = 3.00, $\gamma=0$ and $\lambda$ = $2.91$ for the odd mode (e), $\lambda$ = $2.66$ for even mode (f) and $\lambda$ = $2.28$ for symmetrical mode (g).[]{data-label="fig1"}](fig1_corregida.png){width="100.00000%"} Mobility of localized modes {#mobility} =========================== With the aim to study the interaction between solitons and the linear impurity, it is mandatory to determine the zones in the space $(P,k)$ of powers and momentum, in which coherent mobility is guaranteed. It is well known that the mobility of discrete solitons is restricted by the Peierls-Nabarro (PN) barrier [@kevrekidis2009discrete], which exists due to the non-integrability of the DNLSE [@brazhnyi2013interaction; @peschel2002optical]. This barrier can be estimated as the difference in energy ($H$) between the solutions of the fundamental modes, sketched at top left inset in Figure \[fig1\](a). By applying a power constraint to the Newton-Raphson method we could compare the value of the Hamiltonian (energy) of the modes that have the same power [@lederer2008discrete]. In that way, we can identify those regions where $H$ is similar, both for odd and even modes, which implies that these solutions, previously endowed with momentum, can move across the lattice in an adiabatic way, i. e., they can transform dynamically into the another one almost without radiating energy and preserving their shape. It is well known that for the Kerr nonlinearity, there is a critical power where PN barrier is large enough for the soliton to be confined in the initial guide [@ahufinger2004creation]. In order to determine these zones, solitons with different configurations of $ k $ and $ P $ were propagated. For the case of good mobility, their effective displacement was quantified, in terms of their center of mass $CM \coloneqq \sum_{n = 1}^{N} n |\phi_ {n}|^2/P$, as shown in Figure \[fig2\](a). It is clear that for power greater than $P\approx 3.00 $ the mobility of the soliton is almost zero no matter which was the impinged momentum on the mode. In order to select optimal parameter domain where coherent mobility is guaranteed we calculate the maximum variation of the velocity angle of the center of mass, along the propagation distance with respect to the initial angle. In Figure \[fig2\](b) it can be observed that for values greater than $ P \approx2.50$ the variation of the initial angle is greater than $ 5.00 $ degrees and independent of the momentum. We also refine our procedure by identifying the maximum distance at which the initial angle is conserved, undergoing a change of less than $ 10.00\% $ for different configurations of $ k $ and $ P $. As shown in Figure \[fig2\](c), for powers below $ P \approx 2.30 $ the criterion is fulfilled for the entire propagation distance. The above suggests that the range $ \mid k \mid \leq \pi/9 $ and $ P <2.30 $ ensures a coherence mobility of the information. ![Effective displacement of the center of mass (a), maximum variation of the velocity angle of the center of mass (b) and maximum distance at which change in the velocity angle is below $ 10.00 \% $ (c), as function of $P$ and $k$. []{data-label="fig2"}](fig2_corregida.png){width="100.00000%"} Pulse transmission {#scattering} ================== Let us now consider the interaction between a discrete soliton, endowed with a transverse momentum $k$, and one impurity located at $ n_i = N / 2 $. We numerically integrate the model (\[eq1\]) with a Runge Kutta scheme by taking as initial condition $\phi_n^k=\phi_n\exp{(ikn)}$, being $\phi_n$ a stationary mode of the system. When the soliton reaches the impurity, the radiation can be reflected, transmitted and/or captured. To analyze the soliton-impurity interaction we define the reflectance $ (R) $, transmittance $ (T) $ and the capture fraction $ (C) $ coefficients as $$R\coloneqq \frac{\sum_{n=1}^{n_{i}-\Delta}|\phi_{n}|^2}{\sum_{n=1}^{N}|\phi_{n}|^2}, \hspace{0.5cm} T\coloneqq \frac{\sum_{n=n_{i}+\Delta}^{N}|\phi_{n}|^2}{\sum_{n=1}^{N}|\phi_{n}|^2}, \hspace{0.5cm} C\coloneqq \frac{\sum_{n_{i}-\Delta}^{n_{i}+\Delta}|\phi_{n}|^2}{\sum_{n=1}^{N}|\phi_{n}|^2}. \label{rtc}$$ Here $ \Delta $ is defined as the size of the impurity. As we are dealing with conservative systems, it is clear that the condition $ R + T + C = 1 $ must be fulfilled during beam evolution. It has been observed that when solitons of the same power with different $ k $ are sent toward the impurity, there is a critical momentum $ k_ {c} $ for which the soliton is trapped around the defect [@mejia2019nonlinear]. Therefore, for values $ k <k_ {c} $ the solitons are reflected and for $ k> k_ {c} $ they are transmitted. We observe a similar behavior here, by varying the optical power of the soliton with a fixed $ k $, which interacts with the impurity. As can be seen from Figure \[fig3\](a), it is clear that impurity behaves like a filter once the power of modes is near to $P_c\approx 1.674$, namely, the critical power. Around this value of $P_c$ there exist a narrow region where light can be trapped by the impurity \[cf. Figure \[fig3\](b)\] over significant distances of propagation. Below this critical power ($ P_ {c} $), the radiation becomes transmitted almost entirely as is illustrated in Figure \[fig3\](c). ![Reflection (a), Capture (b) and Transmission (c) coefficients as function of $P$ and $z$ near of the critical power $ P_ {c} = 1.6732 $, for $ k = \pi/18 $, $ \zeta = \gamma = 1.00 $ and $\gamma_{i}=0$.[]{data-label="fig3"}](fig3_corregida.png){width="100.00000%"} With the aim to have a more detailed landscape on how the optical power and the transverse momentum affect the soliton-impurity interaction, we analyze the propagation of solitons with different configurations of $ k $ and $ P $, moving towards the defect, for two different values of $\gamma_{i}$. In order to ensure a consistent estimation of $R$, $C$ and $T$ coefficients, we are going to calculate coefficients at the longitudinal distance $ z $, after the collision with the impurity, equal to the longitudinal distance that they had to travel before to collide with. Figure (\[fig4\]) display these coefficients for two values of nonlinear parameter; upper row for $\gamma_i=0.90$ and lower row for $\gamma_i=0.00$. Domains where total reflection $R$ and transmission $T$ are guaranteed in each case are displayed in first and third column, respectively. Likewise, sub-spaces where solitons can be trapped around the impurity correspond with bright spots in middle column. Finally, comparing the two cases, it is observed that by increasing the value of the nonlinearity of the defect, a shift or increase in the critical values for which the impurity behaves like a filter is obtained, approaching the homogeneous regime when the order between the nonlinear constant of the impurity and the network tends to one. ![Color map of the coefficients (R) Reflectance (C) Capture and (T) Transmittance as function of the transverse momentum $ k $ and the optical power, for $ \zeta = \gamma = 1.00 $ and $\gamma_{i}=0.90$ for the upper row and $\gamma_{i}=0.00$ for the lower row.[]{data-label="fig4"}](fig4.png){width="100.00000%"} Conclusions =========== In this work we address the problem of interaction between discrete solitons and a linear impurity in a photonic lattice composed by a one dimensional waveguides array. To do that, we begun calculating the stationary modes that exist around the defect. The analysis of the mobility of solitons far away from impurity allow us to determine the areas where the high mobility and coherence of information is guaranteed, as function of the transverse momentum and the optical power. The interaction between the impurity and the nonlinear modes displays a corpuscular dynamic. The regions of reflectance and total transmittance in which the impurity behaves as an optical limiter were determined. Besides, domains for critical power and momentum values in which the soliton is trapped around the impurity where identified. Depending of the subspace in the parameter space we observe drastic change in the reflectance and transmittance coefficients as function of $\gamma_{i}$. We hope that these results may be interesting in the design of optical limiters for solitons, that allow optimization of the manipulation and transmission of information within novel photonic chips. 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--- abstract: 'Based on projective representations of smooth Deligne cohomology groups, we introduce an analogue of the space of conformal blocks to compact oriented $(4k+2)$-dimensional Riemannian manifolds with boundary. For the standard $(4k+2)$-dimensional disk, we compute the space concretely to prove that its dimension is finite.' address: ' Graduate school of Mathematical Sciences, The University of Tokyo, Komaba 3-8-1, Meguro-Ku, Tokyo, 153-8914 Japan. ' author: - Kiyonori Gomi title: 'An analogue of the space of conformal blocks in $(4k+2)$-dimensions' --- Introduction ============ As a fundamental ingredient, the *space of conformal blocks* (or the space of vacua) in the Wess-Zumino-Witten model has been investigated by many physicists and mathematicians. While its construction usually appeals to representations of affine Lie algebras [@T-U-Y; @U], the formulation by means of representations of loop groups ([@Bry-M; @S; @W2]) provides schemes for generalizations. The theme of the present paper is an analogue of the space of conformal blocks in $(4k+2)$-dimensions. The idea of introducing such an analogue is to utilize *smooth Deligne cohomology groups* ([@Bry; @D-F; @E-V]), or the groups of *differential characters* ([@C-S]), instead of loop groups. In [@Go1; @Go2], some properties of smooth Deligne cohomology groups, such as projective representations, are studied. In a recent work of Freed, Moore and Segal [@F-M-S], similar representations are also studied in a context of *chiral (or self-dual) $2k$-forms* ([@W2]) on $(4k+2)$-dimensional spacetimes. Our analogue of the space of conformal blocks is a vector space ${\mathbb{V}}(W, \lambda)$ associated to a compact oriented $(4k+2)$-dimensional Riemannian manifold with boundary and an element $\lambda$ in a finite set $\Lambda({\partial}W)$. The finite set $\Lambda({\partial}W)$ is the set of equivalence classes of irreducible *admissible representations* ([@Go2]) of the smooth Deligne cohomology group ${\mathcal{G}}({\partial}W) = H^{2k+1}({\partial}W, \SDC{2k+1})$. As will be detailed in the body of this paper (Section \[sec:analogue\]), ${\mathbb{V}}(W, \lambda)$ consists roughly of (dual) vectors in an irreducible representation realizing $\lambda$ which are invariant under actions of *chiral (or self-dual) $2k$-forms*([@F-M-S; @W1]) on $W$. In the case of $k = 0$, we can interpret ${\mathbb{V}}(W, \lambda)$ as the space of conformal blocks (or modular functor [@S]) based on representations of abelian loop groups. For example, we take $W$ to be the 2-dimensional disk $W = D^2$. In this case, ${\mathcal{G}}(S^1) = H^1(S^1, \SDC{1})$ is isomorphic to the loop group $LU(1)$. Irreducible admissible representations give rise to irreducible *positive energy representations* ([@P-S]) of the loop group $LU(1)$ of level 2, which are classified by $\Lambda(S^1) \cong {\mathbb{Z}}_2$. Then the definition of ${\mathbb{V}}(D^2, \lambda)$ can be read as: $${\mathbb{V}}(D^2, \lambda) = \{ \psi : {\mathcal{H}}_\lambda \to {\mathbb{C}}|\ \mbox{invariant under $\mathrm{Hol}(D^2, {\mathbb{C}}/{\mathbb{Z}})$} \},$$ where ${\mathcal{H}}_\lambda$ is an irreducible representation corresponding to $\lambda$ on which the group $\mathrm{Hol}(D^2, {\mathbb{C}}/{\mathbb{Z}})$ of holomorphic maps $f : D^2 \to {\mathbb{C}}/{\mathbb{Z}}$ acts densely and linearly through the “Segal-Witten reciprocity law” [@Bry-M; @S; @W2]. A property generally desired for ${\mathbb{V}}(W, \lambda)$ is its finite-dimensionality. In the case of $k = 0$, there is a result of Segal regarding the property [@S]. The purpose of this paper is to prove that ${\mathbb{V}}(W, \lambda)$ is finite-dimensional at least in the case where $W$ is the $(4k+2)$-dimensional disk $D^{4k+2} = \{ x \in {\mathbb{R}}^{4k+2} |\ \lvert x \rvert \le 1 \}$. Note that we have $\Lambda(S^{4k+1}) = \{ 0 \}$ for $k > 0$. Then we can show: \[thm:main\] If $k > 0$, then ${\mathbb{V}}(D^{4k+2}, 0) \cong {\mathbb{C}}$. The essential part of the proof is a fact about chiral $2k$-forms on $D^{4k+2}$, which we derive from [@I-T]. (See Section \[sec:disk\] for detail.) The proof of Theorem \[thm:main\] is applicable to the case of $k = 0$, and we have: $${\mathbb{V}}(D^2, \lambda) \cong \left\{ \begin{array}{cc} {\mathbb{C}}, & (\lambda = 0) \\ 0. & (\lambda = 1) \end{array} \right.$$ This result is consistent with the known fact about the dimension of the space of conformal blocks in the $U(1)$ Wess-Zumino-Witten model at level 2 ([@S; @U]). The finite-dimensionality of ${\mathbb{V}}(W, \lambda)$ for general $W$ remains open at present. A possible approach toward the issue is to generalize Segal’s idea (p.431, [@S]), which should be examined in future studies. Analogue of the space of conformal blocks {#sec:analogue} ========================================= In this section, we introduce the vector space ${\mathbb{V}}(W, \lambda)$. For this aim, we summarize some results in [@Go1; @Go2]. In particular, we review central extensions of smooth Deligne cohomology groups, a generalization of the Segal-Witten reciprocity law, and admissible representations. Central extension ----------------- To begin with, we recall the definition of *smooth Deligne cohomology* [@Bry; @D-F; @E-V]. For a non-negative integer $p$ and a smooth manifold $X$, the *(complexified) smooth Deligne cohomology group* $H^*(X, \CSDC{p})$ is defined to be the hypercohomology of the following complex of sheaves on $X$: $$\CSDC{q} : \ {\mathbb{Z}}\longrightarrow {\underline}{A}^0_{\mathbb{C}}\overset{d}{\longrightarrow} {\underline}{A}^1_{\mathbb{C}}\overset{d}{\longrightarrow} \cdots \overset{d}{\longrightarrow} {\underline}{A}^{q-1}_{\mathbb{C}}\longrightarrow 0 \longrightarrow \cdots,$$ where ${\mathbb{Z}}$ is the constant sheaf, and ${\underline}{A}^q_{\mathbb{C}}$ the sheaf of germs of ${\mathbb{C}}$-valued $q$-forms. We fix a non-negative integer $k$, and put ${\mathcal{G}}(X)_{\mathbb{C}}= H^{2k+1}(X, \CSDC{2k+1})$ for a smooth manifold $X$. For a compact oriented $(4k+1)$-dimensional smooth manifold $M$ without boundary, there is a non-trivial central extension $\til{{\mathcal{G}}}(M)_{\mathbb{C}}$ of ${\mathcal{G}}(M)_{\mathbb{C}}$: $$\begin{CD} 1 @>>> {\mathbb{C}}^* @>>> \til{{\mathcal{G}}}(M)_{\mathbb{C}}@>>> {\mathcal{G}}(M)_{\mathbb{C}}@>>> 1. \end{CD}$$ The central extension $\til{{\mathcal{G}}}(M)_{\mathbb{C}}$ is induced from the group 2-cocycle $S_{M, {\mathbb{C}}} : {\mathcal{G}}(M)_{\mathbb{C}}\times {\mathcal{G}}(M)_{\mathbb{C}}\to {\mathbb{C}}/{\mathbb{Z}}$ defined by $S_{M, {\mathbb{C}}}(f, g) = \int_M f \cup g$, where $\int_M$ and $\cup$ are the cup product and the integration in smooth Deligne cohomology. For a smooth manifold $X$, the smooth Deligne cohomology $H^1(X, \CSDC{1})$ is naturally isomorphic to $C^\infty(X, {\mathbb{C}}/{\mathbb{Z}})$. Thus, if $k = 0$ and $M = S^1$, then we can identify ${\mathcal{G}}(S^1)_{\mathbb{C}}$ with the loop group $L{\mathbb{C}}^*$. In this case, $\til{{\mathcal{G}}}(S^1)_{\mathbb{C}}$ is isomorphic to $\widehat{L{\mathbb{C}}^*}/{\mathbb{Z}}_2$, where $\widehat{L{\mathbb{C}}^*}$ is the *universal central extension* of $L{\mathbb{C}}^*$, ([@P-S]). A generalization of the Segal-Witten reciprocity law ---------------------------------------------------- Let $W$ be a compact oriented $(4k+2)$-dimensional Riemannian manifold $W$ possibly with boundary. We denote by $A^{2k+1}(W, {\mathbb{C}})$ the space of ${\mathbb{C}}$-valued $(2k+1)$-forms on $W$. The Hodge star operator $* : A^{2k+1}(W, {\mathbb{C}}) \to A^{2k+1}(W, {\mathbb{C}})$ satisfies $** = -1$. Notice that, in general, the smooth Deligne cohomology ${\mathcal{G}}(X_{\mathbb{C}}) = H^{2k+1}(X, \CSDC{2k+1})$ fits into the following exact sequence. $$0 \to H^{2k}(W, {\mathbb{C}}/{\mathbb{Z}}) \to {\mathcal{G}}(W)_{\mathbb{C}}\overset{\delta}{\to} A^{2k+1}(W, {\mathbb{C}})_{\mathbb{Z}}\to 0,$$ where $A^{2k+1}(W, {\mathbb{C}})_{\mathbb{Z}}\subset A^{2k+1}(W, {\mathbb{C}})$ is the subgroup consisting of closed integral forms. Using $*$ and $\delta$, we define the subgroups ${\mathcal{G}}(W)_{\mathbb{C}}^{\pm}$ in ${\mathcal{G}}(W)_{\mathbb{C}}$ by $${\mathcal{G}}(W)^\pm_{\mathbb{C}}= \{ f \in {\mathcal{G}}(W)_{\mathbb{C}}|\ \delta(f) \mp {\sqrt{\! - \! 1}}* \delta(f) = 0\}.$$ We call ${\mathcal{G}}(W)_{\mathbb{C}}^+$ the *chiral subgroup*, since $2k$-forms $B \in A^{2k}(W, {\mathbb{C}})$ such that $dB = i*dB$ are called *chiral (or self-dual) $2k$-forms*. (See [@F-M-S; @W1] for example.) \[prop:reciprocity\] For a compact oriented $(4k+2)$-dimensional Riemannian manifold $W$ with boundary, the following map is a homomorphism: $$\til{r}^+ : {\mathcal{G}}(W)_{\mathbb{C}}^+ \longrightarrow \til{{\mathcal{G}}}({\partial}W)_{\mathbb{C}}, \quad f \mapsto (f|_{{\partial}W}, 1).$$ In the case of $k = 0$, $W$ is a Riemann surface. Since ${\mathcal{G}}(W)_{\mathbb{C}}^+$ is identified with the group of holomorphic functions $f : W \to {\mathbb{C}}/{\mathbb{Z}}$, Proposition \[prop:reciprocity\] recovers the “Segal-Witten reciprocity law”([@Bry-M; @S; @W2]) for $\widehat{L{\mathbb{C}}^*}/{\mathbb{Z}}_2$. Admissible representations -------------------------- The group ${\mathcal{G}}(X)_{\mathbb{C}}= H^{2k+1}(X, \CSDC{2k+1})$ can be thought of as a complexification of the (real) smooth Deligne cohomology ${\mathcal{G}}(X) = H^{2k+1}(X, \SDC{2k+1})$ defined as the hypercohomology of the following complex of sheaves: $$\SDC{2k+1}: \ {\mathbb{Z}}\longrightarrow {\underline}{A}^0 \overset{d}{\longrightarrow} {\underline}{A}^1 \overset{d}{\longrightarrow} \cdots \overset{d}{\longrightarrow} {\underline}{A}^{2k} \longrightarrow 0 \longrightarrow \cdots,$$ where ${\underline}{A}^q$ is the sheaf of germs of ${\mathbb{R}}$-valued $q$-forms. For a compact oriented $(4k+1)$-dimensional Riemannian manifold $M$ without boundary, *admissible representations* of ${\mathcal{G}}(M)$ are introduced in [@Go2]. An admissible representation $\rho : {\mathcal{G}}(M) \times {\mathcal{H}}\to {\mathcal{H}}$ of ${\mathcal{G}}(M)$ is a certain projective representation on a Hilbert space ${\mathcal{H}}$, and gives a linear representation $\til{\rho} : \til{{\mathcal{G}}}(M) \times {\mathcal{H}}\to {\mathcal{H}}$ of the central extension $\til{{\mathcal{G}}}(M)$ induced from the natural inclusion ${\mathcal{G}}(M) \subset {\mathcal{G}}(M)_{\mathbb{C}}$: $$\begin{CD} 1 @>>> U(1) @>>> \til{{\mathcal{G}}}(M) @>>> {\mathcal{G}}(M) @>>> 1 \\ @. @VVV @VVV @VVV @. \\ 1 @>>> {\mathbb{C}}^* @>>> \til{{\mathcal{G}}}(M)_{\mathbb{C}}@>>> {\mathcal{G}}(M)_{\mathbb{C}}@>>> 1. \end{CD}$$ The set $\Lambda(M)$ of equivalence classes of irreducible admissible representations of ${\mathcal{G}}(M)$ is a finite set [@Go2]. For example, if $H^{2k+1}(M, {\mathbb{Z}})$ is torsion free, then we can identify $\Lambda(M)$ with $H^{2k+1}(M, {\mathbb{Z}}_2)$. We write $(\til{\rho}_\lambda, {\mathcal{H}}_\lambda)$ for the linear representation of $\til{{\mathcal{G}}}(M)$ realizing $\lambda \in \Lambda$. \[prop:complex\_extension\] Let $M$ be a compact oriented $(4k+1)$-dimensional Riemannian manifold $M$ without boundary. For $\lambda \in \Lambda(M)$, there exists an invariant dense subspace $\mathcal{E}_\lambda \subset {\mathcal{H}}_\lambda$, and the representation $\til{\rho}_\lambda : \til{{\mathcal{G}}}(M) \times \mathcal{E}_\lambda \to \mathcal{E}_\lambda$ extends to a linear representation $\til{\rho}_\lambda : \til{{\mathcal{G}}}(M)_{\mathbb{C}}\times \mathcal{E}_\lambda \to \mathcal{E}_\lambda$ of $\til{{\mathcal{G}}}(M)_{\mathbb{C}}$. We notice that $\til{\rho}_\lambda(f) : {\mathcal{E}}_\lambda \to {\mathcal{E}}_\lambda$ is generally unbounded, so that the action of $\til{{\mathcal{G}}}(M)_{\mathbb{C}}$ on ${\mathcal{E}}_\lambda$ does not extends to the whole of ${\mathcal{H}}_\lambda$. In the case of $k = 0$ and $M = S^1$, we can identify ${\mathcal{G}}(S^1)$ with the loop group $LU(1)$, which has ${\mathcal{G}}(S^1)_{\mathbb{C}}\cong L{\mathbb{C}}^*$ as a complexification. Admissible representations of ${\mathcal{G}}(S^1)$ give rise to positive energy representations of level 2. As is known [@P-S], the equivalence classes of irreducible positive energy representations of $LU(1)$ of level 2 are in one to one correspondence with the elements in $\Lambda(S^1) \cong {\mathbb{Z}}_2$. A positive energy representation of $LU(1)$ extends to a representation of $L{\mathbb{C}}^*$ on an invariant dense subspace. Analogue of the space of conformal blocks {#analogue-of-the-space-of-conformal-blocks} ----------------------------------------- We use Proposition \[prop:reciprocity\] and Proposition \[prop:complex\_extension\] to formulate our analogue of the space of conformal blocks: Let $W$ be a compact oriented $(4k+2)$-dimensional Riemannian manifold with boundary. For $\lambda \in \Lambda({\partial}W)$, we define ${\mathbb{V}}(W, \lambda)$ to be the vector space consisting of continuous linear maps $\psi : {\mathcal{E}}_\lambda \to {\mathbb{C}}$ invariant under the action of ${\mathcal{G}}(W)_{\mathbb{C}}^+$ through $\til{r}^+$: $$\begin{split} {\mathbb{V}}(W, \lambda) &= {\mathrm{Hom}}(\mathcal{E}_\lambda, {\mathbb{C}})^{\mathrm{Im}\til{r}^+} \\ &= \{ \psi : \mathcal{E}_\lambda \to {\mathbb{C}}|\ \psi( \tilde{\rho}_\lambda(\til{r}^+(f)) v ) = \psi(v) \ \mbox{for $v \in \mathcal{E}_\lambda$ and $f \in {\mathcal{G}}(W)_{\mathbb{C}}^+$} \}. \end{split}$$ Since the subgroup ${\mathbb{C}}^*$ in $\til{{\mathcal{G}}}(M)_{\mathbb{C}}= {\mathcal{G}}(M)_{\mathbb{C}}\times {\mathbb{C}}^*$ acts on $\mathcal{E}_\lambda$ by the scalar multiplication, we can formulate ${\mathbb{V}}(W, \lambda)$ in terms of the projective representation $(\rho_\lambda, \mathcal{E}_\lambda)$ corresponding to $(\til{\rho}_\lambda, \mathcal{E}_\lambda)$: $$\begin{aligned} {\mathbb{V}}(W, \lambda) &= {\mathrm{Hom}}({\mathcal{E}}_\lambda, {\mathbb{C}})^{\mathrm{Im}r^+} \\ &= \{ \psi : {\mathcal{E}}_\lambda \to {\mathbb{C}}|\ \psi( \rho_\lambda(r^+(f)) v ) = \psi(v) \ \mbox{for $v \in {\mathcal{E}}_\lambda$ and $f \in {\mathcal{G}}(W)_{\mathbb{C}}^+$} \},\end{aligned}$$ where $r^+ : {\mathcal{G}}(W)_{\mathbb{C}}^+ \to {\mathcal{G}}({\partial}W)_{\mathbb{C}}$ is the restriction: $r^+(f) = f|_{{\partial}W}$. One may wonder why we use representations of $\til{{\mathcal{G}}}(M)_{\mathbb{C}}$ on pre-Hilbert spaces to formulate ${\mathbb{V}}(W, \lambda)$, instead of unitary representations of $\til{{\mathcal{G}}}(M)$ on Hilbert spaces. The reason is that we cannot introduce a counterpart of the chiral subgroup ${\mathcal{G}}(W)_{\mathbb{C}}^+$ to ${\mathcal{G}}(W)$. Notice, however, that we can formulate ${\mathbb{V}}(W, \lambda)$ as $${\mathbb{V}}(W, \lambda) = \{ \psi : {\mathcal{H}}_\lambda \to {\mathbb{C}}|\ \psi( \rho_\lambda(r^+(f)) v ) = \psi(v) \ \mbox{for $v \in \mathcal{E}_\lambda$ and $f \in {\mathcal{G}}(W)_{\mathbb{C}}^+$} \},$$ because ${\mathcal{E}}_\lambda$ is dense in ${\mathcal{H}}_\lambda$. Calculation of ${\mathbb{V}}(D^{4k+2}, \lambda)$ {#sec:disk} ================================================ In this section, we prove Theorem \[thm:main\]. As preparations for the proof, we review in some detail the construction of irreducible representations of Heisenberg groups in [@P-S]. We also study chiral $2k$-forms on ${\mathbb{R}}^{4k+2}$ by the help of results in [@I-T]. Representation of Heisenberg group {#subsec:Heisenberg_representation} ---------------------------------- For a compact oriented $(4k+1)$-dimensional Riemannian manifold $M$ without boundary, the group ${\mathcal{G}}(M)_{\mathbb{C}}$ admits the decomposition: $$\begin{aligned} {\mathcal{G}}(M)_{\mathbb{C}}&\cong (A^{2k}(M, {\mathbb{C}})/A^{2k}(M, {\mathbb{C}})_{\mathbb{Z}}) \times H^{2k+1}(M, {\mathbb{Z}}) \\ &\cong ({\mathbb{H}}^{2k}(M, {\mathbb{C}})/{\mathbb{H}}^{2k}(M, {\mathbb{C}})_{\mathbb{Z}}) \times d^*(A^{2k+1}(M, {\mathbb{C}})) \times H^{2k+1}(M, {\mathbb{Z}}),\end{aligned}$$ where ${\mathbb{H}}^{2k}(M, {\mathbb{C}})$ is the group of ${\mathbb{C}}$-valued harmonic $2k$-forms, ${\mathbb{H}}^{2k}(M, {\mathbb{C}})_{\mathbb{Z}}= {\mathbb{H}}^{2k}(M, {\mathbb{C}}) \cap A^{2k}(M, {\mathbb{C}})_{\mathbb{Z}}$ the subgroup of integral harmonic $2k$-forms, and $d^* : A^{2k+1}(M, {\mathbb{C}}) \to A^{2k}(M, {\mathbb{C}})$ the formal adjoint of the exterior differential. Thus, in particular, if $M$ is such that $b^{2k+1}(M) = 0$, then ${\mathcal{G}}(M)_{\mathbb{C}}\cong d^*(A^{2k+1}(M, {\mathbb{Z}}))$. The representations $(\til{\rho}_\lambda, {\mathcal{E}}_\lambda)$ of $\til{{\mathcal{G}}}(M)_{\mathbb{C}}$ in Proposition \[prop:complex\_extension\] are built on a projective representation $(\rho, E)$ of $d^*(A^{2k+1}(M, {\mathbb{C}})$. We review here the construction of $(\rho, E)$ following [@P-S], and give a simple consequence. As in [@Go2], we define the Hermitian inner product $( \ , \ )_V$ on $d^*(A^{2k+1}(M, {\mathbb{C}}))$ by that induced from the Sobolev norm $\p{ \ \cdot \ }_s$ with $s = 1/2$. (Our convention is that $( \ , \ )_V$ is ${\mathbb{C}}$-linear in the first variable, which differs from that in [@P-S].) On the completion $V_{\mathbb{C}}$ of $d^*(A^{2k+1}(M, {\mathbb{C}}))$, we define the linear map $J : V_{\mathbb{C}}\to V_{\mathbb{C}}$ by $J = \til{J}/\abs{\til{J}}$, where $\til{J} : d^*(A^{2k+1}(M, {\mathbb{C}})) \to d^*(A^{2k+1}(M, {\mathbb{C}}))$ is the differential operator $\til{J} = *d$. Then $J$ is a complex structure compatible with $( \ , \ )_V$, and satisfies: $$(\alpha, J\overline{\beta})_V = \int_M \alpha \wedge d \beta$$ for $\alpha, \beta \in d^*(A^{2k+1}(M, {\mathbb{C}}))$. By means of $J$, we decompose $V_{\mathbb{C}}$ into $V_{\mathbb{C}}= W \oplus \overline{W}$, where $J$ acts on $W$ and $\overline{W}$ by ${\sqrt{\! - \! 1}}$ and $-{\sqrt{\! - \! 1}}$, respectively. Then we let $E = {\mathbb{C}}\langle \epsilon_\xi |\ \xi \in W \rangle$ be the vector space generated by the symbols $\epsilon_\xi$ corresponding to $\xi \in W$, and $\langle \ , \ \rangle : E \times E \to {\mathbb{C}}$ the Hermitian inner product $\langle \epsilon_\xi, \epsilon_\eta \rangle = e^{2(\xi, \eta)_V}$. For $v_+ \in W$ and $v_- \in \overline{W}$, we define $\rho(v_+ + v_-) : \ E \to E$ by $$\rho(v_+ + v_-) \epsilon_\xi = \exp\left( - ( v_+, \overline{(v_-)} )_V - 2 ( \xi, \overline{(v_-)} )_V \right) \epsilon_{\xi + v_+}.$$ We can verify $\rho(v)\rho(v') \epsilon_\xi = e^{{\sqrt{\! - \! 1}}(v, J \bar{v}')_V} \rho(v + v') \epsilon_\xi$ for $v, v' \in V_{\mathbb{C}}$, so that we have a projective representation $\rho : V_{\mathbb{C}}\times E \to E$. Because the group 2-cocycle $S_{M, {\mathbb{C}}}$ on $d^*(A^{2k+1}(M, {\mathbb{C}}))$ has the expression : $$S_{M, {\mathbb{C}}} (\alpha, \beta) = \int_M \alpha \wedge d\beta \mod {\mathbb{Z}},$$ we get the projective representation $\rho : d^*(A^{2k+1}(M, {\mathbb{C}})) \times E \to E$. In general, $\rho(\alpha) : E \to E$ is unbounded. However, if $\alpha$ belongs to the real vector space $d^*(A^{2k+1}(M))$ underlying $d^*(A^{2k}(M, {\mathbb{C}}))$, then $\rho(\alpha) : E \to E$ is isometric. Thus, $\rho(\alpha)$ extends to a unitary map on the completion $H = \overline{E}$ of $E$, and we have an irreducible projective unitary representation $\rho : d^*(A^{2k+1}(M)) \times H \to H$. As is shown in [@P-S], we can identify $\overline{E}$ with a completion of the symmetric algebra $S(W)$ by the mapping $\epsilon_\xi \mapsto e^\xi = \sum_{j = 0}^\infty \xi^j/j!$. The next lemma is a simple consequence from the above construction. \[lem:Heisenberg\_representation\] Let $(\rho, E)$ be as above. \(a) The vector space ${\mathrm{Hom}}(E, {\mathbb{C}})^W$ is generated by the continuous linear map $\chi : E \to {\mathbb{C}}$ defined by $\chi(v) = \langle v, \epsilon_0 \rangle$: $${\mathrm{Hom}}(E, {\mathbb{C}})^W = {\mathbb{C}}\langle \chi \rangle.$$ \(b) For a dense subspace $U$ in $W$, we have ${\mathrm{Hom}}(E, {\mathbb{C}})^U = {\mathrm{Hom}}(E, {\mathbb{C}})^W$. To prove (a), we begin with proving the $W$-invariance of $\chi$. Notice that $\chi(\epsilon_\xi) = 1$ for all $\xi \in W$. For $f \in W$ and $v = \sum_j c_j \epsilon_{\xi_j} \in E$, we have: $$\begin{aligned} \chi(v) &= \sum_j c_j \chi(\epsilon_{\xi_j}) = \sum_j c_j, \\ \chi((\rho(f)v) &= \sum_j c_j \chi( \rho(f)\epsilon_{\xi_j} ) = \sum_j c_j \chi(\epsilon_{\xi_j + f}) = \sum_j c_j.\end{aligned}$$ Hence $\chi$ is invariant under the action of $W$, and ${\mathbb{C}}\langle \chi \rangle \subset {\mathrm{Hom}}(E, {\mathbb{C}})^W$. To see ${\mathbb{C}}\langle \chi \rangle \supset {\mathrm{Hom}}(E, {\mathbb{C}})^W$, we show that any $\psi \in {\mathrm{Hom}}(E, {\mathbb{C}})^W$ is of the form $\psi = c \chi$ for some $c \in {\mathbb{C}}$. For $v = \sum_j c_j \epsilon_{\xi_j} \in E$, the invariance of $\psi$ leads to: $$\begin{split} \psi(v) &= \sum_j c_j \psi(\epsilon_{\xi_j}) = \sum_j c_j \psi(\rho(\xi_j) \epsilon_0) = \sum_j c_j \psi(\epsilon_0) \\ &= \psi(\epsilon_0) \sum_j c_j = \psi(\epsilon_0) \chi(v). \end{split}$$ If we put $c = \psi(\epsilon_0)$, then $\psi = c \chi$. For (b), it suffices to prove the inclusion ${\mathrm{Hom}}(E, {\mathbb{C}})^U \subset {\mathrm{Hom}}(E, {\mathbb{C}})^W$. So we will show $\psi \in {\mathrm{Hom}}(E, {\mathbb{C}})^U$ is also invariant under $W$. For $f \in W$, there is a sequence $\{ f_n \}$ in $U$ converging to $f$. Notice that $\rho(\cdot)v : W \to E$ is continuous for $v \in E$. Now, we have: $$\psi(\rho(f)v) = \psi(\rho(\lim_{n \to \infty} f_n) v) = \lim_{n \to \infty} \psi(\rho(f_n)v) = \lim_{n \to \infty} \psi(v) = \psi(v),$$ so that $\psi \in {\mathrm{Hom}}(E, {\mathbb{C}})^W$. The key to Lemma \[lem:Heisenberg\_representation\] (b) is that the map $\rho(\cdot)v : W \to E$ is continuous for each $v \in W$. The representations $\til{\rho}_\lambda : \til{{\mathcal{G}}}(M)_{\mathbb{C}}\times \mathcal{E}_\lambda \to \mathcal{E}_\lambda$ in Proposition \[prop:complex\_extension\] have the same property [@Go2]. Chiral $2k$-forms on ${\mathbb{R}}^{4k+2}$ ------------------------------------------ The Laplacian $\Delta = dd^* + d^*d$ preserves the subspace $d^*(A^{2k+1}(S^{4k+1}, {\mathbb{C}}))$. For an eigenvalue ${\ell}$ of $\Delta$, we define $V_{\ell}$ to be the following eigenspace: $$V_{\ell}= \{ \beta \in d^*(A^{2k+1}(S^{4k+1}, {\mathbb{C}})) |\ \Delta \beta = {\ell}\beta \}.$$ The complex structure $J$, introduced in the previous subsection, preserves $V_{\ell}$. (In particular, $J = *d/\sqrt{{\ell}}$ on $V_{\ell}$.) So we have the decomposition $V_{\ell}= W_{\ell}\oplus \overline{W}_{\ell}$, where $J$ acts on $W_{\ell}$ and $\overline{W}_{\ell}$ by ${\sqrt{\! - \! 1}}$ and $-{\sqrt{\! - \! 1}}$, respectively. \[prop:chiral\_form\_plane\] There is the following relation of inclusion: $$\bigoplus_{{\ell}} W_{{\ell}} \subset \mathrm{Im} \{ i^* : A^{2k}({\mathbb{R}}^{4k+2}, {\mathbb{C}})^+ \to A^{2k}(S^{4k+1}, {\mathbb{C}}) \} \subset W,$$ where $\bigoplus$ means the algebraic direct sum, ${\ell}$ runs through all the distinct eigenvalues, $i : S^{4k+1} \to {\mathbb{R}}^{4k+2}$ is the inclusion, and $A^{2k}({\mathbb{R}}^{4k+2}, {\mathbb{C}})^{\pm}$ are the spaces of chiral and anti-chiral $2k$-forms on ${\mathbb{R}}^{4k+2}$: $$A^{2k}({\mathbb{R}}^{4k+2}, {\mathbb{C}})^\pm = \{ B \in A^{2k}({\mathbb{R}}^{4k+2}, {\mathbb{C}}) |\ d B \mp {\sqrt{\! - \! 1}}*d B = 0 \}.$$ For the proof, we use some results shown by Ikeda and Taniguchi in [@I-T]. To explain the relevant results, we introduce some notations. Let $S^i({\mathbb{R}}^{4k+1})$ and $\Lambda^p({\mathbb{R}}^{4k+1})$ be the spaces of the symmetric tensors of degree $i$ and anti-symmetric tensors of degree $p$. We put $P^p_i = S^i({\mathbb{R}}^{4k+1}) \otimes \Lambda^p({\mathbb{R}}^{4k+1}) \otimes {\mathbb{C}}$, and regard $P^p_i$ as a subspace in $A^p({\mathbb{R}}^{4k+1}, {\mathbb{C}})$. We then define the vector spaces: $$\begin{aligned} H^p_i &= \mathrm{Ker} \Delta \cap {\mathrm{Ker}}d^* \cap P^p_i, \\ {}'\!H^p_i &= \mathrm{Ker}d \cap H^p_i, \\ {}''\!H^p_i &= \mathrm{Ker} i\Big(r \frac{d}{d r} \Big) \cap H^p_i,\end{aligned}$$ where $i\left(r \frac{{\partial}}{{\partial}r} \right)$ is the contraction with the vector field $r \frac{d}{d r} = \sum_{j = 1}^{4k+2}x_j \frac{d}{dx_j}$. Notice that the standard action of $SO(4k+2)$ on ${\mathbb{R}}^{4k+2}$ makes ${}'\!H^p_i$ and ${}''\!H^p_i$ into $SO(4k+2)$-modules. Similarly, $V_{\ell}$ is also an $SO(4k+2)$-module. From [@I-T] (Theorem 6.8, p. 537), we can derive: \[prop:Ikeda\_Taniguchi\] Let ${\ell}_1 < {\ell}_2 < {\ell}_3 < \cdots$ be the sequence of distinct eigenvalues of $\Delta$ on $d^*(A^{2k+1}(S^{4k+1}, {\mathbb{C}}))$. For $i \in {\mathbb{N}}$, we have: \(a) The maps $i^* : A^{2k}({\mathbb{R}}^{4k+2}, {\mathbb{C}}) \to A^{2k}(S^{4k+1}, {\mathbb{C}})$ and $d : A^{2k}({\mathbb{R}}^{4k+2}, {\mathbb{C}}) \to A^{2k+1}({\mathbb{R}}^{4k+2}, {\mathbb{C}})$ induce the following isomorphisms of $SO(4k+2)$-modules: $$\begin{CD} V_{{\ell}_i} @<{i^*}<< {}''\!H^{2k}_i @>{d}>> {}'\!H^{2k+1}_{i-1}. \end{CD}$$ \(b) The $SO(4k+2)$-module $V_{{\ell}_i}$ decomposes into two distinct irreducible modules having the same dimensions. More precisely, the sequence $\{ {\ell}_i \}_{i \in {\mathbb{N}}}$ is given by ${\ell}_i = (2k + i)^2$, and the dimension of the two irreducible modules in $V_{l_i}$ is $\binom{4k + i}{2k} \binom{2k + i -1}{2k}$. We also note the next lemma for later use: \[lem:chiral\_or\_achiral\] Let $( \ , \ )_{L^2}$ be the $L^2$-norm on $A^{2k+1}(D^{4k+2}, {\mathbb{C}})$. \(a) For $B, B' \in A^{2k}({\mathbb{R}}^{4k+2}, {\mathbb{C}})$, we have: $$(i^*B, J i^*B')_V = - (dB, *dB')_{L^2}.$$ \(b) If $B \in A^{2k+1}({\mathbb{R}}^{4k+2}, {\mathbb{C}})$ obeys $(J - \sqrt{-1})i^*B = 0$, then: $$\p{H^+}^2_{L^2} - \p{H^-}^2_{L^2} \ge 0,$$ where $H^{\pm} = (1 \pm \sqrt{-1}*)dB/2$. Similarly, if $(J + \sqrt{-1})i^*B = 0$, then: $$\p{H^-}^2_{L^2} - \p{H^+}^2_{L^2} \ge 0.$$ We can readily show (a) combining properties of $( \ , \ )_V$ and $J$ with Stokes’ theorem. Notice that the eigenspaces $\mathrm{Ker} (1 \pm {\sqrt{\! - \! 1}}*)$ in $A^{2k+1}(D^{4k+2}, {\mathbb{C}})$ are orthogonal to each other with respect to the $L^2$-norm. Then the inequalities in (b) follow from $(i^*B, i^*B)_V \ge 0$ and (a). Proposition \[prop:Ikeda\_Taniguchi\] and the above lemma yield: \[lem:chiral\_extension\_plane\] The map $i^*$ induces the following isomorphisms for $i \in {\mathbb{N}}$: $${}''\!H^{2k}_i \cap A^{2k}({\mathbb{R}}^{4k+2}, {\mathbb{C}})^+ \cong W_{{\ell}_i}, \quad {}''\!H^{2k}_i \cap A^{2k}({\mathbb{R}}^{4k+2}, {\mathbb{C}})^- \cong \overline{W}_{{\ell}_i}.$$ Notice that the action of $SO(4k+2)$ on $V_{{\ell}_i}$ is compatible with $J$. So $W_{{\ell}_i}$ and $\overline{W}_{{\ell}_i}$ are $SO(4k+2)$-modules. The dimensions of $W_{{\ell}_i}$ and $\overline{W}_{{\ell}_i}$ are the same, since they are complex-conjugate to each other. Similarly, since the $SO(4k+2)$-action on ${}'\!H^{2k+1}_{i-1}$ is compatible with the Hodge star operator $*$, the vector spaces $({}'\!H^{2k+1}_{i-1})^{\pm} = {}'\!H^{2k+1}_{i-1} \cap \mathrm{Ker}(1 \mp {\sqrt{\! - \! 1}}*)$ are also $SO(4k+2)$-modules with the same dimensions. Thus, by Proposition \[prop:Ikeda\_Taniguchi\], $W_{{\ell}_i}$ is isomorphic to one of $({}'\!H^{2k+1}_{i-1})^{\pm}$ through $d \circ (i^*)^{-1}$, and $\overline{W}_{{\ell}_i}$ is isomorphic to the other. To settle the case, we appeal to Lemma \[lem:chiral\_or\_achiral\] (b). Then the case of $W_{{\ell}_i} \cong ({}'\!H^{2k+1}_{i-1})^+$ and $\overline{W}_{{\ell}_i} \cong ({}'\!H^{2k+1}_{i-1})^-$ is consistent. Now the isomorphisms $d : {}''\!H^{2k}_i \cap A^{2k}({\mathbb{R}}^{4k+2}, {\mathbb{C}})^\pm \to ({}'\!H^{2k+1}_{i-1})^\pm$ complete the proof. By Lemma \[lem:chiral\_extension\_plane\] we have: $$W_{{\ell}_i} \subset \mathrm{Im} \{ i^* : A^{2k}({\mathbb{R}}^{4k+2}, {\mathbb{C}})^+ \to A^{2k}(S^{4k+1}, {\mathbb{C}}) \},$$ which leads to the first inclusion in Proposition \[prop:chiral\_form\_plane\]. For the second inclusion, we recall that the subspaces $W$ and $\overline{W}$ in $V_{\mathbb{C}}$ are orthogonal with respect to $( \ , \ )_V$. So, it suffices to verify the image $i^*(A^{2k}({\mathbb{R}}^{4k+2}, {\mathbb{C}})^+)$ is orthogonal to $\overline{W}$. By Lemma \[lem:chiral\_extension\_plane\], we also have: $$\overline{W}_{{\ell}_i} \subset \mathrm{Im} \{ i^* : A^{2k}({\mathbb{R}}^{4k+2}, {\mathbb{C}})^- \to A^{2k}(S^{4k+1}, {\mathbb{C}}) \}.$$ Thus, by the help of Lemma \[lem:chiral\_or\_achiral\] (a), we see that $i^*(A^{2k}({\mathbb{R}}^{4k+2}, {\mathbb{C}})^+)$ is orthogonal to each $\overline{W}_{{\ell}_i}$. Because $\bigoplus_i W_{{\ell}_i}$ forms a dense subspace in $W$, the image $i^*(A^{2k}({\mathbb{R}}^{4k+2}, {\mathbb{C}})^+)$ is orthogonal to $\overline{W}$. Proof of the main result ------------------------ We now compute ${\mathbb{V}}(D^{4k+2}, \lambda)$. First, we consider the case of $k > 0$. In this case, we have: $${\mathcal{G}}(S^{4k+1})_{\mathbb{C}}= A^{2k}(S^{4k+1}, {\mathbb{C}})/A^{2k}(S^{4k+1}, {\mathbb{C}})_{\mathbb{Z}}\cong d^*(A^{2k+1}(S^{4k+1}, {\mathbb{C}})).$$ So the projective unitary representation $(\rho, H)$ reviewed in Subsection \[subsec:Heisenberg\_representation\] realizes the unique element in $\Lambda(S^{4k+1}) = \{ 0 \}$, and $E$ gives the invariant dense subspace in Proposition \[prop:complex\_extension\]. If $k > 0$, then ${\mathbb{V}}(D^{4k+2}, 0) \cong {\mathbb{C}}$. Note that ${\mathcal{G}}(D^{4k+2})^+_{\mathbb{C}}= A^{2k}(D^{4k+2}, {\mathbb{C}})^+/A^{2k}(D^{4k+1}, {\mathbb{C}})_{\mathbb{Z}}$. Proposition \[prop:chiral\_form\_plane\] leads to: $U \subset \mathrm{Im} r^+ \subset W$, where the dense subspace $U$ in $W$ is given by $U = \bigoplus_{i \in {\mathbb{N}}} W_{{\ell}_i}$. This relation of inclusion implies: $${\mathrm{Hom}}(E, {\mathbb{C}})^U \supset {\mathrm{Hom}}(E, {\mathbb{C}})^{\mathrm{Im} r^+} \supset {\mathrm{Hom}}(E, {\mathbb{C}})^W.$$ Therefore Lemma \[lem:Heisenberg\_representation\] establishes the theorem. In the case of $k = 0$, we have the familiar decomposition of ${\mathcal{G}}(S^1)_{\mathbb{C}}\cong L{\mathbb{C}}^*$: $${\mathcal{G}}(S^1)_{\mathbb{C}}\cong {\mathbb{C}}/{\mathbb{Z}}\times \{ \phi : S^1 \to {\mathbb{R}}|\ \mbox{$\int \phi(\theta) d\theta = 0$} \} \times {\mathbb{Z}}.$$ As is mentioned, admissible representations of ${\mathcal{G}}(S^1)$ are equivalent to positive energy representations of $LU(1)$ of level 2. For $\lambda \in \Lambda(S^1) = {\mathbb{Z}}_2 = \{ 0, 1 \}$, the invariant dense subspace ${\mathcal{E}}_\lambda$ in Proposition \[prop:complex\_extension\] is given by ${\mathcal{E}}_\lambda = \bigoplus_{\xi \in {\mathbb{Z}}}E_{\lambda + 2\xi}$, where $E_{\lambda + 2\xi} = E$ is the pre-Hilbert space in Subsection \[subsec:Heisenberg\_representation\] and the subgroup of constant loops ${\mathbb{C}}/{\mathbb{Z}}\subset {\mathcal{G}}(S^1)_{\mathbb{C}}$ acts on $E_{\lambda + 2\xi}$ with weight $\lambda + 2\xi$. For $\lambda \in \Lambda(S^1) = {\mathbb{Z}}_2$, we have: $${\mathbb{V}}(D^2, \lambda) \cong \left\{ \begin{array}{cc} {\mathbb{C}}, & (\lambda = 0) \\ 0. & (\lambda = 1) \end{array} \right.$$ Clearly, constant loops $S^1 \to {\mathbb{C}}/{\mathbb{Z}}$ extend to holomorphic maps $D^2 \to {\mathbb{C}}/{\mathbb{Z}}$. So we use Proposition \[prop:chiral\_form\_plane\] to obtain the relation of inclusion: $${\mathbb{C}}/{\mathbb{Z}}\times U \subset \mathrm{Im} r^+ \subset {\mathbb{C}}/{\mathbb{Z}}\times W,$$ where $U = \bigoplus_{i \in {\mathbb{N}}} W_{{\ell}_i}$. Since ${\mathbb{C}}/{\mathbb{Z}}$ acts on $E_{\lambda + 2\xi}$ with weight $\lambda + 2\xi$, we have: $${\mathrm{Hom}}({\mathcal{E}}_\lambda, {\mathbb{C}})^{{\mathbb{C}}/{\mathbb{Z}}} \subset \prod_{\xi \in {\mathbb{Z}}} {\mathrm{Hom}}(E_{\lambda + 2\xi}, {\mathbb{C}})^{{\mathbb{C}}/{\mathbb{Z}}} \cong \left\{ \begin{array}{cc} {\mathrm{Hom}}(E_0, {\mathbb{C}}), & (\lambda = 0) \\ \{ 0 \}. & (\lambda = 1) \end{array} \right.$$ Thus, if $\lambda = 1$, then ${\mathbb{V}}(D^2, \lambda) = \{ 0 \}$. In the case of $\lambda = 0$, we have: $${\mathrm{Hom}}(E_0, {\mathbb{C}})^U \supset {\mathrm{Hom}}(E_0, {\mathbb{C}})^{\mathrm{Im}r^+} \supset {\mathrm{Hom}}(E_0, {\mathbb{C}})^W.$$ Now, Lemma \[lem:Heisenberg\_representation\] completes the proof. 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--- abstract: 'Traffic flow prediction is crucial for urban traffic management and public safety. Its key challenges lie in how to adaptively integrate the various factors that affect the flow changes. In this paper, we propose a unified neural network module to address this problem, called Attentive Crowd Flow Machine (ACFM), which is able to infer the evolution of the crowd flow by learning dynamic representations of temporally-varying data with an attention mechanism. Specifically, the ACFM is composed of two progressive ConvLSTM units connected with a convolutional layer for spatial weight prediction. The first LSTM takes the sequential flow density representation as input and generates a hidden state at each time-step for attention map inference, while the second LSTM aims at learning the effective spatial-temporal feature expression from attentionally weighted crowd flow features. Based on the ACFM, we further build a deep architecture with the application to citywide crowd flow prediction, which naturally incorporates the sequential and periodic data as well as other external influences. Extensive experiments on two standard benchmarks (i.e., crowd flow in Beijing and New York City) show that the proposed method achieves significant improvements over the state-of-the-art methods.' author: - Lingbo Liu - Ruimao Zhang - Jiefeng Peng - Guanbin Li - Bowen Du - Liang Lin bibliography: - 'ACFM-Reference.bib' title: Attentive Crowd Flow Machines --- &lt;ccs2012&gt; &lt;concept&gt; &lt;concept\_id&gt;10010520.10010553.10010562&lt;/concept\_id&gt; &lt;concept\_desc&gt;Computer systems organization Embedded systems&lt;/concept\_desc&gt; &lt;concept\_significance&gt;500&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10010520.10010575.10010755&lt;/concept\_id&gt; &lt;concept\_desc&gt;Computer systems organization Redundancy&lt;/concept\_desc&gt; &lt;concept\_significance&gt;300&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10010520.10010553.10010554&lt;/concept\_id&gt; &lt;concept\_desc&gt;Computer systems organization Robotics&lt;/concept\_desc&gt; &lt;concept\_significance&gt;100&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10003033.10003083.10003095&lt;/concept\_id&gt; &lt;concept\_desc&gt;Networks Network reliability&lt;/concept\_desc&gt; &lt;concept\_significance&gt;100&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;/ccs2012&gt; Introduction ============ Crowd flow prediction is crucial for traffic management and public safety, and has drawn a lot of research interests due to its huge potentials in many intelligent applications, including intelligent traffic diversion and travel optimization. {width="0.600\columnwidth"} {width="0.382\columnwidth"} \[fig:citywide\_crow\_flow\_visual\] Nowadays, we live in an era where ubiquitous digital devices are able to broadcast rich information about human mobility in real-time and at a high rate, which exponentially increases the availability of large-scale mobility data (e.g., GPS signals or mobile phone signals). In this paper, we generate the crowd flow maps from these mobility data and utilize the historical crowd flow maps to forecast the future crowd flow of a city. As shown in Fig. \[fig:citywide\_crow\_flow\_visual\], we partition a city into a grid map based on the longitude and latitude, and measure the number of pedestrians in each region at each time interval with the mobility data. Although the regional scale can vary greatly in different cities, the core problem lies in excavating the evolution of traffic flow in different spatial and temporal regions. Recently, notable successes have been achieved for citywide crowd flow prediction based on deep neural networks coupled with certain spatial-temporal priors [@zhang2016dnn; @zhang2017deep]. Nevertheless, there still exist several challenges limiting the performance of crowd flow analysis in complex scenarios. First, crowd flow data can vary greatly in temporal sequence and capturing such dynamic variations is non-trivial. Second, some periodic laws (e.g., traffic flow suddenly changing due to rush hours or pre-holiday effects) can greatly affect the situation of crowd flow, which increases the difficulty in learning crowd flow representations from data. To solve all above issues, we propose a novel spatial-temporal neural network module, called Attentive Crowd Flow Machine (ACFM), to adaptively exploit diverse factors that affect crowd flow evolution and at the same time produce the crowd flow estimation map end-to-end in a unified module. The attention mechanism embedded in ACFM is designed to automatically discover the regions with primary positive impacts for the future flow prediction and simultaneously adjust the impacts of the different regions with different weights at each time-step. Specifically, the ACFM comprises two progressive ConvLSTM [@xingjian2015convolutional] units. The first one takes input from i) the feature map representing flow density at each moment and ii) the memorized representations of previous moments, to compute the attentional weights, while the second LSTM aims at generating superior spatial-temporal feature representation from attentionally weighted sequential flow density features. The proposed ACFM has the following appealing properties. First, it can effectively incorporate spatial-temporal information in feature representation and can flexibly compose solutions for crowd flow prediction with different types of input data. Second, by integrating the deep attention mechanism [@sharma2015action; @lu2016knowing], ACFM adaptively learns to represent the weights of each spatial location at each time-step, which allows the model to dynamically perceive the impact of the given area at a given moment for the future traffic flow. Third, ACFM is a general and differentiable module which can be effectively combined with various network architectures for end-to-end training. In addition, for forecasting the citywide crowd flow, we further build a deep architecture based on the ACFM, which consists of three components: i) sequential representation learning, ii) periodic representation learning and iii) a temporally-varying fusion module. The first two components are implemented by two parallel ACFMs for contextual dependencies modeling at different temporal scales, while the temporally-varying fusion module is proposed to adaptively merge the two separate temporal representation for crowd flow predictions. The main contributions of this work are three-fold: - We propose a novel ACFM neural network, which incorporates two LSTM modules with spatial-temporal attentional weights, to enhance the crowd flow prediction via adaptively weighted spatial-temporal feature modeling. - We integrate ACFM in our customized deep architecture for citywide crowd flow estimation, which recurrently incorporates various sequential and periodic dependencies with temporally-varying data. - Extensive experiments on two public benchmarks of crowd flow prediction demonstrate that our approach outperforms existing state-of-the-art methods by large margins. Related Work ============ **Crowd Flow Analysis**. Due to the wide application of traffic congestion analysis and public safety monitoring, citywide crowd flow analysis has recently attracted a wide range of research interest [@zheng2014urban]. A pioneer work was proposed by Zheng et al., [@zheng2015trajectory], in which they proposed to represent public traffic trajectories as graphs or tensor structures. Inspired by the significant progress of deep learning on various tasks [@zhang2015bit; @chen2016disc; @li2017face; @zhang2018hierarchical; @liu2018facial], many researchers also have attempted to handle this task with deep neural network. Fouladgar et al. [@fouladgar2017scalable] introduced a scalable decentralized deep neural networks for urban short-term traffic congestion prediction. In [@zhang2016dnn], a deep learning based framework was proposed to leverage the temporal information of various scales (i.e. temporal closeness, period and seasonal) for crowd flow prediction. Following this work, Zhang et al., [@zhang2017deep] further employed a convolution based residual network to collectively predict inflow and outflow of crowds in every region of a city grid-map. To take more efficient temporal modeling into consideration, Dai et al. [@dai2017deeptrend] proposed a deep hierarchical neural network for traffic flow prediction, which consists of an extraction layer to extract time-variant trend in traffic flow and a prediction layer for final crowd flow forecasting. Currently, to overcome the scarcity of crowd flow data, Wang et al. [@wang2018crowd] proposed to learn the target city model from the source city model with a region based cross-city deep transfer learning algorithm. **Memory and attention neural networks**. Recurrent neural networks (RNN) have been widely applied to various sequential prediction tasks [@sutskever2014sequence; @donahue2015long]. As a variation of RNN, Long Short-Term Memory Networks (LSTM) enables RNNs to store information over extended time intervals and exploit longer-term temporal dependencies. It was first applied to the research field of natural language processing [@luong2015effective] and speech recognition [@graves2013speech], while recently many researchers have attempted to combine CNN with LSTM to model the spatial-temporal information for various of computer vision applications, such as video salient object detection [@li2018flow], image caption [@mao2014deep; @wu2018interpretable] and action recognition [@veeriah2015differential]. Visual attention is a fundamental aspect of human visual system, which refers to the process by which humans focus the computational resources of their brain’s visual system to specific regions of the visual field while perceiving the surrounding world. It has been recently embedded in deep convolution networks [@chen2016attention] or recurrent neural networks to adaptively attend on mission-related regions while processing feedforward operation and have been proved effective for many tasks, including machine translation [@luong2015effective], crowd counting [@liu2018crowd], multi-label image classification [@wang2017multi], face hallucination [@cao2017attention], and visual question answering [@xu2016ask]. However, no existing work incorporates attention mechanism in crowd flow prediction. {width="1.99\columnwidth"} \[fig:network-structure\] The most relevant works to us are [@zhang2017fcn; @xiong2017spatiotemporal], which also incorporate ConvLSTM for spatial-temporal modeling. However, they are used for consecutive video frames representation and aims to estimate the crowd counting on a given surveillance image instead of forecasting crowd flow evolution based on mobility data. Moreover, our proposed ACFM is composed of two progressive LSTM modules with learnable attention weights, which is not only adept at modeling spatial-temporal representation, but also efficiently capturing the effect on the global crowd flow evolution caused by the changes of traffic conditions in each particular spatial-temporal region (e.g. a traffic jam caused by an accident). Last but not the least, the attention mechanism embedded in our ACFM module also helps to improve the interpretability of the network process while boosting the performance. Preliminaries ============= In this section, we first describe some basic elements of crowd flow and then define the crowd flow prediction problem. **Region Partition**: There are many ways to divide a city into multiple regions in terms of different granularities and semantic meanings, such as road network  [@deng2016latent] and zip code tabular. In this study, following the previous works [@zhang2017deep; @yao2018deep], we partition a city into $h \times w$ non-overlapping grid map based on the longitude and latitude. Each rectangular grid represents a different geographical region in the city. Figure \[fig:citywide\_crow\_flow\_visual\] illustrates the partitioned regions of Beijing and New York City. **Crowd Flow**: In practical application, we can extract a mass of crowd trajectories from GPS signals or mobile phone signals. With those crowd trajectories, we can measure the number of pedestrians entering or leaving a given region at each time interval, which are called inflow and outflow in our work. For convenience, we denote the crowd flow map at the ${t^{th}}$ time interval of ${d^{th}}$ day as a tensor ${\bm{M_d^t} \in R^{2\times h\times w}}$, of which the first channel is the inflow and the second channel is the outflow. Some crowd flow maps are visualized in Figure \[fig:predicted\_flows\_visual\]. **External Factors**: As mentioned in the previous work [@zhang2017deep], crowd flow can be affected by many complex external factors, such as meteorology information and holiday information. In this paper, we also consider the effect of these external factors. The meteorology information (e.g., weather condition, temperature and wind speed) can be collected from some public meteorological websites, such as Wunderground[^1]. Specifically, the weather condition is categorized into sixteen categories (e.g., sunny and rainy) and it is digitized with One-Hot Encoding [@harris2010digital], while temperature and wind speed are scaled into the range \[0, 1\] with min-max linear normalization. Multiple categories of holiday [^2] (e.g., Chinese Spring Festival and Christmas) can be acquired from calendar and encoded into a binary vector with One-Hot Encoding. Finally, we concatenate all external factors data to a 1D tensor. The external factors tensor at the ${t^{th}}$ time interval of ${d^{th}}$ day is expressed as a ${\bm{E_d^t}}$ in the following sections. **Crowd Flow Prediction**: This problem aims to predict the crowd flow map ${M_d^{t}}$, given historical crowd flow maps and external factors data until the ${(t-1)^{th}}$ time interval of ${d^{th}}$ day. Attentive Crowd Flow Machine ============================ We propose a unified neural network module, named Attentive Crowd Flow Machine (ACFM), to learn the crowd flow spatial-temporal representations. ACFM is designed to adequately capture various contextual dependencies of the crowd flow, e.g., the spatial consistency and the temporal dependency of long and short term. As shown on the left of Fig. \[fig:network-structure\], the ACFM is composed of two progressive ConvLSTM [@xingjian2015convolutional] units connected with a convolutional layer for attention weight prediction at each time step. The first LSTM (bottom LSTM in the figure) models the temporal dependency through original crowd flow feature embedding (extracted from CNN), the output hidden state of which is concatenated with current crowd flow feature and fed to a convolution layer for weight map inference. The second LSTM (upper LSTM in the figure) is of the same structure as the first LSTM but takes the re-weighted crowd flow features as input at each time-step and is trained to recurrently learn the spatial-temporal representations for further crowd flow prediction. For better understanding, we denote the input feature map of the ${i}^{th}$ iteration as ${X_i \in R^{c\times h\times w}}$, with $h$, $w$ and $c$ representing the height, width and the number of channels. Following  [@hochreiter1997long], the hidden state ${H_i^1 \in R^{c\times h\times w}}$ of first LSTM can be formulated as: $$\label{equ:first_lstm} H_i^1 = \textbf{ConvLSTM}(H_{i-1}^1, C_{i-1}^1, X_i),$$ where ${C_{i-1}^1}$ is the memorized cell state of the first LSTM at $(i-1)^{th}$ iteration. The internal hidden state ${H_i^1}$ is maintained to model the dynamic temporal behavior of the previous crowd flow sequences. We concatenate ${H_i^1}$ and ${X_i}$ to generate a new tensor, and feed it to a single convolutional layer with kernel size ${1\times 1}$ to generate an attention map ${W_i}$, which can be expressed as: $$\label{equ:attention} W_i = \textbf{Conv}_{1\times 1}(H_i^1 \oplus X_i, w),$$ where $\oplus$ denotes feature concatenation and $w$ is the parameters of the convolutional layer. And ${W_i}$ indicates the weights of each spatial location on the feature map ${X_i}$. We further reweigh ${X_i}$ with an element-wise multiplication according to ${W_i}$ and take the reweighed map as input to the second LSTM for representation learning, the hidden state ${H_i^2 \in R^{c\times h\times w}}$ of which can be formulated as: $$\label{equ:second_lstm} H_i^2 = \textbf{ConvLSTM}(H_{i-1}^2, C_{i-1}^2, X_i * W_i),$$ where ${*}$ refers to the element-wise multiplication. ${h_i^2}$ encodes the attention-aware content of current input as well as memorizes the contextual knowledge of previous moments. The output of the last hidden state thus encodes the information of the whole crowd flow sequence, and is used as the spatial-temporal representation for evolution analysis of future flow map. In the next section, we will show how to incorporate the proposed ACFM in our crowd flow prediction framework. Citywide Crowd Flow Prediction ==============================  \[sec:ccfp\] We build a deep neural network architecture incorporated with our proposed ACFM to predict citywide crowd flow. As illustrated on the right of Fig. \[fig:network-structure\], the crowd flow prediction framework consists of three components: (1) sequential representation learning, (2) periodic representation learning and (3) a temporally-varying fusion module. For the first two parts of the framework, we employ the ACFM to model the contextual dependencies of crowd flow at different temporal scales. After that, a temporally-varying fusion module is proposed to adaptively merge the different feature embeddings from each component with the weight $r$ learned from the concatenation of respective feature representations and the external information. Finally, the merged feature map is fed to one additional convolution layer for crowd flow map inference. Sequential Representation Learning ----------------------------------  \[sec:srl\] The evolution of citywide crowd flow is usually affected by diverse internal and external factors, e.g., current urban traffic and weather conditions. For instance, a traffic accident occurring on a city main road at 9 am may seriously affect the crowd flow of nearby regions in subsequent time periods. Similarly, a sudden rain may seriously affect the crowd flow in a specific region. To deal with these issues, we take several continuous crowd flow features and their corresponding external factors features as the sequential temporal features, and feed them into our ACFM to recurrently capture the trend of crowd flow in the short term. Specifically, we denote the input sequential temporal features as: $$S_{in} = \{F^{t-k}_d \big| k = n, n-1,...,1\},$$ where ${n}$ is the length of the sequentially related time intervals and ${F_j^i}$ denotes the embedding features of the crowd flow and the external factors at the $i^{th}$ time interval of the ${j^{th}}$ day. The extraction of embedding feature ${F_j^i}$ will be described in Section \[sec:impl\_detail\]. We apply the proposed ACFM to learn sequential representation from temporal features $S_{in}$. As shown on the right of Fig. \[fig:network-structure\], the ACFM recurrently takes each element of $S_{in}$ as input and learns to selectively memorize the context of this specific temporally-varying data. The output hidden state of the last iteration is further fed into a following convolution layer to generate a feature representation of size ${c\times h\times w}$, denoted as ${S_{f}}$, which forms the spatial-temporal feature embedding of the fine-grained sequential data. Periodic Representation Learning -------------------------------- Generally, there exist some periodicities which make a significant impact on the changes of traffic flow. For example, the traffic conditions are very similar during morning rush hours of consecutive workdays, repeating every 24 hours. Similar with sequential representation learning described in Section \[sec:srl\], we take periodic temporal features $$P_{in} = \{F^{t}_{d-k} \big| k = m, m-1,...,1\},$$ to capture the periodic property of crowd flow, where ${n}$ is the length of the periodic days. As shown on the right of Fig. \[fig:network-structure\], we employ ACFM to learn periodic representation with the periodic temporal features ${P}$ as input. The hidden output of the last iteration of ACFM is passed through a convolutional layer to generate a representation ${P_{f} \in R^{c\times h\times w} }$. The ${P_{f}}$ encodes the context of periodic laws, which is essential for crowd flow prediction. Temporally-Varying Fusion -------------------------  \[sec:fusion\] The future crowd flow is affected by the two temporally-varying representations ${S_{f}}$ and ${P_{f}}$. A naive method is to directly merge those two representations, however it is suboptimal. In this subsection, we propose a novel temporally-varying fusion module to adaptively fuse the sequential representation ${S_{f}}$ and the periodic representation ${P_{f}}$ of crowd flow with different weight. Considering that the external factors may affect the importance proportion of two representations, we take the sequential representation ${S_{f}}$, periodic representation ${P_{f}}$ and the external factors integrative feature ${E_{f}}$ to calculate the fusion weight, where ${E_{f}}$ is the element-wise addition of external factors features of all relative time intervals and will be described in Section \[sec:impl\_detail\]. As shown on the right of Fig. \[fig:network-structure\], we first concatenate ${S_{f}}$, ${P_{f}}$ and ${E_{f}}$ and feed them as input to two fully-connected layers (the first layer has 512 neurons and the second has only one neuron) for fusion weight inference. After a sigmoid function, the temporally-varying fusion module outputs a single value $r \in [0,1]$, which reflects the importance of the sequential representation ${S_{f}}$. And $1-r$ is treated as the fusion weight of periodic representation ${P_{f}}$. We then merge these two temporal representations with different weight and further reduce the feature to two channels (input and output flow) with a linear transformation, which can be expressed as: $$\label{equ:fusion} M_f= \mathcal T(r*S_{f} + (1-r)*P_{f}).$$ where ${\mathcal T}$ is the linear transformation implemented by a convolution layer with two filters. The predicted crowd flow map ${\widehat{M}^t_d \in R^{2\times h\times w}}$ can be computed as $$\label{equ:forecast} \widehat{M}^t_d = tanh(M_f).$$ where the hyperbolic tangent ${tan}$ ensures the output values are within $[-1, 1]$ [^3]. Implementation Details ----------------------  \[sec:impl\_detail\] We first detail the method of extracting crowd flow feature as well as external factors feature and then describe our network optimization. **Crowd Flow Feature**: For the crowd flow map ${M^i_j}$ at the ${i^{th}}$ time interval of the ${j^{th}}$ day, we extract its feature ${F^i_j(M)}$ with a customized ResNet [@he2016deep] structure, which is stacked by ${N}$ residual units without any down-sampling operations. Each residual unit contains two convolutional layers followed by two ReLu layers. We set the channel numbers of all convolutional layers as 16 and the kernel sizes as ${3\times3}$. **External Factors Feature**: For the external factors ${E^i_j}$, we extract its feature with a simple neural network implemented by two fully-connected layers. The first FC layer has 256 neurons and the second one has ${16 \times h \times w}$ neurons. The output of the last layer is further reshaped to a 3D tensor ${F^i_j(E) \in R^{16\times h\times w}}$, which is the final feature of ${E^i_j}$. Finally, we concatenate ${F^i_j(M)}$ and ${F^i_j(E)}$ to generate the embedding feature ${F_j^i}$, which can be expressed as $$F_j^i= F^i_j(M) \oplus F^i_j(E),$$ where $\oplus$ denotes feature concatenation. For the external factors integrative feature ${E_{f}}$ described in Section \[sec:fusion\] , it is the element-wise addition of ${\{E^{t-k}_{d} \big| k = n, n-1,...,1\}}$ and ${\{E^{t}_{d-k} \big| k = m, m-1,...,1\}}$. **Network Optimization**: We adopt the TensorFlow [@abadi2016tensorflow] toolbox to implement our crowd flow prediction network. The filter weights of all convolutional layers and fully-connected layers are initialized by Xavier [@glorot2010understanding]. The size of a minibatch is set to 64 and the learning rate is ${10^{-4}}$. We optimize our networks parameters in an end-to-end manner via Adam optimization [@kingma2014adam] by minimizing the Euclidean loss for 270 epochs with a GTX 1080Ti GPU. Experiments =========== In this section, we first conduct experiments on two public benchmarks (e.g., TaxiBJ [@zhang2016dnn] and BikeNYC [@zhang2016dnn]) to evaluate the performance of our model on citywide crowd flow prediction. We further conduct an ablation study to demonstrate the effectiveness of each component in our model. Dataset Setting and Evaluation Metric ------------------------------------- We forecast the inflow and outflow of citywide crowds on two datasets: the TaxiBJ [@zhang2016dnn] dataset for taxicab flow prediction and the BikeNYC [@zhang2016dnn] dataset for bike flow prediction. [**TaxiBJ Dataset:**]{} This dataset contains 22,459 time intervals of crowd flow maps with a size of ${2\times32\times32}$, which are generated with Beijing taxicab GPS trajectory data. The external factors contain weather conditions, temperature, wind speed and 41 categories of holiday. For the fair comparison, we refer to [@zhang2017deep] and take the data in the last four weeks as the testing set and the rest as the training set. In this dataset, we set the sequential length ${\emph n}$ and the periodic length ${\emph m}$ as $3$ and $2$, respectively. As with ST-ResNet [@zhang2017deep], the ResNet described in Section \[sec:impl\_detail\] is composed of 12 residual units. [**BikeNYC Dataset:**]{} This dataset is generated with the NYC bike trajectory data, which contains 4,392 available time intervals crowd flow maps with the size of ${2\times16\times8}$. The data of the last ten days are chosen to be the test set. As for external factors, 20 categories of the holiday are recorded. In this dataset, we set the sequential length ${\emph n}$ as 5 and the periodic length ${\emph m}$ as 7. For a fair comparison with ST-ResNet [@zhang2017deep], we also utilize a ResNet described in Section \[sec:impl\_detail\] with 4 residual units to extract the crowd flow feature. We adopt Root Mean Square Error (RMSE) as evaluation metric to evaluate the performances of all the methods, which is defined as: $$RMSE = \sqrt{{\frac{1}{z}}{\sum_{i=1}^z{(\widehat{Y_i} - Y_i)}^2}},$$ where ${\widehat{Y_i}}$ and ${Y_i}$ represent the predicted flow map and its ground truth map, respectively. $z$ indicates the number of samples used for validation. Comparison with the State of the Art ------------------------------------ [c|c|c]{} ------- Model ------- : Quantitative comparisons on TaxiBJ and BikeNYC using RMSE (smaller is better). Our proposed method outperforms the existing state-of-the-art methods on both datasets with a margin. & -------- TaxiBJ -------- : Quantitative comparisons on TaxiBJ and BikeNYC using RMSE (smaller is better). Our proposed method outperforms the existing state-of-the-art methods on both datasets with a margin. & --------- BikeNYC --------- : Quantitative comparisons on TaxiBJ and BikeNYC using RMSE (smaller is better). Our proposed method outperforms the existing state-of-the-art methods on both datasets with a margin. \ SARIMA [@williams1998urban] & 26.88 & 10.56\ VAR [@lutkepohl2011vector] & 22.88& 9.92\ ARIMA [@box2015time] & 22.78 & 10.07\ ST-ANN [@zhang2017deep] & 19.57 & -\ DeepST [@zhang2016dnn] & 18.18 & 7.43\ ST-ResNet [@zhang2017deep] & 16.69 & 6.33\ Ours & **15.40** & **5.64**\ \[tab:BJ\_NYC\_result\] We compare our method with six state-of-the-art methods, including Auto-Regressive Integrated Moving Average (ARIMA) [@box2015time], Seasonal ARIMA (SARIMA) [@williams1998urban], Vector Auto-Regressive (VAR) [@lutkepohl2011vector], ST-ANN [@zhang2017deep], DeepST [@zhang2016dnn] and ST-ResNet [@zhang2017deep]. For these compared methods, we use the performances provided by Zhang et al. [@zhang2017deep] as their results. Table \[tab:BJ\_NYC\_result\] summarizes the performance of the proposed method and other six methods. On TaxiBJ dataset, our method decreases the RMSE from 16.69 to 15.40 when compared with current best model, and achieves a relative improvement of 7.7%. Our method also boosts the prediction accuracy on BikeNYC, i.e., decreases RMSE from 6.33 to 5.64. Note that some compared methods, e.g., ST-ANN, DeepST and ST-ResNet, also employ deep learning techniques. Experimental results demonstrate that our proposed ACFM is able to explicitly model the spatial-temporal feature as well as the attention weighting of each spatial influence, which greatly outperforms the state-of-the-art. Some crowd flow prediction maps of our full model on TaxiBJ dataset are shown on the second row of the Fig. \[fig:predicted\_flows\_visual\]. As can be seen, our generated crowd flow map is consistently closest to those of the ground-truth, which is accord with the quantitative RMSE comparison. Ablation Study --------------  \[sec:abl\_study\] Our full model for citywide crowd flow prediction consists of three components: sequential representation learning, periodic representation learning and temporally-varying fusion module. For convenience, we denote our full model as **S**equential-**P**eriodic **N**etwork (**SPN**) in the following experiments. To show the effectiveness of each component, we implement seven variants of our full model on the TaxiBJ dataset: - **PCNN:** directly concatenates the periodic features ${P_{in}}$ and feeds them to a convolutional layer with two filters followed by ${tanh}$ to predict future crowd flow; - **SCNN:** directly concatenates the sequential features ${S_{in}}$ and feeds them to a convolutional layer followed by ${tanh}$ to predict future crowd flow; - **PRNN-w/o-Attention:** takes periodic features ${P_{in}}$ as input and learns periodic representation with a LSTM layer to predict future crowd flow; - **PRNN:** takes periodic features ${P_{in}}$ as input and learns periodic representation with the proposed ACFM to predict future crowd flow; - **SRNN-w/o-Attention:** takes sequential features ${S_{in}}$ as input and learns sequential representation with a LSTM layer for crowd flow estimation; - **SRNN:** takes sequential features ${S_{in}}$ as input and learns sequential representation with the proposed ACFM to predict future crowd flow; - **SPN-w/o-Fusion:** directly merges sequential representation and periodic representation with equal weight (0.5) to predict future crowd flow. [c|c]{} ------- Model ------- : Quantitative comparisons (RMSE) of different variants of our model on TaxiBJ dataset for component analysis. & ------ RMSE ------ : Quantitative comparisons (RMSE) of different variants of our model on TaxiBJ dataset for component analysis. \ PCNN & 33.44\ PRNN-w/o-Attention & 32.97\ PRNN & 32.52\ SCNN & 17.48\ SRNN-w/o-Attention & 16.62\ SRNN & 16.11\ SPN-w/o-Fusion & 16.01\ SPN & 15.40\ \[tab:component\_results\] {width="49.00000%"} {width="49.00000%"} \[fig:Periodic-Attention\] {width="49.00000%"} {width="49.00000%"} \[fig:Sequential-Attention\] **Effectiveness of Spatial Attention:** As shown in Table \[tab:component\_results\], adopting spatial attention, SRNN decreases the RMSE by 0.51, compared to SRNN-w/o-Attention. For another pair of variants, PRNN with spatial attention has the similar performance improvement, compared to PRNN-w/o-Attention. Fig. \[fig:Periodic-Attention\] and Fig. \[fig:Sequential-Attention\] show some attentional maps generated by our method as well as the residual maps between the input crowd flow maps and their corresponding ground truth. We can observe that there is a negative correlation between the attentional maps and the residual maps. It indicates that our ACFM is able to capture valuable regions at each time step and make better predictions by inferring the trend of evolution. Roughly, the greater difference a region has, the smaller its weight, and vice versa. We can inhibit the impacts of the regions with great differences by multiplying the small weights on their corresponding location features. With the visualization of attentional maps, we can also get to know which regions have the primary positive impacts for the future flow prediction. According to the experiment, we can see that the proposed model can not only effectively improve the prediction accuracy, but also enhance the interpretability of the model to a certain extent. **Effectiveness of Sequential Representation Learning:** As shown in Table \[tab:component\_results\], directly concatenating the sequential features ${S}$ for prediction, the baseline variant SCNN gets an RMSE of 17.48. When explicitly modeling the sequential contextual dependencies of crowd flow using the proposed ACFM, the variant SRNN decreases the RMSE to 16.11, with 7.8% relative performance improvement compared to the baseline SCNN, which indicates the effectiveness of the sequential representation learning. {width="0.9\columnwidth"} \[fig:predicted\_flows\_visual\] **Effectiveness of Periodic Representation Learning:** We also explore different network architectures to learn the periodic representation. As shown in Table \[tab:component\_results\], the PCNN, which learns to estimate the flow map by simply concatenating all of the periodic features ${P}$, only achieves RMSE of 33.44. In contrast, when introducing ACFM to learn the periodic representation, the RMSE drops to 32.52. This further demonstrates the effectiveness of the proposed ACFM for spatial-temporal modeling. **Effectiveness of Temporally-Varying Fusion:** When directly merging the two temporal representations with an equal contribution (0.5), SPN-w/o-fusion achieves a negligible improvement, compared to SRNN. In contrast, after using our proposed fusion strategy, the full model SPN decreases the RMSE from 16.11 to 15.40, with a relative improvement of 4.4% compared with SRNN. The results show that the significance of these two representations are not equal and are influenced by various factors. The proposed fusion strategy is effective to adaptively merge the different temporal representations and further improve the performance of crowd flow prediction. **Further Discussion:** To analyze how each temporal representation contributes to the performance of crowd flow prediction, we further measure the average fusion weights of two temporal representations at each time interval. As shown in the right of Fig. \[fig:ratio\_rmse\_bj\], the fusion weights of sequential representation are greater than that of the periodic representation at most time excepting for wee hours. Based on this observation, we can conclude that the sequential representation is more essential for the crowd flow prediction. Although the weight is low, the periodic representation still helps to improve the performance of crowd flow prediction qualitatively and quantitatively. Fusing with periodic representation, we can decrease the RMSE of SRNN by 4.4% and generate more precise crowd flow maps, as shown in Table \[tab:component\_results\] and Fig. \[fig:predicted\_flows\_visual\]. {width="0.90\columnwidth"} \[fig:ratio\_rmse\_bj\] Conclusion ========== This work studies the spatial-temporal modeling for crowd flow prediction problem. To incorporate various factors that affect the flow changes, we propose a unified neural network module named Attentive Crowd Flow Machine (ACFM). In contrast to the existing flow estimation methods, our ACFM explicitly learns dynamic representations of temporally-varying data with an attention mechanism and can infer the evolution of the future crowd flow from historical crowd flow maps. A unified framework is also proposed to merge two types of temporal information for further prediction. According to the extensive experiments, we have exhaustively verified the effectiveness of our proposed ACFM on the task for citywide crowd flow prediction. [^1]: <https://www.wunderground.com/> [^2]: The categories of holiday are variational in different datasets. [^3]: When training, we use Min-Max linear normalization method to scale the crowd flow maps into the range $[-1,1]$. When evaluating, we re-scale the predicted value back to the normal values and then compare with the ground truth.

--- author: - 'B. Roussel^^, C. Cabart^^, G. Fève^^, E. Thibierge^^, P. Degiovanni^^' title: Electron quantum optics as quantum signal processing --- Introduction {#sec:introduction} ============ Electron quantum optics is a new perspective on electronic quantum transport that aims at understanding the behavior of electrons in ballistic quantum conductors using paradigms and methods of quantum optics [@Bocquillon:2014-1]. This field has emerged from the development of quantum coherent nanoelectronics during the 90s [@Henny:1999-1; @Oliver:1999-1; @Blanter:2000-1], the demonstration of electronic interferometers [@Ji:2003-1; @Neder:2007-1; @Neder:2007-2] in quantum Hall edge channels and finally the realization of on-demand single-electron sources [@Feve:2007-1; @Dubois:2013-1]. Importantly, the introduction of single-electron sources has catalyzed a shift from questions about the statistics of charge flowing across a quantum conductor [@Levitov:1996-1; @Blanter:2000-1] to questions about the wavefunctions of elementary excitations carrying the charge. This naturally led to the transposition of photon quantum optics concepts introduced in the 60s by Glauber to coherent quantum nanoelectronics, thus giving birth to the central concepts of electronic coherences [@Degio:2011-1; @Haack:2012-2; @Moskalets:2014-1; @Thibierge:2016-1]. Historically, the scattering theory approach to quantum transport [@Landauer:1985-1; @Buttiker:1985-1; @Buttiker:1986-1; @Martin:1992-1; @Buttiker:1992-1] had already emphasized the importance of optics concepts in electronic transport. However, electron quantum optics differs from photon quantum optics because of the Fermi statistics of electrons which changes the nature of the reference “vacuum state" when all sources are switched off. Understanding and extending quantum optics concepts in the presence of such non-trivial vacua was also a motivation for developing electron quantum optics. Even more importantly, electrons are charged particles interacting through Coulomb interactions. As stressed out by M. Büttiker and his collaborators [@Buttiker:1993-1; @Pretre:1996-1; @Christen:1996-1; @Blanter:2000-1], this plays a crucial role in high frequency quantum transport by enforcing charge conservation and gauge invariance [@Pretre:1996-1; @Blanter:2000-1]. Coulomb interactions also raise the basic question of the fate of electronic quasi-particles in a metal, which was the starting point for the Landau-Fermi liquid theory [@Nozieres-Pines]. Electron quantum optics thus offers unprecedented possibilities to study these basic condensed matter physics questions down to the single-electron level. These possibilities are illustrated by recent two-particle interferometry experiments in quantum Hall edge channels which are reviewed in the present volume [@Marguerite:2016-2]. Electron quantum optics also establishes a bridge between quantum coherent nanoelectronics and microwave quantum optics, which now plays an important role in superconducting nanocircuits used for quantum information processing and manipulation [@Eichler:2011-1; @Lang:2013-1; @Bozyigit:2011-1]. Microwave quantum optics is also crucial for understanding the electromagnetic radiation emitted by a quantum conductor, an important problem rising a growing interest [@Beenakker:2004-1; @Grimsmo:2016-1; @Mendes:2015-1; @Forgues:2014-1; @Forgues:2015-1; @Thibaut:2015-1; @Virally:2016-1]. The purpose of this paper is to reconsider electron quantum optics from a more global perspective by interpreting it in the language of signal processing. In its broadest acceptance, signal processing is an enabling technology that aims at processing, transferring and retrieving information carried in various physical formats called “signals" [@Moura:2009-1]. Signal processing involves a huge arsenal of techniques to detect, filter, represent, transmit, and finally extract information or recognize patterns within signals. However, the most famous and historically important examples of signals, such as acoustic or electronic signals, are classical. Here, we would like to emphasize that all electron optics experiments realized so far [@Bocquillon:2013-1; @Jullien:2014-1; @Freulon:2015-1; @Marguerite:2016-1] as well as proposals for accessing two-electron coherence [@Thibierge:2016-1] can be interpreted in the signal processing language as experiments on signals which are no longer classical: namely electron quantum optics coherences. Electronic interferometers realize analog signal processing operations such as “linear filtering" or “overlaps" on these quantum signals and encode the result into experimentally accessible quantities such as average current and current correlations. By emphasizing this point of view, we provide a unified view of the recent developments of electron quantum optics and, as we will show by discussing two-particle interferometry, we gain some inspiration towards envisioning new experimental measurement schemes for electronic coherences. Although this has not really been fully exploited yet, it could also suggest new ways of obtaining data by using suitable sources and data processing for optimizing electronic coherence reconstruction [@Ferraro:2013-1]. It may also rise interest in the signal processing community towards electron quantum optics and lead to new innovative experimental and theoretical ideas. In order to develop this point of view, this paper is organized as follows: after briefly recalling the experimental and theoretical context of electron quantum optics in section \[sec:context\], section \[sec:G1\] reviews the notion of single-electron coherence and the various ways to access this quantity. We will discuss the signal processing operations performed in various single- and two-particle interferometers used to probe single-electron coherence. Our discussion is complementary to that of the review [@Marguerite:2016-2] which discusses two-particle interferometry experiments in quantum Hall edge channels and the underlying theory. Here, the same concepts are reviewed with a strong emphasis on the quantum signal processing point of view. We then turn to two-electron coherence in section \[sec:G2\], and present its definition and its main properties. We introduce its various representations and discuss its non-classical features. We then present its relation to quantum noise of the electrical current, showing the electron quantum optics version of Einstein’s relation between particle number fluctuations and wavefunctions. We show that a whole class of experiments can be interpreted as linear filtering converting the intrinsic second order coherence emitted by a source into current correlations. In the last section, we will connect electronic coherences to quantum information theoretical quantities. For this purpose, we will explain how to derive effective qubit density matrices from a set of orthogonal single-particle states. In particular, we will illustrate this by discussing electron/hole entanglement in the many-body state generated by a mesoscopic capacitor. We will finally sketch how ideas from signal processing lead to a suitable definition of these density matrices for periodically driven systems. The context of electron quantum optics {#sec:context} ====================================== Let us briefly review the main steps that have lead to the development of electron quantum optics. Experiments {#sec:context:experiments} ----------- On the experimental side, the integer quantum Hall effect in high mobility 2-dimensional electron gas (2DEG) in AsGa/AsGaAl heterostructures provides the analogous of optical fibers through chiral propagation of charge carriers within the so-called edge channels. Progresses in nanofabrication not only enabled the fabrication of the quantum point contact (QPC), which plays the role of an ideal electronic beam splitter [@Wharam:1988-1; @vanWees:1988-1; @vanHouten:1992-1], but have also enabled its embedding into complicated geometries such as the electronic Mach-Zehnder interferometer (MZI) [@Ji:2003-1; @Roulleau:2007-2; @Roulleau:2008-1] and the Samuelsson-Büttiker interferometer [@Samuelsson:2004-1; @Neder:2007-2]. These pioneering experiments showed that, at low temperatures, electronic coherence can be maintained over distances comparable to the size of these circuits (from few to ) [@Roulleau:2008-2]. They have also opened the way to the demonstration and study of more complex electronic circuits in which elements such as QPC, quantum dots and single-electron sources could be placed like optical components on an optical table. As mentioned in the Introduction, the demonstration of the mesoscopic capacitor as an on-demand single-electron source [@Feve:2007-1] marked the beginning of electron quantum optics. Now, other single-electron sources have been demonstrated in AsGa/AsGaAl, from turnstiles [@Blumenthal:2007-1; @Fletcher:2013-1], mainly motivated by metrology, to the Leviton source [@Dubois:2013-1] which is reviewed in this volume with great details [@Glattli:2016-1]. Around the same time, progresses in microwave technology and measurement techniques have lead to the exploration of high-frequency transport [@Gabelli:2006-1; @Gabelli:2007-1], thereby confirming the predictions by Büttiker and his collaborators [@Buttiker:1993-2; @Pretre:1996-1]. From a broader perspective, these developments have opened the way to in-depth experimental investigations of high-frequency quantum coherent nanoelectronics. Theory {#sec:context:theory} ------ The electron quantum optics formalism was then developped taking advantage of the chirality of electronic transport in quantum Hall systems: in such systems, the optical analogy is exploited at its best by decomposing the quantum conductor, or more generally the electronic circuit, into simple building blocks such as quantum point contacts or energy filters so that an incoming electronic flow is transformed into an outgoing one. Then, within the measurement stage, the average current or the current noise is measured either at zero or at a given high frequency. The resulting formalism bears a close similarity with the input-output formalism of photon quantum optics [@Gardiner:1985-1]. In particular, it assumes that electronic coherences are probed within a region where interactions can be neglected. In such a region, electrons propagate freely at a Fermi velocity which will be denoted by $v_F$ throughout the present paper. When electronic coherences are probed at a given position, often corresponding to the position of a detector, they depend on time variables. Such a description contains an assumption on the screening of Coulomb interactions. For example, we assume that screening at a quantum point contact is good enough to neglect any capacitive coupling between the incoming and outgoing edge channels. As of today, there has been no experimental evidence contradicting this assumption. Within this framework, interaction effects have been studied extensively, starting with MZI. We will very brielfy discuss some of these works in section \[sec:G1:MZI\] mentioning that interaction effects can lead to a breaking of the paradigm of quantum conductors as linear electron quantum optics components. We also refer the reader interested by interaction induced decoherence effects in Hong Ou Mandel (HOM) experiments to [@Marguerite:2016-2] as well as [@Marguerite:2016-1]. New systems {#sec:context:new-systems} ----------- The rapid development of electron quantum optics has also catalyzed a stream of works whose purpose is to extend its application range to new physical systems. A first line of research deals with extending electron quantum optics to situations in which interactions potentially lead to a drastic change of the ground state such as in superconductivity. This has led to study of electron quantum optics with Bogoliubov quasi-particles which is reviewed in this volume [@Ferraro:2016-1]. Another question is the generalization of electron quantum optics to fractional quantum Hall (FQH) edge channels. Proposals have been made for single quasi-particle and single electron emitted in these systems [@Ferraro:2015-1] and the HOM experiment with Lorentzian pulses has been considered [@Rech:2016-1]. However, a full generalization of electron quantum optics in the FQH regime is still missing, the main problem being the absence of any ideal quasi-particle beam splitter. Nevertheless, perturbative approaches may prove to be useful for experiments whenever the crossover energy scale associated with a constriction [@Fendley:1995-1] is well below the experimentally relevant energy scales. Another line of research focuses on manipulating the spin degree of freedom at the single-electron level. Quantum spintronics has risen a strong interest in the mesoscopic physics community because of the importance of coherent spin transport and manipulation for quantum information processing. Although the $\nu=2$ edge channel system had already been envisioned for quantum spintronics [@Karmakar:2011-1], 2D topological insulators (TI), such as quantum spin Hall systems [@Hasan:2010-1], are now considered as a potentially important class of systems for electron quantum optics. In these materials, such as CdTe/HgTe and InAs/GaSb quantum wells at zero magnetic field, edge channels are topologically protected from backscattering. They come as counterpropagating pairs with opposite spin polarization (spin-momentum locking). The mesoscopic capacitor built from a 2D TI has been proposed as an on-demand single Kramer pair source emitting two single-electron excitations with opposite spins [@Hofer:2013-1; @Inhofer:2013-1]. The HOM experiment has been discussed with such sources [@Ferraro:2014-2]. A variant of this source based on a driven antidot has also recently been proposed [@Dolcetto:2016-1] as well as a different system relying on two quantum dots coupled to a 2D TI via tunneling barriers [@Xing:2014-1]. Coulomb interactions are expected to play an important role in these systems due to the spatial superposition of two counter-propagating edge channels. Comparing to the $\nu=2$ quantum Hall edge channel system, interactions among counterpropagating edge systems are expected to induce new effects ranging from resonance effects [@Sukhorukov:2007-2] to fractionalization [@Calzona:2016-1]. Although these developments go beyond the scope of the present paper, it is important to keep in mind that the basic concepts of electron quantum optics can be extended to quantum spintronics in a relatively simple way. Single-electron coherence {#sec:G1} ========================= Let us now review the concept of single-electron coherence and discuss how it can be accessed through single-particle interferometry and two-particle interferometry. Our main message is that, under certain circumstances, the first type of experiments correspond to linear filtering in the signal processing language whereas the HOM experiment performs an analog computation of the overlap of two single-electron coherences. Definition and representations {#sec:G1:definitions} ------------------------------ Single-electron coherence [@Degio:2011-1] is defined by analogy with first-order coherence in photon quantum optics [@Glauber:1963-1]: $$\label{eq:G1:difinition} \mathcal{G}_{\rho}^{(e)}(1|1')=\mathrm{Tr}(\psi(1)\,\rho\,\psi^\dagger(1'))$$ where $\rho$ denotes the reduced density operator for the electronic fluid, $\psi$ and $\psi^\dagger$ denote the fermionic destruction and creation field operators and $1=(\alpha,x,t)$ and $1'=(\alpha',x',t')$ denote edge channel indices and space time coordinates. For spin polarized edge channels, $\alpha$ and $\alpha'$ correspond to spin indices: $\alpha=\alpha'$ then encodes spin populations whereas $\alpha=-\alpha'$ corresponds to spin coherences, thus showing that the formalism of electron quantum optics can be easily extended to account for the spin. In the following, to simplify notations, channel/spin indices will be dropped out when only one edge channel is involved. single-electron coherence contains all the information on single-electron wavefunctions present in the system. For example, let us consider the $N$ electron state $|\Psi_N\rangle= \prod_{k=1}^N\psi^\dagger[\varphi_k]|\emptyset\rangle $ where the creation operator for an electron in the single-particle state $\varphi_k$ is defined by: $$\psi^\dagger[\varphi_k]=v_F\int_{\mathbb{R}} \varphi_k(t)\psi^\dagger(t)\,{\mathrm{d}}t\,.$$ This state is obtained by filling mutually orthogonal single-particle states $(\varphi_k)_{k\in \{ 1,\dots,N \}}$ on top of the electronic vacuum $|\emptyset\rangle$. Then, a straightforward application of Wick’s theorem shows that, in the space domain at the initial time: $$\mathcal{G}^{(e)}_{|\Psi_N\rangle}(x|y)=\sum_{k=1}^N\varphi_k(x)\,\varphi_k(y)^*\,.$$ In a metallic conductor, at zero temperature and with all electronic sources switched off, the reference state is a Fermi sea vacuum of given chemical potential $\mu(x)$. Therefore, contrary to the case of photon quantum optics, the single-particle coherence does not vanish when sources are switched off but reduces to the Fermi sea contribution $\mathcal{G}^{(e)}_{F_{\mu(x)}}(t|t')=\langle F_{\mu(x)}|\psi^\dagger(x,t')\psi(x,t)|F_{\mu(x)}\rangle$. On the contrary, the inter-channel single-electron coherence vanishes when all electronic sources are switched off: $\mathcal{G}^{(e)}_{F}(\alpha,t|\alpha',t')=0$ for $\alpha\neq\alpha'$. In electron quantum optics, $\mathcal{G}_\rho^{(e)}$ is considered at a given position $x$ within the electronic circuit, thus leading to a two-time function $\mathcal{G}_{\rho,x}^{(e)}(t|t')=\mathcal{G}_{\rho}^{(e)}(x,t|x,t')$. When the sources are switched on, the single-electron coherence is different from the Fermi sea contribution and the excess single-electron coherence is defined by subtracting the Fermi sea contribution: $\Delta\mathcal{G}^{(e)}_{\rho,x}(t|t')= \mathcal{G}_{\rho,x}^{(e)}(t|t')-\mathcal{G}^{(e)}_{F_{\mu(x)}}(t|t')$. The most convenient representation for single-electron coherence is a mixed time-frequency representation called the electronic Wigner distribution function, which captures both temporal evolution and the nature of excitations [@Ferraro:2013-1]: $$\label{eq:G1:Wigner} W^{(e)}_{\rho,x}(t,\omega)=\int_{-\infty}^{+\infty} v_F\,\mathcal{G}^{(e)}_{\rho,x} \left(t+\frac{\tau}{2}\Big\vert t-\frac{\tau}{2}\right)\,{\mathrm{e}}^{{\mathrm{i}}\omega\tau}{\mathrm{d}}\tau$$ where $v_F$ denotes the Fermi velocity at position $x$. The electronic Wigner distribution function is real. Its marginal distributions give access to the time-dependent average electric current and to the electronic distribution function $f_e(\omega)$ at position $x$: \[eq:G1:marginals\] $$\begin{aligned} \label{eq:G1:marginals:occupation} f_e(\omega)&=\overline{W^{(e)}_{\rho,x}(t,\omega)}^t\\ \label{eq:G1:marginals:current} \langle i(x,t)\rangle_\rho &= -e\int_{-\infty}^{+\infty} \Delta W^{(e)}_{\rho,x}(t,\omega)\,\frac{{\mathrm{d}}\omega}{2\pi}\end{aligned}$$ A classicality criterion for the electronic Wigner distribution function has been formulated [@Ferraro:2013-1]: $0\leq W^{(e)}_{\rho,x}(t,\omega)\leq 1$. When verified, it basically means that $W^{(e)}_{\rho,x}(t,\omega)$ can be interpreted as a time-dependent electronic distribution function. This is the case for a quasi-classically driven Ohmic contact, when $|eV_{\mathrm{ac}}|\gg hf$ and $k_B T\gg hf$, $f$ being the driving frequency. In this case, both thermal fluctuations and the ac drive are responsible for multiphoton transitions in the electronic fluid and many electron/hole pairs are generated. On the contrary, for a single-electron excitation, quantum delocalization is responsible for negativities as seen on the example of the MZI [@Ferraro:2013-1]. Accessing single-electron coherence would thus enable to discuss the non classicality of the electronic fluid at single-particle level. Single-particle interferometry as linear filtering {#sec:G1:MZI} -------------------------------------------------- #### Introduction {#introduction} In classical signal processing, linear filters transform time-dependent input signals into output signals under the constraint of linearity. Well known examples include linear circuit elements in classical electronics and linear optics elements such as lenses, beam splitters and other various optical devices. These components act as linear filters on the electromagnetic field classical coherence introduced in the 30s [@Zernicke:1938-1]. This statement also extends to quantum optics by considering quantum optical coherences introduced by Glauber [@Glauber:1963-1]. In this section, we show that the same statement is true in electron quantum optics provided we use quantum conductors in which interaction effects can be neglected. As an example, we will see how the ideal Mach-Zenhder interferometer [@Haack:2011-1] or the measurement of the electronic distribution function using a quantum dot as an energy filter [@Altimiras:2010-1] realize linear filtering of the excess single-electron coherence $\Delta\mathcal{G}_{\rho,x}^{(e)}(t|t')$, which should therefore be seen as a “quantum signal" depending on two times. We will then discuss how Coulomb interactions partly – but not totally – invalidate this statement. In particular, we will explain why measuring finite-frequency currents enables to keep track of interaction effects and to recover information on single-electron coherence. #### Mach-Zehnder interferometry An ideal electronic Mach-Zehnder interferometer, such as the one depicted on Fig. \[fig:MZI\], is characterized by the time of flights $\tau_{1,2}$ along its two arms and the magnetic flux threading it $\Phi_B=\phi_B\times (h/e)$. When an electronic source $S$ is placed on the incoming edge channel $1$, the time-dependent outgoing electric current in channel $1$ is directly proportional to the excess electronic coherence of the source [@Haack:2011-1; @Ferraro:2013-1]: $$\begin{aligned} \label{eq:MZI:classical} &\langle i_{1\mathrm{out}}(t)\rangle = \sum_{j=1,2}\mathcal{M}_{j,j} \langle i_S(t-\tau_j)\rangle\\ \label{eq:MZI:quantum} &-2e|\mathcal{M}_{1,2}|\int_{\mathbb{R}}\cos{(\omega \tau_{12}+\phi)}\,\Delta W_S^{(e)}(t-\bar{\tau},\omega)\,\frac{{\mathrm{d}}\omega}{2\pi}\,\end{aligned}$$ where $\mathcal{M}_{i,j}$ denotes the product $\mathcal{A}_i\mathcal{A}_j^*$, $\mathcal{A}_j$ being the transmission amplitude of the beam splitters along path $j$ of the interferometer. We have introduced $\tau_{12}=\tau_1-\tau_2$, $\bar{\tau}=(\tau_1+\tau_2)/2$ and $\phi=\mathrm{arg}(\mathcal{M}_{1,2}+2\pi\phi_B)$ which is the phase associated with both beam splitters and the magnetic flux. The first line (Eq. ) does not depend on the magnetic flux and therefore corresponds to classical propagation within each of the two arms of the MZI, whereas the second line (Eq. ) corresponds to quantum interferences between propagations within both arms. Because the average electric current is also proportional to the excess single-electron coherence of the source, the outgoing average current is obtained from the excess incoming coherence $\Delta\mathcal{G}^{(e)}_S$ in channel $\mathrm{1in}$ by a linear filter which we write symbolically $\langle i_\text{out,dc}\rangle = \mathcal{L}_{\text{MZI}}[\Delta\mathcal{G}^{(e)}_S]$. Measurements of the $\Phi_B$ dependent part of the average dc current for various $\tau_1-\tau_2$ could then be used to reconstruct single-electron coherence [@Haack:2011-1]. (-1,0) node \[left\] [$A$]{} – (1,0); (7,0) – (9,0) node \[right\] [$B$]{}; (-3,3) node \[above left\] [1in]{} – (0,0) .. controls (2,4) and (6,4) .. (8,0) – ++(3,3) node \[above right\] [1out]{}; (4,3.8) node [1<span style="font-variant:small-caps;">mzi</span>]{}; (4,2.2) node [$\tau_1$]{}; (-3,-3) node \[below left\] [2in]{} – (0,0) .. controls (2,-3) and (6,-3) .. (8,0) – ++(3,-3) node \[below right\] [2out]{}; (4,-3.2) node [2<span style="font-variant:small-caps;">mzi</span>]{}; (4,-1.5) node [$\tau_2$]{}; (4,0) node [$\Phi_B$]{} ; The key ingredient in this derivation is the absence of electronic interactions. Whenever one replaces the Mach-Zehnder interferometer by an ideal ballistic quantum conductor in which interactions are neglected, the outgoing current in the measurement lead would also be proportional to $\Delta \mathcal{G}_S^{(e)}$. Denoting by $\mathcal{S}(t_f,t_i)$ the scattering amplitude for an electron arriving into the conductor at time $t_i$ and going out towards the measurement lead at time $t_f$, then the outgoing average time-dependent current is given by $$\langle i_{\mathrm{out}}(t)\rangle = \int_{\mathbb{R}^2} \mathcal{S}(t,t_+)\,\mathcal{S}^*(t,t_-)\Delta \mathcal{G}_{S}^{(e)}(t_+,t_-)\,{\mathrm{d}}t_+{\mathrm{d}}t_-\,$$ which describes a linear filtering of the incoming single-electron coherence $\Delta\mathcal{G}^{(e)}_S$ associated with time-dependent scattering. In particular, this expression is valid within the framework of Floquet scattering theory [@Moskalets:book]. #### On the role of interactions However, as we shall now discuss, because of Coulomb interactions, the situation in electron quantum optics is subtler than in photon quantum optics. It is therefore important to clarify in which situations the linear filtering paradigm is valid and when it is not valid. To begin with, let us consider a simple situation in which the source is placed at the input of a finite-length interaction region, as depicted on Fig. \[fig:interactions\]. For simplicity, we assume that this interaction region can be described within the edge-magnetoplasmon scattering formalism [@Safi:1999-1; @Degio:2009-1]: the incoming edge-magnetoplasmon mode at a given frequency is scattered elastically between the corresponding outgoing mode and environmental modes associated with other electrical degrees of freedom present in the problem. The environment can involve edge channels or any neighboring linear circuit modeled as a transmission line with a frequency-dependent impedance [@Yurke:1984-1]. Such a modelization is valid as long as all the components of the system are in the linear response regime. This formalism was used to compute the effect of Coulomb interactions on coherent current pulses [@Grenier:2013-1] and single-electron excitations [@Degio:2009-1; @Ferraro:2014-1]. Equivalently, because they induce inelastic processes, Coulomb interactions turn a quantum conductor into a non-linear component from the electron quantum optics point of view. Consequently, in general, the excess outgoing single electron coherence is not a linear filtering of the incoming one! (-2,1) rectangle (2,-1); (-4,2) .. controls (-2.5,0.5) .. (-2,.5) – (2,0.5) .. controls (2.5,0.5) .. (4,2); (-4,-2) .. controls (-2.5,-0.5) .. (-2,-.5) – (2,-0.5) .. controls (2.5,-0.5) .. (4,-2); (0,1) node \[above, align=center\] [Interaction region]{}; (-4,2) node \[above, align=center\] [environment (in)\ $\ket{0}$]{}; (-4,-2) node \[below, align=center\] [edge channel (in)\ $ \bigotimes_\omega \ket{\Lambda_\omega}$]{}; (4,2) node \[above, align=center\] [environment (out)\ $\bigotimes_\omega \ket{r(\omega) \Lambda_\omega}$]{}; (4,-2) node \[below, align=center\] [edge channel (out)\ $\bigotimes_\omega \ket{t(\omega) \Lambda_\omega}$]{}; A criterion for the validity of the electronic scattering theory approach to quantum transport at finite frequencies is that the frequency dependence of the electronic scattering matrix of a quantum conductor can be neglected [@Blanter:2000-1]. Single- to few-electron excitations emitted by electron quantum optics sources such as the mesoscopic capacitor [@Feve:2007-1] or the Leviton source [@Dubois:2013-1] as well as periodic electric currents generated using an advanced waveform generator usually define a frequency scale in the range of one to few tens of . On the other hand, an extended conductor such as a MZI has a scattering matrix varying over frequency scales of the order of the inverse of the time of flight of the conductor. For a $\SI{10}{\micro\meter}$ interferometer, it is of the order of $\SI{10}{\giga\hertz}$. This is why extended quantum conductors such as MZIs fail to satisfy this criterion. The important stream of theoretical work on interaction-induced decoherence [@Chalker:2007-1; @Neder:2007-4; @Levkivskyi:2008-1; @Neuenhahn:2008-1; @Kovrizhin:2009-1] in MZI interferometers illustrate the full complexity of understanding interaction effects in such extended quantum conductors. More recent works [@Tewari:2016-1; @Slobodeniuk:2016-1] dealing with the propagation of individualized energy-resolved single-electron excitations in a MZI are directly relevant for electron quantum optics but also show that this problem is not yet fully understood even at the single-electron level. By contrast, Coulomb interaction effects can be neglected over a much broader frequency range in the QPC which is an almost point-like electronic beam-splitter. As we shall see in the next section, this plays a very important role for the HOM and HBT experiments. Last but not least, the average finite-frequency currents are relatively robust to the effect of Coulomb interactions: whenever all electrical components respond linearily to the incoming excitation, the edge-magnetoplasmon scattering matrix can be used to reconstruct the incoming average finite frequency currents from the outgoing ones. Measurements of finite-frequency average currents have been successfully developped in the 1-11  range to perform the first direct measurement of edge-magnetoplasmon scattering amplitudes [@Bocquillon:2013-2]. Two-particle interferometry as overlap {#sec:G1:HOM} -------------------------------------- #### Motivation {#sec:G1:HOM:motivation} Although amplitude interferometry relies on the measurement of average currents, it does not seem well suited to perform single-electron tomography. First of all, as in optics, it would require a perfect control on electronic optical paths down to the Fermi wavelength. More importantly, as discussed in the previous section, Coulomb interactions prevent reconstructing the incoming single-electron coherence from the experimental signals. This situation is very similar to what happened in astronomy in the 30s and 40s: attempts at directly measuring the diameter of normal stars using amplitude interferometry were plagued by atmospheric turbulences and by the technological challenge of building a large optical interferometer. However, a way to circumvent this bottleneck was found by Hanbury Brown and Twiss (HBT) in the 50s [@Hanbury:1956-1]: their idea was to measure intensity correlations [@Hanbury:1956-2] since these contain interferences between waves emitted by pairs of atoms on the star. In quantum optics, the HBT effect ultimately relies on two-photon interferences [@Fano:1961-1]. In the 80s, the Hong Ou Mandel (HOM) experiment [@Hong:1987-1] also demonstrated two-particle interferences for identical particles (photons). Since then, two-particle interference effects have been observed in many different contexts, from stellar interferometry to nuclear and particle physics [@Baym:1998-1] and more recently with bosonic as well as fermionic cold atoms [@Jeltes:2007-1]. Recent experiments demonstrate a higher degree of control by using independent single-photon [@Beugnon:2006-1] and single atom sources [@Lopes:2015-1]. In this section, we review how the HOM experiment can be used to measure the overlap of the excess single-particle coherences arriving at a beam splitter. Remarkably, this result is true not only for electrons but for any fermionic or bosonic excitation. In photon quantum optics, it forms the basis of homodyne tomography [@Smithey:1993-1; @Lvovsky:2009-1] recently used to characterize few-photon states in the optical domain [@Ourjoumtsev:2006-1]. In the microwave domain, the HOM scheme has been used to access photon quantum optical correlations from electrical measurements [@Bozyigit:2011-1; @Lang:2013-1] and forms the basis of a tomography scheme for itinerant microwave photons [@daSilva:2010-1; @Eichler:2011-1]. #### Theoretical analysis {#sec:G1:HOM:theory} In electron quantum optics, the HBT and HOM experiments are demonstrated by sending electronic excitations generated by one or two electronic sources on an ideal electronic beam splitter, as depicted on Fig. \[fig:HBT-HOM:principle\]. In order to make a precise analogy with photon quantum optics, keeping in mind that in electron quantum optics, the vacuum is the reference Fermi sea and not a true vacuum, we consider that the electronic analogue of the table-top HBT experiment (Fig. \[fig:HBT-HOM:principle\]a) [@Hanbury:1956-2] is realized when one of the incoming channels is fed with the reference Fermi sea ($S_1$ or $S_2$ being off). By the same analogy, the electronic HOM experiment (Fig. \[fig:HBT-HOM:principle\]b) corresponds to situations with both electronic sources in the incoming channels switched on. Finally, whereas in photon quantum optics, the arrival of individual photons can be recorded, in electronics, the quantities of interest are the current correlations at zero frequency in the two outgoing branches. \ Let us focus on the outgoing current noise in channel $1$. A first important point is that the low-frequency current noise does not depend on the distance to the QPC: interaction effects lead to edge-magnetoplasmon scattering among the various outgoing edge channels close to the one considered but the total power remains the same. This is why HBT/HOM interferometry is immune to interaction effects in the measurement stage (beyond the QPC). In the same way, the intensity correlations measured in an optical stellar HBT interferometer are not blurred by atmospheric turbulences. Consequently, what we need is the excess low frequency current noise just after the QPC when both sources $S_1$ and $S_2$ are switched on. It is the sum of three contributions [@Ferraro:2013-1]: $$\begin{aligned} \Delta S_{11}^{(S_1\& S_2)}=\Delta S_{11}^{(S_1)} + \Delta S_{11}^{(S_2)} + \Delta S_{11}^{(\mathrm{HOM})}\end{aligned}$$ where $\Delta S_{11}^{(S_1)}$ and $\Delta S_{11}^{(S_2)}$ are the excess current noise when only the source $S_1$ (resp. $S_2$) is switched on. These contributions correspond to the excess noise in HBT experiments performed on each of the sources. The last term $$\begin{aligned} \Delta & S_{11}^{(\mathrm{HOM})}= \notag \\ &-2e^2RT\int_{\mathbb{R}^2} \Delta W^{(e)}_{1\mathrm{in}}(t,\omega) \Delta W^{(e)}_{2\mathrm{in}}(t,\omega)\,\frac{{\mathrm{d}}t\,{\mathrm{d}}\omega}{2\pi} \label{eq:HOM-overlap}\end{aligned}$$ is the overlap of the excess single-electron coherences arriving at the QPC [@Ferraro:2013-1] ($R$ and $T$ denoting the reflection and transmission probabilities of the QPC). Eq. encodes the effect of two-particle interferences between the excitations emitted by these sources. Note that the time delay of the two sources can be controlled and therefore a single experimental run gives access to the time-shifted overlaps of the excess electronic Wigner functions of the two sources. Finally, the minus sign comes from the fermionic statistics of electrons. The two first terms also involve two-particle quantum interferences. Because they contain information on coherences of each of the sources, they will be discussed in section \[sec:G2\]. Our point here is to emphasize that the electronic HOM experiment automatically encodes into the experimental signal what the signal processing community would call the sliding inner product of the quantum signals formed by the incoming excess single-electron coherences in channels $1$ and $2$. This is why the HOM experiment is so important: it can be used to test for unknown excess electronic Wigner functions by looking at their overlaps with themselves or with the ones generated by controlled and calibrated sources. This idea has been expanded to describe a generic tomography protocol for reconstructing an unknown excess single-electron coherence from its overlaps with coherences generated by suitable ac + dc drives [@Degio:2010-4]. We refer the reader to [@Marguerite:2016-2] in the same volume for a detailed description of this protocol. #### Experimental demonstration {#sec:G1:HOM:experiments} The electronic HBT experiment has been demonstrated in the late 90s using dc sources [@Oliver:1999-1; @Henny:1999-1] and more recently using single-electron sources [@Bocquillon:2012-1] which were then used to perform the electronic HOM experiment [@Bocquillon:2013-1]. These experiments have paved the way to measurements and studies of electron decoherence down to the single-electron level through HOM interferometry. The idea of the tomography protocol has been recently demonstrated by D.C. Glattli’s group [@Jullien:2014-1]: in this experiment, a stream of Lorentzian pulses is sent onto a beam splitter whose other incoming channel is fed with a small ac drive on top of a dc bias. Measurement of the low-frequency noise then enables reconstructing the photo-assisted transition amplitudes which, in this case, contain all the information on single-electron and higher-order electronic coherences [@Dubois:2013-1]. This experiment being performed in a 2DEG at zero magnetic field, interaction effects can be neglected and the experiment leads to the reconstruction of the Leviton single-electron coherence [@Jullien:2014-1]. As reviewed in this volume [@Marguerite:2016-2], the HOM experiment has also recently been used to probe interaction effects within quantum Hall edge channels. In these experiments, performed at filling factor $\nu=2$, two single-electron sources are located at some distance of the QPC and interaction effects are strong enough to lead to quasi-particle destruction, as suggested by energy relaxation experiments [@LeSueur:2010-1]. First, the HOM effect was used to probe how interactions lead to fractionalization of classical current pulses [@Freulon:2015-1] in qualitative agreement with the neutral/charge edge-magnetoplasmon mode model [@Levkivskyi:2008-1] which had been already probed through energy relaxation experiments [@Degio:2010-1] and high-frequency admittance measurements [@Bocquillon:2013-2]. But the real strength of HOM experiment comes from its ability to probe electronic coherence in a time- and energy-resolved way. It was thus recently used to study quantitatively the effect of Coulomb interactions on energy-resolved single-electron excitations (called Landau excitations) [@Marguerite:2016-1]. The experimental data confirm theoretical predictions and validate the decoherence scenario based on edge-magnetoplasmon scattering [@Wahl:2013-1; @Ferraro:2014-1]. Two-electron coherence {#sec:G2} ====================== Let us now turn to two-electron coherence. After briefly reviewing its definition and properties, introducing the two-electron Wigner distribution function will enable us to emphasize its non-classical features arising from Fermi statistics. We will then turn to two-particle interferometry and show that, under appropriate hypotheses, these experiments perform a linear filtering on the intrinsic two-electron coherence, thus generalizing what was discussed in section \[sec:G1:MZI\]. Definition and representations {#sec:G2:definitions} ------------------------------ #### Definition Two-electron coherence is defined by direct analogy with Glauber’s second-order photonic coherence [@Moskalets:2014-1]: $$\label{eq:G2:definition} \mathcal{G}_\rho^{(2e)}(1,2|1'2')=\mathrm{Tr}(\psi(2)\psi(1)\rho\, \psi^\dagger(1')\psi^\dagger(2'))\,$$ where $1=(\alpha_1,x_1,t_1)$ and $2=(\alpha_2,x_2,t_2)$, $\alpha_{1,2}$ being channel indices, and similarly for $1'$ and $2'$. Its physical meaning can be obtained by computing the two-electron coherence for the state $|\Psi_N\rangle$ defined in section \[sec:G1:definitions\]. The result is a sum over all the two-electron wavefunctions $\varphi_{k,l}(x,y)=\varphi_k(x)\varphi_l(y)-\varphi_k(y)\varphi_l(x)$ ($k<l$) that can be extracted from the $N$ single-particle wavefunctions $(\varphi_k)_{k\in \{ 1,\dots,N \}}$ [@Thibierge:2016-1]: $$\label{eq:G2:wavefunctions} \mathcal{G}^{(2e)}_{|\Psi_N\rangle}(x_1,x_2|x'_1,x'_2)= \sum_{k<l}\varphi_{k,l}(x_1,x_2)\,\varphi_{k,l}(x'_1,x'_2)\,.$$ Two-electron coherence at a given position is a function of four times $(t_1,t_2;t'_1,t'_2)$. #### Fermionic statistics and two-electron coherence Although its definition and Eq.  may suggest that two-electron coherence bears similarity with single-electron coherence discussed in the previous section, it already contains all the counter-intuitive aspects of quantum indiscernability. Electrons being fermions, the two-electron coherence satisfies the following antisymmetry properties: \[eq:G2:antisymmetry\] $$\begin{aligned} \mathcal{G}^{(2e)}_\rho(1,2|1'2')&=-\mathcal{G}_\rho^{(2e)}(2,1|1',2')\\ &=-\mathcal{G}_\rho^{(2e)}(1,2|2',1')\end{aligned}$$ Antisymmetry leads to the global symmetry of two-electron coherence $\mathcal{G}_\rho^{(2e)}(1,2|1',2')=\mathcal{G}_\rho^{(2e)}(2,1|2',1')$ expressing that the order of detection of electrons does not matter. It also implies that two-electron coherence vanishes whenever $1=2$ or $1'=2'$: this is the Pauli exclusion principle. Together with the conjugation relation $$\mathcal{G}^{(2e)}_\rho(1,2|1',2')^*=\mathcal{G}^{(2e)}_\rho(1',2'|1,2)\,,$$ antisymmetry implies that two-electron coherence is defined by its values for only $1/8$-th of the possible arguments. The intrinsic two-electron coherence emitted by a source can be defined from the second-order electronic coherence by subtracting not only the Fermi sea contribution but also all processes contributing to two-electron detection and involving the excess single-electron coherence of the source. These involve classical contributions as well as quantum exchange contributions [@Thibierge:2016-1]: \[eq:G2:decomposition\] $$\begin{aligned} \mathcal{G}&_\rho^{(2e)}(1,2|1',2') = \notag \\ \label{eq:G2:decomposition:Fermi} &\mathcal{G}_F^{(2e)}(1,2|1',2')\\ \label{eq:G2:decomposition:classical} +& \mathcal{G}^{(e)}_F(1|1')\,\Delta\mathcal{G}_\rho^{(e)}(2|2')+ \mathcal{G}^{(e)}_F(2|2')\,\Delta\mathcal{G}_\rho^{(e)}(1|1')\\ \label{eq:G2:decomposition:exchange} -&\mathcal{G}^{(e)}_F(1|2')\,\Delta\mathcal{G}_\rho^{(e)}(2|1') -\mathcal{G}^{(e)}_F(2|1')\,\Delta\mathcal{G}_\rho^{(e)}(1|2')\\ \label{eq:G2:decomposition:intrinsic} +& \Delta\mathcal{G}_\rho^{(2e)}(1,2|1',2')\,.\end{aligned}$$ Eq.  should be seen as a definition of the intrinsic two-electron coherence $\Delta\mathcal{G}^{(2e)}_{\rho}$ from the total two-electron coherence, the Fermi sea two-electron coherence and lower-order electronic coherences. The second term is present for classical particles and represent classical correlations in which the origin of the two detected particles can be traced back either to the Fermi sea or the source. Such back-tracking is not possible for the exchange terms whose minus sign comes from the fermionic statistics. Note that Eq.  is fully compatible with Eq. . Moreover, for a state obtained by adding a single-electron or hole excitation to the Fermi sea, the intrinsic two-electron coherence vanishes as expected for a source emitting only one excitation. #### The frequency domain representation of two-electron coherence Exactly as in the case of single-electron coherence, the nature of excitations can be obtained by going to the frequency domain: $$\begin{aligned} \label{eq:G2:frequency-domain} \widetilde{\mathcal{G}}^{(2e)}_\rho&(\boldsymbol{\omega}_+|\boldsymbol{\omega}_-) \notag \\ =&\int_{\mathbb{R}^4} \mathcal{G}^{(2e)}_\rho(\mathbf{t}_+|\mathbf{t}_-)\,{\mathrm{e}}^{{\mathrm{i}}(\boldsymbol{\omega}_+\cdot\mathbf{t}_+- \boldsymbol{\omega}_-\cdot\mathbf{t}_-)}\,{\mathrm{d}}^2\mathbf{t}_+{\mathrm{d}}^2\mathbf{t}_-\,\end{aligned}$$ where $\mathbf{t}_+=(t_1,t_2)$ and $\mathbf{t}_-=(t'_1,t'_2)$ are respectively conjugated to $\boldsymbol{\omega}_+=(\omega_1,\omega_2)$ and $\boldsymbol{\omega}_-=(\omega'_1,\omega'_2)$. Note that antisymmetry properties are also true in the frequency domain. The diagonal of the frequency domain ($\boldsymbol{\omega}_-=\boldsymbol{\omega}_+=(\omega_1,\omega_2)$) can be divided into quadrants depicted on Fig. \[fig:G2:excitations\] that describe the elementary two-particle excitations. When $\omega_1$ and $\omega_2$ are both positive, we have an electronic pair whereas when they are both negative, we have a pair of holes. In the case one is positive and the other negative, we have an electron hole pair. Note that this classification is compatible with the permutation $\omega_1\leftrightarrow \omega_2$. (0,0) – ++(1.5,0) arc (0:90:1.5)– cycle; (0,0) – ++(-1.5,0) arc (-180:-90:1.5)– cycle; (0,0) – ++(1.5,0) arc (0:-90:1.5)– cycle; (0,0) – ++(0,1.5) arc (90:180:1.5)– cycle; (-2,0) – (2,0) node \[above right\] [$\omega_1$]{}; (0,-2) – (0,2) node \[above left\] [$\omega_2$]{}; (-2,0) – (2,0) node \[right\] [$\delta\bar{\omega}$]{}; (0,-2) – (0,2) node \[above\] [$\bar{\omega}$]{}; at (0.4,1) [(2e)]{}; at (0.4,-1) [(2h)]{}; at (-1,.3) [(e+h)]{}; at (1,.3) [(e+h)]{}; The full frequency domain $(\boldsymbol{\omega}_+,\boldsymbol{\omega}_-)$ can then be decomposed into 4D simplexes based on these quadrants for the diagonal. This will naturally be compatible with the antisymmetry properties of the two-electron coherence. Diagonal simplexes are based on $\boldsymbol{\omega}_+$ and $\boldsymbol{\omega}_-$ that both describe the same type of excitation. This leads to a two-electron simplex, a two-hole simplex and two electron/hole pair simplexes respectively containing the contributions of two-electron, two-hole and electron/hole pair excitations. The off-diagonal simplexes where $\boldsymbol{\omega}_+$ and $\boldsymbol{\omega}_-$ do not belong to the same quadrant describe coherences between these four different two-particle excitations. The Wigner representation of two-electron coherence --------------------------------------------------- #### Definition The Wigner representation of two-electron coherence is defined in the same way as for single-electron coherence, that is as a Fourier transform with respect to the time differences $\tau_j=t_j-t'_j$. When considering a diagonal two-electron coherence in the channel indices ($\alpha_j=\alpha'_j$ for all $j=1,2$), this leads to a real function $$\begin{aligned} \label{eq:G2:Wigner:definition} W^{(2e)}_{\rho,x}&(t_1,\omega_1;t_2,\omega_2)= \notag \\ &\int_{\mathbb{R}^2} v_F^2 \mathcal{G}^{(2e)}_{\rho,x}\left(\boldsymbol{t}+\frac{\boldsymbol{\tau}}{2}\Big\vert \boldsymbol{t}-\frac{\boldsymbol{\tau}}{2}\right)\,{\mathrm{e}}^{{\mathrm{i}}\boldsymbol{\omega}\cdot \boldsymbol{\tau}}\,{\mathrm{d}}^2\boldsymbol{\tau}\,.\end{aligned}$$ The Wigner representation of the excess two-electron coherence $\Delta W^{(2e)}_{\rho,x}(t_1,\omega_1;t_2,\omega_2)$ is defined by Eq. from the excess two-electron coherence. Whenever Wick’s theorem applies, the total two-electron coherence can be computed in terms of the single-electron one: $$\begin{aligned} \label{eq:G2:Wick} \mathcal{G}_{\rho,x}^{(2e)}&(1,2|1',2')=\notag \\ &\mathcal{G}^{(e)}_{\rho,x}(1|1')\,\mathcal{G}^{(e)}_{\rho,x}(2|2') -\mathcal{G}^{(e)}_{\rho,x}(1|2')\,\mathcal{G}^{(e)}_{\rho,x}(2|1')\,.\end{aligned}$$ and, using Eqs. , the same equation also describes the intrinsic two-electron coherence in terms of the excess single-electron coherence. The first term contributes to the two-electron Wigner distribution through the product $W^{(e)}_{\rho,x}(t_1,\omega_1)\,W^{(e)}_{\rho,x}(t_2,\omega_2)$ which corresponds to independent classical particles. The second term comes from quantum exchange and, as we shall see now, is responsible for non-classical features of the two-electron Wigner distribution function. #### Non classicality of two-electron coherences In the case of single-electron coherence, a definition of classicality was given [@Ferraro:2013-1] based on the idea of interpreting $W_{\rho,x}^{(e)}(t,\omega)$ as a time-dependent electronic distribution function compatible with the Pauli exclusion principle. It is natural to extend this definition to the two-particle case: $W^{(2e)}_{\rho,x}(t_1,\omega_1;t_2,\omega_2)$ would be called classical if it takes values between $0$ and $1$. Of course, if we consider the inter-channel two-electron Wigner distribution associated with the inter-channel two-electron coherence $\mathcal{G}^{(2e)}_{\rho,x}(1,t_1;2,t_2|1,t'_1;2,t'_2)$, then when the two channels are not correlated, we have $W^{(2e)}_{\rho,x}(1,t_1,\omega_1;2,t_2,\omega_2)= W_1^{(e)}(t_1,\omega_1)\,W^{(e)}_2(t_2,\omega_2)$ as expected for uncorrelated classical objects[^1]. Consequently, if the two-electronic Wigner distribution in channels $1$ and $2$ are classical, the inter-channel two-electron Wigner distribution is also classical. But as we shall see now, because of its antisymmetry properties, the two-electron Wigner distribution in a single channel exhibits non classical features. To illustrate this point, let us consider mutually orthogonal time-shifted wave-packets: $\varphi_1(t)=\varphi_e(t-\tau/2)$ and $\varphi_2(t)=\varphi_e(t+\tau/2)$. The intrinsic two-electron Wigner distribution function associated with the state $|\Psi_2\rangle= \psi^\dagger[\varphi_1]\psi^\dagger[\varphi_2]|F\rangle$ is then $$\begin{aligned} \label{eq:G2:Wigner:2e-example} \Delta & W^{(2e)}_{|\Psi_2\rangle} (t_1,\omega_1;t_2,\omega_2) = \notag \\ &W_{\varphi_1}\left(t_1,\omega_1\right) W_{\varphi_2}\left(t_2,\omega_2\right)\notag \\ + & W_{\varphi_2}\left(t_1,\omega_1\right) W_{\varphi_1}\left(t_2,\omega_2\right)\notag \\ - & 2\cos{((\omega_1-\omega_2)\tau)}\,W_{\varphi_e}(t_1,\omega_1)W_{\varphi_e}(t_2,\omega_2)\,.\end{aligned}$$ When considering a quasi-classical electronic wavepacket, such that $W_{\varphi_e}(t,\omega)$ is almost everywhere positive, we see that the last term contains interference fringes due to the $\cos{((\omega_1-\omega_2)\tau)}$ factor. When $\varphi_1$ and $\varphi_2$ are well separated, negativities appear which reflect the non-classical nature of two-electron wavefunctions within a single edge channel. Note that the dependence in $\omega_1-\omega_2$ comes from the fact that, in the above example, $\varphi_{1}$ and $\varphi_2$ are time-shifted wavepackets. Energy-shifted wavepackets would lead to oscillations in $t_1-t_2$ analogous to Friedel oscillations in solid-state physics. In full generality, the quantum exchange interference terms present both a time and an energy dependence and this prevents $W^{(2e)}$ to be interpreted as a time-dependent two-electron distribution function. Similarly, the two-electron Wigner distribution function of the equilibrium state at electronic temperature $T_{\text{e}}$ and vanishing chemical potential is given by \[eq:W2:Fermi\] $$\begin{aligned} \label{eq:W2:Fermi:1} W^{(2e)}_{\mu=0,T_{\text{e}}}&(t_1,\omega_1;t_2,\omega_2)= f_{T_{\text{e}}}(\omega_1)\,f_{T_{\text{e}}}(\omega_2) \\ \label{eq:W2:Fermi:2} -4\pi k_B&T_{\text{e}}\delta(\omega_1-\omega_2) f_{B,T_{\text{e}}}(\omega_\text{tot})\, \frac{\sin{\left(\omega_\text{tot}t_{12}\right)}}{\sinh{\left(t_{12}/\tau(T_{\text{e}})\right)}}\,\end{aligned}$$ where $\omega_\text{tot}=\omega_1+\omega_2$, $t_{12}=t_1-t_2$ and $\tau(T)=\hbar/k_BT$ denotes the thermal coherence time. Here $f_{T_{\text{e}}}$ is the Fermi-Dirac distribution at temperature $T_{\text{e}}$ and $\mu=0$ whereas $f_{B,T}(\omega)=1/({\mathrm{e}}^{\hbar\omega/k_BT}-1)$ denotes the Bose-Einstein distribution at temperature $T$. The singular second term expresses the Pauli exclusion principle and presents strong oscillations in $t_{12}$. Therefore $W^{(2e)}_{\mu=0,T_{\text{e}}}$ cannot be interpreted as a time-dependent electronic distribution. Relation to current noise {#sec:G2:noise} ------------------------- Let us now describe the precise relation between two-electron coherence and the excess current noise $\Delta S_i(t,t')$ defined as the excess of $$\label{eq:G2:noise:definition} S_i(t,t')=\langle i(t')\,i(t)\rangle - \langle i(t)\rangle\langle i(t')\rangle\,.$$ Since sub-nanosecond time-resolved measurements are not available in electronics, $S_i(t,t')$ is not directly accessible. However, finite-frequency current noise measurements [@Parmentier:2010-1] give access to the noise spectrum which is a time-averaged quantity. More recently, partial measurements of the time-dependent current noise power spectrum have been performed. This quantity is defined as the Wigner function associated with excess current correlations: $$\label{eq:G2:noise:Wigner-Ville} W_{\Delta S_i}(t,\omega)=\int_{\mathbb{R}} \Delta S_i\left(t-\frac{\tau}{2}\Big\vert t+\frac{\tau}{2}\right)\,{\mathrm{e}}^{{\mathrm{i}}\omega\tau}\,{\mathrm{d}}\tau\,.$$ This is achieved through a recently developped homodyne measurement technique [@Gasse:2013-1] which has been used to probe the squeezing of the radiation emitted by a tunnel junction. The canonical anticommutation relations and definition imply that the quantity defined by Eq.  is directly related to the intrinsic two-electron coherence by \[eq:G2:noise:noise-to-G2\] $$\begin{aligned} & W_{\Delta S_i}(t,\omega)+W_{\langle i\rangle}(t,\omega)= \notag \\ &-e\langle i(t)\rangle \label{eq:G2:noise:noise-to-G2:Poisson}\\ \label{eq:G2:noise:noise-to-G2:HBT} &- e^2\int_{\mathbb{R}} h_\mu(\omega,\omega')\,\Delta W^{(e)}_{\rho}(t,\omega')\,\frac{{\mathrm{d}}\omega'}{2\pi}\\ \label{eq:G2:noise:noise-to-G2:G2} &+e^2\int_{\mathbb{R}} v_F^2\Delta\mathcal{G}^{(2e)}_\rho\left(t+\frac{\tau}{2},t-\frac{\tau}{2}\Big\vert t+\frac{\tau}{2},t-\frac{\tau}{2}\right)\,{\mathrm{e}}^{{\mathrm{i}}\omega\tau}{\mathrm{d}}\tau\,\end{aligned}$$ where $h_{\mu}(\omega,\omega')=f_{\mu}(\omega-\omega')+f_{\mu}(\omega+\omega')$ and $W_{\langle i\rangle}(t,\omega)$ denotes the Wigner-Ville transform [@Ville:1948-1] of the average time-dependent current. The first term  is a Poissonian contribution associated with the granular nature of charge carriers. The second term arises from two-particle interferences between excitations generated by the source and electrons within the Fermi sea. These contributions are called the HBT contributions since these two-particle interferences are precisely what is measured in an HBT experiment. Finally, the last term corresponds to the intrinsic two-electron coherence contribution to the current noise. This equation is indeed the electron quantum optics version of the famous relation on fluctuations of particle number in an ideal Bose gas [@Einstein:1925-1] where the term minus $W_{\langle i\rangle}(t,\omega)$ corresponds to Einstein’s quadratic term. Finally, the non triviality of the Fermi sea vacuum is responsible for the term . This equation also relates electronic coherences to the quantum optical properties of the edge magnetoplasmons within the edge channel and therefore of the photons radiated into a transmission line capacitively coupled to the edge channel as in [@Degio:2009-1]. It therefore establishes a bridge between electron quantum optics and the recently studied quantum optics of noise [@Grimsmo:2016-1]. Finally, let us stress that Eq. , which is also valid in the presence of interactions, shows that accessing single-electron coherence as well as the current noise gives access to the diagonal part of two-electron coherence, as expected since the latter contains all the information on time resolved two-electron detection. Directly accessing the intrinsic two-electron coherence without any HBT contribution can be achieved by partitioning the electronic beam onto a beam splitter in an HBT setup (see Fig. \[fig:HBT-HOM:principle\]). These current correlations are directly related to the inter-channel two-electron coherence right after the beam splitter since fermionic fields within different channels anticommute: $$\begin{aligned} \label{eq:G2:HBT:current-correlations} \langle i_{1\mathrm{out}}(t_1)&i_{2\mathrm{out}}(t_2)\rangle = \notag \\ & e^2v_F^2 \Delta\mathcal{G}^{(2e)}_{\mathrm{out}}(1,t_1;2,t_2|1,t'_1;2,t'_2)\end{aligned}$$ Remarkably, this outgoing two-electron coherence is proportional to the incoming two-electron coherence [@Thibierge:2016-1]: $$\begin{aligned} \label{eq:G2:HBT:1source} \Delta\mathcal{G}_{\mathrm{out}}^{(2e)}(1,t_1;2,t_2&|1,t'_1;2,t'_2)= \notag \\ &RT\,\Delta\mathcal{G}^{(2e)}_S(t_1,t_2|t'_1,t'_2)\,.\end{aligned}$$ Measuring outgoing inter-channel current correlations in the HBT setup thus directly probes the intrinsic excess coherence of the source, as in photon quantum optics. Two-particle interferometry as linear filtering {#sec:G2:Franson} ----------------------------------------------- Although current correlations in the HBT geometry only give access to the diagonal part of the intrinsic two-electron coherence in the time domain, Eq.  naturally leads to a general idea for accessing the off-diagonal part $\Delta\mathcal{G}_S^{(2e)}(t_1,t_2|t'_1,t'_2)$ for $(t_1,t_2)\neq (t'_1,t'_2)$. The idea is to use linear filters of single-electron coherence as depicted on Fig. \[fig:Franson-generalized\]. Let us assume that the outgoing current is obtained from a linear filtering of the incoming single-electron coherence $\langle i_A\rangle=\mathcal{L}_A\left[\Delta\mathcal{G}_{1\mathrm{in}}^{(e)}\right] $ with a similar relation for detector $B$. Then, the outgoing current correlations $\langle i_A\,i_B\rangle$ are obtained by applying a linear filter to the incoming two-electron coherence: $$\label{eq:G2:Franson:main} \langle i_A\, i_B\rangle = RT\left(\mathcal{L}_A^{(1)}\otimes \mathcal{L}_B^{(2)}\right)\left[\Delta\mathcal{G}^{(2e)}_S\right]$$ in which has been used to obtain . Despite its compacity, Eq.  unifies many different experiments under a simple physical interpretation: the intrinsic two-electron coherence $\Delta\mathcal{G}_S^{(2e)}$ describing two-particle excitations emitted by the source, is encoded into current correlations $\langle i_A\,i_B\rangle$ via an HBT interferometer and two linear filters for single-electron coherence. In the absence of these filters, the measurement of current correlation gives information on the diagonal part of $\Delta\mathcal{G}_S^{(2e)}$ as seen in section \[sec:G2:noise\]. When $A$ and $B$ are electronic energy filters, and assuming that no electronic relaxation process takes place between the QPC and the filters (see the discussion in section \[sec:G1:MZI\]), we access the diagonal part of $\Delta\mathcal{G}^{(2e)}_S$ in the frequency domain. It also naturally leads to the idea of the Franson interferometer [@Franson:1989-1] originally invented to test photon entanglement [@Brendel:1999-1; @Marcikic:2002-1] and later considered for testing two-particle Aharonov-Bohm effect and electronic entanglement generation [@Splettstoesser:2009-1]. It is a natural way to probe the off-diagonal part of $\Delta\mathcal{G}_S^{(2e)}$ in the time domain since, as explained in section \[sec:G1:MZI\], a MZI converts single-electron coherences in the time domain into electrical currents. \(O) at (0,0); (entree\_MZI1) at (-2.5,-1.5); (sortie\_MZI1) at (-4,-1.5); (entree\_MZI2) at (2.5,-1.5); (sortie\_MZI2) at (4,-1.5); (source1) at (-1.5,1.5); (source2) at (1.5,1.5); (source2) – (O) node \[midway, below right\] [2s]{}; (source1) – (O) node \[midway, below left\] [1s]{}; (0,0) – ++(0,-1) – ++(-.6,0) – ++(-.2,-.4) – ++(.2,.4) – ++(.6,0) – ++(-.2,-.4) – ++(.2,.4) – ++(.6,0) – ++(-.2,-.4) – ++(.2,.4); (source) (0,0) node [$S$]{} (-0.4,-0.4) rectangle (0.4,0.4); ; \(O) – ++(-1.5,-1.5); (-1.5,-1.5) – (entree\_MZI1) node \[midway, above\] [1in]{}; (O) – ++(1.5,-1.5); (1.5,-1.5) – (entree\_MZI2) node \[midway, above\] [2in]{}; (entree\_MZI1)– ++ (0,-0.7) – ++(-1.5,0) – ++(0,1.4) – ++ (1.5,0) –cycle; (entree\_MZI2)– ++ (0,-0.7) – ++(1.5,0) – ++(0,1.4) – ++ (-1.5,0) –cycle; (entree\_MZI1) ++(-0.75,0) node [$\mathcal{L}_{A}$]{}; (entree\_MZI2) ++(0.75,0) node [$\mathcal{L}_{B}$]{}; (sortie\_MZI1) – ++(-0.8,0) node \[left\] [1out]{}; (sortie\_MZI2) – ++(0.8,0) node \[right\] [2out]{}; \(O) ++(0,.7) – ++(0,-1.4) node \[below\] [QPC]{}; From electron quantum optics to quantum information {#sec:QI} =================================================== In this last section, we explain how to relate electron quantum optics quantities to quantum information ones using, once again, signal processing ideas. This enables discussing quantum information quantities in electron quantum optics systems. We illustrate these ideas and procedures by sketching how to get an insight on the many-body state generated by the mesoscopic capacitor. Density matrices from electronic coherences {#sec:QI:density-matrices} ------------------------------------------- Because electrons are fermions, each single-particle state can be viewed as a two-level system, thus raising the question of their use as effective qubits to encode quantum information. Given a normalized wavefunction $\varphi_a$, its average occupation number is obtained from single-electron coherence by $$\label{eq:QI:population} \bar{n}[\varphi_a]= v_F^2\int_{\mathbb{R}^2} \varphi_a(t_+)^*\varphi_a(t_-)\,\mathcal{G}^{(e)}_{\rho,x}(t_+|t_-)\, {\mathrm{d}}t_+\,{\mathrm{d}}t_-\,$$ where $\varphi_a(t)$ denotes the electronic wavefunction for $x=-v_Ft$. The quantity $\bar{n}[\varphi_a]$ being the average value of the number operator $N[\varphi_a]=\psi^\dagger[\varphi_a]\psi[\varphi_a]$, it is between zero and unity. Denoting by $|0_a\rangle$ and $|1_a\rangle$ the states respectively corresponding to the absence or the presence of an electron in the single particle state $\varphi_a$, $\bar{n}_a$ and $1-\bar{n}_a$ can be used as diagonal elements of a $2\times 2$ matrix. However, in the absence of superconductors, charge conservation leads to a superselection rule forbidding quantum superpositions of states of different charges: its off-diagonal elements, which are equal to $\langle \psi^\dagger[\varphi_a]\rangle_\rho$ and its complex conjugate, vanish. Nevertheless, this idea becomes more interesting when considering more than one single-particle state because the various superselection sectors are no longer one dimensional. A first possibility to obtain an effective qubit is to consider two orthogonal single-particle states $\varphi_a$ and $\varphi_b$ and the charge one sector. The basis vector $|0\rangle$ (resp. $|1\rangle$) then corresponds to the state where the electron is totally localized in the electronic state $\varphi_a$ (resp. $\varphi_b$). The reduced density matrix for this railroad electronic qubit [@Ionicioiu:2001-1] is then defined from single and two electron coherences by: $$\begin{aligned} \langle 0|\rho_{\text{qb}}&|0\rangle= \langle \psi^\dagger[\varphi_a]\psi[\varphi_a]\,\psi[\varphi_b]\psi^\dagger[\varphi_b]\rangle_\rho\\ \langle 1|\rho_{\text{qb}}&|1\rangle =\langle \psi^\dagger[\varphi_b]\psi[\varphi_b]\,\psi[\varphi_a]\psi^\dagger[\varphi_a]\rangle_\rho \\ \langle 0|\rho_{\text{qb}}&|1\rangle= \langle \psi^\dagger[\varphi_a]\psi[\varphi_b]\rangle_\rho\end{aligned}$$ This framework can be extended to situations involving more single-particle states. For example, one could consider two pairs of single-electron states $\varphi_{a1}$, $\varphi_{a2}$ and $\varphi_{b1}$ $\varphi_{b2}$ as in the four leads device considered by Samuelsson and Büttiker [@Samuelsson:2006-1]. The two-particle sector would then contain an effective 2-qubit reduced density matrix whose matrix elements can be expressed in terms of electronic coherences. Deriving an effective two-qubit reduced density operator from electron coherences directly enables us to use the results from quantum information on the characterization of entanglement in a bipartite system. Entanglement is well defined and easily characterized for bipartite systems in a pure state using for example the von Neumann entropy of the reduced density matrix of one of the two subsystems [@Book:Nielsen-Chuang]. In cases where the whole system is not in a pure state, its total density matrix is said to represent an entangled state if and only if it is not factorized, that is, if and only if it cannot be written as a statistical mixture of tensor products of density operators relative to each subsystem. In the case of a two qubit system, a single quantity called the concurrence can be used to characterize entanglement. Denoting by $\rho_{\text{2qb}}$ the total density matrix for the two qubits, the concurrence is defined by [@Hill:1997-1]: $$C[\rho_{\text{2qb}}]=\max{(0,\lambda_1-\lambda_2-\lambda_3-\lambda_4)}$$ where $\lambda_1\geq \lambda_2\geq \lambda_3\geq \lambda_4$ are the eigenvalues of the hermitian matrix $R=\sqrt{\sqrt{\rho_{\text{2qb}}}\cdot \tilde{\rho}_{\text{2qb}}\cdot\sqrt{\rho_{\text{2qb}}}}$, and $$\tilde{\rho}_{\text{2qb}}=(\sigma^y\otimes \sigma^y)\,\rho_{\text{2qb}}^*\, (\sigma^y\otimes \sigma^y)\,.$$ The concurrence vanishes if and only if the 2-qubit state is factorized. The entanglement of formation represents the minimum of the average entanglement of an ensemble of pure states representing $\rho_{\text{2qb}}$ [@Wootters:1998-1]. It corresponds to the minimal number of maximally entangled qubits per realization[^2] needed to generate the entangled states described by $\rho_{\text{2qb}}$ [@Bennett:1996-1; @Wootters:2001-1]. Remarkably, for a 2-qubit, it has a closed expression in terms of the concurrence [@Wootters:1998-1]: $$\begin{aligned} \mathcal{E}[\rho_{\text{2qb}}]&=H_2\left(\frac{1+\sqrt{1-\mathcal{C}[\rho_{\text{2qb}}]^2}}{2}\right)\\ H_2(x)& =-x\log_2{(x)}-(1-x)\log_2{(1-x)}\end{aligned}$$ These considerations describe how to obtain quantum information quantities from electron quantum optics concepts. Let us now illustrate these ideas on the study of electron/hole entanglement generated by the mesoscopic capacitor. Electron/hole entanglement from the mesoscopic capacitor {#sec:LPA-source} -------------------------------------------------------- The mesoscopic capacitor depicted on Fig. \[fig:source-LPA\] is a quantum RC circuit [@Gabelli:2006-1] which can be operated in the non-linear regime with a periodic square drive at frequency $f=1/T$ in order to emit a single-electron excitation during the half period $[0,T/2]$ and a single hole excitation during the second half period $[T/2,T]$ [@Feve:2007-1]. Reaching this optimal single-electron source regime requires tuning the QPC transparency $D$ (see Fig. \[fig:source-LPA\]) so that the electron and hole excitations have the time to form and escape during their respective half periods. The source is modeled as a time-dependent single-electron scatterer which amounts to neglecting the effect of Coulomb interactions within the dot. Under this hypothesis, the single-electron coherence can be computed using Floquet’s theory [@Moskalets:book; @Moskalets:2002-1]. Note that in this case, single-electron coherence determines all the electronic coherences emitted by the source since Wick’s theorem is valid. Here we use a Floquet description of [@Degio:2010-4] to obtain the single-electron coherence generated by the mesoscopic capacitor. A program written in C computes the single-electron coherence in the frequency domain. An Haskell code then computes 2-qubit matrices and quantum information theoretical quantities from these data. (1,0.7) – (6.5,0.7) – (6.5,2) – (1,2) – cycle; (4.2,2) – (3.9,2.4) – (4,2.8) arc \[start angle=-60, end angle=240, x radius=1cm, y radius=1cm\] – (3.1,2.4) – (2.8,2); (2.2,2.1) – (2.9,2.3) –(2.9,2.5) – (2.2,2.7) – cycle; (4.8,2.1) – (4.1,2.3) –(4.1,2.5) – (4.8,2.7) – cycle; (2.9,1.7) – (3.2,2.4) – (3,3.1); (4.1,1.7) – (3.8,2.4) – (4,3.1); (3.35,2.1) – (3.35,2.7) node\[midway,right\][$D$]{}; (1,1.3) – (6.5,1.3); (1,1.7) – (2.9,1.7); (4.1,1.7) – (6.5,1.7); (2.9,1.7) – (4.1,1.7); (3.3,1.85) – (3.7,1.85); (3.5,3.7) circle (0.75); (3.5,3.7) circle (0.5); (4.5,2.8) – (4.7,4.7) – (2.3,4.7) – (2.5,2.8) – cycle; (3.5,4.7) – (3.5,5) – (6,5) – (6,4.5); (5.7,4.5) – (6.3,4.5); (5.7,4.35) – (5.8,4.5); (5.85,4.35) – (5.95,4.5); (6,4.35) – (6.1,4.5); (6.15,4.35) – (6.25,4.5); (5.2,5) circle (0.45); at (5.2,5) [$V_d(t)$]{}; at (3.5,1.5) [$1-D$]{}; We have considered the mesoscopic capacitor with the following experimentally relevant parameters: $\Delta/hf =20$. The drive voltage $V_d(t)$ is a square voltage. The optimal operation point with these parameters is found to be at $D_{\text{opt}}\simeq 0.35$. Figure \[fig:source-LPA:We\] presents the electronic Wigner distribution defined by Eq. for $D=0.9$, $D\simeq D_{\mathrm{opt}}$ and $D=0.1$. These plots clearly show that when the dot is fully open, energy resolution is lost, whereas at $D\simeq D_\text{opt}$, we clearly see the single-electron and single-hole excitations emitted during each half period. Decreasing $D$ leads to two phenomena: first of all, the length of the electronic wavepacket increases and we see horizontal interference fringes between two-electronic emissions which suggests delocalization of the electronic excitation beyond the time interval $[0,T]$ (same for the hole). We also see interference fringes between the electronic and hole contributions which correspond to an increase of the weight of $\Delta\mathcal{G}^{(e)}_S$ in the electron/hole coherence quadrant in the frequency domain [@Degio:2010-1]. It was suggested in [@Degio:2010-4] that, in this regime, during each period, the source emits a state of the form $|\Psi_\text{e/h}(u,v)\rangle=(u+v\,\psi[\varphi_h]\psi^\dagger[\varphi_e])|F\rangle$ where $\varphi_e$ and $\varphi_h$ respectively denote the electronic and hole excitations and $(u,v)$ complex amplitudes such that $|u|^2+|v|^2=1$. At the optimal point $(u,v)\simeq (0,1)$ whereas $(u,v)\simeq (1,0)$ when $D\to 0^+$ since the source is shut down in this limit. Consequently, we have $|u|^2\simeq |v|^2\simeq 1/2$ for some intermediate value. In the latter case, this state contains a quantum superposition between the presence and the absence of an elementary electron/hole pair excitation in the edge channel. Elaborating on section \[sec:QI:density-matrices\], we will now test this idea by quantifying the amount of entanglement for an effective 2-qubit built from an electronic and a hole excitation. For this purpose, we define a $4\times 4$ matrix $\rho_{\text{e/h}}$ built from states $|x_h\,x_e\rangle$ associated with the occupation number $x_h\in \{0,1\}$ for a hole excitation based on the single-particle wavefunction $\varphi_h$ and the electronic occupation number $x_e\in \{0,1\}$ associated with the single-particle state $\varphi_e$. All matrix elements that couple different charge sectors vanish. The remaining matrix elements are the diagonal ones: \[eq:ehqb:diagonal\] $$\begin{aligned} \langle 00|\rho_{\text{e/h}}&|00\rangle= \langle \psi^\dagger[\varphi_h]\psi[\varphi_h]\, \psi[\varphi_e]\psi^\dagger[\varphi_e] \rangle_\rho\\ \langle 01|\rho_{\text{e/h}}&|01\rangle= \langle \psi^\dagger[\varphi_h]\psi[\varphi_h]\, \psi^\dagger[\varphi_e]\psi[\varphi_e] \rangle_\rho \\ \langle 10|\rho_{\text{e/h}}&|10\rangle= \langle \psi[\varphi_h]\psi^\dagger[\varphi_h]\, \psi[\varphi_e]\psi^\dagger[\varphi_e] \rangle_\rho \\ \langle 11|\rho_{\text{e/h}}&|11\rangle= \langle \psi[\varphi_h]\psi^\dagger[\varphi_h]\, \psi^\dagger[\varphi_e]\psi[\varphi_e] \rangle_\rho\end{aligned}$$ and the off-diagonal elements coupling the state $|00\rangle$ to the state $|11\rangle$: $$\label{ehqb:non-diagonal:1} \langle 11|\rho_{\text{e/h}}|00\rangle= \langle \psi^\dagger[\varphi_h]\psi[\varphi_e]\rangle_S\,.$$ The diagonal matrix elements are all related to the two-electron coherence $\bar{n}[\varphi_e,\varphi_h]$ whereas the off-diagonal ones are directly the single-electron coherence in the electron/hole coherence quadrant $\mathcal{G}_S^{(e)}[\varphi_e|\varphi_h]= \mathrm{Tr}(\psi[\varphi_e]\,\rho\,\psi^\dagger[\varphi_h])$.  When Wick’s theorem is valid, which is the case in our Floquet modelization of the mesoscopic capacitor, we have: \[eq:ehqb:diagonal:2\] $$\begin{aligned} \langle 00|\rho_{\text{e/h}}|00\rangle &= (1-\bar{n}_e)\bar{n}_h + |\xi|^2 \\ \langle 01|\rho_{\text{e/h}}|01\rangle &= \bar{n}_e\bar{n}_h - |\xi|^2 \\ \langle 10|\rho_{\text{e/h}}|10\rangle &= (1-\bar{n}_e)(1-\bar{n}_h) - |\xi|^2 \\ \langle 11|\rho_{\text{e/h}}|11\rangle &= \bar{n}_e(1-\bar{n}_h) + |\xi|^2\end{aligned}$$ where $\xi=\left\langle 11 \middle| \rho_{\text{e/h}} \middle| 00\right\rangle=\mathcal{G}_S^{(e)}[\varphi_e|\varphi_h]$ is the electron/hole coherence between the electronic and hole wavefunctions and $\bar{n}_{e/h}=\bar{n}[\varphi_{e/h}]$ are their occupation numbers. The Cauchy-Schwarz inequality tells us that $$\label{eq:G1:CS} \left| \mathcal{G}_S^{(e)}[\varphi_e|\varphi_h]\right|^2\leq \bar{n}_e\,\bar{n}_h\,.$$ When the bound is saturated, the matrix $\rho_\text{e/h}$ gets a zero on the diagonal and, in this case, the concurrence can be evaluated analytically: $$\label{eq:QI:C-analytical} \mathcal{C}[\rho_\text{e/h}]=2\sqrt{\bar{n}_e\,\bar{n}_h}\,.$$ It is non zero when we are away from the single-electron source regime ($\bar{n}_e=1$ and $\bar{n}_h=0$). In particular for the state $|\Psi_\text{e/h}(u,v)\rangle$ which can be shown to satisfy Wick’s theorem, considering $|u|^2=|v|^2=1/2$ gives $\bar{n}_e=\bar{n}_h=1/2$ and thus $\mathcal{C}=1$. Then, $\mathcal{E}[\rho_\text{e/h}]=1$, as expected since, in this case, we are producing a pure state with electron/hole entanglement. However, for arbitratry electronic and hole wavepackets $\varphi_{e/h}$, the reduced density operator $\rho_\text{e/h}$ may not be so ideal. We shall now discuss which wavepackets are the best candidates for obtaining non-zero concurrence. Numerical results {#sec:QI:numerics} ----------------- As we shall see now, the choice of $\varphi_{e/h}$ has a strong influence on the result. At fixed $\bar{n}_e$ and $\bar{n}_h$, the concurrence is maximal for $\xi$ saturating the Cauchy-Schwarz bound but its value can be very small when $\bar{n}_e\simeq 0$ and $\bar{n}_h\simeq 1$. We thus have to find wavepackets which (i) bring $\bar{n}_e$ and $\bar{n}_h$ as close as possible to $1/2$ and (ii) maximize $\vert\mathcal{G}_S^{(e)}[\varphi_e|\varphi_h]\vert$. A first guess for $\varphi_e$ is a truncated Lorentzian in energy, centered at $\hbar\omega_e=\Delta/2$ and whose width fits the exponential decay of the average current. Note that its natural width $\gamma_e$ depends on $D$. This wavepacket is expected to lead to $\bar{n}_e\simeq 1$ when $D=D_\text{opt}$. In this case, the natural width of the wavepacket should be less than a half period. However, when $D\rightarrow 0$, it may be relevant to consider electronic wavepackets delocalized over more than one period. Our ansatz is to consider the product of the truncated Lorentzian in energy with a characteristic function in time that selects $n$ half-periods during which the electron is emitted ($t\in [lT,(l+1/2)T]$ for $l=0,\ldots,n-1$). Truncation in the time domain implies that our ansatz are not anymore a purely electronic state. In order to ensure this, we set to zero all values at negative frequencies. This implies that the wavepackets we consider are not strictly zero in time outside of $[lT, (l+1/2)T]$. However, when $D \le D_{\text{opt}}$ the excitations are emitted at well separated energies $\hbar\omega_{e/h}=\pm \Delta/2$ with respect to their natural width $\hbar\gamma_e$, so contributions outside these half periods are very small (and indeed smaller as $D$ decreases). When $D \ge D_{\text{opt}}$, the natural width of the excitation decreases, ensuring it is less than a half period. After normalization, this defines wavefunctions $\varphi_{e_n}$ and in the same way wavefunctions $\varphi_{h_n}$. We present on Fig. \[fig:source-LPA:ne-etc\] the quantities $\bar{n}_{e/h}$, the modulus of the electron/hole coherence $|\xi|$, the Cauchy-Schwarz bound and the von Neumann entropy of the 2-qubit matrix $\rho_{\text{e/h}}$ using specific couples of electronic and hole wavefunctions. We consider $\varphi_{e_n}$ and $\varphi_{h_n}$ for $n=1$, [*id est*]{} truncated to one half period and thus denoted by “Truncated” on Figs. \[fig:source-LPA:ne-etc\] and \[fig:source-LPA:concurrence\] and also $n=\infty$ which corresponds to the “Delocalized” wavefunctions over many half periods.  The von Neumann entropy of the 2-qubit density matrices obtained from these two wavepacket families shows roughly the same behavior for $D\gtrsim 0.1$. First of all, it never vanishes for $D\neq 0$ thus showing that $\rho_\text{e/h}$ does not represent a rank one projector. The von Neumann entropy goes through a minimum close to $D_\text{opt}$ and then re-increases. This comes from the fact that these wavepackets are indeed well suited to describe the excitations emitted by the mesoscopic capacitor around the optimal regime as attested by the fact that $\bar{n}_e\simeq 1$ for this regime. In the vanishing $D$ limit, $\bar{n}_e \approx 0.50$ for the delocalized wavepacket whereas $\bar{n}_e\to 0$ for the truncated ones. This reflects the delocalization over more than one half period of the electronic excitations when $D\rightarrow 0$. For the delocalized wavepackets, the Cauchy-Schwarz bound is not saturated when $D\rightarrow 0$. The effective 2-qubit density matrix $\rho_\text{e/h}$ thus becomes diagonal. Due to the non-vanishing limit of $\bar{n}_e$ when $D\rightarrow 0$, the von Neumann entropy remains close to $2$ (which would be obtained for $\xi=0$ and $\bar{n}_e=\bar{n}_h=1/2$). By contrast, for the truncated wavepackets, $\bar{n}_e$ goes to $0$ and $\bar{n}_h\to 1$ when $D\rightarrow 0$. The Cauchy-Schwarz bound is also not saturated. In this case, $\rho_\text{e/h}$ collapses onto a matrix with only one non-zero element on its diagonal: the von Neumann entropy decreases when $D\rightarrow 0$ for truncated wavepackets. The concurrence as a function of $D$ is depicted on Fig. \[fig:source-LPA:concurrence\]. We see that for delocalized wavepackets, $\rho_\text{e/h}$ exhibits a non-zero concurrence on a finite interval of $D$ that stops before $D$ gets close to zero and starts a little before $D_\text{opt}$. The lower limit of the interval where electron/hole entanglement can be observed with the delocalized wavepackets comes from decay of electron/hole coherences as $D\rightarrow 0$. Although these wavepackets tend to be quite good in terms of their overlaps with $\mathcal{G}_S^{(e)}$, the coherence is too small to get a non-zero entanglement. For the truncated wavepacket, the concurrence is higher than for delocalized wavepackets as $D\rightarrow 0$ as expected since it is only for small $D$ that these two wavepackets become significantly different. ![\[fig:source-LPA:concurrence\] The concurrence $\mathcal{C}[\rho_{\text{e/h}}]$ as function of $D$ for the three families of wavepackets “Truncated”, “Delocalized” and “Martin Landauer”. The mesoscopic capacitor is operated with the parameters used for Fig. \[fig:source-LPA:We\]. ](concurrence.pdf) Although this numerical investigation shows that the mesoscopic capacitor generates some electron/hole pair entanglement in the low $D$ limit, it also shows the importance of using the proper single-particle states to study quantum information theoretical quantities and also to probe the many-body nature of the electronic fluid. We will now sketch a general strategy inspired by signal processing for analyzing electronic coherences generated by a time-periodic source. Electronic atoms of signal {#sec:QI:atoms-of-signal} -------------------------- The above discussion was based on the use of empirical electron and hole wavefunctions without thinking of any detection scheme. However, for a $T$-periodic source, one could envision time-dependent detectors ideally performing repeated detections of a given single-particle template across various periods. Although this is still not demonstrated experimentally, it is important to develop a signal processing framework for the corresponding signals. The idea, borrowed from M. Devoret’s lectures at Collège de France [@Devoret:CDF:2008] consists in using a family of single-particle wavefunctions well suited to $T$-periodicity. We thus consider single-particle wavepackets $\varphi_{k,l}$, which we call electronic atoms of signal or, adapting M. Devoret’s terminology, electronic wavelets such that \[eq:wavelets\] $$\begin{aligned} \label{eq:wavelets:time-shift} \varphi_{k,l}(t)&=\varphi_k(t-lT)\\ \label{eq:wavelets:orthogonality} \langle \varphi_{k',l'}|\varphi_{k,l}\rangle &= \delta_{k,k'}\delta_{l,l'}\,.\end{aligned}$$ Families $\varphi_{k,l}$ obtained by translations in both the frequency and time domains arising from a single wavepacket $\varphi_{0,0}$ are called Gabor bases. An important result for Gabor bases is the Balian-Low theorem which states that there is no orthogonal basis of this type that is well localized both in the time and frequency domain. In the signal processing community, discrete wavelets are families satisfying in which the parameter $k$ corresponds to a scaling: $\varphi_{k,l}(t)=(v_F\tau_0)^{-1/2}s^{-k/2}\varphi((t/s^n-lT)/\tau_0)$. A famous example of electronic atoms of signal has been introduced by Th. Martin and R. Landauer [@Martin:1992-1]. These wavepackets with energy bandwidth $hf$ ($f=1/T$) are defined by: $$\varphi_{n,l=0}(t)=\frac{1}{\sqrt{v_FT}}\,\frac{\sin{(\pi ft)}}{\pi ft}\,{\mathrm{e}}^{-{\mathrm{i}}\omega_nt}$$ where $\omega_n=\pi f(2n+1/2)$. They are called Shannon wavelets in the signal processing community. Remarquably, when a voltage drive of period $T$ is applied, the single-particle state $\varphi_{n,l}$ is scattered among the $\varphi_{n',l}$ with the same time slot index $l$ [@Dubois:2013-1]. More recently, M. Moskalets has found a family of mutually orthogonal electronic wavepackets $(\varphi_l)_{l\in\mathbb{Z}}$ respectively time-shifted by $lT$ and such that a train of single-electron Leviton excitations is represented as the infinite Slater determinant of these $\varphi_l$ on top of the Fermi sea [@Moskalets:2015-1]. It is then tempting to conjecture that there exists a family of electronic atoms of signal $\varphi_{n,l}$ satisfying the orthogonality condition such that a train of charge $n$ Levitons is obtained as the infinite Slater determinant formed by the $\varphi_{k,l}$ for $l\in \mathbb{Z}$ and $k=1,\ldots,n$ on top of the Fermi sea. Electronic atoms of signal can be used to model repeated detection of $\varphi_k$ by considering the overlap of the single-electron coherence of a $T$-periodic source $S$ with a train of $N$ wavepackets $\varphi_{k,l}$ where $k$ is fixed and $l=1,\ldots ,N$. This overlap represents the average cumulated signal after $N$ successive detections of the wavepacket $\varphi_k$ shifted by multiples of $T$. Due to the orthogonality of the $\varphi_{k,l}$ for different values of $l$, these time-shifted wavepackets are perfectly distinguishable: the corresponding number operators $N[\varphi_{k,l}]$ and $N[\varphi_{k,l'}]$ commute. Using the $T$ periodicity of $\mathcal{G}^{(e)}_S$, this overlap scales as $N$ with a prefactor $\bar{n}[\varphi_k]$ defined by Eq. . The same reasoning can be extended to two-particle detection. We then consider $\varphi_a$ and $\varphi_b$ for $a\neq b$ and we perform a repeated detection of the electron pair. The associated wavefunctions are the normalized Slater determinants $\varphi_{(a,b;l)}(t_1,t_2)$ built from $\varphi_{a,l}$ and $\varphi_{b,l}$ for $l=1,\ldots ,N$. Exactly as in the single-particle detection, due to orthogonality of the single-particle wavefunctions for $l\neq l'$, the two-particle states $\varphi_{(a,b;l)}$ are orthogonal for different values of $l$. The cumulated average signal for $N$ periods is then the overlap of $\mathcal{G}^{(2e)}$ with the sum over $l$ of $\varphi_{(a,b;l)}(t_1^+,t_2^+)\,\varphi_{(a,b;l)}(t_1^-,t_2^-)^*$. Once again this overlap scales as $N$ with a prefactor $\bar{n}[\varphi_a,\varphi_b]$ equal to $$v_F^4\int_{\mathbb{R}^4} \varphi_{(a,b)}(\mathbf{t_+})^*\varphi_{(a,b)}(\mathbf{t}_-) \mathcal{G}^{(2e)}_S(\mathbf{t}_+|\mathbf{t}_-)\,{\mathrm{d}}^2\mathbf{t}_+\,{\mathrm{d}}^2\mathbf{t}_-\,.$$ Throught repeated detections of the same wavefunction in different time slots, electronic atoms of signal thus enable us to define effective density matrices in the basis of occupation number for orthogonal single-particle levels. To illustrate this point, Figs. \[fig:source-LPA:ne-etc\] and \[fig:source-LPA:concurrence\] also present the same results as before but for Martin Landauer electronic atoms of signals associated with the period $T/2$ so that $\varphi_e$ is centered in energy at $\omega_e$ and in time on the first half period and $\varphi_h$ is centered in energy at $\omega_h$ and in time on the second half period. As can be seen from the behavior of $\bar{n}_e$, these atoms of signals are not well suited close to $D_\text{opt}$. This was expected since their energy width $2hf$ is less than the natural width of the emitted single-electron excitation. However, the electron/hole coherence $\xi$ is closer to the Cauchy-Schwarz bound than the truncated and delocalized wavepackets. As a result, the non-vanishing interval for the concurrence is indeed concentrated at lower values than for the truncated and delocalized wavepackets. What we are doing here is to analyze the fermionic analogue of inter-mode entanglement in quantum optics. Considering various families of single-particle state amounts to considering various pairs of modes. Naturally, quantum information quantities measuring this entanglement depend on the modes considered. This raises the question of the best description of a quantum signal such as single-electron coherence in terms of single-particle wavefunctions. What are the guidelines for such a choice? Although we do not have a definitive answer to this very general question, we think there are two ways of addressing it. First of all, the experimental setup may impose us a choice. For example, Moskalets wavefunctions and their conjectural generalization are the natural choice when discussing an HOM experiment with a source emitting a periodic train of Leviton excitations in one of the incomming channels. Next, when starting from computational or experimental data on single-electron coherence, the problem is to determine atoms of signals or other wavepackets giving the “simplest" description of single-electron coherence. Although answering this question goes beyond the scope of the present paper, methods inspired from signal processing and exploiting the $T$-periodicity of single-electron coherence are certainly worth investigating. Conclusion and perspectives {#sec:conclusion} =========================== In this paper, we have discussed the recent developments in electron quantum optics from a signal processing perspective. Although part of the material presented here forms a review of our recent works in electron quantum optics [@Bocquillon:2014-1; @Thibierge:2016-1; @Marguerite:2016-2; @Marguerite:2016-1; @Ferraro:2013-1], our discussion aims at showing new interesting perspectives for future work as well as connections to other topics such as quantum optics of current noise or the study of quantum information quantities in electronic systems. From that perspective, our first main message is that single- and two-particle interferometry experiments can be interpreted in the signal processing language as analog operations converting “quantum signals” such as single- and two-electron coherences into experimentally observable quantities which are zero- or finite-frequency average current and current correlations. In particular, we have reviewed how the HOM experiment encodes the overlap of single-electron coherences within low-frequency current noise, thus giving more substance to Landauer’s aphorism “The noise is the signal” [@Landauer:1998-1]. We have also shown that the signal processing point of view is relevant beyond the analysis of HOM experiments. For example, it also unifies the direct probing of intrinsic two-electron coherence within a given edge channel by means of generalized Franson interferometry experiments. Our approach thus suggests that, although the demonstration of a Franson interferometer in electron quantum optics represents a strong challenge due to interaction effects within Mach-Zehnder interferometers, it might be easier to probe two-electron coherences with a simple Hanbury Brown and Twiss interferometer by measuring correlations between finite-frequency currents. Although this would require sophisticated homodyning to bring the finite-frequency components of both outgoing currents in the same low-frequency band, the measurement stage would be rather immune to interaction effects. Interestingly, this would also establish a bridge between electron quantum optics and the recently developped study of quantum properties of the radiation emitted by a quantum conductor [@Grimsmo:2016-1]. Our second message is that signal processing concepts and techniques are useful not only for interpreting electron quantum optics experimental results but also to gain a deeper understanding of the electronic many-body state in these experiments. For this purpose, we have transposed the concept of atoms of signal to electron quantum optics and shown that it can be used to discuss quantum information theory quantities in electron quantum optics experiments. We have illustrated this idea by discussing the electron/hole entanglement generated by the mesoscopic capacitor. However, our discussion is relevant to the study of entanglement in more general setups (see for example [@Hofer:2016-2] in this volume and [@Thomas:2016-1; @Dasenbrook:2016-1] as well as [@Samuelsson:2004-1; @Chtchelkatchev:2002-1; @Samuelsson:2003-1; @Samuelsson:2005-1; @Fazio:2006-1; @Giovannetti:2007-1; @Sherkunov:2012-1] for previous works). By relying only on the use of electron quantum optics coherences, our framework enables discussing Coulomb interaction effects in a very natural way. Second, we think that the pioneering work by M. Moskalets on Leviton trains [@Moskalets:2015-1] raises the question of the “simplest representation” of quantum signals such as single-electron coherences emitted by various electronic sources. We think that combining the arsenal of theoretical techniques (analytical as well as numerical) with signal processing concepts may lead to progresses on this basic question. Last but not least, the rapid development of experiments in electron quantum optics and microwave quantum optics suggests that the perspectives for investigating all these questions on the experimental side are very promising. We warmly thank J. Splettstoesser and R. Haug for setting up this special volume. We acknowledge useful discussions with P. Borgnat and R. Menu. 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[^1]: Here the channel index breaks the indiscernability between electrons within the two different channels. [^2]: As often in quantum information, we are looking for the number of maximally entangled pairs needed to generate $n$ copies of the states contained in $\rho$. For large $n$, this scales as $n$ and we divide by $n$ to obtain the entanglement of formation.

--- abstract: 'Resistivity anomaly, a sharp peak of resistivity at finite temperatures, in the transition-metal pentatellurides $\mathrm{ZrTe_{5}}$ and $\mathrm{HfTe_{5}}$ was observed four decades ago, and more exotic and anomalous behaviors of electric and thermoelectric transport were revealed recent years. Here we present a theory of Dirac polarons, composed by massive Dirac electrons and holes in an encircling cloud of lattice displacements or phonons at finite temperatures. The chemical potential of Dirac polarons sweeps the band gap of the topological band structure by increasing the temperature, leading to the resistivity anomaly. Formation of a nearly neutral state of Dirac polarons accounts for the anomalous behaviors of the electric and thermoelectric resistivity.' author: - Bo Fu - 'Huan-Wen Wang' - 'Shun-Qing Shen' title: 'Dirac Polarons and Resistivity Anomaly in $\mathrm{ZrTe}_{5}$ and $\mathrm{HfTe}_{5}$' --- #### Introduction Resistivity in the transition-metal pentatellurides $\mathrm{ZrTe}_{5}$ and $\mathrm{HfTe}_{5}$ exhibits a sharp peak at a finite temperature $T_{p}$. The peak occurs approximately at a large range of temperatures from 50 to 200K, but the exact value varies from sample to sample. The effect was observed forty years ago [@okada1980giant; @izumi1981anomalous], but has yet to be understood very well. At the beginning, it was thought as a structural phase transition, or occurrence of charge density wave. The idea was soon negated as no substantial evidence is found to support the picture [@disalvo1981possible; @okada1982negative; @bullett1982absence; @fjellvag1986structural]. The measurements of the Hall and Seebeck coefficients showed that the type of charge carriers dominating the electrical transport changes its sign around the peak, which indicates the chemical potential of the charge carriers sweeps band gap around the transition temperature $T_{p}$ [@izumi1982hall; @jones1982thermoelectric; @littleton1999transition; @tritt1999enhancement]. Thus the anomaly is believed to originate in the strong temperature dependence of the chemical potential and carrier mobility. Recent years the advent of topological insulators revives extensive interests to explore the physical properties of $\mathrm{ZrTe}_{5}$ and $\mathrm{HfTe}_{5}$. The first principles calculation suggested that the band structures of $\mathrm{ZrTe}_{5}$ and $\mathrm{HfTe}_{5}$ are topologically nontrivial or very close to the topological transition points [@weng2014transition]. An infrared transmission study provides spectroscopic evidence of the band inversion of the conduction bands and valance bands [@chen2017spectroscopic], and the angle-resolved photoemission spectroscopy (ARPES) measurement also suggested it is a strong topological insulator [@manzoni2016evidence]. Further studies uncover more exotic physics in these compounds [@chen2015optical; @li2016chiral; @Zhou2016Pressure; @Li2018Giant; @Wang2018Discovery; @liang2018anomalous; @wang2019log; @Zhang2019anomalous; @tang2019three; @Hu2019Large; @wang2020quantum], such as the chiral magnetic effect and three-dimensional quantum Hall effect. Other possible causes have been advanced much recently [@Zhao2017anomalous; @PhysRevX.8.021055; @xu2018temperature], but the physical origin of the resistivity anomaly is still unclear. For example, it was suggested that a topological quantum phase transition might occur, and the gap closing and reopening give rise to the resistivity anomaly [@xu2018temperature]. However it contradicts with the observation of the ARPES measurement [@zhang2017electronic; @zhang2017temperature]. Strong temperature dependence of the band structure [@zhang2017electronic] implies that the interaction between the Bloch electrons and the lattice vibrations, *i.e.*, electron-phonon interaction (EPI), is an indispensable ingredient to understand the anomaly [@rubinstein1999hfte]. In this Letter, we consider the topologically nontrivial band structure and EPI in $\mathrm{ZrTe_{5}}$ and $\mathrm{HfTe_{5}}$, and propose a theory of Dirac polarons for the resistivity anomaly at finite temperatures. The Dirac polarons are mixtures of massive Dirac electrons and holes encircling a cloud of phonons, and are the effective charge carriers in the compounds. Increasing temperature will change the overlapping of the Dirac polarons drastically. The chemical potential of Dirac polarons sweeping the band gap from conduction bands to valence bands with increasing the temperature. Consequently, when the chemical potential of Dirac polarons locates around the middle of the band gap, the resistivity is enhanced drastically to form a pronounced peak at a finite temperature. The carriers dominated the charge transport change the sign around the transition. The formation of a nearly neutral state of Dirac polarons accounts for anomalous electric and magneto transport properties in the compounds. ![(a) A comparison of the renormalized energy spectrums according to the theory (the solid lines) and the temperature-dependent band structures from ARPES measurement adopted from Fig. 2(b) in Ref. [@zhang2017electronic]. (b) The renormalized Dirac mass $m$ due to the EPI and (c) the temperature dependent chemical potential $\mu$, the experiment data are extracted from Fig.2(d) and (f) in Ref. [@zhang2017electronic]. The model parameters are set to be $v=4\times10^{5}\mathrm{m}/s$, $b=350\mathrm{meV}\mathrm{nm}^{2}$, $d=-310\mathrm{meV}\mathrm{nm}^{2}$, and $m=15.0\,\mathrm{meV}$ for all the figures if there is no further claiming. The carrier density used here is $n=1.3\times10^{17}\mathrm{cm^{-3}}$.](Fig1){width="8.5cm"} #### Finite temperature spectral function and quasiparticle properties The charge carriers in the conduction and valence bands of the bulk $\mathrm{ZrTe_{5}}$ and $\mathrm{HfTe_{5}}$ are strongly coupled together due to spin-orbit interaction and behave like massive Dirac fermions with nontrivial band topology instead of conventional electrons in semiconductors and metals [@Wu2016evidence; @Li2016Experimental; @manzoni2016evidence]. In the following, we only focus on $\mathrm{ZrTe_{5}}$ for comparison with experimental measurement and theoretical calculation without loss of generality. When the electrons (or holes) are moving through the ionic lattices, the surrounding lattice will be displaced from the original equilibrium positions; consequently, the electrons (or holes) will be encircled by the lattice distortions, or phonons. At finite temperatures the quasiparticles for the coupled electron-lattice systems are composed of both massive Dirac electrons and holes in a cloud of phonons due to the thermal activation when the chemical potential is located around the band edges as illustrated in Fig.1(A). The Hamiltonian describing the EPI in Dirac materials has the form [@Mahan-book], $\mathcal{H}_{tot}=\mathcal{H}_{Dirac}+\mathcal{H}_{ph}+\mathcal{H}_{ep}$. Here the electron system is described by the modified massive Dirac Hamiltonian $\mathcal{H}_{Dirac}$, the phonon part $\mathcal{H}_{ph}$ is in the harmonic approximation, and the EPI part $\mathcal{H}_{ep}$ is dominantly contributed by longitudinal acoustic phonons. The low-energy physics of the electronic states of $\mathrm{ZrTe_{5}}$ and $\mathrm{HfTe_{5}}$ near the $\Gamma$ point can be well described by the Hamiltonian, $$\mathcal{H}_{Dirac}=\sum_{\mathbf{p}}\psi_{\mathbf{p}}^{\dagger}\left[d\mathbf{p}^{2}-\mu+\hbar v\mathbf{p}\cdot\boldsymbol{\alpha}+(m-b\mathbf{p}^{2})\beta\right]\psi_{\mathbf{p}},\label{eq:DiracHamiltonian-1}$$ where $\mathbf{p}=(p_{x},p_{y},p_{z})$ is the momentum in the vicinity of $\Gamma$ point, $\psi_{\mathbf{p}}$ and $\psi_{\mathbf{p}}^{\dagger}$ are the four-component fermionic field operators, $v$ is the effective velocity. $\mu$ is the chemical potential. $d\mathbf{p}^{2}$ breaks the particle-hole symmetry, and plays essential role in the Dirac polaron physics. $m-b\mathbf{p}^{2}$ is the momentum dependent Dirac mass, where the sign of $mb$ determines the topological properties of the system [@SQS]. The Dirac matrices are chosen to be $\boldsymbol{\alpha}=\tau_{x}\otimes(\sigma_{x},\sigma_{y},\sigma_{z})$ and $\beta=\tau_{z}\otimes\sigma_{0}$, where $\sigma$ and $\tau$ are the Pauli matrices acting on spin and orbital space, respectively. The quantitative information about these physical properties, such as the Dirac velocity, the Dirac mass or the energy gap can be extracted from the ARPES data [@Manzoni2015ultrafast; @Moreschini2016nature; @zhang2017electronic]. To explore the EPI effect, we treat $\mathcal{H}_{ep}$ as a perturbation to either electrons or phonons in the Migdal approximation [@Migdal-58jetp] that the self-energy arises from the virtual exchange of a phonon at temperature $T$. Due to the spinor nature of Dirac electrons, the retarded self-energy can be recast in a matrix form as [@Note-on-SM] $$\Sigma_{ep}^{R}(\mathbf{p},\epsilon)=\Sigma_{I}(\mathbf{p},\epsilon)+\lambda_{\alpha}(\mathbf{p},\epsilon)\hbar v\mathbf{p}\cdot\boldsymbol{\alpha}+\Sigma_{\beta}(\mathbf{p},\epsilon)\beta,$$ where $\Sigma_{I}(\mathbf{p},\epsilon)$ is the renormalization to the chemical potential $\mu$,$\lambda_{\alpha}(\mathbf{p},\epsilon)$ is the velocity dressing function and $\Sigma_{\beta}(\mathbf{p},\epsilon)$ is the renormalization to the Dirac mass $m$. The quasiparticle properties can be obtained by the poles of the retarded Green’s function $G^{R}(\mathbf{p},\epsilon)=[\epsilon-\mathcal{H}_{Dirac}(\mathbf{p})-\Sigma^{R}(\mathbf{p},\epsilon)]^{-1}$, which is in the complex plane with its real part gives the spectrum of the quasiparticle and the imaginary part gives its lifetime. The self-energy $\Sigma^{R}(\mathbf{p},\epsilon)=\Sigma_{ep}^{R}(\mathbf{p},\epsilon)+\Sigma_{imp}^{R}(\mathbf{p},\epsilon)$ includes the contribution from the impurities scattering. The spectral function of the quasiparticle properties, i.e., Dirac polarons, is given by the imaginary part of $G^{R}(\mathbf{p},\epsilon)$, $$A_{\zeta}(\mathbf{p},\epsilon)=-\frac{1}{\pi}\langle\zeta s\mathbf{p}|\mathrm{Im}G^{R}(\mathbf{p},\epsilon)|\zeta s\mathbf{p}\rangle\text{,}$$ where $|\zeta s\mathbf{p}\rangle$ are the band states with the band indices $\zeta=\pm$ for the conduction band and valence band and spin indices $s=\pm$. In the absence of disorder and EPI, the spectral function $A_{\zeta}(\mathbf{p},\epsilon)$ is a $\delta$ function reflecting that the wave vector $\mathbf{p}$ is a good quantum number and all its weight ratio is precisely at $\epsilon=\xi_{\mathbf{p}}^{\zeta}$. In the presence of disorder and EPI, at low temperatures, $A_{\zeta}(\mathbf{p},\epsilon)$ exhibits a sharp peak of the Lorentzian type due to a long lifetime. As temperature increases, $A_{\zeta}(\mathbf{p},\epsilon)$ maintains the Lorentzian line shape but becomes broader due to the increasing of the scattering rate, and the peak position moves to the positive energy due to the renormalization of the energy level. The trajectories of the peaks of the spectral function give us the renormalized dispersion $\widetilde{\xi}_{\zeta}(\mathbf{p})$. As shown in Fig. 1(a), we plot the derived energy dispersions $\widetilde{\xi}_{\zeta}(\mathbf{p})$ for different temperatures with the black and red lines corresponding the conduction and valence band respectively. The ARPES data extracted from Ref. [@zhang2017electronic] are also presented as the background for a comparison. The excellent agreement can be found between our theoretical calculations and the experiment data. The overall band structure shifts up to higher energy with increasing temperature. The peak structure of the spectral function can be clearly observed in the temperature range considered, which suggests that a quasiparticle picture is still appropriate at low energy and the EPI largely preserving the weakly perturbed Fermi-liquid behavior. The renormalized Dirac mass $m$ is given by the difference between two energy levels $\widetilde{\xi}_{+}(\mathbf{0})$ and $\widetilde{\xi}_{-}(\mathbf{0})$ for the states at the band edge ($\mathbf{p}=0$):$\bar{m}=\frac{1}{2}[\widetilde{\xi}_{+}(\mathbf{0})-\widetilde{\xi}_{-}(\mathbf{0})]$. At higher temperature, the effective mass varies with the temperature as $\bar{m}\simeq m+g_{m}T$ shown in Fig. 1(b). The coefficient $g_{m}$ are determined by the band structure of massive Dirac fermions and the EPI strength (see the details in Ref.[@Note-on-SM]). For Dirac materials, the renormalization of the energy levels is attributed to the contributions from both intra-band and inter-band scatterings. With increasing the temperature, the more phonon modes with high momenta are active, the larger the renormalization is. The chemical potential is determined by the total number of charge carriers $n=\int_{|\bar{m}|}^{\infty}d\omega\left[\bar{\nu}_{+}(\omega)n_{F}(\omega-\mu)-\bar{\nu}_{-}(-\omega)n_{F}(\omega+\mu)\right]$ where $n_{F}(x)=[\exp(x/k_{B}T)+1]^{-1}$ is the Fermi distribution function and $\bar{\nu}_{\pm}(\omega)$ are the renormalized density of states for the conduction and valence band respectively. In the band structure of $\mathrm{ZrTe_{5}}$, the particle-hole symmetry is broken and the valence band is narrower than the conduction band. At the fixed carrier density $n$, the temperature dependence $\mu(T)$ are plotted in Fig. 1(c). The calculated results demonstrate that the chemical potential sweeps over the energy band gap of the massive Dirac particles with increasing the temperature. At low temperature, the chemical potential $\mu(T)\approx\mu(0)-\frac{\pi^{2}}{6}(k_{B}T)^{2}\frac{d\bar{\nu}_{\pm}(\omega)/d\omega}{\bar{\nu}_{\pm}(\omega)}\Big|_{\omega=\mu(0)}$shows a quadratic temperature dependence by means of the Sommerfeld expansion. $\mu(T)=0$ means the chemical potential is located at the mid-gap, which approximately defines the transition temperature $T_{p}$ around. At high temperatures, due to the strong particle-hole asymmetry and the relatively low electron concentration, the chemical potential shifts into the valence band in a relatively linear fashion with increasing the temperature. ![(a) The zero-field resistivity $\rho$ as a function of temperature for several carrier density $n$. (b) The peak temperature $T_{p}$ as a function of the carrier density. (c) The comparison of the experimental data and theoretical prediction by using the same parameters as Fig. 1. The experimental data are extracted from Fig. 1(d) in Ref. ([@zhang2017electronic]). Both resistivity curves have been normalized to their maximum values $\rho_{\mathrm{peak}}$.](Fig2){width="8.5cm"} #### The resistivity anomaly With the phonon-induced self-energy in hand, we are ready to present the electrical resistivity as a function of temperature by means of the linear response theory [@Mahan-book; @Note-on-SM]. At finite temperatures, the conductivities and the thermoelectric coefficients are contributed from both the electron-like bands and hole-like bands after the phonon-induced renormalization. The two contributions are weighted by the negative energy derivative of the Fermi-Dirac function, whose value is nearly zero except for energies within a narrow window of $k_{B}T$ near the chemical potential $\mu$. Figure 2(a) reproduces the resistivity peak at several initial chemical potentials $\mu(T=0)$, or equivalently carrier densities at $T=0$. For the initial chemical potential locating in the conduction band [\[]{}$\mu(T=0)>0$[\]]{}, as it moves down to the valance band with increasing temperature, it will inevitably sweep over the band gap. When $T=T_{p}$, the effective chemical potential lies around the middle of the effective band gap $\mu(T=T_{p})\simeq0$ and the resistivity reaches the maximum. As the $n$-type carrier concentration is decreased, the resistivity peak will move to the lower temperature with the higher magnitude. The peak temperature as a function of the carrier density is plotted in Fig. 2(b). For a lower carrier concentration, the chemical potential reaches the middle of the band gap with a lower temperature. The height of the resistivity peak is determined by the ratio $\bar{m}(T_{p})/(k_{B}T_{p})$. With increasing the ratio, the peak height increases drastically, and becomes divergent if $\bar{m}(T_{p})\gg k_{B}T_{p}$. It explains why in some experiments with extreme low carrier concentration no resistivity peak is observed [@liang2018anomalous; @mutch2019evidence], which can be regarded as the situation of $T_{p}\sim0$. Thus the sweeping chemical potential over the band gap of Dirac fermions gives rise to the resistivity anomaly at finite temperatures. We use the model parameters in Fig. 1 to calculate the resistivity, which is in a good agreement with the experimental data as shown in Fig. 2(c).The slight deviation at the high temperature might be caused by neglecting the contributions from the optical modes of phonons. {width="8.5cm"} #### Sign change of the Hall and Seebeck coefficients The resistivity anomaly is always accompanied with the sign change of the Hall and Seebeck coefficients around the transition temperature[@jones1982thermoelectric; @chi2017lifshitz; @zhang2020observation; @tang2019three; @Miller2018polycrystalline; @Niemann2019magnetothermoelectric], which can be reproduced in the present theory. As shown in Fig. 3(a), for a positive $\mu(T=0)$ with carrier concentration being $n$-type, with increasing the temperature, the Hall coefficient ($R_{H}=\partial\rho_{xy}/\partial B|_{B=0}$) first maintains its value ($1/en$) at low temperature then decreases down until reaching the minimum. By further increasing the temperature, $R_{H}$ changes from the negative to positive sign at some temperature and then continues to decrease down to nearly zero at high temperature. The sign change of $R_{H}$ indicates the electron-dominated transport is transformed into the hole-dominated as the chemical potential moves from the conduction band to valence band. As the carrier concentration decreases, the Hall coefficient crosses $0$ at a lower temperature with its maximum being larger. As shown in Fig. 3(b), the Seebeck coefficient $S_{xx}$ also reveals a systematic shift in temperature as the carrier density increases. For each curve with fixed carrier density as the temperature increases from zero to room temperature, $S_{xx}$ displays similar nonmonotonic temperature dependence as $R_{H}$, except that $S_{xx}$ starts from absolute zero and exhibits a relative large positive ($p$-type) Seebeck coefficient at high temperature. At low temperature, the chemical potential lies deep in the bulk band, the Mott formula relates the thermoelectric conductivity with the derivative of the electrical conductivity for the thermopower $S_{xx}=\frac{\pi^{2}k_{B}^{2}T}{3e}\frac{d\sigma(\omega)/d\omega}{\sigma(\omega)}|_{\omega=\mu}$ [@Mott1969observation] with $\sigma(\omega)$ is the energy-dependent conductivity. The conductivity $\sigma(\omega)$ is proportional to the square of the group velocity. Hence, as chemical potential locates in conduction band, $S_{xx}$ is negative ($n$-type) and decreases with increasing temperature. $S_{xx}$ attains its largest value when $n$ is tiny but nonvanishing, and varies rapidly with the temperature around $T_{p}$. [@Nolas]. $T_{p}$ decreases with the reduction of the $n$-type carrier concentration at zero temperature qualitatively agrees with previous measurements for single crystals with different carrier concentrations [@chi2017lifshitz]. Near $T=T_{p}$ and if the band gap $\bar{m}(T_{p})$ is comparably smaller than the thermal energy $k_{B}T_{p}$, either $R_{H}$ or $S_{xx}$ is linear in temperature and the system enters a nearly neutral state of Dirac polarons due to the strong thermal activation. {width="8.5cm"} #### Magnetotransport in nearly neutral state of Dirac polarons The presence of an external magnetic field reveals the exotic behaviors of magnetoresistivity near the transition temperature [@tritt1999enhancement; @li2016chiral; @tang2019three; @Zhao2017anomalous; @lv2018tunalbe; @Niemann2019magnetothermoelectric]. Without loss of generality we assume the magnetic field is along the $z$ direction. As shown in Fig. 4(a), the transverse magnetoresistivity $\rho_{xx}(B)$ displays significantly different behaviors for temperature above and below $T_{p}$. Below 120K, a narrow dip is observed around zero magnetic field and above 200K, $\rho_{xx}$ shows a quadratic field dependence. As approaching the peak temperature, $\rho_{xx}$ becomes large and nonsaturating. To see the effect more clear, we plot the resistivity as a function of temperature for different magnetic fields. As shown in Fig. 4(b), $\rho_{xx}(B)$ displays striking resistivity peaks when the temperature crosses the region of the neutral state of Dirac polarons. The peak is strongly enhanced with increasing magnetic field, and even becomes nonsaturated. Its position is observed to shift slightly to a higher temperature with the field increasing, *i.e.* $T_{p}$ is a function of magnetic field. This effect has been reported experimentally in Ref. [@li2016chiral; @tang2019three]. The appearance of giant and nonsaturated transverse magnetoresistivity can be viewed as the electrical signature of the neutral state of Dirac polarons. As shown in Fig. 4(c), the slope of the Hall resistivity $\rho_{xy}$ is negative, indicating a electron-dominated charge transport. As the temperature increases, the nonlinearity of $\rho_{xy}$ becomes more apparent. In the intermediate temperature ($120\sim180$K) around $T_{p}$, due to the formation of the nearly neutral state of Dirac polarons, the slope of the Hall resistivity changes from positive (hole type) at low magnetic field to negative (electron type) at high field, showing a zigzag shaped profile. At high temperature (above 200K), the hole carrier dominates the charge transport thus the slope of $\rho_{xy}$ become positive. The effect of an applied magnetic field on $\rho_{xy}$ as a function of temperature is shown in Fig. 4(d). There is a systematic shift to the higher temperatures with increasing field. The calculated $\rho_{xx}$ and $\rho_{xy}$ as functions of either temperature $T$ or magnetic field $B$ are in an excellent agreement with the experimental measurements in $\text{ZrTe}_{5}$ and $\mathrm{HfTe}_{5}$ [@Zhao2017anomalous; @tang2019three; @Niemann2019magnetothermoelectric; @lv2018tunalbe]. Lastly, we want to point out the differences between the present theory and the two-band model for magnetoresistance [@Pippard1989]. The two-band model commonly requires that the Fermi surface is composed of both electron and hole pockets and predicts a quadratical magnetoresistance, while the present theory only involves a single Dirac band and the multi-carrier transport is activated through thermal excitation and gives a sublinear magnetoresistance over a wide range of temperature. #### Discussion From an experimental standpoint, a temperature-dependent effective carrier density can be deduced from the Hall measurement. The shift of the chemical potential or effective carrier density with the variation of temperature is the key issue to the resistivity anomaly. With no absorption or desorption process through extrinsic doping, the temperature dependent variation of the density of charge carriers or even sign change of charge carriers seems to violate the conservation law of the total charge. However, the relative contribution from each carrier to the total Hall effect also depends on its ability to respond to the applied magnetic field such as velocity and mobility. In Dirac materials with extreme low carrier density and tiny band gap, the strong particle-hole asymmetry will induce a significant temperature variation of the chemical potential, even shifts from conduction band to valence band. The effective carrier density also displays strong temperature dependence. This is a striking feature of the massive Dirac materials with low carrier density at finite temperatures. We thank Li-Yuan Zhang, Nan-Lin Wang and Chen-Jie Wang for helpful discussions. 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--- abstract: 'After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics.' address: 'Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA' author: - Bjorn Poonen date: 'March 18, 2012' title: 'Undecidable problems: a sampler' --- [^1] Two notions of undecidability {#S:undecidability} ============================= There are two common settings in which one speaks of undecidability: 1. Independence from axioms : A single statement is called if neither it nor its negation can be deduced using the rules of logic from the set of axioms being used. (Example: The , that there is no cardinal number strictly between $\aleph_0$ and $2^{\aleph_0}$, is undecidable in the ZFC axiom system, assuming that ZFC itself is consistent [@Godel1940; @Cohen1963; @Cohen1964].) The first examples of statements independent of a “natural” axiom system were constructed by K. Gödel [@Godel1931]. 2. Decision problem : A family of problems with YES/NO answers is called if there is no algorithm that terminates with the correct answer for every problem in the family. (Example: , to decide whether a multivariable polynomial equation with integer coefficients has a solution in integers, is undecidable [@Matiyasevich1970].) In modern literature, the word “undecidability” is used more commonly in sense 2, given that “independence” adequately describes sense 1. To make 2 precise, one needs a formal notion of algorithm. Such notions were introduced by A. Church [@Church1936a] and A. Turing [@Turing1936] independently in the 1930s. From now on, we interpret algorithm to mean , which, loosely speaking, means that it is a computer program that takes as input a finite string of 0s and 1s. The role of the finite string is to specify which problem in the family is to be solved. Often in describing a family of problems, it is more convenient to use higher-level mathematical objects such as polynomials or finite simplicial complexes as input. This is acceptable if these objects can be encoded as finite binary strings. It is not necessary to specify the encoding as long as it is clear that a Turing machine could convert between reasonable encodings imagined by two different readers. One cannot speak of a *single* YES/NO question being undecidable in sense 2, because there exists an algorithm that outputs the correct answer for it, even if one might not know *which* algorithm it is! There is a connection between the two notions of undecidability. Fix a decision problem and an axiom system ${\mathcal{A}}$ such that 1. \[I:generating A\] there is a computer program that generates exactly the axioms of ${\mathcal{A}}$; and 2. \[I:generating Y\_i\] there is a computer program that, when fed an instance $i$ of the decision problem, outputs a statement $Y_i$ in the language of ${\mathcal{A}}$ such that - if $Y_i$ is provable in ${\mathcal{A}}$, then the answer to $i$ is YES, and - if $\lnot Y_i$ is provable in ${\mathcal{A}}$, then the answer to $i$ is NO. Under these assumptions, if the decision problem is undecidable in sense 2, then at least one of its instance statements $Y_i$ is undecidable in sense 1, i.e., independent of ${\mathcal{A}}$. The proof of this is easy: if every $Y_i$ could be proved or disproved in ${\mathcal{A}}$, then the decision problem could be solved by a computer program that generates all theorems deducible from ${\mathcal{A}}$ until it finds either $Y_i$ or $\lnot Y_i$. In fact, under the same assumptions, there must be *infinitely many* $Y_i$ that are independent of ${\mathcal{A}}$, since if there were only finitely many, there would exist a decision algorithm that handled them as special cases. In all the undecidable decision problems we present, the source of the undecidability can be traced back to a single undecidable decision problem, namely the halting problem, or equivalently the membership problem for listable sets (see Sections \[S:halting problem\] and \[S:listable\]). For any of these problems, in principle we can compute a *specific* $i$ for which $Y_i$ is independent of ${\mathcal{A}}$ (cf. the last paragraph of page 294 of [@Post1944]). The value of $i$ depends on ${\mathcal{A}}$; more precisely, $i$ can be computed in terms of the programs in and . Assume that ZFC is consistent, and, moreover, that theorems in ZFC about integers are true. Then, because the undecidability of Hilbert’s tenth problem in sense 2 is proved via the halting problem (see Section \[S:H10\]), there is a specific polynomial $f \in {\mathbb{Z}}[x_1,\ldots,x_n]$ one could write down in principle such that neither $$\label{E:unprovable} (\exists x_1,\ldots,x_n \in {\mathbb{Z}}) \; f(x_1,\ldots,x_n) = 0$$ nor its negation can be proved in ZFC. Moreover, must be false, because if it were true, it could be proved in ZFC by exhibiting the integers satisfying $f(x_1,\ldots,x_n)=0$. (It might seem as if this is a ZFC proof of the negation of , but in fact it is only a ZFC proof of the *implication* > “If ZFC is consistent and proves only true theorems about integers, then the negation of holds.” This observation is related to Gödel’s second incompleteness theorem, which implies that ZFC cannot prove the hypothesis of the implication unless ZFC is inconsistent!) The goal of this survey article is to demonstrate that undecidable decision problems arise naturally in many branches of mathematics. The criterion for selection of a problem in this survey is simply that the author finds it entertaining! We do not pretend that our list of undecidable problems is complete in any sense. And some of the problems we consider turn out to be decidable or to have unknown decidability status. For another survey of undecidable problems, see [@Davis1977]. Logic ===== Gödel’s incompleteness theorems [@Godel1931] provided undecidable statements in sense 1 for a wide variety of axiom systems. Inspired by this, Church and Turing began to prove that certain decision problems were undecidable in sense 2, as soon as they developed their notions of algorithm. The halting problem {#S:halting problem} ------------------- The asks whether it is possible write a debugger that takes as input a computer program and decides whether it eventually halts instead of entering an infinite loop. For convenience, let us assume that each program accepts a natural number as input: input : a program $p$ and a natural number $x$ question : Does $p$ eventually halt when run on input $x$? The halting problem is undecidable. We will use an encoding of programs as natural numbers, and identify programs with numbers. Suppose that there were an algorithm for deciding when program $p$ halts on input $x$. Using this, we could write a new program $H$ such that $$\textup{$H$ halts on input $x$} \quad \iff \quad \textup{program $x$ does not halt on input $x$.}$$ Taking $x=H$, we find a contradiction: $H$ halts on input $H$ if and only if $H$ does not halt on input $H$. To turn the sketch above into a complete proof would require some programming, to show that there is a “universal” computer program that can simulate any other program given its number; this could then be used to construct $H$. Listable sets {#S:listable} ------------- Let ${\mathbb{N}}$ be the set of natural numbers. Let $A$ be a subset of ${\mathbb{N}}$. Call $A$ [^2] if there is an algorithm that takes an input an element $n \in {\mathbb{N}}$ and decides whether or not $n \in A$. On the other hand, call $A$ or   if there is a computer program that when left running forever eventually prints out exactly the elements of $A$. Computable sets are listable. For each listable set $A$, we then have the following decision problem: input : $n \in {\mathbb{N}}$ question : Is $n \in A$? There exists a listable set $A$ for which the membership problem is undecidable. Let $A$ be the set of numbers of programs that halt. Then $A$ is listable (write a program that during iteration $N$ runs each of the first $N$ programs for $N$ steps, and prints the numbers of those that have already halted). But the undecidability of the halting problem implies that $A$ is not computable; in other words, the membership problem for $A$ is undecidable. It would be just as easy to argue in reverse, to use the existence of a non-computable listable set to prove the undecidability of the halting problem. The Entscheidungsproblem ------------------------ Fix a finite set of axioms. Then there are some (first-order) statements that are , meaning that they are true for every mathematical structure satisfying the axioms. By the completeness theorem of first-order logic [@Godel1930], the universally valid statements are exactly the ones that are in the sense that they can be deduced from the axioms using the rules of logic. Can one decide in a finite amount of time whether or not any given statement is universally valid? This is the , proposed by D. Hilbert [@Hilbert-Ackermann1928]\*[Chapter 3, §11]{}. (Entscheidung is the German word for “decision”.) One could try searching for a proof by day and searching for a proof of the negation by night, but such an algorithm might fail to terminate for some input statements since it could be that neither proof exists. More formally, but still without providing full definitions, given a first-order logic ${\mathcal{F}}$, possibly including a finite number of special axioms beyond the basic axioms of first-order logic, one has the following decision problem: input : a first-order sentence $s$ in the language of ${\mathcal{F}}$ question : Is $s$ true in every model of the axioms of ${\mathcal{F}}$? It was known to Hilbert that there is a single first-order logic ${\mathcal{F}}_0$ without special axioms such that if the Entscheidungsproblem for ${\mathcal{F}}_0$ is decidable, so is the Entscheidungsproblem for any other first-order logic. But Church [@Church1936a; @Church1936b] and Turing [@Turing1936]\*[§11]{} independently proved that the Entscheidungsproblem for ${\mathcal{F}}_0$ was undecidable. For more information, see [@Davis1958]\*[Chapter 8, §4]{}. Combinatorics ============= The Post correspondence problem ------------------------------- Imagine a rectangular block with a finite string of $a$’s and $b$’s written along the top and another such string written along the bottom, both upright. When finitely many such blocks are laid side to side, the strings along the top may be concatenated, and the strings along the bottom may be concatenated. E. Post [@Post1946] proved that the following simple-sounding problem is undecidable. input : a finite collection of blocks, labelled as above question : Given an unlimited supply of copies of these particular blocks, can one form a nonempty finite sequence of them for which the concatenation of the top strings equals the concatenation of the bottom strings? The reason that it is undecidable is that one can embed the halting problem in it. Namely, with some work it is possible, given a computer program $p$, to construct an instance of Post correspondence problem that has a positive answer if and only if $p$ halts. Because of its simplicity, the Post correspondence problem is often used to prove the undecidability of other problems, for instance, in the formal theory of languages: see [@Davis1977]. Tiling the plane ---------------- , introduced by H. Wang [@Wang1961]\*[§4.1]{}, are unit squares in the plane, with sides parallel to the axes, such that each side of each square has been assigned a color. Figure \[F:Wang tiles\] shows a collection of $13$ such tiles. They may be translated, but not rotated or reflected. A tiling of the plane into such squares is valid if whenever two squares share an edge, the colors match, as in the game of dominos. Wang proposed the following problem: input : a finite collection of Wang tiles question : Is there a valid tiling of the entire plane using only translated copies of the given tiles? Wang also conjectured [@Wang1961]\*[4.1.2]{} that if a tiling exists for a given finite collection, then there exists a *periodic* tiling, i.e., one that is invariant under translations by the vectors in a finite-index subgroup of ${\mathbb{Z}}^2$, or equivalently by the vectors in $(n{\mathbb{Z}})^2$ for some fixed $n \ge 1$. He observed that this conjecture would imply that the tiling problem was decidable: on the $n^{{{\operatorname{th}}}}$ day one could search for tilings that are invariant under translations in $(n{\mathbb{Z}})^2$, and on the $n^{{{\operatorname{th}}}}$ night one could search for an $n \times n$ square that cannot be tiled (a compactness argument shows that if the entire plane cannot be tiled, then there exists $n$ such that the $n \times n$ square cannot be tiled). But R. Berger [@Berger1966] then proved that the tiling problem was undecidable, by embedding the halting problem as a subproblem of the tiling problem. Combining this with Wang’s observation shows that there exist finite collections that can tile the plane, but only *aperiodically*. Simplifications by R. Robinson [@Robinson1971], J. Kari [@Kari1996], and K. Culik II [@Culik1996] led to the example in Figure \[F:Wang tiles\], with only $13$ tiles. R. Robinson [@Robinson1978] and M. Margenstern [@Margenstern2008] proved similar undecidability results for tilings of the *hyperbolic* plane. Other tiling problems involve polyominos. A is a connected planar region obtained by connecting finitely many unit squares along shared edges. It is unknown whether the following is undecidable (see [@Rhoads2005]\*[p. 330]{}, for instance): input : a polyomino $P$ question : Can one tile the entire plane using translated and rotated copies of $P$? Graph theory ------------ Fix finite graphs $G$ and $H$. Let $V(G)$ be the vertex set of $G$; define $V(H)$ similarly. A from $H$ to $G$ is a (not necessarily injective) map $V(H) \to V(G)$ such that every edge of $H$ maps to an edge of $G$. The $t(H,G)$ is the probability that a uniformly chosen random map $V(H) \to V(G)$ is a homomorphism. If $H_1 \dot{\cup} H_2$ denotes the disjoint union of graphs $H_1$ and $H_2$, then $t(H_1 \dot{\cup} H_2,G) = t(H_1,G) t(H_2,G)$ for any $G$. There are certain known inequalities relating these densities. For instance, for the complete graph $K_n$ on $n$ vertices, elementary counting arguments similar to those in [@Goodman1959] show that $$t(K_3,G) \ge 2 t(K_2,G)^2 - t(K_2,G),$$ or equivalently $$t(K_3,G) - 2 t(K_2 \dot{\cup} K_2,G) + t(K_2,G) \ge 0,$$ for every finite graph $G$. This suggests the following problem: input : $k \in {\mathbb{Z}}_{\ge 0}$, finite graphs $H_1,\ldots,H_k$, and integers $a_1,\ldots,a_k$ question : Does $a_1 t(H_1,G) + \cdots + a_k t(H_k,G) \ge 0$ hold for all finite graphs $G$? H. Hatami and S. Norine proved this problem undecidable by relating it to Hilbert’s tenth problem [@Hatami-Norine2011]\*[Theorem 2.12]{}. Matrix semigroups ================= Matrix mortality ---------------- Given a finite list of square integer matrices, there are many ways to form products, especially if the factors may be repeated. Can one decide whether some product yields the zero matrix $\mathbf{0}$? More formally, we have the following: input : $n \in {\mathbb{Z}}_{\ge 0}$ and a finite set $S$ of $n \times n$ integer matrices question : Does the multiplicative semigroup generated by $S$ contain $\mathbf{0}$? M. Paterson proved that this problem is undecidable, even for sets of $3 \times 3$ matrices, via reduction to the Post correspondence problem [@Paterson1970]. Subsequent work showed that it is undecidable also for sets consisting of seven $3 \times 3$ matrices [@Halava-Harju-Hirvensalo2007]\*[Corollary 1]{} and for sets consisting of two $21 \times 21$ matrices [@Halava-Harju-Hirvensalo2007]\*[Theorem 11]{}. Whether there exists an algorithm for sets of $2 \times 2$ matrices remains an open problem. For a more detailed introduction to the matrix mortality problem, see [@Halava-Harju2001]. Freeness -------- One can ask, given $n$ and $S$, whether distinct finite sequences of matrices in $S$ yield distinct products, i.e., whether the semigroup generated by $S$ is . This turns out to be undecidable too, and already for sets of $3 \times 3$ matrices [@Klarner-Birget-Satterfield1991]. In fact, sets of fourteen $3 \times 3$ matrices suffice for undecidability [@Halava-Harju-Hirvensalo2007]\*[Theorem 13]{}. Finiteness ---------- Can one decide whether the semigroup generated by $S$ is finite? This time the answer turns out to be yes, as was proved independently by G. Jacob [@Jacob1978; @Jacob1977] and by A. Mandel and I. Simon [@Mandel-Simon1977]. Let us outline a proof. The main step consists of showing that there is a computable bound $f(n,s)$ for the size of any finite semigroup of $M_n({\mathbb{Z}})$ generated by $s$ matrices. Now for any $r \ge 1$, let $P_r$ be the set of products of length at most $r$ of matrices in $S$. Start computing $P_1$, $P_2$, and so on. If $\#P_1 < \cdots < \#P_N$ for $N=f(n,s)+1$, then $\#P_N > f(n,s)$, so the semigroup is infinite. Otherwise $P_r = P_{r+1}$ for some $r<N$, in which case the semigroup equals $P_r$ and hence is finite. The algorithm can be extended to decide finiteness of a finitely generated semigroup of $M_n(k)$ for any finitely generated field $k$ presented as an explicit finite extension of ${\mathbb{F}}_p(t_1,\ldots,t_d)$ or ${\mathbb{Q}}(t_1,\ldots,t_d)$. Powers of a single matrix ------------------------- There are even some nontrivial questions about semigroups generated by one matrix! Given $A \in M_k({\mathbb{Z}})$, can one decide whether there exists $n \in {\mathbb{Z}}_{>0}$ such that the upper right corner of $A^n$ is $0$? This problem, whose undecidability status is unknown, is equivalent to the following: input : a linear recursive sequence of integers $(x_n)_{n \ge 0}$, specified by giving $x_0,\ldots,x_{k-1} \in {\mathbb{Z}}$ and $a_0,\ldots,a_{k-1} \in {\mathbb{Z}}$ such that $x_{n+k}= a_{k-1} x_{n+k-1} + \cdots + a_0 x_n$ for all $n \ge 0$ question : Does there exist $n$ such that $x_n=0$? This is known also as Skolem’s problem, since Skolem proved that $\{n:x_n=0\}$ is a union of a finite set and finitely many arithmetic progressions [@Skolem1934]. See [@Halava-Harju-Hirvensalo-Karhumaki-preprint]. Group theory ============ Motivated by topology, M. Dehn [@Dehn1911] asked three questions about groups: 1. Is there an algorithm to recognize the identity of a group? 2. Is there an algorithm to decide whether two given elements of a group are conjugate? 3. Is there an algorithm to decide whether two given groups are isomorphic? Dehn formulated the questions precisely, except for the precise notion of algorithm. Finitely presented groups ------------------------- To make sense of such questions, one must specify how a group is presented and how an element is presented. A natural choice is to describe a group by means of a finite presentation such as $$S_3 = \langle r,t: r^3=1, t^2=1, trt^{-1}=r^{-1} \rangle.$$ This example describes the group of symmetries of an equilateral triangle as being generated by a $120^\circ$ rotation $r$ and a reflection $t$, and lists relations satisfied by $r$ and $t$ such that all other relations are consequences of these. More formally, if $n \in {\mathbb{Z}}_{\ge 0}$, and $F_n$ is the free group on $n$ generators, and $R$ is a finite subset of $F_n$, and $H$ is the smallest normal subgroup of $F_n$ containing $R$, then we may form the quotient group $F_n/H$. Any group arising in this way is called a . An element of an f.p. group can be specified by giving a in the generators, i.e., a finite sequence of the generators and their inverses, such as $rtr^{-1}r^{-1}ttt^{-1}$. The word problem {#S:word problem} ---------------- For each fixed f.p. group $G$ (or more precisely, for each such group equipped with a particular presentation), we have the following: input : word $w$ in the generators of $G$ question : Does $w$ represent $1$ in $G$? The decidability of the word problem depends only on the isomorphism type of the group, and not on the presentation. There are many classes of groups for the word problem is decidable: finite groups, f.p. abelian groups, and free groups on finitely many generators, for instance. (For free groups, one algorithm is to cancel pairs of adjacent inverse symbols repeatedly for as long as possible; the resulting represents $1$ if and only if it is empty.) But in the 1950s, P.S. Novikov [@Novikov1955] and W. Boone [@Boone1959] independently proved that there is an f.p. group for which the word problem is undecidable. The analogue for f.p. semigroups had been proved earlier, by Post [@Post1947] and A. Markov [@Markov1947; @Markov1951]; one proof of this goes through the undecidability of another word problem, namely that for semi-Thue systems, which can also be used to prove undecidability of the Post correspondence problem. Ultimately, the proofs of all these results are via reduction to the halting problem: Novikov and Boone essentially showed, that for a certain f.p. group $G$, one could associate to any computer program $p$ a word $w$ in the generators of $G$ such that $w$ represents $1 \in G$ if and only if $p$ halts. The undecidability of the word problem admits another proof, using the Higman embedding theorem, which we state below after introducing a definition. A finitely generated group is called if it has the form $F_n/H$, where $H$ is the smallest normal subgroup of $F_n$ containing a given subset $R$, which is no longer required to be finite, but is instead required to be listable. Amazingly, it is possible to characterize such groups without mentioning computability: A finitely generated group is recursively presented if and only if it can be embedded in a finitely presented group. The Higman embedding theorem implies the existence of an f.p. group $P$ with undecidable word problem, as we now explain. First, it is rather easy to construct a recursively presented group for which the word problem is undecidable: for instance, if $S$ is any non-computable listable set of positive integers, then one can show that in the recursively presented group $$G_S \colonequals \langle a,b,c,d \mid a^n b a^{-n} = c^n d c^{-n} \textup{ for all $n \in S$} \rangle,$$ $a^n b a^{-n} c^n d^{-1} c^{-n}$ represents $1$ if and only if $n \in S$, so $G_S$ has an undecidable word problem. By the Higman embedding theorem, $G_S$ embeds in some finitely presented group $P$, which therefore has an undecidable word problem too. The conjugacy problem --------------------- For each fixed f.p. group $G$, we have another problem: input : words $w_1,w_2$ in the generators of $G$ question : Do $w_1$ and $w_2$ represent conjugate elements of $G$? The word problem can be viewed as the subproblem of the conjugacy problem consisting of the instances for which $w_2$ is the empty word, which represents $1$. Thus the conjugacy problem for $G$ is at least as hard as the word problem for $G$, which means that it is easier (or at least no harder) to find a $G$ for which the conjugacy problem is undecidable. In fact, P.S. Novikov published a proof of the existence of an f.p. group for which the conjugacy problem is undecidable before publishing the result on the word problem, and this earlier proof is much simpler [@Novikov1954]. The inequality above between the difficulties of the two problems is the only one, in a sense that can be made precise using basic notions of computability theory, namely the notions of and $\le_T$: Given c.e. degrees $W$ and $C$ such that $W \le_T C$, there exists an f.p. group $G$ for which the word problem has degree $W$ and the conjugacy problem has degree $C$. This means that given c.e. subsets $W$ and $C$ of ${\mathbb{N}}$ such that the membership problem for $W$ is decidable given an oracle for the membership problem for $C$, there exists an f.p. group $G$ such that the word problem for $G$ can be solved using an oracle for membership in $W$ and vice versa, and the conjugacy problem for $G$ can be solved using an oracle for membership in $C$ and vice versa. Properties of groups {#S:properties} -------------------- Instead of fixing an f.p. group $G$, one can ask about algorithms that accept a finite presentation as input and try to decide whether the group it defines has a given property. For a wide variety of natural properties, the decision problem turns out to be undecidable. To make this precise, define a to be a property $P$ of f.p. groups, depending only on the isomorphism type of the group, not on the presentation, such that 1. there exists an f.p. group $G_1$ with $P$, and 2. there exists an f.p. group $G_2$ that cannot be embedded in any f.p. group with $P$. Examples are the properties of being trivial, finite, abelian, nilpotent, solvable, free, or torsion-free, because all these properties are inherited by subgroups. The property of having a decidable word problem is yet another Markov property, for the same reason! Using the undecidability of the word problem, S.I. Adian [@Adyan1957a; @Adyan1957b] and M. Rabin [@Rabin1958] proved the following: \[T:Adian-Rabin\] For any Markov property $P$, it is impossible to decide whether an f.p. group $G$ has $P$. \[C:trivial group\] It is impossible to decide whether a finite presentation describes the trivial group. Deciding triviality is a subproblem of the general problem of deciding whether two finite presentations define isomorphic groups, so the isomorphism problem is undecidable too. For a more extended survey of undecidability in group theory, see [@Miller1992]. Topology ======== The homeomorphism problem {#S:homeomorphism problem} ------------------------- Given two manifolds, can one decide whether they are homeomorphic? As usual, to make sense of such a question, we need to specify how a manifold is described. Since every compact smooth manifold can be triangulated, a natural choice is to use finite simplicial complexes to represent manifolds. input : finite simplicial complexes $M$ and $N$ representing smooth manifolds question : Are $M$ and $N$ homeomorphic? (One could alternatively replace homeomorphic by PL-homeomorphic, where PL stands for piecewise-linear.) Given a finite simplicial complex $M$ representing a compact manifold, one obtains a subproblem of the homeomorphism problem by fixing the first input to be $M$: input : a finite simplicial complex $N$ representing a smooth manifold question : Is $N$ homeomorphic to $M$? One can also restrict these problems according to dimension. For $d \le 3$, the homeomorphism problem for $d$-folds turns out to be decidable, because of classification theorems; for $d=3$, this uses the work of G. Perelman on W. Thurston’s geometrization conjecture. But for each $d \ge 4$, the homeomorphism problem for $d$-folds is undecidable, as was proved by Markov [@Markov1958]. Moreover, S.P. Novikov (the son of P.S. Novikov!)  proved that recognizing whether a finite simplicial complex is homeomorphic to the $d$-sphere $S^d$ is an undecidable problem for each $d \ge 5$ (a proof appears in the appendix to [@Volodin-Kuznecov-Fomenko1974]). From this, one can prove that for any fixed compact $d$-fold $M$ with $d \ge 5$, recognizing whether a finite simplicial complex is homeomorphic to $M$ is undecidable. All these results are proved by reduction to the undecidability results for f.p. groups. We now sketch the proofs of the unrecognizability results. (For a survey with more details, see [@Weinberger2005]\*[Chapter 2]{}.) Fix $d \ge 5$. Choose an f.p. group $G$ with undecidable word problem. From $G$ and a word $w$ in the generators of $G$, one can build an f.p. group $G_w$ such that $G_w$ is trivial if and only if $w$ represents $1$, and such that the first and second homology groups $H^1(G_w)$ and $H^2(G_w)$ are trivial. These conditions on $H^1$ and $H^2$ of an f.p. group are necessary and sufficient for there to exist a (a compact $d$-manifold with the same homology as $S^d$) with that fundamental group. In fact, one can effectively construct a finite simplicial complex $X_w$ representing such a homology sphere with fundamental group $G_w$. Now: - If $w$ represents $1$, then $G_w$ is trivial, and $X_w$ is a simply connected homology sphere, but in dimensions $d \ge 5$ a theorem of S. Smale [@Smale1961] implies that any such space is homeomorphic to $S^d$. - If $w$ does not represent $1$, then $X_w$ has nontrivial fundamental group, so $X_w$ is not homeomorphic to $S^d$. Hence, if we had an algorithm to recognize whether a finite simplicial complex is homeomorphic to $S^d$, it could be used to solve the word problem for $G$, a contradiction. Thus recognizing $S^d$ is an undecidable problem. Next suppose that $M$ is [*any*]{} compact $d$-fold for $d \ge 5$. The $M \# X_w$ is obtained by punching a small hole in each of $M$ and $X_w$ and connecting them with a thin cylinder. This construction can be done effectively on finite simplicial complexes. The fundamental group $\pi_1(M \# X_w)$ is the free product of the groups $\pi_1(M)$ and $\pi_1(X_w)$. A group-theoretic theorem states that a free product $G * H$ of finitely generated groups can be isomorphic to $G$ only if $H$ is trivial. Now: - If $w$ represents $1$, then $X_w$ is homeomorphic to $S^d$, and $M \# X_w$ is homeomorphic to $M$. - If $w$ does not represent $1$, then $M \# X_w$ does not even have the same fundamental group as $M$. Hence, if we had an algorithm to recognize $M$, it could be used to solve the word problem for $G$, a contradiction. Is $S^4$ recognizable? P. Seidel used similar ideas to find undecidable problems in *symplectic* geometry [@Seidel2008]\*[Corollary 6.8]{}. Am I a manifold? ---------------- We have seen that it is impossible to recognize whether two manifolds represented by given finite simplicial complexes are homeomorphic. Even worse, one cannot even decide whether a finite simplicial complex represents a manifold! In other words, the following problem is undecidable: input : finite simplicial complex $M$ question : Is $M$ homeomorphic to a manifold? Let us prove the undecidability by embedding the word problem in this problem. Recall that in Section \[S:homeomorphism problem\], we constructed a finite simplicial complex $X_w$, in terms of a word $w$ in the generators of an f.p. group $G$ with unsolvable word problem, such that - if $w$ represents $1$, then $X_w$ is homeomorphic to a sphere $S^d$, and - if $w$ does not represent $1$, then $X_w$ is a manifold with nontrivial fundamental group. The $SX_w$ of $X_w$ is the simplicial complex whose vertices are those of $X_w$ together with two new points $a$ and $b$, and set of faces is ${\bigcup}_{\Delta \in X_w} \{\Delta, \Delta {\cup}\{a\}, \Delta {\cup}\{b\} \}$. Geometrically, one may realize $X_w$ in a hyperplane in ${\mathbb{R}}^n$, and $a$ and $b$ as points on either side of the hyperplane; then $SX_w$ is the union of the line segments connecting a point of $\{a,b\}$ to a point of the realization of $X_w$. Now: - If $w$ represents $1$, then $X_w$ is homeomorphic to a sphere $S^d$, and $SX_w$ is homeomorphic to a sphere $S^{d+1}$. - If $w$ does not represent $1$, then $X_w$ has nontrivial fundamental group, so $SX_w$ contains loops arbitrarily close to $a$ with nontrivial class in the fundamental group of $SX_w - \{a,b\}$, so $SX_w$ is not locally euclidean at $a$. Thus $SX_w$ is homeomorphic to a manifold if and only if $w$ represents $1$. Therefore no algorithm can decide whether a given finite simplicial complex represents a manifold. Knot theory ----------- A is a smooth embedding of the circle $S^1$ in ${\mathbb{R}}^3$. Two knots are if there is an that transforms one into other; loosely speaking, this means that there is a smoothly varying family of diffeomorphisms of ${\mathbb{R}}^3$, parametrized by an interval, starting with the identity and ending with a diffeomorphism that maps one knot onto the other. How do we describe a knot in a way suitable for input into a computer? A knot may be represented by a finite sequence of distinct points in ${\mathbb{Q}}^3$: the knot is obtained by connecting the points in order by line segments, the last of which connects the last point back to the first point (we assume that each segment intersects its neighbors only at its endpoints and intersects other segments not at all, and the piecewise-linear curve should then be rounded at the vertices so as to obtain a smooth curve). input : knots $K_1$ and $K_2$, each represented by a finite sequence in ${\mathbb{Q}}^3$ question : Are $K_1$ and $K_2$ equivalent? W. Haken constructed an algorithm to decide whether a knot was unknotted [@Haken1961], and for the general problem he outlined an approach, the last step of which was completed by G. Hemion [@Hemion1979]. Thus the knot equivalence problem is decidable! One can also consider knots in higher dimension. An is a smooth embedding of $S^n$ in ${\mathbb{R}}^{n+2}$ (or $S^{n+2}$), and one can define equivalence as before. Any embedding equivalent to the standard embedding of $S^n$ as the unit sphere in a hyperplane in ${\mathbb{R}}^{n+2}$ is called . A. Nabutovsky and S. Weinberger prove that the problem of deciding whether an $n$-dimensional knot is unknotted is undecidable for $n \ge 3$ [@Nabutovsky-Weinberger1996]. Since this is a subproblem of the equivalence problem for $n$-dimensional knots, the latter is undecidable too. Nabutovsky and Weinberger leave open the following question: Is the equivalence problem for $2$-dimensional knots decidable? See [@Soare2004] for an exposition of some other undecidable problems in topology and differential geometry. Number theory ============= Hilbert’s tenth problem {#S:H10} ----------------------- One of the 23 problems in a list that Hilbert published after a famous lecture in 1900 asked for an algorithm to decide the solvability of diophantine equations: input : a multivariable polynomial $f \in {\mathbb{Z}}[x_1,\ldots,x_n]$ question : Does there exist $\vec{a}\in {\mathbb{Z}}^n$ with $f(\vec{a})=0$? This was eventually proved undecidable by Yu. Matiyasevich [@Matiyasevich1970]. To explain more, we need a definition. Call a subset $A$ of ${\mathbb{Z}}$ if there exists a polynomial $p(t,\vec{x}) \in {\mathbb{Z}}[t,x_1,\ldots,x_n]$ such that $$A = \{ a \in {\mathbb{Z}}: (\exists \vec{x} \in {\mathbb{Z}}^n) \; p(a,\vec{x})=0\}.$$ In other words, if one views $p(t,\vec{x})=0$ as a family of diophantine equations in the variables $x_1,\ldots,x_n$ depending on a parameter $t$, then $A$ is the set of parameter values that yield a solvable diophantine equation. It is easy to see that diophantine sets are listable. What is remarkable is that the converse holds: \[T:DPRM\] A subset of ${\mathbb{Z}}$ is diophantine if and only if it is listable. Work of M. Davis, H. Putnam, and J. Robinson culminating in [@Davis-Putnam-Robinson1961] proved the analogue for , in which polynomials are replaced by expressions built up from integers using not only addition and multiplication, but also exponentiation. Matiyasevich then showed how to express exponentiation in diophantine terms, to complete the proof of Theorem \[T:DPRM\]. Theorem \[T:DPRM\] immediately yields a negative answer to Hilbert’s tenth problem, because there are listable subsets $A$ of ${\mathbb{Z}}$ for which there is no algorithm to decide whether a given integer belongs to $A$ (see Section \[S:listable\]). The role played by Theorem \[T:DPRM\] for Hilbert’s tenth problem is similar to the role played by the Higman embedding theorem (Section \[S:word problem\]) for the word problem. Hilbert’s tenth problem for other rings --------------------------------------- After the negative answer to Hilbert’s tenth problem, researchers turned to variants in which the ring ${\mathbb{Z}}$ is replaced by some other commutative ring, such as ${\mathbb{Q}}$, or the ring of integers ${\mathcal{O}}_k$ of a fixed number field. ### The field of rational numbers {#S:Q} The problem for ${\mathbb{Q}}$ is equivalent to the problem of deciding whether an algebraic variety over ${\mathbb{Q}}$ has a rational point, because any variety is a finite union of affine varieties, and any system of equations $f_1(\vec{x})=\cdots=f_m(\vec{x})=0$ is solvable over ${\mathbb{Q}}$ if and only if the single equation $f_1(\vec{x})^2+\cdots+f_n(\vec{x})^2=0$ is. It is still not known whether an algorithm exists for this problem. The notion of a subset of ${\mathbb{Q}}$ being can be defined as in the previous section, except with all variables running over ${\mathbb{Q}}$ instead of ${\mathbb{Z}}$. If the subset ${\mathbb{Z}}$ were diophantine over ${\mathbb{Q}}$, then an easy reduction to Matiyasevich’s theorem would prove the undecidability of Hilbert’s tenth problem for ${\mathbb{Q}}$. J. Koenigsmann [@Koenigsmann2010-preprint]\*[Corollary 2]{}, building on [@Poonen2009-ae], proved that the *complement* ${\mathbb{Q}}-{\mathbb{Z}}$ is diophantine over ${\mathbb{Q}}$; a generalization to number fields was recently proved by J. Park [@Park-preprint]. In hopes of finding an undecidable problem, one can make the problem harder, by asking for an algorithm to decide the truth of , such as $$(\exists x) (\forall y) (\exists z) (\exists w) \quad (x \cdot z + 3 = y^2) \; \vee \; \neg ( z = x + w ).$$ Using the theory of quadratic forms over ${\mathbb{Q}}$, J. Robinson [@Robinson1949] proved that the following decision problem is undecidable: input : a first-order sentence $\phi$ in the language of fields question : Is $\phi$ true when the variables run over elements of ${\mathbb{Q}}$? ### Rings of integers Recall that a is a finite extension $k$ of ${\mathbb{Q}}$, and that the ${\mathcal{O}}_k$ of $k$ is the set of $\alpha \in k$ satisfying $f(\alpha)=0$ for some monic $f(x) \in {\mathbb{Z}}[x]$. The problem for ${\mathcal{O}}_k$ is conjectured to have a negative answer for each $k$ [@Denef-Lipshitz1978]. This has been proved for some $k$, namely when $k$ is totally real [@Denef1980], $k$ is a quadratic extension of a totally real number field [@Denef-Lipshitz1978], or $k$ has exactly one conjugate pair of nonreal embeddings [@Pheidas1988; @Shlapentokh1989]. Through arguments of the author and A. Shlapentokh , certain statements about ranks of elliptic curves over number fields would imply a negative answer for every $k$, and such statements have been proved by B. Mazur and K. Rubin [@Mazur-Rubin2010]\*[§8]{} assuming a conjecture of I. Shafarevich and J. Tate. For more about Hilbert’s tenth problem and its variants, see the survey articles [@Davis-Matiyasevich-Robinson1976; @Mazur1994; @Poonen2008-undecidability], the books [@Matiyasevich1993; @H10book; @Shlapentokh2007book], the website [@H10web], and the movie [@H10-movie]. Analysis ======== Inequalities {#S:inequalties} ------------ Given a real-valued function on ${\mathbb{R}}$ or on ${\mathbb{R}}^n$, can one decide whether it is nonnegative everywhere? The answer depends on the kind of functions allowed as input. ### Real polynomials For polynomials in any number of variables, A. Tarski showed that the answer is yes (to make sense of this, one should restrict the input to have coefficients in ${\mathbb{Q}}$ or in the field ${\mathbb{R}}{\cap}{{\overline{{\mathbb{Q}}}}}$ of real algebraic numbers, so that the polynomial admits a finite encoding suitable for a Turing machine). In fact, Tarski [@Tarski1951] gave a decision procedure, based on elimination of quantifiers for ${\mathbb{R}}$ in the language of ordered fields, for the following more general problem: input : a first-order sentence $\phi$ in the language of ordered fields question : Is $\phi$ true when the variables run over elements of ${\mathbb{R}}$? ### Adjoining the exponential function If one tries to extend this by allowing expressions involving also the real exponential function, then one runs into questions of transcendental number theory whose answer is still unknown. For example, can one decide for which rational numbers $r,s,t$ the equation $$e^{e^r} + e^s + t = 0$$ holds? But assuming Schanuel’s conjecture [@LangTranscendental]\*[pp. 30–31]{}, which rules out such “accidental identities”, A. Macintyre and A. Wilkie [@Macintyre-Wilkie1996] gave a decision algorithm for all first-order sentences for ${\mathbb{R}}$ with exponentiation in addition to the usual operations and $\le$. In contrast, for the set ${\mathscr{E}}$ of *complex* functions built up from integers and $z$ using addition, multiplication, and composing with $e^z$, A. Adler proved that it is impossible to decide whether a finite list of functions in ${\mathscr{E}}$ has a common zero in ${\mathbb{C}}$ [@Adler1969]\*[Theorem 1]{}. This can be proved by reduction to Hilbert’s tenth problem, using two observations: 1. One can characterize ${\mathbb{Q}}$ in ${\mathbb{C}}$ as the set of ratios of zeros of $e^z-1$. 2. One can characterize ${\mathbb{Z}}$ as the set of $x \in {\mathbb{Q}}$ such that there exists $z \in {\mathbb{C}}$ with $e^z = 2$ and $e^{zx} \in {\mathbb{Q}}$. ### Adjoining the sine function {#S:adjoining sin} Adjoining most other transcendental functions leads quickly to undecidable problems. For example, consider the following, a variant of a theorem of D. Richardson: \[T:everywhere nonnegative\] There is a polynomial $P \in {\mathbb{Z}}[t,x_1,\ldots,x_n,y_1,\ldots,y_n]$ such that for each $a \in {\mathbb{Z}}$, the real analytic function $$P(a,x_1,\ldots,x_n,\sin \pi x_1,\ldots, \sin \pi x_n)$$ on ${\mathbb{R}}^n$ is either everywhere greater than $1$, or else assumes values less than $-1$ and values greater than $1$, but it is impossible to decide which, given $a$. By Theorem \[T:DPRM\], we can find a polynomial $p(t,\vec{x}) \in {\mathbb{Z}}[t,x_1,\ldots,x_n]$ defining a diophantine subset $A$ of ${\mathbb{Z}}$ that is not computable. A little analysis shows that there is another polynomial $G \in {\mathbb{Z}}[t,x_1,\ldots,x_n]$ whose values are positive and growing so quickly that if we define $$L_a(\vec{x}) \colonequals -2 + 4 p(a,\vec{x})^2 + G(a,\vec{x}) \sum_{i=1}^n \sin^2 \pi x_i,$$ then $L_a(\vec{x}) \le 1$ holds only in tiny neighborhoods of the integer solutions to $p(a,\vec{x})=0$. If $a \in A$, then such integer solutions exist and $L_a$ takes the value $-2$ at those integer solutions and large positive values at some points with half-integer coordinates; otherwise, $L_a(\vec{x}) > 1$ on ${\mathbb{R}}^n$. Richardson’s original statement and proof of Theorem \[T:everywhere nonnegative\] were slightly more involved because they came before Hilbert’s tenth problem had been proved undecidable. Richardson instead had to use the undecidability result for exponential diophantine equations mentioned in Section \[S:H10\]. M. Laczkovich [@Laczkovich2003] found a variant of Theorem \[T:everywhere nonnegative\] letting one use $\sin x_i$ in place of $\sin \pi x_i$ for all $i$. Also, there exist functions $h \colon {\mathbb{R}}\to {\mathbb{R}}^n$ with dense image, such as $$h(x) \colonequals (x \sin x, x \sin x^3, \ldots, x \sin x^{2n-1}),$$ (this function, used by J. Denef and L. Lipshitz in [@Denef-Lipshitz1989]\*[Lemma 3.2]{}, is a simpler version of one used in [@Richardson1968]\*[§1, Theorem Two]{}). By composing a multivariable function with $h$, one obtains analogues of Theorem \[T:everywhere nonnegative\] for functions of *one* variable: Let $\mathscr{S}$ be the set of functions ${\mathbb{R}}\to {\mathbb{R}}$ built up from integers and $x$ using addition, multiplication, and composing with $\sin$. Then it is impossible to decide, given $f \in \mathscr{S}$, whether $f$ is everywhere nonnegative. Deciding whether $f$ is everywhere positive or whether $f$ has a zero are impossible too. For later use, we record the fact that there exist functions $F_a \in \mathscr{S}$, depending in a computable way on an integer parameter $a$, such that either $F_a(x)>1$ on ${\mathbb{R}}$, or else $F_a(x)$ assumes values less than $-1$ and values greater than $1$, but it is impossible to decide which. Equality of functions --------------------- Automatic homework graders sometimes need to decide whether two expressions define the same function. But deciding whether $|f(\vec{x})|$ is the same function as $f(\vec{x})$ amounts to deciding whether $f(\vec{x})$ is everywhere nonnegative, which, by Section \[S:adjoining sin\], is impossible for $f \in {\mathbb{Z}}[x_1,\ldots,x_n,\sin x_1,\ldots, \sin x_n]$ or for $f \in \mathscr{S}$ (cf. [@Richardson1968]\*[§2, Theorem Two]{}). For further undecidability results in analysis deduced from the negative answer to Hilbert’s tenth problem, see [@Adler1969; @Denef-Lipshitz1989; @Stallworth-Roush1997]. Integration ----------- There exists an entire function on ${\mathbb{C}}$ whose derivative is $e^{z^2}$. But work of J. Liouville shows that no such function is expressible by an elementary formula, in the following sense. For a connected open subset $U$ of ${\mathbb{C}}$, let ${\mathcal{M}}(U)$ be the field of meromorphic functions on $U$. Say that a function $g \in {\mathcal{M}}(U)$ is if it belongs to the last field $K_n$ in a tower ${\mathbb{C}}(z) = K_0 \subset K_1 \subset \cdots \subset K_n$ of subfields of ${\mathcal{M}}(U)$ such that each extension $K_{i+1}$ over $K_i$ is either algebraic or obtained by adjoining to $K_i$ either $e^f$ or a branch of $\log f$ defined on $U$ for some $f \in K_i$. For instance, the trigonometric functions and their inverses on a suitable $U$ are elementary functions. Liouville proved a general theorem [@Liouville1835]\*[§VII]{} that implies that there is no elementary antiderivative of $e^{z^2}$ on any $U$. (Earlier, Liouville proved that certain algebraic functions, such as $(1+x^4)^{-1/2}$, have no elementary antiderivative [@Liouville1833].) See [@Rosenlicht1972] for an account of Liouville’s methods. Can one decide whether a given elementary function has an elementary antiderivative? Building on the work of Liouville, R. Risch sketched a positive answer to a precise version of this question [@Risch1970]. To obtain a positive answer, the question must be formulated carefully to avoid having to answer questions about whether a constant or function is identically $0$. For example, it is not clear whether we can decide, given rational numbers $r,s,t$, whether $\int (e^{e^r}+e^s+t) e^{x^2} \, dx$ is an elementary function. Risch avoids this difficulty by restricting attention to functions in a tower of fields in which the constant field is an algebraically closed field of characteristic $0$ with a specified finite transcendence basis, and in which each successive extension in the tower is either an explicit algebraic extension or an extension adjoining $\exp f$ or a branch of $\log f$ that does not change the field of constants. If we try to generalize by allowing expressions involving the absolute value function $|\;|$, we encounter undecidability, as we now explain (cf. [@Richardson1968]\*[§2, Theorem Three]{}). Define $$\label{E:sigma} \sigma(x) \colonequals \frac12 \left( \left| x \right| - \left| x-1 \right| + 1 \right) = \begin{cases} 0, & \textup{ if $x \le 0$} \\ x, & \textup{ if $0 < x < 1$} \\ 1, & \textup{ if $x \ge 1$.} \end{cases}$$ Recall the functions $F_a$ at the end of Section \[S:adjoining sin\]. Then $\sigma(-F_a(x))$ is either $0$ on all of ${\mathbb{R}}$, or it agrees with $1$ on some open interval, but we cannot decide which. Thus we cannot decide whether $\int \sigma(-F_a(x)) e^{x^2} \, dx$ is an elementary function on all of ${\mathbb{R}}$. Deciding whether an improper integral converges is undecidable too, as was observed by P. Wang [@Wang1974]. Specifically, we cannot decide whether $$\int_{-\infty}^\infty \frac{1}{(x^2+1) F_a(x)^2} \, dx$$ converges. Differential equations ---------------------- Consider $$P(x,y,y',y'',\ldots,y^{(n)})=0$$ to be solved by a function $y$ of $x$, where $P$ is a polynomial with integer coefficients. Denef and Lipshitz [@Denef-Lipshitz1989]\*[Theorem 4.1]{} proved that the following problem is undecidable: input : $P \in {\mathbb{Z}}[x,y_1,y_2,\ldots,y_n]$ question : Does $P(x,y',y'',\ldots,y^{(n)})$ admit a real analytic solution on $[0,\infty)$? It remains undecidable even if one restricts the input so as to allow only ADEs that have a unique analytic solution in a neighborhood of $0$. The idea of the proof is to consider a function built up using $\sin$ as in Section \[S:adjoining sin\] for which one cannot decide whether it is either everywhere positive, and then to show that its reciprocal satisfies an ADE. To avoiding having to use $\pi$ in the coefficients of $P$, Denef and Lipschitz observed that $\tan^{-1} x$ is a function satisfying an ADE such that $\lim_{x \to +\infty} \tan^{-1} x = \pi/2$. An alternative would be to use the approach of [@Laczkovich2003] for eliminating $\pi$. ADEs can behave strangely in other ways too. L. Rubel [@Rubel1981] constructed a single explicit ADE whose solutions approximate any continuous function: more precisely, for any continuous functions $f \colon {\mathbb{R}}\to {\mathbb{R}}$ and $\epsilon \colon {\mathbb{R}}\to {\mathbb{R}}_{>0}$, there exists a $C^\infty$ solution $g \colon {\mathbb{R}}\to {\mathbb{R}}$ to the ADE satisfying $|g(x)-f(x)| < \epsilon(x)$ for all $x \in {\mathbb{R}}$. For other results and questions concerning existence and computability of solutions to differential equations, see [@Jaskowski1954; @Adler1969; @Aberth1971; @Pour-El-Richards1979; @Pour-El-Richards1983; @Rubel1983; @Denef-Lipshitz1984; @Rubel1992]. Dynamical systems ================= Many nonlinear dynamical systems are capable of simulating universal Turing machines, and hence they provide undecidable problems. Dynamical systems on ${\mathbb{R}}^n$ ------------------------------------- Call a map ${\mathbb{R}}^n \to {\mathbb{R}}^m$ if it is a linear map plus a constant vector. Call a map $f \colon {\mathbb{R}}^n \to {\mathbb{R}}^m$ if ${\mathbb{R}}^n$ can be partitioned into finitely many subsets $U_i$ each defined by a finite number of affine linear inequalities such that $f|_{U_i}$ agrees with an affine linear map depending on $i$. Call such a map if all the coefficients of the affine linear polynomials involved are rational. Given such map $f$, let $f^k$ be its $k^{{{\operatorname{th}}}}$ iterate. C. Moore [@Moore1990] proved that the following problem is undecidable: input : a rational piecewise affine linear map $f \colon {\mathbb{R}}^2 \to {\mathbb{R}}^2$ and a point $\vec{a} \in {\mathbb{Q}}^2$ question : Does there exist $k$ such that $f^k(\vec{a}) = \vec{0}$? Similarly, H. Siegelmann and E. Sontag proved that neural nets can simulate a universal Turing machine: in particular, if $\sigma \colon {\mathbb{R}}^n \to {\mathbb{R}}^n$ is the map obtained by applying the function  coordinatewise, then there exists $n$ and a specific matrix $A \in M_n({\mathbb{Z}})$, for which it is impossible to decide, given a starting point $\vec{a} \in {\mathbb{Q}}^n$, whether some iterate of $\sigma(A \vec{x})$ maps $\vec{a}$ to $\vec{0}$ [@Siegelmann-Sontag1995]. Instead of asking about the trajectory of one point, one can ask global questions about the dynamical system, such as whether every trajectory converges. V. Blondel, O. Bournez, P. Koiran, and J. Tsitsiklis prove that many such questions are undecidable for piecewise affine linear maps [@Blondel-et-al2001]. For further results relating dynamical systems and computability, see the survey article [@Blondel-Tsitsiklis2000]. Dynamical systems on the set of positive integers {#S:Collatz} ------------------------------------------------- There are also undecidable problems concerning dynamics of maps $f \colon {\mathbb{Z}}_{>0} \to {\mathbb{Z}}_{>0}$ such as $$f(x) \colonequals \begin{cases} 3x+1, &\textup{ if $x$ is odd} \\ x/2, &\textup{ if $x$ is even.} \end{cases}$$ The , still open, asks whether for every $n \in {\mathbb{Z}}_{>0}$, there exists $k$ such that $f^k(n)=1$. Building on work of J. Conway [@Conway1987], computer scientists S. Kurtz and J. Simon [@Kurtz-Simon2007] proved that the following generalization is undecidable: input : $m \in {\mathbb{Z}}_{>0}$, $a_0,\ldots,a_{m-1},b_0,\ldots,b_{m-1} \in {\mathbb{Q}}$ such that the function $f$ given by $f(x) = a_i x + b_i$ for $x \bmod m = i$ maps ${\mathbb{Z}}_{>0}$ to itself question : Is it true that for every $n \in {\mathbb{Z}}_{>0}$ there exists $k$ such that $f^k(n)=1$? Probability =========== Consider a random walk on the set $({\mathbb{Z}}_{\ge 0})^n$ of lattice points in the nonnegative orthant. At each time, the walker takes a step by adding a vector in $\{-1,0,1\}^n$. If $\vec{x}=(x_1,\ldots,x_n)$ is the current position, the vector to add is chosen with respect to a probability distribution $\Lambda_S$ depending only on the set $S = \{i : x_i \ne 0\}$, and $\Lambda_S$ is such that the walker never leaves the orthant. Suppose also that every probability in the description of each $\Lambda_S$ is in ${\mathbb{Q}}$. Say that the random walk starting at $\vec{a}$ is if there exists $C>0$ such that with probability $1$ the walker returns to $\{\vec{x} : |\vec{x}| < C\}$ infinitely often. input : $n \in {\mathbb{Z}}_{\ge 0}$, probability distributions $\Lambda_S$ as above for $S \subseteq \{1,\ldots,n\}$, and $\vec{a} \in ({\mathbb{Z}}_{\ge 0})^n$ question : Is the random walk starting at $\vec{a}$ stable? D. Gamarnik [@Gamarnik2002] proved that this problem is undecidable even if all the probabilities are $0$ or $1$! To do this, he showed that any Turing machine could be simulated by such a deterministic walk. Moreover, many basic questions about the stationary distribution of a random walk as above turn out to be undecidable, even if one assumes that the stationary distribution exists [@Gamarnik2007]. Algebraic geometry ================== Rational sections ----------------- K.H. Kim and F.W. Roush proved the undecidability of Hilbert’s tenth problem for the field ${\mathbb{C}}(t_1,t_2)$ of rational functions in two variables [@Kim-Roush1992]. (Strictly speaking, one should assume that the input has coefficients in ${{\overline{{\mathbb{Q}}}}}(t_1,t_2)$ instead of ${\mathbb{C}}(t_1,t_2)$, for the sake of encoding it for input into a Turing machine, but we will ignore this subtlety from now on.) By the same argument as in Section \[S:Q\], Hilbert’s tenth problem for ${\mathbb{C}}(t_1,t_2)$ is equivalent to the problem of deciding whether a ${\mathbb{C}}(t_1,t_2)$-variety over has a ${\mathbb{C}}(t_1,t_2)$-point (any pair of equations $f=g=0$ can be converted to $f^2+t_1 g^2=0$). K. Eisenträger [@Eisentraeger2004], using work of L. Moret-Bailly [@Moret-Bailly2005], generalized the Kim–Roush result to the function field of any fixed irreducible ${\mathbb{C}}$-variety $S$ of dimension at least $2$. Whether Hilbert’s tenth problem for ${\mathbb{C}}(t)$ is undecidable is an open question, studied in J. Kollár’s article [@Kollar2008]. It is also open for the function field of each other curve over ${\mathbb{C}}$. Let us return to the Kim–Roush result. By interpreting the “constants” $t_1$ and $t_2$ as variables, one can associate to each ${\mathbb{C}}(t_1,t_2)$-variety $X$ a ${\mathbb{C}}$-variety $Y$ equipped with a rational map $\pi \colon Y \dashrightarrow {\mathbb{A}}^2$. The ${\mathbb{C}}(t_1,t_2)$-points of $X$ correspond to of $\pi$, i.e., rational maps $s \colon {\mathbb{A}}^2 \dashrightarrow Y$ such that $\pi \circ s$ is the identity. This dictionary translates the Kim–Roush result into the undecidability of the following problem: input : a ${\mathbb{C}}$-variety $Y$ and a rational map $\pi \colon Y \dashrightarrow {\mathbb{A}}^2$ question : Does $\pi$ admit a rational section? Automorphisms ------------- Using the undecidability of Hilbert’s tenth problem, one can show that it is impossible to decide, given a variety $X$, a point $x \in X$, and a subvariety $Z \subset X$, whether there exists an automorphism of $X$ mapping $x$ into $Z$ [@Poonen2011-automorphism]. In fact, there are fixed $X$ and $x$ for which the problem for a variable input $Z$ is undecidable. More precisely, there is a smooth projective geometrically irreducible ${\mathbb{Q}}$-variety $X$ and a point $x \in X({\mathbb{Q}})$ such that the following problem is undecidable: input : a smooth projective geometrically irreducible subvariety $Z \subset X$ question : Does there exist an automorphism of $X$ mapping $x$ into $Z$? Moreover, $X$ can be chosen so that all its automorphisms over any field extension are already defined over ${\mathbb{Q}}$, so it does not matter whether we require the automorphisms to be defined over the base field. On the other hand, the following question has remained open: \[Q:automorphism\] Is there an algorithm to decide whether a given variety has a nontrivial automorphism? Possibly related to this is the following: \[Q:prescribed automorphism group\] Given an f.p. group $G$, can one effectively construct a variety $X_G$ whose automorphism group is $G$? A positive answer to Question \[Q:prescribed automorphism group\] would yield a negative answer to Question \[Q:automorphism\], since it is impossible to decide whether an f.p. group is trivial (Corollary \[C:trivial group\]). Isomorphism ----------- Given the undecidability of the homeomorphism problem for manifolds, it is natural to ask for the algebraic geometry analogue: input : two varieties $X$ and $Y$ over ${{\overline{{\mathbb{Q}}}}}$ question : Is $X {\simeq}Y$? The question of whether this problem might be undecidable was asked to the author by B. Totaro in 2007. We stated the problem over ${{\overline{{\mathbb{Q}}}}}$, because most algebraic geometry is done over an algebraically closed field and we wanted the input to admit a finite description. Alternatively, we could work over an algebraically closed field $\overline{{\mathbb{Q}}(t_1,t_2,\ldots)}$ of countable transcendence degree over ${\mathbb{Q}}$; this would capture the essence of the problem over ${\mathbb{C}}$, since any pair of varieties over ${\mathbb{C}}$ may be simultaneously defined over a finitely generated subfield of ${\mathbb{C}}$ and the existence of an isomorphism is unaffected by enlarging the ground field from one algebraically closed field to another. One could also consider other fields, such as ${\mathbb{Q}}$, ${{\overline{{\mathbb{F}}}}}_p$, ${\mathbb{F}}_p$, or $\overline{{\mathbb{F}}_p(t_1,t_2,\ldots)}$. There is no field over which it is known whether one can solve the variety isomorphism problem. Birational equivalence ---------------------- It is also unknown whether there is an algorithm to decide whether two given varieties are birationally equivalent. On the other hand, given an explicit rational map, one can decide whether it is a birational map, and whether it is an isomorphism. For algebraic geometers: if at least one of the varieties $X$ and $Y$ over ${{\overline{{\mathbb{Q}}}}}$ is of general type, then the set of birational maps $X \dashrightarrow Y$ is finite and computable (see below), and we can decide which of these birational maps are isomorphisms, and hence solve the variety isomorphism problem in this restricted setting. H. Matsumura proved that the birational automorphism group of a variety of general type is finite [@Matsumura1963]. The set of birational maps $X \dashrightarrow Y$ is either empty or a principal homogeneous space under this group, so it is finite too. Let us sketch an algorithm for computing this set. For $n=1,2,\ldots$, compute the maps determined by the pluricanonical linear system $|nK|$ for $X$ and $Y$ until an $n$ is found for which at least one of the two maps is birational onto its image. Then the other must be too and the linear systems must have the same dimension, say $n$, since otherwise we know already that $X \not{\simeq}Y$. The birational maps are then in bijection with the linear automorphisms of ${\mathbb{P}}^n$ mapping one canonical image to the other, and we can find equations for the locus of these automorphisms as a (finite) subscheme of ${\operatorname{PGL}}_{n+1}$. Algebra ======= Commutative algebra ------------------- If we restrict the variety isomorphism problem to the category of *affine* varieties, we obtain, equivalently, the isomorphism problem for finitely generated ${{\overline{{\mathbb{Q}}}}}$-algebras. Here each algebra can be presented as ${{\overline{{\mathbb{Q}}}}}[x_1,\ldots,x_n]/(f_1,\ldots,f_m)$ by specifying $n$ and explicit polynomials $f_1,\ldots,f_m$. One can replace ${{\overline{{\mathbb{Q}}}}}$ by other rings of constants (whose elements can be encoded for computer input). For example, taking ${\mathbb{Z}}$ yields the following problem: input : two finitely generated commutative rings $A$ and $B$ question : Is $A {\simeq}B$? The undecidability status of this problem is unknown. In fact, the status is unknown also for the isomorphism problem for finitely generated commutative algebras over any fixed nonzero commutative ring (with elements encoded such that addition and multiplication are computable). Noncommutative algebra ---------------------- The noncommutative analogue of the previous problem is undecidable, as we now explain. Let ${\mathbb{Z}}\langle x_1,\ldots,x_n \rangle$ be the noncommutative polynomial ring (free associative algebra with $1$) in $n$ variables over ${\mathbb{Z}}$. A (possibly noncommutative) is the quotient of ${\mathbb{Z}}\langle x_1,\ldots,x_n \rangle$ by the $2$-sided ideal generated by a finite list of elements $f_1,\ldots,f_m$. For an f.p. group $G$, the group ring ${\mathbb{Z}}G$ is an f.p. ${\mathbb{Z}}$-algebra, and ${\mathbb{Z}}G {\simeq}{\mathbb{Z}}$ if and only if $G {\simeq}\{1\}$. So if there were an algorithm to decide whether two f.p. ${\mathbb{Z}}$-algebras were isomorphic, we could use it to decide whether an f.p. group is trivial, contradicting Corollary \[C:trivial group\]. For other undecidable problems concerning noncommutative f.p. algebras, see [@Anick1985]. Games ===== Abstract games -------------- Given $m \ge 1$ and a computable function $W \colon {\mathbb{N}}^m \to \{\textup{A},\textup{B}\}$, consider the two-player game of no chance in which - the players (A and B) alternately choose natural numbers, starting with A, and ending after $m$ numbers $x_1,\ldots,x_m$ have been chosen; - there is perfect information (both players know the rules and can see all previously made choices); and - the winner is $W(x_1,x_2,\ldots,x_m)$. Many games can be fit into this framework. A result of L. Kalmár [@Kalmar1928], building on work of E. Zermelo [@Zermelo1913] and D. König [@Konig1927], states that exactly one of the two players has a winning strategy. But: \[T:who wins\] It is impossible to decide, given $m$ and $W$, *which* player has a winning strategy. Given a program $p$, consider the one-move game in which A chooses a positive integer $x_1$ and wins if $p$ halts within the first $x_1$ steps. Player A has a winning strategy if and only if $p$ halts, which is undecidable. More surprising is the following result of Rabin [@Rabin1957]: There is a three-move game in which B has a winning strategy, but not a *computable* winning strategy (i.e., there is no computable function of $x_1$ that is a winning move $x_2$ for B). Post [@Post1944]\*[§5]{} proved that there exists a , i.e., a c.e. set $S \subset {\mathbb{N}}$ whose complement $\overline{S}$ is infinite but contains no infinite c.e. set. Fix such an $S$. Let $g \colon {\mathbb{N}}\to {\mathbb{N}}$ be a computable function with $g({\mathbb{N}})=S$. Consider the three-move game in which A wins if and only if $x_1+x_2=g(x_3)$. Player B’s winning strategy is to find $t \in \overline{S}$ with $t>x_1$, and to choose $x_2=t-x_1$. A computable winning strategy $x_2=w(x_1)$, however, would yield an infinite c.e. subset $\{x_1+w(x_1) : x_1 \in {\mathbb{N}}\}$ of $\overline{S}$. Using the undecidability of Hilbert’s tenth problem, J.P. Jones gave new proofs of these theorems using games in which $W$ simply evaluates a given polynomial at the $m$-tuple of choices to decide who wins [@Jones1982]. R. Hearn [@Hearn-thesis] proved that team games with imperfect information can be undecidable even if they have only *finitely* many positions! For an account of this work and a complexity analysis of many finite games, see [@Hearn-Demaine2009]. Chess ----- R. Stanley [@Stanley2010mo] asked whether the following problem is decidable: input : A finite list of chess pieces and their starting positions on a ${\mathbb{Z}}\times {\mathbb{Z}}$ chessboard question : Can White force mate? (For a precise specification of the rules, and for related problems, see [@Brumleve-Hamkins-Schlicht-preprint].) It is unknown whether this problem is decidable. On the other hand, D. Brumleve, J. Hamkins, and P. Schlicht [@Brumleve-Hamkins-Schlicht-preprint] showed that one *can* decide whether White can mate in $n$ moves, if a starting configuration and $n$ are given. This statement can be proved quickly by encoding each instance of the problem as a first-order sentence in , which is the theory of $({\mathbb{N}};0,1,+)$. (Presburger arithmetic, unlike the theory of $({\mathbb{N}};0,1,+,\cdot)$, is decidable [@Presburger1929].) Final remarks ============= Each undecidable problem $P$ we presented is at least as hard as the halting problem $H$, because the undecidability proof ultimately depended on encoding an arbitrary instance of $H$ as an instance of $P$. In the other direction, many of these problems could be solved if one could decide whether a certain search terminates; for these $P$, an arbitrary instance of $P$ can be encoded as an instance of $H$. The problems for which both reductions are possible are called , and they are all of exactly the same difficulty with respect to Turing reducibility. For example, an algorithm for deciding whether a finitely presented group is trivial could be used to decide whether multivariable polynomial equations have integer solutions, and vice versa! On the other hand, certain other natural problems are strictly harder than the halting problem. One such problem is the generalized Collatz problem of Section \[S:Collatz\]: see [@Kurtz-Simon2007]\*[Theorem 3]{}. Acknowledgements {#acknowledgements .unnumbered} ================ I thank Henry Cohn, Martin Davis, and Richard Stanley for discussions. [^1]: The writing of this article was supported by the Guggenheim Foundation and National Science Foundation grants DMS-0841321 and DMS-1069236. [^2]: In most twentieth century literature in the subject one finds the terms and . But R. Soare [@Soare1996] has argued in favor of the use of the terms “computable” and “c.e.” instead, and many researchers in the field have followed his recommendation.

--- abstract: 'A mathematical model of Ni-63 source for betavoltaic batteries is presented, based on Monte Carlo calculation. Trajectories of beta particles are simulated in Ni-63 source until their escape or total energy dissipation. Analysis of the effect of physical and technological factors on the performance of a source is carried out. Special attention is given to self-absorption and substrate backscattering because of their impact on power emission. Addition of a protective layer diminishes the source emission because of further absorption. The model has been tested successfully for Ni-63/GaN structure.' author: - Abderrahmane Belghachi^1^ - Kutsal Bozkurt^2^ - Hassane Moughli^1^ - 'Orhan [Ö]{}zdemir^2^' - Benameur Amiri^1^ - Abdelkrim Talhi^1^ title: 'A model for Ni-63 source for betavoltaic application' --- Introduction ============ Conversion of nuclear decay energy into electrical power attracted significance attention since the early 1900s [@1; @2]. During last decades betavoltaic nuclear batteries have been the focus of intensive research work [@3; @4; @5; @6; @7; @8]. Nuclear batteries are best candidates for a number of applications where long-life power sources or low energy consumption are required such as; space applications, pacemakers, microsystems, remote-sensors, etc. A betavoltaic cell consists mainly of a beta particles radioactive source and a semiconducting material with a pn, pin or Schottky junction. The fundamental operating principle of a betavoltaic device is the interaction of beta particles with matter releasing a substantial number of electron-hole pairs. The role of pn junction is the establishment of a build-in electric field insuring the separation of generated free carriers, therefore creating usable electric output power. Several radioisotope substances have been investigated as betavoltaic sources, for instance; H-3, S-35, Ni-63, Kr-85, Y-90, Pm-147, etc. Among these sources, Ni-63 is the most promising choice because of its desirable qualities. Besides, of being pure beta source, Ni-63 has a long half-life (about 100 years), produces low energy beta particles, so minimising radiation damage to semiconductor converter, and can be stopped within few micrometres traveling in solids. As for its abundance, Ni-63 is important in the classification of radioactive waste from nuclear power plants [@9]. In a betavoltaic device, the radioactive source plays a very important role in the determination of the structure performance. Numerous work has been dedicated to betavoltaic batteries but little has been devoted to studying radioactive source component [@10]. The present work edifies the characteristics of beta source by investigating the correlation between physical and technological parameters to source output. The used quantity of radionuclide substance in a source has to be well defined in order to produce maximum power activity. Source thickness (amount of Ni-63) and geometrical form of the emitting surface significantly control the amount power to be delivered to the active semiconducting region. To model a radioactive source all physical phenomena occurring within the structure has to be accounted for. Two major factors affect source output; these are source self-absorption and substrate backscattering. No matter how thin is the active layer, absorption is inevitable, it results in reduction of emitted beta particles energy and number, this is self-absorption. A radioactive source is deposited always on a material that is called source backing material or substrate. This layer is generally a thin film, but no matter how thin, it may backscatter beta particles traveling away from the emitting surface. The suggested source model is obtained as a result of a Monte Carlo technique and assuming an attenuation absorption law. The model predicts Ni-63 source output parameters namely: apparent activity, emitted energy spectra and power density. Similar approach can be implemented to model different radioactive source and with different geometry. The model has been tested successfully for Ni-63/GaN structure, giving results matching well with previously simulated results [@11]. Model ===== Monte Carlo technique have been widely used to simulate stochastic transport phenomena for the last few decades. Particularly with the development of high-level computing skills, reduction of computational times and increase of storage capacity. In order to get more insight into the transport properties of beta particles in matter a simplified MC approach was used. The developed Monte Carlo code simulates the interaction of generated electrons in a radioisotope substance with a solid material using the single scattering approximation. This model is reported to be an accurate representation of electron interaction and is capable of giving excellent results [@12]. For any standard Monte Carlo program, two phenomena must be modelled: elastic collision and energy loss. When beta particles (electron) travels in a solid it could interact elastically with positively charged atomic nucleus or with atomic electrons in which case the electron will be deflected from its initial direction by Coulomb forces while its energy will remain unaltered. Alternatively, electron could interact inelastically with atoms by removing inner-shell electrons from orbit or with valence electrons to produce secondary electrons. In semiconducting materials, this latter scattering mechanism leads to the generation of electron-hole pairs. During their travelling in a solid, electrons are subject to a succession of scatterings, either elastically or inelastically. This process will continue until either the electron gives up all of its kinetic energy to the solid and comes to thermal equilibrium with it or until it manages to escape from the solid across its limiting surface. The trajectory of each electron in a solid comprises of a series of random straight paths, their lengths and directions are determined essentially by scattering mechanisms probability. In the single scattering approximation, only elastic scatterings are implied in the determination of the path and direction taken by any given electron. Electron trajectory is a function of the elastic mean free path $\lambda_{el}$, which depend only upon the scattering rate. This rate is related to the total scattering cross section via $$\lambda^{-1}_{el}=\rho N_{\alpha}\sum^{n}_{i=1}\frac{C_{i}\sigma_{el,i}}{A_{i}}.$$ Here, $N_\alpha$ is Avogadro’s number, $\rho$ is the material density, $A_i$ is the atomic weight of element $i$, $C_i $ is the mass fraction of element $i$, $\sigma_{el,i}$ is the total cross-section for element $i$ and $n$ represents the number of elements forming the material. The free path (distance between two scatterings) is determined by the electron mean free path and a random number $r$ in a range $[0,1]$ $$l=-\lambda_{el}\log (r).$$ The elastic Mott total cross-section for each element is computed using the model given by [@13], $$\sigma^{T}_{M}=5.21\times 10^{-21}\frac{Z^2}{E^2} \frac{4\pi \lambda \Big[1-e^{-\beta \sqrt{E}}\Big]}{\alpha(1+\alpha)}\Bigg [\frac{E+511}{e+1022} \Bigg]^2 ,$$ where $\alpha$ is screening parameter given by $$\alpha=3.4\times10^{-3}\frac{Z^{2/3}}{E} ,$$ $Z$ is the atomic number and E is the electron kinetic energy. The values of $\lambda$ and $\beta$ for each $Z$ are extracted from Ref. [@13] (table 1). In the case of polyatomic material, the element responsible of the scattering has to be determined. To achieve this we draw a random number $r$ and compare $r\times \sum^{n}_{i=1} F_{i}\sigma_{i}$ to the cumulative scattering mechanisms rate for $j$ varying from $1$ to $n$, $$\sum^{j-1}_{i=1} F_{i}\sigma_{i}< r \times \sum^{n}_{i=1} F_{i}\sigma_{i} \leq \sum^{j}_{i=1} F_{i}\sigma_{i} ,~j=2,.....n .$$ Here, $F_i$ is the atomic fraction of element $i$ and $\sigma_i$ is the total cross-section of element $i$. When this inequality is verified, the scattering mechanism $j$ is then selected. The polar angle of an individual elastic collision $\theta$ is determined with the value of the partial cross section of the element $i$. $\theta$ is obtained by solving $$r_{1}=\frac{\int^{\theta}_{0}\frac{d\sigma}{d\omega}\sin\theta d\theta}{\int^{\pi}_{0}\frac{d\sigma}{d\omega}\sin\theta d\theta} .$$ With $r_1$ a random number uniformly distributed between 0 and 1, a solution is found for the case of Rutherford theory [@12] $$\cos\theta=1-\frac{2\times\alpha \times r_1}{(1+\alpha-r_1)} ,$$ with $\alpha$ a screening parameter is given by Eq.(4). The azimuthal angle after the elastic collision $\phi$ is selected by another random number $r_2$, $\phi$ is uniformly distributed between $0$ and $2\pi$ $$\phi=2\pi\times r_2$$ once $\theta$ and $\phi$ are determined, we calculate the new direction $a_x$, $a_y$ and $a_z (cosines)$ as function of previous direction $a_{x0}$, $a_{y0}$ and $a_{z0}$ using spherical coordinate system transformation, $$\begin{aligned} a_x&&=\!a_{x0}cos\theta+\frac{a_{z0}a_{x0}}{\sqrt{1-a_{z0}^2}}\sin\theta\sin\varphi-\frac{a_{y0}}{\sqrt{1-a_{z0}^2}}\sin\theta\sin\varphi \nonumber \\ a_y&&=a_{y0}cos\theta+\frac{a_{z0}a_{y0}}{\sqrt{1-a_{z0}^2}}\sin\theta\cos\varphi-\frac{a_{x0}}{\sqrt{1-a_{z0}^2}}\sin\theta\sin\varphi \nonumber \\ a_z&&=a_{z0}cos\theta-\sqrt{1-a_{z0}^2}\sin\theta\cos\varphi .\end{aligned}$$ The electron energy is assumed to dissipate steadily along its path at a rate governed by the well-known continuous loss approximation described by Bethe relationship [@14] $$\frac{dE}{ds}=\frac{-7.85\times 10^{-3}\rho}{E}\times\sum^{n}_{i=1}\frac{C_{i}Z_i}{A_{i}}ln\Bigg[1.116\Big(\frac{E+k_i J_i}{J_i}\Big)\Bigg] ,$$ where $Z_i$ is the atomic number of element $i$, $C_i$ is the mass fraction of element $i$ $(C_i=m_i/m_{tot} )$, $F_i$ is the atomic fraction of element $i$, $k=0.734\times Z^{0.037}$ [@15] and $n$ is the number of elements in the region (it is the atomic $\%$ of element $i$ in mixture) with $$J= \begin{cases} 11.5\times Z & \text Z<13 \\ 9.76+58.5\times Z^{-0.19} & \text Z\geq13 . \end{cases}$$ Unlike most available Monte Carlo simulators designed to simulate monoenergetic electron beam of a finite section interaction with solids, our program deals with the interaction of a flux electron with an energy spectra impinging a solid through its surface. The program simulates also radioisotope substances where electrons are spontaneously generated within the material bulk. This program is specifically designed for betavoltaic devices, which consist mainly of a radioactive source layer together with a thin film semiconductor. To simulate a betavoltaic battery we chose a common structure, which consists of a radioactive layer laid onto a semiconductor p-n junction rectangular bloc of thickness $t$ and a unit surface $(1cm\times1cm)$. The structure is partitioned into adjacent unit square cells with $1\mu m\times1\mu m$ surface and a thickness $t$ that are assumed identical. This unit cell is sliced into a stack of wafers of $1\mu m\times1\mu m$ area and of thickness $\Delta z$, assuming that the total device thickness is very small compared to its surface, therefore perimeter effect could ignored. The most important parameter for betavoltaic cell is the spatial distribution of energy deposition in semiconductor that is responsible for electron-hole generation. Spatial distribution of energy deposition in the structure is computed as a function of thickness for unit surface. Only electrons generated inside the source region of the cell are followed even across lateral cell boundaries (sidewalls), this ensures continuity between the simulated cell and its neighbour cells. Lateral coordinates $(x, y)$ and normal direction $z$ coordinate describe electron position. Only the $z$ direction is discretised $(z_i=i\times\Delta z)$, normal to the device stacking layers, where deposited energy needs to be known. Energy dissipation rate will change every time electron cross from one region to another. Deposited energy in a wafer $i$ is computed as the sum of the dissipated electrons energy ($\sum_{j}\Delta E^{j}_{i}$ where $j$ represents electron number) while crossing the wafer $\Delta E^{j}_{i}=\frac{dE}{ds}\lambda_{ij}$ where $\lambda_{i,j}$ is part of free path of electron $j$ inside the wafer $i$. The first part of this work we aim to simulate, using an in-house code, the output of Ni-63 source and analyse some factors that affect its behaviour, essentially emitted power density and energy spectra. Most of the available simulators were developed for monoenergetic electron beam, whereas in a real beta source the spectrum of emitted particle energy is rather complex. In our simulation we adopted the beta spectra data obtained by [@16], which reproduced well the measured beta spectra of Ni-63. The simulated source consisted of a thin Ni-63 active layer, with a varying thickness, deposited onto Ni metallic foil as a substrate and a protective Ni layer was optionally added. In the second part, we simulated a Ni-63 source deposited onto a 4 $\mu m$ thick GaN thin film to investigate the effect of source thickness on the deposited energy. In the simulation we used $10^{6}$ electrons. A full energy spectrum of Ni-63 is used (extracted from BetaShape Analytical version: 1.0 (24/06/2016)[@17]) and a specific activity of $2.1\times 10^{12}$ $Bq /g$. The simulated source is $1cm \times 1cm$ and a thickness ranging from 0.01 to 7 $\mu m$. Total history of each electron (position, direction and kinetic energy) is computed from its site of creation inside the radioactive Ni-63 region until it exits from the back of the cell (backscattering is included) or comes to thermal equilibrium (that is when its energy decreases to below 0.05 $keV$). {width="50.00000%"} {width="50.00000%"} Results and Discussion ====================== Source self-absorption effect ----------------------------- To investigate source self-absorption, we simulate a rectangular slab of bare Ni-63 radioisotope material, without substrate nor protective layer, with different thicknesses. Due to the isotropic nature of radioactive substance, the external surface of a uniform mass will radiate equally in all directions. Let us consider a rectangular slab source with a thickness t and with a top and bottom surfaces $S$, so as its volume is $S\times t$ and a total external surface of $2S+4t \sqrt{S} $ ($S$ is assumed square shape); if its specific activity is $A_s$ ($Bq /g$), with the absence of absorption, nominal activity density can be expressed as $$A_{n}=\frac{A_s \rho S t}{2S+4t \sqrt{S}}=A_s \rho t\frac{1}{2+\frac{4t}{ \sqrt{S}}} .$$ So $$A_{n}=A_s \rho t \alpha_0,$$ where $ \alpha_0$ represent a form (geometric) factor in absence of self-absorption, then: $$\alpha_0=\frac{1}{2+\frac{4t}{ \sqrt{S}}}$$ $ \alpha_0$ is a factor always less or equal to 0.5. {width="0.9\linewidth"} In the case of a disc slab source of radius r (cylindrical shape), the form factor is then: $$\alpha_0=\frac{1}{2+\frac{2t}{r}}.$$ For thin film sources, where thickness $t$ is very small compared to $S$, $\alpha_0$ could be approximated by $\alpha_0 \approx 0.5$ then $A_n = 0.5 \times A_s \rho t$. The increase of beta source thickness will increase the amount of emitted electrons, whereas source self-absorption progressively overcomes. Source self-absorption will result in a reduction of kinetic energy of emitted electrons in addition to altering their number. To estimate self-absorption effect we assumed that the source was a uniform deposit of radioisotope Ni-63 emitting beta particles of a specific activity $A_s$ and the emitted particles followed a known attenuation law [@18]. Apparent activity of a source without backscattering is related to source thickness according to the following law $$A(t)=\frac{A_s \rho\alpha}{\mu} (1-e^{-\mu t}),$$ where $\alpha$ represents the new form factor taking into account source self-absorption, we propose an expression for this geometrical factor as follows $$\alpha=\alpha_0+\varepsilon ,$$ then $$A(t)=\frac{A_s \rho(\alpha_0+\varepsilon)}{\mu} (1-e^{-\mu t}).$$ In these equations $\varepsilon$ is assumed constant over the simulated thickness, its value is obtained for the best fit to our calculated activity $A(t)$ curve and $\mu$ is the linear attenuation coefficient of Ni-63. Values of attenuation coefficients are determined, either experimentally or using empirical formulas from literature. These Values are scattered in literature and cover a wide range. Discrepancy of attenuation coefficient in literature could be attributed to material quality; besides geometrical form of structures do have some effect on it [@19]. With Monte Carlo simulation, we computed surface activity of Ni-63 source without backscattering from the backing Ni metal and without protective layer. The variation of source activity against thickness curve when fitted to a common attenuation law (Eq. (18) with $\varepsilon =0$) allow the extraction of linear attenuation coefficient (absorption coefficient) $\mu$, for this case we found $\mu =$ 14080 $cm^{-1}$, this value appears to be high compared to most reported data. The closest value is found in Ref. [@19] where they reported an experimental average of $\mu$ (13183.84 $cm^{-1}$). Using Eq. (18) ($\varepsilon\neq 0$) the best fit to our results gives $\mu = $10650 $cm^{-1}$ which compares well with data obtained from most previously reported attenuation coefficient expressions, as summarised in table 1. In figure 1 we plotted surface activity density obtained by our simulation with both fittings ($\varepsilon =0$ and $\varepsilon\neq 0$) together with a recap of obtained results using Eq. (18) for different values from several $\mu$ formulas (see table 1). The value of $\varepsilon$ corresponding to the best fit was $\varepsilon = - 0.1046$, this value is valid exclusively for our simulated source (i.e. $1cm\times1cm$ profile). Self-absorption in a source is characterised by a factor, $f_{sa}$ defined as a ratio of the number of particles leaving source with self-absorption (apparent activity) to the number of particles leaving source without self-absorption (nominal activity) so: $$f_{sa}=\frac{\alpha}{\mu t \alpha_0} (1-e^{-\mu t}),$$ $$f_{sa}=\frac{(1+2\varepsilon)}{\mu t} (1-e^{-\mu t}).$$ Figure 2 represents the variation of $f_{sa}$ against source thickness. The parameter $\varepsilon$ obtained from activity versus source thickness cure is used to evaluate $f_{sa}$ using two available $\mu$ expressions and an experimental value from references using Eq.(19). =0.4cm \[t3\] Expression of $\frac{\mu}{\rho}$ $\mu$ $(cm^{-1})$ -------------------------------------------------------- ------------------- -- -- -- 15.2 $Z^{4/3}$ $\frac{E^{-1.485}}{A}$ [@20] 10358.32 8 $Z^{0.28}$ $E^{-(1.57-\frac{Z}{160})}$ [@21] 8031.75 Experimental [@19] 13183.84 Monte Carlo calculation      $\varepsilon$ = - 0.1046 10650                                      $\varepsilon$ = 0 14080 : Summary of $\frac{\mu}{\rho}$ extracted from computed curves and values from literature. Substrate back-scattering effect -------------------------------- The substrate or source backing material may backscatter beta particles, therefore contributing to the flux of emitted particles from the source front. To investigate this effect we introduced in our Monte Carlo program the backing material scattering, which is a layer of metallic Ni. The back-scattered beta particles from the substrate or the backing material are reintroduced into the active layer and allowed to diffuse accordingly. In figure 3 we represented source surface activity with and without backscattering effect together with nominal activity. Backscattering effect is observable at low thicknesses where source activity is almost linearly proportional to thickness. As saturation is attained backscattering has no effect and source self-absorption dominates. To separate backscattering effect from self-absorption effect it is necessary to examine the backscattering factor $f_{bs}$ which is defined as the rate of number of particles leaving the source with source backing to the number of particles leaving without source backing. In the case of total reflection and negligible thickness $f_{bs}$ should be equal to 2, for thick sources where self-absorption dominates $f_{bs}$ tends to 1, therefore $1\leq f_{bs}\leq 2$. So the apparent source activity is written as $$A(t)= f_{bs}\frac{A_{s}\rho(\alpha_0+\varepsilon)}{\mu}(1-e^{-\mu t}).$$ For the investigated source, the variation of backscattering factor versus thickness is represented in figure 4. As expected $f_{bs}$ decreases sharply to saturate around unity. In this figure, we plotted a function $f_{bs}(t)$ that best fits Monte Carlo obtained results. This function has two fitting parameters $a$ and $b$, where $$f_{bs}(t)=1+a\times e^{-bt}.$$ Our best fit to the computed results yields: $a$=0.395 and $b$=1.78 $\times$ 10$^4$ $cm^{-1}$. Energy Spectrum --------------- Each radioisotope material have a continuous energy probability distribution known as beta energy spectrum. The beta spectra emitted by Ni-63 radioisotope recently measured down to very low energies, are well reproduced by the calculations of X. Mougeot et al. [@16; @22]. It is characterised by emission of electron ($\beta$-particle) having average energy 17.4 $keV$ and a maximum energy of 66.9 $keV$ diffusing in random directions. Energy probability distribution has been described by an expression of the following form [@23] $$W(E)\propto (E^{2}+2E m_{e}c^{2})^{1/2}(Q-E)^{2}(E+m_{e}c^{2}),$$ where $E$ is the electron kinetic energy, $m_e$ is the electron mass, $Q$ is the decay energy, and $c$ is the speed of light. In our study we proposed an approximation for the probability of beta particles escaping from a Ni-63 source, which can be expressed in the following formula $$W(E)=a_{0}+a_{1}E+a_{2}E^{2}+a_{3}E^{3}+a_{4}E^{4},$$ where $a_{i}$ coefficients depending on source thickness and are determined from $W(E)$ curves. In figure 5, we represented energy beta spectrum at the surface of Ni-63 source for different thicknesses together with a spectrum from the Ni-63 decay. It shows that as the source thickness is increased the maximum of energy distribution is shifted towards higher energies. This is due to source self-absorption; the kinetic energy of emitted electrons is diminished in addition to a reduction of their number. $a_{i}$ coefficients are extracted from calculated curves using best fit to the obtained spectrum using a 4th degree polynomial curve (Eq. (24)). These values saturates as the thickness of source reaches about 4 $\mu m$ where spectra curves coincides, table 2 summarises $a_{i}$ values. At saturation Ni-63 emitted beta particles spectrum at the source surface, independently of source thickness, could be described by the following equation $$\begin{aligned} W(E)&&=\!0.0011+0.0013\times E+4.6\times 10^{-6}\times E^{2} \nonumber \\ &&-1\times 10^{-6}\times E^{3}+1\times 10^{-8}\times E^{4}.\end{aligned}$$ =0.1cm t ($\mu m$)   $a_0$   $a_1$    $a_2$    $a_3$    $a_4$ ------------- --------- ---------- ----------- ----------- ----------- -- -- -- -- 0 0.04476 -0.00154 2.53e-05 -3.31e-07 2.22e-09 0.01 0.02835 0.00015 -3.82e-05 8.25e-07 -5.78e-09 0.05 0.01590 0.00191 -0.00011 2.07e-06 -1.32e-08 0.2 0.00558 0.00280 0.00013 2.14e-06 -1.27e-08 1 0.00068 0.00209 -5.40e-05 3.14e-07 3.09e-10 4 0.00110 0.00127 7.86e-06 -1.08e-06 1.00e-08 7 0.00110 0.00132 4.63e-06 -1.01e-06 9.60e-08 : Summary of $a_i$ values for different source thicknesses extracted from best fit to calculated curves to equation (24).[]{data-label="t2"} Average energy -------------- As the energy spectrum shifted towards higher energies, the average energy increased up to a thickness of about 2 $\mu m$ where it saturates around 30 $keV$, as shown in figure 6. The average energy of emitted beta particles at a source of thickness $t$ $(E_{av} (t))$ is computed using $$E_{av}(t)=\int_{0}^{E_{max}} W(E)EdE,$$ where $W(E)$ is the occupation energy probability and $E_{max}$ is the maximum energy of beta spectrum ($E_{max}$ = 66.9 $keV$). Emitted power density --------------------- Source emitted power is very important for betavoltaic application, it provides the incident power density available for energy conversion. The total emitted power density from a source with a thickness t is computed using the following expression [@11] $$P(t)=A(t) E_{av}(t)q,$$ where $A(t)$ is the density of source surface activity $(Bq/cm^{2})$ and $q$ electron elementary charge. In figure 7 we plotted emitted power density at the source surface against its thickness. The curve shows a linear increase at low thicknesses then saturates at about 3.28 $\mu W/cm^2$ for a thickness above 3.5$\mu m$. Similarly, Ref. [@10] found 3.24 $\mu W/cm^2$ as apparent power density at saturation. The results compare well with reported data of Ref. [@11], where they found 2.85 $\mu W/cm^2$ for 1.5 $\mu m$ source thickness with a specific activity of $2.2\times10^{12}$ $Bq/g$ (our results are 2.75 $\mu W/cm^2$ for the same thickness taking a specific activity of $2.1\times10^{12}$ $Bq/g$). Effect of protective layer on the propriety of Ni-63 source ----------------------------------------------------------- In the next simulation, a protective layer is added to the source so the emitting area will be the external protective layer surface. We investigated the effect of a metallic Ni layer as a protective layer onto a Ni-63 source of a 4 $\mu m$ thickness. The protective layer thickness is varied from 0 to 1000 $nm$. Figure 8 shows the variation of the average energy versus the protective layer thickness. We remark a significant increase of the average energy before saturation, meaning that the spectrum is further shifted towards higher energies. This is due to additional protective layer absorption effect. This effect is confirmed in figure 9, which represents the emitted power of the source versus Ni layer thickness. The emitted power density is seriously reduced if a thick protective layer is used. {width="50.00000%"} Deposited energy in GaN thin film --------------------------------- In this section, we investigate the effect of source thickness (from 0 to 4000 nm) on the absorbed energy in a GaN betavoltaic structure. To simulate Ni-63 source the energy spectrum obtained in section 2.3 is the starting point. Random initial energies for a number $N$ of emitted electrons from source are obtained using a cumulative distribution function CDF (integral of energy probability distribution). The results are plotted in figure 10, the curves show two regions: a)- on the vicinity of source/GaN interface (below 150 nm), as the source thickness is decreased the more percentage energy is transmitted to GaN material, b)- far from the interface (greater than 150 nm) the opposite behaviour is observed. This could be simply explained by the fact that as sources thickness is augmented self-absorption significantly increases. On the other hand, for thicker sources the average energy increases therefore more energy is pumped into GaN. The obtained results, figure 10, are very similar to those obtained by C. E. Munson et al. [@11] using a model, they previously developed [@7]. Conclusions =========== This study presented a mathematical model of Ni-63 beta particles sources. In this model we demonstrated the crucial effect of self-absorption which limits power emission beyond 4$\mu m$ thickness. We observed energy spectra of emitted electrons shifting towards higher energies and convoyed higher average energies up to saturation around 4$\mu m$. The suggested model consists of a modified attenuation law for the source output activity (apparent) to take into account self-absorption. Back-scattering effect from source substrate (backing support) has a remarkable effect for very thin active layers, this is characterised by a factor $f_{bs}$. For ultra-thin Ni-63 layers this factor tends to 2, therefore doubling the source activity. In the case where a protective layer is deposited atop of the source this will reduce significantly source activity and if its thickness is increased it will degrade the source overall performance.The suggested model has been tested to determine deposition energy in GaN and gave results similar to those obtained previously. 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--- abstract: 'Many systems of both theoretical and applied interest display multi-affine scaling at small length scales. We demonstrate analytically and numerically that when vertical discontinuities are introduced into a self-affine surface, the surface becomes multi-affine. The discontinuities may correspond to surface overhangs or to an underlying stepped surface. Two surfaces are numerically examined with different spatial distributions of vertical discontinuities. The multi-affinity is shown to arise simply from the surface of vertical discontinuities, and the analytic scaling form at small length scales for the surface of discontinuities is derived and compared to numerical results.' author: - 'S. J. Mitchell' title: 'Vertical Discontinuities in Self-Affine Surfaces Lead to Multi-affinity' --- Many systems of both theoretical and applied interest display multi-affine scaling at small length scales [@ALB1; @ALB2; @ALB3; @SDS; @CVD; @KK]. Recently, an extensive scaling analysis of surfactant templated hydrogel surfaces as measured by atomic force microscopy (AFM) was performed [@MK]. This analysis indicated that the hydrogel surfaces were self-affine; however, a later numerical study of a frustrated spring-network model of cross-linked hydrogels [@GB] indicated multi-affine scaling. Reconciliation of these two observed behaviors led to an interesting and universal conclusion: introduction of vertical discontinuities into a self-affine surface leads to multi-affine scaling. To our knowledge, this has not previously been reported in the literature, most likely because height-height correlations are usually calculated for the second power of the height increments, in which case the surface constructed only of discontinuities resembles a random walk on all length scales. Here, we provide a discussion which explains this source of multi-affine behavior, and we present both numerical and analytic results. Consider a one-dimensional, real, single-valued surface, $z(x)$, where $x$ is a real number on the interval, $x \in [0,1]$. The generalized height-height correlation function [@ALB1; @ALB2; @ALB3; @SDS; @CVD; @KK; @GB] for this surface is $$C_q(r)=\langle |z(x+r)-z(x)|^q \rangle \;, \label{eq:corr0}$$ where $\langle \cdots \rangle$ denotes an average over all $x$ values, $|r|<1/2$, and $q$ is a positive, non-zero real number. Without loss of generality, we may assume that $r$ is positive, since $C_q(r)=C_q(-r)$ for any function $z(x)$. Often, $C_q(r)$ will display power-law behavior for $r \ll r_{\times}$ and will display a constant value for $r \gg r_{\times}$, where $r_{\times}$ is some cross-over length scale between the two behaviors. Surfaces displaying power-law correlations, $C_q(r) = A_q r^{q \alpha_q}$, fall into one of two categories: $q$-independent scaling, $\alpha_q=\alpha$, called self-affine scaling, and $q$-dependent scaling, called multi-affine scaling [@ALB1; @ALB2; @ALB3; @SDS; @CVD; @KK]. By introducing vertical discontinuities into a self-affine surface, we can cause the surface to become multi-affine. Consider the function $z(x)$, which is self-affine for all $r \ll r_{\times}$. For the numerical results, self-affine surfaces were generated using the method of Ref. [@VOSS]. We introduce a finite number, $N$, of vertical discontinuities into the function $z(x)$ such that the new surface is $$z'(x)=\left\{\sum_{i=1}^N \delta_i\Theta(x-x_i) \right\} +z(x)\; , \label{eq:zprime}$$ where $i$ indexes the discontinuities, $\delta_i$ is the magnitude of the discontinuity at $x=x_i$, and $\Theta(y)$ is a step function which is zero for $y < 0$ and 1 for $y \ge 0$. Without loss of generality, we can assume an order to the set $\{x_i\}$ such that $x_{i-1}<x_i$ and $x_0=0$. ![Multi-affine surface generated from Eq. (\[eq:zprime\]) with discontinuities evenly spaced in $x$. The magnitudes of the discontinuities, $\delta_i$, are drawn independently from a Gaussian distribution with standard deviation 1.0 and mean 0, and $N=41$. The self-affine function $z(x)$ has $\alpha=0.75$. (a) The surface, $z'(x)$, and related functions. (b) $C_q(r)$ for each function shown in (a) as labeled in the plot. The correlation functions for $z_\Theta(x)$ and $z(x)$ have been displaced up by 3 and down by 2 units, respectively, for graphical clarity. No graphical distinction is made between curves with different $q$, but $q \in [0.5,4.0]$ in steps of 0.5, and for the curves shown here, $C_q(r)>C_p(r)$ when $q>p$. The cross-over length scale is $r_{\times}=1/N$, and the cross-over region is very narrow. []{data-label="fig:even"}](even.eps){width="0.75\columnwidth"} ![Multi-affine surface generated from Eq. (\[eq:zprime\]) with discontinuities randomly spaced in $x$. All parameters are the same as in Fig. \[fig:even\]. (a) The surface, $z'(x)$, and related functions. (b) $C_q(r)$ for each function shown in (a) as labeled in the plot. The cross over length scale is $r_{\times}=1/N$, but the cross-over region is much broader than in Fig. \[fig:even\]. []{data-label="fig:uneven"}](uneven.eps){width="0.75\columnwidth"} Physically, $z'(x)$ can be thought of as describing a system with overhangs between regions with self-affine scaling, such as for the spring-network model of Ref. [@GB], or the deposition of a very thin self-affine film onto a stepped surface. A typical stochastic realization of $z'(x)$ with equally spaced $x_i$ is shown in Fig. \[fig:even\](a), and the corresponding generalized height-height correlation function, $C_q(r)$, is shown in Fig. \[fig:even\](b). Figure \[fig:uneven\] shows a similar plot, but the discontinuity positions, $x_i$, are chosen randomly and uniformly on the interval $(0,1)$. The number of discontinuities, $N$, is the same for both Fig. \[fig:even\] and Fig. \[fig:uneven\]. For length scales $r \gg r_{\times}$, the stepped surface, $z_{\Theta} \equiv z'(x)-z(x)$, is expected to be simply a random walk in height with $\alpha_q=0.5$, but this is not obvious in the numerical data because of the relatively small number of discontinuities in the $x$ interval. From examination of the numerical results in Figs. \[fig:even\] and \[fig:uneven\], it is obvious that the multi-affinity is caused by the stepped surface, $z_\Theta(x)$, and the generalized height-height correlation function of $z_\Theta(x)$ can be analytically calculated for $r \ll r_{\times}$, where $r$ is also much smaller than the smallest $x$ separation between discontinuities for finite systems with finite $N$. $$C_q(r)= \int_0^1 | \sum_{i=1}^N \delta_i \left\{ \Theta(x+r-x_i) - \Theta(x-x_i) \right\} |^q dx \; . \label{eq:int0}$$ The argument, $\Theta(x+r-x_i) - \Theta(x-x_i)$, is either 1 ($x_i - r \le x < x_i$) or 0 (otherwise), and thus, in the integration range $x=x_i-r$ to $x=x_i$, only one of the $N$ discontinuities has a non-zero contribution to the integral. Equation (\[eq:int0\]) thus reduces to $$\begin{array}{lll} C_q(r) & = & \sum_{i=1}^N \int_{x_{i-1}}^{x_i} | \sum_{i=1}^N \delta_i \left\{\Theta(x+r-x_i) - \right. \\ & & \left. \Theta(x-x_i)\right\}|^q dx \\ & = & \sum_{i=1}^N \int_{x_i-r}^{x_i} | \delta_i|^q dx \\ & = & r \sum_{i=1}^N | \delta_i|^q \; . \end{array} \label{eq:int1}$$ For comparison with Figs. \[fig:even\] and \[fig:uneven\], $$\frac{1}{q}\log_{10}\{C_q(r)\} = \frac{1}{q}\log_{10}\{r\} + \frac{1}{q}\log_{10}\left\{\sum_{i=1}^N | \delta_i|^q\right\} \; , \label{eq:int2}$$ and thus, $\alpha_q=q^{-1}$ and $\log_{10}\{A_q\} = \log_{10}\left\{N \langle | \delta_i|^q \rangle \right\}$. See Fig. \[fig:stepped\]. Note that the general form is independent of the $x_i$ distribution and depends on the $\delta_i$ distribution only through $\langle | \delta_i|^q \rangle$.  and Fig. \[fig:uneven\](b). (a) The multi-affine scaling exponent. The solid line indicates the analytic solution from Eq. (\[eq:int2\]). (b) The multi-affine scaling prefactor. The solid line indicates the analytic solution from Eq. (\[eq:int2\]). []{data-label="fig:stepped"}](stepped.eps){width="0.75\columnwidth"} ![Dependence of $\alpha_q$ on $q$ and the relative magnitude of the stepped surface. The distribution of the discontinuities, $\delta_i$, is Gaussian with mean zero and standard deviation $\sigma$, and all parameters are the same as in Figs. \[fig:even\] and \[fig:uneven\] unless otherwise indicated. The lines indicate the two asymptotic behaviors, $\alpha_q=1/q$ when $z(x)=0$ and $\alpha_q=0.75$ when $z_{\Theta}(x)=0$. The data indicated by the symbols are each taken from a single realization of the surface, and one can see that the spatial distribution of the discontinuities, $x_i$, has no effect on $\alpha_q$. []{data-label="fig:alpha"}](alpha.eps){width="\columnwidth"} Having derived the small-$r$ scaling behavior for the stepped surface, $z_\Theta(x)$, it remains to be shown how this multi-affine stepped surface influences the multi-affinity of the complete or mixed surface, $z'(x)$. Consider the generalized height-height correlation function for the sum of the two surfaces $$C_q(r)=\langle |\Delta_\Theta^r(x)+\Delta_z^r(x)|^q \rangle \; , \label{eq:twofuncs}$$ where $\Delta_\Theta^r(x)=z_\Theta(x+r)-z_\Theta(x)$ and $\Delta_z^r(x)=z(x+r)-z(x)$. The two extremes of $\langle |z(x)| \rangle \ll \langle |z_\Theta(x)| \rangle$ and $\langle |z_\Theta(x)| \rangle \ll \langle |z(x)| \rangle$ should behave as multi-affine and self-affine surfaces, respectively, but for intermediate mixed surfaces, the behavior is more complex. For the numerical results shown in Fig. \[fig:alpha\], two asymptotic scaling regimes are seen. For large $q$, the mixed surface tends towards multi-affine behavior with $\alpha_q=1/q$, and for small $q$, the mixed surface tends towards self-affine behavior with $\alpha_q=\alpha$. We can derive the two asymptotic scaling behaviors for the mixed function by first considering that for $r \ll r_\times$ $$\begin{array}{lll} C_q(r) & = & \langle |\Delta_\Theta^r(x)+\Delta_z^r(x)|^q \rangle \\ & = & \langle |\Delta_z^r(x) \left(\frac{\Delta_\Theta^r(x)}{\Delta_z^r(x)}+1\right) |^q \rangle \\ & = & \langle |\Delta_z^r(x)|^q | \frac{\Delta_\Theta^r(x)}{\Delta_z^r(x)}+1 |^q \rangle \\ & = & \int_0^1 |\Delta_z^r(x)|^q | \frac{\Delta_\Theta^r(x)}{\Delta_z^r(x)}+1 |^q dx\\ & = & \sum_{i=1}^N \int_{x_{i-1}}^{x_i} |\Delta_z^r(x)|^q | \frac{\Delta_\Theta^r(x)}{\Delta_z^r(x)}+1 |^q dx \; . \end{array}$$ By noticing that $\Delta_\Theta^r(x)=0$ or $\delta_i$ in the interval $x\in (x_{i-1},x_i]$, $$\begin{array}{lll} C_q(r) & = & \sum_{i=1}^N \int_{x_{i-1}}^{x_i-r} |\Delta_z^r(x)|^q | \frac{0}{\Delta_z^r(x)}+1 |^q dx\\ & & + \sum_{i=1}^N \int_{x_i-r}^{x_i} |\Delta_z^r(x)|^q | \frac{\delta_i}{\Delta_z^r(x)}+1 |^q dx\\ & = & \sum_{i=1}^N \int_{x_{i-1}}^{x_i-r} |\Delta_z^r(x)|^q dx\\ & & + \sum_{i=1}^N \int_{x_i-r}^{x_i} |\Delta_z^r(x)|^q | \frac{\delta_i}{\Delta_z^r(x)}+1 |^q dx\; . \end{array}$$ As $q \rightarrow 0$, $| \frac{\delta_i}{\Delta_z^r(x)}+1 |^q \approx 1$, and as $q \rightarrow \infty$, $| \frac{\delta_i}{\Delta_z^r(x)}+1 |^q \approx | \frac{\delta_i}{\Delta_z^r(x)}|^q$. This gives the following approximations for the two asymptotic regimes, $$C_q(r) \approx \left\{ \begin{array}{ll} \langle |\Delta_z^r(x)|^q \rangle = A_q r^{q \alpha} & q \ll 1 \\ A_q r^{q \alpha} + r N \langle |\delta_i|^q \rangle & q \gg 1 \; , \end{array} \right. \label{eq:asymp}$$ where the additional approximation $\int_{x_{i-1}}^{x_i-r} \cdots dx \approx \int_{x_{i-1}}^{x_i} \cdots dx$ is made, which is valid when $r \ll 1$. For large $q$, the scaling may resemble either self-affine scaling or multi-affine scaling, depending on the exact behavior of $A_q$ for $z(x)$ and the behavior of $\langle |\delta_i|^q \rangle$; however, for small $q$, the behavior will always resemble self-affine scaling, provided, of course, that the signal strength of $z(x)$ is sufficiently large compared to the stepped function signal strength to be numerically noticeable. For the mixed functions examined in Fig. \[fig:alpha\], self-affine scaling is seen for small $q$ and multi-affine scaling with $\alpha_q=1/q$ is seen for large $q$, but this is not a universal outcome as indicated in Eq. (\[eq:asymp\]). The scaling behavior of a self-affine surface with vertical discontinuities was investigated numerically and analytically, and it was shown that the surface of discontinuities (the stepped surface) was the source of the multi-affine behavior. It was further shown numerically and analytically, that the general form for the scaling of the stepped surface at small length scales depends on the distribution of discontinuities only through $\langle |\delta_i|^q \rangle$. Two asymptotic scaling behaviors were derived for the self-affine surface with discontinuities, and for the numerical results shown here, self-affine scaling is seen for small $q$ and multi-affine scaling with $\alpha_q=1/q$ is seen for large $q$. The large-$q$ asymptotic behavior is not universal and depends on the detailed $q$ dependence of the mixed function. These results suggest the need to further study scaling and universality for a variety of systems where vertical discontinuities are known or are expected to exist. Such systems include many thin film deposition model and deposition processes onto stepped surfaces. For these processes, the deposition time should have a large effect on the multi-affine scaling behavior. The author thanks G. M. Buendía and P. A. Rikvold for useful discussions and comments on the manuscript. Funding was provided by the Netherlands Organization for Scientific Research (NWO). [9]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , , , ****, (). , ****, (). , , , , , , ****, (). , , , , ****, (). , , , ****, (). , ****, (). , , , , . , , , . , in **, edited by (, , ).

--- abstract: 'Tracking the nonlinear behavior of an RF is challenging. To tackle this problem, we build a connection between residual learning and the nonlinearity, and propose a novel residual neural network structure, referred to as the . Instead of learning the whole behavior of the , the focuses on learning its nonlinear behavior by adding identity shortcut connections between the input and output layer. In particular, we apply the to digital predistortion and measure experimental results on a real . Compared with neural networks recently proposed by Liu *et al*. and Wang *et al*., the achieves the best linearization performance in terms of normalized mean square error and adjacent channel power ratio with less or similar computational complexity. Furthermore, the exhibits significantly faster training speed and lower training error.' author: - | Yibo Wu, Ulf Gustavsson, Alexandre Graell i Amat, and Henk Wymeersch\ Ericsson Research, Gothenburg, Sweden\ Chalmers University of Technology, Gothenburg, Sweden bibliography: - './bibliography/IEEEabrv.bib' - './bibliography/refs.bib' title: | Residual Neural Networks\ for Digital Predistortion --- Introduction ============ wireless systems pose significant challenges to the performance of the  [@kelly2016preparing]. High-frequency and high-bandwidth signals suffer severe distortions from the nonlinear behavior of the , which increases the need for highly linear . Meanwhile, the increasing number of antennas and base-stations require a large number of , which greatly increases the stress on power consumption, so the power efficiency of the is also crucial. In practice, the linearity and efficiency of the becomes a trade-off when both need to be satisfied. This trade-off has triggered intensive research over the past decades [@eun1997new; @kim2001digital; @GMP_2006]. These works aim to preserve the linearity at the high output power region by using , a well-known technique to compensate for the nonlinearity. performs an inverse nonlinear operation before the . This inverse operation can be represented by a parametric model, whose accuracy determines the performance. Conventionally, Volterra series based models [@eun1997new], such as  [@kim2001digital] and  [@GMP_2006], have been widely used for because of their high accuracy. In these models, the behavior of the is represented by a set of Volterra kernels with different nonlinear orders where each kernel also considers memory effects, i.e., past inputs that influence the current output. These memory effects are due to the frequency-dependent behavior of the  [@pedro2005comparative]. However, the performance of Volterra-based models is limited for severely nonlinear even if high-order kernels are used because of the high estimation error for high-order kernels [@orcioni2014improving]. In contrast to model-based approaches, deep learning techniques such as have recently been proposed for  [@isaksson2005wide; @luongvinh2005behavioral; @liu2004dynamic; @mkadem2011physically; @gotthans2014digital; @wang2018augmented; @hongyo2019deep; @tarver2019design]. Among them, the is the most commonly chosen type of for  [@liu2004dynamic; @mkadem2011physically; @gotthans2014digital; @wang2018augmented; @hongyo2019deep; @tarver2019design] because of the simple implementation and training algorithm. Based on the , [@liu2004dynamic] proposed a that separates the complex-valued signal into real in-phase and quadrature components to use a simple real-valued training algorithm. Furthermore, to consider memory effects of the , the input layer of the is fed by both the current instantaneous input and the inputs at previous time instants. To improve the performance of the , many variants have been studied [@mkadem2011physically; @gotthans2014digital; @wang2018augmented], which add more components to the input layer, such as previous samples of the output signal [@mkadem2011physically], future samples of the input signal [@gotthans2014digital], or envelope terms (e.g., amplitude) of the input signal [@wang2018augmented]. However, while these additional components have been shown to improve performance, they also significantly increase the network complexity, which pushes more pressure on the power consumption of . [@tarver2019design] considered a different approach to connect the input and output layer by a linear bypass, which makes the focus on the nonlinear relation. However, this approach is infeasible for a memory input, which limits its performance on with memory. Moreover, the performance comparison between with and without shortcuts for is not discussed in [@tarver2019design]. In this paper, we build a connection between *residual learning* and the . We then propose a residual , referred to as  to learn the nonlinear behavior of the . Unlike  [@liu2004dynamic] and its variants [@mkadem2011physically; @gotthans2014digital; @wang2018augmented] that learn the linear and nonlinear behaviors jointly, the proposed learn them separately. Specifically, the nonlinear behavior is learned by its inner layers, and the linear behavior is added at the end of the inner layers using *identity shortcuts* between the input and output layer. The identity shortcuts introduce no new parameters as well as negligible computational complexity (one element-wise addition). Unlike [@tarver2019design], which excludes memory inputs, the considers memory inputs by applying identity shortcuts between two neurons of the current instantaneous input-output, which also solve the dimension difference between the input and output layer. We apply the proposed  to . Experimental results on a real show that the proposed for achieves a better linearization performance as well as a faster training rate than the in [@liu2004dynamic] and a variant of it in [@wang2018augmented] with similar computational complexity. System Model {#section:model} ============ ![Behavior relation between and the . The power gain of the PA is normalized for simplicity. To compensate the nonlinear behavior $f$ before the saturation point, performs an inverse operation $g$.[]{data-label="fig:dpd_pa_diagram"}](Figures/DPD-PA-Diagram-no-noise.pdf){width="0.75\linewidth"} PA behavior and DPD ------------------- The behaves as a nonlinear system that exhibits static nonlinearity and memory effects. The latter is more obvious in a wideband scenario because of the frequency-dependent gain and phase shift between the input and output signal [@939917]. Memory effects are exhibited in the time domain, which means that the output at any time instant is a function of the current instantaneous input and previous inputs. To take into account memory effects, we consider the as a function $f$: $\mathbb{R}^{L}\rightarrow \mathbb{R}$ with input and output signals $x(n)$ and $y(n)$ for $n\in \mathbb{Z}$, and input memory length $L$. The input-output relation of the can be expressed as $$y(n) = f(x(n-L),\ldots, x(n)). \label{eq:pa_in_out_1}$$ Meanwhile, the is viewed as a function $g$: $\mathbb{R}^{L_1+L_2}\rightarrow \mathbb{R}$, with delayed and advanced memory length $L_1$ and $L_2$, and input signal $u(n)$ for $n\in \mathbb{Z}$, given by $$x(n) = g(u(n-L_1),\ldots, u(n+L_2)). \label{eq:dpd_in_out}$$ As shown in Fig. \[fig:dpd\_pa\_diagram\], is placed before the so as to cancel the distortion introduced by the . Assuming an ideal cancellation, i.e., the perfectly compensates for the distortion introduced by the , we then have the ideal input-output relation of the – system by substituting into , with $\boldsymbol{u}_{n-L} = [u(n-L-L_1),\ldots,u(n-L+L_2)]^{\mathsf{T}}$, $$y(n) = f(g(\boldsymbol{u}_{n-L}),\ldots, g(\boldsymbol{u}_{n})) = u(n). \label{eq:dpd-pa-ideal}$$ In this case, the cascaded – system is distortion-free. However, an ideal cancellation is infeasible in practice because of the saturation region and other non-deterministic factors such as noise. To minimize the distortion at the output of the , various behavioral models explore deterministic functions to approximate $g$ so as to make the – system as linear as possible. Let $\hat{g}$ denote the approximated function, which turns the distortion-free output $u(n)$ in  to a biased output $\hat{u}(n)$, $$\begin{aligned} y(n) &= f(\hat{g}(\boldsymbol{u}_{n-L}),\ldots, \hat{g}(\boldsymbol{u}_{n})) = \hat{u}(n). \label{eq:in-out-actual}\end{aligned}$$ To reduce the output bias, a highly accurate model of $g$ is crucial. Generalized Memory Polynomial (GMP) {#subsec:GMP} ----------------------------------- A popular form of a nonlinear, causal, and finite-memory system (e.g., the ) is described by the Volterra series because of good precision. To ease the high complexity of Volterra series, many simplified Volterra models have been investigated in the literature [@eun1997new; @kim2001digital; @GMP_2006]. In particular, the  [@GMP_2006] behavioral model has been shown to outperform many other models in terms of accuracy versus complexity [@5460970]. Assuming a model with memory depth $M$, nonlinear order $P$, cross-term length $G$, and input signal $x_{\text{in}}(n)$ at time $n$, the output of the at time $n$, $\hat{y}_{\text{out}}(n)$, gives an estimation of the actual output $y_{\text{out}}(n)$ as [@GMP_2006] $$\begin{aligned} \hat{y}_{\text{out}}(n) =& \sum_{p=0}^{P-1}\sum_{m=0}^{M}a_{pm}x_{\text{in}}(n-m)|x_{\text{in}}(n-m)|^{p}\\ &+\sum_{p=1}^{P-1}\sum_{m=0}^{M}\sum_{g=1}^{G} (b_{pmg}x_{\text{in}}(n-m)|x_{\text{in}}(n-m-g)|^{p} \\ &+ c_{pmg}x_{\text{in}}(n-m)|x_{\text{in}}(n-m+g)|^{p}) \label{eq:GMP-formula} \end{aligned}$$ where $a_{pm}$, $b_{pmg}$, and $c_{pmg}$ are complex-valued coefficients. Assuming a total number of coefficients $J$ and total number of input samples $N$, all coefficients can be collected into a $J\times1$ vector $\boldsymbol{w}$. Each element of $\boldsymbol{w}$ corresponds to a $N\times 1$ signal, e.g., coefficient $a_{32}$ corresponds to the $N$ samples signal $x_{\text{in}}(n-2)|x_{\text{in}}(n-2)|^{3}$. Therefore, we can collect these $N\times 1$ input signals into the $N\times J$ matrix $\boldsymbol{X}_{\text{in}}$. Then, can be rewritten in matrix form as $$\hat{\boldsymbol{y}}_{\text{out}} = \boldsymbol{X}_{\text{in}}\boldsymbol{w}. \label{eq:GMP-formula-matrix}$$ To solve for $\boldsymbol{w}$, the least squares algorithm is commonly used by minimizing the between the estimation $\hat{\boldsymbol{y}}_{\text{out}}$ and the observation $\boldsymbol{y}_{\text{out}}$, which gives a solution for $\boldsymbol{w}$, $$\boldsymbol{w} = (\boldsymbol{X}_{\text{in}}^\mathsf{H}\boldsymbol{X}_{\text{in}})^{-1}\boldsymbol{X}_{\text{in}}^\mathsf{H}\boldsymbol{y}_{\text{out}},$$ where $\mathsf{H}$ denotes Hermitian. In a real-time scenario, the running complexity of substantially restricts the system. Assuming $G<M+1$, reference [@5460970] computes the running complexity of , $C_{\text{GMP}}$, for each input sample in terms of the number of , $$\begin{aligned} C_{\text{GMP}} =& 8\left((M+1)(P+2PG) - \frac{G(G+1)}{2}(P-1)\right)\\ &+ 10 + 2P + 2(P-1)G + 2P\text{min}(G,M). \end{aligned}$$ Inverse Structure to identify coefficients ------------------------------------------ Before a behavioral model (e.g., the model) is used to represent the function $g$, we need to identify its coefficients. Since the optimal output signal $x(n)$ is unknown, we cannot directly identify coefficients of a model using $u(n)$ and $x(n)$. Alternatively, we can use an inverse structure, the  [@eun1997new], to indirectly identify parameters. First, an *inverse* model (also known as *post-distorter*) is identified using the output signal $y(n)$ as the input and the input signal $u(n)$ as the output. Once the the post-distorter is identified, its coefficients are copied to to an identical model (known as *pre-distorter*) which is then used as the function $g$. Although the learned post-distorter is not an optimal solution, is still the most used identification method because of simple implementation and good performance. In this paper, we consider the to identify the parameters of a model. Proposed Residual Real-Valued Time-Delay Neural Network {#section:R2TDNN} ======================================================= In this section we build a connection between the residual learning and the behavior, and then propose a residual to learn the nonlinear behavior of the . Residual learning on the PA --------------------------- The behavior consists of a linear and a nonlinear component. If we extract the linear relation, the input-output relation of the   can be rewritten as $$\begin{aligned} y(n) = x(n) + \underbrace{f(x(n-L),\ldots,x(n)) - x(n)}_{=h(x(n-L),\ldots,x(n))}. \label{eq:residual_f} \end{aligned}$$ Here, let us refer to $f(x(n-L),\ldots,x(n))$ as the original function to be learned by the , and the last two terms on the right-hand side of , i.e., $f(x(n-L),\ldots,x(n)) - x(n)$, as the residual function, which is denoted by $h(x(n-L),\ldots,x(n))$. In the field of image recognition, learning a residual function has been shown to be more effective than learning its corresponding original function [@ResNet_Kaiming]. Therefore, we hypothesize that learning the nonlinear behavior of the is easier than learning the whole behavior. We then propose a residual learning to learn the behavior, referred to as . Unlike the  [@liu2004dynamic] and its variants [@mkadem2011physically; @gotthans2014digital; @wang2018augmented] that learn the whole input-output relation of the jointly, i.e., learn the original function $f(x(n-L),\ldots,x(n))$, the proposed learns it separately as in . In particular, the residual function $h(x(n-L),\ldots,x(n))$, i.e., the nonlinear behavior, is learned by inner layers, and $x(n)$, i.e., the linear behavior, is then added to the output of the inner layers by using *shortcut* connections between input and output layers. Specifically, we adopt the identity shortcut, which performs an *identity* mapping between connected layers and introduces no extra parameters. The details of the identity shortcut in the are described in the next subsection. Architecture ------------ ![Architecture of the proposed . Fed by the real in-phase and quadrature components of the input signal, $x_{\text{in}}^{\text{I}}$ and $x_{\text{in}}^{\text{Q}}$, the gives the I and Q output signal estimations $\hat{y}_{\text{out}}^{\text{I}}$ and $\hat{y}_{\text{out}}^{\text{Q}}$. []{data-label="fig:R2TDNN_diagram"}](Figures/R2TDNN_diagram.pdf){width="1\linewidth"} The architecture of the is shown in Fig. \[fig:R2TDNN\_diagram\]. Based on the , the consists of $K$ layers. The number of neurons of layer $k$ is denoted by $D_k$. The input vector of layer $k$ is denoted by $\boldsymbol{z}_k \in \mathbb{R}^{D_{k-1}}$, which is also the output of layer $k-1$. We denote the weight matrix and bias vector of layer $k$ by $\boldsymbol{W}_{k} \in \mathbb{R}^{D_{k} \times D_{k-1}}$ and $\boldsymbol{b}_{k} \in \mathbb{R}^{D_{k}}$, respectively. We consider a real-valued , so the complex-valued input signal $x_{\text{in}}(n) = x_{\text{in}}^{\text{I}}(n) + j x_{\text{in}}^{\text{Q}}(n)$ at time instant $n$ is separated into real in-phase and quadrature components, $x_{\text{in}}^{\text{I}}(n)$ and $x_{\text{in}}^{\text{Q}}(n)$, respectively. To learn memory effects of the , the input signal of the first layer is formed by tapped delay lines, where each delay operator $z^{-1}$ yields one time instant delay, e.g., $x_{\text{in}}^{\text{I}}(n)$ to $x_{\text{in}}^{\text{I}}(n-1)$. We consider memory length $M_1$ and $M_2$ for the I and Q input components, respectively. Thus, the input signal of the first layer at time instant $n$ is given by $$\begin{aligned} \boldsymbol{z}_{\text{1}}(n) = [&x_{\text{in}}^{\text{I}}(n), x_{\text{in}}^{\text{I}}(n-1),...,x_{\text{in}}^{\text{I}}(n-M_1), \\ &x_{\text{in}}^{\text{Q}}(n), x_{\text{in}}^{\text{Q}}(n-1), ..., x_{\text{in}}^{\text{Q}}(n-M_2)], \label{eq:input_vector_R2TDNN} \end{aligned}$$ which yields $(M_1+M_2+2)$ number of neurons for the first layer. When $M_1=M_2=0$, the network neglects memory. The layer $k-1$ and $k$ are fully connected as $$\boldsymbol{z}_{k+1} = \sigma(\boldsymbol{W}_{k} \boldsymbol{z}_{k} +\boldsymbol{b}_{k}),$$ where $\sigma$ is the activation function. The output of the last layer is a $2\times1$ vector which corresponds to the in-phase and quadrature output signal estimations $\hat{y}_{\text{out}}^{\text{I}}(n)$ and $\hat{y}_{\text{out}}^{\text{Q}}(n)$ of the actual complex-valued output signal $y_{\text{out}}(n)$. To output a full range of values, the output layer is considered as a linear layer with no activation function. More importantly, we add the identity shortcut between the input and output layers. Unlike other shortcuts that fully connect two layers, as in [@ResNet_Kaiming], here the identity shortcut connection is performed between neurons. Only the two neurons fed by the current time instant input signal, i.e., $x_{\text{in}}^{\text{I}}(n)$ and $x_{\text{in}}^{\text{Q}}(n)$, are connected to the output neurons. Therefore, the output of layer $K$ can be written as $$[\hat{y}_{\text{out}}^{\text{I}}(n), \hat{y}_{\text{out}}^{\text{Q}}(n)] = [x_{\text{in}}^{\text{I}}(n), x_{\text{in}}^{\text{Q}}(n)] + \boldsymbol{W}_K \boldsymbol{z}_{K} + \boldsymbol{b}_K. \label{eq:r2tdnn_output_1}$$ Note that the last two terms on the right hand side of  represents the residual function $h$ in , whereas the identity shortcut accounts for the linear part. Computation Complexity ---------------------- The identity shortcut connection introduces no new parameters to the , and only two element-wise additions are added to the running complexity. All multiplications and additions are performed between real values, which accounts for one FLOP according to [@5460970 Table I]. The number of needed for the is $$C_{\text{R2TDNN}} = 2\sum_{k=1}^{K-1}D_k D_{k+1} + 2, \label{eq:C_R2TDNN_1}$$ where the first term is the number of for multiplication and addition operations, and the $2$ is for two addition operations contributed by the two identity shortcuts. on ---- The parameters of the can be learned through the back-propagation algorithm by minimizing the between the prediction $\hat{y}_{\text{out}}(n)$ and observation $y_{\text{out}}(n)$, $$(\boldsymbol{W}^*,\boldsymbol{b}^*) = \text{arg }\underset{\boldsymbol{W},\boldsymbol{b}}{\text{min }}\mathbb{E}[(y_{\text{out}}(n)-\hat{y}_{\text{out}}(n))^2],$$ where $\mathbb{E}[\cdot]$ denotes the expectation. Specifically, when the is used as , its parameters can be identified using the , where the output $y(n)$ and input $x(n)$ are fed to the as input $x_{\text{in}}$ and output $y_{\text{out}}$, respectively. Experimental results {#section:results} ==================== We give experimental results of applying different behavioral models to on a real . Evaluation Metrics and Measurement Setup {#section:metric_Weblab} ---------------------------------------- ### Evaluation Metrics To evaluate the performance of , the distortion level of the output signal is generally measured by the between the output signal $y(n)$ (with gain normalization) and input signal $u(n)$, and the of $y(n)$. The is defined as $$\text{NMSE} = \frac{ \sum\limits_{n}|y(n) - u(n)|^2}{\sum\limits_{n}|u(n)|^2}. \label{eq:NMSE}$$ Although the measures the all-band distortion, it can be used to represent the in-band distortion as the power of out-of-band distortion is negligible compared to the in-band distortion. The measures the ratio of the out-of-band leakage to the in-band power, and is defined as $$\text{ACPR} = \frac{\int_{\text{adj.}} |Y(f)|^2\text{d}f}{\int_{\text{ch.}} |Y(f)|^2\text{d}f}, \label{eq:ACPR}$$ where $Y(f)$ denotes the Fourier transform of the output signal. The integration in the numerator and denominator are done over the adjacent channel (the lower or upper one with a larger leakage) and the main channel, respectively. ### Measurement Setup {#section:RF_Weblab} ![Block diagram of the measurement setup. The RF WebLab is remotely accessed by the MATLAB which transmits and receives the pre-distorted and measured signals, respectively. []{data-label="fig:RF_weblab_diagram"}](Figures/weblab_block_diagram.pdf){width="0.9\linewidth"} The experimental setup is based on the RF WebLab[^1] [@landin2015weblab]. Fig. \[fig:RF\_weblab\_diagram\] illustrates how it interacts with the hardware and algorithms, e.g., . In RF WebLab, a (PXIe-5646R VST) transmitter generates analog signals based on the digital signal from MATLAB. Signals are then sent to the Gallium Nitride DUT (Cree CGH40006-TB) driven by a 40 dB linear driver. Then, after a 30 dB attenuator, the receiver obtains the output signals, and eventually measurements are sent back to the MATLAB. We then apply the proposed ,  [@GMP_2006],  [@liu2004dynamic], and  [@wang2018augmented] to with the RF WebLab setup. The learning architecture for all models is the  [@eun1997new] because of simple implementation. To identifiy coefficients, adopts the least squares algorithm, while , , and use the back-propagation algorithm with the loss function. We choose Adam [@kingma2014adam] as the optimizer with a mini-batch size of $256$. The activation function is the leaky with a slope of $0.01$ for a negative input. The input signal $u(n)$ is an signal with length $10^6$, sampling rate $200$ MHz, and signal bandwidth $10$ MHz. We consider a $50$ $\Omega$ load impedance. The measured saturation point and measurement noise variance of the are $24.1$ V ($\approx 37.6$ dBm) and $0.0033$, respectively. To test the performance on the nonlinear region, we consider an average output power of the output signal of $25.36$ dBm, where the corresponding theoretical minimum is $-40.17$ dB according to [@chani2018lower Eq. (10)], and the simulated minimum is $-50.1$ dBc.[^2] Results {#section:results_sub} ------- ### Performance versus Complexity {#subsec:complexity} Fig. \[fig:Flop\_nmse\] and Fig. \[fig:Flop\_acpr\] show the results versus the total number of for the  [@GMP_2006],  [@liu2004dynamic],  [@wang2018augmented], and the proposed . For the and , we consider two scenarios with one and three hidden layers, i.e., $K\in\{3,5\}$. We also plot the results of the  [@wang2018augmented] for $K=5$, where we consider three augmented envelop terms of the signal (amplitude and its square and cube) [@wang2018augmented Tab. II] at the input layer. , , and use the same number of neurons for each layer and identical input memory $M_1=M_2=3$. The number of increases as the number of neurons for each hidden layer increases. For the (blue circle markers), we select the best results with respect to the number of based on an exhaustive search of different values of $P$, $M$, and $G$. Although the model achieves better for a number of $< 500$, the performance flattens around $-33.41$ dB. The proposed allows to reach lower (down to $-38.0$ dB) for a number of $> 500$, i.e., the yields more accurate compensationit can find a better inverse behavior of the . Note that the gap between the and the lower bound may be due to the limitation of the and some stochastic noise, e.g., phase noise. The results in Fig. \[fig:Flop\_acpr\] illustrate similar advantages of the over the for a number of $>600$. For comparison, we also plot the performance of the in [@liu2004dynamic] and in [@wang2018augmented] for $K\in\{3,5\}$. We note that the requires a number of $>3000$ to improve the performance of . However, the proposed achieves lower and with respect to the and for a similar number of . The gain is more considerable for $K=5$ and a number of neurons per hidden layer between $5$ and $12$. ### Convergence speed comparison [X\*[4]{}c]{} & Num. FLOPs & NMSE [\[]{}dB[\]]{} & ACPR [\[]{}dBc[\]]{}\ RVTDNN [@liu2004dynamic] & $504$ & $-31.5$ & $-40.0$\ ARVTDNN [@wang2018augmented] & $720$ & $-31.3$ & $-39.5$\ R2TDNN & $506$ & $\mathbf{-33.7}$ & $\mathbf{-42.5}$ \[tab:training\_comp\_setup\] To further compare the  [@liu2004dynamic],  [@wang2018augmented], and , we plot the training and validation errors during the training procedure in Fig. \[fig:train\_val\]. Based on the same parameter setup in Section \[subsec:complexity\], we select $K=5$ and $D_2=D_3=D_4=9$. The corresponding number of , , and are given in Table \[tab:training\_comp\_setup\]. Compared to the and , the exhibits significantly faster training convergence rate, and eventually achieves lower training and validation errors. This verifies the effectiveness of the proposed residual learning on . Conclusion {#section:conclusion} ========== We applied residual learning to facilitate the learning problem of the behavior, and proposed a novel -based behavioral model, named . By adding shortcuts between the input and output layer, the proposed focus on learning the nonlinear behavior instead of learning its whole behavior. We applied different behavioral models to and evaluated the performance on a real . Results show that the proposed achieves lower and than the and previously proposed in the literature with less or similar computational complexity. Furthermore, it has a faster training convergence rate during the training procedure. Acknowledgment {#acknowledgment .unnumbered} ============== This work was supported by the Swedish Foundation for Strategic Research (SSF), grant no. I19-0021. [^1]: RF WebLab is a PA measurement setup that can be remotely accessed at [www.dpdcompetition.com](www.dpdcompetition.com) [^2]: The simulated minimum represents the of the ideal linear output signal.

--- abstract: | A quantum algorithm is [*exact*]{} if it always produces the correct answer, on any input. Coming up with exact quantum algorithms that substantially outperform the best classical algorithm has been a quite challenging task. In this paper, we present two new exact quantum algorithms for natural problems: - for the problem EXACT$_k^n$ in which we have to determine whether the sequence of input bits $x_1, \ldots, x_n$ contains exactly $k$ values $x_i=1$; - for the problem THRESHOLD$_k^n$ in which we have to determine if at least $k$ of $n$ input bits are equal to 1. author: - | Andris Ambainis, Jānis Iraids, Juris Smotrovs\ University of Latvia, Raiņa bulvāris 19, Riga, LV-1586, Latvia title: Exact quantum query complexity of EXACT and THRESHOLD --- Introduction ============ We consider quantum algorithms in the query model. The algorithm needs to compute a given Boolean function $f:\,\{0,1\}^n\to\{0,1\}$ by querying its input bits until it is able to produce the value of the function, either with certainty, or with some error probability. The complexity of the algorithm is measured as the number of queries it makes (other kinds of computation needed to produce the answer are disregarded). In the [*bounded error*]{} setting where the algorithm is allowed to give an incorrect answer with probability not exceeding a given constant $\epsilon$, $0<\epsilon<\frac12$, many efficient quantum algorithms are known, with either a polynomial speed-up over classical algorithms (e.g., [@Gro96; @Amb07; @FGG08; @RS08; @Bel12]), or, in the case of partial functions, even an exponential speed-up (e.g., [@Sim97; @Shor97]). Less studied is the [*exact*]{} setting where the algorithm must give the correct answer with certainty. Though for partial functions quantum algorithms with exponential speed-up are known (for instance, [@DJ92; @BH97]), the results for total functions up to recently have been much less spectacular: the best known quantum speed-up was just by a factor of 2. Even more, as remarked in [@MJM11], all the known algorithms achieved this speed-up by the same trick: exploiting the fact that XOR of two bits can be computed quantumly with one query, while a classical algorithm needs two queries [@DJ92; @CEMM98; @FGGS98]. A step forward was made by [@MJM11] which presented a new algorithm achieving the speed-up by a factor of 2, without using the “XOR trick”. The algorithm is for the Boolean function EXACT$_2^4$ which is true iff exactly 2 of its 4 input bits are equal to 1. It computes this function with 2 queries, while a classical (deterministic) algorithm needs 4 queries. This function can be generalized to EXACT$_k^n$ in the obvious way. Its deterministic complexity is $n$ (due to its sensitivity being $n$, see [@NS94]). [@MJM11] conjectured that its quantum query complexity is $\max{\{k,n-k\}}$. In this paper we prove the conjecture. We also solve the problem for a similar function, THRESHOLD$_k^n$ which is true iff [*at least*]{} $k$ of the input bits are equal to 1. When $n=2k-1$, this function is well-known as the MAJORITY function. The quantum query complexity of THRESHOLD$_k^n$ turns out to be $\max{\{k,n-k+1\}}$, as conjectured in [@MJM11]. In a recent work [@Amb12], a function $f(x_1, \ldots, x_n)$ with the deterministic query complexity $n$ and the exact quantum query complexity $O(n^{.8675...})$ was constructed. The quantum advantage that is achieved by our algorithms is smaller but we think that our results are still interesting, for several reasons. First, we present quantum algorithms for computational problems that are natural and simple to describe. Second, our algorithms contain new ideas which may be useful for designing other exact algorithms. Currently, the toolbox of ideas for designing exact quantum algorithms is still quite small. Expanding it is an interesting research topic. Technical Preliminaries ======================= We denote $[m]=\{1,2,\ldots,m\}$. We assume familiarity with basics of quantum computation [@NC00]. We now briefly describe the quantum query algorithm model. Let $f:\,\{0,1\}^n\to\{0,1\}$ be the Boolean function to compute, with the input bit string $x=x_1x_2\ldots x_n$. The quantum query algorithm works in a Hilbert space with some fixed basis states. It starts in a fixed starting state, then performs on it a sequence of unitary transformations $U_1$, $Q$, $U_2$, $Q$, …, $U_t$, $Q$, $U_{t+1}$. The unitary transformations $U_i$ do not depend on the input bits, while $Q$, called the [*query transformation*]{}, does, in the following way. Each of the basis states corresponds to either one or none of the input bits. If the basis state ${\left| \psi \right\rangle}$ corresponds to the $i$-th input bit, then $Q{\left| \psi \right\rangle}=(-1)^{x_i}{\left| \psi \right\rangle}$. If it does not correspond to any input bit, then $Q$ leaves it unchanged: $Q{\left| \psi \right\rangle}={\left| \psi \right\rangle}$. For convenience in computations, we denote $\hat x_i=(-1)^{x_i}$. Finally, the algorithm performs a full measurement in the standard basis. Depending on the result of the measurement, it outputs either 0 or 1 which must be equal to $f(x)$. By the principle of delayed measurement, sometimes a measurement performed in the middle of computation is equivalent to it being performed at the end of computation [@NC00]. We will use that in our algorithms, because they are most easily described as recursive algorithms with the following structure: perform unitary $U_1$, query $Q$, unitary $U_2$, then measure; depending on the result of measurement, call a smaller (by 2 input bits) instance of the algorithm. The principle of delayed measurement ensures that such recursive algorithm can be transformed by routine techniques into the commonly used query algorithm model described above. The minimum number of queries made by any quantum algorithm computing $f$ is denoted by $Q_E(f)$. We use $D(f)$ to denote the minimum number of queries used by a deterministic algorithm that computes $f$. Algorithm for EXACT =================== The function $EXACT_k^n$ is a Boolean function of $n$ variables being true iff *exactly* $k$ of the variables are equal to $1$. $$Q_E(EXACT_k^{2k}) \leq k$$ We present a recursive algorithm. When $k=0$ the algorithm returns $1$ without making any queries. Suppose $k=m$. For the recursive step we will use basis states ${\left| 0 \right\rangle}$, ${\left| 1 \right\rangle}$, …, ${\left| n \right\rangle}$ and ${\left| i,j \right\rangle}$ with $i,j\in[2m]$, $i<j$. The $i$-th input bit will be queried from the state ${\left| i \right\rangle}$. We begin in the state ${\left| 0 \right\rangle}$ and perform a unitary transformation $U_1$: $$U_1{\left| 0 \right\rangle} \rightarrow \sum_{i=1}^{2m}{\frac{1}{\sqrt{2m}}{\left| i \right\rangle}}.$$ Next we perform a query: $$\sum_{i=1}^{2m}{\frac{1}{\sqrt{2m}}{\left| i \right\rangle}} \xrightarrow{Q} \sum_{i=1}^{2m}{\frac{\hat x_i}{\sqrt{2m}}{\left| i \right\rangle}}.$$ Finally, we perform a unitary transformation $U_2$, such that $$U_2{\left| i \right\rangle} = \sum_{j>i}{\frac{1}{\sqrt{2m}}{\left| i,j \right\rangle}} - \sum_{j<i}{\frac{1}{\sqrt{2m}}{\left| j,i \right\rangle}} + \frac{1}{\sqrt{2m}}{\left| 0 \right\rangle}$$ One can verify that such a unitary transformation exists by checking the inner products: 1) for any $i\in[2m]$, $${\left\langle i \right|}U_2^\dagger U_2{\left| i \right\rangle} = \sum_{j>i}{\frac{1}{2m}} + \sum_{j<i}{\frac{1}{2m}} + \frac{1}{2m} = 1.$$ 2) for any $i,j\in[2m]$, $i\neq j$, $$\begin{split}{\left\langle j \right|}U_2^\dagger U_2{\left| i \right\rangle} = \left(\sum_{l>j}{\frac{1}{2m}{\left\langle j,l \right|}} - \sum_{l<j}{\frac{1}{2m}{\left\langle l,j \right|}} + \frac{1}{2m}{\left\langle 0 \right|}\right)\cdot \\ \left(\sum_{l>i}{\frac{1}{2m}{\left| i,l \right\rangle}} - \sum_{l<i}{\frac{1}{2m}{\left| l,i \right\rangle}} + \frac{1}{2m}{\left| 0 \right\rangle}\right) = 0 \end{split}$$ The resulting quantum state is $$\sum_{i=1}^{2m}{\frac{\hat x_i}{\sqrt{2m}}{\left| i \right\rangle}} \xrightarrow{U_2} \sum_{i=1}^{2m}{\frac{\hat x_i}{2m}{\left| 0 \right\rangle}} + \sum_{i<j}{\frac{\hat x_i-\hat x_j}{2m}{\left| i,j \right\rangle}}.$$ If we measure the state and get ${\left| 0 \right\rangle}$, then $EXACT_m^{2m}(x)=0$. If on the other hand we get ${\left| i,j \right\rangle}$, then $x_i \neq x_j$ and $EXACT_m^{2m}(x) = EXACT_{m-1}^{2m-2}(x\setminus \{x_i, x_j\})$, therefore we can use our algorithm for $EXACT_{m-1}^{2m-2}$. Note that we can delay the measurements by using ${\left| i,j \right\rangle}$ as a starting state for the recursive call of the algorithm. For the sake of completeness, we include the following corollary already given in [@MJM11]: [@MJM11] $$Q_E(EXACT_k^n) \leq \max{\{k, n-k\}}$$ Assume that $k < \frac{n}{2}$. The other case is symmetric. Then we append the input $x$ with $n-2k$ ones producing $x'$ and call $EXACT_{n-k}^{2n-2k}(x')$. Then concluding that there are $n-k$ ones in $x'$ is equivalent to there being $(n-k)-(n-2k)=k$ ones in the original input $x$. The lower bound can be established by the following fact: \[subfun\] If $g$ is a partial function such that $g(x)=f(x)$ whenever $g$ is defined on $x$, then $Q_E(g) \leq Q_E(f)$. $$Q_E(EXACT_k^n) \geq \max{\{k, n-k\}}$$ Assume that $k \leq \frac{n}{2}$. The other case is symmetric. Define $$g(x_{k+1}, \ldots, x_n) = EXACT_k^n(1, \ldots, 1, x_{k+1}, \ldots, x_n).$$ Observe that $g$ is in fact negation of the $OR$ function on $n-k$ bits which we know [@BBC+98] to take $n-k$ queries to compute. Therefore by virtue of Proposition \[subfun\] no algorithm for $EXACT_k^n$ may use less than $n-k$ queries. Algorithm for THRESHOLD ======================= We will abbreviate THRESHOLD as $Th$. The function $Th_k^n$ is a Boolean function of $n$ variables being true iff *at least* $k$ of the variables are equal to $1$. The function $Th_{k+1}^{2k+1}$ is commonly referred to as $MAJ_{2k+1}$ or $MAJORITY_{2k+1}$ because it is equal to the majority of values of input variables. Remarkably an approach similar to the one used for $EXACT$ works in this case as well. $$Q_E(MAJ_{2k+1})\leq k+1.$$ Again, a recursive solution is constructed as follows. The base case $k=0$ is trivial to perform with one query, because the function returns the value of the single variable. The recursive step $k=m$ shares the states, unitary transformation $U_1$ and the query with our algorithm for $EXACT$, but the unitary $U_2$ is slightly different: $$U_1{\left| 0 \right\rangle} \rightarrow \sum_{i=1}^{2m+1}{\frac{1}{\sqrt{2m+1}}{\left| i \right\rangle}}.$$ $$\sum_{i=1}^{2m+1}{\frac{1}{\sqrt{2m+1}}{\left| i \right\rangle}} \xrightarrow{Q} \sum_{i=1}^{2m+1}{\frac{\hat x_i}{\sqrt{2m+1}}{\left| i \right\rangle}}.$$ $$U_2{\left| i \right\rangle} = \sum_{j>i}{\frac{\sqrt{2m-1}}{2m}{\left| i,j \right\rangle}} - \sum_{j<i}{\frac{\sqrt{2m-1}}{2m}{\left| j,i \right\rangle}} + \sum_{j\neq i}{\frac{1}{2m}{\left| j \right\rangle}}.$$ The resulting state is $$\sum_{i=1}^{2m+1}{\frac{\hat x_i}{\sqrt{2m+1}}{\left| i \right\rangle}} \xrightarrow{U_2} \sum_{i=1}^{2m+1}{\sum_{j\neq i}{\frac{\hat x_j}{2m\sqrt{2m+1}}}{\left| i \right\rangle}} + \sum_{i<j}{\frac{(\hat x_i-\hat x_j)\sqrt{2m-1}}{2m\sqrt{2m+1}}{\left| i,j \right\rangle}}.$$ We perform a complete measurement. There are two kinds of outcomes: 1) If we get state ${\left| i \right\rangle}$, then either a) $x_i$ is the value in the majority which according to the polynomial $\sum_{j\neq i}{\hat x_j}$ not being zero implies that in $x\setminus \{x_i\}$ the number of ones is greater than the number of zeroes by at least 2; or b) $x_i$ is a value in the minority. In both of these cases, for all $j:j\neq i$ it is true that $MAJ_{2m+1}(x)=MAJ_{2m-1}(x \setminus \{x_i, x_j\})$. Therefore, we can solve both cases by removing $x_i$ and one other arbitrary input value and calculating majority from the remaining values. 2) If we get state ${\left| i, j \right\rangle}$, then it is even better: we know that $x_i \neq x_j$ and therefore $MAJ_{2m+1}(x)=MAJ_{2m-1}(x\setminus \{x_i, x_j\})$. If $0 < k < n$, then $$Q_E(Th_k^n)\leq \max{\{k, n-k+1\}}.$$ Assume that $k \leq \frac{n}{2}$. The other case is symmetric. Then we append the input $x$ with $n-2k+1$ ones producing $x'$ and call $MAJ_{2n-2k+1}(x')$. Then $x'$ containing at least $n-k+1$ ones is equivalent to $x$ containing at least $(n-k+1)-(n-2k+1)=k$ ones. $$Q_E(Th_k^n) \geq \max{\{k, n-k+1\}}$$ Assume that $k \leq \frac{n}{2}$. The other case is symmetric. Define $$g(x_k, x_{k+1}, \ldots, x_n) = Th_k^n(1, \ldots, 1, x_k, x_{k+1}, \ldots, x_n).$$ Observe that $g$ is in fact the $OR$ function on $n-k+1$ bits which we know [@BBC+98] takes $n-k+1$ queries to compute. Therefore by virtue of Proposition \[subfun\] no algorithm for $Th_k^n$ may use less than $n-k+1$ queries. Conclusion ========== Coming up with exact quantum algorithms that are substantially better than any classical algorithm has been a difficult open problem. Until a few months ago, no example of total Boolean function with $Q_E(f)<D(f)/2$ was known and the examples of functions with $Q_E(f)=D(f)/2$ were almost all based on one idea: applying 1-query quantum algorithm for $x_1\oplus x_2$ as a subroutine. The first exact quantum algorithm with $Q_E(f)<D(f)/2$ (for a total $f$) was constructed in [@Amb12]. However, no symmetric function with $Q_E(f)<D(f)/2$ is known. It has been proven that if $f(x)$ is a symmetric, non-constant function of $n$ variables, then $Q_E(f)\geq n/2-o(n)$ [@ZGR97; @BdW02]. In this paper, we construct exact quantum algorithms for two symmetric functions: $EXACT$ and $THRESHOLD$. Both of those algorithms achieve $Q_E(f) = D(f)/2$ (exactly or in the limit) and use new ideas. At the same time, our algorithms are quite simple and easy to understand. The main open problem is to come with more algorithmic techniques for constructing exact quantum algorithms. Computer experiments via semidefinite optimization [@MJM11] show that there are many functions for which exact quantum algorithms are better than deterministic algorithms. Yet, in many of those case, the only way to construct these algorithms is by searching the space of all quantum algorithms, using semidefinite optimization as the search tool. For example, from the calculations in [@MJM11] (based on semidefinite optimization) it is apparent that there are 3 symmetric functions of 6 variables for which $Q_E(f)=3$: $PARITY$, $EXACT_3^6$ and $EXACT_{2,4}^6$ (exactly 2 or 4 of 6 variables are equal to 1). Unlike for the first two functions, we are not aware of any simple quantum algorithm or lower bounds for $EXACT_{2,4}^6$. Based on the evidence from semidefinite optimization, we conjecture that if $n$ is even and $2k<n$ then the quantum query complexity of $EXACT_{k,n-k}^n$ is $n-k-1$. In particular, this would mean that the complexity of $EXACT_{n/2-1, n/2+1}^n$ is $\frac{n}{2}$ and this function also achieves a gap of $Q_E(f)=D(f)/2$. At the moment, we know that this conjecture is true for $k=0$ and $k=1$. 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--- author: - Giacomo Meanti - Luigi Carratino - Lorenzo Rosasco - Alessandro Rudi bibliography: - 'main.bib' title: | Kernel methods through the roof:\ handling billions of points efficiently --- Introduction ============ Background {#sec:background} ========== Reformulating kernel solvers for multi-core/multi-GPU architectures {#sec:methods} =================================================================== Large-scale experiments {#sec:exp} ======================= \[sec:large-exp\] Conclusions =========== Acknowledgments =============== This material is based upon work supported by the Center for Brains, Minds and Machines (CBMM), funded by NSF STC award CCF-1231216, and the Italian Institute of Technology. We gratefully acknowledge the support of NVIDIA Corporation for the donation of the Titan Xp GPUs and the Tesla k40 GPU used for this research. Part of this work has been carried out at the Machine Learning Genoa (MaLGa) center, Università di Genova (IT) L. R. acknowledges the financial support of the European Research Council (grant SLING 819789), the AFOSR projects FA9550-17-1-0390 and BAA-AFRL-AFOSR-2016-0007 (European Office of Aerospace Research and Development), and the EU H2020-MSCA-RISE project NoMADS - DLV-777826.

--- abstract: 'We show that at the level of formal expansions, any compact Riemannian manifold is the sphere at infinity of an asymptotically conical gradient expanding Ricci soliton.' address: - | Department of Mathematics\ University of California, Berkeley\ Berkeley, CA 94720-3840\ USA - | Department of Mathematics\ University of California, Berkeley\ Berkeley, CA 94720-3840\ USA author: - John Lott - Patrick Wilson date: 'December 6, 2016' title: Note on asymptotically conical expanding Ricci solitons --- [^1] Introduction ============ When looking at Ricci flow on noncompact manifolds, the asymptotically conical geometries are especially interesting. An asymptotically conical Riemannian manifold $(M,g_0)$ is modelled at infinity by its asymptotic cone $C(Y)$. We take the link $Y$ to be a compact manifold with Riemannian metric $h$. If $\star$ is the vertex of $C(Y)$ then the Riemannian metric on $C(Y) - \star$ is $dr^2 + r^2 h$, with $r \in (0, \infty)$. Suppose that there is a Ricci flow solution $(M, g(t))$ on $M$ that exists for all $t \ge 0$, with $g(0) = g_0$. One can analyze the large time and large distance behavior of the flow by parabolic blowdowns. With a suitable choice of basepoints, there is a subsequential blowdown limit flow $g_\infty(\cdot)$ that is defined at least on the subset of $C(Y) \times [0, \infty)$ given by $\{(r, \theta, t) \in (0, \infty) \times Y \times [0, \infty) \: : \: t \le \epsilon r^2 \}$, for some $\epsilon > 0$ [@Lott-Zhang; @(2013) Proposition 5.6]. For each $t > 0$, the metric $g_\infty(t)$ is asymptotically conical, with asymptotic cone $C(Y)$. Since $g_\infty(\cdot)$ is a blowdown limit, the simplest scenario is that it is self-similar in the sense that it is an expanding Ricci soliton flow coming out of the cone $C(Y)$. This raises the question of whether such an expanding soliton exists for arbitrary choice of $(Y, h)$. Note that the relevant expanding solitons need not be smooth and complete. For example, if $(M, g_0)$ is an asymptotically conical Ricci flat manifold, then the blowdown flow $g_\infty$ is the static Ricci flat metric on $C(Y) - \star$; this is an expanding soliton, although $C(Y)$ may not be a manifold. The equation for a gradient expanding Ricci soliton $(M, g)$, with potential $f$, is $$\label{1.1} {\operatorname{Ric}}+ {\operatorname{Hess}}(f) \: = \: - \: \frac12 \: g.$$ The main result of this paper says that any $(Y, h)$ is the sphere at infinity of an asymptotically conical gradient expanding Ricci soliton, at least at the level of formal expansions. \[1.2\] Given a compact Riemannian manifold $(Y, h)$, there is a formal solution to (\[1.1\]) on $(0,\infty) \times Y$, of the form $$\begin{aligned} \label{1.3} g \: = \: & dr^2 + r^2h + h_{0} + r^{-2}h_{2} + \cdots + r^{-2i}h_{2i} + \cdots \\ f \: = \: & - \frac14 r^2 + f_{0} + r^{-2}f_{2} + \cdots + r^{-2i}f_{2i} + \cdots, \notag\end{aligned}$$ where $h_{2i}$ is a symmetric $2$-tensor field on $Y$ and $f_{2i} \in C^\infty(Y)$. The solution is unique up to adding a constant to $f_0$. When writing (\[1.1\]) in the $(0,\infty) \times Y$ decomposition, one obtains two evolution equations and a constraint equation. The main issue in proving Theorem \[1.2\] is to show that solutions of the evolution equations automatically satisfy the constraint equation. There has been earlier work on asymptotically conical expanding solitons. 1. Schulze and Simon considered the Ricci flow on an asymptotically conical manifold with nonnegative curvature operator [@Schulze-Simon; @(2013)]. They showed that there is a long-time solution and its blowdown limit is an gradient expanding soliton solution. 2. Deruelle showed that if $(Y, h)$ is simply connected and $C(Y) - \star$ has nonnegative curvature operator then there is a smooth gradient expanding Ricci soliton $(M,g,f)$ with asymptotic cone $C(Y)$ [@Deruelle; @(2015)]. 3. In the Kähler case, the analog of Theorem \[1.2\] was proven by the first author and Zhang [@Lott-Zhang; @(2013)]. The Kähler case differs from the Riemannian case in two ways. First, in the Kähler case the Ricci soliton equation reduces to a scalar equation. Second, a Kähler cone has a natural holomorphic vector field that generates a rescaling of the complex coordinates. In [@Lott-Zhang; @(2013) Propositions 5.40 and 5.50] it was shown that there is a formal expanding soliton based on this vector field, and then that the vector field is the gradient of a soliton potential $f$. In the Riemannian case there is no [*a priori*]{} choice of vector field. Instead, we work directly with the gradient soliton equation (\[1.1\]). In what follows, we use the Einstein summation convention freely. We thank the referee for helpful comments. Soliton equations ================= Put $\dim(Y) = n$. Consider a Riemannian metric on $(0, \infty) \times Y$ given in radial form by $g = dr^2 + H(r)$. Here for each $r \in (0, \infty)$, we have a Riemannian metric $H(r)$ on $Y$. Letting $\{x^i\}_{i=1}^n$ be local coordinates for $Y$, the gradient expanding soliton equation (\[1.1\]) becomes the equations $$\begin{aligned} \label{2.1} R^g_{jk} + ({\operatorname{Hess}}_g f)_{jk} + \frac{1}{2}g_{jk} & = 0, \\ R^g_{rr} + ({\operatorname{Hess}}_g f)_{rr} + \frac{1}{2} & = 0, \notag \\ R^g_{rl} + ({\operatorname{Hess}}_g f)_{rl} & = 0. \notag\end{aligned}$$ After multiplying by $2$, these equations can be written explicitly as $$\begin{aligned} \label{2.2} & - H_{jk,rr} + 2R^H_{jk} - \frac{1}{2}H^{il}H_{il,r}H_{jk,r} + H^{il}H_{kl,r}H_{ij,r} \\ & + 2({\operatorname{Hess}}_H f)_{jk} + H_{jk,r}f_{,r} + H_{jk} = 0, \notag\end{aligned}$$ $$\label{2.3} -H^{jk}H_{jk,rr} + \frac{1}{2}H^{ij} H_{jk,r} H^{kl} H_{li,r} + 2f_{,rr} + 1 = 0$$ and $$\label{2.4} H^{im}\left(\nabla_i H_{ml,r} - \nabla_l H_{im,r}\right) + 2f_{,rl} - H^{mn}H_{nl,r}f_{,m} = 0,$$ where the covariant derivatives are with respect to the Levi-Civita connection of $H(r)$. We now write $$\begin{aligned} \label{2.5} H \: = \: & r^2h + h_{0} + r^{-2}h_{2} + \cdots + r^{-2i}h_{2i} + \cdots, \\ f \: = \: & - \frac14 r^2 + f_{0} + r^{-2}f_{2} + \cdots + r^{-2i}f_{2i} + \cdots. \notag\end{aligned}$$ We substitute (\[2.5\]) into (\[2.2\])-(\[2.4\]) and equate coefficients. Using (\[2.4\]), one finds that $f_0$ is a constant. For $i \ge 0$ we can determine $h_{2i}$ in terms of $\{h, h_0, \ldots, h_{2i-2}, f_0, \ldots, f_{2i} \}$ from (\[2.2\]), since the $H_{jk,r} f_{,r}$-term and the $H_{jk}$-term combine to give a factor of $(i+1) r^{-2i} \left( h_{2i} \right)_{jk}$. (When $i = 0$, we determine $h_0$ in terms of $h$ and $f_0$.) And we can determine $f_{2i+2}$ in terms of $\{h, h_0, \ldots, h_{2i} \}$ from (\[2.3\]), thanks to the $f_{,rr}$-term. Iterating this procedure, one finds $$\begin{aligned} \label{2.6} H_{jk} & = r^2h_{jk} -2\left[R_{jk} - (n-1)h_{jk}\right] \\ & \hspace{.5cm} + r^{-2}\left[-\Delta_{L}R_{jk} + \frac{1}{3}(\mbox{Hess}_{h}R)_{jk} + \frac{4}{3}R h_{jk} - 4R_{jk} -4(\frac{n}{3} - 1)(n-1)h_{jk}\right] \notag \\ & \hspace{.5cm} + O(r^{-4}), \notag \\ & \notag \\ f & = -\frac{1}{4}r^2 + {\operatorname{const.}}- \frac{1}{3} r^{-2} \left[\frac{ }{ } R - n(n-1) \right] \notag \\ & \hspace{.5cm} + \frac{1}{5} r^{-4} \left[ \frac{ }{ } - \Delta R - 2 |{\operatorname{Ric}}|^{2}_h + 2(3n-5)R - 4(n-2)(n-1)n \right] + O(r^{-6}), \notag\end{aligned}$$ where all geometric quantities on the right-hand side of each equation are calculated with respect to $h$. Here $\Delta_{L}$ is the Lichnerowicz Laplacian. As $f$ can be changed by a constant without affecting (\[1.1\]), we will assume for later purposes that the $r^0$-term of $f$ is $- (n-1)$. Then the asymptotic expansion is uniquely determined by $h$. By construction, the expressions that we obtain for (\[2.5\]) satisfy (\[2.2\]) and (\[2.3\]) to all orders. It remains to show that (\[2.4\]) is satisfied to all orders. Using (\[2.6\]), one can check that the left-hand side of equation (\[2.4\]) is $O \left( r^{-7} \right)$. Weighted contracted Bianchi identity ==================================== Consider a general Riemannian manifold $(M, g)$ and a function $f \in C^\infty(M)$. We can consider the triple $\left( M,g, e^{-f} {\operatorname{dvol}}_g \right)$ to be a smooth metric-measure space. The analog of the Ricci tensor for such a space is the Bakry-Emery-Ricci tensor ${\operatorname{Ric}}+ {\operatorname{Hess}}(f)$. One can ask if there is a weighted analog of the contracted Bianchi identity $\nabla^a R_{ab} = \frac{1}{2} \nabla_{b} R$, in which the Ricci tensor is replaced by the Bakry-Emery-Ricci tensor. It turns out that $$\label{3.1} \nabla^a\left( R_{ab} + \nabla_a\nabla_b f \right) - (\nabla^a f)\left(R_{ab} + \nabla_a\nabla_b f \right) = \frac{1}{2}\nabla_{b}\left( R + 2\Delta f - |\nabla f|^2 \right).$$ One recognizes $R + 2\Delta f - |\nabla f|^2$ to be Perelman’s weighted scalar curvature [@Perelman; @(2002) Section 1.3]. A slight variation of (\[3.1\]) is $$\begin{aligned} \label{3.2} & \nabla^a\left( R_{ab} + \nabla_a\nabla_b f +\frac{1}{2}g_{ab} \right) - (\nabla^a f)\left(R_{ab} + \nabla_a\nabla_b f + \frac{1}{2}g_{ab}\right) = \\ & \frac{1}{2}\nabla_{b}\left( R + 2\Delta f - |\nabla f|^2 - f \right). \notag \end{aligned}$$ A corollary is the known fact that if $(M,g,f)$ is a gradient expanding Ricci soliton then $R + 2\Delta f - |\nabla f|^2 - f$ is a constant. By adding this constant back to $f$, we can assume that the soliton satisfies $R + 2\Delta f - |\nabla f|^2 - f = 0$. Proof of Theorem \[1.2\] ======================== If we substitute an asymptotic expansion like (\[2.5\]) into (\[3.2\]) then it will be satisfied to all orders. Returning to the variables $\{r, x^1, \ldots, x^n \}$, let us write $X_{ir} = R_{ir} + \nabla_i\nabla_r f$ and $S = R + 2\Delta f - |\nabla f|^2 - f$. If we assume that equations (\[2.2\]) and (\[2.3\]) are satisfied then (\[3.2\]) gives $$\label{4.1} \nabla^i X_{ir} - (\nabla^i f) X_{ir} = \frac{1}{2}\partial_{r} S$$ and $$\label{4.2} \nabla_r X_{ir} - (\partial_r f) X_{ir} = \frac{1}{2}\partial_{i} S,$$ where the covariant derivatives on the left-hand side are with respect to the Levi-Civita connection of $g$. Rewriting in terms of covariant derivatives with respect to the Levi-Civita connection of $H(r)$, the equations become $$\label{4.3} H^{ij} \left[ \nabla_j X_{ir} - (\partial_j f) X_{ir} \right] = \frac{1}{2}\partial_{r}S$$ and $$\label{4.4} \partial_r X_{ir} - \frac12 H^{jk} H_{ki,r} X_{jr} - (\partial_rf) X_{ir} = \frac{1}{2}\partial_i S.$$ \[4.5\] If $S$ vanishes to all orders in $r^{-1}$ then $X_{ir}$ vanishes to all orders in $r^{-1}$. Suppose, by way of contradiction, that $X_{ir} = r^{-N} \phi + O \left( r^{-N-1} \right)$ for some $N \ge 1$ and some nonzero $\phi \in \Omega^1(Y)$. Using the leading order asymptotics for $H$ and $f$ from (\[2.6\]), the left-hand side of (\[4.4\]) is $\frac12 r^{-N+1} \phi_i + O \left( r^{-N} \right)$. As the right-hand side of (\[4.4\]) vanishes to all orders, we conclude that $\phi = 0$, which is a contradiction. This proves the lemma. We now prove Theorem \[1.2\]. It suffices to show that $X_{ir}$ vanishes to all orders. Suppose, by way of contradiction, that $X_{ir} = r^{-N} \phi + O \left( r^{-N-1} \right)$ for some $N \ge 1$ and some nonzero $\phi \in \Omega^1(Y)$. From Lemma \[4.5\], $S$ does not vanish to all orders. Hence $S = r^{-M} \psi + O \left( r^{-M-1} \right)$ for some $M \ge 1$ and some nonzero $\psi \in C^\infty(Y)$. Using the leading order asymptotics for $H$ and $f$ from (\[2.6\]), the left-hand side of (\[4.4\]) is $\frac12 r^{-N+1} \phi_i + O \left( r^{-N} \right)$. The right-hand side of (\[4.4\]) is $\frac12 r^{-M} \partial_i \psi + O \left( r^{-M-1} \right)$. Since $\phi$ is nonzero, we can say that $M \le N-1$. Next, the left-hand side of (\[4.3\]) is $r^{-N-2} h^{ij} \nabla_j \phi_i + O \left( r^{-N-3} \right)$, while the right-hand side of (\[4.3\]) is $- \frac12 M r^{-M-1} \psi + O \left( r^{-M-2} \right)$. Since $\psi$ is nonzero, we can say that $M \ge N+1$. This is a contradiction and proves the theorem. \[4.6\] Consider the quantities $R^g_{rr} + ({\operatorname{Hess}}_g f)_{rr} + \frac12 - \frac12 \left( R^g + 2 \triangle_g f - |\nabla f|_g^2 - f \right)$ and $R^g_{rl} + ({\operatorname{Hess}}_g f)_{rl}$. Without assuming that the gradient expanding soliton equations are satisfied, one finds that these quantities only involve first derivatives of $r$. In this sense, the vanishing of these quantities on a level set of $r$ is like the constraint equations in general relativity. As a nonasymptotic statement, if (\[2.2\]) and (\[2.3\]) hold, and the aforementioned quantities all vanish on one level set of $r$, then from (\[4.3\]) and (\[4.4\]), they vanish identically. Asymptotic expansions can also be constructed for asymptotically conical gradient shrinking solitons. The leading term in the function $f$ becomes $\frac14 r^2$. The (nonasymptotic) uniqueness, in a neighborhood of the end, was shown in [@Kotschwar-Wang; @(2015)]. [10]{} A. Deruelle, “Smoothing out positively curved metric cones by Ricci expanders”, Geom. Funct. Anal. 26, p. 188-249 (2016) B. Kotschwar and L. Wang, “ Rigidity of asymptotically conical shrinking gradient Ricci solitons”, J. Diff. Geom. 100, p. 55-108 (2015) J. Lott and Zhou Zhang, “Ricci flow on quasiprojective manifolds II”, J. Eur. Math Soc. 18, p. 1813-1854 (2016) G. Perelman, “The entropy formula for the Ricci flow and its geometric applications”, preprint, https://arxiv.org/abs/math/0211159 (2002) F. Schulze and M. Simon, “Expanding solitons with non-negative curvature operator coming out of cones”, Math. Z. 275, p. 625-639 (2013) [^1]: Research supported by NSF grants DMS-1344991, DMS-1440140 and DMS-1510192. We thank MSRI for its hospitality during the Spring 2016 program.

--- abstract: 'In this paper, we apply Osgood’s criterion from the theory of ordinary differential equations to detect finite-time singularities in a spatially flat FLRW universe in the context of a perfect fluid, a perfect fluid with bulk viscosity, and a Chaplygin and anti-Chaplygin gas. In particular, we applied Osgood’s criterion to demonstrate singularity behaviour for Type 0/big crunch singularities as well as Type II/sudden singularities. We show that in each case the choice of initial conditions is important as a certain number of initial conditions leads to finite-time, Type 0 singularities, while other precise choices of initial conditions which depend on the cosmological matter parameters and the cosmological constant can avoid such a finite-time singularity. Osgood’s criterion provides a powerful and yet simple way of deducing the existence of these singularities, and also interestingly enough, provides clues of how to eliminate singularities from certain cosmological models.' author: - | Ikjyot Singh Kohli\ isk@mathstat.yorku.ca\ York University - Department of Mathematics and Statistics bibliography: - 'sources.bib' title: 'The Osgood Criterion and Finite-Time Cosmological Singularities' --- Introduction ============ The Osgood criterion [@osgood2] is a classical criterion, due to W.F. Osgood in 1898 which gives conditions for ordinary differential equations to admit unique solutions and also to have singularities where solutions explode in a finite time. Following the conventions in [@leonja], Osgood’s criterion states a solution $x(t)$ of the initial value problem $$\begin{aligned} \label{eq:ivp1} \dot{x} &=& f \left[x(t)\right], \\ \label{eq:ivp2} x\left(t_{0}\right) &=& \xi,\end{aligned}$$ blows up in finite time if and only if $$\label{eq:osgoodcond1} \int_{\xi}^{\infty} \frac{ds}{f(s)} < \infty.$$ This condition can be derived from Barrow’s formula [@arnoldode], which says that the solution $\phi(t)$ of such an ODE is given by solving $$t - t_{0} = \int_{\xi}^{\phi(t)} \frac{ds}{f(s)}.$$ Suppose the solution becomes singular in a finite time, $t_{s}$, that is, $\phi(t_{s}) = \infty$, where $t_{s} < \infty$, then we have that: $$t_{s} = \int_{\xi}^{\infty} \frac{ds}{f(s)} + t_{0} < \infty.$$ Further, a solution to Eqs. - is unique if $$\int_{\xi}^{\phi(t)} \frac{ds}{f(s)} = \infty.$$ Certainly, this is a criterion that is of tremendous importance for physical applications, and as we show in this paper, it is of particular importance in cosmology. In cosmology, assumptions of a spatially homogeneous spacetime, allow Einstein’s field equations to be written as an autonomous system of ordinary differential equations. Divergent solutions, that is, solutions that “blow up” in a finite time are indicative of finite-time physical and curvature singularities that occur in universe models. As we show below, Osgood’s criterion provides a powerful and yet simple way of deducing the existence of these singularities, and also interestingly enough, provides clues of how to eliminate singularities from certain cosmological models. Despite this, we note that Osgood’s criterion has received very little attention in the physics and cosmology community. For this reason, we believe that the work here is unique, and could be of interest in future cosmological studies. In this paper, we will attempt to study the question of whether an expanding spatially flat Friedmann-Lemaîrte-Robertson-Walker (FLRW) universe with cosmological constant and ordinary and exotic matter will expand forever in the future or develop a finite-time singularity. It is understood that the ultimate fate of the universe depends on possible decay of dark energy in the future, and if it does not occur, the universe will expand forever [@elliscosmo]. Further, the concept of the existence of singularities in the context of general relativity has been studied quite extensively. With respect to the cosmological case, there is the fundamental singularity theorem [@elliscosmo] [@tolmanward] [@raych1955] which describes irrotational geodesic singularities, and states that if $\Lambda \leq 0$, $\mu + 3p \geq 0$, and $\mu + p > 0$ in a fluid flow where $\mu$ is the energy density of the fluid, $p$ is the pressure of the fluid, in addition to having $\dot{u} = 0$, $\omega = 0$, and $H_{0} = 0$ at some time $s_{0}$, then a spacetime singularity, where either the expansion scalar goes to zero or the shear scalar diverges occurs at a finite proper time $\tau_{0} \leq 1/H_{0}$ before $s_{0}$. Further elaborations are built upon this singularity theorem. It is of interest to note that in [@elliscosmo] five possible routes to avoid the conclusions of this singularity theorem are discussed in detail. They are a positive cosmological constant, acceleration, vorticity, an energy condition violation, or alternative gravitational equations. We refer the interested reader to [@elliscosmo] for further discussions regarding these issues. Following [@elliscosmo], we also note that singularities occur in cosmology not only in the context of FLRW models, but also for realistic anisotropic and inhomogeneous models of the universe in which the strong energy condition $\mu + 3p > 0$ is satisfied. Related to this, Penrose [@penrose1965] did pioneering work on black hole singularities producing a theorem that allowed one to predict the existence of singularities in realistic gravitational collapse cases. Hawking [@penrosehawking1970] extended these results to the cosmological context leading to the famous Hawking-Penrose singularity theorem which implied that space-time singularities are to be expected if either the universe is spatially closed or there is an ‘object’ undergoing relativistic gravitational collapse (existence of a trapped surface) or there is a point $p$ whose past null cone encounters sufficient matter that the divergence of the null rays through $p$ changes sign somewhere to the past of $p$ (i.e. there is a minimum apparent solid angle, as viewed from $p$ for small objects of given size). The theorem applies if the following four physical assumptions are made: (i) Einstein’s equations hold (with zero or negative cosmological constant), (ii) the energy density is nowhere less than minus each principal pressure nor less than minus the sum of the three principal pressures (the ‘energy condition’), (iii) there are no closed timelike curves, (iv) every timelike or null geodesic enters a region where the curvature is not specially aligned with the geodesic. (This last condition would hold in any sufficiently general physically realistic model.) Further, Bekenstein [@bekenstein1974] studied exact solutions of Einstein-conformal scalar equations and presented a class of FLRW models which contained both incoherent radiation and a homogeneous conformal scalar field which bounce and never pass through a singular state, thereby circumventing the singularity theorems by violating the energy condition. Parker and Fulling [@parkerfulling1973] considered a classical gravitational field minimally coupled to a quantized neutral scalar field possessing mass. They concluded that quantum effects can sometimes lead to the avoidance of the cosmological singularity, at least on the time scale of one Friedmann expansion. Collins and Ellis [@collinsellis1979] examined in detail the singularities that occur in Bianchi cosmologies. Barrow and Matzner [@barrowmatz1977] showed that the singularity corresponding to homogeneous and isotropic universes observationally equivalent to ours must be of simultaneous Robertson-Walker type containing only small curvature fluctuations. In an interesting application of Robertson-Walker singularities, Barrow [@barrow1977] showed that the large-scale velocity and vorticity fields required for the vortex and spinning-core theories of galaxy formation can be generated primordially in a natural way from a FLRW singularity. Related to future *finite-time* singularities that develop in FLRW models is the question of the so-called “ultimate fate of the universe”, there are several possibilities [@2014PhRvD..90f4014F] which we list in Table \[table:table1\]. *Singularity Type* *Description* *T* *K* --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------- -------- -------- Big Crunch (Type 0) $\theta, p, \mu \to \infty, l \to 0$ Strong Strong Big Rip (Type I) [@phantom1] [@2006PhRvD..74f4030F] $l, p, \mu \to \infty$ Strong Strong Sudden Singularities (Type II) [@2004CQGra..21.5619B; @2004PhLB..595....1N; @2004CQGra..21L..79B; @2004CQGra..21L.129L; @2004PhRvD..70j3522N; @2005PhRvD..71j3505D; @2005PhLB..625..184D; @2004MPLA...19.2479C; @2008PhRvD..78l3508B; @2009PhRvD..80d3518B; @2008PhRvD..78d6006N; @2010CQGra..27p5017B; @2012PhRvD..85h3527D; @1986MNRAS.223..835B; @2002CQGra..19L.101S; @2004PhRvD..69l3512G; @2010JCAP...05..034B; @2004PhRvD..70l1503F; @2010PhRvD..82l3534K; @2013PhRvD..88b3535K; @2013PhRvD..88f7301B] $l, \theta, \mu < \infty, p \to \infty$ Weak Weak Big Freeze (Type III) [@2008PhLB..659....1B] $l < \infty, \theta, \mu, p \to \infty$ Weak Strong Generalized Sudden / Big Separation (Type IV) [@2005CQGra..22.1563B] $l, \theta, \mu, p < \infty, \dot{p} \to \infty$ Weak Weak $w$-Singularities (Type V) [@2005PhRvD..71h4018S; @2009PhRvD..79f3521D; @2010PhRvD..82l4004F] $l <\infty, \mu, p \to 0, w \to \infty$ Weak Weak : A classification of known finite-time singularities. $l$ refers to the scale factor in the FLRW metric, $p$ and $\mu$ refer to the pressure and energy density of the matter content in the FLRW spacetime, $w$ is the barotropic index, and $\theta$ is the expansion scalar as defined above. Note that, sudden singularities are weak singularities where the spacetime can be extended after the singular event. The singularities are classified as being strong or weak according to the classification due to Tipler (T) [@1977PhLA...64....8T] and Krolak (K) [@1986CQGra...3..267K].[]{data-label="table:table1"} Further, so-called infinite-time singularities may also occur as well. These include Type $\infty$ or directional singularities, which are of strong type according to both [@1977PhLA...64....8T] and [@1986CQGra...3..267K]. There are also other types of non-singular future behaviour which are also of the infinite-time variety. These include the little rip [@2011PhRvD..84f3003F; @2012PhLB..708..204F], pseudo-rip [@2012PhRvD..85h3001F], and the little sibling of the big rip [@2015IJMPD..2450078B]. These are discussed in more detail in [@2014msu..book..101D] and [@2014PhRvD..90f4014F]. It is also important to note that conditions describing the avoidance of finite-time singularities have been given in [@2004PhLB..595....1N], . In this work, we will consider a spatially homogenous and isotropic $k=0$ FLRW universe with a cosmological constant and different types of matter sources. We will use Osgood’s criterion to establish when such models admit finite-time, Type 0 singularities as discussed below. The Dynamical Equations ======================= It is well-known that covariant derivative of the four-velocity vector $u_{a}$ can be decomposed as [@elliscargese] [@hervik] $$u_{a;b} = \frac{1}{3}\theta h_{ab} + \sigma_{ab} + \omega_{ab} - \dot{u}_{a} u_{b},$$ where $u_a$ is the fluid four-velocity, $\theta$ is the expansion scalar, $\sigma_{ab}$ is the shear tensor, $\omega_{ab}$ is the vorticity tensor, and $h_{ab}$ is the standard projection tensor. For spatially homogeneous universes, we can choose the four-velocity vector to be orthogonal to the space-like surfaces in the $3+1$-spacetime decomposition. With this, the shear tensor $\sigma_{ab}$, the vorticity tensor $\omega_{ab}$, and acceleration $\dot{u}_{a}$ all vanish. One then takes projected, symmetric, and trace-free components of the decomposition equation above in combination with the Einstein field equations $$R_{ab} - \frac{1}{2}Rg_{ab} + \Lambda g_{ab} = \kappa T_{ab},$$ to obtain the Raychaudhuri equation $$\label{eq:raych1} \dot{\theta} + \frac{1}{3}\theta^2 + \frac{\kappa}{2} \left(\mu + 3 p\right) - \Lambda = 0,$$ the energy density evolution equation $$\label{eq:endensity1} \dot{\mu} + \theta \left(\mu + p\right) = 0,$$ and the generalized Friedmann constraint equation $$\label{eq:friedmann1} \frac{1}{3} \theta^2 = \kappa \mu + \Lambda,$$ where $\mu$ denotes the matter energy density and $p$ denotes the matter pressure. In fact, we have that by the Friedmann equation , $$\lim_{t \to t_{s}} \theta_{t} = \infty \Rightarrow \lim_{t \to t_{s}} \frac{1}{3} \theta^2 = \infty \Rightarrow \lim_{t \to t_{s}} \left[\kappa \mu(t) + \Lambda\right] = \infty \Rightarrow \lim_{t \to t_{s}} \mu(t) = \infty.$$ Therefore, the Osgood criterion provides a way of testing for a singularity where $\theta$ and $\mu$ go to infinity in a finite time. This type of singularity is a big crunch or Type 0 singularity. In the context of singularity classification, the Type 0 singularity is considered a strong singularity [@1977PhLA...64....8T; @1986CQGra...3..267K; @2014msu..book..101D]. The advantage of considering spatially flat models is that the dynamical equations , can be reduced to a single dynamical equation via Eq. . We will therefore consider the dynamics of such models to be represented by an evolution equation for $\theta$, although, the analysis in what follows is equally as valid for an evolution equation for $\mu$. Substituting Eq. into Eq. , we obtain $$\label{eq:raych2} \dot{\theta} = -\frac{1}{2} \left(1+w\right) \left(\theta^2 - 3 \Lambda\right),$$ where $w$ is the equation of state parameter. The Existence of Finite-Time Singularities ========================================== With the main evolution equation in hand, we now demonstrate the existence of finite-time singularities. To accomplish this, we will make use of Osgood’s criterion as defined earlier. To apply Osgood’s criterion to our problem, let us consider the initial value problem $$\begin{aligned} \label{eq:raychnew1} \dot{\theta} &=& -\frac{1}{2} \left(1+w\right) \left(\theta^2 - 3 \Lambda\right), \\ \label{eq:raychnew11} \theta\left(t_{0}\right) &\equiv& \theta_{0}.\end{aligned}$$ Following the introductory section and evaluating the integral in , we obtain $$\begin{aligned} t_s - t_{0} &=& \int_{\theta_{0}}^{\infty} -\frac{2 ds}{\left(1+w\right) \left(s^2 - 3 \Lambda\right)} \nonumber \\ \label{eq:integ1} &=& \frac{1}{\sqrt{3}(1+w)\sqrt{\Lambda}} \log \left| \frac{\theta_0 - \sqrt{3}\sqrt{\Lambda}}{\theta_0 + \sqrt{3} \sqrt{\Lambda}} \right|.\end{aligned}$$ Note that $t_s \to \infty$ only if $\theta_{0} = \pm \sqrt{3} \sqrt{\Lambda}$, where the positive value is precisely the value of the de Sitter expansion. Therefore, carefully choosing the initial condition to be equal to the rate of the de Sitter expansion avoids a finite-time singularity where $\theta \to \infty$. The value of Eq. is finite and *real* if either $$\{\Lambda > 0\} \cap \{\theta_{0} > \sqrt{3} \sqrt{\Lambda}\}.$$ Hence, for choices of $\theta_{0}$ that satisfy these inequalities, the integral in Eq. is finite, and hence $\theta \to \infty$ in a finite time. For the sake of completeness, let us consider the case where $\Lambda < 0$. In this case, the integral Eq. is evaluated to be $$t_{s} - t_{0} = \frac{\pi \left(-\frac{1}{\Lambda }\right)^{3/2} \Lambda ^{3/2}-2 \tanh ^{-1}\left(\frac{\theta_{0}}{\sqrt{3} \sqrt{\Lambda }}\right)}{\sqrt{3} \sqrt{\Lambda } (w+1)}.$$ For $\Lambda < 0$, this integral is finite and real if and only if $\theta_{0} = 0$. Therefore, for this choice of initial condition, this universe model will exhibit finite-time, type 0 singularities. Further, the integral diverges if $\theta_{0} = \pm \sqrt{3} \sqrt{\Lambda}$, which, as mentioned above the positive root is that of the de Sitter expansion. Therefore, a finite-time, type 0 singularity can be avoided by choosing the initial condition to be that of the de Sitter expansion. FLRW Models With Bulk Viscosity =============================== We now briefly consider $k=0$ FLRW models with bulk viscosity. Bulk viscous models are typically used in early-universe cosmological models. The role of the bulk viscosity is to simply add a pressure term in the perfect-fluid energy-momentum tensor. The perfect-fluid energy momentum tensor which is imposed by the symmetry of FLRW models for fluid density $\mu$, pressure $p$, and four-velocity vector $u_{a}$ is given by $$T_{ab} = \left(\mu + p \right) u_{a} u_{b} + g_{ab} p.$$ By adding a bulk viscosity term, this energy-momentum tensor is now [@isk1] $$T_{ab} = \left(\mu + p\right) u_{a} u_{b} + g_{ab} p - \xi \theta h_{ab},$$ where $h_{ab}$ is the projection tensor, and $\xi$ is the bulk viscosity, and $p = w \mu$. Following [@0264-9381-12-8-015], we assume an equation of state $$\xi = \xi_{0} \mu^{m},$$ where $m$ is a constant and $\xi_{0}$ is a positive constant. With this definition in mind, let us define the total effective pressure as $$p = \tilde{p} - \xi \theta,$$ where $\tilde{p}$ is defined as $\tilde{p} = w \mu$. Then, by simply substituting this definition of $p$ into Eq. , we see that Eq. becomes $$\label{eq:newraych1} \dot{\theta} = \frac{1}{2} \left[\theta \kappa 3^{1-m} \xi_{0} \left(\frac{\theta ^2-3 \Lambda }{\kappa }\right)^m-\theta ^2 (w+1)+3 \Lambda (w+1)\right].$$ It is not possible to apply Osgood’s criterion in a general sense to Eq. , that is, for an arbitrary $m$ in Eq. . We will therefore consider only specific values for $m$ in our analysis. In fact, the choice of $m$ is very important and can affect the overall dynamics of general FLRW models. This point was extensively studied in [@1986PhLB..180..335B; @1987PhLB..183..285B; @1988NuPhB.310..743B; @1990PhLB..235...40B; @0264-9381-12-8-015; @belkhat]. We will only consider the cases $m=0$ and $m=1$ in what follows as these are physically relevant according to [@0264-9381-12-8-015] and [@belkhat]. Case: $m=0$ ----------- This case corresponds to the case of a constant bulk viscosity. The Raychaudhuri equation becomes $$\label{eq:raych0} \dot{\theta} = -\frac{\theta ^2}{2}+\frac{3 \theta \kappa \xi_{0}}{2}+\frac{3 \Lambda }{2}-\frac{\theta ^2 w}{2}+\frac{3 \Lambda w}{2},$$ where we will use $\theta(t_{0}) = \theta_{0}$ as an initial condition. Then, Osgood’s criterion becomes $$t_s - t_0 = \int_{\theta_{0}}^{\infty} -\frac{2 ds}{\left(1+w\right)s^2 - 3 \left(1+w\right) \Lambda - 3 s\kappa \xi}.$$ Evaluating this integral, we find that it is equal to $$\label{eq:integral2} t_s - t_0 = \frac{2 \log \left|\frac{\sqrt{3} \sqrt{3 k^2 \xi_0^2+4 \Lambda (w+1)^2}}{3 k \xi_0-2 \theta_0(w+1)}+1\right|-\log \left|\frac{\sqrt{3} \sqrt{3 k^2 \xi_0^2+4 \Lambda (w+1)^2}}{2 \theta_0 (w+1)-3 k \xi_0}+1\right|}{\sqrt{3} \sqrt{3 k^2 \xi_0^2+4 \Lambda (w+1)^2}}.$$ This integral only diverges for $$\label{eq:initcond1} \theta_{0} = \frac{3 \kappa \xi_0 \pm \sqrt{3} \sqrt{4(1+w)^2 \Lambda + 3 \kappa^2 \xi_0^2}}{2(1+w)}.$$ Therefore, only for the initial condition as given in Eq. , does a unique solution exist to Eq. where there is no finite-time singularity where $\theta \to \infty$. Finite-time singularities therefore exist, for initial conditions $\theta_0$ (assuming $-1 < w \leq 1$, $\xi_0 > 0$, $\Lambda >0$) if $$\theta_0>\frac{1}{2} \sqrt{\frac{9 k^2 \xi_0^2+12 \Lambda +12 \Lambda w^2+24 \Lambda w}{(w+1)^2}}+\frac{3 k \xi_0}{2 (w+1)}.$$ Case: $m=1$ ----------- The Raychaudhuri equation becomes $$\label{eq:raychbulk} \dot{\theta} = \frac{1}{2}\left(\theta^2-3\Lambda\right)\left(-1-w+\theta \xi_{0}\right), \quad -1 < w \leq 1, \quad \Lambda > 0.$$ Specifically, we now wish to apply Osgood’s criterion to the initial value problem $$\begin{aligned} \dot{\theta} &=& \frac{1}{2}\left(\theta^2-3\Lambda\right)\left(-1-w+\theta \xi_{0}\right), \\ \theta(t_0) &=& \theta_{0}.\end{aligned}$$ Osgood’s criterion then requires us to check the convergence/divergence behaviour of $$\label{eq:integral2} t_s - t_0 = \int_{\theta_{0}}^{\infty} \frac{2 ds}{\left(s^2-3 \Lambda \right) (s \xi_{0}-w-1)}.$$ Evaluating this integral, we find that it is equal to $$\begin{aligned} t_s - t_0 = -\frac{2 \sqrt{3} (w+1) \coth ^{-1}\left(\frac{\theta_0}{\sqrt{3} \sqrt{\Lambda }}\right)-3 \sqrt{\Lambda } \xi_0 \left(2 \log |-\xi_0|+\log \left|\theta_0^2-3 \Lambda \right|-2 \log |-\xi_0 \theta_0+w+1|\right)}{3 \sqrt{\Lambda } \left(-3 \Lambda \xi_0^2+w^2+2 w+1\right)}.\end{aligned}$$ The integral in has finite real values if $$\left\{0 < \Lambda < \frac{1+2w+w^2}{3\xi_{0}^2}\right\} \cap \left\{\theta_{0} > \sqrt{3} \sqrt{\Lambda}\right\},$$ or $$\left\{\Lambda \geq \frac{1 + 2w + w^2}{3 \xi_{0}^2} \right\} \cap \left\{\theta_{0} > \sqrt{3} \sqrt{\Lambda}\right\}.$$ Now, the integral in diverges if $$\theta_{0} \in \left\{\sqrt{3} \sqrt{\Lambda}\right\} \cup \left\{ \pm \sqrt{1 + 3 \Lambda} \right\} \cup \left\{\frac{1+w}{\xi_{0}} \right\}.$$ Therefore, as this calculation shows, the universe model avoids a singularity where $\theta \to \infty$ in finite time under the choice of any of these initial conditions, since with these choices the integral in Eq. diverges. For the sake of completeness, we will also evaluate the integral in for the case of a negative cosmological constant, $\Lambda < 0$. In fact, let us define $\Lambda = -\lambda^2$, where $\lambda \neq 0 \in \mathbb{R}$. We then find that the integral in Eq. has the value t\_s - t\_0 = . This has finite real values if $$\theta_0 > \frac{1+w}{\xi_0}.$$ Therefore, for these values of $\theta_0$, the universe model will admit finite-time singularities. Further, the integral in Eq. will diverge if $\theta_0$ is chosen such that $$\theta_{0} = \frac{1+w}{\xi_0}.$$ For this choice of $\theta_0$, the universe model will not admit any finite-time singularities. FLRW Models With a Chaplygin Gas ================================ In this section, we attempt to use Osgood’s criterion to determine the existence of finite-time, big crunch singularities in FLRW models with what is known as a Chaplygin gas. Unlike barotropic matter, the Chaplygin gas satisfies an exotic equation of state [@elliscosmo], $$\label{eq:chap1} p = -\frac{A}{\mu^{\alpha}}, \quad A \in \mathbb{R}, \quad 0 < \alpha \leq 1,$$ where $A < 0$ describes an anti-Chaplygin gas, and $A > 0$ describes a Chaplygin gas. Finite-time singularities as described above have been studied a number of times in the literature in the context of Chaplygin gas universes . Using Eq. in Eqs. and , we obtain the following version of Raychaudhuri’s equation: $$\label{eq:chap2} \dot{\theta} = \frac{1}{2} \left[3 \left(3^{\alpha } A \kappa \left(\frac{\theta ^2-3 \Lambda }{\kappa}\right)^{-\alpha }+\Lambda \right)-\theta ^2\right].$$ For simplicity, we will consider the case $\alpha = 1$, thereby, obtaining $$\begin{aligned} \label{eq:chap3} \dot{\theta} &=& -\frac{\left(\theta ^2-3 \Lambda \right)^2-9 A \kappa^2}{2 \left(\theta ^2-3 \Lambda \right)}, \\ \theta(t_0) &=& \theta_{0}.\end{aligned}$$ Therefore, applying Osgood’s criterion, we are to test the divergence of $$\label{eq:chaptemp} \int_{\theta_{0}}^{\infty} \frac{2 \left(s ^2-3 \Lambda \right)}{9 A \kappa^2-\left(s ^2-3 \Lambda \right)^2} ds.$$ Evaluating this integral, we get $$\label{eq:chap4} t_s - t_0 = -\frac{1}{2} \pi \sqrt{\frac{1}{3 \sqrt{A} \kappa-3 \Lambda }}+\frac{\tan ^{-1}\left(\frac{\theta_0}{\sqrt{3} \sqrt{\sqrt{A} \kappa-\Lambda }}\right)}{\sqrt{3} \sqrt{\sqrt{A} \kappa-\Lambda }}-\frac{\tanh ^{-1}\left(\frac{\theta_0}{\sqrt{3} \sqrt{\sqrt{A} \kappa+\Lambda }}\right)}{\sqrt{3} \sqrt{\sqrt{A} \kappa+\Lambda }}.$$ Real and finite solutions corresponding to Eq. for the case $A > 0$, that is for the Chaplygin gas, occur for $\Lambda > 0$ and $$\label{eq:chap6} \sqrt{A} \kappa > \Lambda, \quad 0 < \theta_{0} < \sqrt{3} \sqrt{\sqrt{A} \kappa + \Lambda}.$$ Finally, divergent solutions corresponding to Eq. occur when $$\label{eq:chap7} \theta_{0} = \sqrt{3} \sqrt{ \sqrt{A} \kappa + \Lambda}.$$ As can be seen from the above calculations, finite-time singularities where $\theta \to \infty$ occur for the conditions as described in Eq. , while such a singularity is avoided if $\theta_{0}$ is chosen such that Eq. is satisfied for $A > 0$. Therefore, the finite-time singularity is avoided for a Chaplygin gas with such an initial condition. Considering the case of an anti-Chaplygin gas, where $A < 0$, we will apply Osgood’s criterion by setting $A = -a^2$, where $a \neq 0 \in \mathbb{R}$. One can see looking at the expression in Eq. , no real solutions exist irrespective of the choice of principal function. Therefore, we are unable to determine singularity behaviour for an anti-Chaplygin gas. The physical interpretation of this result is perhaps that the expansion never diverges in this case. A further interesting case occurs in the case of a Chaplygin gas, where $A > 0$, but with a vanishing cosmological constant, $\Lambda = 0$. In this case, applying Osgood’s criterion to Eq. , we obtain: $$\label{eq:chap8} t_s - t_0 = \frac{\tan ^{-1}\left(\frac{\theta_0}{\sqrt{3} \sqrt[4]{A} \sqrt{\kappa}}\right)}{\sqrt{3} \sqrt[4]{A} \sqrt{\kappa}}-\frac{\tanh ^{-1}\left(\frac{\theta_0}{\sqrt{3} \sqrt[4]{A} \sqrt{\kappa}}\right)}{\sqrt{3} \sqrt[4]{A} \sqrt{\kappa}}.$$ As can be seen Eq. has finite and real values if $$0 < \theta_0 < \sqrt{3} \sqrt{ \sqrt{A} \kappa}.$$ Therefore, for these values of $\theta_0$, there exist finite-time singularities. Finite-time singularities for a Chaplygin gas with vanishing cosmological constant were demonstrated in [@Kamenshchik2001265]. Further, finite-time singularities can be avoided if $\theta_0$ is chosen such that $$\theta_0 \in \left\{0\right\} \cup \left\{\sqrt{3} \sqrt{ \sqrt{A} \kappa}\right\}.$$ On Sudden Singularities ======================= We now given an example of a situation where Osgood’s criterion can be used to prove the existence of Type II / Sudden Singularities as described in Table \[table:table1\] above. We will consider an anti-Chaplygin gas, with $\alpha = 1$ as per Eq. . We will further assume that the cosmological constant vanishes, $\Lambda = 0$. In the case of a sudden singularity, the pressure diverges, that is, $p \to \infty$, while the energy density and expansion vanish. To apply Osgood’s criterion to this situation, we use Eqs. , , and to formulate a new initial value problem for the pressure, $p(t)$: $$\begin{aligned} \label{eq:sudden1} \dot{p} &=& -\left(\frac{a^2 \kappa}{p}\right)^{-1/2} \sqrt{3} \kappa \left(a^2 + p^2\right), \\ \label{eq:sudden2} p(t_0) &=& p_{0},\end{aligned}$$ where for simplicity, we have denoted the anti-Chaplygin gas constant $A$, by $A = -a^2$, where $a \neq 0 \in \mathbb{R}$. Applying Osgood’s criterion to the initial value problem -, we are to evaluate $$t_s - t_0 = \int_{\theta_0}^{\infty} - \frac{\left( \frac{a^2 \kappa}{s}\right)^{1/2}}{\sqrt{3} \left(a^2 \kappa + \kappa s^2\right)} ds.$$ We find that \[eq:sudden2\] t\_s - t\_0 = --. One can easily confirm that Eq. is real and finite for all $p_0 \in \mathbb{R}$. Therefore, for all $p_0 \in \mathbb{R}$, we have that the universe model will exhibit a finite-time sudden singularity. Conclusions =========== In this paper, we have applied Osgood’s criterion to detect finite-time singularities in a spatially flat FLRW universe in the context of a perfect fluid, a perfect fluid with bulk viscosity, and a Chaplygin and anti-Chaplygin gas. In particular, we applied Osgood’s criterion to demonstrate singularity behaviour for Type 0/big crunch singularities as well as Type II/sudden singularities. We have shown that in each case the choice of initial conditions is important as a certain number of initial conditions leads to finite-time singularities, while other precise choices of initial conditions which depend on the cosmological matter parameters and the cosmological constant can avoid a finite-time singularity. As we have shown, Osgood’s criterion provides a powerful and yet simple way of deducing the existence of these singularities, and also interestingly enough, provides clues of how to eliminate singularities from certain cosmological models. For this reason, we believe that the work here is unique, and could be of interest in future cosmological studies.

--- abstract: 'Two observers, who share a pair of particles in an entangled mixed state, can use it to perform some non-bilocal measurement over another bipartite system. In particular, one can construct a specific game played by the observers against a coordinator, in which they can score better than a pair of observers who only share a classical communication channel. The existence of such a game is an operational implication of an entanglement witness.' author: - Eran Shmaya title: 'Non-Bilocal Measurement via Entangled State' --- Introduction ============ The relationship between entanglement and non-locality of quantum systems has been the subject of extensive research. The most celebrated manifestation of the non-local aspect of entanglement is Bell’s theorem [@bell], that correlations of outcomes of measurements over a pair of particles at singlet state cannot be squared with a local hidden variable model. This theorem was later extended to every pure entangled state [@gissinperes]. The case of mixed states is more challenging. Werner ([@werner]) constructed an entangled bipartite state that admits a local hidden variables model which reproduces all the statistical correlations of von-Neumann (ideal) measurements over the subsystems. In particular, the correlations of outcomes of local ideal measurements on a pair of particles at Werner’s state do not violate any Bell inequality. But it was later discovered that Werner’s states manifest some other non-local aspects [@popescu_tlprt; @popescu_density; @peres_collective]. The question therefore arises, does every entangled state manifest some aspect of non-locality? The concept of entanglement is easily generalized from pure states to mixed states. A nonnegative operator $F$ over a tensor product of Hilbert spaces is called *separable* if it can be written in the form $\Sigma_i K_i\otimes K_i'$, where $K_i,K_i'$ are nonnegative operators. A mixed state of a bipartite quantum system is called *entangled* if the corresponding density operator is non-separable. These are well defined mathematical concepts, which are somehow related to the more vague physical concept of non-locality. As mentioned above, several aspects of non-locality have been suggested in the literature. The purpose of this paper is to present a new facet of non-locality, that is manifested by *any* entangled state: Observers who share a pair of particles in that state can use it to perform non-bilocal measurement over another pair. A measurement (or a POVM measurement) is represented by a $k$-tuple of nonnegative operators $(F_1,\dots,F_k)$ such that $F_1+\dots+F_k=I$. If the state of a system is represented by the density operator $W$ and the POVM measurement $(F_1,\dots,F_k)$ is performed over the system, the outcome is $i$ with probability ${\text{tr}}(W\cdot F_i)$. Particularly important for this paper is the case $k=2$, i.e. measurements with two possible outcomes, ‘yes’ and ‘no’. We call such measurements *yes-no measurements*. A yes-no measurement is given by an operator $F$ such that $0\leq F\leq I$. If the state of a system is $W$ and the yes-no measurement $F$ is applied, the measurement’s outcome is ‘yes’ with probability ${\text{tr}}(W\cdot F)$, and ‘no’ with probability $1-{\text{tr}}(W\cdot F)$. Yes-no measurements are called effects in [@kraus]. A POVM measurement $(F_1,\dots,F_k)$ is called *local* if it can be carried out by Alice or Bob. This means that either $F_i=F_i^{(A)}\otimes I$ for each $i$ (in which case, to perform the measurement Alice alone has to perform the measurement $(F_1^{(A)},\dots,F_k^{(A)})$ on her particle), or that $F_i=I\otimes F_i^{(B)}$ for each $i$. A POVM measurement is called *bilocal* ([@nlclt_entgl]) if it can be performed by a sequence of local measurements and classical communication. Note ([@nlclt_entgl]) that the operator $F$ corresponding to a yes-no bilocal measurement is necessarily separable. In order to get some intuition about how mixed entangled states can be used to perform non-bilocal measurements, we first consider two examples. Assume that all particles have spin $\frac{1}{2}$ and that Alice and Bob share a pair of particles in a singlet state, that is given by the density operator $\rho_{\text{singlet}}=\frac{1}{2}({| 01 \rangle}{\langle 01 |}-{| 01 \rangle}{\langle 10 |}-{| 10 \rangle}{\langle 01 |}+{| 10 \rangle}{\langle 10 |})$. If they are now introduced to another pair of particles at unknown state $W$, they can use the singlet pair $\rho_{\text{singlet}}$ to teleport [@teleportation] Alice’s part of $W$ to Bob. Bob now holds a pair of particles at state $W$, to which he can apply any yes-no measurement. Thus, using the singlet pair, Alice and Bob are able to perform non-bilocal measurements over the new pair. In particular they can deduce more information about the unknown state $W$ than can a pair of observers who can only communicate classically. Consider another example. Suppose that Alice and Bob share a pair of particles at Werner’s state, which is given by the density operator $\rho_W=\frac{1}{2}\rho_{\text{singlet}}+\frac{1}{8}I$, where $I$ is the identity operator. Werner ([@werner]) showed that, even though this state is entangled, there exists a local hidden variables model that reproduces the correlations of all ideal local measurements that can be performed on it. Still, as was shown by Popescu ([@popescu_tlprt]), the non-local aspect of Werner’s state is revealed when one tries to use it in the teleportation scheme instead of the singlet. This yields teleportation with better fidelity than the maximal fidelity that can be achieved by using only classical communication. We now show how this imperfect teleportation can be used to perform some non-bilocal measurement over a pair of particles at state $W$. Assume that Alice and Bob try to transfer Alice’s part of $W$ to Bob using the teleportation scheme with Werner’s state $\rho_W$. Note that Werner’s state can be seen as a mixture of the singlet $\rho_{\text{singlet}}$ with a completely random state $\frac{1}{4}I$. Thus, with probability $0.5$ the teleportation succeeds and Bob holds a pair of particles in state $W$. With probability $0.5$ the teleportation fails, transferring the completely random spin-$\frac{1}{2}$ particle at state $\frac{1}{2}I$ to Bob. Thus after this process Bob holds a pair of particles at state $\frac{1}{2}W+\frac{1}{4}I\otimes {\text{tr}}_A(W)$, where $tr_A(W)$ is the partial trace over subsystem $A$ of $W$ (which represents the state of Bob’s part of $W$ before the measurement). Suppose now that Bob performs the yes-no measurement given by the operator $\rho_{\text{singlet}}$ on this pair. The probability to receive outcome ‘yes’ is given by $${\text{tr}}\left(\left(\frac{1}{2}W+\frac{1}{4}I\otimes {\text{tr}}_A(W)\right)\cdot\rho_{\text{singlet}}\right)= {\text{tr}}(W\cdot\rho_W).$$ Thus using local measurements and classical teleportation, Alice and Bob simulated the yes-no measurement given by the operator $\rho_W$. Since $\rho_W$ is non-separable, this is a non-bilocal measurement. Thus, the non-locality of Werner’s state is revealed by the fact that observers can use it to perform a non-bilocal measurement. The purpose of this paper is to show that *every* entangled state $\rho$ manifests this aspect of non-locality: A pair of observers who share this state can use it to perform some non-bilocal yes-no measurement. In section (\[game\]) the possibility of performing non-bilocal yes-no measurement using an entangled state is given a game theoretic interpretation: We consider game played by a pair of players, Alice and Bob, against a game coordinator, in which Alice and Bob have to guess the state of a bipartite system prepared by the coordinator. It is shown that if Alice and Bob share an entangled state they gain an advantageous guessing strategy by using the non-bilocal measurements. It is interesting to compare the result of this paper with another aspect of non-locality, namely distillation. An entangled state $\rho$ is called *distillable*, if it is possible to create ,with high probability, a singlet state from a large set of copies of $\rho$ using only local operations and classical communication. It is known ([@bennet-dist; @hor-mixed-dist]) that every pure entangled state is distillable, but there exist mixed states which cannot be distilled. These states are sometimes called *bound entangled states*. The fact that for every entangled state $\rho$ there exists a state-guessing game in which sharing $\rho$ is advantageous shows that even bound entangled states are still useful in certain situations. The link between the non-bilocal measurement presented in Section \[protocol\] and the state-guessing game presented in Section \[game\] is an entanglement witness. Entanglement witness can be viewed geometrically as a hyperplane that separates an entangled state from the convex set of separable states. It is known ([@terhal]) that every entangled state $\rho$ admits an entanglement witness and ([@geometric-bell]) that the distance between $\rho$ and the set of separable states in the Euclidian space of Hermitian operators equals the maximal violation of the corresponding “generalized Bell inequality” (see also [@reflections] for relationship between entanglement witnesses and distillation and [@key-distribution] for the use of certain entanglement witnesses to prove the presence of entanglement in order to establish a secure key distribution.) The fact that every entanglement witness gives rise to a specific game, in which the players benefit from sharing $\rho$ is an operational implication of the entanglement witness. Thus, this paper shows that the existence of an entanglement witness is not only necessary for a state to be entangled, but is also sufficient for the state to reveal non-locality. Scheme for Non-Bilocal Measurement {#protocol} ================================== In this section we describe a scheme for performing a non-bilocal measurement using a pre-prepared entangled pair $\rho$. Let $\rho$ be a non-separable density matrix over ${\mathcal{H}}_A\otimes{\mathcal{H}}_B$. Consider a pair of particles at state $\rho$ and assume that Alice has access to the particle that lives in ${\mathcal{H}}_A$ and Bob has access to the particle that lives in ${\mathcal{H}}_B$. Assume now that Alice and Bob are introduced to another pair of particles at the unknown state represented by the density matrix $W$ over ${\mathcal{H}}_A'\otimes{\mathcal{H}}_B'$, such that $\dim({\mathcal{H}}'_A)=\dim({\mathcal{H}}_A)=n$ and $\dim({\mathcal{H}}_B')=\dim({\mathcal{H}}_B)=m$. Thus, the joint state of the $4$ particles is represented by the density matrix $\rho\otimes W$ over ${\mathcal{H}}_A\otimes{\mathcal{H}}_B\otimes{\mathcal{H}}_A'\otimes{\mathcal{H}}_B'$. Alice has access to the subsystem ${\mathcal{H}}_A\otimes{\mathcal{H}}_A'$ and Bob has access to the subsystem ${\mathcal{H}}_B\otimes{\mathcal{H}}_B'$. Let $\{{| i \rangle}\},\{{| i' \rangle}\},\{{| \mu \rangle}\},\{{| \mu' \rangle}\}$ be orthogonal bases for ${\mathcal{H}}_A,{\mathcal{H}}_A',{\mathcal{H}}_B,{\mathcal{H}}_B'$ [[*resp. *]{}]{}Note that Latin indices correspond to the particles held by Alice and Greek indices correspond to the particles held by Bob. Let ${| \phi_A \rangle}=\frac{1}{\sqrt{n}}\Sigma_i{| i \rangle}\otimes{| i' \rangle}$ and ${| \phi_B \rangle}=\frac{1}{\sqrt{m}}\Sigma_\mu{| \mu \rangle}\otimes{| \mu' \rangle}$. Assume that Alice and Bob perform the yes-no measurement ${| \phi_A \rangle}{\langle \phi_A |}\otimes{| \phi_B \rangle}{\langle \phi_B |}$ on the $4$-particle system ${\mathcal{H}}_A\otimes{\mathcal{H}}_A'\otimes{\mathcal{H}}_B\otimes{\mathcal{H}}_B'$. Note that this can be done by local measurements and classical communication: Alice performs the yes-no measurement ${| \phi_A \rangle}{\langle \phi_A |}$ over ${\mathcal{H}}_A\otimes{\mathcal{H}}_A'$, Bob performs the yes-no measurement ${| \phi_B \rangle}{\langle \phi_B |}$ over ${\mathcal{H}}_B\otimes{\mathcal{H}}_B'$, and the outcome of the measurement is given by the logical conjunction of the local outcomes received by Alice and Bob (thus, classical communication is needed to establish the outcome of the measurement from the outcome of the local measurements.) One can verify that, for every density matrix $W$ over ${\mathcal{H}}_A\otimes{\mathcal{H}}_B$, $${\text{tr}}\left(({| \phi_A \rangle}{\langle \phi_A |}\otimes {| \phi_B \rangle}{\langle \phi_B |})\cdot(\rho\otimes W)\right)=\frac{1}{nm}\Sigma_{i,j,\mu,\nu}{\langle i\mu |} \rho{| j\nu \rangle}{\langle i'\mu' |}W{| j'\nu' \rangle}=\frac{1}{nm}{\text{tr}}(W\cdot\rho^t),$$ where $\rho^t$ is the transpose of $\rho$ w.r.t the basis $\{{| i\mu \rangle}\}_{i,\mu}$ of ${\mathcal{H}}_A\otimes{\mathcal{H}}_B$. Thus, this scheme effectively performs the yes-no measurement $\frac{1}{nm}\rho^t$ over $W$. But since $\rho$ is a non-separable matrix, it follows that $\frac{1}{nm}\rho^t$ is also non-separable. Thus using this scheme, Alice and Bob perform a non-separable, and, in particular non-bilocal measurement over the state $W$. A State-Guessing Game {#game} ===================== In this section we try to shed some light on the implications of the non-bilocal measurement constructed above. To do so, we describe a specific *game* that Alice and Bob play against a game coordinator, in which they can use the non-bilocal yes-no measurement $\frac{1}{nm}\rho^t$ to score better than a pair of observers who can only communicate classically. The discussion follows standard game-theoretic arguments. Let $H$ be an *entanglement witness* ([@terhal]), i.e an Hermitian operator such that ${\text{tr}}(H\cdot \rho) < 0$ but ${\text{tr}}(H\cdot D) \geq 0$ for every separable $D$. The existence of such an operator $H$ follows from the inseparability of $\rho$ and the separation theorem for convex cones ([@convex]). Let $H^t$ be the transpose of $H$ w.r.t the basis $\{{| i\mu \rangle}\}_{i,\mu}$ of ${\mathcal{H}}_A\otimes{\mathcal{H}}_B$. We can assume that $H^t=\beta W^2-\alpha W^1$ where $W^1$ and $W^2$ are density operators, and $\beta,\alpha\geq 0$. Since ${\text{tr}}(H^t)={\text{tr}}(H)\geq 0$, it follows that $\beta\geq \alpha$. Suppose that Alice and Bob are engaged in the following game: At the beginning of the game, a pair of particles is prepared by the game coordinator at state $W^1$ or $W^2$ with probabilities $\frac{\alpha}{\alpha+\beta},\frac{\beta}{\alpha+\beta}$ [[*resp. *]{}]{}The first particle is given to Alice and the second to Bob. Alice and Bob, who share a classical communication channel, know the parameters of the game (i.e $W^1,W^2,\alpha,\beta$,) and their goal is to guess which state was actually chosen. They receive payoff $+1$ for a correct guess and $-1$ for an incorrect guess. Every strategy that Alice and Bob can apply in the game corresponds to some yes-no measurement $F$ on the pair of particles: If the outcome of the measurement is ‘yes’ they guess that the state was $W^1$, if the outcome is ‘no’ they guess that the state was $W^2$. Their expected payoff is thus given by $$\begin{gathered} \frac{\alpha}{\alpha+\beta}\left ({\text{tr}}(W^1\cdot F))-{\text{tr}}(W^1\cdot(I-F)\right )\\+\frac{\beta}{\alpha+\beta}\left({\text{tr}}(W^2\cdot(I-F))-{\text{tr}}(W^2 \cdot F)\right)=\frac{\beta-\alpha}{\alpha+\beta}-\frac{2}{\alpha+\beta}{\text{tr}}(H^t\cdot F).\end{gathered}$$ If Alice and Bob can only perform local measurements and communicate classically, the yes-no measurement $F$ they employ is necessarily a separable operator, and their expected payoff is therefore no greater than $\frac{\beta-\alpha}{\alpha+\beta}$. If, on the other hand, Alice and Bob share a bipartite system at state $\rho$, they can implement the scheme described in section \[protocol\] and thus achieve a payoff $\frac{\beta-\alpha}{\alpha+\beta}-\frac{2}{nm(\alpha+\beta)}{\text{tr}}(H^t\cdot\rho^t)$. Since ${\text{tr}}(H^t\cdot\rho^t)={\text{tr}}(H\cdot\rho)<0$ this is strictly greater than the payoff they can achieve without this system. 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--- abstract: 'We prove results about the asymptotic formulae in short intervals for the average number of representations of integers of the forms $n=p_{1}^{\ell_1}+p_{2}^{\ell_2}$, with $\ell_1, \ell_2\in\{2,3\}$, $\ell_1+\ell_2\le 5$ are fixed integers, and $n=p^{\ell_1} + m^{\ell_2}$, with $\ell_1=2$ and $2\le \ell_2\le 11$ or $\ell_1=3$ and $ \ell_2=2$ are fixed integers, $p,p_1,p_2$ are prime numbers and $m$ is an integer.' address: - | Alessandro Languasco\ Università di Padova, Dipartimento di Matematica “Tullio Levi-Civita”, Via Trieste 63, 35121 Padova\ Italy - | Alessandro Zaccagnini\ Università di Parma, Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Parco Area delle Scienze, 53/a, 43124 Parma\ Italy author: - Alessandro Languasco - Alessandro Zaccagnini title: Short intervals asymptotic formulae for binary problems with prime powers --- Introduction ============ Let $N$ be a sufficiently large integer and $1\le H \le N$. In our recent papers [@LanguascoZ2016b] and [@LanguascoZ2017a] we provided suitable asymptotic formulae in short intervals for the number of representation of an integer $n$ as a sum of a prime and a prime square, as a sum of a prime and a square, as the sum of two prime squares or as a sum of a prime square and a square. In this paper we generalise the approach already used there to look for the asymptotic formulae for more difficult binary problems. To be able to formulate or statements in a precise way we need more definitions. Let $\ell_1, \ell_2\ge 1$ be integers, $$\label{density-def-c-def} \lambda: =\frac{1}{\ell_1}+\frac{1}{\ell_2}\le 1 \quad \textrm{and}\quad c(\ell_1,\ell_2):= \frac{\Gamma(1/\ell_1)\Gamma(1/\ell_2)}{\ell_1\ell_2\Gamma(\lambda)} = c(\ell_2,\ell_1).$$ Using these notations we can say that our results in [@LanguascoZ2016b] and [@LanguascoZ2017a] are about $\lambda=3/2$ and $\lambda=1$ while here we are interested in the case $\lambda\le 1$. We also recall that Suzuki [@Suzuki2017b; @Suzuki2017] has recently sharpened our results in [@LanguascoZ2017a] for the case $\lambda=3/2$. Finally let $$\label{A-def} A=A(N,d) := \exp \Bigl( d \Bigl( \frac{\log N}{\log \log N} \Bigr)^{\frac{1}{3}} \Bigr),$$ where $d$ is a real parameter (positive or negative) chosen according to need, and $$\sum_{n=N+1}^{N+H} r^{{{\prime\prime}}}_{\ell_1,\ell_2}(n), \quad \textrm{where} \quad r^{{{\prime\prime}}}_{\ell_1,\ell_2}(n) = \!\!\!\!\!\!\!\! \sum_{\substack {p_{1}^{\ell_1}+p_{2}^{\ell_2}=n\\N/A \le p_{1}^{\ell_1} , \, p_{2}^{\ell_2} \le N}} \!\!\!\!\!\!\!\! \log p_{1} \log p_{2}.$$ Due to the available estimates on primes in almost all short intervals and due to $\lambda\le 1$, we are unconditionally able to get a non-trivial result only for $\ell_1, \ell_2\in\{2,3\}$, $\ell_1+\ell_2\le 5$; in fact, since for this additive problem we can interchange the role of the prime powers involved, such a condition is equivalent to $\ell_1=2, \ell_2\in\{2,3\}$. \[thm-uncond\] Let $N\ge 2$, $1\le H \le N$ be integers. Moreover let $\ell_1=2, \ell_2\in\{2,3\}$. Then, for every ${\varepsilon}>0$, there exists $C=C({\varepsilon})>0$ such that $$\begin{aligned} \sum_{n=N+1}^{N+H} & r^{{{\prime\prime}}}_{2,\ell_2}(n) = c(2,\ell_2) H N^{\lambda-1} + {\mathcal{O}_{\ell_2}\Bigl( H N^{\lambda-1} A(N, -C({\varepsilon})) \Bigr)}, $$ uniformly for $N^{\frac32-\frac{11}{6\ell_2} +{\varepsilon}}\le H \le N^{1-{\varepsilon}}$, where $\lambda$ and $c(2,\ell_2)$ are defined in . Clearly for $\ell_2=2$ Theorem \[thm-uncond\] coincides with the result proved in [@LanguascoZ2016b], but for $\ell_2=3$ it is new. Assuming the Riemann Hypothesis (RH) holds and taking $$\label{r-def} R^{{{\prime\prime}}}_{\ell_1,\ell_2}(n)= \sum_ {p_{1}^{\ell_1}+p_{2}^{\ell_2}=n } \log p_{1} \log p_{2},$$ we get a non-trivial result for $\sum_{n = N+1}^{N + H} R^{{{\prime\prime}}}_{\ell_1,\ell_2}(n)$ uniformly for every $2\le \ell_1 \le \ell_2$ and $H$ in some range. Let further $$\label{aB-def} a(\ell_1,\ell_2):=\frac{\ell_1}{2(\ell_1-1)\ell_2}\in \Bigl(0, \frac12\Bigr] \quad \text{and} \quad b(\ell_1):=\frac{3\ell_1}{2(\ell_1-1)}\in \Bigl(\frac32,3\Bigr].$$ We use throughout the paper the convenient notation $f=\infty(g)$ for $g={{{o}_{}\left(f\right)}}$. \[thm-RH\] Let $N\ge 2$, $1\le H \le N$, $2\le \ell_1\le \ell_2$ be integers and assume the Riemann Hypothesis holds. Then $$\sum_{n = N+1}^{N + H} R^{{{\prime\prime}}}_{\ell_1,\ell_2}(n) = c(\ell_1,\ell_2)HN^{\lambda-1} +{\mathcal{O}_{\ell_1,\ell_2}\Bigl( H^2N^{\lambda-2} + H^{\frac{1}{\ell_1}} N^{\frac{1}{2\ell_2}}( \log N)^{\frac{3}{2}} \Bigr)}$$ uniformly for $\infty(N^{1-a(\ell_1,\ell_2)}( \log N)^{b(\ell_1)})\le H \le{{{o}_{}\left(N\right)}}$, where $\lambda$ and $c(\ell_1,\ell_2)$ are defined in , $a(\ell_1,\ell_2), b(\ell_1)$ are defined in . Clearly for $\ell_1=\ell_2=2$, Theorem \[thm-RH\] coincides with the result proved in [@LanguascoZ2016b] but in all the other cases it is new. To prove Theorem \[thm-RH\] we will have to use the original Hardy-Littlewood generating functions to exploit the wider uniformity over $H$ they allow; see the remark after Lemma \[LP-Lemma-gen\]. A slightly different problem is the one in which we replace a prime power with a power. Letting $$r^{\prime}_{\ell_1,\ell_2}(n) = \!\!\!\!\!\!\!\! \sum_{\substack {p^{\ell_1}+m^{\ell_2}=n\\N/A \le p^{\ell_1} , \, m^{\ell_2} \le N}} \!\!\!\!\!\!\!\! \log p,$$ we have the following \[thm-uncond-HL\] Let $N\ge 2$, $1\le H \le N$. Moreover let $\ell_1, \ell_2\ge2$. Then, for every ${\varepsilon}>0$, there exists $C=C({\varepsilon})>0$ such that $$\begin{aligned} \sum_{n=N+1}^{N+H} & r^{\prime}_{\ell_1,\ell_2}(n) = c(\ell_1,\ell_2)H N^{\lambda-1} + {\mathcal{O}_{\ell_1,\ell_2}\Bigl(H N^{\lambda-1} A(N, -C({\varepsilon})) \Bigr)}, $$ uniformly for $N^{2-\frac{11}{6\ell_1} -\frac{1}{\ell_2}+{\varepsilon}}\le H \le N^{1-{\varepsilon}}$ with $\ell_1=2$ and $2\le \ell_2\le 11$, or $\ell_1=3$ and $ \ell_2=2$, where $\lambda$ and $c(\ell_1,\ell_2)$ are defined in . Clearly for $\ell_1=\ell_2=2$, Theorem \[thm-uncond-HL\] coincides with the result proved in [@LanguascoZ2016b] but in all the other cases it is new. In this case we cannot interchange the role of the prime powers as we can do for the first two theorems we proved; hence the different condition on the size of $H$. In the conditional case, as for the proof of Theorem \[thm-RH\], we need to use the Hardy-Littlewood original functions, but in this case we are forced to restrict our analysis to the $p^\ell+m^2$ problem due to the lack of an analogue of the functional equation in the general case. It is well known that this is crucial in these problems. Letting $$R^{\prime}_{\ell,2}(n)= \sum_ {p^{\ell}+m^2=n } \log p,$$ we have the following \[thm-RH-HL\] Let $N\ge 2$, $1\le H \le N$, $\ell\ge 2$ be integers and assume the Riemann Hypothesis holds. Then $$\begin{aligned} \sum_{n = N+1}^{N + H} R^{\prime}_{\ell,2}(n) &= c(\ell,2) HN^{\frac{1}{\ell}-\frac{1}{2}} \\ &\hskip1cm + {\mathcal{O}_{\ell}\Bigl(\frac{H^2}{N^{\frac{3}{2}-\frac{1}{\ell}}}+ \frac{H N^{\frac{1}{\ell}-\frac{1}{2}}\log \log N} {(\log N)^{\frac{1}{2}}} + H^{\frac{1}{2}}N^{\frac{1}{2\ell}} \log N\Bigr)}\end{aligned}$$ uniformly for $\infty(N^{1-\frac{1}{\ell}}(\log N)^{2}) \le H \le {{{o}_{}\left(N\right)}}$, where $ c(\ell,2) $ is defined in . Clearly for $\ell =2$, Theorem \[thm-RH-HL\] coincides with the result proved in [@LanguascoZ2016b] but in all the other cases it is new. The proof of Theorem \[thm-RH-HL\] needs the use of the functional equation and hence it is different from the one of Theorem \[thm-RH\]. We finally remark that we deal with a similar problem with a $k$-th power of a prime and two squares of primes in [@LZ2018]. **Acknowledgement**. We thank the anonymous referee for his/her precise remarks. Setting ======= Let $\ell \ge 2$, $\ell_1, \ell_2\ge 2$ be integers, $e(\alpha) = e^{2\pi i\alpha}$, $\alpha \in [-1/2,1/2]$, $$\begin{aligned} \notag S_{\ell}(\alpha) &= \sum_{N/A \le m^{\ell} \le N}\Lambda(m)\ e(m^{\ell} \alpha), \quad V_{\ell}(\alpha) = \sum_{N/A \le p^{\ell} \le N} \log p\ e(p^{\ell} \alpha), \\ \label{main-defs} T_{\ell}(\alpha) &= \sum_{N/A \le m^{\ell} \le N} e(m^{\ell} \alpha ), \ f_{\ell}(\alpha) =\frac{1}{\ell}\sum_{N/A \le m\le N} m^{\frac{1}{\ell}-1}\ e(m\alpha), \\ \notag U(\alpha,H) &= \sum_{1\le m\le H}e(m\alpha),\end{aligned}$$ where $A$ is defined in . We also have the usual numerically explicit inequality $$\label{UH-estim} \vert U(\alpha,H) \vert \le \min \bigl(H; \vert \alpha\vert ^{-1}\bigr),$$ see, *e.g.*, on page 39 of Montgomery [@Montgomery1994], and, by Lemmas 2.8 and 4.1 of Vaughan [@Vaughan1997], we obtain $$\label{f-ell-T-ell-estim} f_{\ell}(\alpha) \ll_\ell \min \bigl(N^{\frac{1}{\ell}}; \vert \alpha\vert ^{-\frac{1}{\ell}}\bigr); \quad \vert T_{\ell}(\alpha) - f_{\ell}(\alpha) \vert \ll (1+\vert \alpha \vert N)^{\frac12}.$$ Recalling that ${\varepsilon}>0$, we let $L= \log N$ and $$\label{B-def} B=B(N,c,\ell_1,\ell_2)= N ^{1-\lambda} A(N, c), $$ where $\lambda$ is defined in and $c=c({\varepsilon})>0$ will be chosen later. Lemmas ====== \[UH-average\] Let $H\ge 2$, $\mu\in {\mathbb{R}}$, $\mu\ge 1$. Then $$\int_{-\frac12}^{\frac12} \vert U(\alpha,H)\vert^{\mu} \, {\mathrm{d}}\alpha \ll \begin{cases} \log H& \text{if}\ \mu=1 \\ H^{\mu-1}& \text{if}\ \mu>1. \end{cases}$$ By we can write that $$\int_{-\frac12}^{\frac12} \vert U(\alpha,H)\vert^{\mu} \, {\mathrm{d}}\alpha \ll H^\mu \int_{-\frac1H}^{\frac1H} {\mathrm{d}}\alpha + \int_{\frac1H}^{\frac12} \frac{{\mathrm{d}}\alpha}{\alpha^\mu}$$ and the result follows immediately. \[trivial-lemma\] Let $\ell >0$ be a real number. Then $ \vert S_{\ell}(\alpha)- V_{\ell}(\alpha) \vert \ll_{\ell} N^{\frac{1}{2\ell}} . $ Clearly we have $$\begin{aligned} \vert S_{\ell}(\alpha)- V_{\ell}(\alpha) \vert \le \sum_{k= 2}^{{{\mathcal{O}_{}\left(L\right)}}}\sum_{p^{k\ell}\le N} \log p\ \ll_\ell \int_{2}^{{{\mathcal{O}_{}\left(L\right)}}} N^{1/(t\ell)}\ {\mathrm{d}}t \ll_\ell N^{\frac{1}{2\ell}}, \end{aligned}$$ where in the last but one inequality we used a weak form of the Prime Number Theorem. We need the following lemma which collects the results of Theorems 3.1-3.2 of [@LanguascoZ2013b]; see also Lemma 1 of [@LanguascoZ2016a]. \[App-BCP-Gallagher\] Let $\ell > 0$ be a real number and ${\varepsilon}$ be an arbitrarily small positive constant. Then there exists a positive constant $c_1 = c_{1}({\varepsilon})$, which does not depend on $\ell$, such that $$\int_{-\frac{1}{K}}^{\frac{1}{K}} \vert S_{\ell}(\alpha) - T_{\ell}(\alpha) \vert^2 \, {\mathrm{d}}\alpha \ll_{\ell} N^{\frac{2}{\ell}-1} \Bigl( A(N, - c_{1}) + \frac{K L^{2}}{N} \Bigr),$$ uniformly for $N^{1-\frac{5}{6\ell}+{\varepsilon}}\le K \le N$. Assuming further RH we get $$\int_{-\frac{1}{K}}^{\frac{1}{K}} \vert S_{\ell}(\alpha) - T_{\ell}(\alpha) \vert^2 \, {\mathrm{d}}\alpha \ll_{\ell} \frac{N^{\frac{1}{\ell}} L^{2}}{K} + K N^{\frac{2}{\ell}-2} L^{2},$$ uniformly for $N^{1-\frac{1}{\ell}}\le K \le N$. Combining the two previous lemmas we get \[App-BCP-Gallagher-2\] Let $\ell > 0$ be a real number and ${\varepsilon}$ be an arbitrarily small positive constant. Then there exists a positive constant $c_1 = c_{1}({\varepsilon})$, which does not depend on $\ell$, such that $$\int_{-\frac{1}{K}}^{\frac{1}{K}} \vert V_{\ell}(\alpha) - T_{\ell}(\alpha) \vert^2 \, {\mathrm{d}}\alpha \ll_{\ell} N^{\frac{2}{\ell}-1} \Bigl( A(N, - c_{1}) + \frac{K L^{2}}{N} \Bigr),$$ uniformly for $N^{1-\frac{5}{6\ell}+{\varepsilon}}\le K \le N$. Assuming further RH we get $$\int_{-\frac{1}{K}}^{\frac{1}{K}} \vert V_{\ell}(\alpha) - T_{\ell}(\alpha) \vert^2 \, {\mathrm{d}}\alpha \ll_{\ell} \frac{N^{\frac{1}{\ell}} L^{2}}{K} + K N^{\frac{2}{\ell}-2} L^{2},$$ uniformly for $N^{1-\frac{1}{\ell}}\le K \le N$. By Lemma \[trivial-lemma\] we have that $$\int_{-\frac{1}{K}}^{\frac{1}{K}} \vert S_{\ell}(\alpha) - V_{\ell}(\alpha) \vert^2 \, {\mathrm{d}}\alpha \ll_\ell \frac{N^{\frac{1}{\ell}}}{K}$$ and the result follows using the inequality $\vert a + b \vert^2 \le 2 \vert a \vert^2 + 2\vert b \vert^2 $ and Lemma \[App-BCP-Gallagher\]. \[zac-lemma\] Let $\ell\ge 2$ be an integer and $0<\xi\le \frac{1}{2}$. Then $$\int_{-\xi}^{\xi} \vert T_{\ell}(\alpha) \vert ^2\, {\mathrm{d}}\alpha \ll_\ell \xi N^{\frac{1}{\ell}} + \begin{cases} L & \text{if}\ \ell =2\\ 1 & \text{if}\ \ell > 2, \end{cases}$$ $$\int_{-\xi}^{\xi} \vert S_{\ell}(\alpha) \vert ^2\, {\mathrm{d}}\alpha \ll_\ell N^{\frac{1}{\ell}} \xi L + \begin{cases} L^{2}& \text{if}\ \ell =2\\ 1 & \text{if}\ \ell > 2 \end{cases}$$ and $$\int_{-\xi}^{\xi} \vert V_{\ell}(\alpha) \vert ^2\, {\mathrm{d}}\alpha \ll_\ell N^{\frac{1}{\ell}} \xi L + \begin{cases} L^{2}& \text{if}\ \ell =2\\ 1 & \text{if}\ \ell > 2. \end{cases}$$ The first two parts were proved in Lemma 1 of [@LanguascoZ2017a]. Let’s see the third part. By symmetry we can integrate over $[0,\xi]$. We use Corollary 2 of Montgomery-Vaughan [@MontgomeryV1974] with $T=\xi$, $a_r= \log r$ if $r$ is prime, $a_r= 0$ otherwise and $\lambda_r= 2\pi r^\ell$ thus getting $$\begin{aligned} \int_{0}^{\xi} \vert V_{\ell}(\alpha) \vert ^2\, {\mathrm{d}}\alpha &= \!\!\!\! \sum_{N/A \le r^{\ell} \le N} a(r)^2 \bigl(\xi +{\mathcal{O}\bigl(\delta_r^{-1}\bigr)}\bigr) \ll_{\ell} N^{\frac{1}{\ell}} \xi L + \sum_{p^{\ell} \le N} (\log p)^2 p^{1-\ell},\end{aligned}$$ since $\delta_{r} = \lambda_r - \lambda_{r-1} \gg_{\ell} r^{\ell-1}$. The last error term is $\ll_{\ell}1$ if $\ell >2$ and $\ll L^2$ otherwise. The third part of Lemma \[zac-lemma\] follows. \[media-f-ell\] Let $\ell >0$ be a real number and recall that $A$ is defined in . Then $$\int_{-\frac{1}{2}}^{\frac{1}{2}} \vert f_{\ell}(\alpha) \vert ^2\, {\mathrm{d}}\alpha \ll_{\ell} N^{\frac{2}{\ell}-1} \begin{cases} A^{1-\frac{2}{\ell}} & \text{if}\ \ell > 2\\ \log A & \text{if}\ \ell = 2\\ 1 & \text{if}\ 0<\ell < 2. \end{cases}$$ By Parseval’s theorem we have $$\int_{-\frac12}^{\frac12} \vert f_{\ell}(\alpha) \vert ^2\, {\mathrm{d}}\alpha = \frac{1}{\ell^2} \sum_{N/A \le m \le N} m^{\frac{2}{\ell}-2}$$ and the lemma follows at once. We also need similar lemmas for the Hardy-Littlewood functions since, in the conditional case, we will use them. Let $${\widetilde{S}}_\ell(\alpha) = \sum_{n=1}^{\infty} \Lambda(n) e^{-n^{\ell}/N} e(n^{\ell}\alpha), \quad {\widetilde{V}}_\ell(\alpha) = \sum_{p=2}^{\infty} \log p \, e^{-p^{\ell}/N} e(p^{\ell}\alpha),$$ $$z= 1/N-2\pi i\alpha.$$ We remark that $$\label{z-estim} \vert z\vert ^{-1} \ll \min \bigl(N, \vert \alpha \vert^{-1}\bigr).$$ \[tilde-trivial-lemma\] Let $\ell\ge 1$ be an integer. Then $ \vert {\widetilde{S}}_{\ell}(\alpha)- {\widetilde{V}}_{\ell}(\alpha) \vert \ll_{\ell} N^{\frac{1}{2\ell}} . $ Let $\ell \ge 1$ be an integer, $N \ge 2$ and $\alpha\in [-1/2,1/2]$. Then $${\widetilde{S}}_{\ell}(\alpha) = \frac{\Gamma(1/\ell)}{\ell z^{\frac{1}{\ell}}} - \frac{1}{\ell}\sum_{\rho}z^{-\frac{\rho}{\ell}}\Gamma\Bigl(\frac{\rho}{\ell}\Bigr) + {\mathcal{O}_{\ell}\left(1\right)},$$ where $\rho=\beta+i\gamma$ runs over the non-trivial zeros of $\zeta(s)$. It follows the line of Lemma 2 of [@LanguascoZ2016a]; we just correct an oversight in its proof. In eq. (5) on page 48 of [@LanguascoZ2016a] the term $ - \sum_{m=1}^{\ell \sqrt{3}/4} \Gamma (- 2m/\ell ) z^{2m/\ell} $ is missing. Its estimate is trivially $\ll_{\ell} \vert z \vert^{\sqrt{3}/2} \ll_{\ell} 1$. Hence such an oversight does not affect the final result of Lemma 2 of [@LanguascoZ2016a]. \[Lemma 4 of [@LanguascoZ2016a]\] \[Laplace-formula\] Let $N$ be a positive integer, $z=1/N-2\pi i \alpha$, $\alpha\in [-1/2,1/2]$, and $\mu > 0$. Then $$\int_{-1 / 2}^{1 / 2} z^{-\mu} e(-n \alpha) \, {\mathrm{d}}\alpha = e^{- n / N} \frac{n^{\mu - 1}}{\Gamma(\mu)} + {\mathcal{O}_{\mu}\Bigl(\frac{1}{n}\Bigr)},$$ uniformly for $n \ge 1$. \[LP-Lemma-gen\] Let ${\varepsilon}$ be an arbitrarily small positive constant, $\ell \ge 1$ be an integer, $N$ be a sufficiently large integer and $L= \log N$. Then there exists a positive constant $c_1 = c_{1}({\varepsilon})$, which does not depend on $\ell$, such that $$\int_{-\xi}^{\xi} \, \Bigl\vert {\widetilde{S}}_\ell(\alpha) - \frac{\Gamma(1/\ell)}{\ell z^{\frac{1}{\ell}}} \Bigr\vert^{2} {\mathrm{d}}\alpha \ll_{\ell} N^{\frac{2}{\ell}-1} A(N, - c_{1}) $$ uniformly for $ 0\le \xi < N^{-1 +5/(6\ell) - {\varepsilon}}$. Assuming RH we get $$\int_{-\xi}^{\xi} \, \Bigl\vert {\widetilde{S}}_\ell(\alpha) - \frac{\Gamma(1/\ell)}{\ell z^{\frac{1}{\ell}}} \Bigr\vert^{2} {\mathrm{d}}\alpha \ll_{\ell} N^{\frac{1}{\ell}}\xi L^{2}$$ uniformly for $0 \le \xi \le \frac{1}{2}$. It follows the line of Lemma 3 of [@LanguascoZ2016a] and Lemma 1 of [@LanguascoZ2016b]; we just correct an oversight in their proofs. Both eq. (8) on page 49 of [@LanguascoZ2016a] and eq. (6) on page 423 of [@LanguascoZ2016b] should read as $$\int_{1/N}^{\xi} \Big \vert\sum_{\rho\colon \gamma > 0}z^{-\rho/\ell}\Gamma (\rho/\ell ) \Big \vert^2 {\mathrm{d}}\alpha \le \sum_{k=1}^K \int_\eta^{2\eta} \Big \vert\sum_{\rho\colon \gamma > 0}z^{-\rho/\ell}\Gamma (\rho/\ell ) \Big \vert^2 {\mathrm{d}}\alpha,$$ where $\eta=\eta_k= \xi/2^k$, $1/N\le \eta \le \xi/2$ and $K$ is a suitable integer satisfying $K={{\mathcal{O}_{}\left(L\right)}}$. The remaining part of the proofs are left untouched. Hence such oversights do not affect the final result of Lemma 3 of [@LanguascoZ2016a] and Lemma 1 of [@LanguascoZ2016b]. The main difference between Lemma \[LP-Lemma-gen\] and Lemma \[App-BCP-Gallagher-2\] is the larger uniformity over $\xi$ in the conditional estimate. Hence, under the assumption of RH, Lemma \[LP-Lemma-gen\] will allow us to avoid the unit interval splitting (see below). This will lead to milder conditions on $H$ than something like $N^{1-\frac{1}{\ell_1}}B\le H \le N$ which Lemma \[App-BCP-Gallagher-2\] would require in the conditional analogue of , for example. In conclusion, in the conditional case Lemma \[LP-Lemma-gen\] will give us a wider $H$ and $(\ell_1,\ell_2)$ ranges, while, unconditionally, Lemma \[LP-Lemma-gen\] and Lemma \[App-BCP-Gallagher-2\] are essentially equivalent. \[partial-int-average\] Let $\ell \ge 1$ be an integer, $N$ be a sufficiently large integer and $L= \log N$. Assume RH. We have $$\int_{-\frac12}^{\frac12} \, \Bigl\vert {\widetilde{S}}_\ell(\alpha) - \frac{\Gamma(1/\ell)}{\ell z^{\frac{1}{\ell}}} \Bigr\vert^{2} \vert U(-\alpha,H) \vert \ {\mathrm{d}}\alpha \ll_{\ell} N^{\frac{1}{\ell}} L^{3}.$$ Let ${\widetilde{E}}_{\ell}(\alpha) : ={\widetilde{S}}_\ell(\alpha) - \Gamma(1/\ell)/(\ell z^{\frac{1}{\ell}})$. By we have $$\begin{aligned} \notag \int_{-\frac12}^{\frac12} \, & \vert {\widetilde{E}}_{\ell}(\alpha) \vert^{2} \vert U(-\alpha,H) \vert \ {\mathrm{d}}\alpha \\ \notag &\ll H \int_{-\frac1H}^{\frac1H} \, \vert {\widetilde{E}}_{\ell}(\alpha) \vert^{2} \ {\mathrm{d}}\alpha + \int_{\frac1H}^{\frac12} \, \vert {\widetilde{E}}_{\ell}(\alpha) \vert^{2} \frac{{\mathrm{d}}\alpha}{\alpha} + \int_{-\frac12}^{-\frac1H} \, \vert {\widetilde{E}}_{\ell}(\alpha) \vert^{2} \frac{{\mathrm{d}}\alpha}{\alpha} \\ \label{M-split} & = M_1+M_2+M_3,\end{aligned}$$ say. By Lemma \[LP-Lemma-gen\] we immediately get that $$\label{M1-estim} M_1 \ll_{\ell} N^{\frac{1}{\ell}} L^{2}.$$ By a partial integration and Lemma \[LP-Lemma-gen\] we obtain $$\begin{aligned} \notag M_2 &\ll \int_{-\frac12}^{\frac12} \, \vert {\widetilde{E}}_{\ell}(\alpha) \vert^{2} \ {\mathrm{d}}\alpha + H \int_{-\frac1H}^{\frac1H} \, \vert {\widetilde{E}}_{\ell}(\alpha) \vert^{2} \ {\mathrm{d}}\alpha + \int_{\frac1H}^{\frac12} \Bigl( \int_{-\xi}^{\xi} \vert {\widetilde{E}}_{\ell}(\alpha) \vert^{2} \ {\mathrm{d}}\alpha \Bigr) \frac{\ {\mathrm{d}}\xi}{\xi^2} \\& \label{M2-estim} \ll_{\ell} N^{\frac{1}{\ell}} L^{2} + \int_{\frac1H}^{\frac12} \frac{N^{\frac{1}{\ell}} \xi L^{2}}{\xi^2}\ {\mathrm{d}}\xi \ll_{\ell} N^{\frac{1}{\ell}} L^{3}.\end{aligned}$$ A similar computation leads to $M_3 \ll_{\ell} N^{\frac{1}{\ell}} L^{3}$. By -, the lemma follows. Proof of Theorem \[thm-uncond\] =============================== By now we let $2\le \ell_1\le \ell_2$; we’ll see at the end of the proof how the conditions in the statement of this theorem follow. Assume $H>2B$. We have $$\begin{aligned} \notag \sum_{n=N+1}^{N+H} & r^{{{\prime\prime}}}_{\ell_1,\ell_2}(n) = \int_{-\frac12}^{\frac12} V_{\ell_1}(\alpha) V_{\ell_2}(\alpha) U(-\alpha,H)e(-N\alpha) \, {\mathrm{d}}\alpha \\ \notag &= \int_{-\frac{B}{H}}^{\frac{B}{H}} V_{\ell_1}(\alpha) V_{\ell_2}(\alpha) U(-\alpha,H)e(-N\alpha) \, {\mathrm{d}}\alpha \\ \label{dissect} & \hskip1cm + \int\limits_{I(B,H)} \!\!\!\!\! V_{\ell_1}(\alpha) V_{\ell_2}(\alpha) U(-\alpha,H)e(-N\alpha) \, {\mathrm{d}}\alpha, $$ where $I(B,H):=[-1/2,-B/H]\cup [B/H, 1/2]$. By the Cauchy-Schwarz inequality we have $$\begin{aligned} \int\limits_{I(B,H)} \!\!\! V_{\ell_1}(\alpha) V_{\ell_2}(\alpha) U(-\alpha,H)e(-N\alpha) \, {\mathrm{d}}\alpha & \ll \Bigl( \int\limits_{I(B,H)} \!\!\! \vert V_{\ell_1}(\alpha) \vert ^{2} \vert U(-\alpha,H) \vert \, {\mathrm{d}}\alpha \Bigr)^{\frac12} \\& \times \Bigl( \int\limits_{I(B,H)} \!\!\! \vert V_{\ell_2}(\alpha) \vert ^{2} \vert U(-\alpha,H) \vert \, {\mathrm{d}}\alpha \Bigr)^{\frac12}.\end{aligned}$$ By , Lemma \[zac-lemma\] and a partial integration argument, it is clear that $$\begin{aligned} \notag \int\limits_{I(B,H)} \!\!\! \vert V_{\ell}(\alpha) \vert ^{2} &\vert U(-\alpha,H) \vert \, {\mathrm{d}}\alpha \ll \int_{\frac{B}{H}}^{\frac12} \vert V_{\ell}(\alpha) \vert ^{2} \frac{{\mathrm{d}}\alpha}{\alpha} \\ \label{V-average} & \ll_\ell N^{\frac{1}{\ell}} L + \frac{H L^{2}}{B} + \int_{\frac{B}{H}}^{\frac12} (\xi N^{\frac{1}{\ell}}L + L^{2})\ \frac{{\mathrm{d}}\xi}{\xi^2} \ll_\ell N^{\frac{1}{\ell}} L^{2} + \frac{H L^{2}}{B},\end{aligned}$$ for every $\ell\ge 2$. Hence, recalling , we obtain $$\begin{aligned} \notag \int\limits_{I(B,H)} \!\!\!\! V_{\ell_1}(\alpha) V_{\ell_2}(\alpha) U(-\alpha,H)e(-N\alpha) \, {\mathrm{d}}\alpha & \ll_{\ell_1,\ell_2} N^{\frac{\lambda}{2}} L^{2} + \frac{ H^{\frac12} N^{\frac{1}{2\ell_1}} L^{2}}{B^{\frac12}} +\frac{H L^{2}}{B} \\& \label{V-average-coda} \ll_{\ell_1,\ell_2} \frac{H L^{2}}{B}.\end{aligned}$$ By and we get $$\begin{aligned} \notag \sum_{n=N+1}^{N+H} &r^{{{\prime\prime}}}_{\ell_1,\ell_2}(n) = \int_{-\frac{B}{H}}^{\frac{B}{H}} V_{\ell_1}(\alpha) V_{\ell_2}(\alpha) U(-\alpha,H)e(-N\alpha) \, {\mathrm{d}}\alpha + {\mathcal{O}_{\ell_1,\ell_2}\Bigl(\frac{H L^{2}}{B}\Bigr)} \\ \notag & = \int_{-\frac{B}{H}}^{\frac{B}{H}} f_{\ell_1}(\alpha) f_{\ell_2}(\alpha) U(-\alpha,H)e(-N\alpha) \, {\mathrm{d}}\alpha \\ \notag & \hskip0.5cm + \int_{-\frac{B}{H}}^{\frac{B}{H}} f_{\ell_2}(\alpha) (V_{\ell_1}(\alpha) - f_{\ell_1}(\alpha) ) U(-\alpha,H)e(-N\alpha) \, {\mathrm{d}}\alpha \\ \notag & \hskip0.5cm + \int_{-\frac{B}{H}}^{\frac{B}{H}} f_{\ell_1}(\alpha) (V_{\ell_2}(\alpha) - f_{\ell_2}(\alpha) ) U(-\alpha,H)e(-N\alpha) \, {\mathrm{d}}\alpha \\ \notag & \hskip0.5cm + \int_{-\frac{B}{H}}^{\frac{B}{H}} (V_{\ell_1}(\alpha)-f_{\ell_1}(\alpha) ) (V_{\ell_2}(\alpha) - f_{\ell_2}(\alpha) ) U(-\alpha,H)e(-N\alpha) \, {\mathrm{d}}\alpha \\ \notag & \hskip0.5cm+ {\mathcal{O}_{\ell_1,\ell_2}\Bigl(\frac{H L^{2}}{B}\Bigr)} \\ \label{main-dissection} &= {\mathcal{I}}_1 +{\mathcal{I}}_2 + {\mathcal{I}}_3 + {\mathcal{I}}_4 + E, \end{aligned}$$ say. We now evaluate these terms. Computation of the main term ${\mathcal{I}}_1$ {#comp-main-term} ---------------------------------------------- Recalling Definition and that $I(B,H)=[-1/2,-B/H]\cup [B/H, 1/2]$, a direct calculation and give $$\begin{aligned} \notag {\mathcal{I}}_1 & = \sum_{n=1}^H \int_{-\frac12}^{\frac12} f_{\ell_1}(\alpha) f_{\ell_2}(\alpha) e(-(n+N)\alpha)\, {\mathrm{d}}\alpha +{\mathcal{O}_{\ell_1,\ell_2}\Bigl( \int\limits_{I(B,H)} \!\!\! \frac{ {\mathrm{d}}\alpha}{ \vert \alpha \vert^{1+\lambda}}\Bigr)} \\ \notag & = \frac{1}{\ell_1\ell_2} \sum_{n=1}^H \sum_{\substack {m_{1}+m_{2} =n+N\\ N/A \le m_{1} \le N \\ N/A \le m_{2}\le N}} m_1^{\frac{1}{\ell_1}-1}m_2^{\frac{1}{\ell_2}-1} +{\mathcal{O}_{\ell_1,\ell_2}\Bigl( \Bigl(\frac{H}{B}\Bigr)^{\lambda}\Bigr)} \\ \label{I1-eval} & = M_{\ell_1,\ell_2}(H,N) +{\mathcal{O}_{\ell_1,\ell_2}\Bigl( \Bigl(\frac{H}{B}\Bigr)^{\lambda}\Bigr)}, \end{aligned}$$ say. Recalling Lemma 2.8 of Vaughan [@Vaughan1997] we can see that order of magnitude of the main term $M_{\ell_1,\ell_2}(H,N)$ is $ HN^{\lambda-1}$. We first complete the range of summation for $m_1$ and $m_2$ to the interval $[1, N]$. The corresponding error term is $$\begin{aligned} &\ll_{\ell_1, \ell_2} \!\! \sum_{n=1}^H \sum_{\substack {m_{1}+m_{2} =n+N\\ 1 \le m_{1} \le N/A \\ 1 \le m_{2}\le N}} m_1^{\frac{1}{\ell_1}-1}m_2^{\frac{1}{\ell_2}-1} \ll_{\ell_1, \ell_2} \!\! \sum_{n=1}^H \sum_{m=1}^{N/A} m^{\frac{1}{\ell_2}-1} (n+N-m)^{\frac{1}{\ell_1}-1} \\ &\ll_{\ell_1, \ell_2} H N^{\frac{1}{\ell_1}-1} \sum_{m=1}^{N/A} m^{\frac{1}{\ell_2}-1} \ll_{\ell_1, \ell_2} \!\! H N^{\lambda-1} A^{- \frac{1}{\ell_2}}.\end{aligned}$$ We deal with the main term $M_{\ell_1,\ell_2}(H,N)$ using Lemma 2.8 of Vaughan [@Vaughan1997], which yields the $\Gamma$ factors hidden in $c(\ell_1,\ell_2)$: $$\begin{aligned} \frac{1}{\ell_1\ell_2} & \sum_{n=1}^H \sum_{\substack {m_{1}+m_{2} =n+N\\ 1\le m_{1} \le N \\ 1 \le m_{2}\le N}} m_1^{\frac{1}{\ell_1}-1}m_2^{\frac{1}{\ell_2}-1} = \frac{1}{\ell_1\ell_2} \sum_{n=1}^H \sum_{m=1}^{N} m^{\frac{1}{\ell_2}-1}(n+N-m)^{\frac{1}{\ell_1}-1} \\ &= c(\ell_1,\ell_2) \sum_{n=1}^H \Bigl[ (n+N)^{\lambda-1} + {{\mathcal{O}_{}\left((n+N)^{\frac{1}{\ell_1}-1} + N^{\frac{1}{\ell_2}-1}n^{\frac{1}{\ell_1}}\right)}} \Bigr] \\ &= c(\ell_1,\ell_2) \sum_{n=1}^H (n+N)^{\lambda-1} + {\mathcal{O}_{\ell_1,\ell_2}\Bigl(H N^{\frac{1}{\ell_1}-1} +H^{\frac{1}{\ell_1}+1} N^{\frac{1}{\ell_2}-1}\Bigr)} \\ &= c(\ell_1,\ell_2) HN^{\lambda-1} +{\mathcal{O}_{\ell_1,\ell_2}\Bigl( H^2N^{\lambda-2} + H N^{\frac{1}{\ell_1}-1} +H^{\frac{1}{\ell_1}+1} N^{\frac{1}{\ell_2}-1}\Bigr)}.\end{aligned}$$ Summing up, $$\begin{aligned} \notag M_{\ell_1,\ell_2}(H,N) &= c(\ell_1,\ell_2) HN^{\lambda-1} \\& \label{main-term} +{\mathcal{O}_{\ell_1,\ell_2}\Bigl( H^2N^{\lambda-2} + H N^{\frac{1}{\ell_1}-1} +H^{\frac{1}{\ell_1}+1} N^{\frac{1}{\ell_2}-1} +\frac{HN^{\lambda-1}}{ A^{\frac{1}{\ell_2}}}\Bigr)}.\end{aligned}$$ Estimate of ${\mathcal{I}}_2$ {#estim-I2} ----------------------------- Using we obtain $$\begin{aligned} \notag \vert V_{\ell}(\alpha) - f_{\ell}(\alpha) \vert &\le \vert V_{\ell}(\alpha) - T_{\ell}(\alpha) \vert + \vert T_{\ell}(\alpha) - f_{\ell}(\alpha) \vert \\& \label{V-approx} = \vert V_{\ell}(\alpha) - T_{\ell}(\alpha) \vert + {\mathcal{O}\bigl((1+\vert \alpha \vert N)^{\frac12}\bigr)}.\end{aligned}$$ Hence $$\begin{aligned} \notag {\mathcal{I}}_2 & \ll \int_{-\frac{B}{H}}^{\frac{B}{H}} \vert f_{\ell_2}(\alpha) \vert \vert V_{\ell_1}(\alpha) -T_{\ell_1}(\alpha) \vert\vert U(-\alpha,H)\vert \, {\mathrm{d}}\alpha \\& \notag \hskip1cm + \int_{-\frac{B}{H}}^{\frac{B}{H}} \vert f_{\ell_2}(\alpha) \vert (1+\vert \alpha \vert N)^{\frac12} \vert U(-\alpha,H) \vert \, {\mathrm{d}}\alpha \\ \label{I2-split} &= E_1+E_2, \end{aligned}$$ say. By we have $$\begin{aligned} E_2 &\ll H \int_{-\frac{1}{N}}^{\frac{1}{N}} \vert f_{\ell_2}(\alpha)\vert \, {\mathrm{d}}\alpha + H N^{\frac12} \int_{\frac{1}{N}}^{\frac{1}{H}} \vert f_{\ell_2}(\alpha)\vert \alpha^{\frac12} \, {\mathrm{d}}\alpha \\& \hskip1cm + N^{\frac12}\int_{\frac{1}{H}}^{\frac{B}{H}} \vert f_{\ell_2}(\alpha)\vert\alpha^{-\frac12} \, {\mathrm{d}}\alpha.\end{aligned}$$ Hence, using the Cauchy-Schwarz inequality and Lemma \[media-f-ell\], we get $$\begin{aligned} \notag E_2 & \ll_{ \ell_2} H N^{-\frac12} \Bigl(\int_{-\frac{1}{N}}^{\frac{1}{N}} \vert f_{\ell_2}(\alpha)\vert^2\ {\mathrm{d}}\alpha\Bigr)^{\frac{1}{2}} \\& \notag \hskip1cm + H N^{\frac12} \Bigl(\int_{\frac{1}{N}}^{\frac{1}{H}} \vert f_{\ell_2}(\alpha)\vert^2\ {\mathrm{d}}\alpha\Bigr)^{\frac{1}{2}} \Bigl(\int_{\frac{1}{N}}^{\frac{1}{H}} \alpha \, {\mathrm{d}}\alpha\Bigr)^{\frac{1}{2}} \\ \notag & \hskip1cm + N^{\frac12} \Bigl(\int_{\frac{1}{H}}^{\frac{B}{H}} \vert f_{\ell_2}(\alpha)\vert^2\ {\mathrm{d}}\alpha\Bigr)^{\frac{1}{2}} \Bigl(\int_{\frac{1}{H}}^{\frac{B}{H}} \frac{{\mathrm{d}}\alpha}{\alpha}\Bigr)^{\frac{1}{2}} \\ \notag & \ll_{ \ell_2} \bigl( H N^{\frac{1}{\ell_2}-1} + N^{\frac{1}{\ell_2}} + N^{\frac{1}{\ell_2}} L^{\frac{1}{2}} \bigr) A^{\frac{1}{2}-\frac{1}{\ell_2}} (\log A)^{1/2} \\ \label{E2-estim} & \ll_{ \ell_2} N^{\frac{1}{\ell_2}} A^{\frac{1}{2}-\frac{1}{\ell_2}} L^{\frac{1}{2}} (\log A)^{1/2},\end{aligned}$$ where $A$ is defined in . Using , the Cauchy-Schwarz inequality, and Lemmas \[App-BCP-Gallagher-2\] and \[media-f-ell\] we obtain $$\begin{aligned} \notag E_1&\ll H \Bigl(\int_{-\frac{B}{H}}^{\frac{B}{H}} \vert f_{\ell_2}(\alpha)\vert^2\ {\mathrm{d}}\alpha\Bigr)^{\frac{1}{2}} \Bigl(\int_{-\frac{B}{H}}^{\frac{B}{H}} \vert V_{\ell_1}(\alpha)-T_{\ell_1}(\alpha)\vert^2\ {\mathrm{d}}\alpha\Bigr)^{\frac{1}{2}} \\ \label{E1-estim} & \ll_{\ell_1,\ell_2} H \Bigl( \frac{N}{A}\Bigr)^{\frac{1}{\ell_2}-\frac{1}{2}} ( \log A )^{1/2} N^{\frac{1}{\ell_1}-\frac{1}{2}} A(N, -c_1) \ll_{\ell_1,\ell_2} H N^{\lambda-1} A(N, -C) $$ for a suitable choice of $C=C({\varepsilon})>0$, provided that $N^{-1+\frac{{\varepsilon}}{2}}<B/H<N^{-1+\frac{5}{6\ell_1}-{\varepsilon}}$; hence $N^{1-\frac{5}{6\ell_1}+{\varepsilon}}B\le H \le N^{1-{\varepsilon}}$ suffices. Summarizing, by , - we obtain that there exists $C=C({\varepsilon})>0$ such that $$\label{I2-estim} {\mathcal{I}}_2 \ll_{\ell_1,\ell_2} H N^{\lambda-1} A(N, -C) $$ provided that $N^{-1+\frac{{\varepsilon}}{2}}<B/H<N^{-1+\frac{5}{6\ell_1}-{\varepsilon}}$; hence $N^{1-\frac{5}{6\ell_1}+{\varepsilon}} B \le H \le N^{1-{\varepsilon}}$ suffices. Estimate of ${\mathcal{I}}_3$ {#estim-I3} ----------------------------- It’s very similar to ${\mathcal{I}}_2$’s; we just need to interchange $\ell_1$ with $\ell_2$ thus getting that there exists $C=C({\varepsilon})>0$ such that $$\label{I3-estim} {\mathcal{I}}_3 \ll_{\ell_1,\ell_2} H N^{\lambda-1} A(N, -C) $$ provided that $N^{-1+\frac{{\varepsilon}}{2}}<B/H<N^{-1+\frac{5}{6\ell_2}-{\varepsilon}}$; hence $N^{1-\frac{5}{6\ell_2}+{\varepsilon}} B \le H \le N^{1-{\varepsilon}}$ suffices. Estimate of ${\mathcal{I}}_4$ {#estim-I4} ----------------------------- By and we can write $$\begin{aligned} \notag {\mathcal{I}}_4 & \ll_{\ell_1,\ell_2} \int_{-\frac{B}{H}}^{\frac{B}{H}} \vert V_{\ell_1}(\alpha) -T_{\ell_1}(\alpha) \vert \vert V_{\ell_2}(\alpha) -T_{\ell_2}(\alpha) \vert\vert U(-\alpha,H)\vert \, {\mathrm{d}}\alpha \\ \notag &\hskip1cm + \int_{-\frac{B}{H}}^{\frac{B}{H}} \vert V_{\ell_1}(\alpha) -T_{\ell_1}(\alpha)\vert (1+\vert \alpha \vert N)^{\frac12} \vert U(-\alpha,H) \vert \, {\mathrm{d}}\alpha \\ \notag &\hskip1cm + \int_{-\frac{B}{H}}^{\frac{B}{H}} \vert V_{\ell_2}(\alpha) -T_{\ell_2}(\alpha)\vert (1+\vert \alpha \vert N)^{\frac12} \vert U(-\alpha,H) \vert \, {\mathrm{d}}\alpha \\ \notag &\hskip1cm + \int_{-\frac{B}{H}}^{\frac{B}{H}} (1+\vert \alpha \vert N) \vert U(-\alpha,H) \vert \, {\mathrm{d}}\alpha \\& \label{I4-split} = E_3+E_4+E_5+E_6, \end{aligned}$$ say. By , , the Cauchy-Schwarz inequality and Lemma \[App-BCP-Gallagher-2\] we have $$\begin{aligned} \notag E_3&\ll H \Bigl( \int_{-\frac{B}{H}}^{\frac{B}{H}} \vert V_{\ell_1}(\alpha) -T_{\ell_1}(\alpha) \vert^{2} \, {\mathrm{d}}\alpha \Bigr)^{\frac12} \Bigl( \int_{-\frac{B}{H}}^{\frac{B}{H}} \vert V_{\ell_2}(\alpha) -T_{\ell_2}(\alpha) \vert^{2} \, {\mathrm{d}}\alpha \Bigr)^{\frac12} \\ & \ll_{\ell_1,\ell_2} \label{E3-estim} H N^{\lambda-1} A(N, -C), $$ for a suitable choice of $C=C({\varepsilon})>0$, provided that $N^{-1+\frac{{\varepsilon}}{2}}<B/H<N^{-1+\frac{5}{6\ell_2}-{\varepsilon}}$; hence $N^{1-\frac{5}{6\ell_2}+{\varepsilon}} B \le H \le N^{1-{\varepsilon}}$ suffices. By and the Cauchy-Schwarz inequality we have $$\begin{aligned} \notag E_4&\ll H \int_{-\frac{1}{N}}^{\frac{1}{N}} \vert V_{\ell_1}(\alpha) - T_{\ell_1}(\alpha)\vert \, {\mathrm{d}}\alpha + H N^{\frac12} \int_{\frac{1}{N}}^{\frac{1}{H}} \vert V_{\ell_1}(\alpha) - T_{\ell_1}(\alpha)\vert \alpha^{\frac12} \, {\mathrm{d}}\alpha \\ \notag & \hskip1cm + N^{\frac12}\int_{\frac{1}{H}}^{\frac{B}{H}} \vert V_{\ell_1}(\alpha) - T_{\ell_1}(\alpha)\vert\alpha^{-\frac12} \, {\mathrm{d}}\alpha. \end{aligned}$$ By Lemma \[App-BCP-Gallagher-2\] we obtain $$\begin{aligned} \notag E_4&\ll HN^{-\frac12} \Bigl(\int_{-\frac{1}{N}}^{\frac{1}{N}} \vert V_{\ell_1}(\alpha) - T_{\ell_1}(\alpha)\vert^2 \, {\mathrm{d}}\alpha\Bigr)^{\frac{1}{2}} \\ \notag & \hskip1cm + H N^{\frac12} \Bigl( \int_{\frac{1}{N}}^{\frac{1}{H}} \vert V_{\ell_1}(\alpha) - T_{\ell_1}(\alpha)\vert^2 \, {\mathrm{d}}\alpha \Bigr)^{\frac{1}{2}} \Bigl( \int_{\frac{1}{N}}^{\frac{1}{H}} \alpha \, {\mathrm{d}}\alpha \Bigr)^{\frac{1}{2}} \\ \notag & \hskip1cm + N^{\frac12}\Bigl( \int_{\frac{1}{H}}^{\frac{B}{H}} \vert V_{\ell_1}(\alpha) - T_{\ell_1}(\alpha)\vert^2 \, {\mathrm{d}}\alpha \Bigr)^{\frac{1}{2}} \Bigl( \int_{\frac{1}{H}}^{\frac{B}{H}} \frac{{\mathrm{d}}\alpha}{\alpha} \Bigr)^{\frac{1}{2}} \\ & \ll_{\ell_1,\ell_2} \label{E4-estim} N^{ \frac{1}{\ell_1} } A(N, -C), $$ for a suitable choice of $C=C({\varepsilon})>0$, provided that $N^{-1+\frac{{\varepsilon}}{2}}<B/H<N^{-1+\frac{5}{6\ell_1}-{\varepsilon}}$; hence $N^{1-\frac{5}{6\ell_1}+{\varepsilon}} B \le H \le N^{1-{\varepsilon}}$ suffices. The estimate of $E_5$ runs analogously to the one of $E_4$. We obtain $$\label{E5-estim} E_5 \ll_{\ell_1,\ell_2} N^{ \frac{1}{\ell_2} } A(N, -C), $$ for a suitable choice of $C=C({\varepsilon})>0$, provided that $N^{-1+\frac{{\varepsilon}}{2}}<B/H<N^{-1+\frac{5}{6\ell_2}-{\varepsilon}}$; hence $N^{1-\frac{5}{6\ell_2}+{\varepsilon}} B \le H \le N^{1-{\varepsilon}}$ suffices. Moreover by we get $$\begin{aligned} \label{E6-estim} E_6 \ll H \int_{-\frac{1}{N}}^{\frac{1}{N}} \, {\mathrm{d}}\alpha + H N \int_{\frac{1}{N}}^{\frac{1}{H}} \alpha \, {\mathrm{d}}\alpha + N \int_{\frac{1}{H}}^{\frac{B}{H}} \, {\mathrm{d}}\alpha \ll \frac{NB}{H}.\end{aligned}$$ Hence by and - we obtain for $\ell_1\ge 2$ that $$\label{I4-estim} {\mathcal{I}}_4 \ll_{\ell_1,\ell_2} H N^{\lambda-1} A(N, -C), $$ for a suitable choice of $C=C({\varepsilon})>0$, provided that $N^{-1+\frac{{\varepsilon}}{2}}<B/H<N^{-1+\frac{5}{6\ell_2}-{\varepsilon}}$; hence $N^{1-\frac{5}{6\ell_2}+{\varepsilon}} B\le H \le N^{1-{\varepsilon}}$ suffices. Final words ----------- Summarizing, recalling that $2\le\ell_1\le \ell_2$, by , -, - and , we have that there exists $C=C({\varepsilon})>0$ such that $$\begin{aligned} \sum_{n=N+1}^{N+H} & r^{{{\prime\prime}}}_{\ell_1,\ell_2}(n) = c(\ell_1,\ell_2) H N^{\lambda-1} + {\mathcal{O}_{\ell_1,\ell_2}\Bigl(H N^{\lambda-1} A(N, -C)\Bigr)}, $$ uniformly for $N^{2-\frac{11}{6\ell_2}-\frac{1}{\ell_1}+{\varepsilon}}\le H \le N^{1-{\varepsilon}}$ which is non-trivial only for $\ell_1=2, \ell_2\in\{2,3\}$. Theorem \[thm-uncond\] follows. Proof of Theorem \[thm-RH\] =========================== From now on we assume the Riemann Hypothesis holds. Recalling , we have $$\sum_{n=N+1}^{N+H} e^{-n/N} R^{{{\prime\prime}}}_{\ell_1,\ell_2}(n) = \int_{-\frac{1}{2}}^{\frac{1}{2}} {\widetilde{V}}_{\ell_1}(\alpha) {\widetilde{V}}_{\ell_2}(\alpha) U(-\alpha,H)e(-N\alpha) \, {\mathrm{d}}\alpha .$$ Hence $$\begin{aligned} \notag &\sum_{n=N+1}^{N+H} e^{-n/N} R^{{{\prime\prime}}}_{\ell_1,\ell_2}(n) = \frac{\Gamma(1/\ell_1)\Gamma(1/\ell_2)}{\ell_1\ell_2} \int_{-\frac{1}{2}}^{\frac{1}{2}} z^{-\frac{1}{\ell_1}-\frac{1}{\ell_2}} U(-\alpha,H)e(-N\alpha) \, {\mathrm{d}}\alpha \\ \notag & \hskip0.5cm + \frac{\Gamma(1/\ell_1)}{\ell_1} \int_{-\frac{1}{2}}^{\frac{1}{2}} z^{-\frac{1}{\ell_1}} \Bigl({\widetilde{V}}_{\ell_2}(\alpha)- \frac{\Gamma(1/\ell_2)}{\ell_2 z^{\frac{1}{\ell_2}}} \Bigr) U(-\alpha,H)e(-N\alpha) \, {\mathrm{d}}\alpha \\ \notag & \hskip0.5cm + \frac{\Gamma(1/\ell_2)}{\ell_2} \int_{-\frac{1}{2}}^{\frac{1}{2}} z^{-\frac{1}{\ell_2}} \Bigl({\widetilde{V}}_{\ell_1}(\alpha)- \frac{\Gamma(1/\ell_1)}{\ell_1 z^{\frac{1}{\ell_1}}}\Bigr) U(-\alpha,H)e(-N\alpha) \, {\mathrm{d}}\alpha \\ \notag & \hskip0.5cm + \int_{-\frac{1}{2}}^{\frac{1}{2}} \Bigl({\widetilde{V}}_{\ell_1}(\alpha)- \frac{\Gamma(1/\ell_1)}{\ell_1 z^{\frac{1}{\ell_1}}}\Bigr) \Bigl({\widetilde{V}}_{\ell_2}(\alpha)- \frac{\Gamma(1/\ell_2)}{\ell_2 z^{\frac{1}{\ell_2}}} \Bigr) U(-\alpha,H)e(-N\alpha) \, {\mathrm{d}}\alpha \\ \label{main-dissection-RH-series} & \hskip1cm = {\mathcal{J}}_1 +{\mathcal{J}}_2 + {\mathcal{J}}_3 + {\mathcal{J}}_4, \end{aligned}$$ say. Now we evaluate these terms. Computation of ${\mathcal{J}}_1$ -------------------------------- By Lemma \[Laplace-formula\], and using $e^{-n/N}=e^{-1}+ {{\mathcal{O}_{}\left(H/N\right)}}$ for $n\in[N+1,N+H]$, $1\le H \le N$, a direct calculation gives $$\begin{aligned} \notag {\mathcal{J}}_1 & = c(\ell_1,\ell_2) \sum_{n=N+1}^{N+H} e^{-n/N} n^{\lambda-1} +{\mathcal{O}_{\ell_1,\ell_2}\Bigl(\frac{H}{N}\Bigr)} \\ \notag & = \frac{c(\ell_1,\ell_2)}{e} \sum_{n=N+1}^{N+H} n^{\lambda-1} +{\mathcal{O}_{\ell_1,\ell_2}\Bigl(\frac{H}{N}+H^2N^{\lambda-2}\Bigr)} \\ \label{J1-eval-RH-series} & = c(\ell_1,\ell_2) \frac{HN^{\lambda-1}}{e} +{\mathcal{O}_{\ell_1,\ell_2}\Bigl(\frac{H}{N}+H^2N^{\lambda-2}+N^{\lambda-1}\Bigr)}. \end{aligned}$$ Estimate of ${\mathcal{J}}_2$ ----------------------------- From now on, we denote $$\label{Etilde-def} {\widetilde{E}}_{\ell}(\alpha) : ={\widetilde{S}}_\ell(\alpha) - \frac{\Gamma(1/\ell)}{\ell z^{\frac{1}{\ell}}}.$$ Using Lemma \[tilde-trivial-lemma\] we remark that $$\label{V-tilde-approx} \Bigl\vert {\widetilde{V}}_{\ell}(\alpha)- \frac{\Gamma(1/\ell)}{\ell z^{\frac{1}{\ell}}} \Bigr\vert \le \vert {\widetilde{E}}_{\ell}(\alpha) \vert + \vert {\widetilde{V}}_{\ell}(\alpha)-{\widetilde{S}}_{\ell}(\alpha) \vert = \vert {\widetilde{E}}_{\ell}(\alpha) \vert +{\mathcal{O}_{\ell} (N^{\frac{1}{2\ell}})}.$$ Hence $$\begin{aligned} \notag {\mathcal{J}}_2 &\ll_{\ell_1,\ell_2} \int_{-\frac{1}{2}}^{\frac{1}{2}} \vert z\vert^{-\frac{1}{\ell_1}} \vert {\widetilde{E}}_{\ell_2}(\alpha) \vert \vert U(-\alpha,H) \vert \, {\mathrm{d}}\alpha \\& \label{J2-RH-series-first-estim} \hskip1cm + N^{\frac{1}{2\ell_2}} \int_{-\frac{1}{2}}^{\frac{1}{2}} \vert z\vert^{-\frac{1}{\ell_1}} \vert U(-\alpha,H) \vert \, {\mathrm{d}}\alpha ={\mathcal{A}}+{\mathcal{B}},\end{aligned}$$ say. By and we have $$\begin{aligned} \notag {\mathcal{B}}&\ll_{\ell_1,\ell_2} H N^{\frac{1}{\ell_1}+\frac{1}{2\ell_2}-1} + HN^{\frac{1}{2\ell_2}} \int_{\frac{1}{N}}^{\frac{1}{H}} \alpha^{-\frac{1}{\ell_1}} \, {\mathrm{d}}\alpha + N^{\frac{1}{2\ell_2}} \int_{\frac{1}{H}}^{\frac{1}{2}} \alpha^{-\frac{1}{\ell_1}-1} \, {\mathrm{d}}\alpha \\ \label{B-estim} & \ll_{\ell_1,\ell_2} H N^{\frac{1}{\ell_1}+\frac{1}{2\ell_2}-1} + H^{\frac{1}{\ell_1}}N^{\frac{1}{2\ell_2}}.\end{aligned}$$ By , , the Cauchy-Schwarz inequality, Lemma \[LP-Lemma-gen\] and a partial integration argument similar to the one used in the proof of Lemma \[partial-int-average\] (see the estimate of $M_2$ there), we have $$\begin{aligned} \notag {\mathcal{A}}&\ll_{\ell_1,\ell_2} H N^{\frac{1}{\ell_1}} \int_{-\frac{1}{N}}^{\frac{1}{N}} \vert {\widetilde{E}}_{\ell_2}(\alpha) \vert \, {\mathrm{d}}\alpha + H \int_{\frac{1}{N}}^{\frac{1}{H}} \frac{\vert {\widetilde{E}}_{\ell_2}(\alpha) \vert} {\alpha^{\frac{1}{\ell_1}}} \, {\mathrm{d}}\alpha + \int_{\frac{1}{H}}^{\frac{1}{2}} \frac{\vert {\widetilde{E}}_{\ell_2}(\alpha) \vert} {\alpha^{\frac{1}{\ell_1}+1}} \, {\mathrm{d}}\alpha \\ \notag &\ll_{\ell_1,\ell_2} H N^{\frac{1}{\ell_1}-\frac{1}{2}} \Bigl( \int_{-\frac{1}{N}}^{\frac{1}{N}} \vert {\widetilde{E}}_{\ell_2}(\alpha) \vert^2 \, {\mathrm{d}}\alpha \Bigr)^{\frac{1}{2}} \\ \notag &\hskip1cm + H \Bigl( \int_{\frac{1}{N}}^{\frac{1}{H}} \vert {\widetilde{E}}_{\ell_2}(\alpha) \vert^2 \, {\mathrm{d}}\alpha \Bigr)^{\frac{1}{2}} \Bigl( \int_{\frac{1}{N}}^{\frac{1}{H}} \frac{{\mathrm{d}}\alpha} {\alpha^{\frac{2}{\ell_1}}} \Bigr)^{\frac{1}{2}} \\ \notag &\hskip1cm+ \Bigl( \int_{\frac{1}{H}}^{\frac{1}{2}} \vert {\widetilde{E}}_{\ell_2}(\alpha) \vert^2 \frac{{\mathrm{d}}\alpha} {\alpha^{2}} \Bigr)^{\frac{1}{2}} \Bigl( \int_{\frac{1}{H}}^{\frac{1}{2}} \frac{{\mathrm{d}}\alpha} {\alpha^{\frac{2}{\ell_1}}} \Bigr)^{\frac{1}{2}} \\ \label{A-estim} &\ll_{\ell_1,\ell_2} H N^{\frac{1}{\ell_1}+\frac{1}{2\ell_2}-1}L + H^{\frac{1}{\ell_1}} N^{\frac{1}{2\ell_2}}L^{\frac32}.\end{aligned}$$ By - we have $${\mathcal{J}}_2\ll_{\ell_1,\ell_2} \label{J2-estim-RH-series} H N^{\frac{1}{\ell_1}+\frac{1}{2\ell_2}-1}L + H^{\frac{1}{\ell_1}} N^{\frac{1}{2\ell_2}}L^{\frac32} \ll_{\ell_1,\ell_2} H^{\frac{1}{\ell_1}} N^{\frac{1}{2\ell_2}}L^{\frac32}.$$ Estimate of ${\mathcal{J}}_3$ ----------------------------- The estimate of ${\mathcal{J}}_3$ is very similar to ${\mathcal{J}}_2$’s; we just need to interchange $\ell_1$ with $\ell_2$. We obtain $$\label{J3-estim-RH-series} {\mathcal{J}}_3 \ll_{\ell_1,\ell_2} H N^{\frac{1}{2\ell_1}+\frac{1}{\ell_2}-1}L + H^{\frac{1}{\ell_2}} N^{\frac{1}{2\ell_1}}L^{\frac32} \ll_{\ell_1,\ell_2} H^{\frac{1}{\ell_2}} N^{\frac{1}{2\ell_1}}L^{\frac32}.$$ Estimate of ${\mathcal{J}}_4$ ----------------------------- Using and we get $$\begin{aligned} \notag {\mathcal{I}}_4 & \ll_{\ell_1,\ell_2} \int_{-\frac12}^{\frac12} \vert {\widetilde{E}}_{\ell_1}(\alpha) \vert \vert {\widetilde{E}}_{\ell_2}(\alpha) \vert\vert U(-\alpha,H)\vert \, {\mathrm{d}}\alpha \\ \notag &\hskip1cm + N^{\frac{1}{2\ell_2}} \int_{-\frac12}^{\frac12} \vert {\widetilde{E}}_{\ell_1}(\alpha) \vert \vert U(-\alpha,H) \vert \, {\mathrm{d}}\alpha \\ \notag &\hskip1cm + N^{\frac{1}{2\ell_1}} \int_{-\frac12}^{\frac12} \vert {\widetilde{E}}_{\ell_2}(\alpha) \vert \vert U(-\alpha,H) \vert \, {\mathrm{d}}\alpha + N^{\frac{\lambda}{2}} \int_{-\frac12}^{\frac12} \vert U(-\alpha,H) \vert \, {\mathrm{d}}\alpha \\ \label{J4-split}& = {\mathcal{E}}_1+{\mathcal{E}}_2+{\mathcal{E}}_3+{\mathcal{E}}_4, \end{aligned}$$ say. By the Cauchy-Schwarz inequality, , and Lemma \[partial-int-average\] we obtain $$\begin{aligned} \notag {\mathcal{E}}_1 & \ll_{\ell_1,\ell_2} \Bigl( \int_{-\frac12}^{\frac12} \vert {\widetilde{E}}_{\ell_1}(\alpha) \vert^2 \vert U(-\alpha,H) \vert \, {\mathrm{d}}\alpha \Bigr)^{\frac{1}{2}} \Bigl( \int_{-\frac12}^{\frac12} \vert {\widetilde{E}}_{\ell_2}(\alpha) \vert^2 \vert U(-\alpha,H) \vert \, {\mathrm{d}}\alpha \Bigr)^{\frac{1}{2}} \\& \label{E1-estim-RH-series} \ll_{\ell_1,\ell_2} N^{\frac{\lambda}{2}} L^3.\end{aligned}$$ By the Cauchy-Schwarz inequality, , Lemmas \[UH-average\] and \[partial-int-average\] we obtain $$\begin{aligned} \notag {\mathcal{E}}_2 & \ll_{\ell_1,\ell_2} N^{\frac{1}{2\ell_2}} \Bigl( \int_{-\frac12}^{\frac12} \vert {\widetilde{E}}_{\ell_1}(\alpha) \vert^2 \vert U(-\alpha,H) \vert \, {\mathrm{d}}\alpha \Bigr)^{\frac{1}{2}} \Bigl( \int_{-\frac12}^{\frac12} \vert U(-\alpha,H) \vert \, {\mathrm{d}}\alpha \Bigr)^{\frac{1}{2}} \\& \label{E2-estim-RH-series} \ll_{\ell_1,\ell_2} N^{\frac{\lambda}{2}} L^2.\end{aligned}$$ By the Cauchy-Schwarz inequality, , Lemmas \[UH-average\] and \[partial-int-average\] we obtain $$\begin{aligned} \notag {\mathcal{E}}_3 & \ll_{\ell_1,\ell_2} N^{\frac{1}{2\ell_1}} \Bigl( \int_{-\frac12}^{\frac12} \vert {\widetilde{E}}_{\ell_2}(\alpha) \vert^2 \vert U(-\alpha,H) \vert \, {\mathrm{d}}\alpha \Bigr)^{\frac{1}{2}} \Bigl( \int_{-\frac12}^{\frac12} \vert U(-\alpha,H) \vert \, {\mathrm{d}}\alpha \Bigr)^{\frac{1}{2}} \\& \label{E3-estim-RH-series} \ll_{\ell_1,\ell_2} N^{\frac{\lambda}{2}} L^2.\end{aligned}$$ By we immediately have $$\label{E4-estim-RH-series} {\mathcal{E}}_4 \ll_{\ell_1,\ell_2} N^{\frac{\lambda}{2}} L.$$ Hence by - we finally can write that $$\label{J4-estim-RH-series} {\mathcal{J}}_4 \ll_{\ell_1,\ell_2} N^{\frac{\lambda}{2}} L^3.$$ Final words ----------- Summarizing, recalling $2\le\ell_1\le \ell_2$, by , -, - and , we have $$\begin{aligned} \notag \sum_{n=N+1}^{N+H} e^{-n/N} R^{{{\prime\prime}}}_{\ell_1,\ell_2}(n) &= c(\ell_1,\ell_2) \frac{HN^{\lambda-1}}{e} \\& \label{almost-done} +{\mathcal{O}_{\ell_1,\ell_2}\Bigl( \frac{H}{N} + H^2N^{\lambda-2} + H^{\frac{1}{\ell_1}} N^{\frac{1}{2\ell_2}}L^{\frac32} \Bigr)}\end{aligned}$$ which is an asymptotic formula for $\infty(N^{1-a(\ell_1,\ell_2)}L^{b(\ell_1)})\le H \le {{{o}_{}\left(N\right)}}$, where $a(\ell_1,\ell_2)$ and $b(\ell_1)$ are defined in . From $e^{-n/N}=e^{-1}+ {{\mathcal{O}_{}\left(H/N\right)}}$ for $n\in[N+1,N+H]$, $1\le H \le N$, we get $$\begin{aligned} \sum_{n = N+1}^{N + H} R^{{{\prime\prime}}}_{\ell_1,\ell_2}(n) &= c(\ell_1,\ell_2)HN^{\lambda-1} +{\mathcal{O}_{\ell_1,\ell_2}\Bigl( H^2N^{\lambda-2} + H^{\frac{1}{\ell_1}} N^{\frac{1}{2\ell_2}}L^{\frac32} \Bigr)} \\& + {\mathcal{O}\Bigl(\frac{H}{N}\sum_{n = N+1}^{N + H} R^{{{\prime\prime}}}_{\ell_1,\ell_2}(n) \Bigr)}.\end{aligned}$$ Using $e^{n/N}\le e^{2}$ and , the last error term is $\ll_{\ell_1,\ell_2} H^2N^{\lambda-2}$. Hence we get $$ \sum_{n = N+1}^{N + H} R^{{{\prime\prime}}}_{\ell_1,\ell_2}(n) = c(\ell_1,\ell_2)HN^{\lambda-1} +{\mathcal{O}_{\ell_1,\ell_2}\Bigl( H^2N^{\lambda-2} + H^{\frac{1}{\ell_1}} N^{\frac{1}{2\ell_2}}L^{\frac32} \Bigr)},$$ uniformly for every $2\le \ell_1\le \ell_2$ and $\infty(N^{1-a(\ell_1,\ell_2)}L^{b(\ell_1,\ell_2)})\le H \le{{{o}_{}\left(N\right)}}$, where $a(\ell_1,\ell_2)$ and $b(\ell_1)$ are defined in . Theorem \[thm-RH\] follows. Proof of Theorem \[thm-uncond-HL\] ================================== Assume $H>2B$ and $\ell_1,\ell_2\ge 2$; we’ll see at the end of the proof how the conditions in the statement of this theorem follow; remark that in this case we cannot interchange the role of $\ell_1,\ell_2$. We have $$\begin{aligned} \notag \sum_{n=N+1}^{N+H} & r^{\prime}_{\ell_1,\ell_2}(n)= \int_{-\frac12}^{\frac12} V_{\ell_1}(\alpha) T_{\ell_2}(\alpha) U(-\alpha,H)e(-N\alpha) \, {\mathrm{d}}\alpha \\ \notag &= \int_{-\frac{B}{H}}^{\frac{B}{H}} V_{\ell_1}(\alpha) T_{\ell_2}(\alpha) U(-\alpha,H)e(-N\alpha) \, {\mathrm{d}}\alpha \\& \label{dissect-HL} + \!\! \int\limits_{I(B,H)} \!\!\!\!\! V_{\ell_1}(\alpha) T_{\ell_2}(\alpha) U(-\alpha,H)e(-N\alpha) \, {\mathrm{d}}\alpha,\end{aligned}$$ where $I(B,H):=[-1/2,-B/H]\cup [B/H, 1/2]$. By the Cauchy-Schwarz inequality we have $$\begin{aligned} \notag \int\limits_{I(B,H)} & V_{\ell_1}(\alpha) T_{\ell_2}(\alpha) U(-\alpha,H)e(-N\alpha) \, {\mathrm{d}}\alpha \\ \label{CS-estim} & \ll \Bigl( \int\limits_{I(B,H)} \!\!\! \vert V_{\ell_1}(\alpha) \vert ^{2} \vert U(-\alpha,H) \vert \, {\mathrm{d}}\alpha \Bigr)^{\frac12} \Bigl( \int\limits_{I(B,H)} \!\!\! \vert T_{\ell_2}(\alpha) \vert ^{2} \vert U(-\alpha,H) \vert \, {\mathrm{d}}\alpha \Bigr)^{\frac12}.\end{aligned}$$ A similar computation to the one in leads to $$\begin{aligned} \notag \int\limits_{I(B,H)} \!\!\! \vert T_{\ell}(\alpha) \vert ^{2} &\vert U(-\alpha,H) \vert \, {\mathrm{d}}\alpha \ll \int_{\frac{B}{H}}^{\frac12} \vert T_{\ell}(\alpha) \vert ^{2} \frac{{\mathrm{d}}\alpha}{\alpha} \\ \label{T-average} & \ll_\ell N^{\frac{1}{\ell}} + \frac{H L }{B} + \int_{\frac{B}{H}}^{\frac12} (\xi N^{\frac{1}{\ell}} + L )\ \frac{{\mathrm{d}}\xi}{\xi^2} \ll_\ell N^{\frac{1}{\ell}} L + \frac{H L }{B},\end{aligned}$$ for every $\ell\ge 2$. Hence, by - and recalling and , we obtain $$\begin{aligned} \notag \int\limits_{I(B,H)} V_{\ell_1}(\alpha) & T_{\ell_2}(\alpha) U(-\alpha,H) e(-N\alpha) \, {\mathrm{d}}\alpha \\& \label{V-average-coda-HL} \ll_{\ell_1,\ell_2} N^{\frac{\lambda}{2}} L^{\frac32} + \frac{ H^{\frac12} N^{\frac{1}{2\ell_1}} L^{\frac32}}{B^{\frac{1}{2}}} +\frac{H L^{\frac32}}{B} \ll_{\ell_1,\ell_2} \frac{H L^{\frac32}}{B}.\end{aligned}$$ By and , we get $$\begin{aligned} \notag \sum_{n=N+1}^{N+H} &r^{\prime}_{\ell_1,\ell_2}(n) = \int_{-\frac{B}{H}}^{\frac{B}{H}} V_{\ell_1}(\alpha) T_{\ell_2}(\alpha) U(-\alpha,H)e(-N\alpha) \, {\mathrm{d}}\alpha + {\mathcal{O}_{\ell_1,\ell_2}\Bigl(\frac{H L^{\frac32}}{B}\Bigr)}.\end{aligned}$$ Hence $$\begin{aligned} \notag \sum_{n=N+1}^{N+H} &r^{\prime}_{\ell_1,\ell_2}(n) = \int_{-\frac{B}{H}}^{\frac{B}{H}} f_{\ell_1}(\alpha) f_{\ell_2}(\alpha) U(-\alpha,H)e(-N\alpha) \, {\mathrm{d}}\alpha \\ \notag & \hskip0.5cm + \int_{-\frac{B}{H}}^{\frac{B}{H}} f_{\ell_2}(\alpha) (V_{\ell_1}(\alpha) - f_{\ell_1}(\alpha) ) U(-\alpha,H)e(-N\alpha) \, {\mathrm{d}}\alpha \\ \notag & \hskip0.5cm + \int_{-\frac{B}{H}}^{\frac{B}{H}} f_{\ell_1}(\alpha) (T_{\ell_2}(\alpha) - f_{\ell_2}(\alpha) ) U(-\alpha,H)e(-N\alpha) \, {\mathrm{d}}\alpha \\ \notag & \hskip0.5cm + \int_{-\frac{B}{H}}^{\frac{B}{H}} (V_{\ell_1}(\alpha)-f_{\ell_1}(\alpha) ) (T_{\ell_2}(\alpha) - f_{\ell_2}(\alpha) ) U(-\alpha,H)e(-N\alpha) \, {\mathrm{d}}\alpha \\ \notag & \hskip0.5cm + {\mathcal{O}_{\ell_1,\ell_2}\Bigl(\frac{H L^{\frac32}}{B}\Bigr)} \\ \label{main-dissection-HL} & = {\mathcal{I}}_1 +{\mathcal{I}}_2 + {\mathcal{I}}_3 + {\mathcal{I}}_4 + E, \end{aligned}$$ say. We now evaluate these terms. The main term ${\mathcal{I}}_1$ can be evaluated as in §\[comp-main-term\]; by - it is $$\label{I1-eval-HL} {\mathcal{I}}_1 = c(\ell_1,\ell_2) H N^{\lambda-1} + {\mathcal{O}_{\ell_1,\ell_2}\Bigl( \Bigl(\frac{H}{B}\Bigr)^{\lambda}+ H N^{\lambda-1} A(N, -C)\Bigr)}, $$ for a suitable choice of $C=C({\varepsilon})>0$. ${\mathcal{I}}_2$ can be estimated as in §\[estim-I2\]; by it is $$\label{I2-estim-HL} {\mathcal{I}}_2 \ll_{\ell_1,\ell_2} H N^{\lambda-1} A(N, -C), $$ for a suitable choice of $C=C({\varepsilon})>0$, provided that $N^{-1+\frac{{\varepsilon}}{2}}<B/H<N^{-1+\frac{5}{6\ell_1}-{\varepsilon}}$; hence $N^{1-\frac{5}{6\ell_1}+{\varepsilon}} B \le H \le N^{1-{\varepsilon}}$ suffices. Estimate of ${\mathcal{I}}_3$ {#estimate-of-mathcali_3} ----------------------------- Using we obtain that $$\begin{aligned} \notag {\mathcal{I}}_3 \ll \int_{-\frac{B}{H}}^{\frac{B}{H}} \vert f_{\ell_1}(\alpha) \vert (1+\vert \alpha \vert N)^{\frac12} \vert U(-\alpha,H) \vert \, {\mathrm{d}}\alpha \end{aligned}$$ and the right hand side is equal to $E_2$ of §\[estim-I2\]; hence by we have $$\label{I3-estim-HL} {\mathcal{I}}_3 \ll_{\ell_1,\ell_2} N^{1/\ell_1} A^{1/2-1/\ell_1} L^{1/2} (\log A)^{1/2},$$ where $A$ is defined in . Estimate of ${\mathcal{I}}_4$ {#estim-I4-HL} ----------------------------- By and we can write $$\begin{aligned} \notag {\mathcal{I}}_4 & \ll_{\ell_1,\ell_2} \int_{-\frac{B}{H}}^{\frac{B}{H}} \vert V_{\ell_1}(\alpha) -T_{\ell_1}(\alpha)\vert (1+\vert \alpha \vert N)^{\frac12} \vert U(-\alpha,H) \vert \, {\mathrm{d}}\alpha \\ \label{I4-split-HL} &\hskip1cm + \int_{-\frac{B}{H}}^{\frac{B}{H}} (1+\vert \alpha \vert N) \vert U(-\alpha,H) \vert \, {\mathrm{d}}\alpha = R_1+R_2, \end{aligned}$$ say. $R_1$ is equal to $E_4$ of §\[estim-I4\]; hence we have $$\label{R1-estim} R_1 \ll_{\ell_1,\ell_2} N^{ \frac{1}{\ell_1} } A(N, -C), $$ for a suitable choice of $C=C({\varepsilon})>0$, provided that $N^{-1+\frac{{\varepsilon}}{2}}<B/H<N^{-1+\frac{5}{6\ell_1}-{\varepsilon}}$; hence $N^{1-\frac{5}{6\ell_1}+{\varepsilon}} B \le H \le N^{1-{\varepsilon}}$ suffices. $R_2$ is equal to $E_6$ of §\[estim-I4\]; hence we get $$\label{R2-estim} R_2 \ll \frac{NB}{H}.$$ Summarizing, by and -, we obtain $$\label{I4-estim-HL} {\mathcal{I}}_4 \ll_{\ell_1,\ell_2} H N^{\lambda-1} A(N, -C), $$ for a suitable choice of $C=C({\varepsilon})>0$, provided that $N^{-1+\frac{{\varepsilon}}{2}}<B/H<N^{-1+\frac{5}{6\ell_1}-{\varepsilon}}$; hence $N^{1-\frac{5}{6\ell_1}+{\varepsilon}} B\le H \le N^{1-{\varepsilon}}$ suffices. Final words ----------- Summarizing, recalling that $\ell_1, \ell_2\ge 2$, by , - and , we have that there exists $C=C({\varepsilon})>0$ such that $$\begin{aligned} \sum_{n=N+1}^{N+H} & r^{\prime}_{\ell_1,\ell_2}(n) = c(\ell_1,\ell_2) H N^{\lambda-1} + {\mathcal{O}_{\ell_1,\ell_2}\Bigl(H N^{\lambda-1} A(N, -C)\Bigr)}, $$ uniformly for $N^{2-\frac{11}{6\ell_1} -\frac{1}{\ell_2}+{\varepsilon}}\le H \le N^{1-{\varepsilon}}$ which is non-trivial only for $\ell_1=2$ and $2\le \ell_2\le 11$, or $\ell_1=3$ and $ \ell_2=2$. Theorem \[thm-uncond-HL\] follows. Proof of Theorem \[thm-RH-HL\] =============================== In this section we need some additional definitions and lemmas. Letting $$\label{omega-def} \omega_{\ell}(\alpha) = \sum_{m=1}^{\infty} e^{-m^{\ell}/N} e(m^{\ell}\alpha) = \sum_{m=1}^{\infty} e^{-m^{\ell}z},$$ we have the following \[zac-lemma-series\] Let $\ell\ge 2 $ be an integer and $0<\xi\le 1/2$. Then $$\int_{-\xi}^{\xi} |\omega_{\ell}(\alpha)|^2 \ {\mathrm{d}}\alpha \ll_{\ell} \xi N^{\frac{1}{\ell}} + \begin{cases} L & \text{if}\ \ell =2\\ 1 & \text{if}\ \ell > 2 \end{cases}$$   $$\int_{-\xi}^{\xi} |{\widetilde{S}}_{\ell}(\alpha)|^2 \ {\mathrm{d}}\alpha \ll_{\ell} \xi N^{\frac{1}{\ell}} L + \begin{cases} L^{2} & \text{if}\ \ell =2\\ 1 & \text{if}\ \ell > 2. \end{cases}$$ Recalling the definition of the $\theta$-function $$\theta(z) = \sum_{n=-\infty}^{\infty} e^{-n^{2}/N} e(n^{2}\alpha) = \sum_{n=-\infty}^{\infty} e^{-n^{2}z} = 1+2\omega_{2}(\alpha),$$ its modular relation (see, *e.g.*, Proposition VI.4.3, page 340, of Freitag and Busam [@FreitagB2009]) gives that $ \theta(z) = (\pi/z)^{\frac{1}{2}} \theta ( \pi^2/z) $ for $\Re(z)>0$. Hence we have $$\label{omega-approx} \omega_{2}(\alpha) = \frac12 \left(\frac{\pi}{z}\right)^{\frac{1}{2}} \!\!\!\! - \frac12 + \left(\frac{\pi}{z}\right)^{\frac{1}{2}} \sum_{j=1}^{+\infty} e^{-j^{2}\pi^{2}/z}, \quad \text{for} \ \Re(z)>0.$$ For the series in we have \[Lemma 4 of [@LanguascoZ2016b]\] \[omega-Y\] Let $N$ be a large integer, $z= 1/N-2\pi i\alpha$, $\alpha\in [-1/2,1/2]$ and $Y=\Re(1/z)>0$. We have $$\Bigl\vert \sum_{j=1}^{+\infty} e^{-j^{2}\pi^{2}/z} \Bigr\vert \ll \matrix@check\Biggcases\env@Biggcases e^{- \pi^{2} Y } & \textrm{for} \ Y\ge 1 \\ Y^{-\frac{1}{2}} & \textrm{for} \ 0 <Y\le 1. \endarray $$ Since $$\notag Y = \Re(1/z) = \frac{N}{1+4\pi^2\alpha^2N^2} \ge \frac{1}{5\pi^2} \begin{cases} N & \textrm{if} \ \vert \alpha \vert \le 1/N\\ (\alpha^2N)^{-1} & \textrm{if} \ \vert \alpha \vert > 1/N, \end{cases}$$ from Lemma \[omega-Y\] we get $$\label{omega-Y-estim} \Bigl\vert \sum_{j=1}^{+\infty} e^{-j^{2}\pi^{2}/z} \Bigr\vert \ll \begin{cases} e^{- N/5} & \textrm{if} \ \vert \alpha \vert \le 1/N\\ \exp(- 1/(5\alpha^2N)) & \textrm{if} \ 1/N<\vert \alpha \vert = {o\bigl(N^{-\frac{1}{2}}\bigr)}\\ 1+ N^{\frac{1}{2}} \vert \alpha\vert & \textrm{otherwise}. \end{cases}$$ \[partial-int-average-omega2\] Let $N$ be a sufficiently large integer and $L= \log N$. We have $$\int_{-\frac{1}{2}}^{\frac{1}{2}} \vert \omega_{2}(\alpha) \vert^{2} \vert U(\alpha,H)\vert \, {\mathrm{d}}\alpha \ll N^{\frac{1}{2}} L + HL.$$ By we have $$\begin{aligned} \notag \int_{-\frac{1}{2}}^{\frac{1}{2}} \vert \omega_{2}(\alpha) \vert^{2} \vert U(\alpha,H)\vert \, {\mathrm{d}}\alpha &\ll H \int_{-\frac1H}^{\frac1H} \, \vert \omega_{2}(\alpha) \vert^{2} \ {\mathrm{d}}\alpha + \int_{\frac1H}^{\frac12} \, \vert \omega_{2}(\alpha) \vert^{2} \frac{{\mathrm{d}}\alpha}{\alpha} \\& \notag \hskip1cm + \int_{-\frac12}^{-\frac1H} \, \vert \omega_{2}(\alpha) \vert^{2} \frac{{\mathrm{d}}\alpha}{\alpha} \\ \label{M-split-1} & = M_1+M_2+M_3,\end{aligned}$$ say. By Lemma \[zac-lemma-series\] we immediately get that $$\label{M1-estim-1} M_1 \ll N^{\frac{1}{2}} + HL.$$ By a partial integration and Lemma \[zac-lemma-series\] we obtain $$\begin{aligned} \notag M_2 &\ll \int_{-\frac12}^{\frac12} \, \vert \omega_{2}(\alpha) \vert^{2} \ {\mathrm{d}}\alpha + H \int_{-\frac1H}^{\frac1H} \, \vert \omega_{2}(\alpha) \vert^{2} \ {\mathrm{d}}\alpha + \int_{\frac1H}^{\frac12} \Bigl( \int_{-\xi}^{\xi} \vert \omega_{2}(\alpha) \vert^{2} \ {\mathrm{d}}\alpha \Bigr) \frac{\ {\mathrm{d}}\xi}{\xi^2} \\& \label{M2-estim-1} \ll N^{\frac{1}{2}} + HL + \int_{\frac1H}^{\frac12} \frac{N^{\frac{1}{2}} \xi + L }{\xi^2}\ {\mathrm{d}}\xi \ll N^{\frac{1}{2}} L + HL.\end{aligned}$$ A similar computation leads to $M_3 \ll N^{\frac{1}{2}} L + HL$. By -, the lemma follows. From now on we assume the Riemann Hypothesis holds. Let $1<D=D(N)<H/2$ to be chosen later and $I(D,H):=[-1/2,-D/H]\cup [D /H,1/2]$. By and -, and recalling , it is an easy matter to see that $$\begin{aligned} \notag \sum_{n = N+1}^{N + H} & e^{-n/N} R^{\prime}_{\ell,2}(n) = \int_{-\frac{1}{2}}^{\frac{1}{2}} {\widetilde{V}}_{\ell}(\alpha) \omega_{2}(\alpha) U(-\alpha,H) e(-N\alpha) \, {\mathrm{d}}\alpha \\ \notag &= \int_{-\frac{1}{2}}^{\frac{1}{2}} ({\widetilde{V}}_{\ell}(\alpha)- {\widetilde{S}}_{\ell}(\alpha))\, \omega_{2}(\alpha) U(-\alpha,H) e(-N\alpha) \, {\mathrm{d}}\alpha \\ \notag & \hskip0.5cm + \frac{\Gamma(1/\ell)}{2\ell} \int_{-\frac{D}{H}}^{\frac{D}{H}} \Bigl( \frac{\pi^{\frac{1}{2}}}{ z^{\frac{1}{2}+\frac{1}{\ell}} } - \frac{1}{z^{\frac{1}{\ell}}}\Bigr) U(-\alpha,H) e(-N\alpha)\, {\mathrm{d}}\alpha \\ \notag & \hskip0.5cm + \int_{-\frac{D}{H}}^{\frac{D}{H}} {\widetilde{E}}_{\ell}(\alpha) \omega_{2}(\alpha) U(-\alpha,H) e(-N\alpha)\, {\mathrm{d}}\alpha \\ \notag & \hskip0.5cm + \frac{\pi^{\frac{1}{2}} \Gamma(1/\ell)}{\ell} \int_{-\frac{D}{H}}^{\frac{D}{H}} \frac{1}{z^{\frac{1}{2}+\frac{1}{\ell}} } \Bigl( \sum_{j=1}^{+\infty} e^{-j^{2}\pi^{2}/z} \Bigr) U(-\alpha,H) e(-N\alpha)\, {\mathrm{d}}\alpha \\ \notag & \hskip0.5cm+ \int\limits_{\mathclap{I(D,H)}} {\widetilde{S}}_{\ell}(\alpha)\omega_{2}(\alpha) U(-\alpha,H) e(-N\alpha) \, {\mathrm{d}}\alpha \\ \label{approx-th1-part4} & = I_{0} + I_{1}+I_{2}+I_{3} +I_{4}, \end{aligned}$$ say. Using Lemma \[tilde-trivial-lemma\] and the Cauchy-Schwarz inequality we have $$\begin{aligned} I_{0} \ll_\ell N^{\frac{1}{2\ell}} \Bigl( \int_{-\frac{1}{2}}^{\frac{1}{2}} \vert \omega_{2}(\alpha) \vert^{2} \vert U(\alpha,H)\vert \, {\mathrm{d}}\alpha \Bigr)^{\frac{1}{2}} \Bigl( \int_{-\frac{1}{2}}^{\frac{1}{2}} \vert U(\alpha,H) \vert \, {\mathrm{d}}\alpha \Bigr)^{\frac{1}{2}}.\end{aligned}$$ By Lemmas \[UH-average\] and \[partial-int-average-omega2\] we obtain $$\label{I0-estim-omega} I_{0} \ll_\ell N^{\frac{1}{2\ell}} (N^{\frac{1}{2}}L + H L)^{\frac{1}{2}}L^{\frac12} \ll N^{\frac{1}{4}+\frac{1}{2\ell}}L+ H^{\frac{1}{2}}N^{\frac{1}{2\ell}}L.$$ Now we evaluate $I_{1}$. Using Lemma \[Laplace-formula\], and $e^{-n/N}=e^{-1}+ {{\mathcal{O}_{}\left(H/N\right)}}$ for $n\in[N+1,N+H]$, $1\le H \le N$, we immediately get $$\begin{aligned} \notag I_{1} &= \frac{\Gamma(1/\ell)}{2\ell} \sum_{n = N+1}^{N + H} \Bigl( \frac{\pi^{\frac{1}{2}}}{\Gamma(\frac{1}{2}+\frac{1}{\ell})} n^{\frac{1}{\ell}-\frac{1}{2}} - n^{\frac{1}{\ell}-1} \Bigr) e^{-n/N} + {\mathcal{O}_{\ell}\Bigl(\frac{H}{N}\Bigr)} \\& \notag \hskip1cm + {\mathcal{O}_{\ell}\Bigl(\int_{\frac{D}{H}}^{\frac{1}{2}} \frac{{\mathrm{d}}\alpha}{\alpha^2}\Bigr)} \\ \label{I1-eval-part4} & = \frac{c(\ell,2)}{e} HN^{\frac{1}{\ell}-\frac{1}{2}} + {\mathcal{O}_{\ell}\Bigl(\frac{H}{N^{1-\frac{1}{\ell}}}+\frac{H^2}{N^{\frac{3}{2}-\frac{1}{\ell}}}+\frac{H}{D}\Bigr)}.\end{aligned}$$ To have that the first term in $I_{1}$ dominates in $I_{0} + I_{1}$ we need that $D= \infty(N^{\frac{1}{2}-\frac{1}{\ell}})$, $H={{{o}_{}\left(N\right)}}$ and $H=\infty(N^{1-\frac{1}{\ell}}L^{2})$, $\ell\ge 2$. Now we estimate $I_{3}$. Assuming $H=\infty(N^{\frac{1}{2}}D)$, by and , we have, using the substitution $u=1/(5N\alpha^2)$ in the last integral, that $$\begin{aligned} \notag I_{3} &\ll_\ell \frac{HN^{\frac{1}{2}+\frac{1}{\ell}}}{e^{N/5}} \int_{-\frac{1}{N}}^{\frac{1}{N}} {\mathrm{d}}\alpha + \frac{H}{e^{H^2/(5N)}} \int_{\frac{1}{N}}^{\frac{1}{H}} \frac{{\mathrm{d}}\alpha}{\alpha^{\frac{1}{2}+\frac{1}{\ell}}} + \int_{\frac{1}{H}}^{\frac{D}{H}} \frac{{\mathrm{d}}\alpha}{\alpha^{3/2+\frac{1}{\ell}}e^{1/(5N\alpha^2)}} \\ \notag & \ll_\ell \frac{HN^{\frac{1}{\ell}-\frac{1}{2}}}{e^{N/5}} + \frac{HN^{\frac{1}{\ell}-\frac{1}{2}}L}{e^{H^2/(5N)}} + N^{\frac{1}{4}+\frac{1}{2\ell}} \int_{H^2/(5ND^2)}^{H^2/(5N)} u^{-3/4+\frac{1}{2\ell}}e^{-u}\ {\mathrm{d}}u \\ \label{I3-estim-final-part4} & \ll_\ell \frac{HN^{\frac{1}{\ell}-\frac{1}{2}}L}{e^{H^2/(5N)}} + N^{\frac{1}{4}+\frac{1}{2\ell}} = {o\bigl(HN^{\frac{1}{\ell}-\frac{1}{2}}\bigr)},\end{aligned}$$ provided that $H=\infty(N^{\frac{1}{2}} \log L)$ and $H=\infty(N^{1-\frac{1}{\ell}})$, $\ell\ge 2$. Now we estimate $I_{2}$. Recalling that $H=\infty(N^{\frac{1}{2}}D)$, for every $\vert \alpha \vert \le D/H$ we have, by -, that $\vert\omega_{2}(\alpha) \vert \ll \vert z \vert ^{-\frac{1}{2}}$. Hence $$I_{2} \ll \int_{-\frac{D}{H}}^{\frac{D}{H}} \vert {\widetilde{E}}_{\ell}(\alpha) \vert \frac{\vert U(\alpha,H)\vert}{\vert z \vert ^{\frac{1}{2}}} \, {\mathrm{d}}\alpha.$$ Using and the Cauchy-Schwarz inequality we get $$\begin{aligned} \notag I_{2} &\ll H N^{\frac{1}{2}} \Bigl( \int_{-\frac{1}{N}}^{\frac{1}{N}} {\mathrm{d}}\alpha \Bigr)^{\frac{1}{2}} \Bigl( \int_{-\frac{1}{N}}^{\frac{1}{N}} \vert {\widetilde{E}}_{\ell}(\alpha) \vert^{2} \, {\mathrm{d}}\alpha \Bigr)^{\frac{1}{2}} \\ \notag &\hskip1cm + H \Bigl( \int_{\frac{1}{N}}^{\frac{1}{H}} \frac{{\mathrm{d}}\alpha}{\alpha^{\frac{1}{2}}}\Bigr)^{\frac{1}{2}} \Bigl( \int_{\frac{1}{N}}^{\frac{1}{H}}\!\! \vert {\widetilde{E}}_{\ell}(\alpha) \vert^{2} \frac{{\mathrm{d}}\alpha}{\alpha^{\frac{1}{2}}} \Bigr)^{\frac{1}{2}} \\ \notag &\hskip1cm + \Bigl( \int_{\frac{1}{H}}^{\frac{D}{H}} \frac{{\mathrm{d}}\alpha}{\alpha^{\frac{3}{2}}}\Bigr)^{\frac{1}{2}} \Bigl( \int_{\frac{1}{H}}^{\frac{D}{H}} \vert {\widetilde{E}}_{\ell}(\alpha) \vert^{2} \frac{{\mathrm{d}}\alpha}{\alpha^{\frac{3}{2}}} \Bigr)^{\frac{1}{2}} \end{aligned}$$ By Lemma \[LP-Lemma-gen\] we get $$\begin{aligned} \notag I_{2} & \ll_{\ell} H N^{\frac{1}{2\ell}-\frac{1}{2}} L + H^{3/4} N^{\frac{1}{2\ell}} L \Bigl( \frac{1}{H^{\frac{1}{2}}}+ \int_{\frac{1}{N}}^{\frac{1}{H}} \frac{{\mathrm{d}}\xi}{\xi^{\frac{1}{2}}} \Bigr)^{\frac{1}{2}} \\ \notag &\hskip1cm + H^{\frac{1}{4}} N^{\frac{1}{2\ell}} L \Bigl( H^{\frac{1}{2}} + \int_{\frac{1}{H}}^{\frac{D}{H}} \frac{{\mathrm{d}}\xi}{\xi^{\frac{3}{2}}} \Bigr)^{\frac{1}{2}} \\& \label{I2-estim-final-part4} \ll_{\ell} H^{\frac{1}{2}}N^{\frac{1}{2\ell}} L.\end{aligned}$$ We remark that $I_2={o\bigl(HN^{\frac{1}{\ell}-\frac{1}{2}} \bigr)}$ provided that $H=\infty(N^{1-\frac{1}{\ell}} L^{2})$, $\ell\ge 2$. Now we estimate $I_{4}$. By , Lemma \[zac-lemma-series\] and a partial integration argument we get $$\begin{aligned} \notag I_4 &\ll \int_{\frac{D}{H}}^{\frac{1}{2}} \vert {\widetilde{S}}_{\ell}(\alpha) \omega_{2}(\alpha) \vert \frac{{\mathrm{d}}\alpha}{\alpha} \ll \Bigl( \int_{\frac{D}{H}}^{\frac{1}{2}} \vert {\widetilde{S}}_{\ell}(\alpha)\vert^2 \frac{{\mathrm{d}}\alpha}{\alpha} \Bigr)^{\frac{1}{2}} \Bigl( \int_{\frac{D}{H}}^{\frac{1}{2}} \vert \omega_{2}(\alpha) \vert^2\frac{{\mathrm{d}}\alpha}{\alpha} \Bigr)^{\frac{1}{2}} \\ \notag &\ll_\ell \Bigl( N^{\frac{1}{\ell}} L + \frac{HL^{2}}{D} + L \int_{\frac{D}{H}}^{\frac{1}{2}} (\xi N^{\frac{1}{\ell}} + L) \frac{{\mathrm{d}}\xi}{\xi^2} \Bigr)^{\frac{1}{2}} \\ \notag &\hskip1cm \times \Bigl( N^{\frac{1}{2}} + \frac{HL}{D} + \int_{\frac{D}{H}}^{\frac{1}{2}} (\xi N^{\frac{1}{2}} + L) \frac{{\mathrm{d}}\xi}{\xi^2} \Bigr)^{\frac{1}{2}} \\ \label{I4-estim-final-part4} &\ll_\ell L^{\frac32} \Bigl( N^{\frac{1}{4}+\frac{1}{2\ell}} + \frac{H}{D} \Bigr) .\end{aligned}$$ Clearly we have that $I_4={o\bigl(HN^{\frac{1}{\ell}-\frac{1}{2}}\bigr)}$ provided that $D=\infty(N^{\frac{1}{2}-\frac{1}{\ell}}L^{\frac32})$ and $H=\infty(N^{\frac{3}{4}-\frac{1}{2\ell}}L^{\frac32})$, $\ell\ge 2$. Combining the previous conditions on $H$ and $D$ we can choose $D=N^{\frac{1}{2}-\frac{1}{\ell}}L^{2}/(\log L)$ and $H=\infty(N^{1-\frac{1}{\ell}}L^{2})$. Hence using - we can write $$\begin{aligned} \sum_{n = N+1}^{N + H} e^{-n/N} R^{\prime}_{\ell,2}(n) & = \frac{ c(2,\ell) }{e} HN^{\frac{1}{\ell}-\frac{1}{2}} \\& + {\mathcal{O}_{\ell}\Bigl(\frac{H^2}{N^{\frac{3}{2}-\frac{1}{\ell}}}+ \frac{H N^{\frac{1}{\ell}-\frac{1}{2}}\log L} {L^{\frac12}} + H^{\frac{1}{2}}N^{\frac{1}{2\ell}} L\Bigr)}.\end{aligned}$$ Theorem \[thm-RH-HL\] follows for $\infty(N^{1-\frac{1}{\ell}}L^{2}) \le H \le {{{o}_{}\left(N\right)}}$, $\ell\ge 2$, since the exponential weight $e^{-n/N}$ can be removed as we did at the bottom of the proof of Theorem \[thm-RH\]. [10]{} , *Complex analysis*. Springer-Verlag, second ed., 2009. , *On the [H]{}ardy-[L]{}ittlewood problem in short intervals*. Int. J. Number Theory [**4**]{} (2008), 715–723. , *On a ternary Diophantine problem with mixed powers of primes*. 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[**8**]{} (1974), 73–82. , *A note on the sum of a prime and a prime square*, in Analytic and Probabilistic Methods in Number Theory: Proceedings of the Sixth International Conference, Palanga, Lithuania, 11–17 September 2016, pp.221–226, (2017). , *On the sum of prime number and square number*. Preprint 2017, <http://www.math.sci.hokudai.ac.jp/~wakate/mcyr/2017/pdf/01500_suzuki_yuta.pdf>. . *The [Hardy]{}-[Littlewood]{} method*. Cambridge U. P., second ed., 1997.

--- bibliography: - 'library.bib' --- ${}$\ \ \ \ \ \ \ \ \ \ \ \ \ \ **Roger Lloret-Batlle**\ PhD Student\ University of California, Irvine\ Institute of Transportation Studies\ 4000 Anteater Instruction and Research Bldg (AIRB)\ Irvine, CA 92697-3600\ Email: rlloretb@uci.edu\ \ \ **R. Jayakrishnan**\ Professor\ University of California, Irvine\ Institute of Transportation Studies\ 4000 Anteater Instruction and Research Bldg (AIRB)\ Irvine, CA 92697-3600\ Email: rjayakri@uci.edu\ \ \ \ \ \ \ \ \ \ August 1st, 2017 Abstract ======== This article explores the coalitional stability of a new cooperative control policy for freeways and parallel queuing facilities with multiple servers. Based on predicted future delays per queue or lane, a VOT-heterogeneous population of agents can agree to switch lanes or queues and transfer payments to each other in order to minimize the total cost of the incoming platoon. The strategic interaction is captured by an n-level Stackelberg model with coalitions, while the cooperative structure is formulated as a partition function game (PFG). The stability concept explored is the strong-core for PFGs which we found appropiate given the nature of the problem. This concept ensures that the efficient allocation is individually rational and coalitionally stable. We analyze this control mechanism for two settings: a static vertical queue and a dynamic horizontal queue. For the former, we first characterize the properties of the underlying cooperative game. Our simulation results suggest that the setting is always strong-core stable. For the latter, we propose a new relaxation program for the strong-core concept. Our simulation results on a freeway bottleneck with constant outflow using Newell’s car-following model show the imputations to be generally strong-core stable and the coalitional instabilities to remain small with regard to users’ costs.\ \ *Keywords*: lane-changing, stability, core, dynamic queue routing, parallel queues, connected vehicles Introduction ============ Connected vehicle environments bring new opportunities in the operation of traffic infrastructures. Until now, vehicles traveling on a link selfishly self-organized themselves into a service order which corresponds on average to a First-Come First-Served (FCFS) order, without any mutual exchange of urgency information. For instance, lane changes naturally violate the FCFS service order through a gap acceptance distributed mechanism, with very limited or no precise future delay information. New connectivity can increase the efficiency of lange changes by providing better future delay estimation and by incorporating a new variable: travellers’ Value Of Time (VOT). If it was possible for users to be informed of the downstream traffic conditions for each lane, i.e. the downstream delay on each lane at a particular timestep, and if they could communicate their VOT values to each other, travelers could decide which lane changes are the most beneficial for everybody. Thus, agents can violate the initial order by creating coalitions which give incentives to agents in front to choose longer queues in exchange of a side payment. This would lead to a different level of service for each lane or queue,the less congested lanes or queues then becoming faster than the more congested ones and preferable for the higher-VOT travellers who may be willing to pay. The concept presented in this article is applicable to both facilities with parallel queues (multiple parallel servers) i.e. access gates at ports, traffic intersections, and bottlenecks in a freeway section. We will use the terms “queue” and “lane” indistinctively, as the essential level of performance of each lane is captured by the queue associated with it, for the conceptual purposes of this paper. This article presents a new dynamic queue routing control scheme which violates FCFS and outperforms it in efficiency while being core-stable. The policy is now outlined. Agents can choose which queue or lane they want to switch to. Naturally, vehicles in front get to choose earlier. Agents are assumed to be having perfect knowledge of the delay per lane (or queue) as well as the delay increases due to nearby vehicles’ lane changes. Agents can communicate their values of time to any other vehicle they want to interact with. Agents can form coalitions and exchange payments among them to improve their utility. Our mechanism implements the most efficient allocation and ensures that all agents and any subset of agents present in the system are better off by participating and cooperating with the outcome solution. This is ensured by employing the concept of the core, the pillar of cooperative game theory. Broadly, the core is a feasible set defined by constraints which define the stability of coalitions based on their worth. We hasten to add here that such exchanges may not legally allowed in traffic systems; however, we assume that demonstrated social/system efficiency can lead to regulatory changes in future. Contrary to most common applications in cooperative game theory, traffic operations present externalities. This means that the worth of a particular coalition depends on what other coalitions do. This brings us to the domain of partition function games (PFG), a superset of the more commonly used characteristic function games (CFG), which are not complex enough to express externalities. Equivalent stability concepts to the core are defined for partition function games. In particular, we are going to use the strong core [@ChanderPFG2014], which we believe is small enough to be meaningful and apparently non-empty for the simplified version of the current application. A fundamental result in [@ichiishi1981] establishes an equivalence between strategic games and partition games. This paper will actually relate both approaches, since the strategic-cooperative interaction is modeled as the optimization of the union of n-level Stackelberg games with coalitions. We claim the following contributions in this article. First, we present this novel operational scheme for parallel queues and freeway management. Second, we are the first to use and solve a multiple-discrete-strategies n-level Stackelberg game with coalitions. Third, we found that the problem appears to be always strong-core stable for the vertical queue case, and that it is generally stable for the horizontal queue case as well. Finally, we propose a new relaxation for the strong-core concept found in [@ChanderPFG2014] and generalize it to the dynamic domain. The article is organized as follows: section 2 presents the meaningful literature from microeconomics on queue games and the utilized strategic structures, section 3 presents a static version on the cooperative queuing problem, modeling the queue routing as a parallel static vertical queue, section 4 presents a dynamic version of the problem, modeling the queue routing as parallel horizontal queues, and section 5 presents the conclusions and further research. Literature review ================= Microeconomics literature has extensively explored the stability, fairness and truthfulness of priority queues [@Chun2016book] for single and parallel queues [@Chun2008]. In priority queues, an unordered set of agents with heterogeneous values of time occupies positions on a line valued with linear delay. The efficient queue ordering is the one which places the agents sorted by decreasing value of time. However, queues in transportation systems involve agents with physical dimension and not all queue orderings are possible due to agent obstruction. [@bradford1996] studied pricing and incentives for a multiserver queuing facility. He analyzes both social welfare versus operator’s revenue but does not enforce budget balancedness nor cooperation of any kind. His results are based on steady state and reach standard marginal pricing conclusions in efficiency maximization. [@Yechiali1971] examines stationary equilibria in a GI/M/1 queue policy in which users agree over a common service toll, which is later redistributed among participants. However, it does not examine coalitional stability between agents nor applies a deterministic analysis on individual vehicles as the present article does. Other authors have tackled similar queue problem on the demand side [@Brink2015]. The cooperative interaction between agents for this problem is represented by partition function games. PFG are normally classified by the externality, either positive or negative, that a two coalitions forming creates on a third one. [@Hafalir2007] explores the role of convexity on efficiency and core stability. [@Abe2016] explores PFG with either positive of negative externalities. Similarly to characteristic function games, although less studied, several stability concepts have been developed for this more expressive concept [@hartkurz1983]. These concepts being generally too large when non-empty [@Chander1995; @Chander1997] propose the gamma-core and the strong-core [@ChanderPFG2014]. The strategic interaction between travelers has an inherent arrival ordering. The most adequate equilibrium concept is that of (multilevel) Stackelberg equilibrium. Much has been said about two and three level variants of this concept. Little has been studied on the more general multilevel case. [@Bialas1989a] explore coalition formation in level Stackelberg games for linear resource problems. On a related application, [@lloretbatlle2017ISTTT] proposes a queue jumping mechanism for general purpose freeway operations, in which vehicles coming from upstream can pay queued vehicles for being overtaken. Stability in the problem emanates from envy-freeness, naturally found in position enviornments [@Varian2007]. Parallel vertical queues: Static problem ======================================== We have a section of road which has a bottleneck downstream of constant outflow, this bottleneck can either be highway congestion, a multiserver queue from a port terminal or border crossing point or a saturated intersection. The section downstream has $m$ lanes, and the section upstream has $l$ lanes. $m$ may be larger, equal or smaller than $l$. From each lane $l\in L$, a subset $M_{l}\subseteq M$ is accessible. There is a set $N$ of vehicles approaching the queue from the back. Each lane $l\in L$ has $N_{l}\subseteq N$ vehicles. Each downstream lane $m$ has a queue $Q_{m}\geq0$ built up. Without loss of generality, we assume that $Q_{m^{\prime}}\leq Q_{m},\; \forall m^{\prime}>m$. These queues can represent actual queues of stopped vehicles or congested traffic in the link transmission model fashion. For the analysis in this section, the facility dispatches one vehicle per unit of time per queue. We decompose the ordered set of agents $N\backslash\{i\}$ in two sets. Let $A(i)=\{k\in N\;|\;k<i\}$ be the set of predecessors of agent $i$ and $F(i)=\{k\in N\;|\;k>i\}$ the set of followers of $i$. Let $A(m,i)$ be the set of predecessors of $i$ which choose lane $m$. Let $j_{m}=|A(m,i)|,\;\forall m\in M$. This defines a lane choice set $\sigma:\;N\rightarrow M^{N}$. Contrary to priority queues, in our model, a traveler cannot advance a predecessor unless he joins a queue which is shorter than the queue that predecessor has joined. The delay for agent $i\in N$ joining lane $m$ given lane choice set $\sigma(N)$ is $d_{i}(Q,\sigma(A(i)))=(Q_{m}+j_{m}-1)$ and the valuation experienced by agent $i$, $v_{i}(Q,\sigma(A(i)))=-\theta_{i}d_{i}(Q,\sigma(A(i)))$, where $\theta_{i}$ represents the value of delay in monetary units per unit of time. This variable will also be called the type of agent $i$. Agent $i$ is charged a price $p_{i}$ for bearing the delay $d_{i}(Q,\sigma(A(i)))$. Finally, his utility is $u_{i}=v_{i}(Q,\sigma(A(i)))-p_{i}$. {width="2in"} Vehicles upstream belong to different lanes and are ordered based on their proximity to the downstream boundary. Again without loss of generality, we disregard the lanes $l$ where vehicles are located, which is equivalent to assuming that $M_{l}=M$. If there is no communication between agents, vehicles will join the downstream bottleneck on a First-Come First-Served (FCFS) basis, each vehicle selecting the shortest queue. It is easy to see that if the initial arrival order is not monotonically decreasing on types, the resulting queue ordering will be inefficient, if we view efficiency in a utilitarian social welfare sense. However, if vehicles were to cooperate with each other, that is, forming coalitions to alter this initial ordering, a more efficient ordering for everyone would be achieved. This cooperation would be on terms of multilateral agreements on which lane, every vehicle of the coalition would choose. This defines a multilevel Stackelberg game with coalitions. Of course, such coalition-forming would require communications and decision-making of the kind human drivers in traffic may not accomplish, but apps representing them can accomplish, and the technology for it certainly exists already. [@Bialas1982; @Bialas1989a] explored some sufficient properties for stability for this kind of games, but on a linear resource allocation environment. We apply a similar recursion to our problem, but this time for pure strategies in extensive game. This recursion will give us the value generated by each coalition, given a particular coalition structure (partition of $N$). The recursion starts from the last vehicle and goes up. At each level, the user $i$ attempts to optimize the sum of the valuation of users who are behind it, $j\in N\;|\;j>i$ and which belong to the particular coalition $i$ belongs to. The back recursion to solve the n-level Stackelberg problem with coalitions is, given a partition $P=\{S_{1},...,S_{j}\}\;|\;\cap_{j}S_{j}=\emptyset$: $$V_{S(i)|P}^{*}(h)=\max_{k\in K_{i}(h)}\left\{ v_{i}(h,k)+V_{S(i)|P}^{*}(h\cup k)\right\} \;\forall i=n-1,...,1,\;h\in H_{i}\label{eq:recursion}$$ $$V_{S(n)|P}(h)=\max_{k\in K_{n}(h)}\left\{ v_{n}(h,k)\right\}$$ Where $S(i)$ is the coalition where $i$ belongs. (\[eq:recursion\]) shows that for every user $i$, for every past history $h\in H_{i}$ up to user $i$, the agent selects the action $k$ belonging to the set of available actions given $h$, $K_{i}(h)$, which maximizes the sum of two terms. The optimal value $V_{S(i)}^{*}(h\cup k)$ which emanates from the previous step $i+1$ and, $v_{i}(h,k)$, the valuation of user $i$ from choosing action $k$, given history $h$. {width="3in"}{width="4in"} The computational complexity of the recursion for the vertical queue case is $O(n(n+l)^{l})$: Counting the number of nodes at the final level is equivalent to finding the number of $l$-combinations with repetitions: $\left(\left(\begin{array}{c} l\\ n \end{array}\right)\right)=\left(\begin{array}{c} l+n-1\\ n \end{array}\right)=\frac{\prod_{i=1}^{l}(n+i)}{l-1}<\frac{(n+l)^{l}}{l-1}<(n+l)^{l}$ Since there are $n$ levels, the final complexity is $O(n(n+l)^{l})$. The recursion above needs to be solved for every partition $P\in\mathcal{P}$ to obtain all coalition values. The interaction between all the n-level Stackelberg games with coalitions will be modeled as a cooperative game. Cooperative games are complete-information games in which users are allowed to form coalitions to improve their payoffs. In the absence of coalitional externalities, cooperative games can be represented in a Characteristic Function Form (CFF), which defines a Characteristic Function Game (CFG). However, in the current problem there are externalities, which translates into coalitional payoffs being dependent on which other coalitions are formed. Thus, we will represent the game on a partition function form (PFF) which defines a Partition Function Game (PFG). [@ichiishi1981] shows that any strategic game as can be represented as a partition function game. Thus, we will translate the former strategic games as a single PFG. Let $P=\{S_{1},\dots,S_{k}\}\in\mathcal{P}$ be a partition of $N$ such that $S_{i}\cap S_{j}=\emptyset\;\forall i\neq j$. We define the partition function$v:\;2^{N}\times\mathcal{P}\rightarrow\Re$. That is, $v(S,P)$ represents the value of coalition $S$ when the partition $P$ is formed. $v(S,P)$ is in fact the sum of all the valuations $v_{i}\;\forall i\in S,\forall S\in P,$ coming from the optimal order resulting from the cooperation between agents belonging to $S$ when the partition formed is $P$. The pair$<N,v>$defines a partition function game. There is a coalition of special interest, the grand coalition $S_{G}=N$, which is composed by all members of the participant set. A fundamental question in cooperative game theory is if this total cooperation will eventually occur. This is desirable when the grand coalition is the most efficient coalition. When the partition function game arises from a strategic game, the grand coalition is always efficient since the set of strategies of the grand coalition game includes all the strategies available in the other subgames. In fact, the grand coalition payoff is the shortest path on the graph defined by the recursion (\[eq:recursion\]) when $P=N$. Conversely, a coalition can unanimously dissolve itself into singletons, since every member can still choose the same set of strategies. A fundamental characteristic of PFGs are the externalities that form on a coalition by third coalitions merging or splitting. Literature defines two types of externalities, positive and negative, which we define next. Positive (negative) externalities: $$\forall C,S,T\;|\;C\cap S\cap T=\emptyset,\;\forall\rho\in P(N-(S\cup T\cup C)):\;v(C;\{S\cup T,C\}\cup\rho)>(<)\;v(C;\{S,T,C\}\cup\rho)$$ Basically, a PFG displays positive externalities when two coalitions $S$ and $T$ merging, increases the value of a third coalition $C$, for any complementary partition $\rho$. Conversely, the externalities are negative if the value of $C$ is decreased. It is easy to see that this game has positive externalities. Let $\rho\in P(N\backslash(S\cup T\cup C))$. Suppose all agents forming coalition $C$ precede those of coalitions $S$ and $T$. Then, the merging of $S$ and $T$ has no influence on $v(C,\{C,S\cup T,\rho\})\;\forall\rho\in P(N\backslash(S\cup T\cup C))$ and the externality is zero. If agents forming $C$ go after $S$ and $T$, the externality can only be positive or zero since $S\cup T$ either causes some agents to join longer queues or keep the positions if no improvement is possible, reducing the cost of the members of $C$, which manage to advance some positions or none. However, if members of $C$ are between those of $S,T$ or $S\cup T$, negative externalities may occur. We provide an example next: The static problem with instance $N=\{13,2,14,41\},\;Q=\{4,1\}$ has negative externalties. Let $S=\{1,4\}$, $T=\{3\}$,$C=\{2\}$. Then, by solving the game with the recursion algorithm, we reach the case of $0=v(C|\{S\cup T,C\})<v(C|S,T,C)=2$. A useful property some PFG’s exhibit is superadditivity, which is defined next: $$\forall S,T\subseteq N\;|\;S\cap T=\emptyset,\;\forall\rho\in N-(S\cup T)\;,v(S\cup T;\{S\cup T\}\cup\rho)\geq v(S;\{S,T\}\cup\rho)+v(T;\{S,T\}\cup\rho)\label{eq:def_SA}$$ Superadditive means that if two coalitions $S$ and $T$ merge, their total payoff is larger than when unmerged. For the sake of exposition, the following counterexample shows that this game is not superadditive: $N=\{1,9,5,33\}$ and $h=3$. If $\rho=\{1,3\},\;S=\{2\},\;T=\{4\},$then $v(S\cup T,\{S\cup T,\rho\})=15,\;v(S,\{S,T\}\cup\rho)=9,\;v(T,\{S,T\}\cup\rho)=33$. PFG’s stability concepts and characterizations generally focus on games which have either positive or negative externalities [@Abe2016] or exhibit superadditivity or convexity [@Hafalir2007]. This is not the case of our problem. We found an exception in the literature, which is the strong-core for PFG [@ChanderPFG2014], which is defined next: Strong-core for PFGs [@ChanderPFG2014] $(x_{1},\dots,x_{n})\in\Re^{n}\;|\;\forall P\in\mathcal{P},P=\{S_{1},\dots,S_{p}\}\neq[N],\;\exists S_{i}\in P,|S_{i}|>1\;|\;\sum_{j\in S_{i}}x_{j}\geq v(S_{i},P)\;and\;if\;P=[N],\;x_{i}\geq v(i;[N])\;\forall i\in N$ The definition above states that for every partition which contains non-singleton coalitions, exists at least one coalition of those coalitions which is worse off than in the strong-core imputation $(x_{1},\dots,x_{n})$. Imputation is a term used in game theory to denote the utility agents obtain from a coalitional agreement. Moreover, every agent in the all-singleton partition is worse off. Contrary to other core specifications found in the literature such as the $\alpha,\beta,\gamma,\delta$-cores [@hartkurz1983], the strong-core does not assume any coalition structure for the complementary partition when a given coalition forms. The solution concept is particularly useful for the treatment of the status-quo partition, which, in our case, corresponds to the FCFS queue allocation. If an imputation vector belongs to the strong-core, then all the singleton coalitions belonging to the coalition made of just singletons are better off by merging into the grand coalition. Now the question that remains is to prove non-emptiness of the strong-core. As shown above, our problem has both positive and negative externalities. In order to the strong core to be non-empty, the PFG needs to satisfy two conditions: ([@ChanderPFG2014], Corollary 5) A PFG with general externalities $<N,v>$ has a non-empty strong-core if : 1. $v$ is partially superadditive: $\forall P=\{S_{1},\dots,S_{m}\}\in\mathcal{P},\;|\;|S_{i}|>1\;\forall i=1,\dots,k\;|S_{j}|\;\forall j=k/+1,\dots,m\;k\leq m,\;\sum_{i}^{k}v(S_{i},P)\leq v(S,P^{\prime})\;P^{\prime}=P\backslash\{S_{1},\dots,S_{k}\}\cup\{\cup_{i=1}^{k}S_{i}\}$ 2. and $<N,w^{\gamma}>$is balanced, where $w^{\gamma}(S)=v(S,\{S,[N\backslash S]\}),S\subset N$. Partial superadditivity is weaker than superadditive, which the game does not satisfy. Partial superadditivity is trivially satisfied for games with 3 or 4 players whenever the grand coalition is efficient. The strong-core for the static queuing game as vertical queue is non-empty. We leave the result as a conjecture since it was not possible to prove it. After having run a large number of simulations, we did not find any counterexample, either. The values employed for the simulation study are: $n\sim Unif(1,\bar{n}),\;\overline{n}\in[2,7],\;m\in[1,4],\;\theta\sim logn(\mu=2.16,\sigma=0.7),\;Q_{m}\sim Unif(1,4),\;Q_{i}\sim Unif(1,Q_{i+1})\;\forall i<m$. The semi-empirical distribution used to draw individual Valuation of Delay Savings is developed in [@Lloret-Batlle2016]. Standard proof methods, such as direct proof used in operations research games for Minimum Spanning Tree Games and Shortest Path games [@Borm2001], did not prove successful. Neither did other approaches such as reduction to market games. Moreover, the game did not prove to be convex, which is a sufficient condition for non-emptiness of strong-core. Instead, we are going to evaluate the inclusion of two imputations which are generalizations of the Shapley value for partition function games. These imputations satisfy the four properties of the original Shapley value: efficiency, symmetry, additivity and null player. The two imputations are presented next: [@dohorde2007; @Clippel2008] Externality-free value: $$\phi_{i}^{free}(v)=\sum_{S\subseteq N}\zeta_{S}^{i}v(S,\{S\}\cup\{\{j\}\;|\;j\in N\backslash S\})\;\forall i\in N\label{eq:ext_free}$$ Where: $$\zeta_{S}^{i}=\begin{cases} \begin{array}{cc} \frac{(|S|-1)!(|N|-|S|)!}{|N|!} & if\;i\in S\\ -\frac{|S|!(|N|-|S|-1)!}{|N|!} & if\;i\notin S \end{array}\end{cases}\label{eq:ext_free coeffs}$$ The $\zeta_{S}^{i}$ values arise from the reordering of the marginal increment $v(S\cup\{i\})-v(S)\;\forall S\subseteq N\backslash\{i\}$ expression, often found in the Shapley value definition, in terms of all the partitions $v(S)\;\forall S\subseteq N$. This value represents that an agent leaving the grand coalition always creates a new coalition, that is, a singleton. This value is inline with the $\gamma$-core present in the strong core existence theorem and we expect imputations from this value to generally belong to the strong-core. The second imputation to test is: [@McQuillin2009] McQuillin value: $$\phi_{i}^{McQ}(v)=\sum_{S\subseteq N}\zeta_{S}^{i}v(S,\{S,N\backslash S\})\;\forall i\in N$$ This value entails that an agent always chooses an existing coalition. The following MILP will be used to test the feasibility of $\phi^{free}(v)$ and $\phi^{McQ}(v)$ in the strong-core. We add the following objective function and modify the group rationality condition for the coalitions which have non-singleton partitions $P\in\hat{\mathcal{P}}=\mathcal{P}\backslash\{[N]\cup\{N\}\}$. We call this relaxation the$\epsilon$strong-core for PFG’s, in line with the $\epsilon$-core for characteristic function form games [@ShapleyShubik1966]. $$\min\epsilon$$ $$s.t.$$ $$\sum_{i\in S_{j}}x_{i}\geq v(S_{j},P)-\epsilon-Mz_{jp}\;\forall S_{j}\in\ddot{P}\subseteq P,\forall P\in\hat{\mathcal{P}}\label{eq:score_group_rat_relax-1}$$ $$x_{i}\geq v(\{i\},[N])-\epsilon\;\forall i\in N\label{eq:score_IR}$$ $$\sum_{i\in N}x_{i}=v(N,\{N\})\label{eq:score_eff}$$ **$$\sum_{S_{j}\in\ddot{P}}z_{jp}\leq|\ddot{P}|-1\;\forall\ddot{P}\subseteq P\in\hat{\mathcal{P}}\label{eq:score_bin_bound-1}$$** $$\epsilon\geq0,\;z_{jp}\in\{0,1\}\;\forall S_{j}\in\ddot{P}\subseteq P,\forall P\in\hat{\mathcal{P}}\label{eq:score_var_bounds}$$ Where $\ddot{P}\cup\dot{P}=P\;|\;\ddot{P}\cap\dot{P}\neq\emptyset$ are the collections of non-singleton coalitions $\ddot{P}$ and singleton coalitions $\dot{P}$ of every partition $P$. Essentially, the program consists of the relaxed group rationality constraints (\[eq:score\_group\_rat\_relax-1\]), the individually rational constraints (\[eq:score\_IR\]), the grand coalition efficiency (\[eq:score\_eff\]). The $\epsilon$ variable is the minimal slack for the most constraint coalition $S_{j}\in\ddot{P}\subseteq P$ necessary to make the problem feasible. The binary terms $z_{jp}$ present in (\[eq:score\_group\_rat\_relax-1\]) and (\[eq:score\_bin\_bound-1\]) enforce that at least one non-singleton coalition $S_{j}\in\ddot{P}\subseteq P$ for every $P\in\hat{\mathcal{P}}$ to be group rational. The settings of this simulation are the same ones than for strong-core non-emptiness evaluation. For each of these settings, 250 experiments are run. The next tables show the percent of instances where the imputation was found in the strong-core: $\phi^{free}:\;\overline{n}\backslash M$ 2 3 4 ------------------------------------------ ------- ------- ------- 2 100.0 100.0 100.0 3 100.0 97.2 98.0 4 96.0 85.2 88.4 5 87.6 75.6 72.4 6 84.0 67.6 64.0 7 74.4 52.8 61.6 : Percent of experiments whose imputations are in the strong-core. $\phi^{McQ}:\;\overline{n}\backslash M$ 2 3 4 ----------------------------------------- ------- ------- ------- 2 100.0 100.0 100.0 3 100.0 97.2 98.0 4 96.0 85.2 88.4 5 88.0 75.6 72.4 6 83.6 67.6 64.0 7 74.4 53.2 62.0 : Percent of experiments whose imputations are in the strong-core. We observe that both imputations provide very similar percentages of inclusion into the strong-core, the only differences being due to numerical errors from the solution algorithm. As expected, the larger the vehicle set and the larger the number of lanes, the more the instances of not belonging to the strong core. This result seems to be weaker when increasing the number of lanes than when increasing the maximum platoon size. Both imputations having identical percents suggest that a certain degree of symmetry exists in the violation of the strong-core in the coalition formation process, for both all-singleton blocking coalitions and all-maximum-cardinality complementary blocking coalitions. This can provide some insights for the non-emptiness proof of this vertical queue model. Parallel horizontal queues: Dynamic problem =========================================== Queues in transportation systems have a dynamic nature: they continuously receive arrivals and dispatch departures. Moreover, these queues are “horizontal”, that is, queue length has a physical dimension. Modeling such dynamic process as a succession of static queuing problems may lead to strong inter-temporal inefficiencies, coalitional stability and violation of individual rationality. One of the solutions, employed in [@lloretbatlle2017ISTTT] is to set up a reserve price which prevents low value vehicles from having too much present savings in detriment of further high value vehicles. Another alternative would be to add a terminal cost at the leaves of the static problem tree, however this would require a transfer of payment from former agents to new agents as well as knowledge of further arrivals and their values, which would difficult the dynamic budget balancedness and the efficiency of the system. In this particular problem, we will stick to a control solution which is always budget balanced and efficient, aiming an eventual V2V decentralized implementation. We model again a link with $M$ lanes, but this time with length $\lambda$. Vehicles enter the link upstream at a constant speed $v_{a}$. There is a bottleneck downstream with a constant outflow $q_{out}$ which corresponds to a headway $h_{q}$, spacing $s_{q}$ and speed $v_{q}<v_{a}$. Every time an event happens, vehicle communicate to each other their value of time and positions, including the vehicles at the back of the queue. An imputation which satisfies the minimal $\epsilon$-strong core is found and vehicles are assigned their new lanes. To minimize excessive perturbations due to the lane changes, only vehicles whose incorporation to the back of the queue is imminent will execute the lane change. The rest of vehicles will only execute the lane change once their incorporation becomes imminent. Naturally, the target lane can later change if there are further stability optimizations being executed due to new events happening. Vehicles participate in lane-changing optimizations only as long as their incorporation to the back of the queue is not imminent. Once a vehicle is queued, they do not participate in other cooperative lane changes and are supposed to stay in the queue. The exchange optimization is run at every instant when there is a significant event. We define an event as the arrival of new vehicle to the link or imminent proximity of a moving vehicle to the back of the queue, or any external unpredictable event which could be detected by any of the vehicles. Events happen at time instants $t$, called epochs. With this in mind, and using Newell’s simplified car-following model, vehicle’s $i$ predicted delay at epoch $t$ is the maximum quantity of two situations: arriving to the bottleneck downstream undelayed at free flow speed or being queued behind its predecessor: $$d_{i}^{t}(S,P)=\max\{t_{i-1,t}^{dep}(S,P)+\frac{s_{q}}{v_{q}},a_{i}\}-a_{i}$$ Since vehicles may participate in multiple optimizations, the utility specification is composed of the predicted total cost at epoch $t$ since its arrival to the system minus the price charged at the optimization executed during epoch $t$ minus the accumulated price charged to vehicle $i$ until $t$. $i$’s imputation from being in coalition $S$ and partition $P$ is: $$x_{i}^{t}(S,P)=v_{i}^{t}(S,P)-p_{i}^{t}(S,P)-\pi_{i}^{t-1}\;\forall i\in I_{t},\forall S\subseteq P,\forall P\in\hat{\mathcal{P}}$$ $$v_{i}^{t}(S,P)=-c_{i}^{t}(S,P)=-\theta_{i}d_{i}^{t}(S,P)$$ With the imputations and valuations being defined, the dynamic $\epsilon$-strong core program is defined by replacing $\forall i \in N^t$, $x_i^t$ and $v_i^t$ into equations (\[eq:score\_IR\] - \[eq:score\_bin\_bound-1\]). Naturally, the $\epsilon$ term in equations (\[eq:score\_var\_bounds\]) and in the objective function will be replaced at each program $t$ by the corresponding $\epsilon^t$. Parallel horizontal queues cannot benefit from the polynomial structure employed in the previous section. Vehicles’ costs do not depend this time only on the queue length in front of them, but on the dispatching times of the vehicles’ downstream. Since the departure times depend on the actual sequence of queued vehicles, a particular state is now defined by the sequence of vehicles in front of them and therefore the whole queuing tree needs to be explored. This increases the computational complexity of the problem. For this reason, we will limit the number of participant agents of every optimization to six. Any additional vehicles upstream will stay outside of the exchange and continue advancing through the link in a FCFS basis. The model with the full exponential tree is general enough to include jockeying strategies. That is, agents are allowed to switch queues after having joined a queue. We rule out this possibility in this paper for simplicity in exposition. The computational complexity of the recursion for the horizontal queue case is $O(l^{n})$: For the horizontal queue case, all the tree histories need to be explored. This defines a $l$-ary tree with $n$ levels. The last level has $O(l^{n})$ nodes. The exchange of information in positions also serves to define which lane changes are possible and which ones are obstructed. In our current formulation, if some lane change is not possible at a particular epoch $t$, the cost of it equivalent branch is set to $\infty$ and that strategy does not get explored. However, this is not implemented here. We explore next the core stability of the dynamic problem as horizontal queues. We run 6 1-hour long simulations for each of the scenarios defined by: $\lambda=200\;m,\;L\in\{2,3\},\;q_{in}\in\{360,540,720\}\;veh/h/lane,$ $\;q_{out}=900\;veh/h/lane,\;\theta\sim logn(\mu=2.16,\sigma=0.7)$. Arrivals arise from a binomial distribution for implementation easiness, but the simulation is still event-based: both time and distance are continuous. The simulation is coded in MATLAB and the MILP programs are solved with Gurobi 6.0.5. The next table (left) shows the percent of optimizations which are contained in the strong-score. We observe that for equal vehicle inflow, increasing the number of lanes increases core stability. This can be explained by the incoming platoon gets split into more lanes and their queues and interactions are smaller. Furthermore, increasing the incoming flow seems to increase instability, mostly due to a natural increase of complexity in the strategic interaction between agents. The table on the right displays the average ratio between $\epsilon$ and the average vehicle cost per optimization. This ratio is useful to compare the magnitude of the slack term, i.e. amount of utility that has to be transferred to a blocking coalition, with regard to the average cost of the participating agent. In line with the previous analysis, for equal inflow, a larger number of lanes decreases the magnitude of the instability. However, increasing the inflow seems to decrease the ratio, probably due to a larger increase in average vehicle cost. $\;q_{in} (veh/h) \backslash M$ 2 3 --------------------------------- ------ ------ 360 91.5 99.6 540 88.5 95.1 720 78.7 93.2 : (left) % of strong-core stable optimizations. (right) ratio $\epsilon/av.cost$ $q_{in} (veh/h) \backslash M$ 2 3 ------------------------------- ------ ------ 360 0.29 0.17 540 0.11 0.16 720 0.04 0.02 : (left) % of strong-core stable optimizations. (right) ratio $\epsilon/av.cost$ Conclusions and further research ================================ This article presented a new collaborative control mechanism for freeways and parallel queue facilities. Under this control scheme, agents observe predicted future delays per lane (or queue) and are allowed to collaborate to change lanes such that the total travel cost of their platoon is minimized. High VOT vehicles can pay low VOT vehicles to switch to a more congested lane while they can stay in the same lane or switch to another lane with less vehicles in front. The underlying cooperative principle is the strong-core for partition function games. While aimed as a decentralized, distributed control, the present paper assumes a centralized optimization to evaluate the economic efficiency and stability without excessive technicalities. The control policy has been first explored as a simpler vertical queue model. In this case, the strategic interaction between users structure forms a tree-like structure which is of polynomial-time complexity. While not proved, simulation results suggest that the problem may be strong-core stable. In addition, we have tested two generalizations of the Shapley value for partition function games which we found to generally be strong-core stable. In the next section, we modeled the control policy as a dynamic horizontal queue. We have observed that the policy is generally strong-core stable except for situations when there are sharp increases in the incoming platoon size. However, it is computationally intensive. Further distributed optimization techniques should be used to make it applicable for an eventual real-world implementation. Alternatively, designing an approximation algorithm for the strong-core optimization would serve. As further research we point out the following lines. We believe that developing a formal non-emptiness proof for the vertical case would represent a strong result in cooperative game theory. Also, from a theoretical point of view, exploring the dynamic vertical queue case would be interesting in the sense that dynamic applications in partition function games have never been explored outside of coalition formation in static settings. Concerning the dynamic horizontal queue, we will further explore unstable instances to better determine the source of instability and better understand the problem. Eventually, modeling such policy with a commercial microsimulation software, would provide insights on the efficiency increases of a real world situation, as well as including more realism and efficiency losses due to lane changing obstruction.

--- abstract: | This paper is concerned with the analysis of a class of optimal control problems governed by a time-harmonic eddy current system with a dipole source, which is taken as the control variable. A mathematical model is set up for the state equation where the dipole source takes the form of a Dirac mass located in the interior of the conducting domain. A non-standard approach featuring the fundamental solution of a ${\operatorname{curl}}{\operatorname{curl}}- \mathrm{Id} $ operator is proposed to address the well-posedness of the state problem, leading to a split structure of the state field as the sum of a singular part and a regular part. The aim of the control is the best approximation of desired electric and magnetic fields via a suitable $L^2$-quadratic tracking cost functional. Here, special attention is devoted to establishing an adjoint calculus which is consistent with the form of the state variable and in this way first order optimality conditions are eventually derived. author: - '[^1]' bibliography: - 'biblio.bib' title: | Optimal Control of an Eddy Current Problem\ with a Dipole Source --- Introduction ============ Preliminaries ============= Analysis of the state equation: the eddy current problem with a dipole source ============================================================================= The control problem =================== Necessary and sufficient conditions for optimality -------------------------------------------------- **Acknowledgements.** I am grateful to the PhD school in Mathematics of the University of Trento for its support and funding, and I wish to thank my advisor Alberto Valli for suggesting me this problem as well as for the counteless comments and corrections. [^1]: Department of Mathematics, University of Trento, Via Sommarive 14, Povo (TN), 38123 Italy ([gabriele.caselli@unitn.it]{}).

--- abstract: 'The Maximum Mean Discrepancy (MMD) has found numerous applications in statistics and machine learning, most recently as a penalty in the Wasserstein Auto-Encoder (WAE). In this paper we compute closed-form expressions for estimating the Gaussian kernel based MMD between a given distribution and the standard multivariate normal distribution. We introduce the standardized version of MMD as a penalty for the WAE training objective, allowing for a better interpretability of MMD values and more compatibility across different hyperparameter settings. Next, we propose using a version of batch normalization at the code layer; this has the benefits of making the kernel width selection easier, reducing the training effort, and preventing outliers in the aggregate code distribution. Finally, we discuss the appropriate null distributions and provide thresholds for multivariate normality testing with the standardized MMD, leading to a number of easy rules of thumb for monitoring the progress of WAE training. Curiously, our MMD formula reveals a connection to the Baringhaus-Henze-Epps-Pulley (BHEP) statistic of the Henze-Zirkler test and provides further insights about the MMD. Our experiments on synthetic and real data show that the analytic formulation improves over the commonly used stochastic approximation of the MMD, and demonstrate that code normalization provides significant benefits when training WAEs.' author: - 'Raif M. Rustamov, AT&T Labs Research, Bedminster, NJ' bibliography: - 'biblio.bib' title: 'Closed-form Expressions for Maximum Mean Discrepancy with Applications to Wasserstein Auto-Encoders' --- Introduction ============ The Maximum Mean Discrepancy (MMD) is a measure of divergence between distributions [@mmd] which has found numerous applications in statistics and machine learning; see the recent review [@MMD_review] and citations therein. MMD has a well-established theory, based on which a number of approaches are available for computing the thresholds for hypothesis testing, allowing to make sense of the raw MMD values; however, the whole process can be somewhat intricate. Given the increasing adoption, it is desirable to have closed-form expressions for the MMD so as to make it more accessible to a general practitioner and to streamline its use. Additionally, since the raw MMD values are hard to interpret, it would be important to convert MMD to a more intuitive scale and provide some easy to remember thresholds for testing and evaluating model convergence. To focus the paper, we will concentrate on an application of the MMD in the context of Wasserstein Auto-Encoders, which we now review. MMD quickly entered the neural network arena as a penalty/regularization term in generative modeling—initially within the moment-matching generative networks [@GMMN1; @GMMN2] and later on as a replacement for the adversarial penalty in Adversarial Auto-Encoders [@AdversAE] leading to the MMD version of Wasserstein Auto-Encoders [@InfoVAE; @vegan_cookbook; @WAE]. These WAE-MMDs, to which we will refer simply as WAEs, use an objective that in addition to the reconstruction error includes an MMD term that pushes the latent representation of data towards some reference distribution. Similarly to Variational Auto-Encoders [@VAE], WAEs can be used to generate new data samples by feeding random samples from the reference distribution to the decoder. By making a fundamental connection to optimal transport distances in the data space, [@vegan_cookbook; @WAE] establish theory proving the correctness of this generative procedure. Already in the context of WAEs there has been an effort to replace the MMD with closed-form alternatives. For example, Tabor et al. [@Cramer_Wold_AE] introduce the Cramer-Wold Auto-Encoders inspired by the slicing idea of [@kolouri2018sliced]. While their Cramer-Wold distance has a closed-form expression, it depends on special functions unless one uses an approximation. In addition, similarly to the situation with the MMD, the raw values of the Cramer-Wold distance are not directly interpretable. In this paper, we carry out the analytical computation of the MMD in a special case where the reference distribution is the standard multivariate normal and the MMD kernel is a Gaussian RBF. We are also able to compute the variance of the MMD in closed-form, which allows us to introduce the standardized version of the MMD. This version has the advantage of being more amenable to direct interpretation, which is demonstrated by a number of easy to remember rules of thumb suitable for model evaluation and hypothesis testing. As a curious development, our MMD formula reveals a relationship to the Baringhaus-Henze-Epps-Pulley (BHEP) statistic [@EP; @BH] and the Henze-Zirkler test [@HZ], which also allows making a connection to the Cramer-Wold distance. Focusing on WAEs as an application, we discuss the use of the closed-form standardized MMD as a penalty in the WAE training objective. Estimating the MMD the usual way requires sampling both from the latent code and the target reference distributions. The latter sampling incurs additional stochasticity which has an immediate effect on the gradients for training; using the analytic formula for the MMD essentially integrates out this extra stochasiticty. We also argue that standardization of the MMD induces better compatibility across different hyperparameter settings, which can be advantageous for model selection. As another contribution, we propose using code normalization— a version of the batch normalization [@batchnorm] applied at the code layer—when training WAEs. This has the benefits of making the selection of width for the MMD kernel easier, reducing the training effort, and preventing outliers in the aggregate latent distribution. The paper is organized as follows. Section \[sec:Closed-Form-Expressions\] provides closed-form expressions for the MMD and its variance. In Section \[sec:WAE-Training\], we discuss the standardized MMD and code normalization in the context of WAE training. Section \[sec:Hypothesis-Testing\] discusses thresholds for hypothesis testing, and their application to monitoring the WAE training progress. Section \[sec:Experiments\] provides an empirical evaluation on synthetic and real data. The derivations of the formulas and relevant code are provided in the appendix. \[sec:Closed-Form-Expressions\]Closed-Form Expressions for MMD ============================================================== The maximum mean discrepancy is a divergence measure between two distributions $P$ and $Q$. In the context of WAEs, applying the encoder net to the distribution of the input data (e.g. images) yields the aggregate distribution $Q$ of the latent variables. One of the goals of WAE training is to make $Q$ (which depends on the neural net parameters) as close as possible to some fixed target distribution $P$. This is achieved by incorporating MMD between $P$ and $Q$ as a regularizer into the WAE objective. The computation of the MMD requires specifying a positive-definite kernel; in this paper we always assume it to be the Gaussian RBF kernel of width $\gamma$, namely, $k(x,y)=e^{-\Vert x-y\Vert^{2}/(2\gamma^{2})}$. Here, $x,y\in\mathbb{R}^{d}$, where $d$ is the dimension of the code/latent space, and we use $\Vert\cdot\Vert$ to denote the $\ell_{2}$ norm. The population MMD can be most straight-forwardly computed via the formula [@mmd]: $$\mathrm{MMD}^{2}(P,Q)=\E_{x,x'\sim P}[k(x,x')]-2\E_{x\sim P,y\sim Q}[k(x,y)]+\E_{y,y'\sim Q}[k(y,y')].\label{eq:mmd}$$ In this paper, the target reference distribution $P$ is always assumed to be the standard multivariate normal distribution $\mathcal{N}_{d}(\vec{0},I)$ with the density $p(x)=(2\pi)^{-d/2}e^{-\Vert x\Vert^{2}/2}$, $x\in\mathbb{R}^{d}$. In practical situations, we only have access to $Q$ through a sample. For example, during each step of the WAE training, the encoder neural net will compute the codes $z_{i},i=1,...,n$ corresponding to the input data in the batch (we use “batch” to mean “mini-batch”) and the current values of neural network parameters. Given this sample from $Q$, our goal is to derive a closed-form estimate of $\mathrm{MMD^{2}}(P,Q)$. In this section, we first consider deterministic encoders and derive an analytic formula for an unbiased estimator of the MMD and its variance. Next, we discuss the biased estimator and its connection to BHEP statistics. Finally, we derive a formula for the estimator of the MMD in the case of random encoders. \[subsec:Deterministic-Encoders\]Deterministic Encoders ------------------------------------------------------- #### Unbiased Estimator {#unbiased-estimator .unnumbered} We start with the expression Eq. (\[eq:mmd\]) and using the sample $z_{i},i=1,...,n$, we replace the last two terms by the sample average and the U-statistic respectively to obtain the unbiased estimator: $$\mathrm{MMDU}_{n}^{2}(\mathcal{N}_{d}(\vec{0},I),Q)=\E_{x,x'\sim\mathcal{N}_{d}(\vec{0},I)}[k(x,x')]-\frac{2}{n}\sum_{i=1}^{n}\E_{x\sim\mathcal{N}_{d}(\vec{0},I)}[k(x,z_{i})]+\frac{1}{n(n-1)}\sum_{i=1}^{n}\sum_{j\neq i}^{n}k(z_{i},z_{j}).\label{eq:mmdu}$$ This quantity is denoted by $\mathrm{MMD}_{u}^{2}$ in [@mmd]; our slightly different notation allows including the sample size as a subscript. In Appendix \[appendix:Deterministic-Encoders\] we show that the expectations in this expression can be computed analytically to yield the formula $$\mathrm{MMDU}_{n}^{2}(\mathcal{N}_{d}(\vec{0},I),Q)=\left(\frac{\gamma^{2}}{2+\gamma^{2}}\right)^{d/2}-\frac{2}{n}\left(\frac{\gamma^{2}}{1+\gamma^{2}}\right)^{d/2}\sum_{i=1}^{n}e^{-\frac{\Vert z_{i}\Vert^{2}}{2(1+\gamma^{2})}}+\frac{1}{n(n-1)}\sum_{i=1}^{n}\sum_{j\neq i}^{n}e^{-\frac{\Vert z_{i}-z_{j}\Vert^{2}}{2\gamma^{2}}}.$$ This formula for $\mathrm{MMDU}_{n}^{2}$ reveals two forces at play when optimizing $Q$ to have small divergence from the standard multi-variate normal distribution. One force is pulling the sample points towards the origin, and the other is pushing them apart from each other. Namely, we can see that the second term encourages the sample points $z_{i}$ to be as close as possible to the origin so as to make the exponentials as large as possible. If not for the third term, all of the points would have collapsed onto the origin making the exponentials equal to $1$. However, the third term introduces repealing forces between the sample points, and pushes them away from each other. It is interesting to note that the second and third terms have different widths for the Gaussian kernels. Another observation is that one can compute the optimal translation transform for a given sample, and surprisingly it is not the one that places the center of mass at the origin. In fact, during this shift optimization the third term stays constant, and the second term can be interpreted (up-to a constant factor) as a kernel density estimate with the kernel width of $1+\gamma^{2}$. The optimal shift is the one that places the mode of this density estimate at the origin. Since its computation involves taking a random sample from $Q$, we see that $\mathrm{MMDU}_{n}^{2}$ is a random variable. Thus, even when $Q=P=\mathcal{N}_{d}(\vec{0},I)$, the estimator $\mathrm{MMDU}_{n}^{2}$ will not be identically zero. It is important to understand the behavior of this random variable; using the hypothesis testing terminology, we refer to this as the distribution of $\mathrm{MMDU}_{n}^{2}$ under the null—the null hypothesis being $Q=P=\mathcal{N}_{d}(\vec{0},I)$. By unbiasedness, we have that the null mean is zero: $$\mathbb{E}[\mathrm{MMDU}_{n}^{2}(\mathcal{N}_{d}(\vec{0},I),\mathcal{N}_{d}(\vec{0},I))]=\mathrm{MMD}^{2}(\mathcal{N}_{d}(\vec{0},I),\mathcal{N}_{d}(\vec{0},I))=0,$$ where the expectation is over various realizations of the sample $z_{i},i=1,...,n$ from $Q=\mathcal{N}_{d}(\vec{0},I)$. This immediately means that in contrast to $\mathrm{MMD}^{2}$, the estimator $\mathrm{MMDU}_{n}^{2}$ can take negative values. Next, we would like to obtain the variance $\mathrm{MMDU}_{n}^{2}$ under the null. First, we rewrite $\mathrm{MMDU}_{n}^{2}$ by defining, $$h(z,z')=\left(\frac{\gamma^{2}}{2+\gamma^{2}}\right)^{d/2}-\left(\frac{\gamma^{2}}{1+\gamma^{2}}\right)^{d/2}e^{-\frac{\Vert z\Vert^{2}}{2(1+\gamma^{2})}}-\left(\frac{\gamma^{2}}{1+\gamma^{2}}\right)^{d/2}e^{-\frac{\Vert z'\Vert^{2}}{2(1+\gamma^{2})}}+e^{-\frac{\Vert z-z'\Vert^{2}}{2\gamma^{2}}},$$ and noting that $$\mathrm{MMDU}_{n}^{2}(\mathcal{N}_{d}(\vec{0},I),Q)=\frac{1}{n(n-1)}\sum_{i=1}^{n}\sum_{j\neq i}^{n}h(z_{i},z_{j}).$$ Now according to [@mmd Appendix B.3 ] we have $$\mathbb{E}\left[\left(\mathrm{MMDU}_{n}^{2}(\mathcal{N}_{d}(\vec{0},I),\mathcal{N}_{d}(\vec{0},I))\right)^{2}\right]=\frac{2}{n(n-1)}E_{z,z'\sim\mathcal{N}_{d}(\vec{0},I)}[h^{2}(z,z')].$$ This expression can be computed in a closed form using manipulations similar to those used for computing $\mathrm{MMDU}_{n}^{2}$. Since the mean of $\mathrm{MMDU}_{n}^{2}$ under the null is $0$, the null variance is equal to the second moment, and we obtain the formula, $$\begin{aligned} \mathrm{Var}(\gamma,d,n)\triangleq & \mathbb{E}\left[\left(\mathrm{MMDU}_{n}^{2}(\mathcal{N}_{d}(\vec{0},I),\mathcal{N}_{d}(\vec{0},I))\right)^{2}\right]=\nonumber \\ = & \frac{2}{n(n-1)}\left[\left(\frac{\gamma^{2}}{2+\gamma^{2}}\right)^{d}+\left(\frac{\gamma^{2}}{4+\gamma^{2}}\right)^{d/2}-2\left(\frac{\gamma^{4}}{(1+\gamma^{2})(3+\gamma^{2})}\right)^{d/2}\right]\label{eq:Variance}\end{aligned}$$ #### Biased Estimator and BHEP Statistic {#biased-estimator-and-bhep-statistic .unnumbered} The biased estimator from [@mmd] can be computed in closed form in a similar manner. The only difference is the use of the V-statistic for the third term in Eq. (\[eq:mmd\]); the final expression is as follows: $$\mathrm{MMDB}_{n}^{2}(\mathcal{N}_{d}(\vec{0},I),Q)=\left(\frac{\gamma^{2}}{2+\gamma^{2}}\right)^{d/2}-\frac{2}{n}\left(\frac{\gamma^{2}}{1+\gamma^{2}}\right)^{d/2}\sum_{i=1}^{n}e^{-\frac{\Vert z_{i}\Vert^{2}}{2(1+\gamma^{2})}}+\frac{1}{n^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}e^{-\frac{\Vert z_{i}-z_{j}\Vert^{2}}{2\gamma^{2}}}.$$ Interestingly, this expression is equivalent to a statistic proposed for testing multivariate normality, its history going back to as early as 1983. The Baringhaus-Henze-Epps-Pulley (BHEP) statistic is named after the authors of [@EP; @BH] as coined by [@Csorgo]. This statistic is used in the Henze-Zirkler test of multivariate normality [@HZ]. We give a quick review of this connection since it provides some useful insights about the MMD. ** The BHEP statistic is a measure of divergence between two distributions $P$ and $Q$ that captures how different their characteristic functions are. It is defined as the weighted $L^{2}$-distance: $$W(P,Q)=\int_{\mathbb{R}^{d}}\vert\Psi^{P}(t)-\Psi^{Q}(t)\vert^{2}\varphi(t)dt$$ where $\Psi^{P}(t)$ and $\Psi^{Q}(t)$ are the characteristic functions of the distributions $P$ and $Q$, and $\varphi(t)$ is a weight function. When $P$ is the multivariate normal distribution $\mathcal{N}_{d}(\vec{0},I)$, we have $\Psi^{P}(t)=\exp(-\Vert t\Vert^{2}/2)$. Selecting the weight function to be $\varphi_{\beta}(t)=(2\pi\beta^{2})^{-d/2}e^{-\Vert x\Vert^{2}/(2\beta^{2})}$ and noting that $Q$ is available through a sample $z_{i},i=1,...,n$, the BHEP statistic takes the following form: $$W_{n,\beta}=\int_{\mathbb{R}^{d}}\vert\exp(-\Vert t\Vert^{2}/2)-\Psi_{n}^{Q}(t)\vert^{2}\varphi_{\beta}(t)dt.$$ Here $\Psi_{n}^{Q}(t)$ is the empirical characteristic function of $Q$, $$\Psi_{n}^{Q}(t)=\frac{1}{n}\sum_{i=1}^{n}\exp(\sqrt{-1}\:t\cdot z_{i}),\;t\in\mathbb{R}^{d}.$$ A closed-form formula for $W_{n,\beta}$ can be obtained (see e.g. [@HZ; @HenzeWagner1997]) and *it coincides with the expression for* $\mathrm{MMDB}_{n}^{2}$ *when one sets the Gaussian RBF kernel width* $\gamma=1/\beta$. This connection has a number of useful consequences. Henze and Zirkler [@HZ] show that BHEP statistic can be equivalently obtained as the $L^{2}$-distance between kernel density estimates; in our context, this is a concrete example of the connection described in [@mmd Section 3.3.1]. Based on this equivalence, [@HZ] suggests using a specific value of $\beta$ from optimal density estimation theory. The corresponding $\gamma$ is $$\gamma_{d,n}=\frac{1}{\beta_{d,n}}=\sqrt{2}\left(\frac{4}{2d+1}\right)^{\frac{1}{d+4}}n^{-\frac{1}{d+4}};\label{eq:HZgamma}$$ we will refer to this setting as HZ $\gamma$ in our experimental section. We also note that the one-dimensional distance used in the definition of the Cramer-Wold distance [@Cramer_Wold_AE] is based on exactly the same $L^{2}$-distance between kernel density estimates. As a result, we see that the Cramer-Wold distance is the integral of $\mathrm{MMDB}_{n}^{2}$ over all one-dimensional projections of $Q$. Of course, by similarly integrating $\mathrm{MMDU}_{n}^{2}$ instead, one could introduce a new version of the Cramer-Wold distance that is zero centered under the null. On a conceptual level, this connection allows transferring some insights about the BHEP statistic to the MMD. For example, inspecting the relationship between the MMD and the characteristic function formulation of the BHEP statistic, we see that this formulation more transparently expresses the fact that MMD is performing moment matching. As another example, [@HZ] makes the following qualitative observation: “Choosing a small value of $\beta$ entails that the weight function $\varphi_{\beta}$ puts most of its mass near the origin of $\mathbb{R}^{d}$. Since the tail behavior of a probability distribution is reflected by the behavior of its characteristic function at the origin, the test should be sensitive against alternative distributions with heavy tails.” This intuition is made concrete by studying the limiting behavior of BHEP statistic in [@Henze_limiting] and making connections to Mardia’s kurtosis and skewness statistics[@Mardia], with a summary provided in [@HenzeWagner1997]. When translated to our setting, this means that MMD with a large value of $\gamma$ is useful for distinguishing distributions that have heavy tails. *Remark:* We leave out the computation of the null mean and variance of the $\mathrm{MMDB}_{n}^{2}$; this can be carried out similarly to $\mathrm{MMDU}_{n}^{2}$. Note that the mean and variance of $W_{n,\beta}$ are computed in closed-form in [@HZ]. However, these expressions are based on a different null hypothesis (composite null in Section \[sec:Hypothesis-Testing\]) and have corrections for nuisance parameter estimation. Thus, they should not be used for standardization in Section \[sec:Scaled-Closed-Form-MMD\]. \[subsec:Random-Encoders\]Random Encoders ----------------------------------------- In this section we consider Gaussian random encoders, where instead of one code per input data point, we obtain a distribution of codes given as $z_{i}\sim N(\mu_{i},\Sigma_{i}),i=1,2,...,n$. Here $n$ is the batch size, $\Sigma_{i}$ is a diagonal covariance matrix, $\Sigma_{i}=\mathrm{diag}(\sigma_{i1}^{2},\sigma_{i2}^{2},...,\sigma_{id}^{2})$. Both mean vectors $\mu_{i}\in\mathbb{R}^{d}$ and variance vectors $\sigma_{i}\in\mathbb{R_{+}}^{d}$ are computed by applying neural nets to the input data. Our goals is once again to obtain an estimator for $\mathrm{MMD^{2}}(\mathcal{N}_{d}(\vec{0},I),Q)$. Note that the implied distribution of $Q$ for the current batch is an equally weighted mixture of Gaussians $Q_{\mathrm{batch}}$ with the distribution given by: $$q_{\mathrm{batch}}(z)\sim\frac{1}{n}\sum_{i=1}^{n}\prod_{k=1}^{d}\frac{e^{-(z_{k}-\mu_{ik})^{2}/(2\sigma_{ik}^{2})}}{\sqrt{2\pi\sigma_{ik}^{2}}},$$ where $z_{k}$ is the $k$-th component of the vector $z\in\mathbb{R}^{d}$. We will replace sampling from $Q$ in the formula Eq. (\[eq:mmd\]), by sampling from $Q_{\mathrm{batch}}$, and compute the second and third terms in a closed form. Note that the first term depends only on $P$ and will be the same as before; the computation of the remaining terms is demonstrated in Appendix \[appendix:Random-Encoders\], and yields the following unbiased estimator: $$\begin{aligned} \mathrm{MMDU}_{n}^{2}(\mathcal{N}_{d}(\vec{0},I),Q)= & \left(\frac{\gamma^{2}}{2+\gamma^{2}}\right)^{d/2}-\frac{2}{n}\sum_{i=1}^{n}\prod_{k=1}^{d}\left(\frac{\gamma^{2}}{1+\gamma^{2}+\sigma_{ik}^{2}}\right)^{1/2}e^{-\frac{\mu_{ik}{}^{2}}{2(1+\gamma^{2}+\sigma_{ik}^{2})}}+\\ + & \frac{1}{n^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}\prod_{k=1}^{d}\left(\frac{\gamma^{2}}{\gamma^{2}+\sigma_{ik}^{2}+\sigma_{jk}^{2}}\right)^{1/2}e^{-\frac{(\mu_{ik}-\mu_{jk}){}^{2}}{2(\gamma^{2}+\sigma_{ik}^{2}+\sigma_{jk}^{2})}}.\end{aligned}$$ When the noise is isotropic, namely $\Sigma_{i}=\mathrm{diag}(\sigma_{i}^{2},\sigma_{i}^{2},...,\sigma_{i}^{2})$ with $\sigma_{i}\in\mathbb{R_{+}}$(note that $\sigma_{i}$ was a vector in the general case above, but here it is a single number), we can rewrite this formula in a simpler form: $$\begin{aligned} \mathrm{MMDU}_{n}^{2}(\mathcal{N}_{d}(\vec{0},I),Q)= & \left(\frac{\gamma^{2}}{2+\gamma^{2}}\right)^{d/2}-\frac{2}{n}\sum_{i=1}^{n}\left(\frac{\gamma^{2}}{1+\gamma^{2}+\sigma_{i}^{2}}\right)^{d/2}e^{-\frac{\Vert\mu_{i}\Vert{}^{2}}{2(1+\gamma^{2}+\sigma_{i}^{2})}}+\\ + & \frac{1}{n^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}\left(\frac{\gamma^{2}}{\gamma^{2}+\sigma_{i}^{2}+\sigma_{j}^{2}}\right)^{d/2}e^{-\frac{\Vert\mu_{i}-\mu_{j}\Vert{}^{2}}{2(\gamma^{2}+\sigma_{i}^{2}+\sigma_{j}^{2})}}.\end{aligned}$$ Note that setting the variances $\sigma_{i}^{2}=0$ gives rise to the deterministic encoders where $z_{i}=\mu_{i}$, and the resulting estimator is the same as $\mathrm{MMDB}_{n}^{2}$ and not $\mathrm{MMDU}_{n}^{2}$. The difference is that the last term in the unbiased deterministic estimator includes an average over distinct pairs $(i,j),i\neq j$, whereas for the unbiased random estimator the average runs over all pairs $(i,j)$. The latter is appropriate here because when $\sigma_{i}^{2}\neq0$, in Eq. (\[eq:mmd\]) one can sample $y,y'\sim Q_{\mathrm{batch}}$ independently from the same component of the Gaussian mixture. Doing so in the deterministic case would have resulted in a biased estimate: essentially instead of the U-statistic we would have gotten the upwards biased V-statistic. \[sec:WAE-Training\]WAE Training ================================= \[sec:Scaled-Closed-Form-MMD\]Standardized MMD Penalty ------------------------------------------------------ In the original formulation of the WAE, the MMD penalty enters the objective as the term $\lambda\cdot\mathrm{MMD}^{2}(\mathcal{N}_{d}(\vec{0},I),Q)$, and an estimate is computed by sampling from both $P=\mathcal{N}_{d}(\vec{0},I)$ and $Q$. Obviously, the closed-form formulas for the MMD presented in the previous section can be used instead. In addition, we suggest standardizing the MMD values to make them more directly interpretable. We start by defining the standardized MMD by applying centering and scaling to the unbiased estimator. Since the mean of $\mathrm{MMDU}_{n}^{2}$ under the null is zero, and variance under the null is $\mathrm{Var}(\gamma,d,n)$ as given by Eq. (\[eq:Variance\]), we define for a batch of codes $z_{i},i=1,...,n$ sampled from $Q$: $$\mathrm{SMMDU}_{n}^{2}(Q)\triangleq\frac{\mathrm{MMDU}_{n}^{2}(\mathcal{N}_{d}(\vec{0},I),Q)}{\sqrt{\mathrm{Var}(\gamma,d,n)}}.$$ A Python implementation of this formula is provided in Appendix \[appendix:mmd-python\]. Note that this definition can be used both for deterministic and random encoders, with the only difference being in the formula for the MMD estimator in the numerator. Now, the WAE objective looks as follows, $$\mathrm{Reconstruction\,Loss}+\lambda\mathrm{\cdot SMMDU}_{n}^{2}(Q).$$ Having a closed-form formula for the SMMDU both in deterministic and random case is advantageous for optimization. Computing this penalty the usual way [@mmd; @WAE; @InfoVAE] relies on taking a sample from both $P=\mathcal{N}_{d}(\vec{0},I)$ and $Q$. As a result, this incurs additional stochasticity due to the sampling from $P$. Our formula essentially integrates out this stochasticity, and results in an estimator with a smaller variance. This allows better discrimination between distributions (see Section \[sec:Experiments\]), and, as a result, potentially provides higher quality gradients for training. In some settings, there can also be a computational benefit to using our formula as it requires the computation of $n\times n$ distance matrices in contrast to the $2n\times2n$ ones required by the sampling approach. While at the theoretical level the suggested scaling can be equivalently seen as a re-definition of the regularization coefficient, yet it has a number of benefits in practice. The main advantage of using the SMMDU is that it is amenable to quick inspection when one wants to have a sense of how far the current code distribution is from the target normal multivariate distribution. Indeed, in contrast to the raw MMD numbers, there are easy to remember rules of thumb for SMMDU values that can be used when monitoring the WAE training progress; these will be provided in Section \[sec:Hypothesis-Testing\]. ![\[fig:Variance-curve\]Variance as a function of kernel width $\gamma$ and latent dimensionality $d$. Batch size is fixed to $n=100$.](Figs/variance_curve_plot.pdf){width="100.00000%"} Next, before discussing other benefits of the standardization, we use Figure \[fig:Variance-curve\] to give an insight on the behavior of the scaling term in the denominator of the SMMDU expression. This figure depicts the graphs of $\sqrt{\mathrm{Var}(\gamma,d,n)}$ for different values of the kernel width $\gamma$ and the latent dimension $d$, for a fixed batch size $n=100$. We can clearly see that the scaling varies widely not only across dimensions, but also across kernel width choices at a fixed dimension. The use of the SMMDU as a penalty in the WAE objective is potentially beneficial for model selection.The choice of the best hyperparameters is usually carried out via cross-validation which among others things includes trying out different values of the penalty coefficient $\lambda$, kernel width $\gamma$, and latent dimension $d$. Without the proposed scaling of the MMD term, the values of $\lambda$ are not universal across the choices of $\gamma$ and $d$, which makes cross-validation more difficult. For example, if a small list of $\lambda$’s is used when cross-validating, then disparate regions of the optimization space would be considered across the choices of $\gamma$ and $d$, perhaps resulting in a suboptimal model being chosen. Our scaled formulation can also be beneficial for the commonly used trick of combining kernels of different widths in order to boost the performance of the MMD metric and to avoid search over the kernel width. This is equivalent to adding more MMD terms as a penalty, for example, using a penalty of the form $\lambda\cdot(\mathrm{MMD}^{2}(\mathcal{N}_{d}(\vec{0},I),Q,\gamma=\gamma_{1})+\mathrm{MMD}^{2}(\mathcal{N}_{d}(\vec{0},I),Q,\gamma=\gamma_{2})+\mathrm{MMD}^{2}(\mathcal{N}_{d}(\vec{0},I),Q,\gamma=\gamma_{3}))$. However, when such a combination is performed without the proposed standardization, then MMDs coming from different kernel width choices can be of different orders of magnitude. As a result, one may end up with a single kernel width dominating. In fact, the common choice of including kernels having $s=\gamma^{2}/d\approx1$ together with the ones that have $s=\gamma^{2}/d\ll1$ or $\gg1$ would lead to this issue as can be seen from Figure \[fig:Variance-curve\]. One can see that this observation is also relevant in cases where $\gamma$ is set adaptively per batch, this time leading to various amounts of penalty being applied to each batch. \[subsec:Code-Normalization\]Code Normalization ----------------------------------------------- In this subsection we propose to apply a variant of batch normalization [@batchnorm] on top of the code layer: for each batch, we center and scale the codes so that their distribution has zero mean and unit variance in each dimension; we will refer to this as “code normalization”. Importantly, no scaling or shifting is applied after normalizing (i.e. $\gamma=1$ and $\beta=0$ in the notation of [@batchnorm]) as the decoder network expects a normally distributed input. For deterministic encoders this version of batch normalization is readily available in the existing neural net packages. For example, in Keras [@keras] this would be the layer `BatchNormalization(center=False, scale=False)`. We note, however, that the existing implementations often use the biased estimate of the sample variance which leads to some distortion in the long run; this can be easily remedied by multiplying the variance by $n/(n-1)$. Random encoders require a separate treatment of the mean and variance network outputs. Namely, using the notation of Section \[subsec:Random-Encoders\], code normalization is given in coordinate-wise manner by $$\mu_{\cdot k}\rightarrow(\mu_{\cdot k}-\mathrm{Mean}_{k}(Q_{\mathrm{batch}}))/\mathrm{SD}_{k}(Q_{\mathrm{batch}}),\:\mathrm{and\quad}\sigma_{\cdot k}\rightarrow\sigma_{\cdot k}/\mathrm{SD}_{k}(Q_{\mathrm{batch}}),$$ where subscript $k$ is used to refer to the $k$-th coordinate, $k=1,2,...,d$, and $\mathrm{SD}_{k}(Q_{\mathrm{batch}})=\sqrt{\mathrm{Var}_{k}(Q_{\mathrm{batch}})}$. Since $Q_{\mathrm{batch}}$ is a mixture of Gaussians, closed form expressions for mean and variance are available: $$\mathrm{Mean}_{k}(Q_{\mathrm{batch}})=\frac{1}{n}\sum_{i=1}^{n}\mu_{ik},\:\mathrm{and}\qquad\mathrm{Var}_{k}(Q_{\mathrm{batch}})=\frac{1}{n}\sum_{i=1}^{n}(\mu_{ik}^{2}+\sigma_{ik}^{2})-\left(\frac{1}{n}\sum_{i=1}^{n}\mu_{ik}\right)^{2}.$$ During inference, one uses the population statistics to normalize the codes. This can be achieved by processing multiple mini-batches from the training data at once to compute the required means and standard deviations of the code layer. Another option is to keep a running exponential average of the mean and variance and use them for normalization—this is what the Keras implementation does. Below under separate headings we discuss the benefits of code normalization for the WAE training; we will use the term “MMD penalty” to refer to any kind of penalty based on MMD, including SMMDU. #### Easier Kernel Width Selection {#easier-kernel-width-selection .unnumbered} One advantage of code normalization is that a single setting of the width for the Gaussian RBF kernel, $\gamma$, can be used when computing the MMD penalty. Without code normalization, a fixed choice of $\gamma$ leads to issues. For example, when $\gamma$ is small, and the codes are far away from the origin and from each other, the MMD penalty term has small gradients, which makes learning difficult or even impossible. Indeed, the exponentials become vanishingly small, and since they enter the gradient multiplicatively this makes the gradients small as well. The same issue arises when choosing a large value of $\gamma$ when the codes are not far away from the origin. Thus, one has to use an adaptive choice of $\gamma$ in order to deal with this problem, see e.g. [@WAE] and also our Appendix \[appendix:mmd-python\] for one such particular choice. On the other hand, in the long run, code normalization makes sure that the codes have commensurate distances with $\gamma$ throughout the training process, alleviating the need for an adaptive $\gamma$. This makes possible to decouple the choice of $\gamma$ from the neural network training and to provide practical recommendations as we do in Section \[sec:Experiments\]. #### Reduced Training Effort {#reduced-training-effort .unnumbered} ![\[fig:codenorm-effect\]Code normalization shifts the distribution of $\mathrm{SMMDU}_{n}^{2}$ to lower values. Here, $n=100$, $d=8$, and kernel scale $s=\gamma^{2}/d=1/4$.](Figs/histogram_bnorm_vs_none.pdf){width="100.00000%"} Code normalization shifts and scales codes to be in the “right” part of the space, namely where the target standard multivariate normal distribution lives, and we speculate that this reduces the training effort. The intuition comes from inspecting the relationship between the MMD and the characteristic function formulation of the BHEP statistic. This formulation expresses the fact that at some level MMD is performing moment matching, and so by rendering the first two moments (marginal) equal to those of the standard multivariate normal distribution, code normalization focuses the training effort on matching the higher moments. To illustrate this point, Figure \[fig:codenorm-effect\] shows the distribution of $\mathrm{SMMDU}_{n}^{2}$ values for samples of size $n=100$ taken from $Q=\mathcal{N}_{d}(\vec{0},I)$. The value of $\mathrm{SMMDU}_{n}^{2}$ is computed for the original sample and then for the sample to which code normalization was applied. Note that on average the normalized codes have smaller MMD values compared to the original ones. See Section \[sec:Hypothesis-Testing\] for more about this distribution shift together with an even stronger shift when samples are whitened; in some sense, normalized samples are more “ideal” from the point of the view of the MMD. This means that even if the neural network has converged to the target normal distribution, the gradient for a batch will not be zero but will have components in the direction of shifting and scaling the codes to reduce the MMD for a given batch. Code normalization directly takes care of this reduction, and allows the training process to spend its effort on improving the reconstruction error. Technically, this is achieved by projecting out the components of the gradient corresponding to shifting and scaling which is automatically achieved by normalization, see [@BatchRenorm Section 3, penultimate paragraph]. This observation reveals an interesting aspect of training with the MMD as compared to training in an adversarial manner [@AdversAE]. When training in an adversarial manner, the goal is to make the codes in each batch resemble a sample from the standard multivariate normal distribution. At an intuitive level, we expect this would happen with the MMD penalty as well. However, this is not the case—we see that, on average, the MMD penalty considers normalized/whitened samples more “ideal” than the actual samples from the target distribution. Luckily, the neural network cannot learn batch-wise operations (e.g. it cannot learn to do *batch-wise* normalization or whitening by *itself*) assuming that at inference time the inputs are processed independently of each other. As a result, this phenomenon will not prevent convergence to the target distribution. A rigorous argument follows from unbiasedness, $\mathbb{E}[\mathrm{MMDU}_{n}^{2}]=\mathrm{MMD}^{2}\geq0$ where the expectation is taken over i.i.d. samples and equality holds only at convergence to the target distribution; this makes any overall shift to the left at the inference time impossible. #### Avoiding Outliers {#avoiding-outliers .unnumbered} Another benefit of code normalization is that it provides a solution to outlier insensitivity problem of the MMD penalty, described below. Indeed, scaling by the standard deviation (rather than by a robust surrogate) controls the tail behavior of the code distribution. Due to this control, the code distribution ends up having a light tail and no code falls too far away from the origin. The outlier insensitivity problem is not specific to our closed-form formula or the choice of the kernel (see Section \[sec:Experiments\] for an empirical verification); this problem is relevant to any kernel $k(x,y)=f(\Vert x-y\Vert)$ such that $f(r)\rightarrow0$ as $r\rightarrow0$. Given a sample $z_{i},i=1,...,n$ from the standard multi-variate normal distribution $Q=P$, consider a modified sample $z'_{i}=z_{i},i=2,...,n$ and $z'_{1}$ is far from the origin. Expressing the sum of vanishingly small exponentials via the $O$-notation, we can compute the difference in $\mathrm{MMDU}_{n}^{2}$ incurred by this change: $$\begin{aligned} \Delta\mathrm{MMDU}_{n}^{2} & =\mathrm{MMDU}_{n}^{2}(Q')-\mathrm{MMDU}_{n}^{2}(Q)=\label{eq:mmd_delta}\\ & =\frac{2}{n}\left[\left(\frac{\gamma^{2}}{1+\gamma^{2}}\right)^{d/2}e^{-\frac{\Vert z_{1}\Vert^{2}}{2(1+\gamma^{2})}}-\frac{1}{n-1}\sum_{j\neq1}^{n}e^{-\frac{\Vert z_{1}-z_{j}\Vert^{2}}{2\gamma^{2}}}+O\left(e^{-\frac{\Vert z'_{1}\Vert^{2}}{2(1+\gamma^{2})}}\right)\right]\nonumber \end{aligned}$$ Note that the second term is the sample average approximation of $\mathbb{E}_{x\sim Q=P}[e^{-\Vert z_{1}-x\Vert^{2}/(2\gamma^{2})}]$. This expectation can be computed analytically (in fact it is equivalent to the summand in the second term of Eq. (\[eq:mmdu\])) and it precisely cancels the first term here in Eq. (\[eq:mmd\_delta\]), giving $\mathbb{E}[\Delta\mathrm{MMDU}_{n}^{2}]\approx0$. Thus, MMD changes very little despite the presence of the large outlier. Given the mixed objective and stochasticity inherent in the training process, this issue has an effect on WAE training even before reaching the limits of computer precision. Indeed, in addition to the MMD penalty, the WAE objective contains the reconstruction term. Given the incentive to reconstruct well, the optimizer will realize that it is beneficial to push some of the codes far away from the origin, since the origin is where most of the codes concentrate. If this happens only for a few codes in a batch, the MMD penalty will not be big enough so as to pull these codes back towards the origin. As a result, the training process will result in a distribution $Q$ that has outliers. Our experiments show that the proposed code normalization provides a solution to this issue without a need for using adaptive kernel widths or extra penalties. \[sec:Hypothesis-Testing\] Hypothesis Tests for Multivariate Normality ====================================================================== In this section we discuss hypothesis testing using $\mathrm{SMMDU}_{n}^{2}$ and provide thresholds that can be useful when monitoring progress of WAE code distribution convergence on a single and multiple batch levels. Our initial discussion is set in a broader manner so as to encompass general testing for multivariate normality. We quickly review the hypothesis testing setting following [@HZ] with some notational changes. Let $X_{1},X_{2},...,X_{n}\in\mathbb{R}^{d}$ be i.i.d. random vectors from some underlying distribution. The problem is to test the hypothesis that the underlying distribution is a non-degenerate $d$-variate normal distribution: $X_{i}\sim\mathcal{N}_{d}(\vec{\mu},\Sigma)$, for some mean vector $\vec{\mu}$ and non-degenerate covariance matrix $\Sigma$. Note that the population mean vector and covariance matrix are not known. The test of multivariate normality proceeds as follows. Let $\bar{X}=n^{-1}\sum_{i}X_{i}$ be the sample mean, and $S=(n-1)^{-1}\sum_{i}(X_{i}-\bar{X})(X_{i}-\bar{X})^{T}$ be the sample covariance matrix. Assuming non-degeneracy, define the centered and whitened vectors $$Z_{i}=S^{-1/2}(X_{i}-\bar{X}).$$ Now, the task of testing multivariate normality of $\{X_{i}\}_{i=1}^{n}$ reduces to the simpler problem of testing whether the underlying distribution of $\{Z_{i}\}_{i=1}^{n}$ is $\mathcal{N}_{d}(\vec{0},I)$. While Henze-Zirkler test [@HZ] carries out this last step by using the BHEP statistic, it can also be achieved by using the $\mathrm{SMMDU}_{n}^{2}$. One computes the $\mathrm{SMMDU}_{n}^{2}$ statistic for the sample $\{Z_{i}\}_{i=1}^{n}$ and checks whether it is above the test threshold, and if so, the null hypothesis gets rejected. The most straightforward way to compute the threshold is to run a Monte Carlo simulation: sample $\{Z_{i}\}_{i=1}^{n}$ from the null distribution $\mathcal{N}_{d}(\vec{0},I)$, and compute the corresponding $\mathrm{SMMDU}_{n}^{2}$ value; repeat this many times to obtain the empirical sampling distribution of the statistic and use the $100\cdot(1-\alpha)$-th percentile as the threshold for the $\alpha$-level test. However, this approach is problematic due to the treatment of the nuisance parameters $\vec{\mu}$ and $\Sigma$: the same sample is used both for estimating mean and covariance, and then for testing (this is somewhat like training and testing on the same data). The most apparent consequence is that one introduces dependencies within $\{Z_{i}\}_{i=1}^{n}$, namely, $n^{-1}\sum_{i}Z_{i}=\vec{0}$ and $(n-1)^{-1}\sum_{i}Z_{i}Z_{i}^{T}=I_{d}$, rendering it no longer an i.i.d sample. Thus, when using the Monte Carlo approach with $\{Z_{i}\}_{i=1}^{n}$ sampled directly from $\mathcal{N}_{d}(\vec{0},I)$ we would end up with a wrong null distribution and, so, with the wrong thresholds. Henze-Zirkler test [@HZ] uses appropriate corrections to account for the nuisance parameters when computing the moments under the null. These moments are then used to obtain a log-normal approximation to the null distribution. A similar path can be potentially taken with the $\mathrm{SMMDU}_{n}^{2}$ statistic, but for simplicity we will explain how to correct the issue with Monte Carlo sampling. To fix the problem, the computation of the null distribution should proceed from samples that satisfy the dependency relationships mentioned above. Fortunately, constructing such samples is easy: we sample $\{Z_{i}^{\mathrm{orig}}\}_{i=1}^{n}$ from $\mathcal{N}_{d}(\vec{0},I)$, then apply centering by the mean and whitening by the sample covariance matrix. The resulting sample $\{Z_{i}\}_{i=1}^{n}$ satisfies the relationships $n^{-1}\sum_{i}Z_{i}=\vec{0}$ and $(n-1)^{-1}\sum_{i}Z_{i}Z_{i}^{T}=I_{d}$. This centered and whitened sample is used to compute the $\mathrm{SMMDU}_{n}^{2}$ values and to obtain the thresholds. To prove the correctness of this procedure one has to show that there is a measure preserving and test statistic preserving one-to-one mapping between these samples originating from $\mathcal{N}_{d}(\vec{0},I)$ and samples if they were to originate from $\mathcal{N}_{d}(\vec{\mu},\Sigma)$ with the true $\vec{\mu}$ and $\Sigma$. Using the non-degeneracy of $\Sigma$, with some linear algebra one can show that indeed there is such a mapping given by an orthogonal linear transformation, see Appendix \[subsec:Correctness-of-the\]. The matrix of this transformation depends on $\Sigma$ only, making it measure preserving. Since $\mathrm{SMMDU}_{n}^{2}$ is rotation-invariant, the resulting sampling distributions coincide. Before proceeding, we would like to mention a modification of the above test where the goal is to test whether the sample comes from a normal distribution with a diagonal covariance. This is a test that both checks each dimension for normality and establishes the independence between the dimensions. When conducting the test only the diagonal of the sample covariance matrix is computed and used for transforming $\{X_{i}\}_{i=1}^{n}$ to $\{Z_{i}\}_{i=1}^{n}$ . The corresponding Monte Carlo procedure takes $\{Z_{i}^{\mathrm{orig}}\}_{i=1}^{n}$ from $\mathcal{N}_{d}(\vec{0},I)$, and applies centering by the mean and scaling each dimension by its standard deviation (just like code normalization). Table \[tab:thresholds\] displays the thresholds corresponding to the $0.05$ level test, for sample size of $n=100$ for varying dimensions and kernel scales (we have included dimensions $16$ and $32$ to give an idea about the overall trend; one expects the test to lose power with an increasing dimensionality [@Ramdas_decreasing_power]). The column “Sample Type” indicates what processing was applied to the original sample $\{Z_{i}^{\mathrm{orig}}\}_{i=1}^{n}$ from $\mathcal{N}_{d}(\vec{0},I)$, if any. The “Original” thresholds can be used for testing the following simple hypothesis: given a sample $\{X_{i}\}_{i=1}^{n}$ we would like to test whether the underlying distribution is $\mathcal{N}_{d}(\vec{0},I)$. The “Centered+Scaled” and “Centered+Whitened” rows give the correct thresholds for composite nulls, i.e. testing whether $X_{i}\sim\mathcal{N}_{d}(\vec{\mu},\Sigma)$, for unknown $\vec{\mu}$ and $\Sigma$. “Centered+Scaled” corresponds to the case where $\Sigma$ is assumed to be diagonal, and “Centered+Whitened” correspond to the case of a general non-degenerate $\Sigma$. As expected, dependencies within the sample shift the null distribution of $\mathrm{SMMDU}_{n}^{2}$ to the left considerably; also see Figure \[fig:codenorm-effect\] for side by side histograms of “Centered+Scaled” versus “Original” null distributions. Therefore, using the original thresholds for composite hypotheses would have resulted in tests that are rather liberal. Dimension Sample Type $s=1$ 1/2 1/4 1/8 1/16 HZ ----------- -------------------------- ------- ------- ------- ------- ------- ------- $d=1$ Original 1.97 1.97 1.95 1.92 1.91 1.98 Centered+Scaled/Whitened -0.13 0.32 0.75 1.05 1.24 0.22 $d=2$ Original 1.93 1.94 1.90 1.85 1.83 1.90 Centered+Scaled -0.57 -0.12 0.39 0.86 1.16 0.23 Centered+Whitened -0.79 -0.33 0.22 0.72 1.06 0.03 $d=4$ Original 1.90 1.87 1.83 1.79 1.76 1.83 Centered+Scaled -1.09 -0.66 -0.02 0.63 1.12 0.30 Centered+Whitened -1.60 -1.25 -0.56 0.24 0.85 -0.16 $d=8$ Original 1.85 1.83 1.80 1.77 1.74 1.75 Centered+Scaled -1.76 -1.36 -0.59 0.34 1.11 0.55 Centered+Whitened -2.63 -2.57 -1.88 -0.60 0.58 -0.30 $d=16$ Original 1.81 1.80 1.77 1.74 1.78 1.78 Centered+Scaled -2.65 -2.30 -1.47 -0.15 1.08 1.05 Centered+Whitened -4.00 -4.47 -4.17 -2.31 0.00 -0.03 $d=32$ Original 1.77 1.77 1.74 1.71 1.76 1.30 Centered+Scaled -3.87 -3.59 -2.78 -1.13 0.75 1.02 Centered+Whitened -5.86 -7.12 -7.91 -5.72 -1.24 0.06 : \[tab:thresholds\]Empirical thresholds for hypothesis tests with size $\alpha=0.05$. See text for the details of when each kind of threshold should be used. Here, $n=100$, kernel scale $s=\gamma^{2}/d$. HZ is the $\gamma$ suggested by Henze and Zirkel [@HZ] as given by the formula Eq. (\[eq:HZgamma\]). #### Monitoring WAE Training Progress {#monitoring-wae-training-progress .unnumbered} We will consider two ways of monitoring progress: at a single batch and multi-batch levels. When inspecting the value of $\mathrm{SMMDU}_{n}^{2}$ for a single batch, one can use the above thresholds for hypothesis testing as a guideline. Assuming that this batch is from validation or test set, we can use the above thresholds listed in the “Original” rows of Table \[tab:thresholds\]. By looking at these values, we suggest using $2.0$ as an easy to remember liberal threshold. This applies to code normalized batches as long as the normalization is done using population statistics. However, when the batch is normalized using its own statistics, then the appropriate thresholds are given by the “Centered+Scaled” rows. We should stress again that even upon convergence to the target distribution, one should still expect oscillations of the $\mathrm{SMMDU}_{n}^{2}$ values: it is not the case that samples from the target distribution all have $\mathrm{SMMDU}_{n}^{2}$ equal to zero, instead they follow the appropriate null distribution. *Remark:* With neural nets it is customary to track the training and validation losses during learning. When code normalization is used, the distribution shift exemplified in Figure \[fig:codenorm-effect\] and seen in Table \[tab:thresholds\] will result in an even smaller training loss than the validation loss. This is because code normalization uses the batch statistics during training and population statistics at validation/test time. The difference between these losses can be on the order of several $\lambda$’s; here $\lambda$ is the penalty coefficient. In the multi-batch case, such as when computing the SMMDU for codes corresponding to the validation or test set, one can use the same batch size as used for training and compute the average SMMDU value. This average has a very simple asymptotic distribution under the null as explained below. Assume that the validation/test set contains $m$ batches of size $n$, and the corresponding batches are $\{z_{i}^{b}\}_{i=1}^{n},b=1,2,...,m$. The average SMMDU value is computed as $$B_{m}=\frac{1}{m}\sum_{b=1}^{m}\mathrm{SMMDU}_{n}^{2}(\{z_{i}^{b}\}_{i=1}^{n}).$$ Note that under the null, each summand $\mathrm{SMMDU}_{n}^{2}(\{z_{i}^{b}\}_{i=1}^{n})$ has zero mean and unit variance due to the standardization. Assuming that $m$ is big enough, we can apply the Central Limit Theorem [@CLT_Ref], giving that the null distribution of $B_{m}$ is asymptotically normal with mean $0$ and variance $1/m$. Thus, as a rule of thumb, values of $B_{m}$ that do not fall into the three-sigma interval $[-3/\sqrt{m},3/\sqrt{m}]$ should be considered as an indication that the aggregate code distribution has not converged to the target standard multivariate normal distribution. The raw MMD version of this test together with theoretical results can be found in [@B_test], where it is called the B-test. Thus, we will refer to $B_{m}$ as the B-Statistic. Another popular way of keeping track of progress metrics is exponential moving averaging. The Lyapunov/Lindeberg version of the Central Limit Theorem [@CLT_Ref Chapter 27] can be applied to obtain the corresponding interval. Suppose that the exponential moving average with the momentum of $\alpha$ is used to keep track of a per-batch quantity $S_{b}$. Thus, $E_{b}=\alpha E_{b-1}+(1-\alpha)S_{b}$ is used for $b=1,...,m$. Note that, $E_{m}$ can be written as $$E_{m}=\alpha^{m}E_{0}+(1-\alpha)[\alpha^{m-1}S_{1}+\alpha^{m-2}S_{2}+\cdots\alpha S_{m-1}+S_{m}],$$ here $S_{0}$ is some initial value, usually $0$, which we will use. Assuming that $S_{b}$ are standardized to have zero mean and unit variance, the application of the CLT to random variables $(1-\alpha)\alpha^{m-k}S_{k}$ gives that $E_{m}$ is normally distributed: $$E_{m}\sim\mathcal{N}(0,(1-\alpha^{2m+2})\frac{1-\alpha}{1+\alpha}).$$ By dropping $(1-\alpha^{2m+2})$, we can use $(1-\alpha)/(1+\alpha)$ as an *upper bound* for the variance. This gives the three-sigma interval for the E-Statistic liberally as $[-3\sqrt{(1-\alpha)/(1+\alpha)},3\sqrt{(1-\alpha)/(1+\alpha)}]$. When $\alpha=0.99$ we get the interval as $[-0.212,0.212]$. Once again this interval can be used when monitoring the exponential moving average of $\mathrm{SMMDU}_{n}^{2}$; falling outside this interval should be considered as an indication that the aggregate code distribution has not converged to the target standard multivariate normal distribution. Of course, the single-batch approach above that treats the whole validation/test set as one batch would result in a more powerful test. However, B-statistic or E-statistic tests are simple to state and are computationally inexpensive as they avoid constructing the large pair-wise distance matrices for the overall test. Moreover, neural network packages such as Keras provide these types of averages automatically if one adds the corresponding quantity as a validation metric. At a theoretical level, one should keep in mind that given enough power we will always reject the null: with real-life data one rarely expects the neural net to exactly reproduce the normal distribution. Rejecting the null at high power does not mean that the distributions are easily distinguishable: the practical difference can be so small that a classifier trained to distinguish the two distributions (think of an adversary from an adversarial WAE) would perform at a nearly chance level. Based on these considerations, using the B-Statistic with $m=30-50$ should be a reasonable choice, see the discussion in [@FRISTON20121300 Appendix 1] albeit in a different context; for power calculations for the MMD based tests one can refer to [@Sutherland_model_criticismMMD]. \[sec:Experiments\]Experiments ============================== First we discuss our parameterization for the kernel width used in computation of various MMD measures. A rule of thumb choice of the kernel width is $\gamma^{2}=d$, where $d$ is the dimension of the code space (see e.g. [@InfoVAE; @WAE]). This choice is based on considering the average pair-wise distance between two points drawn from the standard multi-variate normal distribution, and halving it to offset the multiplication by $2$ in the expression for the kernel. We will see that this choice gives rather suboptimal results, yet it provides a good point of reference for defining scale of the kernel as $s=\gamma^{2}/d$. We will experiment with various choices of $s$, where $s>1$ gives wider and $s<1$ gives narrower kernels. #### Validation {#validation .unnumbered} We first experimentally verify that our closed-form formula for $\mathrm{SMMDU}_{n}^{2}$ results in zero mean and unit variance when $Q=\mathcal{N}_{d}(\vec{0},I)$. To this end, we sample $n=100$ points from the standard $d$-variate normal distribution and compute the value of $\mathrm{SMMDU}_{n}^{2}$. This process is repeated 10,000 times to obtain the empirical distribution of the values. Figure \[fig:Variance-curve\] shows the violin plots of these empirical distributions computed for several values of the kernel scale $s$ and dimensionality $d$. The red segments in this plot are centered at the mean, and they extend between mean $\pm$ standard deviation. We observe from the graph that the means are close to zero and the standard deviations are close to 1 as expected. ![\[fig:violin\_plot\]Violin plots verify that $\mathrm{\mathrm{SMMDU}_{n}^{2}}$ has zero mean and unit variance under the null. Here, batch size is $n=100$ and the kernel width is expressed via the scale $s$ as $\gamma^{2}=s\cdot d$](Figs/variance_violin_plot.pdf){width="100.00000%"} #### Discriminative Performance {#discriminative-performance .unnumbered} The goal of the next experiment is to compare our closed-formula estimator of MMD (referred to as “Analytic RBF”) to the commonly used sampling based estimator using the same Gaussian RBF kernel (“Empirical RBF”). We also compare to the sampling based estimator but with the inverse multi-quadratics (IMQ) kernel defined by $k(x,y)=1/\left(1+\Vert x-y\Vert^{2}/(2\gamma^{2})\right)$; we call this “Empirical IMQ”. The IMQ kernel is often claimed to be superior to the RBF kernel due to its slower tail decay. In our first experiment we would like to determine which one of these three methods is most effective at distinguishing the standard $d$-variate normal distribution from the uniform distribution. Since our goal is to train neural networks rather than perform hypothesis testing, we will not use the test power as a metric of interest; instead we will rely on the effect size defined below. In addition, we are not studying the dependence on the latent dimension, so we do not have to worry about the fair choice of alternatives [@Ramdas_decreasing_power]. #### {#section .unnumbered} Method Kernel Scale $d=1$ $d=2$ $d=4$ $d=8$ $d=16$ $d=32$ --------- -------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- an RBF 2 $0.49$ $0.14$ $0.05$ $0.02$ $3.7e-04$ $0.11$ 1 $1.18$ $0.7$ $0.29$ $0.09$ $0.04$ $0.03$ 1/2 $1.85$ $1.64$ $0.97$ $0.45$ $0.17$ $0.03$ 1/4 $1.97$ $2.32$ $2.02$ $1.19$ $0.76$ $0.35$ 1/8 $\mathbf{2.28}$ $\mathbf{2.61}$ $\mathbf{2.56}$ $2.01$ $\mathbf{1.5}$ $0.98$ 1/16 $\mathbf{2.21}$ $\mathbf{2.62}$ $\mathbf{2.49}$ $1.88$ $1.39$ $\mathbf{1.17}$ 1/32 $\mathbf{2.22}$ $\mathbf{2.49}$ $2.1$ $1.16$ $0.51$ $0.44$ HZ $1.46$ $1.99$ $2.26$ $\mathbf{2.13}$ $\mathbf{1.49}$ $0.39$ emp RBF 2 $0.32$ $0.15$ $0.15$ $9.1e-04$ $0.03$ $0.13$ 1 $0.75$ $0.42$ $0.16$ $0.01$ $0.02$ $0.01$ 1/2 $1.07$ $0.95$ $0.5$ $0.23$ $0.09$ $0.01$ 1/4 $1.36$ $1.34$ $1.1$ $0.66$ $0.3$ $0.14$ 1/8 $1.35$ $\mathbf{1.57}$ $1.3$ $\mathbf{0.99}$ $0.66$ $0.51$ 1/16 $\mathbf{1.4}$ $\mathbf{1.54}$ $\mathbf{1.38}$ $\mathbf{1.02}$ $\mathbf{0.71}$ $\mathbf{0.62}$ 1/32 $\mathbf{1.45}$ $1.36$ $1.1$ $0.57$ $0.26$ $0.15$ HZ $1.07$ $1.18$ $1.23$ $\mathbf{0.98}$ $0.64$ $0.25$ emp IMQ 2 $0.51$ $0.29$ $0.08$ $0.06$ $0.01$ $0.03$ 1 $0.74$ $0.48$ $0.27$ $0.12$ $0.06$ $0.03$ 1/2 $1.01$ $0.88$ $0.49$ $0.33$ $0.08$ $0.04$ 1/4 $1.21$ $1.12$ $0.74$ $0.46$ $0.16$ $0.06$ 1/8 $1.23$ $1.36$ $1.1$ $0.45$ $0.2$ $0.1$ 1/16 $\mathbf{1.32}$ $\mathbf{1.41}$ $\mathbf{1.22}$ $0.53$ $0.27$ $0.07$ 1/32 $\mathbf{1.31}$ $\mathbf{1.46}$ $1.19$ $0.61$ $\mathbf{0.32}$ $0.08$ 1/64 $\mathbf{1.33}$ $1.38$ $1.17$ $0.63$ $0.29$ $\mathbf{0.2}$ 1/128 $\mathbf{1.33}$ $\mathbf{1.42}$ $\mathbf{1.26}$ $\mathbf{0.82}$ $\mathbf{0.32}$ $0.08$ 1/256 $\mathbf{1.32}$ $1.26$ $1.19$ $0.63$ $0.28$ $0.13$ 1/512 $1.16$ $1.04$ $1.14$ $0.77$ $0.27$ $0.08$ 1/1024 $1.08$ $1.04$ $1.06$ $0.77$ $\mathbf{0.32}$ $0.11$ {width="100.00000%"} Method Kernel Scale $d=1$ $d=2$ $d=4$ $d=8$ $d=16$ $d=32$ --------- -------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- an RBF 2 $0.17$ $0.12$ $0.11$ $0.11$ $0.1$ $0.1$ 1 $0.35$ $0.33$ $0.43$ $0.5$ $0.39$ $0.31$ 1/2 $0.59$ $0.69$ $1.61$ $1.49$ $1.46$ $1.23$ 1/4 $0.86$ $1.14$ $3.07$ $3.64$ $3.75$ $3.11$ 1/8 $1.1$ $1.42$ $4.6$ $\mathbf{4.95}$ $\mathbf{5.37}$ $\mathbf{4.07}$ 1/16 $1.18$ $1.9$ $\mathbf{4.86}$ $4.71$ $4.24$ $3.21$ 1/32 $\mathbf{1.34}$ $\mathbf{2.1}$ $4.25$ $3.37$ $2.75$ $2.25$ HZ $0.52$ $1$ $3.92$ $\mathbf{5.19}$ $4.18$ $2.5$ emp RBF 2 $0.03$ $0.14$ $0.05$ $0.07$ $0.06$ $0.03$ 1 $0.11$ $0.09$ $0.32$ $0.24$ $0.19$ $0.19$ 1/2 $0.24$ $0.46$ $0.88$ $0.81$ $0.72$ $0.59$ 1/4 $0.53$ $0.67$ $1.69$ $1.98$ $2.42$ $2.16$ 1/8 $0.6$ $0.77$ $2.52$ $3.36$ $\mathbf{4.22}$ $\mathbf{3.68}$ 1/16 $0.73$ $0.98$ $\mathbf{2.68}$ $\mathbf{3.74}$ $3.81$ $3.35$ 1/32 $\mathbf{0.78}$ $\mathbf{1.16}$ $2.53$ $3$ $2.69$ $2.11$ HZ $0.26$ $0.55$ $2.05$ $\mathbf{3.58}$ $3.86$ $2.41$ emp IMQ 2 $4.7e-03$ $0.19$ $0.25$ $0.16$ $0.12$ $0.16$ 1 $0.16$ $0.26$ $0.49$ $0.33$ $0.43$ $0.34$ 1/2 $0.31$ $0.29$ $0.78$ $0.78$ $0.68$ $0.62$ 1/4 $0.4$ $0.49$ $1.21$ $1.26$ $1.25$ $1.06$ 1/8 $0.51$ $0.73$ $1.78$ $1.85$ $1.73$ $1.47$ 1/16 $0.54$ $0.82$ $2.06$ $2.17$ $2.14$ $1.89$ 1/32 $0.75$ $1.08$ $2.18$ $2.49$ $2.32$ $1.97$ 1/64 $0.92$ $\mathbf{1.18}$ $2.3$ $2.7$ $2.43$ $1.95$ 1/128 $0.89$ $\mathbf{1.24}$ $\mathbf{2.49}$ $2.69$ $\mathbf{2.57}$ $2.11$ 1/256 $0.93$ $\mathbf{1.21}$ $\mathbf{2.46}$ $\mathbf{2.82}$ $\mathbf{2.54}$ $\mathbf{2.22}$ 1/512 $\mathbf{1.11}$ $\mathbf{1.22}$ $2.32$ $\mathbf{2.88}$ $\mathbf{2.58}$ $\mathbf{2.26}$ 1/1024 $\mathbf{1.05}$ $1.15$ $2.23$ $\mathbf{2.77}$ $\mathbf{2.61}$ $\mathbf{2.16}$ {width="100.00000%"} The uniform distribution under consideration is $U[-\sqrt{3},\sqrt{3}]^{d}$. Note that this particular uniform distribution has mean 0 and variance 1 in each dimension just like the normal distribution. As a result, distinguishing the two distributions requires going beyond the first two moments. For each of the three methods, for a fixed dimension $d$ and kernel scale $s$, we sample $n=100$ points from the the standard $d$-variate normal distribution and compute the corresponding MMD estimate. Next we sample $n=100$ points from the uniform distribution $U[-\sqrt{3},\sqrt{3}]^{d}$ and compute the corresponding MMD estimate. We repeat this 200 times, and compute the corresponding means $\mathrm{Mean}_{1}$ and $\mathrm{Mean}_{2}$, and the standard deviations $\mathrm{SD}_{1}$ and $\mathrm{SD}_{2}$ corresponding to each of the two sets of 200 MMD values[^1]. Now we can measure the discriminativeness of a given method by computing $$\tau(\mathrm{method},s,d)=\frac{\mathrm{\vert Mean}_{1}-\mathrm{Mean}_{2}\vert}{(\mathrm{SD}_{1}+\mathrm{SD}_{2})/2}.$$ Note that this is the effect size of a two sample t-test as measured by Cohen’s d [@cohen1988spa]. Larger values of $\tau$ mean better discrimination, which potentially translates to better gradients for neural network training. The results are presented in Table \[tab:Discrimination-power-table-unifrom\]. Note that the experiment was done for different values of the kernel scale; due to the heavier tail, we included more scale choices for the IMQ kernel than for the RBF kernel. For each method and dimensionality choice $d$, the best performing choice of the kernel scale corresponds to the maximum value of $\tau$; these $\tau$ values are shown in boldface (we also highlight the $\tau$ values that are within $5\%$ of the maximum). Figure \[fig:boxplot-discrimination\] provides a graphical display for this experiment when $d=8$. In this graph, for each method, the best choice of the kernel scale was used to compute the distributions of MMD values. For each method, the boxes are centered at the corresponding $\mathrm{Mean}_{1/2}$ and the half-height of the box is $\mathrm{SD}_{1/2}$. The whiskers span the range of all of the values; the blue dots correspond to the MMD values. For a given method, when the boxes corresponding to normal and uniform distributions overlap, it means that the method has difficulty discriminating the two distributions. In terms of training neural networks, this means that the corresponding MMD penalty may not be able to provide a strong gradient direction for training because the difference is lost within the stochastic noise. We repeat the same experiment but instead of the uniform distribution we use a distribution obtained from a neural networks. We use the MNIST dataset and train auto-encoders (both encoder and decoder have two hidden layers with 128 neurons each, ReLU activations) with different latent dimensions $d$ with no regularization. The codes corresponding to the test data are extracted and shifted to have zero mean. We observed that with growing $d$ the various latent dimensions were highly correlated (e.g. Pearson correlations as high as $0.4$); thus, to make the task more difficult, we applied PCA-whitening to the latent codes. The resulting discrimination performance is presented in Table \[tab:Discrimination-power-table-mnist\] and Figure \[fig:boxplot-discrimination-mnist\]. By examining both of the tables above, we can see that Analytic RBF method outperforms both the Empirical RBF and IMQ methods in terms of discrimination power. Another observation is that the commonly recommended choice of $\gamma^{2}=d$ (which corresponds to the kernel scale $s=1$) is never a good choice; a similar finding for the median heuristic was spelled out in [@Sutherland_model_criticismMMD]. The kernel width recommended for Henze-Zirkler test gives mixed results, which is somewhat expected—optimality for density estimation does not guarantee optimal discriminative performance. Examining the Analytic RBF results, it seems that kernel scales $s=1/8$ or $s=1/16$ provide a good rule of thumb choices. Finally, in these particular examples we see that despite its having a larger repertoire of kernel scale choices, Empirical IMQ does not perform as well as Empirical RBF. While these results are limited to two datasets, yet they bring into question the commonly recommended choices of the kernel and its width. Of course, our analysis assumes that the alternative distribution has zero mean and unit variance in each dimension. We believe that this is the most relevant setting to WAE learning because during the late stages of WAE training the code distribution starts converging to the normal distribution. #### Outliers {#outliers .unnumbered} ![\[fig:Outlier-discrimination-experimen\]Outlier discrimination experiment carried out for $d=4$. For each method, the most discriminative (i.e. maximum $\tau$) kernel scale is chosen.](Figs/normal_vs_outlier.pdf){width="100.00000%"} Here we experimentally verify the outlier insensitivity of the MMD and demonstrate that the issue is not peculiar to our approach. To this end, we run the discrimination experiment above but this time trying to distinguish a sample from the standard $d$-variate normal distribution from the same but with one of the sample points replaced with a point far away from the origin (namely $z_{1}\rightarrow z'_{1}=100\cdot\vec{1}$). Figure \[fig:Outlier-discrimination-experimen\] shows that all of the three methods fail to distinguish these two distributions in practice. -- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![\[fig:WAE-results\]Qualitative results for WAE trained on MNIST. In (a) odd rows are the real images.](Figs/wae_reconstruct.png "fig:"){width="31.00000%"} ![\[fig:WAE-results\]Qualitative results for WAE trained on MNIST. In (a) odd rows are the real images.](Figs/wae_sample.png "fig:"){width="31.00000%"} ![\[fig:WAE-results\]Qualitative results for WAE trained on MNIST. In (a) odd rows are the real images.](Figs/wae_slice.png "fig:"){width="31.00000%"} ![\[fig:WAE-results\]Qualitative results for WAE trained on MNIST. In (a) odd rows are the real images.](Figs/wae_reconstruct_adaptiveBN.png "fig:"){width="31.00000%"} ![\[fig:WAE-results\]Qualitative results for WAE trained on MNIST. In (a) odd rows are the real images.](Figs/wae_sample_adaptiveBN.png "fig:"){width="31.00000%"} ![\[fig:WAE-results\]Qualitative results for WAE trained on MNIST. In (a) odd rows are the real images.](Figs/wae_slice_adaptiveBN.png "fig:"){width="31.00000%"} ![\[fig:WAE-results\]Qualitative results for WAE trained on MNIST. In (a) odd rows are the real images.](Figs/wae_reconstruct_adaptive.png "fig:"){width="31.00000%"} ![\[fig:WAE-results\]Qualitative results for WAE trained on MNIST. In (a) odd rows are the real images.](Figs/wae_sample_adaptive.png "fig:"){width="31.00000%"} ![\[fig:WAE-results\]Qualitative results for WAE trained on MNIST. In (a) odd rows are the real images.](Figs/wae_slice_adaptive.png "fig:"){width="31.00000%"} a\) Test reconstruction b\) Random samples c\) Slice through code space -- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------- #### WAE results {#wae-results .unnumbered} Here we present the results of training WAE’s on MNIST dataset. The architecture for the neural net is borrowed from RStudio’s “Keras Variational Auto-encoder with Deconvolutions” example[^2]. This network has about 3.5M trainable parameters, almost an order of magnitude less than the the 22M parameter network used by Tolstikhin et al. [@WAE]. We consider three versions: - CodeNorm—code normalization is used, the kernel width is kept fixed. - Adaptive—no code normalization is used, kernel width is chosen adaptively. This has two versions: - AdaptiveBN—since code normalization can have other benefits (e.g. improved optimization [@BatchNormHow]), we add batch normalization as the initial layer of the decoder; - AdaptivePlain—no batch normalization layer added. The CodeNorm version was trained for 60 epochs, but to allow the Adaptive versions to reach a favorable configuration in the code space we trained them for an extra 20 epochs at the initial learning rate. The latent dimension is set to $d=8$ and all versions use the closed-form SMMDU penalty; further details are provided in Appendix \[subsec:Neural-net-architectures\]. Figure \[fig:WAE-results\] (a)-(b) shows the reconstruction of test images and random samples generated from Gaussian noise fed to the decoder. Next we take a planar slice through the origin in the code space and feed the codes at the regular grid along this plane into the decoder. Figure \[fig:WAE-results\] (c) depicts the resulting digit images, giving a taste of the manifold structure captured by the models. Qualitatively, both of the Adaptive results are lower quality than CodeNorm despite the former being trained for more epochs. Quantitative results are presented in Table \[tab:WAEQuantitative\]. CodeNorm achieves the best test reconstruction error. We speculate that the reason for this is that the gradient components of the MMD penalty pointing in the direction of “ideal” samples (see Section \[subsec:Code-Normalization\]) add oscillations that hinder reduction in the reconstruction loss of the Adaptive models. Next, we follow the suggestion of [@Cramer_Wold_AE] to compute Mardia’s multivariate skewness and normalized kurtosis statistics of the latent code distribution of test data; we used the formulas provided in [@Cramer_Wold_AE] and obtained the values as shown in the table. We see that for both measures, the CodeNorm version is better. Skewness is a measure of symmetry, so its small magnitude indicates that the code distribution is symmetrically distributed around the origin. Since kurtosis is a measure of outlier presence [@kurtosis_RIP], its small value indicates that there are no outliers present in the code distribution. We verified experimentally (not presented here) that code normalization is responsible for keeping kurtosis under control. Indeed, removing the code normalization layer from a trained network, modifying the latent layer incoming weights so that the codes have zero mean and unit variance, and continuing to train afterwards leads to increased kurtosis as predicted in Section \[subsec:Code-Normalization\]. Finally, we analyze the results using the B-statistic discussed in Section \[sec:Hypothesis-Testing\] which gives a more in-depth summary of the data than Mardia’s statistics. We computed the B-statistic using $m=50$ batches of size $n=100$ from the test partition of MNIST. The corresponding three sigma interval is $[-3/\sqrt{50},3/\sqrt{50}]=[-0.424,0.424]$. Both CodeNorm and AdaptiveBN look good in terms of this statistic, CodeNorm falling inside the interval; on the other hand AdaptivePlain is somewhat farther away, indicating that its code distribution more noticeably deviates from the target distribution. WAE Version Test MSE Normalized Kurtosis Skewness B-Statistic --------------- ---------- --------------------- ---------- ------------- CodeNorm 0.0156 -0.90 0.56 0.355 AdaptiveBN 0.0244 6.85 2.80 0.449 AdaptivePlain 0.0242 3.81 2.35 0.519 : \[tab:WAEQuantitative\]Quantitative comparison of different WAE versions. Conclusion ========== This paper introduces closed-form formulas for MMD and its variance in the case of the standard multivariate normal target distribution. This allows us to propose a properly normalized and more interpretable standardized version of MMD as a penalty in the WAE training objective. We point out a relationship with the BHEP statistic that provides further insights about the MMD and allows making a connection to the Cramer-Wold distance. In addition, we propose using code normalization when training WAEs; this has the benefits of making the kernel width selection easier, reducing the training effort, and preventing outliers in the aggregate code distribution. Finally, we discuss the appropriate null distributions and provide thresholds for multivariate normality testing with SMMDU. A number of rules of thumb are provided for monitoring the progress of WAE training. Our experiments on synthetic and real data confirm that the analytic formulation improves over the commonly used stochastic approximation of the MMD, and demonstrate that code normalization provides significant benefits when training WAEs. Appendix ======== \[appendix:Deterministic-Encoders\]Deterministic Encoders --------------------------------------------------------- We start with the expression $$\mathrm{MMDU}_{n}^{2}(\mathcal{N}_{d}(\vec{0},I),Q)=\E_{x,x'\sim P}[k(x,x')]-\frac{2}{n}\sum_{i=1}^{n}\E_{x\sim P}[k(x,z_{i})]+\frac{1}{n(n-1)}\sum_{i=1}^{n}\sum_{j\neq i}^{n}k(z_{i},z_{j}),$$ and show that the first two expectations can be computed in closed form. Let us start with the second term, and rewrite each summand as an integral: $$\begin{aligned} \E_{x\sim P}[k(x,z)]= & \int_{\mathbb{R}^{d}}e^{-\Vert x-z\Vert^{2}/(2\gamma^{2})}(2\pi)^{-d/2}e^{-\Vert x\Vert^{2}/2}dx\nonumber \\ = & (2\pi\gamma^{2})^{d/2}\int_{\mathbb{R}^{d}}(2\pi\gamma^{2})^{-d/2}e^{-\Vert x-z\Vert^{2}/(2\gamma^{2})}(2\pi)^{-d/2}e^{-\Vert x\Vert^{2}/2}dx.\label{eq:temp2}\end{aligned}$$ Since $\Vert x-z\Vert^{2}=\Vert z-x\Vert^{2}$, the integral in this expression can be recognized as the probability density function of the sum $Z=U+V$ where $U\sim N(\vec{0},\gamma^{2}I)$ and $V\sim N(\vec{0},I)$. Being a sum of two normal distributions, adding means and variances we get, $Z\sim N(\vec{0},(1+\gamma^{2})I)$, and the above expression computes to $$\E_{x\sim P}[k(x,z)]=(2\pi\gamma^{2})^{d/2}(2\pi(1+\gamma^{2}))^{-d/2}e^{-\frac{\Vert z\Vert^{2}}{2(1+\gamma^{2})}}=\left(\frac{\gamma^{2}}{1+\gamma^{2}}\right)^{d/2}e^{-\frac{\Vert z\Vert^{2}}{2(1+\gamma^{2})}}.\label{eq:mmdu_xz}$$ Next, we compute the first term in the formula Eq. (\[eq:mmdu\]) by rewriting it as an integral: $$\begin{aligned} & \E_{x,x'\sim P}[k(x,x')]=\nonumber \\ = & \int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}e^{-\Vert x-x'\Vert^{2}/(2\gamma^{2})}(2\pi)^{-d/2}e^{-\Vert x\Vert^{2}/2}(2\pi)^{-d/2}e^{-\Vert x'\Vert^{2}/2}dxdx'.\label{eq:mmdu_xx}\end{aligned}$$ In this expression, let us replace $e^{-\Vert x'\Vert^{2}/2}$ by $e^{-\Vert x'-w\Vert^{2}/2}$, and remember that we would get the sought value by setting $w=\vec{0}$. Rewriting this as $$(2\pi\gamma^{2})^{d/2}\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}(2\pi\gamma^{2})^{-d/2}e^{-\Vert x-x'\Vert^{2}/(2\gamma^{2})}(2\pi)^{-d/2}e^{-\Vert x\Vert^{2}/2}dx\right)(2\pi)^{-d/2}e^{-\Vert x'-w\Vert^{2}/2}dx',\label{eq:mmdu_trick}$$ With this replacement, we can recognize the inner integral as the density function of the sum of two multivariate normal variables. Interpreting the outer integral similarly, we can see that the entire double integral captures the probability density function of the sum $W=A+B+C$, where $A\sim N(\vec{0},\gamma^{2}I)$, $B\sim N(\vec{0},I)$ and $C\sim N(\vec{0},I)$. Being a sum of three normal distributions, adding means and variances we get $W\sim N(\vec{0},(2+\gamma^{2})I)$, immediately giving the expression for this integral as $$(2\pi(2+\gamma^{2}))^{-d/2}e^{-\frac{\Vert w\Vert^{2}}{2(2+\gamma^{2})}}$$ Including the multiplier in front of the integral, and setting $w=\vec{0}$, we obtain: $$\E_{x,x'\sim P}[k(x,x')]=\left(\frac{\gamma^{2}}{2+\gamma^{2}}\right)^{d/2}.$$ Putting everything together we obtain the closed-form formula for $\mathrm{MMDU}_{n}^{2}(\mathcal{N}_{d}(\vec{0},I),Q)$ in Section \[subsec:Deterministic-Encoders\]. \[appendix:Random-Encoders\]Random Encoders ------------------------------------------- Let us start by computing the second term in Eq. (\[eq:mmd\]), namely $$\begin{aligned} \E_{x\sim P,y\sim Q_{\mathrm{batch}}}[k(x,y)] & =\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\left(\frac{1}{n}\sum_{i=1}^{n}\prod_{k=1}^{d}\frac{e^{-(y_{k}-\mu_{ik})^{2}/(2\sigma_{ik}^{2})}}{\sqrt{2\pi\sigma_{ik}^{2}}}\right)\cdot e^{-\frac{\Vert x-y\Vert^{2}}{2\gamma^{2}}}(2\pi)^{-d/2}e^{-\frac{\Vert x\Vert^{2}}{2}}dxdy\nonumber \\ = & \int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\left(\frac{1}{n}\sum_{i=1}^{n}\prod_{k=1}^{d}\frac{e^{-(y_{k}-\mu_{ik})^{2}/(2\sigma_{ik}^{2})}}{\sqrt{2\pi\sigma_{ik}^{2}}}\right)\cdot\prod_{k=1}^{d}\frac{e^{-\frac{(x_{k}-y_{k}){}^{2}}{2\gamma^{2}}}e^{-\frac{x_{k}^{2}}{2}}}{\sqrt{2\pi}}dxdy\nonumber \\ = & \frac{1}{n}\sum_{i=1}^{n}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\prod_{k=1}^{d}\frac{e^{-(y_{k}-\mu_{ik})^{2}/(2\sigma_{ik}^{2})}e^{-\frac{(x_{k}-y_{k}){}^{2}}{2\gamma^{2}}}e^{-\frac{x_{k}^{2}}{2}}}{\sqrt{2\pi\sigma_{ik}^{2}}\sqrt{2\pi}}dxdy,\label{eq:temp3}\end{aligned}$$ where $y_{k}$ is the $k$-th coordinate of $y\in\mathbb{R}^{d}$. Note that integrations over the dimensions of $\mathbb{R}^{d}$ are independent, so the main component that we need to compute is $$\sqrt{2\pi\gamma^{2}}\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{e^{-(y_{k}-\mu_{ik})^{2}/(2\sigma_{ik}^{2})}e^{-\frac{(x_{k}-y_{k}){}^{2}}{2\gamma^{2}}}e^{-\frac{x_{k}^{2}}{2}}}{\sqrt{2\pi\sigma_{ik}^{2}\sqrt{2\pi\gamma^{2}}}\sqrt{2\pi}}dx_{k}dy_{k},$$ where we multiplied and divided by the normalizing factor for the kernel. Let us replace $\mu_{ik}$ in the first exponential by $w$, and reasoning as with Eq. (\[eq:mmdu\_trick\]), we see that the integral gives the probability density function of $W=A+B+C$, where $A\sim N(0,\sigma_{ik}^{2})$, $B\sim N(0,\gamma^{2})$, and $C\sim N(0,1)$. Thus, the integral is given by the pdf of $W\sim N(0,1+\gamma^{2}+\sigma_{ik}^{2})$. Including the multiplier in front of the integral, and replacing $w=\mu_{ik}$, we get: $$\left(\frac{\gamma^{2}}{1+\gamma^{2}+\sigma_{ik}^{2}}\right)^{1/2}e^{-\frac{\mu_{ik}{}^{2}}{2(1+\gamma^{2}+\sigma_{ik}^{2})}}.$$ Putting this back into the last expression in Eq. (\[eq:temp3\]), we obtain $$\E_{x\sim P,y\sim Q_{\mathrm{batch}}}[k(x,y)]=\frac{1}{n}\sum_{i=1}^{n}\prod_{k=1}^{d}\left(\frac{\gamma^{2}}{1+\gamma^{2}+\sigma_{ik}^{2}}\right)^{1/2}e^{-\frac{\mu_{ik}{}^{2}}{2(1+\gamma^{2}+\sigma_{ik}^{2})}}.$$ Next we will compute the third term in Eq. (\[eq:mmd\]), namely $$\begin{aligned} & \E_{y,y'\sim Q_{\mathrm{batch}}}[k(y,y')]=\\ = & \int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\left(\frac{1}{n}\sum_{i=1}^{n}\prod_{k=1}^{d}\frac{e^{-(y_{k}-\mu_{ik})^{2}/(2\sigma_{ik}^{2})}}{\sqrt{2\pi\sigma_{ik}^{2}}}\right)\cdot e^{-\frac{\Vert y-y'\Vert^{2}}{2\gamma^{2}}}\cdot\left(\frac{1}{n}\sum_{j=1}^{n}\prod_{k=1}^{d}\frac{e^{-(y'_{k}-\mu_{jk})^{2}/(2\sigma_{jk}^{2})}}{\sqrt{2\pi\sigma_{jk}^{2}}}\right)dydy'.\end{aligned}$$ As before, we can turn the exponential in the middle into a product over the dimensions, and after distributing over the summations and pushing the integrals into products, we obtain, $$\frac{1}{n^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}\prod_{k=1}^{d}\sqrt{2\pi\gamma^{2}}\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{e^{-(y_{k}-\mu_{ik})^{2}/(2\sigma_{ik}^{2})}}{\sqrt{2\pi\sigma_{ik}^{2}}}\cdot\frac{e^{-\frac{(y_{k}-y'_{k})^{2}}{2\gamma^{2}}}}{\sqrt{2\pi\gamma^{2}}}\cdot\frac{e^{-(y'_{k}-\mu_{jk})^{2}/(2\sigma_{jk}^{2})}}{\sqrt{2\pi\sigma_{jk}^{2}}}dy_{k}dy_{k}'.$$ In the double integral, let us replace $\mu_{ik}$ with $w$, keeping $\mu_{jk}$ intact. Now the integral can be split to inner and outer piece, and computed similarly to Eq. (\[eq:mmdu\_trick\]) as the probability distribution of the sum of three one-dimensional Gaussians: $W=A+B+C$, where $A\sim N(0,\sigma_{ik}^{2})$, $B\sim N(0,\gamma^{2}),$and $C\sim N(\mu_{jk},\sigma_{jk}^{2}).$ We immediately get $W\sim N(\mu_{jk},\gamma^{2}+\sigma_{ik}^{2}+\sigma_{jk}^{2})$, and the expression for the integral (multiplied by $\sqrt{2\pi\gamma^{2}}$) in terms of $w$ is $$\left(\frac{\gamma^{2}}{\gamma^{2}+\sigma_{ik}^{2}+\sigma_{jk}^{2}}\right)^{1/2}e^{-\frac{(w-\mu_{jk}){}^{2}}{2(\gamma^{2}+\sigma_{ik}^{2}+\sigma_{jk}^{2})}}.$$ Substituting back $w=\mu_{ik}$ we obtain: $$\E_{y,y'\sim Q_{\mathrm{batch}}}[k(y,y')]=\frac{1}{n^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}\prod_{k=1}^{d}\left(\frac{\gamma^{2}}{\gamma^{2}+\sigma_{ik}^{2}+\sigma_{jk}^{2}}\right)^{1/2}e^{-\frac{(\mu_{ik}-\mu_{jk}){}^{2}}{2(\gamma^{2}+\sigma_{ik}^{2}+\sigma_{jk}^{2})}}.$$ Collecting all the terms in Eq. (\[eq:mmd\]), we obtain the formula for $\mathrm{MMDU}_{n}^{2}(\mathcal{N}_{d}(\vec{0},I),Q)$ in Section \[subsec:Random-Encoders\]. \[subsec:Correctness-of-the\]Correctness of the Monte Carlo Sampling Procedure ------------------------------------------------------------------------------ Here we prove the existence of an orthogonal matrix that establishes a one-to-one measure-preserving mapping between centered-whitened samples from $\mathcal{N}_{d}(\vec{\mu},\Sigma)$ and $\mathcal{N}_{d}(\vec{0},I)$. Consider the diagonalization of the true covariance matrix $\Sigma=ODO^{T}$, where $D$ is a diagonal, and $O$ is an orthogonal matrix—this is possible by the symmetry of $\Sigma$. Given a sample $X_{i}\sim\mathcal{N}_{d}(\vec{\mu},\Sigma)$ we can write $X_{i}=\vec{\mu}+OD^{1/2}Y_{i}$, where $Y_{i}$ distributed as $\mathcal{N}_{d}(\vec{0},I)$. Let $Z_{i}^{X}=S_{X}^{-1/2}(X_{i}-\bar{X})$ be the centered-whitened $X_{i}$, and let $Z_{i}^{Y}=S_{Y}^{-1/2}(Y_{i}-\bar{Y})$ be the centered-whitened $Y_{i}$. We will show that $Z_{i}^{X}=RZ_{i}^{Y}$ for some orthogonal matrix (i.e. rotation) $R$ computed below. We start by noting that $\bar{X}=\vec{\mu}+OD^{1/2}\bar{Y}$, and that the following relationship holds between the sample variance matrices: $$\begin{aligned} S_{X} & =(n-1)^{-1}\sum_{i}(X_{i}-\bar{X})(X_{i}-\bar{X})^{T}\nonumber \\ = & (n-1)^{-1}\sum_{i}(\vec{\mu}+OD^{1/2}Y_{i}-\vec{\mu}-OD^{1/2}\bar{Y})(\vec{\mu}+OD^{1/2}Y_{i}-\vec{\mu}-OD^{1/2}\bar{Y})^{T}\label{eq:sxsy}\\ = & (n-1)^{-1}\sum_{i}OD^{1/2}(Y_{i}-\bar{Y})(Y_{i}-\bar{Y})^{T}(OD^{1/2})^{T}\nonumber \\ = & OD^{1/2}\left[(n-1)^{-1}\sum_{i}(Y_{i}-\bar{Y})(Y_{i}-\bar{Y})^{T}\right](OD^{1/2})^{T}=OD^{1/2}S_{Y}D^{1/2}O^{T}.\nonumber \end{aligned}$$ Now we have, $$\begin{aligned} Z_{i}^{X} & =S_{X}^{-1/2}(X_{i}-\bar{X})=S_{X}^{-1/2}(\vec{\mu}+OD^{1/2}Y_{i}-\vec{\mu}-OD^{1/2}\bar{Y})\\ = & S_{X}^{-1/2}OD^{1/2}(Y_{i}-\bar{Y})=S_{X}^{-1/2}OD^{1/2}S_{Y}^{1/2}S_{Y}^{-1/2}(Y_{i}-\bar{Y})\\ = & RS_{Y}^{-1/2}(Y_{i}-\bar{Y})=RZ_{i}^{Y}.\end{aligned}$$ To finish the proof, we need to show that $R=S_{X}^{-1/2}OD^{1/2}S_{Y}^{1/2}$ is an orthogonal matrix. It is enough to show that $RR^{T}=I$, and indeed: $$\begin{aligned} RR^{T} & =S_{X}^{-1/2}OD^{1/2}S_{Y}^{1/2}S_{Y}^{1/2}D^{1/2}O^{T}S_{X}^{-1/2}=S_{X}^{-1/2}(OD^{1/2}S_{Y}D^{1/2}O^{T})S_{X}^{-1/2}\\ = & S_{X}^{-1/2}S_{X}S_{X}^{-1/2}=I,\end{aligned}$$ where we used that both of the sample variance matrices are symmetric, and replaced the expression in the parenthesis by $S_{X}$ based on Eq. (\[eq:sxsy\]). \[appendix:mmd-python\]Python Implementation of $\mathrm{SMMDU}_{n}^{2}$ ------------------------------------------------------------------------- This implementation uses Tensorflow [@tensorflow2015-whitepaper]. The choice of the kernel width in the adaptive case can be explained as follows. As the neural network converges to the normal distribution, the quantity `mean_norms2`, which captures the mean squared distance to the origin, converges to the latent dimension. This follows from the fact that the mean of $\chi_{d}^{2}$ distribution is $d$. As a result, in this limit, the adaptive and the fixed kernel width would be approximately equal. The reason for using the distance to the origin, and not to the center of mass, is to prioritize convergence of the codes close to the origin. We use a non-robust statistic (mean rather than the median) to have a better control over the outliers. \[subsec:Neural-net-architectures\]WAE Architecture and Training ---------------------------------------------------------------- For the MNIST WAE experiments we use the architecture below. Here `CodeNorm` is the code normalization layer. It can be replaced by `BatchNormalization(center=False, scale=False)` in Keras, with the caveat that the computation of the batch variance in Keras uses the biased sample estimate; to correct for this it needs to be multiplied by $n/(n-1)$. The latent dimension is $d=8$, kernel scale is $s=1/8$, and the regularization weight is $\lambda=0.01$. Adam [@ADAM] is used for 60 epochs with default parameters except for the learning rate. The learning rate for the first 20 epochs is 0.001, then set to 0.001/4 for the next 20, and set to 0.001/16 for the last 20 epochs. No regularization, dropout, noise, or augmentation is used. Adaptive versions use `smmdu` with `adaptive=True`, with further modifications as described in the main text. These are trained for 80 epochs, with 20 extra epochs at the initial rate. [^1]: Of course, we expect $\mathrm{Mean}_{1}\approx0$ since all of the three methods are unbiased. For the Analytic RBF, we also know the theoretical value of $\mathrm{SD}_{1}$ from the closed-form formula for the variance. However, for fairness we will use empirical estimates for all of the three methods. [^2]: https://keras.rstudio.com/articles/examples/variational\_autoencoder\_deconv.html

--- abstract: 'In the context of Standard Model (SM) extensions, the seesaw mechanism provides the most natural explanation for the smallness of neutrino masses. In this work we consider the most economical type-I seesaw realization in which two right-handed neutrinos are added to the SM field content. For the sake of predictability, we impose the maximum number of texture zeros in the lepton Yukawa and mass matrices. All possible patterns are analyzed in the light of the most recent neutrino oscillation data, and predictions for leptonic CP violation are presented. We conclude that, in the charged-lepton mass basis, eight different texture combinations are compatible with neutrino data at $1\sigma$, all of them for an inverted-hierarchical neutrino mass spectrum. Four of these cases predict a CP-violating Dirac phase close to $3\pi/2$, which is around the current best-fit value from global analysis of neutrino oscillation data. If one further reduces the number of free parameters by considering three equal elements in the Dirac neutrino Yukawa coupling matrix, several texture combinations are still compatible with data but only at $3\sigma$. For all viable textures, the baryon asymmetry of the Universe is computed in the context of thermal leptogenesis, assuming (mildly) hierarchical heavy Majorana neutrino masses $M_{1,2}$. It is shown that the flavored regime is ruled out, while the unflavored one requires $M_{1} \sim 10^{14}$ GeV.' author: - 'D. M. Barreiros' - 'R. G. Felipe' - 'F. R. Joaquim' title: 'The minimal type-I seesaw model with maximally-restricted texture zeros' --- Introduction {#Intro} ============ The discovery of neutrino oscillations provided a solid evidence for physics beyond the Standard Model (SM), by confirming the existence of neutrino masses and mixing. From the theory viewpoint, the most straightforward and elegant way of accounting for them consists of adding right-handed (RH) neutrinos to the SM field content. If heavy enough, these states can mediate neutrino masses at the classical level through the well-known seesaw mechanism [@Minkowski:1977sc]. Besides supplying an explanation for small neutrino masses, the addition of RH neutrinos to the SM allows for the leptogenesis mechanism [@Fukugita:1986hr] to work through the out-of-equilibrium decays of the heavy neutrinos in the early Universe (for reviews see e.g. [@Buchmuller:2004nz; @Davidson:2008bu; @Chun:2017spz; @Fong:2013wr]). This offers an answer for another SM puzzle: the baryon asymmetry of the Universe (BAU). Although, in principle, the number of RH neutrinos is arbitrary, at least two are necessary to explain the present neutrino oscillation data, namely, three nonzero neutrino mixing angles and two mass-squared differences. Interestingly, at least two RH neutrinos are also required for leptogenesis to be realized. Therefore, the two RH neutrino seesaw model (2RHNSM) is not only a minimal model for neutrino masses, but also for the generation of the BAU in the context of leptogenesis. Still, even in this scenario, the number of parameters describing the neutrino Lagrangian at high energies is larger than the number of low-energy observables currently (or potentially) measured by experiments. One way of increasing predictability is to consider texture zeros in the lepton Yukawa and mass matrices, which can be motivated, for instance, by imposing U(1) Abelian flavor symmetries [@Grimus:2004hf; @Cebola:2015dwa]. In general, texture zeros imply predictions not only for low-energy neutrino parameters but also for the BAU, since leptogenesis is sensitive to the couplings which control neutrino masses and mixing. Therefore, a complete study of all possible texture zeros in the light of most recent neutrino data is welcome. In particular, since neutrino experiments are starting to deliver some information regarding leptonic CP violation [@Branco:2011zb], predictions for low-energy CP phases are of utmost importance. At the same time, a connection with leptogenesis can also be established in this framework [@Branco:2011zb; @Hagedorn:2017wjy]. These questions have already been partially covered in the literature. For instance, the compatibility of texture-zero hypothesis in the 2RHNSM with neutrino data has been studied in Refs. [@Frampton:2002qc; @Ibarra:2003up; @Harigaya:2012bw; @Rink:2016vvl; @Shimizu:2017fgu] and, in the context of leptogenesis, in Refs. [@GonzalezFelipe:2003fi; @Joaquim:2005zv; @Abada:2006ea; @Zhang:2015tea; @Siyeon:2016wro; @Geib:2017bsw; @Achelashvili:2017nqp; @Shimizu:2017vwi]. In this work, we revisit the 2RHNSM in maximally restricted texture-zero scenarios, i.e. when the maximum number of texture zeros is imposed in the lepton Yukawa and mass matrices. Moreover, we consider cases in which equality relations among the Dirac neutrino Yukawa couplings exist. For textures that reproduce the observed neutrino mass and mixing patterns, we present the predictions for low-energy CP violation, neutrinoless double beta decay and the BAU. Special attention will be paid to the treatment of leptogenesis in the 2RHNSM. Contrary to what is usually done in the literature, where only the decay of the lightest heavy neutrino is considered, we include decays of both heavy neutrinos in our analysis. Moreover, flavor effects which arise from the fact that lepton interactions become out of equilibrium at different temperatures are taken into account. This paper is organized as follows. In Section \[sec1\] we set the basics of the 2RHNSM, by describing the model and identifying the number of parameters at high and low energies. Afterwards, in Section \[sec2\], the maximally-restricted texture zero matrices are identified, and their compatibility with neutrino data is analyzed. Furthermore, the predictions for Dirac and Majorana CP phases are shown, together with those for the effective neutrino mass parameter relevant for neutrinoless double beta decays. We also consider cases with three equal elements in the Dirac neutrino Yukawa coupling matrix in Section \[sec3a\]. We then compute the BAU in the thermal leptogenesis framework in Section \[sec3\], and determine under which conditions its value is compatible with the observed one. Our conclusions are drawn in Section \[sec4\]. The two right-handed neutrino seesaw model {#sec1} ========================================== Considering only Yukawa and mass terms, the lepton Lagrangian density for the SM extended with RH neutrino fields $\nu_{R}$ is $\mathcal{L}=\mathcal{L}_{\ell}+\mathcal{L}_\nu$ with $$\begin{aligned} \mathcal{L}_\nu&=-\overline{\ell_{L}}{\mathbf{Y}^\nu}\tilde{\Phi}\nu_{R} -\frac{1}{2}\overline{(\nu_{R})^c}{\mathbf{M}_R}\nu_{R} + \text{H.c.}\,, \label{LtypeI}\\ \mathcal{L}_\ell&=-\overline{\ell_{L}}{\mathbf{Y}^\ell}\Phi\,e_{R} + \text{H.c.}\,.\label{Lcl}\end{aligned}$$ Here, $\ell_{L}$ and $\Phi$ are the SM lepton and Higgs doublets, respectively, $\tilde{\Phi}=i\sigma_2 \Phi^\ast$, and $e_R$ denote the RH charged-lepton fields. The Dirac neutrino Yukawa couplings and RH neutrino mass matrices are described by ${\mathbf{Y}^\nu}$ and ${\mathbf{M}_R}$. For $N$ RH neutrinos, ${\mathbf{Y}^\nu}$ and ${\mathbf{M}_R}$ are $3\times N$ and $N\times N$ general complex matrices, being ${\mathbf{M}_R}$ symmetric. After integrating out the $\nu_R$’s, the effective Majorana neutrino mass matrix ${\mathbf{M}^\nu}$, obtained upon electroweak symmetry breaking, is given by the seesaw formula [@Minkowski:1977sc] $$\begin{aligned} {\mathbf{M}^\nu}=-v^2{\mathbf{Y}^\nu}{\mathbf{M}_R}^{-1}{{\mathbf{Y}^\nu}}^T\,, \label{Mnuseesaw}\end{aligned}$$ which is valid for ${\mathbf{M}_R}\gg v$, where $v=174\,{\rm GeV}$ is the vacuum expectation value of the neutral component of $\Phi$. This (symmetric) matrix is diagonalized by a unitary matrix $\mathbf{U_\nu}$ as $$\begin{aligned} {\mathbf{U}_\nu}^T {\mathbf{M}^\nu}{\mathbf{U}_\nu}=\text{diag}(m_1,m_2,m_3)\equiv \mathbf{d}_m\,, \label{Mnudiag}\end{aligned}$$ where $m_i$ are the (real and positive) effective neutrino masses. Considering that ${\mathbf{U}_\ell}$ rotates the left-handed (LH) charged-lepton fields to their diagonal mass basis, lepton mixing in charged currents is encoded in the so-called Pontecorvo-Maki-Nakagawa-Sakata (PMNS) unitary matrix ${\mathbf{U}}$ given by $$\begin{aligned} {\mathbf{U}}={\mathbf{U}_\ell}^\dag {\mathbf{U}_\nu}\,. \label{UPMNS}\end{aligned}$$ Throughout this work we will use the standard parametrization [@Patrignani:2016xqp] $$\begin{gathered} \mathbf{U}=\begin{pmatrix} c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\ -s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta}&s_{23}c_{13}\\ s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta}&c_{23}c_{13} \end{pmatrix}\!\!\begin{pmatrix} 1&0&0\\ 0&e^{i\alpha_{21}/2}&0\\ 0&0&e^{i\alpha_{31}/2}\\ \end{pmatrix}\,, \label{Uparam}\end{gathered}$$ where $c_{ij}\equiv \cos\theta_{ij}$, $s_{ij}\equiv \sin\theta_{ij}$ and $\theta_{ij}\,(i < j=1,2,3)$ are the three lepton mixing angles. The phases $\delta$ and $\alpha_{21,31}$ are Dirac and Majorana-type CP-violating phases, respectively. The present values for $\theta_{ij}$, $\delta$ and $\Delta m^2_{ij}=m_i^2-m_j^2$, extracted from global analyses of all neutrino oscillation data [@deSalas:2017kay; @Esteban:2016qun; @Capozzi:2016rtj], are given in Table \[datatable\] for both normally-ordered (NO) and inverted-ordered (IO) neutrino mass spectra defined as: $$\begin{aligned} {\text {NO}:}& m_1 < m_2 < m_3\;\, ({\Delta m^2_{31}}>0)\,,\\ {\text {IO}:}& m_3 < m_1 < m_2\;\, ({\Delta m^2_{31}}<0)\,. \label{NOIO}\end{aligned}$$ Notice that although neutrino mixing angles and mass-squared differences are known with very good precision, the experimental sensitivity to the value of $\delta$ is still limited, and the statistical significance of the presented ranges for that parameter is low. **Parameter** Best Fit $\pm1\sigma$ $3\sigma$ range ---------------------------------------------------------------------- ------------------------ ----------------------- $\theta_{12}\;(^{\circ})$ $34.5_{-1.0}^{+1.1}$ $31.5\rightarrow38.0$ \[0.15cm\] $\theta_{23}\;(^{\circ})$ \[NO\] $41.0\pm1.1$ $38.3\rightarrow52.8$ $\theta_{23}\;(^{\circ})$ \[IO\] $50.5\pm1.0$ $38.5\rightarrow53.0$ \[0.15cm\] $\theta_{13}\;(^{\circ})$ \[NO\] $8.44_{-0.15}^{+0.18}$ $7.9\rightarrow8.9$ $\theta_{13}\;(^{\circ})$ \[IO\] $8.41_{-0.17}^{+0.16}$ $7.9\rightarrow8.9$ \[0.15cm\] $\delta\;(^{\circ})$ \[NO\] $252_{-36}^{+56}$ $0\rightarrow360$ $0\rightarrow31$ $142\rightarrow360$ \[0.15cm\] $\Delta m_{21}^2\;(\times 10^{-5}\;\text{eV}^2)$ $7.56\pm0.19$ $7.05\rightarrow8.14$ \[0.15cm\] $|\Delta m_{31}^2|\;(\times 10^{-3}\;\text{eV}^2)$ \[NO\] $2.55\pm0.04$ $2.43\rightarrow2.67$ $|\Delta m_{31}^2|\;(\times 10^{-3}\;\text{eV}^2)$ \[IO\] $2.49\pm0.04$ $2.37\rightarrow2.61$ : Neutrino oscillation parameters obtained from the global analysis of Ref. [@deSalas:2017kay] (see also Refs. [@Esteban:2016qun] and [@Capozzi:2016rtj]).[]{data-label="datatable"} Let us now consider the simplest type-I seesaw model which can account for the data presented in Table \[datatable\], i.e. the 2RHNSM. In this case, ${\mathbf{Y}^\nu}$ and ${\mathbf{M}_R}$ are $3\times 2$ and $2 \times 2$ matrices, respectively. In the mass-eigenstate basis of $\nu_R$, the free parameters in the Lagrangian (\[LtypeI\]) are the two RH neutrino masses $M_{1,2}$, and the twelve real parameters of ${\mathbf{Y}^\nu}$. By rotating the LH charged-lepton fields, one is able to eliminate three parameters from ${\mathbf{Y}^\nu}$, leaving a total of eleven. Since for the 2RHNSM the effective neutrino mass matrix ${\mathbf{M}^\nu}$ given in Eq. (\[Mnuseesaw\]) is rank two, $m_1=0\,(m_3=0)$ for NO (IO).[^1] Moreover, the diagonal phase matrix in Eq. (\[Uparam\]) must be replaced by $\text{diag}(1,e^{i\alpha/2},1)$ since, in the presence of a massless neutrino, only one Majorana phase is physical. Thus, in the 2RHNSM, the low-energy neutrino sector is described by seven parameters (two masses, three mixing angles and two CP-violating phases), to be compared with the eleven at high energies. One convenient way of parameterizing ${\mathbf{Y}^\nu}$ relies on the so-called Casas-Ibarra parametrization [@Casas:2001sr]. In the basis where both ${\mathbf{M}_R}$ and ${\mathbf{Y}^\ell}$ are diagonal, $$\begin{gathered} \mathbf{Y}^\nu=v^{-1}{\mathbf{U}}^*\,\mathbf{d}_m^{1/2}\,\mathbf{R}\,\mathbf{d}_M^{1/2}\,, \label{CasasandIbarra}\end{gathered}$$ with $\mathbf{d}_M=\text{diag}(M_1,M_2)$. The matrix $\mathbf{R}$ is a $3\times 2$ complex orthogonal matrix which can be parametrized by a single complex angle $z$ in the following way $$\begin{gathered} \mathbf{R}_{\text{NH}}=\begin{pmatrix} 0&0\\ \cos z&-\sin z\\ \xi \sin z&\xi \cos z \end{pmatrix} \;,\; \mathbf{R}_{\text{IH}}=\begin{pmatrix} \cos z &-\sin z \\ \xi \sin z &\xi \cos z\\ 0&0 \end{pmatrix}\; , \label{RmatrixIO}\end{gathered}$$ with $\xi=\pm 1$. Notice that, in the case of a non-diagonal ${\mathbf{M}_R}$, the right-hand side of Eq. (\[CasasandIbarra\]) must be multiplied on the right by ${\mathbf{U}_R}^\dag$, being ${\mathbf{U}_R}$ the unitary matrix which diagonalizes ${\mathbf{M}_R}$ as ${\mathbf{U}_R}^T {\mathbf{M}_R}{\mathbf{U}_R}=\mathbf{d}_M$. Clearly, even in the simplest minimal type-I seesaw model, there are more free independent parameters at high energies than at low energies. In order to reduce the degree of arbitrariness of the 2RHNSM, in the next section we will introduce maximally-restricted texture zeros and study their phenomenological implications. Maximally-restricted texture zeros {#sec2} ================================== In this section we will study the implications of imposing texture zeros in ${\mathbf{Y}^\ell}$, ${\mathbf{Y}^\nu}$ and ${\mathbf{M}_R}$. Our guiding principle is to consider the maximum number of zeros such that the charged-lepton masses and neutrino data can be accommodated. In the former case, this corresponds to having six zeros in ${\mathbf{Y}^\ell}$, which guarantees three non-degenerate masses. There are six textures of this type related among each other by permutations of rows and/or columns applied to ${\mathbf{Y}^\ell}_{\rm diag}={\rm diag}(y_e,y_\mu,y_\tau)$, where $y_{e,\mu,\tau}=m_{e,\mu,\tau}/v$. Textures for ${\mathbf{Y}^\nu}$ with three or more zeros lead to vanishing mixing angles and/or two massless neutrinos, being therefore excluded experimentally. In principle, with two texture zeros in ${\mathbf{Y}^\nu}$, all neutrino data could be reproduced. There are fifteen different types of $3\times 2$ matrices with two vanishing entries. Some of them are automatically excluded by present neutrino data, namely, - [Textures with two zeros placed in the same line $j$ of ${\mathbf{Y}^\nu}$ are excluded since these lead to the case in which the two RH neutrino fields are decoupled from the lepton flavor $j$. Therefore, all elements in line (and column) $j$ of the Majorana neutrino mass matrix ${\mathbf{M}^\nu}$ vanish, implying the existence of two vanishing mixing angles $\theta_{ij}$, which is excluded by the data. In practice, this corresponds to the situation in which one neutrino flavor state coincides with its mass eigenstate.]{} - [If both zeros are placed in lines $(i,j)$ of the same column in ${\mathbf{Y}^\nu}$, then lines (and columns) $(i,j)$ of ${\mathbf{M}^\nu}$ are linearly dependent. Thus, at least one mixing angle $\theta_{ij}$ is zero, leading to the unrealistic case in which one flavor eigenstate is a superposition of only two of the three mass eigenstates.]{} We therefore conclude that the maximally-allowed number of texture zeros in ${\mathbf{Y}^\nu}$ is two. The ${\mathbf{Y}^\nu}$ textures to be analyzed are of the type: $$\begin{gathered} \begin{matrix} {\rm T}_1:\;\begin{pmatrix} 0&\times\\ \times&0\\ \times&\times \end{pmatrix},& {\rm T_2}: \;\begin{pmatrix} 0&\times\\ \times&\times\\ \times&0 \end{pmatrix},& {\rm T_3}:\; \begin{pmatrix} \times&\times\\ 0&\times\\ \times&0 \end{pmatrix},\\ &&&\\ {\rm T_4}:\; \begin{pmatrix} \times&0\\ 0&\times\\ \times&\times \end{pmatrix},& {\rm T_5}: \;\begin{pmatrix} \times&0\\ \times&\times\\ 0&\times \end{pmatrix},& {\rm T_6}:\; \begin{pmatrix} \times&\times\\ \times&0\\ 0&\times \end{pmatrix}, \end{matrix} \label{Tstructures}\end{gathered}$$ where the symbol $\times$ denotes a generic non-vanishing entry. [K[1.5cm]{}|K[1.5cm]{}|c|K[1.2cm]{}|K[1.2cm]{}]{} ${\mathbf{Y}^\nu}$&${\mathbf{M}_R}$&${\mathbf{M}^\nu}$&**NH**&**IH**\ \[-0.15cm\][T$_1$, T$_2$]{}&\[-0.15cm\][R$_2$]{}&&\[-0.15cm\]&\[-0.25cm\]\ \[-0.15cm\][T$_4$, T$_5$]{}&\[-0.15cm\][R$_3$]{}&&\ T$_1$, T$_4$&R$_1$&B: $\begin{pmatrix} \times &0&\times\\ \cdot&\times&\times\\ \cdot&\cdot&\times \end{pmatrix}$&&(${1\sigma}$)\ T$_2$, T$_5$&R$_1$&C: $\begin{pmatrix} \times &\times&0\\ \cdot&\times&\times\\ \cdot&\cdot&\times \end{pmatrix}$&&(${1\sigma}$)\ \[-0.15cm\][T$_3$, T$_4$]{}&\[-0.15cm\][R$_2$]{}&&\[-0.15cm\]&\[-0.25cm\][(${1\sigma}$)]{}\ \[-0.15cm\][T$_1$, T$_6$]{}&\[-0.15cm\][R$_3$]{}&&\ T$_3$, T$_6$&R$_1$&E: $\begin{pmatrix} \times &\times&\times\\ \cdot&\times&0\\ \cdot&\cdot&\times \end{pmatrix}$&&\ \[-0.15cm\][T$_5$, T$_6$]{}&\[-0.15cm\][R$_2$]{}&&\[-0.15cm\]&\[-0.25cm\][(${3\sigma}$)]{}\ \[-0.15cm\][T$_2$, T$_3$]{}&\[-0.15cm\][R$_3$]{}&&\ As for ${\mathbf{M}_R}$, with more that two texture zeros, at least one of the RH neutrinos is massless. On the other hand, with one texture zero, there are three different patterns for ${\mathbf{M}_R}$: $$\begin{aligned} {\rm R}_1:\begin{pmatrix} \times&0\\ \cdot&\times \end{pmatrix}\;,\; {\rm R_2}:\begin{pmatrix} 0&\times\\ \cdot&\times \end{pmatrix}\;,\; {\rm R_3}:\begin{pmatrix} \times&\times\\ \cdot&0 \end{pmatrix}\,, \label{Rstructures}\end{aligned}$$ with the dot $(\cdot)$ indicating the symmetric nature of the matrix. Combining them with the ${\mathbf{Y}^\nu}$ textures (\[Tstructures\]) through the seesaw formula (\[Mnuseesaw\]), one obtains the textures for ${\mathbf{M}^\nu}$ given in the third column of Table \[tabR1R2R3\]. All cases A-F feature the presence of one texture zero in ${\mathbf{M}^\nu}$. Notice that sets of $({\mathbf{Y}^\nu},{\mathbf{M}_R})$ textures related by simultaneous permutations of the columns in ${\mathbf{Y}^\nu}$, and lines and columns in ${\mathbf{M}_R}$, lead to the same ${\mathbf{M}^\nu}$ due to invariance of Eq. (\[Mnuseesaw\]) under $\nu_R$ rotations. Moreover, when ${\mathbf{M}_R}$ is diagonal (texture R$_1$), ${\mathbf{M}^\nu}$ is the same for ${\mathbf{Y}^\nu}$ textures related by a column permutation. For instance, the sets $(\mathrm{T}_1,\mathrm{R}_1)$ and $(\mathrm{T}_4,\mathrm{R}_1)$ lead to the same low-energy predictions since $\mathrm{T}_1$ and $\mathrm{T}_4$ are related by column permutation. The condition ${\mathbf{M}^\nu}_{\alpha\beta}=0$ imposes relations among the neutrino parameters. In particular, from Eq. (\[Mnudiag\]) it is straightforward to conclude that [@Xing:2003ic; @Gautam:2015kya] $$\begin{aligned} \text{NH}:&\dfrac{m_2}{m_3}&\!\!=-\dfrac{{\mathbf{U}}^*_{\alpha 3}{\mathbf{U}}^*_{\beta 3}}{{\mathbf{U}}^*_{\alpha 2}{\mathbf{U}}^*_{\beta 2}}\;, \label{NHrelation}\\ \text{IH}:&\dfrac{m_1}{m_2}&\!\!=-\dfrac{{\mathbf{U}}^*_{\alpha 2}{\mathbf{U}}^*_{\beta 2}}{{\mathbf{U}}^*_{\alpha 1}{\mathbf{U}}^*_{\beta 1}}\;. \label{IHrelation}\end{aligned}$$ Taking into account that neutrino masses $m_i$ are real and positive, $m_2^2={\Delta m^2_{21}}$ ($m_2^2={\Delta m^2_{21}}+|{\Delta m^2_{31}}|$) and $m_3^2={\Delta m^2_{31}}$ ($m_1^2=|{\Delta m^2_{31}}|$) for NH (IH). Thus, we have $$\begin{aligned} \text{NH}:\;\;\;\;\;\;\;r_\nu&=&\left|\dfrac{{\mathbf{U}}^*_{\alpha 3}{\mathbf{U}}^*_{\beta 3}}{{\mathbf{U}}^*_{\alpha 2}{\mathbf{U}}^*_{\beta 2}}\right|^2, \label{NHrelationb}\\ \text{IH}:\dfrac{1}{1+r_\nu}&=&\left|\dfrac{{\mathbf{U}}^*_{\alpha 2}{\mathbf{U}}^*_{\beta 2}}{{\mathbf{U}}^*_{\alpha 1}{\mathbf{U}}^*_{\beta 1}}\right|^2,\;\;\;r_\nu\equiv \dfrac{{\Delta m^2_{21}}}{|{\Delta m^2_{31}}|}\,. \label{IHrelationb}\end{aligned}$$ Given the parametrization in Eq. (\[Uparam\]), and the experimentally-allowed ranges for the mixing angles presented in Table \[datatable\], one can test which textures lead to viable values of $r_\nu$ using the above relations. From all cases, the simplest one to be analyzed is texture A, for which $r_\nu$ is simply given by $$\begin{aligned} {\rm NH:} \,\,r_\nu=\frac{t_{13}^4}{s_{12}^4}\simeq 0.005\;,\; {\rm IH:}\,\, r_\nu=\frac{1}{t_{12}^4}-1\simeq 3.5\,.\end{aligned}$$ These numerical estimates, obtained using the best-fit values given in Table \[datatable\], indicate that texture A is disfavored by data, independently of the value of $\delta$. By varying the mixing angles in their experimentally $1\sigma$ and $3\sigma$ allowed regions,[^2] we plot $r_\nu$ as a function of $\delta$ in Figs. \[NHmassratios\] and \[IHmassratios\] for NH and IH, respectively, using Eqs. (\[NHrelationb\]) and (\[IHrelationb\]) together with Eq. (\[Uparam\]). In light (dark) blue we show the $r_\nu$ regions obtained when all mixing angles vary in their $3\sigma$ ($1\sigma$) experimental ranges. The horizontal pink bands (red line) indicate the $3\sigma$ experimental range (best-fit value) for $r_\nu$. From these results, we conclude that all textures with one zero in ${\mathbf{M}^\nu}$ are incompatible with neutrino data at more than $3\sigma$ level for NH. In the context of the 2RHNSM with texture zeros in ${\mathbf{Y}^\nu}$ and ${\mathbf{M}_R}$, this means that all combinations shown in Table \[tabR1R2R3\] are excluded for that type of neutrino mass spectrum. For IH (Fig. \[IHmassratios\]) and specific ranges of $\delta$, one obtains values for $r_\nu$ compatible with the data at $1\sigma$ for textures B, C and D, and only at $3\sigma$ for texture F. Therefore, all combinations of textures for ${\mathbf{Y}^\nu}$ and ${\mathbf{M}_R}$ leading to textures B, C, D and F for ${\mathbf{M}^\nu}$ are viable. Notice that only textures B and C predict $r_\nu$ values in its $1\sigma$ range, for $\delta$ around its best-fit value.   [ll]{} ${\mathbf{M}^\nu}\;$ &$\hspace{2cm}$**CP-violating phases**\ & $c_\delta=2\dfrac{[s^4_{12}(1+r_\nu)-c^4_{12}]s^2_{23}s^2_{13}+r_\nu c^2_{23}s_{12}^2c^2_{12}}{[s^2_{12}(1+r_\nu)+c^2_{12}]\sin(2\theta_{12})\sin(2\theta_{23})s_{13}}$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ & $c_\alpha=\dfrac{(2+r_\nu)c^2_{23}s^2_{12}c_{12}^2-[s^4_{12}(1+r_\nu)+c^4_{12}]s^2_{23}s^2_{13}}{2\sqrt{1+r_\nu}(c^2_{23}+s^2_{23}s^2_{13})s^2_{12}c^2_{12}}$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ & $c_\delta=-2\dfrac{[s^4_{12}(1+r_\nu)-c^4_{12}]c^2_{23}s^2_{13}+r_\nu s^2_{23}s_{12}^2c^2_{12}}{[s^2_{12}(1+r_\nu)+c^2_{12}]\sin(2\theta_{12})\sin(2\theta_{23})s_{13}}$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ &$c_\alpha=\dfrac{(2+r_\nu)s^2_{23}s^2_{12}c_{12}^2-[s^4_{12}(1+r_\nu)+c^4_{12}]c^2_{23}s^2_{13}}{2\sqrt{1+r_\nu}(s^2_{23}+c^2_{23}s^2_{13})s^2_{12}c^2_{12}}$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ & $c_\delta=2\dfrac{(c^2_{12}\sqrt{1+r_\nu}-s^2_{12})c^2_{23}+(s^2_{12}\sqrt{1+r_\nu}-c^2_{12})s^2_{23}s^2_{13}}{(\sqrt{1+r_\nu}+1)\sin(2\theta_{12})\sin(2\theta_{23})s_{13}}$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ &$c_\alpha\simeq-\dfrac{3+\cos(4\theta_{12})-16s_{13}^2t_{23}^2}{2\sin^2(2\theta_{12})}$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ & $c_\delta=2\dfrac{(s^2_{12}-c^2_{12}\sqrt{1+r_\nu})s^2_{23}+(c^2_{12}-s^2_{12}\sqrt{1+r_\nu})c^2_{23}s^2_{13}}{(\sqrt{1+r_\nu}+1)\sin(2\theta_{12})\sin(2\theta_{23})s_{13}}$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ &$c_\alpha\simeq-\dfrac{3t_{23}^2+t_{23}^2\cos(4\theta_{12})+16s_{13}^2}{2t_{23}^2\sin^2(2\theta_{12})}$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ \[CPtable\] Having identified the compatible textures, we now obtain expressions for $\delta$ in terms of the mixing angles and $r_\nu$ using Eq. (\[IHrelationb\]). By imposing that the right-hand side of Eq. (\[IHrelation\]) is real, we can obtain analytical expressions for the Majorana phase $\alpha$ as a function of $\theta_{ij}$, $r_\nu$ and $\delta$. In Table \[CPtable\], we present the results for $c_\delta\equiv\cos\delta$ and $c_\alpha\equiv\cos\alpha$ for textures B, C, D and F when the neutrino mass spectrum is of IH type. It is worth mentioning that, although of different nature, $\delta$ and $\alpha$ are not independent phases in our case. This is due to the presence of zeros in the effective neutrino mass matrix. Taking $\theta_{ij}$, ${\Delta m^2_{21}}$ and ${\Delta m^2_{31}}$ in their $3\sigma$ $(1\sigma)$ experimental ranges, we show in Fig. \[deltaalpha\] the light (dark) blue allowed regions in the $(\alpha,\delta)$ parameter space for textures B, C, D and F of ${\mathbf{M}^\nu}$. We conclude that, for textures B and C, values of $\delta \simeq 3 \pi/2$ close to the best-fit value are allowed (cf. Table \[datatable\]). For such values of $\delta$, $\alpha \simeq 1.9 \pi\,(0.08\pi)$ is predicted for texture B (C). In fact, for these textures $$\begin{aligned} {\rm B:}\;&c_\delta \simeq \frac{r_\nu \sin(2\theta_{12})}{4s_{13}t_{23}}-\frac{s_{13}t_{23}}{\tan(2\theta_{12})}\,,\\ {\rm C:}\;&c_\delta \simeq -\frac{r_\nu t_{23} \sin(2\theta_{12})}{4s_{13}}+\frac{s_{13}}{t_{23}\tan(2\theta_{12})},\end{aligned}$$ from which we see that $|c_\delta|\ll 1$, implying $\delta \simeq \pm \pi/2$. Instead, for textures D and F $$\begin{aligned} {\rm D:}\;&c_\delta \simeq \frac{1}{2s_{13}t_{23}\tan(2\theta_{12})}\,,\\ {\rm F:}\;&c_\delta \simeq -\frac{t_{23}}{2s_{13}\tan(2\theta_{12})}\,,\end{aligned}$$ one obtains $|c_\delta| \sim \mathcal{O}(1)$ meaning that $\delta$ is far from $\pm \pi/2$. Therefore, as anticipated above, only textures B and C lead to $\delta$ values within the $1\sigma$ range of Table \[datatable\]. For textures D and F, the obtained values for $\delta$ are out of the $1\sigma$ range, but still within the $3\sigma$ one. Presently, attempts to probe the Majorana nature of neutrinos are mainly based on neutrinoless double beta decay ($0\nu\beta\beta$) experiments. The observation of $0\nu\beta\beta$ decay would also provide a measurement of the neutrino mass scale, since the rate of this process is related to the square of the neutrino mass. A relevant quantity for $0\nu\beta\beta$ decay is the effective mass $m_{\beta\beta}$, which, for an IH neutrino mass spectrum, is given by $$\begin{aligned} m_{\beta\beta}&=\left|\sum_{i=1}^{3}m_i {\mathbf{U}}_{1i}^2\right|\nonumber\\ &=c_{13}^2|{\Delta m^2_{31}}|^{1/2}\,\left|c_{12}^2+(1+r_\nu)^{1/2}\,s_{12}^2e^{i\alpha} \right|.\end{aligned}$$ Given that $\alpha$ is a function of $\theta_{ij}$, $\delta$ and $r_\nu$, in Fig. \[mbbdelta\], we show the allowed regions in the $(m_{\beta\beta},\delta)$-plane, taking into account the experimental ranges for the neutrino parameters (the color codes are the same used in previous figures). The results are presented for textures B, C, D and F, where one can see that the value of $m_{\beta\beta}$ is around 50 meV (15 meV) for textures B and C (D and F). These values are compatible with all constraints coming from $0\nu\beta\beta$ decay and cosmological experiments [@Capozzi:2017ipn] for IH, but lie out of the sensitivity range of leading experiments like EXO-200 [@Albert:2017qto], KamLAND-Zen [@KamLAND-Zen:2016pfg], GERDA [@Agostini:2017dxu] and CUORE-0 [@Alduino:2016vtd]. Nevertheless, next-generation experiments will be able to test the IH spectrum (for a general discussion about future prospects and sensitivities of $0\nu\beta\beta$ decay experiments see e.g. Ref. [@Ostrovskiy:2016uyx]). In the above analysis, we have studied the cases with one texture zero in ${\mathbf{M}_R}$. Notice, however, that the maximally-allowed number of zeros in this matrix is actually two, leading to a single possible texture $$\begin{aligned} {\rm R_4:\,}\begin{pmatrix} 0&\times\\ \cdot&0 \end{pmatrix}, \label{R4text}\end{aligned}$$ which is characterized by a spectrum with two degenerate RH neutrinos. Combining through the seesaw formula (\[Mnuseesaw\]) the matrix R$_4$ with all ${\mathbf{Y}^\nu}$ textures presented in Eq. , one obtains the textures for ${\mathbf{M}^\nu}$ given in the third column of Table \[tabR4\]. One can see that in all cases ${\mathbf{M}^\nu}$ contains two zeros, which have been tested individually above.[^3] Moreover, additional relations among the elements of ${\mathbf{M}^\nu}$ (see fourth column of Table \[tabR4\]) arise due to the specific form of ${\mathbf{M}_R}$, which contains a single parameter. For NH, all cases with ${\rm R}_4$ are excluded, since all textures with one zero in ${\mathbf{M}^\nu}$ were already shown to be incompatible with data (see Table \[tabR1R2R3\]). For IH, combinations leading to textures ${\rm A}_1$ and ${\rm A}_2$ for ${\mathbf{M}^\nu}$ are excluded due to the condition ${\mathbf{M}^\nu}_{11}=0$ (see Table \[tabR1R2R3\]). [K[1.2cm]{}|K[0.7cm]{}|c|K[2.1cm]{}|K[0.7cm]{}|K[0.7cm]{}]{} ${\mathbf{Y}^\nu}$&${\mathbf{M}_R}$&${\mathbf{M}^\nu}$&**Relation in** ${\mathbf{M}^\nu}$&**NH**&**IH**\ T$_1$, T$_4$&\[-1.5cm\][R$_4$]{}&A$_1$: $\begin{pmatrix} 0 &\times&\times\\ \cdot&0&\times\\ \cdot&\cdot&\times \end{pmatrix}$&$\dfrac{{\mathbf{M}^\nu}_{33}}{2{\mathbf{M}^\nu}_{23}}=\dfrac{{\mathbf{M}^\nu}_{13}}{{\mathbf{M}^\nu}_{12}}$&&\ T$_2$, T$_5$&&A$_2$: $\begin{pmatrix} 0 &\times&\times\\ \cdot&\times&\times\\ \cdot&\cdot&0 \end{pmatrix}$&$\dfrac{{\mathbf{M}^\nu}_{22}}{2{\mathbf{M}^\nu}_{23}}=\dfrac{{\mathbf{M}^\nu}_{12}}{{\mathbf{M}^\nu}_{13}}$&&\ T$_3$, T$_6$&&D$_1$ : $\begin{pmatrix} \times &\times&\times\\ \cdot&0&\times\\ \cdot&\cdot&0 \end{pmatrix}$&$\dfrac{{\mathbf{M}^\nu}_{11}}{2{\mathbf{M}^\nu}_{12}}=\dfrac{{\mathbf{M}^\nu}_{13}}{{\mathbf{M}^\nu}_{23}}$&&\ ![Predictions for the low-energy phases $\delta$ and $\alpha$ for textures B, C, D and F, using the $3\sigma$ (light blue) and $1\sigma$ (dark blue) ranges given in Table \[datatable\] for the mixing angles and neutrino mass-squared differences. The black dot corresponds to the predictions obtained with the best-fit values of $~\theta_{ij}$, ${\Delta m^2_{21}}$ and $|{\Delta m^2_{31}}|$.[]{data-label="deltaalpha"}](fig3.pdf "fig:")\ ![Predictions for $\delta$ and $m_{\beta\beta}$ for textures B, C, D and F, using the $3\sigma$ (light blue) and $1\sigma$ (dark blue) ranges given in Table \[datatable\] for the mixing angles and neutrino mass-squared differences. The black dot corresponds to the predictions obtained with the best-fit values of $~\theta_{ij}$, ${\Delta m^2_{21}}$ and $|{\Delta m^2_{31}}|$.[]{data-label="mbbdelta"}](fig4.pdf "fig:")\ As for texture D$_1$, although the conditions ${\mathbf{M}^\nu}_{22}=0$ and ${\mathbf{M}^\nu}_{33}=0$ are individually compatible with the data at $3\sigma$, they cannot be simultaneously verified, as one can see in Fig. \[IHmassratios\], comparing the results for textures D and F. Indeed, from these plots one concludes that there is no overlap between the regions allowed by the data for the same values of $\delta$. This seems to contradict previous results obtained in the literature which state that textures with ${\mathbf{M}^\nu}_{22}={\mathbf{M}^\nu}_{33}=0$ are compatible with the data (see e.g. Ref [@Cebola:2015dwa]). Notice, however, that the results in those references were obtained for a general neutrino spectrum with $m_{1,2,3}\neq 0$. One can understand why texture D$_1$ in our case ($m_3=0$) is not valid by inspecting the relations between neutrino masses and ${\mathbf{U}}$ when the conditions ${\mathbf{M}^\nu}_{22}={\mathbf{M}^\nu}_{33}=0$ are imposed, namely [@Fritzsch:2011qv], $$\begin{aligned} \frac{m_3}{m_1}=\left|\frac{{\mathbf{U}}_{22}^2{\mathbf{U}}_{31}^2-{\mathbf{U}}_{21}^2{\mathbf{U}}_{32}^2}{{\mathbf{U}}_{23}^2{\mathbf{U}}_{32}^2-{\mathbf{U}}_{22}^2{\mathbf{U}}_{33}^2}\right|\,,\\ \frac{m_3}{m_2}=\left|\frac{{\mathbf{U}}_{22}^2{\mathbf{U}}_{31}^2-{\mathbf{U}}_{21}^2{\mathbf{U}}_{32}^2}{{\mathbf{U}}_{21}^2{\mathbf{U}}_{33}^2-{\mathbf{U}}_{23}^2{\mathbf{U}}_{31}^2}\right|\,.\end{aligned}$$ Therefore, if $m_3=0$ the condition $$\begin{aligned} \left|{\mathbf{U}}_{22}^2{\mathbf{U}}_{31}^2-{\mathbf{U}}_{21}^2{\mathbf{U}}_{32}^2\right|=0\end{aligned}$$ must be verified for texture D$_1$. The above relation can be approximately written as $$\begin{aligned} c_\delta\simeq \dfrac{2\cos(2\theta_{12})\cos(2\theta_{23})\pm\sqrt{2}\sqrt{\cos(4\theta_{12})+\cos(4\theta_{23})}}{4\sin(2\theta_{12})\sin(2\theta_{23})s_{13}}\,,\end{aligned}$$ which, taking into account the current mixing angle data, always leads to a complex $c_\delta$. In conclusion, we have analyzed all possible textures with six zeros in ${\mathbf{Y}^\ell}$, two zeros in ${\mathbf{Y}^\nu}$, and one or two zeros in ${\mathbf{M}_R}$. The compatibility of all textures is summarized in the last two columns of Tables \[tabR1R2R3\] and \[tabR4\], for NH and IH. We remark that no restriction has been imposed in the non-zero elements of those matrices. The results presented above are valid in the basis where ${\mathbf{Y}^\ell}={\rm diag}(y_e,y_\mu,y_\tau)\equiv {\mathbf{Y}^\ell}_{\rm diag}$ so that the charged lepton mass matrix is $\mathbf{M}^\ell={\rm diag}(m_e,m_\mu,m_\tau)$. One may wonder whether these conclusions hold for any other ${\mathbf{Y}^\ell}$ with six zeros and only three nonzero elements. First, it is straightforward to see that any two nonzero elements in the same line/column lead to a massless charged lepton. This leaves us with six viable textures for ${\mathbf{Y}^\ell}$ with six zeros, which can be obtained from ${\mathbf{Y}^\ell}_{\rm diag}$ by applying permutations of lines and/or columns: $$\begin{gathered} \begin{matrix} {\rm L}_1:\;\begin{pmatrix} \times&0 &0\\ 0&\times &0\\ 0&0 &\times \end{pmatrix},& {\rm L_2}: \;\begin{pmatrix} 0&\times &0\\ \times&0 &0\\ 0&0 &\times \end{pmatrix},& {\rm L_3}:\; \begin{pmatrix} 0&0 &\times\\ 0&\times &0\\ \times&0 &0 \end{pmatrix},\\ &&&\\ {\rm L}_4:\;\begin{pmatrix} \times&0 &0\\ 0&0 &\times\\ 0&\times &0 \end{pmatrix},& {\rm L_5}: \;\begin{pmatrix} 0&0 &\times\\ \times&0 &0\\ 0&\times &0 \end{pmatrix},& {\rm L_6}:\; \begin{pmatrix} 0&\times &0\\ 0&0 &\times\\ \times&0 &0 \end{pmatrix}. \end{matrix} \label{Ylstructures}\end{gathered}$$ Obviously, if only column permutations (rotation of RH charged-lepton fields) are performed, then the results for a specific set of ${\mathbf{Y}^\nu}$ and ${\mathbf{M}_R}$ textures remain unchanged. However, if a permutation of the lines $i$ and $j$ in ${\mathbf{Y}^\ell}$ is involved (rotation of LH charged-lepton fields by the permutation matrix $\mathbf{P}_{ij}$), then the same line permutation has to be performed in ${\mathbf{Y}^\nu}$. At the effective level, this corresponds to permuting the lines and columns $i$ and $j$ in the effective neutrino mass matrix ${\mathbf{M}^\nu}$. Under these rotations, textures T$_1$-T$_6$ of ${\mathbf{Y}^\nu}$ and, consequently, A-F of ${\mathbf{M}^\nu}$, are transformed among themselves. Thus, even if a given texture pair (${\mathbf{Y}^\nu}$,${\mathbf{M}_R}$) is not compatible with data in the ${\mathbf{Y}^\ell}_{\rm diag}$ basis, this may not be the case in another ${\mathbf{Y}^\ell}$ basis obtained from a line permutation $\mathbf{P}_{ij}$. To check the viability of a given set of textures $({\mathbf{Y}^\ell},{\mathbf{Y}^\nu},{\mathbf{M}_R};{\mathbf{M}^\nu})=({\rm L}_i,{\rm T}_i,{\rm R}_i;\text{A-F})$ one has to identify the permutation $\mathbf{P}_{ij}$ which brings $\text{L}_i$ to ${\mathbf{Y}^\ell}_{\rm diag}$, and find the transformed ${\mathbf{M}^\nu}$ texture. For instance, consider the case $({\mathbf{Y}^\ell}_{\rm diag},{\rm T}_3,{\rm R}_1;{\rm E})$, shown in Table \[tabR1R2R3\] to be incompatible with data. Under $\mathbf{P}_{13}$, ${\mathbf{Y}^\ell}_{\rm diag}$ is transformed into ${\rm L}_3$, while texture E becomes texture B, which is compatible with data at $1\sigma$. Therefore, although the set $({\mathbf{Y}^\ell}_{\rm diag},{\rm T}_3,{\rm R}_1;{\rm E})$ is not viable, the set $({\rm L}_3,{\rm T}_3,{\rm R}_1;{\rm E})$ is, since it corresponds to $({\mathbf{Y}^\ell}_{\rm diag},{\rm T}_4,{\rm R}_1;{\rm B})$ under $\mathbf{P}_{13}$. In Table \[permut\] we summarize the transformation properties of each ${\mathbf{M}^\nu}$ texture under line permutations $\mathbf{P}_{ij}$, identifying in each case the compatibility with data taking into account the results obtained for ${\mathbf{Y}^\ell}_{\rm diag}$ given in Table \[tabR1R2R3\]. Notice that when ${\mathbf{M}_R}$ is of type $\text{R}_4$, the results presented in Table \[tabR4\] are valid for any ${\mathbf{Y}^\ell}$ texture of type $\text{L}_i$. This is due to the fact that, under any permutation of lines and/or columns in ${\mathbf{Y}^\ell}$, textures A$_{1,2}$ and D$_1$ (which are all excluded by data) transform among themselves. [K[1.0cm]{}l|K[1cm]{}l|K[1cm]{}l|K[1cm]{}l]{} &&&\ A&&D&($1\sigma$)&F&($3\sigma$)&A&\ B&($1\sigma$)&B&($1\sigma$)&E&&C&($1\sigma$)\ C&($1\sigma$)&E&&C&($1\sigma$)&B&($1\sigma$)\ D&($1\sigma$)&A&&D&($1\sigma$)&F&($3\sigma$)\ E&&C&($1\sigma$)&B&($1\sigma$)&E&\ F&($3\sigma$)&F&($3\sigma$)&A&&D&($1\sigma$)\ Imposing relations among the elements of ${\mathbf{Y}^\nu}$ {#sec3a} ----------------------------------------------------------- We now intend to further restrict the two texture zero patterns analyzed above by imposing equality relations among the elements of ${\mathbf{Y}^\nu}$. The first obvious choice would be to consider all elements in ${\mathbf{Y}^\nu}$ to be equal. However, one can show that the eigenvector associated to $m_3=0$ is always $v_3=(\mp 1,-1,1)/\sqrt{3}$, leading to $s_{13}=\pm 1/\sqrt{3}$, which is excluded by the data. Thus, we move to the analysis of textures with two zeros in ${\mathbf{Y}^\nu}$ and three equal elements. Each case will be denoted by the labels of ${\mathbf{Y}^\nu}$, ${\mathbf{M}_R}$ and corresponding ${\mathbf{M}^\nu}$ (see first column of Table \[Tabrel\]), and indexes of the ${\mathbf{Y}^\nu}$ equal elements (see second column of Table \[Tabrel\]). For instance, the cases with ${\mathbf{Y}^\nu}_{21}={\mathbf{Y}^\nu}_{31}={\mathbf{Y}^\nu}_{12}$ are denoted by $(21,31,12)$. Due to the highly constrained form of the involved matrices, extra relations among the elements of ${\mathbf{M}^\nu}$ arise. These are shown in the third column of Table \[Tabrel\] for all possible combinations. Compatibility with neutrino data is determined by checking whether those relations are verified taking the allowed ranges for the neutrino parameters given in Table \[datatable\]. Also notice that the heavy Majorana neutrino masses and the elements of ${\mathbf{M}^\nu}$ are related. In particular, defining the ratio [K[2.4cm]{}|K[1.6cm]{}|K[3.6cm]{}|K[3.8cm]{}|K[0.8cm]{}|c|c]{} (${\mathbf{Y}^\nu}$, ${\mathbf{M}_R}$, ${\mathbf{M}^\nu}$)&**Equal elements in** ${\mathbf{Y}^\nu}$& **Relations in** ${\mathbf{M}^\nu}$&$r_N\equiv M_2/M_1$&**IH**& ------------------------------------------------------------------- **Low energy predictions** $(\theta_{12},\theta_{23},\theta_{13})^\circ$ $({\Delta m^2_{31}},{\Delta m^2_{21}})\,\times10^{-3}\text{eV}^2$ $(\delta,\alpha)^\circ\;,\;m_{\beta\beta}\,(\text{meV})$ ------------------------------------------------------------------- &$r_N$\ &$(21,31,12)$& ${\mathbf{M}^\nu}_{22}={\mathbf{M}^\nu}_{23}$ &$r_N=\left|\dfrac{{\mathbf{M}^\nu}_{22}}{{\mathbf{M}^\nu}_{11}}\right|$&&&1.91 ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ &$(21,31,32)$&$\dfrac{{\mathbf{M}^\nu}_{11}({\mathbf{M}^\nu}_{33}-{\mathbf{M}^\nu}_{22})}{({\mathbf{M}^\nu}_{13})^2}=1$&$r_N=\left|\dfrac{{\mathbf{M}^\nu}_{11}{\mathbf{M}^\nu}_{22}}{({\mathbf{M}^\nu}_{13})^2}\right|$&&&12.00 ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ &$(21,12,32)$& ${\mathbf{M}^\nu}_{11}={\mathbf{M}^\nu}_{13}$ &$r_N=\left|\dfrac{{\mathbf{M}^\nu}_{22}}{{\mathbf{M}^\nu}_{11}}\right|$&&&$-$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ &$(31,12,32)$&$\dfrac{{\mathbf{M}^\nu}_{22}({\mathbf{M}^\nu}_{33}-{\mathbf{M}^\nu}_{11})}{({\mathbf{M}^\nu}_{23})^2}=1$&$r_N=\left|\dfrac{({\mathbf{M}^\nu}_{23})^2}{{\mathbf{M}^\nu}_{11}{\mathbf{M}^\nu}_{22}}\right|$&&&$-$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ &$(21,31,12)$& ${\mathbf{M}^\nu}_{23}={\mathbf{M}^\nu}_{33}$, &$r_N=\left|\dfrac{{\mathbf{M}^\nu}_{33}}{{\mathbf{M}^\nu}_{11}}\right|$&&&1.91 ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ &$(21,31,22)$&$\dfrac{{\mathbf{M}^\nu}_{11}({\mathbf{M}^\nu}_{22}-{\mathbf{M}^\nu}_{33})}{({\mathbf{M}^\nu}_{12})^2}=1$&$r_N=\left|\dfrac{{\mathbf{M}^\nu}_{11}{\mathbf{M}^\nu}_{33}}{({\mathbf{M}^\nu}_{12})^2}\right|$&&&12.00 ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ &$(21,12,22)$& ${\mathbf{M}^\nu}_{11}={\mathbf{M}^\nu}_{12}$, &$r_N=\left|\dfrac{({\mathbf{M}^\nu}_{23})^2}{{\mathbf{M}^\nu}_{11}{\mathbf{M}^\nu}_{33}}\right|$&&&$-$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ &$(31,12,22)$&$\dfrac{{\mathbf{M}^\nu}_{33}({\mathbf{M}^\nu}_{22}-{\mathbf{M}^\nu}_{11})}{({\mathbf{M}^\nu}_{23})^2}=1$&$r_N=\left|\dfrac{{\mathbf{M}^\nu}_{33}}{{\mathbf{M}^\nu}_{11}}\right|$&&&$-$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ &$(11,31,12)$&&$\dfrac{r_N-1}{r_N-\sqrt{r_N}-1}=\left|\dfrac{{\mathbf{M}^\nu}_{33}}{{\mathbf{M}^\nu}_{13}}\right|$&&&$-$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ &$(11,31,22)$&&$\dfrac{\sqrt{r_N}}{r_N-1}=\left|\dfrac{{\mathbf{M}^\nu}_{23}}{{\mathbf{M}^\nu}_{33}}\right|$&&&$-$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ &$(11,12,22)$&&$\dfrac{\sqrt{r_N}}{r_N-1}=\left|\dfrac{({\mathbf{M}^\nu}_{23})^2}{{\mathbf{M}^\nu}_{12}{\mathbf{M}^\nu}_{33}}\right|$&&&$-$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ &$(31,12,22)$&&$\dfrac{\sqrt{r_N}}{r_N-1}=\left|\dfrac{{\mathbf{M}^\nu}_{23}}{{\mathbf{M}^\nu}_{33}}\right|$&&&$-$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ &$(11,21,12)$&&$\dfrac{r_N-1}{r_N-\sqrt{r_N}-1}=\left|\dfrac{{\mathbf{M}^\nu}_{22}}{{\mathbf{M}^\nu}_{12}}\right|$&&&$-$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ &$(11,21,32)$&&$\dfrac{\sqrt{r_N}}{r_N-1}=\left|\dfrac{{\mathbf{M}^\nu}_{23}}{{\mathbf{M}^\nu}_{22}}\right|$&&&$-$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ &$(11,12,32)$&&$\dfrac{\sqrt{r_N}}{r_N-1}=\left|\dfrac{({\mathbf{M}^\nu}_{23})^2}{{\mathbf{M}^\nu}_{13}{\mathbf{M}^\nu}_{22}}\right|$&&&$-$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ &$(21,12,32)$&&$\dfrac{\sqrt{r_N}}{r_N-1}=\left|\dfrac{{\mathbf{M}^\nu}_{23}}{{\mathbf{M}^\nu}_{22}}\right|$&&&$-$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ &$(21,31,12)$&&$\dfrac{\sqrt{r_N}}{r_N-1}=\left|\dfrac{{\mathbf{M}^\nu}_{12}}{{\mathbf{M}^\nu}_{11}}\right|$&&&1.46 ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ &$(21,31,32)$&&$\dfrac{\sqrt{r_N}}{r_N-1}=\left|\dfrac{({\mathbf{M}^\nu}_{12})^2}{{\mathbf{M}^\nu}_{11}{\mathbf{M}^\nu}_{23}}\right|$&&&1.08 ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ &$(21,12,32)$&&$\dfrac{\sqrt{r_N}}{r_N-1}=\left|\dfrac{{\mathbf{M}^\nu}_{12}}{{\mathbf{M}^\nu}_{11}}\right|$&&&$-$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ &$(31,12,32)$&&$\dfrac{r_N-1}{r_N-\sqrt{r_N}-1}=\left|\dfrac{{\mathbf{M}^\nu}_{11}}{{\mathbf{M}^\nu}_{13}}\right|$&&&$-$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ &$(21,31,12)$&&$\dfrac{\sqrt{r_N}}{r_N-1}=\left|\dfrac{{\mathbf{M}^\nu}_{13}}{{\mathbf{M}^\nu}_{11}}\right|$&&&1.46 ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ &$(21,31,22)$&&$\dfrac{\sqrt{r_N}}{r_N-1}=\left|\dfrac{({\mathbf{M}^\nu}_{13})^2}{{\mathbf{M}^\nu}_{11}{\mathbf{M}^\nu}_{23}}\right|$&&&1.08 ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ &$(21,12,22)$&&$\dfrac{r_N-1}{r_N-\sqrt{r_N}-1}=\left|\dfrac{{\mathbf{M}^\nu}_{11}}{{\mathbf{M}^\nu}_{12}}\right|$&&&$-$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ &$(31,12,22)$&&$\dfrac{\sqrt{r_N}}{r_N-1}=\left|\dfrac{{\mathbf{M}^\nu}_{13}}{{\mathbf{M}^\nu}_{11}}\right|$&&&$-$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ $$\begin{aligned} r_N=\frac{M_2}{M_1}\,, \label{rNdef}\end{aligned}$$ where $M_{2,1}$ are the eigenvalues of ${\mathbf{M}_R}$, we obtain the relations shown in the fourth column of Table \[Tabrel\]. Our analysis shows that only eight combinations are compatible with neutrino data at the $3\sigma$ level (see fifth and sixth columns of Table \[Tabrel\]). The low-energy predictions for the neutrino parameters correspond to the case in which the data is best fitted. It is possible to show analytically that, for all compatible sets of matrices, $\theta_{23}=\pi/4$, which is confirmed by the numerical result. It is worth mentioning that any texture combination obtained from those presented in Table \[Tabrel\] by permuting the columns of ${\mathbf{Y}^\nu}$ will remain valid. For instance, the first case shown in Table \[Tabrel\] becomes (T$_4$,R$_1$,B) with equal elements (11,22,32), leading to the same predictions. Therefore, there are actually sixteen different cases compatible with the data. As mentioned above, the equality among elements of ${\mathbf{Y}^\nu}$ fixes the value of $r_N$, which is indicated in the last column of Table \[Tabrel\]. From inspection of the same table, one can also conclude that none of the texture configurations is compatible with the data at $1\sigma$. As in the analysis presented in the previous section, the results obtained with equal ${\mathbf{Y}^\nu}$ elements correspond to ${\mathbf{Y}^\ell}={\mathbf{Y}^\ell}_{\rm diag}$. For a different ${\mathbf{Y}^\ell}$ texture related to ${\mathbf{Y}^\ell}_{\rm diag}$ by permutations of lines (and columns), the ${\mathbf{Y}^\nu}$ textures transform among themselves, and the equal elements of ${\mathbf{Y}^\nu}$ change position. Thus, combinations which are incompatible with data (see Table \[Tabrel\]) in the charged-lepton mass basis may become compatible for a non-diagonal ${\mathbf{Y}^\ell}$, related to ${\mathbf{Y}^\ell}_{\rm diag}$ by permutation of lines. In Table \[tabrelper\] we summarize the transformation properties of each combination $({\rm T}_i,\text{R}_i,\text{A-F})$ with equal ${\mathbf{Y}^\nu}$ elements under line permutations $\mathbf{P}_{ij}$ (and up to possible column permutation). In each case, we identify the compatibility with data taking into account the results given in Table \[Tabrel\] for ${\mathbf{Y}^\ell}={\mathbf{Y}^\ell}_{\rm diag}$. [K[1.5cm]{}K[0.35cm]{}|K[1.5cm]{}K[0.35cm]{}|K[1.5cm]{}K[0.35cm]{}|K[1.5cm]{}K[0.35cm]{}]{} &&& ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ &&& ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ (21,31,12)&&(21,12,32)&&&&(21,31,12)&\ (21,31,32)&&(31,12,32)&& & &(21,31,22)&\ (21,12,32)&&(21,31,12)&& & &(31,12,22)&\ (31,12,32)&&(21,31,32)&& & &(21,12,22)&\ &&& ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ (21,31,12)&&&&(31,12,22)&&(21,31,12)&\ (21,31,22)&&&&(21,12,22)&& (21,31,32)&\ (21,12,22)&&&&(21,31,22)&&(31,12,32)&\ (31,12,22)&&&&(21,31,12)&& (21,12,32)&\ &&& ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ (11,31,12)&&&&(31,12,32)&&(11,21,12)&\ (11,31,22)&& &&(21,12,32)&& (11,21,32)&\ (11,12,22)&& &&(21,31,32)&& (11,12,32)&\ (31,12,22)&&&&(21,31,12)&& (21,12,32)&\ &&& ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ (11,21,12)&&(21,12,22)&&&&(11,31,12)&\ (11,21,32)&&(31,12,22)&&& &(11,31,22)&\ (11,12,32)&& (21,31,22)&&& &(11,12,22)&\ (21,12,32)&&(21,31,12)&&& &(31,12,22)&\ &&& ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ (21,31,12)&&&&(31,12,22)&&(21,31,12)&\ (21,31,32)&& &&(11,12,22)&&(21,31,22)&\ (21,12,32)&& &&(11,31,22)&& (31,12,22)&\ (31,12,32)&& &&(11,31,12)&& (21,12,22)&\ &&& ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ (21,31,12)&&(21,12,32)&&&&(21,31,12)&\ (21,31,22)&& (11,12,32)&&& &(21,31,32)&\ (21,12,22)&&(11,21,12)&&& &(31,12,32)&\ (31,12,22)&&(11,21,32)&&& &(21,12,32)&\ Leptogenesis in the 2RHNSM with texture zeros {#sec3} ============================================= In the previous sections, several mass matrix patterns were found to be compatible with current neutrino oscillation data at $1\sigma$ and $3\sigma$ C.L., in the framework of the minimal type-I seesaw model with maximally restricted texture zeros. Here, we further analyze these patterns requiring their compatibility with successful leptogenesis [@Fukugita:1986hr]. We recall that the baryon asymmetry of the Universe is parametrized through the baryon-to-photon ratio $$\begin{gathered} \eta_B\equiv\dfrac{n_B-n_{\bar{B}}}{n_\gamma}\; , \label{baryontophotonratio}\end{gathered}$$ where $n_B$, $n_{\bar{B}}$ and $n_\gamma$ are the number densities of baryons, anti-baryons and photons, respectively. From cosmic microwave background (CMB) measurements provided by the Planck collaboration [@Ade:2015xua], the present value of $\eta_B$ is $$\begin{gathered} \eta_B^0=(6.11\pm0.04)\times 10^{-10}\,. \label{presentetab}\end{gathered}$$ In a minimal type-I seesaw context with two right-handed neutrinos, the leptogenesis mechanism may proceed via the out-of-equilibrium decays of the heavy neutrinos $N_1$ and $N_2$ in the early Universe. The generated lepton asymmetry in such decays is partially converted into a baryon asymmetry by ($B+L$)-violating sphaleron processes, leading to [@Antusch:2011nz] $$\begin{gathered} \eta_B=a_\text{sph}\,\dfrac{N_{B-L}}{N_\gamma^\text{rec}}\simeq 9.58\times 10^{-3}\, N_{B-L}\; , \label{finaletab}\end{gathered}$$ where $a_\text{sph} \equiv B/(B-L)=28/79$ is the conversion factor, $N_{B-L}$ is the final asymmetry calculated in a comoving volume, and $N_{\gamma}^\text{rec}$ is the number of photons in the same volume ($N_{\gamma}^\text{rec}\simeq 37.01$) at the recombination temperature. Flavored and unflavored CP asymmetries {#subsecCP} -------------------------------------- An important ingredient in the generation of the BAU is the CP asymmetry produced in the decays of the heavy neutrinos into the lepton flavors $\alpha=e,\mu,\tau$. Working in the mass eigenbasis of the heavy neutrinos $N_i$ and the charged leptons $\ell_\alpha$, the CP asymmetries $\epsilon_i^\alpha$ may be computed as [@Covi:1996wh] $$\begin{gathered} \epsilon_i^\alpha=\dfrac{\Gamma(N_i\rightarrow\Phi\ell_\alpha)-\Gamma(N_i\rightarrow\Phi^\dagger\bar{\ell}_\alpha)}{\sum_\beta[\Gamma(N_i\rightarrow\Phi\ell_\beta)+\Gamma(N_i\rightarrow\Phi^\dagger\bar{\ell}_\beta)]}, \label{generalCP}\end{gathered}$$ where $\Gamma(N_i\rightarrow\Phi\ell_\alpha)\equiv \Gamma_i^\alpha$ and $\Gamma(N_i\rightarrow\Phi^\dagger\bar{\ell}_\alpha)\equiv \overline{\Gamma}_i^\alpha$ are the $N_i$ decay rates into leptons and antileptons, respectively. At tree level, $$\begin{gathered} \Gamma_i^\alpha=\overline{\Gamma}_i^\alpha=M_i\dfrac{|\mathbf{Y}^\nu_{\alpha i}|^2}{16 \pi}, \label{flvdecayrate}\end{gathered}$$ with the sum in the denominator of  running over the three lepton flavors. The leading non-zero contributions to the asymmetry $\epsilon_i^\alpha$ arise from interference of the tree-level process with its one-loop corrections. For the two RH neutrino case, the result is [@Branco:2011zb] $$\begin{aligned} \epsilon_i^\alpha =&\frac{1}{8\pi}\frac{1}{\mathbf{H}_{ii}^\nu}\{\text{Im}[\mathbf{Y}_{\alpha i}^{\nu *}\mathbf{H}_ {ij}^\nu\mathbf{Y}_{\alpha j}^\nu ][f(x_j)+g(x_j)]+\nonumber\\ &\text{Im}[\mathbf{Y}_{\alpha i}^{\nu *}\mathbf{H}_ {ji}^\nu\mathbf{Y}_{\alpha j}^{\nu}]g'(x_j)\}, \label{flavouredcp}\end{aligned}$$ where $j\neq i=1,2$, $x_j=M_j^2/M_i^2$ and $\mathbf{H}^\nu=\mathbf{Y}^{\nu\dagger} \mathbf{Y}^\nu$. The loop functions $f(x)$, $g(x)$ and $g'(x)$ correspond to the one-loop vertex and self-energy corrections, given by $$\begin{aligned} \label{loopfunctions1} f(x)&=\sqrt{x}\left[1-(1-x)\ln\left(1+\frac{1}{x}\right)\right],\\ g(x)&=\sqrt{x}g'(x)=-\frac{\sqrt{x}}{(x-1)}\,.\end{aligned}$$ Summing over the lepton flavors in Eq. , the unflavored CP asymmetry is recovered, $$\begin{gathered} \epsilon_i=\frac{1}{8\pi}\frac{1}{\mathbf{H}_{ii}^\nu}\text{Im}[(\mathbf{H}_{ij}^\nu)^2][f(x_j)+g(x_j)]. \label{unflavouredcp}\end{gathered}$$ In our study, two temperature regimes will be of interest [@Barbieri:1999ma; @Abada:2006fw; @Nardi:2006fx; @Abada:2006ea]. For temperatures above $10^{12}$ GeV in the early Universe, the charged-lepton Yukawa interactions are out of equilibrium. Hence, for this temperature range, the three lepton flavors are indistinguishable (unflavored regime), and the lepton asymmetry may be represented rigorously by a single flavor eigenstate. In this case, the relevant CP asymmetry for leptogenesis is given by Eq. . In the temperature interval $10^{9}\lesssim T \lesssim 10^{12}$ GeV, the $\tau$ Yukawa interactions enter thermal equilibrium and processes involving leptons are able to distinguish between two different flavors: the $\tau$ and a coherent superposition of $e$ and $\mu$ (two-flavored regime). The corresponding CP asymmetries, $\epsilon_i^\tau$ and $\epsilon_i^\gamma\equiv \epsilon_i^e+\epsilon_i^\mu$, are then obtained from Eq. . The CP asymmetries given in Eq.  depend on the Yukawa coupling matrix ${\mathbf{Y}^\nu}$, which can be written in terms of the Casas-Ibarra parametrization presented in Eq. . This allows to rewrite the asymmetry in a more convenient form for leptogenesis analysis, $$\begin{gathered} \epsilon_i^\alpha=-\dfrac{1}{8\pi v^2}\dfrac{M_j}{\sum_k m_k |\mathbf{R}_{ki}|^2}\sum_{k,k'}\sqrt{m_k}m_{k'}\{\sqrt{m_{k'}}\, \text{Im}[\mathbf{U}_{\alpha k}^*\mathbf{U}_{\alpha k'}\mathbf{R}_{ki}\mathbf{R}_{k'i}][f(x_j)+g(x_j)]\nonumber\\ +\sum_{k''}\sqrt{m_{k''}}\,\text{Im}[\mathbf{U}_{\alpha k}^*\mathbf{U}_{\alpha k''}\mathbf{R}_{ki}\mathbf{R}_{k'i}^*\mathbf{R}_{k' j}\mathbf{R}_{k'' j}^*]g'(x_j)\}\, , \label{flavouredcpCI} \end{gathered}$$ where the orthogonal matrix $\mathbf{R}$ is parametrized by a single complex parameter $z$, as shown in Eq. . For an inverted hierarchical neutrino mass spectrum,[^4] the flavored asymmetries generated by $N_1$ and $N_2$ decays are written in terms of $z$ as $$\begin{aligned} \label{flavouredCPtanz1} \epsilon_1^{\alpha}=&-\dfrac{M_2}{8\pi v^2}\dfrac{ A_1^\alpha\left[f(x_2) + g(x_2)\right]+B_1^\alpha g'(x_2)}{m_1|c_z|^2+m_2|s_z|^2}\, ,\\ \epsilon_2^{\alpha}=&-\dfrac{M_1}{8\pi v^2}\dfrac{ A_2^\alpha\left[f(x_1) + g(x_1)\right]+B_2^\alpha g'(x_1)}{m_1|s_z|^2+m_2|c_z|^2}\, , \label{flavouredCPtanz2}\end{aligned}$$ where $c_z\equiv\cos z$, $s_z\equiv \sin z$ and $$\begin{aligned} \label{Afactor} A_1^\alpha=&(m_2^2|\mathbf{U}_{\alpha 2}|^2-m_1^2|\mathbf{U}_{\alpha 1}|^2)\, \text{Im}[s^2_z]+\xi\sqrt{m_1 m_2}\nonumber\\ &\lbrace(m_2-m_1)\text{Im}[\textbf{U}_{\alpha 1}^*\textbf{U}_{\alpha 2}]\text{Re}[c_z s_z]+\nonumber\\ &+(m_2+m_1)\text{Re}[\mathbf{U}^*_{\alpha 1}\mathbf{U}_{\alpha 2}]\text{Im}[c_z s_z]\rbrace\, ,\\ \nonumber\\ B_1^\alpha=&m_1m_2\,(|\mathbf{U}_{\alpha 2}|^2-|\mathbf{U}_{\alpha 1}|^2)\, \text{Im}[c^2_z\, (s^2_z)^*]+\xi\sqrt{m_1 m_2}\nonumber\\ &\lbrace(\,|c_z|^2+|s_z|^2\,)(m_2-m_1)\text{Im}[\textbf{U}_{\alpha 1}^*\textbf{U}_{\alpha 2}]\text{Re}[c_z\, s_z^*]+\nonumber\\ &+(\,|c_z|^2-|s_z|^2\,)(m_2+m_1)\text{Re}[\mathbf{U}^*_{\alpha 1}\mathbf{U}_{\alpha 2}]\text{Im}[c_z\, s_z^*]\rbrace.\nonumber\\ \label{Bfactor}\end{aligned}$$ The factors $A_2^\alpha$ and $B_2^\alpha$ are obtained replacing $s_z\rightarrow c_z$, $c_z\rightarrow s_z$ and $\xi\rightarrow-\xi$ in Eqs.  and , respectively. These factors have the following properties $$\begin{gathered} \sum_{\alpha}A_1^\alpha={\Delta m^2_{21}}\text{Im}[s_z^2], \quad \sum_{\alpha}A_2^\alpha={\Delta m^2_{21}}\text{Im}[c_z^2]\, ,\nonumber\\ \sum_{\alpha}B_i^\alpha=0\, .\end{gathered}$$ Using these relations, the unflavored CP asymmetries  are easily obtained, $$\begin{gathered} \label{unflavouredcpCIz1} \epsilon_1=-\dfrac{M_2}{8\pi v^2}\dfrac{{\Delta m^2_{21}}\text{Im}[s_z^2]}{m_1\,|c_z|^2 + m_2\, |s_z|^2}\left[f(x_2)+g(x_2)\right],\\ \epsilon_2=-\dfrac{M_1}{8\pi v^2}\dfrac{{\Delta m^2_{21}}\text{Im}[c_z^2]}{m_1\,|s_z|^2 + m_2\, |c_z|^2}\left[f(x_1)+g(x_1)\right]. \label{unflavouredcpCIz2}\end{gathered}$$ The presence of a texture zero in ${\mathbf{Y}^\nu}$ allows for the determination of $z$ in terms of low-energy parameters and $M_{1,2}$, as one may see from Eq. . For instance, in the basis where the charged-lepton and RH neutrino mass matrices are diagonal, the condition ${\mathbf{Y}^\nu}_{11}=0$ implies, for IH, $$\begin{gathered} \sqrt{m_1}\,{\mathbf{U}}_{11}^*c_z+\xi\sqrt{m_2}\,{\mathbf{U}}_{12}^*s_z=0\; ,\end{gathered}$$ leading to $$\begin{gathered} \tan z=-\xi\sqrt{\dfrac{m_1}{m_2}}\dfrac{{\mathbf{U}}_{11}^*}{{\mathbf{U}}_{12}^*}\; .\end{gathered}$$ In Table \[tabletan\], we present the expressions for $\tan z$ according to the position of the texture zero in ${\mathbf{Y}^\nu}$ and considering the matrix forms R$_{1,2,3}$ for ${\mathbf{M}_R}$. From this table it is straightforward to see that requiring the presence of two simultaneous zeros in ${\mathbf{Y}^\nu}$ leads to relations among the mixing angles, neutrino masses and the low-energy phases, as expected from Eq. . [K[1cm]{}|K[5cm]{}|K[5cm]{}]{} ${\mathbf{M}_R}$&$\tan z$ for ${\mathbf{Y}^\nu}_{\alpha 1}=0$&$\tan z$ for ${\mathbf{Y}^\nu}_{\alpha 2}=0$\ R$_1$&$-\xi\sqrt{\dfrac{m_1}{m_2}}\dfrac{\mathbf{U}_{\alpha 1}^*}{\mathbf{U}_{\alpha 2}^*}$&$\xi\sqrt{\dfrac{m_2}{m_1}}\dfrac{\mathbf{U}_{\alpha 2}^*}{\mathbf{U}_{\alpha 1}^*}$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ R$_2$&$i$&$\dfrac{- i\, \sqrt{m_1}M_1\mathbf{U}^*_{\alpha 1}+\,\xi\sqrt{m_2}M_2\mathbf{U}^*_{\alpha 2}}{\sqrt{m_1}M_2\mathbf{U}^*_{\alpha 1}+\,i\,\xi\sqrt{m_2}M_1\mathbf{U}^*_{\alpha 2}}$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ R$_3$&$\dfrac{i\, \sqrt{m_1}M_1\mathbf{U}^*_{\alpha 1}+\,\xi\sqrt{m_2}M_2\mathbf{U}^*_{\alpha 2}}{\sqrt{m_1}M_2\mathbf{U}^*_{\alpha 1}-\,i\,\xi\sqrt{m_2}M_1\mathbf{U}^*_{\alpha 2}}$& $i$ ------------------------------------------------------------------------ ------------------------------------------------------------------------ \ Replacing in Eqs.  and the expressions for $\tan z$ given in Table \[tabletan\], and using the low-energy relations of Table \[CPtable\], we obtain predictions for the flavored CP asymmetries $\epsilon_i^\tau$ and $\epsilon_i^\gamma$, for each of the valid texture-zero cases identified in the Section \[sec2\]. ![Flavored CP asymmetries $|\epsilon_{1,2}^\gamma|$ and $|\epsilon_{1,2}^\tau|$ as functions of the low-energy CP-violating phases $\alpha$ and $\delta$, for the texture-zero case ${\mathbf{Y}^\nu}_{11}=0$ and R$_1$. The gray-scale contour regions show the maximum values of $|\epsilon_i^\alpha|$, taking $\theta_{ij}$, ${\Delta m^2_{21}}$ and $|{\Delta m^2_{31}}|$ in the $3\sigma$ experimental range (see Table \[datatable\]) and for $10^9\lesssim M_{1,2}\lesssim 10^{12}$ GeV with $M_2\gtrsim 3M_1$. The colored contour lines are the results obtained for the minimum value of $\chi^2$. In the plot, the triangles and squares correspond to the ($\alpha$,$\delta$) pairs fixed by the conditions ${\mathbf{Y}^\nu}_{11}={\mathbf{Y}^\nu}_{22}=0$ and ${\mathbf{Y}^\nu}_{11}={\mathbf{Y}^\nu}_{32}=0$, respectively (cf. textures B and C in Table \[tabR1R2R3\]).[]{data-label="cpasymflav"}](fig5.pdf "fig:")\ It turns out that, even if one considers a single texture zero in ${\mathbf{Y}^\nu}$, the CP asymmetries are highly suppressed in the flavored regime. As illustration, in Fig. \[cpasymflav\] we show the asymmetries $|\epsilon_i^\gamma|$ and $|\epsilon_i^\tau|$, $i=1,2$, for the case R$_1$ and ${\mathbf{Y}^\nu}_{11}=0$ on the plane ($\alpha$,$\delta$) of the low-energy CP-violating phases. The maximum value for the CP asymmetries (gray scale) is presented for the $3\sigma$ range of the mixing angles and the neutrino mass-squared differences. Notice that we have imposed $M_2\gtrsim 3M_1$ to ensure a nonresonant regime, and $10^9\lesssim M_{1,2}\lesssim 10^{12}$ GeV since $\mu$ and $e$ interactions are in equilibrium. In the same plot, the $|\epsilon_i^\alpha|$ values calculated for the minimum of $\chi^2$ (varying the mixing angles and mass-squared differences) are presented as colored lines. The points marked by triangles and squares correspond to ($\alpha$,$\delta$) fixed by the two-zero conditions ${\mathbf{Y}^\nu}_{11}={\mathbf{Y}^\nu}_{22}=0$ and ${\mathbf{Y}^\nu}_{11}={\mathbf{Y}^\nu}_{32}=0$, respectively, i.e. textures B and C for ${\mathbf{M}^\nu}$ (see Table \[tabR1R2R3\]). We may also see that for the whole $\delta$ and $\alpha$ ranges, the obtained CP asymmetries are highly suppressed being the maximum values below $10^{-6}$. Moreover, $|\epsilon_i^\alpha|\lesssim 10^{-7}$ for ($\alpha$,$\delta$) fixed by textures B and C. Thus, for the case with ${\mathbf{Y}^\nu}_{11}=0$ and $R_1$, the CP asymmetries are too small to ensure efficient leptogenesis. One can show that all other combinations of textures with zeros in ${\mathbf{Y}^\nu}$ and ${\mathbf{M}_R}$ allowed by neutrino data yield similar results. We conclude that thermal leptogenesis in the flavored regime with $10^9\lesssim T\lesssim 10^{12}$ GeV cannot successfully reproduce the observed baryon asymmetry given in Eq. . This conclusion will be corroborated in the next section when the final baryon asymmetry is computed. Let us consider now the unflavored regime. In this case, the CP asymmetries  are enhanced. For each of the valid two-zero textures, the CP asymmetries $\epsilon_1$ and $\epsilon_2$ given in Eqs.  and are computed using the expressions of Tables \[tabletan\] and \[CPtable\]. In Fig. \[cpasymunflav\], we present $|\epsilon_1|$ (blue contour regions) and $|\epsilon_2|$ (gray-scale contour lines) in the $(r_N,M_1)$ plane, for the low-energy neutrino parameters that best fit the 2RHNSM with ${\mathbf{Y}^\nu}$ and ${\mathbf{M}_R}$ textures (T,R). We only show the results for the six combinations (T$_{1,5}$,R$_1$), (T$_{3,4}$,R$_2$), and (T$_{1,6}$,R$_3$), that lead to $\eta_B>0$. From the same plot we see that the maximum values for $|\epsilon_i|$ can now reach $10^{-4}$, which is two orders of magnitude higher than the ones in the flavored regime (cf. Fig \[cpasymflav\]). Furthermore, as the ratio $r_N$ increases, the CP asymmetry $|\epsilon_2|$ gets slightly suppressed with respect to $|\epsilon_1|$. ![Unflavored CP asymmetries $|\epsilon_i|$, $i=1,2$ on the plane ($r_N$, $M_1$), $r_N= M_2/M_1$, for the low-energy neutrino parameters that best fit the texture pairs (T,R). The blue contour regions (gray-scale contour lines) show $|\epsilon_1|$ ($|\epsilon_2|$).[]{data-label="cpasymunflav"}](fig6.pdf "fig:")\ Baryon asymmetry production --------------------------- In the calculation of the final lepton asymmetry we will consider the contributions of both $N_1$ and $N_2$. In the flavored and unflavored regimes, the leptonic CP asymmetries generated in the $N_i$ decays are most likely to be washed out by the out-of-equilibrium inverse decays and scattering processes in which the heavy neutrinos participate. In general, a measure of the washout strength is given by the so-called decay parameter $K_i$, which for a lepton flavor channel $\alpha$ reads $$\begin{gathered} K_i^\alpha=\dfrac{\tilde{m}_i^\alpha}{m_*}, \label{flvdecayparam}\end{gathered}$$ where $\tilde{m}_i^\alpha$ is the flavored effective neutrino mass, $$\begin{gathered} \tilde{m}_i^\alpha=\dfrac{ v^2|{\mathbf{Y}^\nu}_{\alpha i}|^2}{M_i},\end{gathered}$$ and $m_*\simeq 1.09\times 10^{-3}$ eV is the equilibrium neutrino mass. Summing over flavors in Eq. , one obtains the total decay parameter, $$\begin{gathered} K_i=\sum_\alpha K_i^\alpha=\frac{\tilde{m}_i}{m_*},\end{gathered}$$ with $$\begin{gathered} \tilde{m}_i=\sum_\alpha \tilde{m}_i^\alpha=\frac{v^2\,\mathbf{H}^\nu_{ii}}{M_i}.\end{gathered}$$ The relation between $\tilde{m}_i$ and $m_*$ gives a measure of thermal equilibrium for the decays, namely, if $\tilde{m}_i\gg m_*$ ($\tilde{m}_i\ll m_*$) the asymmetry is strongly (weakly) washed out by inverse decays. The fraction of surviving lepton asymmetry can be expressed in terms of efficiency factors $\kappa\in[0,1]$, which are obtained by solving the relevant Boltzmann equations. In our study, we will use instead the simple and accurate analytical approximations for $\kappa_i^\alpha(K_i^\alpha)$ and $\kappa_i(K_i)$ from Refs. [@Antusch:2011nz] and [@Buchmuller:2004nz], respectively. The imposed hierarchy $M_2\gtrsim 3 M_1$ implies $N_{N_1}(T\sim M_2)\simeq N_{N_2}(T\sim M_1)\simeq 0$, so that the computation of the final asymmetry may be split into the $N_1$ and $N_2$ leptogenesis phases. Furthermore, we consider a strong-coupling $N_1$ scenario, where part of the lepton asymmetry generated by $N_2$ decays is projected onto a flavor-direction protected against the washout from $N_1$ interactions [@Antusch:2011nz]. The final ($B-L$)-asymmetry for the flavored temperature regime can be written as [@Antusch:2011nz] $$\begin{gathered} N_{B-L}=N_{\Delta_{\gamma_1}}+N_{\Delta_{\gamma_1^\perp}}+N_{\Delta_{\tau}}, \label{finalasym}\end{gathered}$$ where the $\Delta_\alpha\equiv B/3-L_\alpha$ number densities in each flavor state read $$\begin{aligned} N_{\Delta_{\gamma_1\;}}&\simeq -P_{\gamma_2\gamma_1}\,\epsilon_2^\gamma\,\kappa_2^\gamma\, e^{-\frac{3\pi}{8}K_1^\gamma}-\epsilon_1^\gamma\,\kappa_1^\gamma,\\ N_{\Delta_{\tau\;\;\;}}&\simeq -\epsilon_2^\tau\,\kappa_2^\tau\, e^{-\frac{3\pi}{8}K_1^\tau}-\epsilon_1^\tau\,\kappa_1^\tau,\\ N_{\Delta_{\gamma_1^\perp}}&\simeq -\,(1-P_{\gamma_2\gamma_1})\,\epsilon_2^\gamma\,\kappa_2^\gamma,\end{aligned}$$ in which $\gamma_1$ and $\gamma_1^\perp$ are the parallel and orthogonal flavor components to the interaction channels of $N_1$, respectively. Here, $\kappa_i^\alpha$ are the efficiency factors defined in [@Antusch:2011nz], and $P_{\gamma_2\gamma_1}$ is the probability of flavor $\gamma_2$, generated in the $N_2$ decay, to be transformed into $\gamma_1$ under the $N_1$ decay process, $$\begin{aligned} P_{\gamma_2\gamma_1}=\dfrac{\left|\sum_{\alpha} \mathbf{Y}^{\nu *}_{\alpha 1} {\mathbf{Y}^\nu}_{\alpha 2}\right|^2}{\left(\,\sum_{\alpha} |{\mathbf{Y}^\nu}_{\alpha 1}|^2\right) \left(\, \sum_{\alpha} |{\mathbf{Y}^\nu}_{\alpha 2}|^2 \right)},\end{aligned}$$ where $\alpha=e, \mu$. In the unflavored regime, the lepton flavors are indistinguishable in the primordial plasma and the final $(B-L)$-asymmetry reads [@Blanchet:2011xq] $$\begin{gathered} \label{unflavouredbaryonasym} N_{B-L}\simeq-\epsilon_1\kappa_1-\left(1-P_{21}+P_{21}e^{-\frac{3\pi K_1}{8}}\right)\epsilon_2\kappa_2\; ,\end{gathered}$$ with $\kappa_i$ being defined in [@Buchmuller:2004nz]. Here, $P_{21}$ is the probability of the lepton asymmetry produced in $N_2$ leptogenesis being projected onto the flavor direction of the asymmetry due to $N_1$ interactions, $$\begin{aligned} P_{21}=\dfrac{\left|\mathbf{H}^\nu_{12}\right|^2}{\mathbf{H}^\nu_{11} \mathbf{H}^\nu_{22}}.\end{aligned}$$ After computing the densities $N_{B-L}$, for both flavored and unflavored regimes, using Eqs.  and , the final baryon-to-photon ratio $\eta_B$ is obtained from Eq. . ![Baryon-to-photon ratio $\eta_B$ as a function of the low-energy CP-violating phases $\alpha$ and $\delta$ in the flavored regime, for the texture-zero case ${\mathbf{Y}^\nu}_{11}=0$ and R$_1$. The gray-scale contour regions show the maximum value of $\eta_B$, taking $\theta_{ij}$, ${\Delta m^2_{21}}$ and $|{\Delta m^2_{31}}|$ in the $3\sigma$ experimental range (see Table \[datatable\]) and for $10^9\lesssim M_{1,2}\lesssim 10^{12}$ GeV with $M_2\gtrsim 3M_1$. The colored contour lines are the results obtained for the minimum value of $\chi^2$. In the plot, the triangles and squares correspond to the ($\alpha$,$\delta$) pairs fixed by the conditions ${\mathbf{Y}^\nu}_{11}={\mathbf{Y}^\nu}_{22}=0$ and ${\mathbf{Y}^\nu}_{11}={\mathbf{Y}^\nu}_{32}=0$, respectively (cf. textures B and C in Table \[tabR1R2R3\]).[]{data-label="etabflav"}](fig7.pdf "fig:")\ In Fig. \[etabflav\], we present $\eta_B$ computed for the illustrative case of ${\mathbf{Y}^\nu}_{11}=0$ with R$_1$, for which the flavored CP asymmetries were already analyzed in Section \[subsecCP\]. In that figure, the gray-scale contour regions correspond to the maximum of $\eta_B$ in the $3\sigma$ experimental range of the mixing angles and the neutrino mass-squared differences, taking $10^9\lesssim M_{1,2}\lesssim10^{12}$ GeV. As expected from the small values of $|\epsilon_i^\alpha|$ (see Fig. \[cpasymflav\]), the final baryon asymmetry is suppressed in the whole allowed parameter region. Indeed, the final $\eta_B$ lies between one to two orders of magnitude below the observed value $\eta_B^0$. Moreover, for the ${\mathbf{M}^\nu}$ textures B and C, marked in the figure by a triangle and a square, respectively, $\eta_B\lesssim10^{-12}$ is verified. For all the other combinations of textures T and R that are compatible with neutrino oscillation data, similar results were obtained for the flavored regime, corroborating the fact that thermal leptogenesis in the two-flavor case is not viable. For the unflavored regime, sufficiently large (and positive) values for $\eta_B$ are obtained for six of the twelve pairs (T,R) of textures compatible with neutrino data (see Table \[tabR1R2R3\]). This is shown in Fig. \[etabunflav\], where we present the predicted $\eta_B$ (gray-scale contour regions) as a function of $M_1$ and the mass ratio $r_N$, considering the low-energy neutrino data that best fit the six textures. In fact, for all these cases, the observed baryon-to-photon ratio $\eta_B^0$ (red contour line in Fig. \[etabunflav\]) is achieved for $M_1\sim 10^{14}$ GeV, being $\kappa_i\sim \mathcal{O}(10^{-3})$ (strong washout regime). Hence, one concludes that the texture combinations (T$_{1,5}$,R$_1$), (T$_{3,4}$,R$_2$), and (T$_{1,6}$,R$_3$) lead to successful thermal leptogenesis in the unflavored regime. ![The baryon-to-photon ratio $\eta_B$ on the plane ($r_N$, $M_1$), $r_N= M_2/M_1$, for the unflavored regime and taking the low-energy neutrino parameters that best fit the texture pairs (T,R). The gray-scale contour regions represent the final value of $\eta_B$, while the red contour line corresponds to the observed value $\eta_B^0$ given in Eq. .[]{data-label="etabunflav"}](fig8.pdf "fig:")\ One may wonder whether the above conclusion remains valid, if one considers the more restricted cases discussed in Section \[sec3a\], in which three elements of ${\mathbf{Y}^\nu}$ are equal. We will only consider the cases that were proved to be compatible with neutrino data and, additionally, verify the condition $r_N\gtrsim3$, for which our leptogenesis assumptions hold. From Table \[Tabrel\] and Fig. \[etabunflav\], one can see that only the cases (T$_1$,R$_1$,B) with ${\mathbf{Y}^\nu}_{21}={\mathbf{Y}^\nu}_{31}={\mathbf{Y}^\nu}_{32}$ and (T$_5$,R$_1$,C) with ${\mathbf{Y}^\nu}_{21}={\mathbf{Y}^\nu}_{22}={\mathbf{Y}^\nu}_{32}$ meet those requirements ($r_N\sim 12$) and, simultaneously, yield $\eta_B>0$. In Fig. \[etabcorr\], we present the $\eta_B$ region allowed by the $3\sigma$ experimental interval for the low-energy neutrino parameters (blue region) as a function of the mass $M_1$. Here we also show the results obtained when the contribution of the second neutrino $N_2$ is not taken into account for leptogenesis (gray region). One concludes that for temperatures below $10^{14}$ GeV the effect of the second neutrino $N_2$ is negligible, while for higher temperatures the $N_2$ contribution tends to lower $\eta_B$. The value of $\eta_B^0$ (red horizontal line) is achieved for masses $M_1\sim 10^{14}$ GeV. \ Conclusions {#sec4} =========== In this paper, we have revisited the 2RHNSM considering maximally restricted texture-zero patterns for the lepton Yukawa and mass matrices. Our results are summarized in Table \[tabR1R2R3\]. We conclude that textures B, C and D for the effective neutrino mass matrix ${\mathbf{M}^\nu}$ are compatible with current neutrino data (mixing angles and mass-squared differences) at $1\sigma$, while texture F is compatible at $3\sigma$. In all cases, only an inverted hierarchical neutrino mass spectrum is allowed. A remarkable prediction of textures B and C is that one of the viable solutions for the low-energy CP-violating Dirac phase is $\delta\sim 3\pi/2$, which is very close to the best-fit value obtained from the combined fit of neutrino oscillation data. Aiming at reducing the number of free parameters in the model, we have also explored scenarios in which additional relations (equality) among the Dirac neutrino Yukawa couplings are imposed. The cases with the maximum number of equal elements in ${\mathbf{Y}^\nu}$ which are compatible with neutrino data are presented in Table \[Tabrel\]. As can be seen from the table, compatibility is only verified at the $3\sigma$ confidence level. For the phenomenologically viable textures, we have studied their implications for the BAU in the framework of type-I seesaw thermal leptogenesis. We paid special attention to the treatment of leptogenesis in the 2RHNSM. Contrary to what is customary in the literature, where only the decay of the lightest heavy neutrino is considered, we included the decays of both heavy neutrinos in our analysis. Moreover, flavor effects that arise from the fact that lepton interactions exit thermal equilibrium at different temperatures in the early Universe were taken into account. We considered two temperature regimes for leptogenesis: the two-flavored regime ($10^9\lesssim T\lesssim 10^{12}$ GeV) and the unflavored regime ($T\gtrsim10^{12}$ GeV). Within our assumptions ($M_2\gtrsim 3M_1$), we showed that the CP asymmetries in the flavored regime are too small to generate the required lepton asymmetry for successful leptogenesis. On the other hand, for the unflavored case, the CP asymmetries are enhanced, and the observed baryon-to-photon ratio is achieved in the 2RHNSM for the texture combinations (T$_{1,5}$,R$_1$), (T$_{3,4}$,R$_2$), and (T$_{1,6}$,R$_3$), for $M_1\sim 10^{14}$ GeV. Furthermore, the cases (T$_1$,R$_1$) and (T$_5$,R$_1$), with three equal elements in ${\mathbf{Y}^\nu}$ in the positions (21,31,32) and (21,22,32), respectively, were shown to be also compatible with the present value of the baryon asymmetry for the same leptogenesis temperature $T \sim 10^{14}$ GeV. The nature of the flavor structure of the fermion sector in the standard model and theories beyond it remains puzzling. A common approach to address this problem is to assume certain constraints on the coupling and/or mass matrices in order to reduce the number of free parameters. The lepton textures considered in this work were taken as the simplest and most economical patterns that can be implemented in the framework of the 2RHNSM. We have shown that the maximally constrained 2RHNSM is compatible with current neutrino oscillation data and can also explain the matter-antimatter asymmetry in the Universe via the leptogenesis mechanism. This conclusion holds for several mass matrix textures with the maximal number of allowed zeros and, in a more restricted set, having equal elements in the Dirac Yukawa coupling matrix. It would be interesting to see if such predictive textures could arise from a flavor symmetry principle. 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--- abstract: 'We employ a recently developed model that allows the study of two-dimensional brittle crack propagation under fixed grip boundary conditions. The crack development highlights the importance of voids which appear ahead of the crack as observed in experiments on the nano-scale. The appearance of these voids is responsible for roughening the crack path [*on small scales*]{}, in agreement with theoretical expectations. With increasing speed of propagation one observes the branching instabilities that were reported in experiments. The simulations allow understanding the phenomena by analyzing the elastic stress field that accompanies the crack dynamics.' author: - Itamar Procaccia and Jacques Zylberg title: Propagation Mechanism of Brittle Cracks --- Introduction ============ In laboratory experiments one studies crack propagation in elastic media by holding a stretched slab of material with a grip, and initiating the crack by making a small notch at one end of the sample [@99HHMS]. When the notch exceeds the Griffith’s length [@21Gri], the crack propagates with increasing speed until there is a balance between the release of elastic energy and the creation of surface energy [@98Fre; @99MF]. It turned out that molecular dynamics simulations failed systematically to mimic this set up, forcing simulators to pull continuously on the boundaries to achieve a propagating crack [@11HKL]. With fixed grip boundary conditions crack tended to slow down and stop propagating. The inability to simulate a brittle crack by molecular dynamics gave rise to claims that it would be impossible to advance a brittle crack without continuous stretching. The fundamental reason for this long-standing problem was understood recently. In Ref. [@11DKPZ] it was shown that the solution of this conundrum lies in the range of the inter-particle potential. By changing the potential range one can go from a brittle to a ductile crack, and the latter is stopped by plastic dissipation. Only a brittle crack can support itself with a fixed grip. Brittleness was shown to be guaranteed by choosing a potential range that is of the order of the inter-particle distance. In this paper we present results of molecular dynamics simulations of brittle cracks under a fixed grip that are based on this understanding. One of the interesting issues that can be studied by such simulations is the proposed existence of voids that open up before the crack, acting as pointers for further propagation. This issue had been quite controversial. While some experiments, and in particular those pertaining to ultra-slow crack propagation in glass in wet atmosphere, strongly indicated the appearance of voids ahead of the crack [@Bou], other experiments failed to observe such voids, and claimed that they are irrelevant to the propagation of brittle cracks. In a theoretical study, Ref. [@04BMP] constructed a model of void-dominated crack propagation and attempted to explain the observed roughness of brittle cracks by the randomness associated with the positioning of the voids ahead of the crack. The numerical simulations shown below appear to vindicate this approach almost verbatim. Voids do appear, and their appearance is random as stipulated in Ref. [@04BMP], and see Sect. \[voids\] for more details. Another issue of interest is the instabilities of the propagating crack when its velocity increases. The simulations show very clearly branching instabilities with subsequent competition between the two branches that typically result in the death of one branch in favor of the other. In Sect. \[protocol\] we present the details of the preparation of the samples for the molecular dynamics simulations. In Sect. \[Grif\] we show that our cracks are brittle by demonstrating that they start running precisely when the Griffith’s criterion met. In Sect. \[rough\] we discuss the appearance of voids ahead of the crack and the resulting roughening of the crack path. Finally, in Sect. \[inst\] we present a brief discussion of the observed instabilities. The last section presents a short summary and indicates the road ahead. Molecular Dynamics {#protocol} ================== -1. cm For the numerical experiments we employ a generic glass former in 2-dimensions in the form of a 50-50 binary mixture of ‘small’ and ‘large’ particles, chosen to avoid any crystallization. In fact the particles interact by inter-particle potentials as shown in Fig. \[potentials\], with the analytic form $$\label{potential} \phi\!\left(\!{{\textstyle\frac{r_{ij}}{\lambda_{ij}}}}\!\right) = \left\{\begin{array}{ccl} &&4\varepsilon\left[\left(\frac{\lambda_{ij}}{ r_{ij}}\right)^{12}-\left(\frac{\lambda_{ij}}{r_{ij}}\right)^{6}\right] \ , \hskip 3.5 cm\frac{r_{ij}}{\lambda_{ij}}\le \frac{r_{\rm min}}{\lambda}\\ &&\varepsilon\left[a\left(\frac{\lambda_{ij}}{ r_{ij}}\right)^{12}-b\left(\frac{\lambda_{ij}}{ r_{ij}}\right)^{6} + \displaystyle{\sum_{\ell=0}^{3}}c_{2\ell}\left({{\textstyle\frac{r_{ij}}{\lambda_{ij}}}}\right)^{2\ell}\right] \ , \hskip 0.5 cm\frac{r_{\rm min}}{\lambda}<\frac{r_{ij}}{\lambda_{ij}}<\frac{r_{\rm co}}{\lambda} \\ &&\quad\quad\quad\quad 0 \ , \hskip 5.4 cm \frac{r_{ij}}{\lambda_{ij}} \ge\frac{r_{\rm co}}{\lambda} \end{array} \right.\!\!$$ Here $r_{\rm min}/\lambda_{ij}$ is the length where the potential attain it’s minimum, and $r_{\rm co}/\lambda_{ij}$ is the cut-off length for which the potential vanishes. The coefficients $a,~b$ and $c_{2\ell}$ are chosen such that the repulsive and attractive parts of the potential are continuous with two derivatives at the potential minimum and the potential goes to zero continuously at $r_{\rm co}/\lambda_{ij}$ with two continuous derivatives as well. The interaction length-scale $\lambda_{ij}$ between any two particles $i$ and $j$ is $\lambda_{ij} = 1.0\lambda$, $\lambda_{ij} = 1.18\lambda$ and $\lambda_{ij} = 1.4\lambda$ for two ‘small’ particles, one ‘large’ and one ‘small’ particle and two ‘large’ particle respectively. The unit of length $\lambda$ is set to be the interaction length scale of two small particles, $\varepsilon$ is the unit of energy and $k_B = 1$. We used a modified Berendsen thermostat which couples a constant number of particles to the bath, regardless of the system size [@10KLPZ]. The preparation protocol starts with equilibrating the sample at high temperature and pressure, using periodic boundary conditions. The systems are then cooled to temperature $T=10^{-2}$ while keeping the pressure high, in order to avoid the creation of any holes in the material. Then the pressure is reduced to zero, $P=0.0$, such that the periodic boundary conditions could be removed; particles forming the right and left walls were frozen, but the upper and lower boundaries were rendered free. The brittle crack experiment starts by loading the system uniaxially (with a constant velocity such that $v_{\rm wall} \ll c_s/10$) until a desired stress is reached, and then the side walls are held fixed. A cut is then implemented by the cancelation of forces crossing an imaginary line of desired length which starts at the lower boundary. The evolution of the crack is simulated by molecular dynamics coupled to a heat bath at $T = 0.0$ and requires no further loading of the system. The results presented below were created using two different geometries, one of width 1000$\lambda$ and length 3000$\lambda$, and the other being a square sample of 1600$\lambda\times 1600\lambda$. The density is the zero temperature and zero pressure density $\rho=0.745$. In Fig. \[crack\] we present a typical crack that results from this procedure, in which the loading $\gamma$ was $\gamma=2.5\%$ and an initial crack of length 750$\lambda$. One observes the typical roughness that is discussed below in Sect. \[rough\]. The Griffith Length {#Grif} =================== Upon making the initial cut, the system always has a microscopic plastic response which however stops if the length of the cut is smaller than the Griffith length. To demonstrate that our running cracks are indeed brittle we test the Griffith’s criterion in our simulation. The length of the initial cut that evolves spontaneously into a crack was determined by Griffith [@21Gri], comparing the energy release from the stress field to the energy consumed by the creation of two new fresh surfaces by the advancing crack. With the stress exerted on the slab being $\sigma$, the critical length $L_c$ is determined by the bulk modulus $E$ and the surface energy per unit length $\epsilon$ via the relation $$\sigma \sqrt {L_c }= \sqrt{\frac{2E\epsilon}{\pi}} . \label{griff}$$ In our simulations we are able to measure the critical length $L_c$ for a given loaded system. Since all the other material parameters appearing in Eq. \[griff\] are measured independently, we can show that our model crack propagation agrees with the expected physics of brittle cracks. The stress field $\sigma$ is simply determined by our grip boundary conditions. The bulk modulus $E$ is read from the linear part of a uniaxial straining experiment shown in Fig. \[bulkModulus\]. To estimate the surface energy in our system we computed the total energy before and after making the initial cut of length $L$. Taking the difference and dividing by $L$ we get the estimate $$\epsilon \approx 15 \ .$$ At this point we employed 50 independent samples that were equilibrated at high temperature and quenched to the glassy state. The glassy samples were then loaded quasi-statically to strain values in the range $\gamma= 1.6, 1.8, 2.0, 2.2, 2.4, 2.6$. In order to probe the Griffith’s Length $L_c$ we tracked the behavior of our sample during the first 30 time units of our simulation to see whether a crack has started to run as a result of the introduced cut. For each sample we repeated this procedure for various values of $L$. The results, (averaged over 50 samples) are presented in the bottom panel of Fig. \[Griff\]. We show the length of the evolving crack, denoted as ${{\mathcal{L}}}$, as a function the initial cut size $L$. Where the length ${{\mathcal{L}}}$ begins to run determines $L_c$ for the conditions at hand. In the upper panel of the same figure we plot $\sigma^2$ vs. $1/L_c$. The resulting linear plot is in agreement with Eq. (\[griff\]), with the slope being close to the theoretical expectation of $2E\epsilon/\pi$. Roughening resulting from voids ahead of the crack {#rough} ================================================== The present numerical simulations allows the verification of the role of voids that form ahead of the crack, determining, at least on short length scales, the path chosen by the crack as it develops in the amorphous solid. We note that an amorphous solids is not an ideal elastic material, in which mathematically straight cracks are possible. The randomness of the material must show up in the geometry of the crack in this way or another. The voids ahead of the crack serve as pointers for the forthcoming propagation of the crack, and as is shown below, their random positions determines the roughening of the crack path. In Fig. \[voids\] we show a typical crack tip with the voids that opened ahead. The physical reason why voids may prefer to open ahead of the crack and not on the crack tip was explained in Ref. [@04BMP]. The argument is based on the fact that voids open due to plastic yield, and they do this where the pressure is maximal. Near the crack tip there is a process zone where the pressure is increasing going outward, until one hits the maximal pressure curve which connects with the outer elastic solution, see Fig. \[pressure\]. In this figure we show the average pressure as measured in the numerical simulation in a fixed window tracking in time the crack tip. In particular one should pay attention to the lower right panel in Fig. \[pressure\] which shows that the maximal pressure lies ahead of the tip, at a distance $\xi$ from the tip. This figure should be compared with Fig.2 of Ref. [@04BMP] which it directly vindicates. Obviously, the void that opens ahead of the tip can have a broadly distributed position around the forward direction (where the probability to form a void is maximal). An actual measurement of this distribution for the present simulation is shown in Fig. \[distribution\]. Shown are the actual positions of the voids as they appear ahead of the crack during the time evolution, and in the lower panels the probability distribution for the void to appear at an angle $\theta$ with respect to the forward direction, and at a distance $r$ along the forward direction. These data should be compared with Fig. 3 of Ref. [@04BMP]. Two points are worth stressing: first, the angular distribution of void positions will be the source of roughening of the crack - the probability to fall along the forward direction is not high enough compared to positive or negative angles with the respect to the forward direction. Second, once a void appears on, say, a positive angle, the next void will have an even higher probability for a positive angle, meaning that the rough crack is expected to show a roughening exponent higher than 1/2. Indeed, such a persistent random walk is always expected to show exponents higher than 1/2, whereas anti-persistent random walks are characterized by a roughening exponent smaller than 1/2. The roughening exponent was measure here in the usual way, i.e. considering the crack as a graph $y(x)$ and determining $h(r)$ according to $$h(r) \equiv \langle {\rm max}\{y(\tilde x\}\}_{x<\tilde x<x+r} - {\rm min}\{y(\tilde x\}_{x<\tilde x<x+r}\rangle_x \ .$$ For a self-affine graph the scaling exponent $\zeta$ is defined via the scaling relation $$h(r) \sim r^\zeta \ .$$ We show this quantity in a log-log plot in Fig. \[roughness\]. As expected the function exhibits a persistent scaling exponent of $\zeta\approx 0.66$ for scales $r<30$, and then a cross over to a random graph without persistence or anti-persistence for higher scales. Instabilities {#inst} ============= Crack dynamics are governed by the balance of energy at the crack tip. The influx of elastic energy through the crack-tip is used to create surface energy behind the advancing tip. With increasing crack velocity there is not sufficient surface to store the energy that is released. Therefore the system resorts to instabilities in the form of branching and oscillations in order to increase the amount of surface created per unit length on the axis of propagation. In our simulations we find that when the velocity of the crack tip reaches about $30\%$ of the Rayleigh speed, one begins to observe crack branching. We note that our system never develops two independently propagating cracks, but rather reduced the velocity of propagation through attempted branchings. Oscillations were not observed. A typical branching event is shown in Fig. \[branch\], where we see the two crack tips as they still grow simultaneously. The competition between them always results in the demise of one of the growing cracks in favor of the other, until the next branching event. summary and discussion ====================== In summary, we have demonstrated that molecular dynamics can be usefully employed to study brittle crack propagation by selecting an appropriate interparticle potential. The resulting cracks are growing by nucleating voids ahead of the crack tip in much the same way that was anticipated by Ref. [@Bou] and theorized in Ref. [@04BMP]. This mechanism of growth is responsible for roughening of the crack path on small scales, but as long as side branching does not commence, the crack is globally flat on macroscopic scales, as expected theoretically [@06BBP]. In future work molecular dynamics simulations will be employed in three dimensions where the issue of micro-branching [@99MF] can be studied and compared with experimental results. We thank Laurent Boué for useful discussion. This work had been supported in part by the German Israeli Foundation, the Israel Science Foundation and the European Research Council under an “ideas” grant STANPAS. [99]{} See for example: J. A. Hauch and M. P. Marder, Int. J. Fract. [**90**]{}, 133 (1998); J.A. Hauch, D. Holland, M.P. Marder and H. Swinney, Phys Rev, Lett, [**92**]{}, 3823 (1999). A.A. Griffith, Phil. Trans. Roy. Soc.London, [**221**]{}, 163 (1921). L. B. Freund, [*Dynamic Fracture Mechanics*]{} (Cambridge, London, 1998). M. Marder and J. Fineberg, Phys. Rep. [**313**]{}, 1 (1999). S.I. Heizler, D.A. Kessler and H. Levine, Phys Rev E [**84**]{}, 026102 (2011). O. Dauchot, S. Karmakar, I. Procaccia and J. Zylberg, Phys. Rev. E [**84**]{}, 046105 (2011) E. Bouchaud, J. Phys. Condens. Matter [**9**]{}, 4319 (1997); F. Ce´larie´, S. Prades, D. Bonamy, L. Ferrero, E. Bouchaud, C. Guillot, and C. Marlie‘re, Phys. Rev. Lett. [**90**]{}, 075504 (2003). E. Bouchbinder, J. Mathiesen and I. Procaccia, Phys. Rev. Lett., [**92**]{}, 245505 (2004). S. Karmakar, E. Lerner, I. Procaccia and J. Zylberg,Phys. Rev. E [**82**]{}, 031301 (2010). I. Ben-Dayan, E. Bouchbinder and I. Procaccia, Phys. Rev. E [**74**]{}, 046102 (2006).

--- abstract: 'We develop a simplified method for obtaining higher orders in the perturbative expansion of the singular term $A\left(\alpha_s\right)/[1-x]_{+}$ of non-singlet partonic splitting functions. Our method is based on the calculation of eikonal diagrams. The key point is the observation that the corresponding cross sections exponentiate in the case of two eikonal lines [@George; @gath; @freta], and that the exponent is directly related to the functions $A\left(\alpha_s\right)$ due to the factorization properties of parton distribution functions. As examples, we rederive the one- and two-loop coefficients $A^{(1)}$ and $A^{(2)}$. We go on to derive the known general formula for the contribution to $A^{(n)}$ proportional to $N_f^{n-1}$, where $N_f$ denotes the number of flavors. Finally, we determine the previously uncalculated term proportional to $N_f$ of the three-loop coefficient $A^{(3)}$ to illustrate the method. Our answer agrees with the existing numerical estimate [@vogt]. The exact knowledge of the coefficients $A^{(n)}$ is important for the resummations of large logarithmic corrections due to soft radiation. Although only the singular part of the splitting functions is calculable within our method, higher-order computations are much less complex than within conventional methods, and even the calculation of $A^{(4)}$ may be possible.' --- YITP-SB-02-49\ [**Higher Orders in $A\left(\alpha_s\right)/[1-x]_{+}$ of\ Non-Singlet Partonic Splitting Functions** ]{} [**Carola F. Berger**]{} [*C.N. Yang Institute for Theoretical Physics, SUNY Stony Brook\ Stony Brook, New York 11794 – 3840, U.S.A.*]{} Introduction ============ Parton Distribution Functions ----------------------------- Parton distribution functions (PDFs) are indispensable ingredients in any calculation within perturbative quantum chromodynamics involving hadrons in the initial state. $f_{a/A}(x)$ describes the distribution of parton $a$ in hadron $A$ with a momentum fraction $x$. It cannot be calculated within the framework of perturbation theory. However, its evolution is perturbatively calculable from the renormalization group equation [@DGLAP] f\_[a/A]{} (x,) = \_b \_[x]{}\^1 P\_[a b]{} (,\_s()) f\_[b/A]{} (,), \[evol\] where $P_{a b}$ is the evolution kernel or splitting function, and $\mu$ denotes the factorization scale, usually taken equal to the renormalization scale. This follows from factorization [@pQCD], which enables us to write a large class of physical cross sections as convolutions of these PDFs with perturbatively calculable short-distance functions. Similar considerations apply to partonic cross sections. There the parton-in-parton distribution functions $f_{f_i/f_j}$ describe the probability of finding parton $f_i$ in parton $f_j$. The evolutionary behavior of the partonic PDFs obeys the same equation, (\[evol\]), as for the hadronic PDFs. Thus the splitting functions can be computed in perturbation theory. The convolutions of these PDFs with short-distance functions simplify to simple products when we take moments: \_[f\_1/f\_2]{} (N,) = \_0\^1 d x x\^[N-1]{} f\_[f\_1/f\_2]{} (x,) = \_0\^d x e\^[-N(1-x)]{} f\_[f\_1/f\_2]{} (x,) + (). \[moment\] The first definition in Eq. (\[moment\]) is the Mellin transform which is equivalent to the second definition, the Laplace transform, at large $N$ because then $e^{-N(1-x)} \sim x^N$. Tildes here and below denote quantities in moment space. Up to corrections of order $1/N$ we can neglect flavor mixing, that is, we deal with non-singlet distributions: f\_[NS]{}\^ = f\^\_[q\_a/q]{} - f\^\_[q\_b/q]{},f\^\_[q\_i/q]{} = f\_[q\_i/q]{} f\_[|[q]{}\_i/q]{}. The solution to the evolution equation (\[evol\]) for non-singlet parton-in-parton distributions is in moment space given by $$\tilde{f}_f (N, \mu, \varepsilon) = \exp \left[ \int_0^{\mu^2} \frac{d \mu'^2}{\mu'^2} \gamma_{ff} \left(N,\alpha_s(\mu') \right) \right] + {\mathcal{O}}\left(\frac{1}{N} \right), \label{exppdf}$$ where $\gamma_{ff}(N) = \int_0^1 dx x^{N-1} P_{ff}(x)$ are the moments of the splitting functions, and where we have imposed the boundary condition $\tilde{f}_f(N,\mu = 0,\varepsilon) = 1$. The argument $\varepsilon$ in $\tilde{f}_f$ indicates that we compute in the usual modified minimal subtraction scheme ($\MS$) in $n = 4 - 2 \varepsilon$ dimensions. To leading power in $N$ the moments of the partonic splitting functions, $\gamma_{ff}(N)$, take the simple form [@korch; @alball] $$\gamma_{ff}(N,\alpha_s) = A_f (\alpha_s) \ln N + B_f (\alpha_s) + \mathcal{O}\left(\frac{1}{N} \right),\label{pffN} \ee or in $x$-space, \begin{equation} P_{ff} (x,\alpha_s ) = A_f (\alpha_s) \left[\frac{1}{1-x}\right]_{+} + B_f (\alpha_s) \delta(1-x) + {\mathcal{O}}\left(\left[1-x \right]^0 \right), \label{pff}$$ where the plus distribution $\left[\frac{1}{1-x}\right]_{+}$ is defined by \_z\^1 dx f(x) \_[+]{} = \_z\^1 dx + f(1) (1-z). The term with the plus distribution represents the cancellation of a single overall infrared divergence. The coefficient of $\ln N$ in Eq. (\[pffN\]) can be expanded in the strong coupling, $$A_f (\alpha_s) = \sum_n \left( \frac{ \alpha_s}{\pi} \right)^n A_f^{(n)}. \label{aaa}$$ The exact knowledge of the terms $A^{(n)}$ is important for $x \rightarrow 1$ (large $N$), since there large logarithmic corrections arise due to soft-gluon radiation. These corrections need to be resummed in order to be able to make reliable predictions within perturbation theory. The knowledge of the coefficients $A^{(3)}$ and $B^{(2)}$ is required for threshold resummation [@threshold] at the next-to-next-to-leading logarithmic (NNLL) level [@vogt2]. The anomalous dimensions $\gamma_{ff}(N)$ are currently known to two loops [@2loopknown], and a general formula for the $\alpha_s^n N_f^{n-1}$-terms of $\gamma_{ff}(N)$ was computed by Gracey [@Gracey]. Work on the 3-loop splitting functions, based on the operator product expansion (OPE), is in progress [@3loop]. From the known exact values for some specific moments and the behavior at small $x$ [@moments] a numerical parametrization for the coefficient $A^{(3)}$ was obtained by Vogt [@vogt], although, for the above reasons, the exact knowledge of this term is desirable. The above mentioned calculation by Moch, Vermaseren, and Vogt of the full three-loop splitting functions via the OPE in moment space will be completed in the near future [@MoVe]. Their results for the fermionic contributions are now available [@MVV]. However, the method presented here, although only applicable for the calculation of the coefficients $A$, not of the complete $x$-dependence, is complementary to the OPE method in two ways: we calculate only virtual diagrams, and furthermore, it is much less computationally intensive, thus a computation of the four- or even higher loop coefficients may be feasible. In the following it is convenient to use light-cone coordinates, where our convention is as follows: k\^+ & = & (k\^0 + k\^3),\ k\^- & = & (k\^0 - k\^3 ),\ k\_& = & (k\^1,k\^2). Factorization allows us to define PDFs in terms of nonlocal operators. At leading order in $N$ one can define [@pQCD; @pdfs] the following function f\_[q\_i/q]{} (x) & = & d y\^- e\^[-i x p\^+ y\^-]{}\ & = & d y\^- e\^[-i x p\^+ y\^-]{} \_n \^+ , \[pdfdef\] which describes the distribution of a quark $q_i$, created by the operator $\bar{\psi}_i$, in a quark $q$ with momentum $p$. The operators are separated by a light-like distance, and are joined with a path-ordered exponential, denoted by $P$, to achieve gauge-invariance. This exponential describes the emission of arbitrarily many gluons of polarization in the plus-direction. ${\mathcal{A}}^{(q)}$ is the vector potential in the fundamental representation. In the second line we have inserted a complete set of states, have used the identity P e\^[i g \_0\^ d (n\^)]{} = \^, \[phaseop\] and have defined \_i(y) = P e\^[i g \_0\^ d (y + \^)]{} \_i(y), with the light-like vector $\beta^\mu$ chosen in the minus-direction, $\beta^+ = \beta_\perp = 0$. The Feynman rules for the expansion of an ordered exponential are shown in Fig. \[Frules\], where the double lines represent “eikonal” propagators. Thus, path-ordered exponentials are closely connected to the eikonal, or soft approximation, which results in the same Feynman rules, as we will see below. Gluon parton distribution functions can be constructed analogously [@pQCD; @pdfs], with the vector potentials in the adjoint representation, and appropriate operators for the creation of gluons. Outline of the Paper -------------------- Below we will outline a method which greatly simplifies the calculations of the collinear coefficients $A^{(n)}$ in the non-singlet case. Our method is based on the factorized form of the partonic non-singlet PDFs at leading order, where for $x \rightarrow 1$ the collinear coefficients are factored into an eikonal cross section. It was observed by Sterman that these cross sections exponentiate [@George], not only in abelian theories, but also in nonabelian theories. As we will see, this exponentiation further simplifies the technical calculations, and enables us to go a step further towards the full calculation of the coefficient $A^{(3)}$. To illustrate the method, we compute the as yet undetermined contribution proportional $N_f$ to the three-loop coefficient. A similar observation was made by Korchemsky [@korch], who related the anomalous dimension of PDFs, Eq. (\[pff\]), with the cusp anomalous dimension of a Wilson loop. His work was performed in a noncovariant axial gauge, whereas here we will use Feynman gauge throughout. Korchemsky’s observation was used in [@korchmarch] for the calculation of the two-loop coefficient $A^{(2)}$, which was done in Feynman gauge. The work of Ref. [@korchmarch] was also based on the renormalization properties and exponentiation of Wilson loops (see [@cusp] and references therein). This approach is related to ours. However, the additional observations we make below result in several advantages. The number of diagrams contributing at each order is decreased by working with light-like eikonals. Furthermore, as we will show below, we can restrict ourselves only to virtual graphs. With the help of Ward identities we are also able to justify the eikonal approximation and to show explicitly the absence of infrared (IR) subdivergences and the cancellation of ultraviolet (UV) subdivergences at the eikonal vertex, leaving only the usual QCD UV divergences. The discussion consists of two parts. In the first part, Sections \[sectwebA\] and \[sectexp\], we lay the theoretical foundation of our method, whose technical details are described and explained by means of various examples in the second part, Sections \[sectmeth\] and \[sect3loop\]. We will first derive in Sect. \[sectwebA\] a factorized form for PDFs, and introduce an eikonal factor that absorbs all collinear singularities. This implies that the $A(\alpha_s)$ can be derived from the eikonal factor. In the next section we will review the proof of the exponentiation [@gath; @freta] of this factor, show how to construct the exponent order by order in perturbation theory, and prove the absence of infrared/collinear subdivergences at each order. These properties then allow us to develop an algorithm, using light-cone ordered perturbation theory (LCOPT), for the systematic calculation of the collinear coefficients $A$, which is equivalent to the calculation of the virtual contribution to the anomalous dimension of an eikonal vertex. We summarize the method in Section \[sectmeth\], before going on to rederive as examples the one- and two-loop coefficients $A^{(1)}$ and $A^{(2)}$. In Section \[sect3loop\] we derive a formula for the coefficients of $A^{(n)}$ proportional to $N_f^{n-1}$, which agrees with the corresponding contribution computed by Gracey [@Gracey] using an effective theory. We end by illustrating the steps necessary for the complete calculation of the 3-loop coefficient $A^{(3)}$. The IR structure of $A^{(3)}$ is explored for the graphs contributing at $\alpha_s^3 N_f$, which we calculate exactly and compare to Vogt’s [@vogt] parametrization. In the Appendix we list the Feynman rules for LCOPT. We will work in Feynman gauge, which further simplifies the expressions and reduces the number of diagrams, since then self-energies of light-like eikonal lines vanish. All our explicit calculations are performed in the $\MS$ scheme, but our method is applicable in any minimal subtraction scheme. Non-Singlet Parton Distribution Functions and Eikonal Cross Sections {#sectwebA} ==================================================================== Here we will show that the factorized form of a perturbative non-singlet parton-in-parton distribution function contains an eikonal cross section which absorbs all collinear and infrared singular behavior as $x \rightarrow 1$. Leading Regions {#powercount} --------------- We start with the definition Eq. (\[pdfdef\]) of a perturbative parton-in-parton distribution function, which is shown in Fig. \[partondef\] a) in cut-diagram notation. The part to the left of the cut (the vertical line) represents the amplitude, whereas the complex conjugate amplitude is drawn to the right of the cut. We pick the incoming momentum to flow in the plus direction, $p^\mu = (p^+,0,0_\perp)$. Because the minus and transverse momenta in (\[pdfdef\]) are integrated over, they can flow freely through the eikonal line, whereas no plus momentum flows across the cut in the eikonal line. \ \ We follow the method developed in [@bookTASI; @count] to find the regions in momentum space which can potentially contribute to the leading infrared singular behavior. These regions are shown in Fig. \[partondef\] b) in form of a reduced diagram where all off-shell momenta are contracted. Momentum regions which can contain singularities are found with the help of the Landau equations [@landau], which identify the singularities in Feynman integrals. However, not all regions which satisfy the Landau equations give leading contributions. The leading contributions are found by infrared power counting [@count]. We find the leading regions for the PDF which are associated with the following momentum configurations, scaled relative to the large momentum scale in the problem, $x p^+ \approx p^+$: - Soft momenta, which scale as $k^\mu \sim \lambda\, p^+, \, \lambda \ll 1$, in all components. The corresponding lines are denoted by $S$ in the figure. - Soft momenta with components which scale in the strongly ordered form $k^+ \sim \sigma \, p^+, k^- \sim \lambda \, p^+, k_\perp^2 \sim \lambda \, (p^+)^2$, or $k^- \sim \sigma\, p^+, k^+ \sim \lambda\, p^+, k_\perp^2 \sim \lambda\, (p^+)^2$, respectively, where $\sigma \ll \lambda \ll 1$. These are the so-called Glauber- or Coulomb momenta [@glauber; @coste]. - Momenta collinear to the momenta of initial or final state particles (including the eikonal). The corresponding subdiagrams are denoted as $J_m$, where $m$ labels the external momentum. Momenta collinear to particles moving in the plus direction scale as $k^+ \sim p^+, k^- \sim \lambda\, p^+, k_\perp^2 \sim \lambda\, (p^+)^2$, whereas momenta collinear to the minus direction behave as $k^- \sim p^+, k^+ \sim\lambda\, p^+, k_\perp^2 \sim \lambda\, (p^+)^2$. - These regions are connected by a hard scattering, $H$, where all momenta are far off-shell, and thus scale as $\sim \, p^+$ in all components. In our case we can have a jet $J_p$ collinear to the incoming momentum $p^+$, as well as an arbitrary number of jets $J_i$ emerging from the hard scattering. Furthermore, we can have momenta collinear to the eikonal moving in the minus direction, $\beta^-$, represented by $J_\beta$ in the figure. The jets can be connected by arbitrarily many soft gluons, $S$. Here and below, unless explicitly stated otherwise, we use the term “soft” for both soft and Glauber momenta. A further simplification of the leading behavior occurs when we perform the limit to $x \rightarrow 1$, in which we are interested here. Jet lines having a finite amount of plus momentum in the final state become soft, and only virtual contributions can have large plus momenta. This does not affect the jet collinear to the eikonal, since it is moving in the minus direction. Thus we arrive at the leading regions depicted in Fig. \[partondef\] c), with hard scatterings $H_L$ and $H_R$, which are the only vertices where finite amounts of momentum can be transferred, virtual jets $J_{p,\,L}$ and $J_{p,\,R}$ collinear to the incoming momentum $p^+$, a jet $J_\beta$ collinear to the eikonal $\beta^-$, connected via soft momenta, $S$. Here and below the subscripts $L$ and $R$, respectively, indicate that the momenta and functions are purely virtual, located to the left or to the right of the cut. Infrared power counting [@bookTASI; @count] shows now that the parton distribution function for $x \rightarrow 1$, as shown in Fig. \[partondef\] c), can at most diverge as one inverse power, which corresponds to the $\frac{1}{1-x}$ divergence we are expecting from Eq. (\[pff\]). Since the power counting is straightforward we just state the result here. To get the maximum degree of IR divergence the following conditions have to be fulfilled (see the figure): - No soft vectors directly attach $H_L$ or $H_R$ with $S$. - The jets and the soft part can only be connected through soft gluons, denoted by the sets $\left\{q_{L,\,m_L}\right\}$, $\left\{q_{R,\,m_R}\right\}$, and $\left\{\bar{q}_l\right\}$. - Arbitrarily many scalar-polarized gluons, $\left\{k_{L,\,i_L}\right\}$, $\left\{k_{R,\,i_R}\right\}$, $\{\bar{k}_{L,\,j_L}\}$ and $\{\bar{k}_{R,\,j_R}\}$ , attach the jets with $H_L$ and $H_R$, respectively. The barred momenta are associated with $J_\beta$, the unbarred with the $J_p$’s. - Exactly one scalar, fermion, or physically polarized gluon with momentum $k_L^\mu - \sum\limits_{i_L} k_{L,\,i_L}^\mu$, $k_R^\mu - \sum\limits_{i_R} k_{R,\,i_R}^\mu$, $\bar{k}_L^\mu - \sum\limits_{j_L} \bar{k}_{L,\,j_L}^\mu$ or $\bar{k}_R^\mu - \sum\limits_{j_R} \bar{k}_{R,\,j_R}^\mu$, respectively, connects each of the jets with the hard parts. The momenta $k_L^{\nu_L}$ ($k_R^{\nu_R}$) and $\bar{k}_L^{\rho_L}$ ($\bar{k}_R^{\rho_R}$) denote the *total* momenta flowing into $H_L$ ($H_R$) from $J_{p,\,L}$ ($J_{p,\,R}$) and $J_\beta$, respectively, that is, they are the sum of the scalar, fermion or physically polarized gluon momenta, and the scalar-polarized gluon momenta. In an individual diagram we can have only scalar-polarized gluons connecting the jets with the hard parts, and no scalar, fermion or physically polarized gluon. However, the sum of these configurations vanishes after application of the Ward identity shown in Fig. \[wardeik\] a). - The number of soft and scalar-polarized vector lines emerging from a particular jet is less or equal to the number of 3-point vertices in that jet. In summary, we have found that the regions in momentum space which give leading contributions may be represented as: f\_[f]{}\^[x 1]{}(x) & = & \_[C\_,C\_S]{} \_[i\_L,i\_R,j\_L,j\_R]{}\ & & \_[m\_L,m\_R,l]{} S\^[(C\_S)]{} ( {q\_[L,[m\_L]{}]{}\^[\_[k\_L]{}]{} }; {q\_[R,[m\_R]{}]{}\^[\_[k\_R]{}]{} };{|[q]{}\_l\^[\_l]{}} )\ & & H\_L ( k\_L\^[\_L]{} , {k\_[L,i\_L]{}\^[\_[i\_L]{}]{} }; |[k]{}\_L\^[\_L]{}, {|[k]{}\_[L,j\_L]{}\^[\_[j\_L]{}]{} } ) H\_R ( k\_R\^[\_R]{} , {k\_[R,i\_R]{}\^[\_[i\_R]{}]{} }; |[k]{}\_R\^[\_R]{}, {|[k]{}\_[R,j\_R]{}\^[\_[j\_R]{}]{} } )\ & & J\_[p,L]{} (k\_L\^[\_L]{}, {k\_[L,i\_L]{}\^[\_[i\_L]{}]{} }; {q\_[L,[m\_L]{}]{}\^[\_[m\_L]{}]{} } ) J\_[p,R]{} (k\_R\^[\_R]{}, {k\_[R,i\_R]{}\^[\_[i\_R]{}]{} }; {q\_[R,[m\_R]{}]{}\^[\_[m\_R]{}]{} } )\ & & J\_\^[(C\_)]{} (|[k]{}\_L\^[\_L]{}, {|[k]{}\_[L,j\_L]{}\^[\_[j\_L]{}]{} }; |[k]{}\_R\^[\_R]{}, {|[k]{}\_[R,j\_R]{}\^[\_[j\_R]{}]{} };{|[q]{}\_l\^[\_l]{}} ) \[nonfactor\]\ & & \^n(\_[m\_L]{} q\^\_[L,m\_L]{} + \_[m\_R]{} q\^\_[R,m\_R]{} + \_l |[q]{}\^\_[l]{} ) \^n(k\^\_L + |[k]{}\^\_L - x p\^) \^n(k\^\_R + |[k]{}\^\_R - x p\^)\ & & \^n( k\_L\^-\_[m\_L]{} q\^\_[L,m\_L]{} - p\^) \^n( k\_R\^- \_[m\_R]{} q\^\_[R,m\_R]{} - p\^). The sum in Eq. (\[nonfactor\]) runs over all cuts of jet $J_\beta$, $C_\beta$, and of the soft function $S$, $C_S$, which are consistent with the constraints from the delta-functions due to momentum conservation. The functions in (\[nonfactor\]) are still connected with each other by scalar polarized or soft gluons. This obscures the independent evolution of the functions. In the next subsection we will show how to simplify this result. Factorized Form of the Perturbative Distribution Function --------------------------------------------------------- Following [@pQCD], we will show here that the scalar polarized gluons decouple via the help of a Ward identity, and that we can disentangle the jets and the soft part, which are connected by soft gluon exchanges, via the so-called soft approximation. ### Decoupling of the Hard Part Starting from Eq. (\[nonfactor\]), we use the fact that the leading contributions come from regions where the gluons carrying momenta $\left\{k_{L,\,{i_L}}\right\}$, $\left\{k_{R,\,{i_R}}\right\}$, $\{\bar{k}_{L,\,{j_L}}\}$ and $\{\bar{k}_{R,\,{j_R}}\}$ are scalar polarized. Thus $H_L$, $H_R$, $J_{p,\,L}$, $J_{p,\,R}$, and $J_\beta$ have the following structure: H\_L & = & ( \_[i\_L]{} \^[\_[i\_L]{}]{} ) ( \_[j\_L]{} \^[\_[j\_L]{}]{} ) H\_[L, {\_[i\_L]{} },{\_[j\_L]{}} ]{} (k\_L\^+ \^[\_L]{}, {k\_[L,i\_L]{}\^+ \^[\_[i\_L]{}]{}}; |[k]{}\_L\^- \^[\_L]{}, {|[k]{}\_[L,[i\_L]{}]{}\^- \^[\_[j\_L]{}]{} } ),\ J\_[p,L]{} & = & ( \_[i\_L]{} \^[\_[i\_L]{}]{} ) J\_[p,L {\_[i\_L]{} } ]{} ( k\_L\^[\_L]{}, {k\_[L,i\_L]{}\^[\_[i\_L]{}]{} }; {q\_[L,[k\_L]{}]{}\^[\_[m\_L]{}]{} } ),\ J\_& = & ( \_[j\_L]{} \^[\_[j\_L]{}]{} ) ( \_[j\_R]{} \^[\_[j\_R]{}]{} ) J\_[ {\_[j\_L]{}},{\_[j\_R]{}}]{} ( |[k]{}\_L\^[\_L]{}, {|[k]{}\_[L,j\_L]{}\^[\_[j\_L]{}]{} }; |[k]{}\_R\^[\_R]{}, {|[k]{}\_[R,j\_R]{}\^[\_[j\_R]{}]{} } ;{|[q]{}\_l\^[\_l]{}}), where, as above, for the functions to the right of the cut, we replace the subscripts $L$ with $R$. In these relations the vectors \^& = & \_+\^ ,\ \^& = & \_[-]{}\^ \[lightdef\] are the light-like vectors parallel to $p^\mu$ and parallel to the direction of the eikonal, respectively. We now use the identity depicted in Fig. \[wardeik\] c), where the grey blob denotes the hard part. Fig. \[wardeik\] c) follows from the Ward identity shown in Fig. \[wardeik\] a), and the identity for scalar polarized gluons attaching to an eikonal line in Fig. \[wardeik\] b). The Ward identity says that the sum of all possible attachments of a scalar-polarized gluon to a matrix element vanishes. From this follows Fig. \[wardeik\] a), since, by definition, we do not include the graph into the hard function where the gluon attaches to the physically polarized parton (quark or gluon), shown on the right-hand side of Fig. \[wardeik\] a). The eikonal identity in Fig. \[wardeik\] b) follows trivially from the eikonal Feynman rules in Fig. \[Frules\]. Thus, since the right-hand sides of figures a) and b) are the same (the “empty” eikonal line carries no momentum), the left-hand sides are the same. Repeated application of this identity results in Fig. \[wardeik\] c) [@CSS]. Note that the color factors are included in the Ward identity, resulting in the appropriate color factor for the attachments of the gluons as shown in Fig. c). \ Using the identity in Fig. \[wardeik\] c) for all scalar polarized gluons in the sets $\left\{k_{L,\,{i_L}}\right\}$, $\left\{k_{R,\,{i_R}}\right\}$, $\{\bar{k}_{L,\,{j_L}}\}$ and $\{\bar{k}_{R,\,{j_R}}\}$, we achieve our goal of decoupling the jet functions from the hard function. This decoupling occurs in Feynman gauge only after summation over the full gauge-invariant set of graphs which contribute to the reduced diagram Fig. \[partondef\] c). The result is shown in Fig. \[partonfact1\]. The products over the vectors $\beta$ and $\xi$ are replaced by eikonal factors $\mathcal{E}$, which we can group with the jets. Furthermore, the hard scatterings, by definition far off-shell, become independent of $x$ up to corrections which vanish for $x \rightarrow 1$. Eq. (\[nonfactor\]) then becomes f\_[f]{}\^[x1]{}(x) & = & H\_L(p,;,) H\_R(p,;,) \_[C\_,C\_S]{} \_[m\_L,m\_R,l]{}\ & & \_[p,L]{}(p,;; {q\_[L,[m\_L]{}]{}\^[\_[m\_L]{}]{} } ) \_[p,R]{}(p,;; {q\_[R,[m\_R]{}]{}\^[\_[m\_R]{}]{} })\ & & dy dz S\^[(C\_S)]{} ( y p,; {q\_[L,[m\_L]{}]{}\^[\_[m\_L]{}]{} }; {q\_[R,[m\_R]{}]{}\^[\_[m\_R]{}]{} };{|[q]{}\_l\^[\_l]{}} ) \^[(C\_)]{}\_(z p, ;;{|[q]{}\_l\^[\_l]{}}) \[factform1\]\ & & \^n(\_[m\_L]{} q\^\_[L,m\_L]{} + \_[m\_R]{} q\^\_[R,m\_R]{} + \_l |[q]{}\^\_[l]{} ) (1-x-y-z) , where $\mu$ denotes the renormalization scale, which we set equal to the factorization scale, for simplicity. Corrections are subleading by a power of $1-x$. We define the functions $\tilde{J}_{p,\,L}$, $\tilde{J}_{p,\,R}$, and $\tilde{J}_\beta$ as follows: \_[p,L]{} & = & \_[i\_L]{} (, {k\_[L,i\_L]{}\^+} )\^[{\_[i\_L]{}}]{} J\_[p,L {\_[i\_L]{} } ]{} ( k\_L\^[\_L]{}, {k\_[L,i\_L]{}\^[\_[i\_L]{}]{} }; {q\_[L,[k\_L]{}]{}\^[\_[m\_L]{}]{} } )\ & & \^n (k\_L\^- \_[m\_K]{} q\^\_[L,m\_L]{} - p\^), \[Jp\]\ \^[(C\_)]{}\_& = & \_[j\_L,j\_R]{} (, {|[k]{}\_[L,j\_L]{}\^-} )\^[{\_[j\_L]{}}]{} \^[\*]{} (, {|[k]{}\_[R,j\_R]{}\^-} )\^[{\_[j\_R]{}}]{}\ & & J\^[(C\_)]{}\_[ {\_[j\_L]{}},{\_[j\_R]{}}]{} ( zp; |[k]{}\_L\^[\_L]{}, {|[k]{}\_[L,j\_L]{}\^[\_[j\_L]{}]{} }; |[k]{}\_R\^[\_R]{}, {|[k]{}\_[R,j\_R]{}\^[\_[j\_R]{}]{} } ; {|[q]{}\_l\^[\_l]{}}) , and $\tilde{J}_{p,\,R}$ is defined analogously to $\tilde{J}_{p,\,L}$, with the subscripts $L$ replaced by $R$, and with a complex conjugate eikonal line, since it is to the right of the cut. In (\[factform1\]) the total plus momentum flowing across the cut is restricted to be $(1-x) p^+$, and flows through the soft function and/or the eikonal jet. The plus momenta flowing across the cuts $C_S$ and $C_\beta$, denoted by $y p^+$ and $z p^+$, respectively, are therefore restricted to be $(1-x) p^+$ via the delta-function in Eq. (\[factform1\]). Above we have factorized the hard part from the remaining functions, which are still linked via soft momenta. ### Fully Factorized Form {#softapprox} Here we will use the soft approximation [@coste; @CSS; @back; @pQCD] to factorize the jets $J_p$ from the soft function, which, by power counting, are connected only through soft gluons. We could factorize the eikonal jet $J_\beta$ from $S$ in an analogous way, but we choose not to do so here because eventually we will combine all soft and eikonal functions to form an eikonal cross section. The soft approximation consists of two approximations: the neglect of non-scalar gluon polarizations, and the neglect of soft momenta $q^\mu$ in the numerator compared to jet-momenta $k^\mu$ and $k^2$ compared to $q \cdot k$ in the denominator. After making these approximations we proceed in a fashion similar to the previous subsection. Let us start by decomposing each soft gluon propagator $D_{\mu \nu}$ in the sets $\left\{q_{L,\,m_L}\right\}$ and $\left\{q_{R,\,m_R}\right\}$ into a scalar-polarized contribution and a remainder [@GraYen]: D\_ (q\_[L,m\_L]{}) & = & G\_ (q\_[L,m\_L]{},) + K\_ (q\_[L,m\_L]{},), \[kgdec\] and analogously for the momenta $\left\{q_{R,\,m_R}\right\}$ to the right of the cut, where we define K\_ (q\_[L,m\_L]{},) & & D\_ (q\_[L,m\_L]{}) ,\ G\_ (q\_[L,m\_L]{},) & & D\_ (q\_[L,m\_L]{}) - D\_ (q\_[L,m\_L]{}) , \[kgdef\] The sign of the $i \epsilon$-prescription is chosen in such a way as to not introduce new pinch singularities near the ones produced by the soft gluons under consideration. For momenta to the right of the cut in initial-state jets we also have $+i\epsilon$, since the sign of the momentum flow to the right of the cut is reversed relative to the jet momentum $k$. That is, we have for momenta to the left of the cut $(q+k)^2 + i \epsilon \approx 2 q \cdot k + i \epsilon$, whereas to the right of the cut we obtain $(q-k)^2 - i \epsilon \approx -2 q \cdot k - i \epsilon$. From Eq. (\[kgdef\]) we see, that for the scalar-polarized $K$-gluons the identity shown in Fig. \[wardeik\] c) is immediately applicable, leading to the desired factorized form. So it remains to be shown that the $G$-gluons do not give leading contributions. As mentioned in the previous subsection power-counting shows that only 3-point vertices are relevant for the coupling of soft gluons to jets. Let us consider a 3-point vertex, where a soft $G$-gluon with propagator $G_{\mu \nu}(q)$ couples to a fermion jet-line with momentum $k^\mu$ in the jet $J_p$ moving in the plus-direction: \^ G\_ (q,) 2 k\^G\_ (q,) 0, \[gapprox\] because $k^\nu \approx k^+ \xi^\nu$ with corrections proportional to $\lambda p^+, \lambda \ll 1$, as follows from the power counting described in Sect. \[powercount\], and because $ G_{\mu \nu} \left(q,\xi \right) \xi^\nu = 0$. An analogous observation holds for the coupling of $G$-gluons to jet-lines via triple-gluon vertices. In (\[gapprox\]) we neglect all terms of order $\lambda p^+$ in the numerator, including the momentum $q$, because we assume that the denominator also scales as $\sim \lambda p^+$. This approximation is only valid for soft gluons not in the Glauber or Coulomb region, where the denominator behaves as $\sim \lambda^2 p^+$. The remaining step in order to obtain a fully factorized form of the PDFs is therefore to show that the soft momenta are not pinched in this Glauber/Coulomb region. Then we can deform the integration contours over these momenta away from this dangerous region, into a purely soft region where the above approximations are applicable. Consider again the 3-point vertex in Eq. (\[gapprox\]), where now the gluon with momentum $q$ is in the Glauber/Coulomb region. If $|k^+ q^-|$ is not dominant over $|2 k_\perp \cdot q_\perp + q_\perp^2|$ in the denominator our approximation fails. The poles of the participating denominators are in the $q^-$ complex plane at q\^- & = & ,\ q\^- & = & - k\^- . As long as the jet-line $k$ carries positive plus momentum, we see that the $q^-$-poles are not pinched in the Glauber region. In this case we can deform the contour away from this region into the purely soft region, where $ |k^+ q^-| \gg |2 k_\perp \cdot q_\perp + q_\perp^2|$. In a reduced diagram, which represents a physical process, there must be at every vertex at least one line which flows into the vertex, and at least one line which flows out of the vertex. Thus, at every vertex, we can always find a momentum $k$ for which the above observation holds, and we can always choose the flow of $q$ through the jet to the hard scattering along such lines. Because the soft gluon momenta $\{q_{L,\,m_L}\}$, which we want to decouple, connect only to the purely virtual, initial-state jet $J_p$[^1], this observation remains true throughout the jet. An analogous argument applies to the right of the cut. In summary, soft gluons can be decoupled from initial state jets at leading power in $1-x$. We therefore arrive at the factorized form of the parton distribution function as $x \rightarrow 1$: f\_[f]{}\^[x1]{}(x) & = & H\_L(p,;,) H\_R(p,;,) |[J]{}\_[p,L]{}(p,;; ) |[J]{}\_[p,R]{}(p,;; )\ & & \_[C\_,C\_S]{} \_[m\_L,m\_R,l]{} (, {q\_[L,m\_L]{}\^-} )\^[{\_[m\_L]{}}]{} \^[\*]{} (, {q\_[R,m\_R]{}\^-} )\^[{\_[m\_R]{}}]{}\ & & dy dz S\^[(C\_S)]{} ( y p,; {q\_[L,[m\_L]{}]{}\^[\_[m\_L]{}]{} }; {q\_[R,[m\_R]{}]{}\^[\_[m\_R]{}]{} };{|[q]{}\_l\^[\_l]{}} ) \^[(C\_)]{}\_(z p, ;;{|[q]{}\_l\^[\_l]{}})\ & & \^n(\_[m\_L]{} q\^\_[L,m\_L]{} + \_[m\_R]{} q\^\_[R,m\_R]{} + \_l |[q]{}\^\_[l]{} ) (1-x-y-z) ,\[factform2\] where we have grouped the eikonal factors stemming from the soft approximation with the soft function and the eikonal jet. We define \_[p,L]{}(p,;; {q\_[L,[m\_L]{}]{}\^[\_[m\_L]{}]{} } ) = (, {q\_[L,m\_L]{}\^-} )\^[{\_[m\_L]{}}]{} |[J]{}\_[p,L]{}(p,;; ), \[bardef\] and analogously for the jet to the right of the cut, $J_{p,\,R}$, with a complex conjugate eikonal. ### Fully Factorized Form with an Eikonal Cross Section Although in Eq. (\[factform2\]) the various functions are clearly separated in their momentum dependence, the parton distribution function is not quite in the desired form yet. We want to write the PDF in terms of a color singlet eikonal cross section, built from ordered exponentials: \^[()]{}\_[a a]{}(,\_s(),) & = & e\^[i(1-x)p\^+ y\^-]{}\ & & , \[eiksigdef\] where the product of two non-Abelian phase operators (Wilson lines) in the representation $a$, for quarks, is defined as follows: \^[(aa)]{}(x) & = & \^[(a)]{}\_(,0;x) \^[(a)]{}\_(0,-;x),\ \^[(f)]{}\_(\_2,\_1;x) & = & P e\^[-i g \_[\_1]{}\^[\_2]{} d (+ x )]{}, where the light-like velocities $\xi$ and $\beta$ are defined in (\[lightdef\]), and where ${\mathcal{A}}^{(f)}$ is the vector potential in the representation of a parton with flavor $f$. The trace in (\[eiksigdef\]) is over color indices. The lowest order of the eikonal cross section is normalized to $\delta(1-x)$. This eikonal cross section has ultraviolet divergences which have to be renormalized, as indicated by the renormalization scale $\mu$. Furthermore, the delta-function for the soft momenta in Eq. (\[factform2\]) constrains the momentum of the final state in $\sigma^{(\mbox{\tiny eik})}$ to be $(1-x)p^+$. We can factorize the eikonal cross section (\[eiksigdef\]) in a manner analogous to the full parton distribution function, and obtain \^[()]{}\_[aa]{}(1-x) & = & \_[C\_,C\_S]{} \_[m\_L,m\_R,l]{}\ & & \^[()]{}\_[p,L]{}(,;; {q\_[L,[m\_L]{}]{}\^[\_[m\_L]{}]{} } ) \^[()]{}\_[p,R]{}(,;; {q\_[R,[m\_R]{}]{}\^[\_[m\_R]{}]{} } )\ & & dy dz S\^[(C\_S)]{} (yp,; {q\_[L,[m\_L]{}]{}\^[\_[m\_L]{}]{} }; {q\_[R,[m\_R]{}]{}\^[\_[m\_R]{}]{} };{|[q]{}\_l\^[\_l]{}} ) \^[(C\_)]{}\_(zp,;; {|[q]{}\_l\^[\_l]{}})\ & & \^n(\_[m\_L]{} q\^\_[L,m\_L]{} + \_[m\_R]{} q\^\_[R,m\_R]{} + \_l |[q]{}\^\_[l]{} ) (1-x-y-z). \[eikfact\] The eikonal jets $\tilde{J}^{(\mbox{\tiny eik})}_{p,\,L}$, $\tilde{J}^{(\mbox{\tiny eik})}_{p,\,R}$ moving collinear to the momentum $p$, are defined analogously to Eq. (\[Jp\]), with the fermion line carrying momentum $p$ replaced by an eikonal line in representation $a$ with velocity $\beta$. We can define analogous to Eq. (\[bardef\]) \^[()]{}\_[p,L]{}(,;; {q\_[L,[m\_L]{}]{}\^[\_[m\_L]{}]{} } ) = (, {q\_[L,m\_L]{}\^-} )\^[{\_[m\_L]{}}]{} |[J]{}\^[()]{}\_[p,L]{}(;; ), \[bardefeik\] and similarly for the jet to the right of the cut, $J_{p,\,R}^{(\mbox{\tiny eik})}$, with a complex conjugate eikonal. In the following we suppress the index $a$ for better readability. Combining Eqs. (\[factform2\]), (\[eikfact\]), and (\[bardefeik\]), we arrive at the final form of the factorized parton distribution function, shown in Fig. \[partonfactend\], f\_[f]{}\^[x1]{}(x) & = & H\_L( p, ) H\_R(p, ) J\_[p,L]{}\^R (p,) J\_[p,R]{}\^R (p,) \^[()]{} ((1-x)p,), \[finalform\] suppressing the dependence on the lightlike vectors (\[lightdef\]). The purely virtual jet-remainders are defined by J\_[p,L]{}\^R (p,) = . \[jetvirdef\] The contributions to the various functions in Eq. (\[finalform\]) can be extracted from any Feynman diagram by examining each region of integration separately. In the following, we only need the eikonal cross section in the form (\[eiksigdef\]), as well as the $x$-dependence and renormalization properties of the PDF and the terms constituting its factorized form. These properties are the subject of the next subsection. Renormalization of Parton Distribution Functions ------------------------------------------------ As was shown in [@pdfs], the parton distribution functions, defined in their unrenormalized form in terms of nonlocal operators (\[pdfdef\]), obey the evolution equation (\[evol\]), where the kernel $P_{ab}$ is found from the usual relation [@pdfs; @bookTASI] P\_[ff]{}(\_s,x) = A\_f(\_s) (x) \_+ + …= - g Z\^A\_1 \_+ + …, \[prela\] where $g^2/(4 \pi) = \alpha_s$, and where $\ln\, Z^A_1$ denotes the $\frac{1}{\varepsilon}$-pole of the counterterm which multiplies the plus-distribution, plus scheme dependent constants, if we work in a minimal subtraction scheme with dimensional regularization. Above we only exhibit the term which is singular as $x \rightarrow 1$, since it is this term which we want to extract from the renormalization of our factorized form, Eq. (\[finalform\]). From Eq. (\[finalform\]) we observe that only the eikonal cross section can contribute to the $A$-term proportional to a plus-distribution. This is because the hard functions are off-shell by $\mathcal{O}\left(x p^+\right)$, and the jet-remainders are purely virtual, thus cannot contain plus-distributions. Therefore, their renormalization has to be proportional to $B_f \, \delta(1-x)$, as was observed in [@EGW]. It is thus the renormalization of a color singlet eikonal vertex which we have to study, in order to compute the coefficients $A^{(n)}$ in (\[aaa\]). Although there are, aside from the usual QCD divergences, additional divergences at the eikonal vertex, this is a significant simplification compared to the investigation of the UV behavior of the full PDF. Let us now study the eikonal cross section as defined in Eq. (\[eiksigdef\]), whose perturbative expansion follows the Feynman rules shown in Fig. \[Frules\]. Nonabelian Eikonal Exponentiation {#sectexp} ================================= Although the eikonal approximation simplifies the perturbative calculation already significantly, there are still many diagrams to be calculated at each order. In addition, the eikonal approximation also introduces new infrared divergences. An observation first made by Sterman [@George], then proved by Gatheral [@gath], and Frenkel and Taylor [@freta] solves both problems. This theorem states that a cross section $X$ with two eikonal lines in a nonabelian theory exponentiates, $$\sigma^{(\rm eik)} \equiv X = e^{Y}, \label{eq1}$$ where $Y$ can be given a simple recursive definition. In this section we will recall the proof of Eq. (\[eq1\]) [@gath; @freta], for the sake of completeness, including a few illustrative examples which will be used below for the calculation of the 2-loop coefficient $A^{(2)}$. The exponent $Y$ in (\[eq1\]) has the following properties: 1. $Y$ is a subset of the diagrams contributing to $X$, which we will call “webs” in the following, since, as we will see below, their lines “ \[...\] are all nested \[...\] in a spider’s web pattern” [@George]. 2. The color weights of the diagrams in $Y$ are in general different from those in $X$. 3. For Eq. (\[eq1\]) to hold, the phase-space region should be symmetric in the real gluon momenta. Below we will outline the arguments necessary to prove this theorem. The proof relies on the recursive definition of color-weights and on the iterated application of a well-known eikonal identity. Then we go on to show that IR and UV subdivergences cancel in the exponent at each order. Proof of Exponentiation ----------------------- ### Some Terminology {#defs} In order to specify which subset of diagrams of the original perturbation series $X$ contributes to the exponent $Y$ we need to introduce some terminology. Each diagram will be decomposed into its color part and its Feynman integral in the eikonal approximation. The eikonal Feynman rules were given above in Fig. \[Frules\]. The color part can be represented graphically in a diagram which is similar to an ordinary Feynman diagram, but the vertices represent the color part of the Feynman rules, i.e. the vertices are just the $T^a_{ij}$s and $i\, f_{ijk}$s, for quark or gluon, respectively, and the lines are $\delta_{ij}$s. In addition, all soft lines have to be drawn inside the (cut) eikonal loop for reasons which will become clear shortly. Certain color diagrams are related to each other by use of the commutation relations of the $T^a_{ij}$s and $i\, f_{ijk}$s (Jacobi identity) which are graphically represented in Fig. \[commrel\]. & = & i f\_[abc]{} T\^c\ f\_[ilm]{} f\_[mjk]{} &+ & f\_[jlm]{} f\_[imk]{} + f\_[klm]{} f\_[ijm]{} = 0. \[jacid\] As mentioned above, the diagrams contributing to $Y$ will be called “webs” [@George]. Originally [@gath], a web was defined as a set of gluon lines which cannot be partitioned without cutting at least one of its lines. As already stated above, all soft lines are to be drawn inside the eikonal loop(s). However, at ${\mathcal{O}}\left(\alpha_s^3\right)$ new types of diagrams arise which Frenkel and Taylor [@freta] called connected webs (“c-webs”). c-webs are not included in the original definition for the following reason: If one cuts the horizontal gluon line of the c-web drawn in Fig. \[webexample\] one would get two webs consisting of three-point vertices since real and virtual gluon lines are treated on equal footing in a color-weight diagram. Below we will refer to webs and c-webs just as webs. The definitions given by Gatheral, and Frenkel and Taylor can be unified by the following *definition*: A web is a (sub)diagram consisting of soft gluon lines connecting two eikonal lines which cannot be partitioned into webs of lower order by cutting all eikonal lines exactly once. Stated differently, webs are two-eikonal irreducible diagrams. The *order of a web* is defined to be equal to the powers of $\alpha_s$ it contains, e.g. a web of ${\mathcal{O}}\left(\alpha_s^2\right)$ will be called a web of order 2. Diagrammatic examples are shown in Fig. \[webexample\]. A web has a color factor $\overline{C}$ and a Feynman integral part $\mathcal{F}$. $\mathcal{F}$ contains only those eikonal propagators which are *internal* to the web. The color weight is in general different from the one which one would get from the usual Feynman rules. The color weight of a web of order $m$ is recursively defined as $$\begin{aligned} \overline{C}(W^{(m)}) & \equiv & \frac{1}{\mbox{Tr } \mathbf{1}} C(W^{(m)}) - \sum_d \prod\limits_{n_i} \overline{C} (W_{n_i}^{(i)}), \nonumber \\ \overline{C}(W^{(1)}) & \equiv & \frac{1}{\mbox{Tr } \mathbf{1}} C(W^{(1)}) , \label{colwe}\end{aligned}$$ where $C(W^{(m)})$ is the ordinary color factor, $\frac{1}{\mbox{\tiny Tr } \mathbf{1}}$ is the usual normalization (see Eq. (\[eiksigdef\])), $C^{(0)} = \mbox{Tr } \mathbf{1}$, $\sum_d$ is the sum over the set of all non-trivial decompositions $d$ of $W^{(m)}$ into webs of order $i < m$, and $\prod\limits_{n_i}$ denotes the product of all webs $n_i$ of order $i$ ($1 \leq i < m$) in a particular decomposition $d$. The set of all non-trivial *decompositions* of a given web can be obtained by successively disentangling crossed gluon lines in the web by repeated application of the color identities given in Fig. \[commrel\]. In [@gath] Gatheral showed that webs in the original definition have what he called “maximally nonabelian” color weights $\sim \alpha_s^m C_F C_A^{m-1}$, where the $C_i$s are the Casimir factors in the fundamental and adjoint representation, respectively. This statement, however, is misleading at orders $> \alpha_s^3$ [@freta]. We will see an example below, in the calculation of the $N_f$ term contributing to the coefficient $A$ at three loops. In the following subsection we will clarify the above definitions in an example which shows how to factorize eikonal Feynman diagrams into sums of products of webs. This then leads directly to exponentiation. The recursive definition of the color weights of the webs ensures the factorization of the color parts. For the factorization of the Feynman eikonal integrals $\mathcal{F}$ we will make repeated use of the eikonal identity [@levsuch] $$\frac{1}{p \cdot k_1} \frac{1}{p \cdot (k_1 + k_2)} + \frac{1}{p \cdot k_2} \frac{1}{p \cdot (k_1 + k_2)} = \frac{1}{p \cdot k_1} \frac{1}{p \cdot k_2} \label{eikid},$$ illustrated in Fig. \[rules\] a). This identity can be extended to an arbitrary number of soft gluons in a straightforward way by repeated application of Eq. (\[eikid\]): For two webs $W_1$ and $W_2$ with gluon legs $k_i \,(i = 1,\dots,m)$ and $l_j\,(j=1,\dots, n)$ attached to an eikonal line with velocity $p$ the generalized identity reads $$\begin{aligned} {{\mathcal{F}}}(W_1) {{\mathcal{F}}}(W_2) & \sim & \frac{1}{p \cdot k_1 \, p \cdot (k_1 + k_2) \dots p \cdot (k_1 + \dots + k_m)} \frac{1}{p \cdot l_1 \, p \cdot (l_1 + l_2) \dots p \cdot (l_1 + \dots + l_n)} \nonumber \\ & = & \sum\limits_{\mbox{\tiny perms} (n,m)} F, \label{gen2eikid}\end{aligned}$$ where the sum is over all Feynman diagrams $F$ obtained by permuting the $n+m$ gluon lines such that the order of the $k_i$, and $l_j$, respectively, *within* each web is not changed. A simple example is shown in Fig. \[rules\] b). The extension to more than two webs follows by repeating the above argument: $$\sum\limits_{F \, \mbox{\tiny in } d} F = \prod\limits_{n_i} {\mathcal{F}} \left( W_{n_{i}}^{(i)} \right). \label{geneikid}$$ ### An Example {#examplesubsect} We will show by induction that the terms in the perturbation series $X$ in Eq. (\[eq1\]), normalized by the zeroth order contribution, can be reorganized into a sum of products of webs which can be rewritten as $\exp(Y)$. Therefore it is necessary and also instructive to start with the first nontrivial example, diagrams of ${\mathcal{O}}(\alpha_s^2)$. At this order we have the Feynman diagrams shown in Fig. \[order2\], for quark or antiquark eikonal lines, excluding eikonal and gluon self-energies. Eikonal self-energies vanish if we work in Feynman gauge. The sum of diagrams at a given order in $\alpha_s$ is gauge invariant, of course. The contribution of the first two terms in Fig. \[order2\] can be rearranged by applying Eqs. (\[colwe\]) and (\[eikid\]) as shown in Fig. \[rearr\]. \ Thus we arrive at the following series obtained by rearranging the expansion of $X$ up to ${\mathcal{O}}(\alpha_s^2)$: $$\begin{aligned} X & = & \mathbf{1} + \sum\limits_{\stackrel{\mbox{\tiny all webs }}{\mbox{\tiny of order } 1}} \overline{C} \left(W^{(1)} \right) {\mathcal{F}} \left(W^{(1)} \right) + \nonumber \\ & + & \frac{1}{2 !} \bigg( \sum\limits_{\stackrel{\mbox{\tiny all webs }}{\mbox{\tiny of order } 1}} \overline{C} \left(W^{(1)} \right) {\mathcal{F}} \left(W^{(1)} \right) \bigg)^2 + \sum\limits_{\stackrel{\mbox{\tiny all webs }}{\mbox{\tiny of order } 2}} \overline{C} \left(W^{(2)} \right) {\mathcal{F}} \left(W^{(2)} \right) + \dots, \label{exporder2}\end{aligned}$$ which is illustrated in Fig. \[expgraph\]. The combinatorical factor $\frac{1}{2!}$ is necessary to avoid overcounting since two webs with the same structure are indistinguishable if the integration measure is symmetric in the real gluon momenta. ### Exponentiation Looking at the above example it is now clear that any Feynman diagram with two eikonal lines can be expressed as a sum of products of webs by applying Eqs. (\[colwe\]) and (\[geneikid\]) repeatedly. By induction we arrive at the following equation for the set of Feynman diagrams of ${\mathcal{O}}(\alpha_s^n)$, $F^{(n)}$: $$F^{(n)} = \sum\limits_{\left\{ n_i \right\} } \prod_i \frac{1}{n_i !} \bigg( \sum\limits_{\stackrel{\mbox{\tiny all webs}}{\mbox{\tiny of order }i}} \overline{C}(W^{(i)}) {\mathcal{F}}(W^{(i)}) \bigg)^{n_i}, \label{nfac}$$ where $i$ labels the order of the webs and the sum is over all sets $\left\{ n_i \right\},\, 0 \leq n_i < \infty$ such that $\sum_i i \, n_i = n$. For example, at ${\mathcal{O}} (\alpha_s^3)$ we can have $n_1 = 3$ webs of order 1 ($\{3,0,\dots\}$), or $n_1 = 1$ webs of order 1 and $n_2 = 1$ webs of order 2 ($\{1,1,0,\dots\}$), or $n_3 = 1$ webs of order 3 ($\{0,0,1,0,\dots\}$). The combinatorical factor of $\frac{1}{n!}$ is needed to avoid overcounting because of property 3.) of $X$, stated in the introduction to this section, namely that the integration measure is symmetric in the real gluon momenta, for example ${\mathcal{F}}^{(1)}(k_1) {\mathcal{F}}^{(2)}(k_2, k_3) = {\mathcal{F}}^{(1)}(k_2) {\mathcal{F}}^{(2)}(k_1, k_3)$, which means that webs of the same structure are indistinguishable. Were property 3.) not fulfilled, the perturbation series would not exponentiate. We now rearrange the original perturbation series given in powers of $\alpha_s^n$ $$X = \sum\limits_{n=0}^\infty F^{(n)}, \label{ser}$$ $$\begin{aligned} X & = & \sum\limits_{n=0}^\infty \sum\limits_{\left\{ n_i \right\} } \delta_{n \sum_i i n_i} \prod_i \frac{1}{n_i !} \bigg( \sum\limits_{\stackrel{\mbox{\tiny all webs}}{\mbox{\tiny of order }i}} \overline{C}(W^{(i)}) {\mathcal{F}}(W^{(i)}) \bigg)^{n_i} \nonumber \\ & = & \sum\limits_{\stackrel{\mbox{\tiny all possible }}{ \{n_i\}}} \prod_i \frac{1}{n_i !} \bigg( \sum\limits_{\stackrel{\mbox{\tiny all webs}}{\mbox{\tiny of order }i}} \overline{C}(W^{(i)}) {\mathcal{F}}(W^{(i)}) \bigg)^{n_i} \nonumber \\ & = & \prod_i \left\{ \sum_{n_i} \frac{1}{n_i !} \bigg( \sum\limits_{\stackrel{\mbox{\tiny all webs}}{\mbox{\tiny of order }i}} \overline{C}(W^{(i)}) {\mathcal{F}}(W^{(i)}) \bigg)^{n_i} \right\} \nonumber \\ & = & \prod\limits_{i} \exp \bigg( \sum\limits_{\stackrel{\mbox{\tiny all webs}}{\mbox{\tiny of order }i}} \overline{C}(W^{(i)}) {\mathcal{F}}(W^{(i)}) \bigg),\end{aligned}$$ where we have used the fact that for any function $f(n_i,i)$ \_ \_i f(n\_i,i) = \_i \_[n\_i]{} f(n\_i,i) which is easy to see by comparing the expansions of the left and the right hand sides. So the series exponentiates $$X = e^Y, \quad Y \equiv \sum\limits_{i} \bigg( \sum\limits_{\stackrel{\mbox{\tiny all webs}}{\mbox{\tiny of order }i}} \overline{C}(W^{(i)}) {\mathcal{F}}(W^{(i)}) \bigg). \label{exp2eq}$$ This completes the proof that eikonal cross sections with two eikonal lines can be written as an exponent of an infinite sum of webs. Cancellation of Subdivergences in the Exponent {#wardidproof} ---------------------------------------------- Gatheral, Frenkel, and Taylor showed in [@gafreta] by explicit fixed-order calculations that infrared/ collinear subdivergences cancel in the exponent. Here we will outline the proof of this cancellation, as well as of the cancellation of UV subdivergences involving the eikonal vertex, to all orders with the help of the identities in Fig. \[wardeik\] in the soft approximation. The remaining UV subdivergences are removed via ordinary QCD counterterms, and thus an additional investigation of the renormalizability of the eikonal vertex is unnecessary. To show the absence of subdivergences, let us rewrite Eq. (\[nfac\]) as \_[ n]{} |[C]{}\^[(n)]{} \^[(n)]{} = F\^[(n)]{}\_ + F\^[(n)]{}\_ - \_[{ n\_i }, i &lt; n ]{} \_i ( \_ (W\^[(i)]{}) (W\^[(i)]{}) )\^[n\_i]{}. \[eqsubfact\] Eq. (\[eqsubfact\]) means, that the sum of all webs at order $n$ are given by the original perturbation series at that order where all lower-order webs have been subtracted out. The original perturbation series can be classified into terms without subdivergences, denoted by $F^{(n)}_{\mbox{\tiny conv}}$, and terms which contain subdivergences, $F^{(n)}_{\mbox{\tiny div}}$. In eikonal cross sections, infrared/collinear divergences stem from the same momentum configuration as UV divergences. Since eikonal cross sections are scaleless, when a line becomes collinear to an eikonal it can carry infinite momentum in a light-like direction. But in this case we can employ the soft approximation described in Sect. \[softapprox\] to factorize these jet-like configurations from the rest of the eikonal cross section. The reasoning follows Section \[softapprox\], and we arrive at the equality shown in Fig. \[subfact\]. The grey oval in the figure represents a specific jet-like configuration, collinear to one of the eikonal lines. The displayed equality states that the sum of all webs at a given order, where this jet-like configuration is connected to the rest of the eikonal cross section by soft gluons, can be expressed in the factored form shown on the right-hand side. As in Section \[softapprox\], remainders are non-leading. Due to the definition of the color weights (\[colwe\]), the right-hand side does not constitute a web of the same order, but rather a product of webs of lower orders. The contribution shown on the left-hand side of Fig. \[subfact\] is a contribution to $F^{(n)}_{\mbox{\tiny div}}$ of Eq. (\[eqsubfact\]). In (\[eqsubfact\]), however, we subtract out all products of webs of lower orders, thus cancelling the divergent contributions because of the equality shown in Fig. \[subfact\]. Using the equality in Fig. \[subfact\] and Eq. (\[eqsubfact\]) recursively for every IR/collinear subdivergence, we see that the sum of webs at a given order is free of such subdivergences. To summarize, the collinear configuration does not contribute at the order under consideration after summing over all relevant webs at that order, because this collinear configuration has already been taken into account at a lower order. The only possible collinear and UV vertex divergence can occur in the final, overall integral. Of course, in the original perturbative expansion $X$ of the eikonal cross section in Eq. (\[ser\]) these collinear and UV subdivergent configurations contribute, but in the exponent $Y$ of Eq. (\[exp2eq\]) they only appear as overall divergences. The Method for Obtaining $A^{(n)}_f$ {#sectmeth} ==================================== Renormalization of Webs ----------------------- As was observed in the above sections, webs have at most one collinear/IR divergence, and one overall UV divergence from integration over the final momentum. Using the invariance of eikonal cross sections under rescalings of the eikonal velocities, and the fact that there is at most one overall IR divergence, we can deduce the dependence of the webs on its overall momentum $k$, and on the light-like eikonal momenta. For a color singlet eikonal cross section we have $$W_{aa} \left( k^2, \frac{(k\cdot \beta_1) (k \cdot \beta_2)}{\beta_1 \cdot \beta_2}, \mu^2, \alpha_s(\mu^2), \varepsilon \right) = W_{aa} \left(k^2,k^2 + k_\perp^2, \alpha_s(\mu^2),\varepsilon \right),$$ after integration over internal momenta, in a frame where the eikonal lines are back-to-back [@EGW]. For definiteness, we pick the incoming line $\xi$ moving in the plus direction, and the outgoing eikonal $\beta$ in the minus direction, and since quantities built from eikonal lines are scaleless, we can scale the eikonal velocities to 1. Recall (\[lightdef\]): & = & ( 1,0,0\_)\ & = & ( 0,1,0\_) \[betaframe\] in light-cone coordinates. This choice will simplify the calculations considerably, as we will see below. We can therefore write the contributions from virtual webs of order $n$ to the eikonal cross section as 2 \_0\^ & dk\^2 & W\_[aa]{}\^[(n)]{} (k\^2,k\^2 + k\_\^2, \_s(\^2),)\ & = & 2 |[C]{}\_a\^[(n)]{} ( )\^n (\^2)\^[l ]{} (4 )\^[l ]{} K() \_0\^, \[methodeq\] where $l$ is an integer $\leq n$, and $K$ contains numerical factors (including factors of $\pi$) and is, in general, a function of $\varepsilon$ due to the regulation of infrared and UV (sub)divergences. Above, on the left hand side, all internal momenta have been integrated over, as well as $k^-$, and internal UV divergences have been renormalized. The integration over $k^2$ results in terms $\sim \frac{1}{(k_\perp^2)^{(l-1)\varepsilon}}$. For graphs including (local) counterterms $l < n$, whereas for graphs with $n$ loops $l = n$. Both virtual webs and their complex conjugates contribute to the overall factor of 2. The structure of the integral over $k^+$ follows from boost invariance. In Eq. (\[methodeq\]), this integral is divergent, but these divergences cancel against the corresponding real contributions, and therefore do not affect the anomalous dimension of the eikonal vertex. The $k^+$-integral plays the role of $\frac{dx}{1-x}$ for the full parton-in-parton distribution functions (cf. Eq. (\[eiksigdef\])). The final scaleless $k_\perp$ integral provides the $n$-loop UV counterterm which contributes to the anomalous dimension $P_{ff}$, Eq. (\[pff\]). To isolate the UV pole we temporarily introduce a mass = \^[1-]{} (m\^2 )\^[-l ]{}. \[scalepart\] The counterterm is then given, as usual, by minus the pole terms after expanding in $\varepsilon$. After summing over the contributions of all webs at a given order and their counterterms for subdivergences, all nonlocal terms ($\sim \ln \frac{\mu^2}{m^2}$) cancel as well as UV vertex counterterms and IR divergences, and we obtain the $n$-loop counterterm contributing at $x \rightarrow 1$, which can be written as a series in $\varepsilon$: Z\^[(n)A]{} = \_[m = 1]{}\^n ( )\^n a\_m\^[(n)]{} \_0\^, with purely numerical coefficients $a_m^{(n)}$. Because webs exponentiate, the counterterm for UV divergences in the perturbative expansion of a non-singlet parton distribution is given by Z\^A = { \_[m = 1]{}\^\_[n = 1]{}\^m ( )\^n a\_m\^[(n)]{} \_0\^ } \[counterterm\] in the limit $x \rightarrow 1$, as indicated by the superscript $A$. Now it is trivial to extract the contribution to $P_{ff}$. From (\[prela\]) and (\[counterterm\]) we get A\_f\^[(n)]{} = - n a\_1\^[(n)]{}. \[aendeq\] As emphasized above, internal UV divergences, including the usual QCD divergences and divergences at the eikonal vertex, have to be renormalized. Further complications arise because collinear/IR divergences cancel only after summing over all diagrams at a given order, so an individual diagram has in general UV singularities multiplying IR/collinear singularities. Our method to resolve these technical problems is most transparent in light-cone ordered perturbation theory (LCOPT) [@changma; @kosop], which is equivalent to performing all minus integrals of all loops. The Feynman rules for LCOPT are given in Appendix \[Lrules\] for completeness. Summary of the Method in LCOPT ------------------------------ Our method can be summarized as follows, details will be given below: 1. We start with the expressions in LCOPT for the set of webs at a given order with a fixed coupling. The number of web-diagrams is much less than the number of all possible diagrams at a given order. Moreover, since we work in Feynman gauge, the number of possible webs is further reduced. For example, at order 2, as we will see below, only three diagrams contribute, aside from gluon self-energies. 2. Ultraviolet divergent internal $k_{\perp,\,i}$-integrals are regularized via dimensional regularization, with $\varepsilon > 0$. At this stage we do not yet encounter IR/collinear singularities since all integrals over transverse momenta are performed at fixed plus momenta. 3. We add the necessary QCD counterterms and the counterterms for the eikonal vertex which has to be renormalized as a composite operator. As we showed in Section \[wardidproof\], the sum of the latter cancels because of the recursive definition of webs and a Ward identity. However, in the intermediate stages the vertex counterterms are necessary to make individual diagrams UV finite. 4. After elimination of the UV singularities we dimensionally continue to $\varepsilon < 0$ to regulate the IR/collinear plus-integrals. It follows from the rules for LCOPT that all internal plus momenta are bounded by the total $k^+$ flowing into the minus eikonal. Therefore, the integrals over these internal plus momenta give no UV subdivergences. 5. When we sum over the set of diagrams at a given order the IR divergent parts cancel, as well as the UV counterterms for the vertex, thus also the internally UV divergent vertex parts cancel. 6. The final scaleless $k_\perp$ integral provides the UV counterterm contributing to the anomalous dimension (see Eqs. (\[methodeq\])-(\[aendeq\]) ). The rules for LCOPT can be found in Appendix \[Lrules\] [@changma; @kosop], they can be derived by performing the minus integrals. LCOPT is similar to old-fashioned, or time-ordered, perturbation theory, but ordered along the light-cone, $x^+$, rather than in $x^0$. In a LCOPT diagram all internal lines are on-shell, in contrast to a covariant Feynman diagram, which allows us to identify UV divergent loops in eikonal diagrams more easily. A covariant Feynman diagram is comprised of one or more LCOPT diagrams. We start by writing down all light-cone ordered diagrams of a given covariant Feynman diagram. All momenta in crossed gluon ladders have to be chosen independent of each other, such that they all flow through the eikonal vertex, since we seek the anomalous dimension for this vertex. Because $\xi$ has no minus-component (cf. Eq. (\[betaframe\])), we have $q^- = 0$ in Eq. (\[denom\]) of the appendix when applying the Feynman rules for LCOPT in our case. This can be depicted graphically by contracting all propagators on the minus-eikonal (here $\beta$) to a point, which coincides with the eikonal vertex. Two-loop examples can be found in Fig. \[2loopfig\]. Sometimes, numerators stemming from triple-gluon vertices or quark propagators cancel the corresponding propagators on the plus-eikonal ($\xi \cdot k = k^-$). Graphically, this can again be described by contracting these propagators to a point. Then we can read off easily from the various light-cone ordered diagrams the analytical expressions, whose $k_{\perp,\,i}$-integrals we perform, and renormalize. We need QCD counterterms and counterterms for the effective vertex. More specifically, by QCD counterterms we mean the usual gluon self-energy counterterms, as well as the counterterms for triple-gluon vertices and eikonal-gluon-eikonal vertices. The latter are UV divergent in any covariant diagram, however, this is not necessarily the case for all LCOPT diagrams found from a covariant diagram. Examples will be given below. Self-energies of the light-like eikonal lines vanish in Feynman gauge. Both types of counterterms are found via the (recursive) BPHZ-formalism, and the subdivergences are identified with the help of naive power-counting on a graph-by-graph basis. We will now illustrate our method by the rederivation of the 1- and 2-loop $A$-coefficients. Calculation of the 1- and 2-loop Coefficients $A^{(1)}_f$, $A^{(2)}_f$ {#sectexample} ---------------------------------------------------------------------- The well-known [@2loopknown] coefficients of the collinear parts of the splitting functions to one and two-loop order are given by $$\begin{aligned} A_a^{(1)} & = & C_a, \label{a1number} \\ A_a^{(2)} & = & \frac{1}{2} C_a K \equiv \frac{1}{2} C_a \left[ C_A \left( \frac{67}{18} - \frac{\pi^2}{6} \right) - \frac{10}{9} T_R N_f \right], \label{a2number}\end{aligned}$$ where $C_q = C_F, \, C_g = C_A$, $N_f$ is the number of fermions, and $T_R$ determines the normalization of the generators of fermion representation $R$, $T_F = \frac{1}{2},\, T_A = N_c$, with $N_c$ the number of colors. We will now apply our method to the recalculation of these coefficients. The only web at order 1 in Feynman gauge is a single gluon exchanged between the two eikonal lines. The color weight is $\overline{C}^{(1)} = C_a$ by definition (\[colwe\]), and the web has no internal momenta. Straightforwardly we obtain 2 W\_[aa]{}\^[(1)]{} = \^[(1)]{} ( ) ( )\^(4 )\^ () \_0\^. \[1loopcalc\] So at lowest order we get from Eq. (\[aendeq\]) $A_a^{(1)} = \overline{C}^{(1)} = C_a$, as in (\[a1number\]). At order 2 we have the webs shown in Fig. \[2loops\], where we rotated the eikonal lines in the figure compared to the diagrams shown in Section \[sectexp\], to make the connection to Figs. \[partondef\] a) and \[partonfactend\] more evident. The original color factors are (compare to Fig. \[order2\]) C(W\_b) & = & (C\_F - ) C\_a,\ C(W\_c) & = & C(W\_d) = - C\_a \[2loopcolwe\] for eikonal lines in the $a$-representation. The respective color weights of the webs c) and d) are the same as the original color factors, since they do not have decompositions (these diagrams are “maximally nonabelian”). The decomposition of diagram b) was shown as an example in Section \[examplesubsect\], which resulted in a color weight |[C]{}(W\_b) = - C\_a. \[colweb\] The contribution of Fig. \[2loops\] a) is easily found from Eq. (\[1loopcalc\]) and the well-known finite terms (see e.g. [@bookTASI]) after renormalization of the of the gluon-self energy in the $\MS$ scheme, which is an example of what we called a QCD renormalization in the previous subsection: A\_a\^[(2),a)]{} = C\_A C\_a + C\_A C\_a - T\_R N\_f C\_a = ( C\_A - T\_R N\_f ) C\_a. \[2loopgluon\] The first term in the first equality in Eq. (\[2loopgluon\]) stems from the gluon loop, the second term from the ghost loop. The last term is obviously the fermion loop contribution found from the expression for the graph, 2 W\_[aa,a)N\_f]{}\^[(2)]{} = - T\_R N\_f C\_a ( )\^2 ( )\^[2 ]{} (4 )\^[2]{} 2 B(2-,2-) \_0\^, \[bubblenf\] and its counterterm. The LCOPT diagrams obtained from the webs \[2loops\] b)-d) are shown in Fig. \[2loopfig\]. We see that due to the numerator $(2 k'^- - k^-)$ in the triple-gluon vertex, web d) contains two orderings on the light-cone; the factors of $2$ and $(-1)$ next to the eikonal vertices in the figure come from this numerator. Furthermore, for web b) it is important to route the momenta in the crossed ladder independently of each other, such that both of them flow through the vertex, to separate the subdivergence associated with the upper loop ($k'$) from the overall UV divergence. Now we determine the divergent 1-loop subgraphs for each web by naively counting the powers of transverse momentum components in numerators and denominators. The UV divergent subgraphs are marked with boxes in Fig. \[2loopfig\]. We see that for web c) and the first term of web d) we need a QCD counterterm for the triple-gluon vertex, whereas for the webs a) and d)(ii) we require vertex counterterms, as shown in the second column of Fig. \[2loopfig\]. Web d)(ii) is an example for a LCOPT graph with a triple-gluon vertex which does not need QCD renormalization, in contrast to loop-corrections to 3-gluon-vertices in every covariant diagram. Due to the factor of (-1) in web d)(ii) the two vertex counterterms cancel each other, as announced above. The QCD counterterm, as shown in Fig. \[3gcounter\], is in the $\MS$ scheme for quark eikonal lines $\beta$ given by Z\^[a]{}\_[,ij]{} = - T\^a\_[ij]{} g \^ ( - ), \[3gcount\] where $g$ is the QCD coupling, $\alpha_s = g^2/(4\pi)$, and $\varepsilon > 0$. The next step, after adding the appropriate counterterms to the respective graphs, is to perform the plus-momentum integrals. To do so, we dimensionally continue to $\varepsilon < 0$, that is, to $n > 4$ dimensions. The results for the webs b)-d) and the counterterms for UV subdivergences, denoted by $Z$ (omitting the vertex counterterms which cancel each other) are: 2 W\_[aa, b)]{}\^[(2)]{} & = & - \^[(2)]{} ( )\^2 ()\^[2 ]{} (4 )\^[2 ]{} \_0\^ B(1+,-), \[twoloopa\]\ 2 W\_[aa, c)]{}\^[(2)]{} & = & 2 W\_[aa, d)]{}\^[(2)]{} = \^[(2)]{} ( )\^2 ()\^[2 ]{} (4 )\^[2 ]{} \_0\^\ & & { B(1-,-) - 2 B(1 - ,1-)},\ 2 Z\_[c)]{} & = & 2 Z\_[d)(i)]{} = \^[(2)]{} ( )\^2 ()\^ (4 )\^ \_0\^ ( ) ( - ).\ & & \[twoloopz\] The color weight, as stated in Eqs. (\[2loopcolwe\]) and (\[colweb\]), is $\overline{C}^{(2)} = - \frac{C_A}{2} C_a$ for all diagrams. We notice that diagram d) gives the same contribution as its upside-down counterpart c), as expected, but only after adding different types of counterterms. After summing over the contribution of the webs b)-d) and the counterterms, we see that the infrared poles $1/(-\varepsilon)$ in the Beta-functions cancel, as well as the vertex counterterms, leaving us with A\_a\^[(2),b)-d)]{} = C\_a ( 2 - ) \[remaincont\] according to Eq. (\[aendeq\]). The contributions of all diagrams, (\[2loopgluon\]) and (\[remaincont\]), result in the 2-loop coefficient (\[a2number\]), as announced. Higher Loops {#sect3loop} ============ $N_f^{n-1}$-Terms in $A^{(n)}$ ------------------------------ It is relatively straightforward to obtain a general formula for the $N_f^{n-1}$-contribution to the $n$-loop coefficient $A^{(n)}$, since the only graphs involved are one-loop webs with $n-m-1$ fermion bubbles and $m$ counterterms for the fermion bubbles inserted into the gluon propagator. It is therefore a matter of simple combinatorics to obtain the $\alpha_s^n N_f^{n-1}$ contribution (compare to the one-loop expression Eq. (\[bubblenf\]) ): 2 W\_[aa, N\_f\^[n-1]{}]{}\^[(n)]{} & = & 2 C\_a T\_R\^[n-1]{} N\_f\^[n-1]{} ( )\^n \_0\^\ & & \_[m=0]{}\^[n-1]{} ( [c]{} n-1\ m ) (-1)\^[n-m-1]{}()\^[(n-m) ]{} (4 )\^[(n-m) ]{} 2\^[n-m-2]{}\ & & \^[n-m-1]{} \^m . The $\frac{1}{\varepsilon}$-pole in the expansion of the $\Gamma$- and Beta-functions in the sum is the contribution to the anomalous dimension (cf. Eq. (\[aendeq\]) ). The contributions up to $\alpha_s^6$ are given in Table \[table\]. They coincide with the corresponding values (the $\ln N$-terms, or equivalently, the $S_1(N)$-terms) calculated by Gracey in [@Gracey][^2]. $n$ $C_a (T_R N_f)^{n-1}$-term in $A_a^{(n)}$ ----- ---------------------------------------------------------------------------------------- 2 $-\frac{5}{9}$ 3 $-\frac{1}{27}$ 4 $-\frac{1}{81} + \frac{2}{27} \zeta(3)$ 5 $-\frac{1}{243} - \frac{10}{243} \zeta(3) + \frac{\pi^4}{2430}$ 6 $-\frac{1}{729} - \frac{2}{729} \zeta(3) - \frac{\pi^4}{4374} + \frac{2}{81} \zeta(5)$ : $\alpha_s(\mu^2)^n N_f^{n-1} \left[ \frac{1}{1-x} \right]_+$-contributions to the anomalous dimension $P_{ff}$. The expansion of $A_f$ is performed in terms of $\alpha_s/\pi$ (cf. Eq. (\[aaa\]) ).[]{data-label="table"} Towards the Three-Loop Coefficient $A^{(3)}_f$ ---------------------------------------------- Vogt [@vogt] obtained a numerical parametrization of the $A^{(3)}$ from the known integer moments of the splitting function[^3]: A\^[(3)]{}\_f = C\_F. \[vogtresult\] We obtained the term proportional to $N_f^2$ in the previous subsection, as listed in Table \[table\]. Now we will go on to compute the term proportional to $N_f$. All intermediate expressions are simple enough to be handled by the general algebraic computer program *Mathematica* [@mathematica]. For the calculation of the full $A^{(3)}$ or even higher loops, however, an implementation of the algorithm into a more specialized computer algebra program such as FORM [@form] may be desirable. The diagrams contributing to this term and their QCD counterterms are listed in Table \[graphtab\], labelled in analogy to the two-loop case. We only have to compute the contributions from the $g_{\mu \nu}$ part of the dressed gluon propagator, since the longitudinal parts $\sim k_\mu k_\nu$ cancel due to the Ward identity shown in Fig. \[subfact\]. This cancellation has been verified explicitly. The contributions to the set a) are easily computed to be A\_f\^[(3),a)]{} = C\_A T\_R N\_f C\_F. \[A3a\] The contributions to the $N_f$-part of the two-loop gluon self-energy inserted into a one-loop web (set g) ) give: A\_f\^[(3),g)]{} = - T\_R N\_f C\_F. \[A3g\] To compute the two-loop gluon self-energy, the occurring tensor integrals have been reduced to simple scalar one- and two-loop master integrals using the relations derived in [@relation]. We checked our calculations of the set g) against previous computations of the two-loop gluon self-energy in Feynman gauge, see for example [@davydbraat]. Note that this contribution has a term $\sim C_F^2$, which is not “maximally non-abelian”. The results of [@davydbraat] include the longitudinal terms of the gluon propagator, which is dressed with a fermion bubble. Since these terms in the two-loop gluon self-energy, as stated above, cancel against the longitudinal parts in the remaining webs, Eq. (\[A3g\]) does not contain these contributions. The expressions for the two-loop webs with a one-loop bubble-insertion are found easily from the corresponding two-loop expressions Eqs. (\[twoloopa\])-(\[twoloopz\]), taking into account the proper multiples of $\varepsilon$ in the Gamma- and Beta-functions due to the bubbles. The calculation of the triangles e) and f) is a bit more nontrivial. The resulting contributions can be found in the table. The results for e) and f) have been expanded in terms of $\varepsilon$ and Beta-functions using various identities tabulated in [@polylog]. Since the infrared structure of the graphs is modified by the bubbles, which effectively raise the powers of the corresponding gluon propagators by $\varepsilon$ to a non-integer value, the upside-down counterparts do not give the same contributions. This asymmetry is not surprising, since we compute the coefficients collinear to the plus eikonal, thus introducing an asymmetry in how we treat the eikonal lines and the gluons attaching to them. However, we find that the sum of graphs in set d) gives the same contribution as the sum of graphs in set c), as can be seen from the tabulated expressions. The individual diagrams b)-f) have at most three UV (QCD) divergences and one IR/collinear divergence, in addition to the overall scaleless $k^+$-integral. We observe that the diagrams with a one-loop counterterm for the fermion bubble and the one-loop counterterms for the triangle graphs have the same IR structure as the two-loop webs. Thus their IR divergences cancel separately from the rest of the diagrams. This implies that the collinear divergences have to cancel within the set of remaining diagrams, that is, within the set of webs with bubbles and the triangles. Moreover, we observe that the infrared divergences cancel within certain subsets of these graphs. Namely, they cancel separately between graphs b)(1), c)(1), and d)(1), between graphs b)(2), c)(2), and d)(2), as well as between c)(3), d)(3), e) and f). Summing over all contributions from graphs b)-f) we arrive at A\_f\^[(3),b)-f)]{} = - ( - + ) T\_R N\_f C\_A C\_F. \[A3b\] We performed several checks of our computations. The infrared structure described above is one check of the results listed in Table \[graphtab\]. Another check is the cancellation of non-local logarithms $\sim \log M$. Furthermore, the values of the $1/\varepsilon^3$- and $1/\varepsilon^2$-poles can be predicted from the one- and two-loop calculations performed in Section \[sectexample\] [@tHooft]. The sum of all diagrams contributing at $\alpha_s^3 N_f$ has the following structure: 2 W\^[(3)]{}\_[aa,N\_f]{} & = & { - C\_A C\_F T\_R + . \[poles\]\ & & + . C\_F T\_R + A\_[N\_f]{}\^[(3)]{} } N\_f ( )\^3 \_0\^. The predictions of the higher poles in Eq. (\[poles\]) coincide with the poles obtained from the expansion of the calculated expressions listed in the table. Adding (\[A3a\]), (\[A3g\]), and (\[A3b\]) we obtain the term proportional to $N_f$ contributing to the three-loop coefficient $A^{(3)}$: A\^[(3)]{}\_[N\_f]{} = - T\_R N\_f C\_F = -4.293 T\_R N\_f C\_F, which agrees with the numerical prediction in Eq. (\[vogtresult\]). Table \[graphtab\]: Web Factor Contribution ----- -------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 2 see Eq. (\[A3a\]) 2 $- K M^{3 \varepsilon} \frac{\Gamma(3 \varepsilon)}{2 \varepsilon^2} B(2-\varepsilon,2-\varepsilon) B(1+\varepsilon,-\varepsilon)$ 2 $- K M^{3 \varepsilon} \frac{\Gamma(3 \varepsilon)}{4 \varepsilon^2} B(2-\varepsilon,2-\varepsilon) B(1+2\varepsilon,-2\varepsilon)$ 4 $+ K M^{2 \varepsilon} \frac{\Gamma(2 \varepsilon)}{12 \varepsilon} N_{\varepsilon} B(1+\varepsilon,-\varepsilon)$ 2 $+ K M^{3 \varepsilon} \frac{1}{4} \frac{\Gamma(3 \varepsilon)}{e^2} \frac{\left(\Gamma(1+\varepsilon)\right)^2}{\Gamma(1+2 \varepsilon)} B(2-\varepsilon,2-\varepsilon)$ $\qquad \times \left\{ B(1-\varepsilon,-\varepsilon) - 2 B(1 - \varepsilon,1-\varepsilon)\right\}$ 2 $+ K M^{3 \varepsilon} \frac{1}{4} \frac{\Gamma(3 \varepsilon)}{2 \varepsilon^2} B(2-\varepsilon,2-\varepsilon)$ $\qquad \times \left\{ B(1-\varepsilon,-2 \varepsilon) - 2 B(1 - \varepsilon,1-2\varepsilon)\right\}$ 2 $+ K M^{3 \varepsilon} \frac{1}{4} \frac{\Gamma(3 \varepsilon)}{2 \varepsilon^2} B(2-\varepsilon,2-\varepsilon)$ $\qquad \times \left\{ B(1-2\varepsilon,- \varepsilon) - 2 B(1 - \varepsilon,1-2\varepsilon)\right\}$ 2 $+ K M^{3 \varepsilon} \frac{1}{4} \frac{\Gamma(3 \varepsilon)}{\varepsilon^2} B(2-\varepsilon,2-\varepsilon)$ $\qquad \times \left\{ \frac{\left(\Gamma(1+\varepsilon)\right)^2}{\Gamma(1+2 \varepsilon)} B(1-\varepsilon,-\varepsilon) - B(1 - 2 \varepsilon,1-\varepsilon)\right\}$ 2 $+ K M^{3 \varepsilon} \frac{1}{4} \frac{\Gamma(3 \varepsilon)}{2 \varepsilon^2} B(2-\varepsilon,2-\varepsilon)$ $\qquad \times \left\{ B(1-\varepsilon,-2 \varepsilon) - 4 \frac{\left(\Gamma(1+\varepsilon)\right)^2}{\Gamma(1+2 \varepsilon)} B(1 - \varepsilon,1-2\varepsilon)\right\}$ 2 $+ K M^{3 \varepsilon} \frac{1}{4} \frac{\Gamma(3 \varepsilon)}{2 \varepsilon^2} B(2-\varepsilon,2-\varepsilon)$ $\qquad \times \left\{ B(1-2\varepsilon,- \varepsilon) - 2 B(1 - \varepsilon,1-2\varepsilon)\right\}$ Continuation of Table \[graphtab\]: Web Factor Contribution ----- -------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 12 $- K M^{2 \varepsilon} \frac{1}{24} \frac{\Gamma(2 \varepsilon)}{ \varepsilon} N_{\varepsilon} \left\{ B(1-\varepsilon,-\varepsilon) - 2 B(1 - \varepsilon,1-\varepsilon)\right\}$ 4 $+K M^{2 \varepsilon} \frac{\Gamma(2 \varepsilon)}{2 \varepsilon} N_{\varepsilon} B(2-\varepsilon,2-\varepsilon) $ 4 $-K M^{ \varepsilon} \frac{\Gamma(\varepsilon)}{12} N_{\varepsilon}^2$ 8 $-K M^\varepsilon \frac{\Gamma(\varepsilon)}{2} \left[ \frac{1}{12} N_{\varepsilon}^2 - \frac{1}{18} N_{\varepsilon} \right] $ 2 $- K M^{3 \varepsilon} \frac{\Gamma(3 \varepsilon)}{8 \varepsilon^2 } B(2-\varepsilon,2-\varepsilon) \left\{ B(3-2\varepsilon,-\varepsilon) \right. $ $\quad \left. - B(1-\varepsilon,2-2\varepsilon) + \frac{2 \pi^2}{3} \varepsilon + \left(4 \zeta(3)-2\right) \varepsilon^2 \right\} $ 2 $- K M^{3 \varepsilon} \frac{\Gamma(3 \varepsilon)}{8 \varepsilon^2} B(2-\varepsilon,2-\varepsilon)\left\{ B(3-2\varepsilon,-\varepsilon)\right. $ $\quad \left. - B(1-\varepsilon,2-2\varepsilon) - \frac{\pi^2}{3} \varepsilon + \left(10 \zeta(3) -2 \right) \varepsilon^2 \right\}$ 4 $+ K M^{2 \varepsilon} \frac{1}{24} \frac{\Gamma(2 \varepsilon)}{ \varepsilon} N_{\varepsilon} \left\{ B(1-\varepsilon,-\varepsilon) - 2 B(1 - \varepsilon,1-\varepsilon)\right\} $ 4 $+ K M^\varepsilon \frac{\Gamma(\varepsilon)}{2} \frac{1}{12} \left( N_{\varepsilon}^2 - \frac{5}{12} N_{\varepsilon} \right)$ Continuation of Table \[graphtab\]: Web Factor Contribution ----- -------- ------------------- 2 see Eq. (\[A3g\]) : Webs contributing to the $N_f$-term of the three-loop coefficient $A^{(3)}$ and their counterterms (c.t.s), labelled (C). The cross denotes the counterterm for the fermion bubble. Similarly, the cross in the triple-gluon vertex denotes the counterterm for the fermion triangle. The grey blob represents the counterterm Fig. \[3gcounter\], the grey blob with a cross is the 2-loop counterterm for the triple-gluon vertex with a fermion bubble inside. And finally, the black box denotes the fermion part of the 2-loop counterterm for the triple-gluon vertex. We omitted vertex counterterms which cancel. We refrain from drawing all counterterms which give the same contribution. Instead, we indicate multiple contributions in the column “factor”. A factor of 2 is due to the two complex conjugate contributions, and has already been taken into account in Eqs. (\[A3a\]) and (\[A3g\]). []{data-label="graphtab"} We introduced the following abbreviations: K & & T\_R N\_f C\_F ( )\^3 \_0\^,\ M & & ( ) (4 ),\ N\_ & & - . Conclusions =========== We have developed and proved a method for the calculation of the coefficients proportional to $\left[\frac{1}{1-x}\right]_+$ of the non-singlet parton splitting functions, whose knowledge, for example, is important for NNLL resummations. The method is based on the factorization properties of the splitting functions, and on the exponentiation of eikonal cross sections. We illustrated the method with the rederivation of the 1- and 2-loop coefficients $A^{(1)}$ and $A^{(2)}$, as well as the $N_f^{n-1}$ terms at order $n$. We presented the result for the term proportional to $N_f$ at three loops, which coincides with the approximate result obtained by Vogt [@vogt]. The full splitting functions at three loops are currently being computed by Moch, Vermaseren, and Vogt [@MoVe] with the help of the operator product expansion. Their results for the $N_f$-term [@MVV] provide a further check of our calculations. Although a calculation via the OPE provides the complete $N$, or equivalently, $x$-dependence of the splitting functions, it involves a large number of diagrams and complex expressions at higher orders. A computation at three loops is a formidable task, and it seems unlikely that higher loop calculations will be completed in the near future. However, for certain observables, large logarithms due to soft and/or collinear radiation become numerically important, and need to be resummed to as high a level in logarithms as possible. Our method, although limited to only the computation of $A$, has the advantage that higher-order computations are much less complex than within conventional methods, because the number of graphs is greatly reduced, and the expressions involved are relatively simple in LCOPT. Moreover, a fully computerized implementation of the algorithm should be straightforward. Therefore, the computation of the coefficients $A$ at four or even higher loops may be within reach. Rules for LCOPT {#Lrules} =============== As already stated above, these rules can be obtained by performing first all minus integrals [@changma; @kosop]. - We start with forming all possible light-cone time orderings of a given covariant diagram. - Only those configurations are kept which describe possible physical processes once the energy flow is specified. - For every loop we have a factor if we work in $n = 4 - 2 \varepsilon$ dimensions. - For every internal line we have a factor , corresponding to the flow of plus momentum through the graph. - Every intermediate virtual state contributes a factor , \[denom\] where we sum over all momenta comprising that virtual state, and where $q^-$ is the external minus-momentum of the incoming state(s). - Every intermediate real state gives a momentum conserving delta-function: 2 (q\^- - \_j ), where the sum is over all momenta in that real state. - Since all lines are on-shell, we replace for every Fermion numerator its minus component by its on-shell value: l\^- = . 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--- abstract: 'We have performed angle-resolved photoemission spectroscopy on transition-metal dichalcogenide 1$T$-HfTe$_2$ to elucidate the evolution of electronic states upon potassium (K) deposition. In pristine HfTe$_2$, an in-plane hole pocket and electron pockets are observed at the Brillouin-zone center and corner, respectively, indicating the semimetallic nature of bulk HfTe$_2$, with dispersion perpendicular to the plane. In contrast, the band structure of heavily K-dosed HfTe$_2$ is obviously different from that of bulk, and resembles the band structure calculated for monolayer HfTe$_2$. It was also observed that lightly K-dosed HfTe$_2$ is characterized by quantized bands originating from bilayer and trilayer HfTe$_2$, indicative of staging. The results suggest that the dimensionality-crossover from 3D (dimensional) to 2D electronic states due to systematic K intercalation takes place via staging in a single sample. The study provides a new strategy for controlling the dimensionality and functionality of novel quantum materials.' author: - 'Y. Nakata,$^1$ K. Sugawara,$^{1,2,3}$ A. Chainani,$^4$ K. Yamauchi,$^{5}$ K. Nakayama,$^{1}$ S. Souma,$^{2,3}$ P.-Y. Chuang,$^4$ C.-M. Cheng,$^4$ T. Oguchi,$^5$ K. Ueno,$^6$ T. Takahashi,$^{1,2,3}$ and T. Sato$^{1,2,3}$' title: 'Dimensionality reduction and band quantization induced by potassium intercalation in 1$T$-HfTe$_2$' --- Controlling the dimensionality of materials is one of the key challenges in condensed matter physics because dimensionality plays a crucial role in determining exotic physical properties and quantum phenomena [@Novoselov1; @Klitzing; @Cao; @Zeng; @Mak1; @Mak2; @Wang; @Xi1; @Lu; @Xi2; @Saito; @Mak3]. It is often seen that reduction of dimensionality from 3D to 2D causes emergence of novel physical properties absent in 3D bulk, as represented by massless Dirac fermions in graphene [@Novoselov1], quantum Hall effect in semiconductor heterostructures [@Klitzing], valley-selective circular dichroism [@Cao; @Mak1; @Zeng], and valley Hall effect [@Mak2]. Moreover, the 3D-to-2D crossover sometimes leads to more gigantic physical properties, as exemplified by a drastic increase in the superconducting transition temperature in monolayer FeSe [@Wang], the enhancement of charge-density-wave (CDW) transition temperature [@Xi1] and upper-critical field [@Lu; @Xi2; @Saito] in transition-metal dichalcogenides (TMDs). These drastic changes in physical properties upon reducing the dimensionality are known to be inherently linked to the modification of the electronic band structure such as the spin splitting caused by the inversion-symmetry breaking [@Zhu; @Nakata1] and the indirect-to-direct transition nature of band gap due to the absence of 3D chemical bonding [@Mak3]. Also, 2D systems are very sensitive to external stimuli like strain, charge doping, and electric field, so that the engineering of band structure is more feasible, making 2D systems a promising platform to explore novel physical properties. Given the importance of controlling the dimensionality, a next important issue is how to control it. A widely used approach to reduce the dimensionality of materials from 3D to 2D is to mechanically or chemically exfoliate the sample [@Novoselov2; @Joesen] (top-down approach). This can lead to an exotic change in properties as highlighted by the realization of quantum Hall effect in graphene exfoliated from graphite [@Novoselov2]. A bottom-up approach such as deposition of atoms or molecules on a substrate using molecular-beam-epitaxy method [@Koma; @Zhang; @Nakata1; @Nakata2; @Sugawara] or chemical-vapor-deposition technique [@Sutter; @Kim], is also useful. While these approaches have been successfully employed to explore unconventional physical properties associated with the reduction of dimensionality, a systematic control of dimensionality in these approaches needs great efforts because it requires one-by-one fabrication of various films with different thickness. As reported here, we have found a rather unexpected and effective way to systematically control the dimensionality and band structure of a TMD system. In this Rapid Communication, we carried out a simple and useful approach to control the dimensionality and electronic structure on the surface of the TMD 1$T$-HfTe$_2$. Using angle-resolved photoemission spectroscopy (ARPES) for visualizing the band structure of 1$T$-HfTe$_2$, we first show that the pristine bulk compound is a typical semimetal with hole and electron pockets at the $\Gamma$ and M points in the Brillouin zone. Very surprisingly, our results also show that the original 3D electronic structure in bulk pristine 1$T$-HfTe$_2$ converts into a purely 2D electronic structure upon potassium (K) intercalation. Intriguingly, while the observed valence-band (VB) structure in pristine HfTe$_2$ is well reproduced by the band calculations for bulk, those in lightly and heavily K-dosed HfTe$_2$ well follow the calculated bands for $bilayer/trilayer$ and $monolayer$, respectively, providing evidence for dimensionality reduction and band quantization driven by K intercalation. {width="1.65in"} We have chosen HfTe$_2$ as a test material to carefully monitor the evolution of electronic states upon K deposition, without complications from various orders (such as CDW and superconductivity) [@Brattas; @Hodul1; @Hodul2; @Klipstein; @Aminalragia-Giamini; @Mangelsen] which are known to exist in many other TMDs. High-quality single crystals of 1$T$-HfTe$_2$ \[see Fig. 1(a) for the crystal structure\] were grown by the chemical-vapor-transport method [@Ueno]. ARPES measurements were performed at the beamlines BL28 of Photon Factory, KEK and BL21B1 of Taiwan Light Source, NSRRC. First-principles band-structure calculations were carried out by a projector augmented wave method [@VASP] with generalized gradient approximation (GGA) [@GGA]. For details of experiments and calculations, see sections 1-3 of Supplemental Material. First, we show the electronic states of pristine HfTe$_2$. Figure 1(c) displays the energy distribution curves (EDCs) in the VB region measured along the $\Gamma$M cut in the bulk Brillouin zone (BZ) \[Fig. 1(b)\] at photon energy ($hv$) of 85 eV. We recognize several dispersive bands such as holelike bands centered at the $\Gamma$ point. One of these holelike bands crosses the Fermi level ($E_{\rm_F}$), forming a small hole pocket at the $\Gamma$ point. A low-intensity but clear Fermi-edge cut-off is seen around the M point \[see the inset in Fig. 1(c) and Fig. 2(c)\], indicating a small electron pocket around the M point. These results confirm the semimetallic nature of HfTe$_2$, consistent with previous transport measurements [@Klipstein; @Mangelsen] and first-principles band-structure calculations [@Aminalragia-Giamini; @Mangelsen]. To see the VB structure in more detail, we show in Figs. 1(d) and 1(e) the experimental band dispersions obtained by plotting the ARPES intensity as a function of wave vector ($k$) and binding energy ($E_{\rm_B}$) along high-symmetry cuts in the $\Gamma$KM ($k{_z}$ = 0) and AHL ($k{_z}$ = $\pi$) planes, respectively. We also show the corresponding band dispersions calculated for bulk 1$T$-HfTe$_2$ for comparison. As shown in Fig. 1(d), the overall experimental VB dispersion such as the location of dispersive holelike bands at the $\Gamma$ point is well reproduced in the calculation when the calculated bands are shifted downward as a whole by 0.47 eV. These holelike bands are attributed to the Te 5$p$ orbital. On the other hand, the electron pocket at the M point stemming from the Hf 5$d$ orbital [@Aminalragia-Giamini; @Mangelsen] is significantly larger in the calculation. This is not due to the imperfect compensation of electrons and holes in the experiment, but due to the overestimation of the semimetallic band overlap ($i.e.$ negative band gap) in the calculations ($\sim$ 1 eV in contrast to 0.05 eV in experiments). Such overestimation was also recognized in previous band calculations of HfTe$_2$ [@Aminalragia-Giamini; @Mangelsen] and also in other TMDs such as TiSe$_2$ [@Chen]. While one may attribute such overestimation to the unoptimized interlayer coupling, we found that the change in the $c$-axis lattice constant ($i.e.$ layer spacing) in the calculation is insufficient to correctly reproduce the experimental band structure, suggesting that there are other factors to overestimate the band gap in the calculation (see section 4 of Supplemental Material for details). As shown in Figs. 1(d) and 1(e), while the overall experimental VB dispersion looks similar between the $k{_z}$ = 0 and $\pi$ plane, a closer look reveals some characteristic differences such as the number of holelike bands and presence/absence of a flat dispersion at $E{\rm_B}$ $\sim$ 2.6 eV for $k{_z}$ = 0 / $k{_z}$ = $\pi$, indicating the 3D nature of the band structure. It is noted that the agreement of band structure between experiments and calculations is relatively poor in the $k{_z}$ = $\pi$ plane compared to that in the $k{_z}$ = 0 plane. In fact, it was necessary to shift the calculated VB dispersions upward by 0.28 eV to make a better match to the experimental data for $k{_z}$ = $\pi$. Now that the band structure of pristine HfTe$_2$ is established, next we demonstrate how the electronic states are influenced by K deposition. One would naturally expect that K atoms donate electrons into the HfTe$_2$ top layers. This is clearly visible as a downward shift of the electronlike band at the M point, as shown in Figs. 2(a) and 2(b). We have estimated the energy shift ($E{\rm_{shift}}$) to be 165 meV by comparing the peak position of EDCs as shown in Fig. 2(c). The electron doping with K deposition is also seen in the change of Fermi surface \[Figs. 2(d) and 2(e)\]. Upon K deposition, the hole pocket at the $\Gamma$ point disappears and at the same time the ellipsoidal electron pocket at the M point expands. Hereafter we label each sample with the size of electron pocket at the M point (See Section 5 of Supplemental Material for the detailed procedure), which corresponds to the electron density in a unit layer. For example, samples in Figs. 2(d) and 2(e) are labeled $n{\rm_e}$=0.02 and $n{\rm_e}$=0.19 samples, respectively. {width="1.55in"} ![(color online): (a), (b) VB-ARPES intensity along the M$\Gamma$K cut measured at $h{\nu}$ = 85 eV for $n{\rm_e}$ = 0.02 and 0.19, respectively. Bottom panel shows the MDC at $E{\rm_B}$ = 1.3 eV (dashed line in the top panel), with $k$ position of bands indicated by arrows. Comparison of EDC at the $\Gamma$ point between $n{\rm_e}$ = 0.02 (dashed curve) and 0.19 (solid curve) is also shown in (b). (c), (d) First-principles band-structure calculations for bulk ($k{_z}$ = 0) and monolayer, respectively, compared with the ARPES intensity for $n{\rm_e}$ = 0.19 \[same as (b) but plotted with gray scale\]. Calculated bands were shifted downward by 700 and 925 meV, respectively. (e) Calculated band structure along the $\Gamma$A line for bulk HfTe$_2$. (f), (g) Normal-emission ARPES intensity as a function of $h{\nu}$ and $E{\rm_B}$ for $n{\rm_e}$ = 0.02 and 0.19, respectively.](fig3.eps){width="1.6in"} {width="2.15in"} Figures 3(a) and 3(b) show a side-by-side comparison of experimental band dispersions between pristine ($n{\rm_e}$=0.02) and heavily K-deposited ($n{\rm_e}$=0.19) sample. Although the electron pocket at the M point is shifted downward by K deposition as expected, the data show several anomalous changes in the band structure which cannot be explained within a simple rigid-band scheme. For example, in the $n{\rm_e}$=0.02 sample, we observe three holelike bands centered at the $\Gamma$ point; two topped at around $E{\rm_F}$ and one topped at $\sim$ 1 eV. On the other hand, only two holelike bands (topped at $E{\rm_B}$ $\sim$ 0.1 and 0.75 eV) exist in the $n{\rm_e}$=0.19 sample. Such a difference in the number of bands is also highlighted by the representative momentum distribution curve (MDC) in the bottom panels of Figs. 3(a) and 3(b) which signify the presence of broad three peaks and sharp two peaks in the $\Gamma$M cut (also in the $\Gamma$K cut) for $n{\rm_e}$ = 0.02 and 0.19, respectively. As shown in Figs. 3(a) and 3(b), while a relatively flat band at $E{\rm_B}$ $\sim$ 2.6 eV appears to shift downward upon K deposition, a new M-shaped band emerges slightly above this band ($E{\rm_B}$ $\sim$ 2.5 eV) in the $n{\rm_e}$=0.19 sample. To clarify the origin of such anomalous variation of the band structure, we compare in Figs. 3(c) and 3(d) the experimental band structure for $n{\rm_e}$ = 0.19 with the calculated band structure for bulk and monolayer HfTe$_2$, respectively. One can see that the calculated band structure for monolayer shows a good agreement with the experimental band structure (except for the electron pocket at the M point), while the calculated band structure for the bulk apparently shows some disagreements such as in the location of the holelike bands and the absence of the M-shaped band. This suggests that the originally bulk-like band dispersion is “converted" into the monolayer-like one upon K deposition. Such a change to the monolayer-like behavior is also confirmed by performing photon-energy-dependent ARPES measurements that signify a finite energy dispersion along $k{_z}$ in the pristine ($n{\rm_e}$=0.02) sample \[Fig. 3(f)\] in line with the band calculation \[Fig. 3(e)\], in contrast to no discernible $k{_z}$ dispersion in the $n{\rm_e}$=0.19 sample \[Fig. 3(g)\]. These results indicate that the K deposition switches the dimensionality of electronic structure from 3D to 2D. We found that such a 3D-2D transition is accompanied by a sharpening of the spectral line shape highlighted by the comparison of EDC at the $\Gamma$ point between $n{\rm_e}$ = 0.02 and 0.19 in Fig. 3(b), which likely reflects absence of $k{_z}$-broadening effect and reduced contribution of photoelectron lifetime for $n{_e}$ = 0.19. To gain further insight into the origin of observed dimensionality change, we investigated the evolution of electronic states for various K-deposition time and show the ARPES intensities for a series of $n{\rm_e}$ in Figs. 4(a)-4(d). This plot again confirms that the number of bands are apparently different between $n{\rm_e}$ = 0.02 and 0.19, reflecting the dimensionality change. One can also see that the band structures for $n{\rm_e}$ = 0.19 and 0.25 are very similar to each other. This suggests that the 2D nature of electronic states is more or less established at $n{\rm_e}$ = 0.19 and further K deposition simply leads to extra electron doping with a small constant shift of the overall band structure to higher binding energies. We comment here that, while a previous study for epitaxy-grown atomic-layer HfTe$_2$ thin film on AlN [@Aminalragia-Giamini] reported the Dirac-semimetal phase characterized by the Dirac-cone band at the $\Gamma$ point, the present result shows no upper Dirac-cone-like band for monolayer HfTe$_2$ \[Fig. 4(d)\] (see Section 5 of Supplemental Material for details). This difference may be attributed to a substrate induced effect in epitaxial atomic-layer HfTe$_2$ film on AlN compared to the present study. Finally, we present another very important finding. We found that the experimental band structure for $n{\rm_e}$ = 0.05 shows unexpected behavior at energies away from $E{\rm_F}$. As highlighted by the area enclosed by red rectangle in Fig. 4(b), there exist three holelike bands topped at the $E{\rm_B}$ range of 1-1.7 eV. Such multiple bands are absent in other samples, and hence they are likely a characteristic of an intermediate state between 3D ($n{\rm_e}$ = 0.02) and 2D ($n{\rm_e}$ = 0.19 - 0.25). To obtain further insight into the origin of such subband feature, we compare the ARPES-derived band dispersion \[Fig. 4(e)\] with the calculations for multilayer HfTe$_2$ \[Figs. 4(f), 4(g), and 4(h)\]. One can see that band A is due to the bulk band since its shape and energy position are well reproduced by the calculation for bulk HfTe$_2$ as shown in Fig. 4(f). This assignment is also corroborated with the observation of a similar band in the pristine sample ($n{\rm_e}$ = 0.02) \[Fig. 4(a)\]. On the other hand, bands B and C show a reasonable agreement with the calculated topmost quantized bands for trilayer and bilayer HfTe$_2$, respectively. This implies that the surface of the $n{\rm_e}$ = 0.05 sample is inhomogeneous in terms of the K concentration, and different domains coexist at the surface. It is thus likely that we simultaneously detect three domains ($i.e.$, bulk, trilayer, and bilayer) in the ARPES data \[Fig. 4(b)\] for $n{\rm_e}$ = 0.05. It is noted that a weak feature labeled D in Fig. 4(e) may be ascribed to a mixture of the second and/or third quantized $p_z$ orbital in trilayer and bilayer domains. We discuss the origin of observed intriguing change in the band structure. We found no obvious change in the LEED (low-energy-electron-diffraction) pattern upon K deposition, in particular, regarding the location of the LEED spot. This suggests that the in-plane lattice parameter does not change and no surface reconstruction takes place (see section 7 of Supplemental Material for details). Thus, the observed evolution of band structure upon K deposition in Figs. 2-4 is not ascribed to the structural modulation of the HfTe$_2$ layer itself. A plausible explanation for the observation of monolayer-like band dispersion for $n{\rm_e}$ = 0.19 and 0.25 is that K atoms are intercalated into the van der Waals gap of HfTe$_2$ layers around the surface, as naively understood by referring to stage-one graphite intercalation compounds (GICs) where atoms are intercalated in all the available van der Waals gaps. Since K atoms are randomly placed in HfTe$_2$ as inferred from the absence of band folding and additional LEED spots after K deposition, the K atoms do not enhance the interlayer coupling unlike the case of GICs where the periodic arrangement of intercalant atoms would promote the 3D nature of materials. In contrast, in the case of HfTe$_2$, each layer is effectively isolated from adjacent layers due to the increased layer spacing, leading to enhancement of monolayer-like nature. A similar behavior has been observed in K-intercalated MoS$_2$ [@Eknapakul1] and H-intercalated graphene on SiC [@H-SiC]. On the other hand, besides the monolayer-like feature, we found that a multiple staging from stage-two to stage-three takes place at the surface of a single sample and gives rise to emergence of several quantized bands in the lightly K-deposited regime ($n{\rm_e}$ = 0.05). This finding is of particular significance since we could experimentally demonstrate that the dimensionality of the electronic states (in other words, staging of the intercalation) around the surface can be $systematically$ and $easily$ controlled by the simple K-deposition technique. The above quantization picture is further corroborated by considering the orbital character for the observed subbands. As visible in Fig. 4(b), the band quantization is well resolved only for the bands located at $\sim$1-2 eV. These bands originate from the $p{_z}$ orbital which is highly dispersive along the $k{_z}$ direction, as seen in Fig. 3(e). This situation is favorable for forming the quantized bands since the quantum confinement occurs along the $z$ direction (perpendicular to the surface). In contrast, the $p{_x}$ and $p{_y}$ orbitals are unlikely to be well quantized because of their weak $k{_z}$ dispersion; this is indeed inferred from the absence of subbands for the $p{_{x,y}}$-derived flat band at $\sim$2.6 eV in Figs. 4(a)-4(d) \[see also Fig. 3(e)\]. It is worthwhile to note here that the previous study reported similar subbands after Na intercalation in HfSe$_2$ [@Eknapakul2]. However, the mechanism of subband formation is totally different from the present case since the subbands of Na-intercalated HfSe$_2$ have a $p{_{x,y}}$ character and they originate from the in-plane lattice strain [@Eknapakul2]. This is also consistent with the ARPES and LEED data (see section 7 of Supplemental Material) of HfTe$_2$ which show no discernible variation of the in-plane lattice constant upon K intercalation in support of a weak strain effect. The present study demonstrates for the first time that the band quantization and dimensionality of electronic states can be manipulated by a simple deposition of atoms on the sample surface, as highlighted in Fig. 4(i). We emphasize that the method proposed here to control the dimensionality and visualize the electronic states is useful since it can be performed on a single sample; this could be contrasted to the so-far established exfoliation and MBE techniques in which systematic control is rather difficult because they require one-by-one fabrication of various films with different thickness. Also, it is expected that our method can be widely applicable to other layered materials including TMDs if the condition of intercalation such as the species of alkali metals and evaporation temperature is optimized for each material (this point is important since the intercalation/adsorption condition would strongly depend on the combination of alkali-metal elements and constituent elements of TMDs [@Biswas; @Kang]). While we selected HfTe$_2$ to monitor the evolution of electronic states to avoid complications from various orders [@Brattas; @Hodul1; @Hodul2; @Klipstein; @Aminalragia-Giamini; @Mangelsen] and to effectively demonstrate controllability of dimensionality, a choice of other TMDs would provide us a precious opportunity to study in a systematic way the interplay between dimensionality and various exotic physical properties, such as unconventional superconductivity [@Lu; @Xi2; @Saito], ferromagnetism [@Bonilla], topological phase transition, and quantum spin Hall effect [@Qian; @Tang; @Fei; @Wu]. Experiments combining surface spectroscopies and magneto-transport measurements in K-deposited TMDs would be highly desired in future. In conclusion, our ARPES study on HfTe$_2$ revealed a rich variation of electronic structure upon K intercalation associated with the 3D-to-2D crossover. We proposed a new technique to control the dimensionality of electronic states at the surface by simple K deposition. The present result would serve as a foundation for investigating the interplay between dimensionality and exotic physical properties in TMDs and other layered quantum materials. We thank T. Kato, K. Hori, K. Owada, T. Nakamura, H. Oinuma, K. Shigekawa, and D. Takane for their assistance in the ARPES measurements. We also thank KEK-PF for access to beamline BL28 (Proposal number: 2018S2-001) and NSRRC-TLS for beamline BL21A2. This work was supported by JST-CREST (No: JPMJCR18T1), MEXT of Japan (Innovative Area “Topological Materials Science" JP15H05853), JSPS (JSPS KAKENHI No: JP17H01139, JP26287071, JP18H01160, JP18H01821, 18K18986, JP25107003, JP25107004, and 18J10038), Grant for Basic Science Research Projects from the Sumitomo Foundation, and Murata Science Foundation. Y. N. acknowledges support from GP-Spin at Tohoku University. [50]{} K. S. Novoselov, A. K. 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--- abstract: | We investigate how accurately phase space distribution functions (DFs) in galactic models can be reconstructed by a made-to-measure (M2M) method, which constructs $N$-particle models of stellar systems from photometric and various kinematic data. The advantage of the M2M method is that this method can be applied to various galactic models without assumption of the spatial symmetries of gravitational potentials adopted in galactic models, and furthermore, numerical calculations of the orbits of the stars cannot be severely constrained by the capacities of computer memories. The M2M method has been applied to various galactic models. However, the degree of accuracy for the recovery of DFs derived by the M2M method in galactic models has never been investigated carefully. Therefore, we show the degree of accuracy for the recovery of the DFs for the anisotropic Plummer model and the axisymmetric Stäckel model, which have analytic solutions of the DFs. Furthermore, this study provides the dependence of the degree of accuracy for the recovery of the DFs on various parameters and a procedure adopted in this paper. As a result, we find that the degree of accuracy for the recovery of the DFs derived by the M2M method for the spherical target model is a few percent, and more than ten percent for the axisymmetric target model. date: 17 August 2016 --- \[firstpage\] galaxies: kinematics and dynamics – galaxies: structure – astrometry – methods: numerical – Galaxy: formation Introduction {#sec_1} ============ Investigating the internal dynamical structures of galaxies is important to infer their formations and evolutions. On the other hand, the phase space distribution function (DF) for the total of matters in a galaxy leads to physical quantities that characterize the internal dynamical structure of the galaxy such as the mass distribution, the gravitational potential and the velocity distribution. Hence it is very important to get the DFs of the total of matter that has gravity in galaxies. The DF of a galaxy has the following characteristics: galaxies can be regarded as collisionless systems because the thermal relaxation time of the self-gravitational system consisting of a large number of stars ($\gtrsim10^9$) is known to be much longer than the Hubble time. The DFs of such collisionless stellar systems obey the collisionless Boltzmann equation. In addition, galaxies are often regarded as quasi steady states in the first approximation since the mass and velocity distributions of a certain type of galaxies such as elliptical galaxies and spiral galaxies are observed to be almost independent of their ages. Therefore, galaxies are supposed to be quasi steady state for a long time. Any steady-state solution of the collisionless Boltzmann equation depends on the phase-space coordinates only through integrals of motion by the Jeans theorem [@jea15]. Hence the DFs of the present galaxies can be mostly represented by a few integrals of motion instead of seven variables, which are the phase space coordinates of positions, velocities and time. In this way, we can reduce the number of the variables of the DFs and so the DFs can be handled easily. We should get the DFs for the total of matter including the dark matter and all stars that have gravity to infer the actual internal dynamical structures of a galaxy. However, the construction of the DFs from observational data is a difficult task. First, this is because we cannot obtain the information of the distance and proper motion of stars only by the photometric and spectroscopic observations [@mor13]. These observations usually give the information of the surface density and the line-of-sight velocity distribution (LOSVD) as functions of position on the plane of the sky. However, for the Milky Way galaxy, the difficulty for the construction of the DF from this point of view can be overcome because future high-precision astrometric observations will give us the additional information such as the parallaxes and proper motions of stars. Furthermore, actually, the era of great Galactic astrometric survey mission in the Milky Way Galaxy is coming. The ongoing space satellite projects such as Gaia [e.g. @per01] which was launched on December $19$, $2013$ and began routine operations in August, $2014$, and Small-JASMINE [Japan Astrometry Satellite Mission for INfrared Exploration, e.g. @gou12] will bring the highly precise astrometric data. Combining the astrometric data with the spectroscopic observations, we will directly obtain the six-dimensional phase space coordinates of observed stars. Another difficulty for the construction of the DFs for the total of matter is caused by the fact that we can obtain the DF only for finite observed stars. We cannot directly get the DF for the total of matter in a galaxy because we cannot observe the dark matter and very faint stars. Therefore, it is necessary to infer the DF for the total of matter by the development of methods. We will show a method for the construction of the DFs for the total of matter by use of the DF for the observed stars. We first assume a pair of the gravitational potential and corresponding density distribution of the total of matter as a target galactic model. Second, we theoretically construct the DF for the total of matter in the assumed galactic model. In this paper, we call this constructed DF “a template". Third we assume some other target galactic models that have different gravitational potentials and constructed the DFs in these galactic models (templates). The DF in the most plausible galactic model will give the best fit for the DF for the observed stars. Hence, finally, we select the best fit model of the DF (best fit template) by the use of statistical techniques such as the maximum likelihood method in the comparison of the templates with observational data. In this way, we can infer the DF for the total of matter that reflects the real galaxy with high possibility. Here it should be remarked that in the comparison of the template with the observations we should take into account the observational noise, selection effects of a finite sample of observed stars and so on. It should be stressed here that it is very important to construct the templates sufficiently accurately so that we can infer the real DF for the total of matter with high possibility by the comparison with accurate observations. Several methods are used to construct the templates as described below. A moment-based method solves the Jeans equation so as to find most plausible DF (template) which is best matched to the observed mass and velocity distribution [e.g. @you80; @bin90; @mag94; @mag95; @cap08; @cap09]. To solve the Jeans equation, it is necessary to neglect and/or assume higher order velocity moments. The main drawback of this method is that the positive DFs is not guaranteed. Furthermore this method can usually only be used for spherically symmetric models. A DF-based method prepares models of DFs that are functions of the integrals of motion. Additionally, this method constrains parameters included in the function of a DF. These parameters are determined so that the density and/or velocity distributions derived from the assumed DF can match with the observed distributions. This method was applied for spherical models [@dej87; @ger91a; @car95], axisymmetric models [@deh94], integrable systems [@dej88; @hun92], nearly integrable potentials [@deh93; @bin10], and action-based models [@bov13; @pif14; @san15; @tri16]. This method is restricted in the case that a target system has analytic integrals of motion. However, the integrals of motion cannot be obtained analytically for most systems. The general systems require torus construction methods [@mcg90; @mcm08; @ued14] to construct the integrals of motion as functions of six-dimensional coordinates. An orbit-based method calculates a weight of each orbit (the occupancy ratio of a stellar orbit to all stellar orbits) so that the mass and/or velocity distributions constructed from the convolution of the weight of each orbit are the best fit to the assumed or observed mass and/or velocity distributions in a target galactic model [@sch79; @sch93]. The best fitting orbital weights represent the DF of the target galaxy. To calculate the weight of each orbit with good accuracy requires that a large number of orbits should be evolved over many orbital periods in a fixed potential of the target galaxy. This method is used for various galactic models [e.g. @bos08]. However, the number of the orbits is severely constrained by computer memory capacity. If there are $N$ orbits (or particles) and $J$ observables, this orbit-based method has to store $O(NJ)$ variables, while a particle-based method shown below stores only $O(N)$ variables. A particle-based method (hereafter, M2M method) varies the weights of particles (stars) while each particle is evolved in a gravitational potential of a target galaxy, until the constructed mass and/or velocity distributions are best matched to the observed mass and velocity distribution [@sye96 hereafter ST96]. The advantages of this method are the absence of need of the assumption of the spatial symmetries of target galaxies, and the number of stellar orbits can be stored under the constraint of the capacities of computer memories in the M2M method is much larger than that in the orbit-based method. The M2M method was first applied to the Milky Way’s bulge and disk in @bis04, and most recently applied to Milky Way in @por15a and @por15b. Thereafter, the M2M algorithm has been improved by @del07 [hereafter DL07], @deh09, @lon10, @hun13. The M2M method is also applied to various galactic models [@del08; @das11; @lon12; @mor13]. In this paper, we use the M2M method so as to investigate how accurately templates (DFs for the total of matter in galactic models) can be constructed. Here, it is notable that the accuracies of astrometric observations will be improved before long. Thereby, the application of the M2M method to Gaia mock data is only tried in @hun14b. However, it is not clear whether the accuracies of the templates are less than those of the observed six-dimensional coordinates of stars. If the uncertainties of the templates are larger than those of the observed six-dimensional coordinates of stars, it is not meaningful to obtain the more accurate six-dimensional observational coordinates of stars to find the best fit DF of the target galaxy. Therefore, the accuracies of templates that are compared with observational data should be improved. Hence, it is important to examine the degrees of accuracy for the recovery of the DFs (templates). In this examination, as the target models, we use two analytic models that are the anisotropic Plummer model and the axisymmetric Stäckel model, which depends on three integrals of motion. The reason of choice of these models is that the DFs of these models are given analytically. Hence we can get the degrees of accuracy with accurate quantities by the comparison of the constructed template with the exact solution. Hitherto, the degrees of accuracy for the recovery of a DF derived by the M2M method were presented only for spherical target models. In this study, “the solution" is also constructed by the M2M method [@mor12] and so this solution is not guaranteed to be exact. Thus, in this paper, we show the degrees of accuracy for the recovery of the DFs derived by the M2M method for two specified models to estimate how accurately the templates can be constructed with accurate quantities. Furthermore, this study provides the dependence of the degrees of accuracy for the recovery of the DFs on various parameters and a procedure adopted in the M2M method. The parameters we investigate are the number of the particles used in the M2M method (particle number), the number of the constraints such as the density profiles and/or velocity fields (data number), an initial particle distribution (initial condition), higher order velocity moments, the entropy parameter, and the configurations for the grids of the kinematic observable. This paper is organized as follows. In Section 2, we describe the M2M method used in this paper. In Section 3, we show the conditions for the construction of the DFs. In Section 4, we present how accurately the DFs can be reconstructed by the M2M method. In Section 5, we discuss the dependence of the degrees of accuracy for the recovery of the DFs. In Section 6, we summarize this paper. THE M2M METHOD {#sec_2} ============== The goal of the M2M method is evolving weights of $N$-body particles orbiting in a gravitational potential given as target systems or calculated self-consistently by the particle distribution [@deg10; @hun13] until the constructed mass and/or velocity distributions are best matched to the observed mass and velocity distribution. In this section, we describe the M2M algorithm used in this paper. More detailed descriptions for the M2M technique are written in ST96, DL07, @deh09, and @lon10. THE M2M ALGORITHM {#subsec_the} ----------------- The observables of a target system characterized by the phase space DF $f(\mbox{\boldmath$z$})$ of the target system are defined by $$Y_{k,j}=\int K_{k,j}(\mbox{\boldmath$z$})f(\mbox{\boldmath$z$})\mathrm{d}^6\mbox{\boldmath$z$},$$ where $\mbox{\boldmath$z$}=(\mbox{\boldmath$r$},\mbox{\boldmath$v$})$ are the phase space coordinates of the particles, and $K_{k,j}$ is known as a kernel, which represents the degree that an orbit at $z$ contributes to a kind of an observable $k$ at the grid $j$. Examples of typical observable $Y_{k,j}$ is mass distributions $M_j$, or velocity dispersions. The corresponding observables for the model that is constructed by the M2M method (model observables) are given as $$y_{k,j}(t)=\sum_{i=1}^{N}w_{i}(t)K_{k,j}[\mbox{\boldmath$z$}_i(t)]. \label{y_give}$$ In the case of mass distributions, $w_{i}K_{k,j}[\mbox{\boldmath$z$}_i(t)]=M\delta_{ij}w_{i}$, where $M$ is the total mass of the system, and $\delta_{ij}$ is the selection function, which takes 1 if the $i$-th particle exists at the grid $j$ and takes 0 otherwise. Here individual particles have masses $m_i=w_i M/\sum_{i=1}^N w_i$. In the M2M method, the weight of $i$-th particle $w_{i}(t)$ evolves until model observables $y_{k,j}$ agree with the target observables $Y_{k,j}$. To reduce temporal fluctuations, and to increase the number of effective particles that contribute to the model observables, the model observables $y_{k,j}(t)$ are commonly replaced as $$\tilde y_{k,j}(t)=\alpha \int _0^\infty y_{k,j}(t-\tau)e^{-\alpha \tau}{\mathrm d}\tau, \label{eq_smooth}$$ where $\alpha$ is the smoothing parameter, which controls the degree of a temporal smoothing. The number of effective particles is increased according to the degree of the temporal smoothing since the weight of each particle contributes to the backward spatial regions along its trajectory. This temporal smoothing makes the number of effective particles increases from $N$ to $$N_\mathrm{eff}=N\frac{t_{1/2}}{\Delta t}, \label{eff_num}$$ where $\Delta t$ is the time step of the weight evolution equation (\[eq:foc\]), and $t_{1/2}=(\mathrm{ln}2)/\alpha$ is the half lifetime of the ghost particles. The weights of the orbital particles needs to be varied so as to match the modelling observables $\tilde y_{k,j}$ with the target observables $Y_{k,j}$. This is archived by solving the differential equation called the ’force-of-change’: $$\label{eq:foc} \frac{{\mathrm d}w_i(t)}{{\mathrm d}t}=\epsilon w_i(t){\left(\mu \frac{\partial S}{\partial w_i}-\sum_k^K \sum_j^{J_k}\lambda_k \frac{K_{k,j}[\mbox{\boldmath$z$}_i(t)]}{\sigma(Y_{k,j})}\Delta_{k,j}(t)\right)},$$ where $\epsilon$ is the parameter to control the rate of a change of the weight shown in the equation (\[eq:foc\]), $\sigma(Y_{k,j})$ is the error in the target observable $Y_{k,j}$, $\lambda_k$ is the parameter that allows us to control the contribution of the observable $k$ to the force of change [@hun13], $J_k$ is the number of the observable $k$, and $K$ is the number of the kinds of the observables. The entropy function $S$ is defined as $$S=-\sum_{i=1}^N w_i \mathrm{log}(w_i/\hat w_i-1) % S=-\sum_{i=1}^N w_i \mathrm{log}(w_i/\hat w_i), \label{regularization_term}$$ [@mor12], where $\hat{w}_i$ is called priors, and traditionally set to $\hat w_i=1/N$. The entropy function $S$ is used for the regularization, and the degree of regularization is controlled by the parameter $\mu$. The regularization makes the distribution of the weight smooth. Equation (\[eq:foc\]) maximizes the merit function $$F=-\frac{1}{2}\chi^2+\mu S, %%+\sum_i^Q C_i \label{eq_merit}$$ where $$\chi^2=\sum_k^K \lambda_k \chi_k^2, \label{eq_chi}$$ $$\chi^2_k=\sum_j^{J_k}\Delta_{k,j}^2,$$ and $$\label{eq:eq1} \Delta_{k,j}(t)=\frac{\tilde y_{k,j}-Y_{k,j}}{\sigma(Y_{k,j})}.$$ To avoid excessive temporal smoothing, ST96 indicated that the smoothing parameter $\alpha$ should satisfy $2\epsilon A<\alpha$, where $A$ is approximately averaged value of $\lambda_k K_{k,j}\Delta_{k,j}/\sigma(M_{j})$. To satisfy this relation roughly, $\epsilon$ is given by $\epsilon=\epsilon'\epsilon''$, where $\epsilon'$ is set to be $2\epsilon'<\alpha$, and $\epsilon''=10/\mathrm{max}_{i,j}(\lambda_k K_{k,j}\Delta_{k,j}/\sigma(Y_{k,j}))$ as DL07 and @hun13. CONCLUSION ========== We have shown the degree of accuracy for the recovery of the distribution functions (DFs) to investigate the validation of the M2M method. Hitherto, the degree of accuracy for the recovery of the DFs ($f_\mathrm{dif}$) using the M2M method was presented only for spherical target models. In this previous study, the solution is also constructed by the M2M method [@mor12] and so this solution is not guaranteed to be exact. In this paper, we show the degree of accuracy for the recovery of the mass, velocity dispersion distribution and DFs for the anisotropic Plummer model and the axisymmetric Stäckel model, which depends on three integrals of motion. Furthermore, we provide the dependence of $f_\mathrm{dif}$ on the several parameters. Consequently, our main results are summarized as follows. - For the isotropic spherical target model, we set the number of the mass constraints ($N_\mathrm{m}$) and kinematic constraints ($N_\mathrm{k}$) at $100$, and the number of particles used in the M2M method ($N$) at $10^6$. As a result, the average of the RMS values normalized by the target values for the mass distribution ($\overline{\mathrm{RMS}}(m)$) is $0.36\%$. The averages of the RMS values normalized by the target values for the radial and tangential velocity dispersion distributions are $0.811\%$ and $1.04\%$. The average of the absolute values of the differences between the modelling and target DF ($f_\mathrm{dif}$) is $1.55\%$. - For the axisymmetric Stäckel target model, we set that $N_\mathrm{m}=32$, $N_\mathrm{k}=16$, and $N=10^6$. As a result, $\overline{\mathrm{RMS}}(m)$ is $1.17\%$, the averages of the RMS values normalized by the target values for the velocity dispersion distributions of the radial, azimuthal and $z$ directions are $3.74\%$, $6.76\%$, and $3.29\%$, and $f_\mathrm{dif}$ is $19.9\%$. - We represent the dependences of $f_\mathrm{dif}$ on $N$ for the spherical and the axisymmetric target models. Consequently, we find that the increase of $N$ from $\sim10^6$ to $\sim10^7-10^8$ reduces $f_\mathrm{dif}$ by a few percent. - We show the dependence of $f_\mathrm{dif,min}$, which is $f_\mathrm{dif}$ for a sufficiently large $N$, on the data number $N_\mathrm{d}$ ($N_\mathrm{m}$). As a result, we give the relations as $f_\mathrm{dif,min}=6.5\times10^2~N_\mathrm{d}^{-1.6}\%$ ($N_\mathrm{d}\leq80$) and $f_\mathrm{dif,min}=0.80\%$ ($N_\mathrm{d}\ge80$) for the isotropic Plummer target model, and $f_\mathrm{dif,min}=24.3~N_\mathrm{m}^{-0.075}\%$ for the axisymmetric Stäckel model with $N_\mathrm{k}=16$. - The results for the isotropic spherical target model indicate that $f_\mathrm{dif}$ is limited at a few percent according to the particle initial condition. To identify the cause of the existence of the lower limit, we investigated effects as the higher order velocity moments for the LOSVD, the entropy parameter, the temporal smoothing effect, and the configuration of the kinematic observables. However, the cause of the lower limits of $f_\mathrm{dif,min}$ remains uncertain. We have shown how accurately templates (DFs) can be reconstructed. Our results suggest that the uncertainties of the templates for the axisymmetric three integrals model ($\sim$ a few tens percent) are larger than those of the six-dimensional coordinates of stars that will be observed by Gaia or Small-JASMINE ($\sim$ a ten percent). Note that the effects of the dust extinction may reduce the accuracy of the templates as indicated in @hun14b. Furthermore, although we investigated the reconstruction for the simplistic targets models, the accuracy of the templates will be reduced for real galaxies, which are non-axisymmetric. We will investigate the influence of such effects on the recovery of the DFs in the future. Thus, since our results are thought to be problematic, any methods that construct the templates more accurately are desired. Acknowledgments {#acknowledgments .unnumbered} =============== We thank the referee for providing useful comments. We are also thankful to Masaki Yamaguchi for useful comments on the manuscript. 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E., 2008, MNRAS, 385, 614 Young P.,1980, ApJ, 242, 1232 The distribution function of the anisotropic Plummer model ========================================================== We show the DF for the spherical anisotropic Plummer model [@dej86]. A potential-density pair of the Plummer model can be written as $$\psi(r)=1/\sqrt{1+r^2},$$ and $$\rho=\frac{3}{4\pi}\psi^5.$$ The anisotropic model DF that corresponds to the potential-density pair is given as $$\label{eq:eq87} F_q(E,L)=\frac{3\Gamma(6-q)}{2(2\pi)^{5/2}\Gamma(q/2)}E^{7/2-q} \mathrm{H}\left(0,q/2,9/2-q,1;\frac{L^2}{2E} \right).$$ where $q$ is the parameter, $$\begin{aligned} \mathrm{H}(a,b,c,d;x)= %\begin{cases} \frac{\Gamma(a+b)}{\Gamma(c-a)\Gamma(a+d)}x^a\; _{2}F_{1}(a+b,1+a-c;a+d;x) &x\leq1,\\ \frac{\Gamma(a+b)}{\Gamma(d-b)\Gamma(b+c)}x^{-b}\; _{2}F_{1}(a+b,1+b-d;b+c;\frac{1}{x}) &x\geq1,\\ %\end{cases}\end{aligned}$$ and $$_{2}F_{1}(a,b,c;z)=\sum_{n=0}^{\infty}\frac{(a)_n(b)_n}{(c)_n n\!}z^n.$$ This model gives the velocity dispersions as $$\sigma_r^2=\frac{1}{6-q}\frac{1}{\sqrt{1+r^2}},$$ and $$\sigma_{\phi}^2=\sigma_{\theta}^2=\frac{1}{6-q} \frac{1}{\sqrt{1+r^2}}\left(1-\frac{q}{2}\frac{r^2}{1+r^2}\right),$$ and so the anisotropic parameter $\beta$ is represented by $$\beta=1-\frac{\sigma_{\phi}^2}{\sigma_r^2}= 1-\frac{\sigma_{\theta}^2}{\sigma_r^2}=\frac{q}{2}\frac{r^2}{1+r^2}.$$ Thus, $q$ gives an anisotropy of the model. If $q=0$, $q>0$, and $q<0$, the models are isotropic, radially anisotropic, and tangentially anisotropic [@dej87]. The distribution function of Stäckel models with three integrals of motion ========================================================================== We show the DF for the axisymmetric Stäckel model, which depends on three integrals of motion [@dej88]. This model is special case that the DF, which depends on three integrals of motion, can be written in an analytical form. In this model, the total DF is given by the sum of the DF that depends on two integrals of motion ($F_1(E,I_2)$) and the DF that depends on three integrals of motion ($F_2(E,I_2,I_3)$). $F_1(E,I_2)$ is obtained by integrating density $\rho_1$ in three dimension velocity space, where $$\rho_1=\rho-\rho_2$$ $\rho$ corresponds density for the total of the model, and $\rho_1$ and $\rho_2$ are given by $$\rho_1=\int\int\int F_1(E,I_2) \mathrm{d}^3 \mbox{\boldmath$v$},~ \rho_2=\int\int\int F_2(E,I_2,I_3) \mathrm{d}^3 \mbox{\boldmath$v$}.$$ We use the Kuzumin-Kutuzov model [@kuz62] as a potential-density pair, which is given by $$\psi(R,z)=\frac{GM}{(R^2+z^2+a^2+c^2+2\sqrt{a^2 c^2+c^2 R^2+a^2 z^2})^{1/2}},$$ and $$\rho (R,z)=\frac{Mc^2}{4\pi} \frac{(a^2+c^2)R^2+2a^2 z^2+2a^2 c^2+a^4+3a^2\sqrt{a^2 c^2+c^2 R^2+a^2 z^2}}{(a^2 c^2+c^2 R^2+a^2 z^2)^{3/2}(R^2+z^2+a^2+c^2+2\sqrt{a^2 c^2+c^2 w^2+a^2 z^2})^{3/2}}. \label{kk_rho}$$ The DF that depends on two integrals of motion and represents a potential-density pair is given as $$\begin{aligned} %%uation} \label{eq:eq97_} F(E,I_2)=\frac{1}{(2\pi)^{5/2}}\frac{c^2}{4a}E^{5/2}&\sum_{k=0}^\infty(k+1)\frac{\Gamma(k+5)}{k+7/2}(aE)^k\Biggl( 2 \; _3F_2\left(\frac{k}{2}+\frac{5}{2},\frac{k}{2}+3,\frac{k}{2}+\frac{1}{2};k+\frac{7}{2},\frac{1}{2};2AEL_z^2 \right)\nonumber\\ &+(k+2) \; _3F_2\left(\frac{k}{2}+\frac{5}{2},\frac{k}{2}+3,\frac{k}{2}+\frac{3}{2};k+\frac{7}{2},\frac{1}{2};4AEI_2 \right) \Biggr).\end{aligned}$$The model can add the DF that depends on three integrals of motion ($F_2(E,I_2,I_3)$). $F_2(E,I_2,I_3)$ is a room to change a velocity dispersion of the model. To give the total density by equation (\[kk\_rho\]), it is required to subtract the DF that depends on two integrals of motion whose corresponding density distribution is same as $F_2(E,I_2,I_3)$ from the sum of $F_2(E,I_2,I_3)$ and $F(E,I_2)$ that is corresponding to the density of equation (\[kk\_rho\]). . In these models, $F_2(E,I_2,I_3)$ can be given by $$\label{eq:eq95_2} F_2(E,I_2,I_3)=\sum_{l,m,n}a_{lmn}E^lI_2^m(I_2+I_3)^n,$$ and we use the corresponding density distribution $$\rho(\lambda,\nu)=\sum_{l,m,n}a_{lmn}\rho_{lmn}(\lambda,\nu),$$ where the third integral of motion ($I_3$) in this potential is given as $$I_3=\frac{1}{2}(L^2-2I_2)+(a^2-c^2)\left(\frac{1}{2}v_z^2-z^2\frac{G(\lambda)-G(\nu)}{\lambda-\nu}\right),$$ and $$G(\tau)=\frac{GM}{c+\sqrt{\tau}}.$$ The density distribution that corresponds to the $F=E^lI_2^m(I_2+I_3)^n$ is given as $$\begin{aligned} \label{eq:eq99} \rho_{lmn}(R,z)&=&2^{3/2-n}\sqrt{\pi}n!\Gamma(l+1)R^{2m}\psi^{l+m-n+3/2}\sum_{k=0}^{n}\frac{1}{\Gamma(3/2+m+k)}\nonumber \\ &\times&\sum_{i=0}^{k/2}\frac{\Gamma(i+1/2)\Gamma(1/2+m+k-2i)}{i!(k-2i)!}\sum_{i_1=0}^{i}\left(\begin{array}{lcr}i\\i_1\end{array}\right)(2a)^{2i_1}\sum_{i_2=0}^{i_1}\left(\begin{array}{lcr}i_1\\i_2\end{array}\right)(-4a)^{i_2}\nonumber \\ &\times&\sum_{i_3=0}^{k-2i}\left(\begin{array}{lcr}k-2i\\i_3\end{array}\right)(-2a)^{i_3}\psi^{2i_1-i_2+i_3}(1-AR^2\psi^2)^{(i_2+i_3)/2}\nonumber \\ &\times&\sum_{j=0}^{n-k}\frac{\Gamma(m+n+3/2-j)}{\Gamma(l+m+n+5/2-j)}\frac{(-2c)^j}{j!(n-k-j)!}\sum_{j_1=0}^j\left(\begin{array}{lcr}j\\j_1\end{array}\right)\left(\frac{a^2-c^2}{c}\right)^{j_1}\nonumber \\ &\times&\sum_{j_2=0}^{j_1}\left(\begin{array}{lcr}j_1\\j_2\end{array}\right)(\frac{a}{c^2}-{a^2})^{j_2}\sum_{j_3=0}^{n-j-k}\left(\begin{array}{lcr}n-j-k\\j_3\end{array}\right)[2(a^2-c^2)]^{j_3}\nonumber \\ &\times&\sum_{j_4=0}^{j_3}\left(\begin{array}{lcr}j_3\\j_4\end{array}\right)(-a)^{j_4}\psi^{j+j_1-j_2+2j_3-j_4}(1-AR^2\psi^2)^{(j_2+j_4)/2}.\end{aligned}$$ Moreover, equation (\[eq:eq99\]) can be rewritten as $$\rho_{lmn}(R,\psi)=\sum_{p_1=l+m-n+3/2}^{l+m+n+3/2} \sum_{p_2=0}^{n}a_{lmn}B_{p_1,p_2}R^{2m}\psi^{p_1}(1-AR^2\psi^2)^{p_2/2}.$$ The DF that depends on two integral of motions and corresponds to this density is given as $$\begin{aligned} \label{eq:eq96_} F_{lmn}(E,I_2)=\frac{1}{\sqrt{2}\pi}\sum_{p1,p2}B_{p_1,p_2}\frac{\Gamma(p_1+1)} {\Gamma(p_1-m-1/2)\Gamma(m+1/2)}E^{p_1-m-3/2}I_2^m\nonumber\\ \; _3F_2\left(\frac{1+p_1}{2},1+\frac{p_1}{2},\frac{-p_2}{2};p_1-\frac{m-1}{2},\frac{1}{2}+m;4AEI_2\right).\end{aligned}$$ $F_1(E,I_2)$ is given by equation (\[eq:eq97\_\]) minus equation (\[eq:eq96\_\]), and $F_2(E,L_z,I_3)$ is given by equation (\[eq:eq95\_2\]). Thus, a self-consistent model depending on three integrals of motion can be represented analytically. \[lastpage\]

--- abstract: 'We report the results of experiments with Bose-Einstein condensates of rubidium atoms in a triaxial TOP-trap, presenting measurements of the condensate fraction and the free expansion of a condensate released from the trap. The experimental apparatus and the methods used to calibrate the magnetic trapping fields are discussed in detail. Furthermore, we compare the performance of our apparatus with other TOP-traps and discuss possible limiting factors for the sizes of condensates achievable in such traps.' address: 'INFM, Dipartimento di Fisica, Università di Pisa, Via Buonarroti 2, I-56127 Pisa, Italy' author: - 'J H Müller, D Ciampini, O Morsch, G Smirne, M Fazzi, P Verkerk, F Fuso, and E Arimondo' title: 'Bose-Einstein condensation of rubidium atoms in a triaxial TOP-trap' --- Introduction ============ Since the first observations of Bose-Einstein condensation (BEC) in dilute alkali gases [@wieman; @hulet; @ketterle], experimental as well as theoretical studies of degenerate quantum gases have been published at an astonishing rate [@ketterle2; @stringari]. Far beyond the mere realization and detection of BEC, experimenters have investigated the static and dynamic properties of Bose-Einstein condensates and have gained considerable control over these macroscopic quantum objects, up to the point of creating coherent beams of matter waves - atom lasers, in other words. In spite of these early successes, experimental BEC is still a growing and thriving field, and much research needs to be done in order to test the vast number of theoretical predictions made in the last few years.\ In this paper, we present the experimental apparatus used to create BECs of rubidium atoms in a triaxial time-orbiting-potential (TOP) trap [@toptrap]. To the best of our knowledge, while the triaxial TOP trap has been used in BEC experiments on sodium [@hagley], no previous application to rubidium has been reported. We describe in some detail the experimental parameters of our system and compare the performance of our apparatus with those of other groups using similar setups. Section 2 presents the experimental set-up, with emphasis on the original parts for the rubidium cooling and transfer between the two magneto-optical traps. Section 3 reports the parameters for the loading and evaporative cooling phases required to produce the condensate. Moreover, the gain in phase-space density achieved during the evaporation phases has been measured. In the following sections the results of various measurements on the condensate are reported. The final phase-space density, number of atoms and temperatures associated to the different condensates are presented. Furthermore, the expansion of the condensate following a switch-off of the magnetic trap has been studied and compared to different theoretical models. Finally, we describe different methods used for precise measurements of the magnetic fields. In this way, we obtained an accurate calibration which was needed as an input parameter for a theoretical model simulating the motion of the atomic cloud [@micromopaper]. Experimental setup ================== Our experimental apparatus is based on a double-MOT system with a TOP-trap. The design of the vacuum system and the positioning of the coils are shown in figure \[setup\]. Owing to the arrangement of the quadrupole coils and the TOP-coils, our trap is triaxial without cylindrical symmetry. In the following, we give a brief overview of the specifications of our system.\ [*Vacuum system:*]{} Our vacuum system is composed of two quartz cells connected by a glass tube of inner diameter $12\,\mathrm{mm}$ and length $20\,\mathrm{cm}$ (see figure \[setup\]). At the upper end of the glass tube, a graphite tube of length $6\,\mathrm{cm}$ and inner diameter $5\,\mathrm{mm}$ is inserted in order to enhance differential pumping. The upper cell is connected to a $20\,\mathrm{ls^{-1}}$ ion pump, whereas the lower cell is pumped on by a $40\,\mathrm{ls^{-1}}$ ion pump in conjunction with a Ti-sublimation pump. In this way, a pressure gradient is created between the two cells with the pressure in the upper cell being of the order of $10^{-8}\,\mathrm{Torr}$ and that of the lower cell below $10^{-10}\,\mathrm{Torr}$. The upper cell also contains two Rb dispensers (SAES getters) which we operate at $3.0\,\mathrm{A}$.\ [*Lasers:*]{} The laser light for the upper and lower MOTs is derived from a MOPA (tapered amplifier) injected in turn by a $50\,\mathrm{mW}$ diode laser. Under typical conditions we extract up to $320\,\mathrm{mW}$ of useful output from this system, which is then frequency-shifted by acousto-optic modulators (AOMs) and mode-cleaned by optical fibres. In this way, we create up to $60\,\mathrm{mW}$ of laser power for the upper MOT and $15\,\mathrm{mW}$ for the lower MOT. The repumping light for both the upper and the lower MOT is derived from a $75\,\mathrm{mW}$ diode laser, yielding about $9\,\mathrm{mW}$ of total power after passage through all the optical elements. The injecting laser for the MOPA and the repumping laser are both injected by $50\,\mathrm{mW}$ grating stabilized diode lasers locked to Rb absorption lines.\ [*Magnetic trap:*]{} Our TOP-trap consists of a pair of quadrupole coils capable of producing field gradients $2b^{\prime}$ (along the symmetry axis) in excess of $1000\,\mathrm{Gcm^{-1}}$ for maximum currents of about $230\,\mathrm{A}$, and two pairs of TOP-coils. The quadrupole coils are water-cooled and are oriented horizontally (along the $x$-axis, see fig. \[setup\]) about the lower glass cell of our apparatus. A combination of IGBTs and varistors is used for fast switching of the current provided by a programmable current source (HP6882) whilst protecting the circuits from damage due to high voltages induced during switch-off. In this way we are able to switch off the quadrupole field within less than $50\,\mathrm{\mu s}$ even for the largest field gradients. The rotating bias field $B_{0}$ is created by two pairs of coils: One (circular) pair is incorporated into the quadrupole coils, whilst the other (rectangular) pair is mounted along the $y$-axis. Within the adiabatic and harmonic approximations, for an atom with mass $m$ and magnetic moment $\mu$ this results in a triaxial time-orbiting potential $V_{TOP}$ given by $$V_{TOP}=\frac{4 \pi^{2}m}{2}\left(\nu_{x}^{2}x^{2}+\nu_{y}^{2}y^{2}+\nu_{z}^{2}z^{2}\right)$$ with the following frequencies along the three axes of the trap in the ratio $2:1:\sqrt{2}$, as introduced in [@hagley]: $$\begin{aligned} \nu_{x}&=&\frac{1}{2\pi} \sqrt{\frac{2\mu} {mB_{0}} } b^{\prime}\\ \nu_{y}&=&\frac{1}{2\pi} \sqrt{\frac{\mu} {2mB_{0}} } b^{\prime}\\ \nu_{z}&=&\frac{1}{2\pi}\ \sqrt{\frac{\mu} {mB_{0}} } b^{\prime}. \label{frequencies}\end{aligned}$$ The anharmonic and gravitational effects neglected in this approximation will be discussed in section 5. The TOP-coils in our experiment can produce a bias field $B_{0}$ of up to $30\,\mathrm{G}$ and are operated at a frequency of $10\,\mathrm{kHz}$.\ [*Imaging:*]{} Detection of the condensates is done by shadow imaging using a near-resonant probe beam. The absorptive shadow cast by the atoms is imaged onto a CCD-camera. With a camera pixel size of $9\,\mathrm{\mu m}$ and a magnification of about $1.2$, we achieve a resolution of just over $7\,\mathrm{\mu m}$. Most of our measurements are made after a few milliseconds of free fall of the released condensate, when typical dimensions are of the order of $10-30\,\mathrm{\mu m}$. Evaporative cooling and creation of the condensate ================================================== A typical experimental cycle from the initial collection of atoms in the upper MOT to the creation of a BEC is as follows. First, we load about $5\times10^7$ $\mathrm{Rb}$ atoms into the lower MOT by repeatedly (up to 200 times) loading the upper MOT for $\approx 160\,\mathrm{ms}$ and then flashing on a near-resonant push beam that accelerates the atoms down the connecting tube. Once the lower MOT has been filled, a $30\,\mathrm{ms}$ compressed-MOT phase increases the density of the cloud, which is then cooled further to about $15\,\mathrm{\mu K}$ by a molasses phase of a few milliseconds. At this point, the molasses beams are switched off and an optical pumping beam is flashed on five times for $20\,\mathrm{\mu s}$, synchronized with the rotating bias field of $1\,\mathrm{G}$ to define a quantization axis, in order to transfer the atoms into the $|F=2,m_F=2\rangle$ Zeeman substate desired for magnetic trapping. Transfer into the TOP-trap is then effected by simultaneously switching on the rotating bias field (at its maximum value of about $25\,\mathrm{G}$) and the quadrupole field (at a value for the gradient chosen such as to achieve mode-matching between the initial cloud of atoms and the resulting magnetic trap frequencies). The subsequent evaporative cooling ramps for the quadrupole and the bias fields are shown schematically in figure \[ramps\]. After an adiabatic compression phase, during which the quadrupole gradient is increased to its maximum value, the bias field amplitude is ramped down linearly. In this way, we perform circle-of-death evaporative cooling down to a bias field of around $4\,\mathrm{G}$. Next, at a constant bias field, we switch on a radio-frequency field, scanning its frequency exponentially from $6.5\,\mathrm{MHz}$ down to around $3.2\,\mathrm{MHz}$, which we find to be the threshold for condensation for our system. At threshold, we have up to $3\times10^4$ atoms in the condensate/thermal cloud-conglomerate. Continuing rf-evaporation still further yields pure condensates of up to $1-2\times10^4$ atoms with no discernible thermal fraction. The value for the bias field at which we switch from circle-of-death to rf-evaporation was chosen by maximizing the final condensate number. The approach to BEC is illustrated graphically in figure \[phasedens\], in which the phase-space density is plotted as a function of the number of atoms.\ Before imaging the condensate, we adiabatically change the trap frequency by ramping the bias field and the quadrupole gradient in $200\,\mathrm{ms}$. In this way, we can choose the frequency of the trap in which we wish to study the condensate. Thereafter, both fields are switched off on a timescale of $20-50\,\mathrm{\mu s}$ for the quadrupole field and $100-200\,\mathrm{\mu s}$ for the bias field. Owing to these short timescales, the change in trap frequency during the switching can essentially be neglected as typical oscillation periods in the trap are larger than $10\,\mathrm{ms}$. In fact, we were able to observe non-adiabatic motion of the trapped condensates at the frequency of the rotating bias field [@micromopaper]. Experimental results ==================== In the following, we briefly summarize some initial measurements made on the condensates obtained with our apparatus. Evidence for condensation and condensate fraction ------------------------------------------------- In order to find the threshold for condensation, the RF-frequency in the final evaporation step is lowered whilst monitoring the properties of the atom cloud (through shadow imaging after $3\,\mathrm{ms}$ of free expansion). At the threshold, the tell-tale signs of condensation, namely a sudden increase in peak density and the onset of a bimodal distribution, begin to appear. Figure \[condfrac\] shows plots of the peak density normalized with respect to the number of atoms (which removes the considerable experimental jitter especially in the condensed regime) and the condensate fraction as a function of the final RF-frequency. The condensate fraction is determined from a bimodal fit to single pixel rows of the absorption picture, and it is evident in the two plots that condensation sets in at a final frequency of about $3.2\,\mathrm{MHz}$, corresponding to a temperature of $365\,\mathrm{nK}$ as calculated from the ballistic expansion of the cloud, and a peak density of $\approx 5\times 10^{11}\,\mathrm{cm^{-3}}$. From this, we calculate a phase-space density of $2.5$ at the threshold, in agreement with theoretical predictions. Using the expression $k_B T_0=\hbar\bar{\omega}(N/\zeta(3))^{1/3}$ (valid in the non-interacting approximation and with $\bar{\omega}$ equal to the geometric mean of the three trap frequencies) with $N=10^4$ atoms at the threshold [@kasevich], we find $T_0\approx 400\,\mathrm{nK}$ in good agreement with our observed threshold temperature.\ We note here that, unlike in the case of a static trap, for a TOP-trap there is no strict proportionality between $\nu_{cut}-\nu_0$ and $k_B T_{cut}$, where $\nu_{cut}$ is the frequency of the RF-field, $\nu_0$ is the resonance frequency at the bottom of the trap, and $T_{cut}$ is the equivalent cut temperature. A simple calculation considering the maximum instantaneous field at the resonance shell shows that, for low temperatures, $$\nu_{cut}-\nu_0 = \frac{g_F }{h} \bigl(2k_B T_{cut}\mu B_0\bigr)^{1/2}.$$ This geometric average between the thermal cut energy $k_B T_{cut}$ and the magnetic energy in the bias field $\mu B_0$ of the TOP-trap leads to a considerably more accurate control of the cut energy in a TOP-trap. For instance, at a bias field of $B_0=4\,\mathrm{G}$, a frequency difference $\nu_{cut}-\nu_0$ of $350\,\mathrm{kHz}$ corresponds to a cut energy $T_{cut}$ of only $1.2\,\mathrm{\mu K}$, whereas the same frequency difference in a static trap leads to $T_{cut}=34\,\mathrm{\mu K}$. Free expansion of the condensate -------------------------------- One way of obtaining information on the properties of a Bose-Einstein condensate is to investigate its behaviour after it is released from the trap. Its subsequent evolution is then monitored by taking absorption images after a variable time-of-flight. The results of such measurements on a condensate released from a trap with $\nu_z = 363\,\mathrm{Hz}$ are shown in figure \[expansion\]. Theoretically, the expansion of a condensate has been investigated by several authors, and analytical expressions for the condensate width and its aspect ratio as a function of time can be found in special cases. Figure \[expansion\] shows the predictions of a model based on the Thomas-Fermi approximation [@castin+dum], in which the energy of the condensate is dominated by the mean-field interaction between the atoms, as well as the theoretical expansion of a ground-state harmonic oscillator wavefunction, for which interactions are neglected entirely. Clearly, our experimental data agree with neither of these two extremes. This is to be expected, as the sizes of our condensates, with typically a few thousand atoms in a pure condensate, are rather small and therefore do not fully satisfy the conditions for a Thomas-Fermi treatment. It is, therefore, necessary to compare our data with a numerical integration of the full Gross-Pitaevskii equation. The results of such an integration are also plotted in figure \[expansion\]. As expected, they lie between the two extreme models and fit our data reasonably well. It is clear, however, that our condensate number is so low that the interaction term in the Gross-Pitaevskii equation is almost negligible and the numerical results are close to the pure harmonic oscillator case. Calibration of the magnetic fields ================================== In many applications of magnetic traps, it is sufficient to describe the trap by its characteristic frequencies for dipolar oscillations of atomic clouds. In such a measurement, one applies a magnetic field along a chosen axis for a short time, thus giving a kick to the (initially stationary) atomic cloud, and monitors the subsequent oscillations of the atoms. With a judicious choice of the points in time at which the position of the cloud is sampled, one can achieve frequency measurements with uncertainties well below the percent level. Deducing absolute values of the magnetic field gradient and the bias field from these measurements with similar accuracy, however, is not so straightforward. The main incentive for us to accurately measure these absolute values was that we needed them as input parameters for numerical simulations of non-adiabatic motion in the TOP-trap [@micromopaper]. In the following, we shall briefly describe several methods we used to measure absolute values for both the quadrupole gradient and the bias field and indicate the uncertainties associated with these measurements. For the most part, the measurements were carried out with condensates, which facilitated the determination of the position of the atomic cloud.\ In the first method, we measure the vibrational frequencies $\tilde{\nu_x}$ and $\tilde{\nu_z}$ along the $x$- and $z$-axes, respectively, exciting the dipolar modes along these two directions simultaneously. In order to be able to use theoretical formulas derived in the harmonic approximation taking into account the effect of gravity, we have calculated the anharmonic corrections up to fourth order, including cross-terms, following the scheme presented by Ensher [@ensher]. The results reported in Appendix A allow us to deduce from our measured frequencies the corresponding values in the harmonic limit (equivalent to infinitesimal oscillation amplitudes; typical amplitudes in our experiment are between $20\,\mathrm{\mu m}$ and $60\,\mathrm{\mu m}$.). Those anharmonic corrections can be up to $1\%$ of the measured values and are, therefore, essential if an accuracy in the magnetic field below the percent level is desired. The quadrupole gradient can be calculated directly from the ratio $\tilde{\nu_x}/{\tilde\nu_z}$ given by in the harmonic approximation with the gravitational corrections by $$\frac{\tilde{\nu_x}}{\tilde{\nu_z}}=\sqrt{2}\sqrt{\frac{1+\eta^2}{1-\eta^2}}$$ Here, $\eta$, defined by $$\eta=\frac{\mu b'}{mg}$$ measures the ratio of magnetic and gravitational forces along the $z$-axis. It is interesting to note that in the triaxial TOP the gravity corrections are equal to those derived for a cylindrically symmetric TOP-trap [@ensher]. Re-substitution of the value for $b'$ thus retrieved along with either of the two frequencies into the expression for $\tilde{\nu_x}$ or $\tilde{\nu_z}$ then yields a value for $B_0$. For instance, $\tilde{\nu_z}$ is given by $$\tilde{\nu_z}=\frac{1}{2\pi}\sqrt{\frac{\mu}{mB_{0}}}b'\left(1-\eta^2\right)^{3/ 4}.$$ In order to check the obtained values for $B_0$ and $b'$ by independent methods not relying on the calculated frequencies for a TOP-trap, we use two separate strategies. In one method, the quadrupole gradient is measured by first trapping and evaporatively cooling atoms in the presence of both the quadrupole and the bias fields. Then, the bias field is switched off, which shifts the centre of the quadrupole potential with respect to the TOP-potential. The quadrupole gradient is subsequently determined by measuring the acceleration of the atoms and subtracting the acceleration due to gravity. In this way, $b'$ can be determined with a relative error of less than $1\%$. An independent measurement of the bias field $B_0$ is made by switching off the quadrupole field after the atoms have been cooled in the TOP whilst leaving the bias field on. A short ($100-500\,\mathrm{\mu s}$) RF-pulse is then applied to the atoms at a given frequency, and the number of atoms remaining in the original trapped state is measured after turning the quadrupole field back on (about $1\,\mathrm{ms}$ after switching it off). When the frequency of the RF-pulse matches the Zeeman-splitting due to the bias field, atoms are transferred into untrapped Zeeman-substates and hence lost from the trap. Using this method, we found two different values of the RF-pulses for which atoms were lost from the trap, indicating that there is a slight asymmetry between the magnetic fields produced by the two pairs of TOP-coils. Measuring $B_0$ with this method proved to be less reliable than with the method described above, but yielded the same value for the bias field to within $5\%$.\ Condensate numbers in TOP-traps =============================== In our experimental apparatus, we obtain condensates containing up to a few $10^4$ atoms, starting from typical MOT numbers of about $5\times10^7$. Extrapolating this linearly, one would expect to achieve condensate numbers of up to $10^6$ for an initial number of $5\times10^9$ atoms in the MOT. In the literature, however, one typically finds reports of some $10^5$ atoms in the condensate under such circumstances. In figure \[loading\] we have plotted typical figures for the MOT and the condensate numbers for a few groups using rubidium TOP-traps. Evidently, the reported condensate numbers do not scale linearly with the MOT numbers. Instead, they can be fitted roughly by a square-root law. Varying the MOT numbers in our own experiment, we find a similar behaviour on a smaller scale. We discovered this when trying to increase the size of our condensates and found that the main limiting factor comes from the compression phase after loading the magnetic trap. Above a certain number of atoms loaded into the MOT, we saw next to no increase in the atom number after compression (or, for that matter, in the condensate) when increasing the initial number of atoms. As in [@han], we attribute this to an unfavourable ratio of the size of the initial cloud and the circle-of-death radius. When the cloud becomes too big, the circle-of-death cuts into it during compression and thus any increase in the atom number is eaten up by this cutting. This may be a limiting mechanism for most groups and could explain the law of diminishing returns that is evident in figure \[loading\]. In this context it is interesting to note that, for instance, the JILA group uses a much higher bias field ($50\,\mathrm{G}$) than most other groups and achieves a much better transfer efficiency from the MOT to the condensate [@cornell], obtaining condensates of $\approx 10^6$ atoms for initial numbers of the order of $2\times 10^8$. Although this may suggest that a larger bias field is the answer, it is not clear whether there are other effects that limit the transfer efficiencies achievable in TOP-traps. ![\[loading\]Typical condensate numbers of various groups as a function of initial MOT numbers. The data were taken from publications of the groups at Austin [@han], Otago [@otago], and Oxford [@oxford].](loading2.eps){width="80.00000%"} Conclusion ========== We have presented the results of preliminary measurements on Bose-Einstein condensates of rubidium atoms obtained in a triaxial TOP-trap. Our experimental data for the condensation threshold and the free expansion of the condensate agree well with theoretical predictions. Increasing the number of atoms in our condensates will allow us to further improve on the quality of our data and investigate the properties of the condensates in more detail. Acknowledgments {#acknowledgments .unnumbered} =============== O.M. gratefully acknowledges financial support from the European Union (TMR Contract-Nr. ERBFMRXCT960002). This work was supported by the INFM ’Progetto di Ricerca Avanzata’ and by the CNR ’Progetto Integrato’. The participation of G. Memoli and D. Wilkowski in the early stages of the experiment is gratefully acknowledged. The authors are grateful to R. Mannella for the numerical integration of the Gross-Pitaveskii equation and to M. Anderlini for help in the calculation of the anharmonic corrections. Anharmonic corrections in the TOP-trap ====================================== For our calibration measurements, we deduced the frequencies in the harmonic limit from the anharmonic corrections in the triaxial TOP-trap. Terms containing the amplitude of the oscillations along the $y$-axis have not been calculated as we do not excite oscillations along that direction, but can be obtained in the same manner. The expressions for the frequencies along the axis $i$ ($i=x,z$) are then $$\nu_i^{\rm anh}=\nu_{i}+\Delta\nu_i\quad ; \quad \Delta\nu_i=\left(\frac{b'}{B_0}\right)^2\sum_j\alpha_{ij}a^2_j$$ where $\nu_{i}$ is the frequency in the harmonic approximation, as given by Eqs. \[frequencies\] and $a_j$ is the amplitude of the oscillation in the $j$-direction. The elements $\alpha_{ij}$ of the anharmonic correction matrix are given by $$\begin{aligned} \alpha_{xx}=\frac{\nu_{x}}{4}\left[6\frac{1-\eta^2}{1+\eta^2}\left(2-3\eta^2 -\frac{15}{8}(1-\eta^2)^2\right)-\frac{\eta^2(1-3\eta^2)^2}{18(1+\eta^2)}\right] \\ \alpha_{xz}=\frac{\nu_{x}}{4}\left[\frac{7-\eta^2(18-15\eta^2)}{12}-\frac{9 \eta^2(3\eta^2-1)(14+8\eta^2)-8\eta^2}{36(7+9\eta^2)}\right] \\ \alpha_{zx}=\frac{\nu_{z}}{2}\left[\frac{1-3\eta^2}{2}-\frac{2\eta^2(1-3\eta ^2)}{9(3+5\eta^2)}+\frac{7-3\eta^2-15\eta^2(1-\eta^2)}{12}\right] \\ \alpha_{zz}=\frac{\nu_{z}}{2}\left[-\frac{3}{8}(1-\eta^2)(1-5\eta^2)-\frac{1 5}{8}\eta^2(1-\eta^2)\right].\end{aligned}$$ [100]{} Anderson M H, Ensher J R, Matthews M R, Wieman C E and Cornell E A 1995 [*Science*]{} [**269**]{} 198 Bradley C C, Sackett C A, Tollett J J and Hulet R G 1995 Phys. Rev. Lett. [**75**]{} 1687 Davis K B, Mewes M O, Andrews M R, van Druten N J, Durfee D S, Kurn D M and Ketterle W 1995 Phys. Rev. Lett. [**75**]{} 3969 for reviews see Ketterle W, Durfee D S, and Stamper-Kurn D M, in [*Proceedings of Int. School “Enrico Fermi”, Course CXL*]{}, eds. Inguscio M, Stringari S, and Wieman C E, (IOS, Amsterdam 1999) pag. 67; Cornell E A, Ensher J R, and Wieman CE, in [*ibidem*]{} pag. 177 Dalfovo F, Giorgini S, Pitaevskii L P and Stringari S 1999 Rev. Mod. Phys. [**71**]{} 463 Petrich W, Anderson M H, Ensher J R and Cornell E A 1995 Phys. Rev. Lett. [**74**]{} 3353 Hagley E W, Deng L, Kozuma M, Wen J, Helmerson K, Rolston S L and Phillips W D 1999 [*Science*]{} [**283**]{} 1706 Müller J H, Morsch O, Ciampini D, Anderlini M, Mannella R and Arimondo E 2000 [*to be submitted*]{} Anderson B P and Kasevich M A 1999 Phys. Rev. A [**59**]{} R938 Castin Y and Dum R 1996 Phys. Rev. Lett. [**77**]{} 5315 Ensher J R, Ph.D. Thesis 1998, University of Colorado Han D J, Wynar R H, Courteille Ph and Heinzen D J 1998 Phys. Rev. A [**57**]{} R4114 Cornell E 2000 [*private communication*]{} Martin J L, McKenzie C R, Thomas N R, Sharpe J C, Warrington D M, Manson P J, Sandle W J and Wilson A C 1999 J. Phys.B: At. Mol. Opt. Phys. [**32**]{} 3065 Arlt J, Maragò O, Hodby E, Hopkins S A, Hechenblaikner G, Webster S and Foot C J 1999 J. Phys.B: At. Mol. Opt. Phys. [**32**]{} 5861

--- abstract: 'Dihadron spectra in high-energy heavy-ion collisions are studied within the next-to-leading order perturbative QCD parton model with modified jet fragmentation functions due to jet quenching. High-$p_T$ back-to-back dihadrons are found to originate mainly from jet pairs produced close and tangential to the surface of the dense matter. However, a substantial fraction also comes from jets produced at the center with finite energy loss. Consequently, high-$p_T$ dihadron spectra are found to be more sensitive to the initial gluon density than the single hadron spectra that are more dominated by surface emission. A simultaneous $\chi^2$-fit to both the single and dihadron spectra can be achieved within a range of the energy loss parameter $\epsilon_0=1.6-2.1$ GeV/fm. Because of the flattening of the initial jet production spectra at $\sqrt{s}=5.5$ TeV, high $p_T$ dihadrons are found to be more robust as probes of the dense medium.' author: - 'Hanzhong Zhang$^a$, Joseph F. Owens$^b$, Enke Wang$^a$ and Xin-Nian Wang$^c$' title: | Dihadron Tomography of High-Energy Nuclear Collisions\ in Next-to-Leading Order perturbative QCD --- Jet quenching as discovered in high-energy heavy-ion collisions at the Relativistic Heavy-ion Collider (RHIC) is manifested in both the suppression of single inclusive hadron spectra at large transverse momentum ($p_T$) [@phenix] and the disappearance of the typical back-to-back jet structure in dihadron correlation in vacuum [@star]. Since jet quenching is caused by parton energy loss which in turn depends on the gluon density and transport coefficient of the medium, detailed study of the suppression of large $p_T$ hadron spectra and correlations can shed light on the properties of the dense medium [@review] . In heavy-ion collisions, the spatial distribution of the initial jet production points is given by the nuclear overlap, $T_{AB}({\bf b},{\bf r})=t_A(r)t_B(|{\bf b}-{\bf r}|)$ with $t_A(r)$ being the thickness function of each nucleus. The suppression factor of the leading hadrons from jet fragmentation will depend on the total parton energy loss which in turn is related to the weighted gluon density integrated along the jet propagation path. Therefore, measurements of large $p_T$ hadron suppression can be directly related to the averaged gluon density and medium transport coefficient. As the average gluon density increases with colliding energy and centrality, one should expect continued decrease of the suppression factor until particle production in the outer corona of the medium dominates the single hadron spectra [@eskola]. In this case, the suppression factor of high-$p_T$ hadron spectra will lose its effectiveness as a sensitive probe since it only depends on the thickness of the outer corona which varies very slowly with the initial gluon density. In this Letter we will employ for the first time a next-to-leading order (NLO) perturbative QCD (pQCD) parton model to study the suppression of both single and dihadron spectra due to jet quenching. In particular, we will investigate the robustness of back-to-back dihadron spectra as a probe of the initial gluon density when single hadron spectra suppression become fragile. Within the NLO pQCD parton model, large $p_T$ hadron production cross section in $p+p$ collisions can be expressed as a convolution of NLO parton-parton scattering cross sections, parton distributions inside nucleons and parton fragmentation functions (FF). The calculations discussed in this Letter are carried out within a NLO Monte Carlo based program [@owens] which utilizes two-cutoff parameters, $\delta_s$ and $\delta_c$, to isolate the soft and collinear divergences in the squared matrix elements of the $2\rightarrow 3$ processes. The regions containing the divergences are integrated over in n-dimensions and the results are combined with the squared matrix elements for the $2\rightarrow 2$ processes. This results in a set of two-body and three-body weights, each of which depends on the cut-offs. However, this dependence cancels when the weights are combined in the calculation of physical observables. For the study of large $p_T$ single and dihadron production in $A+A$ collisions, we assume that the initial hard scattering cross sections are factorized as in $p+p$ collisions. As in the lowest-order (LO) pQCD parton model study [@Wang:2003mm], we further assume that the effect of final-state interaction between produced parton and the bulk medium can be described by the effective medium-modified FF’s, $$\begin{aligned} D_{h/c}(z_c,\Delta E_c,\mu^2) &=&(1-e^{-\langle \frac{L} {\lambda}\rangle}) \left[ \frac{z_c^\prime}{z_c} D^0_{h/c}(z_c^\prime,\mu^2) \right. \nonumber \\ & &\hspace{-0.9in} \left. + \langle \frac{L}{\lambda}\rangle \frac{z_g^\prime}{z_c} D^0_{h/g}(z_g^\prime,\mu^2)\right] + e^{-\langle\frac{L}{\lambda}\rangle} D^0_{h/c}(z_c,\mu^2), \label{eq:modfrag}\end{aligned}$$ where $z_c^\prime=p_T/(p_{Tc}-\Delta E_c)$, $z_g^\prime=\langle L/\lambda\rangle p_T/\Delta E_c$ are the rescaled momentum fractions, $\Delta E_c$ is the average radiative parton energy loss and $\langle L/\lambda\rangle$ is the number of scatterings. The FF’s in vacuum $D^0_{h/c}(z_c,\mu^2)$ are given by the KKP parameterization [@KKP]. According to recent theoretical studies [@gvw; @ww; @sw02] the total parton energy loss in a finite and expanding medium can be approximated as a path integral, $$\Delta E \approx \langle \frac{dE}{dL}\rangle_{1d} \int_{\tau_0}^{\infty} d\tau \frac{\tau-\tau_0}{\tau_0\rho_0} \rho_g(\tau,{\bf b},{\bf r}+{\bf n}\tau),$$ for a parton produced at a transverse position ${\bf r}$ and traveling along the direction ${\bf n}$. $\langle dE/dL\rangle_{1d}$ is the average parton energy loss per unit length in a 1-d expanding medium with an initial uniform gluon density $\rho_0$ at a formation time $\tau_0$ for the medium gluons. The energy dependence of the energy loss is parameterized as $$\langle\frac{dE}{dL}\rangle_{1d}=\epsilon_0 (E/\mu_0-1.6)^{1.2} /(7.5+E/\mu_0), \label{eq:loss}$$ from the numerical results in Ref. [@ww] in which thermal gluon absorption is also taken into account. The parameter $\epsilon_0$ should be proportional to $\rho_0$. Neglecting transverse expansion, the gluon density distribution in a 1-d expanding medium in $A+A$ collisions at impact-parameter ${\bf b}$ is assumed to be proportional to the transverse profile of participant nucleons , $$\rho_g(\tau,{\bf b},{\bf r})=\frac{\tau_0\rho_0}{\tau} \frac{\pi R^2_A}{2A}[t_A({\bf r})+t_A(|{\bf b}-{\bf r}|)].$$ The average number of scatterings along the parton propagating path is $$\langle L/\lambda\rangle =\int_{\tau_0}^{\infty} \frac{d\tau}{\rho_0\lambda_0} \rho_g(\tau,{\bf b},{\bf r}+{\bf n}\tau).$$ Here, we neglect the time-dependence of the cross section and characterize it by the mean free path $\lambda_0$ via $\sigma_0=1/(\rho_0\lambda_0)$ and a hard-sphere nuclear distribution is used. The parton distributions per nucleon $f_{a/A}(x_a,{\bf r})$ inside the nucleus are assumed to be factorizable into the parton distributions in a nucleon given by CTEQ6M parameterization [@CTEQ] and the new HIJING parameterization [@lw02] of the impact-parameter dependent nuclear modification factor, including the isospin dependence. The calculated single inclusive $\pi^0$ and $\pi^+ + \pi^-$ spectra in the NLO pQCD parton model in $p+p$ collisions agree well with the experimental data at the RHIC energy with the factorization scale in the range $\mu=0.9 \sim 1.5 p_T$ [@zoww]. We use the same factorization scale in both $p+p$ and $A+A$ collisions in our calculation. Shown in Fig. \[fig:rau0-10\] are the nuclear modification factors, $$\begin{aligned} R_{AA}=\frac{d\sigma_{AA}/dp_T^2dy} {\int d^2b\, T_{AA}({\bf b})d\sigma_{NN}/dp_T^2dy}\,, \label{eqn:modifactoer}\end{aligned}$$ for single inclusive $\pi^0$ spectra calculated in both leading-order (LO) and NLO pQCD parton model as compared to the PHENIX data on central $Au+Au$ collisions at $\sqrt{s}=200$ GeV. The factorization scale in the NLO result is set $\mu=1.2 p_T$ though the nuclear modification factor $R_{AA}$ is not at all sensitive to the choice of $\mu$. However, $R_{AA}$ in NLO calculation is always smaller than the LO result because of the relative larger ratio of gluon/quark jets in NLO than in LO calculation and gluon energy loss is assumed to be 9/4 larger than that of a quark. In both calculations, we have chosen the parameters as $\mu_0=1.5$ GeV, $\epsilon_0\lambda_0=0.5$ GeV and $\tau_0=0.2$ fm/$c$. The results are not sensitive to small values of $\tau_0$. ![\[fig:rau0-10\] (color online). Nuclear modification factors for $\pi^0$ spectra in LO (dashed) and NLO pQCD (solid) in $Au+Au(0-10\%)$ collisions at $\sqrt{s}=200$ GeV with different values of energy loss parameter $\epsilon_0=1.48$, 1.68 and 2.08 GeV/fm (from top to bottom) compared with data [@akiba].](fig1.eps){width="85mm"} Because of jet quenching, the dominant contribution to the measured single hadron spectra at large $p_T$ comes from those jets that are initially produced in the outer corona of the overlap region toward the direction of the detected hadron. This is clearly illustrated in Fig. \[fig:con-sin-y0\] by the spatial distribution of the production points of those jets that have survived the interaction with the medium and contribute to the measured spectra. Jets produced in the region away from the detected hadron are severely suppressed due to their large energy loss and don’t contribute much to the final hadron spectra. As pointed out in Ref. [@eskola], when jets produced in the inner part of the overlapped region are completely suppressed due to large initial gluon density, the final large $p_T$ hadron production is dominated by “surface emission”. High-$p_T$ hadron yield via such surface emission should be proportional to the thickness of the outer corona which decreases with the initial gluon density. Therefore, the suppression factor for single hadron spectra should continue to decrease with the initial gluon density as shown in Fig. \[fig:eps-dep\]. The variation, nevertheless, is very weak when surface emission becomes dominant and single hadron suppression is no longer a sensitive probe of the initial gluon density. ![\[fig:con-sin-y0\] (color online). Spatial transverse distribution (arbitrary normalization) of the initial parton production points that contribute to the final $\pi^0$ at $8 < p_T < 15$ GeV along $\phi=\pi/2$. The insert is the same distribution projected onto the $y$-axis.](fig2.eps){width="85mm"} To find robust probes of the high initial gluon density when single hadron spectra suppression becomes fragile, we will study back-to-back dihadron spectra within NLO pQCD parton model in this Letter. To quantify dihadron spectra, a hadron-triggered fragmentation function, $$D_{AA}(z_T,p_T^{\rm trig}) \equiv p_T^{\rm trig}\frac{ d\sigma^{h_1h_2}_{AA}/dy^{\rm trig} dp^{\rm trig}_T dy^{\rm asso}dp_T^{\rm asso}} {d\sigma^{h_1}_{AA}/dy^{\rm trig}dp^{\rm trig}_T}\,, %\label{Daa-tmp}$$ was introduced [@Wang:2003mm] as a function of $z_T=p^{\rm asso}_T/p^{\rm trig}_T$, which is essentially the away-side hadron spectrum associated with a triggered hadron. Both hadrons are limited to the central rapidity region $|y^{\rm trig, asso}|<0.5$ and the azimuthal angle relative to the triggered hadron is integrated over $|\Delta\phi|>2.5$. As in the NLO calculation of single hadron spectra, one also has to fix the factorization scale in the NLO calculation of dihadron spectra, which is chosen to be the invariant mass of the dihadron $M^2=(p_1+p_2)^2$. The NLO results on associated hadron spectra $D_{pp}(z_T,p_T^{\rm trig})$ in $p+p$ collisions are compared to the $d+Au$ data at $\sqrt{s}=200$ GeV in Fig. \[fig:Daa\], assuming no nuclear effects in $d+Au$ collisions. The overall normalization of NLO results is quite sensitive to the factorization scale as indicated by the shaded region corresponding to $\mu=0.8 - 1.8 M$. We will use $\mu=1.2 M$ in this study with which NLO results describe the experimental data well. We use the same scale in the calculation of dihadron spectra in $Au+Au$ collisions and the results for $D_{AA}(z_T,p_T^{\rm trig})$ agree well with STAR data as shown in Fig. \[fig:Daa\] using the same energy loss parameter $\epsilon_0=1.68-2.08$ GeV/fm. The centrality dependence of the data is also reproduced by the NLO calculation [@zoww]. The nuclear modification factor of the hadron-triggered fragmentation function $$I_{AA}=\frac{D_{AA}(z_T,p_T^{\rm trig})}{D_{pp}(z_T,p_T^{\rm trig})}$$ as plotted in the lower panel is coincidentally similar to the modification factor for single hadron spectra $R_{AA}$. ![\[fig:eps-dep\] (color online). The suppression factors for single ($R_{AA}$), dihadron ($I_{AA}$) spectra at fixed transverse momentum and $\chi^2 /d.o.f.$ (curves with filled squares) in fitting experimental data on single [@akiba] ($p_T=4 - 20 $ GeV/$c$) and away-side spectra [@Adams:2006yt] ($p_T^{\rm trig}=8-15$ GeV, $z_T=0.45-0.95$) in central $Au+Au$ collisions at $\sqrt{s}=200$ GeV as functions of the initial energy loss parameter $\epsilon_0$.](fig3.eps){width="85mm"} ![(color online). Hadron-triggered fragmentation functions $D_{AA}(z_T)$ and the medium modification factors $I_{AA}(z_T)$ in NLO pQCD as compared to the data [@Adams:2006yt].[]{data-label="fig:Daa"}](fig4.eps){width="85mm"} We have adjusted the energy loss parameter $\epsilon_0$ to fit experimental data on both single ($d\sigma_{AA}/dp_Tdy$ between $p_T=4 - 20$ GeV/$c$) and away-side spectra \[$D_{AA}(z_T)$\] in the most central $Au+Au$ collisions at $\sqrt{s}=200$ GeV. The best fits occur for $\epsilon_0=1.6 - 2.1$ GeV/fm as shown by the $\chi^2$ in Fig. \[fig:eps-dep\]. The fact that both $\chi^2$’s reach their minima in the same region for two different measurements is highly nontrivial, providing convincing evidence for the jet quenching description. Because of trigger bias, most of the contribution to dihadron spectra comes from dijets produced close and tangential to the surface of the overlapped region, as shown in Fig. \[fig:di-x0\]. However, there are still about 25% of the contribution coming from dijets near the center of the overlapped region. These jets truly punch through the medium and emerge after finite amount of energy loss. This is why dihadron spectra is slightly more sensitive to $\epsilon_0$ than the single spectra as was noted in Refs. [@dainese; @renk]. As one further increases the initial gluon density, the fraction of these punch-through jets will also diminish and the final dihadron spectra will be dominated by the tangential jets in the outer corona. Dihadron spectra at RHIC will also lose its sensitivity to the initial gluon density of the medium as shown in Fig. \[fig:eps-dep\] Even if one includes the effect of transverse expansion of the bulk matter, the above results remain qualitatively the same [@renk]. ![\[fig:di-x0\] (color online). The same as Fig. \[fig:con-sin-y0\] except for dihadrons along the direction $\phi=0$ and $\pi$.](fig5.eps){width="85mm"} Also shown in Fig. \[fig:eps-dep\] are the single and dihadron suppression factors at the LHC energy as a function of $\epsilon_0$ for fixed values of $p_T^{\rm trig}=20$ GeV and $p_T^{\rm asso}=10$ GeV. Because of the flattening of the overall spectra shape at the LHC energy, back-to-back dihadron spectra at a given $p_T$ have more contribution from dijets with higher initial energy. They are therefore less suppressed than at the RHIC energy and are more sensitive to the initial gluon density. The single hadron spectra on the other hand are increasingly dominated by surface emission and the suppression factor $R_{AA}$ becomes much smaller than $I_{AA}$ for dihadrons. From a model estimate of the bulk hadron production at LHC [@lw02], the energy loss parameter is about $\epsilon_0\approx 5$ GeV/fm in central $Pb+Pb$ collisions. The dihadron suppression factor $I_{AA}$ at around this value of $\epsilon_0$ is significantly larger than the single hadron suppression factor $R_{AA}$ and is more sensitive to the initial gluon density. In summary, we have used the NLO pQCD parton model with effective modified FF’s due to radiative parton energy loss to study both single and dihadron spectra in high-energy heavy-ion collisions. We found that the surface emission dominates the single hadron production process within the range of initial gluon density in the most central $Au+Au$ collisions at RHIC. However, there is still a significant fraction of dijets that are produced in the center of the dense medium and contribute to the final dihadron spectra after losing finite amount of energy. Therefore, dihadron spectra are more robust probes of the initial gluon density in the most central $Au+Au$ collisions at RHIC while single hadron spectra become less sensitive. A simultaneous $\chi^2$-fit to both the single and dihadron spectra can be achieved within a narrow range of the energy loss parameters $\epsilon_0=1.6-2.1$ GeV/fm. If the initial gluon density at RHIC were to increase further, dihadron spectra at a fixed $p_T$ would also be dominated by the surface emission and become insensitive to the initial gluon density. At LHC, however, the flattening of the initial jet production spectra leads to an increase in the dihadron suppression factor. Dihadrons will therefore remain more robust probes within a large range of initial gluon density. We thank P. Jacobs for helpful discussions. This work was supported by DOE under contracts No. DE-AC02-05CH11231 and No. DE-FG02-97IR40122, by NSFC under Project No. 10440420018, No. 10475031 and No. 10635020, and by MOE of China under projects No. NCET-04-0744, No. SRFDP-20040511005 and No. IRT0624. [99]{} K. Adcox [*et al.*]{}, Phys. Rev. Lett.  [**88**]{}, 022301 (2001); C. Adler [*et al.*]{}, Phys. Rev. Lett.  [**89**]{}, 202301 (2002). C. Adler [*et al.*]{}, Phys. Rev. Lett.  [**90**]{}, 082302 (2003). 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--- abstract: 'We study Hamiltonian systems with point interactions and give a systematic description of the corresponding boundary conditions and the spectrum properties for self-adjoint, PT-symmetric systems and systems with real spectra. The integrability of one dimensional many body systems with these kinds of point (contact) interactions are investigated for both bosonic and fermionic statistics.' title: 'Exactly Solvable Many-Body Systems and Pseudo-Hermitian Point Interactions' --- The complex generalization of conventional quantum mechanics has been investigated extensively in recent years. In particular it is shown that the standard formulation of quantum mechanics in terms of Hermitian Hamiltonians is overly restrictive and a consistent physical theory of quantum mechanics can be built on a complex Hamiltonian that is not Hermitian but satisfies the less restrictive and more physical condition of space-time reflection symmetry (PT symmetry) [@bender]. It is proven that if PT symmetry is not spontaneously broken, the dynamics of a non-Hermitian Hamiltonian system is still governed by unitary time evolution. A number of models with PT-symmetric and continuous interaction potentials have been constructed and studied [@models]. In this article we study Hamiltonian systems with singular interaction potentials at a point. We give a systematic and complete description of the boundary conditions and the spectra properties for self-adjoint, PT-symmetric systems and systems with real spectra. We then study the integrability of one dimensional many body systems with these kinds of point interactions. Self-adjoint point interactions =============================== Self-adjoint quantum mechanical models describing a particle moving in a local singular potential concentrated at one or a discrete number of points have been extensively discussed in the literature, see e.g. [@agh-kh; @gaudin; @AKbook] and references therein. One dimensional problems with contact interactions at, say, the origin ($x=0$) can be characterized by separated or nonseparated boundary conditions imposed on the wave function $\psi$ at $x=0$ [@kurasov]. The first model of this type with the pairwise interactions determined by $\delta$-functions was suggested and investigated in [@mcguire]. Intensive studies of this model applied to statistical mechanics (particles having boson or fermion statistics) are given in [@y; @y1]. Nonseparated boundary conditions correspond to the cases where the perturbed operator is equal to the orthogonal sum of two self-adjoint operators in $L_{2} (-\infty,0]$ and $L_{2} [0,\infty)$. The family of point interactions for the one dimensional Schrödinger operator $ - \frac{d^2}{dx^2}$ can be described by unitary $ 2 \times 2 $ matrices via von Neumann formulas for self-adjoint extensions of symmetric operators. The non-separated boundary conditions describing the self-adjoint extensions have the following form $$\label{bound} \left( \begin{array}{c} \psi(+0)\\ \psi '(+0)\end{array} \right) = e^{i\theta} \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \left( \begin{array}{c} \psi(-0)\\ \psi '(-0)\end{array} \right),$$ where $ad-bc = 1$, $\theta, a,b,c,d \in \Rb$, $\psi(x)$ is the wave function of a spinless particle with coordinate $x$. The values $\theta = b=0$, $a=d=1$ in (\[bound\]) correspond to the case of a positive (resp. negative) $\delta$-function potential for $c>0$ (resp. $c<0$). For general $a,b,c$ and $d$, the properties of the corresponding Hamiltonian systems have been studied in detail, see e.g. [@kurasov; @ch; @abd]. The separated self-adjoint boundary conditions are described by \[bounds\] \^(+0) = h\^+ (+0) ,     \^(-0) = h\^- (-0), where $h^{\pm} \in \Rb \cup \{ \infty\}$. $ h^+ = \infty$ or $ h^- = \infty$ correspond to Dirichlet boundary conditions and $ h^+ = 0$  or  $ h^- = 0$ correspond to Neumann boundary conditions. PT-symmetric point interactions =============================== An operator is said to be PT-symmetric if it commutes with the product operator of the parity operator P and the time reversal operator T. It can be shown that the family of PT-symmetric second derivative operators with point interactions at the origin coincides with the set of restrictions of the second derivative operator to the domain of functions satisfying the boundary conditions at the origin [@afkpt]: $$\label{bcond1} \left(\begin{array}{c} \psi (+0) \\ \psi' (+0) \end{array}\right) = B \left(\begin{array}{c} \psi (-0) \\ \psi' (-0) \end{array}\right)$$ for non-separated type, where $$B = e^{i \theta} \left( \begin{array}{cc} \sqrt{1 +bc} \; e^{i\phi} & b \\ c & \sqrt{1+bc} \; e^{-i\phi} \end{array} \right),$$ the real parameters $ b \geq 0, c \geq -1/b$[^1], $ \theta, \phi \in [0, 2 \pi)$; or corresponding to the separated type $$\label{bcond2} h_0 \psi' (+0) = h_1 e^{i\theta} \psi (+0),~~~ h_0 \psi' (+0) = - h_1 e^{-i\theta} \psi (-0)$$ with the real phase parameter $ \theta \in [0,2\pi)$ and the parameter $ {\bf h} = (h_0, h_1)$ taken from the (real) projective space $ {\bf P}^1$. The spectrum of any PT-symmetric second derivative operator with point interactions at the origin consists of the branch $[0,\infty)$ of the absolutely continuous spectrum and at most two (counting multiplicity) eigenvalues, which are real negative or are (complex) conjugated to each other. The eigenvalues corresponding to PT-symmetric eigenfunctions are real and negative. Every eigenfunction corresponding to any real eigenvalue can be chosen either PT-symmetric or -antisymmetric. The spectrum of the PT-symmetric second derivative operator with non-separated type point interaction at the origin is pure real if and only if the parameters appearing in (\[bcond1\]) satisfy in addition at least one of the following conditions: $$\label{bcondreal1} bc \sin^2 \phi \leq \cos^2 \phi;$$ $$\label{bcondreal2} bc \sin^2 \phi \geq \cos^2 \phi \; \; {\rm and}\; \; \cos \phi \geq 0.$$ Point interactions with real spectra ==================================== We consider further non-separated type boundary conditions at the origin leading to second derivative operators with real spectrum. A general form of the boundary condition can be written as $$\label{generalb} \left(\begin{array}{c} \psi (+0) \\ \psi' (+0) \end{array}\right) = \left( \begin{array}{cc} \alpha & \beta \\ \gamma & \delta \end{array} \right) \left(\begin{array}{c} \psi (-0) \\ \psi' (-0) \end{array}\right),$$ where $\alpha$, $\beta$, $\gamma$ and $\delta\in\Cb$. We suppose that the matrix $ B = \left( \begin{array}{cc} \alpha & \beta \\ \gamma & \delta \end{array} \right) $ appearing in the boundary conditions (\[generalb\]) is non degenerate (from $GL(2, \Cb)$). Again it is easy to prove that the operator has branch of absolutely continuous spectrum $ [0, \infty)$. To study its discrete spectrum we use the following [*Ansatz*]{} for the eigenfunction $$\psi (x) = \left\{ \begin{array}{ll} c_1 e^{-ikx}, & x < 0 \\ c_2 e^{ikx}, & x > 0 \end{array} \right. , \; \; \Im k > 0,$$ corresponding to the energy $\lambda = k^2$. Substituting this function into the boundary conditions (\[generalb\]) we get the dispersion equation $k^2 \beta + ik (\alpha + \delta) - \gamma = 0$. The set of coefficients $\alpha,~ \beta,~ \gamma$, and $\delta$ satisfying the condition $\Im k_{1,2} \leq 0$ can be parameterized by $8$ real parameters and leads to operators with pure absolutely continuous spectrum $ [0, \infty)$. Pure imaginary solutions to the dispersion equation leads to nontrivial discrete spectrum. Set $ \tau = \alpha + \delta$. The solutions are pure imaginary if and only if the following conditions are satisfied: $$\label{betaneq0} \tau = t e^{i\theta},~ \beta = b e^{i\theta},~ \gamma = c e^{i \theta},~4 \frac{c}{b} \leq \frac{t^2}{b^2},$$ for $\beta\neq 0$, where $ t,b,c $ are real numbers. If $\beta=0$, the spectrum is guaranteed to be real when $\alpha + \delta$ and $\gamma$ have the same phases, i.e., $$\label{beta0} \tau = t e^{i\theta},~ \gamma = c e^{i\theta}.$$ The real spectrum point interaction (\[betaneq0\]) is parameterized by $6$ real parameters. The four-parameter family of self-adjoint (non-separated) boundary conditions (\[bound\]) is contained in this $6$-parameter family. The family of PT-symmetric (non-separated) boundary conditions leading to operators with real spectrum is also included in the family of boundary conditions (\[betaneq0\]) or (\[beta0\]). Integrable many-body systems ============================ The self-adjoint boundary conditions (\[bound\]) and (\[bounds\]), PT-symmetric boundary conditions (\[bcond1\]) and (\[bcond2\]), and the real spectrum boundary conditions (\[generalb\]) with the parameters satisfying (\[betaneq0\]) or (\[beta0\]) also describe two spinless particles moving in one dimension with contact interaction when they meet (i.e. the relative coordinate $x=0$). When the particles have spin $s$ but without any spin coupling among the particles, $\psi$ represents any one of the components of the wave function. In the following we study the integrability of one dimensional systems of $N$-identical particles with general contact interactions described by the non-separated boundary conditions that are imposed on the relative coordinates of the particles. We first consider the case of two particles ($N=2$) with coordinates $x_1$, $x_2$ and momenta $k_1$, $k_2$ respectively. Each particle has $n$-‘spin’ states designated by $s_1$ and $s_2$, $1\leq s_i\leq n$. For $x_1\neq x_2$, these two particles are free. The wave functions $\varphi$ are symmetric (resp. antisymmetric) with respect to the interchange $(x_1,s_1)\leftrightarrow(x_2,s_2)$ for bosons (resp. fermions). In the region $x_1<x_2$, from the Bethe ansatz the wave function is of the form, \[w1\] =\_[12]{}e\^[i(k\_1x\_1+k\_2x\_2)]{}+\_[21]{}e\^[i(k\_2x\_1+k\_1x\_2)]{}, where $\alpha_{12}$ and $\alpha_{21}$ are $n^2\times 1$ column matrices. In the region $x_1>x_2$, \[w2\] =(P\^[12]{}\_[12]{})e\^[i(k\_1x\_2+k\_2x\_1)]{} +(P\^[12]{}\_[21]{})e\^[i(k\_2x\_2+k\_1x\_1)]{}, where according to the symmetry or antisymmetry conditions, $P^{12}=p^{12}$ for bosons and $P^{12}=-p^{12}$ for fermions, $p^{12}$ being the operator on the $n^2\times 1$ column that interchanges $s_1\leftrightarrow s_2$. Set $k_{12} = (k_1 -k_2)/2$. In the center of mass coordinate $X=(x_1+x_2)/2$ and the relative coordinate $x=x_2-x_1$, we get, by substituting (\[w1\]) and (\[w2\]) into the boundary conditions (\[generalb\]) at $x=0$, \[a1\] { [l]{} \_[12]{}+\_[21]{} =P\^[12]{}(\_[12]{}+\_[21]{})+ i k\_[12]{}P\^[12]{}(\_[12]{}-\_[21]{}),\ ik\_[12]{}(\_[21]{}-\_[12]{}) = P\^[12]{}(\_[12]{}+\_[21]{})+ik\_[12]{}P\^[12]{} (\_[12]{}-\_[21]{}). . Eliminating the term $P^{12}\alpha_{12}$ from (\[a1\]) we obtain the relation \[2112\] \_[21]{} = Y\_[21]{}\^[12]{} \_[12]{} , where \[a21a12\] Y\_[21]{}\^[12]{} =. For $N\geq 3$ and $x_1<x_2<...<x_N$, the wave function is given by \[psi\] &=&\_[12...N]{}e\^[i(k\_1x\_1+k\_2x\_2+...+k\_Nx\_N)]{} +\_[21...N]{}e\^[i(k\_2x\_1+k\_1x\_2+...+k\_Nx\_N)]{}\ &&+(N!-2) other terms. The columns $\alpha$ have $n^N\times 1$ dimensions. The wave functions in the other regions are determined from (\[psi\]) by the requirement of symmetry (for bosons) or antisymmetry (for fermions). Along any plane $x_i=x_{i+1}$, $i\in 1,2,...,N-1$, from similar considerations we have \[a1n\] \_[l\_1l\_2...l\_il\_[i+1]{}...l\_N]{}=Y\_[l\_[i+1]{}l\_i]{}\^[ii+1]{} \_[l\_1l\_2...l\_[i+1]{}l\_i...l\_N]{}, where \[y\] Y\_[l\_[i+1]{}l\_i]{}\^[ii+1]{}= . Here $k_{l_il_{i+1}}=(k_{l_i}-k_{l_{i+1}})/2$ play the role of spectral parameters. $P^{ii+1}=p^{ii+1}$ for bosons and $P^{ii+1}=-p^{ii+1}$ for fermions, with $p^{ii+1}$ the operator on the $n^N\times 1$ column that interchanges $s_i\leftrightarrow s_{i+1}$. For consistency $Y$ must satisfy the Yang-Baxter equation with spectral parameter [@y; @ma], i.e., $$Y^{m,m+1}_{ij}Y^{m+1,m+2}_{kj}Y^{m,m+1}_{ki} =Y^{m+1,m+2}_{ki}Y^{m,m+1}_{kj}Y^{m+1,m+2}_{ij},$$ or \[ybe1\] Y\^[mr]{}\_[ij]{}Y\^[rs]{}\_[kj]{}Y\^[mr]{}\_[ki]{} =Y\^[rs]{}\_[ki]{}Y\^[mr]{}\_[kj]{}Y\^[rs]{}\_[ij]{} if $m,r,s$ are all unequal, and \[ybe2\] Y\^[mr]{}\_[ij]{}Y\^[mr]{}\_[ji]{}=1,       Y\^[mr]{}\_[ij]{}Y\^[sq]{}\_[kl]{}=Y\^[sq]{}\_[kl]{}Y\^[mr]{}\_[ij]{} if $m,r,s,q$ are all unequal. These Yang-Baxter relations are satisfied when \[realsp\] -=1,     =0,     =, i.e., $\beta=0$, $\alpha=\delta=\pm 1$. Therefore for self-adjoint contact interactions (\[bound\]), the $N$-body system is integrable when $\theta =0$, $a=d=\pm 1$, $b=0$, $c$ arbitrary. The case $a=d=1$, $\theta =b=0$ corresponds to the usual $\delta$-function interactions, which has been investigated in [@y; @y1]. The case $a=d=-1$, $\theta =b=0$, is related to a kind of anti-$\delta$ interactions [@adf]. For $N$-body systems with PT-symmetric contact interactions (\[bcond1\]), the integrable condition (\[realsp\]) implies that $\theta=\phi=b=0$, which is just the usual self-adjoint $\delta$-interaction. From (\[beta0\]) and (\[realsp\]), we also have that an many-body system with contact interaction and real spectra is integrable only when $\alpha=\delta=\pm 1$, $\gamma=c$ for some $c\in\Rb$, which is again the self-adjoint $\delta$-type interactions, see Fig.1 for the relations among point interactions according to integrability. We have presented a complete picture of self-adjoint, PT-symmetric and real spectrum point interactions, and their corresponding integrability. What we concerned here are just the case of particles with only pure contact interactions, and the possible contact coupling of the spins of two particles are not taken into account [@spin]. 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--- abstract: 'We have analyzed $Kepler$ light curves for 849 stars with T$_{\rm eff}$ $\leq$ 5200 K from our Cycle 1 Guest Observer program. We identify six new eclipsing binaries, one of which has an orbital period of 29.91 d, and two of which are probably W UMa variables. In addition, we identify a candidate “warm Jupiter” exoplanet. We further examine a subset of 670 sources for variability. Of these objects, 265 stars clearly show periodic variability that we assign to rotation of the low-mass star. At the photometric precision level provided by $Kepler$, 251 of our objects showed no evidence for variability. We were unable to determine periods for 154 variable objects. We find that 79% of stars with T$_{\rm eff}$ $\leq$ 5200 K are variable. The rotation periods we derive for the periodic variables span the range 0.31 $\leq$ P$_{\rm rot}$ $\leq$ 126.5 d. A considerable number of stars with rotation periods similar to the solar value show activity levels that are 100 times higher than the Sun. This is consistent with results for solar-like field stars. As has been found in previous studies, stars with shorter rotation periods generally exhibit larger modulations. This trend flattens beyond P$_{\rm rot}$ = 25 d, demonstrating that even long period binaries may still have components with high levels of activity and investigating whether the masses and radii of the stellar components in these systems are consistent with stellar models could remain problematic. Surprisingly, our modeling of the light curves suggests that the active regions on these cool stars are either preferentially located near the rotational poles, or that there are two spot groups located at lower latitudes, but in opposing hemispheres.' author: - 'T. E. Harrison, J. L. Coughlin, N. M. Ule' - 'M. L$\acute{o}$pez-Morales' title: '$Kepler$ Cycle 1 Observations of Low Mass Stars: New Eclipsing Binaries, Single Star Rotation Rates, and the Nature and Frequency of Starspots' --- [*Key words:*]{} stars: low-mass — stars: late-type — binaries: eclipsing — stars: spots Introduction ============ The $Kepler$ mission was designed to discover and characterize transiting exoplanetary systems (Borucki et al. 2010), and it has been quite successful with more than 1200 candidate systems already identified (Borucki et al. 2011). Perhaps equally exciting to the discovery and observation of exoplanets is the impact of high precision, long-term photometric observations on our understanding of ordinary stars. These new data are providing a wealth of information on the asteroseismology of stars similar to the Sun (e.g., Verner et al. 2011), as well as those objects in the classical regions of the instability strip (e.g., Benko et al. 2010). It is also possible to use $Kepler$ to explore important, outstanding issues that remain for low-mass stars. One of the most important of these is that some of the measured fundamental parameters of low-mass stars (masses, radii, and T$_{\rm eff}$) appear to be in conflict with values predicted by models. For example, analysis by L$\acute{o}$pez-Morales (2007) shows that the observed radii of low-mass stars are 10 to 20% larger than predicted by the stellar models of Baraffe et al. (1998). L$\acute{o}$pez-Morales found that there was a clear correlation between the activity levels of short period (P$_{\rm orb}$ $\sim$ 3 d) binaries and the discrepancy in the radii of the stellar components. Morales et al. (2008, 2010) suggest that either the models are flawed, or that differences in metallicity, magnetic activity, or the presence/distribution of star spots make such comparisons moot. If the larger radii observed for the components in short period binaries is indeed due to enhanced magnetic activity as a result of tidal locking, then it is suspected that the low mass stars in binaries with orbital periods longer than $\sim$ 10 d should have radii in agreement with models. The main issue that existed before the launch of the $Kepler$ mission was the lack of long period (P$_{\rm orb}$ $>$ 10 d), low-mass eclipsing binaries. The samples used to arrive at the results above relied on a handful of shorter period eclipsing binaries. It is well known (Bopp 1987, Radick et al. 1987) that rapidly rotating low-mass stars are intrinsically more active, and thus if the components in short period eclipsing binaries are spun-up by gravitational interactions (Zahn 1977, 1994), then their value for comparison to more slowly rotating isolated field dwarfs is diminished. It would be extremely useful to discover additional low-mass eclipsing binaries with longer periods to investigate whether these more slowly rotating objects might be used to constrain stellar models. This was the genesis for our Cycle 1 and 3 $Kepler$ programs that proposed to obtain light curves for 2400 cool (T$_{\rm eff}$ $\leq$ 5200 K) stars to search for previously unknown long period, low-mass eclipsing binaries (LMEBs). As we discuss below, we have been successful in identifying an LMEB that has a period of 29.91 d, along with five shorter period LMEBs in our Cycle 1 data. This success rate is in line with our assumption that the mean binary star fraction is 35% for K and M dwarfs (Mayor et al. 1992, Leinert et al. 1997, and Fischer & Marcy 1992). Along with the discovery of new LMEBs, the $Kepler$ light curves from our program objects provides an unprecedented data set to investigate the variability of an unbiased sample of low-mass main sequence stars on timescales of $\sim$ 90 d. For example, we can use the modulation of the light curves induced by starspots to determine the rotation rates for these stars, and compare them with predictions for the angular momentum evolution of such objects (e.g., Krishnamurthi et al. 1997). It is also possible to investigate the nature of stellar activity for low-mass stars to levels not previously attainable using ground-based photometry. With a large sample of objects with continuous light curves spanning 90 (or more) days, we can probe the longevity of the active regions on these low-mass stars, and attempt to unravel the latitudinal (and/or longitudinal) distribution of starspots on these objects. In the following we discuss the selection of our targets and the reduction of the Cycle 1 $Kepler$ light curves in section 2, discuss the discovery of the new LMEBs and derivation of the stellar characteristics in section 3, and discuss our conclusions in section 4. Observations ============ Our sample of objects for observation in Cycle 1 (which ran from 2009 June 20, until 2010 June 23) was selected using the stellar parameters available in the [*Kepler Input Catalog*]{} (the “KIC”, Brown et al. 2011). With multi-band photometry ($griz$, “DDO51”, and $JHK$ from 2MASS), Brown et al. were able to assign values of T$_{\rm eff}$, log($g$), A$_{\rm V}$, and estimate the metallicity for more than 4 million objects in the $Kepler$ field-of-view (note: see the discussion in Brown et al. for important caveats on each of these derived parameters). From the KIC for Cycle 1, we constructed a list of all of the stars that had T$_{\rm eff}$ $\leq$ 5200 K, $r$ $\leq $ 17.0, log($g$) $\geq$ 4.0, A$_{\rm V}$ $\leq$ 0.5, and that were not already designated as $Kepler$ program stars. In addition, we only selected targets that had no neighboring objects within 6" of our sources. There were 1239 objects that met these criteria. Dropping the faintest 39 sources, we ended up with a list of 1200 targets. The brightnesses of our observed targets spanned the range 15.16 $\leq$ $r$ $\leq$ 16.97. Our proposal sought to obtain “long cadence” (30 min) light curves spanning 90 days (a single $Kepler$ “Quarter”) for 1200 late-type dwarfs. Unfortunately, due to a mission programming error, only 849 unique targets were observed during Cycle 1. This did mean, however, that 260 targets were accidentally re-observed in a second quarter. In nearly all these cases we obtained two quarters of data that [*spanned*]{} three quarters, skipping the middle quarter (e.g., quarter 2 + 4, or quarter 3 + 5). This separation allowed us to examine the light curves for coherent variations on timescales of up to 270 days, and is especially useful in searching for very slowly rotating stars. Extraction of the Light Curves ------------------------------ The “Pre-search Data Conditioned” (PDC) light curves released to Guest Observers (GOs) have been processed through the $Kepler$ pipeline (see Jenkins et al. 2010). After careful examination, the process used by the $Kepler$ team appears to introduce artifacts, and/or remove valid photometric variations. Part of the reason for this is that the drift for some stars during a quarter was large enough that significant flux was lost out of the optimal photometric aperture used by the $Kepler$ pipeline to produce the light curves. While the fine guidance sensors on $Kepler$ generally keep the pointing to within $\pm$ 0.05 pixels, the effects of differential velocity aberration as the telescope orbits the Sun introduces an annual motion of 6" on the $Kepler$ field-of-view. Subsequent polynomial fits to remove such trends appears to both introduce, and eliminate, photometric variations. Since it is impossible for a GO to reconstruct the process for any of the program targets, we decided to build our own processing pipeline that starts with the complete set of calibrated, sky-subtracted postage stamp images released to the GOs in early 2011. The default photometric aperture assigned to each target depends on the source brightness, and usually spans several pixels. This central aperture is then surrounded by a “sky” region. For the faint stars of our program, the resulting postage stamp images were usually 3 $\times$ 5 or 4 $\times$ 6 pixels in size. Preliminary investigations showed, however, that a significant portion of the stellar flux was not contained within the pre-defined apertures for our targets. Thus, some of the stellar flux could often be detected in the “sky” pixels that surrounded every target’s aperture. This is partially due to the large point spread function of the defocussed images produced by $Kepler$, but mostly due to positional drift over the quarter. The result was that the PDC light curves have systematic variations due to improper “sky” removal, and too small of a photometric aperture. As a result, we decided to sum-up the flux in [*all*]{} of the pixels in each postage stamp image to get a better handle on the light curve of a source. We found this process alone dramatically reduced the presence of the large scale systematics noted above. The main drawback to this method is that flux from nearby stars can often contaminate the resulting light curve. In the cases where this background severely contaminates the light curve of our program object, we define those light curves as suffering from “third light” contamination. As noted below, many times such contamination can be non-varying, and be accounted for in extracting the target light curve. In other cases, it is impossible to determine which object in the FOV is the source of the variations. To better avoid such issues we would recommend that future observers of faint targets increase the nearest neighbor radius to $>$ 12" if there are any nearby stars that might provide sufficient contaminating flux (i.e., stars as bright, or brighter than the target). The light curves generated by the summing process are then (only) used to determine the mean flux value for that source during the quarter. After the flux summing process, the systematics in the light curve due to positional drift are greatly reduced, but pixel-to-pixel sensitivity, both in spectral response, as well as quantum efficiency, and intrapixel variations, can produce significant systematics. To account for these, we performed a Principal Component Analysis (PCA; Murtagh & Heck 1987) on the ensemble of post stamp images for each source. The PCA process correlates the variation of the flux in each pixel with the fluxes in all of the other pixels in the image. We then subtract the most significant principal component from the light curve and scale it using the mean of the summed light curve to conserve flux. This latter step is done so that we can obtain light curve amplitudes that are calibrated to the $Kepler$ system. In this way, the effects noted above are largely accounted for, and the result is a relatively clean light curve. To demonstrate the process we present light curves for three targets in Fig. 1. In the top panels of this figure are the PDC “calibrated” pipeline processed $Kepler$ light curves as delivered to GOs. It is clear from the pipeline processed light curve that Kepler ID\#8016381 (Fig. 1a) shows periodic variations, but they are somewhat irregular. The summed and PCA-processed light curves are much more symmetric. In the case of K8016381, these latter two light curves are identical. This result is due to the fact that the centroid of this star only moved a tiny fraction of a pixel over the 90 d quarter. In contrast to K8016381 is the light curve for K5428432 (Fig. 1b). The PDC pipeline processed light curve suggests a possible slow, periodic variation with a significant amplitude. Our PCA processing reveals what appears to be a much more rapid oscillation of lower amplitude. As shown in the centroid plot, however, this star appeared to have numerous jumps in its position, and nearly every one of these is imprinted on its light curve. These glitches have a variety of origins, from focus changes, to cosmic ray hits, all of which are discussed in Jenkins et al. (2010). In the following, all light curves from sources with large and choppy centroid motion have been removed from our rotation/starspot analysis. Such light curves remain useful for discovering LMEBs, exoplanet transits, or for short period asteroseismology investigations (see Garcia et al. 2011 for one possible method to correct these types of light curves). As mentioned above, some of our objects suffer from third light contamination. Such contamination is easily identified by examining the motion of the centroid. As shown in Fig. 1c, if the motion of the centroid of the star reflects the variability seen in the light curve, there is significant third light contamination. The amount of such contamination varied from an insignificant level, up to to the point where it was so dominant as to render the light curve useless. Where we felt that a light curve was only marginally affected, it was included in the analysis program, otherwise it was discarded. There was no defined rule for such decisions, given the great range in behavior exhibited by our program objects but, generally, if the third light imparted centroid motion on order of $\pm$ 0.01 pixels, it was deemed to be problematic. Of the 849 light curves that comprised our Cycle 1 survey, 173 were dropped from further analysis either due to third light contamination (33), or from multiple large, abrupt changes in the centroid motion (140). Light Curve Classification -------------------------- The $Kepler$ PDC pipeline-processed light curves, even with the flaws noted above, were sufficient to search for eclipsing binaries. From the 849 targets of our sample we identified six eclipsing binaries (two of which are almost certainly W UMa variables). These objects, along with their derived parameters, are listed in Table 1. After discarding the unusable light curves, we proceeded to analyze the light curves to classify the objects as periodic variables, probable periodic variables of long period, aperiodic variables, and objects showing no variability. We define periodic variables as objects that display either two similar, symmetric maxima with one minimum, or two minima with one maximum in their light curves (which either spanned 90, 180, or 270 d). Long period variables were objects that clearly showed evidence for both a minimum and a maximum, but the light curves were of insufficient duration to reveal a second maximum (or minimum). Aperiodic variables were objects with complex light curves that ranged from quasi-periodic, to chaotic. Non-variables were obviously objects that showed no significant variability. We found that we could generally detect symmetric oscillations that had amplitudes of $\sim$ 10 counts peak-to-peak in light curves that have mean fluxes of $\sim$ 2000 counts. Thus, non-variables are objects whose [*periodic*]{} variability has a total amplitude below $\sim$ 0.5%. These objects could be aperiodic, but at this level, tiny centroid changes can impart aperiodic features in the light curves, making such classification impossible. Of the 670 light curves analyzed for variability, we found 265 periodic variables (Table 2), 126 long period variables (Table 3), 28 aperiodic variables (Table 4), and 251 non-variables (Table 5). The Sample ---------- After classification, we can examine the types of stars found in each group. A histogram of the temperature distribution of our entire sample of 849 objects can be found in the top-left panel of Fig. 2. Our sample is dominated by early K dwarfs. Our hottest stars have T$_{\rm eff}$ = 5200 (K0V), and our coolest objects have T$_{\rm eff}$ = 3500 K (M2V). Histograms of the temperature distributions of our long period variables, and the non-variable objects, are shown in the top-right, and bottom-left hand panels, respectively, of Fig. 2. The histogram for the long period variables closely resembles that of the overall sample, while the histogram for the non-variable objects is dominated by hotter stars. It is clear that nearly all stars with T$_{\rm eff}$ $\leq$ 4000 K are variables at the level detectable with $Kepler$. This is reflected in the histogram for the sample of periodic variables (bottom-right panel of Fig. 2) that demonstrates periodicities are more likely to be found in cooler stars. Basri et al. (2011) have performed a statistical analysis of the entire sample of 150,000+ program stars observed during Q1, and found that 87% of their (6522) stars with temperatures below 4500 K were variable. We find that 79% of our sample of (298) stars with T$_{\rm eff}$ $\leq$ 4500 K are intrinsic variables. Since the stellar activity level increases as one descends the main sequence, it is obvious that rotational modulation of the light curve due to starspots is what is driving the majority of the periodic variations. We will discuss this assumption in the next section. To better characterize the periodic sample, we present a color-color plot of these objects in Fig. 3. Even with the caveats in the assignment of T$_{\rm eff}$ in the KIC ($\pm$ 200 K, or roughly one spectral type) noted by Brown et al. (2011), our set of periodic variables clearly has the desired high temperature cutoff near a spectral type of K0V. An HR diagram (Fig. 4) of the periodic variable sample indicates that most of these objects are early to mid-K dwarfs with distances near 1 kpc. The height above the galactic plane of our periodic variable sample is shown in Fig. 5. The mean scale height of our sample is 249.9 pc. This is consistent with an “intermediate disk population” (e.g., Ng et al. 1997) with a mean age near 5.75 Gyr. Thus, this group of periodic variables should be dominated by objects similar to the Sun in both age and metallicity. ### Determining Rotation Periods We will assume that the periodic modulations in the observed light curves of our sample of late-type stars is due to rotation. The association of photometric modulations in late-type stars with starspots dates back to at least Kron (1947). Vaughan et al. (1981) obtained long term Ca II H and K photometry of a sample of ninety one F to M stars that showed days, to month long variations, in nearly all of their stars. Twenty of these stars clearly showed evidence for rotational modulation with periods ranging from 2.5 to 54 days. They found that these rotation periods were consistent with spectroscopic determinations of the ($v$sin$i$) rotation rates for their stars, confirming rotation as the source of the photometric variability. These results demonstrate that, unlike the Sun, large regions of stellar surface activity can last for several rotation periods. Dorren & Guinan (1983) observed several of the stars from Vaughan et al. in both narrow-band and intermediate-band filters to sample both the blue continuum, and H-alpha spectral regions. For their most variable object (HD149661, K0V), total variations of 4% in the blue, and 2% in the red continuum filters were observed. Interestingly, they found that both the H-alpha and Ca II emission were anticorrelated to the continuum fluxes. They conclude that the photometric modulations were due to dark spots on these stars (all of which probably have significantly higher levels of chromospheric activity than seen on the Sun). To determine the rotation rates for our set of periodic variables we have used Period04 (Lenz & Breger 2005) to perform Fourier analysis to identify the dominant frequencies in each light curve. One of the nice features of Period04 is the ability to remove low frequency modulations of the light curves to produce “pre-whitened” light curves to precisely identify the period of the larger amplitude modulations. There are numerous examples in our periodic variable data set where the dominant periodic amplitudes are slowly modulated by a longer term variation that is intrinsic to the star. In some cases this appears to be due to changing starspot size, while in others it is simply a global change in the continuum level. For most objects, such slow modulations were fit with a single minimum or maximum of a sinusoidal variation that had a period that was many times longer than the duration of the light curve. In addition, there are a number of objects that show more complex light curves, with a second period that could be identified. We believe that for the majority of objects, this is due to a separate starspot group that is present on the star. In some cases, the periods for these second modulations are identical to the primary modulation period, but are simply offset in phase. In other sources, however, the second period is different to the primary period. We believe that the best explanation for such period differences is differential rotation. Results ======= The main goal of our program was to identify new LMEBs with periods in excess of 10 days so as to test whether rotational spin-up was the genesis for the discrepancy between the observed radii of low-mass stars, and predictions from stellar models. From our Cycle 1 data we identified six new LMEBs. Two of these objects, K636722 and K8086234, are almost certainly W UMa variables, and technically not LMEBs. One of these six, K6431670, has a period in excess of 10 days (see Table 1). Given that before the launch of $Kepler$ the longest known period of an LMEB was 8.4 d (Devor et al., 2008), this new object certainly met the program goals. This finding, however, is overshadowed by the results from analysis of the $Kepler$ public release data from Q1. Coughlin et al. (2011, see also Pr$\check{s}$a et al. 2011) identified 231 eclipsing binaries in this data set where the primary star is cooler than the Sun. Twenty nine of those LMEBs have periods longer than 10 d. As the majority of those systems are several magnitudes brighter than K6431670, they are much better binaries for determining whether the components in long period LMEBs have radii in line with the predictions from models. We are currently obtaining radial velocity and multi-wavelength light curves for a subset of these 231 objects to allow for further investigation of this issue. As noted above, the six LMEBs we have discovered are within expectations for our sample size. Mayor et al. (1992) estimate a binary fraction of 45% for K dwarfs, while for M dwarfs the binary fraction estimate ranges from 25% (Leinert et al. 1997) to 42% (Fischer & Marcy 1992). In a complete, and unbiased survey, Delfosse et al. (2004) found a binary fraction of 26 $\pm$ 3% for M dwarfs. Since our sample draws from a mixture of these two spectral types, we would expect binary fractions of $\approx$ 35%. The orbital period distribution for late type stars is not very well known, but Fischer & Marcy (1992) found a peak centered at 9 yr for M dwarfs. Their result resembled that for G stars, which have a Gaussian distribution centered at $\langle$P$_{\rm orb}$$\rangle$ = 173 $\pm$ 8.6 yr (Duquennoy & Mayor 1991). Putting these data together, assuming $\langle$P$_{\rm orb}$$\rangle$ = 9 $\pm$ 8.6 yr, and a main sequence radius relationship, we estimate that in a sample of 849 stars, we would expect to find five low mass eclipsing binaries. In addition to the six LMEBs detected in our survey, we also discovered a candidate exoplanet system: K5164255 ($r$ = 16.37, designated as KOI824.01 by the $Kepler$ team). This object is probably a “warm” ($\sim$ 650 K) Jupiter due to its moderate orbital period (P$_{\rm orb}$ = 15.4 d), and the fact that the host star primary is a K3V (T$_{\rm eff}$ = 4829). Our modeling of the $Kepler$ light curve for this object, presented in Fig. 6, using JKTEBOP (Southworth et al. 2004a,b) leads to the parameters for the host star and planet listed in Table 6. Given the large number of such objects detected by $Kepler$, this object is of little interest for further investigation due to the faintness of its host star, which is below the capabilities of current radial velocity studies. Rotation Rates of Low Mass Stars -------------------------------- We have assumed that the periodic modulation seen in the light curves (e.g., Fig. 1) is the rotation period of those stars. In Table 5 we list the periods of the dominant modulation found from fitting the light curves using Period04. The derived periods range from 0.31 d to 126.5 d, with a mean of 32.12 d. We present a plot of rotation period vs. T$_{\rm eff}$ in Fig. 7. The resulting distribution is remarkably flat (the means in 200 K bins are also plotted). As discussed earlier, rotation of a spotted star is the obvious interpretation for the modulations we detect in the light curves of our late-type dwarfs. As shown by Christensen-Dalsgaard & Frandsen (1983), solar-like oscillations in late-type dwarf stars have similar pulsation periods as the Sun ($\sim$ 5 min), with amplitudes of a few parts-per-million. Such oscillations could never be seen in the $Kepler$ long duration light curves. Gilliland (2008) has shown that red giants have variability on these time scales, but those variations have a recognizable photometric signature. Basri et al. (2011) investigated this aperiodic signature to see if it is possible to select giants vs. dwarfs based upon their $Kepler$ light curves alone. They found that the selection process was robust, with only a few high gravity objects showing up in their analysis as red giants. Basri et al. suggest that it is possible that these few objects are actually red giants that were misclassified as dwarfs in the KIC. This process demonstrates that the gravities listed in the KIC are fairly reliable for late-type stars. The light curves of red giants can show quasi-periodic behavior that are similar in nature to the oscillations seen in the Sun, except for their longer periods and much greater amplitudes (see Huber et al. 2010). We attempted to fit periods to all objects whose light curves might be periodic. Only after we could not identify a period consistent with the [*entire*]{} light curve were those objects re-classified as aperiodic. It is highly likely that [*no*]{} red giants remain in our “rotation” sample. It is possible, however, that the period we determine for the rotation rate is an even multiple, or fraction, of the true period. Shown in Fig. 8 is an example of an object (K10200948) that has one of the more complicated, and rapidly evolving light curves of any of the sources in our sample. In the Q3 data, analysis using Period04 finds a period of 7.197 d. In the Q5 data, we find the dominant period to be 14.45 d. Analysis of the combined light curves results in a best-fit period of 14.006 d. This latter period is what is over-plotted on the light curves shown in Fig. 8, and is what we assign as the rotation period for this object. Clearly, additional spots are present on K10200948 that are moving/evolving relative to the “main spot group” responsible for the coherent oscillation that retains the exact same phasing over 270 d. Fortunately for this target we have two quarters of data that allows us to isolate the “correct” period for this target. It is obvious, however, that there could be a few cases where similar sets of starspot groups are found to be centered on opposite hemispheres so as to create confusion about the true period. It is also obvious that if differential rotation is present, as suggested by the changing shapes of the maxima in the light curve of K10200948, the period we derive could be incorrect due to the slow migration of the main spot group. Since we have no secondary information as to the inclination of the rotational axis of these stars, it is impossible to know the exact latitude of these active regions, or whether the large spot groups are actually changing position with time, or are simply evolving in size and/or shape. Another possible effect that could cause errant periods is evolution of a starspot group on a timescale similar to the rotation period. If the stellar activity was somehow constrained to evolve at certain discrete longitudes, then the appearance, or disappearance, of spot groups with lifetimes similar to the rotation period could lead to erroneous period determinations. This is most true for the longest period systems. Instead of rotation, these light curves could be modeled by the repeated growth and decay of a fixed single spot group on a non-rotating star. Hopefully, nature is not this cruel, but since our knowledge of stellar activity cycles and the rotation rates of low-mass field stars is still primitive, it is impossible to rule out such behavior. Amplitude of the Spot Modulation -------------------------------- Tabulated in Table 5 are the amplitudes of the modulations for our sample of periodic objects. The amplitude listed there is the [*largest peak-to-peak*]{} variation seen in the light curve of that object. We plot the distribution of these amplitudes with respect to temperature (as well as their means) in Fig. 9. Surprisingly, this distribution is flat, with the early K dwarfs showing a greater range in modulation than the M-type dwarfs. The temperature-dependent means (in 200 K bins), however, are consistent with a single value (the sample mean was 1.3%). The level of variation seen here is consistent with the observations of Dorren & Guinan (1983), suggesting that our sample of rotating stars has a higher level of activity than exhibited by the Sun. However, it is important to realize that a nearly equal number of stars in our survey were found to exhibit no variations, though even those stars could be more active than the Sun. Treating the Sun as a variable star using the VIRGO instrument on SOHO, Lanza et al. (2004) show that the white light (similar to $Kepler's$ filterless response function, see below) variations during solar maximum are of order of 500 ppm (see also Pagano et al. 2005), a factor of ten smaller than our detection limit of $\sim$ 0.5%. It is also interesting to examine the rotation period-activity relationship. In Fig. 10 we plot the amplitude of the variations vs. the rotation period for our sample of periodic variables. It has been well established that younger stars rotate more quickly, and that these objects display a higher level of activity (c.f., Radick et al. 1987). This trend is observed in our sample, where the most rapid rotators generally show larger photometric modulations. At periods longer than $\sim$ 25 d, the trend flattens dramatically. We discuss the implications of this result in the next section. Spot Modeling of the Light Curves --------------------------------- Without additional observations, it is difficult to derive the inclination angles of the rotational axes for any of the stars in our sample. Statistically, the mean inclination angle for a random sampling of rotating stars is $i$ = 57$^{\circ}$. There are two light curve morphologies that can be generated using a [*single spot*]{}: continuously variable (sinusoidal), and flat maxima. Continuously variable light curves occur when the starspot is “circumpolar” in the sense that it is always in view for the observer. Flat-maxima light curves occur when the spot is out of view for some fraction of the rotation period. It is interesting to examine what the break-down into these two categories might imply for the latitudinal distribution of spots. We have examined the light curves of all our rotation targets, and have classified them into the following groups: flat-maxima (90 objects), continuously variable (119), flat-minima (40), and complex (16). A flat maximum light curve simply has maxima that are broader than the minima seen in that light curve. Flat minima light curves are the opposite to flat maxima. Continuously variable light curves have minima and maxima that have nearly identical shapes. Complex light curves have such dramatically changing spot groups that it is not possible to identify a consistent portion of the light curve that allows classification into the one of the other three categories. Complex and flat minima light curves cannot be explained using a single spot. A light curve with a flat minimum suggests that a second spot, with similar properties, rotates into view as the first spot is passing out of view. The flat-minima light curves in our sample always have non-flat maxima. The fact that 57% of our rotation sample have continuously variable light curves suggests that for the majority of our objects the sum of the inclination angle ($i$) and the starspot co-latitude ($b$, formally defined below) is less than 90$^{\circ}$. This relationship simply states that the starspot remains in view at all times (especially given that a starspot must subtend an appreciable angle to be detectable, see below). If we assume a random set of rotational inclination angles (0.0 $\leq$ $i$ $\leq$ 90.0), and a random value for the starspot’s co-latitude (0.0 $\leq$ $b$ $\leq$ 180.0), we find that statistically, only 21% of our stars should have continuously variable light curves. Assuming single, dominant spot groups, this result would strongly argue for “polar” spots. The only alternative to this conclusion is that many of our stars have two spot groups in diametrically opposite hemispheres with similar enough properties to create continuously variable light curves. The fact that 40 of our stars have flat minima, an indicator for two very similar spot groups, indicates that this latter conclusion may have some validity. We can attempt to model the light curves of our rotating stars to investigate starspot sizes and distributions, but before we do so, it is important to establish what effect the broad bandpass of the $Kepler$ mission has on the detection of cool spots. Basri et al. (2010) have investigated the variability of solar-like stars with an emphasis on comparison to the Sun. They found that the $g+r$ light curves from the VIRGO instrument on SOHO essentially reproduces the broad $Kepler$ bandpass. Thus, the minima in the $Kepler$ light curves of cool stars are due to dark spots on the surface, and thus we are not seeing maxima due to bright faculae (that actually cover a larger fraction of the solar photosphere than spots). As discussed in Knaack et al. (2001), faculae have a higher contrast near the limb of the Sun in the visual bandpass. In contrast, sunspots have a higher contrast when located near the center of the solar disk. Knaack et al. investigated whether viewing the Sun from higher inclination angles would result in a greater photometric variation, making the Sun more consistent with the larger variability observed for field stars [*like*]{} the Sun. They found that changing the Sun’s inclination has only a modest ($\sim$ +6%) effect on the solar irradiance. Thus, the $Kepler$ light curves should be useful for constraining the properties of starspots for our sample of randomly inclined, rotating stars. To investigate the starspot parameters for the objects in our rotational sample we have modeled the $Kepler$ light curves using PHOEBE (Pr$\check{s}$a & Zwitter 2005), a graphical interface to the Wilson-Divinney binary star light curve modeling code (see Kallrath et al. 1998). We use PHOEBE for modeling single stars due to the fact that the most recent version has been adapted for the $Kepler$ mission (with new stellar atmosphere models and limb darkening coefficients). We simply set the orbital period of the binary to the rotation period, and turn-off the light from the companion star. With limited details about our stars, there are few constraints on the input parameters used in light curve modeling. In the following, we have assumed that our objects are main sequence dwarfs with the masses, radii, and log$g$ expected for stars with the T$_{\rm eff}$ listed in the $KIC$. There are four relevant parameters for adding a spot to generate a light curve using PHOEBE: the spot longitude, co-latitude, radius, and temperature (input as the ratio T$_{\rm spot}$/T$_{\rm eff}$). Co-latitude has its normal meaning in that it is the angle between the pole of the star, and the center of the spot. Radius is the angular radius of the spot as measured at the center of the star. The minima in light curves for stars with a single spot are driven mostly by the interplay between the inclination angle and co-latitude, and less by spot size and temperature contrast, which are somewhat, but not fully, degenerate. PHOEBE assumes round spots of uniform temperature. To demonstrate the difficulties with ascertaining the exact spot distributions for any one light curve, we return to K10200948. First, if we were to classify the Q3 light curve for this object, we would have deemed it “continuously variable”. The Q5 light curve, however, shows it to have flat-maxima for the last four maxima (and a flat minimum for the first part of the light curve!). Obviously, this type of issue could erupt for every object in our sample through the appearance and/or disappearance of a second spot. In Fig. 11a we present the Q5 light curve for K10200948 phased to its rotation period. It is clear that the phasing cleans-up most of the light curve, and it emphasizes the flat-topped nature of its maxima. We found that the best fit, one spot model (middle panel) for this light curve occurs with an inclination angle of $i$ = 70$^{\circ}$, and a co-latitude of $b$ = 45$^{\circ}$. For this model the spot has a radius of 10$^{\circ}$, and a temperature ratio $w$ = 0.89. At larger co-latitudes the shoulders on the minima are too square, at smaller co-latitudes the model light curves are too sinusoidal. There is a set of models with $i$ $\approx$ 50$^{\circ}$ and $b$ $\approx$ 60$^{\circ}$ that fit equally well. In both families of models, the spot transits a similar line-of-sight chord ($i$ + $b$ $\sim$ 110$^{\circ}$). In all spot models the shoulder on the light curves disappears for $i$ + $b$ $\leq$ 90$^{\circ}$. As noted above, these are circumpolar spots that never fully disappear for the viewer, leading to continuously changing light curves. In Fig. 11b we present the phased Q3 light curve for K10200948. The rapidly changing maxima during this quarter results in a messier phased light curve, but shows relatively symmetric minima and maxima at roughly twice the rotation period. To model this light curve we added a second spot at a longitude that is located 175$^{\circ}$ from the spot used to model the Q5 light curve. For the resulting models we assumed that both spots had identical co-latitudes. Again, the best fitting model was one with an inclination angle of $i$ = 70$^{\circ}$, and a co-latitude of $b$ = 45$^{\circ}$. Both spots had the same radius, 12$^{\circ}$, slightly larger than found for the one spot model while at the same time having slightly higher temperature ratios: $w$ = 0.91. Changing spot size affects the total continuum level. Close inspection of Fig. 8 shows that the peaks of the maxima in Q5 were slightly larger, and thus the size of the spots needed to be increased to lower the overall flux level seen in the maxima of the Q3 light curve (note: small changes in the normalization between the two quarters could be the source of this issue). Meanwhile, the flux level of the minima remained similar, so the spots needed to be “less dark” (hotter) to fit these minima. As with the single spot model, an additional family of models with $i$ $\sim$ 50$^{\circ}$, $b$ $\sim$ 60$^{\circ}$ fit equally well. Given the number of parameters that have no apriori constraints, it is likely that there are other two spot models that might exist that can explain the Q3 light curve of K10200948 equally well, but the near-identical morphologies of the minima strongly suggests two spots with similar parameters. It is also difficult to obtain the observed light curve without employing spots that are centered in opposite hemispheres (or nearly so). That two spots with similar parameters would form on opposite hemispheres is surprising, but a similar result was found for the host star of CoRoT-2 by Lanza et al. (2009). Unfortunately, we do not have the Q4 data that would allow us to investigate exactly how the two spot phase transitioned to just a single spot. To further confuse the issue, there is some evidence in these light curves for a third spot that appears to modify some of the maxima seen in the Q3 and Q5 light curves for K10200948. The issues we have encountered in modeling this object could obviously be true for nearly every other object in our sample, and thus we defer modeling the light curves of additional objects to a future effort. As Neff et al. (1995) point out, any photometric modulations due to starspots is the [*asymmetric*]{} component of the starspot coverage. Thus, the spots that lead to the observed modulations could be isolated on an unspotted disk, or just be the largest features on a disk that is randomly covered by numerous smaller active regions. We do not yet have a good idea of the fractional coverage of stellar photospheres, nor the exact temperature(s) one should ascribe to starspots. Obviously, we need to limit one of these parameters to derive the other. It is not possible to do this with photometry, but spectroscopic observations have shown promise in constraining spot temperatures. Using the onset of TiO features in the red end of the visual spectrum, Neff et al. find that for the RS CVn binary II Peg, the best fit spot temperature is 3500 K. They also derive a “quiet” photospheric temperature of 4800 K for this star, leading to a temperature factor for the starspots of $w$ = 0.73. Neff et al. find that this value is consistent with large sunspots ($w$ = 0.70), and what has been found in other active stars (0.65 $\leq$ $w$ $\leq$ 0.85). To get a 2% change in the light curve of a cool star, we need to employ spots that have radii of $r$ $\sim$ 10$^{\circ}$, and temperature factors of $w$ $\sim$ 0.85. Obviously, to get the same effect with bigger spots will require higher temperature factors, while smaller spots have to be cooler. The smallest spot radius we can employ for modeling the Q5 data for K10200948 is 6$^{\circ}$, assuming a completely black spot. At the opposite extreme, a spot with a temperature factor of 0.99 needs to be 32$^{\circ}$ in radius to get a proper fit to the minima. Note that the fit of a model with this enormous of a spot is only slightly poorer than the solution arrived at using a spot with a radius of 10$^{\circ}$. We could make the larger spot model fit equally well by simply increasing its co-latitude so as to sharpen the shoulders of its light curve maxima. Fortunately, with the high precision photometry emanating from both $Kepler$ and $CoRoT$, it is becoming possible to directly measure spot sizes using exoplanet transits. Silva-Valio & Lanza (2011) found for the rapidly rotating (P$_{\rm rot}$ = 4.46 d), active G7V host of the transiting exoplanet CoRoT-2, the average spot radius was $\sim$ 2.6 $\pm$ 1$^{\circ}$, with the largest spots having radii of about twice this value. It is interesting that the more slowly rotating (P$_{\rm rot}$ = 23.6 d) and cooler (G9V) host star for CoRoT-7 appears to have much larger spot groups, with radii on order of $\sim$ 20$^{\circ}$ (Lanza et al. 2010). As shown earlier, the average amplitude of the variations in our sample of stars is 1.5%. If we assume single spots with $r$ $\sim$ 10$^{\circ}$ and $w$ = 0.89, the fractional photospheric coverage of such a spot is 0.76%. Solanki & Unruh (2004) discuss the spot coverage for the Sun, and find that it ranges by a factor of ten, with a mean near 0.165%. The sizes of individual sunspots has a lognormal distribution, with the largest spots covering about 0.01% of the visible photosphere. If cool dwarfs have a similar lognormal spot distribution, then the mean of the actual, fractional spot coverage will be closer to 10% for our rotating sample of stars. Discussion and Conclusions ========================== The original goal of our program was to identify new LMEBs of long period so as to investigate whether the fundamental parameters for the components in those systems might more closely resemble the predictions from stellar models. We have been successful in this quest by finding a new long period LMEB. Unfortunately, this object is quite faint, and this result has now been superseded by the findings of Coughlin et al. (2011) where numerous such objects, all brighter than K6431670, were discovered. Follow-up of those objects is better suited to investigating whether rapid rotation is playing an important role in creating the discrepancy in radii between observations and models. While we do find that activity levels in single stars decrease with increasing rotation period, it can still remain quite high for rotation periods of P$_{\rm rot}$ $\leq$ 25 d, so even some long period binaries may harbor components with significant rotation-induced activity levels. The light curves of low-mass stars exhibit quite a large range in behavior. Prior to $Kepler$, it was difficult to obtain long duration observations of the required precision to explore this variability. At a level of 1%, the majority of these late type stars are variable, though there is a strong trend for the cooler stars to show a higher incidence of variability. For example, we find that less than half of the K0 dwarfs in our sample are variable, while 88% of stars with T$_{\rm eff}$ $\leq$ 4000 K are variable. Early K dwarfs thus make excellent targets for exoplanet searches. Dinissenkov et al. (2010) have modeled the angular momentum transport in solar-like stars and found that after 4 Gyr, all low-mass stars should rotate with periods similar to that of the Sun. The mean rotation period that we find, $\langle$ P$_{\rm rot}$ $\rangle$ = 32.12 d, is completely consistent with this prediction, especially given that our sample of periodic variables should have roughly same age and metallicity as the Sun. If ages could somehow be established for our objects, it would provide useful insight into the angular momentum loss process in a region of parameter space that is currently only occupied by the Sun. Obviously, age determinations for isolated late-type dwarfs are extremely difficult, but it is possible to measure the kinematic motions of a large sample of such objects and arrive at a statistically useful age vs. rotation rate determination. Equally interesting is the pursuit of the longer period variables. As Vaughan et al. (1981) have shown, active regions on solar-like stars seem to be able to persist for several rotation cycles. Our results bear-out this finding with several objects having rotation periods in excess of 100 d. Is there an upper limit to the rotation period of late-type stars, or conversely, the lifetime of active regions on these objects? Out of our 670 target sample, we found 134 objects that appeared to have sinusoidal light curves with periods in excess of 90 d. The amplitudes of these variations were generally similar to the rotational sample, though few showed variations in excess of 1.5%. These objects warrant further $Kepler$ follow-up to enable the determination of whether these light curves are consistent with rotation and, if so, what are the upper bounds on the rotation rates of late-type dwarfs? Solar active regions rarely remain intact for more than a single rotation period. The fact that some objects appear to have much longer-lived features is intriguing, suggesting that some other factor besides rotation influences magnetic activity in solar-like stars. It remains difficult to extract significant insight into the nature of starspots from the broad-band light curves of late-type stars, even those with the high precision afforded by $Kepler$. The results we have found for the sizes and relative temperatures of the spots on these stars are fully consistent with those found by others. Future investigations into starspot parameters using exoplanet transits will be considerably more useful, though they will only probe a rather small range in latitude for the exoplanet host stars. Our results do strongly suggest, however, that the majority of the stars in our sample either have polar spots, or they have two spots of similar size and temperature that are separated by $\sim$ 180$^{\circ}$ in longitude. This conclusion is further strengthened by the fact that we have numerous objects with flat minima. Such objects must have more than one spot to attain such a light curve, though it is actually quite hard to produce models with flat minima using round spots. This probably indicates that the active regions on these stars have complex shapes that are quite extended in longitude. Kepler was competitively selected as the tenth Discovery mission. Funding for this mission is provided by NASA’s Science Mission Directorate. The authors have been partially supported from NASA grant NNX10AC40G. JLC is supported through an NSF Graduate Research Fellowship. [**References**]{} Baraffe, I., Chabrier, G., Allard, F., & Hauschildt, P. H. 1998, A&A. 337, 403\ Basri, G., et al. 2011, AJ, 141, 20\ Basri, G., et al. 2010, ApJ, 713, 155\ Benko, J. M., et al. 2010, MNRAS, 409, 1585\ Bessell, M. S. 1991, AJ, 101, 662\ Bopp, B. W. 1987, ApJ, 317, 781\ Borucki, W. 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--- abstract: 'Generating entanglement by simply cooling a system into a stationary state which is highly entangled has many advantages. Schemes based on this idea are robust against parameter fluctuations, tolerate relatively large spontaneous decay rates, and achieve high fidelities independent of their initial state. A possible implementation of this idea in atom-cavity systems has recently been proposed by Kastoryano [*et al.*]{} \[Phys. Rev. Lett. [**106**]{}, 090502 (2011)\]. Here we propose an improved entanglement cooling scheme for two atoms inside an optical cavity which achieves higher fidelities for comparable single-atom cooperativity parameters $C$. For example, we predict fidelities above $90 \%$ even for $C$ as low as $20$ without requiring individual laser addressing and without having to detect photons.' address: | $^1$The School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom\ $^2$Department of Physics, Sofia University, James Bourchier 5 blvd, 1164 Sofia, Bulgaria author: - 'J. Busch,$^1$ S. De,$^1$ S. S. Ivanov,$^{1,2}$ B. T. Torosov,$^{1,2}$ T. P. Spiller,$^1$ and A. Beige$^1$' title: 'Cooling atom-cavity systems into entangled states' --- Introduction ============ Current atom-cavity experiments with coupling constants $g$, cavity decay rates $\kappa$, and atomic decay rates $\Gamma$ operate in a parameter regime where the single-atom cooperativity parameter $C$, $$\begin{aligned} \label{C} C \equiv {g^2 \over \kappa \Gamma} \, ,\end{aligned}$$ is at most one or two orders of magnitude larger than one [@Rempe0; @Walther; @Chapman; @Meschede; @Hinds; @Reichel]. However, the practical realisation of atom-cavity quantum computing schemes usually requires $C$’s above $200$ to achieve single-operation fidelities above $90 \, \%$ [@Pellizzari; @zoller2; @Beige; @zheng; @Pachos; @you; @Rempe; @zoller3]. The only alternative are probabilistic quantum computing schemes. These promise fidelities above $90 \, \%$ even when $C = 10$ but rely either on the detection of single photons [@Lim; @sean] or on the observation of macroscopic fluorescence signals [@Metz]. Because of being conditional, they require relatively high photon detection efficiencies and cavity mirrors with low absorption coefficients. Using currently available experimental setups to entangle atoms in optical cavities with a very high fidelity therefore requires a different approach than previously proposed in the literature. Recently it has been pointed out by several authors [@Milburn; @Kraus-Cirac; @Kraus:2008p5470; @Verstraete:2009p3815; @Diehl:2008p7796; @Vacanti:2009p5419; @Wang; @Viola; @Cho] that it is possible to generate entanglement in a controlled way by simply [*cooling*]{} qubits into well-defined, highly entangled states. The main idea behind this approach is to design laser fields such that the target state becomes the stationary state of the system. State preparation schemes based on this idea are expected to tolerate much higher spontaneous decay rates than proposals which do not use dissipation in this way. Moreover, when cooling a system into an entangled state, the fidelity of the state preparation no longer depends on the initial state of the system which makes the entanglement generation more robust against errors. Although being very promising, this approach has only recently been studied as a tool to entangle two atoms trapped inside the same optical cavity. The only examples are Kastoryano [*et al.*]{} [@Sorensen] and Wang and Schirmer [@Wang2]. In this paper we follow similar ideas as Refs. [@Sorensen; @Wang2] and design an entangling scheme to cool two atoms inside an optical cavity into a maximally entangled state. As proposed in Ref. [@Vacanti:2009p5419], and in close analogy to the laser sideband cooling technique of trapped ions [@cool], we employ level shifts and apply laser fields such that only the target state experiences off-resonant driving. Every ground state of the system other than the target state couples resonantly and sufficiently strong to rapidly decaying excited states. Doing so, the target state becomes the stationary state of the quantum system. It is reached independently of the initial state of the system after a certain transition time. As in laser sideband cooling, the fidelity of the final state reaches one when the detuning of the target state becomes much larger than the relevant laser Rabi frequencies and decay rates. The concrete experimental setup which we consider in this paper is shown in Fig. \[scheme\]. It consists of two atoms simultaneously trapped inside an optical cavity. The main decay channels in this system are the spontaneous emission of photons from the excited atomic state $|2 \rangle$ with decay rate $\Gamma$ and the leakage of photons through the cavity mirrors with decay rate $\kappa$. Both atoms are driven by external laser fields which couple respectively to the 0–1 and the 1–2 transition. In the following, we design the detunings and Rabi frequencies of these laser fields such that the stationary state of the atom-cavity system is to a very good approximation given by the maximally entangled atomic ground state $$\begin{aligned} \label{target} {\left\vert+ \right\rangle\!} &=& ({\left\vert01 \right\rangle\!} + {\left\vert10 \right\rangle\!}) / \sqrt{2} \, ,\end{aligned}$$ while there is no photon inside the cavity. As we shall see below, individual laser addressing of the atoms is not required. The entangling scheme proposed in this paper uses an energy shift of the target state which is due to a non-zero atom-cavity coupling constant $g$ as well as spontaneous emission from excited states. This makes it possible to prepare the state in Eq. (\[target\]) with a fidelity above $90 \, \%$ even when $C$ is as low as 20 and without having to detect photons. One advantage of the state preparation scheme presented in this paper is that it predicts higher fidelities than the recently proposed entangling schemes in Refs. [@Sorensen; @Wang2] although they employ similar level shifts to cool two the atoms into a maximally entangled state. Ref. [@Sorensen] uses a similar atomic level scheme as the one shown in Fig. \[scheme\] but with the addition of a driven microwave transition between the triplet states. Ref. [@Wang2] relies on the presence of an external magnetic field gradient to produce the required level splittings. There are five sections in this paper. In the next section, we introduce a four-level toy-model and discuss how to cool it into one of its ground states. The reason for the introduction of this toy-model is that the entangling scheme proposed in this paper cannot be modelled easily analytically. There is no interaction picture in which the system Hamiltonian becomes time-independent. Although being much simpler, the toy-model in Section \[toy\_model\] captures all the basic features of the proposed state preparation scheme, provides much insight into its cooling mechanism, but is nevertheless analytically tractable. In Section \[full\], we present all the details of our entangling scheme, draw analogies to the toy-model, and support our claims about the parameter dependence of its fidelity with the help of numerical simulations. We finally summarize our findings in Section \[conc\]. State preparation in a toy-model {#toy_model} ================================ In this section we consider a simple four-level system and pose the task to prepare it in one of its two ground states. The role of the experimental parameters in this simple model, i.e. its laser Rabi frequencies, detunings, and spontaneous decay rates, can later be mapped onto the atom-cavity coupling constant $g$, the cavity and the atom decay rates $\kappa$ and $\Gamma$, and laser parameters $\Omega_i$ and $\delta_i$ in the entangling scheme proposed in Section \[full\]. Our understanding of the toy-model will allow us to correctly predict the general dependence of the fidelity and the cooling rate of the proposed entangling scheme on these experimental parameters. Theoretical model {#theoretical_model} ----------------- The toy-model contains only two ground states $|0 \rangle$ and $|1 \rangle$ and two excited states $|2 \rangle$ and $|3 \rangle$, as shown in Fig. \[toy\_levels\]. For simplicity, we assume that the decay rates for all four decay channels are the same and denote the overall spontaneous decay rate of level 2 and 3 by $\Gamma$. Moreover, we assume that the system is driven by a single laser field of frequency $\omega_{\rm L}$ and Rabi frequency $\Omega$. This laser is in resonance with the 1–3 transition but detuned from the 0–2 transition by a detuning $\Delta$. The spontaneous emission of photons is in the following taken into account using the master equation $$\begin{aligned} \label{t_master} \dot{\varrho}(t) &=& -\frac{{\rm i}}{\hbar} \left[H_{\rm cond} \varrho - \varrho H_{\rm cond}^{\dagger}\right] + \mathcal{R}(\varrho) \, ,\end{aligned}$$ where $\varrho $ is the density matrix of the system. The conditional Hamiltonian $H_{\rm cond}$ describes the time evolution under the condition of no photon emission, while ${R}(\varrho) $, $$\begin{aligned} \mathcal{R}( \varrho ) &=& \sum_{i=0,1} \sum_{j=2,3} \frac{1}{2} \Gamma {\left\verti \right\rangle\!}{\!\left\langle j\right\vert}\varrho{\left\vertj \right\rangle\!}{\!\left\langle i\right\vert} \, ,\end{aligned}$$ relates to the reset state in case of a photon emission. Within the rotating wave approximation and in the interaction picture with respect to the free Hamiltonian $$\begin{aligned} \label{toy_H0} H_0 &=& \sum_{i=0}^3 \hbar \omega_i {\left\verti \right\rangle\!}{\!\left\langle i\right\vert} - \hbar \Delta {\left\vert2 \right\rangle\!}{\!\left\langle 2\right\vert} \, ,\end{aligned}$$ where $\hbar \omega_i$ is the energy of level $i$, $H_{\rm cond}$ equals $$\begin{aligned} \label{toy_Hcond} H_{\rm cond} &=& \frac{1}{2} \hbar\Omega \left({\left\vert0 \right\rangle\!}{\!\left\langle 2\right\vert} + {\left\vert1 \right\rangle\!}{\!\left\langle 3\right\vert} + {\rm H.c.} \right) + \hbar\Delta {\left\vert2 \right\rangle\!}{\!\left\langle 2\right\vert} \nonumber \\ && - {1 \over 2} {\rm i}\hbar\Gamma \left( {\left\vert2 \right\rangle\!}{\!\left\langle 2\right\vert} + {\left\vert3 \right\rangle\!}{\!\left\langle 3\right\vert} \right) \, ,\end{aligned}$$ since $\omega_{\rm L} = \omega_2 - \Delta$. The master equation in Eq. (\[t\_master\]) can now be used to calculate the fidelity of the proposed state preparation scheme, i.e. the stationary state population of its target state $|0 \rangle$, and its cooling rate. This is most easily done using rate equations which are a complete set of differential equations for the time evolution of expectation values. The time derivative of an expectation value of a time-independent operator $A$ equals $$\begin{aligned} \langle \dot A \rangle &=& \mbox{Tr} (A \dot{\varrho}) \, .\end{aligned}$$ The above master equation hence implies that $$\begin{aligned} \label{dotA0} \langle \dot A \rangle &=&- {1 \over 2} {\rm i} \Omega \left \langle \, \left[ A , |0 \rangle \langle 2| + |1 \rangle \langle 3| + {\rm H.c.} \right] \, \right \rangle \nonumber \\ && - {\rm i} \Delta \left \langle \, \left[ A , |2 \rangle \langle 2| \, \right] \, \right \rangle - \sum_{j=2,3} {1 \over 2} \Gamma \left \langle \, A |j \rangle \langle j| + |j \rangle \langle j| A \, \right \rangle \nonumber \\ && + \sum_{i=0,1} \sum_{j=2,3} {1 \over 2} \Gamma \left \langle \, |j \rangle \langle i| \, A \, |i \rangle \langle j| \, \right \rangle \, . ~~\end{aligned}$$ In the following we consider the Hermitian operators $|i \rangle \langle i|$, $|i \rangle \langle j| + |j \rangle \langle i|$, and ${\rm i} (|i \rangle \langle j| - |j \rangle \langle i| )$ and denote their (real) expectation values by $$\begin{aligned} P_i &=& {\!\left\langle i\right\vert}\varrho{\left\verti \right\rangle\!} \, , \nonumber \\ k_{ij} &=& 2{\rm Im} \, {\!\left\langle i\right\vert}\varrho{\left\vertj \right\rangle\!} \, , \nonumber \\ l_{ij} &=& 2{\rm Re} \, {\!\left\langle i\right\vert}\varrho{\left\vertj \right\rangle\!} \, .\end{aligned}$$ Substituting these operators into Eq. (\[dotA0\]) yields $$\begin{aligned} \label{rateeqns} \dot{P}_0 &=& - {1 \over 2} \Omega \, k_{02} + {1 \over 2} \Gamma \left( P_2 + P_3 \right) \, , \nonumber \\ \dot{P}_1 &=& - {1 \over 2} \Omega \, k_{13} + {1 \over 2} \Gamma \left( P_2 + P_3 \right) \, , \nonumber \\ \dot{P}_2 &=& {1 \over 2} \Omega \, k_{02} - \Gamma P_2 \, , \nonumber \\ \dot{P}_3 &=& {1 \over 2} \Omega \, k_{13} - \Gamma P_3 \, , \nonumber \\ \dot{k}_{02} &=& \Omega (P_0 - P_2) + \Delta l_{02} - {1 \over 2} \Gamma k_{02} \, , \nonumber \\ \dot{k}_{13} &=& \Omega (P_1 - P_3) - {1 \over2} \Gamma k_{13} \, , \nonumber \\ \dot{l}_{02} &=& -\Delta k_{02} - {1 \over 2} \Gamma l_{02} \, .\end{aligned}$$ These seven equations form a complete set of rate equations and are sufficient to analyse the time evolution of the population $P_0$ in the target state $|0 \rangle$ which equals the fidelity ${\rm F}$. The basic idea {#basic_idea} -------------- Suppose we aim to transfer the toy-model in Fig. \[toy\_levels\] into one of its ground states, for example the $|0 \rangle$ state, with a very high fidelity and without having to control the initial state of the system. This is possible when laser driving is applied such that the $|0 \rangle$ state becomes the stationary state of the toy-model as it applies when the laser detuning for the $0$–$2$ transition is much larger than the other system parameters, i.e. when $$\begin{aligned} \label{toy_regime} \Delta &\gg & \Omega, \, \Gamma \, .\end{aligned}$$ This condition guarantees that it is much more likely for the system to spontaneously decay into the $|0 \rangle$ state when being in one of the other three states, than being driven out of it [@Vacanti:2009p5419]. Once the system has reached its stationary state it therefore remains there with a very high probability. This is confirmed by Fig. \[toy\_omega\] which shows the time dependence of $1-{\rm F}$, i.e. of the total population in states other than the target state $|0 \rangle$, for a wide range of experimental parameters. The solid lines in this Fig. \[toy\_omega\] are the result of a numerical integration of the rate equations in Eq. (\[rateeqns\]) which assumes the worst case scenario with the toy-model initially in $|1 \rangle$. The plots show exponential cooling towards the target state until the system reaches a stationary state. The fidelity of the state preparation equals the population of the $|0 \rangle$ state and is indeed very close to unity, as long as condition (\[toy\_regime\]) applies. The cooling rate and the fidelity of the state preparation both depend on the relative size of $\Omega$ and $\Gamma$ with respect to the detuning $\Delta$. Stationary state fidelity {#toy_fidelity} ------------------------- To identify the best way of preparing the target state, we now derive approximate analytical expressions for the stationary state fidelity, ${\rm F}$, and the cooling rate, $\gamma_{\rm c}$, which is a measure for the time it takes the system to reach its stationary state. Since ${\rm F}$ is the stationary state population $P_0$ in the ${\left\vert0 \right\rangle\!}$ state, it can be calculated simply by setting the time derivatives of the expectation values in Eq. (\[rateeqns\]) equal to zero. Doing so, we find that $$\begin{aligned} \label{t_stationary} {\rm F} &=& 1 - \frac{3\Omega^2 + \Gamma^2}{4\Delta^2 + 4\Omega^2 + 2\Gamma^2} \, .\end{aligned}$$ For relatively large detunings $\Delta$, as in Eq. (\[toy\_regime\]), this equation simplifies to $$\begin{aligned} \label{t_stationary2} {\rm F} &=& 1 - \frac{3\Omega^2 + \Gamma^2}{4\Delta^2} \, .\end{aligned}$$ This result confirms that the fidelity is close to one in the parameter regime given by Eq. (\[toy\_regime\]). Fig. \[toy\_contour\] illustrates the effects of finite $\Omega$ and $\Gamma$. It also shows that an increase in the Rabi frequency $\Omega$ reduces the stationary state fidelity more rapidly than an increase in the decay rate $\Gamma$. Heating and cooling rates {#cooling_rate} ------------------------- To see how quickly the toy-model reaches its stationary state, we now introduce the notion of a cooling and a heating rate which we denote $\gamma_{\rm c}$ and $\gamma_{\rm h}$, respectively. For simplicity, and since we are anyway only interested in the general scaling of these rates with the experimental parameters, we assume that these rates do not depend on the current state $\varrho $ of the system. Invoking the conservation of probability flux, we then find that $$\begin{aligned} \label{gammas} \dot{P}_0 = \gamma_{\rm c} \, \left( 1-P_0 \right) - \gamma_{\rm h} \, P_0 \, .\end{aligned}$$ The principle here is that the rate at which the fidelity, i.e. the current population in the $|0 \rangle$ state, changes in time is equal to the rate at which population is cooled into the target state minus the rate at which population is heated out of the target state. When the system reaches its stationary state, the fidelity remains constant. The above equation hence implies $$\begin{aligned} \label{gamma_stationary} \gamma_{\rm h} \, {\rm F} = \gamma_{\rm c} \, \left( 1 - {\rm F} \right) \, .\end{aligned}$$ Since we already know ${\rm F}$ (cf. Eq. (\[t\_stationary\])), this relation can be used to obtain the cooling rate after obtaining an estimate for the heating rate. As we shall see below, it is easier to derive an approximate expression for $\gamma_{\rm h}$, than calculating $\gamma_{\rm c}$ directly. Considering the parameter regime in Eq. (\[toy\_regime\]), the rate equations in Eq. (\[rateeqns\]) can be simplified via an adiabatic elimination. Only the coherences ${k}_{02}$ and ${l}_{02}$ evolve on the fast time scale given by $\Delta$. Setting their time derivatives equal to zero, we find that $$\begin{aligned} k_{02} = \frac{2 \Gamma\Omega}{\Gamma^2 + 4 \Delta^2}\left( P_0 - P_2 \right) \, .\end{aligned}$$ Assuming that the toy-model is in $|0 \rangle$, i.e. that $P_0 = 1$ and $P_1 = P_2 = P_3 = 0$, and substituting the above expression for $k_{02}$ into the rate equation for $P_0$ yields $$\begin{aligned} \label{toy_P_0} \dot{P}_0 = - \frac{\Gamma\Omega^2}{\Gamma^2 + 4\Delta^2}\, P_0 \, .\end{aligned}$$ Comparing this equation with Eq. (\[gammas\]) for $P_0 =1$, we find that the heating rate is to a very good approximation given by $$\begin{aligned} \label{gamma_h} \gamma_{\rm h} = \frac{\Gamma\Omega^2}{\Gamma^2 + 4\Delta^2} \, .\end{aligned}$$ This is the rate at which the target state $|0 \rangle$ loses its population. Substituting this result and Eq. (\[t\_stationary\]) into Eq. (\[gamma\_stationary\]), we get $$\begin{aligned} \label{gamma_c} \gamma_{\rm c} &=& \frac{\Gamma \Omega^2 \left( 4\Delta^2 + \Omega^2 + \Gamma^2 \right)}{\left( 4\Delta^2 + \Gamma^2 \right)\left( 3\Omega^2 + \Gamma^2 \right)} \, .\end{aligned}$$ Fig. \[toy\_CoolRate\] shows this cooling rate $\gamma_{\rm c}$ for a wide range of experimental parameters. For relatively small Rabi frequencies $\Omega$, the cooling process becomes faster with increasing $\Omega$. However, it is not worth increasing $\Omega$ beyond a certain size which saturates $\gamma_{\rm c}$. In the parameter regime given by Eq. (\[toy\_regime\]), the cooling rate $\gamma_{\rm c}$ simplifies to $$\begin{aligned} \label{gamma_c2} \gamma_{\rm c} &=& \frac{\Gamma \Omega^2}{3\Omega^2+\Gamma^2} \, .\end{aligned}$$ which no longer depends on $\Delta$ but only holds for sufficiently large detunings. In order to get a feeling for the accuracy of the cooling rate $\gamma_{\rm c}$ in Eq. (\[gamma\_c\]), we now solve Eq. (\[gammas\]) analytically and compare the result with exact numerical solutions of the rate equations in Eq. (\[rateeqns\]). Doing so and assuming $P_0(0) = 0$ we find that $$\begin{aligned} \label{F(t)} P_0(t) = \frac{\gamma_{\rm c}}{\gamma_{\rm c}+\gamma_{\rm h}}\left( 1 - {\rm e}^{-\left(\gamma_{\rm c} + \gamma_{\rm h}\right)t} \right) \, .\end{aligned}$$ Fig. \[toy\_omega\] compares this analytical result with numerical solutions of $P_0(t)$ for different experimental parameters $\Omega/\Delta$ and $\Gamma/\Delta$. It shows that Eq. (\[gamma\_c\]) reflects the general parameter dependence of the cooling rate on $\Omega/\Delta$ and $\Gamma/\Delta$ correctly. The above approximate solution is in general slightly higher than the actual cooling rate. The reason for this is that the heating rate in Eq. (\[gamma\_h\]) has been calculated for the case, where the system is initially in $|0 \rangle$, i.e. when it is the highest. Choosing experimental parameters {#choosing} -------------------------------- Fig. \[toy\_contour\] shows that maximising the stationary state fidelity ${\rm F}$ requires a Rabi frequency $\Omega$ as small as possible. However, from Fig. \[toy\_CoolRate\] we see that we only obtain high cooling rates when $\Omega$ is relatively large. To minimise the state preparation time while maintaining a high fidelity, we therefore suggest using a laser pulse with a time-dependent Rabi frequency to prepare the target state. This laser pulse should be large initially and should reach zero by the end of the cooling process. For example one could choose $$\begin{aligned} \label{toy_Ot} \Omega(t) = {3 \Omega_0 \over \left( 1 + \gamma_{\rm c}(0) t \right)^{2}} \end{aligned}$$ with $\gamma_{\rm c}(0)$ being the cooling rate in Eq. (\[gamma\_c2\]) for the initial Rabi frequency $\Omega(0) = \Omega_0$. Alternatively, one could choose an exponentially decreasing Rabi frequency. However, in this case, $\Omega$ would drop off too rapidly, thereby resulting in a fidelity that is far from optimal. Here we do not discuss how to optimise the spontaneous decay rate, since $\Gamma$ is in general fixed. Fig. \[toy\_varOmega\] confirms that choosing the Rabi frequency $\Omega$ as in Eq. (\[toy\_Ot\]) indeed yields a significant speed up compared to time-independent Rabi frequencies. We also observe a stationary state fidelity which is close to the theoretical maximum obtained when setting $\Omega = 0$. From Eq. (\[t\_stationary2\]) we see that this maximum is to a very good approximation given by $$\begin{aligned} \label{opt} {\rm F}(\Omega=0) = 1 - \frac{\Gamma^2}{4 \Delta^2 + 2 \Gamma^2} \, .\end{aligned}$$ It is indicated by a dashed line in Fig. \[toy\_varOmega\]. In the next Section we use similar time-dependent laser pulses to prepare two atoms inside an optical cavity relatively fast and with a high fidelity in a maximally entangled state. Entangling Scheme {#full} ================= The toy-model described in the previous section is based on a simple principle: driving populations out of all undesired states resonantly while driving the target state off-resonantly. This approach can indeed be used to prepare target states with a high fidelity [@Vacanti:2009p5419]. In this section, we use this idea to prepare two atoms inside an optical cavity (cf. Fig. \[scheme\]) in the maximally entangled state $|+ \rangle$ in Eq. (\[target\]). The first half of this section presents a theoretical description of the atom-cavity system. After identifying its dressed states, we select appropriate laser Rabi frequencies and detunings. As we shall see below, the state preparation requires three different driving lasers but there is no need to address atoms individually. A comparison with the toy-model introduced in the previous section allows us to predict the dependence of fidelity and cooling rate of the proposed state preparation scheme on the experimental parameters with a very high accuracy. System Hamiltonian without laser driving {#AtomCav_theoretical_model} ---------------------------------------- The experimental setup which we consider in this paper consists of two atoms placed inside an optical cavity as shown in Fig. \[scheme\]. Each atom contains a $\Lambda$-type level configuration with $\hbar \omega_j$ and $|j \rangle$ denoting the corresponding energies and energy eigenstates $(j=0,1,2)$. Suppose the 1–2 transition of each atom couples resonantly with coupling strength $g$ to the quantised cavity field mode with frequency $\omega_{\rm c}$. Then the Hamiltonian $H_{\rm sys}$ of this system equals $$\begin{aligned} \label{Hsys} H_{\rm sys} &=& \sum_{i=1}^{2}\hbar g \, |1\rangle_{ii}\langle2| c^{\dagger}+{\rm H.c.} + \sum_{i=1}^{2} \sum_{j=0}^2 \hbar\omega_{j} \, |j\rangle_{ii}\langle j\vert \nonumber \\ && + \hbar \omega_{\rm c} \, c^{\dagger}c \end{aligned}$$ in the absence of external laser driving. Here $c$ and $c^\dagger $ are the cavity photon annihilation and creation operators for a single photon inside the optical cavity. As we shall see below, it is important that the atomic states $|0 \rangle$ and $|1 \rangle$ differ in energy by an amount which is significantly larger than $\hbar g$.    Energy eigenstate       Energy    --------------------------------------------------------------------------------------------------------------------------------------------- -------------------------- ${\left\vert00,0 \right\rangle\!}$ 0    ${\left\vert+,0 \right\rangle\!} \equiv \left({\left\vert01,0 \right\rangle\!} + {\left\vert10,0 \right\rangle\!}\right) / \sqrt{2} $    $\hbar \omega_1$    ${\left\vert-,0 \right\rangle\!} \equiv \left({\left\vert01,0 \right\rangle\!} - {\left\vert10,0 \right\rangle\!}\right) / \sqrt{2} $    $\hbar \omega_1$ ${\left\vert11,0 \right\rangle\!}$    $2 \hbar \omega_1$    : Energy eigenstates and energy eigenvalues of the system Hamiltonian $H_{\rm sys}$ in Eq. (\[Hsys\]) for the ${\cal H}_0$ subspace with no atom in $|2 \rangle$ and no photons in the cavity.[]{data-label="g_states"} In the following, we identify the relevant energy eigenstates of this Hamiltonian, since this will allow us to identify appropriate laser drivings and detunings for the proposed state preparation scheme. To do so, we denote states with atom $1$ in ${\left\vertj_1 \right\rangle\!}$, atom $2$ in ${\left\vertj_2 \right\rangle\!}$ and $n$ photons in the cavity by ${\left\vertj_1j_2,n \right\rangle\!}$. Moreover we notice that the Hamiltonian $H_{\rm sys}$ preserves the total amount of population in the excited atomic state $|2 \rangle$ and the cavity field mode. It therefore acts on fixed excitation subspaces ${\cal H}_n$ of the complete Hilbert space whose energy eigenstates can be calculated separately. The eigenstates and eigenvalues of the subspace ${\cal H}_0$ of states with no population of the excited atomic state $|2 \rangle$ and no photons in the cavity are summarised in Table \[g\_states\]. Table \[e\_states\] shows the eight energy eigenstates and the corresponding energy eigenvalues of the subspace ${\cal H}_1$ of states with either one atom in $|2 \rangle$ or one photon in the cavity and adopts the notation $$\begin{aligned} \label{eigenvecs} {\left\vert\mu_{1} \right\rangle\!} &\equiv & \left({\left\vert21,0 \right\rangle\!} - {\left\vert12,0 \right\rangle\!}\right) / \sqrt{2} \, , \nonumber \\ {\left\vert\mu_{0},\pm \right\rangle\!} &\equiv & \left({\left\vert02,0 \right\rangle\!} - {\left\vert20,0 \right\rangle\!} \pm {\left\vert01,1 \right\rangle\!} \mp {\left\vert10,1 \right\rangle\!}\right) / 2 \, , \nonumber \\ {\left\vert\lambda_{0},\pm \right\rangle\!} &\equiv & \left({\left\vert02,0 \right\rangle\!} + {\left\vert20,0 \right\rangle\!} \pm {\left\vert01,1 \right\rangle\!} \pm {\left\vert10,1 \right\rangle\!}\right) / 2 \, , \nonumber \\ {\left\vert\lambda_{1},\pm \right\rangle\!} &\equiv & \left({\left\vert12,0 \right\rangle\!} + {\left\vert21,0 \right\rangle\!} \pm \sqrt{2}{\left\vert11,1 \right\rangle\!}\right) / 2 \, .\end{aligned}$$ Fortunately, there is no need to identify the energy eigenstates of the atom-cavity system of the subspace of states with more than one excitation in the atomic state $|2 \rangle$ and the cavity field mode. The reason for this is that these states do not couple directly to the states in Table \[g\_states\] in case of laser driving. Therefore they do not have to be taken into account when choosing laser parameters such that only the $|+,0 \rangle$ state experiences off-resonant laser driving. Laser driving {#laserdriving} ------------- As we shall see below, the state preparation of the maximally entangled atomic state $|+ \rangle$ in Eq. (\[target\]) requires the simultaneous excitation of the two atoms with three different laser fields. In the following we assume that the 0–2 transition of each atom is driven by two different lasers with Rabi frequencies $\Omega_{0}^{(k)}$ and frequencies $\omega_{0}^{(k)}$ respectively $(k=1,2)$. The 1–2 transition of each atom should moreover be driven by a laser field with Rabi frequency $\Omega_1$ and frequency $\omega_{\rm L1}$. The laser Hamiltonian in the Schrödinger picture and the usual rotating wave approximation is then given by $$\begin{aligned} \label{HL} H_{\rm L}(t) &=& \sum_{i=1}^{2} \sum_{k=1}^{2} \frac{1}{2} \hbar \Omega^{(k)}_{0} \, {\rm e}^{{\rm i} \omega^{(k)}_{\rm L0} t} \, |0 \rangle_{ii} \langle 2| + {\rm H.c.} \nonumber \\ && + \sum_{i=1}^{2} \frac{1}{2} \hbar \Omega_{1} \, {\rm e}^{{\rm i} \omega_{\rm L1} t} \, |1 \rangle_{ii} \langle 2| + {\rm H.c.} \end{aligned}$$ The realisation of this Hamiltonian does not require individual laser addressing, since both atoms experience exactly the same laser driving.    Energy eigenstate       Energy    ----------------------------------------------- ------------------------------------------------------------ ${\left\vert00,1 \right\rangle\!}$    $\hbar \omega_{\rm c} = \hbar (\omega_2 - \omega_1)$    ${\left\vert\mu_{1} \right\rangle\!}$ $\hbar (\omega_1 + \omega_2)$ ${\left\vert\mu_{0},\pm \right\rangle\!}$ $\hbar (\omega_2 \pm g)$ ${\left\vert\lambda_{0},\pm \right\rangle\!}$ $\hbar (\omega_2 \pm g)$ ${\left\vert\lambda_{1},\pm \right\rangle\!}$ $\hbar (\omega_1 + \omega_2 \pm \sqrt{2}g)$ : Energy eigenstates and energy eigenvalues of the system Hamiltonian $H_{\rm sys}$ in Eq. (\[Hsys\]) for the ${\cal H}_1$ subspace of states with either one atom in $|2 \rangle$ or one photon in the cavity. The table uses the notation introduced in Eq. (\[eigenvecs\]).[]{data-label="e_states"} In order to see how to best choose the laser frequencies $\omega_{\rm L0}^{(k)}$ and $\omega_{\rm L1}$, we now consider the effect of this laser Hamiltonian on the ${\cal H}_0$ subspace. This effect can be described by the restricted laser Hamiltonian $\tilde H_{\rm L}$ defined as $$\begin{aligned} \tilde H_{\rm L}(t) &\equiv & P \, H_{\rm L}(t) \, P \end{aligned}$$ with the projector $P$ being the projector on ${\cal H}_0$ and ${\cal H}_1$ given by $$\begin{aligned} P &=& \sum_{x=+,-} |\mu_0,x \rangle \langle \mu_0,x| + \sum_{j=0,1} \sum_{x=+,-} |\lambda_j,x \rangle \langle \lambda_j,x| \nonumber \\ && + |\mu_1 \rangle \langle \mu_1|\, .\end{aligned}$$ Using the eigenvectors of the undriven atom-cavity system Hamiltonian which can be found in Tables \[g\_states\] and \[e\_states\] one can show that this Hamiltonian equals $$\begin{aligned} \label{H_laser} \tilde H_{\rm L}(t) &=& \sum_{k=1}^{2} \sum_{x=+,-} {1 \over 2 \sqrt{2}} \hbar \Omega_0^{(k)} \, {\rm e}^{{\rm i}\omega_{{\rm L0}}^{(k)}t} \, \Big[ {\left\vert00,0 \right\rangle\!} {\!\left\langle \lambda_0,x\right\vert} \nonumber \\ && + {\left\vert+,0 \right\rangle\!} {\!\left\langle \lambda_1,x\right\vert} \Big] + \sum_{k=1}^{2} {1 \over 2} \hbar \Omega_0^{(k)} \, {\rm e}^{{\rm i}\omega_{{\rm L0}}^{(k)}t} \, {\left\vert-,0 \right\rangle\!}{\!\left\langle \mu_1\right\vert} \nonumber \\ && + \sum_{x=+,-} \frac{1}{2 \sqrt{2}} \hbar \Omega_1 \, {\rm e}^{{\rm i}\omega_{\rm L1} t} \, \Big[ {\left\vert+,0 \right\rangle\!} {\!\left\langle \lambda_0,x\right\vert} \nonumber \\ && + {\left\vert-,0 \right\rangle\!} {\!\left\langle \mu_0,x\right\vert} + {\left\vert11,0 \right\rangle\!} {\!\left\langle \lambda_1,x\right\vert} \Big] + {\rm H.c.} \end{aligned}$$ in the Schrödinger picture. Changing into an interaction picture in which $\tilde H_{\rm L}(t)$ becomes time independent is not possible, since there are more laser fields than atomic transitions. ---------------------------------------------- --------------------------------------------- ----------------------------- -------------------------------------------------------- Ground Excited Rabi Effective \[-0.15cm\] state state frequency detuning \[0.1cm\] ${\left\vert00,0 \right\rangle\!}$ ${\left\vert\lambda_0,\pm \right\rangle\!}$ $\Omega_0^{(1)} / \sqrt{2}$ $\omega_{\rm L0}^{(1)} - \omega_2 \pm g$ ${\left\vert\lambda_0,\pm \right\rangle\!}$ $\Omega_0^{(2)} / \sqrt{2}$ $\omega_{\rm L0}^{(2)} - \omega_2 \pm g$ ${\left\vert+,0 \right\rangle\!}$ ${\left\vert\lambda_1,\pm \right\rangle\!}$ $\Omega_0^{(1)} / \sqrt{2}$ $\omega_{\rm L0}^{(1)} - \omega_2 \pm \sqrt{2} g$ ${\left\vert\lambda_1,\pm \right\rangle\!}$ $\Omega_0^{(2)} / \sqrt{2}$ $\omega_{\rm L0}^{(2)} - \omega_2 \pm \sqrt{2} g$ ${\left\vert\lambda_0,\pm \right\rangle\!}$ $\Omega_1$ $\omega_{\rm L1} + \omega_1 - \omega_2 \mp g$ ${\left\vert-,0 \right\rangle\!}$ ${\left\vert\mu_1 \right\rangle\!}$ $\Omega_0^{(1)}$ $\omega_{\rm L0}^{(1)} - \omega_2$ ${\left\vert\mu_1 \right\rangle\!}$ $\Omega_0^{(2)}$ $\omega_{\rm L0}^{(2)} - \omega_2$ ${\left\vert\mu_0,\pm \right\rangle\!}$ $\Omega_1$ $\omega_{\rm L1} + \omega_1 - \omega_2 \mp g$ ${\left\vert11,0 \right\rangle\!}$ ${\left\vert\lambda_1,\pm \right\rangle\!}$ $\Omega_1$ $\omega_{\rm L1} + \omega_1 - \omega_2 \mp \sqrt{2} g$ ---------------------------------------------- --------------------------------------------- ----------------------------- -------------------------------------------------------- : Most relevant laser-driven transitions of the atom-cavity system in the dressed state picture. The table shows the respective ground and excited states and indicates the corresponding laser parameters.[]{data-label="t_off_res2"} To make it nevertheless easy to identify the relevant laser Rabi frequencies and detunings, we now transform the laser Hamiltonian $\tilde H_{\rm L} (t)$ for the subspace ${\cal H}_0 \oplus {\cal H}_1$ into the interaction picture with respect to the system Hamiltonian in Eq. (\[Hsys\]). Taking into account the eigenvalues of this Hamiltonian which can be found in Tables \[g\_states\] and \[e\_states\] we obtain another time-dependent Hamiltonian from which we can directly read off the information which is relevant for the construction of an entangling scheme via cooling. The result of this calculation is summarised in Table \[t\_off\_res2\] which shows all laser-driven transitions and states the corresponding relevant laser parameters. Effect of spontaneous emission ------------------------------ As has been illustrated already in Section \[toy\_model\], dissipation is an essential component of state preparation via cooling. In the atom-cavity system analysed in this section, dissipation can occur via the photon emission from the excited atomic state $|2 \rangle$ with the spontaneous decay rate $\Gamma$ and via the leakage of a photon through the cavity mirrors with the spontaneous decay rate $\kappa$. The conditional Hamiltonian that describes the time evolution of the atom-cavity system between photon emissions equals $$\begin{aligned} \label{Hcond} H_{\rm cond} &=& H_{\rm sys} + H_{\rm L}(t) - {{\rm i} \over 2} \hbar \Gamma \sum_{i=1}^{2} \vert 2 \rangle_{ii} \langle 2 \vert - {{\rm i} \over 2} \hbar \kappa \, c^{\dagger}c \, . \nonumber \\ \end{aligned}$$ The first two terms in this equation are the system Hamiltonian $H_{\rm sys}$ in Eq. (\[Hsys\]) and the laser Hamiltonian $H_{\rm L}(t)$ in Eq. (\[HL\]). In case of an emission, the density matrix of the atom-cavity system changes up to normalisation into $$\begin{aligned} \label{RR} \mathcal{R}(\varrho) &=& \sum_{j=0,1}\sum_{i=1,2} \Gamma_{j} \, \vert j \rangle_{ii} \langle2\vert \varrho \vert2\rangle_{ii}\langle j \vert + \kappa c\varrho c^{\dagger} \, , ~~\end{aligned}$$ where $\Gamma_j$ denotes the spontaneous decay rate of the atomic 2–$j$ transition. The overall decay rate of the excited atomic state is given by $\Gamma = \Gamma_0 + \Gamma_1$. Overall, the time evolution of the system in the presence of spontaneous emission is described by master equations which are of exactly the same form as the master equations in Eq. (\[t\_master\]). Appropriate laser parameters {#main2} ---------------------------- ------------------------------------ ---------------------------------------------- ----------------------------- -------------------------------    Ground       Excited    Rabi    Effective    \[-0.15cm\] state state    frequency       detuning    ${\left\vert00,0 \right\rangle\!}$ ${\left\vert\lambda_0,+ \right\rangle\!}$ $\Omega_0^{(1)} / \sqrt{2}$ $-2g$ ${\left\vert\lambda_0,- \right\rangle\!}$ $\Omega_0^{(1)} / \sqrt{2}$ $0$ ${\left\vert\lambda_0,\pm \right\rangle\!}$ $\Omega_0^{(2)} / \sqrt{2}$ $\mp g$ ${\left\vert+,0 \right\rangle\!}$ ${\left\vert\lambda_1,\pm \right\rangle\!}$ $\Omega_0^{(1)} / \sqrt{2}$    $\pm (\sqrt{2} \pm 1)g$    ${\left\vert\lambda_1,\pm \right\rangle\!}$ $\Omega_0^{(2)} / \sqrt{2}$ $\mp \sqrt{2}g$ ${\left\vert\lambda_0,\pm \right\rangle\!}$ $\Omega_1$ $-(\sqrt{2} \pm 1)g$ ${\left\vert-,0 \right\rangle\!}$ ${\left\vert\mu_1 \right\rangle\!}$ $\Omega_0^{(1)}$ $-g$ ${\left\vert\mu_1 \right\rangle\!}$ $\Omega_0^{(2)}$ $0$ ${\left\vert\mu_0,\pm \right\rangle\!}$ $\Omega_1$ $-(\sqrt{2} \pm 1)g$ ${\left\vert11,0 \right\rangle\!}$ ${\left\vert\lambda_1,+ \right\rangle\!}$ $\Omega_1$ $-2\sqrt{2}g$ ${\left\vert\lambda_1,- \right\rangle\!}$ $\Omega_1$ 0 ------------------------------------ ---------------------------------------------- ----------------------------- ------------------------------- : Transitions between dressed states driven near resonance by the application of three lasers with Rabi frequency $\Omega_0^{(1)}$, $\Omega_0^{(2)}$ and $\Omega_1$.[]{data-label="t_off_res"} As already mentioned above, the target state of the state preparation which we propose here is the maximally entangled atomic state $|+ \rangle$ in Eq. (\[target\]). In order to assure that this state becomes the stationary state of the atom-cavity system in Fig. \[scheme\], we need to choose the laser frequencies $\omega_{\rm L0}^{(1)}$, $\omega_{\rm L0}^{(2)}$, and $\omega_{\rm L1}$ such that the $|+,0 \rangle$ experiences only off-resonant driving, while the states $|00,0 \rangle$, $|-,0 \rangle$, and $|11,0 \rangle$ couple resonantly to at least one of the three driving lasers. Having a closer look at Table \[t\_off\_res2\], we see that this applies, if we choose $$\begin{aligned} \label{main} \omega_{\rm L0}^{(1)} &=& \omega_2 - g \, , \nonumber \\ \omega_{\rm L0}^{(2)} &=& \omega_2 \, , \nonumber \\ \omega_{\rm L1} &=& \omega_2 - \omega_1 - \sqrt{2} g \, .\end{aligned}$$ Table \[t\_off\_res\] shows the effect of this choice of laser frequencies on the sixteen transitions which need to be taken into account when designing the state preparation scheme proposed in this paper. The system Hamiltonian $H_{\rm sys}$ treats both atoms in exactly the same way. Its eigenvectors are therefore either symmetric or antisymmetric with respect to an exchange of the two atoms. The same applies to the effective laser Hamiltonian $\tilde H_{\rm L}(t)$. Since both atoms experience exactly the same Rabi frequencies, the lasers excite either transitions between two symmetric states or two anti-symmetric states. This allows us to consider the symmetric and the antisymmetric state space separately when analysing the effect of the laser driving in the dressed state picture of the atom-cavity system. There are three symmetric ground states and one antisymmetric ground state. These are $\{ {\left\vert00,0 \right\rangle\!}, \, {\left\vert+,0 \right\rangle\!}, \, {\left\vert11,0 \right\rangle\!} \}$ and $\{ {\left\vert-,0 \right\rangle\!} \}$ respectively. Fig. \[off\_res\] illustrates the laser driving experienced by the target state $|+,0 \rangle$. Since this state is a symmetric state, the relevant level configuration involves only the target state and the four symmetric states with one excitation in $|2 \rangle$ or the cavity mode. As one can see from Table \[t\_off\_res\], in the dressed state picture, these three lasers involve $|+, 0 \rangle$ in six different transitions. For simplicity, we show only the least detuned couplings for each laser. We see that the target state $|+,0 \rangle$ experiences indeed only off-resonant driving. The smallest and therefore most relevant detuning is given by $$\begin{aligned} \label{delta_min} \delta_{\rm min} &=& (\sqrt{2} - 1) g \, ,\end{aligned}$$ as long as the frequency $\omega_1$ is sufficiently larger than the atom-cavity coupling constant $g$. All other states with no excitation are resonantly driven by one laser field. This is illustrated in Fig. \[sym\] which shows the resonant transitions in the symmetric and the antisymmetric subspace separately. One laser couples ${\left\vert00,0 \right\rangle\!}$ to ${\left\vert\lambda_0,- \right\rangle\!}$ with Rabi frequency $\Omega_0^{(1)}/\sqrt{2}$. Another laser couples ${\left\vert-,0 \right\rangle\!}$ to ${\left\vert\mu_1 \right\rangle\!}$ with Rabi frequency $\Omega_0^{(2)}/\sqrt{2}$, while a third laser drives ${\left\vert11,0 \right\rangle\!}$ into ${\left\vert\lambda_1,- \right\rangle\!}$ with Rabi frequency $\Omega_1$. In principle, we would like these lasers which empty unwanted states to be relatively strong. However, it is not possible to increase them without increasing also the Rabi frequencies for the off-resonant driving of the target state shown in Fig. \[off\_res\]. Fidelities and cooling rates for constant laser driving {#main3} ------------------------------------------------------- Eq. (\[delta\_min\]) shows that the minimum detuning experienced by the target state $\delta_{\rm min}$ depends only on the atom-cavity coupling constant $g$. Eq. (\[toy\_regime\]) in Section \[toy\_model\] therefore suggests that the stationary state of the atom-cavity system in Fig. \[scheme\] is to a very good approximation given by the state $|+,0 \rangle$ as long as $$\begin{aligned} \label{condi} \omega_1 \, \gg \, g \, \gg \, \Gamma, \, \kappa, \, \Omega_0^{(1)}, \, \Omega_0^{(2)}, \, \Omega_1 \, .\end{aligned}$$ In other words, for this parameter regime we can expect the atoms to be with a very high fidelity in the maximally entangled state $|+ \rangle$ in Eq. (\[target\]) after a certain transition time $t$. A comparison with the toy-model state preparation scheme in Section \[toy\_model\] even yields approximate solutions for the fidelity ${\rm F}$ and the cooling rate $\gamma_{\rm c}$ of the proposed entangling scheme. For simplicity, we assume in the following that all three cooling lasers have the same Rabi frequency $\Omega $ and that the two atomic decay rates, $\Gamma_0$ and $\Gamma_1$, are equal. The only remaining spontaneous decay rates are the spontaneous atom decay rate $\Gamma$ and the cavity photon leakage rate $\kappa $. For example, the symmetric state $({\left\vert\lambda_0,+ \right\rangle\!} + {\left\vert\lambda_0,- \right\rangle\!})/\sqrt{2}$ with one atom in $| 2 \rangle$ has the spontaneous decay rate $\Gamma$, whilst the state $({\left\vert\lambda_0,+ \right\rangle\!} - {\left\vert\lambda_0,- \right\rangle\!})/\sqrt{2}$ with one photon in the cavity decays with $\kappa$. We infer from this that the fidelity of the proposed entangling scheme depends on the size of both decay rates. Taking this into account, we replace the spontaneous decay rate $\Gamma$ in Eq. (\[gamma\_c2\]) in the following with the average of $\kappa$ and $\Gamma$, $$\begin{aligned} \Gamma &\longrightarrow & {1 \over 2} (\kappa + \Gamma) \, .\end{aligned}$$ The analog of the laser detuning $\Delta$ in the toy-model state preparation scheme is the detuning $\delta_{\rm min}$ in Eq. (\[delta\_min\]) which is the minimum laser detuning experienced by the target state $|+,0 \rangle$ during the cooling process. Taking into account that $$\begin{aligned} \Delta &\longrightarrow & \delta_{\rm min} \, ,\end{aligned}$$ Eqs. (\[gamma\_c2\]) and (\[t\_stationary2\]) suggest that $\gamma_{\rm c}$ and ${\rm F}$ are to a very good approximation given by $$\begin{aligned} \label{gamma_c3} \gamma_{\rm c} &=& \frac{2 \Omega^2 (\kappa + \Gamma)}{12\Omega^2+(\kappa+\Gamma)^2} \, , \nonumber \\ {\rm F} &=& 1 - \frac{12 \Omega^2 + (\kappa + \Gamma)^2}{16 (\sqrt{2}-1)^2 g^2} \, .\end{aligned}$$ This result confirms Eq. (\[condi\]) which suggests that fidelities ${\rm F}$ close to one are only obtained when $g$ is much larger than all other system parameters. The remainder of this paper confirms these approximate solutions with the help of a numerical analysis of the proposed entangling scheme. The following analysis is based on the quantum jump approach [@Hegerfeldt] which allows us to simulate all the possible trajectories of the atom-cavity system in Fig. \[scheme\]. By averaging over many trajectories, we obtain an approximate solution of the master equation in Eq. (\[t\_master\]). To calculate the no-photon time evolution of the system we use the conditional Hamiltonian $H_{\rm cond}$ in Eq. (\[Hcond\]). In case of a photon emission we reset the atom-cavity system such that its state after a photon emission is on average given by $\mathcal{R}(\varrho) \Delta t$ with $\mathcal{R}(\varrho)$ as in Eq. (\[RR\]). For simplicity, we consider only relatively small Rabi frequencies. In this way we avoid the population of highly excited states. The population of such states is not expected to decrease the fidelity of the final state since they decay relatively rapidly. However, in this way we can restrict the size of the Hilbert space which has to be taken into account during simulations to states with at most three photons in the cavity. Let us first have a closer look at how changing the relative size of $\kappa$ with respect to $\Gamma$ affects the fidelity ${\rm F}$ of the state preparation. Fig. \[Kvar\] shows ${\rm F}$ for different spontaneous decay rates $\kappa$ and $\Gamma$. These are chosen such that the single-atom cooperativity parameter $C$ in Eq. (\[C\]) remains constant at $C = 25$. Each data point represents the average fidelity calculated from a time series like the one shown in Fig. \[C25\_compare\]. As a result we find that the proposed state preparation scheme works best when $\Gamma - \kappa = 0.15 \, g$. This implies that ideally one should have $$\begin{aligned} \label{O(t)4} \kappa & = &2 \Gamma \, ,\end{aligned}$$ when $C = 25$. We therefore assume that this applies in the remainder of this section. The main result of this subsection is an estimation of the stationary state fidelity of the maximally entangled atomic state $|+ \rangle$ which can be achieved with the proposed entangling scheme. To establish this numerically, we use a series of time evolutions such as those shown in Fig. \[C25\_compare\] and average over the fidelity once the system is approximately in its stationary state. Fig. \[fidelity\] shows the stationary state fidelity of the target state as a function of the cooperativity parameter $C$ for a constant laser Rabi frequency $\Omega $. The numerical results are compared with the analytical result for ${\rm F}$ in Eq. (\[gamma\_c3\]). Indeed we find very good agreement between analytical and numerical results. It is clear from Fig. \[fidelity\] that the achievable fidelity ${\rm F}$ increases rapidly with increasing cooperativity parameter $C$. However, fidelities above $90\%$ are possible, even for a cooperativity parameter $C$ as low as 20. Minimising the state preparation time {#main10} ------------------------------------- As in Section \[toy\_model\], we find that a relatively large cooling rate $\gamma_{\rm c}$ requires relatively large Rabi frequencies. At the same time, we only obtain a fidelity ${\rm F}$ close to unity for very small Rabi frequencies. In order to maximise the fidelity of the state preparation while maintaining a substantial cooling rate, we therefore proceed in the following as in Section \[choosing\] and assume a time dependent Rabi frequency $\Omega$. Similarly as in Eq. (\[toy\_Ot\]), we assume in the following that $$\begin{aligned} \label{O(t)2} \Omega(t) &=& {6 \Omega_0 \over \left( 1 + \gamma_{\rm c}(0) t \right)^{2}} \, ,\end{aligned}$$ where $\gamma_{\rm c}(0)$ denotes the cooling rate of the entangling scheme for the initial Rabi frequency $\Omega(0) =\Omega_0$. Fig. \[C25\_compare\] confirms that choosing a time-dependent Rabi frequency $\Omega$ indeed improves the speed of the entangling scheme without sacrificing much of its quality. Conclusions {#conc} =========== In this paper, we propose an entangling scheme for two atoms trapped inside an optical cavity. Each atom should contain a $\Lambda$-like level configuration with the ground states $|0 \rangle$ and $|1 \rangle$ forming one qubit and an excited state $|2 \rangle$ (cf. Fig. \[scheme\]). Three laser fields should be applied simultaneously. Two of them continuously drive the 0–2 transition which is in resonance with the cavity mode, while the third laser drives the 1–2 transition. Individual laser addressing of the atoms is not required. Most importantly, the laser detunings should be chosen as proposed in Eq. (\[main\]) in Section \[main2\]. As a result, the maximally entangled atomic ground state $|+,0 \rangle$ with no photons in the cavity becomes the stationary state of the atom-cavity system. To complete the state preparation, the laser fields should be turned off after a certain transition time. The presence of non-zero spontaneous decay channels, i.e. the leakage of photons through the cavity mirrors and direct spontaneous emission from the atoms, are essential for the scheme to work. The proposed state preparation scheme is a concrete realisation of a recent proposal to cool atoms into entangled state [@Vacanti:2009p5419]. Choosing the laser detunings as proposed in Eq. (\[main\]) guarantees that only the target state $|+,0 \rangle$ experiences off-resonant driving. All other states with no population in $|2 \rangle$ and in the cavity mode interact resonantly with one of the three applied laser fields. As in laser sideband cooling [@cool], this makes it much more likely for the atom-cavity system to decay into the target state than being driven out of it. As a result, most of the population of the system accumulates in $|+,0 \rangle$ with both atoms in a well-defined, maximally entangled state. Since the relevant detuning of this state is essentially given by the atom-cavity coupling constant $g$, the analogy to laser sideband cooling suggests that the scheme works best when all other system parameters are much smaller than $g$. Due to laser driving with three different laser fields, it is not possible to solve the time evolution of the proposed entangling scheme analytically. The reason is that there is no interaction picture in which the Hamiltonian of the system becomes time independent. To obtain at least approximate analytical solutions for the cooling rate $\gamma_{\rm c}$ and the stationary state fidelity ${\rm F}$ (cf. Eq. (\[gamma\_c3\])), Section \[toy\_model\] discusses a closely related state preparation scheme for a much simpler analytically tractable toy-model. A comparison with this toy-model provides much insight into the state preparation via cooling as well as analytical results. These are confirmed in Section \[main3\] by extensive numerical solutions of the time evolution of the atom-cavity system in Fig. \[scheme\]. Section \[main10\] finally suggests a method to speed up the state preparation without sacrificing its fidelity by using time-dependent laser fields with rapidly decreasing Rabi frequencies (cf. Eq. (\[O(t)2\])). Compared to other recent entangling schemes for atom-cavity systems [@Sorensen; @Wang2], the scheme proposed here predicts higher fidelities for the same experimental parameters. As illustrated in Fig. \[fidelity\] in Section \[main3\], it can achieve fidelities above $90\%$ even when the single-atom cooperativity parameter $C$ is as low as 20. With a cooperativity parameter $C = 25$ we can achieve a fidelity of $93\%$, while Ref. [@Sorensen] predicts fidelities above $92\%$ only for $C > 50$. Ref. [@Sorensen] uses a similar level scheme as our proposal but with the addition of a driven microwave transition between the triplet states. Ref. [@Wang2] requires the presence of a magnetic field gradient to produce the required level splittings to cool atoms into an entangled state. Compared to other quantum computing schemes [*using*]{} dissipation [@Pellizzari; @zoller2; @Beige; @Pachos; @Rempe; @zoller3; @sean; @Lim; @Metz], the state preparation scheme discussed here no longer relies on the detection of single photons or macroscopic fluorescence signals to herald the success of the state preparation. 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--- author: - | Susumu Osawa\ [High Energy Accelerator Research Organization (KEK), ]{}\ [Tsukuba, Ibaraki 305-0801, Japan.]{}\ [ E-mail: osawa@post.kek.jp]{}\ [and]{}\ Hiroshi Nagaoka\ [Graduate School of Information Systems,]{}\ [University of Electro-Communications,]{}\ [Chofu, Tokyo 182–8585, Japan.]{}\ [ E-mail: nagaoka@is.uec.ac.jp]{} title: Numerical Experiments on The Capacity of Quantum Channel with Entangled Input States --- [H]{} §[[S]{}]{} The capacity of quantum channel with product input states was formulated by the quantum coding theorem. However, whether entangled input states can enhance the quantum channel is still open. It turns out that this problem is reduced to a special case of the more general problem whether the capacity of product quantum channel exhibits additivity. In the present study, we apply one of the quantum Arimoto-Blahut type algorithms to the latter problem. The results suggest that the additivity of product quantum channel capacity always holds and that entangled input states cannot enhance the quantum channel capacity [^1]. quantum entanglement, quantum channel capacity, quantum coding theorem, quantum information theory Introduction {#intro} ============ The coding theorem for quantum channels[^2] was proved in recent publications [@Holevo2; @Schumacher1] combined with pioneering works such as [@Holevo0; @Holevo1]. This theorem gives the formula of the capacity of quantum channel with product (not entangled) input states. On the other hand, the use of entangled states in quantum communications provides us with another interesting aspect, which was already pointed out in [@Bennett]. Even though lots of theoretical attempts have been made so far in this direction (see section \[sec:related\_works\] for recent works on this subject), we are still far from deep understanding of the role of entanglement in quantum communications. In particular, whether the use of entangled input states can increase the capacity of quantum channel is a big open problem, which, as is seen from Theorem 1 below, can be reduced to a special case of another open problem whether the capacity of product quantum channel exhibits additivity. In the present study, we examine the latter problem numerically by means of a quantum version of Arimoto-Blahut algorithm [@nagaoka1], and observe that the additivity seems to always hold. Capacity of quantum channel and entangled states ================================================ Quantum channel with product input states ----------------------------------------- In this section, we give a brief review of the standard notion of quantum channel with product input states and its capacity. Let ${\cal H}$ be a Hilbert space which corresponds to a quantum system. A quantum state is represented by a density operator on ${\cal H}$, i.e. non-negative operator with unit trace. We denote by ${\cal S}({\cal H})$ the totality of density operators on ${\cal H}$. Letting ${\cal H}_1$ and ${\cal H}_2$ be input and output systems, a quantum channel is described by a completely positive [@Stinespring] trace preserving linear map $$\Gamma \; : \;{\cal T}({\cal H}_{1})\to{\cal T}({\cal H}_{2})$$ where ${\cal T} ({\cal H}_1)$ and ${\cal T} ({\cal H}_2)$ are the totalities of the trace class operators on ${\cal H}_1$ and ${\cal H}_2$. Note that the complete positivity and the trace preservation jointly characterize the physical realizability of quantum channels [@Kraus]. A quantum communication system in which a quantum channel $\Gamma$ is used $n$ times is described as follows. A message set ${\cal M}_n :=\{1,2, \cdots , M_{n}\}$ denotes the totality of the messages which are to be transmitted. Each message $k\in{\cal M}_n$ is encoded to a codeword which is a product state in the form $\rho^{(n)}(k):=\rho_{1}(k) \otimes \cdots \otimes \rho_{n}(k)$ on ${\cal H}_{1} ^{\otimes n},$ where ${\cal H}_{1} ^{\otimes n}$ denotes the tensor product Hilbert space ${\cal H}_{1}\otimes \cdots \otimes {\cal H}_{1}$. The sender transmits the codeword by multiple use of a quantum channel $\Gamma$. Then the received state is also a product state $\Gamma^{\otimes n}(\rho^{(n)}(k))= \Gamma (\rho_{1}(k)) \otimes \cdots \otimes \Gamma(\rho_{n}(k))$ on ${\cal H}_{2} ^{\otimes n}$. Here $\Gamma ^{\otimes n}$ denotes the n-fold tensor product channel $\Gamma \otimes \cdots \otimes \Gamma$ acting on ${\cal T}({\cal H}_{1} ^{\otimes n})$. The receiver estimates which codeword has been actually transmitted by performing an ${\cal M}_n$-valued measurement. Mathematically, this measurement is represented by a positive operator valued measure (POVM) $X^{(n)}=\{X_1^{(n)}, \cdots ,X_{M_n}^{(n)} \}$ on ${\cal H}_{2} ^{\otimes n}$, i.e. $X_k^{(n)} \ge 0 \, (k=1, \cdots ,M_{n})$ and $\sum_{k=1}^{M_n}X_k^{(n)}=I$, where $I$ denotes the identity operator on ${\cal H}_{2} ^{\otimes n}$. Given a [*coding system*]{} ${\Phi_n}$ consisting of codewords $\{\rho ^{(n)}(k)\}_{k=1}^{M_n}$ and a measurement $X^{(n)}$, the error probability averaged over all codewords is given by $$\label{Per} P_{er}(\Phi_n,\Gamma)=1-\frac{1}{M_n}\sum_{k=1}^{M_n}{\mbox{\rm Tr\,}}[\Gamma^{\otimes n}( \rho ^{(n)}(k)) X_k^{(n)}],$$ and the quantity $R_n(\Phi_n):= \log M_n /n$ is called the [*rate*]{} of the coding system $\Phi_{n}$. Now the capacity of the quantum channel $\Gamma$ with product input states is defined as $$C(\Gamma ) := \sup_{\{\Phi_n\}} \{ \lim_{n \to \infty } R_n(\Phi_n) \; ; \, \lim_{n \to \infty } P_{er}(\Phi_n, \Gamma)=0 \}. \label{capdef}$$ Next let us introduce the quantum mutual information. Let $$\begin{aligned} \Pi_{n} &:=& \{ (\lambda_{1},\cdots ,\lambda_{n}\, ;\, \sigma_{1},\cdots ,\sigma_{n})\;; \\ && \quad 0 \le \lambda_i \in {\bf R}, \; \sum_{i=1}^{n} \lambda_{i} =1,\; \sigma_i \in {\cal S}({\cal H}_{1})\}, \\ \Pi &:=& \bigcup_{n}^{\infty} \Pi_{n}.\end{aligned}$$ An element $\pi =(\lambda_{1},\cdots ,\lambda_{n}\, ;\, \sigma_{1},\cdots ,\sigma_{n})\in\Pi$ is considered as a discrete probability distribution on ${\cal S}({\cal H}_1)$ assigning probability $\lambda_i$ to the state $\sigma_i$ for each $i$. The quantum mutual information for $\pi$ and $\Gamma$ is then defined as $${{\rm I}}( \pi ; \Gamma):= \sum_{j}\lambda_{j} {{\rm D}}(\Gamma (\sigma_j) \|\Gamma(\rho)),$$ where $\rho :=\sum_{j}\lambda_{j}\sigma_j$ is a convex combination of the states in $\pi$ and ${{\rm D}}( \rho \|\sigma ):={\mbox{\rm Tr\,}}[\rho (\log \rho - \log \sigma )]$ is the quantum relative entropy. Now, the quantum channel coding theorem [@Holevo0; @Holevo1; @Holevo2; @Schumacher1] states that $$C(\Gamma) = \sup_{\pi \in \Pi} {{\rm I}}( \pi ; \Gamma). \label{qcapacity}$$ In addition, supremization is reduced to maximization on certain finite-dimensional compact set [@fujiwara]. That is, $$C(\Gamma)=\max_{\pi \in \Pi_{n}} {{\rm I}}(\pi ; \Gamma)=\max_{\pi \in \Pi_{n}^{e}}{{\rm I}}(\pi ; \Gamma)$$ where $n = \dim \Gamma ({\cal S}({\cal H}_1))+1$, $ \Pi_{n}^{e}:=\{ (\lambda_i ;\sigma_i)\in \Pi_n \;;\; \sigma_i \in \partial _{e}{\cal S}({\cal H}_1),\, i=1, \cdots ,n \}$. Here $\partial _{e}{\cal S}({\cal H}_1)$ is the totality of extreme points (pure states) of ${\cal S}({\cal H}_1)$. Quantum channel with entangled input states ------------------------------------------- Some states on a tensor product Hilbert space cannot be represented as product states or their convex combinations. These states are called [*entangled states*]{}. In the formulation given in the previous section we treated only product states as inputs to (and, consequently, outputs from) a quantum channel. Now let us consider communication systems in which we are allowed to use entangled input states. The capacity of the quantum channel $\Gamma$ with entangled input states $\tilde{ C}(\Gamma)$ is defined in the same way as (\[capdef\]) except that arbitrary states on ${\cal H}_1^{\otimes n}$, not necessarily product states, can be codewords. It is obvious by definition that $$\tilde{C}(\Gamma) \ge C(\Gamma). \label{entgenoent}$$ However, neither example of channel exhibiting the strict inequality nor proof that the equality $$\tilde{C}(\Gamma) = C(\Gamma) \label{enteqalnoent}$$ always holds have been reported yet[^3]. This problem can be reduced to the additivity problem for the capacity of product channels as described in the next section. Capacity of product quantum channel {#product_channel} ----------------------------------- Let $\Gamma^{(i)} \, : \,{\cal T}({\cal H}_{1}^{(i)})\to{\cal T}({\cal H}_{2}^{(i)})$ for $i=1, 2$ be quantum channels, and let $\Gamma^{(1)} \otimes \Gamma^{(2)} \, : \,{\cal T}({\cal H}_{1}^{(1)} \otimes {\cal H}_1^{(2)})\to{\cal T}({\cal H}_{2}^{(1)}\otimes {\cal H}_{2}^{(2)})$ be their product channel. The capacity $C(\Gamma^{(1)} \otimes \Gamma^{(2)})$ is defined as (\[capdef\]) by replacing $\Gamma$ with $ \Gamma ^{(1)}\otimes \Gamma^{(2)}$ in which each input state (code word) is written in the form $\rho^{(n)}(k)=\rho_{1}(k) \otimes \cdots \otimes \rho_{n}(k)$, where $\rho_{i}(k)\;(i=1,\cdots ,n)$ are arbitrary states on ${\cal H}_1^{(1)} \otimes {\cal H}_1^{(2)}$. Then it is easy to see that the superadditivity $$C(\Gamma^{(1)} \otimes \Gamma^{(2)})\ge C(\Gamma^{(1)}) + C(\Gamma^{(2)}) \label{superadditive}$$ holds. However, as in the case of (\[entgenoent\]), we have no example of the strict inequality nor proof that the additivity $$C(\Gamma^{(1)} \otimes \Gamma^{(2)})= C(\Gamma^{(1)}) + C(\Gamma^{(2)}) \label{additive}$$ always holds. Actually, this problem includes the previous one as is seen from the following theorem, whose proof is given in Appendix A. [**Theorem 1:**]{} $$\tilde{C}(\Gamma)=\lim_{N \to \infty} \frac{C(\Gamma ^{\otimes N})}{N}=\sup_{N}\frac{C(\Gamma ^{\otimes N})}{N}$$ holds. Here $C(\Gamma ^{\otimes N})$ is defined as (\[capdef\]) by replacing $\Gamma$ with $ \Gamma ^{\otimes N}$ in which each input state is written in the form $\rho^{(n)}(k)=\rho_{1}(k) \otimes \cdots \otimes \rho_{n}(k)$, where $\rho_{i}(k)\;(i=1,\cdots ,n)$ are arbitrary states on ${\cal H}_1^{\otimes N}$. If the additivity (\[additive\]) always holds, we have $C(\Gamma ^{\otimes N}) = N C(\Gamma)$, which, combined with Theorem 1, leads to the equality (\[enteqalnoent\]). In other words, the additivity implies that entanglement of input states cannot increase the capacity of quantum channel. The aim of the present paper and related works {#sec:related_works} ---------------------------------------------- In the last few years the additivity (\[additive\]) has gradually been receiving recognition as a difficult but important problem in the quantum information theory, and has been proved for several special cases. At the moment, proofs are known for the cases when $\Gamma^{(1)}$ is arbitrary and $\Gamma^{(2)}$ is the identity [@Schumacher2], when $\Gamma^{(1)}$ is arbitrary and $\Gamma^{(2)}$ is either a Holevo’s classical-quantum or quantum-classical channel [@King3] and when $\Gamma^{(1)}$ is arbitrary and $\Gamma^{(2)}$ is a certain class of unital binary channels [@King4]. See also [@Bruss; @Holevo3], whose results are now included in some of the above-mentioned ones. On the other hand, no example of channel violating the additivity have been found so far, and naturally the conjecture that the additivity always holds is arising [@Amosov1; @Holevo3; @King2; @King3; @nagaoka2; @osawa; @Schumacher2]. The aim of the present paper is to show that an efficient algorithm for computing quantum channel capacity, which was recently introduced by one of the authors, is applicable to verification of the conjecture and to report that all the randomly chosen channels have exhibited the additivity [^4]. Numerical experiments on the additivity ======================================= Quantum version of Arimoto-Blahut algorithm {#sec:Arimoto-Blahut} ------------------------------------------- The Arimoto-Blahut algorithm is known for computing the capacity of classical channel [@Arimoto; @Blahut]. Recently, one of the authors proposed some algorithms of this type for computing the capacity of quantum channel [@nagaoka1; @nagaoka2]. We use one of these, which is called the [*boundary algorithm*]{} since its recursion works on the extreme boundary set $\partial _{e}{\cal S}({\cal H}_{1})$. The outline of the theoretical basis is as follows. Let us introduce a two-variable extension of ${{\rm I}}(\pi;\Gamma)$: $$J(\pi, \pi^{\prime}) := -{{\rm D}}(\lambda \| \lambda ^{\prime})+\sum_{i=1}^{n} \lambda_{i}{\mbox{\rm Tr\,}}[ \Gamma (\sigma_{i})\Phi (\sigma_{i}^{\prime},\rho^{\prime})],$$ where $$\pi =(\lambda_i; \sigma_i),\;\pi^{\prime} =(\lambda_i^{\prime}; \sigma_{i}^{\prime}) \in \Pi_n,$$ $${{\rm D}}(\lambda \| \lambda ^{\prime}):= \sum_{i=1}^{n}\lambda_i \log \frac{\lambda_i}{\lambda_i^{\prime}}, \; \rho^{\prime}:=\sum_{i=1}^{n}\lambda_i^{\prime}\sigma_i^{\prime},$$ $$\Phi (\sigma_{i}^{\prime},\rho^{\prime}):= \log (\Gamma (\sigma_{i}^{\prime}))-\log (\Gamma (\rho^{\prime})).$$ Then it holds that $${{\rm I}}( \pi ; \Gamma) = J(\pi , \pi)= \max_{\pi^{\prime}}J(\pi , \pi^{\prime}). \label{j1}$$ In addition, we can compute $\displaystyle\hat{\pi}=(\hat{\lambda_i}$, $\hat{ \sigma_i}) := \argmax_{\pi} J(\pi,\pi^{\prime}) $ by the following equations. $$\hat{ \sigma_i}= \argmax_{\sigma \in {\cal S}({\cal H}_{1})} {\mbox{\rm Tr\,}}[ \Gamma (\sigma)\Phi (\sigma_{i}^{\prime},\rho^{\prime})],$$ $$\textstyle \hat{\lambda_i}=\lambda_i^{\prime} \exp ( {\mbox{\rm Tr\,}}[ \Gamma (\hat{\sigma_{i}})\Phi (\sigma_{i}^{\prime},\rho^{\prime}) ])/\hat{Z},$$ where $\hat{Z}$ is the normalizing constant: $$\hat{Z}:= \sum_{i=1}^{n}\lambda_i^{\prime} \exp ( {\mbox{\rm Tr\,}}[ \Gamma (\hat{\sigma_{i}})\Phi (\sigma_{i}^{\prime},\rho^{\prime}) ] ).$$ Note that, since $ {\mbox{\rm Tr\,}}[ \Gamma (\sigma)\Phi (\sigma_{i}^{\prime},\rho^{\prime})]$ is linear in $\sigma$, we can always choose $\hat{\sigma_i}$ to be an extreme point of ${\cal S}({\cal H}_1)$, i.e. a pure state $| \psi_i\rangle \langle \psi_i|$, where $|\psi_i\rangle$ is a normalized eigenvector of $\Gamma^{\ast}(\Phi (\sigma_{i}^{\prime},\rho^{\prime}))$ corresponding to the maximum eigenvalue. Here $\Gamma^{\ast}\;:\;{\cal B}({\cal H}_2) \to {\cal B}({\cal H}_1)$ denotes the dual map of $\Gamma$ defined by ${\mbox{\rm Tr\,}}[\Gamma(X)Y]={\mbox{\rm Tr\,}}[X\Gamma^{\ast}(Y)]$ for $\forall X \in {\cal T}({\cal H}_1)$ and $\forall Y \in {\cal B}({\cal H}_2)$, where ${\cal B}({\cal H}_i)\;(i=1,2)$ are the totalities of bounded operators on ${\cal H}_i$. Given a number $n\le\dim \Gamma ({\cal S}({\cal H}_1))+1$ and an arbitrary initial element $ \pi^{(1)} \in \Pi_{n}$, let the sequence $\{\pi^{(k)}\}_{k=1}^{\infty}$ be defined by $$\pi^{(k+1)}:= \argmax_{\pi}J(\pi, \pi^{(k)}).$$ Note that the sequence $\{{{\rm I}}(\pi ^{(k)}; \Gamma)\}_{k=1}^{\infty}$ is monotonous, since $${{\rm I}}(\pi ^{(k)}; \Gamma) \le J(\pi ^{(k+1)}, \pi ^{(k)}) \le {{\rm I}}(\pi ^{(k+1)}; \Gamma)$$ holds. Therefore we can efficiently compute the limit value $\lim_{k \to \infty}{{\rm I}}(\pi ^{(k)}; \Gamma)$. Unfortunately, it is not necessarily the quantum channel capacity since the quantum version of Arimoto-Blahut algorithm does not assure the global maximum. Thus we make several convergent sequences and adopt the maximum limit value as an estimate of the capacity. We judge that a sequence reaches the limit value when ten successive numerical values are the same to six places of decimals. Setting of the experiments -------------------------- = 3pt --------------------------------------------------------------------------------------------------- A b ------------ --------------------------------------------------------- ---------------------------- $\Gamma_1$ $ \pmatrix{ 0.5 & 0 & 0 \cr $\pmatrix{0.2 \cr 0 & 0.4 & 0 \cr 0 \cr 0 & 0 & 0.2 \cr 0 \cr }$ } $ $\Gamma_2$ $ \pmatrix{ 0.05 & -0.2 & 0.4 \cr $\pmatrix{0 \cr -0.2 & -0.05 & -0.2 \cr 0 \cr 0.2 & 0 & -0.5 \cr 0.1 \cr }$ } $ $\Gamma_3$ $ \pmatrix{ 1/\sqrt{2} &-1/\sqrt{6} &1/\sqrt{3} \cr $\pmatrix{0.2 \cr 1/\sqrt{2} & 1/\sqrt{6} & -1/\sqrt{3}\cr -0.2 \cr 0 & 2/\sqrt{6} &1/\sqrt{3} \cr 0.2 \cr }\cdot } $ \pmatrix{ -0.45 & 0 & 0 \cr 0 & 0.6 & 0 \cr 0 & 0 & -0.6 \cr }\cdot \pmatrix{ 0.8 & 0.6 & 0 \cr 0.6 & -0.8 & 0 \cr 0 & 0 & 1 \cr }$ $\Gamma_4$ $ \pmatrix{ 0.1 & -0.3 & 0 \cr $\pmatrix{0 \cr -0.3 & -0.1 & -0.2 \cr 0.2 \cr 0 & 0 & -0.05 \cr 0.55 \cr }$ } $ --------------------------------------------------------------------------------------------------- \[chara1\] = 3pt --------------------------------------------------------------------------------------------------------------------------- $V_1$ $V_2$ ------------ ------------------------------------------------------------ ------------------------------------------------- $\Gamma_5$ $ \pmatrix{ 0.2 & 0.3 & 0.4 \cr $\pmatrix{0.1-0.3i & 0 & 0 \cr 0 & 0.5i & 0 \cr 0 & -0.3i & 0.1-0.2i \cr 0.1i & 0.4i & 0.5i \cr 0.3-0.3i & 0.2+0.1i & 0\cr }$ } $ $\Gamma_6$ $ \pmatrix{ 0.19 & 0.7 & -0.1+0.3i \cr $\pmatrix{0.3 & -0.1 & 0.1\cr 0.4i & 0.06 & -0.1+0.05i \cr 0.2 & 0.3 & 0.02i \cr 0.2 & 0.39 & 0.4-0.4i \cr 0.1 & 0.2 & 0.1i \cr }$ } $ --------------------------------------------------------------------------------------------------------------------------- \[chara2\] We apply the quantum Arimoto-Blahut algorithm to various quantum channels $\Gamma^{(1)}$ and $\Gamma^{(2)}$ as well as their product channels $\Gamma^{(1)}\otimes\Gamma^{(2)}$ to examine whether the additivity (\[additive\]) always holds. Here we restrict ourselves to the case when ${\cal H}_1 ={\cal H}_2 ={\bf C}^2$ or ${\cal H}_1 ={\cal H}_2 ={\bf C}^3$ to reduce the computational complexity. Tables \[chara1\] and \[chara2\] show representative examples of channels used as $\Gamma^{(1)}$ and $\Gamma^{(2)}$ in the experiments. Here the first four channels $\Gamma_1, \cdots, \Gamma_4$ are quantum binary channels in the sense that ${\cal H}_1 ={\cal H}_2 ={\bf C}^2$, and are expressed in terms of the coefficients $(A,b)$ of the corresponding affine transformations on ${\bf R}^3$ (see Appendix C), while the other channels $\Gamma_5$ and $\Gamma_6$ are of ${\cal H}_1 ={\cal H}_2 ={\bf C}^3$ and are expressed by the generators $\{V_1, \cdots, V_m\}$ of their operator-sum representations (see Appendix B), restricting ourselves to the case when $m=3$ and $V_3= \sqrt{I-V_1^{\ast}V_1-V_2^{\ast}V_2}$. Note that these channels do not belong to the special classes mentioned in section \[sec:related\_works\] for which the additivity has been proved. These examples are chosen basically in random manners so that they are as generic as possible, not being intended to have any special properties or to represent any concrete physical processes, except that some extra points are taken into consideration in view of computational efficiency and generality, as explained below. In the case of quantum channels with ${\cal H}_1 ={\cal H}_2 ={\bf C}^2$, the necessary and sufficient condition for the coefficients $(A,b)$ to represent a pseudoclassical channel is known [@fujiwara]. Here a quantum channel is said to be [*pseudoclassical*]{} when its capacity is unchanged even if the measurements $X^{(n)}$ on ${\cal H}_2^{\otimes n}$ in equation (\[Per\]) are restricted to separable measurements which are constructed from the tensor products of measurements on ${\cal H}_2$. Considering the fundamental importance of the pseudoclassicality in classification of quantum channels, we choose $\Gamma_1$ and $\Gamma_3$ to be pseudoclassical, while $\Gamma_2$ and $\Gamma_4$ to be non-pseudoclassical, and examine various combinations of these channels. In the case of quantum channels with ${\cal H}_1 ={\cal H}_2 ={\bf C}^3$, on the other hand, we do not care about the pseudoclassicality, since no practical criterion for this property is known. The general operator-sum representation of channel in this case is given by generators $\{V_1, \cdots, V_m\}$ satisfying $\sum_{k=1}^m V_k^* V_k =I$ with $m\leq 9$, while our setting of $\Gamma_5$ and $\Gamma_6$ is much more restrictive. This restriction simply comes from a demand to reduce computational complexity. Nevertheless, the choice of channels may be considered sufficiently generic in the sense that it does not assume any special structure in view of the additivity. As we mentioned in section \[sec:Arimoto-Blahut\], the quantum version of Arimoto-Blahut algorithm does not assure the global maximum, and the limit value of ${{\rm I}}(\pi^{(k)} ; \Gamma^{(1)})$ or ${{\rm I}}(\pi^{(k)} ; \Gamma^{(1)} \otimes \Gamma^{(2)})$ may depend on the initial element $\pi^{(1)}\in \Pi_n$. Therefore, we repeatedly apply the algorithm to a channel with several different initial elements, and adopt the maximum of the convergent values as the estimate of the capacity. However, it empirically appears that the algorithm is not so sensitive to the initial condition. Indeed, we have observed that randomly chosen initial conditions mostly yield the same convergent value as far as the number $n$ of the states in $\pi^{(1)}$ is chosen to be sufficiently large (i.e. $n \approx \dim \Gamma ({\cal S}({\cal H}_1))+1$). The following is an example of $\pi^{(1)}$ for which the convergent value has attained the capacity when applied to each of the quantum binary channels $\Gamma_1, \cdots, \Gamma_4$: $$\pi^{(1)}=(\lambda_1, \cdots, \lambda_4 \, ;\, \sigma_1 ,\cdots , \sigma_4)$$ with $\lambda_1 = \cdots = \lambda_4 = 1/4$ and $$\sigma_1 = \frac{1}{2}\pmatrix{ 1 & 1 \cr 1 & 1 \cr }, \quad \sigma_2 = \frac{1}{2}\pmatrix{ 1 & i \cr -i & 1 \cr },$$ $$\sigma_3 = \pmatrix{ 1 & 0 \cr 0 & 0 \cr }, \quad \sigma_4 = \frac{1}{2 \sqrt{3}} \pmatrix{ \sqrt{3}-1 & -1-i \cr -1+i & \sqrt{3}+1 \cr} .$$ Results ------- = 3pt $\Gamma^{(1)}$ $\Gamma^{(2)}$ $C(\Gamma^{(1)})$ $C(\Gamma^{(2)})$ $C(\Gamma^{(1)})+C(\Gamma^{(2)})$ $C(\Gamma ^{(1)} \otimes \Gamma^ {(2)})$ ---------------- ---------------- ------------------- ------------------- ----------------------------------- ------------------------------------------ $\Gamma_1$ $\Gamma_1$ 0.138166 0.138166 0.276311 0.276311 $\Gamma_2$ $\Gamma_2$ 0.258679 0.258679 0.517358 0.517358 $\Gamma_1$ $\Gamma_3$ 0.138166 0.243068 0.381233 0.381233 $\Gamma_3$ $\Gamma_2$ 0.243068 0.258679 0.501747 0.501746 $\Gamma_2$ $\Gamma_4$ 0.258679 0.0898225 0.348501 0.348501 $\Gamma_5$ $\Gamma_5$ 0.677358 0.677358 1.354716 1.354716 $\Gamma_6$ $\Gamma_6$ 0.829580 0.829580 1.659160 1.659160 $\Gamma_5$ $\Gamma_6$ 0.677358 0.829580 1.506938 1.506938 \[results\] = 3pt --------------------------------------------------------------------------------------------------------------------------------- $ \pi^{\ast}=(\lambda_i^{\ast}\; ; \; \sigma_i^{\ast} )$ ------------ -------------------------------------------------------------------------------------------------------------------- $\Gamma_1$ ($0.521046, 0.478954 \;;\;$ $\pmatrix{0.500 & 0.500\cr 0.500 &0.500 \cr}$, $\pmatrix{0.500 & -0.500\cr -0.500 &0.500 \cr})$ $\Gamma_2$ ($0.512423, 0.487577 \;;\;$ $ \pmatrix{0.009 & 0.055-0.079i\cr 0.055+0.079i &0.991 \cr}$, $ \pmatrix{0.991, & -0.049+0.082i\cr -0.049-0.082i &0.009 \cr}$) $\Gamma_3$ ($0.271288, 0.728711 \;;\;$ $\pmatrix{0.00 & 0.00\cr 0.00 &1.00 \cr}$, $\pmatrix{1.00 & 0.00\cr 0.00 &0.00 \cr}$) $\Gamma_4$ ($0.47431, 0.52569 \;;\;$ $\pmatrix{0.772 & 0.398-0.133i\cr 0.398+0.133i &0.228 \cr}$, $\pmatrix{0.218 & -0.392+0.131i\cr -0.392-0.131 &0.782 \cr}$) $\Gamma_5$ ($0.31721, 0.383025, 0.299764 \;;\;$ $\pmatrix{0.690& -0.365-0.260i & 0.111-0.034i \cr -0.365+0.260i & 0.290 & -0.046+0.060i \cr 0.111+0.034i &-0.046-0.060i&0.019 \cr}$, $\pmatrix{0.023 & 0.079+0.024i&-0.121-0.035i \cr 0.079-0.024i &0.294&-0.448+0.004i \cr -0.121+0.035i& -0.448-0.004i&0.683 }$, $\pmatrix{0.157 &0.273-0.004i&0.233-0.055i \cr 0.273+0.004i & 0.476&0.408-0.090i \cr 0.233+0.055i &0.408+0.090i&0.367 }$) $\Gamma_6$ ($0.327542, 0.285361, 0.387097 \;;\;$ $\pmatrix{0.535 &-0.335+0.328i&0.020-0.168i \cr -0.335-0.328i& 0.411&-0.116+0.093i\cr 0.020+0.168i &-0.116-0.093i&0.054 }$, $\pmatrix{0.392& 0.375-0.250i& 0.152+0.113i \cr 0.375+0.250i&0.516 &0.073+0.204i \cr 0.152-0.113i &0.073-0.204i&0.091 \cr},$ $\pmatrix{0.039 &-0.009-0.038i&-0.189+0.002i\cr -0.009+0.038i&0.039&0.041-0.186i \cr -0.189-0.002i &0.041+0.186i&0.922 })$ --------------------------------------------------------------------------------------------------------------------------------- \[syujyotai\] We have observed that the additivity exactly holds for all the cases we examined, as is seen in Table \[results\] for the representative examples. Table \[piast\] shows the probability distributions which maximize the quantum mutual information of the quantum channels $ \Gamma_i \; (i=1, \cdots ,6)$. In the case of product channels, the probability distribution $\displaystyle \pi^{\ast} :=\argmax _{\pi}{{\rm I}}(\pi ; \Gamma ^{(1)} \otimes \Gamma ^{(2)})$ has turned out to be the product probability distribution of $\displaystyle \pi_{1}^{\ast} = (\lambda_{i1}^{\ast} ; \sigma_{i1}^{\ast} ) :=\argmax _{\pi} {{\rm I}}(\pi ; \Gamma ^{(1)} )$ and $\displaystyle \pi_{2}^{\ast}=(\lambda_{j2}^{\ast} ; \sigma_{j2}^{\ast}) :=\argmax _{\pi}{{\rm I}}(\pi ; \Gamma ^{(2)} )$, which assigns probability $\lambda_{i1}^{\ast}\lambda_{j2}^{\ast}$ to the state $\sigma_{i1}^{\ast} \otimes \sigma_{j2}^{\ast}$. Fig. 1 illustrates the change in the quantum mutual information ${{\rm I}}(\pi^{(k)}, \Gamma^{(1)} \otimes \Gamma^{(2)})$ for $\Gamma^{(1)}=\Gamma^{(2)} = \Gamma_2$ in the process of recursive computation $\pi^{(k)} \rightarrow \pi^{(k+1)}$ starting from some entangled states in ${\cal S}({\cal H}_1^{(1)} \otimes {\cal H}_1^{(2)})$. In addition, we measure the entanglement [^5] of the states in $\pi^{(k)}=(\lambda_{i}^{(k)};\sigma_{i}^{(k)})$ by $${{\rm Ent}}(\pi^{(k)}):= \sum_{i}\lambda_{i}^{(k)}{{\rm D}}(\sigma_{i}^{(k)} \|\sigma_{i1}^{(k)}\otimes \sigma_{i2}^{(k)}),$$ where $\sigma_{i1}^{(k)}$ and $ \sigma_{i2}^{(k)}$ are the marginal states of $\sigma_{i}^{(k)}$ defined by partial trace. Fig. 2 shows how the states get disentangled through the recursion. (1500,900)(0,0) =cmr10 at 10pt (181,123)[(0,0)\[r\][0.5]{}]{} (201.0,307.0) ------------------------------------------------------------------------ (181,307)[(0,0)\[r\][0.505]{}]{} (201.0,492.0) ------------------------------------------------------------------------ (181,492)[(0,0)\[r\][0.51]{}]{} (201.0,676.0) ------------------------------------------------------------------------ (181,676)[(0,0)\[r\][0.515]{}]{} (181,860)[(0,0)\[r\][0.52]{}]{} (328.0,123.0) ------------------------------------------------------------------------ (328,82)[(0,0)[5]{}]{} (487.0,123.0) ------------------------------------------------------------------------ (487,82)[(0,0)[10]{}]{} (645.0,123.0) ------------------------------------------------------------------------ (645,82)[(0,0)[15]{}]{} (804.0,123.0) ------------------------------------------------------------------------ (804,82)[(0,0)[20]{}]{} (963.0,123.0) ------------------------------------------------------------------------ (963,82)[(0,0)[25]{}]{} (1122.0,123.0) ------------------------------------------------------------------------ (1122,82)[(0,0)[30]{}]{} (1280.0,123.0) ------------------------------------------------------------------------ (1280,82)[(0,0)[35]{}]{} (1439,82)[(0,0)[40]{}]{} (201.0,123.0) ------------------------------------------------------------------------ (1439.0,123.0) ------------------------------------------------------------------------ (201.0,860.0) ------------------------------------------------------------------------ (90,940)[(0,0)[${{\rm I}}(\pi^{(k)}; \Gamma^{(1)}\otimes \Gamma^{(2)})$]{}]{} (820,21)[(0,0)[$k$]{}]{} (201.0,123.0) ------------------------------------------------------------------------ (233,253) (264,591) (296,666) (328,705) (360,728) (391,741) (423,750) (455,755) (487,758) (518,760) (550,761) (582,762) (614,762) (645,762) (677,762) (709,762) (741,763) (772,763) (804,763) (836,763) (868,763) (899,763) (931,763) (963,763) (995,763) (1026,763) (1058,763) (1090,763) (1122,763) (1153,763) (1185,763) (1217,763) (1249,763) (1280,763) (1312,763) (1344,763) (1376,763) (1407,763) (1500,900)(0,0) =cmr10 at 10pt (141,123)[(0,0)\[r\][-30]{}]{} (161.0,246.0) ------------------------------------------------------------------------ (141,246)[(0,0)\[r\][-25]{}]{} (161.0,369.0) ------------------------------------------------------------------------ (141,369)[(0,0)\[r\][-20]{}]{} (161.0,492.0) ------------------------------------------------------------------------ (141,492)[(0,0)\[r\][-15]{}]{} (161.0,614.0) ------------------------------------------------------------------------ (141,614)[(0,0)\[r\][-10]{}]{} (161.0,737.0) ------------------------------------------------------------------------ (141,737)[(0,0)\[r\][-5]{}]{} (141,860)[(0,0)\[r\][0]{}]{} (259.0,123.0) ------------------------------------------------------------------------ (259,82)[(0,0)[2]{}]{} (456.0,123.0) ------------------------------------------------------------------------ (456,82)[(0,0)[4]{}]{} (653.0,123.0) ------------------------------------------------------------------------ (653,82)[(0,0)[6]{}]{} (849.0,123.0) ------------------------------------------------------------------------ (849,82)[(0,0)[8]{}]{} (1046.0,123.0) ------------------------------------------------------------------------ (1046,82)[(0,0)[10]{}]{} (1242.0,123.0) ------------------------------------------------------------------------ (1242,82)[(0,0)[12]{}]{} (1439,82)[(0,0)[14]{}]{} (161.0,123.0) ------------------------------------------------------------------------ (1439.0,123.0) ------------------------------------------------------------------------ (161.0,860.0) ------------------------------------------------------------------------ (90,940)[(0,0)[$\log{{\rm Ent}}(\pi^{(k)})$]{}]{} (800,21)[(0,0)[$k$]{}]{} (161.0,123.0) ------------------------------------------------------------------------ (161,750) (259,567) (358,409) (456,323) (554,304) (653,291) (751,279) (849,267) (947,255) (1046,243) (1144,231) (1242,219) (1341,207) (1439,183) Conclusions =========== We have applied the quantum Arimoto-Blahut algorithm to various quantum channels $\Gamma^{(1)}$ and $\Gamma^{(2)}$ with ${\cal H}_1 ={\cal H}_2 ={\bf C}^2$ and ${\cal H}_1 ={\cal H}_2 ={\bf C}^3$ as well as their product channels $\Gamma^{(1)}\otimes\Gamma^{(2)}$, and have verified that the additivity (\[additive\]) holds for all the examples investigated. Note that the additivity (\[additive\]) has been proved only for some special classes of channels so far as explained in section \[sec:related\_works\] and that the examined channels do not belong to them. Needless to say, this is not a theoretical analysis but a numerical one, applied to only a limited number of channels on low-dimensional Hilbert spaces, using an algorithm which does not ensure the global maximum. Therefore we cannot rely upon the obtained results too much. Nevertheless, it seems very unlikely that all the randomly chosen examples happen to satisfy the additivity by a coincidence or that dimensions $2$ and $3$ are special in a property like the additivity which is not explicitly related to the dimension. We are thus naturally led to conclude that the results suggest that the additivity always holds. Acknowledgement {#acknowledgement .unnumbered} =============== We would like to thank Dr. A.S. Holevo of Steklov Mathematical Institute and Dr. A. Fujiwara of the Osaka University for giving crucial comments on this work. Proof of Theorem 1 ================== Since $C(\Gamma ^{\otimes (N+M)}) \ge C(\Gamma ^{\otimes N})+C(\Gamma ^{\otimes M})$ holds, $\displaystyle \lim_{N \to \infty}\frac{C(\Gamma ^{\otimes N})}{N}$ exists and is proved to be $\displaystyle \sup_{N}\frac{C(\Gamma ^{\otimes N})}{N}$. Let $\{\Phi_N\}_{N=1}^{\infty}$ be a sequence of coding systems satisfying $\displaystyle \lim_{N \to \infty} P_{er}( \Phi_N , \Gamma ) = 0$, where $\Phi_N$ consists of codewords $\{\sigma^{(N)}(i)\}_{i=1}^{M_N}$ which are arbitrary states on ${\cal H}_1^{\otimes N}$ and a measurement $X^{(N)} = \{X^{(N)}_i\}_{i=1}^{M_N}$. Let $Y_N$ be the classical random variable on the message set $\{1, \cdots, M_N\}$ which takes each value with equal probability $1/M_N$, and let $\hat {Y}_N$ be the classical random variable on the same set which represents the decoded message obtained by performing the measurement $X^{(N)}$ to the output state $\Gamma^{\otimes N} (\sigma^{(N)}(i))$, where the transmitted message $i$ is assumed to be $Y_N$. Then the Fano inequality (see e.g. [@Cover]) implies $$1+P_{er} ( \Phi_N , \Gamma )\log M_N \ge \log M_N-{{\rm I}}(Y_N ; \hat{Y}_N),$$ where ${{\rm I}}(Y_N ; \hat{Y}_N)$ is the classical mutual information between $Y_N$ and $\hat{Y}_N$. This leads to $$(1-P_{er}( \Phi_N , \Gamma ))\frac{1}{N} \log M_N \le \frac{1}{N}+ \frac{1}{N}{{\rm I}}(Y_N ; \hat{Y}_N). \label{fano2}$$ In addition, we have $$\begin{aligned} \lefteqn{{{\rm I}}(Y_N;\hat{Y}_N)} \nonumber\\ &&= \frac{1}{M_N} \sum_{i=1}^{M_N}{{\rm D}}_{X^{(N)}}(\Gamma ^{\otimes N}(\sigma^{(N)}(i))\| \Gamma ^{\otimes N}(\rho^{(N)} )) \nonumber\\ && \le \frac{1}{M_N} \sum_{i=1}^{M_N} {{\rm D}}(\Gamma ^{\otimes N}(\sigma^{(N)}(i))\|\Gamma ^{\otimes N}(\rho^{(N)} ))\nonumber \\ &&\le \max_{\pi} {{\rm I}}(\pi;\Gamma ^{\otimes N}) \nonumber \\ &&= C(\Gamma ^{\otimes N}), \label{cgamma}\end{aligned}$$ where ${{\rm D}}_{X^{(N)}}(\Gamma ^{\otimes N}(\sigma^{(N)}(i))\| \Gamma ^{\otimes N} ( \rho^{(N)} ))$ is the classical relative entropy between the conditional probability $P_{\hat{Y}_N|Y_N} (\cdot|i ):={\mbox{\rm Tr\,}}[\Gamma ^{\otimes N}(\sigma^{(N)}(i))X.^{(N)}]$ and the probability $P_{\hat{Y}}(\cdot):={\mbox{\rm Tr\,}}[\Gamma ^{\otimes N}(\rho^{(N)})X.^{(N)}]$ with $\rho^{(N)}:=\frac{1}{M_N}\sum_{i}\sigma^{(N)}(i)$, and the first inequality follows from the monotonicity of the relative entropy. Substituting (\[cgamma\]) into (\[fano2\]), letting N $\to \infty $ and taking supremum with respect to $\{ \Phi_N\}_{N=1}^{\infty}$, we come to the inequality $\displaystyle \tilde{C}(\Gamma) \le \lim_{N \to \infty} \frac{C(\Gamma ^{\otimes N})}{N}$. Conversely, since $C(\Gamma ^{\otimes N})$ is the supremum of the limit values of the rates of asymptotically error-free coding systems whose codewords are restricted to product states of the form $\rho_{1}\otimes \cdots \otimes \rho_{n} \in {\cal S}({\cal H}_1^{\otimes Nn})$, where $\rho_{i}\;(i=1,\cdots ,n)$ are arbitrary states on ${\cal H}_{1} ^{\otimes N}$, it cannot be greater than $N \tilde{C}(\Gamma )$ by the definition of $\tilde{C}(\Gamma )$. Hence we have $\displaystyle \tilde{C}(\Gamma) \ge \lim_{N \to \infty} \frac{C(\Gamma ^{\otimes N})}{N}$. Operator-sum representation =========================== An arbitrary completely positive trace preserving linear map $\Gamma \, : \,{\cal T}({\cal H}_{1})\to{\cal T}({\cal H}_{2})$ can be written in the form $$\Gamma (\rho)=\sum_{k=1}^m V_k \rho V_k^{\ast}$$ where ${\cal V}=\{V_k\}_{k=1}^m$ is a collection of bounded operators from ${\cal H}_1$ to ${\cal H}_2$ satisfying $\sum_{k=1}^m V_k^{\ast}V_k ={{\rm I}}$ [@Kraus] and $m$ can be taken at most $\dim {\cal H}_1\dim {\cal H}_2$ [@fujiwara2]. This is called the [*operator-sum representation*]{} or the [*Kraus decomposition*]{} of $\Gamma$ with the [*generator*]{} ${\cal V}$. Quantum binary channel ====================== A quantum channel whose input and output systems are both ${\bf C}^2 $ is called a [*quantum binary channel*]{}. Since every density operator on ${\bf C}^2 $ is represented in the form $$\rho_{\theta} = \frac{1}{2} \pmatrix{ 1+\theta_3 & \theta_1 - i\theta_2 \cr \theta_1+ i\theta_2 & 1-\theta_3 \cr }$$ with $\theta = (\theta_1, \theta_2, \theta_3)^t $ lying in the unit ball $${\cal V}=\{ \theta \in {\bf R}^3\; ;\; \parallel\theta \parallel^2 = \theta_1^2 + \theta_2^2 +\theta_3^2 \le 1\},$$ an arbitrary quantum binary channel is represented as $\Gamma(\rho_{\theta})=\rho_{A \theta +b}$ by a $3 \times 3 $ real matrix $A$ and a 3-dimensional real column vector $b$. We denote such a channel by $\Gamma = (A,b)$. For representing a completely positive map, they should satisfy the following condition [@fujiwara2] $$\pmatrix{ \frac{1}{2}+p & x & r & w \cr \overline{x} & \frac{1}{2}-p & y & -r \cr \overline{r} & \overline{y} & \frac{1}{2}+q & z \cr \overline{w} & -\overline{r} & \overline{z} & \frac{1}{2}-q \cr } \ge 0$$ when A and b are represented as $$A= \pmatrix{ y_R +w_R & y_I+ w_I & x_R-z_R \cr y_I - w_I & -y_R+w_R & -x_I+ z_I \cr 2r_R & 2r_I & p-q \cr },$$ $$b= \pmatrix{ x_R+z_R \cr -x_I-z_I \cr p+q \cr }.$$ (The subscripts $R$ and $I$ denote the real and imaginary parts, i.e. $x=x_R+i x_I$, etc.) [99]{} G.G. Amosov, A.S. Holevo, and R.F. Werner, “On some additivity problems in quantum information theory,” [*Probl. Inform. Transm.*]{}, vol.36, pp.25–34, 2000. (Originally appeared in LANL archive math-ph/0003002.) S. Arimoto, “An algorithm for calculating the capacity of an arbitrary discrete memoryless channel, ”[*IEEE Trans. Inform. Theory*]{}, vol.18, pp.14–20, 1972. C.H. Bennett, C.A. Fuchs, and J.A. Smolin, “Entanglement-enhanced classical communication on a noisy quantum channel,” [*Quantum Communication, Computing and Measurement,*]{} eds. O. Hirota, A.S. Holevo, and C.M. Caves, pp.79–88, Plenum, 1997. (Originally appeared in LANL archive quant-ph/9611006.) R. Blahut, “Computation of channel capacity and rate distortion functions,” [*IEEE Trans. Inform. Theory*]{}, vol.18, pp.460–473, 1972. D. Bruss, L. Faoro, C. Macchiavello, and G.M. Palma, “Quantum entanglement and classical communication through a depolarising channel," [*J. Mod. Opt.*]{}, vol.47, pp.325–332, 2000. (Originally appeared in LANL archive quant-ph/9903033.) T.M. Cover and J.A. Thomas, “[*Elements of Information Theory*]{},” Wiley, 1991. A. Fujiwara and H. Nagaoka, “Operational capacity and pseudoclassicality of a quantum channel,” [*IEEE Trans. Inform. Theory*]{}, vol.44, pp.1071–1086, 1998. A. Fujiwara and P. Algoet, “One-to-one parametrization of quantum channels,” [*Phys. Rev. A*]{}, vol.59, pp.3290–3294, 1999. A.S. Holevo, “Bounds for the quantity of information transmitted by quantum communication channel,” [*Probl. Inform. Transm.*]{}, vol.9, no.3, pp.177–183, 1973. A.S. Holevo, “On the capacity of quantum communication channel," [*Probl. Inform. Transm.*]{}, vol.15, no.4, pp.247–253, 1979. A.S. Holevo, “The capacity of the quantum channel with general signal states,” [*IEEE Trans. Inform. Theory*]{}, vol.44, pp.269–273, 1998. (Originally appeared in LANL archive quant-ph/9611023.) A.S. Holevo, “Coding theorems for quantum channels,” LANL archive quant-ph/9809023. C. King and M.B. Ruskai, “Minimal entropy of states emerging from noisy quantum channels,” [*IEEE Trans. Inform. Theory*]{}, vol.47, pp.192–209, 2001. (Originally appeared in LANL archive quant-ph/9911079.) C. King and M.B. Ruskai, “Capacity of quantum channels using product measurements,” [*J. Math. Phys.*]{}, vol.42, pp.87–98, 2001. (Originally appeared in LANL archive quant-ph/0004062. C. King, “Maximization of capacity and $l_p$ norms for some product channels,” LANL archive quant-ph/0103086. C. King, “Additivity for a class of unital qubit channels,” LANL archive quant-ph/0103056. K. Kraus, “[*States, Effects, and Operations: Fundamental Notions of quantum Theory*]{},” Springer, 1983. H. Nagaoka, “Algorithms of Arimoto-Blahut type for computing quantum channel capacity,” [*Proc. of 1998 IEEE International Symposium on Information Theory*]{}, p.354, 1998. H. Nagaoka and S. Osawa, “Theoretical basis and applications of the quantum Arimoto-Blahut algorithms,” [*Proc. of the second QIT*]{}, pp.107–112, 1999. S. Osawa and H. Nagaoka, “Numerical experiments on the quantum channel capacity when input states can be entangled,” [*Proc. of the 22nd Symposium on Information Theory and Its Applications*]{}, pp.387–390, 1999. B. Schumacher and M.D. Westmoreland, “Sending classical information via noisy quantum channels,” [*Phys. Rev. A*]{}, vol.56, pp.131–138, 1997. B. Schumacher and M.D. Westmoreland, “Relative entropy in quantum information theory,” LANL archive quant-ph/0004045. W.F. Stinespring, “Positive functions on $C^{\ast}$-algebras,” [*Proc. Amer. Math. Soc.*]{}, vol.6, pp.211–216, 1955. V. Vedral and M.B. Plenio, “Entanglement measures and purification procedures," [*Phys. Rev. A*]{}, vol.57, pp.1619–1633, 1998. [^1]: The content of this paper was partly presented at the second QIT [@nagaoka2] and the 22nd SITA [@osawa]. [^2]: In this paper we treat only quantum memoryless channels and simply call them quantum channels. [^3]: An earlier example of statement of this problem is found in the concluding remarks of [@fujiwara]. [^4]: In several references such as [@Amosov1; @King1; @King2] it is shortly mentioned, without any detail, that some numerical works relating the conjecture have been made. [^5]: According to the criterion of [@Vedral], the relative entropy ${{\rm D}}(\sigma_{i}^{(k)} \|\sigma_{i1}^{(k)}\otimes \sigma_{i2}^{(k)})$ is inappropriate as a measure of entanglement in $\sigma_{i}^{(k)}$ since it takes a positive value even when $\sigma_{i}^{(k)}$ is a classical mixture (convex combination) of several product states. Nevertheless, it does not mean that its use is inappropriate for our study.

--- abstract: 'Given a set of $n$ points in the Euclidean plane, such that just $k$ points are strictly inside the convex hull of the whole set, we want to find the shortest tour visiting every point. The fastest known algorithm for the version when $k$ is significantly smaller than $n$, i.e., when there are just few inner points, works in ${\mathcal{O}}(k^{11\sqrt{k}}k^{1.5}n^{3})$ time \[Knauer and Spillner, WG 2006\], but also requires space of order $k^{\Theta(\sqrt{k})}n^{2}$. The best linear space algorithm takes ${\mathcal{O}}(k!kn)$ time \[Deineko, Hoffmann, Okamoto, Woeginer, Oper. Res. Lett. 34(1), 106-110\]. We construct a linear space ${\mathcal{O}}(nk^2+k^{{\mathcal{O}}(\sqrt{k})})$ time algorithm. The new insight is extending the known divide-and-conquer method based on planar separators with a matching-based argument to shrink the instance in every recursive call. This argument also shows that the problem admits a quadratic bikernel.' author: - Paweł Gawrychowski and Damian Rusak bibliography: - 'biblio.bib' title: | Euclidean TSP with few inner points\ in linear space --- Introduction ============ The traveling salesman problem is one of the most natural optimization questions. Already proven to be NP-hard in the classical book by Garey and Johnson, it remains to be NP-hard even in the most natural Euclidean version [@npcomplete]. A simple ${\mathcal{O}}(2^{n}n^{2})$ dynamic programming can be used to solve the general version, where $n$ is the number of points, but one can do much better by exploiting the additional properties of the Euclidean variant. This was independently observed by Smith [@Smith], Kann [@kann1992approximability], and Hwang, Chang, and Lee [@separators], who all applied a similar reasoning, which we will call the strategy of searching over separators, to achieve an ${\mathcal{O}}(n^{{\mathcal{O}}(\sqrt{n})})$ running time. Even though the problem is NP-hard, we might try to construct an algorithm whose running time depends exponentially only on some parameter $k$ of the input instead of the whole $n$. We say that a problem is *fixed-parameter tractable*, if it is possible to achieve a running time of the form ${\mathcal{O}}(f(k)n^{c})$, where $k$ is the parameter. A closely connected notion is the one of admitting a *bikernel*, which means that we can reduce in polynomial time any its instance to an instance of a different problem, whose size is bounded by a function of $k$.[^1] In case of the Euclidean traveling salesman problem, a natural parameterization is to choose $k$ to be the number of inner points, where a point is inner if it lies strictly inside the convex hull of the input. A result of Deineko, Hoffman, Okamoto and Woeginger [@parametrized] is that in such setting ${\mathcal{O}}(2^{k}k^{2}n)$ time is possible (see their paper for an explanation why such parameterization is natural). This was subsequently improved to ${\mathcal{O}}(k^{11\sqrt{k}}k^{1.5}n^{3})$ by Knauer and Spillner [@fasterparametrized].[^2] The space consumption of their method (and the previous method) is superpolynomial, as they apply a dynamic programming on $k^{\Theta(\sqrt{k})}n^{2}$ states. #### Contribution. Our goal is to construct an efficient linear space algorithm. As the previously mentioned exact algorithm for the non-parametrized version [@separators] requires polynomial space, a natural approach is to apply the same strategy. In our case we want the total running time to depend mostly on $k$, though, so we devise a technique of reducing the size of current instance by applying a matching-based argument, which allows us to show that the problem admits a bikernel of quadratic size. By applying the same strategy of searching over separators on the bikernel, we achieve ${\mathcal{O}}(nk^2+k^{{\mathcal{O}}(k)})$ running time. To improve on that, we extend the strategy by using weighted planar separators, which give us a better handle on how the number of inner points decreases in the recursive calls. The final result is an ${\mathcal{O}}(nk^{2}+k^{{\mathcal{O}}(\sqrt{k})})$ time linear space algorithm. #### Overview. As in the previous papers, we start with the simple observation that the optimal traveling salesman tour visits the points on the convex hull in the cyclic order. In other words, we can treat subsequent points on the convex hull as the start and end points of subpaths of the whole tour that go only through the inner points. Obviously, no more than $k$ of such potential subpaths include any inner points. We call them *important* and show that we can quickly (in polynomial time) reduce the number of pairs of subsequent points from the convex hull that can create such important subpath to $k^{2}$, and for the remaining pairs we can fix the corresponding edge of the convex hull to be a part of the optimal tour, which shows that the problem admits a bikernel of quadratic size. The reduction shown in Section \[sec:reduction\] is based on a simple (weighted) matching-based argument and works in ${\mathcal{O}}(nk^2 + k^6)$ time and linear space. The second step is to generalize the *Generalized Euclidean Traveling Salesman Problem* [@separators] as to use the properties of the convex hull more effectively. In Section \[sec:searching\] we modify the strategy of searching over separators, so that its running time depends mostly on the number of inner points. More specifically, we use the weighted planar separator theorem of Miller [@miller] to prove that there exists a separator whose size is proportional to the square root of the number of inner points, irrespectively of the number of outer points. Now if the number of outer points is polynomial, which can be ensured by extending the aforementioned matching-based reduction, we can iterate over all such separators. Having the separator, we guess how the solution intersects with it, and recurses on the two smaller subproblems. The separator is chosen so that the number of inner points decreases by a constant factor in each subproblem, so then assuming the reduction is performed in every recursive call, we obtain ${\mathcal{O}}(k^{{\mathcal{O}}(\sqrt{k})})$ running time in linear space. #### Assumptions. We work in the Real RAM model, which ignores the issue of being able to compute distances only up to some accuracy. By $d(p,q)$ we denote the Euclidean distance between $p$ and $q$. In the rest of the paper, by planar graph we actually mean its fixed straight-line embedding, as the nodes will be always known points in the plane. Whenever we are talking about sets of points, we want distinct points, which can be ensured by perturbing them. The reduction {#sec:reduction} ============= We want to construct an efficient algorithm for a variant of the *Euclidean Traveling Salesman Problem*, called ${k\mbox{-}\textrm{ETSP}}$, in which we are given a set $V$ of $n$ points such that exactly $k$ of them lie strictly inside $\operatorname{CH}(V)$, which is the convex hull of the whole set. The algorithm first reduces the problem in ${\mathcal{O}}(nk^{2}+k^{6})$ time to an instance of *Generalized Euclidean Traveling Salesman Problem* of size at most ${\mathcal{O}}(k^2)$, and then solves the instance in ${\mathcal{O}}(k^{{\mathcal{O}}(\sqrt{k})})$ time. By the size we mean the value of $n+2m$, where $n$ and $m$ are defined as below. ***Generalized Euclidean Traveling Salesman Problem*** ${(\ifthenelse{\equal{}{}}{V}{},T)\mbox{-}\textrm{GETSP}}$\ Given a set $V = \left\{v_{1},\dots,v_{n}\right\}$ of inner points and a set $T = \left\{(t_{1}, t'_{1}),\dots,(t_{m},t'_{m})\right\}$ of terminal pairs of points, find a set of $m$ paths with the smallest total length such that: 1. the $i$-th path is built on $(t_{i},t'_{i})$, i.e., it starts from $t_{i}$ and returns to $t'_{i}$, 2. every $v_{i}$ is included in exactly one of these paths, assuming that in any optimal solution the paths have no self-intersections, and no path intersects other path, except possibly at the ends. It is well-known that in an optimal solution to an instance of ${k\mbox{-}\textrm{ETSP}}$ the outer points are visited in order in which they appear on $\operatorname{CH}(V)$ (otherwise the solution intersects itself and can be shortened). Hence we can reduce ${k\mbox{-}\textrm{ETSP}}$ to ${(\ifthenelse{\equal{V'}{}}{V}{V'},T)\mbox{-}\textrm{GETSP}}$ by setting $V'=V\setminus\operatorname{CH}(V)$ and $T=\left\{ (x_{1},x_{2}),\dots,(x_{n-k},x_{1})\right\}$, where $\operatorname{CH}(V)=\langle x_{1}, \dots, x_{n-k} \rangle$. As any optimal solution to ${k\mbox{-}\textrm{ETSP}}$ has no self-intersections, the paths in any optimal solution to the resulting instance have no self-intersections and do not intersect each other, except possibly at the ends. We will show that given any instance of ${(\ifthenelse{\equal{}{}}{V}{},T)\mbox{-}\textrm{GETSP}}$, we can quickly reduce the number of terminal pairs to ${\mathcal{O}}(n^{2})$. A path in a solution to such instance is *important* if it includes at least one point from $V$, and *redundant* otherwise. Obviously, a redundant path consists of just one edge $(t_{i},t'_{i})$ for some $i$, and the number of important paths in any solution is at most $n$. What is maybe less obvious, we can efficiently determine a set of at most $n^{2}$ terminal pairs such that the paths built on the other terminal pairs are all redundant in some optimal solution. To prove this, we will notice that every solution to an instance of ${(\ifthenelse{\equal{}{}}{V}{},T)\mbox{-}\textrm{GETSP}}$ corresponds to a matching, and apply a simple combinatorial lemma. The idea is that every important path $\left\langle u_{0},u_{1},\ldots,u_{\ell},u_{\ell+1}\right\rangle$ consists of the middle part $\left\langle u_{1},\ldots,u_{\ell} \right\rangle$ containing only inner points, and the endpoints $u_{0}=t$, $u_{\ell+1}=t'$ for some terminal pair $(t,t')$. We create a weighted complete bipartite graph, where every possible pair of inner points $(u,u')$ corresponds to a left vertex, and every terminal pair corresponds to a right vertex. The weight of an edge between $(u,u')$ with $(t,t')$ is $d(t,u)+d(u',t')-d(t,t')$, see Fig. \[fig:paths\](c). First we present a simple combinatorial lemma. Given a weighted complete bipartite graph $G=(U\cup V,U\times V,c)$, where $c(u,v)$ is the weight of an edge $(u,v)$, $\operatorname{cost}(X,Y)$ denotes the weight of a cheapest matching of $X\subseteq U$ to $Y\subseteq V$, if $|X|\leq |Y|$. If $M$ is a matching of $X$ to $Y$, then we denote by $M[X]$ and $M[Y]$ the subsets of $X$ and $Y$ matched by $M$. \[matching\] Let $G = (U \cup V, U\times V)$ be a weighted complete bipartite graph, where $|U| \leq |V|$. If $M_{\min}$ is a cheapest matching of $U$ to $V$, then for every $A \subseteq U$ we have $\operatorname{cost}(A,M_{\min}[V])=\operatorname{cost}(A,V)$. Assume the opposite, i.e., there is some $A \subseteq U$ such that for any cheapest matching $M$ of $A$ to $V$ we have $M[V]\not\subseteq M_{\min}[V]$. Fix such $A$ and take any cheapest matching $M$ of $A$ to $V$. If there are multiple such $M$, take the one with the largest $|M[V]\cap M_{\min}[V]|$. Then look at $M\oplus M_{\min}$, which is the set of edges belonging to exactly one of $M$ and $M_{\min}$. It consists of node-disjoint alternating cycles and alternating paths, and the alternating paths can be of either odd or even length. Because $M[V]\not\subseteq M_{\min}[V]$, there is a vertex $x\in V$ such that $x\in M[V]\setminus M_{\min}[V]$. It is clear that there is a (nontrivial) path $P$ starting at $x$, as $x\in M[V]$ but $x\notin M_{\min}[V]$. We want to argue that its length is even. Its first edge comes from $M$, so if the total length is odd, then the last edge comes from $M$ as well, so the path ends at a vertex $y\in A$. But all such $y$ are matched in $M_{\min}$, so $P$ cannot end there. Hence it ends at a vertex $y\in M_{\min}[V]\setminus M[V]$, and its length is even. Now we consider three cases depending on the sign of $\operatorname{cost}(P)$, which is the total weight of all edges in $P\cap M_{\min}$ minus the total weight of all edges in $P\cap M$: 1. if $\operatorname{cost}(P)>0$ then $M_{\min}\oplus P$ is cheaper than $M_{\min}$, so $M_{\min}$ was not a cheapest matching of $U$ to $V$, 2. if $\operatorname{cost}(P)<0$ then $M\oplus P$ is cheaper than $M$, so $M$ was not a cheapest matching of $A$ to $V$, 3. if $\operatorname{cost}(P)=0$, then $M'=M\oplus P$ is a cheapest matching of $A$ to $V$, and $|M'[V]\cap M_{\min}[V]|>|M[V]\cap M_{\min}[V]|$, so $M$ was not a cheapest matching of $A$ to $V$ with the largest $|M[V]\cap M_{\min}[V]|$ in case of a tie. Hence there is a cheapest matching $M$ of $A$ to $V$ such that $M[V]\subseteq M_{\min}[V]$. \[matching-alg\] With a read-only constant-time access to a weighted complete bipartite graph $G=(U\cup V,U\times V,c)$, where $|U|\leq |V|$, we can find a cheapest matching of $U$ to $V$ in ${\mathcal{O}}(|U|^{3}+|U||V|)$ time and ${\mathcal{O}}(|V|)$ space. Let $p=|U|$ and $q=|V|$. The naive approach would be to find a cheapest matching using $p$ iterations of Dijkstra’s algorithm implemented with a Fibonacci heap [@FredmanFibonacci], which uses ${\mathcal{O}}(p+q)$ space and ${\mathcal{O}}(p((p+q)\log(p+q)+pq))$ total time. We want to smaller time complexity when $p$ is significantly smaller than $q$. The first straightforward observation is that we can remove all but at most $p^{2}$ nodes from $V$, because we only need to keep, for every $u\in U $, its $p$ cheapest neighbors from $V$. By running the aforementioned algorithm on such truncated graph, the total time becomes ${\mathcal{O}}(p^{4})$, but the space complexity changes to ${\mathcal{O}}(p^{2})$, which might be larger than ${\mathcal{O}}(p+q)$, so we need an additional idea. We briefly recap how to use the Dijkstra’s algorithm to compute a cheapest matching. We start with an empty matching and iteratively extend the current matching $M$ using the cheapest augmenting path. An augmenting path connects an unmatched vertex $u\in U$ with an unmatched vertex $v\in V$ and alternates between the nodes of $U$ and $V$. To find the cheapest augmenting path, for every $(u,v)\not\in M$ we create an edge $u\rightarrow v$ with a cost of $c(u,v)-\pi_{u}+\pi_{v}$, and for every $(u,v)\in M$ we create an edge $v\rightarrow u$ with a cost of $-c(u,v)+\pi_{u}-\pi_{v}$. The [ *potentials*]{} $\pi_{u}$ and $\pi_{v}$ are initially all equal to zero, and then maintained so that the costs of all directed edges are nonnegative, so that we can apply the Dijkstra’s algorithm to find the cheapest augmenting path. Now consider a single iteration. Let $E_{V}$ be the set of edges incident to the already matched vertices of $U$. To correctly find the cheapest augmenting path, it is enough to consider, for every $u\in U$, only the cheapest incident edge which does not belong to $E_{V}$. This reduces the complexity of a single iteration to ${\mathcal{O}}(p\log p+|E_{V}|)={\mathcal{O}}(p^{2})$, assuming that we can quickly extract that cheapest edge for every $u\in U$. To accelerate the extraction, for every $u\in U$ we generate a list $E_{u}$ of $q/p$ cheapest edges incident to $u$ and not belonging to $E_{V}$. The lists are recalculated every $q/p$ iterations. Then, in every iteration, for every $u\in U$ we know that the cheapest incident edge which does not belong to $E_{V}$ belongs to the current $E_{u}$, hence it is enough to run the Dijkstra’s algorithm on $|E_{V}+\cup_{u\in U}E_{u}|={\mathcal{O}}(p^{2}+q)$ edges, which takes ${\mathcal{O}}(p\log p+p^{2}+q)={\mathcal{O}}(p^{2}+q)$ time and requires ${\mathcal{O}}(p+q)$ space. Because in every iteration exactly one node $v\in V$ becomes matched, recalculating the lists $E_{u}$ every $q/p$ iterations is enough. Now we analyze how much time do we need to generate every $E_{u}$. We claim that every $E_{u}$ can be found in ${\mathcal{O}}(q)$ time and ${\mathcal{O}}(q/p)$ space. We partition the sequence of all (at most) $q$ edges incident to $u$ and not belonging to $E_{V}$ into blocks of length $q/p$ and process the blocks one-by-one. After processing the first $k$ blocks, we know the $q/p$ smallest elements in the corresponding prefix of the sequence. To process the next block, we take these $q/p$ known smallest elements, add all elements in the current block, and use the linear time median selection algorithm [@median] to select the $q/p$ smallest elements in the resulting set of $2q/p$ numbers. After all blocks are processed, we have exactly the $q/p$ smallest elements of the whole original sequence. The total time complexity is ${\mathcal{O}}(q/p)$ per every block, so ${\mathcal{O}}(q)$ in total, and we clearly need only ${\mathcal{O}}(q/p)$ space. In every iteration we spend ${\mathcal{O}}(p^{2}+q)$ time to run the Dijkstra’s algorithm. Additionally, every $q/p$ iterations we need ${\mathcal{O}}(q)$ time to recompute the lists $E_{u}$. Hence the total time is ${\mathcal{O}}(p^{3}+pq)$. The space usage is clearly ${\mathcal{O}}(p+q)$. \[GETSP\] Given an instance of ${(\ifthenelse{\equal{}{}}{V}{},T)\mbox{-}\textrm{GETSP}}$ with $m \geq n^2$, we can find $T_{0}\subseteq T$ of size $m-n^2$, such that there is an optimal solution in which the paths built on pairs from $T_{0}$ are all redundant, in ${\mathcal{O}}(mn^2 + n^6)$ time and ${\mathcal{O}}(m)$ space. Let $W = V \times V$ and $G = (W \cup T, W\times T,c)$ be a weighted complete bipartite graph with $W$ and $T$ as the left and right vertices, respectively. The weight of an edge connecting $(v,v')$ and $(t,t')$ is defined as $c((v,v'),(t, t'))=d(t,v)+d(v',t')-d(t, t')$. Informally, given a path $\left\langle v,\ldots,v'\right\rangle$ consisting of inner points, $c((v,v'),(t,t'))$ is the cost of replacing a redundant path $\left\langle t,t'\right\rangle$ with an important path $\left\langle t,v,\ldots,v',t'\right\rangle$, assuming that we have already taken into the account the length of the inner part $\left\langle v,\ldots,v'\right\rangle$, see Fig. \[fig:paths\](c). Now any solution corresponds to a matching in $G$, because for every important path $\left\langle t_{i},v_{i},\ldots,v_{i}',t_{i}'\right\rangle$ we can match $(v_{i},v_{i}')$ to $(t_{i},t_{i}')$. More precisely, if we denote by $i_{1}<\ldots<i_{s}$ the indices of all these important paths and fix their inner parts $\langle v_{i_{j}},\ldots,v'_{i_{j}}\rangle$, then the solution corresponds to a matching of $W'=\{(v_{i_{1}},v'_{i_{1}}),\ldots,(v_{i_{s}},v'_{i'_{s}})\}$ to $T$, and the cost of the solution is equal to the total length of all inner parts plus $\sum_{i}d(t_{i},t'_{i})$ plus the cost of the matching. In the other direction, any matching of $W'$ to $T$ corresponds to a solution with the given set of inner parts (but possibly different indices of important paths). The cost of that solution is, again, equal to the total length of all inner parts plus $\sum_{i}d(t_{i},t'_{i})$ plus the cost of the matching, so any cheapest matching corresponds to an optimal solution. By Lemma \[matching\] we know, that $\operatorname{cost}(W',M_{\min}[T])=\operatorname{cost}(W',T)$, so there always is a cheapest matching of $W'$ to $T$ which uses only the nodes in $M_{\min}[T]$, where $M_{\min}$ is a cheapest matching of $W$ to $T$ in the whole $G$. Therefore, we can set $T_{0} =T\setminus M_{\min}[T]$, because there is at least one optimal solution, where the paths built on pairs from such $T_{0}$ are all redundant. Clearly, $|T_{0}| = m-n^2$. Finally, we can use Lemma \[matching-alg\] to find a cheapest matching, as we can implement read-only access to any $c((v,v'),(t,t'))$ without explicitly storing the graph, so the total space usage is ${\mathcal{O}}(n)$ and the total time complexity is ${\mathcal{O}}(mn^{2}+n^{6})$ as claimed. Searching over separators {#sec:searching} ========================= In this section we briefly recap the method of searching over separators used in [@separators] to solve the *Euclidean Traveling Salesman Problem*. At a high level, it is a divide-and-conquer algorithm. We know that an optimal solution has no self-intersections, hence we can treat it as a planar graph. Every planar graph has a small simple cycle separator, which is a simple cycle, which can be removed as to split the whole graph into smaller pieces. Such separator can be used to divide the original problem into smaller subproblems, which are then solved recursively. \[Miller\] In any $2$-connected planar graph with nonnegative weights summing up to $1$ assigned to nodes, there exists a simple cycle, called a simple cycle separator, on at most $2\sqrt{2\left \lfloor{d/2} \right \rfloor N}$ vertices, dividing the graph into the interior and the exterior part, such that the sum of the weights in each part is at most $\frac{2}{3}$, where $d$ is the maximum face size and $N$ is the number of nodes. Consider an instance of ${(\ifthenelse{\equal{}{}}{V}{},T)\mbox{-}\textrm{GETSP}}$ and its optimal solution, which by the assumption has no self-intersections, so there exists a planar graph such that any edge of the solution appears there. Then by Theorem \[Miller\] there is a simple cycle on at most $2\sqrt{2\lfloor d/2\rfloor (n+2m)}$ nodes such that any edge of the solution is either completely outside, completely inside, or lies on the cycle, and furthermore there are at most $\frac{2}{3}(n+2m)$ points in either the exterior and the interior part. We want the cycle to be small, so we need to bound $d$. For all inner faces, this can be ensured by simply triangulating them. To ensure that the outer face is small, we add three enclosing points, see Fig. \[fig:three\]. Now consider how the paths in the solution intersect with the simple cycle separator. Each path is either completely outside, completely inside, or intersects with one of the nodes of the separator. Any such intersecting path can be partitioned into shorter subpaths, such that the endpoints of the subpaths are either the endpoints of the original paths or the nodes of the separator, and every subpath is outside or inside, meaning that all of its inner nodes are completely outside or completely inside. This suggest that we can create two smaller instances of ${(\ifthenelse{\equal{}{}}{V}{},T)\mbox{-}\textrm{GETSP}}$ corresponding to the interior and the exterior part of the graph, such that the solutions of these two smaller subproblems can be merged to create the solution for the original problem, see Fig. \[fig:tourcycle\]. Of course we don’t know the solution, so we cannot really find a simple cycle separator in its corresponding triangulated planar graph. But the size of the separator is at most $c\sqrt{n+2m+3}$ for some constant $c$, so we can iterate over all possible simple cycles of such length, and for every such cycle check if it partitions the instance into two parts of sufficiently small sizes. The number of cycles is at most $c\sqrt{n+2m+3} \binom{n+2m+3}{c\sqrt{n+2m+3}}(c\sqrt{n+2m+3})!$, which is ${\mathcal{O}}((n+2m)^{{\mathcal{O}}(\sqrt{n+2m})})$. Similarly, because we don’t know the solution, we cannot check how it intersects with our simple cycle separator. But, again, we can iterate over all possibilities. To bound the number of possibilities, we must be a little bit more precise about what intersecting with the separator means. We create a number of new terminal pairs. Every node of the separator appears in one or two of these new terminal pairs. Additionally, the new terminal pairs might contain some of the original terminal points, under the restriction that for any original terminal pair, either none of its points are used in the new terminal pairs, or both are (and in the latter case, we remove the original terminal pair). Additional, there cannot exist a sequence of new terminal pairs creating a cycle, i.e., $(p_{1},p_{2}), \ldots, (p_{\ell-1},p_{\ell}), (p_{\ell},p_{1})$ with $\ell\geq 3$. Then, for every new terminal pair $(p,p')$, we decide if its path lies fully within the exterior or the interior part (if it directly connects two consecutive points on the cycle, we can consider it as belonging to either part). Notice that if $p$ is one of the original terminal points, and $p'$ is a new terminal point, then the corresponding path lies fully within the part where $p$ belongs to. One can see that such a choice allows us to partition the original problem into two smaller subproblems, so that their optimal solutions can be merged to recover the whole solution, and that the subproblems are smaller instances of ${(\ifthenelse{\equal{}{}}{V}{},T)\mbox{-}\textrm{GETSP}}$. Hence iterating over all choices and choosing an optimal solution in every subproblem allows us to find an optimal solution for the original instance. To bound the number of choices, the whole process can be seen as partitioning the nodes of the separator into ordered subsets, selecting two of the original terminal points for every of these subsets, and finally guessing, for every two nodes subsequent in one of the subsets, whether the path connecting them belongs to the exterior or the interior part. We must also check if it holds that for any original pair $(p,p')$ it holds that either none of its points was selected, or both of them were, but even without this last easy check the number of possibilities is bounded by $B_{c\sqrt{n+2m+3}}(c\sqrt{n+2m+3})!\binom{2m}{2c\sqrt{n+2m+3}} 2^{c\sqrt{n+2m+3}}$, where $B_{s}$ is the $s$-th Bell number. This is, again, ${\mathcal{O}}((n+2m)^{{\mathcal{O}}(\sqrt{n+2m})})$. The algorithm iterates over all separators and over all possibilities of how the solution intersects with each of them. For each choice, it recurses on the resulting two smaller subproblems, and combines their solutions. Even though we cannot guarantee that all optimal solutions in these subproblems have no self-intersections, any optimal solution to the original problem has such property, so for at least one choice the subproblems will have such property, which is enough for the correctness. Because the size of every subproblem is at most $b=\frac{2}{3}(n+2m)+c\sqrt{n+2m+3}$, the recurrence for the total running time is $T(n+2m) = {\mathcal{O}}((n+2m)^{{\mathcal{O}}(\sqrt{n+2m})}) \cdot 2T(b)$. For large enough $n+2m$, we have that $b \leq \frac{3}{4}(n+2m)$, and the recurrence solves to $T(n+2m)={\mathcal{O}}((n+2m)^{{\mathcal{O}}(\sqrt{n+2m})})$. The space complexity is linear, because we only need to generate the subproblems, which requires iterating over all subsets and all partitions into ordered subsets, and this can be done in linear space. ${(\ifthenelse{\equal{}{}}{V}{},T,H)\mbox{-}\textrm{GETSP}}$ ============================================================ To extend the divide-and-conquer algorithm described in the previous section, we need to work with a slightly extended version of ${(\ifthenelse{\equal{}{}}{V}{},T)\mbox{-}\textrm{GETSP}}$, which is more sensitive to the number of terminal pairs such that both points belong to the convex hull. We call the extended version ${(\ifthenelse{\equal{}{}}{V}{},T,H)\mbox{-}\textrm{GETSP}}$, and define its size to be $n+2m+2\ell$. Given an instance of ${k\mbox{-}\textrm{ETSP}}$, we can reduce the problem to solving an instance of ${(\ifthenelse{\equal{}{}}{V}{},T,H)\mbox{-}\textrm{GETSP}}$ with $|V|=k$, $|T|=0$, and $|H|=n-k$. ***GeneralizedEuclideanTravelingSalesmanProblem***${(\ifthenelse{\equal{}{}}{V}{},T,H)\mbox{-}\textrm{GETSP}}$\ Given a set $V = \left\{v_{1},\dots,v_{n}\right\}$ of inner points, a set $T = \left\{(t_{1}, t'_{1}),\dots,(t_{m},t'_{m})\right\}$ of terminal pairs of points, and a set $H = \left\{(h_{1}, h'_{1}),\dots,(h_{\ell},h'_{\ell})\right\}$ of hull pairs of points, where for any $i$ the point $h_{i}$ and $h'_{i}$ are neighbors on the convex hull of the set of all points, find a set of $m+\ell$ paths with the smallest total length such that: 1. the $i$-th path is built on $(t_{i},t'_{i})$, for $i=1,2,\ldots,m$, 2. the $m+i$-th path is built on $(h_{i},h'_{i})$, for $i=1,2,\ldots,\ell$, 3. every $v_{i}$ is included in exactly one of these paths, assuming that in any optimal solution the paths have no self-intersections, and no path intersects other path, except possibly at the ends. We will show that if $\ell={\text{poly}}(n)$, then ${(\ifthenelse{\equal{}{}}{V}{},T,H)\mbox{-}\textrm{GETSP}}$ can be solved in ${\mathcal{O}}((n+2m)^{{\mathcal{O}}(\sqrt{n+2m})})$ time and linear space using an extension of the method from the previous section. Combined with Theorem \[GETSP\], this gives an ${\mathcal{O}}(nk^2+k^{{\mathcal{O}}(\sqrt{k})})$ time and linear space solution for ${k\mbox{-}\textrm{ETSP}}$. First we extend Theorem \[GETSP\]. \[GETSPH\] Take an instance of ${(\ifthenelse{\equal{}{}}{V}{},T,H)\mbox{-}\textrm{GETSP}}$ with $n = |V|$, $m = |T|$, and $\ell = |H|$. If $m+\ell > n^2$ then in ${\mathcal{O}}((m+\ell)n^{2} + n^6)$ time and ${\mathcal{O}}(m+\ell)$ space we can find $T_{0}\subseteq T$ and $H_{0}\subseteq H$ such that $|T_{0}| + |H_{0}| = m+\ell-n^2$ and there is an optimal solution in which the paths built on pairs from $T_{0} \cup H_{0}$ are all redundant. Now applying the divide-and-conquer method described in the previous section directly together with the above lemma gives us a running time of ${\mathcal{O}}((n+2m+2\ell)^{{\mathcal{O}}(\sqrt{n+2m+2\ell)}})={\mathcal{O}}(n^{{\mathcal{O}}(n)})$, and we want to improve on that to get ${\mathcal{O}}(n^{{\mathcal{O}}(\sqrt{n})})$. Recall that the recursive method described in the previous section iterates over simple cycle separators. Because now the (unknown) graph is on $n+2m+2\ell$ vertices, the best bound on the length of the separator that we could directly get from Theorem \[Miller\] is $c\sqrt{n+2m+2\ell+3}$, which is too large. But say that we can show that there exists a simple cycle separator of length ${\mathcal{O}}(\sqrt{n+2m+2})$, such that the value of $n+2m$ decreases by a constant factor in both parts. Iterating over all such simple cycle separators takes ${\mathcal{O}}((n+2m+2\ell)^{{\mathcal{O}}(\sqrt{n+2m})})$ time, and iterating over all possibilities of how the separator intersects with the solution then takes $B_{{\mathcal{O}}(\sqrt{n+2m})}{\mathcal{O}}(\sqrt{n+2m}!)\binom{n+2m+2\ell}{{\mathcal{O}}(\sqrt{n+2m})} 2^{{\mathcal{O}}(\sqrt{n+2m})}$ time. All in all, the total number of possibilities becomes ${\mathcal{O}}((n+2m+2\ell)^{{\mathcal{O}}(\sqrt{n+2m})})$, which assuming that $\ell={\text{poly}}(n)$ is ${\mathcal{O}}((n+2m)^{{\mathcal{O}}(\sqrt{n+2m})})$. Applying this reasoning in a recursive manner as in the previous section results in Algorithm \[algorithm\]. Compared to the algorithm from the previous section, the changes are as follows: 1. we reduce the number of terminal and hull pairs using Lemma \[GETSPH\] in line \[line:reduce\], 2. we add just two enclosing points (instead of three) in line \[line:enclosing\], 3. when forming the subproblems in line \[line:form\], we might connect both some terminal points and some hull points with the nodes of the separator, and in the latter case, the new pair always becomes a terminal pair in the subproblem. If $\ell={\text{poly}}(n)$ in the original instance, then we can maintain such invariant in all recursive calls without increasing the running time, because the (polynomial) cost of the reduction in a subproblem can be charged to its parent. Therefore, because the value of $n+2m$ decreases by a constant factor in both subproblems, the total time is ${\mathcal{O}}((n+2m)^{{\mathcal{O}}(\sqrt{n+2m})})$ by the same recurrence as previously. \[1\][**if** \#1 **then**]{} \[1\][**if** \#1 **then**]{} \[1\][**for each** \#1 **do**]{} \[1\][**for each** \#1 **do**]{} **return** all edges directly connecting the pairs in $T \cup H$ \[alg:reducestart\] \[line:reduce\] Apply Lemma \[GETSPH\] to find $T_{0}$ and $H_{0}$. Directly connect the redundant pairs in $T_{0}\cup H_{0}$. Add two enclosing points $I_{1}$ and $I_{2}$. \[line:enclosing\] Check if $C$ forms a simple cycle. Check if there are at most $\frac{2}{3}(n+2m)$ inner and terminal points in either part. Form the exterior subproblem and the interior subproblem. \[line:form\] Recursively solve the exterior subproblem. Recursively solve the interior subproblem. Combine the solutions for the subproblems and update the best solution. **return** the best solution found in the whole process. Now the goal is to prove that it is enough to consider simple cycle separators of length $c\sqrt{n+2m+2}$. To this end, we will prove that there exists a planar graph with the following properties: (a) its set of nodes includes all inner and terminal points together with the two enclosing points, and possibly some hull points,\[prop:1\] (b) any edge from the solution is either an edge in the graph, or lies within one of its faces,\[prop:2\] (c) all of its faces are of size at most $4$ and its size is ${\mathcal{O}}(n+2m)$.\[prop:3\] If such a graph exists, then by Theorem \[Miller\] it has a simple cycle separator of size ${\mathcal{O}}(\sqrt{n+2m})$ due to (\[prop:3\]). Furthermore, by assigning equal weights summing up to one to all inner and terminal points, which by (\[prop:1\]) are nodes of the graph, and zero weights to the remaining nodes, we get a simple cycle separator which, by (\[prop:2\]), divides the original problem into two subproblems, such that the optimal solution to the subproblems can be combined to form an optimal solution to the original problem, and there are at most $\frac{2}{3}(n+2m)$ inner and terminal points in every subproblem, so Algorithm \[algorithm\] is correct. Before we show that such a graph exists, we provide the details of how to choose the enclosing points. \[enclosing\] For any set of points $A$ we can find two enclosing points $I_{1}$, $I_{2}$, lying outside $\operatorname{CH}(A)$, and two nodes of $\operatorname{CH}(A)$ called $v_{up}, v_{down}$, such that: 1. all points of $\operatorname{CH}(A)$ between $v_{up}$ and $v_{down}$ (clockwise) lie inside $\bigtriangleup I_{2}v_{up}v_{down}$, and all points of $\operatorname{CH}(A)$ between $v_{down}$ and $v_{up}$ lie inside $\bigtriangleup I_{1}v_{up}v_{down}$, 2. for any point $w$ of $\operatorname{CH}(A)$ between $v_{up}$ and $v_{down}$, $I_{1}w$ has no common point with $\operatorname{CH}(A)$ except for $w$, and for any $w$ between $v_{down}$ and $v_{up}$, $I_{2}w$ has no common point with $\operatorname{CH}(A)$ except for $w$. If $A$ contains less than two points, any two $I_{1}$ and $I_{2}$ are fine. For any larger $A$, we can find two distinct parallel lines $k_{1}$ and $k_{2}$, such that each of them has exactly one common point with $\operatorname{CH}(A)$. Call these common points $v_{up}$ and $v_{down}$, respectively. Let $y,z'$ be the neighbors of $v_{up}$ on $\operatorname{CH}(A)$ and $z,y'$ be neighbors of $v_{down}$, such that $y,y'$ are on the other side of $v_{up}v_{down}$ than $z,z'$. Consider the angle $\beta_{1}$ obtained by extending segments $v_{up}y$ and $v_{down}y'$, and the angle $\beta_{2}$ obtained by extending $v_{up}z'$ and $v_{down}z$. Finally, let $\alpha_{1}$ and $ \alpha_{2}$ be angles vertically opposite to $\beta_{1}$ and $\beta_{2}$, respectively, see Fig. \[fig:zones\]. Now we choose $I_{1}$ as any point strictly inside $\alpha_{1}$ and $I_{2}$ as any point strictly inside $\alpha_{2}$. We must show that for such a choice both properties hold. Because of the symmetry, it is enough to prove the first part of each of them. 1. One of the properties of a convex hull is that all of its points lie inside the intersection of the halfplanes, which are defined by its segments. The intersection of the halfplanes defined by the segments $v_{up}y$ and $v_{down}y'$ is precisely $\beta_{1}$. All points between $v_{up}$ and $v_{down}$ in the counterclockwise order lie on the same side of $v_{up}v_{down}$ as $I_{1}$. Therefore they all lie inside the part of $\beta_{1}$ bounded by the segment $v_{up}v_{down}$. Due to our choice of $I_{1}$ this part lies inside $\bigtriangleup I_{1}v_{up}v_{down}$, and so the first propery holds. 2. Assume the opposite, i.e., the segment $I_{1}w$ has a common point with $CH(A)$ other that $w$, call it $u$. Clearly, $\bigtriangleup wv_{up}v_{down}$ lies inside $\bigtriangleup uv_{up}v_{down}$ and due to the convexity of the hull all points strictly inside $\bigtriangleup uv_{up}v_{down}$ are strictly inside $CH(A)$ as well. But that is in contradiction with $w$ being a node of $CH(A)$, and so the second property holds. We say that $(U,H,S)$ is a *hull structure* if: 1. $U$ and $H$ are two sets of points in the plane with $H \subseteq \operatorname{CH}(U \cup H)$, 2. $S$ is a collection of segments connecting the points in $U\cup H$ such that no segment intersects other segment, except possibly at the ends, 3. any point from $U\cup H$ is an endpoint of at most two segments in $S$, 4. every segment in $S$ connecting two points from $H$ lies on $\operatorname{CH}(U \cup H)$. One can easily see that any optimal solution to an instance of ${(\ifthenelse{\equal{}{}}{V}{},T,H)\mbox{-}\textrm{GETSP}}$ corresponds to a hull structure $(U,H,S)$, where $U$ consists of all inner and terminal points, $H$ contains all hull points, and $S$ is a collection of segments constituting the paths. Furthermore, for any hull structure the following holds. \[small\_triang\] If $(U,H,S)$ is a hull structure, and $I_{1}, I_{2}$ are the points enclosing $U\cup H$, then there exists a planar graph, such that: 1. the nodes are all points from $U\cup\{I_{1},I_{2}\}$ and possibly some points from $H$, 2. any segment from $S$ is either an edge of the graph, or lies within its face, 3. all of its faces are of size at most $4$, 4. the size of the graph is ${\mathcal{O}}(|U|)$. The enclosing points $I_{1}, I_{2}$ are defined by applying Lemma \[enclosing\] on $U\cup H$, same for $v_{up}$ and $v_{down}$. The subsets of $U\cup H$ on the same side of the line going through $v_{up}v_{down}$ as $I_{1}$ and $I_{2}$ will be called $V_{1}$ and $V_{2}$, respectively. The subset of $S$ containing all segments with at least one endpoint in $U$ will be called $S'$. Because any point of $U$ is an endpoint of at most two segments, $|S'|={\mathcal{O}}(|U|)$. We define $U'$ to be the whole $U$ together with the points of $H$ which are an endpoint of some segment in $S'$, and create the first approximation of the desired planar graph using $U'$ as its set of nodes, and $S'$ as its set of edges. We triangulate this planar graph, so that its inner faces are of size $3$. Notice that all of its edges are inside or on $\operatorname{CH}(U')$, and all remaining segments in $S\setminus S'$ lie on $\operatorname{CH}(U\cup H)$, see Fig. \[fig:Usteps\]. So far, the size of the planar graph is ${\mathcal{O}}(|U|)$, its faces are small, and the nodes are all points from $U$ and possible some points from $H$, and any segment from $S'$ is an edge there. Therefore, we just need to make sure that any remaining segment is either an edge, or lies within a face. To deal with the remaining segments, we add $I_{1}$ and $I_{2}$ to the set of nodes. Fix any point $P$ strictly inside $\operatorname{CH}(U')$ and, for every node $P'$ of $\operatorname{CH}(U')$, draw a ray starting in $P$ and going through $P'$. All these rays partition the region outside $\operatorname{CH}(U')$ into convex subregions $R_{1},R_{2},\ldots,R_{|\operatorname{CH}(U')|}$. The intersection of $R_{i}$ with $\operatorname{CH}(U\cup H)$, called $T_{i}$, contains exactly two vertices of $\operatorname{CH}(U')$, call them $y_{i}$ and $y'_{i}$, see Fig \[fig:problem13\](a). We will process every such $T_{i}$ separately, extending the current graph by adding new triangles. Consider the sequence of points $v_{a_{i}},v_{a_{i}+1},\ldots,v_{b_{i}}$ of $\operatorname{CH}(U\cup H)$, which belong to $T_{i}$. If the sequence is empty, there is nothing to do. Otherwise we have two cases: 1. if $a_{i} = b_{i}$, create a new triangle $\bigtriangleup v_{b_{i}}y_{i}y'_{i}$ to $\mathcal{T}$, see Fig. \[fig:problem13\](b), 2. if $a_{i} \neq b_{i}$, create two new triangles $\bigtriangleup v_{a_{i}}y_{i}y'_{i}$, $\bigtriangleup v_{a_{i}}v_{b_{i}}y'_{i}$. Then add a triangle $\bigtriangleup I_{j}v_{a_{i}}v_{b_{i}}$ if both $v_{a_{i}}$ and $v_{b_{i}}$ belong to the same $V_{j}$, see Fig. \[fig:problem5\](a). Otherwise either $v_{up}$ or $v_{down}$ is in $v_{a_{i}}, \dots, v_{b_{i}}$, and we add three triangles as in Fig. \[fig:problem5\](b). Now any remaining segment which lies within a single $T_{i}$ is inside one of the new triangles. To deal with the other remaining segments, for each such segment $v_{j}v_{j+1}$ we simply add either $\bigtriangleup I_{1}v_{j}v_{j+1}$ or $\bigtriangleup I_{2}v_{j}v_{j+1}$, see Fig. \[fig:problem5\](c). This is correct because any two consecutive points on $\operatorname{CH}(H)$ always either both belong to $V_{1}$ or both belong to $V_{2}$. One ray can cross at most one edge, so the number of triangles created in this step is at most $|U'|$. By the construction, the insides of any new triangles are disjoint. Also, they all lie outside $\operatorname{CH}(U')$. Hence we can add the new triangles to the initial planar graph to form a larger planar graph. Because we created ${\mathcal{O}}(U')$ new triangles, the size of the new planar graph is still ${\mathcal{O}}(U)$, though. Now some of its faces might be large, though, so we include $I_{1}, I_{2}, v_{up}, v_{down}$ in its set of nodes, and all $I_{1}v_{up}$, $I_{1}v_{down}, I_{2}v_{up}, I_{2}v_{down}$ in its set of edges. Finally, we triangulate the large inner faces, if any. The size of the final graph is ${\mathcal{O}}(U)$ and we ensured that any segment from $S$ is either among its edges, or lies within one of its faces. Lemma \[small\_triang\] shows that it is indeed enough to iterate over separators of size $c\sqrt{n+2m+2}$, hence Algorithm \[algorithm\] is correct. The remaining part is to argue that it needs just linear space. By Lemma \[GETSPH\], the reduction in line \[line:reduce\] uses ${\mathcal{O}}(m+\ell)$ additional space which can be immediately reused. Iterating through all ordered subsets of size at most $c\sqrt{n+2m+2}$ can be easily done with ${\mathcal{O}}(\sqrt{n+2m})$ additional space. Bounding the space necessary to iterate over all possibilities of how the solution intersects with the separator is less obvious, but the same bound can be derived by looking at how the possibilities were counted. The ${\mathcal{O}}(\sqrt{n+2m})$ additional space must be stored for every recursive call. Additionally, for each call we must store its arguments $V, T$ and $H$, which takes ${\mathcal{O}}(n+2m+2\ell)$ additional space. As $n+2m$ decreases by a constant factor in every recursive call, the recursion depth is ${\mathcal{O}}(\log(n+2m))$, which in turn implies ${\mathcal{O}}(n+2m+2\ell\log(n+2m))$ overall space consumption. Even though we always reduce the instance so that $\ell \leq n^{2}$, this bound might be superlinear, and we need to add one more trick. Recall that the hull pairs in the subproblems are disjoint subsets of all hull pairs in the original problem. Hence, instead of copying the hull pairs to the subproblems, we can store them in one global array. All hull pairs in the current problem are stored in a contiguous fragment there. Before the recursive calls, we rearrange the fragment so that the hull pairs which should be processed in both subproblems are, again, stored in contiguous fragments of the global array. The rearranging can be done in linear space and constant additional space. There is one problem, though. When we return from the subproblems, the fragment containing the hull pairs might have been arbitrarily shuffled. This is a problem, because we are iterating over the ordered subsets of all points, which requires operating on their indices. Now the order of the hull pairs might change, so we cannot identify a hull point by storing the index of its pair. Nevertheless, we can maintain an invariant that all hull pairs in the current problem are lexicographically sorted. In the very beginning, we just sort the global array. Then, before we recurse on a subproblem, we make sure that its fragment is sorted. After we are done with both subproblems, we re-sort the fragment of the global array corresponding to the current problem. This doesn’t increase the total running time and decreases the overall space complexity to linear. Together with Theorem \[GETSP\], this gives the final result. ${k\mbox{-}\textrm{ETSP}}$ can be solved in ${\mathcal{O}}(nk^2+k^{\mathcal{O}}(\sqrt{k}))$ time and linear space. [^1]: This notion is usually used for decision problem, while we will be working with an optimization question, but this is just a technicality. [^2]: The authors state the result for minimum weight triangulation, but the companion technical report shows that the same strategy works for our problem.

--- abstract: 'We study the large time fluctuations of entropy production in Markov processes. In particular, we consider the effect of a coarse-graining procedure which decimates [*fast states*]{} with respect to a given time threshold. Our results provide strong evidence that entropy production is not directly affected by this decimation, provided that it does not entirely remove loops carrying a net probability current. After the study of some examples of random walks on simple graphs, we apply our analysis to a network model for the kinesin cycle, which is an important biomolecular motor. A tentative general theory of these facts, based on Schnakenberg’s network theory, is proposed.' author: - 'A. Puglisi' - 'S. Pigolotti' - 'L. Rondoni' - 'A. Vulpiani' bibliography: - 'fluct.bib' title: 'Entropy production and coarse-graining in Markov processes' --- [**To our friend and colleague Massimo Falcioni, on his 60th birthday**]{} Introduction ============ The coarse-graining procedure is a fundamental ingredient of the statistical description of physical systems [@M85; @K00; @CFLV08]. By coarse-graining we mean a procedure which reduces the number of observables to simplify the physical description. For instance, it is used to describe the behaviour of the physically relevant quantities, or [*slow variables*]{}, which depends on the coupling among all variables characterizing the system of interest, including the so-called [*fast variables*]{}. The archetype of such a procedure is the treatment of Brownian colloidal particles, immersed in a fluid, in terms of the Langevin equation. In this sense any model meant to represent a real phenomenon may be thought of as a coarse-grained, i.e. reduced, description. The purpose of a model is, indeed, to advance our understanding of the object under investigation, by highlighting its interesting features and discarding the irrelevant ones. In turn, the roles of relevant and irrelevant characteristics depend on the purpose of the analysis to be performed. Furthermore, it isn’t always obvious which quantities should be listed as interesting, and which ones should be neglected, especially if a new problem is to be tackled [@M85; @K00; @CFLV08; @AURIGA; @DGRBC]. Therefore, it is critical to understand how specific physical observables depend on the coarse-graining procedure. Examples of coarse-grained descriptions at different resolution levels include the steps meant to connect the microscopic descriptions of systems of physical interest to the macroscopic ones, for instance the passage from the deterministic $\Gamma$-space description (positions and momenta of the $N$ particles) to the stochastic $\mu$-space description (position and momentum of one particle), up to macroscopic descriptions such as hydrodynamics, Fourier law, Navier-Stokes equations, etc. Other methods use the coarse-graining procedure in order to reduce the number of variables, e.g. by a decimation method which suppresses the fast variables, or perform a spatial coarse-graining, as in the renormalization group approach. In these methods the coarse-graining is parametrized by some threshold, here denoted as coarse-graining level (CGL). This paper is devoted to the investigation of the impact of variations of CGL on the entropy production of non-equilibrium systems. In the last decades, the introduction of the so-called Fluctuation Relations (FR) for deterministic dynamics, by Evans, Cohen, Morriss, Gallavotti, Jarzynski and other authors brought about important developements in the physics of far from equilibrium systems [@ECM; @GC; @CJ]. In the specific context of Markov processes, here discussed, Lebowitz and Spohn [@LS99] showed that the “entropy production” per unit time, measured on a time-interval $t$, ${\mathcal{W}}_t$ say, is described by a large deviation theory whose Cramer function, $C$, enjoys the following symmetry property: $$C({\mathcal{W}}_t)-C(-{\mathcal{W}}_t)=-{\mathcal{W}}_t ~.$$ This relation is the stochastic counterpart of the deterministic steady state FR, and we call it Lebowitz-Spohn FR, or simply FR. We remark that the FR does not provide any specific information about the shape of $C$. Therefore, a coarse-graining procedure which preserves the Markovian character of the model should preserve the validity of the FR as well, although it may change the shape of the Cramer function. As a matter of fact, Rahav and Jarzynski [@RJ07] argue that the validity of the FR is little affected by the coarse-graining procedure, even in nontrivial cases, such as those in which the decimation (or blocking) of variables results in the loss of the Markovian property. In the present paper, at variance with Ref.[@RJ07], we do not address the question of validity of the FR (which is always satisfied by our models), but focus our attention on the effects of the decimation procedures on the behaviour of the Cramer function. Understanding how $C$ changes under variations of the CGL is relevant, e.g. to interpret experimental results, since they are always obtained at finite resolution (for instance in frequency, [@AURIGA]). Likewise, any model describing a real system is necessarily affected by some degree of approximation or of idealism. For instance, the entropy production defined by Lebowitz and Spohn appears to be a rather abstract quantity, depending on the direct and inverse trajectories in the state space, as well as on their probabilities in the stationary state, cf. Section II. Furthermore, such a quantity cannot be measured in a direct way. Therefore, it needs to be connected to directly measurable quantities, for its properties to be assessed. Naively, one may expect the entropy production computed through a model which encompasses lots of details of the system of interest to be higher than that computed through less detailed models. Consider for example the Markov chain depicted in Fig. \[fig:4states\]a, with transition probabilities $2 \to 1$ and $3 \to 4$ much larger than the remaining ones. One may impose that a net current flows from $3$ to $1$ and from $2$ to $4$, by choosing transition probabilities $P_{3 \to 1} \ll P_{1 \to 3}$ and $P_{2 \to 4} \ll P_{4 \to 2}$ and by tuning the other parameters so that all states have the same stationary probability. ![A simple example of decimation which results in vanishing entropy production. \[fig:4states\]](fourstates.eps){width="10cm"} In this system, detailed balance does not hold, hence the mean entropy production is positive. However, the mean entropy production vanishes if the fast states $2$ and $3$ are decimated, and the Markov chain is reduced to the one represented in Fig. \[fig:4states\]b, where $A$ corresponds to the old state $1$ and $B$ to the old state $4$. Indeed, detailed balance, which implies a vanishing entropy production, holds in any Markov chain with two states. In the following we consider Markovian systems described by a Master equation: $$\label{eq:master} \frac{d P_n}{dt}=\sum_{l \neq n} P_l W_{l \to n} - W_n^0 P_n$$ where $W_{l \to n}$ is the transition rate from state $l$ to state $n$, with $l,n\in[1,N]$, $N$ being the number of possible states of the process, $P_n(t)$ is the probability to stay in the state $n$ at time $t$, and $$W_n^0=\sum_{l \neq n} W_{n \to l}.$$ We adopt the coarse graining procedure introduced in Ref.[@PV08] and described in Appendix \[ap:dec\], which amounts to a decimation of all states whose characteristic times $\tau_n=1/W_n^0$ are smaller than a given $\Delta \tau > 0$. The resulting master equation for the surviving states may be written as $$\frac{d {\tilde{P}}_j}{dt}=\sum_{i \neq j} {\tilde{P}}_i {\tilde{W}}_{i \to j} - {\tilde{W}}_j^0 {\tilde{P}}_j$$ with transition rates ${\tilde{W}}_{i \to j}$ as prescribed by [@PV08], $i,j\in[1,{\tilde{N}}]$, ${\tilde{N}}<N$, and ${\tilde{W}}_j^0<1/\Delta \tau$ for all $j\in[1,{\tilde{N}}]$. Varying $\Delta \tau$ the number of the slow states and the shape of the Cramer function $C_{\Delta \tau}({\mathcal{W}}_t)$ may in principle change. In Section II, we present some numerical results for differently decimated Markov processes. In section III, we discuss the possibility of constructing a general theory, not yet available, of the decimation effects. In Section IV, we draw some conclusions and discuss open problems. Appendix A illustrates the decimation procedure; Appendix B reports analytical results about the effect of decimation on the current in a single loop; Appendix C recalls the graph analysis of currents in a Markov system, based on Schnakenberg’s theory; Appendix D describes in detail the kinesine model mentioned at the end of section II; Appendix E lists the main symbols used in the text. Some numerical results ====================== Entropy production on a trajectory ---------------------------------- While the concept of entropy production, or energy dissipation, dates a long time back [@DEGM], only recently have the [*fluctuations*]{} of entropy production attracted particular interest, thanks to important theoretical and numerical results, supported by some experimental evidence. For a trajectory of duration $t$ of a continuous time Markov process, in which $m$ transitions $\omega_0 \to \omega_1 ... \omega_{m-1} \to \omega_m$ are observed, $\omega_i$ being the $i$-th visited state, the following definition of entropy production has been given by Lebowitz and Spohn [@LS99]: $$\label{ls_def} {{\mathcal{W}}_t}=\frac{1}{t}\ln \frac{W_{\omega_0 \to \omega_1}W_{\omega_1 \to \omega_2}...W_{\omega_{m-1} \to \omega_m}}{W_{\omega_{1} \to \omega_0}W_{\omega_2 \to \omega_1}...W_{\omega_m \to \omega_{m-1}}}.$$ It can be shown that the times $t_i$ at which the transitions occur affect the numerical value of ${{\mathcal{W}}_t}$ only with corrections of order $\mathcal{O}(1/t)$, negligible in the $t \to \infty$ limit. In the present paper, we consider this entropy production, introduced by Lebowitz and Spohn. Clearly, this quantity does not need to represent any real thermodynamic observable, since it can be defined independently of the physical relevance of the Markov process at hand. Nevertheless, as commonly done in the literature, we will refer to it merely as to “entropy production”. Even if in this paper we consider the case of continuous time, in the discrete time case (Markov chains) one can use the same definition , by replacing $W_{i \to j}$ with the probability of a transition in a time step $\delta t$, $P_{i\to j}$. The relation between continuous and discrete time quantities is incorporated in the equalities $P_{i \to j}=W_{i \to j}\delta t$ for $i \neq j$, and $P_{i \to i}=1-W_i^0\delta t$. The connections of ${{\mathcal{W}}_t}$ with other definitions of entropy production rate are discussed in Appendix \[ap:graph\]. Here, it suffices to recall that $\langle {{\mathcal{W}}_t}\rangle=0$ in the steady state, if the invariant probability $P^{inv}_{\omega}$ of the process satisfies the detailed balance condition $$\label{detbal} \frac{W_{\omega \to \omega'}}{W_{\omega' \to \omega}}=\frac{P^{inv}_{\omega'}}{P^{inv}_{\omega}}.$$ The system is in equilibrium if eq.  holds. If detailed balance does not hold, one has $\langle {{\mathcal{W}}_t}\rangle >0$. Let $C$ be the Cramer function of the probability density function (pdf) $f$ of ${{\mathcal{W}}_t}$, in the steady state, i.e. let $C$ be defined by $$C({{\mathcal{W}}_t})=-\lim_{t \to \infty} \frac{1}{t} \log[f({{\mathcal{W}}_t})].$$ In numerical calculations, the Cramer function $C$ must be approximated by its finite time counterparts, $C({{\mathcal{W}}_t}) \approx -\log[f({{\mathcal{W}}_t})]/t$. Therefore, in our calculations, we have chosen times $t$ large enough that further growths of the averaging times practically do not affect our results. Lebowitz and Spohn have shown that the condition $$\label{fr} C({\mathcal{W}}_t)-C(-{\mathcal{W}}_t)=-{\mathcal{W}}_t,$$ is better and better approximated as the time $t$ grows [@LS99]. The $t \to \infty$ limit of relation  is known as a Steady State Fluctuation Relation (SSFR). It does not provide the shape of $C$, but only a symmetry property of $C$. Remarkably, $C$ is system-dependent [@BPRV08], while (\[fr\]) holds quite in general. In the following, we address the question of the dependence of $C$ on the CGL which, in the decimation procedure of [@PV08], is parametrized by the threshold time $\Delta t$. The protocol of Ref.[@PV08], eliminates all states $i$ with average exit time $\tau_i<\Delta t$, and requires the surviving states to have re-normalized transition rates ${\tilde{W}}_{\omega \to \omega'}$. Denote by $f_{\Delta t}$ and by $C_{\Delta t}$ the pdf of the entropy production and its Cramer function, for the decimated process with threshold time $\Delta t$. The present investigation suggests the conjecture that the entropy production does not depend sensibly on the precise properties of fast and slow variables of the Markov process: it only depends on the currents flowing in the system. Results on $1d$ and $2d$ regular lattices ----------------------------------------- Let us begin focusing on continuous time random walks on simple topologies, i.e. on regular lattices with periodic boundary conditions, with random transition rates restricted to nearest neighbours. To simplify the procedure, we require every state $n$, $n=1,...,N$, to have characteristic (exit) time $\tau_n=1/\sum_{l \neq n} W_{n \to l}$, belonging to a set $\{\tau^{(1)},...,\tau^{(M)}\}$ such that $\tau^{(\alpha-1)} \ll \tau^{(\alpha)} \ll \tau^{(\alpha+1)}$. This condition corresponds to a separation of time-scales which represents a mild requirement for the decimation protocol of [@PV08] to apply. Transition rates may then be chosen to have, or not to have, a preferential direction, in order to allow, or to prevent, a positive entropy production. For instance, entropy production can be positive in $1d$ lattices, only if some rates obey $W_{i \to i+1}/W_{i \to i-1} \neq 1$ (cf. Appendix \[ap:graph\], for a more precise condition, based on the notion of affinities). Simulations for regular lattices show a striking robustness of the entropy production Cramer function with respect to decimation. The numerically computed Cramer function $-\log f_{\Delta t}({{\mathcal{W}}_t})/t$ for $1d$ chains is plotted in Figure \[fig:1d\] for different CGL. The figure shows that decimating $90\%$ of the system, i.e. leaving only the slowest states, the fluctuations of entropy production remain substantially the same. The result does not seem to depend on the details of the transition rates, but only on the separation of time-scales. In the following sections, we show that the entropy production is not directly related to the properties of fast and slow states [*per se*]{} either, while it seems reasonable to conjecture that it is closely related to the [*currents*]{} flowing in the system. These are global quantities, rather than local ones, which depend on the topology and on the interplay among all transition rates, an idea that may be understood in simple terms, as follows. In the case of a random walk on a ring (a 1-dimensional lattice with periodic boundary conditions), one may write $$\frac{W_{\omega_0 \to \omega_1}W_{\omega_1 \to \omega_2}...W_{\omega_{n-1} \to \omega_n}}{W_{\omega_{1} \to \omega_0}W_{\omega_2 \to \omega_1}...W_{\omega_n \to \omega_{n-1}}}=\left(\frac{W_{forw}}{W_{back}}\right)^m\times \mathcal{R}$$ where $W_{forw}=W_{1 \to 2}W_{2 \to 3}...W_{N-1 \to N}W_{N \to 1}$, $W_{back}=W_{1 \to N}W_{N \to N-1}...W_{3 \to 2}W_{2 \to 1}$, $m$ is an integer and $\mathcal{R}\approx\mathcal{O}(1)$ is a correction term. In presence of a mean current $J>0$, one has $m=[G(t) t]$ where $G(t)$ is the current computed in the time window $(0,t)$ (i.e. $\lim_{t \to \infty} \langle G(t) \rangle=J$) and $[ a ]$ indicates the integer part of $a$. This leads to the relation $${{\mathcal{W}}_t}\approx G(t) \log\frac{W_{forw}}{W_{back}}+\mathcal{O}(1/t).$$ Since the decimation protocol eliminates fast states and modifies transition rates in order to leave basically unaltered the currents connecting the surviving states, the fluctuations of entropy production should not be sensibly affected by the decimation procedure. Observe, however, that this current conservation is only approximate, not exact. The modification of the current produced by the decimation can indeed be estimated analytically in simple cases, as in systems whose states form a single loop. In Appendix \[ap:singleloop\], we argue that the correction should be generally small and related to the ratio of the times spent in the fast states and in the slow ones. ![Numerically computed Cramer function for entropy production distribution in $1D$ continuous time random walks with p.b.c. In each plot a comparison among different coarse graining levels is shown. In both panels, each state of the non-decimated system may take one out of three possible average exit times: $1$ ($10\%$ of states), $0.1$ ($20\%$ of states) and $0.01$ ($70\%$ of states). In the left panel, the probability of jumping to the right is $0.4$ ($60\%$ of states) or $0.6$ ($40\%$ of states). In the right panel, the probability of jumping to the right is $0.4$. In the left frame we have $N=100$ and $t=2\cdot 10^3$. In the right frame we have $N=300$ and $t=10^3$. The numerical computation has been performed with the Gillespie algorithm [@gill], where the actual probability of a transition is the product of the transition rate by the characteristic time. \[fig:1d\]](pdf_1d_seitipi.eps "fig:"){width="7cm"} ![Numerically computed Cramer function for entropy production distribution in $1D$ continuous time random walks with p.b.c. In each plot a comparison among different coarse graining levels is shown. In both panels, each state of the non-decimated system may take one out of three possible average exit times: $1$ ($10\%$ of states), $0.1$ ($20\%$ of states) and $0.01$ ($70\%$ of states). In the left panel, the probability of jumping to the right is $0.4$ ($60\%$ of states) or $0.6$ ($40\%$ of states). In the right panel, the probability of jumping to the right is $0.4$. In the left frame we have $N=100$ and $t=2\cdot 10^3$. In the right frame we have $N=300$ and $t=10^3$. The numerical computation has been performed with the Gillespie algorithm [@gill], where the actual probability of a transition is the product of the transition rate by the characteristic time. \[fig:1d\]](pdf_1d_tretipi.eps "fig:"){width="7cm"} Similar results are reported in Figure \[fig:2d\] for 2D regular square lattices, where jumps occur among nearest neighbours: even with this topology, the fluctuations of entropy production appear not to be affected by the CGL, although the result is not as robust as in the $1D$ case. Indeed, a substantial change in entropy production can be observed if the system contains a very large number of fast states to be decimated. Nevertheless, it still is interesting to realize that the Cramer function has not changed substantially, even after $50\%$ of the original system has been decimated. Note that the square lattice topology is drastically altered by decimation: states which have not been decimated remain connected by chains of transitions, but the system is not planar nor a regular lattice anymore. Currents in this case may still be defined within a more general graph theory [@S76], such as that discussed in Appendix \[ap:graph\]. ![Approximated Cramer function for entropy production distribution in continuous time random walks on a $2D$ squared lattice (nearest neighbours) with p.b.c. In each plot a comparison among different coarse graining levels is shown. In both cases the probabilities of jumping to one of the four nearest neighbours are biased to give a net current in one direction. The left and the right panels differ by the values of the exit times. Left: the states have exit times $1$ ($50\%$ of states), $0.1$ ($20\%$ of states) and $0.01$ ($30\%$ of states). Right: the states have exit times $1$ ($70\%$ of states), $0.1$ ($20\%$ of states) and $0.01$ ($10\%$ of states). In both cases $N=100$ and $t=1000$ \[fig:2d\]](pdf_2d_meglio.eps "fig:"){width="7cm"} ![Approximated Cramer function for entropy production distribution in continuous time random walks on a $2D$ squared lattice (nearest neighbours) with p.b.c. In each plot a comparison among different coarse graining levels is shown. In both cases the probabilities of jumping to one of the four nearest neighbours are biased to give a net current in one direction. The left and the right panels differ by the values of the exit times. Left: the states have exit times $1$ ($50\%$ of states), $0.1$ ($20\%$ of states) and $0.01$ ($30\%$ of states). Right: the states have exit times $1$ ($70\%$ of states), $0.1$ ($20\%$ of states) and $0.01$ ($10\%$ of states). In both cases $N=100$ and $t=1000$ \[fig:2d\]](pdf_2d.eps "fig:"){width="7cm"} Results on graphs with fast and slow loops ------------------------------------------ Guided by the conjecture that the fundamental ingredient for entropy production is the current flowing in a circuit, we construct Markov processes composed of independent loops joined by a single interchange state. The general structure of this graph is illustrated in Fig. \[fig:topi\]. The main slow loop is decorated by fast loops (first level), which are on their turn decorated by faster loops (second level), etc. After decimation, one may encounter different situations: 1. the new and the old structures have the same topology, i.e. only pieces of loops have been suppressed but the number and position of loops is the same; 2. all loops of the faster (outer) level are suppressed; 3. all loops of the two fastest levels are suppressed; 4. and so on; Loops at the same level have similar properties and, in particular, are chosen to have, or not to have, a positive entropy production, i.e. to have or not to have a preferential direction in their transition rates. ![Sketch of a graph made of three levels of nested loops: in this example, the main level loop has no preferential direction, while the second and third levels are made of smaller loops with faster states and preferential directions. \[fig:topi\]](gra.eps){width="7cm"} The computed Cramer function for the entropy production of the case of Figure \[fig:topi\] is reported in Figures \[fig:gra1\] and \[fig:gra2\]. In Figure \[fig:gra1\] the states of the fast loop have slightly different characteristic times, allowing a progressive decimation of the fast loop. At a decimation threshold such that the fast loop is still alive, even if made of only three states (blue curve), we are in situation 1 and the Cramer function of the entropy production is very close to that of the non-decimated system (black curve). A further increase of the decimation threshold makes the fast loop disappear: it remains with only two states and the system falls in situation 2, where the Cramer function of the entropy production has a sudden macroscopic change (red curve). The inset of Figure \[fig:gra1\] shows $\langle {\mathcal{W}}_t \rangle$ as a function of the decimated percentage of the fast loop: neglecting a very weak growth, $\langle {\mathcal{W}}_t \rangle$ appears practically constant, until the fast loop is not reduced to a $2$-states branch. If the main loop is configured to have a non-zero current, this is what remains at that point, otherwise the fluctuations of ${{\mathcal{W}}_t}$ are reduced to a very narrow and symmetric peak around zero. Figure \[fig:gra2\] shows other cases with three levels of loops. The Cramer function of the entropy production always changes when a level of current-carrying loops is entirely removed. ![Approximate Cramer function for entropy production distribution in continuous time random walks on a graph similar to that of Figure \[fig:topi\], with two hierarchical levels. A comparison among different coarse graining levels is shown. The main loop is made of $100$ states with average exit time $1$ and preferential direction given by balanced (left) or unbalanced (right) transition rates. In the case of unbalanced rates, they are $0.6$ toward left and $0.4$ toward right. The second level loops have $30$ states and a bias in the transition probabilities ($0.8$ vs. $0.2$) chosen to give a preferential direction, while their characteristic times range from $0.1$ to $0.7$. In all simulations $t=1000$. \[fig:gra1\]](gra_progressive2.eps "fig:"){width="7cm"} ![Approximate Cramer function for entropy production distribution in continuous time random walks on a graph similar to that of Figure \[fig:topi\], with two hierarchical levels. A comparison among different coarse graining levels is shown. The main loop is made of $100$ states with average exit time $1$ and preferential direction given by balanced (left) or unbalanced (right) transition rates. In the case of unbalanced rates, they are $0.6$ toward left and $0.4$ toward right. The second level loops have $30$ states and a bias in the transition probabilities ($0.8$ vs. $0.2$) chosen to give a preferential direction, while their characteristic times range from $0.1$ to $0.7$. In all simulations $t=1000$. \[fig:gra1\]](gra_progressive.eps "fig:"){width="7cm"} ![Approximate Cramer function for entropy production distribution in continuous time random walks on the topologies of Figure \[fig:topi\], with three hierarchical levels of nested loops and $t=10^4$. In each plot a comparison among different coarse grained levels is shown. The main loop is made of $100$ states with average exit time $1$ and no preferred direction. The second level loops (each with $10$ states) have average exit time $0.1$ and a bias in the transition probability chosen to give a preferential direction. The third level loops (each with $5$ states) have average exit time $0.01$. The difference between the left and right frames is in the transition rates of this third level. Left: third level loops have a preferential direction. Right: third level loops without preferential direction. \[fig:gra2\]](pdf_gra2.eps "fig:"){width="7cm"} ![Approximate Cramer function for entropy production distribution in continuous time random walks on the topologies of Figure \[fig:topi\], with three hierarchical levels of nested loops and $t=10^4$. In each plot a comparison among different coarse grained levels is shown. The main loop is made of $100$ states with average exit time $1$ and no preferred direction. The second level loops (each with $10$ states) have average exit time $0.1$ and a bias in the transition probability chosen to give a preferential direction. The third level loops (each with $5$ states) have average exit time $0.01$. The difference between the left and right frames is in the transition rates of this third level. Left: third level loops have a preferential direction. Right: third level loops without preferential direction. \[fig:gra2\]](pdf_gra.eps "fig:"){width="7cm"} A model from molecular biology: coarse graining of the Kinesin’s network {#sec:kin} ------------------------------------------------------------------------ The examples in the previous sections suggest that the entropy production is weakly affected by the coarse graining procedure, apart from the cases in which loops contributing significantly to the entropy are destroyed and the entropy production undergoes an abrupt decrease. A natural question is whether this phenomenology is a peculiarity of the model introduced here, or similar behaviors pertain to other realistic models of non-equilibrium systems. Biochemical reactions are often characterized by non-equilibrium processes acting over different timescales and thus afford an ideal benchmark for the ideas proposed here. In this subsection, we study the effect of coarse graining on a recent network model of the kinesin motor cycle [@LL07]. Kinesins are a common category of motor proteins [@H01] that are used for transport on microtubules in eucaryotic cells. Like many other non-equilibrium reactions inside cells, kinesin is powered by ATP. A number of experiments during the last decades elucidated many structural and dynamical details of this systems. In particular, it is now understood that kinesins are made of two identical heads, that walk on microtubules with a “hand-over-hand” mechanism [@YTVS04], alternating their position at the front. The model proposed in [@LL07] describes both the ATP-driven chemical reactions and the mechanical step in which the two heads swap. The multiple cycle structure of the reaction is given by this chemomechanical nature and by considering the fact that ATP may be burned by both heads. The scheme of the reaction and the possible transitions are illustrated in Fig.(\[kinesin\_scheme\]). ![Scheme of the transition network of the kinesin model [@LL07]. a) Reaction network. The states, numbered from $1$ to $6$, are characterized by the two heads bound to ATP (A), ADP (D) or free. The molecules bound (or released) during the chemical transitions are shown as connected to the arrows. The dashed arrow represents the mechanical transition, in which kinesin makes its step on the microtubule. b) Kinesin network after coarse graining the fastest states, first state $2$, then state $5$. Dot-dashed arrows represent the new transitions appearing as a consequence of the coarse graining procedure.[]{data-label="kinesin_scheme"}](scheme_new.eps){width="6cm"} Each of the two heads may be in three different configurations: free, bound to ATP and bound to ADP, resulting in $3\times3=9$ possible states. However, the motor is believed to work “out of phase”, i.e. states in which the two heads are in the same configuration are unlikely to be observed. This reduces the model to the $6$ states represented in the scheme of Fig. (\[kinesin\_scheme\]). Of all the possible transition among the states, only those which are consistent with experimental observations are considered in the model and shown in the diagram. Clearly, the assumptions above (in particular that of considering only $6$ states) already imply some level of coarse graining with respect to the complete problem. However, the effect of these assumptions on the entropy production is hard to determine, since it would be difficult to construct a more detailed model, from the available experimental results. We then take the model of Fig. (\[kinesin\_scheme\]) as our starting point, and study the effect of decimating the states of the system. The transition rates of the model depend on the ADP, ATP and P concentration and on the load force $F$ of the molecular motor. Moreover, the parameters determining these rates have a slight dependence on the kind of the experiment one wants to reproduce, since different experiments work in different conditions and may use different kinds of proteins in the kinesin family. We determined the rates by choosing the parameters fitting the experiment of Ref.[@CC05] and assumed fixed concentrations of $[ADP]=[ATP]=[P]=1\mu M$ (micromoles) for simplicity. We then consider two different cases: one without work load and one with a work force equal to $F\approx 5 pN$ (piconewton). Details on the derivation of the rates and numerical values are given in Appendix (\[app\_kin\]). In both cases (with and without load), state $2$ is the fastest and state $5$ is the second fastest. We compare then the entropy production of the complete model, of the model in which state $2$ has been adiabatically eliminated and of the model in which both $2$ and $5$ have been eliminated. The pdf of finite-time-averaged entropy production $W_{t_{max}}/t_{max}$, obtained from $10000$ realizations of trajectories of length $t_{max}$ is shown in Figure \[fig:kin\], without load in the left panel, with load in the right panel. Decimation of state 2 leaves the pdf unchanged. In the case without load, further decimation of state 5 changes abruptly the pdf to a close-to-zero peaked pdf. ![Pdf’s of the entropy production per unit of time. Black curve is for the original model with 6 states. Red is after decimation of state 2. Green is after decimation of states 2 and 5. Left panel: the case without load. Right panel: the case with load.[]{data-label="fig:kin"}](pdfload.eps "fig:"){width="7cm"} ![Pdf’s of the entropy production per unit of time. Black curve is for the original model with 6 states. Red is after decimation of state 2. Green is after decimation of states 2 and 5. Left panel: the case without load. Right panel: the case with load.[]{data-label="fig:kin"}](pdfnoload.eps "fig:"){width="7cm"} This model reproduces the same scenario of the “fast loop-slow loop” model of the previous section. The first coarse graining strongly alters the structure of the network, but its effect on the entropy production and its fluctuations is barely noticeable. Conversely, decimating one more state drastically reduces the entropy production. Notice also that the most irreversible transition in the original model is the “mechanical” transition between state $2$ and $5$, since $W_{25}/W_{52}=3\ 10^3$ for the load-free case and $\approx 57$ for the loaded case. In the sense specified in the next section, the information about the irreversibility of this transition is lost, when the level of coarse graining is too large. Tentative theory ================ Consider a continuous time Markov process: each state $n$ can be seen as a vertex of a graph, and transitions $n\to n'$ with a positive rate correspond to edges (also called links) between $n$ and $n'$. As in [@LS99], we assume that the transition $n \to n'$ has a positive rate whenever the inverse transition $n' \to n$ does. As illustrated in Section 2.B, the entropy production of random walks on 1D rings is closely related to the current flowing in the ring. In this Section, we attempt to generalize this observation to generic graphs [@bc05]. The main tool for this purpose is a decomposition in fundamental cycles, which is illustrated in Appendix \[ap:graph\]. In the example of Figure \[fig:topi\], the fundamental cycles are nothing but the loops. Let us introduce a different functional $Q_t$: $$Q_t=\sum_\alpha A(\vec{C}_\alpha)G_\alpha(t)$$ which depends only on a few “structural properties” of the process: the [*fundamental cycles*]{} $\vec{C}_\alpha$ (i.e. a property of the graph), their [*affinities*]{} $A(\vec{C}_\alpha)$ and their fluctuating [*currents*]{} $G_\alpha$, averaged over a time interval $(0,t)$, which depend also on the transition rates. ![Equivalence of the Cramer function for ${\mathcal{W}}_t$ and $Q_t$ at $t=10^4$. Left: random walk on a 2d lattice with same parameters as in Figure \[fig:2d\] (right). Right: random walk on a nested loop graph with same parameters as that in Figure \[fig:gra2\] (right). \[fig:equiv\]](w_q_2d.eps "fig:"){width="7cm"} ![Equivalence of the Cramer function for ${\mathcal{W}}_t$ and $Q_t$ at $t=10^4$. Left: random walk on a 2d lattice with same parameters as in Figure \[fig:2d\] (right). Right: random walk on a nested loop graph with same parameters as that in Figure \[fig:gra2\] (right). \[fig:equiv\]](w_q_ring.eps "fig:"){width="7cm"} In various numerical simulations, we have verified that the fluctuations of $Q_t$ and those of ${{\mathcal{W}}_t}$ are practically indistinguishable, at large times: cf. Figure \[fig:equiv\] for two examples. It is also known that $\langle Q_t \rangle = \langle {{\mathcal{W}}_t}\rangle$ for large times [@AG07]. To obtain a complete theory, it remains to show that the “structural properties” are not affected by the decimation; this task can be subdivided in three steps: 1. examine the fate of fundamental cycles after decimation of one fast state; there are three possibilities: [**i.**]{} cycles may be destroyed, [**ii.**]{} transformed into different ones, [**iii.**]{} new cycles may be created; 2. derive the corresponding variation of affinities; 3. obtain the values of currents after decimation. Concerning task $2$, it is easy to realize that affinities do not change in transformed cycles, while new cycles have zero affinity, so they do not contribute to the entropy production. Disappearing cycles pose, instead, a difficult question: numerical simulations show that they are usually small and that the affinity lost with their removal is equally small. At the moment, however, we do not have an analytical estimate of this quantity. Task 3 is a hard problem too: the stationary value of currents must satisfy many coupled Kirchhoff equations and depends on the properties of the whole graph. Numerical simulations suggest that average currents are not drastically influenced by our decimation procedure. One rough explanation of this fact can be given for a system $\Sigma$ with small entropy production, obtained from a perturbation of an equilibrium system $\Sigma^0$. Indeed, one may assume a linear relation between the affinities $A$ and the average currents $J_\beta$ of $\Sigma$, of the form $A(\vec{C}_\alpha)=\sum_{\beta} L_{\alpha\beta} J_\beta$, with coefficients $L_{\alpha\beta}$ determined only by the properties of $\Sigma^0$. If the decimation procedure, which replaces $\Sigma$ with a new system $\Sigma'$, leaves substantially unaltered the invariant probability of the surviving states, it is reasonable to assume that the decimated system $\Sigma'$ is another small perturbation of $\Sigma^0$. Then, the linear relation between affinities and currents of $\Sigma'$ retains the same coefficients $L_{\alpha\beta}$, leading to the conclusion that the currents are conserved under decimation, if affinities are. We now discuss the consequences of decimation on cycles and their affinities. A maximal (also called “spanning”) tree $T$ is found on the original graph $G$. This tree includes all $N$ vertices (states) and only a part ($N-1$) of the original number $E$ of edges. All pairs of vertices are connected by a unique path on this tree. All edges left out from the tree (a number $\nu=E-(N-1)$) are called “chords”. A chord connecting vertices $i$ and $j$, attached to the unique path connecting $i$ and $j$ along the tree, is a closed loop. All loops generated in this way constitute the set of fundamental loops, which become “cycles” when orientation is taken into account. These fundamental cycles determine the statistics of $Q_t$ and therefore of ${{\mathcal{W}}_t}$, cf. Appendix \[ap:graph\] for the details. In the cases discussed below, the removal of a vertex using our decimation procedure preserves almost exactly the fundamental cycles and their affinities; the small variations observed are due to the possibile reduction of whole $3$-loops to $2$-loops (i.e. simple links) corresponding to a total loss of the affinity of the original $3$-loop. The impact of this unfortunate event is difficult to estimate, because it depends on the topology of the graph: the removal of a vertex may lead to a crunch of a number of loops smaller than or equal to the degree of the removed vertex. The amount of lost affinity for each reduced loop is expected to be small, since it is associated with a small loop, and correspondingly small should be the loss in current and in entropy production. It is remarkable that the exit times of states do not affect the affinities, although they can affect the currents. Nevertheless, decimation may affect the large loops as well; a progressive and repeated removal of vertices may eventually reduce a large loop to a $2$-loop. Unfortunately, controlling these events goes beyond our mathematical ability, therefore, the size of the error in the conservation of fundamental cycles under decimation remains an open question. In the figures, all black objects (vertices and links) are related to the original graph, red objects are the new ones formed after decimation. Solid links are part of the maximal tree, dashed links are chords. When the state labelled by $0$ is removed, it is linked to some other states collectively denoted as $j$: $j$-states (linked to $0$) are in number of $n$. These links are broken and all pairs of states $j$ and $j'$ (previously connected to $0$) are connected among each other with a new transition rate $W_{j->0}W_{0->j'}/W_0^0$ (or, if the link $j-j'$ already exists, its transition rate is updated adding that amount). We consider the simplest case where one chord at most is involved in the decimation procedure. With this assumption, three possibilities can be encountered: ![An example of state-removal where no chords belong to the subgraph involving the removed state. Black objects (vertices and links) are relative to the original graph, red objects are formed by the decimation protocol. Solid links are part of the maximal tree, dashed links are chords. \[fig:noprob\]](noprob.eps){width="6cm"} 1. [**new loops with no entropy production**]{}: The simplest case is realized when no chords connect any $j$ to $0$ and no chords connect any $j$ to any $j'$, cf. Fig. \[fig:noprob\] for one example. In this case, all original links $j\to0$ ($A$, $B$, $C$ and $D$ in Fig. \[fig:noprob\]) are on the spanning tree and no links join any $j$ to any $j'$. After the links and the central state have been removed, the red links are created (a, b, c, d, e, f in the example): they are in a number $n(n-1)/2$. A number $(n^2-3n+2)/2>0$ (for any $n \ge 2$) are [*new chords*]{} (a, d and e in the example), while the remaining $n-1$ are links of the new spanning tree (b, c and f). Therefore new loops have been created (in the example they are $3-4-2-3$ with chord a, $3-1-2-3$ with chord e and $1-2-4-1$ with chord d). It is immediate to verify that the affinity of the new loops vanishes: for instance the loop $3-1-2-3$ has forward transition rate given by $3\to 0, 0\to 1, 1\to 0, 0\to 2, 2\to 0, 0\to 3$ and backward transition rate given by $3 \to 0, 0\to 2, 2\to 0, 0\to 1, 1\to 0, 0\to 3$ and they exactly cancel out (the exit rates are omitted, but they cancel out trivially): these new loops do not contribute to the entropy production. ![An example of state-removal where a chord is removed by decimation. Black objects (vertices and links) concern the original graph, before decimation, red objects are the new ones formed after decimation. Solid links are part of the maximal tree, dashed links are chords. \[fig:cordaint\]](cordaint.eps){width="6cm"} 2. [**loop-shortening**]{}: Another possibility (see Fig. \[fig:cordaint\]) is that some link $0 \to j^*$ is a chord in the original graph, which means that it is not in the spanning tree: e.g. link A in the example, with $j^*=4$. Then, state $j^*$ is connected to 0 through some other unique path on the tree, possibly passing through a state $j'$ (the unique path on the tree is also represented in the figure, terminating with the link $3-0$), forming a loop $0-j^*-tree-j'-0$. In this case, the decimation of state $0$ creates the link $j^* \to j'$ (link “a” in the example) as a chord of the loop $j^*-tree-j'-j^*$, which is two steps shorter than the orginal loop. It is immediate to see that the affinity of the new loop is the same as that of the old one. In this case, all new links starting from $j^*$, or from $j'$, and ending in another $j$, must be chords, since $j^*$ and $j'$ are joined by a unique path on the tree which has not been touched by decimation: the number of new chords is larger than in the previous case, but all their loops have zero affinities. There is also the possibility that the loop passing through chord $A$ is simply given by $j^*-0-j-j^*$, i.e. that it is a $3$-loop, originally belonging to the graph: this case can be put in the last category, simply exchanging the roles of the chords A and a: we call loop-crunching this case. ![An example of state-removal where a chord connects two states which are directly linked to the removed state. Black objects (vertices and links) pertain to the original graph, before decimation, red objects are formed by decimation. Solid links are part of the maximal tree, dashed links are chords. \[fig:cordaext\]](cordaext.eps){width="6cm"} 3. [**loop-crunching**]{}: The last possibility is that some link $j \to j'$ already existed in the original graph, which means that it is a chord of the loop $A-j-j'-A$, cf. Fig. \[fig:cordaext\], where chord a connects states 4 and 3. In this case the removal of state $A$ leads us to crunch the $3$-loop, making it a simple link $j-j'$ with a new transition rate. The original loop and its contribution to entropy production are then lost. We stress that a mathematical proof of the above considerations is still lacking, although our arguments are strongly supported by numerical results. Concluding remarks and open problems ==================================== In this paper we support, both numerically and theoretically, the idea that fluctuations of the entropy production are essentially insensitive to a coarse-graining based on decimation of fast states, provided that decimation does not remove fundamental loops carrying net currents. The threshold of coarse-graining level which trigger the removal of such loops is not fully understood, but our investigation suggests that entropy production fluctuations are generally quite robust with respect to decimation. Moreover, this robustness does not appear directly related to the characteristic times of the removed states. Robustness or fragility of loops appears mostly related to the global structure of the network at hand. We applied this analysis to the network model of the biomolecule known as kinesin, discovering that no entropy production is lost if a coarse-graining from six to five states is performed. This observation is potentially interesting in biophysics, since entropy production is a fundamental property of irreversible chemical reactions, such as those fueling the kinesin motor protein. On the contrary, the decimation of the model from six to four states is catastrophic, making the model unsuitable to produce work. More detailed studies are necessary to quantify the entropy production variations induced by coarse graining: the main missing ingredient is the evaluation of the effect of decimation on the currents of the surviving loops. This will lead to a better understanding of the role and meaning of ${{\mathcal{W}}_t}$ as a definition of entropy production. In particular, understanding the relation between ${{\mathcal{W}}_t}$ and macroscopic observable properties of the system may help in modelling non-equilibrium systems. A. P. acknowledges the support of the “Granular-Chaos” project, funded by the Italian MIUR under the FIRB-IDEAS grant number RBID08Z9JE. S. P. wishes to thank M. Mueller for suggesting the kinesin example. L. R. acknowledges the contribution of the European Research Council within the 7th Framework Programme (FP7) of the European Community (EC), ERC Grant Agreement n. 202680. The EC is not responsible for any use that might be made of the data appearing herein. The decimation procedure {#ap:dec} ======================== In this Appendix, we summarize the coarse graining method introduced in [@PV08]. Consider a master equation of the form of eq. (\[eq:master\]). Due to the Markovian nature of the process, the time spent in a generic state $n$ is exponentially distributed with average $\tau_n=1/W^0_n$. One may wish to decimate all states having an average permanence time smaller than a prescribed threshold $\Delta \tau$. To do that, Ref.[@PV08] sets to 0 the time spent in these states. In this way, the fast states disappear from the description and transitions to them are redirected to other states with proper statistical weights. In formulae, if a state $i$ is linked to a state $j$ via a fast state $n$ that must be eliminated, the transition rate $W_{i\to j}$ from $i$ to $j$ is renormalized to yield the rate: $$\label{eq:decimate} {\tilde{W}}_{i\to j}=W_{i \to j}+W_{i \to n}W_{n \to j}/W^0_n$$ If $W_{i \to j}=0$, the decimation creates a direct connection between the surviving states, which is reminscent of the states that disappeared from the model under consideration. This procedure corresponds to an adiabatic approximation and is commutative, if the prescription of [@PV08] is followed. Once the set of states to be decimated is determined by the threshold, they can be decimated in any order without affecting the final result, as long as the set itself is not modified during the decimation procedure. It may happen, indeed, that the permanence time of some of the states selected for decimation becomes larger than $\Delta \tau$, while other states are decimated. The recipe of [@PV08] requires that this state be eventually decimated nonetheless. Effect of decimation on the current in a single loop {#ap:singleloop} ==================================================== In this Appendix, we investigate the effect of decimation on the current of a single loop consisting of $N$ states. For convenience, let us rewrite the master equation: $$\frac{d}{dt} P_n(t)= W_{n-1\to n}P_{n-1}+W_{n+1\to n}P_{n+1}-P_n (W_{n\to n+1}+W_{n\to n-1}) ~,$$ with $P_0=P_N$, as: $$\frac{d}{dt} P_n(t)= J_n-J_{n-1}$$ where the [*local*]{} current $J_n$ is given by: $$J_n=P_{n-1}W_{n-1\to n}-P_nW_{n\to n-1}.$$ In a stationary state, the current is site-independent and one may write $J_n=J$. In particular, detailed balance and equilibrium hold if $J=0$. The set of equation $J_n=J$, together with the normalization condition $\sum_n P_n=1$, can be solved for both $J$ and the invariant measure $P_n^{inv}$. For instance, let us proceed iteratively, as follows: $$\begin{aligned} P_n&=&P_{n-1} \frac{W_{n-1\to n}}{W_{n\to n-1}}-\frac{J}{W_{n\to n-1}}=\nonumber\\ &=&P_{n-2}\frac{W_{n-2\to n-1}W_{n-1\to n}}{W_{n\to n-1}W_{n-1\to n-2}} -J\left(\frac{1}{W_{n\to n-1}}+\frac{1}{W_{n-1\to n-2}}\frac{W_{n-1\to n}} {W_{n\to n-1}} \right)=\dots\nonumber\\ &=&P_n\prod\limits_{k=1}^N \frac{W_{k-1\to k}}{W_{k\to k-1}} -J\left(\sum\limits_{j=0}^{N-1} \frac{1}{W_{n-j\to n-j-1}}\prod\limits_{k=0}^j\frac{W_{n-j+k-1\to n-j+k}} {W_{n-j+k\to n-j+k-1}}\right).\end{aligned}$$ We obtain $P_n$ from the last expression $$P_n=\frac{-J\left(\sum\limits_{j=0}^{N-1}\frac{1}{W_{n-j\to n-j-1}} \prod\limits_{k=0}^j \frac{W_{n-j+k-1\to n-j+k}}{W_{n-j+k\to n-j+k-1}}\right)} {1-\prod\limits_{k=1}^N \frac{W_{k-1\to k}}{W_{k\to k-1}}}$$ and by means of the normalization condition $\sum P_n=1$, we reach the following closed expression for $J$: $$\label{eqcurrent} J=\frac{\left(\prod\limits_{k=1}^N \frac{W_{k-1\to k}}{W_{k\to k-1}}\right)-1} {\sum\limits_{n=1}^N \sum\limits_{j=0}^{N-1}\frac{1}{W_{n-j\to n-j-1}}\prod\limits_{k=0}^j \frac{W_{n-j+k-1\to n-j+k}}{W_{n-j+k\to n-j+k-1}}}.$$ Let us now decimate one fast state, $n^*$ say, and consider the current. It is easy to show that the numerator is not affected by the decimation protocol defined by eq. (\[eq:decimate\]). Conversely, the denominator decreases by an amount $\Delta D$ which can be espressed as follows: $$\label{eq:deltaD} \Delta D= D_o-D_d=\frac{1}{W_{n^*\to n^*-1}}+ \sum\limits_{j=0}^{N-1}\frac{1}{W_{n^*+1\to n^*}} \prod\limits_{k=0}^j\frac{W_{n^*+k\to n^*+k+1}}{W_{n^*+k+1\to n^*+k}}$$ where $D_0$ and $D_d$ are the denominator in (\[eqcurrent\]) for the original and the decimated system, respectively. As $\Delta D$ is positive, the current in the decimated system is larger than in the original one, the difference being $$\Delta J=J_d-J=J \frac{\Delta D}{D_0-\Delta D}.$$ This allows us to check what happens in simple cases. For instance, eq. (\[eq:deltaD\]) leads to $\Delta D/D=1/N$, if all state have same left and right jump rates (the two must be different to have a non-trivial current). If the rate of the decimated state is much faster than the others, eq. (\[eq:deltaD\]) also shows that the correction decreases linearly with the separation of time scales, i.e. with the ratio of the average rates of the fast states and that of the other states. This is consistent with the picture of the current correction being essentially due to a rescaling of the times related to the elimination of the fast state. In other words, the magnitude of the correction seems to be always related to the ratio of the time spent in the fast state(s) and the time spent in the slow ones. Graphs and currents {#ap:graph} =================== We consider a Markovian (continuous time) process on $N$ states. The $N$ states are considered as nodes of a graph. The transitions between different states are considered as links (edges) between nodes. Fundamental cycles ------------------ Graph theory simplifies the classification of closed loops on a graph [@S76], identifying a set of fundamental “cycles”. Given a graph $G$ with $N$ vertices (nodes) and $E$ edges (links between nodes), the strategy - exemplified in Figure \[fig:circuits\] - is the following: ![An example of graph with $5$ states and three fundamental loops: all transitions (links) have a given orientation. A possible maximal tree is the one made of only solid links, with the three dashed links representing the remaining chords, which individuate three fundamental loops. Any other possible loop, e.g. $\vec{C}=1 \to 2 \to 4 \to 5 \to 1$, can be decomposed in the sum of fundamental loops, e.g. $\vec{C}=\vec{C}_1+\vec{C}_3$. \[fig:circuits\]](snack.eps){width="6cm"} - identify a maximal tree $T(G)$, i.e. a set containing all $N$ vertices and part of the $E$ edges, which is connected and does not contain circuits. It is easy to show that $T(G)$ has $N-1$ edges. Many maximal trees can be identified, but one suffices; - given an arbitrary maximal tree $T(G)$, the edges of $G$ which do not belong to $T(G)$ are called chords of $T(G)$, they are in a number $\nu=E-N+1$; - if only one chord $s_\alpha$, $\alpha \in [1,\nu]$, is added to $T(G)$, the new graph contains only one circuit, $C_\alpha$, obtained by $T(G)+s_\alpha$ removing all edges which are not part of the circuit; therefore, from a maximal tree, $\nu$ circuits can be generated adding the $\nu$ chords; - the set of $\nu$ circuits obtained from the $\nu$ chords of a maximal graph is called a fundamental set of circuits, denoted by $\{C_1,...,C_\nu \}$ - orientation of edges must be introduced: each edge is assumed to be oriented in an arbitrary direction, giving the oriented version of $G$, denoted as $\vec{G}$; then one can take a subgraph with oriented edges $\vec{P}$, which may have different orientations with respect to the original orientations of the edges of $\vec{G}$. The function $S_e(\vec{P})$ is introduced for these cases: it returns $1$ if the edge $e$ is in $\vec{P}$ and has the original orientation, $-1$ if it is in $\vec{P}$ and has opposite orientation, and $0$ if $e$ is not in $\vec{P}$. - a cycle is an oriented circuit, e.g. $\vec{C}$; a fundamental cycle is denoted by $\vec{C}_\alpha$: for simplicity we always choose the orientation of a fundamental circuit to be parallel to the orientation of its chord $\alpha$, i.e. $S_\alpha(\vec{C}_\alpha)=1$; - the scalar product among cycles is defined as $$(\vec{C},\vec{C}_\alpha)=S_\alpha(\vec{C})S_\alpha(\vec{C}_\alpha)\equiv S_\alpha(\vec{C})$$ where $\alpha$ is the chord which generates the circuit $C_\alpha$; this scalar product can only take three values: $0$, $1$ or $-1$. - a decomposition of cycles is finally achieved: any cycle (oriented circuit) of the graph $G$ can be linearly decomposed using the fundamental set as a basis: $$\vec{C}=\sum_{\alpha=1}^\nu(\vec{C},\vec{C}_\alpha)\vec{C}_\alpha$$ Currents -------- The current for the $\omega \to \omega'$ transition is $$\label{cur_def} J(\omega \to \omega',t)=P_\omega(t)W_{\omega \to \omega'}-P_{\omega'}(t)W_{\omega' \to \omega}.$$ The stationary state value is denoted by $J(\omega \to \omega')$. The stationarity condition $dP_\omega^{inv}/dt=0$ is equivalent to $$\sum_{\omega'} J(\omega' \to \omega)=0 \;\;\;\;\;\;\; \forall \omega$$ which is known as [*Kirchhoff current law*]{}. If the transition $\omega \to \omega'$ corresponds to the oriented edge $e$, its steady state current is also denoted as $J_e$. The current (or flux) on a fundamental circuit is defined as the steady state transtion current flowing in the chord $\alpha$ in the original direction and is denoted by $J_\alpha$. For instance if $\alpha$ is the oriented edge corresponding to the transition $\omega \to \omega'$, then $J_\alpha=J(\omega \to \omega')$ and the flux of the associated cycle $\vec{C}_\alpha$ is equal to $+J_\alpha$. The Kirchhoff law for the steady state guarantees that a current on any edge is the sum of the currents going through the cycles which intersect the edge, i.e. $$\label{edgecur} J_e=\sum_{\alpha=1}^\nu S_e(\vec{C}_\alpha)J_\alpha.$$ An edge of the graph $G$ can be oriented in a different direction with respect to the edges of the cycles, therefore the sign function $S_e$ is used. The fluctuating instantaneous current $J_\alpha$ depends on the particular realization of the Markov process; it is measured on a chord $\alpha$ as: $$j_\alpha(t)=\sum_{n=-\infty}^{+\infty}S_\alpha(e_n)\delta(t-t_n)$$ where $t_n$ is the time of the random transition $e_n$ (an oriented edge of the graph) during a trajectory of the stochastic process. In brief, $j_\alpha$ is the instantaneous and oriented rate of the transitions in the chord $\alpha$, for a particular realization of the process. It is a stochastic variable. Its time-average (in a finite time $t$) is denoted by $$G_\alpha(t)=\frac{1}{t}\int_0^t dt' j_\alpha(t'),$$ which is still a stochastic variable. Some properties of $j_\alpha$ and $G_\alpha$ have been studied in [@AG07]. Affinities ---------- The affinity of a transition $\omega \to \omega'$ is defined as $$A(\omega \to \omega',t)=\ln \frac{P_\omega(t)W_{\omega \to \omega'}}{P_\omega'(t)W_{\omega' \to \omega}}$$ The affinity of a cycle $C$ is defined as $A(C)=\sum_e S_e(C)A(e)$, but it can also be defined as $A(C)=\sum_e S_e(C)B(e)$. where $B(\omega \to \omega')=\ln (W_{\omega \to \omega'}/W_{\omega' \to \omega})$. The equivalence of these two forms is due to the fact that all $P_\omega(t)$ cancel out, in a cycle. For this reason the affinity of a cycle does not depend upon time, but only on the transition rates, which come from the “external physical constraints”, e.g. mechanical, chemical and thermodynamical forces. Thanks to the decomposition of cycles described above, one can linearly decompose the affinity of any cycle in terms of affinities of a “fundamental set of cycles” $\{\vec{C}_\alpha\}$: $$A(\vec{C})=\sum_\alpha (\vec{C},\vec{C}_\alpha) A(\vec{C}_\alpha)$$ where $(.,.)$ is the previously defined scalar product between cycles. Entropy production ------------------ Having defined the Gibbs entropy as $${\mathcal{S}}(t)=-\sum_\omega P_\omega(t) \ln P_\omega(t),$$ its time derivative can be decomposed in two parts $d{\mathcal{S}}/dt=d_e {\mathcal{S}}/dt+d_i {\mathcal{S}}/dt$, where the bilinear form $$\frac{d_i {\mathcal{S}}}{dt}=\frac{1}{2}\sum_{\omega,\omega'}J(\omega \to \omega',t)A(\omega \to \omega',t) \ge 0$$ is considered as the internal entropy production, and the rest $d_e {\mathcal{S}}/dt$ is the entropy flux through the boundaries of the system of interest. In the steady state one has $d_e {\mathcal{S}}/dt=-d_i {\mathcal{S}}/dt$. A definition of entropy production per trajectory is given by Lebowitz and Spohn [@LS99], see Eq. . It depends on a particular realization, i.e. it is a stochastic variable. It can also be written as: $$\label{ls2} {{\mathcal{W}}_t}=\frac{1}{t}\sum_eB(e)\int_0^tdt'j_e(t').$$ Lebowitz and Spohn have noticed that $$\lim_{t \to \infty} \langle {{\mathcal{W}}_t}\rangle=\left.\frac{d_i {\mathcal{S}}}{dt}\right|_{st}$$ in the stationary state. The following relation has instead been noticed in Ref.[@AG07]: $$\label{res} {{\mathcal{W}}_t}=Q_t+R_t$$ with $$\begin{aligned} Q_t&=\sum_\alpha A(\vec{C}_\alpha)G_\alpha(t)\\ R_t&=\frac{1}{t}\sum_{e \neq {\alpha}} B(e)\left[\int_0^t dt' \left(j_e(t')- \sum_\alpha S_e(\vec{C}_\alpha)j_\alpha(t')\right) \right].\end{aligned}$$ The $Q_t$ term is the contribution due to the fundamental set of cycles. The “remainder” $R_t$ has zero average (thanks to the Kirchhoff law Eq. ). This implies that $$\left.\frac{d_i {\mathcal{S}}}{dt}\right|_{st}=\lim_{t \to \infty} \langle {{\mathcal{W}}_t}\rangle= \lim_{t \to \infty} \langle Q_t \rangle=\sum_\alpha A(\vec{C}_\alpha)J_\alpha,$$ since $$\lim_{t \to \infty}\langle G_\alpha(t) \rangle=\lim_{t \to \infty} \frac{1}{t}\left\langle \int_0^t j_\alpha(t') dt' \right\rangle=J_\alpha.$$ Numerical comparison of the fluctuations of $Q_t$ and those of ${{\mathcal{W}}_t}$ show that they have identical Cramer functions (see Figure \[fig:equiv\]), in many examples of continuous time Markov processes. From eq. , detailed balance, with respect to the invariant measure, is equivalent to $$J(\omega\to\omega')=0 \;\;\;\;\; \forall (\omega \to \omega'),$$ which implies that the probability of any trajectory is equal to the probability of its time reversal. Detailed balance also implies that the flux on any cycle vanishes, $J_\alpha=0$, and that affinities vanish on a single edge as well as on any cycle, eg. $A(\vec{C}_\alpha)=0$. As an immediate consequence, the internal entropy production vanishes: $$\left.\frac{d_i {\mathcal{S}}}{dt}\right|_{st}=0.$$ Appendix: parameters in the kinesin model {#app_kin} ========================================= As sketched in Section \[sec:kin\], the rates in the kinesin network model of Ref.[@LL07] are adjusted to the parameters obtained by specific experiments. Moreover, they depend on the concentrations of the chemical species entering the reaction (ADP, ATP and P), as well as on the load force $F$. More formally, one has: $$\label{kin_rate_def} W_{i \to j}=k_{ij}\ I_{ij}([X])\ \Phi_{ij}(F)$$ where the $k$’s are the experiment-specific parameters. The functions $I_{ij}$ and $\Phi_{ij}$ express the dependence of the reaction rates on a generic chemical species $X$ and/or on the load force $F$. If the transition from $i$ to $j$ does not involve chemical binding, we define $I_{ij}\equiv 1$. Assuming diluted solutions, all reactions are diffusion-limited, so that we can assume $I([X])\sim [X]$. The $\Phi$’s are adimensional functions, with the convention $\Phi(0)=1$. Theoretical considerations lead to $\Phi_{ij}(F)=\Phi_{ji}(F)=2/(1+e^{\chi_{ij}\bar{F}})$ for the chemical transitions, i.e. all but those between states $2$ and $5$. Mechanical transitions are parametrized by $\Phi_{25}=e^{-\theta \bar{F}}$ and $\Phi_{52}=e^{(1-\theta) \bar{F}}$. The $\chi$’s and $\theta$ are additional parameters obtained by experiments, while $\bar{F}=lF/kT$ is the adimensional force ($l\approx 8 nm$ being the average kinesin step length and $k$ the Boltzmann constant). With these choices, the $k$’s are dimensionally different depending on whether they multiply a concentration (dimensions of rate divided by concentration, $[(\mu M\ s)^{-1}]$) or not (dimension of a rate, $[s^{-1}]$). The parameters we used in the simulations are derived from those reproducing the results of the experiment [@CC05]: - The values of $k$’s describing the experiment [@CC05] according to [@LL07] are $k_{25}=3\cdot10^5$, $k_{52}=0.24$, $k_{12}=k_{45}=2.0$, $k_{21}=100$, $k_{56}=k_{61}=k_{23}=k_{34}=100$, $k_{65}=k_{32}=0.02$, $k_{16}=k_{43}=0.02$. The upper and lower cycle in Fig. \[kinesin\_scheme\] are assumed to have same parameters, apart from the transition from $5$ to $4$, which is determined from theoretical considerations as $k_{54}=k_{21}(k_{52}/k_{25})^2=6.4 \ 10^{-11}$. - Typical concentrations in the experiment are $0.5\mu M$. For simplicity, we assume all of them to be kept constant and equal to $[P]=[ADP]=[ATP]=1\mu M$. - The mechanical parameters reproducing the results of experiment [@CC05] are: $\theta=0.65$, $\chi_{12}=\chi_{45}=0.25$, $\chi_{23}=\chi_{56}=0.15$, $\chi_{34}=\chi_{61}=0.15$. In all cases, we have $\chi_{ij}=\chi_{ji}$. In section \[sec:kin\], we considered two instances of the model. The first one is without load, $F=0$. In this case and with the assumptions above, it is easy to obtain the transition rates: all the $\Phi$’s and concentrations are equal to $1$, so from Eq. (\[kin\_rate\_def\]) we obtain $W_{i\to j}=k_{ij}$: the rates are just the $k$’s listed above. About the load case, the unit of the adimensional force is equal to $kT/l\approx 0.5 pN$. Experiments are performed with forces of the order of piconewton. We took a value $\bar{F}=10$: substituting this value in the expression for the $\Phi$’s leads to the following values of the transition rates, which are those used in the simulations of the model with load: $W_{2\to 5}=451$, $W_{5\to 2}=7.95$, $W_{1\to 2}=W_{4\to 5}=0.3$, $W_{2\to 1}=15$, $W_{5\to 6}=W_{6\to 1}=W_{2\to 3}=W_{3\to 4}=36.5$, $W_{6\to 5}=W_{3\to 2}=0.007$, $W_{1\to 6}=W_{4\to 3}=0.007$, $W_{5\to 4}=0.5\ 10^{-11}$. List of the main symbols ======================== - $W_{i \to j}$ is the transition rates from $i$ to $j$ - $W_i^0$ is the exit rate from state $i$ - $\tau_i=1/W_i^0$ is the characteristic time of state $i$ - $P_n(t)$ is the probability of being in $n$ at time $t$ - $P_n^{inv}$ is the invariant probability of being in $n$ - $\Delta \tau$ is the time threshold for decimation - ${\tilde{W}}_{i \to j}$ are the new transition rates in the decimated process - ${{\mathcal{W}}_t}$ is the Lebowitz-Spohn entropy production integrated on time $t$ and divided by $t$ - $C()$ is the Cramer’s function of the entropy production - $f({{\mathcal{W}}_t}) \sim e^{-tC({{\mathcal{W}}_t})}$ is the probability density of ${{\mathcal{W}}_t}$ - $C_{\Delta \tau}({{\mathcal{W}}_t})$ is the Cramer function in the decimated process with a time threshold $\Delta \tau$. - $\vec{C}_\alpha$ is an oriented cycle of the graph - $G_\alpha(t)$ is the current on cycle $\alpha$ averaged on a finite time $t$ - $A(\vec{C}_\alpha)$ is the affinity associated to the oriented cycle $\vec{C}_\alpha$.

--- abstract: | In this paper we describe the prime ideals of some important classes of skew $PBW$ extensions, using the classical technique of extending and contracting ideals. Skew $PBW$ extensions include as particular examples Weyl algebras, enveloping algebras of finite-dimensional Lie algebras (and its quantization), Artamonov quantum polynomials, diffusion algebras, Manin algebra of quantum matrices, among many others. *Key words and phrases.* Prime ideals, semiprime rings, nilradical, skew polynomial rings, $PBW$ extensions, quantum algebras, skew $PBW$ extensions. 2010 *Mathematics Subject Classification.* Primary: 16D25, 16N60, 16S80. Secondary: 16S36, 16U20. author: - | Oswaldo Lezama\ `jolezamas@unal.edu.co`\ Juan Pablo Acosta & Milton Armando Reyes Villamil\ Seminario de Álgebra Constructiva - SAC$^2$\ Departamento de Matemáticas\ Universidad Nacional de Colombia, Sede Bogotá title: '**Prime ideals of skew $PBW$ extensions**' --- Introduction ============ In this paper we describe the prime ideals of some important classes of skew $PBW$ extensions. For this purpose we will consider the techniques that we found in [@Bell], [@Goodearl] and [@Goodearl2]. However, in several of our results, some modifications to these techniques should be introduced. In this section we recall the definition of skew $PBW$ (Poincaré-Birkhoff-Witt) extensions defined firstly in [@LezamaGallego], and we will review also some elementary properties about the polynomial interpretation of this kind of non-commutative rings. Weyl algebras, enveloping algebras of finite-dimensional Lie algebras (and its quantization), Artamonov quantum polynomials, diffusion algebras, Manin algebra of quantum matrices, are particular examples of skew $PBW$ extensions (see [@lezamareyes1]). \[gpbwextension\] Let $R$ and $A$ be rings. We say that $A$ is a *skew $PBW$ extension of $R$* $($also called a $\sigma-PBW$ extension of $R$$)$ if the following conditions hold: 1. $R\subseteq A$. 2. There exist finite elements $x_1,\dots ,x_n\in A$ such $A$ is a left $R$-free module with basis ${\rm Mon}(A):= \{x^{\alpha}=x_1^{\alpha_1}\cdots x_n^{\alpha_n}\mid \alpha=(\alpha_1,\dots ,\alpha_n)\in \mathbb{N}^n\}$. In this case it says also that *$A$ is a left polynomial ring over $R$* with respect to $\{x_1,\dots,x_n\}$ and $Mon(A)$ is the set of standard monomials of $A$. Moreover, $x_1^0\cdots x_n^0:=1\in Mon(A)$. 3. For every $1\leq i\leq n$ and $r\in R-\{0\}$ there exists $c_{i,r}\in R-\{0\}$ such that $$\label{sigmadefinicion1} x_ir-c_{i,r}x_i\in R.$$ 4. For every $1\leq i,j\leq n$ there exists $c_{i,j}\in R-\{0\}$ such that $$\label{sigmadefinicion2} x_jx_i-c_{i,j}x_ix_j\in R+Rx_1+\cdots +Rx_n.$$ Under these conditions we will write $A:=\sigma(R)\langle x_1,\dots ,x_n\rangle$. The following proposition justifies the notation and the alternative name given for the skew $PBW$ extensions. \[sigmadefinition\] Let $A$ be a skew $PBW$ extension of $R$. Then, for every $1\leq i\leq n$, there exists an injective ring endomorphism $\sigma_i:R\rightarrow R$ and a $\sigma_i$-derivation $\delta_i:R\rightarrow R$ such that $x_ir=\sigma_i(r)x_i+\delta_i(r)$, for each $r\in R$. See [@LezamaGallego], Proposition 3. A particular case of skew $PBW$ extension is when all derivations $\delta_i$ are zero. Another interesting case is when all $\sigma_i$ are bijective and the constants $c_{ij}$ are invertible. We recall the following definition (cf. [@LezamaGallego]). \[sigmapbwderivationtype\] Let $A$ be a skew $PBW$ extension. 1. $A$ is quasi-commutative if the conditions [(]{}iii[)]{} and [(]{}iv[)]{} in Definition \[gpbwextension\] are replaced by 1. For every $1\leq i\leq n$ and $r\in R-\{0\}$ there exists $c_{i,r}\in R-\{0\}$ such that $$x_ir=c_{i,r}x_i.$$ 2. For every $1\leq i,j\leq n$ there exists $c_{i,j}\in R-\{0\}$ such that $$x_jx_i=c_{i,j}x_ix_j.$$ 2. $A$ is bijective if $\sigma_i$ is bijective for every $1\leq i\leq n$ and $c_{i,j}$ is invertible for any $1\leq i<j\leq n$. Some extra notation will be used in the paper. \[1.1.6\] Let $A$ be a skew $PBW$ extension of $R$ with endomorphisms $\sigma_i$, $1\leq i\leq n$, as in Proposition \[sigmadefinition\]. 1. For $\alpha=(\alpha_1,\dots,\alpha_n)\in \mathbb{N}^n$, $\sigma^{\alpha}:=\sigma_1^{\alpha_1}\cdots \sigma_n^{\alpha_n}$, $|\alpha|:=\alpha_1+\cdots+\alpha_n$. If $\beta=(\beta_1,\dots,\beta_n)\in \mathbb{N}^n$, then $\alpha+\beta:=(\alpha_1+\beta_1,\dots,\alpha_n+\beta_n)$. 2. For $X=x^{\alpha}\in {\rm Mon}(A)$, $\exp(X):=\alpha$ and $\deg(X):=|\alpha|$. 3. If $f=c_1X_1+\cdots +c_tX_t$, with $X_i\in Mon(A)$ and $c_i\in R-\{0\}$, then $\deg(f):=\max\{\deg(X_i)\}_{i=1}^t.$ The skew $PBW$ extensions can be characterized in a similar way as was done in [@Gomez-Torrecillas2] for $PBW$ rings. \[coefficientes\] Let $A$ be a left polynomial ring over $R$ w.r.t. $\{x_1,\dots,x_n\}$. $A$ is a skew $PBW$ extension of $R$ if and only if the following conditions hold: 1. For every $x^{\alpha}\in {\rm Mon}(A)$ and every $0\neq r\in R$ there exist unique elements $r_{\alpha}:=\sigma^{\alpha}(r)\in R-\{0\}$ and $p_{\alpha ,r}\in A$ such that $$\label{611} x^{\alpha}r=r_{\alpha}x^{\alpha}+p_{\alpha , r},$$ where $p_{\alpha ,r}=0$ or $\deg(p_{\alpha ,r})<|\alpha|$ if $p_{\alpha , r}\neq 0$. Moreover, if $r$ is left invertible, then $r_\alpha$ is left invertible. 2. For every $x^{\alpha},x^{\beta}\in {\rm Mon}(A)$ there exist unique elements $c_{\alpha,\beta}\in R$ and $p_{\alpha,\beta}\in A$ such that $$\label{612} x^{\alpha}x^{\beta}=c_{\alpha,\beta}x^{\alpha+\beta}+p_{\alpha,\beta},$$ where $c_{\alpha,\beta}$ is left invertible, $p_{\alpha,\beta}=0$ or $\deg(p_{\alpha,\beta})<|\alpha+\beta|$ if $p_{\alpha,\beta}\neq 0$. See [@LezamaGallego], Theorem 7. We remember also the following facts from [@LezamaGallego]. \[identities\] (i) We observe that if $A$ is quasi-commutative, then $p_{\alpha,r}=0$ and $p_{\alpha,\beta}=0$ for every $0\neq r\in R$ and every $\alpha,\beta \in \mathbb{N}^n$. \(ii) If $A$ is bijective, then $c_{\alpha,\beta}$ is invertible for any $\alpha,\beta\in \mathbb{N}^n$. \(iii) In $Mon(A)$ we define $x^{\alpha}\succeq x^{\beta}\Longleftrightarrow \begin{cases} x^{\alpha}=x^{\beta}\\ \text{or} & \\ x^{\alpha}\neq x^{\beta}\, \text{but} \, |\alpha|> |\beta| & \\ \text{or} & \\ x^{\alpha}\neq x^{\beta},|\alpha|=|\beta|\, \text{but $\exists$ $i$ with} & \alpha_1=\beta_1,\dots,\alpha_{i-1}=\beta_{i-1},\alpha_i>\beta_i. \end{cases}$ It is clear that this is a total order on $Mon(A)$. If $x^{\alpha}\succeq x^{\beta}$ but $x^{\alpha}\neq x^{\beta}$, we write $x^{\alpha}\succ x^{\beta}$. Each element $f\in A$ can be represented in a unique way as $f=c_1x^{\alpha_1}+\cdots +c_tx^{\alpha_t}$, with $c_i\in R-\{0\}$, $1\leq i\leq t$, and $x^{\alpha_1}\succ \cdots \succ x^{\alpha_t}$. We say that $x^{\alpha_1}$ is the *leader monomial* of $f$ and we write $lm(f):=x^{\alpha_1}$ ; $c_1$ is the *leader coefficient* of $f$, $lc(f):=c_1$, and $c_1x^{\alpha_1}$ is the *leader term* of $f$ denoted by $lt(f):=c_1x^{\alpha_1}$. A natural and useful result that we will use later is the following property. \[1.1.10a\] Let A be a skew PBW extension of a ring R. If R is a domain, then A is a domain. See [@lezamareyes1]. The next theorem characterizes the quasi-commutative skew $PBW$ extensions. \[1.3.3\] Let $A$ be a quasi-commutative skew $PBW$ extension of a ring $R$. Then, 1. $A$ is isomorphic to an iterated skew polynomial ring of endomorphism type, i.e., $A\cong R[z_1;\theta_1]\cdots [z_{n};\theta_n]$. 2. If $A$ is bijective, then each endomorphism $\theta_i$ is bijective, $1\leq i\leq n$. See [@lezamareyes1]. \[1.3.2\] Let $A$ be an arbitrary skew $PBW$ extension of $R$. Then, $A$ is a filtered ring with filtration given by $$\label{eq1.3.1a} F_m:=\begin{cases} R & {\rm if}\ \ m=0\\ \{f\in A\mid {\rm deg}(f)\le m\} & {\rm if}\ \ m\ge 1 \end{cases}$$ and the corresponding graded ring $Gr(A)$ is a quasi-commutative skew $PBW$ extension of $R$. Moreover, if $A$ is bijective, then $Gr(A)$ is a quasi-commutative bijective skew $PBW$ extension of $R$. See [@lezamareyes1]. \[1.3.4\] Let $A$ be a bijective skew $PBW$ extension of $R$. If $R$ is a left $($right$)$ Noetherian ring then $A$ is also a left $($right$)$ Noetherian ring. See [@lezamareyes1]. Invariant ideals ================ Let $A=\sigma(R)\langle x_1,\dots,x_n\rangle$ be a skew $PBW$ extension of a ring $R$. By Proposition \[sigmadefinition\], we know that $x_ir-\sigma_i(r)x_i=\delta_i(r)$ for all $r\in R$, where $\sigma_i$ is an injective endomorphism of $R$ and $\delta_i$ is a $\sigma_i$-derivation of $R$, $1\leq i\leq n$. This motivates the following general definition. \[1.4.1\] Let $R$ be a ring and $(\Sigma,\Delta)$ a system of endomorphisms and $\Sigma$-derivations of $R$, with $\Sigma:=\{\sigma_1,\dots,\sigma_n\}$ and $\Delta:=\{\delta_1,\dots,\delta_n\}$. 1. If $I$ is an ideal of $R$, $I$ is called $\Sigma$-invariant if $\sigma_i(I)\subseteq I$, for every $1\leq i\leq n$. In a similar way are defined the $\Delta$-invariant ideals. If $I$ is both $\Sigma$ and $\Delta$-invariant, we say that $I$ is $(\Sigma,\Delta)$-invariant. 2. A proper $\Sigma$- invariant ideal $I$ of $R$ is $\Sigma$-prime if whenever a product of two $\Sigma$-invariant ideals is contained in $I$, one of the ideals is contained in $I$. $R$ is a $\Sigma$-prime ring if the ideal $0$ is $\Sigma$-prime. In a similar way are defined the $\Delta$-prime and $(\Sigma,\Delta)$-prime ideals and rings. 3. The system $\Sigma$ is commutative if $\sigma_i\sigma_j=\sigma_j\sigma_i$ for every $1\leq i\leq n$. In a similar way is defined the commutativity for $\Delta$. The system $(\Sigma,\Delta)$ is commutative if both $\Sigma$ and $\Delta$ are commutative. The following proposition describe the behavior of these properties when we pass to a quotient ring. \[1.4.2b\] Let $R$ be a ring, $(\Sigma,\Delta)$ a system of endomorphisms and $\Sigma$-derivations of $R$, $I$ a proper ideal of $R$ and $\overline{R}:=R/I$. 1. If $I$ is $(\Sigma,\Delta)$-invariant, then over $\overline{R}:=R/I$ is induced a system $(\overline{\Sigma},\overline{\Delta})$ of endomorphisms and $\overline{\Sigma}$-derivations defined by $\overline{\sigma_i}(\overline{r}):=\overline{\sigma_i(r)}$ and $\overline{\delta_i}(\overline{r}):=\overline{\delta_i(r)}$, $1\leq i\leq n$. If $\sigma_i$ is bijective and $\sigma_i(I)=I$, then $\overline{\sigma_i}$ is bijective. 2. Let $I$ be $\Sigma$-invariant. If $\Sigma$ is commutative, then $\overline{\Sigma}$ is commutative. Similar properties are valid for $\Delta$ and $(\Sigma,\Delta)$. 3. Let $I$ be $\Sigma$-invariant. $I$ is $\Sigma$-prime if and only if $\overline{R}$ is $\overline{\Sigma}$-prime. Similar properties are valid for $\Delta$ and $(\Sigma,\Delta)$. All statements follow directly from the definitions. According to the properties of $\Sigma$ and $\Delta$, we need to introduce some special classes of skew $PBW$ extensions. Let $A$ be a skew $PBW$ extension of a ring $R$ with system of endomorphisms $\Sigma:=\{\sigma_1,\dots,\sigma_n\}$ and $\Sigma$-derivations $\Delta:=\{\delta_1,\dots,\delta_n\}$. 1. If $\sigma_i=i_R$ for every $1\leq i\leq n$, we say that $A$ is a skew $PBW$ extension of derivation type. 2. If $\delta_i=0$ for every $1\leq i\leq n$, we say that $A$ is a skew $PBW$ extension of endomorphism type. In addition, if every $\sigma_i$ is bijective, $A$ is a skew $PBW$ extension of automorphism type. 3. $A$ is $\Sigma$-commutative if the system $\Sigma$ is commutative. In a similar way are defined the $\Delta$ and $(\Sigma,\Delta)$-commutativity of $A$. Related with the previous definition, we have the following two interesting results. The second one extends Lemma 1.5. (c) in [@Goodearl2]. Let $A$ be a skew $PBW$ extension of derivation type of a ring $R$. Then, for any $\theta,\gamma,\beta\in \mathbb{N}^n$ and $c\in R$, the following identities hold: $c_{\gamma,\beta}c_{\theta,\gamma+\beta}=c_{\theta,\gamma}c_{\theta+\gamma,\beta}$, $cc_{\theta,\gamma}=c_{\theta,\gamma}c$. In particular, the system of constants $c_{i,j}$ are central. This is a direct consequence of Remark \[identities\], part (ii). \[1.4.9\] Let $A$ be a skew $PBW$ extension of a ring $R$. If for every $1\leq i\leq n$, $\delta_i$ is inner, then $A$ is a skew $PBW$ extension of $R$ of endomorphism type. Let $a_i\in R$ such that $\delta_i=\delta_{a_i}$ is inner, $1\leq i\leq n$. We will prove that $A=\sigma(R)\langle z_1,\dots,z_n\rangle$, where $z_i:=x_i-a_i$, the system of endomorphisms coincides with the original system $\Sigma$ and every $\sigma_i$-derivation is equal zero. We will check the conditions in Definition \[gpbwextension\]. It is clear that $R\subseteq A$. Let $r\in R$, then $z_ir=(x_i-a_i)r=x_ir-a_ir=\sigma_i(r)x_i+\delta_{a_i}(r)=\sigma_i(r)x_i+a_ir-\sigma_i(r)a_i-a_ir=\sigma_i(r)(x_i-a_i)=\sigma_i(r)z_i$. Thus, the systems of constants $c_{i,r}$ of $\sigma(R)\langle z_1,\dots,z_n\rangle$ coincides with the original one, and the same is true for the system of endomorphisms. Note that the system of $\Sigma$-derivations is trivial, i.e., each one is equal zero. This means that $\sigma(R)\langle z_1,\dots,z_n\rangle$ is of endomorphism type. $z_jz_i=(x_j-a_j)(x_i-a_i)=x_jx_i-x_ja_i-a_jx_i+a_ja_i=c_{ij}x_ix_j+r_0+r_1x_1+\cdots+r_nx_n-x_ja_i-a_jx_i+a_ja_i$, for some $r_0,r_1,\dots,r_n\in R$. Replacing $x_i$ by $z_i+a_i$ for every $1\leq i\leq n$, we conclude that $z_jz_i-c_{ij}z_iz_j\in R+Rz_1+\cdots+Rz_n$. Finally, note that $Mon\{z_1,\dots,z_n\}:=\{z^{\alpha}=z_1^{\alpha_1}\cdots z_n^{\alpha_n}|\alpha=(\alpha_1,\dots ,\alpha_n)\in \mathbb{N}^n\}$ is a left $R$-basis of $A$. In fact, it is clear that $Mon\{z_1,\dots,z_n\}$ generates $A$ as left $R$-module. Let $c_1,\dots, c_t\in R$ such that $c_1z^{\alpha_1}+\cdots+c_tz^{\alpha_t}=0$ with $z^{\alpha_i}\in Mon\{z_1,\dots,z_n\}$, $1\leq i\leq n$. Then, using the deglex order in Remark \[identities\], we conclude that $c_1x^{\alpha_1}+\cdots+c_tx^{\alpha_t}$ should be zero, whence $c_1=\cdots=c_t=0$. In the next proposition we study quotients of skew $PBW$ extensions by $(\Sigma,\Delta)$-invariant ideals. \[1.4.2a\] Let $A$ be a skew $PBW$ extension of a ring $R$ and $I$ a $(\Sigma,\Delta)$-invariant ideal of $R$. Then, 1. $IA$ is an ideal of $A$ and $IA\cap R=I$. $IA$ is proper if and only if $I$ is proper. Moreover, if for every $1\leq i\leq n$, $\sigma_i$ is bijective and $\sigma_i(I)=I$, then $IA=AI$. 2. If $I$ is proper and $\sigma_i(I)=I$ for every $1\leq i\leq n$, then $A/IA$ is a skew $PBW$ extension of $R/I$. Moreover, if $A$ is of automorphism type, then $A/IA$ is of automorphism type. If $A$ is bijective, then $A/IA$ is bijective. In addition, if $A$ is $\Sigma$-commutative, then $A/IA$ is $\overline{\Sigma}$-commutative. Similar properties are true for the $\overline{\Delta}$ and $(\overline{\Sigma},\overline{\Delta})$ commutativity. 3. Let $A$ be of derivation type and $I$ proper. Then, $IA=AI$ and $A/IA$ is a skew $PBW$ extension of derivation type of $R/I$. 4. Let $R$ be left $($right$)$ Noetherian and $\sigma_i$ bijective for every $1\leq i\leq n$. Then, $\sigma_i(I)=I$ for every $i$ and $IA=AI$. If $I$ is proper and $A$ is bijective, then $A/IA$ is a bijective skew $PBW$ extension of $R/I$. \(i) It is clear that $IA$ is a right ideal, but since $I$ is $(\Sigma,\Delta)$-invariant, then $IA$ is also a left ideal of $A$. It is obvious that $IA\cap R=I$. From this last equality we get also that $IA$ is proper if and only if $I$ is proper. Using again that $I$ is $(\Sigma,\Delta)$-invariant, we get that $AI\subseteq IA$. Assuming that $\sigma_i$ is bijective and $\sigma_i(I)=I$ for every $i$, then $IA\subseteq AI$. \(ii) According to (i), we only have to show that $\overline{A}:=A/IA$ is a skew $PBW$ extension of $\overline{R}:=R/I$. For this we will verify the four conditions of Definition \[gpbwextension\]. It is clear that $\overline{R}\subseteq \overline{A}$. Moreover, $\overline{A}$ is a left $\overline{R}$-module with generating set $Mon\{\overline{x_1},\dots,\overline{x_n}\}$. Next we show that $Mon\{\overline{x_1},\dots,\overline{x_n}\}$ is independent. Consider the expression $\overline{r_1}\ \overline{X_1}+ \dotsb + \overline{r_n}\overline{X_n}=\overline{0}$, where $X_i\in {\rm Mon}(A)$ for each $i$. We have $r_1X_1+\dotsb + r_nX_n\in IA$ and hence $$r_1X_1+\dotsb + r_nX_n=r_1'X_1+\dotsb +r_n'X_n,\ \ \text{for some }\ r_i'\in I,\ i=1,\dotsc,n.$$ Thus, $(r_1-r_1')X_1+\dotsb + (r_n-r_n')X_n=0$, so $r_i\in I$, i.e., $\overline{r_i}=\overline{0}$ for $i=1,\dotsc, n$. Let $\overline{r}\neq \overline{0}$ with $r\in R$. Then $r\notin IA$, and hence, $r\notin I$, in particular, $r\neq 0$ and there exists $c_{i,r}:=\sigma_i(r)\neq 0$ such that $x_ir=c_{i,r}x_i+\delta_i(r)$. Thus, $\overline{x_i}\,\overline{r}=\overline{c_{i,r}}\,\overline{x_i}+\overline{\delta_i(r)}$. Observe that $\overline{c_{i,r}}\neq \overline{0}$, contrary $c_{i,r}=\sigma_i(r)\in IA\cap R=I=\sigma_i(I)$, i.e, $r\in I$, a contradiction. This completes the proof of condition (iii) in Definition \[gpbwextension\]. In $A$ we have $x_jx_i-c_{i,j}x_ix_j\in R+\sum_{t=1}^n Rx_t$, with $c_{i,j}\in R-\{0\}$, so in $\overline{A}$ we get that $\overline{x_j}\, \overline{x_i}-\overline{c_{i,j}}\, \overline{x_i}\, \overline{x_j}\in \overline{R}+\sum_{t=1}^n \overline{R}\, \overline{x_t}$. Since $I$ is proper and $c_{i,j}$ is left invertible for $i<j$ and right invertible for $i>j$, then $\overline{c_{i,j}}\neq \overline{0}$. This completes the proof of condition (iv) in Definition \[gpbwextension\]. By Proposition \[1.4.2b\], if $\sigma_i$ is bijective, then $\overline{\sigma_i}$ is bijective. It is obvious that if every constant $c_{ij}$ is invertible, then $\overline{c_{ij}}$ is invertible.. The statements about the commutativity follow from Proposition \[1.4.2b\]. \(iii) This is direct consequence of (i) and (ii). \(iv) Considering the Noether condition and the ascending chain $I\subseteq \sigma_{i}^{-1}(I)\subseteq \sigma_{i}^{-2}(I)\subseteq \cdots$ we get that $\sigma_i(I)=I$ for every $i$. The rest follows from (i) and (ii). Extensions of derivation type ============================= Now we pass to describe the prime ideals of skew $PBW$ extensions of derivation type. Two technical propositions are needed first. The total order introduced in Remark \[identities\] will be used in what follows. \[Terminator\] Let $A$ be a skew $PBW$ extension of a ring $R$ such that $\sigma_i$ is bijective for every $1\leq i\leq n $. Let $J$ be a nonzero ideal of $A$. If $f$ is a nonzero element of $J$ of minimal leader monomial $x^{\alpha_t}$ and $\sigma^{\alpha_t}(r)=r$ for any $r\in {\rm rann}_R({\rm lc}(f))$, then ${\rm rann}_A(f)=({\rm rann}_R({\rm lc}(f)))A$. Consider $0\neq f=m_1X_1+\dotsb +m_tX_t$ an element of $J$ of minimal leader monomial $X_t=x^{\alpha_t}$, with $X_t\succ X_{t-1}\succ \cdots \succ X_1$. By definition of the right annihilator, ${\rm rann}_R({\rm lc}(f)):=\{r\in R\mid m_tr=0\}$. From Theorem \[coefficientes\] we have $$fr=m_1X_1r+\dotsb + m_t(\sigma^{\alpha_t}(r)x^{\alpha_t}+p_{\alpha_t,r}),$$ where $p_{\alpha_t,r}=0$ or $\deg(p_{\alpha_t,r})<|\alpha_t|$ if $p_{\alpha_t, r}\neq 0$. Note that if $r\in {\rm rann}_R({\rm lc}(f))$, then $fr=0$. In fact, if the contrary is assumed, since $\sigma^{\alpha_t}(r)=r$, we get $lm(fr)\prec X_t$ with $fr\in J$, but this is a contradiction since $X_t$ is minimal. Thus, $f{\rm rann}_R({\rm lc}(f))=0$ and $f{\rm rann}_R({\rm lc}(f))A=0$. Therefore ${\rm rann}_R({\rm lc}(f))A\subseteq {\rm rann}_A(f)$. Next we will show that ${\rm rann}_A(f) \subseteq {\rm rann}_R({\rm lc}(f))A$. Let $u=r_1Y_1+\dotsb + r_kY_k$ an element of ${\rm rann}_A(f)$, then $$fu=(m_1X_1+\dotsb +m_tX_t)(r_1Y_1+\dotsb + r_kY_k)=0,$$ which implies that $m_tX_tr_kY_k=0$, whence $m_t\sigma^{\alpha_t}(r_k)X_tY_k=0$, that is, $m_t\sigma^{\alpha_t}(r_k)=0$ which means $\sigma^{\alpha_t}(r_k)\in {\rm rann}_R(m_t)$. Let $\sigma^{\alpha_t}(r_k):=s$. Then $s\in {\rm rann}_R({\rm lc}(f))$. Note that $r_k=\sigma^{-\alpha_t}(s)=s$; moreover $\sigma^{\alpha_t}(s)=s$ implies $s=\sigma^{-\alpha_t}(s)$ and hence $r_k\in {\rm rann}_R({\rm lc}(f))$. This shows that $r_kY_k\in {\rm rann}_R({\rm lc}(f))A\subseteq {\rm rann}_A(f)$, but since $u\in {\rm rann}_A(f)$ then $u-r_kY_k\in {\rm rann}_A(f)$. Continuing in this way we obtain that $r_{k-1}Y_{k-1}, r_{k-2}Y_{k-2},\dotsc, r_1Y_1\in {\rm rann}_R({\rm lc}(f))A$, which guarantees that $u\in {\rm rann}_R({\rm lc}(f))A$. Thus, we have proved that ${\rm rann}_A(f)\subseteq {\rm rann}_R({\rm lc}(f))A$. \[Terminator1\] Let $A$ be a skew $PBW$ extension of derivation type of a ring $R$ and let $K$ be a non zero ideal of $A$. Let $K'$ be the ideal of $R$ generated by all coefficients of all terms of all polynomials of $K$. Then $K'$ is a $\Delta$-invariant ideal of $R$. Let $k\in K'$, then $k$ is a finite sum of elements of the form $rcr'$, with $r,r'\in R$ and $c$ is the coefficient of one term of some polynomial of $K$. It is enough to prove that for every $1\leq i\leq n$, $\delta_i(rcr')\in K'$. We have $\delta_i(rcr')=rc\delta_i(r')+r\delta_i(c)r'+\delta_i(r)cr'$. Note that $rc\delta_i(r'),\delta_i(r)cr'\in K'$, so only rest to prove that $\delta_i(c)\in K'$. There exists $p\in K$ such that $p=cx^{\alpha}+p'$, with $p'\in A$ and $x^{\alpha}$ does not appear in $p'$. Note that the coefficients of all terms of $p'$ are also in $K'$. Observe that $x_ip\in K$ and we have $x_ip=x_icx^{\alpha}+x_ip'=cx_ix^{\alpha}+\delta_i(c)x^{\alpha}+x_ip'$; from the previous expression we conclude that the coefficient of $x^{\alpha}$ in $x_ip$ is $\delta_i(c)+cr+r'$, where $r$ is the coefficient of $x^{\alpha}$ in $x_ix^{\alpha}$ and $r'$ is the coefficient of $x^{\alpha}$ in $x_ip'$. Since $c\in K'$ we only have to prove that $r'\in K'$. Let $p'=c_1x^{\beta_1}+\cdots+c_tx^{\beta_t}$, then $c_1,\dots,c_t\in K'$ and $x^{\alpha}\notin \{x^{\beta_1},\dots, x^{\beta_t}\}$. We have $x_ip'=(c_1x_i+\delta_i(c_1))x^{\beta_1}+\cdots +(c_tx_i+\delta_i(c_t))x^{\beta_t}=c_1x_ix^{\beta_1}+\delta_i(c_1)x^{\beta_1}+\cdots+c_tx_ix^{\beta_t}+\delta_i(c_t)x^{\beta_t}$; from the previous expression we get that $r'$ has the form $r'=c_1r_1+\cdots+c_tr_t$, where $r_j$ is the coefficient of $x^{\alpha}$ in $x_ix^{\beta_j}$, $1\leq j\leq t$. This proves that $r'\in K'$. The following theorem gives a description of prime ideals of skew $PBW$ extensions of derivation type without assuming any conditions on the ring of coefficients. This result generalizes the description of prime ideals of classical $PBW$ extensions given in Proposition 6.2 of [@Bell]. Compare also with [@McConnell], Proposition 14.2.5 and Corollary 14.2.6. \[Terminator2\] Let $A$ be a skew $PBW$ extension of derivation type of a ring $R$. Let $I$ be a proper $\Delta$-invariant ideal of $R$. $I$ is a $\Delta$-prime ideal of $R$ if and only if $IA$ is a prime ideal of $A$. In such case, $IA=AI$ and $IA\cap R=I$. \(i) By Proposition \[1.4.2a\] we know that $A/IA$ is a skew $PBW$ extension of $R/I$ of derivation type, $IA=AI$ and $IA\cap R=I$. Then we may assume that $I=0$. Note that if $R$ is not $\Delta$-prime, then $A$ is not prime. Indeed, there exist $I,J\neq 0$ $\Delta$-invariant ideals of $R$ such that $IJ=0$, so $IA,JA\neq 0$ and $IAJA=IJA=0$, i.e., $A$ is not prime. Suppose that $R$ is $\Delta$-prime. We need to show that if $J,K$ are nonzero ideals of $A$, then $JK\neq 0$. Let $K'$ as in Proposition \[Terminator1\], then $K'\neq 0$ and it is $\Delta$-invariant. Now let $j$ be a nonzero element of $J$ of minimal leader monomial. If $jK=0$, then taking $M=R$ and $T=J$ in Proposition \[Terminator\] we get $$K\subseteq {\rm rann}_A(j)={\rm rann}_R({\rm lc}(j))A.$$ Therefore ${\rm lc}(j)K'=0$ and hence ${\rm lann}_R(K')\neq 0$. We have ${\rm lann}_R(K')K'=0$, and note that ${\rm lann}_R(K')$ is also $\Delta$-invariant. In fact, let $a\in {\rm lann}_R(K')$ and $k'\in K'$, then $\delta_i(a)k'=\delta_i(ak')-a\delta_i(k')=\delta_i(0)-a\delta_i(k')=0$ since $\delta_i(k')\in K'$. Thus, $R$ is not $\Delta$-prime, a contradiction. In this way $jK\neq 0$ and so $JK\neq 0$, which concludes the proof. Extensions of automorphism type =============================== In this section we consider the characterization of prime ideals for extensions of automorphism type over commutative Noetherian rings. \[5.2.6N\] Let $A$ be a bijective skew $PBW$ extension of a ring $R$. Suppose that given $a,b\in R-\{0\}$ there exists $\theta\in \mathbb{N}^n$ such that either $aR\sigma^\theta(b)\neq 0$ or $aR\delta^\theta(b)\neq 0$. Then, $A$ is a prime ring. Suppose that $A$ is not a prime ring, then there exist nonzero ideals $I,J$ of $A$ such that $IJ=0$. We can assume that $I:={\rm lann}_A(J)$ and $J:={\rm rann}_A(I)$. Let $u$ be a nonzero element of $I$ with minimal leader monomial $x^{\alpha}$ and leader coefficient $c_u$. We will prove first that $\sigma^{-\alpha}(c_u)\in I$, i.e., $\sigma^{-\alpha}(c_u)J=0$. Since ${\rm rann}_A(I)\subseteq {\rm rann}_A(u)$, then it is enough to show that $\sigma^{-\alpha}(c_u){\rm rann}_A(u)=0$. Suppose that $\sigma^{-\alpha}(c_u){\rm rann}_A(u)\neq 0$, let $v\in {\rm rann}_A(u)$ of minimal leader monomial $x^{\beta}$ and leader coefficient $c_v$ such that $\sigma^{-\alpha}(c_u)v\neq 0$. Since $uv=0$ and $c_{\alpha,\beta}$ is invertible (see Theorem \[coefficientes\] and Remark \[identities\]), then $c_u\sigma^{\alpha}(c_v)=0$, whence $lm(uc_v)\prec x^{\alpha}$. The minimality of $x^{\alpha}$ implies that $uc_v=0$, and hence $u(v-c_vx^{\beta})=0$. Moreover, $v-c_vx^{\beta}\in {\rm rann}_A(u)$ and $lm((v-c_vx^{\beta})\prec x^{\beta}$, so $\sigma^{-\alpha}(c_u)(v-c_vx^{\beta})=0$. However, $c_u\sigma^{\alpha}(c_v)=0$, so we have $\sigma^{-\alpha}(c_u)c_v=0$ and hence $\sigma^{-\alpha}(c_u)v=0$, a contradiction. Thus, $I\cap R\neq 0$, and by symmetry, $J\cap R\neq 0$. Let $0\neq a\in I\cap R$ and $0\neq b\in J\cap R$, by the hypothesis there exists $\theta\in \mathbb{N}^n$ and $r\in R$ such that $ar\sigma^\theta(b)\neq 0$ or $ar\delta^\theta(b)\neq 0$. If $\theta=(0,\dots,0)$, then $arb\neq 0$, and hence $IJ\neq 0$, a contradiction. If $\theta\neq (0,\dots,0)$, then $arx^{\theta}b=ar(\sigma^{\theta}(b)x^{\theta}+p_{\theta,b})$, but note that the independent term of $p_{\theta,b}$ is $\delta^{\theta}(b)$ (see Theorem \[coefficientes\], part (b)). Thus, $arx^{\theta}b\neq 0$, i.e., $IJ\neq 0$, a contradiction. If $R$ is a prime ring and $A$ is a bijective skew $PBW$ extension of $R$, then $A$ is prime and $rad(A)=0$. Since $R$ is prime and $A$ is bijective, given $a,b\in R-\{0\}$, then $aR\sigma^{\theta}(b)\neq 0$ for every $\theta\in \mathbb{N}^n$. Thus, from the previous proposition, $A$ is prime. \[3.3.3b\] Let $A$ be a bijective $\Sigma$-commutative skew $PBW$ extension of automorphism type of a left $(right)$ Noetherian ring $R$. Let $I$ be a proper ideal of $R$ $\Sigma$-invariant. $I$ is a $\Sigma$-prime ideal of $R$ if and only if $IA$ is a prime ideal of $A$. In such case, $IA=AI$ and $IA\cap R=I$. By Proposition \[1.4.2a\], $IA=AI$ is a proper ideal of $A$, $I=IA\cap R$ and $\overline{A}:=A/IA$ is a bijective skew $PBW$ extension of $\overline{R}:=R/I$ of automorphism type. In addition, observe that $I$ es $\Sigma$-prime if and only if $\overline{R}$ is $\overline{\Sigma}$-prime (Proposition \[1.4.2b\]). Thus, we can assume that $I=0$, and hence, we have to prove that $R$ is $\Sigma$-prime if and only if $A$ is prime. $\Rightarrow)$: Suppose that $R$ is $\Sigma$-prime, i.e., $0$ is $\Sigma$-prime. According to Proposition \[5.2.6N\], we have to show that given $a,b\in R-\{0\}$ there exists $\theta\in \mathbb{N}^n$ such that $aR\sigma^\theta(b)\neq 0$. Let $L$ be the ideal generated by the elements $\sigma^{\theta}(b)$, $\theta\in \mathbb{N}^n$; observe that $L\neq 0 $, and since $A$ is $\Sigma$-commutative, $L$ is $\Sigma$-invariant. But $R$ is Noetherian and $A$ is bijective, then $\sigma_i(L)=L$ for every $1\leq i\leq n$ (see Proposition \[1.4.2a\]). This implies that $Ann_R(L)$ is $\Sigma$-invariant, but $0$ is $\Sigma$-prime, therefore $Ann_R(L)=0$. Thus, $aL\neq 0$, so there exists $\theta\in \mathbb{N}^n$ such that $a\sigma^\theta(b)\neq 0$. $\Leftarrow)$: Note that if $R$ is not $\Sigma$-prime, then $A$ is not prime. Indeed, there exist $K,J\neq 0$ $\Sigma$-invariant ideals of $R$ such that $KJ=0$, so $KA,JA\neq 0$ and since $J$ is $\Sigma$-invariant, then $AJ=JA$ and hence $KAJA=KJA=0$, i.e., $A$ is not prime. \[1.4.21\] Any skew $PBW$ extension $A$ of a commutative ring $R$ is $\Sigma$-commutative. For $i=j$ it is clear that $\sigma_j\sigma_i=\sigma_i\sigma_j$. Let $i\neq j$, say $i<j$, then for any $r\in R$ we have $lc(x_jx_ir)=\sigma_j\sigma_i(r)c_{ij}=c_{ij}\sigma_i\sigma_j(r)$, but since $R$ is commutative and $c_{ij}$ is invertible, then $\sigma_j\sigma_i(r)=\sigma_i\sigma_j(r)$. \[3.3.5a\] Let $A$ be a bijective skew $PBW$ extension of automorphism type of a commutative Noetherian ring $R$. Let $I$ be a proper $\Sigma$-invariant ideal of $R$. $I$ is a $\Sigma$-prime ideal of $R$ if and only if $IA$ is a prime ideal of $A$. In such case, $IA=AI$ and $IA\cap R=I$. This follows from Lemma \[3.3.3b\] and Propositions \[1.4.2a\] and \[1.4.21\]. Extensions of mixed type ======================== Our next task is to give a description of prime ideals of bijective skew $PBW$ extensions of mixed type, i.e., when both systems $\Sigma$ and $\Delta$ could be non trivial. We will assume that the ring $R$ is commutative, Noetherian and semiprime. The proof of the main theorem (Theorem \[5.2.4N\]) is an in Lemma \[3.3.3b\], but anyway we have to show first some preliminary technical propositions. Let $R$ be a commutative ring, $I$ a proper ideal of $R$ and $\overline{R}:=R/I$, we define $S(I):=\{a\in R|\overline{a}:=a+I \ \text{is regular }\}$. By regular element we mean a non zero divisor. Note that $S(0)$ is the set of regular elements of $R$. Next we will describe the behavior of the properties introduced in Definition \[1.4.1\] when we pass to the total ring of fractions. \[1.4.12\] Let $R$ be a commutative ring with total ring of fractions $Q(R)$ and let $(\Sigma,\Delta)$ be a system of automorphisms and $\Sigma$-derivations of $R$. Then, 1. Over $Q(R)$ is induced a system $(\widetilde{\Sigma},\widetilde{\Delta})$ of automorphisms and $\widetilde{\Sigma}$-derivations defined by $\widetilde{\sigma_i}(\frac{a}{s}):=\frac{\sigma_i(a)}{\sigma_i(s)}$ and $\widetilde{\delta_i}(\frac{a}{s}):=-\frac{\delta_i(s)}{\sigma_i(s)}\frac{a}{s}+\frac{\delta_i(a)}{\sigma_i(s)}$. 2. $Q(R)$ is $\widetilde{\Sigma}$-prime if and only if $R$ is $\Sigma$-prime. The same is valid for $\Delta$ and $(\Sigma,\Delta)$. \(i) This part can be proved not only for commutative rings but also in the noncommutative case (see [@Lezama-OreGoldie]). \(ii) $\Rightarrow )$: Let $I,J$ be $\Sigma$-invariant ideals of $R$ such that $IJ=0$, then $IJS(0)^{-1}=IS(0)^{-1}JS(0)^{-1}=0$, but note that $IS(0)^{-1},JS(0)^{-1}$ are $\widetilde{\Sigma}$-invariant, so $IS(0)^{-1}=0$ or $JS(0)^{-1}=0$, i.e., $I=0$ or $J=0$. $\Leftarrow )$: $K,L$ be two $\widetilde{\Sigma}$-invariant ideals of $Q(R)$ such that $KL=0$, then $K=IS(0)^{-1}$ and $L=JS(0)^{-1}$, with $I:=\{a\in R|\frac{a}{1}\in K\}$ and $J:=\{b\in R|\frac{b}{1}\in L\}$. Note that $IJ=0$, moreover, $I,J$ are $\Sigma$-invariant. In fact, if $a\in I$, then $\frac{a}{1}\in K$ and hence $\widetilde{\sigma_i}(\frac{a}{1})=\frac{\sigma_i(a)}{1}\in K$, i.e., $\sigma_i(a)\in I$, for every $1\leq i\leq n$. Analogously for $L$. Since $R$ is $\Sigma$-prime, then $I=0$ or $J=0$, i.e., $K=0$ or $L=0$. The proofs for $\Delta$ and $(\Sigma,\Delta)$ are analogous. Next we will use the following special notation: Let $m\geq 0$ be an integer, then $\sigma(m)$ will denote the product of $m$ endomorphisms taken from $\Sigma$ in any order, and probably with repetitions, i.e., $\sigma(m)=\sigma_{i_1}\cdots \sigma_{i_m}$, with $i_1,\dots, i_m\in \{1,\dots, n\}$. For $m=0$ we will understand that this product is the identical isomorphism of $R$. \[1.4.14N\] Let $R$ be a commutative ring and $(\Sigma,\Delta)$ a system of endomorphisms and $\Sigma$-derivations of $R$. Let $I$ be a $\Sigma$-invariant ideal of $R$. Set $I_0:=R$, $I_1:=I$, and for $j\geq 2$, $I_j:=\{r\in I|\delta_{i_1}\sigma(m(1))\delta_{i_2}\sigma(m(2))\cdots \delta_{i_l}\sigma(m(l))(r)\in I,\ \text{for all}\ l=1,\dots,j-1;\, i_1,\dots, i_l\in \{1,\dots,n\} \text{and}\ m(1),\dots, m(l)\geq 0\}$. Then, 1. $I_0\supseteq I_1\supseteq I_2\supseteq \cdots $. 2. $\delta_i(I_j)\subseteq I_{j-1}$, for every $1\leq i\leq n$ and any $j\geq 1$. 3. $I_j$ is a $\Sigma$-invariant ideal of $R$, for any $j\geq 0$. 4. $II_j\subseteq I_{j+1}$, for any $j\geq 0$. \(i) This is evident. \(ii) It is clear that for every $i$, $\delta_i(I_1)\subseteq I_0$. Let $j\geq 2$ and let $r\in I_j$, then $\delta_{i_1}\sigma(m(1))\delta_{i_2}\sigma(m(2))\cdots \delta_{i_l}\sigma(m(l))(r)\in I$, for all $l=1,\dots, j-1$. From this obtain that $\delta_{i_1}\sigma(m(1))\delta_{i_2}\sigma(m(2))\cdots \delta_{i_l}\sigma(m(l))\delta_i\sigma(0)(r)\in I$ for all $l=1,\dots, j-2$. This means that $\delta_i(r)\in I_{j-1}$. \(iii) It is clear that $I_0,I_1$ are $\Sigma$-invariant ideals. Let $j\geq 2$ and let $r\in I_j$, then for every $\sigma_i$ we have $\delta_{i_1}\sigma(m(1))\delta_{i_2}\sigma(m(2))\cdots \delta_{i_l}\sigma(m(l))(\sigma_i(r))=\delta_{i_1}\sigma(m(1))\delta_{i_2}\sigma(m(2))\cdots \delta_{i_l}\sigma(m(l)+1)(r)\in I$. This means that $\sigma_i(r)\in I_j$, i.e., $I_j$ is $\Sigma$-invariant. Let us prove that $I_j$ is an ideal of $R$. By induction we assume that $I_j$ is an ideal. It is obvious that if $a,a'\in I_{j+1}$, then $a+a'\in I_{j+1}$; let $r\in R$, then $a\in I_j$ and hence $ra\in I_j$, therefore $\delta_{i_1}\sigma(m(1))\delta_{i_2}\sigma(m(2))\cdots \delta_{i_k}\sigma(m(k))(ra)\in I$ for all $k<j$. Consider any $\sigma(m(j))$, we have $\sigma(m(j))(a)\in I_{j+1}$, so $\sigma(m(j))(a),\delta_i(\sigma(m(j))(a))\in I_j$ for every $i$. Therefore, $\delta_i(\sigma(m(j))(ra))=\delta_i(\sigma(m(j))(r)\sigma(m(j))(a))=\sigma_i\sigma(m(j))(r)\delta_i(\sigma(m(j))(a))+ \delta_i(\sigma(m(j))(r))\sigma(m(j))(a)\in I_j$. This implies that $\delta_{i_1}\sigma(m(1))\delta_{i_2}\sigma(m(2))\cdots \delta_{i_j}\sigma(m(j))(ra)\in I$. This means that $ra\in I_{j+1}$. This proves that $I_{j+1}$ is an ideal. \(iv) Of course $II_0\subseteq I_1$. Let $j\geq 1$, suppose that for $II_{j-1}\subseteq I_j$, we have to prove that $II_j\subseteq I_{j+1}$. Let $a\in I$ and $b\in I_j$, then $b\in I_{j-1}$ and $ab\in I_j$. Therefore, $\delta_{i_1}\sigma(m(1))\delta_{i_2}\sigma(m(2))\cdots \delta_{i_k}\sigma(m(k))(ab)\in I$ for $i<j$. For any $\sigma(m(j))$ and every $\delta_i$ we have, as above, $\delta_i(\sigma(m(j))(ab))=\delta_i(\sigma(m(j))(a)\sigma(m(j))(b))=\sigma_i\sigma(m(j))(a)\delta_i(\sigma(m(j))(b))+ \delta_i(\sigma(m(j))(a))\sigma(m(j))(b)\in II_{j-1}+RI_j\subseteq I_j$. From this we conclude that $\delta_{i_1}\sigma(m(1))\delta_{i_2}\sigma(m(2))\cdots \delta_{i_j}\sigma(m(j))(ab)\in I$, and this means that $ab\in I_{j+1}$. This completes the proof. \[1.4.15\] Let $R$ be a commutative Noetherian ring and $\Sigma$ a system of automorphisms of $R$. Then, any $\Sigma$-prime ideal of $R$ is semiprime. Let $I$ be a $\Sigma$-prime ideal of $R$. Since $R$ is Noetherian, $\sqrt{I}$ is finitely generated, so there exists $m\geq 1$ such that $(\sqrt{I})^m\subseteq I$. Since $I$ is $\Sigma$-invariant and $R$ is Noetherian, $\sqrt{I}$ is $\Sigma$-invariant, whence $\sqrt{I}\subseteq I$, i.e., $\sqrt{I}=I$. This means that $I$ is intersection of prime ideals, i.e., $I$ is semiprime. \[1.4.13aN\] Let $R$ be a commutative Noetherian ring and $(\Sigma,\Delta)$ a system of automorphisms and $\Sigma$-derivations of $R$. Let $rad(R)$ be the prime radical of $R$. If $R$ is $(\Sigma,\Delta)$-prime, then 1. $rad(R)$ is $\Sigma$-prime. 2. $S(0)=S(rad(R))$. 3. $Q(R)$ is Artinian. \(i) The set of $\Sigma$-invariant ideals $I$ of $R$ such that $Ann_R(I)\neq 0$ is not empty since $0$ satisfies these conditions. Since $R$ is assumed to be Noetherian, let $I$ be maximal with these conditions. Let $K,L$ be $\Sigma$-invariant ideals of $R$ such that $I\varsubsetneq K$ and $I\varsubsetneq L$, then $Ann_R(K)=0=Ann_R(L)$, and hence $Ann_R(KL)=0$. This implies that $KL\nsubseteq I$. This proves that $I$ is $\Sigma$-prime. We will prove that $I=rad(R)$. By Proposition \[1.4.14N\], we have the descending chain of $\Sigma$-invariant ideals $I_0\supseteq I_1\supseteq \cdots$, and the ascending chain $Ann_R(I_0)\subseteq Ann_R(I_1)\subseteq \cdots $. There exists $m\geq 1$ such that $Ann_R(I_m)=Ann_R(I_{m+1})$ (since $I_0=R$ and $I_1=I$, see Proposition \[1.4.14N\], $m\neq 0$). Note that $Ann_R(I_m)$ is $\Sigma$-invariant since $\sigma_i(I_m)=I_m$ for every $i$ (here we have used again that $R$ is Noetherian, Proposition \[1.4.2a\]). Let $b\in Ann_R(I_m)$. For $a\in I_{m+1}$ we have $a\in I_m$, moreover, by Proposition \[1.4.14N\], for every $i$, $\delta_i(a)\in I_m$, so $ab=0=\delta_i(a)b$, therefore $0=\delta_i(ab)=\sigma_i(a)\delta_i(b)$. From this we obtain that $\sigma_i(I_{m+1})\delta_i(b)=0$, i.e., $I_{m+1}\delta_i(b)=0$. Thus, $\delta_i(b)\in Ann_R(I_{m+1})=Ann_R(I_m)$. We have proved that $Ann_R(I_m)$ is $(\Sigma,\Delta)$-invariant. Let $H:=Ann_R(Ann_R(I_m))$, we shall see that $H$ is also $(\Sigma,\Delta)$-invariant. In fact, let $x\in H$, then $xAnn_R(I_m)=0$ and for every $i$ we have $\sigma_i(x)\sigma_i(Ann_R(I_m))=0=\sigma_i(x)Ann_R(I_m)$, thus $\sigma_i(x)\in H$. Now let $y\in Ann_R(I_m)$, then $xy=0$ and for every $i$ we have $\delta_i(xy)=0=\sigma_i(x)\delta_i(y)+\delta_i(x)y$, but $\delta_i(y)\in Ann_R(I_m)$, so $\sigma_i(x)\delta_i(y)=0$. Thus, $\delta_i(x)y=0$, i.e., $\delta_i(x)\in H$. Since $R$ is $(\Sigma,\Delta)$-prime and $HAnn_R(I_m)=0$, then $H=0$ or $Ann_R(I_m)=0$. From Proposition \[1.4.14N\], $Ann_R(I_m)\supseteq Ann_R(I)\neq 0$, whence $H=0$. Since $I_m\subseteq H$, then $I_m=0$. Again, from Proposition \[1.4.14N\] we obtain that $I^m\subseteq I_m$, so $I^m=0$, and hence $I\subseteq rad(R)$. On the other hand, since $I$ is $\Sigma$-prime, then $I$ is semiprime (Proposition \[1.4.15\]), but $rad(R)$ is the smallest semiprime ideal of $R$, whence $I=rad(R)$. Thus, $rad(R)$ is $\Sigma$-prime. \(ii) The inclusion $S(0)\subseteq S(rad(R))$ is well known (see [@McConnell], Proposition 4.1.3). The other inclusion is equivalent to prove that $R$ is $S(rad(R))$-torsion free. Since $I_m=0$, it is enough to prove that every factor $I_j/I_{j+1}$ is $S(rad(R))$-torsion free. In fact, in general, if $M$ is an $R$-module and $N$ is an submodule of $M$ such that $M/N$ and $N$ are $S$-torsion free (with $S$ an arbitrary system of $R$), then $M$ is $S$-torsion free. Thus, the assertion follows from $R=I_0/I_m$, $I_0/I_2/I_1/I_2\cong I_0/I_1$, $I_0/I_3/I_2/I_3\cong I_0/I_2$, $\dots,$ $I_0/I_m/I_{m-1}/I_m\cong I_0/I_{m-1}$. $I_0/I_1=R/rad(R)$, it is clearly $S(rad(R))$-torsion free. By induction, we assume that $I_{j-1}/I_j$ is $S(rad(R))$-torsion free. Let $a\in I_j$ and $r\in S(rad(R))$ such that $r\widehat{a}=\widehat{0}$ in $I_j/I_{j+1}$, i.e., $ra\in I_{j+1}$. From Proposition \[1.4.14N\] we get that for every $i$ and any $\sigma(m)$, $\sigma(m)(a)\in I_j$, $\sigma(m)(ra)\in I_{j+1}$ and $\delta_i(\sigma(m)(ra))\in I_j$, then $\delta_i(\sigma(m)(ra))=\delta_i(\sigma(m)(r)\sigma(m)(a))=\sigma_i(\sigma(m)(r))\delta_i(\sigma(m)(a)) +\delta_i(\sigma(m)(r))\sigma(m)(a)\in I_j$, whence, $\sigma_i(\sigma(m)(r))\delta_i(\sigma(m)(a))\in I_j$. For every $k$, $\sigma_k(rad(R))=rad(R)$, then we can prove that $\sigma_k(S(rad(R)))=S(rad(R))$, so $\sigma_i(\sigma(m)(r))\in S(rad(R))$; moreover, $\delta_i(\sigma(m)(a))\in I_{j-1}$, then by induction $\delta_i(\sigma(m)(a))\in I_j$. But this is valid for any $i$ and any $\sigma(m)$, then $a\in I_{j+1}$. This proves that $I_{j-1}/I_j$ is $S(rad(R))$-torsion free. \(iii) This follows from (ii) and Small’s Theorem (see [@McConnell], Corollary 4.1.4). \[5.2.4aN\] Let $R$ be a commutative Noetherian semiprime ring and $(\Sigma,\Delta)$ a system of automorphisms and $\Sigma$-derivations of $R$. If $R$ is $(\Sigma,\Delta)$-prime, then $R$ is $\Sigma$-prime. By Proposition \[1.4.13aN\], the total ring of fractions $Q(R)$ of $R$ is Artinian. Then, by Proposition \[1.4.12\], we can assume that $R$ is Artinian. Applying again Proposition \[1.4.13aN\], we get that $0=rad(R)$ is $\Sigma$-prime, i.e., $R$ is $\Sigma$-prime. \[5.2.4N\] Let $R$ be a commutative Noetherian semiprime ring and $A$ a bijective skew $PBW$ extension of $R$. Let $I$ be a semiprime $(\Sigma,\Delta)$-invariant ideal of $R$. $I$ is a $(\Sigma,\Delta)$-prime ideal of $R$ if and only if $IA$ is a prime ideal of $A$. In such case, $IA=AI$ and $I=IA\cap R$. The proof is exactly as in Lemma \[3.3.3b\], anyway we will repeat it. By Proposition \[1.4.2a\], $IA=AI$ is a proper ideal of $A$, $I=IA\cap R$ and $\overline{A}:=A/IA$ is a bijective skew $PBW$ extension of the commutative Noetherian semiprime ring $\overline{R}:=R/I$. In addition, observe that $I$ is $(\Sigma,\Delta)$-prime if and only if $\overline{R}$ is $(\overline{\Sigma},\overline{\Delta})$-prime (see Propositions \[1.4.2b\] and \[1.4.2a\]). Thus, we can assume that $I=0$, and hence, we have to prove that $R$ is $(\Sigma,\Delta)$-prime if and only if $A$ is prime. $\Rightarrow)$: From Corollary \[5.2.4aN\], we know that $R$ is $\Sigma$-prime, i.e., $0$ is $\Sigma$-prime. Let $L$ be the ideal generated by the elements $\sigma^{\theta}(b)$, $\theta\in \mathbb{N}^n$; observe that $L\neq 0 $, and since $A$ is $\Sigma$-commutative (Proposition \[1.4.21\]), $L$ is $\Sigma$-invariant. But $R$ is Noetherian and $A$ is bijective, then $\sigma_i(L)=L$ for every $1\leq i\leq n$ (see Proposition \[1.4.2a\]). This implies that $Ann_R(L)$ is $\Sigma$-invariant, therefore $Ann_R(L)=0$. Thus, $aL\neq 0$, so there exists $\theta\in \mathbb{N}^n$ such that $a\sigma^\theta(b)\neq 0$. From Proposition \[5.2.6N\] we get that $A$ is prime. $\Leftarrow)$: Note that if $R$ is not $(\Sigma,\Delta)$-prime, then $A$ is not prime. Indeed, there exist $I,J\neq 0$ $(\Sigma,\Delta)$-invariant ideals of $R$ such that $IJ=0$, so $IA,JA\neq 0$ and $IAJA=IJA=0$, i.e., $A$ is not prime. Examples ======== Some important examples of skew $PBW$ extensions covered by Theorems \[Terminator2\], \[3.3.5a\] or \[5.2.4N\] are given in the following table. The definition of each ring or algebra can be found in [@lezamareyes1]. For each example we have marked with $\checkmark$ the theorem that can be applied. Theorem \[3.3.5a\] gives a description of prime ideals for the ring of skew quantum polynomials over commutative Noetherian rings. Skew quantum polynomials were defined in [@lezamareyes1], and represent a generalization of Artamonov’s quantum polynomials (see [@Artamonov], [@Artamonov2]). They can be defined as a localization of a quasi-commutative bijective skew $PBW$ extension. We recall next its definition. Let $R$ be a ring with a fixed matrix of parameters $\textbf{q}:=[q_{ij}]\in M_n(R)$, $n\geq 2$, such that $q_{ii}=1=q_{ij}q_{ji}=q_{ji}q_{ij}$ for every $1\leq i,j\leq n$, and suppose also that it is given a system $\sigma_1,\dots,\sigma_n$ of automorphisms of $R$. The ring of *skew quantum polynomials over $R$*, denoted by $R_{\textbf{q},\sigma}[x_1^{\pm 1 },\dots,x_r^{\pm 1}, x_{r+1},\dots,x_n]$, is defined as follows: 1. $R\subseteq R_{\textbf{q},\sigma}[x_1^{\pm 1},\dots,x_r^{\pm 1}, x_{r+1},\dots,x_n]$; 2. $R_{\textbf{q},\sigma}[x_1^{\pm 1 },\dots,x_r^{\pm 1}, x_{r+1},\dots,x_n]$ is a free left $R$-module with basis $$\label{equ1.4.2} \{x_1^{\alpha_1}\cdots x_n^{\alpha_n}|\alpha_i\in \mathbb{Z} \ \text{for}\ 1\leq i\leq r \ \text{and} \ \alpha_i\in \mathbb{N}\ \text{for}\ r+1\leq i\leq n\};$$ 3. the variables $x_1,\dots,x_n$ satisfy the defining relations $x_ix_i^{-1}=1=x_i^{-1}x_i$, $1\leq i\leq r$, $x_jx_i=q_{ij}x_ix_j$, $x_ir=\sigma_i(r)x_i$, $r\in R$, $1\leq i,j\leq n$. When all automorphisms are trivial, we write $R_{\textbf{q}}[x_1^{\pm 1 },\dots,x_r^{\pm 1}, x_{r+1},\dots,x_n]$, and this ring is called the ring of *quantum polynomials over $R$*. If $R=K$ is a field, then $K_{\textbf{q},\sigma}[x_1^{\pm 1 },\dots,x_r^{\pm 1}, x_{r+1},\dots,x_n]$ is the *algebra of skew quantum polynomials*. For trivial automorphisms we get the *algebra of quantum polynomials* simply denoted by $\mathcal{O}_\textbf{q}$ (see [@Artamonov]). When $r=0$, $R_{\textbf{q},\sigma}[x_1^{\pm 1},\dots,x_r^{\pm 1}, x_{r+1},\dots,x_n]=R_{\textbf{q},\sigma}[x_1,\dots,x_n]$ is the *$n$-multiparametric skew quantum space over $R$*, and when $r=n$, it coincides with $R_{\textbf{q},\sigma}[x_1^{\pm 1},\dots,x_n^{\pm 1}]$, i.e., with the *$n$-multiparametric skew quantum torus over $R$*. Note that $R_{\textbf{q},\sigma}[x_1^{\pm 1},\dots,x_r^{\pm 1}, x_{r+1},\dots,x_n]$ can be viewed as a localization of the $n$-multiparametric skew quantum space, which, in turn, is a a skew $PBW$ extension. In fact, we have the quasi-commutative bijective skew $PBW$ extension $$\label{equ2.2.3} A:=\sigma(R)\langle x_1,\dots, x_n\rangle, \, \text{with} \, x_ir=\sigma_i(r)x_i \, \text{and}\, x_jx_i=q_{ij}x_ix_j, 1\leq i,j\leq n;$$ observe that $A=R_{\textbf{q},\sigma}[x_1,\dots,x_n]$. If we set $$S:=\{rx^{\alpha}\mid r\in R^*, x^{\alpha}\in {\rm Mon}\{x_1,\dots,x_r\}\},$$ then $S$ is a multiplicative subset of $A$ and $$\label{equ2.7.4} S^{-1}A\cong R_{\textbf{q},\sigma}[x_1^{\pm 1},\dots,x_r^{\pm 1}, x_{r+1},\dots,x_n]\cong AS^{-1}.$$ Thus, if $R$ is a commutative Noetherian ring, then Theorem \[3.3.5a\] gives a description of prime ideals for $A$. With this, we get a description of prime ideals for $R_{\textbf{q},\sigma}[x_1^{\pm 1},\dots,x_r^{\pm 1}, x_{r+1},\dots,x_n]$ since it is well known that there exists a bijective correspondence between the prime ideals of $S^{-1}A$ and the prime ideals of $A$ with empty intersection with $S$ (recall that $A$ is left (right) Noetherian, Theorem \[1.3.4\]). [200]{} **Acosta, J.P., Chaparro, C., Lezama, O., Ojeda, I., and Venegas, C.**, *Ore and Goldie theorems for skew PBW extensions*, Asian-European Journal of Mathematics, 6 (4), 2013, 1350061-1; 1350061-20. **Artamonov, V.**, *Quantum polynomials*, WSPC Proceedings, 2008. **Artamonov, V.**, *Serre’s quantum problem*, Russian Math. Surveys, 53(4), 1998, 657-730. **Bell, A. and Goodearl, K.**, *Uniform rank over differential operator rings and Poincaré-Birkhoff-Witt extensions*, Pacific Journal of Mathematics, 131(1), 1988, 13-37. **Bueso, J., Gómez-Torrecillas, J. and Verschoren, A.**, *Algorithmic Methods in noncommutative Algebra: Applications to Quantum Groups*, Kluwer, 2003. **Goodearl, K. and Warfield, R. Jr.**, *An Introduction to Non-commutative Noetherian Rings*, London Mathematical Society, ST 61, 2004. **Goodearl, K.**, *Prime ideals in skew polynomial rings and quantized Weyl algebras*, J. of Algebra, 150, 1992, 324-377. **Lezama, O. and Gallego, C.**, [*Gröbner bases for ideals of sigma-PBW extensions*]{}, Communications in Algebra, 39 (1), 2011, 50-75. **Lezama, O. & Reyes, M.**, [*Some homological properties of skew $PBW$ extensions*]{}, Comm. in Algebra, 42, (2014), 1200-1230. **McConnell, J. and Robson, J.**, *Non-commutative Noetherian Rings*, Graduate Studies in Mathematics, AMS, 2001.

--- abstract: | Trusting simulation output is crucial for Sandia’s mission objectives. We rely on these simulations to perform our high-consequence mission tasks given national treaty obligations. Other science and modeling applications, while they may have high-consequence results, still require the strongest levels of trust to enable using the result as the foundation for both practical applications and future research. To this end, the computing community has developed workflow and provenance systems to aid in both automating simulation and modeling execution as well as determining exactly how was some output was created so that conclusions can be drawn from the data. Current approaches for workflows and provenance systems are all at the user level and have little to no system level support making them fragile, difficult to use, and incomplete solutions. The introduction of container technology is a first step towards encapsulating and tracking artifacts used in creating data and resulting insights, but their current implementation is focused solely on making it easy to deploy an application in an isolated “sandbox” and maintaining a strictly read-only mode to avoid any potential changes to the application. All storage activities are still using the system-level shared storage. This project explores extending the container concept to include storage as a new container type we call *data pallets*. Data Pallets are potentially writeable, auto generated by the system based on IO activities, and usable as a way to link the contained data back to the application and input deck used to create it. author: - | Jay Lofstead\ Sandia National Labs [^1] - | Joshua Baker\ Sandia National Labs - | Andrew Younge\ Sandia National Labs bibliography: - 'sample-bibliography.bib' title: 'Data Pallets: Containerizing Storage For Reproducibility and Traceability' --- [^1]: This manuscript has been authored by National Technology and Engineering Solutions of Sandia, LLC. under Contract No. DE-NA0003525 with the U.S. Department of Energy/National Nuclear Security Administration. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes.

--- abstract: 'We investigate the particle and heat transport in quantum junctions with the geometry of star graphs. The system is in a nonequilibrium steady state, characterized by the different temperatures and chemical potentials of the heat reservoirs connected to the edges of the graph. We explore the Landauer-Büttiker state and its orbit under parity and time reversal transformations. Both particle number and total energy are conserved in these states. However, the heat and chemical potential energy are in general not separately conserved, which gives origin to a basic process of energy transmutation among them. We study both directions of this process in detail, introducing appropriate efficiency coefficients. For scale invariant interactions in the junction our results are exact and explicit. They cover the whole parameter space and take into account all nonlinear effects. The energy transmutation depends on the particle statistics.' --- §[S]{} H [[U]{}]{}[U]{} Ł[L]{} Ø[O]{} [**v**]{} [**Energy transmutation in nonequilibrium\ quantum systems**]{}\ [Mihail Mintchev]{}\ [Luca Santoni]{}\ [Paul Sorba]{}\ Introduction ============ The study of non-equilibrium quantum systems is among the most rapidly expanding areas of theoretical physics. Triggered by the remarkable experimental progress in manipulating trapped ultra-cold atomic gases, there is recently great interest in the search for universal properties of such systems (see e.g. [@pssv] for a review). Much attention is devoted to the behavior of quantum systems after a quench and in particular, on the nature of the equilibrium state which is approached in this case. Other interesting studies concern the impact of both internal and space-time symmetries (e.g. scale invariance) on the quantum transport in Non-Equilibrium Steady States (NESS). In this paper we investigate some general features of the energy transport in NESS’s, representing non-equilibrium extensions of a Gibbs State (GS). The physical models we focus on are schematically shown in Fig. 1. We are dealing with a multicomponent system represented by the $n\geq 2$ semi-infinite leads (edges) $L_i$ of a star graph $\Gamma$. Each lead $L_i$ is attached at infinity to a heat reservoir $R_i$. The interaction between the $n$ components of the system is localized in the vertex of the graph and is defined by a scattering matrix $\S$. Each heat reservoir $R_i$ is described by a GS characterized by inverse temperature $\beta_i$ and chemical potential $\mu_i$. From Fig. 1 it is evident that the system is away from equilibrium if $\S$ admits a nontrivial transmission coefficient between two reservoirs $R_i$ and $R_j$ with $(\beta_i,\mu_i)\not=(\beta_j,\mu_j)$. (550,110)(20,255) ![A star graph $\Gamma$ with scattering matrix $\S$ at the vertex and leads $L_i$ connected at infinity to thermal reservoirs $R_i$.[]{data-label="junction"}](Fig-1n.eps "fig:") We start our investigation by shortly reviewing the explicit construction of a NESS $\Omega_{\beta, \mu}$, induced by the scattering matrix $\S$ from the tensor product of the GS’s relative to the heat reservoirs $R_i$. The state $\Omega_{\beta, \mu}$ is fully determined by $\S$ and $(\beta_i,\mu_i)$. We show that $\Omega_{\beta, \mu}$ is not invariant under parity and time-reversal transformations and generates therefore a nontrivial orbit $O_{\beta,\mu}$, consisting of four nonequivalent NESS’s. The total energy of the system in these states has two components: heat energy (parametrized by $\beta_i$) and chemical potential energy (parametrized by $\mu_i$). Provided that the dynamics of the system is invariant under time translations, the total energy is conserved. For generic values of $(\beta_i,\mu_i)$ the two energy components are however not separately conserved and an energy transmutation occurs. The process is controlled by a single parameter ${{\dot Q}}$, describing the heat flow in the junction. For ${{\dot Q}}<0$ heat energy is transformed in chemical potential energy. The opposite transformation takes place for ${{\dot Q}}>0$. The regime ${{\dot Q}}<0$ has been investigated mostly by means of linear response theory. For the details we refer to the review paper [@bcps] and the references therein, observing only that in these studies the junction is usually compared to a heat engine, whose efficiency represents the main point of interest. This comparison is very suggestive, keeping always in mind that the junction acts actually as a converter between heat and chemical potential energy. The case ${{\dot Q}}>0$ is quite subtle and has been explored only partially in the domain where the junction can be compared to a refrigerator or heat pump. The main goal of the present paper is to study the system in Fig. 1 as an energy converter, thus providing a systematic and unified description of both regimes ${{\dot Q}}\lessgtr 0$, which are characterized by appropriate efficiency coefficients. Since the energy transport in the junction is strongly influenced by nonlinear phenomena, the well known linear response approximation is not suitable for our investigation because it covers only a little part of the parameter space. For this reason a fundamental objective of our study is to simplify as much as possible the dynamics in order to solve the problem in exact form, preserving at the same time a nontrivial quantum transport. In this respect we demonstrate that assuming free propagation along $L_i$ and scale-invariant point-like interactions in the vertex is enough for this purpose. We study both fermionic and bosonic systems, showing that the particle statistics affects the energy transmutation. The detailed comparison between the two cases is very instructive, because the bosonic case is poorly investigated. The impact of parity and time reversal on the quantum transport is discussed in detail as well. We proceed as follows. In the next section we briefly recall the construction of the NESS $\Omega_{\beta, \mu}$ and its orbit $O_{\beta, \mu}$ under parity and time reversal transformations. In section 3 we fix the Schrödinger dynamics on the leads and the interaction at the vertex of $\Gamma$. Studying the local conserved currents and the relative densities, we show that the junction converts heat energy in chemical potential energy or vice versa, depending on the values of the parameters characterizing the reservoirs $R_i$. The basic features of this process of energy transmutation are investigated in section 4, where we introduce the efficiency coefficients for both cases ${{\dot Q}}\lessgtr 0$. We describe here also the scale invariant (critical) interactions in the junction. The efficiency of the quantum transport for a system with two leads is investigated in section 5 both for the LB state $\Omega_{\beta, \mu}$ and the orbit $O_{\beta, \mu}$. We also relate here our approach to some exact results, concerning other systems and NESS’s. In section 6 we describe the main features of a junction with 3 leads. Section 7 is devoted to the bosonic case. The conclusions and some future developments form the content of Section 8. The appendices collect some technical results about the Onsager matrix and the 3-lead junction. Non equilibrium states on $\Gamma$ ================================== Following the pioneering work of Landauer [@la-57] and Büttiker [@bu-86], non-equilibrium systems of the type in Fig. 1 have been extensively investigated. In this section we briefly recall an explicit quantum field theory construction [@Mintchev:2011mx] of the Landauer-Büttiker (LB) state $\Omega_{\beta, \mu}$, which adapts to the star graph case some general ideas [@els-96]-[@st-06] about NESS’s. This construction is very convenient for deriving correlation functions and for studying the orbit $O_{\beta, \mu}$ of $\Omega_{\beta, \mu}$ under parity and time reversal transformations. As mentioned in the introduction, the elements of this orbit represent new nonequivalent NESS’s with interesting physical properties. The first step in constructing the LB state $\Omega_{\beta, \mu}$ is to describe the asymptotic dynamics at $t=-\infty$ (i.e. before the interaction) in terms of the operators $$\{a^*_i(k),\, a_i(k)\, :\, k >0, \, i=1,...,n\}\, , \label{in}$$ which create and annihilate the particle excitations with momentum $k$ in the reservoir $R_i$. In agreement with the orientation of the leads in Fig.1, the condition $k>0$ implies that (\[in\]) create and annihilate incoming particles. For fermionic systems (\[in\]) generate therefore an incoming Canonical Anti-commutation Relations (CAR) algebra $\A^\inc$. We denote by $\Omega_{\beta_i,\mu_i}$ be the GS associated with the heat reservoir $R_i$ (see e.g. [@BR]) and consider the tensor product $\otimes_{i=1}^n \Omega_{\beta_i,\mu_i}$. The next step is to relate $\A^\inc$ with the CAR algebra of outgoing excitations $\A^\out$, generated still by the creation and annihilation operators (\[in\]), but with $k<0$. For this purpose we have to introduce the interaction, connecting different heat reservoirs and driving the system away from equilibrium. We consider the simple case where the incoming particles propagate freely along the leads towards the vertex of the graph, where they are reflected or transmitted with some probability in the rest of the graph. This process is codified in the reflection-transmission equations $$a_i(k) = \sum_{j=1}^n \S_{ij} (k) a_j (-k) \, , \qquad a^\ast_i (k) = \sum_{j=1}^n a^\ast_ j(-k) \S^*_{ji} (k) \qquad k<0 \, , \label{constr1}$$ which relate $\A^\inc$ and $\A^\out$. Here $\S(k)$ is the scattering matrix describing the point-like interaction in the vertex of the graph. We assume unitarity and Hermitian analyticity $$\S(k) \S(k)^* = {\mbox{${\mathbb I}$}}\, , \qquad \S(k)^*=\S(-k) \, , \label{unitha}$$ the star ${}^*$ indicating Hermitian conjugation. These conditions imply that $\S(k) \S(-k) = {\mbox{${\mathbb I}$}}$, which ensures the consistency of the constraints (\[constr1\]). The whole algebra $\A$, generated by polynomials involving generators of both $\A^\inc$ and $\A^\out$, is a deformed[^1] CAR algebra, where $[a_i(k)\, ,\, a_j(p)]_+ = [a^*_i (k)\, ,\, a^*_j (p)]_+ = 0$ and $$[a_i(k)\, ,\, a^*_j (p)]_+ = 2\pi [\delta (k-p)\delta_{ij} + \S_{ij}(k)\delta(k+p)] \, . \label{rta2}$$ Because of (\[unitha\]), the right hand side of (\[rta2\]) defines the kernel of an integral projection (instead of the usual identity) operator. Hermitian analyticity implies [@Liguori:1996xr] that the $*$-operation in $\A$ is a conjugation. At this point the LB state $\Omega_{\beta, \mu}$ is the extension of $\otimes_{i=1}^n \Omega_{\beta_i,\mu_i}$ from $\A^\inc$ to the whole algebra $\A$, performed by linearity via the reflection-transmission relations (\[constr1\]). This construction, which may look at the first sight a bit abstract, is in fact very efficient for deriving the correlation functions defining the LB representation $\H_{\rm LB}$ of $\A$. We stress that $\H_{\rm LB}$ describes non-equilibrium physics and is not equivalent to the Fock and Gibbs representations $\H_{\rm F}$ and $\H_{\rm G}$ of $\A$, known [@Liguori:1996xr; @Mintchev:2004jy] from the equilibrium case. Denoting by $(\cdot\, ,\, \cdot)$ the scalar product in the Hilbert space $\H_{\rm LB}$, one has [@Mintchev:2011mx] $$\begin{aligned} (\Omega_{\beta, \mu}\, ,\, a_j^*(p)a_i(k) \Omega_{\beta, \mu}) \equiv \langle a_j^*(p)a_i(k)\rangle_{\beta, \mu} = \qquad \qquad \qquad \nonumber \\ 2\pi \delta (k-p)\left [\theta(k)\delta_{ij}d_i(k) + \theta(-k)\sum_{l=1}^n \S_{il}(k)\, d_l(-k)\, \S^*_{lj}(k)\right ] \qquad \nonumber \\ +2\pi \delta (k+p)\left [\theta(k) d_i(k) \S^*_{ij}(-k) + \theta(-k)\S_{ij}(k) d_j(-k) \right ] \, , \qquad \; \; \, \label{cor1}\end{aligned}$$ were $$d_i(k) = \frac{\e^{-\beta_i \left [\omega_i (k) -\mu_i\right ]}} {1+ \e^{-\beta_i \left [\omega_i (k) -\mu_i \right ]}} \label{fbd1}$$ is the Fermi distribution in the heat reservoir $R_i$ with dispersion relation $\omega_i(k)$. Notice that the construction allows for different dispersion relations in the different reservoirs. The explicit form of $\langle a_i(k)a_j^*(p)\rangle_{\beta, \mu}$ is obtained from (\[cor1\]) by the substitution $$d_i(k) \longmapsto c_i(k) = \frac{1}{1+ \e^{-\beta_i \left [\omega_i (k) -\mu_i\right ]}} \, . \label{fbd2}$$ As well known, employing the CAR algebra, one can express a generic $n$-point correlation function as a polynomial of the two-point correlator (\[cor1\]). The non-equilibrium features of the correlation functions are encoded in the mixing of the Fermi distributions, associated with the different heat reservoirs $R_i$, via the scattering matrix $\S$, which is evident already from (\[cor1\]). An advantage of the above framework is that it allows to investigate directly the behavior of the LB state $\Omega_{\beta, \mu}$ under parity and time reversal. We first recall that these operations are implemented by a unitary operator $P$ and an anti-unitary operator $T$, acting in the algebra $\A$ in the standard way: $$P a_i(k) P^{-1} = \chi_P\, a_i(-k)\, ,\qquad Ta^*_i(k) T^{-1} = \chi_T\, a_i(-k)\, , \qquad |\chi_P|=|\chi_T|=1\, . \label{pt}$$ Apart from the multiplicative phase factors, the action of $P$ and $T$ seems to be the same, but one should remember that $P$ is a linear operator, whereas $T$ is anti-linear. Using the explicit form (\[cor1\]) of the two-point function and (\[pt\]), one can explicitly verify that $$\left (X\Omega_{\beta , \mu}\, ,\, a_j^*(p) a_i(k) X\Omega_{\beta , \mu} \right ) \equiv \langle a_j^*(p) a_i(k) \rangle^X_{\beta, \mu} \not= \langle a_j^*(p)a_i(k)\rangle_{\beta, \mu}\, , \qquad X= P,\, T,\, PT\, , \label{xcorr}$$ if $(\beta_i,\mu_i)\not=(\beta_j,\mu_j)$ for some reservoirs. Therefore, $\Omega_{\beta, \mu}$ is not invariant under parity and time reversal and generates the nontrivial orbit $$O_{\beta,\mu}=\{\Omega_{\beta, \mu},\, \Omega^X_{\beta, \mu} = X \Omega_{\beta, \mu}\, :\, X= P,\, T,\, PT\}\, . \label{orbit}$$ The property (\[xcorr\]) captures the fundamental difference of the LB state $\Omega_{\beta, \mu}$ with respect to the Fock vacuum $\Omega_{\rm F}\in \H_{\rm F}$ and the Gibbs state $\Omega_{\rm G}\in \H_{\rm G}$, which are both $P$- and $T$-invariant. The result (\[xcorr\]) suggests also that the four states in the orbit $O_{\beta,\mu}$ have different transport properties, which is confirmed by the analysis in section 5.2 below. Since time reversal symmetry is the quantum counterpart of classical reversibility, one can interpret the breakdown of this symmetry in the LB representation $\H_{\rm LB}$ as quantum irreversibility. We conclude by observing that above construction can be easily generalized to bosons by replacing the CAR algebra with a Canonical Commutation Algebra (CCA), which implies the substitution of the Fermi distribution in (\[cor1\]) with the Bose distribution: $$d_i(k) \longmapsto b_i(k) = \frac{\e^{-\beta_i \left [\omega_i (k) -\mu_i\right ]}}{1- \e^{-\beta_i \left [\omega_i (k) -\mu_i\right ]}}\, . \label{bd2}$$ As one can expect on general grounds and as shown in section 7, the quantum transport is influenced by the statistics. The Schrödinger junction ======================== Quantum systems away from equilibrium behave usually in a complicated way. In most of the cases the linear response or other approximations are not enough for fully describing the complexity of this behavior. For this reason the existence of models which incorporate the main non-equilibrium features, while being sufficiently simple to be analyzed exactly, is conceptually very important. An interesting family of such models in $s$ space dimensions is characterized by requiring that the interaction, which drives the system away from equilibrium, is localized on a $(s-1)$-dimensional sub-manifold, whereas the propagation in the complementary orthogonal direction is free. In what follows we will consider a special case of this scenario, focussing on a one-dimensional space with the geometry of a star graph $\Gamma$. Each point $P\in \Gamma$ is parametrized by the coordinates $\{(x,i)\, ,:\, x<0,\, i=1,...,N\}$, where $|x|$ is the distance of $P$ from the vertex and $i$ labels the lead. Since $s=1$, the interaction is implemented by a point-like defect in the vertex of $\Gamma$ and the propagation along the leads $L_i$ is free. We assume in this paper that it is governed by the Schrödinger equation $$\left (\ri {\partial}_t +\frac{1}{2m} {\partial}_x^2\right )\psi (t,x,i) = 0\, , \label{eqm1}$$ but other types of time evolution can be considered [@Mintchev:2011mx; @Mintchev:2012pe] in the same way. The field $\psi$ is complex and the system has a global $U(1)$-invariance, generating particle number conservation. In the fermionic case $\psi$ satisfies the standard equal-time CAR’s. The scattering matrix $\S$ in the vertex is fixed by requiring that the bulk Hamiltonian $-{\partial}_x^2$ admits a self-adjoint extension on the whole graph. These extensions are defined [@ks-00]-[@k-08] by $$\lim_{x\to 0^-}\sum_{j=1}^n \left [\lambda ({\mbox{${\mathbb I}$}}-\UU)_{ij} +\ri ({\mbox{${\mathbb I}$}}+\UU)_{ij}{\partial}_x \right ] \psi (t,x,j) = 0\, , \label{bc1}$$ where $\UU$ is a $n\times n$ unitary matrix and $\lambda \in {\mbox{${\mathbb R}$}}$ is a parameter with dimension of mass. Eq. (\[bc1\]) guaranties unitary time evolution of the system on the graph. The matrices $\UU={\mbox{${\mathbb I}$}}$ and $\UU=-{\mbox{${\mathbb I}$}}$ define the Neumann and Dirichlet boundary conditions respectively. The explicit form of the scattering matrix is [@ks-00]-[@k-08] $$\S(k) = -\frac{[\lambda ({\mbox{${\mathbb I}$}}- \UU) - k({\mbox{${\mathbb I}$}}+\UU )]}{[\lambda ({\mbox{${\mathbb I}$}}- \UU) + k({\mbox{${\mathbb I}$}}+\UU)]} \, . \label{S1}$$ The diagonal element $\S_{ii}(k)$ represents the reflection amplitude from the vertex on the lead $L_i$, whereas $\S_{ij}(k)$ with $i\not=j$ equals the transmission amplitude from $L_i$ to $L_j$. One easily verifies that (\[S1\]) satisfies (\[unitha\]) and therefore defines an algebra $\A$ of the type introduced in the previous section. Moreover, $\S(k)$ is a meromorphic function in the complex $k$-plane with finite number of simple poles on the imaginary axis. For simplicity we consider in this paper the case without bound states (poles in the upper half plane), referring for the general case to [@Mintchev:2004jy], [@Bellazzini:2010gs]. In this case the solution of equation (\[eqm1\]) is fixed uniquely by (\[bc1\]) and takes the following simple form $$\psi (t,x,i) = \sum_{j=1}^n \int_{0}^{\infty} \frac{dk}{2\pi } \e^{-\ri \omega (k)t}\left [\e^{-\ri kx}\, \delta_{ij} + \e^{\ri kx}\, \S_{ij}(-k)\right ] a_j (k) \, , \qquad \omega(k) = \frac {k^2}{2m} \, . \label{psi1}$$ As expected, the parity and time-reversal transformations (\[pt\]) in the algebra $\A$ imply $$P \psi (t,x,i) P^{-1} = \chi_P \psi (t,-x,i)\, , \qquad T \psi (t,x,i) T^{-1} = \chi_T \psi (-t,x,i)\, . \label{pt1}$$ Let us describe now the basic local observables, whose behavior away from equilibrium is the main topic of this paper. The local particle density and relative current are $$j_t (t,x,i)= \left [ \psi^* \psi \right ] (t,x,i)\, , \qquad j_x(t,x,i)= \frac{\ri }{2m} \left [ \psi^* (\partial_x\psi ) - (\partial_x\psi^*)\psi \right ] (t,x,i) \, , \label{curr1}$$ The total energy density is $$\theta_{tt} (t,x,i) = -\frac{1}{4m} \left[ \psi^* \left (\partial_x^2 \psi \right )+ \left (\partial_x^2 \psi^* \right )\psi \right] (t,x,i) \, , \label{endens1}$$ with energy flow $$\theta_{xt} (t,x,i) = \frac{1}{4m} [\left (\partial_t \psi^* \right )\left (\partial_x \psi \right ) + \left (\partial_x \psi^* \right )\left (\partial_t \psi \right ) \\ - \left (\partial_t \partial_x \psi^* \right ) \psi - \psi^*\left (\partial_t \partial_x \psi \right ) ](t,x,i) \, . \label{en1}$$ The equations of motion lead to the local conservation laws $$\left (\partial_t j_t - \partial_x j_x \right )(t,x,i) = \left (\partial_t \theta_{tt} - \partial_x \theta_{xt} \right )(t,x,i) = 0 \, . \label{cons1}$$ The relation between local conservation laws and the associated charges on a star graph has been investigated in [@Bellazzini:2008mn]. In the presence of a defect, like the junction in our case, the local conservation (\[cons1\]) alone is not enough [@Bellazzini:2008mn] to ensure the conservation of the relative quantum numbers. One needs in addition the Kirchhoff rules $$\sum_{i=1}^n j_x(t,0,i) = 0\, , \qquad \sum_{i=1}^n \theta_{xt}(t,0,i) = 0 \, . \label{KK1}$$ It is worth stressing that the explicit form (\[S1\]) of the $\S$-matrix is fundamental for proving [@Mintchev:2011mx] the operator form (\[KK1\]) of the Kirchhoff rules, which guaranties the charge conservation in the whole state space $\H_{\rm LB}$ of the system. Combining (\[cons1\]) and (\[KK1\]) one concludes that for these $\S$-matrices the particle number and the total energy in our system are conserved. The heat density $q_t$ in the lead $L_i$ is obtained (see e.g. [@call]) by subtracting from the total energy density the energy density relative to the chemical potential $\mu_i$, namely $$q_t(t,x,i) = \theta_{tt} (t,x,i) - \mu_i j_t (t,x,i) \, . \label{hd}$$ Accordingly, the heat current is $$q_x(t,x,i) = \theta_{xt} (t,x,i) - \mu_i j_x (t,x,i) \, . \label{hc}$$ Local heat conservation $$\left (\partial_t q_t - \partial_x q_x \right )(t,x,i) = 0 \, , \label{cons2}$$ is a direct consequence of (\[cons1\]), but in general the relative Kirchhoff rule is not at all satisfied. In fact, a key observation is that the heat current obeys the operator Kirchhoff rule if and only if $\mu_i = \mu_j$ for all $i,j=1,...,N$. Otherwise, the heat current violates the Kirchhoff rule and the heat energy is therefore not conserved. The chemical potential energy shares the same property, because the total energy is conserved. Summarizing, if $\mu_i \not= \mu_j$ for some $i$ and $j$, the system converts heat energy in chemical potential energy or vice versa. This is the basic physical process which takes place in the junction. In what follows we will study in detail this phenomenon of energy transmutation. Transport in the state $\Omega_{\beta,\mu}$ =========================================== Currents and efficiency ----------------------- In order to study the non-equilibrium features of our system, we derive now the expectation values in the state $\Omega_{\beta,\mu}$ of the charge densities and currents introduced above. Since these observables are quadratic, the basic input is the non-equilibrium two-point correlation function $$\begin{aligned} \langle \psi^*(t_1,x_1,i) \psi (t_2,x_2,j)\rangle_{\beta, \mu} = \int_0^{\infty} \frac{{{\rm d}}k}{2\pi} \e^{\ri \omega(k) t_{12}} \Bigl [\delta_{ji} d_i(k) \e^{\ri k x_{12}} + \quad \nonumber \\ d_j(k) \S_{ji}(k) \e^{-\ri k {\widetilde{x}}_{12}} + \S^\ast_{ji}(k) d_i(k) \e^{\ri k {\widetilde{x}}_{12}} + \sum_{l=1}^n \S^\ast_{jl}(k) d_l(k) \S_{li}(k) \e^{-\ri k x_{12}} \Bigr ] , \label{corr11}\end{aligned}$$ following from (\[cor1\],\[psi1\]). Here $t_{12}=t_1-t_2$, $x_{12}=x_1-x_2$ and ${\widetilde{x}}_{12}=x_1+x_2$. The invariance of (\[corr11\]) under time translations explicitly confirms that the total energy of the system is conserved. Combining (\[corr11\]) with (\[curr1\],\[en1\]) and performing the limits $t_1 \to t_2=t$ and $x_1 \to x_2 =x$ one gets the following current expectation values $$J_i^N \equiv \langle j_x(t,x,i) \rangle_{\beta, \mu} = \sum_{j=1}^n \int_0^{\infty} \frac{{{\rm d}}k}{2\pi} \frac{k}{m} \left [\delta_{ij} - |\S_{ij}(k)|^2 \right ] d_j(k)\, , \label{ccurr1}$$ $$J_i^E \equiv \langle \theta_{tx}(t,x,i) \rangle_{\beta, \mu} = \sum_{j=1}^n \int_0^{\infty} \frac{{{\rm d}}k}{2\pi} \frac{k}{m} \left [\delta_{ij} - |\S_{ij}(k)|^2 \right ] \omega(k) d_j (k)\, . \label{ecurr1}$$ The right hand sides of (\[ccurr1\],\[ecurr1\]) are the Landauer-Büttiker expressions [@la-57; @bu-86; @si-86] for the particle and energy currents in our case specified by the $\S$ matrix (\[S1\]). This is the main reason for referring to $\Omega_{\beta,\mu}$ as the LB state. Notice that for $\beta_1=\cdots =\beta_n$ and $\mu_1=\cdots =\mu_n$ the system is in equilibrium and the currents (\[ccurr1\], \[ecurr1\]) vanish due to the unitarity of $\S$. From (\[hc\]) one gets the heat current $$J_i^Q = J_i^E -\mu_i J_i^N\, . \label{hc1}$$ As already observed, in general the heat and chemical potential currents $J_i^Q$ and $\mu_i J_i^N$ do not satisfy separately the Kirchhoff rule. In fact, the heat flow $\dot Q$ from the junction is given by $${\dot Q} + \sum_{i=1}^n J_i^Q = 0\, . \label{hf0}$$ Taking into account the Kirchhoff rule (\[KK1\]) for the energy current one gets $${\dot Q} = -\sum_{i=1}^n J_i^Q = \sum_{i=1}^n \mu_i J_i^N \, , \label{hf1}$$ If ${\dot Q} <0$ the junction transforms heat energy in chemical potential energy. The efficiency of this process can be characterized as follows. Let us denote by ${{\mathcal K}}_\out$ the subset of heat currents leaving the reservoirs $R_i$. With our choice for the orientation of the leads (see Fig. 1), these currents are positive. With this convention the efficiency of the junction to transform heat energy in chemical potential energy is defined by $$\eta = \frac{\sum_{i=1}^n J_i^Q}{\sum_{i\in {{\mathcal K}}_\out} J_i^Q}=\frac{-{{\dot Q}}}{\sum_{i\in {{\mathcal K}}_\out} J_i^Q}\, , \label{eta1}$$ which is a direct extension of the cases $n=2,3$ [@bcps; @sns-14] to a generic $n$. Combining (\[hf1\]) with ${\dot Q} <0$ one concludes that $\sum_{i\in {{\mathcal K}}_\out} J_i^Q>0$, which implies that $\eta$ is well defined and satisfies $0<\eta\leq 1$. In the regime ${\dot Q} >0$ of converting chemical potential energy to heat energy the junction is usually compared (see e.g. [@bcps]) to a refrigerator or a heat pump, thus involving in the description the known performance coefficients of these devises. We show below that this approach works in parts of the parameter space, but can not be applied globally to the whole domain with ${{\dot Q}}>0$. In order to avoid this problem, for ${{\dot Q}}>0$ we propose and adopt below the efficiency $${\widetilde{\eta}}= \frac{\sum_{i=1}^n \mu_iJ_i^N}{\sum_{i\in \L_\out} \mu_iJ_i^N} =\frac{{{\dot Q}}}{\sum_{i\in \L_\out} \mu_iJ_i^N}\, , \label{etat1}$$ where now the sum in the denumerator runs over the subset $\L_\out$ of outgoing (positive) chemical potential energy currents $\mu_iJ_i^N$. The general form of (\[etat1\]) is analogous to that of (\[eta1\]), but refers to the chemical potential energy. By construction $0<{\widetilde{\eta}}\leq 1$ in this case as well. For ${{\dot Q}}=0$ there is no energy transmutation. We conclude this subsection by observing that the heat flow in the system generates the following entropy production [@call] $${{\dot S}}= -\sum_{i=1}^n \beta_i J_i^Q \, . \label{entr1}$$ Employing (\[ccurr1\],\[ecurr1\]), ${{\dot S}}$ can be written in the form $${{\dot S}}= \sum_{i,j=1}^n \int_0^{\infty} \frac{{{\rm d}}k}{2\pi} \frac{k}{m} |\S_{ij}(k)|^2 \left [\sigma_i(k) -\sigma_j(k)\right ]d_j(k)\, , \label{nentr1}$$ with $$\sigma_i(k) = \beta_i\left [\omega(k) -\mu_i\right ] \, . \label{nentr2}$$ Except in section 5.3, we will always assume in this paper that the bulk theory and the heat reservoirs have the same dispersion relation, namely $$\omega_i(k) = \omega(k) \, . \label{nentr3}$$ With this assumption $$d_j(k) = \frac{1}{\e^{\sigma_j(k)}+1} \, . \label{nentr4}$$ Now, using that (\[nentr4\]) is a strictly decreasing function of $\sigma_j$, one can prove [@gn] that the integrand of (\[nentr1\]) is nonnegative[^2], implying the second law of thermodynamics ${{\dot S}}\geq 0$ in the LB state for all $(\beta_i,\mu_i)$ and scattering matrices (\[S1\]). As explained in section 5.2 below, this argument does not apply to the other states of the orbit $O_{\beta,\mu}$ and the entropy production there behaves indeed differently. Summarizing, the process of energy transmutation in the junction is controlled by the two parameters ${{\dot Q}}$ and ${{\dot S}}$ and is characterized by the efficiency coefficients $\eta$ and ${\widetilde{\eta}}$. We derive in what follows the explicit form of these physical quantities, assuming that the interaction at the junction is scale invariant. The scale invariant junction ---------------------------- The great advantage of considering the non-equilibrium Schrödinger junction defined in section 3 is that it provides both exact and explicit results with nontrivial transport. In order to keep this property also after the integration in $k$, we select in what follows the scale-invariant elements among the scattering matrices (\[S1\]). They preserve the basic features of the system, while being simple enough to allow the explicit $k$-integration in (\[ccurr1\],\[ecurr1\]). The requirement of scale invariance implies that the interaction is $k$-independent and leads to [@Calabrese:2011ru] $$\S = {{\cal U}}\, \S_d \, {{\cal U}}^*\, , \qquad {{\cal U}}\in U(n)\, , \qquad \S_d = {\rm diag}(\pm 1,\pm 1,...,\pm 1)\, . \label{sinvS}$$ The family (\[sinvS\]) is the orbit of the diagonal matrix $\S_d$ under the adjoint action of $U(n)$. We can always enumerate the leads $L_i$ in such a way that the first $i$ eigenvalues of $\S$ are $+1$ and the remaining $n-i$ are $-1$. For $i=n$ and $i=0$ one gets $\S={\mbox{${\mathbb I}$}}$ and $\S=-{\mbox{${\mathbb I}$}}$, which imply vanishing transport because the leads $L_i$ are disconnected. One has a nontrivial transport for $0<i<n$. The scattering matrices of the type (\[sinvS\]) are in general not symmetric. One can easily prove however that the relative amplitudes are symmetric, namely $$|\S_{ij}|^2 = |\S_{ji}|^2 \, . \label{symS}$$ This property of the critical points (\[sinvS\]) simplifies considerably the study of junctions with $n\geq 3$ leads. The $k$-integration in (\[ccurr1\],\[ecurr1\]) with the constant scattering matrices (\[sinvS\]) can be performed explicitly. One finds $$J^N_i = \frac{1}{2\pi} \sum_{j=1}^n \left [\delta_{ij} - |\S_{ij}|^2 \right ] \frac{1}{\beta_j} \ln \left (1+\e^{\beta_j\mu_j} \right ) \, , \label{ccurr2}$$ $$J^E_i = -\frac{1}{2\pi} \sum_{j=1}^n \left [\delta_{ij} - |\S_{ij}|^2 \right ] \frac{1}{\beta_j^2} \li_2 \left (-\e^{\beta_j\mu_j} \right )\, , \label{ecurr2}$$ where $\li_s$ is the polylogarithm function. These expressions are the building blocks for the parameters ${{\dot Q}}$ and ${{\dot S}}$ and the efficiency coefficients $\eta$ and ${\widetilde{\eta}}$. They depend on $(\beta_i,\mu_i)$ and $|\S_{ij}|^2$, which (due to the unitarity of $\S$) leads to $n(n+3)/2$ independent real parameters. We observe in particular that the currents depend on $\mu_i$ separately and not on differences $\mu_i-\mu_j$. In order to reduce this large number it is instructive to start by considering the system with two leads shown in Fig. \[junction2\]. (700,40)(80,355) ![Two heat reservoirs $(\beta_1,\mu_1)$ and $(\beta_2,\mu_2)$ connected by $\S$.[]{data-label="junction2"}](Fig-2.eps "fig:") The case of two leads ===================== The LB state ------------ In the case $n=2$ there is only one transmission probability $|\S_{12}|^2=|\S_{21}|^2$ and the currents (\[ccurr2\],\[ecurr2\]) take the form $$J^N_1 = -J^N_2= \frac{|\S_{12}|^2}{2\pi \beta_1} \left [\ln \left (1+\e^{-\lambda_1} \right ) - r \ln \left (1+\e^{-\lambda_2} \right )\right ]\, , \label{ccurr4}$$ $$J^E_1 = -J^E_2 = -\frac{|\S_{12}|^2}{2\pi \beta_1^2} \left [\li_2 \left (-\e^{-\lambda_1} \right ) - r^2 \li_2 \left (-\e^{-\lambda_2} \right ) \right ] \, , \label{ecurr4}$$ where $$r=\frac{\beta_1}{\beta_2}\, , \qquad \lambda_i = -\beta_i \mu_i \, , \quad i=1,2\, . \label{not1}$$ The factorization of $|\S_{12}|^2$ implies that the signs of ${{\dot Q}}$ and ${{\dot S}}$ depend exclusively on the parameters of the two heat reservoirs $R_{1,2}$, which greatly simplifies the study of the energy transmutation in the junction. We focus first on ${{\dot Q}}$ denoting by ${{\cal D}}_\mp$ the domains in the space of parameters $(\beta_i, \mu_i)$ where ${{\dot Q}}\lessgtr 0$ respectively. Without loss of generality one can assume $0\leq r \leq 1$. The direct investigation of $$\dot Q (\lambda_1,\lambda_2,r) = \frac{|\S_{12}|^2}{2\pi \beta^2_1} (\lambda_1 - r \lambda_2) \left [r \ln \left (1+\e^{-\lambda_2} \right )-\ln \left (1+\e^{-\lambda_1} \right )\right ] \, , \label{hf2}$$ shows that $${{\cal D}}_- = D_1 \cup D_2 \cup D_3 \, , \label{dom1}$$ with $$\begin{aligned} D_1 &=& \{0<\lambda_1 \leq \lambda_2\, ,\, 0\leq r < r_1\}\, , \label{domains1} \\ D_2 &=& \{\lambda_1 > \lambda_2\, ,\, \lambda_1>0 \, ,\, 0\leq r < r_2\}\, , \label{domains2} \\ D_3 &=& \{0\geq \lambda_1 > \lambda_2 \, ,\, r_1< r < r_2\}\, , \label{domains3}\end{aligned}$$ where $$r_1 = \frac{\lambda_1}{\lambda_2}\, , \qquad r_2 = \frac{\ln \left (1+\e^{-\lambda_1} \right )}{\ln \left (1+\e^{-\lambda_2} \right )}\, . \label{r12}$$ The domain ${{\cal D}}_-$ has a complicated structure due to the dependence of $r_{1,2}$ on $\lambda_i$. We are ready at this point to compute the efficiency $\eta$. Using that $J^Q_1>0$ and $J_2^Q<0$ in ${{\cal D}}_-$, one gets from (\[eta1\]) $$\eta(\lambda_1,\lambda_2;r) = \frac{(\lambda_1-r\lambda_2) \left [\ln \left (1+\e^{-\lambda_1} \right ) - r \ln \left (1+\e^{-\lambda_2} \right )\right ]} {\lambda_1\left [\ln \left (1+\e^{-\lambda_1} \right ) - r \ln \left (1+\e^{-\lambda_2} \right )\right ]-\left [\li_2 \left (-\e^{-\lambda_1} \right ) - r^2 \li_2 \left (-\e^{-\lambda_2} \right ) \right ]}\, . \label{eta2}$$ This is an exact and explicit result for the efficiency of the Schrödinger junction in transforming heat to chemical potential energy at criticality. The expression (\[eta2\]) shows the power of scale invariance and makes evident the advantage of the above approach with respect to the linear response approximation, which gives information about (\[eta2\]) only in the neighborhood of $\lambda_1 \sim \lambda_2$ and $r\sim 1$. We demonstrate in appendix A that in this neighborhood the efficiency (\[eta2\]) reproduces exactly the result of the linear response theory in [@bcps]. The analysis of (\[eta2\]) in ${{\cal D}}_-$ shows that the maximal efficiency is obtained in the limit $\lambda_1=\lambda_2\equiv \lambda \to +\infty$. In fact one has $$\eta_{\rm max}(r) = \lim_{\lambda \to +\infty} \eta (\lambda,\lambda; r) = 1-r \equiv \eta_C\, , \label{eta3}$$ which is the well known Carnot efficiency. According to (\[hf2\]), in this limit the heat energy conversion vanishes $\lim_{\lambda \to \infty} {{\dot Q}}(\lambda,\lambda,r)=0$. (260,100)(80,25) ![The efficiency $\eta$ compared to $\eta_C$ (left) and the heat $-{{\dot Q}}$ converted to chemical energy (right) with $\lambda=1$ (dotted), $\lambda=3$ (dashed) and $\lambda=5$ (continuous) .[]{data-label="plot1"}](Fig-3a.eps "fig:") ![The efficiency $\eta$ compared to $\eta_C$ (left) and the heat $-{{\dot Q}}$ converted to chemical energy (right) with $\lambda=1$ (dotted), $\lambda=3$ (dashed) and $\lambda=5$ (continuous) .[]{data-label="plot1"}](Fig-3b.eps "fig:") Fig. \[plot1\] illustrates the behavior of $\eta$ and the heat converted to chemical energy $-{{\dot Q}}$ for some values of the parameter $\lambda\equiv \lambda_1=\lambda_2$. The plots in this figure show that the chemical energy production decreases with increasing the efficiency. (80,100)(80,25) ![The efficiency $\eta^*$ and the Curzon-Ahlborn bound $\eta_{CA}$.[]{data-label="plot2"}](Fig-4.eps "fig:") Another physically interesting regime is obtained by maximizing the chemical energy production $-{{\dot Q}}(\lambda_1,\lambda_2,r)$ with respect to $\lambda_i$. A simple analysis shows that this function reaches its maximum at $\lambda_1=\lambda_2 \equiv \lambda^*$, where $$\lambda^* - (1+\e^{\lambda^*}) \ln (1+\e^{-\lambda^*}) =0\, . \label{sol1}$$ The solution of (\[sol1\]) is $\lambda^*=1.14455...$ and the efficiency (\[eta2\]) takes the form $$\eta^*(r) \equiv \eta(\lambda^*,\lambda^*;r) = \frac{(1-r) \lambda^* \ln \left (1+\e^{-\lambda^*} \right )} {\lambda^* \ln \left (1+\e^{-\lambda^*} \right ) - (1+r) \li_2 \left (-\e^{-\lambda^*} \right )}\, . \label{eta4}$$ The quantity $\eta^*(r)$ is the counterpart of the concept of efficiency at maximal power, used in the context of heat engines. The plot in Fig. 4 shows that (\[eta4\]) satisfies the Curzon-Ahlborn bound $$\eta^*(r)\leq 1-\sqrt r \equiv \eta_{CA}\, , \label{CAb}$$ for all $r\in [0,1]$, which was known previously from linear response theory only for $r\sim 1$. We recall that this bound has been proposed for heat engines in the framework of endoreversible thermodynamics in [@ca]. The rigorous proof [@vb] covers the linear response regime, but it is known [@tu-08; @bl-12; @sb-12] that away of this regime the bound is not universal and can be violated. The possibility to enhance $\eta^*$ above $\eta_{CA}$ in our context is discussed in section 5.3. We turn now to the case when chemical energy is transformed in heat, namely $${{\dot Q}}= \mu_1 J_1^N + \mu_2 J_2^N = (\mu_1-\mu_2)J_1^N >0\, . \label{ch1}$$ It is convenient to use in the domain ${{\cal D}}_+$ the coordinates $(\beta_i,\mu_i)$. Keeping $\beta_i$ arbitrary, one can assume without loss of generality that $\mu_1>\mu_2$ and set ${{\cal D}}_+=\{\beta_1,\beta_2,\mu_1>\mu_2\}$. From (\[ch1\]) and the Kirchhoff rule one infers that $J_1^N > 0 > J_2^N$ on ${{\cal D}}_+$. Using this information one finds that the efficiency ${\widetilde{\eta}}$, defined by (\[etat1\]), can be expressed in terms of the parameter $u\equiv \mu_2/\mu_1$ in the simple form $${\widetilde{\eta}}(u) = \begin{cases} 1-u\, , & \qquad \mu_1 > \mu_2 \geq 0\, , \\ 1\, , & \qquad \mu_1 \geq 0 > \mu_2\, , \\ 1-1/u\, , & \qquad 0>\mu_1 > \mu_2\, , \\ \end{cases} \qquad \quad u=\frac{\mu_2}{\mu_1}\, , \label{etat2}$$ fully covering the domain ${{\cal D}}_+$. The formula (\[etat2\]) describes in exact form the conversion of chemical potential energy in heat. For $n=2$ the efficiency ${\widetilde{\eta}}$ does not depend on the temperatures and the explicit form of the currents $\mu_i J_i^N$, but only on the values of the chemical potentials $\mu_i$. The first line of (\[etat2\]) resembles the Carnot formula, where the temperature ratio $r$ is substituted by the chemical potential ratio $u$. The third line instead takes into account that differently from the temperatures, the chemical potentials can take also negative values. Finally, we observe that for $\mu_1 > 0 > \mu_2$ both chemical potential currents $\mu_i J_i^N$ are flowing towards the junction, transforming the chemical energy in heat completely (${\widetilde{\eta}}=1$). Since in this case the domains of $\mu_i$ are separated by the point $\mu=0$, this regime of energy transmutation in the junction can not be reached by linear response theory. (80,100)(80,25) ![The flow ${{\dot Q}}>0$ and the currents $J_1^Q$ and $J_2^Q$ for $\lambda_1=-1.5$ and $\lambda_2=-3$.[]{data-label="fig5"}](Fig-5.eps "fig:") We have seen that for ${{\dot Q}}<0$ the junction can be compared to a heat engine. In analogy, one can attempt to interpret the system for ${{\dot Q}}>0$ as a refrigerator or a heat pump. A careful analysis shows that this possible only partially because there are subsets in ${{\cal D}}_+$ where both heat currents have the same sign, i.e. they are both leaving or entering the heat reservoirs. A typical situation is shown in Fig. \[fig5\], where $J_1^Q$ and $J_2^Q$ become both negative (i.e. entering the heat reservoirs) for $r>0.75$, which is not the case of conventional refrigerators and heat pumps. For this reason the standard coefficients of performance for a refrigerator and a heat pump can not be applied in the whole domain ${{\cal D}}_+$. As already mentioned, this is our main motivation to introduce the efficiency ${\widetilde{\eta}}$ by (\[etat1\]), which has the advantage of working everywhere in ${{\cal D}}_+$. Let us consider finally the entropy production. Substituting (\[ccurr4\],\[ecurr4\]) in (\[entr1\]) one obtains $$\begin{aligned} {{\dot S}}= \frac{|\S_{12}|^2}{2\pi r\beta_1} \Bigl \{(1-r) \left [ r^2 \li_2 \left (-\e^{-\lambda_2} \right ) - \li_2 \left (-\e^{-\lambda_1} \right ) \right ] \nonumber \\ +\, r (\lambda_1-\lambda_2) \left [r \ln \left (1+\e^{-\lambda_2} \right )-\ln \left (1+\e^{-\lambda_1} \right ) \right ]\Bigr \} > 0\, , \label{entr2}\end{aligned}$$ confirming the general statement [@gn] about the entropy production in the LB state. Transport in the orbit $O_{\beta,\mu}$ -------------------------------------- We consider here the quantum transport in the states $\{\Omega^X_{\beta,\mu}\, :\, X=P,T,PT\}\subset \H_{\rm LB}$. Using (\[pt1\]), the current expectation values in these states are simply expressed in terms of (\[ccurr1\],\[ecurr1\]) as follows: $$\begin{aligned} P\, &:&\, J_i^N \longmapsto -J_i^N\, \qquad J_i^E \longmapsto -J_i^E\, , \label{P}\\ T\, &:&\, J_i^N \longmapsto J_i^N\, \qquad \; \; \; J_i^E \longmapsto -J_i^E\, , \label{T}\\ P\,T\, &:&\, J_i^N \longmapsto -J_i^N\, \qquad J_i^E \longmapsto J_i^E\, . \label{PT}\end{aligned}$$ Let us observe in passing that the action of the charge conjugation $$C\, : \, J_i^N \longmapsto -J_i^N\, \qquad J_i^E \longmapsto J_i^E\, , \label{C}$$ coincides with the $PT$ operation (\[PT\]). In fact, $CPT$ on the above currents is the identity transformation. The minus sign appearing in some currents affects the quantum transport. Moreover, the entropy production ${{\dot S}}^X$ in the state $\Omega^X_{\beta,\mu}$ differs from that (\[entr2\]) in the LB state. In fact, combining (\[entr1\]) and (\[P\]) we conclude that ${{\dot S}}^P \leq 0$ for all values of the heat reservoir parameters $(\beta_i,\mu_i)$. We see that the state $\Omega^P_{\beta,\mu}$, which from the microscopic point of view is a well defined state of the system, violates the second law of thermodynamics and therefore has no admissible macroscopic behavior. The situation with the states $\Omega^T_{\beta,\mu}$ and $\Omega^{PT}_{\beta,\mu}$ is more subtle. The analysis shows that for these two states there exist domains in the whole parameter space $(\beta_i,\mu_i)$, where ${{\dot S}}^T>0$ and ${{\dot S}}^{PT}>0$. In these domains the quantum transport is consistent with the laws of thermodynamics. We consider for illustration the family of states $\{\Omega^{PT}_{\beta,\mu}\, :\, \lambda_1=\lambda_2\equiv \lambda\}$. The entropy production there is $${{\dot S}}^{PT} = -\frac{|\S_{12}|^2}{2\pi r\beta_1} (1-r)^2(1+r) \li_2 \left (-\e^{-\lambda} \right ) > 0\, . \label{entr3}$$ Moreover $${{\dot Q}}^{PT}(\lambda,\lambda,r)= \frac{|\S_{12}|^2}{2\pi \beta^2_1} (1-r)^2\lambda \ln \left (1+\e^{-\lambda} \right ) <0\, \qquad {\rm for}\quad \lambda <0\, , \label{qpt}$$ and the efficiency of transforming heat into chemical potential energy is given by $$\eta^{PT}(\lambda,r) = \frac{(1-r) \lambda \ln \left (1+\e^{-\lambda} \right )} {\lambda \ln \left (1+\e^{-\lambda} \right ) + (1+r) \li_2 \left (-\e^{-\lambda} \right )}\, , \qquad \lambda < 0\, . \label{eta5}$$ This result resembles (\[eta4\]), but for a sign in the denumerator. The relative maximum is $$\eta_{\rm max}^{PT}(r)= \lim_{\lambda \to -\infty} \eta^{PT}(\lambda,r) = \frac{2(1-r)}{r+3} \not=\eta_C \, , \label{eta6}$$ which coincides also with the efficiency at maximal chemical energy production $$\eta^{*PT}(r)= \frac{2(1-r)}{r+3} < \eta_{CA}\, . \label{eta6star}$$ These results show that the efficiency in the state $\Omega_{\beta,\mu}^{PT}$ differs from that in $\Omega_{\beta,\mu}$. We will elaborate more on the value of $\eta^{*PT}$ few lines below. Concerning the regime ${{\dot Q}}>0$, it is easy to deduce from (\[P\]-\[PT\]) that all four states in $O_{\beta,\mu}$ have the same efficiency ${\widetilde{\eta}}$ given by (\[etat2\]). Summarizing, parity and time reversal have an important impact on the quantum transport and efficiency. Indeed, we have shown that there are regions in the parameter space of the states $\Omega^T_{\beta,\mu}$ and $\Omega^{PT}_{\beta,\mu}$, where the second law of thermodynamics is satisfied and the efficiency $\eta$ has a physically acceptable value, which differs from that in the LB state $\Omega_{\beta,\mu}$. Comments about $\eta^*$ ----------------------- It is instructive to compare now (\[eta4\],\[eta6star\]) with some exact results about the efficiency at maximal power obtained for other systems. Applying stochastic thermodynamics to a simple model of classical particle transport, the following explicit expression $$\eta_{cp}^*=\dfrac{\eta_C^2}{\eta_C-(1-\eta_C)\ln(1-\eta_C)} \label{bl}$$ has been derived in [@bl-12]. The same expression has been obtained [@tu-08] for the Feynman’s ratchet model as a heat engine. For a Brownian particle undergoing a Carnot cycle it has been found [@ss-08] that $$\eta_{Bp}^*=\dfrac{2\eta_C}{4-\eta_C}\, , \label{ss}$$ which, remarkably enough, coincides with the Schrödinger junction efficiency $\eta^{*PT}$ given by (\[eta6star\]). The case of electron transport through a quantum dot has been treated in [@ebl-09; @ssj]. It has been observed in [@tu-08; @bl-12; @ssj] that away ($r<0.5$) from the linear response regime, the efficiency (\[bl\]) exceeds the Curzon-Ahlborn bound. One can wonder if a mechanism exists to enhance the efficiency (\[eta4\]) in the LB state $\Omega_{\beta,\mu}$ above $\eta_{CA}$ as well. This question attracted recently some attention [@bcps], the proposal being to couple the system with an appropriate external potential. In [@bcps] the effect of a classical magnetic field has been explored in the linear response approximation. We describe here an alternative, which simplifies our previous construction in [@mss-13] both from the technical and physical points of view. The main idea is based on the fact that in our general setting the heat reservoir dispersion relations $\omega_i$ need not to be equal and/or to coincide with the bulk dispersion relation $\omega$. So, let us perform the shift $\omega \longmapsto \omega -v$, which is equivalent to the introduction of a constant potential $V=-v$ in the bulk equation of motion (\[eqm1\]). If one performs the same shift in $\omega_i$ the efficiency will not change. One can imagine however to screen the reservoirs $R_i$ from the potential $V$, thus keeping $\omega_i = k^2/2m$ invariant. This operation does not affect the particle currents $J_i^N$ which, according to (\[ccurr1\]), depend only on $\omega_i$. From (\[ecurr1\]) one infers however that the shift in $\omega$ modifies the energy transport in the following simple way $$J_i^E \longmapsto J_i^E - v J_i^N\, . \label{shift1}$$ In the domain ${{\cal D}}_-$, which is still given by (\[dom1\]), one finds the efficiency $$\eta(\lambda_1,\lambda_2;r,a) = \frac{(\lambda_1-r\lambda_2) \left [\ln \left (1+\e^{-\lambda_1} \right ) - r \ln \left (1+\e^{-\lambda_2} \right )\right ]} {(\lambda_1-a)\left [\ln \left (1+\e^{-\lambda_1} \right ) - r \ln \left (1+\e^{-\lambda_2} \right )\right ]-\left [\li_2 \left (-\e^{-\lambda_1} \right ) - r^2 \li_2 \left (-\e^{-\lambda_2} \right ) \right ]}\, , \label{eta2a}$$ with $a\equiv \beta_1 v$ being dimensionless. The maximal efficiency is attained at $\lambda_1=\lambda_2 \to +\infty$ and equals $\eta_C$ as before. For $\eta^*$ one finds instead $$\eta^*(r,a) \equiv \eta(\lambda^*,\lambda^*;r,a) = \frac{(1-r) \lambda^* \ln \left (1+\e^{-\lambda^*} \right )} {(\lambda^*-a) \ln \left (1+\e^{-\lambda^*} \right ) - (1+r) \li_2 \left (-\e^{-\lambda^*} \right )}\, , \label{eta4a}$$ which reproduces (\[eta4\]) for $a=0$. For the moment $a$ is a free real parameter, but the condition (\[nentr3\]) is violated and the sign of ${{\dot S}}$ needs to be investigated. Imposing ${{\dot S}}>0$ for all $r\in [0,1]$, one obtains the constraint $$a< \frac{-\li_2 \left (-\e^{-\lambda^*} \right )}{\ln \left (1+\e^{-\lambda^*} \right )} = 1,07122... \equiv a^*\, . \label{enh1}$$ (80,100)(80,25) ![The efficiency $\eta^*(r,a)$ for different values of $a$ compared to $\eta_{CA}$.[]{data-label="fig.6"}](Fig-6.eps "fig:") It is easy to deduce from (\[eta4a\]) that for $a\in (0,a^*)$ the efficiency $\eta^*(r,a)$ is enhanced above $\eta^*(r,0)=\eta^*(r)$ given by (\[eta4\]). The opposite effect is observed for $a\in (-\infty ,0)$. This behavior of the efficiency is explained by the following intuitive physical argument. Since $\omega_i-\omega=a/\beta_1$, positive values of $a$ favor the particle emission from the heat reservoirs $R_i$ to the leads $L_i$, thus improving the efficiency. Negative values of $a$ are instead damping this emission, causing the opposite effect. Fig. \[fig.6\] illustrates the basic properties of $\eta^*(r,a)$. In particular, we see that for $a\sim 1$ the Curzon-Ahlborn bound is exceeded. Finally, concerning the experimental realization of the above theoretical setup, one possibility could be to use the electric field produced by a wire with constant linear static charge distribution, which is located parallel at a finite distance from the two-terminal devise in Fig. \[junction2\]. This charged wire produces a constant electric field along the devise. When the heat baths are screened by metallic boxes, the electric field provides the constant shift needed in the bulk dispersion relation. Junctions with three leads ========================== The new element in the treatment of the case with $n>2$ leads is that the transmission amplitudes $|\S_{ij}|^2$ do no longer factorize in front of the currents. For $n=3$ the transport properties depend on 9 parameters. The linear response approximation has been studied recently in [@sns-14; @ewa]. Here we pursue further the exact analysis of the scale invariant case for $$\beta_1 \mu_1 = \beta_2 \mu_2 = \beta_3 \mu_3 \equiv -\lambda \, , \qquad r=\frac{\beta_1}{\beta_2}\, , \qquad s=\frac{\beta_1}{\beta_3}\, . \label{fug}$$ Using (\[symS\]), one obtains from (\[ccurr2\],\[ecurr2\]) the heat currents $$\begin{aligned} J^Q_1 = \frac{1}{2\pi \beta^2_1} \Bigl \{ \lambda \ln \left (1+\e^{-\lambda } \right )\left [ |\S_{12}|^2 (1-r) +|\S_{13}|^2 (1-s) \right ] \nonumber \\ - \li_2 \left (-\e^{-\lambda} \right ) \left [ |\S_{12}|^2 (1-r^2) +|\S_{13}|^2 (1-s^2) \right ] \Bigr \} \, , \label{TJ1}\end{aligned}$$ $$\begin{aligned} J^Q_2 = \frac{1}{2\pi \beta^2_1} \Bigl \{ r \lambda \ln \left (1+\e^{-\lambda } \right )\left [ |\S_{23}|^2 (r-s) +|\S_{12}|^2 (r-1) \right ] \nonumber \\ - \li_2 \left (-\e^{-\lambda} \right ) \left [ |\S_{23}|^2 (r^2-s^2) +|\S_{12}|^2 (r^2-1) \right ] \Bigr \} \, , \label{TJ2}\end{aligned}$$ $$\begin{aligned} J^Q_3 = \frac{1}{2\pi \beta^2_1} \Bigl \{ s \lambda \ln \left (1+\e^{-\lambda } \right )\left [ |\S_{23}|^2 (s-r) +|\S_{13}|^2 (s-1) \right ] \nonumber \\ - \li_2 \left (-\e^{-\lambda} \right ) \left [ |\S_{23}|^2 (s^2-r^2) +|\S_{13}|^2 (s^2-1) \right ] \Bigr \} \, . \label{TJ3}\end{aligned}$$ For the entropy production and heat flow on gets $${{\dot S}}= -\frac{\li_2 \left (-\e^{-\lambda} \right )}{2\pi s r\beta_1} \left [|\S_{12}|^2 s (1+r)(1-r)^2 + |\S_{13}|^2 r (1+s)(1-s)^2 + |\S_{23}|^2(r+s)(r-s)^2 \right ] > 0\, , \label{3entr3}$$ $${{\dot Q}}= -\frac{1}{2\pi \beta^2_1} \lambda \ln \left (1+\e^{-\lambda} \right ) \left [|\S_{12}|^2 (1-r)^2 +|\S_{13}|^2 (1-s)^2 + |\S_{23}|^2 (r-s)^2 \right ] \, . \label{qpt3}$$ As expected, the entropy production in the LB state is always positive. For $\lambda > 0$ one has ${{\dot Q}}<0$ and one can study the efficiency (\[eta1\]) of transforming heat to chemical potential energy. In order to do that one has to determine first the sign of the currents (\[TJ1\]-\[TJ3\]). Without loss of generality one can assume that $$r<1\, , \qquad s<1\, ,\qquad r<s\, , \label{cond1}$$ which implies the following ordering $T_2<T_3<T_1$ among the temperatures of the heat reservoirs. For simplifying the analysis we assume to end of this section that $|\S_{13}|^2=0$, devoting the appendix B to the case of generic transmission amplitudes. Then (\[TJ1\]-\[TJ3\]) imply $J^Q_1>0,\, J^Q_2 <0,\, J^Q_3 >0$ for $\lambda >0$. With this information one obtains from (\[eta1\]) $$\begin{aligned} \eta (\lambda;r,s) = \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \nonumber\\ \frac{\lambda \ln \left (1+\e^{-\lambda} \right )\left[|\S_{12}|^2 (1-r)^2 + |\S_{23}|^2 (s-r)^2\right ]} {\lambda \ln \left (1+\e^{-\lambda} \right ) \left[|\S_{12}|^2 (1-r) + |\S_{23}|^2 s (s-r)\right ] - \li_2 \left (-\e^{-\lambda} \right )\left[|\S_{12}|^2 (1-r^2) + |\S_{23}|^2 (s^2-r^2)\right ]}\, . \nonumber \\ \label{eta7}\end{aligned}$$ The relative maximum is $$\eta_{\rm max}(r,s) = \lim_{\lambda \to +\infty} \eta (\lambda;r,s) = 1- r \frac{|\S_{12}|^2 (1-r) + |\S_{23}|^2 (s-r)}{|\S_{12}|^2 (1-r) + |\S_{23}|^2 s (s-r)}\, , \label{eta8}$$ which suggests to introduce an effective $r$-parameter $$r_{\rm eff}(s) = r \frac{|\S_{12}|^2 (1-r) + |\S_{23}|^2 (s-r)}{|\S_{12}|^2 (1-r) + |\S_{23}|^2 s (s-r)} \leq 1 \qquad {\rm for} \quad r<1\, , r<s\, \label{eff}$$ with the following physical meaning. The hotter reservoirs of our system are $R_1$ and $R_3$ because $T_1>T_2$ and $T_3>T_2$. Both $R_1$ and $R_3$ communicate with the cold reservoir $R_2$ via $\S_{12}$ and $\S_{23}$. Suppose now we replace $R_1$ and $R_3$ with one heat bath $R^\prime$ and ask about its temperature $T^\prime$, which gives the same efficiency. It turns out that the answer is $T^\prime=T_2/r_{\rm eff}$. In fact, one can express the 3-lead efficiency (\[eta7\]) in terms of the 2-lead formula (\[eta2\]) simply as $$\eta (\lambda;r,s) = \eta(\lambda,\lambda;r_{\rm eff}(s)) \, . \label{eff1}$$ We turn now to the regime of heat production ${{\dot Q}}>0$ which, according to (\[qpt3\]), takes place for $\lambda<0$. The corresponding efficiency is defined by (\[etat1\]). Let us introduce the parameters $$u = \frac{\mu_2}{\mu_1}\, , \qquad v = \frac{\mu_3}{\mu_1}\, . \label{chem}$$ Because of (\[fug\]), the conditions (\[cond1\]) can be rewritten in the form $$u<1\, , \qquad v<1\, ,\qquad u<v\, . \label{cond2}$$ One can show now that $\mu_1J^N_1>0,\, \mu_2 J^N_2 <0,\, \mu_3J^N_3 >0$. Therefore, using (\[etat1\]) one obtains $${\widetilde{\eta}}(u,v) = 1- u \frac{|\S_{12}|^2 (1-u) + |\S_{23}|^2 (v-u)}{|\S_{12}|^2 (1-v) + |\S_{23}|^2 v (v-u)}\, . \label{etat8}$$ The effective $u$-parameter now reads $$u_{\rm eff}(v) = u \frac{|\S_{12}|^2 (1-u) + |\S_{23}|^2 (v-u)}{|\S_{12}|^2 (1-v) + |\S_{23}|^2 v (v-u)} \leq 1 \qquad {\rm for} \quad u<1\, , u<v\, \label{eff3}$$ and $${\widetilde{\eta}}(u,v) = {\widetilde{\eta}}(u_{\rm eff}(v)) \, , \label{eff4}$$ whose right hand side is given by the first line (since $u<1$) of the two-lead expression (\[etat2\]). We refer to appendix B for the analysis of the general case, in which all three transmission amplitudes are nontrivial. Bosonic junctions ================= We illustrate in this section the influence of the statistics on the transport and efficiency of the Schrödinger junction. Substituting in (\[ccurr1\],\[ecurr1\]) the Fermi distribution $d_j(k)$ with the Bose one $b_j(k)$, the corresponding integrands develop a singularity at $k^2=2m \mu_i$. This singularity signals condensation like phenomena, whose consideration is beyond the scope of the present paper. For this reason we assume in this section $\mu_i < 0$. Focussing on the case $n=2$, the bosonic counterparts of (\[ccurr4\],\[ecurr4\],\[hf2\],\[entr2\]) are $$J_1^N = - J_2^N = \frac{\vert \mathbb{S}_{12}\vert^2}{2\pi\beta_1}\left[-\ln(1-\e^{-\lambda_1})+ r\ln(1-\e^{-\lambda_2})\right]\, , \label{JNbos2}$$ $$J_1^E = - J_2^E = \frac{\vert \mathbb{S}_{12}\vert^2}{2\pi\beta_1^2}\left[\mathrm{Li}_2(\e^{-\lambda_1})- r^2\mathrm{Li}_2(\e^{-\lambda_2})\right]\, , \label{JEbos2}$$ $${{\dot Q}}^{(b)}(\lambda_1,\lambda_2, r) = \frac{\vert \mathbb{S}_{12}\vert^2}{2\pi\beta_1^2} (\lambda_1-r \lambda_2) \left[\ln(1-\e^{-\lambda_1})- r\ln(1-\e^{-\lambda_2})\right] \, , \label{Qbos}$$ $$\dot{S}^{(b)}=\frac{\vert \mathbb{S}_{12}\vert^2}{2\pi r \beta_1} \left\lbrace (1-r) \left[\mathrm{Li}_2(\e^{-\lambda_1})- r^2\mathrm{Li}_2(\e^{-\lambda_2})\right] + r(\lambda_1 - \lambda_2) \left[\ln(1-\e^{-\lambda_1})- r\ln(1-\e^{-\lambda_2})\right] \right\rbrace \, , \label{EntropyBos}$$ where one should keep in mind that $\lambda_i \equiv -\beta_i\mu_i > 0$. The apex $(b)$ in (\[Qbos\],\[EntropyBos\]) and below is added in order to distinct the bosonic from the fermionic expressions. The domains ${{\cal D}}^{(b)}_\mp$, where ${{\dot Q}}^{(b)}\lessgtr 0$, can be determined like in fermion case. One has $$\mathcal{D}^{(b)}_- = D^{(b)}_1\cup D^{(b)}_2 \label{dombos}$$ with $$\begin{gathered} D^{(b)}_1 = \lbrace 0 < \lambda_1 \leq \lambda_2 , \ 0\leq r < r_1 \rbrace \, , \label{D1bos} \\ D^{(b)}_2 = \lbrace 0 < \lambda_2 < \lambda_1 , \ 0 \leq r < r_2 \rbrace \, , \label{D2bos}\end{gathered}$$ $$r_1 = \dfrac{\lambda_1}{\lambda_2}\, , \ \ \ r_2= \dfrac{\ln(1-\e^{-\lambda_1})}{\ln(1-\e^{-\lambda_2})}\, . \label{r12bos}$$ In ${{\cal D}}^{(b)}_-$ one finds the bosonic efficiency $$\eta^{(b)}(\lambda_1,\lambda_2, r)= \frac{(\lambda_1-r \lambda_2) \left[ r\ln(1-\e^{-\lambda_2})-\ln(1-\e^{-\lambda_1})\right]}{\lambda_1 \left[ r\ln(1-\e^{-\lambda_2})-\ln(1-\e^{-\lambda_1})\right]+ \left[ \mathrm{Li}_2(\e^{-\lambda_1})- r^2\mathrm{Li}_2(\e^{-\lambda_2})\right]}\, . \label{EffBos}$$ In the overlap of ${{\cal D}}_-$ and ${{\cal D}}^{(b)}_-$ one can compare (\[EffBos\]) to the fermion efficiency (\[eta2\]). For the same values of $\lambda_i$ in $D_1^{(b)}=D_1$ one finds that the bosonic efficiency exceeds the fermionic one. The same holds for the amount of heat transformed in chemical energy. This feature persists in the overlap of $D_2^{(b)}$ with $D_2$, except in the neighborhood of $r_2$, where one has the opposite behavior. The maximal bosonic efficiency is obtained for $\lambda_1=\lambda_2\equiv\lambda\rightarrow +\infty$ and coincides with the Carnot efficiency, $$\eta^{(b)}_{\rm{max}}(r)=\lim_{\lambda\rightarrow +\infty}\eta^{(b)}(\lambda,\lambda, r)= 1-r \equiv \eta_C\, . \label{EffMaxBos}$$ (80,100)(80,25) ![Comparing the bosonic and fermionic efficiencies $\eta^{*(b)}$ and $\eta^*$.[]{data-label="plot3"}](Fig-7.eps "fig:") Let us derive now the bosonic efficiency at maximal chemical energy production. The maximum of the function $-{{\dot Q}}^{(b)}(\lambda_1,\lambda_2, r)$ is reached at $\lambda_1=\lambda_2\equiv \lambda_b^*$, where $$\lambda_b^* - (1-\e^{\lambda_b^*})\ln(1-\e^{-\lambda_b^*})=0\, . \label{MaxQbos}$$ The solution of (\[MaxQbos\]) is $\lambda_b^* = 0.69314 \ldots$ and the efficiency reads $$\eta^{*(b)} (r)\equiv \eta^{(b)}(\lambda_b^*,\lambda_b^*, r)= \frac{(1-r)\lambda_b^* \ln \left (1-\e^{-\lambda_b^*}\right )} {\lambda_b^* \ln\left (1-\e^{-\lambda_b^*}\right )- (1+r) \mathrm{Li}_2\left (\e^{-\lambda_b^*}\right )}\, . \label{EffQmaxBos}$$ The study of (\[EffQmaxBos\]) reveals that at maximal chemical energy production the bosonic junction is slightly less efficient then the fermion one, as shown in Fig. \[plot3\]. The enhancement mechanism for fermions, described in section 5.3, applies to bosonic junctions as well. We conclude by observing that in the regime ${{\dot Q}}^{(b)}>0$ the bosonic efficiency ${\widetilde{\eta}}^{(b)}$ of transforming chemical energy in heat precisely coincides with the fermionic one (\[etat2\]). Conclusions =========== We described in this paper a basic process of transformation of heat in chemical potential energy and vice versa, which takes place in systems away from equilibrium. The phenomenon is universal and stems from the fact that even if the total energy of the system is conserved, the heat and chemical potential energies are in general not separately conserved. Both directions of the process of energy transmutation are characterized by their own efficiency coefficient. The specific features of the phenomenon depend on the particle statistics and on the choice of nonequilibrium state. We illustrated this fact by studying the fermionic and bosonic Schrödinger junctions in the LB state and its orbit under parity and time reversal. The relative quantum transport depends in a complicated nonlinear way on the temperatures and chemical potentials, which parametrize the nonequilibrium states. In order to control these characteristics of the system, we avoided the use of any approximation and in particular, of the linear response theory. Assuming that the interaction which drives the system away from equilibrium is scale invariant, we described the quantum transport exactly and derived the explicit expressions of the efficiency coefficients. Exploring the orbit of the LB state under parity and time reversal transformations, we have shown that space-time symmetries have an essential impact on the quantum transport and efficiencies. Concerning the internal symmetries, our model has a global $U(1)$-symmetry associated with the particle number. For systems with a larger internal symmetry group, the process of energy transmutation becomes even more involved, due to the presence of several types of chemical potentials. In this case the Landauer-Büttiker state is induced not by a Gibbs state, but by a generalized Gibbs ensemble [@j-57] generated by a complete set of commuting charges of the extended symmetry group. The study of such nonequilibrium states may provide an important insight in the role of internal symmetries in quantum transport. M.M. would like to thank the Laboratoire de Physique Théorique d’Annecy-le-Vieux for the kind hospitality during the preparation of the manuscript. Contact with linear response theory =================================== We show here that the exact efficiency (\[eta2\]) reproduces in the linear response regime the result of [@bcps]. The meeting point of the two frameworks is the Onsager matrix $X$ [@call]. Setting $$\beta_1=\beta \, ,\quad \beta_2 = \beta + \delta \beta\, , \quad \mu_1=\mu\, , \quad \mu_2 = \mu + \delta \mu\, , \quad \lambda = - \beta \mu \, , \label{X0}$$ the entries $X_{ij}$ of $X$ are defined by [@call] $$\begin{aligned} -J_1^N = X_{11}\, \beta\, \delta \mu + X_{12}\, \delta \beta + \cdots \, , \nonumber \\ J_1^Q =J_1^E -\mu J_1^N = X_{21}\, \beta\, \delta \mu + X_{22}\, \delta \beta + \cdots \, , \label{X1}\end{aligned}$$ where the dots stand for higher orders of the expansion in $\delta \mu$ and $\delta \beta$. For the Onsager matrix of our system with scale invariant interaction in the junction one gets from [@bcps] $$\begin{aligned} X_{11} &=& \frac{|\S_{12}|^2}{2\pi \beta} \frac{1}{1+\e^{\lambda}}\, , \nonumber \\ X_{12} &=& X_{21} = -\frac{|\S_{12}|^2}{2\pi \beta^2} \left [\frac{\lambda}{1+\e^{\lambda}}+ \ln \left (1+\e^{-\lambda} \right ) \right ] \, , \nonumber \\ X_{22} &=& \frac{|\S_{12}|^2}{2\pi \beta^2} \left [\frac{\lambda^2}{1+\e^{\lambda}}+ 2\lambda \ln \left (1+\e^{-\lambda} \right )- \li_2 \left (-\e^{-\lambda} \right )\right ] \, . \label{X2} \end{aligned}$$ Moreover, according to [@bcps] the linear response efficiency $\eta_{{}_{\rm LR}}$ is given by $$\eta_{{}_{\rm LR}} = \frac{-(X_{11}\, \beta\, \delta \mu + X_{12}\, \delta \beta)\, \delta \mu} {X_{21}\, \beta\, \delta \mu + X_{22}\, \delta \beta} \, . \label{X3}$$ Inserting (\[X2\]) in (\[X3\]) and expanding in $\delta \mu$ and $\delta \beta$ one obtains to the first order $$\begin{aligned} \eta_{{}_{\rm LR}} = \frac {\e^{-\lambda}} {(1+\e^{-\lambda})\ln \left (1+\e^{-\lambda} \right ) +\lambda \e^{-\lambda}}\, \beta\, \delta \mu - \qquad \qquad \qquad \; \nonumber \\ \frac {(1+\e^{-\lambda})\ln^2 \left (1+\e^{-\lambda} \right ) +2 \e^{-\lambda}\li_2 \left (-\e^{-\lambda} \right )} {\beta [(1+\e^{-\lambda})\ln \left (1+\e^{-\lambda} \right ) +\lambda \e^{-\lambda}]^2}\, \delta \beta + \cdots \, . \label{X4}\end{aligned}$$ This result coincides precisely with the expansion of the exact efficiency (\[eta2\]) (expressed in terms of the variables (\[X0\])) to the first order in $\delta \mu$ and $\delta \beta$, which concludes the proof. The above argument has a direct generalization to the case $n>2$. The 3-lead junction with generic $\S$-matrix amplitudes ======================================================= Combining (\[TJ1\],\[TJ2\]) with (\[cond1\]), we conclude that $J_1^Q>0$ and $J_2^Q<0$. So, one is left with the study of the sign of $J_3^Q$ given by (\[TJ3\]). The coefficients of the logarithm and the polylogarithm are both positive when $$s > \dfrac{r\vert \mathbb{S}_{23}\vert^2 + \vert \mathbb{S}_{13}\vert^2} {\vert \mathbb{S}_{23}\vert^2 + \vert\mathbb{S}_{13}\vert^2} \equiv s_1 \, ,\qquad s^2 > \dfrac{r^2\vert \mathbb{S}_{23}\vert^2 + \vert \mathbb{S}_{13}\vert^2} {\vert \mathbb{S}_{23}\vert^2 + \vert\mathbb{S}_{13}\vert^2} \equiv s_2^2\, . \label{A1}$$ Using (\[cond1\]) it is not difficult to show that $s_2 > s_1 > r$. Therefore, $$\begin{aligned} J_3^Q <0 \, , & \quad {\rm for} \quad r<s<s_1\, , \label{A2}\\ J_3^Q >0 \, , & \quad {\rm for} \quad s_2 < s<1\, . \label{A3}\end{aligned}$$ For $s\in (s_1,s_2)$ the sign of $J_3^Q$ depends on $\lambda $ as well. In this way one finds $$\eta (\lambda;r,s) = \begin{cases} \frac{A}{B_1} & \quad {\rm for} \quad r<s<s_1\, , \\ \frac{A}{B_2} & \quad {\rm for} \quad s_2<s<1\, , \label{A4} \end{cases}$$ where $$A=\lambda \ln \left (1+\e^{-\lambda} \right )\left[|\S_{12}|^2 (1-r)^2 + |\S_{13}|^2(1-s)^2+|\S_{23}|^2 (s-r)^2\right ]\, , \label{A5}$$ $$\begin{aligned} B_1=\lambda \ln \left (1+\e^{-\lambda} \right ) \left[|\S_{12}|^2 (1-r) + |\S_{13}|^2 (1-s)\right ] - \nonumber \\ \li_2 \left (-\e^{-\lambda} \right )\left[|\S_{12}|^2 (1-r^2) + |\S_{13}|^2 (1-s^2)\right ]\, , \qquad \label{A6}\end{aligned}$$ and $$\begin{aligned} B_2 = \lambda \ln \left (1+\e^{-\lambda} \right ) \left[|\S_{12}|^2 (1-r) + |\S_{13}|^2 (1-s)^2+|\S_{23}|^2 s(s-r)\right ] - \nonumber \\ \li_2 \left (-\e^{-\lambda} \right )\left[|\S_{12}|^2 (1-r^2) + |\S_{23}|^2 (s^2-r^2)\right ]\, . \qquad \qquad \qquad \quad \label{A7}\end{aligned}$$ [99]{} A. Polkovnikov, K. Sengupta, A. Silva and M. Vengalatorre, Rev. Mod. Phys. 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--- abstract: 'We revisit the hypothesis of a possible line structure in the Hawking evaporation spectrum of black holes. Because of nonperturbative quantum gravity effects, this would take place arbitrarily far away from the Planck mass. We show, based on a speculative but consistent hypothesis, that this naive prediction might in fact hold in the specific context of loop quantum gravity. A small departure from the ideal case is expected for some low-spin transitions and could allow us to distinguish several quantum gravity models. We also show that the effect is not washed out by the dynamics of the process, by existence of a mass spectrum up to a given width, or by the secondary component induced by the decay of neutral pions emitted during the time-integrated evaporation.' author: - Aurélien Barrau bibliography: - 'refs.bib' title: Evaporation Spectrum of Black Holes from a Local Quantum Gravity Perspective --- Introduction and model ====================== The Hawking radiation of black holes [@Hawking:1974sw], characterized by a Planck law slightly modulated by grey-body factors [@Page:1976df], is one of the most robust predictions of quantum field theory on a curved background. It is also the perfect phenomenon to investigate possible deviations from the semiclassical dynamics due to nonperturbative quantum gravity effects. Bekenstein and Mukhanov [@Bekenstein:1995ju] have suggested that an interesting way to account for quantum gravity at the effective level could be to assume that the area of a black hole (BH) can only take values proportional to a fundamental area assumed to be of the order of the Planck area. This has interesting consequences. In particular, this leads to the appearance of emission lines instead of a continuous spectrum, as we will explain in more details later. However, it was then shown that in the specific setting of loop quantum gravity (LQG) the use of the actual eigenstates of the area operator does [*not*]{} lead to a Bekestein-Mukhanov-like spectrum [@Barreira:1996dt]. The density of energy levels instead reads as $\rho(M)\sim \rm{exp}(M\sqrt{4\pi G/3})$ which means that the spectral lines are virtually dense in frequency for large masses. The phenomenology of evaporating black holes in LQG has, therefore, focused so far on the very last stages of the emission where lines are distinguishable. This is theoretically interesting but probably out of reach of any reasonable phenomenological approach.\ We would like to revisit this conclusion, somewhat in the line of [@Yoon:2012cq]. Basically, what was assumed in virtually all LQG studies (see, [*e.g.*]{}, [@Barrau:2011md; @Barrau:2015ana]) on evaporating black holes is that “something" independent from quantum gravity triggers the emission of a particle from the black hole. The exact energy of this particle is determined by one of the area eigenvalues of the hole. The selected value is usually taken to be as close as possible to the one favored by the semiclassical quantum process. Alternatively, one can just calculate the transition probabilities between black-hole states by weighting them by exp$(S_2-S_1)$ where $S_1$ and $S_2$ are the entropies associated to the initial and final states. What is a black hole in LQG (see, [*e.g.*]{}, [@Engle:2009vc; @Krasnov:2009pd; @Ashtekar:1998sp; @FernandoBarbero:2009ai; @Ghosh:2011fc]) ? It is basically an isolated horizon punctured by the edges of a spin network, that is a graph with edges labeled by $SU(2)$ representations and nodes characterized by intertwiners. An edge with spin representation $j$ carries an area of eigenvalue $$A_j=8\pi \gamma l_{Pl}^2\sqrt{j(j+1)}, \label{eq1}$$ where $j$ is a half-integer and $\gamma$ is the Barbero-Immirzi parameter. A surface punctured by $N$ edges has a spectrum given by $$A_j=8 \pi \gamma l_{Pl}^2\sum_{n=1}^N\sqrt{j_n(j_n+1)},$$ where the sum is carried out over all intersections of the edges with the surface. Each state with spin $j$ has a degeneracy $(2j+1)$. We believe that there might be two problems with the usual view of the Hawking evaporation in this framework. When one considers the transition from a state with mass $M_1$ to a state with mass $M_2$ which is, in general, very close to $M_1$ if the black hole is macroscopic, the actual final quantum state is completely different from the initial one most of the time. Even if the masses typically differ by much less than the Planck mass (and the areas differ by approximately the Planck area), the second quantum state corresponds to values of the spins that are in general completely different from the initial state. Using the quasidense distribution of states requires a complete reassigning of the quantum numbers to each puncture for every single transition. This is in tension with a quantum gravitational origin of the evaporation process itself. If one considers the evaporation as due to a change of state of a given “elementary area cell" (or, more precisely, to the settling down of the BH following this transition), there is no reason for all of the other elementary surfaces to change their quantum state at the same time. This was considered in, for example, [@Makela:2011vd]. In addition, this raises an obvious second problem about causality: how can a “far away" elementary cell know the way it has to change to adjust to the others? Our hypothesis here is that each particle evaporated by a black hole is basically due to the relaxation of the black hole following a change of state of a single elementary cell. We call this a local quantum gravity process and investigate it in the well-developed framework of LQG. In principle, however, the idea is quite general. It should be made clear that our hypothesis is nothing more than a reasonable alternative view that deserves to be studied until we have a fully dynamical quantum gravity description of the process. Ideally, one would build a model in which loop quantum gravity is coupled directly to quantum electrodynamics and compute the actual changes in the gravitational state upon emission of Hawking radiation. This is obviously beyond the scope of this study. The key point here is that a purely local change of state of an “elementary cell", with a small or moderate change in quantum number, is the fundamental quantum gravity process taking place. This does [*not*]{} inconsistently assume that local physics knows the global BH quantities like temperature and mass. Once the quantum transition has taken place, without any [*a priori*]{} knowledge of the global picture, the BH relaxes through semiclassical processes according to the energy made available by the quantum transition. This automatically leads to a spectrum whose main classical properties agree with the Hawking evaporation. Of course, in the future, it would be important to investigate the settling down in a fully consistent way, going beyond the isolated horizon considered here which is, by construction, stationary. Low energy case =============== Let us first consider the simpler case of a low-energy evaporation signal. It is easier for two reasons: first because the dynamics can then be neglected and, second, because there is no secondary emission associated with decaying particles in the sense that the black hole does not emit hadrons leading to gamma-rays. The main image is very simple and relies on the fact that the structure of a Schwarzshild black hole is such that (in Planck units) $dA=32\pi MdM$. If a quantum of energy $E\sim T$, with $T=1/(8 \pi M)$, is emitted, it induces a BH mass change $dM=E$. The area change will then be $dA\sim4$. This is the main interest of the Bekenstein-Mukhanov hypothesis: assuming a discrete area spectrum with regularly spaced eigenvalues induces a line structure in the energies of the evaporation spectrum, even arbitrarily far away from the Planck mass. The very same change of area $dA$ (of order of the Planck area) will indeed lead to a relative line separation in the spectrum $dE/T$ which does [*not*]{} depend on the mass.\ Following our hypothesis that the evaporation is due to a local change of state of a quantum of area, the spectrum will still exhibit lines in LQG. However, the eigenvalues of the area operator given by Eq. (\[eq1\]) are a bit more subtle. In the large-$j$ limit, one recovers a regular line spectrum but the first eigenvalues are not equally spaced. If the resulting line spectrum is to have, as expected, the Hawking spectrum as an envelope, the favored transition is always one between two states $A_j$ and $A_{j-n}$ with $n$ a half-integer not much greater than unity: 95% of the transitions will have $n\leq 3/2$. The first studies of black holes in LQG (see, [*e.g.*]{}, [@Ashtekar:1997yu]) claimed that the punctures are mostly “low-spin" ones. A black hole is then expected to have most of his $j$’s close to 0. Transitions do exhibit generically a regular line structure. However, in some cases, largely corresponding either to transitions between $A_{j}$ and $A_{0}$ or between $A_j$ and $A_{1/2}$, there will be a deviation with respect to what would be expected from a regular line structure. This is what is shown in Fig. \[error\], which displays the relative energy difference between some $A_j\rightarrow A_0$ and $A_j\rightarrow A_{1/2}$ transitions and what would be expected from the same transitions in the regular Bekenstein-Mukhanov spectrum. The other way round, new models of holographic black holes were developed (see, [*e.g.*]{}, [@Ghosh:2013iwa]). Here, one uses the qualitative behavior of matter degeneracy suggested by standard QFT with a cutoff at the vicinity of the horizon – [*i.e.*]{}, an exponential growth of vacuum entanglement in terms of the BH area. In this case, large $j$’s should dominate and the prediction is clearly that the line structure will be nearly perfect, as $A_{j}-A_{j-n}$ is very close to $n$ as soon as $j$ is much greater than unity. This opens up a very interesting possibility: not only should this effect allow to observe quantum gravity features at high masses, but it should also allow us to distinguish between BH models in LQG.\ An important question arises. Obviously, nearly equally spaced area eigenvalues and regular jumps between those values do [*not*]{} lead to the emission of quanta at the same energy as long as the evaporation goes on. When the black-hole area decreases by the same amount $dA$, the emitted energy varies like $1/M$. So, one should ensure that the change of area during the evaporation does not destroy the very possibility of observing lines. Let us evaluate the energy shift between two successive emissions associated with an identical area variation. Let us call $pA_0$ the area variation induced by the transition where $p$ is a half-integer and $A_0$ is the fundamental area of order $A_{Pl}$. It is easy to show that the relative variation of energy of the emitted quanta between consecutive emissions is, at lower order, $$\frac{\Delta E}{E}\approx\frac{p}{2}\frac{A_0}{A}.$$ This ensures that as long as the area is much larger than the Planck area, which is definitely the case for macroscopic black holes, the change in energy is negligible and the line structure can be observed if it exists: the fact that the BH mass changes during its evaporation does not wash out this interesting feature.\ In practice, however, evaporating black holes are low-mass black holes, at least when compared to a solar mass. This means that they must be primordial black holes (PBHs), except in some exotic low-Planck-scale models where they could be formed by collisions of particles in the contemporary Universe [@Barrau:2005zb]. The modes of production of PBHs are hypothetical (see, [*e.g.*]{}, [@Carr:1975qj; @Carr:2009jm]). Various mechanisms have been considered. If we were to observe a single evaporating black hole, we would not care about its origin as far as the phenomenon studied here is concerned. Let us estimate the maximum distance at which this can be efficiently measured. The lifetime of the black hole is of order $M^3$, where $M$ is its initial mass. It is mostly determined by the emission of quanta of energy $E\sim T\sim 1/M$. There are, therefore, roughly $M^2$ quanta emitted in a time $M^3$, which means that the mean time between two emissions is of order $M$. The percentage of emitted quanta reaching a detector of surface $S$ is of order $S/R^2$ if the PBH is at distance $R$. The correct criterion for detection and identification of the signal consists of requiring a mean time $\Delta t$ between two measured photons from the same PBH to be smaller that a reference interval $\Delta t_0$ (otherwise the signal is lost in the background). This leads to the requirement $MR^2/S<\Delta t_0$, that is, a maximum detectable distance of $$R_{max}\approx\sqrt{\frac{S\Delta t_0}{M}}.$$ The most interesting case is, however, the signal emitted by a distribution of PBHs, whose masses are necessarily not exactly the same. Does the global line structure remains? This is not obvious, as different masses will induce different line energies and this might make the phenomenon experimentally invisible. By studying the energy of an emitted quantum in a given transition for two different BH masses, one can see that the relative energy variation is actually given by $\Delta M/M$. It is a quite general prediction of PBH formation mechanisms that their initial mass is roughly equal to the cosmological horizon mass at the formation time, $M\sim M_H \propto t$. As $t\propto T^{-2}$, where $T$ is the temperature of the Universe, $$\left| \frac{\Delta E}{E} \right|=2\left| \frac{\Delta T}{T} \right|.$$ So if the relative change is to remain smaller than, say, 10%, it is enough that the relative change in temperature during the formation period remains smaller than 5%. This is reasonable for a PBH production associated, for example, with a phase transition (see, [*e.g.*]{} [@Cline:1996mk; @Jedamzik:1999am]). ![Relative difference in the emitted energy between a purely regular line structure and the actual LQG line structure (in the local point of view of this study) in the $A_j\rightarrow A_0$ transitions (upper curve) and $A_j\rightarrow A_{1/2}$ transitions (lower curve).[]{data-label="error"}](error.pdf){width="85mm"} High energy case ================ When the temperature of the black hole is greater than the QCD confinement scale (but still not much above the energies probed by accelerators), we conservatively assume that quarks are emitted and fragmentate into subsequent hadrons. This is a purely semiclassical prediction that does not rely on the underlying theory of quantum gravity. Some of those hadrons are unstable and will eventually decay into gamma-rays [@MacGibbon:1990zk; @MacGibbon:1991tj]. Most gamma-rays emitted in this way come from the decay of neutral pions. We call this the secondary component. The problem is that even if the quarks are emitted with a spectrum made of lines, the resulting gamma-rays will obviously be distributed according to a continuum and the previously mentioned approach might not hold any longer. In addition, the instantaneous spectrum emitted by a black hole at a given temperature contains many more photons du to the secondary component than associated with the primary component (direct emission). We have investigated this point into the details by using the “Lund Monte Carlo" PYTHIA code (with some scaling approximations in the low-energy range) [@Sjostrand:2014zea] to determine the normalized differential fragmentation functions $dg(Q,E)/dE$, where $Q$ is the quark energy and $E$ is the photon energy. It takes into account a large number of physics aspects, including hard and soft interactions, parton distributions, initial- and final-state parton showers, multiple interactions, fragmentation and decay. With this tool, we derived an analytical fit for the resulting fragmentation functions which describe the number of gamma rays generated between $E$ and $E+dE$ by the decay of the hadronization product of a quark with energy between $Q$ and $Q+dQ$. The secondary spectrum of gamma-rays reads as $$\begin{aligned} \frac{d^2N_{\gamma}}{dEdt}&=& \sum_j\int_{Q=E}^{\infty}\alpha_j\Gamma_j(Q,T) \left(e^{\frac{Q}{T}}-(-1)^{2s_j}\right)^{-1}\\ &\times&\frac{dg_{j\gamma}(Q,E)}{dE}{dQ},\end{aligned}$$ where $j$ is the type of quark and $s_j=1/2$. The time-integration of this spectrum can obtained by writing $$\frac{dN_{\gamma}}{dE}=\int_{M_i}^{M_f}\frac{d^2N_{\gamma}}{dEdt}\frac{dt}{dM}dM.$$ The primary component of the instantaneous spectrum is a quasi-Planckian law (either a continuous one for the usual case or as the envelope of the lines for the quantum gravity case) but the secondary spectrum is more complicated. Mostly due to the decay of neutral pions, it can be roughly approximated by a Cauchy distribution near its maximum, and then an $E^{-1}$ power law followed by an exponential cutoff around the initial quark energy. It is continuous even if the primary emission is discrete. The time-integrated signal associated with the primary emission can easily be analytically shown to lead to a differential spectrum scaling as $E^{-3}$. We have performed the numerical integration of the secondary component. The very interesting point is that, as shown in Fig. \[time\], the time-integrated signal is nearly the same for both components. This is quite unexpected as the physics involved depends on the details of nongravitational processes (subtleties of the hadronization, cross sections for decays into gamma-rays, etc.). At a given BH temperature – and, therefore, at a given mass – the number of secondary photons is much higher than the number of primary photons. However, the mean energy of the secondary component is much smaller than for the primary component. This means that the primary emission was peaked at this energy when the BH mass was higher and, due the dynamics of the process $(dM/dt\propto -M^{-2})$, it has spent a “longer time" in this mass region (between $M$ and $M+dM$). Those phenomena compensate each other and the neat result is that both components have the same order of magnitude. ![Time integrated primary (upper curve) and secondary (lower curve) gamma-ray emission from an evaporating BH.[]{data-label="time"}](bh2.pdf){width="85mm"} The important consequence of this calculation is that, even in this case, the interesting quantum gravity line structure remains, in principle, detectable. It is not “diluted" in a huge continuous signal due to the secondary component. The secondary component only induces a reasonable “self-background" and as soon as the detector resolution is better than the line spacing, the phenomenon is easy to identify, if it exists at all. The relative energy difference between lines can easily be shown to be $\Delta E/E \sim \pi \gamma \sim 1$. As a typical detector resolution is of the order of $10\%-20\%$, the structure is clearly visible, even when taking into account this secondary component. Conclusion ========== A quantum-gravity “local" perspective on the horizon structure of black holes might lead to a new view of the Hawking process: the evaporation would then be associated with field quanta emitted by the settling down of the black hole after the transition of a single “surface element" between two area eigenstates. This is a speculative hypothesis that requires a more detailed theoretical investigation. However, if correct, we have shown that this would lead to a line structure in the spectrum, even for masses arbitrarily larger than the Planck mass. This is not washed out by the fact that black holes might be formed over a nonvanishing interval of masses. It also remains during the dynamics of the process in the sense that the energy variation between consecutive emissions is very small when compared with the separation between lines. More importantly it also remains visible when the secondary component, associated with the decay of unstable hadrons, is also taken into account. Finally, beyond being a “smoking gun" candidate probe for quantum gravity, this would open interesting perspectives to discriminate between detailed loop quantum gravity models: high-spin models have a perfectly regular line structure whereas low-spin models exhibit some deviations with respect to the ideal case. Acknowledgments =============== A.B. would like to thank Marrit Schutten for the PYTHIA fits.

--- abstract: 'We study theoretically the sound propagation in charge- and spin-density waves in the hydrodynamic regime. First, making use of the method of comoving frame, we construct the stress tensor appropriate for quasi-one dimensional systems within tight-binding approximation. Taking into account the screening effect of the long-range Coulomb interaction, we find that the increase of the sound velocity below the critical temperature is about two orders of magnitude less for longitudinal sound than for transverse one. It is shown that only the transverse sound wave with displacement vector parallel to the chain direction couples to the phason of the density wave, therefore we expect significant electromechanical effect only in this case.' address: - | Research Institute for Solid State Physics, H-1525 Budapest, P. O. Box 49, Hungary\ and Institute of Physics, Technical University of Budapest, H-1521 Budapest, Hungary - 'Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484' author: - Attila Virosztek - Kazumi Maki title: | Sound propagation in density wave conductors and the effect of\ long-range Coulomb interaction --- Introduction ============ Several aspects of the collective transport associated with the phason in charge- or spin-density waves (CDW or SDW) are still not well understood. One of the intriguing phenomena is the electromechanical effect observed in CDW[@Brill; @Moz; @Xi1; @Xi2; @Jac] and more recently in SDW[@Brown]. First of all, most of the elastic moduli increase upon entrance into the CDW or SDW state, often with a sharp dip at $T_c$, the transition temperature. Second, some of the elastic moduli in CDW or SDW soften when the density wave is depinned by an external electric field in excess of the depinning threshold field $E_T$. Third, the change in the elastic moduli due to depinning of the density wave depends on the frequency $\omega$ of the flexural vibration[@Xi3], and decreases like $\omega^{-p}$ with $p\approx 1$. This behavior is similar to the frequency dependence of the change in the dielectric constant upon depinning in CDW and SDW[@Cava]. We have shown earlier[@MV1] in the collisionless limit that the hardening of the elastic constants can be understood in terms of the reduction in the quasiparticle screening of the ion potential due to the formation of the density wave state. The electromechanical effect was interpreted as an additional screening contribution from the collective mode of the density wave condensate (phason) liberated by depinning. A later extension of that theory to the experimentally more relevant hydrodynamic limit[@VM1] did not modify the above picture qualitatively. However in these papers it was assumed that the phonon simply couples to the electronic density, and the effect of the long-range Coulomb interaction was neglected. The purpose of the present paper is twofold. First, we shall develop the theory of the electron-phonon coupling for a strongly anisotropic system, which will enable us to distinguish between the behavior of transverse and longitudinal sound waves propagating in various directions. Second, we shall include the effect of the long-range Coulomb interaction following Kadanoff and Falko[@KF]. In Section II we construct the electronic stress tensor (which couples to the deformation tensor of the sound wave) for a quasi-one dimensional system following the method of comoving frame[@Tsu]. In Section III we concentrate on the quasiparticle contribution to the stress tensor correlation functions corresponding to the pinned case. Section IV is devoted to the examination of the coupling of the stress tensor to the phason, which is relevant to the electromechanical effect. Our conclusions are summarized in Section V. A preliminary report on this work has already been published elsewhere[@VM2]. Electron-phonon coupling ======================== Following Tsuneto[@Tsu] let us assume that both the ionic potential and the electronic wavefunction are deformed in the slowly varying sound field ${\bf u}({\bf r},t)={\bf u}\cos({\bf qr}-\omega t)$ imposed externally (extreme tight-binding limit): $$\begin{aligned} V({\bf r})\rightarrow&V[{\bf r}-{\bf u}({\bf r},t)]\nonumber\\ \psi({\bf r})\rightarrow&\psi[{\bf r}-{\bf u}({\bf r},t)] (1+\nabla{\bf u})^{-1/2}.\end{aligned}$$ This displacement is generated by the unitary operator $U=\exp[-{\bf u}({\bf r},t)\nabla]$. Therefore the deformed wavefunction is an eigenfunction of the transformed Hamiltonian $h=Uh_0U^+$, where $h_0=\varepsilon(-i\nabla)$ with $\varepsilon ({\bf p})$ being the zone-periodic electronic energy spectrum. The transformed Hamiltonian $h$ is then expanded in terms of the deformation tensor $\nabla_iu_j$, which is much smaller than one, even though the displacement ${\bf u}$ itself may be many times of the lattice constant for sound propagation. We obtain $h=h_0+h_{el-ph}$, where the Hamiltonian for the electron-phonon coupling is given by $$h_{el-ph}=\sum_{i,j}(\nabla_iu_j)i\nabla_jv_i(-i\nabla),$$ with ${\bf v}({\bf p})=\partial\varepsilon({\bf p})/\partial {\bf p}$, the velocity of the Bloch electron. The matrix element of $h_{el-ph}$ between Bloch states is evaluated as $$\langle {\bf p}+{\bf q}|h_{el-ph}|{\bf p}\rangle = -i\sum_{i,j}q_ju_i\tau_{ij}({\bf p}),$$ where the stress tensor $$\tau_{ij}({\bf p})=mv_i({\bf p})v_j({\bf p}),$$ and $m$ is the bare electron mass. This expression generalizes the stress tensor used for an isotropic metal[@KF]. In orthorombic symmetry the sound wave polarized in the ${\bf i}$ direction and propagating in the ${\bf j}$ direction couples to the $\tau_{ij}$ component of the stress tensor, and in order to determine the effect of that coupling on the frequency (or velocity) of the sound, we have to evaluate the appropriate stress tensor correlation function $\langle [\tau_{ij},\tau_{ij}]\rangle$. Once this is known, the renormalized sound velocity can be calculated in the weak coupling limit as $$c=c_0\{1-\langle [\tau,\tau ]\rangle /2Mc_0^2\},$$ where $c_0$ is the sound velocity without electron-phonon coupling, $M$ is the ion mass, and for clarity we have suppressed the indeces both for the stress tensor component and for the sound velocity. For a highly anisotropic ($t_a\gg t_b\gg t_c$) tight-binding dispersion $$\varepsilon({\bf p})=-2t_a\cos(ap_x)-2t_b\cos(bp_y) -2t_c\cos(cp_z)-\mu,$$ widely used for CDW and SDW materials[@Yam], the velocity and stress tensor components are easily obtained, and their values on the open Fermi surface can conveniently be expressed by the component of ${\bf p}$ perpendicular to the chains (${\bf x}$ direction). However, since the Green‘s functions for CDW and SDW are usually written in the left-right spinor representation[@VM1], involving measuring momenta from $\pm{\bf Q}/2$ with the density wave wavevector ${\bf Q}=(2p_F,\pi/b,\pi/c)$, we should express the stress tensor elements in a compatible manner. It turns out, that for each stress tensor component the term proportional to the unit matrix dominates, therefore we have $$\begin{aligned} \tau_{xx}=&mv_F^2\{1-[t_b/t_a\sin(ap_F)]^2 \sin^2(bp_y)\}\nonumber\\ \tau_{yy}=&mv_y^2[1+\cos(2bp_y)]\nonumber\\ \tau_{xy}=&mv_Fv_y\sqrt{2}\cos(bp_y).\end{aligned}$$ Here $p_F$ is the Fermi momentum, $v_F=2at_a\sin(ap_F)$ is the Fermi velocity in the chain direction, while $v_y=\sqrt{2}bt_b\ll v_F$ is a typical velocity in the perpendicular direction. In the followings we restrict our study to the $x-y$ plane, since behavior involving the $z$ direction should be similar to that of the $y$ direction. We note, that all components of the stress tensor depend on momentum through the combination $\varphi=bp_y$ only. Pinned density waves ==================== In this section we consider sound propagation with no applied electric field. The density wave is pinned, therefore the condensate is unable to contribute to correlation functions, including that for the stress tensor. Mathematically this situation can be simulated by setting the coupling of the stress tensor (and of the density) to the phason to zero. Then the stress tensor couples only to the density fluctuations, resulting in the well known Coulomb screening[@KF]: $$\langle [\tau,\tau]\rangle=\langle[\tau,\tau]\rangle_0- {\langle[\tau,n]\rangle_0\langle[n,\tau]\rangle_0\over q^2/4\pi e^2+\langle[n,n]\rangle_0}.$$ Here $n$ stands for the electronic particle density, and $\langle[A,B]\rangle_0$ denotes a correlation function, in which only the effect of impurity scattering is taken into account. The density correlator $\langle[n,n]\rangle_0$ in the presence of impurity scattering was evaluated in [@VM1]. A straightforward extension of that calculation confirms that under the circumstances of the sound experiment ($lq\ll 1$, where $l$ is the mean free path) the stress tensor correlator has two distinct contributions: $$\langle[\tau,\tau]\rangle_0=\langle\tau(\varphi)\rangle_{\varphi}^2 \langle[n,n]\rangle_0+\langle[\delta\tau(\varphi)]^2\rangle_{\varphi} \langle[n,n]\rangle_0^{no vertex}.$$ The first contribution features only the average of the stress tensor $\langle\tau(\varphi)\rangle_{\varphi}= (2\pi)^{-1}\int_{-\pi}^\pi d\varphi\tau(\varphi)$, and is proportional to the diffusive (vertex corrected) density correlator. The second contribution containes only the fluctuating part of the stress tensor component $\delta\tau(\varphi)=\tau(\varphi)-\langle\tau(\varphi) \rangle_{\varphi}$, and therefore it is proportional to the density correlator $\langle[n,n]\rangle_0^{no vertex}$ calculated without vertex corrections. According to the same argument, only the average of the stress tensor couples to the density, therefore $$\langle[\tau,n]\rangle_0= \langle[n,\tau]\rangle_0=\langle\tau(\varphi)\rangle_{\varphi} \langle[n,n]\rangle_0.$$ Combining Eqs.(8)-(10) we see that in the long wavelength limit appropriate for the sound experiment ($q\approx 1/L$, where $L$ is the sample size), the average part of the stress tensor ($s$-wave component, proportional to density) is completely screened out by the Coulomb interaction, and only the fluctuating part contributes to the correlation function: $$\langle[\tau,\tau]\rangle=\langle[\delta\tau(\varphi)]^2 \rangle_{\varphi}\langle[n,n]\rangle_0^{no vertex}.$$ The situation here is the same as in the electronic Raman scattering, where the long-range Coulomb interaction suppresses the density (charge) fluctuations, and only non $s$-wave channels survive[@AG]. The evaluation of $\langle[n,n]\rangle_0^{no vertex}$ can be done starting with the results of [@VM1]. The calculation is rather technical, therefore we delegate it to the Appendix, and we give here the results only. Without vertex corrections there is no diffusion pole, and both the wavenumber ${\bf q}$ and the frequency $\omega$ could be set to zero. However we keep a finite (but small) frequency for finite imaginary part of the correlator: $$\langle[n,n]\rangle_0^{no vertex}=N_F{i\tilde\Gamma_{qp}(1-\tilde f) \over \omega+i\tilde\Gamma_{qp}},$$ where $N_F$ is the density of states at the Fermi surface. The corresponding “unrenormalized” condensate density $\tilde f$ (for the general formula see Eq.(32) in the Appendix) and quasiparticle damping $\tilde\Gamma_{qp}$ are evaluated in two limiting cases, close to $T_c$ and close to zero temperature as $$\begin{aligned} \tilde f(T\rightarrow T_c)=&-2({\Delta\over 4\pi T})^2 \psi^{\prime\prime}({1\over 2}+{\Gamma\over 2\pi T})\approx 7\zeta(3)({\Delta\over 2\pi T})^2\nonumber\\ \tilde f(T\rightarrow 0)=&1-3\pi\alpha/16,\end{aligned}$$ and $$\begin{aligned} \tilde\Gamma_{qp}(T\rightarrow T_c)=&2\Gamma\nonumber\\ \tilde\Gamma_{qp}(T\rightarrow 0)=&{9\pi\over 32}{\Delta \alpha^{8/3}\over u_0^2(5-4\alpha^{2/3})}{G\over T}e^{G/T}.\end{aligned}$$ Here $\Delta$ is the density wave order parameter, $\Gamma= \Gamma_F+\Gamma_B/2$ is a combination of the impurity forward and backscattering rate, $\alpha=\Gamma/\Delta$, $u_0^2=1- \alpha^{2/3}$ and $G=\Delta u_0^3$ is the density wave gap. As it is seen from Eq.(13), $\tilde f$ increases linearly in $(T_c-T)$ below $T_c$, but it is slightly less than one at $T=0$ ($\Gamma$ is usually an order of magnitude smaller than $T_c$). Nevertheless, the temperature dependence of the sound velocity in the pinned case will still be qualitatively the same as in the collisionless limit[@MV1]. The relative change of the sound velocity compared to the normal state ($c_n$) is easily obtained from Eqs.(5), (11) and (12) as $$(c-c_n)/c_0=\lambda\tilde f,$$ where the effective coupling $$\lambda={N_F\over 2Mc_0^2}\langle[\delta\tau(\varphi)]^2 \rangle_{\varphi}.$$ Using Eq.(7), these effective couplings for the various sound waves are $$\begin{aligned} \lambda_{xx}=&{N_F(mv_F^2)^2\over 16Mc_0^2}[{t_b\over t_a\sin( ap_F)}]^4\nonumber\\ \lambda_{yy}=&{N_F(mv_y^2)^2\over 4Mc_0^2}\nonumber\\ \lambda_{xy}=&{N_F(mv_Fv_y)^2\over 2Mc_0^2}.\end{aligned}$$ Since $v_y/v_F\approx t_b/t_a\approx 1/10$ in many quasi one dimensional materials, we expect that the relative increase of the sound velocity below $T_c$ will be a factor $10^2$ smaller for longitudinal sound than for transverse sound. Electromechanical effect ======================== If an external electric field in excess of the threshold field $E_T$ of the nonlinear conductivity is applied in the chain direction, then the condensate is depinned and is able to contribute to various correlation functions[@MV1]. The best known example is of course the conductivity itself, but the situation is the same for the stress tensor correlator as well. The collective contribution to $\langle[\tau,\tau]\rangle$ can be obtained if we allow both the stress tensor and the density to couple to the phason (in the previous section this coupling was blocked due to pinning).In this case the stress tensor correlator has another contribution $\langle[\tau,\tau]\rangle^{coll}$ in addition to the one calculated in the previous section, namely: $$\langle[\tau,\tau]\rangle^{coll}={U\langle[\tau,\delta\Delta] \rangle^{Coul}\langle[\delta\Delta,\tau]\rangle^{Coul}\over 1-U\langle[\delta\Delta,\delta\Delta]\rangle^{Coul}}.$$ Here $U$ is the on-site Coulomb repulsion responsible for the formation of the SDW state (for CDW it should be replaced by the phonon propagator, but that does not affect our conclusions), $\delta\Delta$ is the phase fluctuation of the order parameter, and $\langle[A,B]\rangle^{Coul}$ is the correlation function of quantities $A$ and $B$ including the effect of the long-range Coulomb interaction (like in Eq.(8)). As we have seen earlier, in the long wavelength limit this yields: $$\langle[A,B]\rangle^{Coul}=\langle[A,B]\rangle_0-\langle[A,n] \rangle_0\langle[n,B]\rangle_0/\langle[n,n]\rangle_0.$$ First we consider if the allowed coupling to the phason actually takes place for various sound waves. According to Eq.(18) we need to examine $\langle[\tau,\delta\Delta]\rangle^{Coul}$, which is given by Eqs.(19) and (10) as $$\langle[\tau,\delta\Delta]\rangle^{Coul}=\langle[\tau,\delta \Delta]\rangle_0-\langle\tau(\varphi)\rangle_{\varphi}\langle [n,\delta\Delta]\rangle_0.$$ The density-phason correlator $\langle[n,\delta\Delta]\rangle_0$ was evaluated in [@VM1]. Here we only reiterate that result in the limit of experimental interest $lq\ll c_0/v_F$ (dynamic limit): $$\langle[n,\delta\Delta]\rangle_0=iN_F{\langle\zeta(\varphi) \rangle_{\varphi}\over 2\Delta}f_d,$$ where in our two dimensional geometry the wavenumber ${\bf q}$ appears in $\zeta(\varphi)=v_Fq_x+\sqrt{2}v_yq_y\cos\varphi$, and the condensate density in the dynamic limit is given by: $$\begin{aligned} f_d(T\rightarrow T_c)=&{\Delta^2\over 2\pi T\Gamma_B} \psi^\prime({1\over 2}+{\Gamma\over 2\pi T})\approx {\pi\Delta^2\over 4T\Gamma_B}\nonumber\\ f_d(T\rightarrow 0)=&1.\end{aligned}$$ We recall that the same $f_d$ appears in the current-phason correlator $\langle[j,\delta\Delta]\rangle_0$, as well as in the phason propagator[@MV2]. Note that $f_d$ increases from zero much faster below $T_c$ than $\tilde f$ does, and that at zero temperature it saturates exactly to $1$. The stress tensor-phason correlator can be calculated similarly: $$\langle[\tau,\delta\Delta]\rangle_0=iN_F{\langle\tau(\varphi) \zeta(\varphi)\rangle_{\varphi}\over 2\Delta}f_d.$$ Now we shall examine Eq.(20) for different sound waves in order to determine if there is a collective contribution to the corresponding stress tensor correlator. We consider longitudinal and transverse sound waves propagating in the ${\bf x}$ and ${\bf y}$ directions. Clearly, the second (screening) term in Eq.(20) is nonzero only for the longitudinal sound propagating in the chain direction (${\bf q}\parallel {\bf u} \parallel {\bf x}$), in which case it completely cancels the first term, leading to no collective contribution. The other longitudinal sound propagating perpendicular to the chains (${\bf q}\parallel {\bf u}\parallel {\bf y}$) does not couple to the phason either, because $\langle\tau_{yy}(\varphi)\cos\varphi\rangle_{\varphi} =0$ (see Eq.(7)). The coupling of the transverse wave propagating in the chain direction (${\bf q}\parallel {\bf x}$ and ${\bf u}\parallel {\bf y}$) is controlled by $\langle \tau_{xy}(\varphi)\rangle_{\varphi}=0$, yielding again no collective contribution. This means that in all of the above three cases there will be no electromechanical effect. For the rest of this section we will concentrate on the only interesting case, when the transverse sound propagates perpendicular to the chains (${\bf q}\parallel {\bf y}$ and ${\bf u}\parallel {\bf x}$). In this case there will be coupling to the phason, since $$\langle[\tau_{xy},\delta\Delta]\rangle^{Coul}= \langle[\tau_{xy},\delta\Delta]\rangle_0= iN_F{f_d\over 2\Delta}mv_Fv_y^2q_y$$ is nonzero. Now we have to consider the denominator in Eq.(18). Since $\langle[n,\delta\Delta]\rangle_0=0$ for $q_x=0$, therefore $\langle[\delta\Delta,\delta\Delta]\rangle^{Coul}= \langle[\delta\Delta,\delta\Delta]\rangle_0$, and we can use the result for the phason propagator calculated in [@VM1], which in our case reduces to $$1-U\langle[\delta\Delta,\delta\Delta]\rangle_0= {UN_Ff_d\over (2\Delta)^2}[(v_yq_y)^2-i\omega\Gamma_{ph}].$$ Here $\Gamma_{ph}$ is the phason damping rate and is given by $$\begin{aligned} \Gamma_{ph}(T\rightarrow T_c)=&2\Gamma_B\nonumber\\ \Gamma_{ph}(T\rightarrow 0)=&{8u_0^2T\Gamma_B\over 3\alpha^{4/3}G}e^{-G/T}.\end{aligned}$$ The phason damping freezes out for low temperature, and approaches $2\Gamma_B$ at $T_c$. Note the discontinuity in $\Gamma_{ph}$ at $T_c$ (approaching from above $\Gamma_{ph} \approx 2\pi^3T/7\zeta(3)$), which is the consequence of the finite order parameter $\Delta$ below $T_c$ exceeding almost immediately the energy scale set by $\omega$ and $vq$. We are now able to write down the total correlation function for this sound wave in the unpinned case. Using Eqs.(11),(18), (24) and (25) we obtain $$\begin{aligned} \langle[\tau_{xy},\tau_{xy}]\rangle=&N_F(mv_Fv_y)^2 \times\nonumber\\ \times&\left [ {i\tilde\Gamma_{qp}(1-\tilde f)\over \omega+i\tilde \Gamma_{qp}}+{(v_yq_y)^2f_d\over(v_yq_y)^2-i\omega\Gamma_{ph}} \right ].\end{aligned}$$ The above equation means that the collective contribution (the second term on the right hand side) due to the moving condensate recovers some of the screening of the ion motion lost because of the decrease in the number of quasi particles. In fact, at zero temperature it overcompensates somewhat, since $f_d>\tilde f$. Therefore we expect the electromechanical effect for the transverse sound polarized in the chain direction only. According to Eq.(27), close to $T_c$ the electromechanical effect on the sound velocity should be much smaller than the temperature effect, while at low temperatures the softening is somewhat bigger than the hardening was upon cooling. Although the collective contribution in Eq.(27) does have a frequency dependence, it does not appear to describe the suppresion of the electromechanical effect when the frequency is increased[@Xi3]. This is rather puzzling, although the above results for sound propagation may not translate literally to the flexural experiment. Conclusions =========== We have derived for the first time the appropriate stress tensor for quasi-one dimensional electron systems. Under conditions of a sound velocity measurement ($lq\ll 1$) the long-range Coulomb interaction has a simple role to suppress the $s$-wave channel as in the theory of electronic Raman scattering. In highly anisotropic systems like Bechgaard salts the increase of the sound velocity in the density wave state is two orders of magnitude smaller for longitudinal sound waves than for transverse ones. We also find, that a sound wave with polarization perpendicular to the chain direction can not couple to the phason, because the density wave condensate can only move parallel to the chains. The coupling of the longitudinal sound propagating in the chain direction is screened away by the Coulomb interaction, which leaves us only the transverse sound wave propagating perpendicular to the chains as the one which does couple to the phason, and shows the electromechanical effect. Recently Britel [*et.al.*]{} measured the elastic constant $c_{44}$ of (Ta$_{1-x}$Nb$_x$Se$_4$)$_2$I at 15MHz in the geometry ${\bf u}\parallel {\bf x}$[@Mon], and found a relative reduction of order $10^{-4}$ in the presence of an electric field approximately $10E_T$. This seems to be consistent with our analysis, although the observed effect appears to be a little too small. Acknowledgments {#acknowledgments .unnumbered} =============== This publication is sponsored by the U.S.-Hungarian Science and Technology Joint Fund in cooperation with the National Science Foundation and the Hungarian Academy of Sciences under Project No. 264/92a, which enabled one of us (A.V.) to enjoy the hospitality of USC. This work is also supported in part by the National Science Foundation under Grant No. DMR92-18317 and by the Hungarian National Research Fund under Grants No. OTKA15552 and T4473. Appendix {#appendix .unnumbered} ======== We evaluate here the density correlation function $\langle[n,n] \rangle_0^{no vertex}$ (Eqs.(12)-(14) in the text). We start with the corresponding thermal product (See [@VM1]): $$\langle[n,n]\rangle_0^{no vertex}=N_F[1-\pi T\sum_n F(iu_n,iu_n^\prime)],$$ where $N_F$ is the density of states, and $u_n$ and $u_n^\prime$ are related to the Matsubara frequencies $\omega_n$ and $\omega_n^\prime= \omega_{n-\nu}$ by $$\omega_n/\Delta=u_n[1-\alpha(u_n^2+1)^{-1/2}],$$ with $\Delta$ the order parameter, $\alpha=\Gamma/\Delta$ and $\Gamma=\Gamma_F+\Gamma_B/2$ a combination of the impurity forward and backscattering rates. Neglecting vertex corrections in the relevant formulas in [@VM1] leads to: $$F(u,u^\prime)={1+{1+uu^\prime\over (1-u^2)^{1/2}(1-u^{\prime 2}) ^{1/2}}\over \Delta[(1-u^2)^{1/2}+(1-u^{\prime 2})^{1/2}]},$$ where $u$ and $u^\prime$ are analytic continuations of $iu_n$ and $iu_n^\prime$, and in the absence of the diffusion pole the wavenumber ${\bf q}$ was already set to zero. While evaluating the correlation function we follow the standard method[@Zit]. Expanding up to linear order in $\omega$ we obtain $$\langle[n,n]\rangle_0^{no vertex}=N_F(1-\tilde f+i\omega I),$$ where $$\tilde f={\pi T\over\Delta}\sum_n(u_n^2+1)^{-3/2},$$ and $$\begin{aligned} I=&{1\over 2\Delta}\int_G^\infty{dE\over 2T}\cosh^{-2}\left ({E\over 2T}\right )\times\nonumber\\ \times&\left [{h^\prime\over {\rm Re}(1-u^2)^{1/2}}-{\rm Re} (1-u^2)^{-3/2}\right ].\end{aligned}$$ Here $h^\prime=(1/2)[1+(|u|^2+1)/|u^2-1|]$, and $G=\Delta u_0^3$ is the gap with $u_0^2=1-\alpha^{2/3}$. The above equations can be evaluated in two limiting cases, with temperature close to $T_c$ and close to zero, and can be brought to the form of Eqs.(12)-(14) in the text. J. W. Brill and W. Roark, Phys. Rev. Lett. [**53**]{}, 846 (1984); J. W. Brill, W. Roark and G. Minton, Phys. Rev. B [**33**]{}, 6831 (1986). G. Mozurkewich, P. M. Chaikin, W. G. Clark and G. Gruner, Sol. State Commun. [**56**]{}, 421 (1985). X.-D. Xiang and J. W. Brill, Phys. Rev. B [**36**]{}, 2969 (1987). X.-D. Xiang and J. W. Brill, Phys. Rev. B [**39**]{}, 1290 (1989). R. L. Jacobsen and G. Mozurkewich, Phys. Rev. B [**39**]{}, 10913 (1989). S. E. Brown, B. Alavi, G. Gruner and K. Bartholomew, Phys. Rev. B [**46**]{}, 10483 (1992). X.-D. Xiang and J. W. Brill, Phys. Rev. Lett. [**63**]{}, 1853 (1989). R. J. Cava [*et.al.*]{}, Phys. Rev. B [**30**]{}, 3228 (1984); S. Tomic, N. Biskup and A. Omerzu (unpublished). K. Maki and A. Virosztek, Phys. Rev. B [**36**]{}, 2910 (1987). A. Virosztek and K. Maki, Phys. Rev. B [**41**]{}, 7055 (1990). L. P. Kadanoff and I. I. Falko, Phys. Rev. [**136**]{}, 1170 (1964). T. Tsuneto, Phys. Rev. [**121**]{}, 402 (1961). A. Virosztek and K. Maki, Synth. Metals [**70**]{}, 1283 (1995). See for example K. Yamaji, J. Phys. Soc. Jpn. [**51**]{}, 2787 (1982). A. A. Abrikosov and V. M. Genkin, Zh. Eksp. Teor. Fiz. [**65**]{}, 842 (1973) \[Sov. Phys. JETP [**38**]{}, 417 (1974)\]. K. Maki and A. Virosztek, Phys. Rev. B [**39**]{}, 2511 (1989). In this reference $f_d$ was called $\tilde f_0$. R. Britel, M. Saint-Paul, P. Monceau and F. Levy, J. de Physique IV, Colloque 3, C2-75 (1993). J. Zittartz, Phys. Rev. [**165**]{}, 605 (1968).

--- abstract: 'We consider the dynamics of a small spherical particle driven through an unbounded viscoelastic shear flow by an external force. We give analytical solutions to both the mobility problem (velocity of forced particle) and the resistance problem (force on fixed particle), valid to second order in the dimensionless Deborah and Weissenberg numbers, which represent the elastic relaxation time of the fluid relative to the rate of translation and the imposed shear rate. We find a shear-induced lift at $O({\ensuremath{\textrm{Wi}}})$, a modified drag at $O({\ensuremath{\textrm{De}}}^2)$ and $O({\ensuremath{\textrm{Wi}}}^2)$, and a second lift that is orthogonal to the first, at $O({\ensuremath{\textrm{Wi}}}^2)$. The relative importance of these effects depends strongly on the orientation of the forcing relative to the shear. We discuss how these forces affect the terminal settling velocity in an inclined shear flow. We also describe a new basis set of symmetric Cartesian tensors, and demonstrate how they enable general tensorial perturbation calculations such as the present theory. In particular this scheme allows us to write down a solution to the inhomogenous Stokes equations, required by the perturbation expansion, by a sequence of algebraic manipulations well suited to computer implementation.' author: - Jonas Einarsson - Bernhard Mehlig title: Spherical particle sedimenting in weakly viscoelastic shear flow --- Introduction ============ In this paper we consider the mobility of a small spherical particle driven through an unbounded viscoelastic shear flow by an external force. In a Newtonian fluid the velocity of the particle is determined by the balance between the Stokes drag and the external force, and it is unaffected by the shear flow because of the linearity of Stokes equations. But in a viscoelastic fluid the disturbance flow around the particle interacts non-linearly with the shear flow to induce viscoelastic stresses. As a consequence the mobility depends non-linearly on the forcing and the shear flow. A viscoelastic shear flow can reduce the terminal velocity of a sphere when the applied shear flow is perpendicular to gravity [@van_den_brule_effects_1993; @housiadas_drag_2012; @padhy_simulations_2013; @davino2015]. This so-called cross-shear flow is a model system for transport of particles in vertical cracks induced by hydraulic fracturing [@barbati_complex_2016]. Experiments by @van_den_brule_effects_1993 first demonstrated that a cross-shear flow strongly reduces the settling velocity, and that fluid elasticity is the dominant mechanism. Recently, numerical simulations by @padhy_simulations_2013 [@padhy_effect_2013] verified an increased drag on a sphere translating through a cross-shear flow, and showed that the experimental observation is explained by a combination of viscoelasticity and the effects of the nearby walls in the experiment. Calculations by @housiadas_drag_2012 [@housiadas_rheological_2014] demonstrate that the drag is increased also in an unbounded viscoelastic cross-shear flow. These studies concern settling in a cross-shear flow in which the gravity acts along the vorticity axis. In this case the physical system is invariant under a 180$^\circ$ rotation around the vorticity axis. This symmetry was exploited in both the analytical and numerical calculations to reduce the number of variables [@housiadas_drag_2012; @housiadas_rheological_2014; @padhy_simulations_2013]. In particular the only relevant force is the drag force, and the particle only rotates around the vorticity axis. The effect of the shear on settling is substantial in this symmetrical case, and this fact raises new questions: How are the dynamics affected when gravity acts at an angle to the vorticity? Are there additional forces and torques when the symmetry is broken? How does the drag change as the angle between gravity and flow vorticity changes? In this paper we calculate the effect of an unbounded shear flow on the terminal particle velocity for any orientation of the external force relative to the shear. We must consequently abandon the simplifications of the symmetrical case. We must allow for lift forces in the flow-shear plane, and for rotation around any axis. Further, since viscoelasticity is a non-linear effect, it is not possible to construct the general result as a linear combination of results for two independent directions. Therefore we must solve the general perturbation problem for the flow velocity ${\boldsymbol{u}}$ and viscoelastic stress tensor ${\boldsymbol{\Pi}}$ around a translating and rotating particle in a shear flow. Our calculation relies on a perturbation theory for weak elasticity, valid to second order in the Deborah and Weissenberg numbers. These dimensionless numbers relate the elastic relaxation time of the fluid relative to the rate of translation and the shear rate. @brunn1977 [@brunn1977a] and @vishnampet2012 considered the first order of this problem, and both found lateral migration, although their detailed results do not agree with each other. The first part of our calculation is an independent check of their results, which we return to in [Section \[sec:mobility\]]{}. @housiadas2011a and @davino2008 considered the angular velocity of the sphere in the absence of an external force, and @leslie1961 calculated the drag on a sphere in absence of shear flow. These two results coincide with our theory in their respective limits. We solve the problem in tensorial form. Our solution does not refer to any coordinate representation such as spherical coordinates. Since the governing equations and boundary conditions are tensorial in nature, this substantially simplifies the calculations. All steps of the calculation are algebraic, and therefore well suited to computer implementation. To achieve this we introduce a new basis set of symmetric, rank-$n$ Cartesian tensors. We describe these tensors and how to calculate with them in some detail in [Section \[sec:Ttensors\]]{}, because we expect that they will be useful for treating other problems too. We present two related calculations. The first is the mobility problem, where we impose an external force ${\boldsymbol{F}}^{\mathrm{ext}}$ on the particle, and compute the resulting particle velocity ${\boldsymbol{v}}({\boldsymbol{F}}^{\mathrm{ext}})$. This corresponds directly to the experimental protocol of for example @van_den_brule_effects_1993, where they release a sphere in a cylindrical Couette device and measure the steady settling velocity. The other question is the resistance problem, where we prescribe the particle velocity ${\boldsymbol{v}}$, and compute the resulting force ${\boldsymbol{F}}({\boldsymbol{v}})$ exerted by the fluid on the particle. This approach corresponds to the calculations by @housiadas_rheological_2014 and numerical simulations by @padhy_effect_2013. The two are related, because given the solution to the mobility problem ${\boldsymbol{v}}({\boldsymbol{F}}^{\mathrm{ext}})$, and the solution of the resistance problem ${\boldsymbol{F}}({\boldsymbol{v}})$, it must hold that ${\boldsymbol{F}}({\boldsymbol{v}})=-{\boldsymbol{F}}^{\mathrm{ext}}$. In this paper we solve both the mobility problem and the resistance problem for a freely rotating spherical particle in an unbounded viscoelastic shear flow, with no restriction on the direction of ${\boldsymbol{v}}$ or ${\boldsymbol{F}}^{\mathrm{ext}}$ relative to the shear. The rest of this paper is organized as follows. In [Section \[sec:problem\]]{} we describe the problem, give the governing equations, and describe how we apply the Lorentz reciprocal theorem. In [Section \[sec:Ttensors\]]{} we explain our algebraic solution of the inhomogeneous Stokes equation in terms of Cartesian tensors, and summarise their algebraic properties. We summarise our calculation and give the final result in [Section \[sec:results\]]{}. We discuss the results and conclude in [Section \[sec:discussion\]]{}. Problem formulation =================== [\[sec:problem\]]{} Equation of motion and dimensionless parameters ----------------------------------------------- We consider the steady-state motion of a spherical particle of radius $a$, suspended in a viscoelastic fluid and subject to an external force ${{\boldsymbol{F}}}^{\mathrm{ext}}$. For concreteness we may think of the gravitational force ${{\boldsymbol{F}}}^{\mathrm{ext}} = 4\pi a^3(\rho_p-\rho_f){\boldsymbol{g}}/3$. The particle moves with center-of-mass velocity ${{\boldsymbol{v}}}$, and rotates with angular velocity ${{\boldsymbol{\omega}}}$. Far away from the particle the flow is a simple shear flow $$\begin{aligned} {{\boldsymbol{u}}}^\infty&={{\boldsymbol{\Omega}}}\times{{\boldsymbol{r}}} + {{\ensuremath{\amsbb{S}}}}{{\boldsymbol{r}}}\,,\end{aligned}$$ where ${{\boldsymbol{\Omega}}}$ is half the flow vorticity, and the symmetric tensor ${{\ensuremath{\amsbb{S}}}}$ is the rate of strain. In a simple shear flow the vorticity and strain are related by ${{\ensuremath{\amsbb{S}}}} {{\boldsymbol{\Omega}}}=0$ and $2|{{\boldsymbol{\Omega}}}|^2={\ensuremath{\textrm{Tr}\,}}{{\ensuremath{\amsbb{S}}}}{{\ensuremath{\amsbb{S}}}}$, in contrast to a general linear flow. We work in dimensionless variables. The length scale is given by the particle radius $a$. The time scale is given by the reciprocal of the imposed shear rate $s=\sqrt{2{\ensuremath{\textrm{Tr}\,}}{{\ensuremath{\amsbb{S}}}} {{\ensuremath{\amsbb{S}}}}}$, which also determines the scale of ${{\boldsymbol{u}}}^\infty$ to $sa$. The particle and disturbance flow velocities are nondimensionalized by the characteristic flow velocity $v_c$ past the particle. In the resistance problem, $v_c$ is simply the magnitude $|{{\boldsymbol{v}}}|$ of the imposed velocity ${{\boldsymbol{v}}}$. In the mobility problem we estimate the characteristic speed by $v_c={F}^\mathrm{ext}/a\mu$, related to the terminal velocity in Stokes flow under an external force of magnitude ${F}^\mathrm{ext}$. Here $\mu$ is the total viscosity, defined precisely in conjunction with the constitutive equations below. Stresses are made dimensionless by $v_c\mu/a$, and forces by $v_c \mu a$. In the remainder of this paper all quantities are dimensionless: $t'=s t$, ${\boldsymbol{r}}'={{\boldsymbol{r}}}/a$, ${\ensuremath{\amsbb{S}}}' = {{\ensuremath{\amsbb{S}}}}/s$, ${\boldsymbol{\Omega}}' = {{\boldsymbol{\Omega}}}/s$, ${\boldsymbol{F}}' = {{\boldsymbol{F}}}/(v_c\mu a)$, and so forth. We drop the primes since all quantities are dimensionless. It follows that there are two dimensionless parameters that govern this problem, corresponding to the translational and rotational motion of the particle compared to the relaxation time $\lambda$ of the viscoelastic fluid. The Deborah number ${\ensuremath{\textrm{De}}}= \lambda v_c/a$ is associated with the time scale of convective flow over the particle size. The Weissenberg number ${\ensuremath{\textrm{Wi}}}=\lambda s$ is associated with the shear rate. The ratio $\alpha={\ensuremath{\textrm{Wi}}}/{\ensuremath{\textrm{De}}}$ measures the relative importance of the imposed shear to the translational motion. The perturbation theory in this paper is valid in the limit ${\ensuremath{\textrm{De}}}\ll1$ and ${\ensuremath{\textrm{Wi}}}\ll1$. This implies that the elastic part of the fluid relaxes quickly relative to the rate at which it is deformed by the moving particle and the shear flow. In the following we expand in ${\ensuremath{\textrm{De}}}$, and treat the ratio $\alpha$ as an $O(1)$-quantity. But in the end we give the result in terms of ${\ensuremath{\textrm{De}}}$ and ${\ensuremath{\textrm{Wi}}}$. We note that the choice between $1/s$ and $a/v_c$ for the characteristic timescale is arbitrary up to factors of $\alpha = {\ensuremath{\textrm{Wi}}}/{\ensuremath{\textrm{De}}}$, which we assume to be $O(1)$. We neglect the effects of fluid inertia. This requires that the viscous relaxation time of the fluid is shorter than the elastic relaxation time $\lambda$, so that we can neglect effects of inertio-elastic coupling. More precisely the particle Reynolds number ${\ensuremath{\textrm{Re}_{\rm p}}}=\rho_fv_ca/\mu\ll{\ensuremath{\textrm{De}}}$. This condition is equivalent to $\rho_fa^2/\mu\ll\lambda$. We write down the dimensionless governing equations for with respect to a frame moving with the steady center-of-mass velocity ${\boldsymbol{v}}$. In this frame the fluid pressure $p$ and velocity ${\boldsymbol{u}}$ satisfy [\[eqn:floweq1\]]{} $$\begin{aligned} \nabla \cdot {\boldsymbol{\sigma}}&=0\,,\quad {\boldsymbol{\sigma}} = -p {\ensuremath{\amsbb{I}}} + (1-\mu_r) (\nabla {\boldsymbol{u}}+ (\nabla{\boldsymbol{u}}){\ensuremath{^{\rm T}}})+\mu_r{\boldsymbol{\Pi}}\,, {\label{eqn:floweq1a}}\\ \nabla\cdot{\boldsymbol{u}}&=0\,.{\label{eqn:floweq1b}}\end{aligned}$$ The viscoelastic stress tensor ${\boldsymbol{\Pi}}$ is modeled by the steady Oldroyd-B constituitive equations [@larson_constitutive_2013]. They describe a suspension of elastic dumbbells, which is one of the simplest models of an elastic polymer suspension that exhibits a normal stress difference in a shear flow. The equations are $$\begin{aligned} {\boldsymbol{\Pi}} + {\ensuremath{\textrm{De}}}\left[({\boldsymbol{u}}\cdot\nabla){\boldsymbol{\Pi}} - (\nabla {\boldsymbol{u}}){\boldsymbol{\Pi}} - {\boldsymbol{\Pi}} (\nabla {\boldsymbol{u}}){\ensuremath{^{\rm T}}}\right]=\nabla {\boldsymbol{u}} + (\nabla {\boldsymbol{u}}){\ensuremath{^{\rm T}}}{\label{eqn:oldBdef}}\,.\end{aligned}$$ The parameter $\mu_r=\mu_p/(\mu_s+\mu_p)$ is the relative contribution to the total viscosity from the elastic polymers, relative to the solvent viscosity $\mu_s$. We denote the total viscosity $\mu=\mu_s+\mu_p$. The flow problem in Eqns. [(\[eqn:floweq1\])]{} and [(\[eqn:oldBdef\])]{} is completed by the no-slip boundary condition on the particle surface $S_p$ and that it approaches ${\boldsymbol{u}}^\infty$ as $|{\boldsymbol{r}}|\to\infty$: $$\begin{aligned} {\boldsymbol{u}} &= \alpha {\boldsymbol{\omega}} \times {\boldsymbol{r}},\quad{\boldsymbol{r}} \in S_p\,,{\nonumber}\\ {\boldsymbol{u}} &\to \alpha {\boldsymbol{u}}^\infty-\, {\boldsymbol{v}}\,,\quad|{\boldsymbol{r}}|\to \infty\,. {\label{eqn:flowbc1}}\end{aligned}$$ The force and torque on the particle are given by $$\begin{aligned} {\boldsymbol{F}}-{\boldsymbol{F}}^{\mathrm{ext}} &= \int_{S_p} {\boldsymbol{\sigma}} \cdot {\boldsymbol{{\ensuremath{\textrm{d}}}S}}\,,\\ {\boldsymbol{T}} &= \int_{S_p} {\boldsymbol{r}} \times {\boldsymbol{\sigma}} \cdot {\boldsymbol{{\ensuremath{\textrm{d}}}S}}\,.{\label{eqn:forceandtorque}}\end{aligned}$$ We do not consider any external torques in this paper and so ${\boldsymbol{T}}^\mathrm{ext}=0$. We compute the forces on the particle, or the resulting velocities, with the Lorentz reciprocal theorem [@kim1991]. We outline how to apply it to our problem in the following Section. Reciprocal theorem ------------------ In the context of perturbation theory, the Lorentz reciprocal theorem relates certain integral quantities such as the force or torque to order $n+1$, given the detailed flow solution to only order $n$ [@kim1991; @leal1979]. For non-Newtonian flows in particular, the method has for example been used to calculate lateral migration [@ho_migration_1976], and orbit drift of non-spherical particles [@leal1975]. In this Section we state the theorem as applicable to our problem and notation. We denote the Newtonian part of the stress by ${\boldsymbol{\sigma}}^{N}=-p {\ensuremath{\amsbb{I}}} + (\nabla {\boldsymbol{u}}+ (\nabla{\boldsymbol{u}}){\ensuremath{^{\rm T}}})$, and the “extra stress” ${\boldsymbol{\sigma}}^{E}=\mu_r({\boldsymbol{\Pi}} - (\nabla {\boldsymbol{u}}+ (\nabla{\boldsymbol{u}}){\ensuremath{^{\rm T}}}))$, so that ${\boldsymbol{\sigma}} = {\boldsymbol{\sigma}}^{N}+{\boldsymbol{\sigma}}^{E}$. The flow equation of motion [(\[eqn:floweq1a\])]{} is therefore $$\begin{aligned} \nabla \cdot {\boldsymbol{\sigma}}^{N}=-\nabla \cdot {\boldsymbol{\sigma}}^{E}\,. \end{aligned}$$ The Lorentz reciprocal theorem for an arbitrary Stokes flow $({\boldsymbol{\tilde u}},{\boldsymbol{\tilde \sigma}})$ and the flow defined in [Eq. (\[eqn:floweq1\])]{} reads [@kim1991] $$\begin{aligned} \int_S {\boldsymbol{\tilde u}} \cdot {\boldsymbol{\sigma}}^{N} \cdot {\boldsymbol{{\ensuremath{\textrm{d}}}S}} = \int_S {\boldsymbol{u}} \cdot {\boldsymbol{\tilde\sigma}} \cdot {\boldsymbol{{\ensuremath{\textrm{d}}}S}} + \int_V {\boldsymbol{\tilde u}}\cdot \nabla \cdot {\boldsymbol{\sigma}}^{N}{{\ensuremath{\textrm{d}}}V}\,.{\label{eqn:rt01}}\end{aligned}$$ Here $V$ is any volume outside the particle, and $S$ denotes the surfaces bounding $V$. The vector ${\boldsymbol{{\ensuremath{\textrm{d}}}S}}={\boldsymbol{n}} {\ensuremath{\textrm{d}}}S$, where ${\boldsymbol{n}}$ is the surface normal pointing out of $V$. Using ${\boldsymbol{\sigma}}={\boldsymbol{\sigma}}^{N}+{\boldsymbol{\sigma}}^{E}$, it follows from [(\[eqn:rt01\])]{} that $$\begin{aligned} \int_S {\boldsymbol{\tilde u}} \cdot {\boldsymbol{\sigma}} \cdot {\boldsymbol{{\ensuremath{\textrm{d}}}S}} = \int_S {\boldsymbol{u}} \cdot {\boldsymbol{\tilde\sigma}} \cdot {\boldsymbol{{\ensuremath{\textrm{d}}}S}} + \int_S {\boldsymbol{\tilde u}} \cdot {\boldsymbol{\sigma}}^{E} \cdot {\boldsymbol{{\ensuremath{\textrm{d}}}S}} - \int_V {\boldsymbol{\tilde u}}\cdot \nabla \cdot {\boldsymbol{\sigma}}^{E}{{\ensuremath{\textrm{d}}}V}{\label{eqn:rt02}}\,.\end{aligned}$$ In the first two surface integrals in [(\[eqn:rt02\])]{} we identify the hydrodynamic force and torque on the particle, as given in [Eq. (\[eqn:forceandtorque\])]{}. We take ${\boldsymbol{\tilde u}}$ to be the Stokes flow around a spherical particle translating with velocity ${\boldsymbol{\tilde v}}$ and rotating with angular velocity ${\boldsymbol{\tilde \omega}}$ in an otherwise quiescent fluid. We write this auxiliary flow as ${\boldsymbol{\tilde u}}={\ensuremath{\amsbb{M}}}_{{\boldsymbol{v}}}{\boldsymbol{\tilde v}}+{\ensuremath{\amsbb{M}}}_{{\boldsymbol{\omega}}}{\boldsymbol{\tilde \omega}}$, and we also know that ${\boldsymbol{\tilde F}}=-6\pi\tilde{{\boldsymbol{v}}}$ and ${\boldsymbol{\tilde T}}=-8\pi\tilde{{\boldsymbol{\omega}}}$. Finally, upon taking the size of the volume $V$ to infinity, and inserting the boundary conditions [(\[eqn:flowbc1\])]{} into [Eq. (\[eqn:rt02\])]{}, we find $$\begin{aligned} {\boldsymbol{\tilde v}}\cdot({\boldsymbol{F}}-{\boldsymbol{F}}^{\mathrm{ext}})+{\boldsymbol{\tilde \omega}}\cdot{\boldsymbol{T}} = -6\pi{\boldsymbol{\tilde v}}\cdot{\boldsymbol{v}} + 8\pi\alpha{\boldsymbol{\tilde \omega}}\cdot({\boldsymbol{\Omega}}-{\boldsymbol{\omega}}) + {\boldsymbol{\tilde v}}\cdot\int_{S_p} {\boldsymbol{\sigma}}^{E} \cdot {\boldsymbol{{\ensuremath{\textrm{d}}}S}} + {\boldsymbol{\tilde \omega}}\cdot\int_{S_p} {\boldsymbol{r}} \times ({\boldsymbol{\sigma}}^{E} \cdot {\boldsymbol{{\ensuremath{\textrm{d}}}S}}) {\nonumber}\\ - {\boldsymbol{\tilde v}}\cdot\int_V {\ensuremath{\amsbb{M}}}_{{\boldsymbol{v}}}{\ensuremath{^{\rm T}}}\nabla \cdot {\boldsymbol{\sigma}}^{E}{{\ensuremath{\textrm{d}}}V} - {\boldsymbol{\tilde \omega}}\cdot\int_V {\ensuremath{\amsbb{M}}}_{{\boldsymbol{\omega}}}{\ensuremath{^{\rm T}}}\nabla \cdot {\boldsymbol{\sigma}}^{E}{{\ensuremath{\textrm{d}}}V}\,.{\label{eqn:rt}}\end{aligned}$$ The surface integrals at infinity do not contribute, and the remaining surface integrals are only over the particle surface. This is well known for the disturbance quantities, say ${\boldsymbol{\sigma}}^E({\boldsymbol{u}})-{\boldsymbol{\sigma}}^E({\boldsymbol{u}}^\infty)$, because the integrands decay faster than $1/r^2$ [@leal1979]. The potentially problematic terms are those from ${\boldsymbol{\sigma}}^E({\boldsymbol{u}}^\infty)$ that are independent of ${\boldsymbol{r}}$. But ${\ensuremath{\amsbb{M}}}_{{\boldsymbol{v}}}$ is an even function of ${\boldsymbol{r}}$, so that surface integral vanishes by symmetry. On the other hand, ${\ensuremath{\amsbb{M}}}_{{\boldsymbol{\omega}}}$ integrated over the sphere is an antisymmetric tensor that vanishes upon contraction with the symmetric stress tensor. Because ${\boldsymbol{\tilde v}}$ and ${\boldsymbol{\tilde \omega}}$ may be chosen arbitrarily, we have two separate theorems for the force and torque: $$\begin{aligned} ({\boldsymbol{F}}-{\boldsymbol{F}}^{\mathrm{ext}}) = -6\pi{\boldsymbol{v}} + \int_{S_p} {\boldsymbol{\sigma}}^{E} \cdot {\boldsymbol{{\ensuremath{\textrm{d}}}S}} - \int_V {\ensuremath{\amsbb{M}}}_{{\boldsymbol{v}}}{\ensuremath{^{\rm T}}}\nabla \cdot {\boldsymbol{\sigma}}^{E}{{\ensuremath{\textrm{d}}}V}{\label{eqn:rt_trans}}\,.\end{aligned}$$ $$\begin{aligned} {\boldsymbol{T}} = 8\pi\alpha({\boldsymbol{\Omega}}-{\boldsymbol{\omega}}) + \int_{S_p} {\boldsymbol{r}} \times ({\boldsymbol{\sigma}}^{E} \cdot {\boldsymbol{{\ensuremath{\textrm{d}}}S}}) - \int_V {\ensuremath{\amsbb{M}}}_{{\boldsymbol{\omega}}}{\ensuremath{^{\rm T}}}\nabla \cdot {\boldsymbol{\sigma}}^{E}{{\ensuremath{\textrm{d}}}V}{\label{eqn:rt_torque}}\,.\end{aligned}$$ In this paper we do not consider external torques, and therefore ${\boldsymbol{T}}=0$ implies $$\begin{aligned} {\boldsymbol{\omega}} = {\boldsymbol{\Omega}} + \frac{1}{8\pi\alpha}\int_{S_p} {\boldsymbol{r}} \times ({\boldsymbol{\sigma}}^{E} \cdot {\boldsymbol{{\ensuremath{\textrm{d}}}S}}) - \frac{1}{8\pi\alpha}\int_V {\ensuremath{\amsbb{M}}}_{{\boldsymbol{\omega}}}{\ensuremath{^{\rm T}}}\nabla \cdot {\boldsymbol{\sigma}}^{E}{{\ensuremath{\textrm{d}}}V}{\label{eqn:rt_rot}}\,.\end{aligned}$$ The reciprocal theorem may be used to solve either the resistance problem, or the mobility problem. For the resistance problem we take ${\boldsymbol{F}}^{\mathrm{ext}}=0$ and use [Eq. (\[eqn:rt\_trans\])]{}. For the mobility problem we require the total force ${\boldsymbol{F}}=0$ and find $$\begin{aligned} {\boldsymbol{v}} = \frac{1}{6\pi}{\boldsymbol{F}}^{\mathrm{ext}} + \frac{1}{6\pi}\int_{S_p} {\boldsymbol{\sigma}}^{E} \cdot {\boldsymbol{{\ensuremath{\textrm{d}}}S}} - \frac{1}{6\pi}\int_V {\ensuremath{\amsbb{M}}}_{{\boldsymbol{v}}}{\ensuremath{^{\rm T}}}\nabla \cdot {\boldsymbol{\sigma}}^{E}{{\ensuremath{\textrm{d}}}V}{\label{eqn:rt_v}}\,.\end{aligned}$$ The integrands in Eqs. (\[eqn:rt\_trans\]-\[eqn:rt\_v\]) are functions of the yet unknown ${\boldsymbol{\sigma}}^{E}$. In [Section \[sec:results\]]{} we evaluate these integrals by calculating ${\boldsymbol{\sigma}}^E$ perturbatively. As a final note, [Eq. (\[eqn:rt\_trans\])]{} is equivalent to the integral theorem @ho_migration_1976 used to compute the lateral drift of a spherical particle in wall-bounded flow. Their Eq. (2.22) follows from our [Eq. (\[eqn:rt\_trans\])]{} because $$\begin{aligned} \int_S {\boldsymbol{\tilde u}} \cdot {\boldsymbol{\sigma}}^{E} \cdot {\boldsymbol{{\ensuremath{\textrm{d}}}S}} - \int_V {\boldsymbol{\tilde u}}\cdot \nabla \cdot {\boldsymbol{\sigma}}^{E}{{\ensuremath{\textrm{d}}}V} = \int_V {\boldsymbol{\sigma}}^{E}:\nabla{\boldsymbol{\tilde u}}{{\ensuremath{\textrm{d}}}V}\,.\end{aligned}$$ However, we evaluate the two integral contributions in [Eq. (\[eqn:rt\_trans\])]{} separately, because they give the contributions from two different physical mechanisms. The surface integral represents the extra polymer stress acting directly on the particle surface (see [Eq. (\[eqn:v2polymer\])]{} and [(\[eqn:f2polymer\])]{} in [Section \[sec:results\]]{}). The volume integral represents the indirect effect that the polymer stress modifies the flow field, which in turn modifies the viscous stress on the particle. Method: T-tensors ================= [\[sec:Ttensors\]]{} In this Section we introduce a basis set of symmetric rank-$n$ Cartesian tensors $T^{nl}_{i_1i_2..i_n}$. Each of these basis tensors is a linear combination of spherical harmonics $Y_l^m$ with a particular value of the angular momentum quantum number $l$, but different modes $m$. Therefore the Cartesian tensors share many useful properties with the spherical harmonics. For example, surface integrals vanish unless $l=0$, tensors with different values of $l$ are orthogonal with respect to integration over the unit sphere, and they have known Fourier transforms. Because of their direct relation to the spherical harmonics, the $T$-tensors are an alternative basis for Lamb’s general solution for Stokes flow [@kim1991]. But as explained in detail below, a rank-$n$ $T$-tensor is also closely related to the rank-$n$ polyad $\hat r_{i_1}\hat r_{i_2}..\hat r_{i_n}$ of a unit vector ${\boldsymbol{\hat r}}$. Together with the radial functions $1/r^m$ these polyads are the building blocks of the familiar multipole expansion for Stokes flow, for example the Stokeslet $\delta_{ij}/r + \hat r_i\hat r_j/r$, or the rotlet ${\ensuremath{\varepsilon}}_{ijk}\hat r_j/r^2$. Therefore the $T$-tensors stand as a new alternative between Lamb’s general solution in spherical coordinates, and the Cartesian multipole expansion. Although any calculation may in principle be performed in any of these representations, we found that the basis described here is suitable for implementation in computer algebra. In particular it enables us to write down particular solutions to inhomogenous Stokes equations in tensorial form, without any explicit coordinate representation, and without explicitly solving differential equations. In this Section we use index notation to avoid any ambiguity. When appropriate we use the vector notation ${\boldsymbol{T}}^{nl}$, remembering that ${\boldsymbol{T}}^{nl}$ is rank $n$ and symmetric in all indices. Definition ---------- We consider the rank-$n$ polyad $\hat r_{i_1}\hat r_{i_2}..\hat r_{i_n}$ of a unit vector ${\boldsymbol{\hat r}}$. Any given polyad is a smooth function defined on the sphere, and may be expanded in the spherical harmonics $$\begin{aligned} \hat r_{i_1}\hat r_{i_2}..\hat r_{i_n} &= \sum_{l=0}^{\infty}\sum_{m=-l}^{m=l}c^{nlm}_{i_1i_2..i_n}Y_l^m\,.{\label{eqn:Tdefexpansion1}}\end{aligned}$$ For our present purposes it is not necessary to calculate the expansion coefficients $c^{nlm}_{i_1i_2..i_n}$, but we may deduce the following important fact. The left hand side is a polynomial of order $n$, and every term in the sum on the right hand side is a polynomial of order $l$, so we conclude that $c^{nlm}_{i_1i_2..i_n}=0$ if $l>n$. Therefore $$\begin{aligned} \hat r_{i_1}\hat r_{i_2}..\hat r_{i_n} &= \sum_{l=0}^{n}\sum_{m=-l}^{m=l}c^{nlm}_{i_1i_2..i_n}Y_l^m\,,{\label{eqn:Tdefexpansion2}}\end{aligned}$$ We define the tensor $T^{nl}_{i_1i_2..i_n}$ as the inner sum in [Eq. (\[eqn:Tdefexpansion2\])]{}, so that $$\begin{aligned} T^{nl}_{i_1i_2..i_n} &\equiv \sum_{m=-l}^{m=l}c^{nlm}_{i_1i_2..i_n}Y_l^m\,,{\label{eqn:Tdefinition}} \intertext{and therefore by construction} \hat r_{i_1}\hat r_{i_2}..\hat r_{i_n} &= \sum_{l=0}^{n}T^{nl}_{i_1i_2..i_n}\,.{\label{eqn:polyadexpansion}}\end{aligned}$$ We complete the definition by $T^{00} = 1$. Properties of the $T$-tensors ----------------------------- [*Symmetry.*]{} From the expansion [(\[eqn:polyadexpansion\])]{} it is clear that any $T$-tensor is symmetric in all indices. By definition only tensors with $l \leq n$ are non-zero: $$\begin{aligned} T^{nl}_{i_1i_2..i_n}&=0\,,\quad l > n\,. \end{aligned}$$ Further, the polynomial in the left hand side of [Eq. (\[eqn:polyadexpansion\])]{} has parity $(-1)^n$ under inversion of ${\boldsymbol{\hat r}}$, and every term on the right hand side has parity $(-1)^l$. Therefore ${\boldsymbol{T}}^{nl}$ is non-zero only if both $n$ and $l$ are even, or if both $n$ and $l$ are odd. [*Integrals.*]{} Any two $T$-tensors are orthogonal with respect to integrals over the unit sphere $S$, because the spherical harmonics enjoy this property: $$\begin{aligned} \int_S T^{nl_1}_{i_1i_2..i_n}T^{ml_2}_{i_1i_2..i_m}{\ensuremath{\textrm{d}}}S = 0,\quad l_1 \neq l_2\,.{\label{eqn:orthogonality}}\end{aligned}$$ It follows that $$\begin{aligned} \int_S T^{nl}_{i_1i_2..i_n}{\ensuremath{\textrm{d}}}S = \int_S T^{nl}_{i_1i_2..i_n}T^{00}{\ensuremath{\textrm{d}}}S = 0\,,\quad l\neq0\,.{\label{eqn:l0integral}}\end{aligned}$$ [*Cartesian rank.*]{} Taking a trace, i.e. contracting any two indices, of a $T$-tensor lowers its rank $n$ by two: $$\begin{aligned} T^{nl}_{i_1i_2..i_n}\delta_{i_{n-1}i_n} &= T^{n-2,l}_{i_1i_2..i_{n-2}}\,.{\label{eqn:loweringn}}\end{aligned}$$ This follows from [Eq. (\[eqn:polyadexpansion\])]{} and the orthogonality property [(\[eqn:orthogonality\])]{}. An important consequence is that $T^{ll}_{i_1i_2..i_l}$ is traceless: $$\begin{aligned} T^{ll}_{i_1i_2..i_l}\delta_{i_{l-1}i_l} &= 0\,.{\label{eqn:traceless}}\end{aligned}$$ Conversely, for $l \leq n-2$, we raise the Cartesian rank $n$ by $$\begin{aligned} T^{nl}_{i_1i_2..i_n} &= \frac{2}{n(n+1)-l(l+1)}\big(\delta_{i_1i_2}T^{n-2,l}_{i_3i_4..i_n}+...+\delta_{i_{n-1}i_n}T^{n-2,l}_{i_1i_2..i_{n-3}}\big){\label{eqn:raisingn}}\end{aligned}$$ The parenthesis in [Eq. (\[eqn:raisingn\])]{} contains $n(n-1)/2$ terms, one for each unique pairing of the $n$ indices. We have not proven [Eq. (\[eqn:raisingn\])]{} for general values of $n$ and $l$, but it is straightforward to work out the cases $l=n-2$, $l=n-4$, and so on, by taking a trace of [Eq. (\[eqn:raisingn\])]{} and using Eqns. [(\[eqn:loweringn\])]{} and [(\[eqn:traceless\])]{} repeatedly. We have checked all values of $n$ and $l$ that are used in our calculations in this paper. [*Dimensionality.*]{} The tensor ${\boldsymbol{T}}^{ll}$ is a rank-$l$ Cartesian tensor. In general it could have $3^l$ unique elements (in three spatial dimensions). In contrast, there are only $2l+1$ spherical harmonics $Y_l^m$ of degree $l$, corresponding to the values $m=-l...\,l$. But we have shown that ${\boldsymbol{T}}^{ll}$ is symmetric and traceless. A symmetric tensor of rank $l$ has $(l+1)(l+2)/2$ unique elements, and it has $l(l-1)/2$ unique traces that we require to vanish. These conditions leave exactly $2l+1$ degrees of freedom for a symmetric and traceless rank-$l$ Cartesian tensor. This is the reason we refer to the $T$-tensors as a basis set. The coefficients $c^{llm}_{i_1i_2..i_l}$ in [Eq. (\[eqn:Tdefinition\])]{} are the elements of the rotation matrix between the two basis sets ${\boldsymbol{T}}^{ll}$ and $Y_l^m$. We claim that this transformation is unitary for a certain choice of normalization of the spherical harmonics. In other words, it is in fact a proper rotation. However, we have not proven this for general values of $l$, but we confirmed that it is true up to $l=8$ by brute force calculation of $c^{llm}$ from the definition [Eq. (\[eqn:Tdefinition\])]{}, see [Appendix \[app:Tquestions\]]{}. [*Multiplication.*]{} The product of two $T$-tensors follows directly from [(\[eqn:polyadexpansion\])]{} as a recurrence relation: $$\begin{aligned} T^{l_1 l_1}_i T^{l_2 l_2}_j &= \sum_{J=0}^{l_1+l_2} T^{l_1+l_2,J}_{{\boldsymbol{i}} {\boldsymbol{j}}} -\sum_{j_1=0}^{l_1-2}\sum_{j_2=0}^{l_2-2} T^{l_1j_1}_{{\boldsymbol{i}}} T^{l_2j_2}_{{\boldsymbol{j}}} -\sum_{j_1=0}^{l_1-2} T^{l_1j_1}_{{\boldsymbol{i}}} T^{l_2l_2}_{{\boldsymbol{j}}} -\sum_{j_2=0}^{l_2-2} T^{l_1l_1}_{{\boldsymbol{i}}} T^{l_2j_2}_{{\boldsymbol{j}}}\,.{\label{eqn:Tmul}}\end{aligned}$$ Here ${\boldsymbol{i}}$ and ${\boldsymbol{j}}$ are short-hand for $i_1..i_n$ and $j_1..j_n$. We use [Eq. (\[eqn:Tmul\])]{} in practical calculations, but there is also a largely unexplored connection to quantum angular momentum algebra and Clebsch-Gordan coefficients. In particular, it can be shown ([Appendix \[app:Tquestions\]]{}) that $$\begin{aligned} T^{l_1 l_1}_{{\boldsymbol{i}}} T^{l_2 l_2}_{{\boldsymbol{j}}} &= \sum_{J=|l_1-l_2|}^{l_1+l_2}A_{{\boldsymbol{i}}{\boldsymbol{j}}{\boldsymbol{k}}}^{l_1l_1l_2l_2J}T^{JJ}_{{\boldsymbol{k}}}\,,{\label{eqn:TmulCG}}\end{aligned}$$ for some coupling tensor $A$ independent of ${\boldsymbol{\hat r}}$. [*Relation to polyads.*]{} We convert any polyadic expression into $T$-tensors by replacing $\hat r_i\to T^{11}_i$, and applying [Eq. (\[eqn:Tmul\])]{} until no products remain. Conversely any $T$-tensor is expressed as a polyadic by recursively using $$\begin{aligned} T^{nn}_{i_1i_2..i_n} &= \hat r_{i_1}\hat r_{i_2}..\hat r_{i_n} - \sum_{l=0}^{n-2}T^{nl}_{i_1i_2..i_n}\,,\end{aligned}$$ and [Eq. (\[eqn:raisingn\])]{}. The first few tensors are $$\begin{aligned} T^{00} &= 1\,,\quad T^{11}_i = \hat r_i\,{\nonumber}\\ T^{20} &= \frac{1}{3}\delta_{ij}\,,\quad T^{22}_{ij}=\hat r_i\hat r_j-T^{20}_{ij}{\nonumber}\\ T^{31}_{ijk}&=\frac{1}{5}\left(\delta_{ij}\hat r_k+\delta_{ik}\hat r_j+\delta_{jk}\hat r_i\right)\,,\quad T^{33}_{ijk}= \hat r_i\hat r_j\hat r_k - T^{31}_{ijk}{\nonumber}\\ T^{40}_{ijkl} &=\frac{1}{15} \left(\delta _{i l} \delta _{j k}+\delta _{i k} \delta _{j l}+\delta _{i j} \delta _{k l}\right){\nonumber}\\ T^{42}_{ijkl} &= \frac{1}{7} \left(\hat r_k \hat r_l \delta _{i j}+\hat r_j \hat r_l \delta _{i k}+\hat r_i \hat r_l \delta _{j k}+\hat r_j \hat r_k \delta _{i l}+\hat r_i \hat r_k \delta _{j l}+\hat r_i \hat r_j \delta _{k l}\right)-\frac{2}{21} \left(\delta _{i l} \delta _{j k}+\delta _{i k} \delta _{j l}+\delta _{i j} \delta _{k l}\right){\nonumber}\\ T^{44}_{ijkl} &= \hat r_i\hat r_j\hat r_k\hat r_l - T^{42}_{ijkl} - T^{40}_{ijkl}\,.{\nonumber}\\\end{aligned}$$ [*Differentiation.*]{} In order to calculate using only algebraic manipulations on the $T$-tensors, we must know how the gradient operator $\nabla$, where $\nabla_i = \partial/\partial r_i$, acts on them. We will show that the action of $\nabla$ is to scatter a tensor of degree $l$ into a linear combination of tensors with degrees $l-1$ and $l+1$ (given in [Eq. (\[eqn:difffinal\])]{}). We will briefly describe how to compute the coefficients of this linear combination, for any $l$. Consider the differential operator $T^{nl}_{i}(\nabla)$, defined by taking the polynomial $r^l T^{nl}_i({\boldsymbol{\hat r}})$ and replacing the components of ${\boldsymbol{r}}$ with the partial derivatives $\partial/\partial r_i$. Hobson’s theorem on differentiation [@hobson_theorem_1892; @weniger_spherical_2005] explains that any such differential operator built from a harmonic polynomial acts on radial functions in a particularly simple way. In our case we use his result to find $$\begin{aligned} T^{nl}_{i}(\nabla)\,r^a &= b_a^lr^{a-l}T^{nl}_i({\boldsymbol{\hat r}})\,, {\label{eqn:radialdiff}}\end{aligned}$$ with $$\begin{aligned} b_a^l &= \prod_{k=0}^{l-1}(a-2k)\,.{\label{eqn:radialdiffb}}\end{aligned}$$ Therefore the general $J$-th order derivative is $$\begin{aligned} T^{NJ}_{i}(\nabla)\, r^m T^{nl}_j({\boldsymbol{\hat r}})&= \frac{1}{b_{m+l}^l}T^{NJ}_{i}(\nabla)\,T^{nl}_{j}(\nabla)\,r^{m+l}\,.{\label{eqn:diffgeneral}}\end{aligned}$$ The product $T^{NJ}_{i}(\nabla)\,T^{nl}_{i}(\nabla)$ is given by [Eq. (\[eqn:TmulCG\])]{}, and the general formula follows from [Eq. (\[eqn:radialdiff\])]{}. In this paper we only consider first order derivatives which correspond to $J=1$, because $\partial/\partial r_i = T_i^{11}(\nabla)$. For $J=1$ the general formula [(\[eqn:diffgeneral\])]{} and [Eq. (\[eqn:TmulCG\])]{} give $$\begin{aligned} \frac{\partial}{\partial r_i} r^m T^{nl}_j({\boldsymbol{\hat r}})&= \frac{1}{b_{m+l}^l}\nabla^2 A_{ijk}^{11nl,l-1}T^{l-1,l-1}_k(\nabla)\,r^{m+l} + \frac{1}{b_{m+l}^l} A_{ijk}^{11nl,l+1}T^{l+1,l+1}_k(\nabla)\,r^{m+l}\,,\end{aligned}$$ which becomes, using [Eq. (\[eqn:radialdiff\])]{} and [Eq. (\[eqn:radialdiffb\])]{} , $$\begin{aligned} = (m+l+1)A_{ijk}^{11nl,l-1}T^{l-1,l-1}_k({\boldsymbol{\hat r}})r^{m-1} + (m-l)A_{ijk}^{11nl,l+1}T^{l+1,l+1}_k({\boldsymbol{\hat r}})r^{m-1}\,.{\label{eqn:difffinal}}\end{aligned}$$ To evaluate [Eq. (\[eqn:difffinal\])]{} in our computer program we compute the product $r^{m-1}{\boldsymbol{T}}^{11}{\boldsymbol{T}}^{nl}\equiv\alpha r^{m-1} {\boldsymbol{T}}^{l-1,l-1} + \beta r^{m-1}{\boldsymbol{T}}^{l+1,l+1}$ using [(\[eqn:Tmul\])]{}, and replace the coefficients by $\alpha \rightarrow(m+l+1)\alpha$, and $\beta\rightarrow(m-l)\beta$. [*Fourier transform.*]{} We use a symmetric convention for the Fourier transform: $$\begin{aligned} {\ensuremath{\mathcal F f}}({\boldsymbol{k}}) &\equiv \frac{1}{(2\pi)^{3/2}}\int_{\mathbb R^3}{\ensuremath{\textrm{d}}}^3{\boldsymbol{r}} e^{-i {\boldsymbol{r}}\cdot {\boldsymbol{k}}}f({\boldsymbol{r}})\,,\\ {\ensuremath{\mathcal F^{-1} f}}({\boldsymbol{r}}) &\equiv \frac{1}{(2\pi)^{3/2}}\int_{\mathbb R^3}{\ensuremath{\textrm{d}}}^3{\boldsymbol{k}} e^{i {\boldsymbol{r}}\cdot {\boldsymbol{k}}}f({\boldsymbol{k}})\,.\end{aligned}$$ The Fourier transform of $r^m T^{nl}_i({\boldsymbol{\hat r}})$ follows directly from that of the functions $r^m Y_l^\mu(\theta,\varphi)$ given in Ref. [@samko_fourier_1978]. For all values of $m$ and $l$ that appear in the present calculation $$\begin{aligned} {\ensuremath{\mathcal F r}}^m T^{nl}({\boldsymbol{\hat r}}) &= \frac{\Psi_{ml}}{k^{m+3}}T^{nl}({\boldsymbol{\hat k}})\,,\quad m\neq l+2j \textrm{ and } m\neq -(l+3)-2j\,,\quad j=0,1,... {\label{eqn:fourier}}\\ \Psi_{ml}&= (-i)^l 2^{m+3/2}\frac{\Gamma(\frac{m+l+3}{2})}{\Gamma(\frac{l-m}{2})}\end{aligned}$$ When $m=l+2j$, $r^m T^{nl}({\boldsymbol{\hat r}})$ is a polynomial in the components of ${\boldsymbol{r}}$, and its Fourier transform is the Dirac delta function and its derivatives. The case $m= -(l+3)-2j$ is more complicated, involving logarithms [@samko_fourier_1978]. Neither of these cases arise in this paper. Particular solution for the inhomogenous Stokes equation -------------------------------------------------------- [\[sec:Tparticular\]]{} Consider the inhomogenous Stokes problem $$\begin{aligned} -\partial_i p + \nabla^2 u_i &= f_i\,,\quad \partial_i u_i = 0, {\label{eqn:inhomogenous1}}\end{aligned}$$ where we assume that $f_i$ is a linear combination of $T$-tensors. The Fourier transform of [Eq. (\[eqn:inhomogenous1\])]{} is $$\begin{aligned} -ik_i{\ensuremath{\mathcal F p}}-k^2{\ensuremath{\mathcal F u_i}} &= {\ensuremath{\mathcal F f}}_i\,,\quad k_i {\ensuremath{\mathcal F u}}_i=0\,,\end{aligned}$$ where $k=|{\boldsymbol{k}}|$. This algebraic equation is solved by $$\begin{aligned} {\ensuremath{\mathcal F p}} &= -\frac{ k_j{\ensuremath{\mathcal F f}}_j}{ik^2}\,,{\label{eqn:pufourierp}}\\ {\ensuremath{\mathcal F u}}_i &= -\frac{1}{k^2}(\delta_{ij} - \hat k_i\hat k_j){\ensuremath{\mathcal F f}}_j\,,{\label{eqn:pufourieru}}\end{aligned}$$ where $\hat k_i\equiv k_i/k$ is a unit vector. In terms of $T$-tensors, the Fourier space Green’s function is $$\begin{aligned} -\frac{1}{k^2}(\delta_{ij} - \hat k_i\hat k_j) &= \frac{1}{k^2}\left(T^{22}_{ij}({\boldsymbol{\hat k}})-\frac{2}{3}\delta_{ij}\right)\,.{\label{eqn:greensk}}\end{aligned}$$ The procedure to find the solution $u_i$ is therefore 1. compute the Fourier transform of $f_i$ using [Eq. (\[eqn:fourier\])]{}, 2. multiply with the Greens function [(\[eqn:greensk\])]{} using [Eq. (\[eqn:Tmul\])]{}, 3. Inverse Fourier transform the product again using [Eq. (\[eqn:fourier\])]{}. In this paper we never need an explicit expression for the pressure $p$, but if needed it is computed in the analogous way from [Eq. (\[eqn:pufourierp\])]{}. Results ======= [\[sec:results\]]{} In this Section we give the solutions to both the mobility problem ([Section \[sec:mobility\]]{}) and the resistance problem ([Section \[sec:resistance\]]{}) for a freely rotating spherical particle in an unbounded viscoelastic shear flow, with no restriction on the direction of ${\boldsymbol{v}}$ or ${\boldsymbol{F}}^{\mathrm{ext}}$ relative to the shear. The mobility problem -------------------- [\[sec:mobility\]]{} Here we consider a particle moving under the effect of an external force. The particle velocity ${\boldsymbol{v}}$ is a function of ${\ensuremath{\textrm{De}}}$ and ${\ensuremath{\textrm{Wi}}}$ to be determined, and to that end we require that the total force ${\boldsymbol{F}}=0$. We proceed with a regular perturbation expansion in ${\ensuremath{\textrm{De}}}$: $$\begin{aligned} {\boldsymbol{u}} &= {\boldsymbol{u}}^{(0)}+{\ensuremath{\textrm{De}}}\, {\boldsymbol{u}}^{(1)}+{\ensuremath{\textrm{De}}}^2\, {\boldsymbol{u}}^{(2)}+...{\nonumber}\\ p &= p^{(0)}+{\ensuremath{\textrm{De}}}\, p^{(1)}+{\ensuremath{\textrm{De}}}^2\, p^{(2)}+...{\nonumber}\\ {\boldsymbol{\omega}} &= {\boldsymbol{\omega}}^{(0)}+{\ensuremath{\textrm{De}}}\, {\boldsymbol{\omega}}^{(1)}+{\ensuremath{\textrm{De}}}^2\, {\boldsymbol{\omega}}^{(2)}+...{\nonumber}\\ {\boldsymbol{v}} &= {\boldsymbol{v}}^{(0)}+{\ensuremath{\textrm{De}}}\, {\boldsymbol{v}}^{(1)}+{\ensuremath{\textrm{De}}}^2\, {\boldsymbol{v}}^{(2)}+...{\nonumber}\\ {\boldsymbol{\Pi}} &= {\boldsymbol{\Pi}}^{(0)}+{\ensuremath{\textrm{De}}}\, {\boldsymbol{\Pi}}^{(1)}+{\ensuremath{\textrm{De}}}^2\, {\boldsymbol{\Pi}}^{(2)}+...{\nonumber}\end{aligned}$$ At each order ${\boldsymbol{\Pi}}^{(k)}$ is given by an algebraic equation, and ${\boldsymbol{u}}^{(k)}$ by an inhomogenous Stokes equation (except for the lowest order, which is homogenous). To lowest order ${\ensuremath{\textrm{De}}}^0$ we have from [Eq. (\[eqn:oldBdef\])]{} $$\begin{aligned} {\boldsymbol{\Pi}}^{(0)} &= \nabla {\boldsymbol{u}}^{(0)}+(\nabla {\boldsymbol{u}}^{(0)}){\ensuremath{^{\rm T}}}\,,\end{aligned}$$ and therefore ${\boldsymbol{\sigma}}^{E(0)}=0$, and therefore from Eqs. [(\[eqn:rt\_v\])]{} and [(\[eqn:rt\_rot\])]{} we have $$\begin{aligned} {\boldsymbol{v}}^{(0)} &= \frac{1}{6\pi}{\boldsymbol{F}}^{\mathrm{ext}}\,,{\nonumber}\\ {\boldsymbol{\omega}}^{(0)} &= {\boldsymbol{\Omega}}\end{aligned}$$ To order ${\ensuremath{\textrm{De}}}^0$ the flow satisfies $$\begin{aligned} &-\nabla p^{(0)} + \nabla^2 {\boldsymbol{u}}^{(0)} = 0\,,\\ {\boldsymbol{u}}^{(0)} &= \alpha {\boldsymbol{\omega}}^{(0)} \times {\boldsymbol{r}},\quad{\boldsymbol{r}} \in S{\nonumber}\\ {\boldsymbol{u}}^{(0)} &\to \alpha{\boldsymbol{u}}^\infty-{\boldsymbol{v}}^{(0)}\,,\quad|{\boldsymbol{r}}|\to \infty\,. \end{aligned}$$ At order ${\ensuremath{\textrm{De}}}^1$ [Eq. (\[eqn:oldBdef\])]{} gives $$\begin{aligned} \label{eq:eqPi1} {\boldsymbol{\Pi}}^{(1)} &= -\left[({\boldsymbol{u}}^{(0)}\cdot\nabla){\boldsymbol{\Pi}}^{(0)} - (\nabla {\boldsymbol{u}}^{(0)}){\boldsymbol{\Pi}}^{(0)} - {\boldsymbol{\Pi}}^{(0)} (\nabla {\boldsymbol{u}}^{(0)}){\ensuremath{^{\rm T}}}\right] + \nabla {\boldsymbol{u}}^{(1)} + (\nabla {\boldsymbol{u}}^{(1)}){\ensuremath{^{\rm T}}}\,,\end{aligned}$$ where ${\boldsymbol{u}}^{(0)}$ is known, but ${\boldsymbol{u}}^{(1)}$ is still unknown. Consequently $$\begin{aligned} {\boldsymbol{\sigma}}^{E(1)} &= -\left[({\boldsymbol{u}}^{(0)}\cdot\nabla){\boldsymbol{\Pi}}^{(0)} - (\nabla {\boldsymbol{u}}^{(0)}){\boldsymbol{\Pi}}^{(0)} - {\boldsymbol{\Pi}}^{(0)} (\nabla {\boldsymbol{u}}^{(0)}){\ensuremath{^{\rm T}}}\right] \,.{\label{eqn:sigmae1}}\end{aligned}$$ The reciprocal theorem [(\[eqn:rt\_v\])]{} gives $$\begin{aligned} {\boldsymbol{v}}^{(1)} &= \frac{1}{6\pi}\int_{S_p} {\boldsymbol{\sigma}}^{E(1)} \cdot {\boldsymbol{{\ensuremath{\textrm{d}}}S}} - \frac{1}{6\pi}\int_V {\ensuremath{\amsbb{M}}}_{{\boldsymbol{v}}}{\ensuremath{^{\rm T}}}\nabla \cdot {\boldsymbol{\sigma}}^{E(1)}{{\ensuremath{\textrm{d}}}V} =-\frac{\alpha\mu_r}{6\pi} {\boldsymbol{\Omega}}\times {\boldsymbol{F}}^{\mathrm{ext}}{\label{eqn:mobilityvorder1}}\\ {\boldsymbol{\omega}}^{(1)} &= \frac{1}{8\pi\alpha}\int_{S_p} {\boldsymbol{r}} \times ({\boldsymbol{\sigma}}^{E(1)} \cdot {\boldsymbol{{\ensuremath{\textrm{d}}}S}}) - \frac{1}{8\pi\alpha}\int_V {\ensuremath{\amsbb{M}}}_{{\boldsymbol{\omega}}}{\ensuremath{^{\rm T}}}\nabla \cdot {\boldsymbol{\sigma}}^{E(1)}{{\ensuremath{\textrm{d}}}V} = 0\,.{\label{eqn:mobilityomegaorder1}}\end{aligned}$$ At this order the shear flow and the disturbance from the external forcing interact to create a lateral drift perpendicular to both the direction of forcing and the vorticity. The drift arises from both the extra stress on the particle surface, and the viscous stress induced by the viscoelastic medium, in proportion $2:3$. The lateral drift [Eq. (\[eqn:mobilityvorder1\])]{} was first calculated by @brunn1977. Our [Eq. (\[eqn:mobilityvorder1\])]{} agrees with his when accounting for the erratum [@brunn1977a] and letting his $\kappa_0^{11}=-2\kappa_0^{22}$, which corresponds to the second-order fluid limit of the Oldroyd-B model. The mobility derived in Ref.  is different from [Eq. (\[eqn:mobilityvorder1\])]{}. In particular they report a contribution proportional to ${\ensuremath{\amsbb{S}}} {\boldsymbol{F}}^{\mathrm{ext}}$. The $O({\ensuremath{\textrm{Wi}}})$ contribution to the angular velocity vanishes, in agreement with all the previous results [@brunn1977; @brunn1977a; @vishnampet2012]. With ${\boldsymbol{\sigma}}^{E(1)}$, ${\boldsymbol{v}}^{(1)}$, and ${\boldsymbol{\omega}}^{(1)}$ given by Eqs. (\[eqn:sigmae1\]-\[eqn:mobilityomegaorder1\]) we can write down the inhomogenous Stokes problem for ${\boldsymbol{u}}^{(1)}$: $$\begin{aligned} -\nabla p^{(1)} + \nabla^2 {\boldsymbol{u}}^{(1)} &= -\nabla \cdot {\boldsymbol{\sigma}}^{E(1)}\,,{\label{eqn:order1floweq1}}\end{aligned}$$ subject to $$\begin{aligned} {\boldsymbol{u}}^{(1)} &= \alpha {\boldsymbol{\omega}}^{(1)} \times {\boldsymbol{r}},\quad{\boldsymbol{r}} \in S{\nonumber}\\ {\boldsymbol{u}}^{(1)} &\to -{\boldsymbol{v}}^{(1)}\,,\quad|{\boldsymbol{r}}|\to \infty\,. {\label{eqn:order1flowbc1}} \end{aligned}$$ First we compute a particular solution ${\boldsymbol{u}}^{(1)p}({\boldsymbol{r}})$ as explained in [Section \[sec:Tparticular\]]{}. The flow field ${\boldsymbol{u}}^{(1)p}$ satisfies the inhomogenous [Eq. (\[eqn:order1floweq1\])]{}, but not the boundary conditions [Eq. (\[eqn:order1flowbc1\])]{}. We next solve for a Stokes flow ${\boldsymbol{u}}^{(1)h}$ that satisfies the homogenous equation $$\begin{aligned} &-\nabla p^{(1)h} + \nabla^2 {\boldsymbol{u}}^{(1)h} = 0\,,\end{aligned}$$ and the boundary conditions $$\begin{aligned} {\boldsymbol{u}}^{(1)h} &= {\boldsymbol{\omega}}^{(1)} - {\boldsymbol{u}}^{(1)p} \times {\boldsymbol{r}},\quad{\boldsymbol{r}} \in S{\nonumber}\\ {\boldsymbol{u}}^{(1)h} &\to -{\boldsymbol{u}}^{(1)p}\,,\quad|{\boldsymbol{r}}|\to \infty\,. \end{aligned}$$ By construction ${\boldsymbol{u}}^{(1)}={\boldsymbol{u}}^{(1)h}+{\boldsymbol{u}}^{(1)p}$. At order ${\ensuremath{\textrm{De}}}^2$ we have from [Eq. (\[eqn:oldBdef\])]{} $$\begin{aligned} {\boldsymbol{\Pi}}^{(2)} &= {\boldsymbol{\sigma}}^{E(2)}+ \nabla {\boldsymbol{u}}^{(2)} + (\nabla {\boldsymbol{u}}^{(2)}){\ensuremath{^{\rm T}}}\,,\end{aligned}$$ with $$\begin{aligned} {\boldsymbol{\sigma}}^{E(2)} &=-\bigg[ ({\boldsymbol{u}}^{(0)}\cdot\nabla){\boldsymbol{\Pi}}^{(1)} - (\nabla {\boldsymbol{u}}^{(0)}){\boldsymbol{\Pi}}^{(1)} - {\boldsymbol{\Pi}}^{(1)} (\nabla {\boldsymbol{u}}^{(0)}){\ensuremath{^{\rm T}}}+ ({\boldsymbol{u}}^{(1)}\cdot\nabla){\boldsymbol{\Pi}}^{(0)} - (\nabla {\boldsymbol{u}}^{(1)}){\boldsymbol{\Pi}}^{(0)} - {\boldsymbol{\Pi}}^{(0)} (\nabla {\boldsymbol{u}}^{(1)}){\ensuremath{^{\rm T}}}\bigg] \,,\end{aligned}$$ where ${\boldsymbol{u}}^{(0)}$, ${\boldsymbol{u}}^{(1)}$, ${\boldsymbol{\Pi}}^{(0)}$ and ${\boldsymbol{\Pi}}^{(1)}$ are all known. The reciprocal theorem at order ${\ensuremath{\textrm{De}}}^2$ gives $$\begin{aligned} {\boldsymbol{v}}^{(2)} &= \frac{1}{6\pi}\int_{S_p} {\boldsymbol{\sigma}}^{E(2)} \cdot {\boldsymbol{{\ensuremath{\textrm{d}}}S}} - \frac{1}{6\pi}\int_V {\ensuremath{\amsbb{M}}}_{{\boldsymbol{v}}}{\ensuremath{^{\rm T}}}\nabla \cdot {\boldsymbol{\sigma}}^{E(2)}{{\ensuremath{\textrm{d}}}V} {\label{eqn:v2rt}}\\ {\boldsymbol{\omega}}^{(2)} &= \frac{1}{8\pi\alpha}\int_{S_p} {\boldsymbol{r}} \times ({\boldsymbol{\sigma}}^{E(2)} \cdot {\boldsymbol{{\ensuremath{\textrm{d}}}S}}) - \frac{1}{8\pi\alpha}\int_V {\ensuremath{\amsbb{M}}}_{{\boldsymbol{\omega}}}{\ensuremath{^{\rm T}}}\nabla \cdot {\boldsymbol{\sigma}}^{E(2)}{{\ensuremath{\textrm{d}}}V} {\label{eqn:omega2rt}}\end{aligned}$$ We first consider the angular velocity ${\boldsymbol{\omega}}^{(2)}$. The angular velocity due to the surface integral in [Eq. (\[eqn:omega2rt\])]{} vanishes, so the induced viscous stress alone explains the rotation rate at this order. The full result for the angular velocity, to second order in ${\ensuremath{\textrm{De}}}$ and ${\ensuremath{\textrm{Wi}}}$, takes the form $$\begin{aligned} {\boldsymbol{\omega}} &= {\boldsymbol{\Omega}} - {\ensuremath{\textrm{Wi}}}^2\,\frac{\mu_r}{2} {\boldsymbol{\Omega}} + {\ensuremath{\textrm{De}}}^2\, \frac{5\mu_r (910 \mu_r-1941)}{96096} \frac{1}{(6\pi)^2}{\boldsymbol{F}}^\mathrm{ext} \times {\ensuremath{\amsbb{S}}} {\boldsymbol{F}}^\mathrm{ext}\,.{\label{eqn:mobilityomega}}\end{aligned}$$ The numerically largest contribution is the $O({\ensuremath{\textrm{Wi}}}^2)$ slowdown of the rotation around vorticity. This contribution agrees with an earlier analytical result [@housiadas2011a] that also explains numerical simulations [@davino2008]. The $O({\ensuremath{\textrm{De}}}^2)$ contribution shows a coupling between the external force and rotation rate, through the strain. It is numerically small, but may be important because it describes a rotation around another axis than ${\boldsymbol{\Omega}}$. For the particle velocity ${\boldsymbol{v}}^{(2)}$ the surface integral of the extra stress on the particle evaluates to $$\begin{aligned} \int_{S_p} {\boldsymbol{\sigma}}^{E(2)} \cdot {\boldsymbol{{\ensuremath{\textrm{d}}}S}}&=\alpha^2\mu_r\bigg( \frac{2}{3}(\mu_r-1) {\boldsymbol{\Omega}}\times{\boldsymbol{\Omega}}\times{\boldsymbol{F}}^\mathrm{ext} -\frac{1}{3}\,{\boldsymbol{\Omega}} \times {\ensuremath{\amsbb{S}}} {\boldsymbol{F}}^\mathrm{ext}{\label{eqn:v2polymer}} \bigg)\,.\end{aligned}$$ We see that this contribution may affect the velocity along ${\boldsymbol{F}}^\mathrm{ext}$, but in particular it gives another lateral drift in a direction perpendicular to the $O({\ensuremath{\textrm{Wi}}})$ lateral drift calculated above. Next we evaluate the volume integral in [(\[eqn:v2rt\])]{}, and this gives the final result for ${\boldsymbol{v}}$ to second order in ${\ensuremath{\textrm{De}}}$ and ${\ensuremath{\textrm{Wi}}}$: $$\begin{aligned} 6\pi {\boldsymbol{v}} &= {\boldsymbol{F}}^\mathrm{ext} - {\ensuremath{\textrm{Wi}}}\, \mu_r{\boldsymbol{\Omega}}\times {\boldsymbol{F}}^\mathrm{ext} + \mu_r\bigg( {\ensuremath{\textrm{De}}}^2\, \frac{143 \mu_r+258}{25025}\frac{|{\boldsymbol{F}}^\mathrm{ext}|^2}{(6\pi)^2} + {\ensuremath{\textrm{Wi}}}^2\, \frac{5 (237005 \mu_r-291618)}{378378}|{\boldsymbol{\Omega}}|^2 \bigg){\boldsymbol{F}}^\mathrm{ext}{\nonumber}\\ &\qquad+{\ensuremath{\textrm{Wi}}}^2\mu_r\bigg((\mu_r-1) {\boldsymbol{\Omega}}\times{\boldsymbol{\Omega}}\times{\boldsymbol{F}}^\mathrm{ext} + \frac{3}{2}\,{\boldsymbol{\Omega}} \times {\ensuremath{\amsbb{S}}} {\boldsymbol{F}}^\mathrm{ext} + \frac{183339-286735 \mu_r}{126126} {\ensuremath{\amsbb{S}}} {\ensuremath{\amsbb{S}}} {\boldsymbol{F}}^\mathrm{ext}\bigg) {\label{eqn:mobilityv}}\end{aligned}$$ The induced viscous stress, given by the volume integral, contributes to the same terms as the extra stress on the surface shown in [Eq. (\[eqn:v2polymer\])]{}. In addition there is yet another velocity proportional to ${\ensuremath{\amsbb{S}}}{\ensuremath{\amsbb{S}}}{\boldsymbol{F}}^\mathrm{ext}$, and a component in the direction of ${\boldsymbol{F}}^\mathrm{ext}$. The second order contribution to the velocity directly proportional to ${\boldsymbol{F}}^\mathrm{ext}$ consists of one term proportional to ${\ensuremath{\textrm{De}}}^2$, and one proportional to ${\ensuremath{\textrm{Wi}}}^2$. The velocity along ${\boldsymbol{F}}^\mathrm{ext}$ increases as ${\ensuremath{\textrm{De}}}$ increases, but the numerical prefactor is small. The important result is that the velocity decreases with increasing shear rate, as observed in experiment [@van_den_brule_effects_1993]. We discuss this effect for an inclined shear flow in [Section \[sec:discussion\]]{}. In the next Section we solve the resistance problem that can be directly compared with earlier calculations for the cross-shear flow. The resistance problem ---------------------- [\[sec:resistance\]]{} The calculation for this problem is very similar to that of the mobility problem, so we omit most details. Here we consider a freely rotating sphere moving at velocity ${\boldsymbol{v}}$ through a shear flow, and calculate the resulting hydrodynamic force ${\boldsymbol{F}}$ and angular velocity ${\boldsymbol{\omega}}$. The crucial differences to the mobility problem are that ${\boldsymbol{F}}^{\mathrm{ext}}=0$, and ${\boldsymbol{v}}$ is a constant, independent of ${\ensuremath{\textrm{De}}}$. The zeroth order problem is the corresponding Stokes problem, which determines ${\boldsymbol{u}}^{(0)}$ and ${\boldsymbol{\sigma}}^{E(1)}$. The reciprocal theorem [(\[eqn:rt\_rot\])]{} gives $$\begin{aligned} {\boldsymbol{\omega}}^{(0)} &= {\boldsymbol{\Omega}}\,, {\nonumber}\\ {\boldsymbol{\omega}}^{(1)} &= \frac{1}{8\pi\alpha}\int_{S_p} {\boldsymbol{r}} \times ({\boldsymbol{\sigma}}^{E(1)} \cdot {\boldsymbol{{\ensuremath{\textrm{d}}}S}}) - \frac{1}{8\pi\alpha}\int_V {\ensuremath{\amsbb{M}}}_{{\boldsymbol{\omega}}}{\ensuremath{^{\rm T}}}\nabla \cdot {\boldsymbol{\sigma}}^{E(1)}{{\ensuremath{\textrm{d}}}V} = 0\,.\end{aligned}$$ The first order flow problem has a different boundary condition to that of the mobility problem, because ${\boldsymbol{v}}$ is independent of ${\ensuremath{\textrm{De}}}$. The first order equations are $$\begin{aligned} -\nabla p^{(1)} + \nabla^2 {\boldsymbol{u}}^{(1)} &= -\nabla \cdot {\boldsymbol{\sigma}}^{E(1)}\,,{\label{eqn:order1floweqres}}\end{aligned}$$ subject to $$\begin{aligned} {\boldsymbol{u}}^{(1)} &= \alpha {\boldsymbol{\omega}}^{(1)} \times {\boldsymbol{r}},\quad{\boldsymbol{r}} \in S{\nonumber}\\ {\boldsymbol{u}}^{(1)} &\to 0\,,\quad|{\boldsymbol{r}}|\to \infty\,. {\label{eqn:order1flowbcres}} \end{aligned}$$ The reciprocal theorem [(\[eqn:rt\_trans\])]{} gives $$\begin{aligned} {\boldsymbol{F}}^{(0)} &= -6\pi{\boldsymbol{v}}\,, {\nonumber}\\ {\boldsymbol{F}}^{(1)} &= \int_{S_p} {\boldsymbol{\sigma}}^{E(1)} \cdot {\boldsymbol{{\ensuremath{\textrm{d}}}S}} - \int_V {\ensuremath{\amsbb{M}}}_{{\boldsymbol{v}}}{\ensuremath{^{\rm T}}}\nabla \cdot {\boldsymbol{\sigma}}^{E(1)}{{\ensuremath{\textrm{d}}}V}\,, {\nonumber}\\ {\boldsymbol{F}}^{(2)} &= \int_{S_p} {\boldsymbol{\sigma}}^{E(2)} \cdot {\boldsymbol{{\ensuremath{\textrm{d}}}S}} - \int_V {\ensuremath{\amsbb{M}}}_{{\boldsymbol{v}}}{\ensuremath{^{\rm T}}}\nabla \cdot {\boldsymbol{\sigma}}^{E(2)}{{\ensuremath{\textrm{d}}}V} \,,{\nonumber}\\ {\boldsymbol{\omega}}^{(2)} &= \frac{1}{8\pi\alpha}\int_{S_p} {\boldsymbol{r}} \times ({\boldsymbol{\sigma}}^{E(2)} \cdot {\boldsymbol{{\ensuremath{\textrm{d}}}S}}) - \frac{1}{8\pi\alpha}\int_V {\ensuremath{\amsbb{M}}}_{{\boldsymbol{\omega}}}{\ensuremath{^{\rm T}}}\nabla \cdot {\boldsymbol{\sigma}}^{E(2)}{{\ensuremath{\textrm{d}}}V}\,. {\label{eqn:rtF2}}\end{aligned}$$ The contributions from the surface and volume integrals are similar to those of the mobility problem. Specifically, $$\begin{aligned} \frac{1}{6\pi}\int_{S_p} {\boldsymbol{\sigma}}^{E(2)} \cdot {\boldsymbol{{\ensuremath{\textrm{d}}}S}} &= \alpha^2\,\mu_r\bigg( -\frac{2}{3}{\boldsymbol{\Omega}}\times{\boldsymbol{\Omega}}\times{\boldsymbol{v}} -\frac{\mu_r}{3}{\boldsymbol{\Omega}}\times{\ensuremath{\amsbb{S}}}{\boldsymbol{v}} \bigg)\,.{\label{eqn:f2polymer}}\end{aligned}$$ After evaluating the volume integrals in [Eq. (\[eqn:rtF2\])]{}, we find for the resistance problem with velocity ${\boldsymbol{v}}$ $$\begin{aligned} \frac{{\boldsymbol{F}}}{6\pi} &= -{\boldsymbol{v}}-{\ensuremath{\textrm{Wi}}}\,\mu_r {\boldsymbol{\Omega}} \times {\boldsymbol{v}} +\mu_r\bigg({\ensuremath{\textrm{De}}}^2\,\frac{143 \mu_r+258}{25025}|{\boldsymbol{v}}|^2+{\ensuremath{\textrm{Wi}}}^2\,\frac{5 (237005 \mu_r-291618)}{378378}|{\boldsymbol{\Omega}}|^2\bigg){\boldsymbol{v}} {\nonumber}\\ &\quad+{\ensuremath{\textrm{Wi}}}^2\,\mu_r\bigg( -{\boldsymbol{\Omega}}\times{\boldsymbol{\Omega}}\times{\boldsymbol{v}} +\frac{3}{2}{\boldsymbol{\Omega}}\times{\ensuremath{\amsbb{S}}}{\boldsymbol{v}} +\frac{183339-286735 \mu_r}{126126}{\ensuremath{\amsbb{S}}}{\ensuremath{\amsbb{S}}}{\boldsymbol{v}} \bigg){\label{eqn:resistanceF}}\\ {\boldsymbol{\omega}} &= {\boldsymbol{\Omega}} - {\ensuremath{\textrm{Wi}}}^2\,\frac{\mu_r}{2}{\boldsymbol{\Omega}} + {\ensuremath{\textrm{De}}}^2\,\frac{5\mu_r (910 \mu_r-1941)}{96096}{\boldsymbol{v}}\times{\ensuremath{\amsbb{S}}}{\boldsymbol{v}}\end{aligned}$$ As expected, the expression for the resistance force [(\[eqn:resistanceF\])]{} is similar to the expression for the mobility velocity [(\[eqn:mobilityv\])]{} with ${\boldsymbol{F}}^\mathrm{ext}$ replaced by $-6\pi{\boldsymbol{v}}$. However, they differ in a term ${\ensuremath{\textrm{Wi}}}^2 \mu_r^2\,{\boldsymbol{\Omega}}\times{\boldsymbol{\Omega}}\times{\boldsymbol{v}}$. This difference arises because of the lateral force at $O({\ensuremath{\textrm{Wi}}})$ for the following reason. In the mobility problem the particle is allowed to relax this lateral hydrodynamic force by drifting sideways. But in the resistance problem we essentially force the fluid, through the boundary conditions, with the lateral force required to keep the particle moving with the prescribed velocity ${\boldsymbol{v}}$. This forcing, or lack thereof, at $O({\ensuremath{\textrm{Wi}}})$ is what gives the differing term at $O({\ensuremath{\textrm{Wi}}}^2)$. Upon substitution of the mobility velocity [Eq. (\[eqn:mobilityv\])]{} into the expression for the resistance force [Eq. (\[eqn:resistanceF\])]{} we find ${\boldsymbol{F}} = -{\boldsymbol{F}}^{\mathrm{ext}}$ to second order in ${\ensuremath{\textrm{De}}}^2$ and ${\ensuremath{\textrm{Wi}}}^2$, as advertised in the Introduction. The drag term proportional to ${\ensuremath{\textrm{De}}}^2$ is the drag in absence of shear. This term was first calculated by @leslie1961 in the context of sedimentation in a quiescent fluid. Our coefficient matches theirs when $\mu_r=1$, and their $\epsilon=\beta=0$. For the cross-shear flow, ${\boldsymbol{v}}$ is parallel to ${\boldsymbol{\Omega}}$, so that ${\boldsymbol{\Omega}} \times {\boldsymbol{v}}={\ensuremath{\amsbb{S}}}{\boldsymbol{v}}=0$. For this case only the first and third terms on the right-hand side of [Eq. (\[eqn:resistanceF\])]{} remain. This expression agrees with the analytical result of @housiadas_rheological_2014, and therefore with @padhy_simulations_2013 as shown in their Fig. 7. Summary and Discussion ====================== [\[sec:discussion\]]{} [Fig1.pdf]{} ![[\[fig:v\_tilty\]]{} Shear-dependent velocity of a sphere forced by an external force in the flow-vorticity plane, for different angle of attack $\varphi$ between the forcing ${\boldsymbol{F}}^\mathrm{ext}$ and the vorticity axis. The cross-shear flow corresponds to $\varphi=0$, whereas the force is along the flow direction when $\varphi=90^\circ$. (a) Velocity along ${\boldsymbol{F}}^\mathrm{ext}$. (b) Lateral drift in the shear direction ${\boldsymbol{\hat y}}$. (c) Lateral drift perpendicular to the shear direction. Parameters: $\mu_r = 0.3$, ${\ensuremath{\textrm{De}}}=0.1$.](Fig2.pdf) ![[\[fig:stress\_tilty\]]{} The trace of the first order viscoelastic stress ${\boldsymbol{\Pi}}^{(1)}$ around a particle driven by an external force through a shear flow. All panels show a center cross-section of the particle, viewed along the direction of external forcing. In these panels $y$ indicates the shear direction, and $x$ indicates the direction perpendicular to both ${\boldsymbol{F}}^\mathrm{ext}$ and ${\boldsymbol{\hat y}}$, see [Fig. \[fig:inclinedshear\]]{}. Left column shows full stress field, right column shows its asymmetric part under inversion of ${\boldsymbol{r}}$. Top row shows stress field when the external forcing is aligned with vorticity, bottom row shows same when the external forcing is at angle $\varphi = 45^\circ$ to vorticity. Parameters: $\alpha=1$ (implying ${\ensuremath{\textrm{Wi}}}={\ensuremath{\textrm{De}}}$). The trace of ${\boldsymbol{\Pi}}^{(1)}$ is independent of $\mu_r$, which follows from Eq. (\[eq:eqPi1\]) because the flow field ${\boldsymbol{u}}$ is incompressible. ](Fig3.pdf) We have derived analytical results for the linear and angular velocities of a particle driven through a viscoelastic shear flow by an external force, valid to second order in ${\ensuremath{\textrm{De}}}$ and ${\ensuremath{\textrm{Wi}}}$, given in Eqs. [(\[eqn:mobilityv\])]{} and [(\[eqn:mobilityomega\])]{}. We found three qualitatively different corrections to the predicted velocity in a Newtonian fluid. First, at $O({\ensuremath{\textrm{Wi}}})$ there is a drift proportional to ${\boldsymbol{\Omega}} \times {\boldsymbol{F}}^\mathrm{ext}$, that is, perpendicular to the forcing and vorticity. Second, the resulting velocity along the forcing is modified at $O({\ensuremath{\textrm{De}}}^2)$ and $O({\ensuremath{\textrm{Wi}}}^2)$. The numerical prefactor of the ${\ensuremath{\textrm{De}}}^2$-contribution is small, so in practice only the $O({\ensuremath{\textrm{Wi}}}^2)$ effect is important. These terms correspond to the effect of the imposed shear flow. Third, at $O({\ensuremath{\textrm{Wi}}}^2)$ there is yet another lateral drift, perpendicular to the first one. Even for ${\ensuremath{\textrm{Wi}}}=0.5$ this second drift may be as strong as the $O({\ensuremath{\textrm{Wi}}})$-drift, but it points in another direction. The relative importance of these three effects depends strongly on the direction of external forcing relative to the orientation of the shear flow. There are two mechanisms that contribute to these corrections. First, the extra stress acts directly on the particle, giving a force and a torque. Secondly, the extra stress acts on the fluid, which modifies the flow and indirectly gives a force and a torque via the viscous and pressure terms. The lateral drift at $O({\ensuremath{\textrm{Wi}}})$ is a combination of these two mechanisms \[[Eq. (\[eqn:mobilityomegaorder1\])]{}\]. The decreased velocity of a sphere sedimenting in a cross-shear flow, however, is due to the indirect increase of viscous stress, and the direct contribution from the extra stress vanishes \[[Eq. (\[eqn:v2polymer\])]{}\]. This observation is in qualitative agreement with the observations of numerical simulations [@padhy_simulations_2013]. When the forcing is at an angle to the vorticity vector, the correction is typically a combination of the two mechanisms. The angular velocity around the vorticity slows down at $O({\ensuremath{\textrm{Wi}}}^2)$, in agreement with earlier results [@housiadas2011a; @davino2008]. But at $O({\ensuremath{\textrm{De}}}^2)$we also find a coupling between the strain and translation that induces a rotation around the axis ${\boldsymbol{F}}^{\mathrm {ext}}\times{\ensuremath{\amsbb{S}}}{\boldsymbol{F}}^{\mathrm {ext}}$. The prefactor is quite small, but the effect could be important because it describes rotation around another axis than ${\boldsymbol{\Omega}}$. [*Settling in inclined shear flow.*]{} [Eq. (\[eqn:mobilityv\])]{} is valid for any orientation of the forcing relative to the shear flow, described for instance by two angles relative to the vorticity and flow directions. In the remainder of this discussion we focus on the concrete example of a particle settling under gravity, ${\boldsymbol{F}}^\mathrm{ext}=m{\boldsymbol{g}}$, with the particular set of orientations so that gravity lies in the plane spanned by the vorticity axis and the flow direction, see [Fig. \[fig:inclinedshear\]]{}. This situation corresponds to settling between two far-apart shearing walls, parallel to the walls, but where the shearing is at an angle to gravity. We denote by $\varphi$ the angle between ${\boldsymbol{g}}$ and the vorticity, see [Fig. \[fig:inclinedshear\]]{}. We show the resulting settling velocity as a function of ${\ensuremath{\textrm{Wi}}}$ in [Fig. \[fig:v\_tilty\]]{}, for $\varphi=0$, $45^\circ$, and $90^\circ$. When $\varphi=0$ we recover the cross-shear result, and the lateral drift vanishes. As the angle of inclination increases the settling velocity increases, diminishing the shear-induced drag increase described for $\varphi=0$ in Refs. [@van_den_brule_effects_1993; @housiadas_rheological_2014; @padhy_simulations_2013]. When gravity acts along the flow direction, $\varphi=90^\circ$, the settling velocity is almost the same as that given by Stokes law, only slightly higher. The direction of the $O({\ensuremath{\textrm{Wi}}})$ lateral drift is along the $-{\boldsymbol{\hat y}}$ direction, see [Fig. \[fig:inclinedshear\]]{}. For finite $\varphi$ and small ${\ensuremath{\textrm{Wi}}}$ this drift is the dominant feature of the particle velocity. But even for larger ${\ensuremath{\textrm{Wi}}}$ the magnitude of this drift is comparable to the reduction in settling velocity when $\varphi=45^\circ$. The additional $O({\ensuremath{\textrm{Wi}}}^2)$ drift is in the third independent direction, given by $ {\boldsymbol{F}}^\mathrm{ext}\times{\boldsymbol{\hat y}}$ ([Fig. \[fig:inclinedshear\]]{}). For ${\ensuremath{\textrm{Wi}}}\approx0.5$ it is comparable in magnitude to both the reduction in settling velocity and the $O({\ensuremath{\textrm{Wi}}})$ drift. The direction of the $O({\ensuremath{\textrm{Wi}}})$ lateral drift can be understood by considering how the elastic fluid is stretched in the vicinity of the sphere. In [Fig. \[fig:stress\_tilty\]]{} we show the trace of the first order elastic stress tensor, ${\ensuremath{\textrm{Tr}\,}}{\boldsymbol{\Pi}}^{(1)}({\boldsymbol{r}})$, around the sphere. This trace indicates how strongly the dumbells are stretched by the lowest-order Newtonian flow. The Figure shows a center cross-section of the particle, viewed along the direction of external forcing. In the cross-shear flow, $\varphi=0$, the stretching is a complicated function of the spatial variables, but perfectly symmetric around the sphere (top row in [Fig. \[fig:stress\_tilty\]]{}). Therefore there is no net force on the particle. But as the external forcing is tilted, the particle is forced to move along the flow direction. Now the particle surface moves opposite to the undisturbed flow on one side, and along the undisturbed flow on the other. This asymmetry results in different stretching of the dumbbells on the two sides, as illustrated for $\varphi=45^\circ$ in the second row of [Fig. \[fig:stress\_tilty\]]{}. This stress contributes to the particle drift both directly, by forcing the particle surface, and indirectly by forcing the suspending fluid and thereby inducing additional viscous drag. [*Method.*]{} In this paper we also introduced the tensors $T^{nl}_{i_1i_2..i_n}$ ([Section \[sec:Ttensors\]]{}). These tensors are a basis suitable for symbolic calculations of tensorial quantities in spherical geometry. In particular they allow us to write down solutions to inhomogenous Stokes equations in tensorial form, without any explicit coordinate representation, and without explicitly solving differential equations. The calculation in this paper demonstrates the power of our method for treating tensorial equations such as the coupled rank-$2$ constitutive [Eq. (\[eqn:oldBdef\])]{} and the rank-$1$ flow [Eq. (\[eqn:floweq1\])]{}. Nevertheless, there are many open questions regarding the algebraic properties of the $T$-tensors. Most importantly, we have shown that the product ${\boldsymbol{T}}^{l_1l_1}{\boldsymbol{T}}^{l_2l_2}$ is given by a linear combination of ${\boldsymbol{T}}^{JJ}$ with $|l_1-l_2|\leq J\leq l_1+l_2$, analogous to the product of two spherical harmonics (see [Appendix \[app:Tquestions\]]{}). But further work must be done to determine the properties of the coefficients in this linear combination, to determine the general expression for differentiation, and to prove the general case of [Eq. (\[eqn:raisingn\])]{}. Our tensor formalism can also be extended to other geometries. Nearly spherical geometry can be treated by perturbation theory. Other geometries can be treated by the method of images [@blake_note_1971; @chwang_hydromechanics_1975]. For example, the flow around a spheroid in unbounded flow is given by a finite distribution of multipoles [@chwang_hydromechanics_1975]. However, the radial functions are no longer simply $r^m$, but integrals $I_m^n=\int_{-c}^c \xi^n/|{\boldsymbol{r}}-\xi {\boldsymbol{n}}|^m\,{\ensuremath{\textrm{d}}}\xi$, where ${\boldsymbol{n}}$ is the direction of the spheroid and $c$ is a shape-dependent constant [@chwang_hydromechanics_1975; @einarsson_rotation_2015]. Their algebraic properties must be worked out in order to use the formulae in [Section \[sec:Ttensors\]]{} e.g. for the Fourier transform. For wall interactions, or other many-center problems, it is possible to derive a translation theorem that expresses $|{\boldsymbol{r}}-{\boldsymbol{r}}'|^m {\boldsymbol{T}}^{nl}({\boldsymbol{r}}-{\boldsymbol{r}}')$ as an infinite series of $|{\boldsymbol{r}}|^m {\boldsymbol{T}}^{nl}({\boldsymbol{r}})$ and $|{\boldsymbol{r}}'|^m {\boldsymbol{T}}^{nl}({\boldsymbol{r}}')$ [@weniger_spherical_2005], which restores the linearity of the problem. [*Acknowledgements*]{}. This work was supported by Vetenskapsrådet, and by the grant [*Bottlenecks for particle growth in turbulent aerosols*]{} from the Knut and Alice Wallenberg Foundation, grant number 2014.0048. [24]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****,  ()](\doibase 10.1016/0377-0257(93)85026-7) [****,  ()](\doibase 10.1016/j.jnnfm.2012.07.002) [****,  ()](\doibase 10.1016/j.jnnfm.2013.02.003) [****,  ()](\doibase 10.1016/j.jnnfm.2014.09.014) [****,  ()](\doibase 10.1146/annurev-chembioeng-080615-033630) [****,  ()](\doibase 10.1016/j.jnnfm.2013.07.007) [****,  ()](\doibase 10.1063/1.4862483) [****,  ()](\doibase 10.1017/S0022112077000822) [****, ()](http://link.springer.com/article/10.1007/BF01523744) [****,  ()](\doibase 10.1063/1.4733700) [****,  ()](\doibase 10.1063/1.3583376) [****,  ()](\doibase 10.1122/1.2998219) [****,  ()](\doibase 10.1093/qjmam/14.1.36) @noop [**]{} (, ) @noop [**]{}, Butterworth-Heinemann series in chemical engineering (, , ) [****,  ()](\doibase 10.1016/0377-0257(79)85004-1) [****, ()](\doibase 10.1017/S002211207600089X) [****,  ()](\doibase 10.1017/S0022112075001450) [****,  ()](\doibase 10.1112/plms/s1-24.1.55) [****, ()](\doibase 10.1135/cccc20051225) @noop [****,  ()]{} [****,  ()](\doibase 10.1017/S0305004100049902) [****,  ()](\doibase 10.1017/S0022112075000614) [****,  ()](\doibase 10.1063/1.4921543) Multiplication of $T$-tensors ============================= [\[app:Tquestions\]]{} We use following normalization of the spherical harmonics, $$\begin{aligned} Y_l^m(\theta, \varphi)&=\sqrt{\sqrt{\pi}\frac{2^{-l} l! \Gamma (l-m+1)}{\Gamma \left(l+\frac{1}{2}\right) \Gamma (l+m+1)}}P_l^m(\cos\theta)e^{im\varphi}\,. \end{aligned}$$ With this normalization we conjecture that the transformation between $T^{ll}_i$ and $Y_l^m$ is unitary: $$\begin{aligned} T_i^{ll} = \sum_m c^{llm}_{{\boldsymbol{i}}} Y_l^m\,,\quad Y_l^m = \sum_{{\boldsymbol{i}}} \overline{c_{{\boldsymbol{i}}}^{llm}}T_{{\boldsymbol{i}}}^{ll}\,. \end{aligned}$$ We have checked this relation up to $l=8$ by explicitly calculating the $c^{llm}$. The most convenient way to calculate is to express $\hat{{\boldsymbol{r}}}$ in the complex basis ${\hat{{\boldsymbol{e}}}_{-1}, \hat{{\boldsymbol{e}}}_{0}, \hat{{\boldsymbol{e}}}_{1}}$, related to ${\hat{{\boldsymbol{e}}}_{x}, \hat{{\boldsymbol{e}}}_{y}, \hat{{\boldsymbol{e}}}_{z}}$ by the complex rotation $$\begin{aligned} \begin{array}{lll} R_{x, -1} = \frac{1}{\sqrt 2} & R_{x, 0} = 0 & R_{x, 1} = -\frac{1}{\sqrt 2} {\nonumber}\\ R_{y, -1} = i \frac{1}{\sqrt 2}& R_{y, 0} = 0 & R_{y, 1} = i \frac{1}{\sqrt 2} {\nonumber}\\ R_{z, -1} = 0 & R_{z, 0} = 1 & R_{z, 1} = 0 \,, \end{array}\end{aligned}$$ so that $$\begin{aligned} c^{nlm}_{i_1..i_n}=\sum_{\nu_i=-1,0,1}R_{i_1\nu_1}..R_{i_n\nu_n}c^{nlm}_{\nu_1..\nu_n}\,.\end{aligned}$$ In this basis $$\begin{aligned} \hat r_\nu=T^{11}_{\nu}&=Y_1^\nu\,,\end{aligned}$$ and therefore $$\begin{aligned} \hat r_{\nu_1}..\hat r_{\nu_n} = \sum_{L=0}^{n}\sum_{m=-L}^lc^{nJM}_{\nu_1..\nu_n}Y_J^M &= \sum_{l=0}^{n-1}\sum_{m=-l}^l c^{n-1,lm}_{\nu_1..\nu_{n-1}}Y_l^m Y_1^{\nu_n}\,.\end{aligned}$$ Because of the orthogonality of the spherical harmonics, this leads to a recurrence relation for the coefficients: $$\begin{aligned} c^{nJM}_{\nu_1..\nu_n} &= \sum_{l=0}^{n-1}\sum_{m=-l}^l c^{n-1,lm}_{\nu_1..\nu_{n-1}} g(l,m,1,\nu_n,J,M)\,,\end{aligned}$$ where $g$ is the Gaunt coefficient for integrals of the spherical harmonics, $$\begin{aligned} g(l_1,m_1,l_2,m_2,J,M) &= \int_S Y_{l_1}^{m_1}Y_{l_2}^{m_2}\overline Y_{J}^{M}\,{\ensuremath{\textrm{d}}}S\,.\end{aligned}$$ Provided this unitary transformation, we have $$\begin{aligned} T^{l_1l_1}_{{\boldsymbol{i}}}T^{l_2l_2}_{{\boldsymbol{j}}}=\sum_{J=|l_1-l_2|}^{l_1+l_2}A^{l_1l_1l_2l_2J}_{{\boldsymbol{i}}{\boldsymbol{j}}{\boldsymbol{k}}}T^{JJ}_{{\boldsymbol{k}}}\,, \end{aligned}$$ where $$\begin{aligned} A^{l_1 l_1 l_2 l_2 J}_{{\boldsymbol{i}}{\boldsymbol{j}}{\boldsymbol{k}}} &= \sum_{m_1=-l_1}^{l_1} \sum_{m_2=-l_2}^{l_2} \sum_{M=-J}^J c^{l_1 l_1 m_1}_{{\boldsymbol{i}}}c^{l_2 l_2 m_2}_{{\boldsymbol{j}}}\overline{c^{J J M}_{{\boldsymbol{k}}}}g(l_1,m_1,l_2,m_2,J,M)\,. \end{aligned}$$

--- abstract: 'We investigate the evolution of differential emission measure distribution (DEM\[T\]) in various phases of a B8.3 flare, which occurred on July 04, 2009. We analyze the soft X-ray (SXR) emission in 1.6-8.0 keV range, recorded collectively by Solar Photometer in X-rays (SphinX; Polish) and Solar X-ray Spectrometer (SOXS; Indian) instruments. We make a comparative investigation of the best-fit DEM\[T\] distributions derived by employing various inversion schemes viz. single gaussian, power-law, functions and Withbroe-Sylwester (W-S) maximum likelihood algorithm. In addition, SXR spectrum in three different energy bands viz. 1.6-5.0 keV (low), 5.0-8.0 keV (high) and 1.6-8.0 keV (combined) is analyzed to determine the dependence of the best-fit DEM\[T\] distribution on the selection of energy interval. The evolution of DEM\[T\] distribution, derived using W-S algorithm, reveals the plasma of multi-thermal nature during the rise to the maximum phase of the flare, while of isothermal nature in the post-maximum phase of the flare. Thermal energy content is estimated considering the flare plasma to be of 1) iso-thermal and 2) multi-thermal nature. We find that the energy content during the flare, estimated from the multi-thermal approach, is in good agreement with that derived using the iso-thermal assumption except during the maximum of the flare. Further, (multi-) thermal energy estimated employing low-energy band of the SXR spectrum result in higher values than that derived from the combined-energy band. On the contrary, the analysis of high-energy band of SXR spectrum lead to lower thermal energy than that estimated from the combined-energy band.' author: - Arun Kumar Awasthi - Barbara Sylwester and Janusz Sylwester - Rajmal Jain title: 'Thermal characteristics and the differential emission measure distribution during a B8.3 flare on July 04, 2009' --- Introduction ============ Solar flare is one of the most energetic phenomena occurring in the atmosphere of our Sun, releasing typically $\mathrm{10^{27}}$-$\mathrm{10^{32}}$ ergs of energy in $\sim$ $10^3$ s. This immense energy release is understood to be powered by the magnetic energy via the process of magnetic reconnection [@Shibata1999; @Jain2011a; @Choudhary2013; @Aschwanden2014; @Dalmasse2015]. A typical M-class solar flare can be observed across almost entire electromagnetic spectrum [@Benz2008; @Fletcher2011]. Therefore, various energy release processes occurring at various heights in the solar atmosphere can be probed by the investigation of the observed multi-wavelength flare emission. X-ray emission during solar flares mainly originates from the corona and upper chromosphere. Moreover, X-ray emission recorded during a flare can serve as the best probe of studying various plasma processes of thermal and non-thermal character [@Li2005; @Saint-Hilaire2005; @Jain2008; @Awasthi2014]. Low-energy X-ray emission ($<$ 10 keV), also known as soft X-ray (SXR), is understood to be originated in the process of free-free, free-bound and bound-bound emission due to collision of charged particles (mostly electrons) having thermal (Maxwell-Boltzmann) distribution. On the other hand, high energy X-ray emission (hard X-rays) is known to be produced as a consequence of thick-target bremsstrahlung of non-thermal electron beam with the dense plasma in the chromosphere [@Brown1971; @Kulinova2011]. Moreover, SXR emission during a flare is understood to be produced by multi-thermal plasma [@Aschwanden2007; @Jain2011b; @Sylwester2014; @Aschwanden2015]. The study of thermal characteristics of the flare plasma is made by the inversion of observed X-ray spectrum through postulating an empirical functional form of differential emission measure distribution (DEM\[T\]). Although DEM\[T\] plays a key role in deriving thermal characteristics and in turn energetics of the flare plasma, it is less accurately known owing to the fact that inversion of observed radiation needs to be performed which is very ill-posed problem [@Craig1976]. Moreover, several DEM\[T\] schemes which postulate certain functional dependence of DEM on T viz. single gaussian, bi-gaussian, power-law etc. have been proposed (see @Aschwanden2015 for exhaustive list of schemes). Further, a Withbroe-Sylwester (W-S) maximum likelihood DEM inversion algorithm has been established by @Sylwester1980 where the functional form of DEM\[T\] is not a-priori defined. In this regards, a comparative survey of the aforesaid DEM schemes in the form of derived thermal characteristics of the flare plasma is very necessary provided application of various inversion schemes result in similar outcome. In addition, an inevitable restriction on deriving the complete thermal characteristics of the flare plasma is posed by the availability of observations from different instruments in certain specific energy-bands only. Thermal emission can be best studied by measuring the X-ray spectrum in typically 1-12 keV energy band. Observations in aforesaid energy band with high spectral and temporal cadence is very difficult to achieve from a single instrument due to huge difference of flux during a flare across the said energy band. Therefore, we examine temperature dependence of differential emission measure from the analysis of multi-instrument data for a B8.3 flare which occurred on July 04, 2009. As the flare selected for the analysis is the only event common between Solar Photometer in X-Rays (SphinX; a Polish instrument) and Solar X-ray spectrometer (SOXS; an Indian instrument), combined data-set provides a unique opportunity for exhaustive study of the complete thermal characteristics of a small flare. Both the instruments make use of Si PIN detectors for observing solar atmosphere in X-ray waveband. Section \[sec:obs\] presents observations used for the present study and the specification of respective instruments. In Section \[sec:data-analysis\], we present the study of DEM\[T\] distribution derived by employing different inversion schemes and it’s dependence on the selection of energy band of input SXR spectrum. In Section \[sec:e-th\], thermal energetics of the flare, estimated from the parameters derived from various schemes, is presented. Section \[sec:sum-conc\] is comprised of the summary and conclusions. Observations {#sec:obs} ============ We investigate a B8.3 intensity class flare which occurred on July 04, 2009 in an active region AR11024. AR 11024 appeared on the disk on July 3, 2009 and rotated off the disk on July 15, 2009. More than 500 flares or small brightenings have been observed with SphinX mission on the soft X-ray light curve during that time. Figure \[light-curve-all-sphinx\] shows the temporal evolution of X-ray emission recorded by SphinX during the aforesaid period. {height="90.00000%"} The SOL2009-07-04T04:37 flare, selected for the present study, is the only event observed in common by SphinX and SOXS missions due to the fact that SOXS mission usually observed the Sun in X-rays for only 2-3 hours in a day. We analyze the X-ray spectra in 1.6-5.0 keV and 5.0-8.0 keV energy bands, recorded from SphinX and SOXS missions, respectively. We briefly discuss the data and respective instruments’ specifications as following: Solar Photometer in X-Rays (SphinX) mission ------------------------------------------- We analyze X-ray spectra in 1.6-5.0 keV (hereafter low-energy band) from the SphinX instrument [@Gburek2011; @Sylwester2012; @Gburek2013]. SphinX, a spectrophotometer designed to observe solar corona in soft X-rays, was flown on-board the Russian *CORONAS-PHOTON* satellite on January 30, 2009. SphinX employed three Si PIN diode detectors to record X-rays in the energy range $\sim$ 1.2-15.0 keV. The temporal and spectral cadence of SphinX observations are as good as 6 $\mu$s and 0.4 keV, respectively. Detailed information regarding the observations, procedure for calibration and data warehouse may be referred from @Gburek2013 and from the SphinX instrument homepage[^1]. Solar X-ray Spectrometer (SOXS) mission --------------------------------------- X-ray spectra in 5.0-8.0 keV (hereafter high-energy band) during the flare are obtained by the Solar X-ray Spectrometer (SOXS) instrument [@Jain2005; @Jain2008]. SOXS employed two semiconductor devices, viz. silicon (Si) PIN detector for recording X-ray observations in the energy range 4-25 keV while Cadmium Zinc Telluride (CZT) detector for that in the energy range 4-56 keV. The energy resolution of the Si detector is $\sim$ 0.8 keV while that for CZT detector is $\sim$ 1.7 keV. The temporal cadence of the observations obtained from both the detectors is 3 sec during the quiet and gradual phase of the flare. However, on-board automated algorithm allowed to record the observations with 100 msec cadence during the rise to peak phase of the flare. The data obtained during the entire observing-span (May 2003- April 2011) of SOXS mission and the analysis procedures are available on the instrument homepage[^2]. In the present study, we employ the observations obtained from Si detector in view of it’s better energy resolution and sensitivity in comparison to that of CZT detector. Left panel of the Figure \[x-ray-sphinx-soxs-goes-lc\] presents the evolution of X-ray emission as observed by SphinX (top row) and SOXS (bottom row) missions during the flare in various energy bands plotted with different colors. Intensity curves shown by black and red colors represent the X-ray emission recorded by SphinX in 1.6-3.0 keV and 3.0-5.0 keV, respectively. Further, X-ray emission in 5.0-7.0 keV and 7.0-8.0 keV, drawn by blue and green colors, respectively are obtained from the SOXS. It may be noted from the Figure \[x-ray-sphinx-soxs-goes-lc\] that SOXS receives higher background than that seen by SphinX. On the contrary, a comparison of the count rates recorded by SphinX and SOXS in 4-6 keV, the energy band commonly covered by both the instruments, revealed that SOXS observations are lower by a factor of $\sim$2.5. This may be attributed to the systematic difference of sensitivities between the two instruments. @Mrozek2012 reported higher flux in SphinX 3-8 keV energy band in comparison to the observations obtained in the same energy range from *RHESSI* mission by a factor varying in the range of 2-6. On the other hand, the comparison of SOXS and *RHESSI* observations in 6-12 keV energy band has been performed by @Caspi2010, which resulted in the agreement of the spectra, obtained from both the instruments, within 5%-10%. In this study, we prepared combined-data by applying the aforesaid ‘empirical normalization factor’ in the records obtained from SOXS. On the other hand, we consider the flux recorded by SphinX as the true flux owing to the fact that this is the only instrument available to observe X-ray emission in the energy less than 4 keV during the flare. Therefore, the difference of inter-instrument sensitivity and hence normalization factor in the aforesaid energy range can’t be established. However, in order to study the effect of this approximation, we have also carried out the investigation of emission measure distribution by applying the inverse normalization factor on the SphinX records while retaining the SOXS counts as such. We discuss the effect of both the aforesaid cases on the thermal energy estimates as presented in the Section \[sec:e-th\]. Geostationary Operational Environmental Satellite (*GOES*) ---------------------------------------------------------- Geostationary Operational Environmental Satellite (*GOES*) refers to a series of satellites dedicated to observe X-ray emission from Sun-as-a-star in two wavelength bands viz. 1.0-8.0 $\mathrm{\AA}$ and 0.5-4.0 $\mathrm{\AA}$. Right column of the Figure \[x-ray-sphinx-soxs-goes-lc\] shows the background subtracted flux in 1.0-8.0 $\mathrm{\AA}$ and 0.5-4.0 $\mathrm{\AA}$ bands plotted by black and red colors, respectively. We treat the observations averaged during 04:20-04:25 UT as the background. Temperature and emission measure (EM) estimated from the flux-ratio technique adopted for *GOES* data are also plotted in the middle and bottom rows of the right panel of Figure \[x-ray-sphinx-soxs-goes-lc\], respectively. It may be noted that temperature is found to be varying in the range of 7-12 MK while the EM in the range of 0.003-0.08 $ \times 10^{49} \mathrm{cm^{-3}}$. We use these T & EM estimates to calculate the thermal energy during the flare (cf. Section \[sec:e-th\]). {height="90.00000%"} Morphology of the flare in Extreme Ultraviolet (EUV) emission ------------------------------------------------------------- Temporal and morphological evolution of the flaring region is studied from the observations obtained from EUV Imaging Telescope (EIT) [@Delaboudiniere1995] on-board *Solar and Heliospheric Observatory (SOHO)*. Moreover, images in 171, 284 and 304 $\mathrm{\AA}$ wavelength, recorded by *STEREO* twin-satellites, are also processed. In Figure \[stereo-twin\], we present the morphological evolution of the flaring region in 171 and 304 $\mathrm{\AA}$ during the flare as obtained from *STEREO-A* & *B*. -- -- -- -- -- -- -- -- From the time sequence of the EUV images presented in Figure \[stereo-twin\], we note that although the flare event considered is of a small B8.3 intensity class, it is associated with an eruption. The study of the eruption is out of the scope of the aim of this paper. We estimate the volume of the emitting region from the EUV images to derive thermal energetics of the flare as presented in section \[sec:e-th\]. DEM\[T\] distribution from the application of various inversion schemes {#sec:data-analysis} ======================================================================= In order to study thermal characteristics of the flare plasma, we make an exhaustive investigation of evolution of DEM\[T\] relationship employing the X-ray spectra observed from SphinX and SOXS. We explore the dependence of DEM\[T\] distribution, which is derived by employing various inversion schemes on the SXR spectra of various energy bands viz. 1.6-5.0 keV (low-energy), 5.0-8.0 keV (high-energy) and 1.6-8.0 keV (hereafter combined-energy). This study aims to understand the dependence of the best-fit DEM\[T\] representing a selective part of the SXR emission, which in turn presents the consequence of restrictions posed by the co-temporal observations recorded in separate energy bands from different instruments viz. SphinX and SOXS. In this study, we employ the DEM inversion schemes which postulate (1) single-gaussian and, (2) power-law functional relationship of DEM with T. In addition, we also employ (3) a well-established Withbroe-Sylwester (W-S) maximum likelihood inversion algorithm which is independent of a-priori assumption of a functional form of DEM\[T\]. In the following, we present the thermal characteristics of X-ray emission during the flare as derived by applying aforesaid inversion schemes. DEM varying as a single-gaussian function over temperature ---------------------------------------------------------- We investigate the best-fit DEM\[T\] distributions, obtained by employing the scheme of single-gaussian functional dependence of DEM on $T$, on the observed SXR spectrum in the low, high and combined energy bands. However, firstly, we also employ this DEM scheme on synthesized model multi-thermal spectrum. Below we discuss the aforesaid two cases. ### DEM\[T\] distribution of a synthesized model multi-thermal spectrum We synthesize multi-thermal photon spectra by the model photon flux arrays corresponding to iso-thermal plasma in the temperature range of 1-23 MK with a temperature bin of $log~T$=0.1 MK. Iso-thermal photon spectrum at specific temperature and emission measure is calculated by using the iso-thermal model (f\_vth.pro) available in SPectral EXecutive (SPEX) package within *SolarSoftWare (SSW)*. We derive EM values corresponding to a temperature from the emission measure model of @Dere1979, and also available in CHIANTI atomic database [@Landi2012; @Del; @Zanna2015]. In addition we consider the abundance to be 0.1 times the coronal abundance available in the CHIANTI distribution. Next, the multi-thermal photon spectrum is derived by the weighted sum of iso-thermal spectra in following manner. $$\label{eq-F-multi-thermal} F_{MT}=\sum_{k=T_{min}}^{T_{max}} w_{k} F(T_k,EM_k)$$ Here $F(T_k,EM_k)$ is the iso-thermal photon flux and shown by grey colour (dotted) plots in Figure \[theory-multithermal-spectra\] while the multi-thermal flux ($F_{MT}$) synthesized in such a way is over-plotted with red color. Further, $w_k$ is the weight factor which is assumed to be a normalized gaussian function of temperature with the maximum at $T$=5.6 MK and FWHM of $\sim$5 MK as shown in panel \[d\] of the Figure \[fit-theory-spectra\]. Such integration allows more realistic scheme of synthesizing theoretical multi-thermal spectra than the one adopted in @Aschwanden2007 and @Jain2011b. They use direct sum of iso-thermal fluxes with equal weight, in which the synthesized multi-thermal photon spectrum is dominated by the contribution from iso-thermal spectrum corresponding to the peak temperature. ![The spectra plotted in grey color (dotted) represent the iso-thermal photon flux corresponding to the temperature range 1-23 MK with the interval log T=0.1 MK. The plot drawn with red color represents the integrated (multi-thermal) photon flux.[]{data-label="theory-multithermal-spectra"}](iso_multithermal_theory_spectra.ps){height="45.00000%"} Next, we forward fit the synthesized multi-thermal spectrum using DEM varying as a single-gaussian function of T. The form of the DEM\[T\] is considered as $$\label{eq-dem-gauss} DEM[T] \propto exp(\frac{-(log T_{p}- log T )^2 }{2\sigma^2})$$ where $T_{p}$ is the temperature at the peak of DEM \[T\] while $\sigma$ is the gaussian width. By iteratively varying the independent variables of Equation \[eq-dem-gauss\] viz. $T_{p}$, $\sigma$ etc., a photon flux best-fit to the input synthesized multi-thermal photon spectra is derived. It is to be noted that the minimum and maximum temperature values for deriving the DEM\[T\] are fixed to 0.087 keV (1 MK) and 8.5 keV (100 MK), respectively. The best-fit is assessed by estimating reduced $\chi^2$ in each step of iteration, which converges to a small value. Following the aforesaid procedure, we derive the best-fit parameters for the synthesized multi-thermal photon spectrum in the low- and high-energy bands. Next, we apply the similar procedure to derive the best fit parameters for the input photon spectrum in the combined-energy band. Panels \[a\], \[b\] and \[c\] of Figure \[fit-theory-spectra\] show the synthesized input multi-thermal photon spectrum (grey color) overlaid by best-fit model flux drawn in black, blue and red colors for low-, high- and combined-energy bands, respectively. In addition, normalized residuals are also plotted with the respective panels. Moreover, panel \[d\] presents the DEM evolution corresponding to the best-fit model photon spectra. -- -- -- -- From the aforesaid analysis, presented in Figure \[fit-theory-spectra\], we find that the best-fit flux derived for the input photon spectrum in low-energy band (panel \[a\]) does not provide a good fit (over-estimation) to the higher energy part of the spectrum. On the other hand, the best-fit flux obtained for the high-energy band (panel \[b\]) does not completely follow (under-estimates) the low-energy part of the spectrum. This trend is clearly represented in the form of normalized residuals, plotted with the respective panels. Panel \[d\] of the Figure \[fit-theory-spectra\] shows the DEM\[T\] for the best-fit model photon flux estimated for input photon spectrum corresponding to the low, high and combined energy ranges. This enable us to make a comparative study of the DEM\[T\] dependence on input energy bands selection. We note that the best-fit DEM\[T\] curves for the low and high-energy bands yield high peak values of DEM ($DEM_p$)=1.72 $\times$ $10^{49}$ $cm^{-3} MK^{-1}$, however at low temperature $T_p$=7.8 MK for the former (low-energy band) case while relatively lower $DEM_p$=1.66 $\times$ $10^{49}$ $cm^{-3} MK^{-1}$ at higher $T_p$=8.2 MK for the latter case. Moreover, moderate $DEM_p$ at $T_p$ best fits the input spectra of combined-energy band. ### DEM\[T\] distribution derived from observed X-ray emission during the flare We analyze X-ray emission in the low (1.6-5.0 keV) and high (5.0-8.0 keV) energy bands, obtained from SphinX and SOXS, respectively during the flare. In this regard, we prepare a time-series of the spectra by integrating the observed X-ray emission into 120 second time intervals during 04:27-04:33 UT and 04:38-05:00 UT, corresponding to the rise and decay phase of the flare, respectively. On the other hand, better count statistics during 04:33-04:38 UT, corresponding to the impulsive phase of the flare enabled us to integrate the observation in 60 second time intervals. Time sequence of the spectra obtained in such a way serves as the input to the inversion scheme. We forward-fit the observed SXR spectrum in the low-, high- and combined-energy bands with a model photon flux, which is derived by the inversion scheme employing a single-gaussian functional dependence of the DEM on $T$ (cf. Equation \[eq-dem-gauss\]). Panel \[a\] of the Figure \[fit-dem-gauss-spectra\] presents the observed count rate in low-energy band (black color) during 04:36-04:37 UT, corresponding to the maximum of impulsive phase of the flare. Similarly, X-ray emission in high-energy band (blue color) is analyzed, as shown in panel \[b\] of the Figure \[fit-dem-gauss-spectra\] for the aforesaid time interval. Panel \[c\] presents the analysis of combined-energy band data (from SphinX and SOXS), for the time intervals same as that of panels \[a\] and \[b\]. Best fit model count rates are over-plotted by red color in the respective panels and the derived values are also shown. Panel \[d\] presents the DEM\[T\] distribution corresponding to the best-fit model obtained for SXR spectra of different energy bands. -- -- -- -- From Figure \[fit-dem-gauss-spectra\], it may be noted that $DEM_{p}$ estimated from spectral fitting of the low, high and combined-energy band data are 1.40, 0.15 and 0.38 ($ \times 10^{49} \mathrm{cm^{-3} MK^{-1}}$), respectively. On the other hand, $T_p$ is estimated to be 6.37, 8.93 and 7.57 MK, respectively. It may be noted that the trend of the best-fit parameters viz. $DEM_{p}$ and $T_p$ for three cases of input energy band is in good agreement with that revealed by the study of model multi-thermal spectrum employing the same inversion scheme as presented in the previous section. DEM varying as power-law function of T -------------------------------------- We derive the DEM\[T\] distribution for the X-ray spectra corresponding to various energy bands, similar to the analysis made in the previous section, however with different functional dependence of DEM on $T$. In this multi-thermal model, DEM is approximated to be varying with $T$ in the form of power-law and can be expressed as: $$DEM(T) \propto (\frac{2}{T})^\gamma$$ Next, employing this DEM scheme, we forward-fit the observed flare X-ray spectrum in low-energy band obtained from SphinX (black color) as shown in panel \[a\] of the Figure \[fit-dem-power-law-spectra\]. During the iterative procedure for obtaining the best-fit model, the low-temperature value is fixed to be 0.5 keV (5.8 MK) while the maximum temperature is determined as one of the outputs. All the spectra during various time intervals of the flare are analyzed, however here we present only the results from the observations during 04:36-04:37 UT, same as that presented in the previous section. X-ray emission in high-energy band, obtained from SOXS (blue color) is presented in panel \[b\] of the Figure \[fit-dem-power-law-spectra\]. Next, we also fit the observed X-ray spectrum in combined-energy band as shown in panel \[c\]. The best-fit models are over-plotted by red color lines in the respective plots. Parameters of the best-fit viz. $T_{max}$, DEM (at T=2 keV) and the power-law index ($\gamma$) are also shown in respective plots. Panel \[d\] shows the derived DEM\[T\] curves corresponding to the best-fit models obtained for different energy ranges. -- -- -- -- From Figure \[fit-dem-power-law-spectra\], it may be noted that at the peak of the impulsive phase of the flare, $T_{max}$ estimated from the observation recorded by SphinX, SOXS as well as combined observations is 23.33, 19.58 and 18.47 MK, respectively. It may be noted that $T_{max}$ and DEM values estimated in such a way follow the same trend as that resulted in the previous DEM scheme. Moreover, the negative power-law index ($\gamma$) of the best-fit DEM\[T\] distribution corresponding to the SXR spectrum in low, high and combined-energy bands is estimated to be 5.46, 4.45 and 4.17, respectively. The less-negative (steeper) value of ‘$\gamma$’ for the high and combined-energy cases suggest enhanced contribution of high-temperature plasma than that obtained from the analysis of SXR emission in the low-energy band only. Withbroe-Sylwester (W-S) maximum likelihood DEM inversion algorithm ------------------------------------------------------------------- We employ Withbroe-Sylwester (W-S) maximum likelihood DEM inversion algorithm [@Sylwester1980; @Kepa2006; @Kepa2008] on the X-ray spectra observed during the flare. The W-S algorithm is a Bayesian numerical technique which employs maximum likelihood approach in which the DEM distribution in one step of iteration ‘j’ ($DEM_{j}[T]$) is estimated from that derived in the preceding iteration ($DEM_{j-1}[T]$), and by employing a correction factor ($c_i$) as well as weight factor ($w_i$) in the form given below. $$DEM_{j}[T]=DEM_{j-1}[T] \frac{\sum_{i=1}^{k}c_i w_i(T)}{\sum_{i=1}^{k}w_i(T)}$$ Here, the correction factor, $c_i$, is estimated from the ratio of the observed flux with the calculated flux, which is derived using previous DEM distribution form and can be expressed mathematically as: $$c_{i}=\frac{F_{obs, i}}{F_{cal, i}}$$ where, $F_{cal,i}$ is the calculated model flux and obtained by, $$F_{cal,i}=\int_{j=0}^ \infty f_i(T) DEM_j(T) dT$$ In the aforesaid function, $f_i(T)$ is the theoretical emission function for energy ‘i’ and is derived using the CHIANTI package [@Del; @Zanna2015]. The weight factor $w_i$ is estimated as: $$\begin{split} w_i(T)=f_i(T) DEM_j(T) dT \frac{\int_{j=0}^ \infty f_i(T) DEM_j(T) dT}{\int_{j=0}^ \infty [f_i(T) DEM_j(T)]^2 dT} \times \\ [\frac{|F_{obs,i}-F_{cal,i}|}{\delta_i}+1]^a \end{split}$$ Here, $\delta_i$ is the uncertainty corresponding to the observations for energy ‘i’ and ‘a’ is termed as the speed convergence parameter. We apply the aforesaid W-S DEM inversion algorithm on the X-ray spectra obtained from SphinX and SOXS missions during the flare to obtain the best-fit photon flux and corresponding DEM\[T\] distribution. Coronal abundances from CHIANTI atomic database have been adopted while calculating theoretical dependence of spectral shapes. Top and the middle rows of the Figure \[w-s-dem-fit-total-sphinx-soxs\] present the results of the application of W-S algorithm on the X-ray emission measured by SphinX, SOXS, respectively, during 04:27:30-05:00:00 UT, covering the entire flare duration. Moreover, bottom panel of the Figure \[w-s-dem-fit-total-sphinx-soxs\] shows the same, however, corresponding to the combined data-set. Left panel shows the DEM\[T\] distributions obtained from the best-fit model (red) for the observations, shown in the right column. [c]{}\ \ \ From the application of W-S scheme, as shown in Figure \[w-s-dem-fit-total-sphinx-soxs\], we find the peak temperature ($T_{p}$)= 10.0, 9.5 and 10.0 MK, and the total emission measure (EM) log(EM)=47.42, 47.17 and 47.41 ($cm^{-3}$), corresponding to the SphinX, SOXS and combined-energy band data, respectively. This suggests that the trend of the parameters obtained by the W-S scheme for the three cases of the input energy bands is in agreement with that obtained from the previous schemes. The aforesaid analysis is made for the spectra obtained by integrating the emission in the whole flare duration. Next, we derive the temporal evolution of DEM\[T\] distribution during various phases of the flare by applying W-S algorithm on the X-ray emission observed during various time intervals of the flare, as presented in Figure \[w-s-dem-fit-sphinx-soxs\]. Left panels of the Figure \[w-s-dem-fit-sphinx-soxs\] show the temporal evolution of best-fit DEM\[T\] distribution derived over various time intervals of the flare, while the respective right panels show the observed X-ray spectra in the combined-energy band (1.6-5.0 and 5.0-8.0 keV, observed by SphinX and SOXS) overlaid by best-fit model (red). [c]{} From Figure \[w-s-dem-fit-sphinx-soxs\], it may be noted that the best fit DEM\[T\] curve, obtained from the analysis of the X-ray emission measured during the flare onset time (04:27:30-04:29:45 UT), can be well approximated by a single gaussian function of T with a width $\sim$1 MK. Moreover, the peak temperature ($T_p$) is estimated to be 5.62 MK. On the contrary, the best-fit DEM\[T\] curve obtained by analysing the spectra during 04:31-04:34 UT, corresponding to the rise phase of the flare, resembles the double-peak gaussian with the increased widths (in comparison to that during the flare onset) $\sim$1.5 MK . Moreover, $T_p$ is estimated to be varying in the range of 6.3-14.1 MK. This reveals the signature of contribution of high temperature plasma in this phase in addition to the low-temperature component, which was present during the flare onset. Further, DEM\[T\] derived for the spectra obtained during 04:32-05:00 UT, corresponding to the peak of the impulsive phase and decay phase of the flare, resulted in single peak gaussian nature, however, with peak temperature varying in the range of $\sim$ 13.0-5.5 MK. It is intriguing to note that the best-fit DEM\[T\] distribution, obtained by integrating the emission in the whole flare duration, as shown in Figure \[w-s-dem-fit-total-sphinx-soxs\], can be well approximated to iso-thermal nature. On the contrary, temporal evolution of DEM\[T\] distribution over various phases of the flare suggests the presence of multi-thermal plasma during the rise phase of the flare. This apparent inconsistency may be explained by the fact that if the X-ray spectrum is integrated for the whole flare duration, it is dominated by the emission at the peak of the impulsive phase. Now, it may be noted that the best-fit DEM\[T\], derived for the spectrum during 04:36:00-04:38:30 UT (corresponding to the peak of the impulsive phase, see Figure \[w-s-dem-fit-sphinx-soxs\]) is iso-thermal in nature. In Figure \[dem-3d-sphinx+combined\], we present the different visualisation of temporal evolution of DEM\[T\] distribution over the flare duration, from the SphinX observations in the left panel while that from the combined-energy band observations in the right panel. The comparison of DEM\[T\] derived from the SphinX observations alone with that obtained from combined observations reveals the signature of high temperature component in the latter analysis during the rise-phase of the flare. -- -- -- -- Thermal energetics of the flare {#sec:e-th} =============================== We estimate thermal energy content during various phases of the flare. We denote the energy content which is estimated employing the iso-thermal approach as the ‘iso-thermal energy’ while that derived considering flare plasma to be of multi-thermal nature is termed as ‘multi-thermal energy’. Next, we make a comparative study of the multi-thermal energy content derived by the application of various DEM inversion schemes. In order to estimate the iso-thermal energy content of the flare plasma, we derive the temperature ($T$) and emission measure ($EM$) by employing the technique presented in @Gburek2013 on the high temporal cadence SphinX spectra. $T$ and $EM$, derived in such a way vary in the range of 2.7-15.7 MK and 1.15-28.66 ($\times 10^{47}$ $cm^{-3}$), respectively. Next, we derive the thermodynamic measure ($\eta$, see @Sylwester1995, @Sylwester2006) which is associated with the thermal energy as follows: $$\label{eth} E_{th}=3 k_b\eta \sqrt{V}$$ Here, $V$ is the volume of the emitting plasma. Thermodynamic measure, $\eta$, defined as $T\sqrt{EM}$, characterizes the thermal energy of the plasma for the case of the constant volume of the emitting region. In this study, we derive the volume of the emitting region from the EUV images in 284 $\mathrm{\AA}$, the hottest channel, as obtained from *STEREO* twin satellites. Figure \[st-a-b-eit-contour\] shows the sequence of images in 284 $\mathrm{\AA}$ during the flare, recorded by *STEREO* as well as in 195 $\mathrm{\AA}$ by EIT/*SOHO*. The contours drawn on the images are 5, 10 and 20% of the maximum intensity of respective images. -- -- -- -- -- -- From Figure \[st-a-b-eit-contour\], it may be noted that the contour of green color, corresponding to 20% of the maximum intensity best represents the emitting region. Volume of the region, assuming spherical geometry, is estimated as follows: $$\label{eq-volume} V=\frac{4}{3} \pi (R)^3$$ where ‘$R$’ is the equivalent radius \[=$(A/\pi)^{1/2}$\] of a circle having area ($A$) equal to that of the region within the iso-contour of 20% of the maximum intensity (green color) of the images presented in Figure \[st-a-b-eit-contour\]. We estimate the temporal evolution of the flare volume from the images in several EUV wavelengths (171, 195 and 284 $\mathrm{\AA}$) made available by *STEREO* and EIT/*SOHO* satellites, in the aforesaid manner. However, the volume estimated from the images of 284 $\mathrm{\AA}$ (representing the hot plasma region) is used for deriving thermal energetics of the flare. As *STEREO* provides the images of the region with a time cadence of 20 minutes, we interpolate the flare volume at intermediate times using cubic spline interpolation technique. The flare volume, derived in the aforesaid manner, varies in the range of 1.2-5.4 ($\times 10^{28}$ $cm^3$). Using the volume estimated above, we derive the time evolution of the iso-thermal energy content ($E_{th}$) during the flare using Equation \[eth\] as plotted (black color) in Figure \[E-th\]. The iso-thermal energy content estimated in such a way varies in the range of 2-9 $\times 10^{29}$ ergs. In a similar fashion, we also estimate the iso-thermal energy content employing the $T$ and $EM$ derived from the *GOES* observations (cf. Figure \[x-ray-sphinx-soxs-goes-lc\]). Thus the iso-thermal energy, as derived from the *GOES* observations, is found to be varying in the range of 1.5-6.5$\times 10^{29}$ ergs during the flare and shown by yellow color plot in the Figure \[E-th\]. The estimation of source size may contain various kinds of uncertainties which is crucial to investigate as it is subsequently propagated to the thermal energy estimates. As the EUV source sizes are used with the DEM\[T\] distribution (derived from X-ray observations) while estimating the thermal energy content, a disagreement between the co-temporal source sizes within EUV and X-ray waveband may become one of the major contributor to the uncertainty. Unfortunately imaging mode observations in X-ray waveband have not been available for this flare, and hence EUV images have been used in this study for source size estimation. Although the 284 $\mathrm{\AA}$ filter provides the peak temperature response (at $\sim$2MK), maximum among the other EUV wavelengths available from *STEREO* satellites, it is still quite far from flare plasma temperatures in which X-ray emission is obtained. In this regard, we estimate the co-temporal X-ray and EUV source sizes of thirteen flares of intensity class B1.1 - C1.0 which have occurred during July 04-06, 2009 in the active region AR11024. It may be noted that SOL2009-07-04T04:37, the flare of our present study, is also produced from the same active region. In this statistical investigation, we have used X-ray images obtained from *HINODE*/XRT while EUV images *STEREO* twin satellites in 195 $\mathrm{\AA}$. The source size in both the aforesaid wavelengths is estimated employing the same approach as discussed previously. The comparative investigation has revealed that the source size estimated from the X-ray images are systematically smaller than that derived from the EUV images whereas the ratio is varying in the range of 1.1-9.0 with a median value of 6. Next, imaging a asymmetric flaring region with the instruments that observe the Sun from different angles e.g. the observation of AR11024 with *STEREO*-A and B satellites (Figure \[stereo-twin\]), may contribute to additional uncertainty in the source size estimation. In view of the same, we have also made a comparison of EUV source sizes estimated from 195 $\mathrm{\AA}$ images obtained from *STEREO*-A and B satellites for the aforesaid thirteen flares. This study revealed that the orthogonal view of the flaring region systematically results in larger source size by a factor varying in the range of 1.2-1.5 than that calculated from the images having on-disk view. As the uncertainty in the source size, arising due to the difference in EUV and X-ray source sizes is larger than than that occurred due to observing the region in different angles, the latter may be neglected while calculating the uncertainties in the thermal energy estimates. Thus, in conclusion, considering the fact that EUV source sizes are systematically larger than X-ray sources by a factor of 6, the volume derived from the same suffers an overestimation by a factor of 4. Employing the scheme of error propagation, the application of aforesaid uncertainty in the volume estimates may result in the overestimation of thermal energy content (Equation \[eq-volume\]) by a factor of $\sim$2. We show the aforesaid uncertainty in the iso-thermal energy estimate in the form of associated filled area (light red color) in Figure \[E-th\]. ![Temporal evolution of the iso-thermal and multi-thermal energy content during the flare. Iso-thermal energy, estimated from SphinX observations, is plotted in black (smoothed in red) while that derived from *GOES* observations in shown in yellow color. Multi-thermal energy, derived by applying W-S algorithm on the SphinX & SOXS combined data is shown in the form of blue color histogram. The uncertainty in the estimation of iso-thermal energy content is shown by filled area (light red) while the same corresponding to the multi-thermal energy content is shown in the form of error bars (blue).[]{data-label="E-th"}](sphinx_Eth.ps){width="45.00000%" height="45.00000%"} Next, we estimate the multi-thermal energy content of the flare with the help of the DEM\[T\] distribution, derived from the W-S inversion scheme, as per the following equation [@Sylwester2014; @Aschwanden2015]: $$\label{eq-multi-thermal-energy} E_{th}=3 k_B V^{1/2} \sum_k T_k DEM_k^{1/2}$$ The multi-thermal energy content derived in the aforesaid manner varies in the range of 1-7$\times 10^{29}$ ergs as plotted in the form of blue color histogram in Figure \[E-th\]. In the estimation of multi-thermal energetics, we have employed the combined data-set which is prepared from the X-ray emission in 1.6-5.0 keV as obtained from SphinX and in 5.0-8.0 keV (with the application of a normalization factor of ‘2.5’) from SOXS (cf. Section \[sec:data-analysis\]). It may be argued that this scheme of normalization is biased. In this regard, we made a parallel case study in which the best-fit DEM\[T\] distribution is derived using combined data-set which, however, is prepared by applying the inverse normalization factor on the SphinX observations while considering SOXS observations to be true. This investigation resulted in the DEM values systematically lowered by a factor of 2.5 in comparison to that estimated in the previous case. On the other hand, the best-fit plasma temperature values remain unchanged (also see @Mrozek2012). Therefore, considering the fact that DEM values in the former case are larger by the aforesaid factor i.e. 2.5, multi-thermal energy content is resulted to be overestimated by a factor of 4 (Equation \[eq-multi-thermal-energy\]). In this calculation, we have also included the uncertainty in the volume estimation obtained previously. In Figure \[E-th\], we show the uncertainty in the multi-thermal energy estimates (blue) with the low error bars. The comparison of iso-thermal and multi-thermal energy content for this flare (Figure \[E-th\]) revealed that multi-thermal energy matches well with the iso-thermal energy during the rise and the decay phase of the flare. However, during the maximum of the impulsive phase, minor disagreement in the form of lower values of multi-thermal energy in comparison to the iso-thermal energy is noted. Next, we derive the multi-thermal energy from the best-fit DEM\[T\] distribution, obtained by employing various DEM schemes on the observed SXR spectrum in 1.6-5.0 keV (low-energy), 5.0-8.0 keV (high-energy) and 1.6-8.0 keV (combined-energy) bands, during the peak of impulsive phase of the flare (04:36:00-04:38:30 UT). Multi-thermal energy for the low-energy band SXR is estimated to be 177, 225 and 4.1 $\times 10^{29}$ ergs, corresponding to the the single-gaussian, power-law and W-S DEM schemes, respectively. On the other hand, the energy content derived by employing the aforesaid DEM schemes on the high-energy band SXR is obtained to be 64, 65 and 1.5 $\times 10^{29}$ ergs, respectively. Further, the multi-thermal energy is resulted to be 85, 91 and 4.4 $\times 10^{29}$ ergs by applying the aforesaid DEM schemes on the combined-energy band SXR, respectively. By comparing the above mentioned energy estimates, we note that the flare energetics, estimated from the parameters derived only from the spectral inversion of the low-energy band of SXR spectrum leads to higher values than that obtained from combined-energy band case. On the other hand, the multi-thermal energies, resulted by applying various DEM schemes on the high-energy part of SXR spectrum, are estimated to be lower than that obtained from combined-energy band SXR spectrum. This trend is consistently noted in the energetics estimated employing all the aforesaid DEM schemes. On the contrary, we find that the best-fit DEM\[T\] distribution, obtained with the DEM schemes which postulate either the single gaussian or power-law functional dependence of DEM, leads to the overestimation of multi-thermal energy by approximately one order in comparison to that estimated from W-S algorithm. Summary and Conclusions {#sec:sum-conc} ======================= We investigate the thermal characteristics of the flare plasma by analysing X-ray emission in the energy band 1.6-8.0 keV observed during SOL2009-07-04T04:37 flare, the only common event observed by SphinX and SOXS instruments. We derive the evolution of the best-fit DEM\[T\] distribution during the flare by employing various DEM inversion algorithms. In addition, we have also studied the dependence of the best-fit DEM\[T\] corresponding to various input energy bands within SXR emission. Following are the key points of our study: 1. Best-fit DEM\[T\] distribution for the low (1.6-5.0 keV), high (5.0-8.0 keV) and combined-energy band (1.6-8.0 keV) of X-ray emission during the flare resulted in higher values of $DEM_p$, however at low $T_p$ for the low-energy band in comparison to the relatively lower values of $DEM_p$ at higher $T_p$ obtained by analyzing the high-energy band of the SXR. 2. We derive the time evolution of DEM\[T\] distribution during various phases of the flare by employing Withbroe-Sylwester maximum likelihood DEM inversion algorithm on the individual as well as combined observations of SphinX and SOXS during the flare. The results are summarised as follows: - The best-fit DEM\[T\] distribution corresponding to the X-ray emission during the flare onset can be well represented by a single gaussian function with a width of $\sim$1 MK, which suggests flare plasma to be of iso-thermal nature in this phase. - Analysis of X-ray emission during the rise to the peak of impulsive phase of the flare revealed the presence of multi-thermal plasma as the corresponding best-fit DEM\[T\] curves show double gaussian form with the widths of $\sim$ 1.5 MK. - Temporal evolution of the best-fit DEM\[T\] distribution corresponding to the post-maximum phase of the flare can be well represented by a single gaussian function, however, with the peak temperature varying in the range of $\sim$ 13.0-5.5 MK. 3. Iso-thermal and multi-thermal energy content is estimated during the flare. We find that the multi-thermal energy estimates are in close agreement with the iso-thermal energy values except during the peak of the impulsive phase of the flare where iso-thermal energy is estimated to be larger than the multi-thermal energy content. 4. Multi-thermal energy is determined from the best-fit DEM\[T\] distribution resulted from the application of various inversion schemes on the X-ray emission measured during the peak of the impulsive phase of the flare. We find that the energy content estimated from the parameters derived only from spectral inversion of the low-energy band (1.6-5.0 keV) of SXR spectrum result in larger values than that obtained from the analysis of the SXR emission in combined-energy band. On the contrary, the same derived from only the high-energy band of SXR spectrum leads to lower estimates when comparing with the energy values calculated from combined-energy band analysis. This trend is consistently resulted in the thermal energetics determined from all the DEM schemes. This suggests that the observations of SXR emission during a flare in the combined-energy band with high temporal and energy cadence is very important to derive the complete thermal energetics of the flare. 5. The best-fit DEM\[T\] distribution obtained for the DEM schemes which postulate either single-gaussian or power-law functional form of DEM-T curve, lead to the estimation of thermal energy content much higher by approximately one order than that estimated from the W-S scheme. This can be understood by the fact that the width of the best-fit DEM\[T\] distribution, obtained by employing single-gaussian approach (see Figure \[fit-dem-gauss-spectra\]) is larger than that resulted by the application of W-S scheme (Figure \[w-s-dem-fit-sphinx-soxs\]). It may be noted that this disagreement between various DEM inversion schemes, and hence thermal energy estimates can have significant impact in the context of coronal heating from low intensity class (micro- and nano-) flares. However, as X-ray emission covers only high-temperature corona, recent studies focussing coronal heating energized by small intensity flares also combine multi-wavelength observations with the X-ray emission during flares [@Testa2014]. Moreover, several advanced schemes of DEM inversion viz. ‘DEM\_manual’ [@Schmelz2015], ‘EM Loci approach’ [@Cirtain2007], combination of gaussian and power-law functional form of DEM [@Guennou2013; @Aschwanden2015SoPh] etc. have also been employed in deriving thermal characteristics of emission measure during small intensity class flares. Therefore, in future, we plan to extrapolate the application of W-S DEM inversion scheme on the combined EUV and X-ray observations during small flares in order to make a comparative survey of thermal energy content derived by W-S method and other DEM inversion schemes. This research has been supported by Polish NCN grant 2011/01/B/ST9/05861 and from the European Commission’s Seventh Framework Programme under the grant agreement No. 284461 (eHEROES project). Moreover, the research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 606862 (F-CHROMA). Authors also acknowledge the open data policy of the SphinX, SOXS, *SOHO*, *HINODE* and *STEREO* missions. SAO/ADS abstract service is duly acknowledged for providing the up-to-date and well-organized bibliography. Additionally, the Coyote’s IDL programming support is acknowledged. Authors also thank the anonymous referee for constructive comments which improved the manuscript. 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--- abstract: 'Between the BICEP2 and Keck Array experiments, we have deployed over 1500 dual polarized antenna coupled bolometers to map the Cosmic Microwave Background’s polarization. We have been able to rapidly deploy these detectors because they are completely planar with an integrated phased-array antenna. Through our experience in these experiments, we have learned of several challenges with this technology- specifically the beam synthesis in the antenna- and in this paper we report on how we have modified our designs to mitigate these challenges. In particular, we discus differential steering errors between the polarization pairs’ beam centroids due to microstrip cross talk and gradients of penetration depth in the niobium thin films of our millimeter wave circuits. We also discuss how we have suppressed side lobe response with a Gaussian taper of our antenna illumination pattern. These improvements will be used in Spider, Polar-1, and this season’s retrofit of Keck Array.' author: - | R. O’Brient, P. A. R. Ade, Z. Ahmed, R.W. Aikin, M. Amiri, S. Benton, C. Bischoff, J.J. Bock, J. A. Bonetti, J. A. Brevik, B. Burger, G. Davis, P. Day, C.D. Dowell, L. Duband, J. P. Filippini, S. Fliescher, S.R. Golwala, J. Grayson, M. Halpern, M. Hasselfield, G. Hilton, V.V. Hristov, H. Hui, K. Irwin, S. Kernasovskiy, J. M. Kovac, C. L. Kuo, E. Leitch, M. Lueker, Megerian, K, L. Moncelsi, C.B. Netterfield, H. T. Nguyen, R. W. Ogburn IV, C. L. Pryke, C. Reintsema, J.E. Ruhl, M.C. Runyan, R. Schwarz, C. D. Sheehy, Z. Staniszewski, R. Sudiwala, G. Teply, J. E. Tolan, A. D. Turner, R.S. Tucker, A. Vieregg, D. V. Wiebe, P. Wilson, C. L. Wong, W.L.K. Wu, K.W. Yoon California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125 USA;\ Jet Propulsion Laboratory, 4800 Oak Grove Dr., Pasadena, CA 91109, USA;\ Dept. of Physics and Astronomy, University of Wales, Cardiff, CF24 3YB, Wales, UK;\ Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, BC V6T1Z1, Canada;\ Department of Physics, University of Toronto, Toronto, ON M5S 1A7, Canada;\ Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138;\ Service des Basses Temperatures, DRFMC, CEA-Grenoble, 17 rue des Martyrs, 38054 Grenoble Cedex 9, France;\ School of Physics and Astronomy, University of Minnesota, 116 Church Street S.E.,Minneapolis, MN 55455;\ NIST Quantum Devices Group, 325 Broadway, Boulder, CO 80305, USA;\ University of Chicago, KICP, 933 E. 56th St., Chicago, IL 60637 USA;\ Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305, USA;\ Kavli Institute for Particle Astrophysics and Cosmology (KIPAC), Sand Hill Road 2575, Menlo Park, CA 94025, USA;\ Physics Department, Case Western Reserve University, Cleveland, OH 44106 USA;\ bibliography: - 'SPIE\_bib.bib' title: 'Antenna-coupled TES bolometers for the Keck Array, Spider, and Polar-1' --- INTRODUCTION {#sec:intro} ============ The Cosmic Microwave Background’s (CMB) temperature anisotropies and curl-free E-mode polarization anisotropies, both generated by scalar inflationary potentials, have been mapped by numerous experiments and used to constrain a multitude of cosmological parameters. To date, divergence-free B-modes in the CMB, which would have been seeded by inflationary tensor potentials, have not been detected. However, current data favor a scalar index $n_s<1$, which suggests that the tensor-to-scalar ratio $r$ may be non-zero [@WMAP_seven_year]. A B-mode detection at degree angular scales would provide strong confirmation of inflation, but even upper limits on $r$ will constrain the energy scale of inflation, thus making it a fundamental parameter of interest. Additionally, gravitational lensing can shear E-modes into B-modes, creating a smaller angular scale non-primordial anisotropy. The details of this peak can constrain the summed mass of the different neutrino species and potentially inform us of the existence of sterile neutrinos[@Dodelson_white_paper]. Our team has developed a low cost camera design with just enough resolution to detect B-modes from primordial tensor perturbations. This design builds off the success of BICEP-1, using refracting optics with a 30 arcmin resolution[@Yoon_B1_SPIE][@Aiken_SPIE]. BICEP2 and Keck Array’s five BICEP2 style cameras are both currently collecting data at the South Pole with nearly 1500 pixels (3000 detectors) collectively [@Keck_array_SPIE]. The balloon-borne Spider will fly in 2013 from the McMurdo station in Antarctica with six cameras and a comparable pixel count[@Filippini_SPIE_SPIDER]. Finally, we will deploy the 1.5m crossed-Dragone telescope Polar-1 to the South Pole in 2013 with 1900 pixels. Its 5 arcmin resolution will be sufficient to detect lensed B-modes. All four of these experiments use phased-array antenna-coupled TES bolometers developed at Caltech and JPL. Our detectors are entirely planar, and hence scalable to large monolithic arrays. This has provided our team with a crucial advantage over competing teams, allowing early deployment of high sensitivity focal planes. As a result, our technology has matured to a point where we have identified several failure modes from both lab and field data and we have refined our original design with these measurements in mind. This paper describes some of the efforts to improve our detectors. The second section summarizes our detector design, the third describes how we corrected differential pointing between polarization pairs, and the last outlines how we have suppressed sidelobe response with a tapered illumination pattern in anticipation of Polar. DETECTOR AND CAMERA ARCHITECTURE ================================ Figure \[B2\_rays\] shows the BICEP2 optics design, which we used in Keck Array and Spider with only minor modifications. Each camera is a refracting telescope with two HDPE lenses that image the sky onto a focal plane while forming an aperture just on the sky-side of the objective lens. We place a cold stop at this point to terminate side lobes of the antenna coupled bolometers in our focal plane. We also use a series of Teflon, nylon, and metal-mesh IR blockers to filter away higher frequency out-of-band power[@Aiken_SPIE]. Figure \[FPU\_pic\] shows a a BICEP2 focal plane; those for the Keck Array are nearly identical, while those for SPIDER employ modifications for enhanced magnetic shielding. Each contains four tiles, each of which have a monolithic imaging array with 64 pixels. We couple optical power onto each pixel (Figure \[pixel\_pic\]) with a planar dual-polarized phased-array antenna[@Kuo_antenna_coupled_TES]. All antennas deployed to date use a 12x12 array of subradiators, where each subradiator contains echelon pairs of 412 $\mu$m x 12 $\mu$m slots etched into a 0.15 $\mu$m thick superconducting niobium (Nb) ground plane. The opposite polarization’s slot pairs interlock with these with a common center, such that the two polarizations’ antennas in each pixel are co-located. Each pixel’s antenna is 7.2 mm on side, providing bare beams (i.e. without any gain from optics) with 14$^o$ FWHM that nicely match the f/2.1 camera optics. Each slot couples to a microstrip summing network at a pair of feedpoints placed close enough to the slot ends to maintain a low impedance of 47+j15 $\Omega$. We tune away the reactance with shunt capacitors and match the radiation resistance with 0.93 $\mu$m wide microstrip. This upper layer is also Nb that is 0.4$\mu$m thick and separated from the ground by 0.3 $\mu$m of sputtered silicon dioxide (SiO2). The summing network combines the waves from the slots in a series of microstrip-tees, first summing slots across different columns of each rows, then summing the total waves in each row. Each polarization has its own independent summing tree whose horizontal arms must interlock with the other polarization. After all the slots’ waves are summed onto one microstrip line per polarization, we pass each through a band-defining 150GHz filter with 25% bandwidth. The filters have three-poles with three short stretches of CPW acting as series inductors that are separated by T-networks of capacitors. Neither the summing tree nor filter require any microstrip vias or microstrip-cross-overs. We terminate power in a lossy meandered gold microstrip line in close thermal contact with the bolometer’s TES. The gold is deposited before the Nb and cleaned with an Ar ion mill to provide a reliable electrical contact with the superconducing microstrip. The meander’s length is 2.2mm, which provides a return loss of less than -20dB. We pattern two TESs in series on each bolometer made of Aluminum (Al) and Titanium (Ti) which respectively have nominal superconducting transition temperatures $T_c$ of 1.2K and 0.5K. We voltage bias into the Al transition for lab testing where we need to expose the camera to 300K and into the Ti transition for science observations, where the effective sky temperature is much lower. The bolometer itself is suspended on a 1$\mu$m thick film of low stress nitride released by a XeF$_2$ etch. We select the legs’ length and geometry to control the thermal conductance (G), and hence the thermal carrier noise and saturation power. We read the current through the bolometers with a SQUID-based time domain multiplexer system developed by NIST and UBC. Multiplexing chips and Nyquist filter chips are visible on the perimeter of the focal planes in Figure \[FPU\_pic\]. DIFFERENTIAL POINTING OF POLARIZATION PAIRS =========================================== BEAM SYSTEMATICS ---------------- Improving on BICEP-1’s constraint of $r<0.72$[@BICEP1_Emodes] requires large imaging arrays of detectors to attain higher sensitivity. Additionally, the two cross-polarized detectors in each pixel should view the sky through the same optics path; this way, differencing within each detector pair should greatly reduce sky noise. While our monolithic and planar arrays of dual-polarized bolometers have provided an expedient way to meet these needs, we have a significant burden to synthesize useable beams with the phased-array antennas. Ideally, beam patterns of the two polarizations in a pair would subtract perfectly. In practice, this subtraction often leaves residual structure that can leak temperature anisotropies into the polarization channels[@Shimon_systematics]. Between beams with a Gaussian profile, (**a**) differential widths can generate a monopole difference pattern, (**b**) differential pointing of centroids can generate a dipole difference pattern, and (**c**) differential ellipticity can generate a quadrupole difference pattern. Of these defects, the dipoles (**b**) are by far the most dominant (See Figure \[dipole\_pic\]), with an angular separation between centroids as high as 10% of the FWHM as measured in the far field. ![Polarization Pair and difference from BICEP2 illustrating the dipole structure in near field maps. [**We have since reduced the peak-to-peak amplitude to from $\pm7$% of peak power to $\pm1$%**]{}[]{data-label="dipole_pic"}](beams.eps) Dipoles can leak the much brighter temperature gradients into the polarization channels. BICEP2 has mitigated this problem by co-adding maps taken at different boresight rotations and they are currently developing an analysis technique to remove leaked B-modes by regression against temperature maps. Additionally, Spider will see little sky noise, obviating their need to difference the polarization pairs’ time-streams. Nonetheless, the BICEP-1 team concluded that for their scan strategy and analysis pipeline at the time, they would need differential pointing less than 1.9% to avoid a false B-mode signal of $r=0.1$, and we have taken this figure as a benchmark for our detector design. HORIZONTAL DIPOLES ------------------ BICEP2 has seen differential steering of 7% of the beam width in the camera’s near field (i.e. near the aperture) on the axis parallel to the antenna’s summing tree rows (horizontal), but very little in the direction along columns (vertical). They have measured comparably large dipoles in the far field, although with an orientation that varies between detectors and not strictly horizontal [@Aiken_SPIE]. While the offsets in the near-field should not generate offsets in the far-field, an optical misalignment that moves the beam aperture locations from the location of the stop could allow such leakage. We have not yet succeeded in finding a misalignment, but in an effort to make our cameras robust against such alignments errors, we have redesigned the detectors to remove the near-field dipole. The near-field dipoles from BICEP2 always maintain the same horizontal orientation in every tile with a very narrow spread in magnitude, which suggested a problem related to the microstrip artwork. Cross talk between adjacent parallel microstrip lines can in fact generate horizontal differential steering. Figure \[mid\_tree\_pic\] shows the interlocking horizontal summing trees from both polarizations, focusing on one of the few places that our design breaks left-right symmetry. The lines that connect the end branches of the vertical trees to the top of the horizontal trees run parallel to top branch of the horizontal trees for a length of 3mm, or 3.75 wavelengths at band-center. In early designs, the lines had an average separation of 8 $\mu$m, center-to-center. These microstrip pairs effectively form a forward-wave directional coupler, but only on the half of the horizontal tree closest to the vertical tree. The fields of the odd mode on the coupled lines (same current, opposite phase) fringe more out of the silicon oxide than those of the even mode (same current, same phase), allowing the modes to propagate with different wave speeds. In reference to the port labels in Figure \[mid\_tree\_pic\], a wave launched from the vertical tree towards the horizontal (from port 1) is a superposition of these even and odd modes that cancel in the adjacent line at the end nearest the vertical tree (Port 2). However, the modes’ different wave speeds guarantee that they will not cancel in the adjacent line at the end nearest the pixel center (Port (3)). We label ports (3) and (4) in parenthesis because they are subsequently cascaded in the microstrip tee, thus becoming internal ports. If we explicitly sum the scattering parameters[@Coupled_lines_book], $$\begin{aligned} S_{(4)1}=(S^e_{(4)1}+S^o_{(4)1})/2=(e^{-jk_e\ell}+e^{-jk_o\ell})/2 &=& \quad e^{-j(k_e+k_o) \ell/2}\cos ((k_e-k_o)\ell/2) \nonumber \\ S_{(3)1}=(S^e_{(3)1}+S^o_{(3)1})/2=(e^{-jk_e\ell}-e^{-jk_o\ell})/2 &=& -je^{-j(k_e+k_o) \ell/2}\sin ((k_e-k_o)\ell/2)\end{aligned}$$ \ then we see that this coupled wave is small in amplitude and lags the through wave (in Port (4)) by 90$^o$. We split the through wave with even power in a microstrip tee, but the coupled wave adds in quadrature to waves on the side opposite the vertical tree, retarding the phase. This phase step steers the two polarizations’ beams away from the vertical tree, thus generating differential pointing in the horizontal direction. Simulations using a variety of commercial packages (HFSS, Sonnet, ADS Momentum) show that the coupled wave amplitude is a function of dielectric permittivity and thickness, the spacing between the lines, and the thickness of the metal conductors. The upper Nb film thickness of 0.4 $\mu$m exceeds that of the dielectric (0.3 $\mu$m) and is a substantial fraction of the line separation, so it can contribute to the capacitive coupling in a substantial way. If we neglect this thickness, then we underestimate the coupling by a factor of 40%. Accounting for the Nb microstrip thickness in simulation provides horizontal differential steering that agrees with measurements as shown in the black dashed line in Figure \[histo\_nfbm\]. This understanding has proven useful for the SuperSpec team that uses microstrip couplers to drive their spectrometer resonators[@Shiro_SPIE_Superspec]. The spacing between horizontal lines is constrained by the need to route between slots in the tree. Nonetheless, we have managed to rearrange the summing tree in ways that increase the average separation between lines. Figure \[histo\_nfbm\] shows the splitting from several data tiles with different average spacing, and we have reduced the steering to less than 2% of FWHM[@LTD14_OBrient]. Finally, we have fabricated tiles with residual microstrip coupling but have tuned the steering to zero by introducing intentional phase lags (extra line length) on the sides furthest from the vertical trees. Figure \[histo\_nfbm\] also shows that those have a steering reduced to an average within a standard deviation of zero on the horizontal axis. We are currently fabricating detectors with summing trees that have an average separation of 50 $\mu$m between the top lines of the horizontal tree; we expect this to have no measurable horizontal offset, which will obviate the phase-lag correction. ![Polarization Pair Centroid displacement normalized to FWHM in the near field. We show histograms of the tiles with two different spacings as well as one where built phase lags into the summing tree to manually tune this to zero.[]{data-label="histo_nfbm"}](Spacing_histogram_SPIE.eps) VERTICAL DIPOLES ---------------- The Keck Array cameras have shown evidence of vertical near field components in addition to the horizontal ones. Unlike the horizontal components described above, these have a large scatter in magnitude, although a constant orientation. This large variance suggested that the problem was not due to the microstrip artwork but was probably related to non-uniform film properties. Between the BICEP2 and Keck Array deployments, we switched to a Nb sputter system with a larger gun to attain better uniformity. However, this system ran at significantly higher power than the old one and lacked grounding to shunt away higher energy particles. Until recently, we have used liftoff to define our microstrip artwork, and we now believe that the elevated temperatures in the sputter system allowed contaminants to leach out of the photoresist and into the Nb films during sputtering. RRR measurements of lines defined by etching exceed those defined by liftoff by nearly 40%. In a few extreme cases, we have seen visible blackening of the lifted-off microstrip lines. A superconductor is considered dirty when the mean-free path between scattering from impurities $\ell$ is much less than the London depth $\lambda_L$ or the Cooper-pair coherence length $\xi_o$. In this limit, the effective penetration depth is [@Tinkham_book] $$\lambda_{eff}=\lambda_L\sqrt{\frac{\xi_o}{\ell}} \label{eqn_pen}$$ Niobium has $\lambda_L=52$nm $\xi_o=$39nm[@Poole_book]. We have fabricated test devices that include stretches of transmission line mismatched in impedance from the surrounding lines[@Kuo_antenna_coupled_TES]. These form Fabrey-Perot cavities whose standing wave period is sensitive to the niobium’s penetration depth. Our films appear to have $\lambda_{eff}$=110nm, which is similar to measurements of other researchers in the field[@Kerr_ALMAmemo]. Equation \[eqn\_pen\] suggests that the mean free path between scatterings is $\ell=10$nm, which puts us well into the dirty limit in which film cleanliness can impact the circuit performance. Non-uniform contamination can produce non-uniform kinetic inductance, which in turn can spatially perturb the wavespeeds in the microstrip summing tree. Variations in wavespeed can steer beams off boresight. Additionally, the summing tree does not treat the two polarizations identically, and as a result, they can steer differentially. We have observed that the tiles with the largest scatter in beam centroid position correspond to those with the largest vertical dipole components. We also expect the tree to induce larger steering in the vertical than horizontal because slots along rows combine immediately in the horizontal tree resulting in less integrated phase error than those along columns that combine in the vertical tree after horizontal summing. To understand the effects of nonuniform niobium contamination, we have written a circuit model of our summing tree. We use a quasi-static microstrip model[@Hammerstad_Jensen] for the lines and simple power division models for the microstrip tees. The code uses user-provided arrays of film properties as a function of physical position to locally compute the impedances and wavespeeds for differential (1$\mu$m long) line segments. If we cascade two sequential ($m+\ell$) port and ($\ell+n$) port circuits with scattering matrices $S^1$ and $S^2$ respectively through common ports $\ell$, then the combined circuit matrix is[@S_matrix_cascade]: $$S^{1+2}= \begin{bmatrix} S^1_{m,m}+ S^1_{m,\ell}S^2_{\ell,\ell}\left(I_{\ell,\ell}-S^1_{\ell,\ell}S^2_{\ell,\ell}\right)^{-1}S^1_{\ell,m}& S^1_{m, \ell}\left( I_{\ell,\ell}-S^2_{\ell, \ell} S^1_{\ell, \ell} \right)^{-1}S^2_{\ell, n} \\ S^1_{m, \ell}\left( I_{\ell,\ell}-S^2_{\ell, \ell} S^1_{\ell, \ell} \right)^{-1}S^2_{n, \ell} & S^2_{n,n}+ S^2_{n,\ell}S^2_{n,n}\left(I_{\ell,\ell}-S^2_{\ell,\ell}S^1_{\ell,\ell}\right)^{-1}S^2_{\ell,n} \end{bmatrix}$$ We implemented this algorithm in software that is functionally similar to commercial packages like Agilent’s ADS circuit simulator, except that it includes spatially varying film properties. Figure \[gradient\_plots\] shows the computed dipoles that arise from linear gradients in the penetration depth $\lambda_{eff}$ versus orientation of that dipole. Each curve corresponds to a different film gradient magnitude, and a gradient of 5% in penetration depth across a pixel can generate measured vertical dipoles similar to our measured levels. In reality, the spatial variations of $\lambda_{eff}$ may be more complicated than this, but these computations demonstrate that, to first order, differential steering can arise from film gradients. ![Simulated Magnitude of differential pointing as a function of orientation of $\lambda_{eff}$ gradient orientation for different gradient magnitudes . The steering is a minimum near angles 45$^o$ from the polarization planes, where the gradient effects both polarizations equally and it is twice as large for the vertical as horizontal. The magnitude is over twice as large when oriented on the vertical axis than horizontal[]{data-label="gradient_plots"}](Computed_gradient_dipoles.eps) We have changed our niobium processing from lift-off to etching to reduce this contamination. This simple change has created a dramatic improvement in polarization-pair centroid alignment. Figure \[2D\_histograms\] shows displacements of best fit centroids in both the near and far-fields of a Spider testbed. At this point, the alignment is better than the BICEP-1 benchmark. SIDELOBE SUPRESSION =================== As mentioned above, BICEP2 and Keck Array cameras all have stops at their apertures cooled to 4K (Figure \[B2\_rays\]); the Spider team cools theirs to 2K. This has let the three projects use the simplest antenna design where each slot provides equal power to the bolometer. The summing tree for this antenna is relatively easy to design, but the uniform (tophat) illumination generates a sinc-function pattern with side lobes at -13dB of peak power. These side lobes terminate on the cold stop in these cameras. However, large throughput crossed-Dragone telescopes like Polar-1 shown in Figure \[P1\_rays\] have an aperture beyond the large primary where it is difficult to be made cold. In lieu of a traditional absorbing cold stop, Polar-1 will use an aluminum Gaussian scattering surface and a larger Winston cone to direct the spillover onto the sky. This will create an effective stop with a temperature of 15K. In parallel to this novel optical design, we have redesigned the detectors to reduce spillover. We do this by illuminating the subradiator slots with a Gaussian illumination pattern, which synthesizes a beam with a lower sidelobe level, albeit with a wider main lobe [@Dolph_array]. We can ultimately construct an arbitrary illumination pattern by choosing the line widths, and hence impedances, at each of the mictostrip tees in the feed network. In the uniform illumination tree, most junctions split power evenly (in a time reversed picture). However, most of the junctions in the Gaussian illuminated tree split power asymmetrically. Only a subset of the the subradiating slots are on the sky-side of a given tee-junction. This subset further partitions into “inside” slots connected to the microstrip line directed to the antenna’s interior, and “outside” slots connected to the microstrip line directed to the antenna’s exterior. The required power division at each junction is: $$P_{inside}=\left( 1-\frac{\displaystyle \sum_{outside}P_{slot}}{\displaystyle \sum_{inside}P_{slot}}\right)^{-1} \label{taper_design}$$ \ and $P_{outside}=1-P_{inside}$. The ratio of impedances $Z_{inside}/Z_{outside}$ on the slot side lines must be the reciprocal of the power ratio i.e. $P_{outside}/P_{inside}$, subject to the requirement that the impedance be matched when looking into the junction from the bolometer side. Once the desired distribution of powers in the slots has been chosen, we use these formulae to design each of the microstrip tees in the feed network. We have optimized our feed to minimize spillover onto Polar’s stop of f/1.52. For pixels that are 7.2mm on a side (12x12 subradiators), the spillover can be as low as 2.6%, while for a more ambitious 6mm pixel (10x10 subradiators), the spillover should be 6.2%. In both cases, the optimal waist of the Gaussian feed pattern is 3.6mm, but in the 6mm pixel case, the beam truncates more abruptly, generating larger sidelobes. We fabricated several test prototypes pixels with an illumination waist of 4.4mm and measured their beam patterns in a test cryostat without refracting optics or a stop. Figure \[tapered\_pattern\] shows measurements of a pair of patterns taken from a tophat and Gaussian illumination, both 7.2mm on a side. In this case, the sidelobes are supressed from -12dB to -17dB while the main lobes have increased from 14$^o$ FWHM to 16.5$^o$ FWHM. For this tile, we found that the tophat antennas had 47% total receiver efficiency while the guassian antennas had 49% [@LTD14_OBrient]. This confirms that the tapered illumination is generated by receiving less power at the edge, but more in the center, rather than less power overall. ![Beam Patterns of tophat and Gaussian illuminated antennas, where Polar-1’s stop is shown in the dashed circle. \[tapered\_pattern\]](both_patterns.eps) Using the measured pattern waist and sidelobe level, the total spillover onto f/1.52 from these prototype pixels should be 7.1%. We can also directly integrate the total spillover beyond the stop ring shown in Figure \[tapered\_pattern\], and we find 7.5$\pm$ 3.0 %, which agrees with the figure computed from the beam width and side lobe levels. Note that the tophat would spill 12.8% (see Figure \[spillover\_plot\]). ![Spillover histograms for tophat and spillover antennas, each 7.2mm on a side (12.x12 format)[]{data-label="spillover_plot"}](spillover_1p5.eps) We have since fabricated devices with the optimal waist sizes for three different pixel sizes: 7.2mm (12x12), 6.0mm (10x10), and 4.8mm (8x8). Due to a fabrication error, the efficiency of that batch was low, leading to a low signal to noise measurement and a diffuse response of -17dB per map pixel that we suspect is from direct stimulation of the bolometer. In a focal with a higher efficiency, this direct stimulation is below -25dB per map pixel. As a result, it is difficult to characterize the side-lobe level in these devices. However, the measured beam widths of 16.8$^o$, 19.0$^o$, and 22.4$^o$ FWHM match the expected widths, so we anticipate that they will have spillovers of roughly 2.5%, 6.2%, and 14%. Figure \[cuts\_plot\] shows cuts from 36 stacked maps on a linear scale to emphasize the FWHM of the main lobe, and co-plots the model in dashed lines to emphasize the agreement. We also show the direct stimulation floor to emphasize how we cannot really characterize the side-lobes aside from placing an upper bound of -17dB. ![E-plane cuts for the three different sized pixels. The colored dashed lines are the expected beams from simulation and the crosses are the half-power points interpolated from the measurement[]{data-label="cuts_plot"}](Cuts_correct_feed.eps) Future Work =========== We have demonstrated differential steering that is below the 1% of FWHM standard set by the BICEP-1 team. We will upgrade detectors from the Keck Array this winter, and focal planes being made now for Spider’s 2013 flight have these fixes. We will refabricate the detectors with the tapered illumination to check the side lobe levels before we start populating Polar-1’s focal plane this fall. R. O’Brient and Z. Staniszewski would like to thank Oak Ridge Associated Universities for funding through the NASA Post-doctoral Program. Keck Array is partially funded by the Keck Foundation and the detector development through the Betty and Gordon Moore Foundation. All devices were fabricated in the Microdevices Laboratory (MDL) at NASA’s Jet Propulsion Laboratory.

--- abstract: 'Peccati, Solè, Taqqu, and Utzet recently combined Stein’s method and Malliavin calculus to obtain a bound for the Wasserstein distance of a Poisson functional and a Gaussian random variable. Convergence in the Wasserstein distance always implies convergence in the Kolmogorov distance at a possibly weaker rate. But there are many examples of central limit theorems having the same rate for both distances. The aim of this paper is to show this behaviour for a large class of Poisson functionals, namely so-called U-statistics of Poisson point processes. The technique used by Peccati et al. is modified to establish a similar bound for the Kolmogorov distance of a Poisson functional and a Gaussian random variable. This bound is evaluated for a U-statistic, and it is shown that the resulting expression is up to a constant the same as it is for the Wasserstein distance.' author: - 'Matthias Schulte[^1]' title: '**Normal approximation of Poisson functionals in Kolmogorov distance**' --- **Key words:** Central limit theorem, Malliavin calculus, Poisson point process, Stein’s method, U-statistic, Wiener-Itô chaos expansion\ **MSC (2010):** Primary: 60F05; 60H07 Secondary: 60G55 Introduction and results ======================== Let $\eta$ be a Poisson point process over a Borel space $(X,{\mathcal{X}})$ with a $\sigma$-finite non-atomic intensity measure $\mu$ and let $F=F(\eta)$ be a random variable depending on the Poisson point process $\eta$. In the following, we call such random variables Poisson functionals. Moreover, we assume that $F$ is square integrable (we write $F\in L^2(\P_\eta)$) and satisfies $\E F=0$. By $N$ we denote a standard Gaussian random variable. In [@Peccatietal2010], Peccati, Solé, Taqqu, and Utzet derived by a combination of Stein’s method and Malliavin calculus the upper bound $$\label{eqn:Peccatietal} d_W(F,N)\leq \E|1-\langle DF,-DL^{-1}F\rangle_{L^2(\mu)}|+\int_{X}\E(D_zF)^2 \, |D_zL^{-1}F| \, \mu({{\operatorname{d}}}z)$$ for the Wasserstein distance of $F$ and $N$. Here, $\langle\cdot,\cdot\rangle_{L^2(\mu)}$ stands for the inner product in $L^2(\mu)$, and the difference operator $D$ and the inverse of the Ornstein-Uhlenbeck generator $L$ are operators from Malliavin calculus. The underlying idea of these operators is that each square integrable Poisson functional has a representation $$F=\E F+\sum_{n=1}^\infty I_n(f_n),$$ where the $f_n$ are square integrable functions supported on $X^n$, $I_n$ stands for the $n$-th multiple Wiener-Itô integral, and the right-hand side converges in $L^2(\P_\eta)$. This decomposition is called Wiener-Itô chaos expansion, and the Malliavin operators of $F$ are defined via their chaos expansions. The operators $D_zF$ and $D_zL^{-1}F$ that occur in are given by $$D_zF=\sum_{n=1}^\infty n \, I_{n-1}(f_n(z,\cdot))\ \text{ and }\ D_zL^{-1}F=-\sum_{n=1}^\infty I_{n-1}(f_n(z,\cdot))\ \text{ for }z\in X.$$ Here, $f_n(z,\cdot)$ stands for the function on $X^{n-1}$ we obtain by taking $z$ as first argument. For exact definitions including the domains and more details on the Malliavin operators we refer to Section \[sec:Preliminaries\]. The Wasserstein distance between two random variables $Y$ and $Z$ is defined by $$d_W(Y,Z)=\sup\limits_{h\in {\operatorname{Lip}}(1)}|\E h(Y)-\E h(Z)|,$$ where ${\operatorname{Lip}}(1)$ is the set of all functions $h:\R\rightarrow\R$ with Lipschitz constant less than or equal to one. Another commonly used distance for random variables is the Kolmogorov distance $$d_K(Y,Z)=\sup\limits_{s\in\R}|\P(Y\leq s)-\P(Z\leq s)|,$$ which is the supremum norm of the difference of the distribution functions of $Y$ and $Z$. Because of this straightforward interpretation, one is often more interested in the Kolmogorov distance than in the Wasserstein distance. For the important case that $Z$ is a standard Gaussian random variable $N$ it is known (see [@ChenShao Theorem 3.1]) that $$\label{eqn:dKdW} d_K(Y,N)\leq 2\sqrt{d_W(Y,N)}.$$ This inequality gives us for the Kolmogorov distance a weaker rate of convergence than for the Wasserstein distance. But for many classical central limit theorems, one has actually the same rate of convergence for both metrics. In order to overcome the problem that a detour around the Wasserstein distance and the inequality often gives a suboptimal rate of convergence for the Kolmogorov distance, we derive a similar bound as for the Kolmogorov distance by a modification of the proof in [@Peccatietal2010]. \[thm:Theorem1\] Let $F\in L^2(\P_\eta)$ with $\E F=0$ be in the domain of $D$ and let $N$ be a standard Gaussian random variable. Then $$\begin{aligned} \label{eqn:BoundNormalApproximation} d_K(F,N) &\leq& \E|1-\langle DF,-DL^{-1}F\rangle_{L^2(\mu)}|+2\E\langle (DF)^2,|DL^{-1}F| \rangle_{L^2(\mu)}\\ && + 2\E\langle (DF)^2,|F \ DL^{-1}F| \rangle_{L^2(\mu)} + 2\E\langle (DF)^2,|DF \ DL^{-1}F| \rangle_{L^2(\mu)} \notag\\ && +\sup\limits_{s\in\R}\E\langle D\1(F>s),DF \, |DL^{-1}F|\rangle_{L^2(\mu)}\notag\\ & \leq & \E|1-\langle DF,-DL^{-1}F\rangle_{L^2(\mu)}|+2c(F) \sqrt{\E\langle (DF)^2,(DL^{-1}F)^2\rangle_{L^2(\mu)}} \notag\\ &&+\sup\limits_{s\in\R}\E\langle D\1(F>s),DF \, |DL^{-1}F|\rangle_{L^2(\mu)}\notag\end{aligned}$$ with $$c(F)=\sqrt{\E\langle (DF)^2,(DF)^2\rangle_{L^2(\mu)}}+\left(\E\langle DF,DF\rangle_{L^2(\mu)}^{2}\right)^{\frac 1 4}\left(\left(\E F^4\right)^{\frac{1}{4}}+1\right).$$ Comparing and , one notes that both terms of the Wasserstein bound also occur in , which means that the bound for the Kolmogorov distance is always larger. We apply our Theorem \[thm:Theorem1\] to two situations, where we obtain the same rate of convergence for the Kolmogorov distance and the Wasserstein distance. At first we derive the classical Berry-Esseen inequality with the optimal rate of convergence for the normal approximation of a classical Poisson random variable. As another application of Theorem \[thm:Theorem1\], we consider so-called U-statistics of Poisson point processes, which are defined as $$F=\sum_{(x_1,\hdots,x_k)\in\eta_{\neq}^k}f(x_1,\hdots,x_k)$$ with $k\in\N$, $\eta_{\neq}^k=\left\{(x_1,\hdots,x_k)\in\eta^{k}: x_i\neq x_j \, \forall \, i\neq j\right\}$, and $f\in L^1(\mu^k)$. In [@LachiezeReyPeccati2011; @LachiezeReyPeccati2012] and [@ReitznerSchulte2011], Lachièze-Rey and Peccati and Reitzner and Schulte used the bound for the Wasserstein distance to derive central limit theorems with explicit rates of convergence for such Poisson functionals occurring in stochastic geometry and random graph theory. Now Theorem \[thm:Theorem1\] allows us to replace the Wasserstein distance by the Kolmogorov distance without changing the rate of convergence, which means that the inequality is not sharp for this class of Poisson functionals. These applications are discussed in Section \[sec:Ustatistics\], and the result for U-statistics is shown in Section \[sec:ProofUstatistic\]. Before we prove our main result Theorem \[thm:Theorem1\] in Section \[sec:Proof\], we introduce some facts from Malliavin calculus and Stein’s method in Section \[sec:Preliminaries\]. In this paper, we use the following notation. By $L^p(\P_\eta)$, $p>0$, we denote the set of random variables $Y$ depending on a Poisson point process $\eta$ such that $\E |Y|^p<\infty$. Let $L^p(\mu^n)$, $p>0$, be the set of functions $f:X^n\rightarrow\overline{\R}:=\R\cup\left\{\pm\infty\right\}$ satisfying $\int_{X^n} |f|^p \, {{\operatorname{d}}}\mu^n=\int_{X^n} |f(x_1,\hdots,x_n)|^p \, \mu^n({{\operatorname{d}}}(x_1,\hdots,x_n))<\infty$ and let $||\cdot||_n$ and $\langle\cdot,\cdot\rangle_{L^2(\mu^n)}$ be the norm and the inner product in $L^2(\mu^n)$, respectively. By $L^p_s(\mu^n)$ we denote the set of all functions $f\in L^p(\mu^n)$ that are symmetric, i.e. invariant under permutations of their arguments. Preliminaries {#sec:Preliminaries} ============= #### Malliavin calculus for Poisson functionals. In the sequel, we briefly introduce three Malliavin operators and some properties of them that are necessary for the proofs in this paper. For more details on Malliavin calculus for Poisson functionals we refer to [@LP; @NualartVives1990; @Peccatietal2010; @Privault2009] and the references therein. By $I_n(\cdot)$, $n\geq 1$, we denote the $n$-th multiple Wiener-Itô integral, which is defined for all functions $f\in L_s^2(\mu^n)$ and satisfies $\E I_n(f)=0$. The multiple Wiener-Itô integrals are orthogonal in the sense that $$\label{eqn:orthogonality} \E I_m(f) I_n(g)=\begin{cases} n! \, \langle f,g \rangle_{L^2(\mu^n)}, & m=n\\ 0, & m\neq n\end{cases}$$ for all $f\in L_s^2(\mu^m), g\in L^2_s(\mu^n), m,n\geq 1$. We use the convention $I_0(c)=c$ for $c\in\R$. It is known (see [@LP] for a proof) that every Poisson functional $F\in L^2(\P_\eta)$ has a unique so-called **Wiener-Itô chaos expansion** $$\label{eqn:WienerChaos} F=\E F+\sum_{n=1}^{\infty}I_n(f_n)$$ with $f_n\in L^2_s(\mu^n)$, where the series converges in $L^2(\P_\eta)$. In the following, we call the functions $f_n$ kernels of the Wiener-Itô chaos expansion of $F$ and say that $F$ has a chaos expansion of order $k$ if $f_n=0$ for all $n>k$. Combining and , we obtain $${\operatorname{Var}}F=\sum_{n=1}^\infty n! \, ||f_n||_n^2.$$ The representation allows us to define the difference operator $D$, the Ornstein-Uhlenbeck generator $L$, and the Skorohod integral $\delta$ in the following way: Let $F\in L^2(\P_\eta)$ with the Wiener-Itô chaos expansion . If\ $\sum_{n=1}^{\infty}n \, n! \, ||f_n||_n^2<\infty,$ then the random function $z\mapsto D_zF$ defined by $$D_zF=\sum_{n=1}^{\infty}n \, I_{n-1}(f_n(z,\cdot)),\ z\in X$$ is called the **difference operator** of $F$. For $\sum_{n=1}^{\infty}n^2 \, n! \, ||f_n||_n^2<\infty$ the **Ornstein-Uhlenbeck generator** of $F$, denoted by $LF$, is given by $$LF=-\sum_{n=1}^{\infty}n \, I_{n}(f_n).$$ Let $z\mapsto g(z)$ be a random function with a chaos expansion $$g(z)=g_0(z)+\sum_{n=1}^{\infty}I_{n}(g_n(z,\cdot)), \ g_n(z,\cdot)\in L_s^2(\mu^n)$$ for $\mu$-almost all $z\in X$ and $\sum_{n=0}^{\infty}(n+1)! \, ||g_n||_{n+1}^2<\infty$. Then the **Skorohod integral** of $g$ is the random variable $\delta(g)$ defined by $$\delta(g)=\sum_{n=0}^{\infty}I_{n+1}(\tilde{g}_n),$$ where $\tilde{g}_{n}$ is the symmetrization $\tilde{g}_{n}(x_1,\hdots,x_{n+1})=\frac{1}{(n+1)!}\sum_{\sigma}g_n(x_{\sigma(1)},\hdots,x_{\sigma(n+1)})$ over all permutations $\sigma$ of the $n+1$ variables. We denote the domains of these operators by ${\operatorname{dom}}D$, ${\operatorname{dom}}L$, and ${\operatorname{dom}}\delta$. The difference operator also has the geometric interpretation $$\label{eqn:addone} D_zF=F(\eta+\delta_z)-F(\eta)$$ a.s. for $\mu$-almost all $z\in X$, where $\delta_z$ stands for the Dirac measure concentrated at the point $z\in X$, whence it is sometimes called add-one-cost operator (see Theorem 3.3 in [@LP]). If $F\notin{\operatorname{dom}}D$, we can define the difference operator by . If we iterate this definition and put $D^n_{x_1,\hdots,x_n}F=D_{x_n}D^{n-1}_{x_1,\hdots,x_{n-1}}F$, the kernels of the Wiener-Itô chaos expansion of $F$ in are given by the formula $$f_n(x_1,\hdots,x_n)=\frac{1}{n!}\E D^n_{x_1,\hdots,x_n}F =\frac{1}{n!} \E \sum_{I\subset\{1,\hdots,n\}}(-1)^{n+|I|} F(\eta+\sum_{i\in I}\delta_{x_i})$$ for $x_1,\hdots,x_n\in X$ (see Theorem 1.3 in [@LP]). For centred random variables $F\in L^2(\P_\eta)$, i.e. $\E F=0$, the inverse Ornstein-Uhlenbeck generator is given by $$L^{-1}F=-\sum_{n=1}^{\infty}\frac{1}{n}I_n(f_n).$$ The following lemma summarizes how the operators from Malliavin calculus are related. - For every $F\in{\operatorname{dom}}L$ it holds that $F\in{\operatorname{dom}}D$, $DF\in{\operatorname{dom}}\delta$, and $$\label{eqn:IdentityMalliavin} \delta D F=-LF.$$ - Let $F\in {\operatorname{dom}}D$ and $g \in {\operatorname{dom}}\delta$. Then $$\label{eqn:IntegrationByParts} \E\langle DF, g\rangle_{L^2(\mu)}=\E[F \, \delta(g)].$$ For proofs we refer to [@Peccatietal2010] and [@NualartVives1990], respectively. Equation is sometimes called **integration by parts formula**. Because of this identity, one can see the difference operator and the Skorohod integral as dual operators. For our applications in Section \[sec:Ustatistics\] we need a special integration by parts formula, where it is not required that the first Poisson functional is in ${\operatorname{dom}}D$. In this case, the difference operator is given by . \[lem:Integration\] Let $F\in L^2(\P_\eta)$, $s\in\R$, and $g\in{\operatorname{dom}}\delta$ such that $g(z)$ has a Wiener-Itô chaos expansion of order $k$ for $\mu$-almost all $z\in X$. Moreover, assume that $D_z\1(F>s) \, g(z)\geq 0$ a.s. for $\mu$-almost all $z\in X$. Then $$\E\langle D\1(F>s),g\rangle_{L^2(\mu)}=\E[\1(F>s) \, \delta(g)].$$ It is easy to see that $\1(F>s)\in L^2(\P_\eta)$, whence it has a Wiener-Itô chaos expansion $$\1(F>s)=\sum_{n=0}^\infty I_n(h_n)$$ with $h_0=\E\1(F>s)$ and kernels $h_n\in L^2_s(\mu^n)$, $n\geq 1$, that are given by $$h_n(x_1,\hdots,x_n)=\frac{1}{n!} \, \E D^n_{x_1,\hdots,x_n}\1(F>s).$$ For a fixed $z\in X$ the expression $D_z\1(F>s)$ is bounded, and its chaos expansion has the kernels $$\frac{1}{n!} \, \E D_{x_1,\hdots,x_n}^n D_z\1(F>s)=\frac{1}{n!} \, \E D^{n+1}_{z,x_1,\hdots,x_n}\1(F>s)=(n+1) \, h_{n+1}(z,x_1,\hdots,x_{n}).$$ Hence, we obtain the representation $$D_z\1(F>s)=\sum_{n=1}^\infty n \, I_{n-1}(h_n(z,\cdot))$$ for all $z\in X$. From Fubini‘s theorem and the orthogonality of the multiple Wiener-Itô integrals it follows that $$\label{eqn:lhsIntegration} \begin{split} & \E\langle D\1(F>s),g\rangle_{L^2(\mu)}\\ & = \int_X \E \left[D_z\1(F>s) \, g(z)\right] \, \mu({{\operatorname{d}}}z)\\ & = \int_{X} \E \left[\sum_{n=1}^\infty n \, I_{n-1}(h_n(z,\cdot)) \sum_{n=0}^k I_n(g_n(z,\cdot))\right] \, \mu({{\operatorname{d}}}z)\\ & = \int_X \sum_{n=1}^{k+1} n! \int_{X^{n-1}} h_n(z,x_1,\hdots,x_{n-1}) \, g_{n-1}(z,x_1,\hdots,x_{n-1}) \, \mu^{n-1}({{\operatorname{d}}}(x_1,\hdots,x_{n-1})) \, \mu({{\operatorname{d}}}z). \end{split}$$ On the other hand, we have $$\label{eqn:rhsIntegration} \begin{split} \E\left[\1(F>s)\,\delta(g)\right] & = \E\left[\sum_{n=0}^\infty I_n(h_n) \sum_{n=0}^k I_{n+1}(\tilde{g}_n)\right]\\ & = \sum_{n=1}^{k+1} n! \int_{X^n} h_n(x_1,\hdots,x_n) \, \tilde{g}_{n-1}(x_1,\hdots,x_n) \, \mu^n({{\operatorname{d}}}(x_1,\hdots,x_n))\\ & = \sum_{n=1}^{k+1} n! \int_{X^n} h_n(x_1,\hdots,x_n) \, g_{n-1}(x_1,\hdots,x_n) \, \mu^n({{\operatorname{d}}}(x_1,\hdots,x_n)). \end{split}$$ Here, we use the symmetry of $h_n$ in the last step. Comparing and concludes the proof. \ The next lemma provides an upper bound for the second moment of a Skorohod integral, that is used in Section 5. \[lem:SkorohodSquare\] Let $f\in L^2(\mu^{k+1})$ be symmetric in its last $k$ arguments and let $g(z)=I_k(f(z,\cdot))$. Then $$\E\left[\delta(g)^2\right]\leq (k+1) \, \E\int_X I_k(f(z,\cdot))^2 \, \mu({{\operatorname{d}}}z).$$ By the definition of $\delta$, we obtain $\delta(g)=I_{k+1}(\tilde{f})$ with the symmetrization $$\tilde{f}(x_1,\hdots,x_{k+1})=\frac{1}{(k+1)!}\sum_{\sigma}f(x_{\sigma(1)},\hdots,x_{\sigma(k+1)})$$ as above. From the Cauchy-Schwarz inequality, it follows that $||\tilde{f}||^2_{k+1}\leq ||f||^2_{k+1}$. Combining this with Fubini’s Theorem, we have $$\E\left[\delta(g)^2\right]=(k+1)! \, ||\tilde{f}||^2_{k+1}\leq (k+1)! \, ||f||^2_{k+1}=(k+1) \, \E\int_X I_k(f(z,\cdot))^2 \, \mu({{\operatorname{d}}}z),$$ which concludes the proof. #### Stein’s method. Besides Malliavin calculus our proof of Theorem \[thm:Theorem1\] rests upon Stein’s method, which goes back to Charles Stein [@Stein1972; @Stein1986] and is a powerful tool for proving limit theorems. For a detailed and more general introduction into this topic we refer to [@ChenGoldsteinShao2011; @ChenShao; @Stein1986]. Very fundamental for this approach is the following lemma (see Chapter II in [@Stein1986]): The function $$\label{eqn:definitiongt} g_s(w)=e^{\frac{w^2}{2}}\int_{-\infty}^{w}\left(\1(u\in(-\infty,s])-\P(N\leq s)\right)e^{-\frac{u^2}{2}} \, {{\operatorname{d}}}u$$ is a solution of the differential equation $$\begin{aligned} \label{eqn:Steinsequation} g_s'(w)-w \, g_s(w)=\1(w\in(-\infty,s])-\P(N\leq s)\end{aligned}$$ and satisfies $$\label{eqn:gt1} 0<g_s(w)\leq\frac{\sqrt{2\pi}}{4},\ \ |g_s'(w)|\leq 1,\ \ \text{ and }\ \ |w \, g_s(w)|\leq 1$$ for any $w\in\R$. Equation is usually called **Stein’s equation**. The function $g_s$ is infinitely differentiable on $\R\setminus\left\{s\right\}$, but it is not differentiable in $s$. We denote the left-sided and right-sided limits of the derivatives in $s$ by $g_s^{(m)}(s-)$ and $g_s^{(m)}(s+)$, respectively. For the first derivative, a direct computation proves $$\label{eqn:gtt} g_s'(s+)=-1+g_s'(s-),$$ and we define $g_s'(s):=g_s'(s-)$. By replacing $w$ by a random variable $Z$ and taking the expectation in , one obtains $$\E[g_s'(Z)-Z \, g_s(Z)]=\P(Z\leq s)-\P(N\leq s)$$ and as a consequence of the definition of the Kolmogorov distance $$\label{eqn:dKgt} d_{K}(Z,N)=\sup\limits_{s\in\R}|\E[g_s'(Z)-Z \, g_s(Z)]|.$$ The identity will be our starting point in Section \[sec:Proof\]. Note furthermore, that we obtain, by combining and , the upper bound $$\label{eqn:gt3} |g_s''(w)|\leq \frac{\sqrt{2\pi}}{4}+|w|$$ for $w\in \R\setminus\left\{s\right\}$. Proof of Theorem \[thm:Theorem1\] {#sec:Proof} ================================= By a combination of Malliavin calculus and Stein’s method similar to that in [@Peccatietal2010], we derive the upper bound for the Kolmogorov distance. #### **Proof of Theorem \[thm:Theorem1\]**: {#proof-of-theorem-thmtheorem1} Using the identity and the integration by parts formula , we obtain $$\label{eqn:TrickMalliavin} \E[F \, g_s(F)]=\E[LL^{-1}F \, g_s(F)]=\E[\delta(-DL^{-1}F) \, g_s(F)]=\E\langle -DL^{-1}F,Dg_s(F)\rangle_{L^2(\mu)}.$$ In order to compute $D_zg_s(F)$, we fix $z\in X$ and consider the following cases: 1. $F,F+D_zF\leq s$ or $F,F+D_zF> s$; 2. $F\leq s<F+D_zF$; 3. $F+D_zF\leq s<F$. For $F,F+D_zF\leq s$ or $F,F+D_zF> s$, it follows from and Taylor expansion that $$\begin{aligned} D_z g_s(F) &=&g_s(F+D_zF)-g_s(F)=g_s'(F)D_z F+\frac{1}{2}g_s''(\tilde{F})(D_zF)^2\\ &=:& g_s'(F)D_zF+r_1(F,z,s),\end{aligned}$$ where $\tilde{F}$ is between $F$ and $F+D_zF$. For $F\leq s<F+D_zF$, we obtain by , Taylor expansion, and $$\begin{aligned} D_z g_s(F) &=& g_s(F+D_zF)-g_s(F)=g_s(F+D_zF)-g_s(s)+g_s(s)-g_s(F)\\ &=& g_s'(s+)(F+D_zF-s)+\frac{1}{2}g_s''(\tilde{F}_1)(F+D_zF-s)^2\\ &&+g_s'(F)(s-F)+\frac{1}{2}g_s''(\tilde{F}_2)(s-F)^2\\ &=& g_s'(F)D_zF+ (g_s'(s-)-1-g_s'(F))(F+D_zF-s)\\ && +\frac{1}{2}g_s''(\tilde{F}_1)(F+D_zF-s)^2+\frac{1}{2}g_s''(\tilde{F}_2)(s-F)^2\\ &=& g_s'(F)D_zF-(F+D_zF-s)+g_s''(\tilde{F}_0)(s-F)(F+D_zF-s)\\ && +\frac{1}{2}g_s''(\tilde{F}_1)(F+D_zF-s)^2+\frac{1}{2}g_s''(\tilde{F}_2)(s-F)^2\\ &=:& g_s'(F)D_zF-(F+D_zF-s)+r_2(F,z,s)\end{aligned}$$ with $\tilde{F}_0,\tilde{F}_1,\tilde{F}_2\in \left(F,F+D_zF\right)$. For $F+D_zF\leq s<F$, we have analogously $$\begin{aligned} D_z g_s(F) &=& g_s(F+D_zF)-g_s(F)=g_s(F+D_zF)-g_s(s)+g_s(s)-g_s(F)\\ &=& g_s'(s-)(F+D_zF-s)+\frac{1}{2}g_s''(\tilde{F}_1)(F+D_zF-s)^2\\ &&+g_s'(F)(s-F)+\frac{1}{2}g_s''(\tilde{F}_2)(s-F)^2\\ &=& g_s'(F)D_zF+ (g_s'(s+)+1-g_s'(F))(F+D_zF-s)\\ && +\frac{1}{2}g_s''(\tilde{F}_1)(F+D_zF-s)^2+\frac{1}{2}g_s''(\tilde{F}_2)(s-F)^2\\ &=& g_s'(F)D_zF+(F+D_zF-s)+g_s''(\tilde{F}_0)(s-F)(F+D_zF-s)\\ && +\frac{1}{2}g_s''(\tilde{F}_1)(F+D_zF-s)^2+\frac{1}{2}g_s''(\tilde{F}_2)(s-F)^2\\ &=:& g_s'(F)D_zF+(F+D_zF-s)+r_2(F,z,s)\end{aligned}$$ with $\tilde{F}_0,\tilde{F}_1,\tilde{F}_2\in \left(F+D_zF,F\right)$. Thus, $D_zg_s(F)$ has a representation $$\label{eqn:RepresentationDifferenceOperator} D_zg_s(F)=g_s'(F)D_zF+R(F,z,s),$$ where $R(F,z,s)$ is given by $$\begin{split} & R(F,z,s)\\ & =\left(\1(F,F+D_zF\leq s)+\1(F,F+D_zF>s)\right)r_1(F,z,s)\\ & \quad +\left(\1(F\leq s<F+D_zF)+\1(F+D_zF\leq s<F)\right)(r_2(F,z,s)-|F+D_zF-s|). \end{split}$$ Combining and yields $$\E\left[g_s'(F)-Fg_s(F)\right]=\E\left[g_s'(F)-\langle g_s'(F)DF+R(F,\cdot,s), -DL^{-1}F\rangle_{L^2(\mu)}\right].$$ Thus, the triangle inequality and $|g_s'(F)|\leq 1$ lead to $$\begin{aligned} \label{eqn:boundStein} |\E\left[g_s'(F)-Fg_s(F)\right]| & \leq & |\E\left[g_s'(F)\left(1-\langle DF,-DL^{-1}F\rangle_{L^2(\mu)}\right)\right]|\\ &&+|\E\langle R(F,\cdot,s),DL^{-1}F \rangle_{L^2(\mu)}|\notag\\ & \leq & \E |1-\langle DF, -DL^{-1}F\rangle_{L^2(\mu)}|+\E\langle |R(F,\cdot,s)|,|DL^{-1}F|\rangle_{L^2(\mu)}.\notag\end{aligned}$$ In $r_2(F,z,s)$, we assume that $s$ is between $F$ and $F+D_zF$ so that $$|F+D_zF-s|\leq |D_zF|\ \text{ and }\ |F-s|\leq |D_zF|.$$ The inequality and the fact that $\tilde{F}_i$ is between $F$ and $F+D_zF$ allow us to bound all second derivatives in $R(F,z,s)$ by $$|g''_s(\tilde{F}_i)|\leq\frac{\sqrt{2\pi}}{4}+|F|+|D_zF|.$$ Now it is easy to see that $$\begin{split} & |R(F,z,s)| \\ & \leq \left(\1(F,F+D_zF\leq s)+\1(F,F+D_zF>s)\right)\frac{1}{2}\left(\frac{\sqrt{2\pi}}{4}+|F|+|D_zF|\right)(D_zF)^2\\ & \quad +\left(\1(F\leq s<F+D_zF)+\1(F+D_zF\leq s<F)\right)|D_zF|\\ & \quad +\left(\1(F\leq s<F+D_zF)+\1(F+D_zF\leq s<F)\right)2\left(\frac{\sqrt{2\pi}}{4}+|F|+|D_zF|\right)(D_zF)^2\\ & \leq 2\left(\frac{\sqrt{2\pi}}{4}+|F|+|D_zF|\right)(D_zF)^2\\ & \quad +\left(\1(F\leq s < F+D_zF)+\1(F+D_zF\leq s<F)\right)|D_zF|. \end{split}$$ By , the last summand can be rewritten as $$\left(\1(F\leq s < F+D_zF)+\1(F+D_zF\leq s<F)\right)|D_zF|= D_z\1(F>s) \, D_zF.$$ Hence, it follows directly that $$\begin{split} &\E\langle |R(F,\cdot,s)|,|DL^{-1}F|\rangle_{L^2(\mu)}\\ & \leq 2\E\langle (DF)^2,|DL^{-1}F|\rangle_{L^2(\mu)}+2\E\langle (DF)^2,|F \ DL^{-1}F|\rangle_{L^2(\mu)}\\ & \quad +2\E\langle (DF)^2,|DF \ DL^{-1}F|\rangle_{L^2(\mu)}+\E\langle D\1(F>s) \, DF, |DL^{-1}F|\rangle_{L^2(\mu)}. \end{split}$$ Putting this in concludes the proof of the first inequality in . The second bound in is a direct consequence of the Cauchy-Schwarz inequality.$\Box$ In [@Peccatietal2010], the right-hand side of is evaluated for twice differentiable functions $f:\R\rightarrow\R$ with $\sup_{x\in\R}|f'(x)|\leq 1$ and $\sup_{x\in\R}|f''(x)|\leq 2$ (for the Wasserstein distance the solutions of Stein’s equation must have these properties) instead of the functions $g_s$ as defined in . For such a function $f$ it holds that $$D_zf(F)=f'(F) D_zF +\tilde{r}(F)$$ with $|\tilde{r}(F)|\leq (D_zF)^2$. Since this representation is easier than the representation we obtain for $D_zg_s(F)$, the bound for the Wasserstein distance in is shorter and easier to evaluate than the bound for the Kolmogorov distance in . Applications of Theorem \[thm:Theorem1\] {#sec:Ustatistics} ======================================== #### Normal approximation of a Poisson random variable. As a first application of Theorem \[thm:Theorem1\], we compute an upper bound for the Kolmogorov distance between a Poisson random variable $Y$ with parameter $t>0$ and a normal distribution. In Example 3.5 in [@Peccatietal2010], the bound is used to compute a bound for the Wasserstein distance, and the known optimal rate of convergence $t^{-\frac{1}{2}}$ is obtained. $Y$ has the same distribution as $F_t=\sum_{x\in\eta_t}1$, where $\eta_t$ is a Poisson point process on $[0,1]$ with $t$ times the the restriction of the Lebesgue measure on $[0,1]$ as intensity measure $\mu_t$. In the following, we denote by $I_{n,t}(\cdot)$ the $n$-th multiple Wiener-Itô integral with respect to $\eta_t$. The representation $$I_{1,t}(f)=\sum_{x\in\eta_t}f(x)-\int_X f(x) \, \mu_t({{\operatorname{d}}}x)$$ for a Wiener-Itô integral of a function $f\in L^1(\mu_t)\cap L^2(\mu_t)$ and the fact that $$F_t=t \int_0^1 1\, {{\operatorname{d}}}x +\sum_{x\in\eta_t} 1 - t \int_0^1 1\, {{\operatorname{d}}}x$$ imply that $F_t$ has the Wiener-Itô chaos expansion $F_t=\E F_t+I_{1,t}(f_1)=t+I_{1,t}(1)$. Hence, the standardized random variable $$G_t=\frac{F_t-\E F_t}{\sqrt{{\operatorname{Var}}F_t}}=\frac{F_t-t}{\sqrt{t}}$$ has the chaos expansion $G_t=I_{1,t}(1)/\sqrt{t}$ and $D_zG_t=-D_zL^{-1}G_t=1/\sqrt{t}$ for $z\in\left[0,1\right]$. It is easy to see that $$\E|1-\langle DG_t,-DL^{-1}G_t\rangle_{L^2(\mu_t)}|=|1-\frac{1}{t}\langle1,1\rangle_{L^2(\mu_t)}|=|1-\frac{t}{t}|=0.$$ We obtain $$\E\langle (DG_t)^2, (DL^{-1}G_t)^2\rangle_{L^2(\mu_t)}=\E\langle (DG_t)^2,(DG_t)^2\rangle_{L^2(\mu_t)}=\frac{1}{t},$$ $\E\langle DG_t,DG_t \rangle_{L^2(\mu_t)}^2=1$, and $\E G_t^4= 3+1/t$ by analogous computations. Since $D_z\1(G_t>s)\,D_zG_t \, |D_zL^{-1}G_t|\geq 0$ for $s\in\R$ and $z\in [0,1]$ and $D_zG_t\,|D_zL^{-1}G_t|=1/t$ for $z\in[0,1]$, it follows from Lemma \[lem:Integration\] and the Cauchy-Schwarz inequality that $$\begin{split} & \sup\limits_{s\in\R}\E\langle D\1(G_t>s), DG_t \, |DL^{-1}G_t|\rangle_{L^2(\mu_t)}\\ & = \sup\limits_{s\in\R} \E[ \1(G_t>s) \, \delta(DG_t \, |DL^{-1}G_t|)] \leq \E[\delta(DG_t \, |DL^{-1}G_t|)^2]^{\frac{1}{2}} =\frac{1}{t} \, \E[ I_{1,t}(1)^2]^{\frac{1}{2}} =\frac{1}{\sqrt{t}}. \end{split}$$ Now Theorem \[thm:Theorem1\] yields $$d_K\left(\frac{Y-t}{\sqrt{t}},N\right)\leq 2 \left(\frac{1}{\sqrt{t}}+\left(3+\frac{1}{t}\right)^{\frac{1}{4}}+1\right)\frac{1}{\sqrt{t}}+\frac{1}{\sqrt{t}}\leq\frac{8}{\sqrt{t}}$$ for $t\geq 1$, which is the classical Berry-Esseen inequality with the optimal rate of convergence (up to a constant). #### Normal approximation of U-statistics of Poisson point processes. As a second application of Theorem \[thm:Theorem1\], we discuss U-statistics of the form $$F=\sum_{(x_1,\hdots,x_k)\in\eta^k_{\neq}}f(x_1,\hdots,x_k)$$ with $k\in\N$ and $f\in L_s^1(\mu^k)$. Here, $\eta^k_{\neq}$ stands for the set of all $k$-tuples of distinct points of $\eta$. We denote $k$ as the order of the U-statistic $F$. From now on, we always assume that $F\in L^2(\P_\eta)$. In [@ReitznerSchulte2011], the chaos expansions of such Poisson functionals are investigated, and the bound is used to prove a central limit theorem with a rate of convergence for the Wasserstein distance. From there it is known that the kernels of the chaos expansion of a U-statistic $F$ are $$\label{eqn:kernelsUstatistic} f_i(x_1,\hdots,x_i)=\binom{k}{i}\int_{X^{k-i}}f(x_1,\hdots,x_i,y_1,\hdots,y_{k-i}) \, \mu^{k-i}({{\operatorname{d}}}(y_1,\hdots,y_{k-i}))$$ for $i=1,\hdots,k$ and $f_i=0$ for $i>k$. An application of the bound to such Poisson functionals yields (see Theorem 4.1 in [@ReitznerSchulte2011]) $$\label{eqn:dWUstatisticBasic} d_W\left(\frac{F-\E F}{\sqrt{{\operatorname{Var}}F}},N\right)\leq k\sum_{i,j=1}^k\frac{\sqrt{R_{ij}}}{{\operatorname{Var}}F}+k^{\frac{7}{2}}\sum_{i=1}^k\frac{\sqrt{\tilde{R}_i}}{{\operatorname{Var}}F},$$ where $R_{ij}$ and $\tilde{R}_i$ are given by $$\begin{aligned} R_{ij}&=&\E \left(\int_X I_{i-1}\left(f_i(z,\cdot)\right)I_{j-1}\left(f_j(z,\cdot)\right) \, \mu({{\operatorname{d}}}z)\right)^2\\ &&-\left(\E\int_X I_{i-1}\left(f_i(z,\cdot)\right) I_{j-1}\left(f_j(z,\cdot)\right) \, \mu({{\operatorname{d}}}z)\right)^2\\ \tilde{R}_i&=&\E\int_X I_{i-1}\left(f_i(z,\cdot)\right)^4 \, \mu({{\operatorname{d}}}z) $$ for $i,j=1,\hdots,k$. In [@ReitznerSchulte2011], the right-hand side of is bounded by a sum of deterministic integrals depending on $f$. Due to technical reasons it is assumed that the U-statistic $F$ is absolutely convergent, which means that the U-statistic $\overline{F}$ given by $$\overline{F}=\sum_{(x_1,\hdots,x_k)\in\eta^k_{\neq}} |f(x_1,\hdots,x_k)|$$ is in $L^{2}(\P_\eta)$. The U-statistic $\overline{F}$ has a finite Wiener-Itô chaos expansion with kernels $$\overline{f}_i(x_1,\hdots,x_i)=\binom{k}{i}\int_{X^{k-i}}|f(x_1,\hdots,x_i,y_1,\hdots,y_{k-i})| \, \mu^{k-i}({{\operatorname{d}}}(y_1,\hdots,y_{k-i}))$$ for $i=1,\hdots,k$ and $\overline{f}_i=0$ for $i>k$. In order to bound the right-hand side of by a sum of deterministic integrals, we use the following notation. For $i,j=1,\hdots,k$ let $\overline{\Pi}_{\geq 2}(i,i,j,j)$ be the set of all partitions $\pi$ of $$x_1^{(1)},\hdots,x_i^{(1)},x_1^{(2)},\hdots,x_i^{(2)},x_1^{(3)},\hdots,x_j^{(3)},x_1^{(4)},\hdots,x_j^{(4)}$$ such that - two variables with the same upper index are in different blocks of $\pi$; - each block of $\pi$ includes at least two variables; - there are no sets $A_1,A_2\subset \{1,2,3,4\}$ with $A_1\cup A_2=\{1,2,3,4\}$ and $A_1\cap A_2=\emptyset$ such that each block of $\pi$ either consists of variables with upper index in $A_1$ or of variables with upper index in $A_2$. Let $|\pi|$ stand for the number of blocks of a partition $\pi$. For functions $g_1,g_2: X^i\rightarrow\overline{\R}$ and $g_3,g_4: X^j\rightarrow\overline{\R}$ the tensor product $g_1\otimes g_2 \otimes g_3 \otimes g_4: X^{2i+2j}\rightarrow\overline{\R}$ is given by $$\begin{split} & (g_1\otimes g_2 \otimes g_3 \otimes g_4)(x_1^{(1)},\hdots,x_{j}^{(4)})\\ & = g_1(x_1^{(1)},\hdots,x_i^{(1)}) \, g_2(x_1^{(2)},\hdots,x_i^{(2)}) \, g_3(x_1^{(3)},\hdots,x_j^{(3)}) \, g_4(x_1^{(4)},\hdots,x_j^{(4)}). \end{split}$$ For $\pi\in \overline{\Pi}_{\geq 2}(i,i,j,j)$ we define the function $(g_1\otimes g_2 \otimes g_3 \otimes g_4)_\pi: X^{|\pi|}\rightarrow\overline{\R}$ by replacing all variables that are in the same block of $\pi$ by a new common variable. Since we later integrate over all these new variables, their order does not matter. Using this notation, we define $$\begin{aligned} M_{ij}(f) & = & \sum_{\pi\in \overline{\Pi}_{\geq 2}(i,i,j,j)} \int_{X^{|\pi|}} (\overline{f}_i\otimes\overline{f}_i\otimes \overline{f}_j\otimes \overline{f}_j)_\pi \, {{\operatorname{d}}}\mu^{|\pi|}\end{aligned}$$ for $i,j=1,\hdots,k$. Now we can state the following upper bound for the Wasserstein distance (see Theorem 4.7 in [@ReitznerSchulte2011]): \[prop:dWUstatistic\] Let $F\in L^2(\P_\eta)$ be an absolutely convergent U-statistic of order $k$ and let $N$ be a standard Gaussian random variable. Then $$\label{eqn:dWUstatistic} d_W\left(\frac{F-\E F}{\sqrt{{\operatorname{Var}}F}},N\right)\leq 2k^{\frac{7}{2}}\sum_{1\leq i\leq j\leq k}\frac{\sqrt{M_{ij}(f)}}{{\operatorname{Var}}F}.$$ In this situation, we can use Theorem \[thm:Theorem1\] to prove the following bound analogous to for the Kolmogorov distance between a standardized U-statistic and a standard Gaussian random variable: \[thm:dKUstatistic\] Let $F\in L^2(\P_\eta)$ be an absolutely convergent U-statistic of order $k$ and let $N$ be a standard Gaussian random variable. Then $$\label{eqn:dKUstatistic} d_K\left(\frac{F-\E F}{\sqrt{{\operatorname{Var}}F}},N\right)\leq 19 k^{5} \sum_{i,j=1}^k\frac{\sqrt{M_{ij}(f)}}{{\operatorname{Var}}F}.$$ Before we prove this theorem in Section \[sec:ProofUstatistic\], we discuss some of its consequences. Let us consider a family of Poisson point processes $\eta_t$ with intensity measures $\mu_t$ and U-statistics $$F_t=\sum_{(x_1,\hdots,x_k)\in(\eta_t)^k_{\neq}}f_t(x_1,\hdots,x_k)$$ with $f_t\in L^1_s(\mu^k_t)$ and $F_t\in L^2(\P_{\eta_t})$ such that $$\frac{\sqrt{M_{ij}(f_t)}}{{\operatorname{Var}}F_t}\rightarrow 0 \ \text{ as }\ t\rightarrow\infty \ \text{ for all } \ i,j=1,\hdots,k.$$ Here, we integrate with respect to $\mu_t$ in $M_{ij}(f_t)$. Comparing the right-hand sides in and for the U-statistics $F_t$, we see that the bounds for the Wasserstein and Kolmogorov distance have the same rates of convergence and differ only by constants. An important special case of the described setting is that the Poisson point process depends on a real valued intensity parameter. The following corollary deals with this situation and is the counterpart of Theorem 5.2 in [@ReitznerSchulte2011] for the Kolmogorov distance. Let $\eta_t$ be a Poisson point process with intensity measure $\mu_t=t\mu$ with $t\geq 1$ and a fixed $\sigma$-finite non-atomic measure $\mu$ and let $N$ be a standard Gaussian random variable. We consider U-statistics $F_t\in L^2(\P_{\eta_t})$ of the form $$F_t=g(t)\sum_{(x_1,\hdots,x_k)\in(\eta_t)^k_{\neq}} f(x_1,\hdots,x_k)$$ with $g: (0,\infty)\rightarrow (0,\infty)$ and $f\in L^1_s(\mu^k)$ independent of $t$. Moreover, we assume that $$\int_X \left(\int_{X^{k-1}}f(x,y_1,\hdots,y_{k-1}) \, \mu^{k-1}({{\operatorname{d}}}(y_1,\hdots,y_{k-1}))\right)^2 \, \mu({{\operatorname{d}}}x)>0$$ and $M_{ij}(f)<\infty$ for $i,j=1,\hdots,k$. Then there is a constant $C>0$ such that $$d_K\left(\frac{F_t-\E F_t}{\sqrt{{\operatorname{Var}}F_t}},N\right)\leq C \, t^{-\frac{1}{2}}$$ for all $t\geq 1$. This corollary follows from bounding $\sqrt{M_{ij}(f_t)}/{\operatorname{Var}}F_t$ in the same way as in the proof of [@ReitznerSchulte2011 Theorem 5.2]. In [@LachiezeReyPeccati2011], a so-called fourth moment condition for Poisson functionals with finite Wiener-Itô chaos expansion and non-negative kernels satisfying some integrability conditions is derived. More precisely, for such Poisson functionals it is proven that $$d_W\left(\frac{F-\E F}{\sqrt{{\operatorname{Var}}F}},N\right)\leq C_{W,k} \sqrt{\E\left(\frac{F-\E F}{\sqrt{{\operatorname{Var}}F}}\right)^4-3}$$ with a constant $C_{W,k}>0$ only depending on $k$. U-statistics $F\in L^2(\P_\eta)$ of the form $$F=\sum_{(x_1,\hdots,x_k)\in\eta^k_{\neq}}f(x_1,\hdots,x_k)\ \text{ with }f\in L^1_s(\mu^k)\text{ and }f\geq 0$$ such that $ M_{ij}(f)<\infty$ for $i,j=1,\hdots,k$ belong to this class and satisfy $$\frac{M_{ij}(f)}{({\operatorname{Var}}F)^2}\leq \E \left(\frac{F-\E F}{\sqrt{{\operatorname{Var}}F}}\right)^4-3.$$ Then can be modified to $$d_K\left(\frac{F-\E F}{\sqrt{{\operatorname{Var}}F}},N\right)\leq C_k \sqrt{\E\left(\frac{F-\E F}{\sqrt{{\operatorname{Var}}F}}\right)^4-3}$$ with a constant $C_k>0$ only depending on $k$. Proof of Theorem \[thm:dKUstatistic\] {#sec:ProofUstatistic} ===================================== In our proof of Theorem \[thm:dKUstatistic\], we make use of the following property of U-statistics: \[lem:Linverse\] For a U-statistic $F\in L^2(\P_\eta)$ of order $k$ the inverse of the Ornstein-Uhlenbeck generator has a representation $$\label{eqn:Linverse} \begin{split} & -L^{-1}(F-\E F)\\ & = \sum_{m=1}^k\frac{1}{m}\sum_{(x_1,\hdots,x_m)\in\eta^m_{\neq}}\int_{X^{k-m}} f(x_1,\hdots,x_m,y_1,\hdots,y_{k-m}) \, \mu^{k-m}({{\operatorname{d}}}(y_1,\hdots,y_{k-m}))\\ & \quad -\sum_{m=1}^k\frac{1}{m}\int_{X^k} f(y_1,\hdots,y_k) \, \mu^k({{\operatorname{d}}}(y_1,\hdots,y_k)). \end{split}$$ We define $\widehat{f}_i:X^i\rightarrow\overline{\R}$ by $\widehat{f}_i(x_1,\hdots,x_i)=\binom{k}{i}^{-1}f_i(x_1,\hdots,x_i)$ for $i=1,\hdots,k$. Using this notation and formula for the kernels of a U-statistic, we obtain the chaos expansion $$\begin{aligned} &&\sum_{(x_1,\hdots,x_m)\in\eta^m_{\neq}}\int_{X^{k-m}}f(x_1,\hdots,x_m,y_1,\hdots,y_{k-m}) \, \mu^{k-m}({{\operatorname{d}}}(y_1,\hdots,y_{k-m}))\notag\\ &&=\int_{X^k}f(y_1,\hdots,y_k) \, \mu^k({{\operatorname{d}}}(y_1,\hdots,y_{k}))+\sum_{i=1}^{m}\binom{m}{i}I_i(\widehat{f}_i)\end{aligned}$$ for $m=1,\hdots,k$. Combining this with an identity for binomial coefficients, we see that the right-hand side in equals $$\begin{aligned} \sum_{m=1}^k\frac{1}{m}\sum_{i=1}^{m}\binom{m}{i}I_i(\widehat{f}_i)&=&\sum_{m=1}^k\sum_{i=1}^{k}\frac{1}{m}\binom{m}{i}I_i(\widehat{f}_i)=\sum_{i=1}^k\sum_{m=1}^k\frac{1}{m}\binom{m}{i}I_i(\widehat{f}_i)\\ &=&\sum_{i=1}^k\frac{1}{i}\binom{k}{i}I_i(\widehat{f}_i)=\sum_{i=1}^k\frac{1}{i}I_i(f_i),\end{aligned}$$ which is the chaos expansion of $-L^{-1}(F-\E F)$ by definition. In order to deal with expressions as $R_{ij}$ and $\tilde{R}_i$ in the previous section, one needs to compute the expectation of products of multiple Wiener-Itô integrals. This can be done by using Proposition 6.5.1 in [@PeccatiTaqqu2011] (see also [@Sur Theorem 3.1], [@Privault2009 Proposition 4.5.6], or [@LPST2012 Theorem 3.1]). This so-called product formula gives us the Wiener-Itô chaos expansion of a product of two multiple Wiener-Itô integrals. Together with , one obtains that the expectation of a product of four multiple Wiener-Itô integrals is a sum of deterministic integrals depending on the integrands of the stochastic integrals and partitions of their variables as used for the definition of $M_{ij}(f)$. By using this product formula, one can prove in a similar way as in [@ReitznerSchulte2011 Subsection 4.2] that $$\label{eqn:inequalityRM} R_{ij}\leq M_{ij}(f)\ \text{ and }\ \int_X \E I_{i-1}(f_i(z,\cdot))^2I_{j-1}(f_j(z,\cdot))^2 \, \mu({{\operatorname{d}}}z)\leq M_{ij}(f)$$ for $i,j=1,\hdots,k$. Moreover, we prepare the proof of Theorem \[thm:dKUstatistic\] by the following two lemmas: \[lem:fourthMoment\] Let $F\in L^2(\P_\eta)$ be an absolutely convergent U-statistic with $M_{ij}(f)<\infty$ for $i,j=1,\hdots,k$. Then $$\E (F-\E F)^4 \leq k^2\sum_{i,j=1}^k M_{ij}(f)+3k^2 ({\operatorname{Var}}F)^2.$$ Using the Wiener-Itô chaos expansion of $F$ and the Cauchy-Schwarz inequality, we obtain $$\E (F-\E F)^4 = \E \left(\sum_{i=1}^k I_i(f_i)\right)^2 \left(\sum_{j=1}^k I_j(f_j)\right)^2 \leq k^2 \, \E \sum_{i=1}^k I_i(f_i)^2 \sum_{j=1}^k I_j(f_j)^2.$$ Now the previously mentioned product formula for multiple Wiener-Itô integrals and ${\operatorname{Var}}F=\sum_{n=1}^k n! \, \|f_n\|_n^2$ yield that $$\E I_i(f_i)^2I_j(f_j)^2 \leq M_{ij}(f)+ 3 \, i! \, ||f_i||_i^2 \, j! \, ||f_j||_j^2 \leq M_{ij}(f)+ 3 ({\operatorname{Var}}F)^2.$$ In the first inequality, we have equality for $i=j$ and $f\geq 0$. \[lem:sup\] Let $F\in L^2(\P_\eta)$ be an absolutely convergent U-statistic with $M_{ij}(f)<\infty$ for $i,j=1,\hdots,k$. Then $$\begin{split} & \sup\limits_{s\in\R}\E\langle D\1(F>s),DF \, |DL^{-1}(F-\E F)|\rangle_{L^2(\mu)}\\ & \leq \sqrt{(2k-1) \, \E\langle (D\overline{F})^2, (DL^{-1}\left(\overline{F}-\E\overline{F}\right))^2\rangle_{L^2(\mu)}}. \end{split}$$ We can write the U-statistic $F$ as $$F=\sum_{(x_1,\hdots,x_k)\in\eta^k_{\neq}}f(x_1,\hdots,x_k)=\underbrace{\sum_{(x_1,\hdots,x_k)\in\eta^k_{\neq}}f^{+}(x_1,\hdots,x_k)}_{=F^{+}}-\underbrace{\sum_{(x_1,\hdots,x_k)\in\eta^k_{\neq}}f^{-}(x_1,\hdots,x_k)}_{=F^{-}}$$ with $f^{+}=\max\left\{f,0\right\}$ and $f^{-}=\max\left\{-f,0\right\}$ and have $\overline{F}=F^++F^-$. As a consequence of , we know that $D_zV\geq 0$ for a U-statistic $V$ with non-negative summands. Combining this with $f^+,f^-\geq 0$ and Lemma \[lem:Linverse\], we see that $$-D_zL^{-1}\left(F^{+}-\E F^{+}\right)\geq 0\quad \text{ and }\quad -D_zL^{-1}\left(F^{-}-\E F^{-}\right)\geq 0.$$ Moreover, it holds that $D_z\1(F>s) \, D_zF\geq 0$. Proposition 6.5.1 in [@PeccatiTaqqu2011] implies that the product $D_zF\ D_zL^{-1}\left(\overline{F}-\E\overline{F}\right)$ has a finite chaos expansion with an order less than or equal to $2k-2$. Together with Lemma \[lem:Integration\], the Cauchy-Schwarz inequality, and Lemma \[lem:SkorohodSquare\], we obtain $$\begin{split} &\sup\limits_{s\in\R}\E\langle D\1(F>s),DF \, |DL^{-1}(F-\E F)|\rangle_{L^2(\mu)}\\ &= \sup\limits_{s\in\R}\E\langle D\1(F>s),DF \, |DL^{-1}\left(F^{+}-\E F^{+}-F^{-}+\E F^{-}\right)|\rangle_{L^2(\mu)}\\ &\leq \sup\limits_{s\in\R}\E\langle D\1(F>s),DF \left(-DL^{-1}(F^{+}-\E F^{+})-DL^{-1}(F^{-}-\E F^{-})\right)\rangle_{L^2(\mu)}\\ &\leq \E\left[\delta\left(DF\, DL^{-1}\left(\overline{F}-\E\overline{F}\right)\right)^2\right]^{\frac{1}{2}}\\ & \leq \sqrt{(2k-1)\E\langle (DF)^2, (DL^{-1}\left(\overline{F}-\E\overline{F}\right))^2\rangle_{L^2(\mu)}}. \end{split}$$ Now the fact that $(D_zF)^2\leq (D_z\overline{F})^2$ concludes the proof. #### **Proof of Theorem \[thm:dKUstatistic\]**: {#proof-of-theorem-thmdkustatistic} In the following, we can assume that $M_{ij}(f)<\infty$ for $i,j=1,\hdots,k$ since is obviously true, otherwise. We consider the standardized random variable $G=(F-\E F)/\sqrt{{\operatorname{Var}}F}$, whose Wiener-Itô chaos expansion has the kernels $g_i=f_i/\sqrt{{\operatorname{Var}}F}$ for $i=1,\hdots,k$. In order to simplify our notation, we use the abbreviation $$S=\sum_{i,j=1}^k \frac{\sqrt{M_{ij}(f)}}{{\operatorname{Var}}F}.$$ Exactly as in the proof of Theorem 4.1 in [@ReitznerSchulte2011], we obtain $$\E\left|1-\langle DG,-DL^{-1}G\rangle_{L^2(\mu)}\right|\leq k\sum_{i,j=1}^k\frac{\sqrt{R_{ij}}}{{\operatorname{Var}}F}\leq k\sum_{i,j=1}^k\frac{\sqrt{M_{ij}(f)}}{{\operatorname{Var}}F} = k S.$$ From straightforward computations using Fubini’s Theorem, the Cauchy-Schwarz inequality, and , it follows that $$\begin{aligned} \E\langle (DG)^2,(DL^{-1}G)^2\rangle_{L^2(\mu)}&=&\int_X \E (D_zG)^2(D_zL^{-1}G)^2 \, \mu({{\operatorname{d}}}z)\\ &=& \int_X \E \left(\sum_{i=1}^{k} i \, I_{i-1}(g_i(z,\cdot))\right)^2\left(\sum_{i=1}^{k} I_{i-1}(g_i(z,\cdot))\right)^2 \, \mu({{\operatorname{d}}}z)\\ &\leq & k^4 \int_X \E \sum_{i=1}^{k} I_{i-1}(g_i(z,\cdot))^2 \sum_{j=1}^{k} I_{j-1}(g_j(z,\cdot))^2 \, \mu({{\operatorname{d}}}z)\\ & \leq & k^4 \sum_{i,j=1}^k \frac{M_{ij}(f)}{({\operatorname{Var}}F)^2} \leq k^4 S^2,\end{aligned}$$ $$\begin{aligned} \E\langle DG^2,DG^2\rangle_{L^2(\mu)} &=& \int_X \E\left(\sum_{i=1}^{k}i \, I_{i-1}(g_i(z,\cdot))\right)^4 \, \mu({{\operatorname{d}}}z)\\ & \leq & k^6 \int_X \E \sum_{i=1}^k I_{i-1}(g_i(z,\cdot))^2 \sum_{j=1}^k I_{j-1}(g_j(z,\cdot))^2 \, \mu({{\operatorname{d}}}z)\\ & \leq & k^6 \sum_{i,j=1}^k \frac{M_{ij}(f)}{({\operatorname{Var}}F)^2}\leq k^6 S^2,\end{aligned}$$ and $$\begin{aligned} \E\langle DG,DG\rangle_{L^2(\mu)}^2 &\leq& \E\left(\sum_{i,j=1}^k i j\int_X I_{i-1}(g_i(z,\cdot)) I_{j-1}(g_j(z,\cdot)) \, \mu({{\operatorname{d}}}z)\right)^2\\ &\leq & k^2 \sum_{i,j=1}^k i^2 j^2 \E\left(\int_X I_{i-1}(g_i(z,\cdot)) I_{j-1}(g_j(z,\cdot)) \, \mu({{\operatorname{d}}}z)\right)^2\\ &\leq & k^6 \sum_{i,j=1}^k \frac{R_{ij}}{({\operatorname{Var}}F)^2}+k^4\sum_{i=1}^k \frac{(i!)^2||f_i||_i^4}{({\operatorname{Var}}F)^2}\\ &\leq& k^6 \sum_{i,j=1}^k \frac{R_{ij}}{({\operatorname{Var}}F)^2}+k^4\leq k^6 \sum_{i,j=1}^k \frac{M_{ij}(f)}{({\operatorname{Var}}F)^2}+k^4\leq k^6S^2+k^4.\end{aligned}$$ As a consequence of Lemma \[lem:fourthMoment\], we have that $$\E\left(\frac{F-\E F}{\sqrt{{\operatorname{Var}}F}}\right)^4\leq k^2\sum_{i,j=1}^k\frac{M_{ij}(f)}{({\operatorname{Var}}F)^2}+3 k^2=k^2 S^2 + 3k^2.$$ Combining the last three inequalities, we obtain $$\begin{split} & 2\sqrt{\E\langle (DG)^2,(DG)^2\rangle_{L^2(\mu)}}+2\left(\E\langle DG, DG\rangle_{L^2(\mu)}^2\right)^{\frac{1}{4}} \left((E G^4)^{\frac{1}{4}}+1\right)\\ & \leq 2 k^3 S+2(k^{\frac{3}{2}}\sqrt{S}+k)(\sqrt{k}\sqrt{S}+3^{\frac{1}{4}}\sqrt{k}+1)\leq 16k^3 \end{split}$$ for $S\leq 1$. Lemma \[lem:sup\] together with a similar computation as for $\E\langle (D G)^2, (DL^{-1}G)^2\rangle_{L^2(\mu)}$ implies that $$\sup_{s\in\R}\E\langle D\1(G>s),DG \, |DL^{-1}G|\rangle_{L^2(\mu)} \leq \sqrt{(2k-1)k^4\sum_{i,j=1}^k\frac{M_{ij}(f)}{({\operatorname{Var}}F)^2}}\leq \sqrt{2}k^{\frac{5}{2}} S.$$ Thus, it follows from Theorem \[thm:Theorem1\] that $$d_K\left(\frac{F-\E F}{\sqrt{{\operatorname{Var}}F}},N\right) \leq k S+16 k^3 \, k^2 S +\sqrt{2} k^{\frac{5}{2}} S\leq 19k^5 S$$ for $S\leq 1$. Otherwise, this inequality still holds since the Kolmogorov distance is at most one. $\Box$ In a similar way, one can also obtain an upper bound for the Kolmogorov distance between a Gaussian random variable and a finite sum of Poisson U-statistics. This class of Poisson functionals is interesting since Theorem 3.6 in [@ReitznerSchulte2011] tells us that each Poisson functional $F\in L^2(\P_\eta)$ of order $k$ with kernels $f_i\in L^1_s(\mu^i)\cap L_s^2(\mu^i)$ for $i=1,\hdots,k$ is a finite sum of Poisson U-statistics (and a constant). For such a Poisson functional the upper bounds for the inner products in Theorem \[thm:dKUstatistic\] that depend on $R_{ij}$ and $$\int_X \E I_{i-1}(f_i(z,\cdot))^2I_{j-1}(f_j(z,\cdot))^2\mu({{\operatorname{d}}}z)$$ for $i,j=1,\hdots,k$ still hold. Moreover, we can compute a similar bound as in Lemma \[lem:sup\] using the representation of $F$ as a sum of Poisson U-statistics. Together with the fourth centred moment of $F$, one can obtain an upper bound for the Kolmogorov distance between $(F-\E F)/\sqrt{{\operatorname{Var}}F}$ and a standard Gaussian random variable. 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[^1]: *Karlsruhe Institute of Technology, Institute of Stochastics, D-76128 Karlsruhe, e-mail: matthias.schulte\[at\]kit.edu*

--- abstract: 'Let $A$ and $C$ be self-adjoint operators such that the spectrum of $A$ lies in a gap of the spectrum of $C$ and let $d>0$ be the distance between the spectra of $A$ and $C$. We prove that under these assumptions the sharp value of the constant $c$ in the condition $\|B\|<c d$ guaranteeing the existence of a (bounded) solution to the operator Riccati equation $XA-CX+XBX=B^*$ is equal to $\sqrt{2}$. We also prove an extension of the Davis-Kahan $\tan\Theta$ theorem and provide a sharp estimate for the norm of the solution to the Riccati equation. If $C$ is bounded, we prove, in addition, that the solution $X$ is a strict contraction if $B$ satisfies the condition $\|B\|<d$, and that this condition is sharp.' address: - | Fraunhofer-Institut für Lasertechnik\ Steinbachstra[ß]{}e 15\ Aachen, D-52074, Germany - | Department of Mathematics\ University of Missouri\ Columbia, MO 65211, USA - | Department of Mathematics\ University of Missouri\ Columbia, MO 65211, USA. BLTP, JINR\ 141980 Dubna\ Moscow Region\ Russia author: - 'V. Kostrykin' - 'K. A. Makarov' - 'A. K. Motovilov' title: 'On the existence of solutions to the operator Riccati equation and the tan$\boldsymbol{\Theta}$ theorem' --- Introduction {#Intro} ============ In this paper we consider the operator Riccati equation $$\label{Ric0} XA-CX+XBX=B^*$$ associated with the self-adjoint $2\times 2$ block operator matrix $$\label{cHintro} {H}=\begin{pmatrix} A & B \\ B^* & C \end{pmatrix}$$ on the orthogonal sum ${{\mathcal H}}={{\mathcal H}}_A \oplus {{\mathcal H}}_C$ of separable Hilbert spaces ${{\mathcal H}}_A$ and ${{\mathcal H}}_C$. Here $A$ is a bounded self-adjoint operator on the Hilbert space ${{\mathcal H}}_A$, $C$ possibly unbounded self-adjoint operator on the Hilbert space ${{\mathcal H}}_C$, and $B$ is a bounded operator from ${{\mathcal H}}_C$ to ${{\mathcal H}}_A$. Solving the Riccati equation appears to be an adequate tool in the study of the invariant subspaces of the operator $H$ that are the graphs of bounded operators from ${{\mathcal H}}_A$ to ${{\mathcal H}}_C$, the so-called graph subspaces. It is well known that given a bounded solution $X:{{\mathcal H}}_A\to {{\mathcal H}}_C$ to the Riccati equation (with $\operatorname{\mathrm{Ran}}X\subset {\mathrm{Dom}}(C)$), the graph $${{\mathcal G}}(X)=\left\{x\oplus X x|\ x\in{{\mathcal H}}_A \right\}$$ of the operator $X$ reduces the operator $H$ (see, e.g., [@AMM Section 5]) and in the framework of this approach the following two problems naturally arise. The first problem is to study the spectrum of the part of $H$ associated with the reducing subspace ${{\mathcal G}}(X)$ (respectively its orthogonal complement ${{\mathcal G}}(X)^\perp$), and the second one is to estimate the operator angle $\Theta$ (see, e.g., [@KMMgeom] for discussion of this notion) between the subspaces ${{\mathcal G}}(X)$ and ${{\mathcal H}}_A$ (respectively ${{\mathcal G}}(X)^\perp$ and ${{\mathcal H}}_C$). Both of these problems can efficiently be solved if a bounded solution $X$ to is known: the operator $H$ appears to be similar to the diagonal block operator matrix $$\begin{pmatrix} A + B X & 0 \\ 0 & C- B^\ast X^\ast \end{pmatrix}$$ associated with the decomposition ${{\mathcal H}}={{\mathcal H}}_A\oplus {{\mathcal H}}_C$ (in particular, $\sigma(H)=\sigma(A + B X)\cup \sigma (C- B^\ast X^\ast )$), and the operator angle $\Theta$ between the subspaces ${{\mathcal G}}(X)$ and ${{\mathcal H}}_A$ has the representation $$\Theta = \arctan \sqrt{X^\ast X}.$$ If the spectra of $A$ and $C$ overlap, the Riccati equation may have no solution at all (cf., e.g., [@AMM Example 3.2]). At the same time the spectra separation requirement alone does not guarantee the existence of solutions either (see, e.g., [@AMM Lemma 3.11]). Under the spectra separation hypothesis $$\label{spsep} {{\ensuremath{\mathrm{dist}}}}(\sigma(A), \sigma(C)) > 0,$$ a natural sufficient condition for the existence of solutions to the Riccati equation requires a smallness assumption on the operator $B$ of the form $$\label{cdest} \|B\|<c_{\text{best}}\,{{\ensuremath{\mathrm{dist}}}}\bigl({\sigma}(A),{\sigma}(C)\bigr)$$ with a constant $c_{\text{best}}>0$ independent of the distance between the spectra $\sigma(A)$ and $\sigma(C)$ of the operators $A$ and $C$, respectively. The best possible constant $c_{\mathrm{best}}$ in is still unknown. However, $c_{\mathrm{best}}$ is known to be in the interval $\left[{\pi}^{-1},\sqrt{2}\right]$ (see [@AMM]). If both $A$ and $C$ are bounded, then $c_{\mathrm{best}}\in\left[c_\pi,\sqrt{2}\right]$ with $c_\pi=\frac{3\pi-\sqrt{\pi^2+32}}{\pi^2-4}=0.503288...$ (see [@KMMgamma]). In [@KMMgamma] the best possible constant $c_{\mathrm{best}}$ has been conjectured to be $\sqrt{3}/2$. Some earlier results in this direction can be found in [@AdLT], [@MeMo99], [@Motovilov:SPb:91], and [@Motovilov:95]. In some particular cases the optimal solvability condition can be relaxed provided that some additional assumptions upon mutual disposition of the spectra of $A$ and $C$ are posed. For instance, if the spectra of $A$ and $C$ are subordinated, e.g., $$\label{subord} \sup{\sigma}(A)<\inf{\sigma}(C),$$ the Riccati equation is known to have a strictly contractive solution for any bounded $B$ (see, e.g., [@AL95]). To some extent abusing the terminology one may say that in this case the best possible constant in inequality is infinite: No smallness assumptions on $B$ are needed. In the limiting case of , $$\sup{\sigma}(A)=\inf{\sigma}(C),$$ the existence of contractive solutions has been established in [@AdLT] under some additional assumptions which have been dropped in [@KMMalpha]. See also [@MenShk] where the spectra separation condition has also been somewhat relaxed and the existence of a bounded but not necessarily contractive solution has been established. Our *first principal result* concerns the case where the operator $C$ has a finite spectral gap containing the spectrum of $A$. Recall that by a finite spectral gap of a self-adjoint operator $T$ one understands an *open* finite interval on the real axis lying in the resolvent set of $T$ such that both of its end points belong to the spectrum of $T$. \[thm:1\] Assume that the self-adjoint operator $C$ has a finite spectral gap $\Delta$ containing the spectrum of the bounded self-adjoint operator $A$. \(i) Suppose that $$\label{korint} \|B\|<\sqrt{d|\Delta|} \quad \text{where} \quad d={{\ensuremath{\mathrm{dist}}}}({\sigma}(A),{\sigma}(C)),$$ with $|\cdot|$ denoting Lebesgue measure on ${\mathbb R}$. Then the spectrum of the block operator matrix $H=\begin{pmatrix}A & B\\ B^* & C\end{pmatrix}$ in the gap $\Delta$ is a (proper) closed subset of (the open set) $\Delta$. The spectral subspace of the operator $H$ associated with the interval $\Delta$ is the graph of a bounded solution $X:\ {{\mathcal H}}_A\rightarrow{{\mathcal H}}_C$ to the Riccati equation . Moreover, the operator $X$ is the unique solution to the Riccati equation in the class of bounded operators with the properties $$\label{Uniq} {\sigma}(A+B X)\subset\Delta\quad\text{and}\quad {\sigma}(C-B^* X^*)\subset{\mathbb{R}}\setminus\Delta.$$ \(ii) If in addition the operator $C$ is bounded and $$\label{korints} \|B\|<\sqrt{d(|\Delta|-d)},$$ then the solution $X$ is a strict contraction and $$\label{XestFin2} \|X\|\leq \tan\frac{1}{2} \arctan\left(\frac{2\|AB+BC\|}{d(|\Delta|-d)-\|B\|^2}\right)<1.$$ As a corollary, under the assumption that the operator $C$ has a finite spectral gap $\Delta$ containing the spectrum of $A$, we prove that $c=\sqrt{2}$ is best possible in condition ensuring the existence of a bounded solution to the Riccati equation (see Remark \[coptim\]) while $c=1$ is best possible in to ensure that the solution is a contraction (see Remark \[controptim\]). The proof of the part (i) of Theorem \[thm:1\] will be given in Section \[SecExist\] and that of the part (ii) in Section \[SecEst\]. Our *second principal result* holds with no [*a priori*]{} assumption upon the mutual disposition of the spectra of $A$ and $C$ (in particular, the spectra of $A$ and $C$ may overlap). \[thm:2\] Assume that the self-adjoint operator $C$ has a spectral gap $\Delta$ (finite or infinite) and the self-adjoint operator $A$ is bounded. Assume that the Riccati equation has a bounded solution $X$ and hence the graph subspace ${{\mathcal G}}(X)$ reduces the block operator matrix $H$. Suppose that the spectrum of the part $H|_{{{\mathcal G}}(X)}$ of the operator $H$ associated with the reducing subspace ${{\mathcal G}}(X)$ is a (proper) closed subset of (the open set) $\Delta$. Then the the following norm estimate holds: $$\label{NXest} \|X\|\leq \frac{\|B\|}{\delta} \quad \text{with} \,\,\, \delta={{\ensuremath{\mathrm{dist}}}}\bigl({\sigma}(H|_{{{\mathcal G}}(X)}),{\sigma}(C)\bigr).$$ Equivalently, $$\label{TanTheta} \|\tan\Theta\|\leq\frac{\|B\|}{\delta},$$ where $\Theta$ is the operator angle between the subspaces ${{\mathcal H}}_A$ and ${{\mathcal G}}(X)$. Estimate extends the Davis-Kahan $\tan\Theta$ theorem [@Davis:Kahan], a result previously known only in the case where the spectra of $C$ and $H|_{{{\mathcal G}}(X)}$ are subordinated, that is, the operator $C$ is semibounded and the spectrum of the part $H|_{{{\mathcal G}}(X)}$ lies in the *infinite* spectral gap of $C$. This generalization extends the list of the celebrated Davis-Kahan $\sin\Theta$ and $\sin 2\Theta$ theorems, proven in the case where the operator $C$ has a gap of finite length [@Davis:Kahan]. The proof of Theorem \[thm:2\] will be given in Section \[SecTan\]. Our main techniques are based on applications of the Virozub-Matsaev factorization theorem for analytic operator-valued functions [@ViMt] (in the spirit of the work [@MenShk] (cf. [@LMMT])) and the Daletsky-Krein factorization formula [@DK]. Under the hypothesis of Theorem \[thm:1\] we prove that - for $\lambda\notin{\sigma}(C)$ the operator-valued Herglotz function $M(\lambda)=\lambda I-A+B(C-\lambda I)^{-1}B^*$ admits a factorization $$\label{ZAint} M(\lambda)=W(\lambda)(Z-\lambda I),$$ with $W$ being an operator-valued function holomorphic on the resolvent set of the operator $C$ and $Z$ is a bounded operator with the spectrum in the spectral gap $\Delta$ of the operator $C$, - the Riccati equation has a bounded solution of the form $$\label{Xintro} X=-\frac{1}{2\pi{\mathrm i}}\int_\Gamma d\lambda(C-\lambda I)^{-1}B^* (Z-\lambda I)^{-1},$$ where $\Gamma$ is an appropriate Jordan contour encircling the spectrum of the operator $Z$, - the spectral subspace of the $2\times 2$ block operator matrix $H$ associated with the interval $\Delta$ is the graph of the operator $X$, - the spectrum of the operator $H$ in the interval $\Delta$ coincides with that of the operator $Z$, that is, ${\sigma}(H)\cap\Delta={\sigma}(Z)$. In Section \[SecFact\] we recall the concept of invariant graph subspaces for linear operators as well as their relation to the Riccati equation. Theorem \[RicSol\] below presents a general result linking the factorization property of the operator valued function $M(\lambda)$ with the existence of a spectral subspace for the $2\times2$ self-adjoint block operator matrix admitting representation as the graph of the operator . In Section \[SecExist\] under hypothesis we prove factorization formula and give bounds on the location of the spectrum of the operator $Z$ (Theorem \[Zexist\]), and finally prove the part (i) of Theorem \[thm:1\]. The proof of Theorem \[thm:2\] is given in Section \[SecTan\]. In Section \[SecEst\] combining the results of Theorems \[thm:1\] (i) and \[thm:2\] under assumptions and respectively we provide norm estimates on the solution $X$ of the Riccati equation and prove Theorem 1 (ii). We conclude the section by an example showing that condition ensuring the strict contractivity of the solution $X$ is sharp. Few words about notations used throughout the paper. Given a Hilbert space ${{\mathcal K}}$ by $I_{{{\mathcal K}}}$ we denote the identity operator on ${{\mathcal K}}$. If it does not lead to any confusion we will simply write $I$ instead of more pedantic notation $I_{{\mathcal K}}$. The set of all bounded linear operators from the Hilbert space ${{\mathcal K}}$ to a Hilbert space ${{\mathcal L}}$ will be denoted by ${{\mathcal B}}({{\mathcal K}},{{\mathcal L}})$. If ${{\mathcal L}}={{\mathcal K}}$ the shorthand ${{\mathcal B}}({{\mathcal K}})$ will be used for this set. Let $K$ and $L$ be self-adjoint operators on a Hilbert space ${{\mathcal K}}$. We say $K< L$ (or, equivalently, $L> K$) if there is a number $\gamma>0$ such that $L-K>\gamma I$. The notation $\rho(T)$ will be used for the resolvent set of a closed operator $T$. After completing this work we learned that a result similar to the part (ii) of Theorem \[thm:1\] has been recently obtained within a different approach by A.V. Selin (private communication). Invariant graph subspaces and block diagonalization {#SecFact} =================================================== In this section we collect some results related to the invariant graph subspaces of a linear operator as well as to the closely related problem of block diagonalization of block operator matrices. First, recall the definition a graph subspace. Let ${{\mathcal K}}$ be a closed subspace of a Hilbert space ${{\mathcal N}}$ and $X\in{{\mathcal B}}({{\mathcal K}},{{\mathcal K}}^\perp)$. Denote by $P_{{\mathcal K}}$ and $P_{{{\mathcal K}}^\perp}$ the orthogonal projections in ${{\mathcal N}}$ onto the subspace ${{\mathcal K}}$ and orthogonal complement ${{\mathcal K}}^\perp$, respectively. The set $${{\mathcal G}}(X)=\{x\in{{\mathcal N}}\,|\, P_{{{\mathcal K}}^\perp}x=XP_{{{\mathcal K}}}x\}$$ is called the graph subspace associated with the operator $X$. For notational setup we assume the following \[Hmatr\] Let $H_0$ be a self-adjoint operator in a Hilbert space ${{\mathcal H}}$ and ${{\mathcal H}}_A\subset{{\mathcal H}}$ a reducing subspace of $H_0$. Assume that with respect to the decomposition $$\label{decom} {{\mathcal H}}={{\mathcal H}}_A\oplus{{\mathcal H}}_C \quad ({{\mathcal H}}_C={{\mathcal H}}\ominus{{\mathcal H}}_A)$$ the operator $H_0$ reads as the block diagonal operator matrix $$H_0=\mathop{\mathrm{diag}}(A,C)$$ with $A$ being a bounded self-adjoint operator in ${{\mathcal H}}_A$, $C$ a possibly unbounded self-adjoint operator in ${{\mathcal H}}_C$, and ${\mathrm{Dom}}(H_0)={{\mathcal H}}_A\oplus{\mathrm{Dom}}(C)$. Assume, in addition, that with respect to the decomposition the self-adjoint operator $H$ reads as $$\label{twochannel} H=\begin{pmatrix} A & B \\ B^* & C \end{pmatrix},\quad{\mathrm{Dom}}(H)={{\mathcal H}}_A\oplus{\mathrm{Dom}}(C),$$ where $B$ is a bounded operator from ${{\mathcal H}}_C$ to ${{\mathcal H}}_A$. Under Hypothesis \[Hmatr\], a bounded operator $X$ from ${{\mathcal H}}_A$ to ${{\mathcal H}}_C$ is said to be a solution to the Riccati equation if $\operatorname{\mathrm{Ran}}(X)\subset{\mathrm{Dom}}(C)$ and holds as an operator equality. The existence of a bounded solution to the Riccati equation is equivalent to a possibility of the block diagonalization of the operator matrix $H$ with respect to the decomposition ${{\mathcal H}}={{\mathcal G}}(X)\oplus{{\mathcal G}}(X)^\perp$. The precise statement is as follows (see Lemma 5.3 and Theorem 5.5 in [@AMM]; also cf. [@AdLT], [@Daughtry], [@KMMgeom], [@Motovilov:95]). \[thHi2\] Assume Hypothesis \[Hmatr\]. Then a bounded operator $X$ from ${{\mathcal H}}_A$ to ${{\mathcal H}}_C$ is a solution to the Riccati equation iff the graph ${{\mathcal G}}(X)$ of $X$ reduces the $2\times2$ block operator matrix $H$. Moreover, if $X\in{{\mathcal B}}({{\mathcal H}}_A,{{\mathcal H}}_C)$ is a solution to then: [(i)]{} The operator $V^{-1}HV$ with $$V=\begin{pmatrix} I & -X^* \\ X & I \end{pmatrix}$$ is block diagonal with respect to decomposition . Furthermore, $$V^{-1}HV=\begin{pmatrix} Z & 0 \\ 0 & \widehat{Z} \end{pmatrix},$$ where $Z=A+BX$ with ${\mathrm{Dom}}(Z)={{\mathcal H}}_A$ and $\widehat{Z}=C-B^*X^*$ with ${\mathrm{Dom}}(\widehat{Z})={\mathrm{Dom}}(C)$. [(ii)]{} The operator $$\label{HA} \Lambda =(I+X^*X)^{1/2}Z (I+X^*X)^{-1/2}$$ and possibly unbounded operator $$\label{HC} \widehat{\Lambda} = (I+XX^*)^{1/2}\widehat{Z} (I+XX^*)^{-1/2}$$ with ${\mathrm{Dom}}(\widehat \Lambda)=(I+XX^*)^{1/2}({\mathrm{Dom}}(C))$ are self-adjoint operators in ${{\mathcal H}}_A$ and ${{\mathcal H}}_C$, respectively. Theorem \[thHi2\] yields the following uniqueness result as a corollary. \[Xuniq\] Assume Hypothesis \[Hmatr\]. Suppose that $\Sigma$ and $\widehat{\Sigma}$ are disjoint Borel subsets of ${\mathbb R}$ such that ${{\ensuremath{\mathrm{dist}}}}(\Sigma,\widehat{\Sigma})>0$. Let ${{\mathcal X}}={{\mathcal X}}(A,B,\Sigma,\widehat{\Sigma})$ be the set of all bounded operators $X$ from ${{\mathcal H}}_A$ to ${{\mathcal H}}_C$ with the properties $$\begin{aligned} \label{sZA} &{\sigma}(A+BX)\subset\Sigma ,\\ \label{sZC} &{\sigma}(C-B^*X^*)\subset\widehat{\Sigma}, \quad {\mathrm{Dom}}(C-B^*X^*)={\mathrm{Dom}}(C).\end{aligned}$$ Then if $X,Y\in{{\mathcal X}}$ satisfy the Riccati equation , then $X=Y$. Suppose $X$ and $Y$ are two bounded solutions to both satisfying and . Then by Theorem \[thHi2\] the graphs of $X$ and $Y$ both coincide with the spectral subspace of the $2\times2$ operator matrix associated with the set $\Sigma$, and hence, $X=Y$. Under Hypothesis \[twochannel\] introduce the operator-valued Herglotz function $$\label{M1def} M(\lambda)=\lambda I -A+B(C-\lambda I)^{-1}B^*, \quad \lambda\in\rho(C).$$ By definition the spectrum ${\sigma}(M)$ of the function $M$ is the set of all $\lambda\in{\mathbb C}$ such that either the operator $M(\lambda)$ is not invertible or the inverse $[M(\lambda)]^{-1}$ is an unbounded operator. It is well known (see, e.g., [@MeMo99]) that the resolvent of the operator $H$ can be represented as the following $2\times2$ operator matrix $$\begin{aligned} \nonumber (H-\lambda I)^{-1}=&\begin{pmatrix} 0 & 0 \\ 0 & (C-\lambda I)^{-1} \end{pmatrix}\\ \label{ResH} &-\left(\begin{array}{c} I \\ -(C-\lambda I)^{-1}B^* \end{array}\right) M(\lambda)^{-1} \begin{pmatrix} I &\,\, -B(C-\lambda I)^{-1} \end{pmatrix}, \\ \nonumber & \quad \lambda\in\rho(H),\end{aligned}$$ where $M$ is the Herglotz function given by . Representation shows that for $\lambda\in\rho(C)$ the operator $H-\lambda I$ has a bounded inverse iff $M(\lambda)$ does, which means that $$\label{HM0} {\sigma}(H)\cap\rho(C)={\sigma}(M)\cap\rho(C).$$ We will also need the following general result (cf. [@AdLT Theorem 2.2], [@MenShk Proposition 2.4 and Theorem 2.5], and [@LMMT Theorems 4.4 and 5.1]). \[RicSol\] Assume Hypothesis \[Hmatr\] and suppose that the Herglotz function admits the factorization $$\label{M1fact} M(\lambda)=W(\lambda)(Z-\lambda I),\quad\lambda\in\Omega,$$ where $Z$ is a bounded operator in ${{\mathcal H}}_A$ such that ${\sigma}(Z)\cap{\sigma}(C)=\emptyset$, $\Omega$ is a domain in $\rho(C)$ such that ${\sigma}(Z)\subset\Omega$, and $W$ is a holomorphic ${{\mathcal B}}({{\mathcal H}}_A)$-valued function on $\Omega$, such that for any $\lambda\in{\sigma}(Z)$ the operator $W(\lambda)$ has a bounded inverse. Then ${\sigma}(Z)$ is an isolated part of the spectrum of the operator $H$ and the spectral subspace $\operatorname{\mathrm{Ran}}{\mathsf{E}}_H(\sigma(Z))$ of $H$ associated with the set ${\sigma}(Z)$ is the graph of the bounded operator $X$ from ${{\mathcal H}}_A$ to ${{\mathcal H}}_C$ given by $$\label{Xdeff} X=-\frac{1}{2\pi{\mathrm i}} \int_\Gamma d\lambda(C-\lambda I)^{-1}B^* (Z-\lambda I)^{-1}.$$ Here, $\Gamma$ is an arbitrary Jordan contour in $\rho(Z)\cap\rho(C)$ (maybe consisting of several simple Jordan contours) encircling ${\sigma}(Z)$ in the clockwise direction and having winding number $0$ with respect to the spectrum of $C$. Moreover, the operator $Z$ can be written in terms of the operator $X$ given by as $$\label{ZvX} Z=A+BX$$ and the factor $W(\lambda)$ admits analytic continuation to the whole resolvent set of the operator $C$ by the following formula $$\label{W} W(\lambda)=I-B(C-\lambda)^{-1}X, \quad \lambda\in\rho(C).$$ By hypothesis the function $W(\lambda)$ is holomorphic on an open set $\Omega\subset\rho(C)$ containing the closed subset ${\sigma}(Z)$ and the operator $W(\lambda)$ has a bounded inverse for any $\lambda\in{\sigma}(Z)$. Hence there is an open neighborhood ${\widetilde{\Omega}}$ of ${\sigma}(Z)$ in $\Omega$ where the operator $W(\lambda)$ is boundedly invertible, i.e. $W(\lambda)^{-1}\in{{\mathcal B}}({{\mathcal H}}_A)$ for any $\lambda\in{\widetilde{\Omega}}$. By $$\label{MZinv} M(\lambda)^{-1}=(Z-\lambda I)^{-1}W(\lambda)^{-1}, \quad \lambda\in {\widetilde{\Omega}}\setminus{\sigma}(Z).$$ Taking into account one infers that the spectrum of $H$ in ${\widetilde{\Omega}}$ coincides with that of $M$ and, thus, with that of $Z$, that is, $$\label{HUZ} {\sigma}(H)\cap{\widetilde{\Omega}}={\sigma}(Z),$$ Since ${\widetilde{\Omega}}$ is an open set, ${\sigma}(Z)$ is a closed set, and ${\sigma}(Z)\subset{\widetilde{\Omega}}$, one also concludes that ${\sigma}(Z)$ is isolated from the remaining part of the spectrum of $H$, i.e. $${{\ensuremath{\mathrm{dist}}}}({\sigma}(Z),{\sigma}(H)\setminus{\sigma}(Z))>0.$$ Using representation , for the spectral projection ${\mathsf E}_H\bigl({\sigma}(Z)\bigr)$ of the operator $H$ associated with the set ${\sigma}(Z)$ the Riesz integration yields $$\begin{aligned} {\mathsf E}_H\bigl({\sigma}(Z) \bigr)=& {\mathsf E}_H\bigl({\sigma}(Z)\bigr)=\frac{1}{2\pi{\mathrm{i}}}\int\limits_{\Gamma} d\lambda(H-\lambda I)^{-1}\\ =&-\frac{1}{2\pi{\mathrm{i}}}\int\limits_{\Gamma} d\lambda\begin{pmatrix} I \\ -(C-\lambda I)^{-1}B^* \end{pmatrix} M(\lambda)^{-1} \begin{pmatrix} I & \,\quad -B(C-\lambda I)^{-1} \end{pmatrix},\end{aligned}$$ where $\Gamma$ stands for an arbitrary Jordan contour (possibly consisting of several simple Jordan contours) in ${\widetilde{\Omega}}$ encircling the spectrum of $Z$ in the clockwise direction and having winding number $0$ with respect to the spectrum of $C$. Hence, $$\label{EH} {\mathsf E}_H\bigl({\sigma}(Z)\bigr)= \begin{pmatrix} E & G^*\\ G & F\\ \end{pmatrix},$$ where $$\begin{aligned} \label{Eint} E&=-\frac{1}{2\pi{\mathrm{i}}}\int_\Gamma d\lambda M(\lambda)^{-1}, \\ \nonumber F&=-\frac{1}{2\pi{\mathrm{i}}}\int_\Gamma d\lambda(C-\lambda I)^{-1}B^* M(\lambda)^{-1}B(C-\lambda I)^{-1},\end{aligned}$$ and $$\begin{aligned} \nonumber G&=\frac{1}{2\pi{\mathrm{i}}}\int_\Gamma d\lambda(C-\lambda I)^{-1}B^* M(\lambda)^{-1}\\ &=\frac{1}{2\pi{\mathrm{i}}}\int_\Gamma d\lambda (C-\lambda I)^{-1}B^* (Z-\lambda I)^{-1} W(\lambda)^{-1},\end{aligned}$$ using factorization formula . Since both the operator-valued functions $(C-\lambda)^{-1}$ and $W(\lambda)^{-1}$ are holomorphic in ${\widetilde{\Omega}}$, applying the Daletsky-Krein formula (see [@DK Lemma I.2.1]) we get $$\begin{aligned} \nonumber G=&\left[\frac{1}{2\pi{\mathrm{i}}} \int_\Gamma d\lambda (C-\lambda I)^{-1}B^*(Z-\lambda I)^{-1}\right]\\ \label{G} &\times \left[\frac{1}{2\pi{\mathrm{i}}}\int_\Gamma d\lambda (Z-\lambda I)^{-1}W(\lambda)^{-1}\right].\end{aligned}$$ Hence, combining and proves the representation $$\label{XE} G=XE,$$ where $X$ (the first factor on the r.h.s. part of ) is given by $$\label{XinT} X=-\frac{1}{2\pi{\mathrm i}} \int_\Gamma d\lambda(C-\lambda I)^{-1}B^* (Z-\lambda I)^{-1}.$$ In an analogous way one also proves that $$\label{XEX} F=XEX^*.$$ Clearly, for $\lambda\in\Gamma$ we have $\operatorname{\mathrm{Ran}}(C-\lambda I)^{-1}\subset{\mathrm{Dom}}(C)$ and hence $$\operatorname{\mathrm{Ran}}(X)\subset{\mathrm{Dom}}(C),$$ which immediately follows from . Multiplying both sides of by $B$ from the left yields $$\label{BX1} BX=-\frac{1}{2\pi{\mathrm i}}\int_\Gamma d\lambda B(C-\lambda I)^{-1}B^*(Z-\lambda I)^{-1}.$$ Meanwhile, $$B(C-\lambda I)^{-1}B^*=A-\lambda I+M(\lambda)= A-\lambda I +W(\lambda)(Z-\lambda I), \quad \lambda\in\Gamma,$$ and, hence, using $$\begin{aligned} \label{BXint} BX&=-\frac{1}{2\pi{\mathrm i}}\int_\Gamma d\lambda \left[W(\lambda)+A(Z-\lambda I)^{-1}-\lambda(Z-\lambda I)^{-1}\right].\end{aligned}$$ The function $W(\lambda)$ is holomorphic in the domain bounded by the contour $\Gamma$ and, thus, the first term in the integrand on the r.h.s. of gives no contribution. Since $\Gamma$ encircles the spectrum of $Z$, the integration of the remaining two terms in can be performed explicitly using the operator version of the residue theorem, which yields $BX=-A+Z$ and hence $$\label{Xgr} Z=A+BX$$ proving . Since the spectra of the operators $C$ and $Z$ are disjoint and $Z$ is a bounded operator, it is straightforward to show (see, e.g., [@D53] or [@R56]) that the operator $X$ given by is the unique solution to the operator Sylvester equation $$XZ-CX=B^*,$$ which by proves that $X$ solves the Riccati equation . Now applying Theorem \[thHi2\] and using , , and we arrive at the series of equalities $$\begin{aligned} \nonumber I&= \frac{1}{2\pi{\mathrm{i}}}\int_\Gamma d\lambda(Z-\lambda I)^{-1}\\ \nonumber &=\frac{1}{2\pi{\mathrm{i}}}\int_\Gamma d\lambda{{\mathsf P}}_{A} \begin{pmatrix} (Z-\lambda I)^{-1} & 0 \\ 0 & (\widehat{Z}-\lambda I)^{-1} \end{pmatrix} {{\mathsf P}}_{A}^*\\ \nonumber &=\frac{1}{2\pi{\mathrm{i}}}\int_\Gamma d\lambda {{\mathsf P}}_{A} V^{-1}(H-\lambda I)^{-1}V {{\mathsf P}}_{A}^*\\ \nonumber &={{\mathsf P}}_{A} V^{-1} {\mathsf E}_H\bigl({\sigma}(Z)\bigr) V{{\mathsf P}}_{A}^*\\ \label{IXX} &=E(I+X^*X),\end{aligned}$$ where $\widehat{Z}=C-B^*X^*$ with ${\mathrm{Dom}}(\widehat{Z})={\mathrm{Dom}}(C)$, $$V=\begin{pmatrix} I & -X^* \\ X & I \end{pmatrix},$$ and ${{\mathsf P}}_A$ is the canonical projection from ${{\mathcal H}}$ to ${{\mathcal H}}_A$ (i.e., ${\mathsf P}_A(f_A\oplus f_C)=f_A$ for $f_A\in{{\mathcal H}}_A$ and $f_C\in{{\mathcal H}}_C$). Combining , and one concludes that the spectral projection ${\mathsf E}_{H}({\sigma}(Z))$ admits the representation $$\label{sEQ} {\mathsf E}_{H}({\sigma}(Z))= \begin{pmatrix} (I+X^*X)^{-1} & (I+X^*X)^{-1}X^* \\ X(I+X^*X)^{-1} & X(I+X^*X)^{-1}X^* \end{pmatrix}.$$ Note that the contour $\Gamma$ in can be replaced by an arbitrary Jordan contour in $\rho(Z)\cap\rho(C)$ (possibly consisting of several simple Jordan contours) encircling the set ${\sigma}(Z)$ in the clockwise direction and having winding number $0$ with respect to ${\sigma}(C)$. Then observing that the r.h.s. of is nothing but the orthogonal projection in ${{\mathcal H}}={{\mathcal H}}_A\oplus{{\mathcal H}}_C$ onto the graph of the operator $X$ proves that $\operatorname{\mathrm{Ran}}{\mathsf{E}}_H({\sigma}(Z))={{\mathcal G}}(X)$. We conclude with the proof of the representation . First, by we notice that $$\begin{aligned} [I-B(C-\lambda)^{-1}X](Z-\lambda)&=[I-B(C-\lambda)^{-1}X](A+BX-\lambda)\\ &=A+BX-\lambda -B(C-\lambda)^{-1}(XA+XBX-\lambda X), \\ &\quad \lambda\in\rho(C).\end{aligned}$$ Since $X$ solves the Riccati equation one infers that $XA+XBX=B^*+CX$ which implies $$\begin{aligned} \nonumber [I-B(C-\lambda)^{-1}X](Z-\lambda)&=M(\lambda)+BX-B(C-\lambda)^{-1}(C-\lambda)X\\ \label{dopM} &=M(\lambda), \quad \lambda\in\rho(C).\end{aligned}$$ Hence combining and yields $$W(\lambda)=M(\lambda)(Z-\lambda)^{-1}=I-B(C-\lambda)^{-1}X, \quad \lambda\in\Omega\setminus{\sigma}(Z).$$ Then the analytic continuation completes the proof. \[RemAnal\] By analytic continuation of both parts of one concludes that the factorization formula holds for any $\lambda\in\rho(C)$ with $W(\lambda)$ given by . Moreover, the representation $$[W(\lambda)]^{-1}=(Z-\lambda I)[M(\lambda)]^{-1}, \quad \lambda\in{\widetilde{\Omega}}\setminus{\sigma}(M)={\widetilde{\Omega}}\setminus{\sigma}(Z),$$ and the fact that ${\sigma}(M)\subset{\sigma}(H)$ imply that $[W(\lambda)]^{-1}$ admits analytic continuation as a ${{\mathcal B}}({{\mathcal H}}_A)$-valued holomorphic function to the domain $\rho(H)\cup{\sigma}(Z)$. An existence result. Proof of Theorem 1 (i) {#SecExist} =========================================== As we have already mentioned in Introduction our main technical tool in proving the solvability of the Riccati equation is the Virozub-Matsaev factorization theorem [@ViMt] (also see [@MrMt]). For convenience of the reader we reproduce the corresponding statement following Propositions 1.1 and 1.2 in [@MenShk]. \[Matsa\] Let ${{\mathcal K}}$ be a Hilbert space and $F(\lambda)$ a holomorphic ${{\mathcal B}}({{\mathcal K}})$-valued function on a simply connected domain $\Omega\subset{\mathbb C}$. Assume that $\Omega$ includes an interval $[a,b]\subset{\mathbb R}$ such that $$F(a)< 0, \quad F(b)> 0,\quad\text{and}\quad\frac{d}{d\lambda}F(\lambda)> 0\quad \text{for all}\quad\lambda\in[a,b].$$ Then there exist a domain $\widetilde{\Omega}\subset\Omega$ containing $[a,b]$ and a unique bounded operator $Z$ on ${{\mathcal K}}$ with ${\sigma}(Z)\subset(a,b)$ such that $F(\lambda)$ admits the factorization $$F(\lambda)=G(\lambda)(Z-\lambda I), \quad \lambda\in\widetilde{\Omega},$$ where $G(\lambda)$ is a holomorphic operator-valued function on $\widetilde{\Omega}$ whose values are bounded and boundedly invertible operators in ${{\mathcal K}}$, that is, $$G(\lambda)\in{{\mathcal B}}({{\mathcal K}})\quad\text{and}\quad [G(\lambda)]^{-1}\in{{\mathcal B}}({{\mathcal K}})\quad \quad\lambda\in\widetilde{\Omega}.$$ Based on Theorem \[Matsa\] we obtain the following factorization result, the cornerstone for our further considerations. \[Zexist\] Assume Hypothesis \[Hmatr\]. Assume, in addition, that $C$ has a finite spectral gap $\Delta =(\alpha,\beta)$, $\alpha<\beta$, the spectrum of $A$ lies in $\Delta$, i.e., ${\sigma}(A)\subset \Delta$, and $$\|B\|<\sqrt{d|\Delta|},$$ where $$d={{\ensuremath{\mathrm{dist}}}}\bigl({\sigma}(A),{\sigma}(C)\bigr).$$ Then there is a unique operator $Z\in{{\mathcal B}}({{\mathcal H}}_A)$ with ${\sigma}(Z)\subset\Delta$ such that the operator-valued function $M(\lambda)$ given by admits the factorization $$\label{fact} M(\lambda)=W(\lambda)(Z-\lambda I),\quad\lambda\in\rho(C),$$ with a holomorphic ${{\mathcal B}}({{\mathcal H}}_A)$-valued function $W$ on $\rho(C)$. Moreover, for any $$\lambda\in\bigl({\mathbb C}\setminus {\sigma}(M)\bigr)\cup\Delta$$ the operator $W(\lambda)$ has a bounded inverse and $$\label{inclusion} {\sigma}(Z)={\sigma}(H)\cap\Delta \subset[\inf{\sigma}(A)-\delta_-,\sup{\sigma}(A)+\delta_+],$$ where $$\begin{aligned} \label{dBm} \delta_-&=\|B\|\tan\left(\frac{1}{2}\arctan\frac{2\|B\|} {\beta-\inf{\sigma}(A)}\right)<\inf{\sigma}(A)-\alpha ,\\ \label{dBp} \delta_+&=\|B\|\tan\left(\frac{1}{2}\arctan\frac{2\|B\|} {\sup{\sigma}(A)-\alpha }\right)<\beta-\sup{\sigma}(A).\end{aligned}$$ By the spectral theorem $$B(C-\lambda I)^{-1}B^*=\int\limits_{{\mathbb{R}}\setminus \Delta} B{\mathsf{E}}_C(d\mu)B^*\frac{1}{(\mu-\lambda)},\quad\lambda\in\rho(C),$$ where ${\mathsf{E}}_C(\mu)$ stands for the spectral family of the self-adjoint operator $C$. Hence $$\label{DerE} \displaystyle\frac{d}{d\lambda}M(\lambda)=I + \int\limits_{{\mathbb{R}}\setminus \Delta} B{\mathsf{E}}_C(d\mu)B^*\frac{1}{(\mu-\lambda)^2}, \qquad\lambda\in\rho(C).$$ For $\lambda\in\Delta$ the integral in is a non-negative operator. Therefore, the derivative of $M(\lambda)$ is a strictly positive operator: $$\label{Der} \displaystyle\frac{d}{d\lambda}M(\lambda)\geq I>0, \quad \lambda\in \Delta.$$ Next we estimate the quadratic form of $M(\lambda)$. Let $f\in{{\mathcal H}}_A$, $\|f\|=1$. Then $$\begin{aligned} \label{Fflam} {\langle}M(\lambda)f,f{\rangle}&= \lambda-{\langle}A f,f{\rangle}+{\langle}(C-\lambda)^{-1}B^*f,B^*f{\rangle}\nonumber\\ &=\lambda-{\langle}A f,f{\rangle}+\int\limits_{-\infty}^{\alpha } \frac{1}{\mu-\lambda}{\langle}{\mathsf{E}}_C(d\mu)B^*f,B^*f{\rangle}\\ &\quad+\int\limits_{\beta}^{+\infty} \frac{1}{\mu-\lambda}{\langle}{\mathsf{E}}_C(d\mu)B^*f,B^*f{\rangle}, \quad\lambda\in\rho(C). \nonumber\end{aligned}$$ Since for $\lambda\in \Delta$ the integral in the second line of is non-positive and the one in the third line is non-negative, one obtains the two-sided estimate $$\bigg ( \lambda-\sup {\sigma}(A)-\frac{\|B\|^2}{\lambda-\alpha }\bigg )I \le M(\lambda) \le \bigg ( \lambda-\inf {\sigma}(A)-\frac{\|B\|^2}{\lambda-\beta} \bigg )I, \quad\lambda\in \Delta.$$ Now, a simple calculation shows that $$\label{Ier} M(\lambda)< 0 \quad \mbox{for}\quad \lambda\in\bigl(\alpha ,\inf{\sigma}(A)-\delta_-\bigr)$$ and $$\label{Iel} M(\lambda)> 0 \quad \mbox{for}\quad \lambda\in\bigl(\sup{\sigma}(A)+\delta_+,\beta\bigr),$$ where $\delta_-$ and $\delta_+$ are given by and , respectively. Thus, the function $F(\lambda)=M(\lambda)$ satisfies assumptions of Theorem \[Matsa\] for any $a\in(\alpha ,\inf{\sigma}(A)-\delta_-)$ and any $b\in(\sup{\sigma}(A)+\delta_+, \beta)$, proving the existence of the unique bounded operator $Z\in{{\mathcal B}}({{\mathcal H}}_A)$ such that and hold taking into account . It follows from Theorem \[Matsa\] that the factor $W(\lambda)$ in has a bounded inverse in a complex neighborhood $U\subset{\mathbb C}$ of the interval $[\inf{\sigma}(A)-\delta_-,\sup{\sigma}(A)+\delta_+]$. Moreover, the operator $W(\lambda)$ has a bounded inverse for any $\lambda\in\bigl({\mathbb C}\setminus {\sigma}(M)\bigr)\cup\Delta$ by Remark \[RemAnal\]. The proof is complete. We are ready to prove the part (i) of Theorem 1. By Theorem \[Zexist\] the Herglotz function admits the factorization with $W(\lambda)$ and $Z$ satisfying hypothesis of Theorem \[RicSol\]. Therefore, the Riccati equation has a bounded solution $X$ given by . Theorem \[RicSol\] also shows that the graph ${{\mathcal G}}(X)$ of the operator $X$ coincides with the spectral subspace $\operatorname{\mathrm{Ran}}{\mathsf E}_H\bigl({\sigma}(Z)\bigr)$ for the block operator matrix $H$ given by . By the subspace $\operatorname{\mathrm{Ran}}{\mathsf E}_H\bigl({\sigma}(Z)\bigr)$ coincides with $\operatorname{\mathrm{Ran}}{\mathsf E}_H\bigl(\Delta\bigr)$, proving that $$\label{EHG} \operatorname{\mathrm{Ran}}{\mathsf E}_H\bigl(\Delta\bigr)={{\mathcal G}}(X).$$ To prove that $X$ possesses the properties we proceed as follows. Let $$V=\left(\begin{array}{cc} I & -X^* \\ X & I \end{array}\right), \quad S=(I+X^*X)^{1/2}, \quad\text{and}\quad \widehat{S}=(I+XX^*)^{1/2}.$$ From Theorem \[thHi2\] it follows that $$\label{EHZZ} {\mathsf{E}}_H(\Delta)=V\left(\begin{array}{cc} S^{-1}{\mathsf{E}}_\Lambda(\Delta)S & 0 \\ 0 & \widehat{S}^{-1}{\mathsf{E}}_{\widehat{\Lambda}}(\Delta)\widehat{S} \end{array}\right)V^{-1},$$ where $\Lambda$ and $\widehat{\Lambda}$ are the self-adjoint operators defined by and , respectively; ${\mathsf{E}}_\Lambda(\Delta)$ and ${\mathsf{E}}_{\widehat{\Lambda}}(\Delta)$ denote the spectral projections for $\Lambda$ and $\widehat{\Lambda}$ associated with the interval $\Delta$. Since $\Lambda$ is similar to $Z$ ($\Lambda=SZS^{-1}$) and by Theorem \[Zexist\] the inclusion ${\sigma}(Z)\subset\Delta$ holds, one concludes that ${\sigma}(\Lambda)\subset\Delta$. Hence, ${\mathsf{E}}_\Lambda(\Delta)=I$ and then implies the equality $$\label{EHZZ1} {\mathsf{E}}_H(\Delta)= V \left( \begin{array}{cc} I & 0 \\ 0 & 0 \end{array}\right) V^{-1} + V\left(\begin{array}{cc} 0 & 0 \\ 0 & \widehat{S}^{-1}{\mathsf{E}}_{\widehat{\Lambda}}(\Delta)\widehat{S} \end{array}\right)V^{-1}.$$ The first summand on the r.h.s. of $$V \left( \begin{array}{cc} I & 0 \\ 0 & 0 \end{array}\right) V^{-1}=\begin{pmatrix} (I+X^*X)^{-1} & (I+X^*X)^{-1}X^* \\ X(I+X^*X)^{-1} & X(I+X^*X)^{-1}X^* \end{pmatrix}$$ coincides with the orthogonal projection onto the graph subspace ${{\mathcal G}}(X)$ and, hence, by it equals ${\mathsf E}_H\bigl(\Delta\bigr)$. Thus, the second summand on the r.h.s. of vanishes, $$V\left(\begin{array}{cc} 0 & 0 \\ 0 & \widehat{S}^{-1}{\mathsf{E}}_{\widehat{\Lambda}}(\Delta)\widehat{S} \end{array}\right)V^{-1}=0,$$ which means that ${\mathsf{E}}_{\widehat{\Lambda}}(\Delta)=0$ and, therefore, ${\sigma}(\widehat{\Lambda})\cap\Delta=\emptyset$. By it is now straightforward to see that $X$ satisfies . It remains to conclude that by Corollary \[Xuniq\] the operator $X$ is the unique bounded solution to satisfying which completes the proof of Theorem 1 (i). \[coptim\] Obviously, under hypothesis of Theorem \[thm:1\] the inequality $|\Delta|\geq 2d$ holds, with the equality sign occurring only if the spectrum of the operator $A$ is a one point set. Hence, the condition $\|B\|<\sqrt{2}d$, which is stronger than , implies the existence of a bounded solution to the Riccati equation . This means that in the case where $C$ has a finite spectral gap $\Delta$ and ${\sigma}(A)\subset\Delta$ the best possible constant $c_{\mathrm{best}}$ in condition ensuring the solvability of the Riccati equation satisfies the inequality $c_{\mathrm{best}}\geq \sqrt{2}$. On the other hand in [@AMM Lemma 3.11 and Remark 3.12] it is shown that $c_{\mathrm{best}}\leq \sqrt{2}$. Thus, $c=\sqrt{2}$ is best possible in inequality which ensures the solvability of the Riccati equation under the additional hypothesis that ${{\ensuremath{\mathrm{dist}}}}(\sigma(A), \sigma (C))>0$ and that the spectrum of $ A$ lies in a spectral gap of the operator $C$. The $\text{tan}{\,\Theta}$ Theorem {#SecTan} ================================== We start out by recalling a concept of the operator angle between two subspaces in a Hilbert space going back to the works by Friedrichs [@Friedrichs], M. Krein, Kransnoselsky, and Milman [@Krein:Krasnoselsky], [@Krein:Krasnoselsky:Milman], Halmos [@Halmos:69], and Davis and Kahan [@Davis:Kahan]. A comprehensive discussion of this concept can be found, e.g., in [@KMMgeom]. Given a closed subspace ${{\mathcal Q}}$ of the Hilbert space ${{\mathcal H}}={{\mathcal H}}_A\oplus{{\mathcal H}}_C$, introduce the operator angle $\Theta$ between the subspaces ${{\mathcal H}}_A\oplus\{0\}$ and ${{\mathcal Q}}$ by $$\label{sinT} \Theta=\arcsin\sqrt{I_{{{\mathcal H}}_A}-{{\mathsf P}}_A{\mathsf Q}{{\mathsf P}}_A^*},$$ where ${{\mathsf P}}_A$ is the canonical projection from ${{\mathcal H}}$ onto ${{\mathcal H}}_A$ and ${\mathsf Q}$ the orthogonal projection in ${{\mathcal H}}$ onto ${{\mathcal Q}}$. If the subspace ${{\mathcal Q}}$ is the graph ${{\mathcal G}}(X)$ of a bounded operator $X$ from ${{\mathcal H}}_A$ to ${{\mathcal H}}_C$, then (see [@KMMgeom]; cf. [@Davis:Kahan] and [@Halmos:69]) $$\label{tan-X} \tan\Theta=\sqrt{X^*X}$$ and $$\label{sinTPQ} \|\sin\Theta\|=\|{\mathsf Q}-{\mathsf P}\|,$$ where ${\mathsf P}={{\mathsf P}_A}^{\!\!\!*}{\mathsf P}_A$ denotes the orthogonal projection in ${{\mathcal H}}$ onto the subspace ${{\mathcal H}}_A\oplus\{0\}$. Note that the common definition of the operator angle (see, e.g., [@KMMgeom]) slightly differs from . Usually, the operator angle is defined as the restriction of the operator onto the maximal subspace of ${{\mathcal H}}_A$ where it has a trivial kernel. Clearly, the difference in these two definitions does not effect the value of the norm $\|\tan\Theta\|$. Now we are ready to prove the second principal result of the paper, a generalization of the Davis-Kahan $\tan\Theta$ Theorem [@Davis:Kahan]. By hypothesis the Riccati equation has a bounded solution $X$. Then, by Theorem \[thHi2\] the operator $Z=A+BX$ is similar to the bounded self-adjoint operator $\Lambda$ given by and hence $$\label{spZA} {\sigma}(Z)={\sigma}(\Lambda)\subset{\mathbb R}\,.$$ The Riccati equation can be rewritten in the form $$\label{SylHelp} X(Z-\gamma I)-(C-\gamma I)X=B^*,$$ where $$\label{lamcen1} \gamma=\frac{1}{2}\bigl(\sup{\sigma}(Z)+\inf{\sigma}(Z)\bigr) =\frac{1}{2}\bigl(\sup{\sigma}(\Lambda)+\inf{\sigma}(\Lambda)\bigr).$$ By hypothesis ${{\ensuremath{\mathrm{dist}}}}\bigl({\sigma}(H|_{{{\mathcal G}}(X)}),{\sigma}(C)\bigr)=\delta>0$ and then Theorem \[thHi2\] implies that $${\sigma}(H|_{{{\mathcal G}}(X)})={\sigma}(Z)={\sigma}(\Lambda),$$ which by hypothesis proves the inclusion $$\label{spC} {\sigma}(C)\subset\bigl(-\infty,\inf{\sigma}(\Lambda)-\delta\bigr]\cup \bigl[\sup{\sigma}(\Lambda)+\delta,\infty\bigr).$$ Hence, combining , , and proves that $\gamma\in\rho(C)$ and $$\label{CmA} \|(C-\gamma I)^{-1}\|=\frac{1}{\|\Lambda-\gamma I\|+\delta}.$$ Multiplying both sides of by $(C-\gamma I)^{-1}$ from the left one gets the representation $$\label{SylZC} X=(C-\gamma I)^{-1}\bigl(X(Z-\gamma I)-B^*\bigr).$$ Using Theorem \[thHi2\] (ii) one obtains the estimate $$\begin{aligned} \nonumber \|X(Z-\gamma I)\|=&\|X(I+X^*X)^{-1/2}(\Lambda-\gamma I) (I+X^*X)^{1/2}\| \\ \label{NX1} &\leq \|X(I+X^*X)^{-1/2}\|\, \|\Lambda-\gamma I\| (1+\|X\|^2)^{1/2}.\end{aligned}$$ Clearly, by the spectral theorem $$\|X(I+X^*X)^{-1/2}\|=\sqrt{\|X^*X(I+X^*X)^{-1}\|} =\frac{\|X\|}{(1+\|X\|^2)^{1/2}}.$$ Hence implies the estimate $$\|X(Z-\gamma I)\|\leq \|X\|\, \|(\Lambda-\gamma I)\|,$$ which together with proves the norm inequality $$\label{XN2} \|X\|\leq \|(C-\gamma I)^{-1}\|\bigl(\|\Lambda-\gamma I\|\|X\|+\|B\|\bigr).$$ Solving inequality with respect to $\|X\|$ and taking into account proves the norm estimate . Finally, since $\bigl\|\sqrt{X^*X}\bigr\|=\|X\|$, by the definition of the operator angle one gets $$\|\tan\Theta\|=\|X\|.$$ Hence, is equivalent to . It is natural to ask whether estimate remains to hold if one replaces the distance $\delta={{\ensuremath{\mathrm{dist}}}}(\sigma(H|_{{{\mathcal G}}(X)}),\sigma(C))$ by $$\widehat{\delta}={{\ensuremath{\mathrm{dist}}}}\bigl({\sigma}(H|_{{{\mathcal G}}(X)^\perp},{\sigma}(A)\bigr) ={{\ensuremath{\mathrm{dist}}}}\bigl({\sigma}(C-B^*X^*),{\sigma}(A)\bigr).$$ The answer is “No”: Example 6.1 in [@Davis:Kahan] shows that the inequality $\widehat{\delta}\|\tan\Theta\|\leq\|B\|$ fails to hold in general. Norm Estimates of Solutions. Proof of Theorem 1 (ii) {#SecEst} ==================================================== The existence result of Theorem 1 (i) by itself gives no clue for estimating the norm of the corresponding solution $X$ to the Riccati equation. To the contrary, Theorem 2 provides such an estimate whenever some additional information on the spectrum location of the “perturbed” operator matrix $H=\begin{pmatrix} A& B\\B^*&C\end{pmatrix}$ is available. In turn, the bounds on the spectrum of the part $H|_{{{\mathcal G}}(X)}$ of the operator matrix $H$ associated with the reducing subspace ${{\mathcal G}}(X)$ (needed to satisfy the hypotheses of Theorem \[thm:2\]) can be obtained by combining the results of Theorems \[RicSol\] and \[Zexist\]. As a result of performing this program one gets an *a priori* estimate on the norm of the solution $X$. \[TanFin\] Assume hypothesis of Theorem \[thm:1\] (i) with $\Delta=(\alpha,\beta)$, $\alpha<\beta$. Let $X$ be the unique solution to the Riccati equation referred to in Theorem \[thm:1\]. Then $$\label{XestFin} \|X\|\leq\frac{\|B\|}{\widetilde{\delta}},$$ where $$\label{dtilde} \widetilde{\delta}=\min\{\inf{\sigma}(A)-\alpha-\delta_-,\,\, \beta-\sup{\sigma}(A)-\delta_+\}>0$$ with $\delta_\pm$ given by and . Under hypothesis of Theorem \[thm:1\] (i) with $\Delta=(\alpha,\beta)$, $\alpha<\beta$ one can apply Theorem \[Zexist\] and using the same strategy of proof as that of Theorem \[thm:1\] (i) one concludes that $$Z=A+BX,$$ where $Z$ is the unique operator with ${\sigma}(Z)\subset\Delta$ referred to in Theorem \[Zexist\] and $X$ is the unique solution to the Riccati equation referred to in Theorem \[thm:1\] (i). Hence, by Theorem \[Zexist\] $$\label{2st} {\sigma}(A+BX)\subset[\inf{\sigma}(A)-\delta_-,\sup{\sigma}(A)+\delta_+].$$ By hypothesis (of Theorem \[thm:1\] (i)) $$\label{3st} {\sigma}(C)\subset\bigl(-\infty,\alpha \bigr]\cup \bigl[\beta,\infty\bigr),$$ which together with yields the inequality $${{\ensuremath{\mathrm{dist}}}}\bigl({\sigma}(A+BX),{\sigma}(C)\bigr)\geq\widetilde{\delta}.$$ By Theorem \[RicSol\] one observes that ${\sigma}(H)|_{{{\mathcal G}}(X)}={\sigma}(Z)$ which proves using Theorem \[thm:2\]. The proof of Theorem \[thm:1\] (ii) needs complementary considerations. Our reasoning is based on the celebrated Davis-Kahan $\tan 2 \Theta$-Theorem [@Davis:Kahan]. For convenience of the reader we reproduce the corresponding result (cf. Theorem 2.4 (iii) and Remark 2.8 in [@KMMalpha] and Corollary 6.4 in [@KMMgeom]). \[DaKa\] Assume Hypothesis \[Hmatr\]. Suppose that the operator $C$ is bounded and $\sup{\sigma}(A)<\inf{\sigma}(C)$. Then the open interval $(\sup{\sigma}(A),\inf{\sigma}(C))$ is a spectral gap of the operator $H$ and the spectral subspace ${\mathsf{E}}_H\bigl((-\infty,\sup{\sigma}(A)]\bigr)$ is the graph of a contractive operator $X$ from ${{\mathcal H}}_A$ to ${{\mathcal H}}_C$. Moreover, the operator $X$ is the unique contractive solution to the Riccati equation and its norm satisfies the estimate $$\|X\|\le \tan \left(\frac{1}{2}\arctan \frac{2\|B\|}{d}\right) < 1,$$ where $$d={{\ensuremath{\mathrm{dist}}}}({\sigma}(A),{\sigma}(C)).$$ Now we are prepared to prove Theorem 1 (ii). Let $X$ be the solution to the Riccati equation referred to in Theorem \[thm:1\] (i) and thus the spectral subspace of the operator $H$ associated with the interval $\Delta$ is the graph ${{\mathcal G}}(X)$ of the operator $X$. Then ${{\mathcal G}}(X)$ is also the spectral subspace of the operator $(H-\gamma I)^2$ associated with the interval $[0,|\Delta|^2/4)$ where $\gamma$ is the center of the interval $\Delta$, that is, $$\label{kon} {{\mathcal G}}(X)=\operatorname{\mathrm{Ran}}\big ({\mathsf{E}}_H(\Delta)\big )=\operatorname{\mathrm{Ran}}\big ({\mathsf{E}}_{(H-\gamma I)^2} [0,{|\Delta |^2}/{4})\big ).$$ By inspection one obtains that with respect to the decomposition ${{\mathcal H}}={{\mathcal H}}_A\oplus{{\mathcal H}}_A$ the non-negative operator $(H-\gamma I)^2$ reads $$\label{kvadrat} (H-\gamma I)^2=\begin{pmatrix} \widehat{A} & \widehat{B}\ \\ \widehat{B}^* & \widehat{C} \end{pmatrix},$$ where $\widehat{A}=(A-\gamma I)^2+BB^*$, $\widehat{B}=AB+BC$, and $\widehat{C}=(C-\gamma I)^2+B^*B$. The hypothesis that the spectrum of $C$ lies in ${\mathbb R}\setminus\Delta$ implies the operator inequality $$\widehat{C}\geq \frac{|\Delta|^2}{4}I+BB^*\geq \frac{|\Delta|^2}{4}\,I.$$ The hypothesis ${\sigma}(A)\subset\Delta$ yields $$\label{puta} 0\le\widehat{A}\leq \left(\frac{|\Delta|}{2}-d\right)^2I+BB^*\leq \left[\left(\frac{|\Delta|}{2}-d\right)^2+\|B\|^2\right]I,$$ taking into account that ${{\ensuremath{\mathrm{dist}}}}({\sigma}(A),{\sigma}(C))=d$. Hence, under hypothesis one concludes that $$\label{dura} {{\ensuremath{\mathrm{dist}}}}\bigl({\sigma}(\widehat{A}),{\sigma}(\widehat{C})\bigr)\geq d(|\Delta|-d)-\|B\|^2>0$$ and that the spectra ${\sigma}(\widehat{A})$ and ${\sigma}(\widehat{C})$ of the entries $\widehat{A}$ and $\widehat{C}$ are subordinated, that is, $\sup{\sigma}(\widehat{A})<\inf{\sigma}(\widehat{C})$. By Theorem \[DaKa\] (cf. Theorem 2.1 in [@AL95]) one infers that the interval $\bigl(\sup{\sigma}(\widehat{A}),\inf{\sigma}(\widehat{C})\bigr)$ lies in the resolvent set of the operator $(H-\gamma I)^2$. In particular, the following inclusion holds $$\label{gap2} \bigl((|\Delta|/2-d)^2+\|B\|^2,|\Delta|^2/4\bigr)\subset \rho\bigl((H-\gamma I)^2\bigr).$$ Therefore, the spectral subspaces of the operator $(H-\gamma I)^2$ associated with the intervals $[0,|\Delta|^2/4)$ and $[0,|(\Delta|/2-d)^2+\|B\|^2]$, respectively, coincide, that is, $$\label{ruba} \begin{split} {{\mathcal G}}(X) &=\operatorname{\mathrm{Ran}}\bigl({\mathsf{E}}_{(H-\gamma I)^2}\bigl([0,|\Delta|^2/4)\bigr)\bigr)\\ & =\operatorname{\mathrm{Ran}}\bigl({\mathsf{E}}_{(H-\gamma I)^2}([0,(|\Delta|/2-d)^2+\|B\|^2])\bigr). \end{split}$$ From one concludes that the operator matrix $(H-\gamma I)^2$ is an off-diagonal perturbation of the matrix ${{\ensuremath{\mathrm{diag}}}}\{\widehat{A}, \widehat{C}\}$ diagonal with respect to the decomposition ${{\mathcal H}}={{\mathcal H}}_A\oplus {{\mathcal H}}_C$. Applying again Theorem \[DaKa\] proves that the spectral subspace of the operator $(H-\gamma I)^2$ associated with the interval $[0,|(\Delta|/2-d)^2+\|B\|^2]$ is the graph of a contraction $\widehat{X}$ satisfying the norm-estimate . By and this subspace coincides with ${{\mathcal G}}(X)$ and, therefore, $\widehat{X}=X$ and hence holds for $X$, completing the proof. \[controptim\] Condition ensuring the strict contractivity of the solution $X$ is sharp. This can be seen as follows. Let ${{\mathcal H}}_A={\mathbb C}$, ${{\mathcal H}}_C={\mathbb C}^2$, $A=0$, $$C=\left(\begin{array}{rr} -d & 0 \\ 0 & d \end{array}\right),\quad d>0,$$ and $$B=\left(\frac{b}{\sqrt{2}},\frac{b}{\sqrt{2}}\right), \quad b\in{\mathbb R}.$$ By inspection one proves that the $2\times1$ matrix $$\label{Xexem} X=\left(\begin{array}{r} -\frac{b}{\sqrt{2}\,d} \\[0.5em] \frac{b}{\sqrt{2}\,d} \end{array}\right)$$ solves the Riccati equation $$XA-CX+XBX=B^*.$$ Moreover, $$A+BX=0$$ and $X$ possesses the properties . Clearly, $\|B\|=b$ and $ \|X\|=\frac{b}{d}=\frac{\|B\|}{d}. $ For $b\in [d, \sqrt{2}d)$ hypothesis is satisfied with $\Delta=(-d, d)$, condition fails to hold, and $\|X\|\ge 1$, that is, estimate is sharp. [00]{} V. M. Adamjan and H. Langer, *Spectral properties of a class of rational operator valued functions*, J. Operator Theory **33** (1995), 259 – 277. V. Adamjan, H. Langer, and C. Tretter, *Existence and uniqueness of contractive solutions of some Riccati equations*, J. Funct. Anal. **179** (2001), 448 – 473. S.Albeverio, K.A.Makarov, and A.K.Motovilov, *Graph subspaces and the spectral shift function*, Canad. J. Math. (to appear); arXiv:math.SP/0105142. Yu. L. Daleckiĭ, On the asymptotic solution of a vector differential equation, *Dokl. Akad. Nauk SSSR* **92** (1953), 881 – 884 (Russian). Ju. L. Daleckiĭ and M. G. Kreĭn, *Stability of Solutions of Differential Equations in Banach Spaces*, Translations of Mathematical Monographs, Vol.43, AMS, Providence, Rhode Island, 1974. 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M. G. Krein and M. A. Krasnoselsky, *Fundamental theorems about extensions of Hermite operators and some applications to the theory of orthogonal polynomials and to the moment problem*, Uspekhi Mat. Nauk **2** (1947), 60 – 106 (Russian). M. G. Krein, M. A. Krasnoselsky, and D. P. Milman, *On defect numbers of linear operators in Banach space and some geometric problems*, Sbornik Trudov Instituta Matematiki Akademii Nauk Ukrainskoy SSR, **11** (1948), 97 – 112 (Russian). H. Langer, A. Markus, V. Matsaev, and C. Tretter, *A new concept for block operator matrices: the quadratic numerical range*, Linear Algebra Appl. **330** (2001), 89 – 112. A. S. Markus and V. I. Matsaev, *Spectral theory of holomorphic operator-functions in Hilbert space*, Funct. Anal. Appl. **9** (1975), 73 – 74. R. Mennicken and A. K. Motovilov, *Operator interpretation of resonances arising in spectral problems for $2\times2$ operator matrices*, Math. Nachr. **201** (1999), 117 – 181; arXiv:funct-an/9708001. 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Acknowledgments {#acknowledgments .unnumbered} --------------- V. Kostrykin in grateful to V. Enss, A. Knauf, H. Leschke, and R. Schrader for useful discussions. K. A. Makarov is indebted to Graduiertenkolleg “Hierarchie und Symmetrie in mathematischen Modellen" for kind hospitality during his stay at RWTH Aachen in summer 2002. A. K. Motovilov acknowledges the kind hospitality and support by the Department of Mathematics, University of Missouri, Columbia, MO, USA. He was also supported in part by the Russian Foundation for Basic Research.

--- abstract: 'Deep learning has shown high performances in various types of tasks from visual recognition to natural language processing, which indicates superior flexibility and adaptivity of deep learning. To understand this phenomenon theoretically, we develop a new approximation and estimation error analysis of deep learning with the ReLU activation for functions in a Besov space and its variant with mixed smoothness. The Besov space is a considerably general function space including the [Hölder ]{}space and Sobolev space, and especially can capture spatial inhomogeneity of smoothness. Through the analysis in the Besov space, it is shown that deep learning can achieve the minimax optimal rate and outperform any non-adaptive (linear) estimator such as kernel ridge regression, which shows that deep learning has higher adaptivity to the spatial inhomogeneity of the target function than other estimators such as linear ones. In addition to this, it is shown that deep learning can avoid the curse of dimensionality if the target function is in a [*mixed smooth*]{} Besov space. We also show that the dependency of the convergence rate on the dimensionality is tight due to its minimax optimality. These results support high adaptivity of deep learning and its superior ability as a feature extractor.' author: - | Taiji Suzuki\ The University of Tokyo, Tokyo, Japan\ Center for Advanced Intelligence Project, RIKEN\ Japan Digital Design\ `taiji@mist.i.u-tokyo.ac.jp`\ bibliography: - 'main.bib' - 'main\_colt.bib' title: | Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces:\ optimal rate and curse of dimensionality --- Introduction ============ Deep learning has shown great success in several applications such as computer vision and natural language processing. As its application range is getting wider, theoretical analysis to reveal the reason why deep learning works so well is also gathering much attention. To understand deep learning theoretically, several studies have been developed from several aspects such as approximation theory and statistical learning theory. A remarkable property of neural network is that it has universal approximation capability even if there is only one hidden layer [@cybenko1989approximation; @hornik1991approximation; @sonoda2015neural]. Thanks to this property, deep and shallow neural networks can approximate any function with any precision (of course, the meaning of the terminology “any” must be rigorously defined like “any function in $L^1(\Real)$”). A natural question coming next to the universal approximation capability is its expressive power. It is shown that the expressive power of deep neural network grows exponentially against the number of layers [@NIPS2014_5422; @bianchini2014complexity; @cohen2016expressive; @ICML:Cohen+Shashua:2016; @NIPS:Poole+etal:2016] where the “expressive power” is defined by several ways. The expressive power of neural network can be analyzed more precisely by specifying the target function’s property such as smoothness. @barron1993universal [@barron1994approximation] developed an approximation theory for functions having limited “capacity” that is measured by integrability of their Fourier transform. An interesting point of the analysis is that the approximation error is not affected by the dimensionality of the input. This observation matches the experimental observations that deep learning is quite effective also in high dimensional situations. Another typical approach is to analyze function spaces with smoothness conditions such as the [Hölder ]{}space. In particular, deep neural network with the ReLU activation [@nair2010rectified; @glorot2011deep] has been extensively studied recently from the view point of its expressive power and its generalization error. For example, [@DBLP:journals/corr/Yarotsky16] derived the approximation error of the deep network with the ReLU activation for functions in the [Hölder ]{}space. [@2017arXiv170806633S] evaluated the estimation error of regularized least squared estimator performed by deep ReLU network based on this approximation error analysis in a nonparametric regression setting. [@petersen2017optimal] generalized the analysis by [@DBLP:journals/corr/Yarotsky16] to the class of [*piece-wise*]{} smooth functions. [@arXiv:Imaizumi+Fukumizu:2018] utilized this analysis to derive the estimation error to estimate the piece-wise smooth function and concluded that deep leaning can outperform linear estimators in that setting; here, the [*linear method*]{} indicates an estimator which is linearly dependent on the output observations $(y_1,\dots,y_n)$ (it could be nonlinearly dependent on the input $(x_1, \dots,x_n)$; for example, the kernel ridge regression depends on the output observations linearly, but it is nonlinearly dependent on the inputs). Although these error analyses are standard from a nonparametric statistics view point and the derived rates are known to be (near) minimax optimal, the analysis is rather limited because the analyses are given mainly based on the [Hölder ]{}space. However, there are several other function spaces such as the Sobolev space and the space of finite total variations. A comprehensive analysis to deal with such function classes from a unified view point is required. In this paper, we give generalization error bounds of deep ReLU networks for a [*Besov space*]{} and its variant with [*mixed smoothness*]{}, which includes the [Hölder ]{}space, the Sobolev space, and the function class with total variation as special cases. By doing so, (i) we show that deep learning achieves the minimax optimal rate on the Besov space and notably it outperforms [*any linear estimator*]{} such as the kernel ridge regression, and (ii) we show that deep learning can [*avoid the curse of dimensionality*]{} on the mixed smooth Besov space and achieves the minimax optimal rate. As related work, [@mhaskar1992approximation; @mhaskar1993approximation; @chui1994neural; @mhaskar1996neural; @pinkus1999approximation] also developed an approximation error analysis which essentially leads to analyses for Besov spaces. However, the ReLU activation is basically excluded and comprehensive analyses for the Besov space have not been given. Consequently, it has not been clear whether ReLU neural networks can outperform another representative methods such as kernel methods. As a summary, the contribution of this paper is listed as follows: - To investigate adaptivity of deep learning, we give an explicit form of approximation and estimation error bounds for deep learning with the ReLU activation where the target functions are in the Besov spaces ($B_{p,q}^s$) for $s > 0$ and $0 < p,q \leq \infty$ with $s > d(1/p - 1/r)_+$ where $L^r$-norm is used for error evaluation. In particular, deep learning outperforms any linear estimator such as kernel ridge regression if the target function has highly spatial inhomogeneity of its smoothness. See Table \[tab:CompDeepLinearApprox\] for the overview. - To investigate the effect of dimensionality, we analyze approximation and estimation problems in so-called the mixed smooth Besov space by ReLU neural network. It is shown that deep learning with the ReLU activation can avoid the curse of dimensionality and achieve the near minimax optimal rate. The theory is developed on the basis of the [*sparse grid*]{} technique [@smolyak1963quadrature]. See Table \[tab:Summary\] for the overview. Model Deep learning Linear method -------------------------- ------------------------------------- ------------------------------------------------------------------------------------------ -- Approximation error rate $\tilde{O}(N^{-\frac{s}{d}})$ $\tilde{O}\left(N^{-\frac{s}{d} + (\frac{1}{p} - \frac{1}{r})_+}\right)$ Estimation error rate $\tilde{O}(n^{-\frac{2s}{2s + d}})$ $\Omega\big(n^{- \frac{2s - (2/(p \wedge 1) - 1)}{2s + 1 - (2/(p \wedge 1) - 1)}}\big)$ : Comparison between the performances achieved by deep learning and linear methods. Here, $N$ is the number of parameters to approximate a function in a Besov space ($B^s_{p,q}([0,1]^d)$), and $n$ is the sample size. The approximation error is measured by $L^r$-norm. The $\tilde{O}$ symbol hides the poly-log order. []{data-label="tab:CompDeepLinearApprox"} \ [|p[2cm]{}|p[3cm]{}|p[2.2cm]{}|p[2.1cm]{}|p[2.7cm]{}|]{}Function class & [ [Hölder ]{}]{} & Barron class & [ m-Sobolev ($0 < \beta \leq 2$)]{} & [m-Besov ($0 < \beta$)]{}\ \ Author & [ @DBLP:journals/corr/Yarotsky16, @liang2016deep]{} & @barron1993universal & @MontanelliDu2017 & This work\ Approx. error & ${\tilde{O}}(N^{-\frac{\beta}{d}})$ & ${\tilde{O}}(N^{- 1/2})$ & ${\tilde{O}}(N^{- \beta})$ & $ {\tilde{O}}(N^{-\beta})$\ \ Author & @2017arXiv170806633S & @barron1993universal & & This work\ Estimation error & ${\tilde{O}}(n^{-\frac{2\beta}{2\beta + d}})$ & ${\tilde{O}}(n^{-\frac{1}{2}})$ & & ${\tilde{O}}(n^{-\frac{2\beta}{2\beta + 1}} \times \log(n)^{\frac{2(d-1)(u + \beta)}{1+2\beta}})$\ Set up of function spaces {#sec:FunctionClassDefinitions} ========================= In this section, we define the function classes for which we develop error bounds. In particular, we define the Besov space and its variant with mixed smoothness. The typical settings in statistical learning theory is to estimate a function with a [*smoothness*]{} condition. There are several ways to characterize “smoothness.” Here, we summarize the definitions of representative functional spaces that are appropriate to define the smoothness assumption. Let $\Omega \subset \Real^d$ be a domain of the functions. Throughout this paper, we employ $\Omega = [0,1]^d$. For a function $f:\Omega \to \Real$, let $\|f\|_p := \|f\|_{L^p(\Omega)} := (\int_\Omega |f|^p \dd x )^{1/p}$ for $0 < p < \infty$. For $p=\infty$, we define $\|f\|_\infty :=\|f\|_{L^\infty(\Omega)} := \sup_{x \in \Omega} |f(x)|$. For $\alpha \in \Real^d$, let $|\alpha| = \sum_{j=1}^d |\alpha_j|$. Let ${\mathcal{C}}^0(\Omega)$ be the set of continuous functions equipped with $L^\infty$-norm: ${\mathcal{C}}^0(\Omega) := \{ f : \Omega \to \Real \mid \text{$f$ is continuous and $\|f\|_\infty < \infty$} \}$ [^1]. For $\alpha \in \Natural_+^d$, we denote by $ D^\alpha f(x) = \frac{\partial^{|\alpha|} f}{\partial^{\alpha_1} x_1 \dots \partial^{\alpha_d} x_d}(x) $ [^2]. Let $\beta > 0$ with $\beta \not \in \Natural$ be the smoothness parameter. For an $m$ times differentiable function $f:\Real^d \to \Real$, let the norm of the [Hölder ]{}space ${\mathcal{C}}^\beta(\Omega)$ be $ \|f\|_{{\mathcal{C}}^\beta} := \max_{|\alpha| \leq m} \big\|D^\alpha f\|_{\infty} + \max_{|\alpha| = m} \sup_{x,y \in\Omega} \frac{|\partial^\alpha f(x) - \partial^\alpha f(y)|}{|x-y|^{\beta -m}}, $ where $m=\lfloor \beta \rfloor$. Then, ($\beta$-)[Hölder ]{}space ${\mathcal{C}}^\beta(\Omega)$ is defined as ${\mathcal{C}}^\beta(\Omega) = \{f \mid \|f\|_{{\mathcal{C}}^\beta} < \infty\}$. The parameter $\beta > 0$ controls the “smoothness” of the function. Along with the [Hölder ]{}space, the [*Sobolev space*]{} is also important. Sobolev space $(W^k_p(\Omega))$ with a regularity parameter $k \in \Natural$ and a parameter $1 \leq p \leq \infty$ is a set of functions such that the Sobolev norm $ \textstyle \|f\|_{W^k_p} := ( \sum_{|\alpha| \leq k}\|D^\alpha f\|^p_{p} )^{\frac{1}{p}} $ is finite. There are some ways to define a Sobolev space with fractional order, one of which will be defined by using the notion of [*interpolation space*]{} [@adams2003sobolev], but we don’t pursue this direction here. Finally, we introduce [*Besov space*]{} which further generalizes the definition of the Sobolev space. To define the Besov space, we introduce the modulus of smoothness. For a function $f \in L^p(\Omega)$ for some $p \in (0,\infty]$, the $r$-th modulus of smoothness of $f$ is defined by $$w_{r,p}(f,t) = \sup_{\|h\|_2 \leq t} \|\Delta_h^r(f)\|_{p},$$ where $ \Delta_h^r(f)(x) = \begin{cases} \sum_{j=0}^r {r \choose j} (-1)^{r-j} f(x+jh)~& ( x\in \Omega, ~ x+r h\in \Omega), \\ 0 & (\text{otherwise}). \end{cases} $ Based on the modulus of smoothness, the Besov space is defined as in the following definition. For $0 < p,q \leq \infty$, $\alpha > 0$, $r:= \lfloor \alpha \rfloor + 1$, let the seminorm $|\cdot|_{B^\alpha_{p,q}}$ be $$|f|_{B^\alpha_{p,q}} := \begin{cases} \left(\int_0^\infty (t^{-\alpha} w_{r,p}(f,t))^q \frac{\dd t}{t} \right)^{\frac{1}{q}} & (q < \infty), \\ \sup_{t > 0} t^{-\alpha} w_{r,p}(f,t) & (q = \infty). \end{cases}$$ The norm of the Besov space $B_{p,q}^\alpha(\Omega)$ can be defined by $\|f\|_{B_{p,q}^\alpha} := \|f\|_{p} + |f|_{B^\alpha_{p,q}}$, and we have $B^\alpha_{p,q}(\Omega) = \{f \in L^p(\Omega) \mid \|f\|_{B_{p,q}^\alpha} < \infty\}$. Note that $p, q < 1$ is also allowed. In that setting, the Besov space is no longer a Banach space but a quasi-Banach space. The Besov space plays an important role in several fields such as nonparametric statistical inference [@GineNickl2015mathematical] and approximation theory [@Book:Temlyakov:1993]. These spaces are closely related to each other as follows [@triebel1983theory]: - For $m \in \mathbb{N}$, $B^m_{p,1}(\Omega) \hookrightarrow W_p^m(\Omega) \hookrightarrow B^m_{p,\infty}(\Omega),$ and $B^m_{2,2}(\Omega) = W_2^m(\Omega)$. - For $0 < s < \infty$ and $s \not \in \mathbb{N}$, $ {\mathcal{C}}^s(\Omega) = B^s_{\infty,\infty}(\Omega). $ - For $0 < s,p,q,r \leq \infty$ with $s > \delta := d(1/p - 1/r)_+$, it holds that $ B_{p,q}^s(\Omega) \hookrightarrow B_{r,q}^{s - \delta}(\Omega). $ In particular, under the same condition, from the definition of $\|\cdot\|_{B_{p,q}^s}$, it holds that $$\begin{aligned} B_{p,q}^s(\Omega) \hookrightarrow L^r(\Omega). \label{eq:BesovLrEmbedding}\end{aligned}$$ - For $0 < s,p,q \leq \infty$, if $s > d/p$, then $$\begin{aligned} B_{p,q}^s(\Omega) \hookrightarrow {\mathcal{C}}^0(\Omega). \label{eq:BesovContEmbedding}\end{aligned}$$ Hence, if the smoothness parameter satisfies $s > d/p$, then it is continuously embedded in the set of the continuous functions. However, if $s < d/p$, then the elements in the space are no longer continuous. Moreover, it is known that $B_{1,1}^1([0,1])$ is included in the space of bounded total variation [@peetre1976new]. Hence, the Besov space also allows spatially inhomogeneous smoothness with spikes and jumps; which makes difference between linear estimators and deep learning (see Sec. \[sec:EstErrorAnalysisBesov\]). It is known that the minimax rate to estimate $\ftrue$ is lower bounded by $ n^{- 2s/(2s + d)}, $ [@GineNickl2015mathematical]. We see that the [*curse of dimensionality*]{} is unavoidable as long as we consider the Besov space. This is an undesirable property because we easily encounter high dimensional data in several machine learning problems. Hence, we need another condition to derive approximation and estimation error bounds that are not heavily affected by the dimensionality. To do so, we introduce the notion of [*mixed smoothness*]{}. The Besov space with mixed smoothness is defined as follows [@schmeisser1987unconditional; @sickel2009tensor]. To define the space, we define the coordinate difference operator as $$\Delta_h^{r,i} (f)(x) =\Delta_h^{r}(f(x_1,\dots,x_{i-1},\cdot,x_{i+1},\dots,x_d))(x_i)$$ for $f:\Real^d \to \Real$, $h \in \Real_+$, $i \in [d]$, and $r \geq 1$. Accordingly, the mixed differential operator for $e \subset \{1,\dots,d\}$ and $h \in \Real^d$ is defined as $$\textstyle \Delta_h^{r,e}(f) = \left(\prod_{i \in e} \Delta_{h_i}^{r,i} \right) (f),~~\Delta_h^{r,\emptyset}(f) = f.$$ Then, the mixed modulus of smoothness is defined as $$\textstyle w_{r,p}^e(f,t) := \sup_{|h_i| \leq t_i, i \in e} \|\Delta_h^{r,e}(f)\|_{p}$$ for $t \in \Real_+^d$ and $0 < p \leq \infty$. Letting $0 < p,q \leq \infty$, $\alpha \in \Real_{++}^d$ and $r_i := \lfloor \alpha_i \rfloor + 1$, the semi-norm $|\cdot|_{{{M\!B}_{p,q}^{\alpha,e}}}$ based on the mixed smoothness is defined by $$|f|_{{M\!B}_{p,q}^{\alpha,e}} := \begin{cases} \left\{ \int_\Omega [ (\prod_{i \in e} t_i^{-\alpha_i}) w_{r,p}^e(f,t) ]^q \frac{ \dd t}{\prod_{i \in e} t_i} \right\}^{1/q} & (0 < q < \infty), \\ \sup_{t \in \Omega} (\prod_{i \in e} t_i^{-\alpha_i} ) w_{r,p}^e(f,t) & (q=\infty). \end{cases}$$ By summing up the semi-norm over the choice of $e$, the (quasi-)norm of the mixed smooth Besov space (abbreviated to m-Besov space) is defined by $$\|f\|_{{M\!B}^\alpha_{p,q}} := \|f\|_{p} + \sum_{e \subset \{1,\dots,d\}} |f|_{{M\!B}_{p,q}^{\alpha,e}},$$ and thus ${M\!B}^\alpha_{p,q}(\Omega) := \{f \in L^p(\Omega) \mid \|f\|_{{M\!B}^\alpha_{p,q}} < \infty \}$ where $0 < p,q \leq 1$ and $\alpha \in \Real_{++}^d$. In this paper, we assume that $\alpha_1 = \dots = \alpha_d$. With a slight abuse of notation, we also use the notation $MB^\alpha_{p,q}$ for $\alpha > 0$ to indicate $MB^{(\alpha,\dots,\alpha)}_{p,q}$. For $\alpha \in \Real_+^d$, if $p=q$, the m-Besov space has an equivalent norm with the [*tensor product*]{} of the one-dimensional Besov spaces: $$\begin{aligned} {M\!B}_{p,p}^{\alpha} & = B_{p,p}^{\alpha_1} \otimes_{\delta_p} \cdots \otimes_{\delta_p} B_{p,p}^{\alpha_d}, $$ where $\otimes_{{\delta_p}}$ is a [*tensor product with respect to the $p$-nuclear tensor norm*]{} (see [@sickel2009tensor] for its definition and more details). We can see that the following models are included in the m-Besov space: - Additive model [@AS:Meier+Geer+Buhlmann:2009]: if $f_j \in B^{\alpha_j}_{p,q}([0,1])$ for $j=1,\dots,d$, $$f(x) = \sum_{r=1}^d f_d(x_d) \in {M\!B}^\alpha_{p,q}(\Omega),$$ - Tensor model [@TechRepo:Signoretto+etal:2010]: if $f_{r,j} \in B^{\alpha_j}_{p,q}([0,1])$ for $r=1,\dots,R$ and $j=1,\dots,d$, $$f(x) = \sum_{r=1}^R \prod_{j=1}^d f_{r,j}(x_j) \in {M\!B}^\alpha_{p,q}(\Omega).$$ (m-Besov space allows $R \to \infty$ if the summation converges with respect to the quasi-norm of $\|\cdot\|_{MB_{p,q}^\alpha}$). It is known that an appropriate estimator in these models can avoid curse of dimensionality [@AS:Meier+Geer+Buhlmann:2009; @raskutti2012minimax; @ICML:Kanawaga+etal:2016; @suzuki2016minimax]. What we will show in this paper supports that this fact is also applied to deep learning from a unifying viewpoint. The difference between the (normal) Besov space and the m-Besov space can be informally explained as follows. For regularity condition $\alpha_i \leq 2~ (i=1,2)$, the m-Besov space consists of functions for which the following derivatives are “bounded”: $$\frac{\partial f}{\partial x_1},\frac{\partial f}{\partial x_2}, \frac{\partial^2 f}{\partial x_1^2},\frac{\partial^2 f}{\partial x_2^2},\frac{\partial^2 f}{\partial x_1 \partial x_2}, \frac{\partial^3 f}{\partial x_1 \partial x_2^2}, \frac{\partial^3 f}{\partial x_1^2 \partial x_2}, \frac{\partial^4 f}{\partial x_1^2\partial x_2^2}.$$ That is, the “max” of the orders of derivatives over coordinates needs to be bounded by 2. On the other hand, the Besov space only ensures the boundedness of the following derivatives: $$\frac{\partial f}{\partial x_1},\frac{\partial f}{\partial x_2}, \frac{\partial^2 f}{\partial x_1^2},\frac{\partial^2 f}{\partial x_2^2},\frac{\partial^2 f}{\partial x_1 \partial x_2},$$ where the “sum” of the orders needs to be bounded by 2. This difference directly affects the rate of convergence of approximation accuracy. Further details about this space and related topics can be found in a comprehensive survey [@dung2016hyperbolic]. #### Relation to Barron class. [@barron99l99lb; @barron1993universal; @barron1994approximation] showed that, if the Fourier transform of a function $f$ satisfies some integrability condition, then we may avoid curse of dimensionality for estimating neural networks with sigmoidal activation functions. The integrability condition is given by $$\int_{\mathbb{C}^d} \|\omega \| |\hat{f}(\omega)| \dd \omega < \infty,$$ where $\hat{f}$ is the Fourier transform of a function $f$. We call the class of functions satisfying this condition [*Barron class*]{}. A similar function class is analyzed by [@klusowski2016risk] too. We cannot compare directly the m-Besov space and Barron class, but they are closely related. Indeed, if $p=q=2$ and $s = \alpha_1 = \cdots = \alpha_d$, then m-Besov space ${M\!B}^{s}_{2,2}(\Omega)$ is equivalent to the tensor product of Sobolev space [@sickel2011spline] which consists of functions $f: \Omega \to \Real$ satisfying $$\int_{\mathbb{C}^d} \prod_{i=1}^d (1 + |\omega_i|^2)^{s} |\hat{f}(\omega)|^2 \dd \omega < \infty.$$ Therefore, our analysis gives a (similar but) different characterization of conditions to avoid curse of dimensionality. Approximation error analysis ============================ In this section, we evaluate how well the functions in the Besov and m-Besov spaces can be approximated by neural networks with the ReLU activation. Let us denote the ReLU activation by $\eta(x) = \max\{x,0\}~(x \in \Real)$, and for a vector $x$, $\eta(x)$ is operated in an element-wise manner. Define the neural network with height $L$, width $W$, sparsity constraint $S$ and norm constraint $B$ as $$\begin{aligned} & \Phi(L,W,S,B) := \{ ({W^{(L)}} \eta( \cdot) + {b^{(L)}}) \circ \dots \circ ({W^{(1)}} x + {b^{(1)}}) \\ & \mid {W^{(\ell)}} \in \Real^{W \times W},~{b^{(\ell)}} \in \Real^W,~ \sum_{\ell=1}^L (\|{W^{(\ell)}}\|_0+ \|{b^{(\ell)}}\|_0) \leq S, \max_{\ell} \|{W^{(\ell)}}\|_\infty \vee \|{b^{(\ell)}}\|_\infty \leq B \},\end{aligned}$$ where $\|\cdot\|_0$ is the $\ell_0$-norm of the matrix (the number of non-zero elements of the matrix) and $\|\cdot\|_\infty$ is the $\ell_\infty$-norm of the matrix (maximum of the absolute values of the elements). We want to evaluate how large $L,W,S,B$ should be to approximate $\ftrue \in {M\!B}^\alpha_{p,q}(\Omega)$ by an element $f \in \Phi(L,W,S,B)$ with precision $\epsilon >0$ measured by $L^r$-norm: $\min_{f \in \Phi}\|f - \ftrue\|_r \leq \epsilon$. Approximation error analysis for Besov spaces --------------------------------------------- Here, we show how the neural network can approximate a function in the Besov space which is useful to derive the generalization error of deep learning. Although its derivation is rather standard as considered in [@chui1994neural; @bolcskei2017optimal], it should be worth noting that the bound derived here cannot be attained any [*non-adaptive*]{} method and the generalization error based on the analysis is also unattainable by any [*linear*]{} estimators including the kernel ridge regression. That explains the high adaptivity of deep neural network and how it outperforms usual linear methods such as kernel methods. To show the approximation accuracy, a key step is to show that the ReLU neural network can approximate the [*cardinal B-spline*]{} with high accuracy. Let $\calN(x) = 1~ (x \in [0,1]),~0 ~ (\text{otherwise})$, then the [*cardinal B-spline of order $m$*]{} is defined by taking $m+1$-times convolution of $\calN$: $$\calN_m(x) = (\underbrace{\calN * \calN * \dots * \calN}_{\text{$m+1$ times}})(x),$$ where $f* g(x) := \int f(x -t) g(t) \dd t$. It is known that $\calN_m$ is a piece-wise polynomial of order $m$. For $k=(k_1,\dots,k_d) \in \Natural^d$ and $j=(j_1,\dots,j_d)\in \Natural^d$, let $M_{k,j}^d(x) = \prod_{i=1}^d \calN_m(2^{k_i} x_i - j_i)$. Even for $k \in \Natural$, we also use the same notation to express $M_{k,j}^d(x) = \prod_{i=1}^d \calN_m(2^{k} x_i - j_i)$. Here, $k$ controls the spatial “resolution” and $j$ specifies the location on which the basis is put. Basically, we approximate a function $f$ in a Besov space by a super-position of $M_{k,j}^m(x)$, which is closely related to wavelet analysis [@Mallat99a]. [@mhaskar1992approximation; @chui1994neural] have shown the approximation ability of neural network for a function with bounded modulus of smoothness. However, the activation function dealt with by the analysis does not include ReLU but it deals with a class of activation functions satisfying the following conditions, $$\begin{aligned} & \lim_{x \to \infty} \eta(x)/x^k \to 1,~~\lim_{x \to -\infty} \eta(x)/x^k = 0, ~~ \text{$\exists K > 1$ s.t. $|\eta(x)| \leq K (1 + |x|)^k~(x \in \Real)$}, \label{eq:activation_kterm}\end{aligned}$$ for $k=2$ which excludes ReLU. [@mhaskar1993approximation] analyzed deep neural network under the same setting but it restricts the smoothness parameter to $s = k+1$. [@mhaskar1996neural] considered the Sobolev space $W^{m}_p$ with an infinitely many differentiable “bump” function which also excludes ReLU. However, approximating the cardinal B-spline by ReLU can be attained by appropriately using the technique developed by [@DBLP:journals/corr/Yarotsky16] as in the following lemma. \[lemm:Mnapproximation\] There exists a constant $c_{(d,m)}$ depending only on $d$ and $m$ such that, for all $\epsilon > 0$, there exists a neural network $ \check{M} \in \Phi(L_0,W_0,S_0,B_0) $ with $L_0 := 3 + 2 \left\lceil \log_2\left( \frac{3^{d\vee m} }{\epsilon c_{(d,m)}} \right)+5 \right\rceil \left \lceil\log_2(d \vee m)\right\rceil$, $W_0 := 6 d m(m+2) + 2 d$, $S_0 := L_0 W_0^2$ and $B_0 := 2 (m+1)^m$ that satisfies $$\|M_{0,0}^d - \check{M}\|_{L^\infty(\Real^d)} \leq \epsilon,$$ and $\check{M}(x) = 0$ for all $x \not \in [0,m+1]^d$. The proof is in Appendix \[sec:ProofBsplineApprox\]. Based on this lemma, we can translate several B-spline approximation results into those of deep neural network approximation. In particular, combining this lemma and the B-spline interpolant representations of functions in Besov spaces [@devore1988interpolation; @devore1993wavelet; @dung2011optimal], we obtain the optimal approximation error bound for deep neural networks. Here, let $U(\calH)$ be the unit ball of a quasi-Banach space $\calH$, and for a set $\calF$ of functions, define the worst case approximation error as $$R_r(\calF,\calH) :=\sup_{\ftrue \in U(\calH)} \inf_{f \in \calF} \|\ftrue - f\|_{L^r([0,1]^d)}.$$ \[prop:BesovApproxByNN\] Suppose that $0 < p,q,r \leq \infty$ and $0 < s < \infty$ satisfy the following condition: $$\begin{aligned} s > d(1/p - 1/r)_+. \label{eq:sprConditions} $$ Assume that $m \in \Natural$ satisfies $0 < s < \min( m , m -1 + 1/p)$. Let $\nu = (s - \delta)/(2\delta)$. For sufficiently large $N \in \Natural$ and $\epsilon = N^{-s/d - (\nu^{-1} + d^{-1})(d/p -s)_+}\log(N)^{-1}$, let L &= 3 + 2 \_2( )+5 \_2(d m), & W &= N W\_0 , &\ S &= (L-1) W\_0\^2 N + N, & B &= O(N\^[(\^[-1]{} + d\^[-1]{}) (1(d/p - s)\_+) ]{}), & then it holds that $$R_r(\Phi(L,W,S,B),B^s_{p,q}([0,1]^d)) \lesssim N^{ - s/d}.$$ By , the condition indicates that $\ftrue \in B_{p,q}^s$ satisfies $\ftrue \in L^r(\Omega)$. If we set $p = q = \infty$ and $r=\infty$, then $B_{p,q}^s(\Omega) = C^s(\Omega)$ which yields the result by [@DBLP:journals/corr/Yarotsky16] as a special case. The proof is in Appendix \[sec:ProofPropBesovApp\]. An interesting point is that the statement is valid even for $p \neq r$. In particular, the theorem also supports non-continuous regime $(s < d/p)$ in which $L^\infty$-convergence does no longer hold but instead $L^r$-convergence is guaranteed under the condition $s > d(1/p - 1/r)_+$. In that sense, the convergence of the approximation error is guaranteed in considerably general settings. [@pinkus1999approximation] gave an explicit form of convergence when $1 \leq p=r$ for the activation functions satisfying which does not cover ReLU and an important setting $p\neq r$. [@petrushev1998approximation] considered $p=r=2$ and activation function with where $s$ is an integer and $s \leq k+1+(d-1)/2$. [@chui1994neural] and [@bolcskei2017optimal] dealt with the smooth sigmoidal activation satisfying the condition with $k \geq 2$ or a “smoothed version” of the ReLU activation which excludes ReLU; but [@bolcskei2017optimal] presented a general strategy for neural-net approximation by using the notion of best $M$-term approximation. [@mhaskar1992approximation] gives an approximation bound using the modulus of smoothness, but the smoothness $s$ and the order of sigmoidal function $k$ in is tightly connected and $\ftrue$ is assumed to be continuous which excludes the situation $s < d/p$. On the other hand, the above proposition does not require such a tight connection and it explicitly gives the approximation bound for Besov spaces. [@williamson1992splines] derived a spline approximation error bound for an element in a Besov space when $d=1$, but the derived bound is only $O(N^{-s + (1/p-1/r)_+})$ which is the one of non-adaptive methods described below, and approximation by a ReLU activation network is not discussed. We may also use the analysis of [@cohen2001tree] which is based on compactly supported wavelet bases, but the cardinal B-spline is easy to handle through quasi-interpolant representation as performed in the proof of Proposition \[prop:BesovApproxByNN\]. It should be noted that the presented approximation accuracy bound is not trivial because it can not be achieved by a [*non-adaptive method*]{}. Actually, the [*linear $N$-width*]{} [@tikhomirov1960diameters] of the Besov space is lower bounded as $$\begin{aligned} \label{eq:LinearWidthBesov} \inf_{L_N} \sup_{f \in U(MB_{p,q}^s)} \|f - L_N(f)\|_r \gtrsim \begin{cases} N^{- s/d + (1/p - 1/r)_+} & \begin{cases}\text{either} & (0 < p \leq r \leq 2), \\ \text{or} & (2 \leq p \leq r \leq \infty), \\ \text{or} & (0 < r \leq p \leq \infty), \end{cases} \\ N^{- s/d + 1/p - 1/2} & (0 < p <2 < r < \infty,~s > d\max(1-1/r,1/p), \end{cases}\end{aligned}$$ where the infimum is taken over all linear oprators $L_N$ with rank $N$ from $B_{p,q}^s$ to $L^r$ (see [@vybiral2008widths] for more details). Similarly, the [*best $N$-term approximation error*]{} (Kolmorogov width) of the Besov space is lower bounded as $$\begin{aligned} \label{eq:KolmorogororovBesov} \inf_{S_N \subset B_{p,q}^s} \sup_{f \in U(B_{p,q}^s)} \inf_{\fcheck \in S_N} \|f - \fcheck\|_{L^r(\Omega)} \gtrsim \begin{cases} N^{- s/d + (1/p - 1/r)_+} & (1 < p < r \leq 2,~s > d(1/p - 1/r)), \\ N^{- s/d + 1/p - 1/2} & (1 < p <2 < r \leq \infty,~s > d/p), \\ N^{- s/d } & (2 \leq p < r \leq \infty,~s > d/2), \end{cases}\end{aligned}$$ if $1 < p < r \leq \infty$, $1 \leq q < \infty$ and $1 < s$, where $S_N$ is any $N$-dimensional subspace of $B_{p,q}^s$ (see [@vybiral2008widths], and see also [@Romanyuk2009Kolmogorovwidth; @myronyuk2016kolmogorov] for a related space). That is, any linear/non-linear approximator with [*fixed*]{} $N$-bases does not achieve the approximation error $N^{- \alpha/d}$ in some parameter settings such as $0 < p < 2 < r $. On the other hand, adaptive methods including deep learning can improve the error rate up to $N^{- \alpha/d}$ which is rate optimal [@dung2011optimal]. The difference is significant when $p < r$. This implies that deep neural network possesses high adaptivity to find which part of the function should be intensively approximated. In other words, deep neural network can properly extracts the feature of the input (which corresponds to construct an appropriate set of bases) to approximate the target function in the most efficient way. Approximation error analysis for m-Besov space ---------------------------------------------- Here, we deal with m-Besov spaces instead of the ordinary Besov space. The next theorem gives the approximation error bound to approximate functions in the m-Besov spaces by deep neural network models. Here, define $D_{k,d} := \left(1 + \frac{d-1}{k}\right)^k \left(1 + \frac{k}{d-1}\right)^{d-1}.$ Then, we have the following theorem. \[eq:mBesovApproxByNN\] Suppose that $0 < p,q,r \leq \infty$ and $s < \infty$ satisfies $s > (1/p - 1/r)_+.$ Assume that $m \in \Natural$ satisfies $0 < s < \min( m , m -1 + 1/p)$. Let $\delta = (1/p - 1/r)_+$ and $\nu = (s - \delta)/(2\delta)$. For any $K \geq 1$, let $K^* = \lceil K(1 + \frac{2\delta}{\alpha - \delta}) \rceil $. Then, for $N = (2 + (1 - 2^{- \nu})^{-1} ) 2^K D_{K^*,d} $, if we set & L = 3 + 2 \_2( )+5 + (s + ( -s)\_+ + 1) K\^\* + (\[e(m+1)\]\^d (1+K\^\*)) \_2(d m),\ & W =N W\_0,    S = (L-1) N W\_0\^2 + N,   B = O(N\^[(\^[-1]{} + 1) (1(1/p - s)\_+) ]{}), then it holds that\ \[eq:NeuralBoundmBesov\] &        R\_r((L,W,S,B),MB\^s\_[p,q]{}(\[0,1\]\^d)) 2\^[-K s ]{} D\_[K,d]{}\^[(1/(r,1) - 1/q)\_+]{}, R\_r((L,W,S,B),MB\^s\_[p,q]{}(\[0,1\]\^d)) 2\^[-K s ]{} D\_[K,d]{}\^[(1/r - 1/q)\_+]{} & (r &lt; ),\ 2\^[-K s ]{} D\_[K,d]{}\^[(1 - 1/q)\_+]{} & (r = ). The proof is given in Appendix \[sec:ProofmBesovApproxNN\]. Now, the number $S$ of non-zero parameters for a given $K$ is evaluated as $S = \Omega(N) \simeq 2^{K} D_{K,d}$ in this theorem. It holds that $N \simeq 2^K K^{(d-1)}$, which implies $2^{- K} \simeq N^{-1} \log^{d-1}(N)$ if $N\gg d$ (see also the discussion right after Theorem \[eq:mBesoApproxSpline\] in Appendix \[sec:SparseGridBound\] for more details of calculation). Therefore, when $r \gg q$, the approximation error is given as $O(N^{-s} \log^{s(d-1)}(N))$ in which the effect of dimensionality $d$ is much milder than that of Proposition \[prop:BesovApproxByNN\]. This means that the curse of dimensionality is much eased in the mixed smooth space. The obtained bound is far from obvious. Actually, it is better than any linear approximation methods as follows. Let the linear $M$-width introduced by [@tikhomirov1960diameters] be $$\lambda_N(MB_{p,q}^s,L^r) := \inf_{L_N} \sup_{f \in U(MB_{p,q}^s)} \|f - L_N(f)\|_r,$$ where the infimum is taken over all linear oprators $L_N$ with rank $N$ from $MB_{p,q}^s$ to $L^r$. The linear $N$-width of the m-Besov space has been extensively studies as in the following proposition (see Lemma 5.1 of [@Complexity:Dung:2011], and [@Romanyuk2001]). \[eq:LinearWidthmBesov\] Let $1 \leq p,r \leq \infty$, $0 < q \leq \infty$ and $s > (1/p - 1/r)_+$. Then we have the following asymptotic order of the linear width for the asymptotics $N \gg d$:\ (a) For $p \geq r$, $$\begin{aligned} \lambda_N(MB_{p,q}^s,L^r) \simeq \begin{cases} (N^{-1} \log^{d-1}(N))^s & {\small \begin{cases} (q \leq 2 \leq r \leq p < \infty), \\ (q \leq 1, ~p = r = \infty), \\ (1 < p = r \leq 2,~q \leq r), \end{cases} } \\ (N^{-1} \log^{d-1}(N))^s (\log^{d-1}(N))^{1/r - 1/q} & (1 < p = r \leq 2,~q > r), \\ (N^{-1} \log^{d-1}(N))^s (\log^{d-1}(N))^{(1/2 - 1/q)_+} & (2 \leq q,~1 < r < 2 \leq p < \infty), \end{cases}\end{aligned}$$ (b) For $1 < p < r < \infty$, $$\begin{aligned} \lambda_M(MB_{p,q}^s,L^r) \simeq \begin{cases} (N^{-1} \log^{d-1}(N))^{s + 1/r - 1/p} & (2 \leq p,~2 \leq q \leq r), \\ (N^{-1} \log^{d-1}(N))^{s + 1/r - 1/p}(\log^{d-1}(N))^{(1/r - 1/q)_+} & (r \leq 2). \\ \end{cases}\end{aligned}$$ Therefore, the approximation error given in Theorem \[eq:mBesovApproxByNN\] achieves the optimal linear width ($(N^{-1}\log^{d-1}(N))^s$) for several parameter settings of $p,q,s$. In particular, when $p < r$, the bound in Theorem \[eq:mBesovApproxByNN\] is better than that of Proposition \[eq:LinearWidthmBesov\]. This is because to prove Theorem \[eq:mBesovApproxByNN\], we used an adaptive recovery technique instead of a linear recovery method. This implies that, by constructing a deep neural network accurately, we achieve the same approximation accuracy as the adaptive one which is better than that of linear approximation. Estimation error analysis {#sec:EstErrorAnalysis} ========================= In this section, we connect the approximation theory to generalization error analysis (estimation error analysis). For the statistical analysis, we assume the following nonparametric regression model: $$y_i = \ftrue(x_i) + \xi_i~~~~(i=1,\dots,n),$$ where $x_i \sim P_X$ with density $0 \leq p(x) < R$ on $[0,1]^d$, and $\xi_i \sim N(0,\sigma^2)$. The data $D_n = (x_i,y_i)_{i=1}^n$ is independently identically distributed. We want to estimate $\ftrue$ from the data. Here, we consider a regularized learning procedure: $$\fhat = \argmin_{\bar{f}: f \in \Phi(L,W,S,B)} \sum_{i=1}^n (y_i - \bar{f}(x_i))^2$$ where $\bar{f}$ is the [*clipping*]{} of $f$ defined by $\bar{f} = \min\{\max\{f, - F\},F\}$ for $F > 0$ which is realized by ReLU units. Since the sparsity level is controlled by $S$ and the parameter is bounded by $B$, this estimator can be regarded as a regularized estimator. In practice, it is hard to exactly compute $\fhat$. Thus, we approximately solve the problem by applying sparse regularization such as $L_1$-regularization and optimal parameter search through Bayesian optimization. The generalization error that we present here is an “ideal” bound which is valid if the optimal solution $\fhat$ is computable. Estimation error in Besov spaces {#sec:EstErrorAnalysisBesov} -------------------------------- In this subsection, we provide the estimation error rate of deep learning to estimate functions in Besov spaces by using the approximation error bound given in the previous sections. \[thm:EstimationErrorNNBesov\] Suppose that $0 < p,q \leq \infty$ and $s > d(1/p - 1/2)_+$. If $\ftrue \in B^{s}_{p,q}(\Omega) \cap L^\infty(\Omega)$ and. $\|\ftrue\|_{B^{s}_{p,q}} \leq 1$ and $\|\ftrue\|_\infty \leq F$ for $F \geq 1$, then letting $(W,L,S,B)$ be as in Proposition \[prop:BesovApproxByNN\] with $N \asymp n^{\frac{d}{2s+d}}$, we obtain $$\begin{aligned} \EE_{D_n}[\|\ftrue - \fhat\|_{\LPi(P_X)}^2] \lesssim n^{- \frac{2s}{2s + d}} \log(n)^{2},\end{aligned}$$ where $\EE_{D_n}[\cdot]$ indicates the expectation w.r.t. the training data $D_n$. The proof is given in Appendix \[sec:ProofsOfEstimationErrorBounds\]. The condition $\|\ftrue\|_\infty \leq F$ is required to connect the empirical $L^2$-norm $\frac{1}{n} \sum_{i=1}^n (\fhat(x_i) - \ftrue(x_i))^2$ to the population $L^2$-norm $\|\fhat - \ftrue\|_{\LPi(P_X)}^2$. It is known that the convergence rate $n^{-\frac{2s}{2s + d}}$ is mini-max optimal [@donoho1998minimax; @GineNickl2015mathematical]. Thus, it cannot be improved by any estimator. Therefore, deep learning can achieve the minimax optimal rate up to $\log(n)^2$-order. The term $\log(n)^2$ could be improved to $\log(n)$ by using the construction of [@petersen2017optimal]. However, we don’t pursue this direction for simplicity. Here an important remark is that this minimax optimal rate cannot be achieved by any [*linear estimator*]{}. We call an estimator [*linear*]{} when the estimator depends on $(y_i)_{i=1}^n$ linearly (it can be non-linearly dependent on $(x_i)_{i=1}^n$). Several classical methods such as the kernel ridge regression, the Nadaraya-Watson estimator and the sieve estimator are included in the class of linear estimators (e.g., kernel ridge regression is given as $\fhat(x) = k_{x,X}(k_{XX} + \lambda \mathrm{I})^{-1}Y$). The following proposition given by [@donoho1998minimax; @zhang2002wavelet] states that the minimax rate of linear estimators is lower bounded by $n^{- \frac{2s - 2(1/p - 1/2)_+}{2s + 1 - 2(1/p - 1/2)_+}}$ which is larger than the minimax rate $n^{- \frac{2s}{2s + 2}}$ if $p < 2$. \[Prop:LinearMinimax\] Suppose that $d=1$ and the input distribution $P_X$ is the uniform distribution on $[0,1]$. Assume that $s > 1/p$, $1 \leq p, q \leq \infty$ or $s=p=q=1$. Then, $$\inf_{\text{$\fhat$: linear}} \sup_{\ftrue \in U(B_{p,q}^s)}\EE_{D_n}[\|\ftrue - \fhat\|_{\LPi(P_X)}^2] \gtrsim n^{- \frac{2s - v}{2s + 1 - v}}$$ where $v = 2/(p \wedge 2) - 1$ and $\fhat$ runs over all linear estimators, that is, $\fhat$ depends on $(y_i)_{i=1}^n$ linearly. When $p < 2$, the smoothness of the Besov space is somewhat inhomogeneous, that is, a function in the Besov space contains spiky/jump parts and smooth parts (remember that when $s = p = q = 1$ for $d=1$, the Besov space is included in the set of functions with bounded total variation). Here, the setting $p < 2$ is the regime where there appears difference between non-adaptive methods and deep learning in terms of approximation accuracy (see ). On the other hand, the linear estimator captures only global properties of the function and cannot capture variability of local shapes of the function. Hence, the linear estimator cannot achieve the minimax optimal rate if the function has spatially inhomogeneous smoothness. However, deep learning possesses adaptivity to the spatial inhomogeneity. We would like to remark that The shrinkage estimator proposed in [@donoho1998minimax; @zhang2002wavelet] achieves the minimax optimal rate for $s > 1/p$ with $d = 1$ and $1 \leq p,q \leq \infty$ which excludes an interesting setting such as $s = p = q = 1$. However, the result of Theorem \[thm:EstimationErrorNNBesov\] also covers more general settings where $d \geq 1$ and $s > d(1/p - 1/2)_+$ with $0 < p,q \leq \infty$. [@arXiv:Imaizumi+Fukumizu:2018] has pointed out that such a discrepancy between deep learning and linear estimator appears when the target function is [*non-smooth*]{}. Interestingly, the parameter setting $s > 1/p$ assumed in Proposition \[Prop:LinearMinimax\] ensures smoothness (see ). This means that non-smoothness is not necessarily required to characterize the superiority of deep learning, but [*non-convexity*]{} of the set of target functions is essentially important. In fact, the gap is coming from the property that the [*quadratic hull*]{} of the model $U(B_{p,q}^s)$ is strictly larger than the original set [@donoho1998minimax]. Estimation error in mixed smooth Besov spaces --------------------------------------------- Here, we provide the estimation error rate of deep learning to estimate functions in mixed smooth Besov spaces. \[thm:EstimationErrorNN\] Suppose that $0 < p,q \leq \infty$ and $s > (1/p - 1/2)_+$. Let $u = (1 - 1/q)_+$ for $p \geq 2$ and $u = (1/2 - 1/q)_+$ for $p < 2$. If $\ftrue \in {M\!B}^{s}_{p,q}(\Omega) \cap L^\infty(\Omega)$ and $\|\ftrue\|_{{M\!B}^{s}_{p,q}} \leq 1$ and $\|\ftrue\|_\infty \leq F$ for $F \geq 1$, then letting $(W,L,S,B)$ be as in Theorem \[eq:mBesovApproxByNN\], we obtain $$\begin{aligned} \EE_{D_n}[\|\ftrue - \fhat\|_{\LPi(P_X)}^2] \lesssim n^{- \frac{2s}{2s + 1}}\log(n)^{\frac{2(d-1)(u + s)}{1+2s}} \log(n)^2. $$ Under the same assumption, if $s > u\log_2(e)$ is additionally satisfied, we also have $$\begin{aligned} \EE_{D_n}[\|\ftrue - \fhat\|_{\LPi(P_X)}^2] \lesssim n^{- \frac{2s - 2u\log_2(e)}{2s + 1 + (1-2u)\log_2(e)}} \log(n)^2. $$ The proof is given in Appendix \[sec:ProofsOfEstimationErrorBounds\]. The risk bound (Theorem \[thm:EstimationErrorNN\]) indicates that the curse of dimensionality can be eased by assuming the mixed smoothness compared with the ordinary Besov space ($n^{-\frac{2s}{2s + d}}$). We show that this is almost minimax optimal in Theorem \[eq:MinimaxOptimalboundOfmBesov\] below. In the first bound, the dimensionality $d$ comes in the exponent of ${\mathrm{poly}}\log(n)$ term. If $u=0$, then the effect of $d$ can be further eased. Actually, in this situation ($u=0$), the second bound can be rewritten as $$n^{- \frac{2s}{2s + 1 + \log_2(e)}} \log(n)^2, $$ where the effect of the dimensionality $d$ completely disappears from the exponent. This explains partially why deep learning performs well for high dimensional data. [@MontanelliDu2017] has analyzed the mixed smooth [Hölder ]{}space with $s < 2$. However, our analysis is applicable to the m-Besov space which is more general than the mixed smooth [Hölder ]{}space and the covered range of $s,p,q$ is much larger. Here, we again remark the adaptivity of deep learning. Remind that this rate cannot be achieved by the linear estimator for $p < 2$ when $d=1$ by Proposition \[Prop:LinearMinimax\]. This explains the adaptivity ability of deep learning to the spatial inhomogeneity of the smoothness. #### Minimax optimal rate for estimating a function in the m-Besov space Here, we show the minimax optimality of the obtained bound as follows. \[eq:MinimaxOptimalboundOfmBesov\] Assume that $0 < p,q \leq \infty$ and $s > (1/p - 1/2)_+$ and $P_X$ is the uniform distribution over $[0,1]^d$. Regarding $d$ as a constant, the minimax learning rate in the asymptotics of $n \to \infty$ is lower bounded as follows: There exists a constant $\widehat{C}_1$ such that $$\begin{aligned} \inf_{\fhat} \sup_{\ftrue \in U(MB_{p,q}^s)}\EE_{D_n}[\|\fhat - \ftrue \|_{\LPiPx}^2] \geq \widehat{C}_1 n^{- \frac{2s}{2s + 1}} \log(n)^{\frac{2(d-1)(s + 1/2 - 1/q)_+}{2s+1}} \label{eq:minimaxLp}\end{aligned}$$ where “inf” is taken over all measurable functions of the observations $(x_i,y_i)_{i=1}^n$ and the expectation is taken for the sample distribution. The proof is given in Appendix \[sec:MinimaxMixedSmooth\]. Because of this theorem, our bound given in Theorem \[thm:EstimationErrorNN\] achieves the minimax optimal rate in the regime of $p < 2$ and $1/2 - 1/q > 0$ up to $\log(n)^2$ order. Even outside of this parameter setting, the discrepancy between our upper bound and the minimax lower bound is just a poly-$\log$ oder. See also [@neumann2012multivariate] for some other related spaces and specific examples such as $p=q=2$. Conclusion ========== This paper investigated the learning ability of deep ReLU neural network when the target function is in a Besov space or a mixed smooth Besov space. Based on the analysis for the Besov space, it is shown that deep learning using the ReLU activation can achieve the minimax optimal rate and outperform the linear method when $p < 2$ which indicates the spatial inhomogeneity of the shape of the target function. The analysis for the mixed smooth Besov space shows that deep learning can adaptively avoid the curse of dimensionality. The bound is derived by sparse grid technique. All analyses in the paper adopted the cardinal B-spline expansion and the adaptive non-linear approximation technique, which allowed us to show the minimax optimal rate. The consequences of the analyses partly support the superiority of deep leaning in terms of adaptivity and ability to avoid curse of dimensionality. From more high level view point, these favorable property is reduced to its high feature extraction ability. This paper did not discuss any optimization aspect of deep learning. However, it is important to investigate what kind of practical algorithms can actually achieve the optimal rate derived in this paper in an efficient way. We leave this important issue for future work. Acknowledgment {#acknowledgment .unnumbered} ============== TS was partially supported by MEXT Kakenhi (25730013, 25120012, 26280009, 15H05707 and 18H03201), Japan Digital Design, and JST-CREST. Proof of Lemma \[lemm:Mnapproximation\] {#sec:ProofBsplineApprox} ======================================= First note that $\calN_m(x) = \frac{1}{m!} \sum_{j=0}^{m+1} (-1)^j {m+1 \choose j} ( x-j)_+^m$ (see Eq. (4.28) of [@mhaskar1992approximation] for example). Thus, if we can make an approximation of $\eta(x)^m$, then by taking a summation of those basis, we obtain an approximate of $\calN_m(x)$. It is shown by [@DBLP:journals/corr/Yarotsky16] that, for $D \in \Natural$ and any $\epsilon > 0$, there exists a neural network $\phi_{\mathrm{mult}} \in \Phi(L,W,S,B)$ with $L = \lceil \log_2\left( \frac{3^D}{\epsilon} \right)+5 \rceil \lceil\log_2(D)\rceil$, $W = 6d $, $S = LW^2$ and $B = 1$ such that $$\sup_{x \in [0,1]^d} \left | \phi_{\mathrm{mult}}(x_1,\dots,x_D) - \prod_{i=1}^D x_ i \right | \leq \epsilon,$$ and $\phi_{\mathrm{mult}}(0,\dots,0) = 0$ for $y \in \Real^d$ such that $\prod_{j=1}^d y_j=0$. Moreover, for any $M > 0$, we can realize the function $\min\{M,\max\{x,0\}\}$ by a single-layer neural network $\phi_{(0,M)}(x) := \eta(x) - \eta(x-M) (=\min\{M,\max\{x,0\}\})$. Thus, for $x \in \Real$, it holds that $$\sup_{x \in [0,M]} \left| \phi_{\mathrm{mult}}(\phi_{(0,1)}(x/M),\dots,\phi_{(0,1)}(x/M)) - (\phi_{(0,1)}(x/M))^m \right| \leq \epsilon.$$ Now, since $\calN_m(x) = 0$ for $x \not\in [0,m+1]$, it also holds $ \calN_m(x) = \frac{1}{m!} \sum_{j=0}^{m+1} (-1)^j {m+1 \choose j}\phi_{(0,m+1 -j)}(x - j)^m = \frac{1}{m!} \sum_{j=0}^{m+1} (-1)^j {m+1 \choose j} (m+1)^m \phi_{(0,1 -j/(m+1))}((x-j)/(m+1))^m. $ Therefore, letting $$f(x) = \frac{1}{m!} \sum_{j=0}^{m+1} (-1)^j (m+1)^m {m+1 \choose j} \phi_{\mathrm{mult}}\Bigg(\underbrace{\phi_{(0,1 -\frac{j}{m+1})}\left( \frac{x-j}{m+1}\right),\dots,\phi_{(0,1 -\frac{j}{m+1})}\left(\frac{x-j}{m+1}\right)}_{\text{$m$-times}}\Bigg),$$ we have that $f(x) = 0$ for all $x \leq 0$ and $$\begin{aligned} & \sup_{0 \leq x \leq m+1} | \calN_m(x) - f(x) | \leq \frac{1}{m!} \sum_{j=0}^{m+1} {m+1 \choose j} (m+1)^m \epsilon \leq \frac{(m+1)^m}{\sqrt{2\pi} m^{m+1/2} e^{-m}} 2^{m+1} \epsilon \\ &~~~~~~~~~~~ \leq e \frac{ (2e)^m}{\sqrt{m}} \epsilon =: \epsilon',\end{aligned}$$ where we used $ \sum_{j=0}^{m+1} {m+1 \choose j} = 2^{m+1}$ and Stirling’s approximation $m! \geq \sqrt{2\pi} m^{m+1/2} e^{-m}$ in the second inequality. Hence, we also have $$\begin{aligned} & f(x) = \frac{1}{m!} \sum_{j=0}^{m+1} (-1)^j {m+1 \choose j} (m+1)^m \\ & ~~~~~~~~~~~~~~~~~~~~~~~~\times \phi_{\mathrm{mult}}\left(\phi_{(0,1 -\frac{j}{m+1})}\left(\frac{m+1-j}{m+1}\right),\dots, \phi_{(0,1 -\frac{j}{m+1})}\left(\frac{m+1-j}{m+1}\right)\right) \\ & ~~~~~~~~=: \delta'~~(\forall x > m+1).\end{aligned}$$ It holds that $|\delta'| \leq \epsilon'$. Because of this and noting $0 \leq \calN_m(x) \leq 1$, we see that $g(x) := \phi_{(0,1)}(f(x) - \frac{\delta'}{m+1} \phi_{(0,m+1)}(x)) $ yields $$\sup_{x \in \Real}|\calN_m(x) - g(x)| \leq 2 \epsilon',$$ $\sup_{x \in \Real} |g(x)| \leq 1$, and $g(x) =0$ for all $x \not \in [0,m+1]$. Hence, by applying $\phi_{\mathrm{mult}}$ again, we finally obtain that $$\begin{aligned} & \sup_{x \in [0,1]^d} |M_{0,0}^d(x) - \phi_{\mathrm{mult}}(g(x_1),\dots,g(x_d))| \\ \leq & \sup_{x \in [0,1]^d} \left|M_{0,0}^d(x) - \prod_{j=1}^d g(x_j) \right| + \sup_{x \in [0,1]^d} \left |\prod_{j=1}^d g(x_j) - \phi_{\mathrm{mult}}(g(x_1),\dots,g(x_d)) \right| \\ \leq & 2 d \epsilon' + \epsilon.\end{aligned}$$ We again applying $\phi_{(0,1)}$, we obtain that $h = \phi_{(0,1)} \circ \phi_{\mathrm{mult}}(g(x_1),\dots,g(x_d))$ satisfies $\|M_{0,0}^d - h\|_{L^\infty(\Real^d)} \leq 2 d \epsilon' + \epsilon$, $h(x) = 0$ for all $x \not \in [0,m+1]^d$, and $\|h\|_\infty \leq 1$. Finally, by carefully checking the network construction, it is shown that $h \in \Phi(L,W,S,B)$ with $L = 3 + 2 \lceil \log\left( \frac{3^{d\vee m} }{\epsilon} \right)+5 \rceil \lceil\log_2(d \vee m)\rceil$, $W = 6 d m(m+2) + 2 d$, $S = L W^2$ and $B = 2 (m+1)^m$. Hence, resetting $\epsilon \leftarrow 2d \epsilon' + \epsilon = (1 + 2d e \frac{(2e)^m}{\sqrt{m}}) \epsilon$, this $h$ is the desired $\check{M}$. Proof of Proposition \[prop:BesovApproxByNN\] {#sec:ProofPropBesovApp} ============================================= For the order $m \in \Natural$ of the cardinal B-spline bases, let $J(k) = \{- m, - m +1, \dots, 2^k-1, 2^k \}^d$ and the quasi-norm of the coefficient $(\alpha_{k,j})_{k,j}$ for $k \in \Natural_+$ and $j \in J(k)$ be $$\| (\alpha_{k,j})_{k,j}\|_{b^s_{p,q}}. = \left\{\sum_{k \in \Natural_+} \left[ 2^{k(s - d/p)}\Big(\sum_{j \in J(k)} |\alpha_{k,j}|^p\Big)^{1/p} \right]^q\right\}^{1/q}.$$ \[lemm:BSplineInterpolation\] Under one of the conditions in Proposition \[prop:BesovApproxByNN\] and the condition $0 < s < \min( m , m -1 + 1/p)$ where $m \in \Natural$ is the order of the cardinal B-spline bases, for any $f \in B^s_{p,q}(\Omega)$, there exists $f_N$ that satisfies $$\begin{aligned} \label{eq:ffNoptimalAdaptiveApprox} \|f - f_N\|_{L^r(\Omega)} \lesssim N^{-s /d} \|f\|_{B^s_{p,q}}\end{aligned}$$ for $N \gg 1$, and has the following form: $$\begin{aligned} \label{eq:fNformat} f_N(x) = \sum_{k=0}^K \sum_{j \in J(k)} \alpha_{k,j} M_{k,j}^d(x) + \sum_{k=K+1}^{K^*} \sum_{i=1}^{n_k} \alpha_{k,j_i} M_{k,j_i}^d(x),\end{aligned}$$ where $(j_i)_{i=1}^{n_k}\subset J(k)$, $K = \lceil C_1 \log(N)/d \rceil$, $K^* = \lceil \log(\lambda N) \nu^{-1} \rceil + K + 1$, $n_k = \lceil \lambda N 2^{-\nu (k - K)}\rceil~ (k=K+1,\dots,K^*)$ for $\delta = d(1/p - 1/r)_+$ and $\nu = (s - \delta)/(2\delta)$, and the real number constants $C_1 > 0$ and $\lambda > 0$ are chosen to satisfy $\sum_{k=1}^K (2^k+m )^d + \sum_{k=K+1}^{K^*} n_k \leq N$ independently to $N$. Moreover, we can choose the coefficients $(\alpha_{k,j})$ to satisfy $$\|(\alpha_{k,j})_{k,j}\|_{b^s_{p,q}} \lesssim \|f\|_{B_{p,q}^s}.$$ [@devore1988interpolation] constructed a linear bounded operator $P_k$ having the following form: $$\begin{aligned} P_k(f)(x) = \sum_{j \in J(k)} a_{k,j} M_{k,j}^d(x) \label{eq:PkDecomposition}\end{aligned}$$ where $\alpha_{k,j}$ is constructed in a certain way, where for every $f \in L^p([0,1]^d)$ with $0 < p \leq \infty$, it holds $$\begin{aligned} \| f - P_k(f) \|_{L^p} \leq C w_{r,p}(f,2^{-k}). \label{eq:fPkwkpIneq}\end{aligned}$$ Let $$p_k(f) := P_k(f) - P_{k-1}(f),~~P_{-1}(f) = 0.$$ Then, it is shown that for $0 < p, q \leq \infty$ and $0 < s < \min( m , m -1 + 1/p)$, $f$ belongs to $B_{p,q}^s$ if and only if $f$ can be decomposed into $$f = \sum_{k=0}^\infty p_k(f),$$ with the convergence condition $\|(p_k(f))_{k=0}^{\infty} \|_{b_p^s(L^p)} < \infty$; in particular, $\|f\|_{B_{p,q}^s} \simeq \| (p_k(f))_{k=0}^{\infty} \|_{b_p^s(L^p)} =: (\sum_{k \in \Natural_+} (2^{s k} \|p_k\|_{L^p} )^q)^{1/q} $. Here, each $p_k$ can be expressed as $p_k(x) = \sum_{j \in J(k)} \alpha_{k,j} M_{k,j}^d(x)$ for a coefficient $(\alpha_{k,j})_{k,j}$ (which could be different from $(a_{k,j})_{k,j}$ appearing in ). Hence, $f \in B^s_{p,q}$ can be decomposed into $$\begin{aligned} \label{eq:DevorePopovExpansion} f = \sum_{k=0}^\infty \sum_{j \in J(k)} \alpha_{k,j} M_{k,j}^d(x)\end{aligned}$$ with convergence in the sence of $L^p$. Moreover, it is shown that $\|p_k\|_{L^p} \simeq( 2^{-kd}\sum_{j \in J(k)} |\alpha_{k,j}|^p )^{1/p}$ and thus $$\begin{aligned} \|f\|_{B_{p,q}^s} \simeq \| (\alpha_{k,j})_{k,j}\|_{b^s_{p,q}}. \label{eq:BpqEquivalence}\end{aligned}$$ Based on this decomposition, [@dung2011optimal] proposed an optimal adaptive recovery method such that the approximator has the form under the conditions for $K,K^*,n_k$ given in the statement and satisfies the approximation accuracy . This can be proven by applying the proof of Theorem 3.1 in [@dung2011optimal] to the decomposition instead of Eq. (3.8) of that paper. See also Theorem 5.4 of [@dung2011optimal]. Moreover, the equivalence gives the norm bound of the coefficient $(\alpha_{k,j})$. Basically, we combine Lemma \[lemm:Mnapproximation\] and Lemma \[lemm:BSplineInterpolation\]. We substitute the approximated cardinal B-spline basis $\check{M}$ into the decomposition of $f_N$ . Let the set of indexes $(k,j) \in \Natural \times \Natural$ that consists $f_N$ given in be $E_{N}$: $f_N = \sum_{(k,j)\in E_N} \alpha_{k,j} M_{k,j}^d$. Accordingly, we set $\fcheck := \sum_{(k,j)\in E_N} \alpha_{k,j} \check{M}_{k,j}^d$. For each $x \in \Real^d$, it holds that $$\begin{aligned} |f_N(x) - \fcheck(x)| & \leq \sum_{(k,j) \in E_N} |\alpha_{k,j} | |M_{k,j}^d(x) - \check{M}_{k,j}^d(x)| \\ & \leq \epsilon \sum_{(k,j) \in E_N} |\alpha_{k,j} | \boldone\{M_{k,j}^d(x) \neq 0\} \\ & \leq \epsilon (m+1)^d (1+K^*) 2^{ K^* (d/p-s)_+} \|f\|_{B^s_{p,q}} \\ & \lesssim \log(N) N^{(\nu^{-1} + d^{-1})(d/p-s)_+} \epsilon \|f\|_{B^s_{p,q}}, $$ where we used the definition of $K^*$ in the last inequality. Therefore, for each $f \in U(B^s_{p,q}([0,1]^d))$, it holds that $$\|f - \fcheck\|_{L^r} \lesssim \|f - f_N\|_{L^r} + \|f_N- \fcheck\|_{L^r} \lesssim \log(N) N^{(\nu^{-1} + d^{-1})(d/p-s)_+}\|f\|_{B^s_{p,q}} \epsilon + N^{-s/d}.$$ By taking $\epsilon$ to satisfy $\log(N) N^{(\nu^{-1} + d^{-1})(d/p-s)_+}\epsilon \leq N^{-s/d}$ (i.e., $\epsilon \leq N^{-s/d - (\nu^{-1} + d^{-1})(d/p -s)_+}\log(N)^{-1}$), then we obtain the approximation error bound. Next, we bound the magnitude of the coefficients. Each coefficient $\alpha_{j,k}$ satisfies $|\alpha_{j,k}| \lesssim 2^{k(d/p - s)_+}\|f\|_{B^s_{p,q}} \leq 2^{k(d/p - s)_+} \lesssim N^{(\nu^{-1} + d^{-1})(d/p - s)_+}$ for $k \leq K^*$. Finally, the magnitudes of the coefficients hidden in $\check{M}_{k,j}^d$ are evaluated. Remembering that $\check{M}_{k,j}^m(x) = \check{M}(2^{k} x_1 - j_1,\dots, 2^{k} x_d - j_d)$, we see that we just need to bound the quantity $2^{k}~(k \leq K^*)$. However, this is bounded by $2^k \leq N^{\nu^{-1} + d^{-1}}$ for $k \leq K^*$. Hence, we obtain the assertion. Proof of Theorem \[eq:mBesovApproxByNN\] {#sec:ProofmBesovApproxNN} ======================================== Let $\Natural_+^d(e) := \{ s \in \Natural_+^d \mid s_i = 0, i \not \in e \}$ and for $ k \in \Natural_+^d(e), $ we define $2^{-k}:=(2^{-k_{i_1}}, \dots, 2^{-k_{i_{|e|}}}) \in \Real_+^{|e|}$ where $(i_1,\dots,i_{|e|}) = e$. By defining $\|(g_k)_k\|_{b_{q}^{\alpha,e}} := \left(\sum_{k \in \Natural_+^d(e)} (2 ^{\alpha \|k\|_1} |g_k|)^q \right)^{1/q} $ for a sequence $(g_k)_{k \in \Natural_+^d(e)}$, then it holds that $$|f|_{MB_{p,q}^{\alpha,e}} = \sum_{e \subset \{1,\dots,d\}} \|(w_{r,p}^e(f,2^{-k}))_k\|_{b_{q}^{\alpha,e}}.$$ The result is immediately follows from Theorem \[eq:mBesoApproxSpline\]. Let the set of indexes of $(k,j)$ consisting of $R_K$ be $E_K$: $R_K(f) = \sum_{(k,j) \in E_K} \alpha_{k,j} M_{k,j}^d(x)$. As in the proof of Proposition \[prop:BesovApproxByNN\], we approximate $R_K(f)$ by a neural network given as $$\fcheck(x)= \sum_{(k,j) \in E_K} \alpha_{k,j} \check{M}_{k,j}^d(x).$$ Each coefficient $\alpha_{j,k}$ satisfies $|\alpha_{j,k}| \lesssim 2^{\|k\|_1(1/p - s)_+}\|f\|_{MB^s_{p,q}} \lesssim 2^{K^* (1/p - s)_+}.$ The difference between $$\begin{aligned} |R_K(f) - \fcheck(x)| & \leq \sum_{(k,j) \in E_K} |\alpha_{k,j}| |M_{k,j}^d(x) - \check{M}_{k,j}^d(x)| \\ & \leq \epsilon \sum_{(k,j) \in E_K} |\alpha_{k,j}| \boldone\{M_{k,j}^d(x) \neq 0 \} \\ & \lesssim \epsilon (m+1)^d (1+K^*)D_{K^*,d} 2^{K^*(1/p -s )_+} \|f\|_{MB_{p,q}^s}.\end{aligned}$$ Therefore, by taking $\epsilon$ so that $\epsilon (m+1)^d (1+K^*)D_{K^*,d} 2^{K^*(1/p - s)_+} \leq 2^{-K s}$ is satisfied, it holds that $$|R_K(f) - \fcheck(x)| \lesssim 2^{-K s}.$$ By the inequality $D_{K^*,d} \leq e^{K^* + d-1}$, it suffices to let $\epsilon \leq \frac{e^{- K^*(s + (1/p -s)_+ + 1)}}{ [e(m+1)]^d (1+K^*) }$. The cardinality of $E(K)$ is bounded as $$\begin{aligned} & \sum_{\kappa=0,\dots,K} 2^\kappa {\kappa + d-1 \choose d-1} + \sum_{k: K < \|k\|_1 \leq K^*} n_k \\ \leq & 2^{K+1} {K + d-1 \choose d-1} + \sum_{K < \kappa \leq K^*} 2^{K - \frac{s - \delta}{2 \delta}(\kappa -K)} {\kappa + d-1 \choose d-1} \\ \leq & 2^{K+1} D_{K,d} + 2^K (1 - 2^{- \frac{s - \delta}{2 \delta}})^{-1} D_{K^*,d} \leq (2 + (1 - 2^{- \frac{s - \delta}{2 \delta}})^{-1} ) 2^K D_{K^*,d} = N. $$ Since each unit $\check{M}_{k,j}^d$ requires width $W_0$, the whole width becomes $W = N W_0$. The number of nonzero parameters to construct $\check{M}_{k,j}^d$ is bounded by $S = (L-1)W_0^2 N + N$. Finally, the magnitudes of the coefficients hidden in $\check{M}_{k,j}^d$ are evaluated. Remembering that $\check{M}_{k,j}^d(x) = \check{M}(2^{k_1} x_1 - j_1,\dots, 2^{k_d} x_d - j_d)$, here maximum of $2^{k_j}$ is bounded by $2^{K^*} \lesssim N^{(1 + 1/\nu)}$. Hence, we obtain the assertion. Similarly, it holds that $|\alpha_{j,k}| \lesssim N^{(1 + 1/\nu) \{1 \vee (1/p -s)_+\}}$. Proof of Theorem \[eq:mBesoApproxSpline\] ========================================= Preparation: sparse grid {#sec:SparseGridBound} ------------------------ Here, we give technical details behind the approximation bound. The analysis utilizes the so called [*sparse grid*]{} technique [@smolyak1963quadrature] which has been developed in the function approximation theory field. As we have seen in the above, in a typical B-spline approximation scheme, we put the basis functions $M_{k,j}^m(x)$ on a “regular grid” for $k=1,\dots,K$ and $(j_1,\dots,j_d) \in J(k)$, and take its superposition as $f(x) \approx \sum_{k =1,\dots,K} \sum_{j \in J(k)} \alpha_{k,j} M_{k,j}^m(x)$, which consists of $O(2^{Kd})$ terms (see ). Hence, the number of parameters $O(2^{Kd})$ is affected by the dimensionality $d$ in an exponential order. However, to approximate functions with mixed smoothness, we do not need to put the basis on the whole range of the regular grid. Instead, we just need to put them on a [*sparse grid*]{} which is a subset of the regular grid and has much smaller cardinality than the whole set. The approximation algorithm utilizing sparse grid is based on Smolyak’s construction [@smolyak1963quadrature] and its applications to mixed smooth spaces [@dung1990recovery; @ICM-Satellite:DinhDung:1991; @dung1992optimal; @MathUSSRSob:Temlyakov:1982; @Book:Temlyakov:1993; @JComplexity:Temlyakov:1993]. [@Complexity:Dung:2011] studied an optimal non-adaptive linear sampling recovery method for the mixed smooth Besov space based on the cardinal B-spline bases. We adopt this method, and combining this with the adaptive technique developed in [@dung2011optimal], we give the following approximation bound using a non-linear adaptive method to obtain better convergence for the setting $p < r$. Before we state the theorem, we define an quasi-norm of a set of coefficients $\alpha_{k,j} \in \Real$ for $k \in \Natural_+^d$ and $j \in J_{m}^d(k) := \{- m, - m +1, \dots, 2^{k_1}-1, 2^{k_1} \} \times \dots \times \{- m, - m +1, \dots, 2^{k_d}-1, 2^{k_d} \} $ as $$\|(\alpha_{k,j})_{k,j}\|_{mb_{p,q}^{\alpha}} := \left(\sum_{k \in \Natural_+^d} \left[2^{(\alpha - 1/p)\|k\|_1} \Big( \sum_{j \in J_m^d(k)} |\alpha_{k,j}|^p \Big)^{1/p} \right]^q\right)^{1/q}.$$ \[eq:mBesoApproxSpline\] Suppose that $0 < p,q,r \leq \infty$ and $\alpha > (1/p - 1/r)_+$. Assume that the order $m \in \Natural$ of the cardinal B-spline satisfies $0 < s < \min( m , m -1 + 1/p)$. Let $\delta = (1/p - 1/r)_+$. Then, for any $f \in MB_{p,q}^s(\Omega)$ and $K > 0$, there exists $R_K(f)$ such that $R_K(f)$ can be represented as $$R_K(f)(x) = \sum_{\substack{k \in \Natural_+^d: \\ \|k\|_1 \leq K}} \sum_{j \in J_m^d(k)} \alpha_{k,j} M_{k,j}^d(x) + \sum_{\substack{k \in \Natural_+^d: \\ K < \|k\|_1 \leq K^*}} \sum_{i=1}^{n_k} \alpha_{k,j_i^{(k)}} M_{k,j_i^{(k)}}^d(x),$$ where $K^* = \lceil K(1 + \frac{2\delta}{\alpha - \delta}) \rceil $, $(j_i^{(k)})_{i=1}^{n_k} \subset J_m^d(k)$, and $n_k = \lceil 2^{K -\frac{\alpha - \delta}{2 \delta} (\|k\|_1 - K)} \rceil$, and has the following properties: - For $p \geq r$, $$\begin{aligned} \|f - R_K(f)\|_r \lesssim 2^{-K \alpha } D_{K,d}^{(1/\min(r,1) - 1/q)_+} \|f\|_{MB_{p,q}^s}.$$ - For $p < r$, $$\begin{aligned} \|f - R_K(f)\|_r \lesssim \begin{cases} 2^{-K \alpha } D_{K,d}^{(1/r - 1/q)_+} \|f\|_{MB_{p,q}^s}& (r < \infty), \\ 2^{-K \alpha } D_{K,d}^{(1 - 1/q)_+} \|f\|_{MB_{p,q}^s}& (r = \infty). \end{cases}\end{aligned}$$ Moreover, the coefficients $(\alpha_{k,j})_{k,j}$ can be taken to hold $\|(\alpha_{k,j})_{k,j}\|_{mb_{p,q}^{\alpha}} \lesssim \|f\|_{MB_{p,q}^\alpha}$. The proof is given in Appendix \[sec:ProofOfTheoremmBesoApproxSpline\]. The total number of cardinal B-spline bases consisting of $R_K(f)$ can be evaluated as $$\begin{aligned} & 2^{K+1} {K + d-1 \choose d-1} + \sum_{k: K < \|k\|_1 \leq K^*} n_k \\ \lesssim & 2^K D_{K,d} + 2^K D_{K^*,d} \lesssim 2^K D_{K,d}~~~~~~~~(\because \text{\Eqref{eq:Dkd_upperbound}}).\end{aligned}$$ Here, $D_{K,d}$ can be evaluated as $$D_{K,d} \lesssim K^{d-1} ~~~\text{or}~~~D_{K,d} \lesssim d^K. $$ Therefore, the total number of bases can be evaluated as $$2^K \min\{K^{d-1}, d^K\}$$ which is much smaller than $2^{Kd}$ which is required to approximate functions in the ordinal Besov space (see Lemma \[lemm:BSplineInterpolation\]). In this proposition, $K$ controls the resolution and as $M$ goes to infinity, the approximation error goes to 0 exponentially fast. A remarkable point in the proposition is in the construction of $R_K(f)$ in which the superposition is taken over $\|k\|_1 \leq M$ instead of $\|k\|_\infty \leq K^* = O(K)$. Hence, the number of terms appearing in the summation is at most $O(2^K K^{d-1})$ while the full grid takes $O(2^{K d})$ terms. This represents how the mixed smoothness is important to ease the curse of dimensionality. Several aspects of the m-Besov space such as the optimal $N$-term approximation error and Kolmogorov widths have been extensively studied in the literature (see a comprehensive survey [@dung2016hyperbolic]). An analogous result is already given by [@Complexity:Dung:2011] in which $\alpha > 1/p$ is assumed and a linear interpolation method is investigated. However, our result only requires $\alpha > (1/p - 1/q)_+$. This difference comes from a point that our analysis allows nonlinear adaptive interpolation instead of (linear) non-adaptive sampling considered in [@Complexity:Dung:2011]. Because of this, our bound is better than the optimal rate of linear methods [@Galeev1996; @Romanyuk2001] and non-adaptive methods [@dung1990recovery; @ICM-Satellite:DinhDung:1991; @dung1992optimal; @MathUSSRSob:Temlyakov:1982; @Book:Temlyakov:1993; @JComplexity:Temlyakov:1993] especially in the regime of $p < r$ ([@dung1992optimal] also deals with adaptive method but does not cover $p < r$ for adaptive method). See Proposition \[eq:LinearWidthmBesov\] for comparison. Proof of Theorem \[eq:mBesoApproxSpline\] {#sec:ProofOfTheoremmBesoApproxSpline} ----------------------------------------- Now we are ready to prove Theorem \[eq:mBesoApproxSpline\]. For $k=(k_1,\dots,k_d) \in \Natural_+^d$, let $ P_{k_i}^{(i)} f(x) $ be the function operating $P_k$ defined in to $f$ as a function of $x_i$ with other components $x_j~(j \neq i)$ fixed, and let $$\begin{aligned} p_k := \prod_{i=1}^d (P_{k_i}^{(i)} - P_{k_{i}-1}^{(i)})f. \label{eq:pkdefinition}\end{aligned}$$ Then, $p_k$ can be expressed as $p_k(x) = \sum_{j \in J_m^d(k) }\alpha_{k,j} M_{k,j}^d(x)$. Let $T_{k_i}^{(i)} = I - P^{(i)}_{k_i}$ and $\|f\|_{p,i}$ be the $L^p$-norm of $f$ as a function of $x_i$ with other components $x_j~(j \neq i)$ fixed (i.e., if $p < \infty$, $\|f\|_{p,i}^p = \int |f(x)|^p \dd x_i$), then gives $$\|T_{k_i}^{(i)} f \|_{p,i} \lesssim \sup_{|h_i| \leq 2^{-k_i}} \|\Delta_{h_i}^{r,i}(f) \|_{p,i}.$$ Thus, by applying the same argument again, it also holds $$\begin{aligned} \| \|T_{k_i}^{(i)}T_{k_j}^{(j)} f \|_{p,i} \|_{p,j} & \lesssim \| \sup_{|h_i| \leq 2^{-k_i}} \|\Delta_{h_i}^{r,i}(T_{k_j} f) \|_{p,i} \|_{p,j}\\ & = \sup_{|h_i| \leq 2^{-k_i}} \|\| T_{k_j} \Delta_{h_i}^{r,i}(f) \|_{p,j}\|_{p,i} ~~~(\text{$\because$ the definition of $\Delta_{h_i}^{r,i}$ and Fubini's theorem})\\ & \lesssim \sup_{|h_i| \leq 2^{-k_i}} \sup_{|h_j| \leq 2^{-k_j}} \| \| \Delta_{h_i}^{r,i}(\Delta_{h_j}^{r,j} (f)) \|_{p,j}\|_{p,i}, \end{aligned}$$ for $i \neq j$. Thus, applying the same argument recursively, for $u \subset [d]$, it holds that $$\left\| \prod_{i \in u}T_{k_i}^{(i)} f \right\|_p \lesssim w_{r,p}^u(f,2^{-k})$$ for $k \in \Natural_+^d(u)$. Therefore, since $p_k = \prod_{i=1}^d (T_{k_i - 1}^{(i)} - T_{k_{i}}^{(i)})f = \sum_{u \subset [d]} (-1)^{|u|} \left( \prod_{i \in u} T_{k_i}^{(i)} \prod_{i \not \in u} T_{k_{i-1}}^{(i)} \right)f$, by letting $e = \{i \mid k_i > 0\}$, we have that $$\|p_k\|_p \lesssim \sum_{u \subset [d]} \left\|\left( \prod_{i \in u} T_{k_i}^{(i)} \prod_{i \not \in u} T_{k_{i-1}}^{(i)} \right)f \right\|_{p} \lesssim \sum_{u \subset [d]} w_{r,p}^{\hat{u}}(f,2^{-(k^u)_{\hat{u}}}) \lesssim \sum_{e \subset u} w_{r,p}^{u}(f,2^{-k^u}) $$ where $k^u_i := k_i~(i \in u)$ and $k^u_i := k_i -1~(i \not \in u)$, $\hat{u} = \{ i \mid k^u_i \geq 0\}$, and $(k^u)_{\hat{u}}$ is a vector such that $(k^u)_{\hat{u},i} = k^u_i$ for $i \in \hat{u}$ and $(k^u)_{\hat{u},i} = 0$ for $i \not \in \hat{u}$. Now let $$\|(p_k)_k\|_{b_q^{\alpha}(L^p)} = \Bigg(\sum_{k \in \Natural_+^d} \big (2^{\alpha\|k\|_1} \|p_k\|_{L^p} \big)^q\Bigg)^{1/q}$$ for $p_k \in L^p(\Omega) ~(k \in \Natural_+^d)$. Hence, if we set $a_k = \sum_{e \subset u} w_{r,p}^{u}(f,2^{-k^u})$ for $k \in \Natural^d$ and $e = \{ i \mid k_i > 0\}$, we have that $$\|(p_k)_k\|_{b_q^{\alpha}(L^p)} \lesssim \|(a_k)\|_{b_q^{\alpha}(L^p)} \simeq \|f\|_{MB_{p,q}^s}.$$ On the other hand, following the same line of Theorem 2.1 (ii) of [@Complexity:Dung:2011], we also obtain the opposite inequality $\|f\|_{MB_{p,q}^s} \simeq \|(a_k)\|_{b_q^{\alpha}(L^p)} \lesssim \|(p_k)_k\|_{b_q^{\alpha}(L^p)}$ (note that the analogous inequality to Lemma 2.3 of [@Complexity:Dung:2011] also holds in our setting by replacing $q_s$ with $p_s$ and $\omega_r^e(f,2^{-k})_p$ by $w_{r,p}^e$). Therefore, $f \in MB_{p,q}^\alpha$ if and only if $(p_k)_{k \in \Natural_+^d}$ given by satisfies $\|(p_k)_k\|_{b_q^{\alpha}(L^p)} < \infty$ and $f$ can be decomposed into $ f = \sum_{k \in \Natural_+^d} p_k $ where convergence is in $MB_{p,q}^\alpha$. Moreover, it holds that $\|f\|_{MB^\alpha_{p,q} } \simeq \|(p_k)_k\|_{b_q^{\alpha}(L^p)}$. This can be shown by Theorem 2.1 of [@Complexity:Dung:2011]. Moreover, by the quasi-norm equivalence $\|p_k\|_p \simeq 2^{-\|k\|_1/ p} (\sum_{j \in J_m^d(k)} |\alpha_{k,j}|^p)^{1/p}$, we also have $\|(\alpha_{k,j})_{k,j}\|_{mb_{p,q}^\alpha} \simeq \|f\|_{MB^\alpha_{p,q} } $. If $p \geq r$, the assertion can be shown in the same manner as Theorem 3.1 of [@Complexity:Dung:2011]. For the setting of $p < r$, we need to use an adaptive approximation method. In the following, we assume $p < r$. For a given $K$, by choosing $K^*$ appropriately later, we set $$R_K(f)(x) = \sum_{k \in \Natural_+^d: \|k\|_1 \leq K} p_k + \sum_{k \in \Natural_+^d: K < \|k\|_1 \leq K^*} G_k(p_k),$$ where $G_k(p_k)$ is given as $$G_k(p_k) = \sum_{1 \leq i \leq n_k} \alpha_{k,j_i} M_{k,j_i}^d(x)$$ where $(\alpha_{k,j_i})_{i=1}^{|J_m^d(k)|}$ is the sorted coefficients in decreasing order of their absolute value: $|\alpha_{k,j_1}| \geq |\alpha_{k,j_2}| \geq \dots \geq |\alpha_{k,j_{|J_m^d(k)|}}|$. Then, it holds that $$\|p_k - G_k(p_k)\|_{r} \leq \|p_k\|_{p} 2^{\delta \|k\|_1} n_k^{-\delta},$$ where $\delta := (1/p - 1/r)$ (see the proof of Theorem 3.1 of [@dung2011optimal] and Lemma 5.3 of [@Complexity:Dung:2011]). Moreover, we also have $$\|p_k \|_{r} \leq \|p_k\|_{p} 2^{\delta \|k\|_1}$$ for $k \in \Natural_+^d$ with $\|k\|_1 > K^*$. Here, we define $N$ as $$N = \lceil \log_2(K) \rceil.$$ Let $\epsilon = (\alpha - \delta)/(2\delta)$, and $$K^* = \lceil K(1 + 1/\epsilon) \rceil, $$ and $n_k = \lceil 2^{K -\epsilon(\|k\|_1 - K)}\rceil$ for $k \in \Natural_+^d$ with $K+1 \leq \|k\|_1 \leq K^*$. Then, by Lemma 5.3 of [@Complexity:Dung:2011], we have that $$\begin{aligned} \|f - R_K(f)\|_{L^r}^r & \lesssim \sum_{K < \|k\|_1 \leq K^*} \|p_k - G_k(p_k)\|_{L^r}^r + \sum_{K^* < \|k\|_1 } \|p_k \|_{L^r}]^r \notag \\ & \lesssim \sum_{K < \|k\|_1 \leq K^*} [ \|p_k\|_{p} 2^{\delta \|k\|_1} n_k^{- \delta}]^r + \sum_{K^* < \|k\|_1 } [2^{\delta \|k\|_1} \|p_k \|_{L^p}]^r. \label{eq:fRNsubtraction}\end{aligned}$$ In the following, we require an upper bound of ${k + d-1 \choose d-1 }$. Hence, we evaluate this quantity beforehand. This can be upper bounded by the Stering’s formula as $${k + d-1 \choose d-1 } \leq \frac{\sqrt{2}e}{2 \pi} \underbrace{\left(1 + \frac{d-1}{k}\right)^k \left(1 + \frac{k}{d-1}\right)^{d-1}}_{= D_{k,d}} \leq D_{k,d}.$$ Let $\xi > 0$ be a positive real number satisfying $1 + \xi \geq K^*/K$. We can see that $\xi$ can be chosen as $\xi = 1/\epsilon + o(1)$. Then, we have that $$\begin{aligned} D_{K^*,d} & = D_{K,d} \frac{(1 + \frac{d-1}{K^*})^{K^*}}{(1 + \frac{d-1}{K})^{K}} \frac{(1 + \frac{K^*}{d-1})^{d-1}}{(1 + \frac{K}{d-1})^{d-1}} \leq D_{K,d} \frac{(1 + \frac{d-1}{K^*})^{K^*}}{(1 + \frac{d-1}{K^*})^{K}} \left(\frac{1}{1 + \frac{K}{d-1}} + \frac{K^*}{(d-1) (1 + \frac{K}{d-1})}\right)^{d-1} \notag \\ & \leq D_{K,d} \left(1 + \frac{d-1}{K^*}\right)^{K^*-K} \left(\frac{d -1 + K^*}{d-1 + K}\right)^{d-1} = D_{K,d} \left(1 + \frac{d-1}{K}\right)^{\xi K} \left(1 + \xi\right)^{d-1} \notag \\ & \leq D_{K,d} e^{(d-1)\xi} (1 + \xi)^{d-1} \simeq D_{K,d}. \label{eq:Dkd_upperbound}\end{aligned}$$ \(a) Suppose that $q \leq r$ and $r<\infty$. Then $$\begin{aligned} & \|f - R_K(f)\|_{L^r}^q =\|f - R_K(f)\|_{L^r}^{r \frac{q }{r}} \\ & \lesssim \left\{ \sum_{K < \|k\|_1 \leq K^*} [2^{\delta \|k\|_1 } n_k^{-\delta} \|p_k \|_{L^p}]^r + \sum_{K^* < \|k\|_1 } [2^{\delta \|k\|_1} \|p_k \|_{L^p}]^r \right\}^{\frac{q}{r}} ~~~~~~~~(\text{$\because$ \Eqref{eq:fRNsubtraction}}) \\ & \lesssim \sum_{K < \|k\|_1 \leq K^*} [2^{\delta \|k\|_1 } n_k^{-\delta} \|p_k \|_{L^p}]^q + \sum_{K^* < \|k\|_1 } [2^{\delta \|k\|_1} \|p_k \|_{L^p}]^q \\ & \leq N^{- \delta q} 2^{-(\alpha - \delta)K q} \sum_{K < \|k\|_1 \leq K^*} [\underbrace{2^{-(\alpha - \delta - \delta \epsilon)( \|k\|_1 - K)}}_{\leq 1} 2^{\alpha \|k\|_1} \|p_k \|_{L^p}]^q + 2^{-q (\alpha - \delta) K^*} \sum_{K^* < \|k\|_1 } [2^{\alpha \|k\|_1} \|p_k \|_{L^p}]^q \\ & \lesssim (N^{- \delta} 2^{-(\alpha - \delta)K} +2^{- (\alpha - \delta) K^*} )^q \|f\|_{MB^\alpha_{p,q}}^q \\ & \leq (N^{- \alpha} )^q \|f\|_{MB^\alpha_{p,q}}^q.\end{aligned}$$ (b) Suppose that $q > r$ and $r < \infty$. Then, letting $\nu = q/r ( > 1)$ and $\nu' = 1/(1-1/\nu) = q/(q - r)$, we have $$\begin{aligned} & \|f - R_K(f)\|_{L^r}^r \lesssim \sum_{K < \|k\|_1 \leq K^*} [2^{\delta \|k\|_1 } n_k^{-\delta} \|p_k \|_{L^p}]^r + \sum_{K^* < \|k\|_1 } [2^{\delta \|k\|_1} \|p_k \|_{L^p}]^r ~~~~~(\text{$\because$ \Eqref{eq:fRNsubtraction}})\\ & \leq N^{- \delta r} 2^{-(\alpha - \delta)K r} \sum_{K < \|k\|_1 \leq K^*} [2^{-(\alpha - \delta - \delta \epsilon)( \|k\|_1 - K)} 2^{\alpha \|k\|_1} \|p_k \|_{L^p}]^r + \sum_{K^* < \|k\|_1 } [2^{\alpha \|k\|_1} \|p_k \|_{L^p}]^r (2^{- (\alpha - \delta)\|k\|_1} )^r \\ & \leq (N^{- \delta} 2^{-(\alpha - \delta)K} +2^{- (\alpha - \delta) K^*} )^r \Big\{ \sum_{K < \|k\|_1 \leq K^*} [2^{-(\alpha - \delta - \delta \epsilon)( \|k\|_1 - K)} 2^{\alpha \|k\|_1} \|p_k \|_{L^p}]^r \\ & + \sum_{K^* < \|k\|_1 } [2^{\alpha \|k\|_1} \|p_k \|_{L^p}]^r 2^{-(\alpha -\delta)(\|k\|_1- K^*)r} \Big\} \\ & \leq (N^{- \delta} 2^{-(\alpha - \delta)K} +2^{- (\alpha - \delta) K^*} )^r \left\{ \sum_{K < \|k\|_1 \leq K^*} [ 2^{\alpha \|k\|_1} \|p_k \|_{L^p}]^{r\nu} + \sum_{K^* < \|k\|_1 } [2^{\alpha \|k\|_1} \|p_k \|_{L^p}]^{r\nu} \right\}^{1/\nu}\\ & \times \left\{ \sum_{K < \|k\|_1 \leq K^*} [2^{-(\alpha - \delta - \delta \epsilon)( \|k\|_1 - K) }]^{r \nu'} + \sum_{K^* < \|k\|_1 } [2^{-(\alpha -\delta)(\|k\|_1 - K^*) }]^{r \nu'} \right\}^{1/\nu'} \\ & \lesssim (N^{- \delta} 2^{-(\alpha - \delta)K} +2^{- (\alpha - \delta) K^*} )^r \|f\|_{MB^\alpha_{p,q}}^r D_{K,d}^{r(1/r - 1/q)} ~~~~~~~~~(\because \text{\Eqref{eq:Dkd_upperbound}}) \\ & \lesssim (N^{- \alpha} D_{K,d}^{1/r - 1/q})^r \|f\|_{MB^\alpha_{p,q}}^r.\end{aligned}$$ \(c) Suppose that $r=\infty$. Then, similarly to the analysis in (b), we can evaluate $$\begin{aligned} & \|f - R_K(f)\|_{L^r} \\ & \lesssim N^{- \delta} 2^{-(\alpha - \delta)K} \sum_{K < \|k\|_1 \leq K^*} [2^{-(\alpha - \delta - \delta \epsilon)( \|k\|_1 - K)} 2^{\alpha \|k\|_1} \|p_k \|_{L^p}] + \sum_{K^* < \|k\|_1 } [2^{\alpha \|k\|_1} \|p_k \|_{L^p}] (2^{- (\alpha - \delta)\|k\|_1} ) \\ & \lesssim (N^{- \delta} 2^{-(\alpha - \delta)K} +2^{- (\alpha - \delta) K^*} ) D_{K,d}^{(1 - 1/q)_+} \|f\|_{MB^\alpha_{p,q}} \\ & \lesssim N^{- \alpha} D_{K,d}^{(1 - 1/q)_+} \|f\|_{MB^\alpha_{p,q}} .\end{aligned}$$ Proofs of Theorems \[thm:EstimationErrorNNBesov\] and \[thm:EstimationErrorNN\] {#sec:ProofsOfEstimationErrorBounds} =============================================================================== We use Proposition \[prop:RiskBoundCovering\]. We just need to evaluate the covering number of $\hat{\calF} = \{\fbar \mid f \in \Psi(L,W,S,B)\}$ for $(L,W,S,B)$ given in Theorem \[eq:mBesovApproxByNN\] where $\fbar$ is the clipped function for a given $f$. Note that the covering number of $\hat{\calF}$ is not larger than that of $\Psi(L,W,S,B)$. Thus, we may evaluate that of $\Psi(L,W,S,B)$. From Lemma \[lemm:CovNPhi\], the covering number is obtained as $$\log N(\delta,\hat{\calF},\|\cdot\|_\infty) \lesssim N [\log(N)^2 + \log(\delta^{-1}) ].$$ From Proposition \[prop:BesovApproxByNN\], it holds that $$\|\ftrue - R_K(\ftrue)\|_2 \lesssim N^{-s/d}.$$ Note that $$\|f - \ftrue\|_{\LPi(P_X)}^2 \lesssim \|f - \ftrue\|_2^2.$$ for any $f:[0,1]^d \to \Real$ because $p(x) \leq R$. Therefore, by applying Proposition \[prop:RiskBoundCovering\] with $\delta = 1/n$, we have that $$\begin{aligned} \EE_{D_n}[\|\fhat - \ftrue\|_{\LPi(P_X)}^2] \lesssim N^{-2 s/d} + \frac{N (\log(N)^2 + \log(n)) }{n} + \frac{1}{n}. \label{eq:DnHatBound}\end{aligned}$$ Here, the right hand side is minimized by setting $N \asymp n^{\frac{d}{2s + d}}$ up to $\log(n)^2$-order, and then have an upper bound of the RHS as $$n^{- \frac{2s}{2s + d}} \log(n)^2.$$ This gives the assertion. The proof follows the almost same line as the proof of Theorem \[thm:EstimationErrorNNBesov\]. By noting $S = O(2^{K} D_{K,d})$, $L = O(K)$ and $W = O(2^{K} D_{K,d})$, Lemma \[lemm:CovNPhi\] gives an upper bound of the covering number as $$\log N(\delta,\hat{\calF},\|\cdot\|_\infty) \lesssim 2^{K} D_{K,d} [ K \log(2^{K} D_{K,d}) + \log(\delta^{-1}) ] \lesssim 2^{K} D_{K,d} (K^2 + \log(1/\delta)).$$ Letting $r = 2$, we have that $$\|\ftrue - R_K(\ftrue)\|_2 \lesssim 2^{-s K} D_{K,d}^u$$ where $u = (1 - 1/q)_+$ for $p \geq 2$ and $u = (1/2 - 1/q)_+$ for $p < 2$. Then, by noting that $$\|f - \ftrue\|_{\LPi(P_X)}^2 \lesssim \|f - \ftrue\|_2^2,$$ for any $f:[0,1]^d \to \Real$, and by applying Proposition \[prop:RiskBoundCovering\] with $\delta = 1/n$, we have that $$\begin{aligned} \EE_{D_n}[\|\fhat - \ftrue\|_{\LPi(P_X)}^2] \lesssim 2^{-2 s K}D_{K,d}^{2u} + \frac{2^K D_{K,d} (K^2 + \log(\delta^{-1})) }{n} + \frac{1}{n}. \label{eq:DnHatBound}\end{aligned}$$ Here, we use the following evaluations for $D_{K,d}$: (a) $D_{K,d} \lesssim K^{d-1}$, and (b) $D_{K,d} \lesssim [e(1 + \frac{d}{K})]^{K}$. \(a) For the evaluation, $D_{K,d} \lesssim K^{d-1}$, we have an upper bound of the right hand side of as $$2^{-2sK} K^{2u(d-1)} + \frac{2^K K^{d-1}(K^2 + \log(n)) }{n},$$ which is minimized by setting $K = \lceil \frac{1}{1+2 s} \log_2(n) + \frac{(2u-1)(d - 1)}{1+2 s} \log_2 \log(n) \rceil$ up to $\log\log(n)$-order. In this situation, we have the generalization error bound as $$n^{- \frac{2s}{2s+1}} \log(n)^{\frac{2 (d-1)(u + s)}{1+2s}} \log(n)^2. $$ \(b) For the evaluation, $D_{K,d} \lesssim [e(1 + \frac{d}{K})]^{K} \leq e^K e^d$, gives an upper bound of $$2^{-2 sK} e^{2uK} + \frac{2^K e^K (K^2 + \log(n)) }{n}.$$ Then, the right hand side is minimized by $K =\lceil \frac{1}{1+2s + (1-2u)\log_2(e)} \log_2(n) \rceil$. Then, we have that $$n^{-\frac{2s - 2 u \log_2(e)}{1+2s+(1-2u) \log_2(e)}} \log(n)^2. $$ This gives the assertion. Minimax optimality {#sec:MinimaxMixedSmooth} ================== First note that since $P_X$ is the uniform distribution, it holds that $\|\cdot\|_{\LPiPx} = \|\cdot\|_{\LPi([0,1]^d)}$. The $\epsilon$-covering number $\calN(\epsilon,\calG,\LPiPx)$ with respect to $\LPiPx$ for a function class $\calG$ is the minimal number of balls with radius $\epsilon$ measured by $\LPiPx$-norm needed to cover the set $\calG$ [@Book:VanDerVaart:WeakConvergence]. The $\delta$-packing number $\calM(\delta,\calG,L^2(P_X))$ of a function class $\calG$ with respect to $\LPiPx$ norm is the largest number of functions $\{f_1, \dots, f_{\calM} \} \subseteq \calG$ such that $\|f_i - f_j\|_{\LPiPx} \geq \delta$ for all $i\neq j$. It is easily checked that $$\calN(\delta/2,\calG,\LPiPx) \leq \calM(\delta,\calG,\LPiPx) \leq \calN(\delta,\calG,\LPiPx). \label{eq:NMrelation}$$ For a given $\delta_n > 0$ and $\varepsilon_n > 0$, let $Q$ be the $\delta_n$ packing number $\calM(\delta_n,U(MB_{p,q}^s),\LPi(P_X))$ of $U(MB_{p,q}^s)$ and $N$ be the $\varepsilon_n$ covering number of that. [@JMLR:Raskutti+Martin:2012] utilized the techniques developed by [@AS:Yang+Barron:99] to show the following inequality in their proof of Theorem 2(b) : $$\begin{aligned} \inf_{\fhat} \sup_{\fstar \in U(MB_{p,q}^s)}\EE_{D_n}[\|\fhat - \fstar \|_{\LPiPx}^2] & \geq \inf_{\fhat} \sup_{\fstar \in U(MB_{p,q}^s)} \frac{\delta_n^2}{2} P[\|\fhat - \fstar \|_{\LPiPx}^2 \geq \delta_n^2/2] \\ & \geq \frac{\delta_n^2}{2}\left(1-\frac{\log(N) + \frac{n}{2\sigma^2}\varepsilon_n^2 + \log(2)}{\log(Q)}\right).\end{aligned}$$ Thus by taking $\delta_n$ and $\varepsilon_n$ to satisfy $$\begin{aligned} \frac{n}{2\sigma^2}\varepsilon_n^2 &\leq \log(N), \label{eq:epsilonnbound} \\ 8 \log(N) &\leq \log(Q), \label{eq:NQbound} \\ 4 \log(2) &\leq \log(Q), \label{eq:2Qbound} \end{aligned}$$ the minimax rate is lower bounded by $\frac{\delta_n^2}{4}$. This can be achieved by properly setting $\varepsilon_n \simeq \delta_n$. Now, for given $N$ with respect to $\delta_n > 0$, $M = \log(N)$ satisfies $$\delta_n \gtrsim M^{-s} \log(M)^{(d-1)(s+ 1/2 - 1/q)_+} $$ (Theorem 6.24 of [@dung2016hyperbolic]). Hence, it suffices to take $$\begin{aligned} \label{eq:epsboundfinal} & M \simeq n^{\frac{1}{2s+1}} \log(n)^{\frac{2(d-1)(s + 1/2 - 1/q)_+}{2s+1}}, \\ & \varepsilon_n \simeq \delta_n \simeq n^{- \frac{2s}{2s+1}} \log(n)^{\frac{2(d-1)(s + 1/2 - 1/q)_+}{2s+1}},\end{aligned}$$ which gives the assertion. Auxiliary lemmas ================ Let the $\epsilon$-covering number with respect to $\LPiPx$ for a function class $\calG$ be $\calN(\epsilon,\calG,\LPiPx)$ as defined in the proof of Theorem \[eq:MinimaxOptimalboundOfmBesov\]. \[prop:RiskBoundCovering\] Let $\calF$ be a set of functions. Let $\fhat$ be any estimator in $\calF$. Define $$\Delta_n := \EE_{D_n}\left[\frac{1}{n}\sum_{i=1}^n (y_i - \fhat(x_i))^2 - \inf_{f\in \calF} \frac{1}{n}\sum_{i=1}^n (y_i - f(x_i))^2 \right].$$ Assume that $\|\ftrue \|_\infty \leq F$ and all $f \in \calF$ satisfies $\|f\|_\infty \leq F$ for some $F \geq 1$. If $0 < \delta < 1$ satisfies $\calN(\delta,\calF,\|\cdot\|_\infty) \geq 3$, then there exists a universal constant $C$ such that $$\begin{aligned} & \EE_{D_n}[\|\fhat - \ftrue\|_{\LPi(P_X)}^2] \\ & \leq C (1+\epsilon)^2 \left[ \inf_{f \in \calF} \|f-\ftrue\|_{\LPi(P_X)}^2 + F^2 \frac{\log \calN(\delta,\calF,\|\cdot\|_\infty) - \log(\delta)}{n \epsilon} + \delta F^2 + \Delta_n \right],\end{aligned}$$ for any $\epsilon \in (0,1]$. This is almost direct consequence of Lemma 8 of [@2017arXiv170806633S][^3]. The only difference is the assumption of $\|f\|_\infty \leq F$ for $f \in \calF$ and $f = \ftrue$ while Lemma 8 of [@2017arXiv170806633S] assumed $ 0 \leq f(x) \leq F'$ for $F' > 1$. However, this can be easily fixed by shifting the function value by $+ F$ then the range of $f$ is modified to $[0,2F]$. Then, our situation is reduced to that of Lemma 8 of [@2017arXiv170806633S] by substituting $F' \leftarrow 2F$. \[lemm:CovNPhi\] The covering number of $\Phi(L,W,S,B)$ can be bounded by $$\begin{aligned} \log \calN(\delta,\Phi(L,W,S,B),\|\cdot\|_\infty) & \leq S \log(\delta^{-1} L (B \vee 1)^{L -1} (W+1)^{2L} ) \\ & \leq 2 S L \log(\delta^{-1} L (B \vee 1) (W+1) ).\end{aligned}$$ Given a network $f \in \Phi(L,W,S,B)$ expressed as $$f(x) = ({W^{(L)}} \eta( \cdot) + {b^{(L)}}) \circ \dots \circ ({W^{(1)}} x + {b^{(1)}}),$$ let $$\calA_k(f)(x) = \eta \circ ({W^{(k-1)}} \eta( \cdot) + {b^{(k-1)}}) \circ \dots \circ ({W^{(1)}} x + {b^{(1)}}),$$ and $$\calB_k(f)(x) = ({W^{(L)}} \eta( \cdot) + {b^{(L)}}) \circ \dots \circ ({W^{(k)}} \eta(x) + {b^{(k)}}),$$ for $k=2,\dots,L$. Corresponding to the last and first layer, we define $\calB_{L+1}(f)(x) = x$ and $\calA_{1}(f)(x) = x$. Then, it is easy to see that $f(x) = \calB_{k+1}(f) \circ ({W^{(k)}} \cdot + {b^{(k)}}) \circ \calA_{k}(f)(x)$. Now, suppose that a pair of different two networks $f, g \in\Phi(L,W,S,B)$ given by $$f(x) = ({W^{(L)}} \eta( \cdot) + {b^{(L)}}) \circ \dots \circ ({W^{(1)}} x + {b^{(1)}}),~~ g(x) = ({{W^{(L)}}}' \eta( \cdot) + {{b^{(L)}}}') \circ \dots \circ ({{W^{(1)}}}' x + {{b^{(1)}}}'),$$ has a parameters with distance $\delta$: $\|{W^{(\ell)}} - {{W^{(\ell)}}}'\|_\infty \leq \delta$ and $\|{b^{(\ell)}} - {{b^{(\ell)}}}'\|_\infty \leq \delta$. Now, not that $\|\calA_k(f) \|_\infty \leq \max_j \|{W^{(k-1)}}_{j,:}\|_1 \|\calA_{k-1}(f) \|_\infty + \|{b^{(k-1)}}\|_\infty \leq W B \|\calA_{k-1}(f) \|_\infty + B \leq (B \vee 1) (W +1 )\|\calA_{k-1}(f) \|_\infty \leq (B \vee 1)^{k-1} (W +1 )^{k-1}$, and similarly the Lipshitz continuity of $\calB_k(f)$ with respect to $\|\cdot\|_\infty$-norm is bounded as $ (B W)^{L - k + 1}. $ Then, it holds that $$\begin{aligned} & |f(x) - g(x)| \\ = & \left|\sum_{k=1}^L \calB_{k+1}(g) \circ ({W^{(k)}} \cdot + {b^{(k)}}) \circ \calA_{k}(f)(x) - \calB_{k+1}(g) \circ ({{W^{(k)}}}' \cdot + {{b^{(k)}}}') \circ \calA_{k}(f)(x) \right| \\ \leq & \sum_{k=1}^L (B W)^{L - k } \| ({W^{(k)}} \cdot + {b^{(k)}}) \circ \calA_{k}(f)(x) - ({{W^{(k)}}}' \cdot + {{b^{(k)}}}') \circ \calA_{k}(f)(x) \|_\infty \\ \leq & \sum_{k=1}^L (B W)^{L - k } \delta [W (B \vee 1)^{k-1} (W +1 )^{k-1} + 1] \\ \leq & \sum_{k=1}^L (B W)^{L - k } \delta (B \vee 1)^{k-1} (W +1 )^{k} \leq \delta L (B \vee 1)^{L -1} (W+1)^{L}.\end{aligned}$$ Thus, for a fixed sparsity pattern (the locations of non-zero parameters), the covering number is bounded by $ \left( \delta/ [L (B \vee 1)^{L -1} (W+1)^{L}] \right)^{-S}$. There are the number of configurations of the sparsity pattern is bounded by ${(W+1)^L \choose S} \leq (W+1)^{LS}$. Thus, the covering number of the whole space $\Phi$ is bounded as $$ (W+1)^{LS} \left\{ \delta/ [L (B \vee 1)^{L -1} (W+1)^{L}] \right\}^{-S} = [\delta^{-1} L (B \vee 1)^{L -1} (W+1)^{2L} ]^S,$$ which gives the assertion. [^1]: Since $\Omega = [0,1]^d$ in our setting, the boundedness automatically follows from the continuity. [^2]: We let $\Natural_+ := \{0,1,2,3,\dots\}$, $\Natural_+^d := \{(z_1,\dots,z_d) \mid z_i \in \Natural_+ \}$, $\Real_+ := \{x \geq 0 \mid x \in \Real\}$, and $\Real_{++} := \{x > 0 \mid x \in \Real\}$. [^3]: We noticed that there exit some technical flaws in the proof of the lemma, e.g., an incorrect application of the uniform bound to derive the risk of an estimator. However, these flaws can be fixed and the statement itself holds with a slight modification. Accordingly, there appears $-\log(\delta)$ term and $\delta F$ is replaced by $\delta F^2$

--- abstract: 'Stochastic models for the development of cracks in 1 and 2 dimensional objects are presented. In one dimension, we focus on particular scenarios for interacting and non-interacting fragments during the breakup process. For two dimensional objects, we consider only non-interacting fragments, but analyze isotropic and anisotropic development of fissures. Analytical results are given for many observables. Power-law size distributions are predicted for some of the fragmentation pictures considered.' address: 'Instituto de Física, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, 21941-972 Rio de Janeiro - RJ, Brazil' author: - 'F.P.M. dos Santos' - 'R. Donangelo' - 'S.R. Souza' bibliography: - 'artigo.bib' title: Schematic models for fragmentation of brittle solids in one and two dimensions --- , , and Fracture ,size distributions ,power-law 89.75.-k ,05.90.+m ,46.50.+a Introduction ============ The breakup of a system into many pieces, [*i.e.*]{} fragmentation, is a subject of intensive investigation in different areas of science and engineering [@dynFragReview1999; @Bondorf; @Gross; @fragMolecules]. The interest in this phenonemon ranges from studies of the observed size distributions of atomic nuclei [@Bondorf; @Gross] and chains of molecules [@fragMolecules] to asteroids [@asteroidFrag2000], ice floes [@ice12004; @ice22005; @ice32005], brittle solids [@fragBrittle1; @fragBrittle2; @fragBrittle3; @fragBrittle4; @fragBrittleFluid1], thin glass plates[@fragGlass2002], eggshells [@egg1; @egg2], frozen potatoes [@FragBohr1993], fluids [@fragBrittleFluid1], drops in general [@fragPauloMurilo1; @fragPauloMurilo2], etc. The size distributions observed in the breakup of these systems exhibit, in general, a power-law behavior, suggesting, in some cases, at least, that the process is related to critical phenomena. This peculiarity has led to the development of many schematic models which, to a large extent, disregard microscopic details of the fragmenting macroscopic objects. The fact that the size distributions seem to be fairly insensitive to the constituent elements of the fragmenting body [@FragBohr1993] gives strong support to this approach. Therefore, different treatments, based on quite general assumptions, have been proposed to explain the characteristics of the fragmentation of brittle solids. These approximations include, for instance, mean-field treatments [@meanField1; @meanField2; @meanField3; @meanField4], fractal analysis [@fragGlass2002; @fractalKim1995], dynamical models of granular solids [@granularSolids1], random forces stopping models [@fragRNDForces1], and schematic branching-merging models [@unifrag; @dynFrag2] (for a recente review see [@dynFragReview1999] and references therein). In this work, we present schematic models for the breakup of rods and flat brittle objects. For the latter case, our models are intended to describe the fragmentation of objects which suffer a strong impact on one of its sides, in contrast to those proposed in [@meanField1; @meanField2; @meanField3; @meanField4; @unifrag], in which the stress is uniformly distributed inside the body. Effects associated with anisotropy are also investigated. In the one-dimensional case, we compare the properties of the fragments produced when the interaction between them can be neglected, [*i.e.*]{} when they are ejected from the parent body, with the case in which they continue interacting during the breakup process. In sect. \[sect:models\] we present the models and make analytical predictions related to the size distributions, whereas in sect. \[sect:results\] we discuss and interpret the results. Conclusions are drawn in sect. \[sect:conclusions\]. The models {#sect:models} ========== The main characteristic of the models presented below is that they can be viewed as stochastic processes, in which strict mass conservation is imposed in each event. We present below their detailed formulation. 1-dimensional models {#sec:mod1d} -------------------- Objects whose lengths are large compared with their cross-sections are represented by one-dimensional segments of a line. Thus, in this subsection we consider fractures on a line of unitary length. The number of fractures $N$ is related to the violence of the impact on the object. More specifically, $N$ points $\{x_i\}$, $i=1\dots N$, are associated with the fractures and two adjacents points delimit a fragment, so that $N+1$ fragments are formed at the end of the process. We concentrate on two different fragmentation models, which aim at describing distinct breakup scenarios: [**i-**]{} Non-Interacting Fragments (NIF1), where fracture points are sequentially generated and the $(i+1)$-[*th*]{} crack can only appear at the right hand side of the $i$-[*th*]{} fissure. It is numerically implemented by selecting the first random point $x_1$ uniformly in the interval $(0$,$1)$. Then, the next crack point $x_2$ is uniformly chosen in the interval $(x_1$,$1)$, and so on. This corresponds to a simplified picture for the breakup of an object when the impact zone is concentrated close to its left edge, similar to the (almost) perpendicular fall of a rod. In this case, the dynamics cannot produce further fractures in fragments that have already been released from the parent fragment. [**ii-**]{} Interacting Fragments (IF1), in which the broken pieces exert stress on the others, leading to further cracks. For simplicity, at the $i$-[*th*]{} step, one of the $i+1$ fragments is chosen with equal probability and a fissure point is sampled uniformly inside the selected fragment. The analytical construction of the NIF1 model can be achieved through the following considerations: [**1-**]{} When the $i$-[*th*]{} crack point is made, there exists an equivalence between the fragment just formed and the remaining of the parent body. [**2-**]{} Let us assume that, after the $i$-[*th*]{} fracture, the parent body has length $l$. Then, as the $(i+1)$-[*th*]{} fissure is produced, the probability of it having length $\chi$ must be inversely proportional to its parent’s length $l$. Upon denoting by $P_{i}(\chi)$ the probability density of creating a fragment of size $0<\chi<1$ at the $i$-[*th*]{} crack, one may then write: $$P_{i+1}(\chi)=\int_\chi^1\,\frac{1}{l}P_i(l)\,dl\;. \label{eq:recnif1}$$ By noticing that $P_1(\chi)=1$, this recurrence relation can be iterated, leading to: $$P_i(\chi)=\frac{\left[-\log(\chi)\right]^{i-1}}{(i-1)!}\;, \label{eq:probnif1}$$ so that the probability density of observing a fragment of size $\chi$, produced at any stage of the fragmentation process is: $$P(\chi)=\frac{1}{N}\sum_{i=1}^NP_i(\chi)=\frac{1}{N}\sum_{i=1}^N \frac{\left[-\log(\chi)\right]^{i-1}}{(i-1)!}\;. \label{eq:spronif1}$$ For $N$ large, the above expression can then be approximated by: $$P(\chi)\approx\frac{1}{N}\exp[-\log(\chi)]=\frac{1}{N}\chi^{-1}\;. \label{eq:sumprobnif1}$$ It is worth mentioning that normalization is apparently lost in this approximation, since the integral of $1/\chi$ from 0 to 1 diverges. However, the above expression is strictly valid only for $N\rightarrow \infty$ and, therefore, the $1/N$ factor would cancel out the divergency. Thus, this expression should be considered as an approximation for finite (but large) $N$. Nevertheless, the results shown in the next section reveal that this formula is fairly accurate for modest values of $N$. One should notice that these results are qualitative different from those corresponding to uniform fragmentation [@unifrag], which should take place when the object suffers a violent impact equally distributed over its length. In this case, the probability density of finding a fragment of size $\chi$, after $N$ fractures is [@unifrag]: $$P_N(\chi)=N(1-\chi)^{N-1}\;. \label{eq:probhom}$$ The analysis of the IF1 model is more involved. Although we could not obtain closed expressions, we found useful recurrence relations which allow one to compute the properties of the model very easily. First, one should notice that when a fragment is cracked, the probability distribution for the size of the generated fragment is equal to that of the remnant. Then, the probability density of having a fragment of size $\chi$ after the first crack is made, $\Psi_1(\chi)$, can be expressed as: $$\Psi_1(\chi)=\frac{1}{2}[P_1(\chi)+P_1(\chi)]\;, \label{eq:p1if1}$$ where $P_i(\chi)$ is given by Eq. (\[eq:probnif1\]). The normalization factor $1/2$ is associated with the number of fragments formed after the first crack of the system. When the second fracture is made, there are two contributions which should be taken into account: [**a)**]{} a fragment of size $\chi$ can be produced by the crack of a fragment of size $l$, $\chi < l < 1$, or [**b)**]{} the cracking of a selected fragment may lead to a piece of any size, but the unbroken fragment has size $\chi$. Since (a) can also lead to a fragment of size $1-\chi$ and a remaining part of size $\chi$, which are statistically equivalent, one may write: $$\Psi_2(\chi) = \frac{1}{3}\left[2\int_\chi^1\,\frac{1}{l}\Psi_1(l)\,dl +(2-1)\Psi_1(\chi)\right]\;, \label{eq:pp2_1}$$ where the first term accounts for the contribution of (a) and the second one is associated with (b). The factor $(2-1)$ accounts for the degeneracy of the remaining fragments in (b), recalling their statistical equivalence. The above expression can be further expanded through the use of Eqs. (\[eq:recnif1\]) and (\[eq:p1if1\]), and one obtains: $$\Psi_2(\chi) = \frac{1}{6}\left[4P_2(\chi)+2P_1(\chi)\right]\;. \label{eq:pp2_2}$$ The general expression for $\Psi_i(\chi)$ may be obtained, iteratively, through: $$\Psi_i(\chi)=\frac{1}{i+1}\left[2\left[\int_\chi^1\, \frac{1}{l}\Psi_{(i-1)}(l)\,dl \right]+(i-1)\Psi_{(i-1)}(\chi)\right]\;. \label{eq:ppi}$$ More especifically, if $\{a_i^{(k)}\}$ denote the coefficients of the expansion of $\Psi_i(\chi)$ at the $i$-[*th*]{} step: $$\Psi_i(\chi)=\sum_{k=1}^ia_i^{(k)}P_k(\chi)\;, \label{eq:ppexp}$$ the above recurrence relation gives: $$a_i^{(k)}=\frac{1}{i+1} \left[2(1-\delta_{k,1})a_{(i-1)}^{(k-1)}+(i-1)a_{(i-1)}^{(k)} \right]\;, \label{eq:aexp}$$ where Eq. (\[eq:recnif1\]) has been used and $\delta_{i,j}$ is the usual Kronecker Delta function. Starting from $a_1^{(1)}=1$ and $a_1^{(k)}=0$, $1 < k \le N$, one may thus generate all the coefficients associated with the $N$-[*th*]{} fracture of the system. This procedure is extremely fast and requires a very small amount of computational effort. 2-dimensional models {#sec:mod2d} -------------------- The 2-dimensional objects considered in this sub-section are squares of unitary sides. We assume that, during the fracture process, the broken pieces detach from the body and do not fragment afterwards. The models we discuss aim at describing some of the gross features of the fragmentation of a square plate caused by a strong impact on one of its sides. Therefore, straight fissures always start at the left edge of the square and end up at one of its borders or at another crack, whichever is found first. As in the 1-dimensional case, the number of fractures $N$ is associated with the strength of the impact on the object. The position of each starting point of a crack is uniformly selected along the left edge of the object. We consider two different pictures for the propagation of the cracks: [**a)**]{} Non-Interacting Isotropic Fissure (NIIF2): in which a crack grows as a straight line with equal probability along any direction inside the square. [**b)**]{} Non-Interacting Anisotropic Fissure (NIAF2): where a straight line representing a crack propagates only along selected directions. This version of the model aims at describing the fragmentation of objects, such as cristaline solids, whose internal structure leads to preferential cracking directions. For the sake of simplicity, only two possible directions are considered, $-\pi/4$ or $+\pi/4$, with respect to the perpendicular to the left edge of the square. Each of these two directions is chosen with equal probability. These scenarios lead to qualitatively different fragmentation patterns. This is illustrated in Fig. \[fig:fragpatterns\], which shows samples of the patterns obtained for different values of $N$ for the two cases we have considered. ![Fragmentation patterns produced by the isotropic (upper panel) and anisotropic (lower panel) propagation of the fractures, for $N=10$, $N=50$, and $N=100$ cracks.[]{data-label="fig:fragpatterns"}](pattern.eps){width="10cm"} In contrast to the 1-dimensional cases, analytical predictions are much more difficult in the present situation. However, particular considerations can be made for very small areas. Among the complex geometric objects that may be formed, triangled shaped pieces are much more likely to contribute in this area region, at least for $N$ not too large. The area of the triangles is $\chi^2/2$, where $\chi$ denotes the size of their bases. Since the distances between two adjacent cracks $\chi$ are statistically distributed according to Eq. (\[eq:probhom\]), the probability density $P_N(A)$ of observing a fragment of area $A$, if the system undergoes $N$ fractures, is given by: $$\begin{aligned} P_N(A) & = &\int_0^1\,\delta(A-\frac{\chi^2}{2})N(1-\chi)^{N-1}\,d\chi\nonumber\\ & = &\frac{N}{\sqrt{2A}}\left[1-\sqrt{2A}\right]^{N-1}\;,\;\;A<<1\;. \label{eq:pamsall}\end{aligned}$$ In this expression, we have neglected contributions from other geometric shapes. Similar considerations also hold for the NIAF2 model, but thin trapezoidal shapes, such as those which appear in Fig. \[fig:fragpatterns\], also must be considered. However, the corresponding area should also be proportional to $\chi^2$. Therefore, the area distribution, for small values of $A$, should also be a power-law with exponent $-1/2$, and a similar behavior is expected to be observed in both models. Results {#sect:results} ======= Computational experiments have been carried out for all the models described in the previous section. The corresponding results are presented below, together with their interpretations, accompanied by the analytical explanations we have obtained. 1-dimensional models {#sect:1d} -------------------- The probability density of observing a fragment of size $\chi$, when the violence of the impact on it is such that $N$ fractures are made on the system, has been predicted in the last section, for the two 1-dimensional scenarios we considered. In Fig. \[fig:comp1dm\], we compare our predictions to the results obtained in our computer simulations. ![(Color online) Simulation of 1-dimensional models compared to the analytical predictions of Eqs. (\[eq:spronif1\]) and (\[eq:ppi\]). Results corresponding to the NIF1 model are shown at the left panel, whereas those associated with the IF1 are displayed on the right panel.[]{data-label="fig:comp1dm"}](comp1dm.eps){width="11cm"} It shows that the size distributions corresponding to the different fragmentation modes are qualitatively different. Particularly, one notices that the interaction among the fragments leads to smaller fragment sizes, compared to the NIF1 model. This is quite reasonable, on physical grounds, since any further interaction, after their formation, would reduce their average size, compared to the case in which they do not interact. A striking feature of the NIF1 model is that the approximation, given by Eq. (\[eq:sumprobnif1\]), holds for 10 decades, for $N$ as small as $N=50$, as anticipated in the last section. As a matter of fact, the analytical predictions reproduce the data to a great degree of accuracy. The difference between the two fragmentation scenarios may be further illustrated by considering the average size of the fragments generated at the $i$-[*th*]{} step. In the NIF1 model, the average size of the fragment created at the $i$-[*th*]{} step is given by: $$\langle\chi_i\rangle = \int_0^1\,\chi P_i(\chi)\d\chi = \frac{1}{2^i}\;. \label{eq:avesizeNIF1}$$ To arrive at this expression, we have used Eq. (\[eq:probnif1\]). If the fragments are allowed to interact, as in the case of the IF1 model, the average size can be estimated in the following way. The $i$-[*th*]{} fissure is made with equal probability on each of the $i$, already existent, fragments. The average size of these fragments is equal to $1/i$. Therefore, the size of the $(i+1)$-[*th*]{} fragment, created at the $i$-[*th*]{} step, corresponds to: $$\langle\chi_i\rangle = \frac{1}{2i}\;. \label{eq:avesizeIF1}$$ These results show that the average size of the fragments produced at a given step in the IF1 model is, in general, much larger than in the case of the NIF1 model. This prediction is illustrated in Fig. \[fig:compavesize1d\], as well as the corresponding simulation data. The agreement between the analytical formulae and the numerical simulations is, once again, remarkably good. ![(Color online) Comparison between the average size of the fragments produced at the $i$-[*th*]{} step in the 1-dimensional models.[]{data-label="fig:compavesize1d"}](aveSize1d.eps){width="11cm"} This conclusion apparently contradicts that drawn by the analysis of the size distributions displayed in Fig. \[fig:comp1dm\], [*i.e.*]{} that the interacting fragment picture gives much smaller fragments than in the non-interacting scenario. Nevertheless, the apparent inconsistency disappears if one realizes that the survival probability of the fragment produced in the NIF1 model is 1, whereas the fragments are not preserved during the breakup process in the IF1 model. Therefore, large fragments formed at any step in the NIF1 model remain during the whole process, in contrast to those produced in the IF1 model. The characteristics of the two models may be better understood by analizing the Dalitz plot associated with their final size distributions. This is constructed by selecting the 3 largest fragments within each event, and calculating: $$x_k=\chi_k/\sum_{k=1}^3\chi_k\;,\;\;\;\;\;\;(k=1,\, 2,\,{\rm and}\, 3) \label{eq:Dalitz}$$ which represent the perpendicular distances to the $k$-[*th*]{} side of an equilateral triangle. In this equation, $\chi_k$ corresponds to the size of one of the selected fragments. Thus, a point associated with a given event is plotted inside the triangle using the above expression. To eliminate artificial structures, the indices $\{k\}$ in Eq. (\[eq:Dalitz\]) are randomized in each event. --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Dalitz plots associated with the size distributions of the NIF1 (left panels) and IF1 (right panels) models. For details, see text.[]{data-label="fig:Dalitz"}](Dalitz_1_10.eps "fig:"){width="6cm"} 1.0cm ![Dalitz plots associated with the size distributions of the NIF1 (left panels) and IF1 (right panels) models. For details, see text.[]{data-label="fig:Dalitz"}](Dalitz_2_10.eps "fig:"){width="6cm"} ![Dalitz plots associated with the size distributions of the NIF1 (left panels) and IF1 (right panels) models. For details, see text.[]{data-label="fig:Dalitz"}](Dalitz_1_50.eps "fig:"){width="6cm"} ![Dalitz plots associated with the size distributions of the NIF1 (left panels) and IF1 (right panels) models. For details, see text.[]{data-label="fig:Dalitz"}](Dalitz_2_50.eps "fig:"){width="6cm"} ![Dalitz plots associated with the size distributions of the NIF1 (left panels) and IF1 (right panels) models. For details, see text.[]{data-label="fig:Dalitz"}](Dalitz_1_100.eps "fig:"){width="6cm"} ![Dalitz plots associated with the size distributions of the NIF1 (left panels) and IF1 (right panels) models. For details, see text.[]{data-label="fig:Dalitz"}](Dalitz_2_100.eps "fig:"){width="6cm"} --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- By construction, all the points must lie inside the triangle. When the size distribution is dominated by a large fragment, whereas the others are much smaller, one should observe bumps close to the vertices. On the other hand, if the 3 largest fragments have approximately the same size, the peak of the distribution is found near the center of the triangle. Finally, if the size of two of the selected fragments is similar, but much larger than that of the third one, bumps close to the middle point of the triangle sides should be observed. The results shown in Fig. \[fig:Dalitz\] reveal that a large fragment is always surounded by smaller ones in the NIF1 model, in contrast to the IF1 model in which the three largest fragments have approximately the same size. Thus, the two fragmentation pictures lead to qualitatively different size distributions. 2-dimensional models {#sect:2d} -------------------- Computer simulations have also been carried out for the fragmentation of square objects, described in sect. \[sec:mod2d\]. The results associated with the NIIF2 model are depicted in Fig. \[fig:areaDC\], for different values of $N$. It is clear that the area distribution exhibits two power-law regimes. One of them has been already predicted in the last section, corresponding to $A^{-1/2}$, and is represented by the full lines in this figure. The agreement with the analytical prediction is very good in all cases. However, one notices that deviations appear as $N$ increases. This behavior should be expectedi since, as already mentioned, when more cracks are made on the system, the contribution of objects more complex than triangles becomes non-negligible for small areas. ![(Color online) Probability density of observing a fragment of area $A$, obtained with the NIIF2 model for different values of $N$. For details, see text.[]{data-label="fig:areaDC"}](areaDC2.eps){width="11cm"} One may also notice that another power-law regime, of expoment approximately equal to $-1.4$, appears after the breakdown of Eq. (\[eq:pamsall\]), in the area regions where no particular geometric shape should dominate the distribution. This is illustrated by the dashed lines in this figure. Since we could not obtain any analytical explanation for this power-law behavior, this is an empirical finding. Nevertheless, it agrees qualitatively with the fractal analysis made in ref. [@fractalKim1995] as well as with experimental studies [@FragBohr1993] on the fragmentation of rectangular objects, which have demonstrated that the fragment mass distribution (of relatively large pieces) obeys a power-law with exponent approximately equal to 1.15 – 1.20. However, those experiments focused on impacts equally distributed over the surface of the plate. Since our model is intended to describe a different picture, the small discrepancy between our results and the experimental observations is quite reasonable. The qualitative difference between our results and those just mentioned is the existence of two power-law regimes in our model, whereas only one has been observed in the experiments. Unfortunately, it is rather difficult to investigate our prediction experimentally since the second power-law regime should appear for very small values of the area. ![(Color online) Probability density of observing a fragment of area $A$, obtained with the NIAF2 model for different values of $N$. The inserts correspond to log-linear plots of the respective area distributions. For details, see text.[]{data-label="fig:areaDDin"}](areaDDin2.eps){width="11cm"} The behavior observed in the anisotropic model (NIAF2) is qualitatetively different from the former case, as is illustrated in Fig. \[fig:areaDDin\]. The power-law regime, with exponent $-1/2$, predicted in the last section is indeed observed, but the area distribution tends to develop a gap for $1/3 < A < 1/2$, whereas the production of fragments whose $A>0.75$ is hindered. This trend becomes more and more pronounced as $N$ increases. This feature of the area distribution has a simple explanation, based on the geometric constraints imposed by the anisotropy. In the limit of large $N$, only a few particular shapes may contribute to large areas. More specifically, only those corresponding to the hatched areas in Fig. \[fig:survivingAreas\] would not undergo further fractures and survive until the end of the process. ![Fragments of large areas that could be observed in the end of the process for $N$ large.[]{data-label="fig:survivingAreas"}](survivingAreas.eps){width="10cm"} The area corresponding to the pentagon shown in the left panel of this figure is: $$A(\chi)=1-\frac{\chi^2}{2}-\frac{(1-\chi)^2}{2}\;, \label{eq:areaTrap1}$$ whereas that associated with the trapezoid displayed in the right panel reads: $$A(\chi)=\frac{1}{2}-\frac{\chi^2}{4}-\frac{(1-\chi)^2}{2}\;. \label{eq:areaTrap2}$$ The area of the former lies between $1/2 \le A \le 3/4$, whereas for the latter one has $0 \le A \le 1/3$. The only big triangle which would not be cut into pieces for large $N$ is the one shown in the right panel, but contributes to the area distribution at the right hand side of the gap. Other shapes have vanishing small survival probability, for large $N$, and would, therefore, be broken into pieces, contributing only to small areas. Thus, the preferred cracking directions give rise to the peculiarities observed in the area distributions. We have checked that qualitatively similar results are also obtained for other cracking directions. In a more realistic model, fractures should develop preferentially along the selected directions but, with lower probability, other directions should also be allowed. However, one still finds fingerprints on the area distributions. This is illustrated in Fig. \[fig:areaD3A\], where the area distributions for different values of $N$ are shown, in the case where one allows 3 equiprobable cracking directions: $\theta=-\frac{\pi}{4}$, $\theta=0$, and $\theta=+\frac{\pi}{4}$. Although the gap disappears, due to fissures corresponding to $\theta=0$, the first derivative of $P(A)$ shows a singularity at $A=1/2$. One should also notice that, for large areas, but $A<1/2$, the distribution follows the same power-law found in the isotropic case. ![(Color online) Probability density of observing a fragment of area $A$, obtained with the NIAF2 model for different values of $N$, in the case that 3 cracking directions are allowed. The inserts correspond to log-linear plots of the respective area distributions. For details, see text.[]{data-label="fig:areaD3A"}](areaDDin3_2.eps){width="11cm"} Concluding remarks {#sect:conclusions} ================== The schematic models developed in this work for the fragmenation of one dimensional objects revealed that qualitatively different size distributions should be observed whether the fragments interact or not during the breakup process. Size distributions obeying a power-law are only obtained in the non-interacting scenario. Intuitively, this finding is quite reasonable since the interaction among the fragments should lead to the appearance of a scale. The area distributions, for not too small areas, predicted by our isotropic models are in reasonable agreement with those predicted in [@fractalKim1995] and obtained experimentally in [@FragBohr1993]. However, from simple considerations, we predict that the probability density of finding fragments of small areas should be a power-law with exponent 1/2, in both isotropic and anisotropic fragmenting pictures. Owing to the difficulties in measuring small fragments, we are not aware of experimental observations in this area region, which could be confronted with our predictions. Our results also suggest that clear signatures of anisotropy effects should be found in the area distributions, for not too small values of $A$. Therefore, we suggest that the fragmentation of anisotropic materials be studied to provide more insight in this phenomenon. **Acknowledgements** We would like to thank Dr. K. Sneppen, Dr. P.M.C. de Oliveira, and Dr. C.E. Aguiar for many fruitful discussions. We would like to acknowledge CNPq, FAPERJ, and PRONEX(CNPq-FAPERJ) under contract 26.171.176.2003, for partial finantial support.

--- abstract: 'We propose two alternative entanglement concentration protocols (ECPs) using the Faraday rotation of photonic polarization. Through the single-photon input-output process in cavity QED, it is shown that the maximally entangled atomic (photonic) state can be extracted from two partially entangled states. The distinct feature of our protocols is that we can concentrate both atomic and photonic entangled states via photonic Faraday rotation, and thus they may be universal and useful for entanglement concentration in the experiment. Furthermore, as photonic Faraday rotation works in low-Q cavities and only involves virtual excitation of atoms, our ECPs are insensitive to both cavity decay and atomic spontaneous emission.' author: - 'Zhao-Hui Peng,$^{1,2,}$ Jian Zou,$^{3}$ Xiao-Juan Liu,$^{2}$ Yong-Jun Xiao,$^{1}$ and Le-Man Kuang$^{1,}$[^1]' title: Atomic and photonic entanglement concentration via photonic Faraday rotation --- Entanglement is the key resource in quantum information processing (QIP), such as quantum teleportation [@1], quantum key distribution [@2] and quantum dense coding [@3]. In order to complete such QIP protocols perfectly, the maximally entangled states are usually required. However, the entanglement will inevitably degrade in the process of distribution and storage due to the interaction between system and its external environment. To overcome the dissipation and decoherence, Bennett *et al.* proposed the protocols of entanglement purification [@4] and entanglement concentration [@5]. By use of entanglement purification protocols (EPPs), one can distill a set of mixed entangled states into a subset of highly entangled states with local operation and classical communication [@4]. However, EPPs can only improve the quality of the mixed state and can not get the maximally entangled state. On the other hand, entanglement concentration protocols (ECPs) [@5] can be used to convert the partially entangled pairs to the maximally entangled ones. In the early days, many efforts have been devoted to photonic ECPs with linear [@6; @7] or nonlinear [@8] optical elements. Recently, ECPs of solid state qubits (such as atomic [@9; @10; @11] or electric qubits [@12]) have also been investigated frequently. Cavity quantum electrodynamics (QED) system [@13] is an excellent platform for understanding the fundamental principle of quantum mechanics and investigating QIP. In most of QIP protocols based on cavity QED, they usually require that atoms strongly interact with high-Q cavity field, which guarantees not only entanglement preparation but also further implementation of QIP tasks. However, as the high-Q cavity is well isolated from the environment, it seems unsuitable for efficiently accomplishing the input-output process of photons, which is the key step to implement long-distance QIP in a scalable fashion. Recently, An *et al.* [@14] proposed a novel scheme to implement QIP with a single photon by an input-output process with respect to low-Q cavities. It is shown that the different polarized photon can gain different phase shift when it interacts with the atom trapped in the low-Q cavity, which is known as Faraday rotation [@15]. Due to the fact that photonic Faraday rotation works in low-Q cavities and only involves virtual excitation of atoms, it is insensitive to both cavity decay and atomic spontaneous emission. Following this scheme, various works including entanglement generation [@16], quantum logic gate [@17] and quantum teleportation [@18] have been presented. To our best knowledge, ECPs in cavity QED mainly focused on atomic entanglement concentration [@9; @10; @11], but there is no report on concentration of photonic entanglement. The main reason is that in most cases we are only interested in high-Q cavities not the low-Q ones, and in some cases the cavity mode is even adiabatically eliminated and thus has no contribution to the system evolution in the case of large detuning between the cavity field and atoms [@9; @19]. Inspired by Ref. [@14], we investigate ECPs using the Faraday rotation of photonic polarization. The low-Q cavity and single-photon pulse (three-level atom) are introduced to assist concentration of atomic (photonic) entangled state. Through the single-photon input-output process in cavity QED, we can extract the maximally entangled atomic (photonic) state from two partially entangled states. The distinct feature of our proposals is that we can concentrate both atomic and photonic entangled states via photonic Faraday rotation, and thus they may be universal and useful in the experiment. Furthermore, as our ECPs work in low-Q cavities and only involve virtual excitation of atoms, they are insensitive to both cavity decay and atomic spontaneous emission, and may be feasible with current technology. \[0.22\][]{} Firstly, we briefly review photonic Faraday rotation. Consider a three-level atom interacting with a low-Q cavity (one-side) driving by an input photon pulse, as shown in Fig. 1. The atom has two degenerate ground states ($|g_{L}\rangle$ and $|g_{R}\rangle$) and an excited state ($|e\rangle$). The transitions $|g_{L}\rangle{\leftrightarrow}|e\rangle$ and $|g_{R}\rangle{\leftrightarrow}|e\rangle$ for the atom are assisted respectively by left-circularly ($L$) and right-circularly ($R$) polarized photons and the transition frequency is $\omega_{0}$. We consider the low-Q cavity limit and the weak excitation limit, then can solve the Langevin equations of motion for cavity and atomic lowering operators analytically. Adiabatically eliminating the cavity mode, we obtain the reflection coefficient for the atom-field system as follows [@14] $$\begin{split} r_{j}(\omega_{p}){=}\frac{[i(\omega_{c}{-}\omega_{p}){-}\frac{\kappa}{2}][i(\omega_{0}{-}\omega_{p}){+}\frac{\gamma}{2}]{+}g^{2}} {[i(\omega_{c}{-}\omega_{p}){+}\frac{\kappa}{2}][i(\omega_{0}{-}\omega_{p}){+}\frac{\gamma}{2}]{+}g^{2}}, (j{=}L,R) \end{split}$$ where $\omega_{c}$ and $\omega_{p}$ are the frequencies of the cavity and photon pulse, $\kappa$ and $\gamma$ are the cavity damping rate and atomic decay rate respectively, and $g$ is the atom-cavity coupling strength. Due to the large damping rate of cavity, the absolute value of $r_{j}(\omega_{p})$ is verified to be close to unity [@14]. This implies that the photon experiences a very weak absorption, and thereby we may approximately consider that the output photon only experiences a pure phase shift, i.e., $r_{j}(\omega_{p})=e^{i\phi}$, without any absorption. On the other hand, considering the case $g$=$0$ (the atom uncoupled to the cavity or an empty cavity) we have $r_{j0}(\omega_{p})=\frac{i(\omega_{c}{-}\omega_{p}){-}\frac{\kappa}{2}} {i(\omega_{c}{-}\omega_{p}){+}\frac{\kappa}{2}}$ which can be rewritten as a pure phase shift, i.e., $r_{j0}(\omega_{p}){=}e^{i\phi_{0}}$. If the parameters of atom-field system satisfy $\omega_{0}{=}\omega_{c}$, $\omega_{p}{=}\omega_{c}{-}\kappa/2$ and $g{=}\kappa/2$, we can obtain $\phi{=}\pi$ and $\phi_{0}{=}\pi/2$, corresponding to the evolution of atom and photon as $$\begin{split} &|L\rangle|g_{L}\rangle\rightarrow -|L\rangle|g_{L}\rangle, \ \ |R\rangle|g_{L}\rangle\rightarrow i|R\rangle|g_{L}\rangle,\\ &|L\rangle|g_{R}\rangle\rightarrow i|L\rangle|g_{R}\rangle, \ \ \ |R\rangle|g_{R}\rangle\rightarrow -|R\rangle|g_{R}\rangle. \end{split}$$ In the following ECPs, we will straightforwardly utilize the evolution as shown in Eq. (2) without further illustration. \[0.23\][]{} We now discuss concentration of atomic entangled states via photonic Faraday rotation and the schematic setup is sketched in Fig. 2. Assume that there are two pairs of non-maximally entangled three-level atoms $1$, $2$ and $3$, $4$ as follows $$\begin{split} |\psi\rangle_{12}=a_{1}|g_{L}\rangle_{1}|g_{R}\rangle_{2}+b_{1}|g_{R}\rangle_{1}|g_{L}\rangle_{2},\\ |\psi\rangle_{34}=a_{2}|g_{L}\rangle_{3}|g_{R}\rangle_{4}+b_{2}|g_{R}\rangle_{3}|g_{L}\rangle_{4}, \end{split}$$ where $a_{i}$ and $b_{i}$ $(i{=}1, 2)$ are the normalized coefficients such that $|a_{i}|^{2}{+}|b_{i}|^{2}{=}1$, and we assume that they are all real numbers without loss of generality. In principle, the entangled states $|\psi\rangle_{12}$ and $|\psi\rangle_{34}$, which are prepared by the same experimental setup, have the identical amount of entanglement, i.e., $a_{1}{=}a_{2}$ and $b_{1}{=}b_{2}$. However, the experimental imperfections or the effect of communication channels in the preparation and distribution processes will lead to a tiny deviation between $a_{1}(b_{1})$ and $a_{2}(b_{2})$. For simplicity, we firstly omit the deviation and will discuss its effect to the fidelity of our ECP later. We assume that three spatially separate users, say Alice, Bob and Charlie, share entangled states $|\psi\rangle_{12}$ and $|\psi\rangle_{34}$ where atoms $1$, $4$ are in the hands of Alice and Bob respectively, and atoms $2$, $3$ are all in the hand of Charlie. To extract maximally entangled state from the pair of non-maximally entangled states via photonic Faraday rotation, two low-Q cavities $C_{2}$ and $C_{3}$, where atoms $2$ and $3$ are trapped respectively, are introduced at Charlie’s station. A single-photon pulse with the initial state $|\psi\rangle_{p}{=}\frac{1}{\sqrt{2}}(|L\rangle{+}|R\rangle)$ will be sent through the cavities $C_{2}$ and $C_{3}$ sequentially. Then Charlie performs the Hadamard operation on atoms 2, 3 and photon respectively. Note that atomic Hadamard gate can be implemented by driving the atom with an external classical field (polarized lasers), and the quarter-wave plate (QWP) acts as the role of photonic Hadamard gate. To be concrete, atomic and photonic Hadamard operations can be expressed as $|g_{L}\rangle {\rightarrow} \frac{1}{\sqrt{2}}(|g_{L}\rangle+|g_{R}\rangle)$, $|g_{R}\rangle {\rightarrow} \frac{1}{\sqrt{2}}(|g_{L}\rangle{-}|g_{R}\rangle)$, $|L\rangle{\rightarrow} \frac{1}{\sqrt{2}}(|L\rangle{+}|R\rangle)$ and $|R\rangle{\rightarrow} \frac{1}{\sqrt{2}}(|L\rangle{-}|R\rangle)$. After this evolution process, the quantum state of whole system is $$\begin{split} \sum_{j,k{=}L,R}\frac{1}{2}[{\mp}i|L\rangle|g_{j}\rangle_{2}|g_{k}\rangle_{3} (a_{1}a_{2}|g_{L}\rangle_{1}|g_{R}\rangle_{4}{\pm} b_{1}b_{2}|g_{R}\rangle_{1}|g_{L}\rangle_{4})&\\{+} |R\rangle|g_{j}\rangle_{2}|g_{k}\rangle_{3}(\mp a_{1}b_{2}|g_{L}\rangle_{1}|g_{L}\rangle_{4}{+} b_{1}a_{2}|g_{R}\rangle_{1}|g_{R}\rangle_{4}).& \end{split}$$ Finally, Charlie performs measurement on the states of photon and atoms at his side, and thus the atomic state at Alice’s and Bob’s sides will collapse into one of the corresponding components in Eq. (4). To be explicit, if Charlie’s measurement outcome is $|L\rangle|g_{j}\rangle_{2}|g_{k}\rangle_{3}$ $(j, k{=}L, R)$, the quantum state of atoms $1$, $4$ will be $|\psi\rangle_{14}{=} a_{1}a_{2}|g_{L}\rangle_{1}|g_{R}\rangle_{4}{\pm} b_{1}b_{2}|g_{R}\rangle_{1}|g_{L}\rangle_{4}$ (unnormalized). On the other hand, if Charlie’s measurement outcome is $|R\rangle|g_{j}\rangle_{2}|g_{k}\rangle_{3}$ $(j, k{=}L, R)$, the quantum state of atoms $1$, $4$ will be $|\psi'\rangle_{14}{=}\mp a_{1}b_{2}|g_{L}\rangle_{1}|g_{L}\rangle_{4}+ b_{1}a_{2}|g_{R}\rangle_{1}|g_{R}\rangle_{4}$. Obviously, $|\psi'\rangle_{14}$ are maximally entangled under the previous condition of the initial states, and thus the total successful probability of our ECP is $P{=}2a_{1}^{2}(1{-}a_{1}^{2})$, which is the same as that in Ref. [@9]. In this atomic ECP, two atoms $1$, $4$, which never interacted with each other before, are left in a pure maximally entangled state after the whole operation process. In other words, Alice and Bob are completely passive in the whole concentration process. Starting from this point of view, we can generalize this ECP to reconstruct multi-atom Greenberger-Horne-Zeilinger (GHZ) state from the partially entangled atomic GHZ-class states as follows $$\begin{split} |\Psi\rangle_{1}=a_{1}|g_{L}g_{R}{\cdots}g_{R}\rangle_{C_{1}A_{1}{\cdots}A_{N}}{+}b_{1}|g_{R}g_{L}{\cdots}g_{L}\rangle_{C_{1}A_{1}{\cdots}A_{N}},\\ |\Psi\rangle_{2}=a_{2}|g_{L}g_{R}{\cdots}g_{R}\rangle_{C_{2}B_{1}{\cdots}B_{N}}{+}b_{2}|g_{R}g_{L}{\cdots}g_{L}\rangle_{C_{2}B_{1}{\cdots}B_{N}}, \end{split}$$ where the subscripts $A_{j}$, $B_{j}$$(j{=}1,...,N)$, $C_{1}$ and $C_{2}$ represent atoms held by Alice, Bob and Charlie, respectively. If we define $|g_{L}'\rangle_{A}{=}|g_{L}{\cdots}g_{L}\rangle_{A_{1}{\cdots} A_{N}}$, $|g_{R}'\rangle_{A}{=}|g_{R}{\cdots}g_{R}\rangle_{A_{1}{\cdots} A_{N}}$, $|g_{L}'\rangle_{B}{=}|g_{L}{\cdots}g_{L}\rangle_{B_{1}{\cdots} B_{N}}$ and $|g_{R}'\rangle_{B}{=}|g_{R}{\cdots}g_{R}\rangle_{B_{1}{\cdots} B_{N}}$, the GHZ-class states can be rewritten as $|\Psi\rangle_{1}=a_{1}|g_{L}\rangle_{C_{1}}|g_{R}'\rangle_{A}{+}b_{1}|g_{R}\rangle_{C_{1}}|g_{L}'\rangle_{A}$ and $|\Psi\rangle_{2}=a_{2}|g_{L}\rangle_{C_{2}}|g_{R}'\rangle_{B}{+}b_{2}|g_{R}\rangle_{C_{2}}|g_{L}'\rangle_{B}$. By inspection, they just have the same forms as the entangled states in Eq. (3). Thus, we can adopt the same procedure as that in the case of two-atom entangled state, and reconstruct $2N$-atom GHZ state between Alice and Bob with the successful probability $P{=}2a_{1}^{2}(1{-}a_{1}^{2})$. \[0.24\][]{} Benefitting from the single-photon input-output process in the cavity QED system, we show that photonic Faraday rotation can also be used to concentrate photonic entangled states. The concrete schematic setup for photonic ECP is depicted in Fig. 3. We assume that there are two pairs of partially entangled photonic states as follows $$\begin{split} |\phi\rangle_{12}{=}a_{1}|L\rangle_{1}|R\rangle_{2}{+}b_{1}|R\rangle_{1}|L\rangle_{2},\\ |\phi\rangle_{34}{=}a_{2}|L\rangle_{3}|R\rangle_{4}{+}b_{2}|R\rangle_{3}|L\rangle_{4}, \end{split}$$ where photons $1$, $4$ are in the hands of Alice and Bob respectively, and Charlie holds photons $2$ and $3$. To implement photonic ECP, a three-level atom (labeled as $a$), which is trapped in a low-Q cavity $C_{a}$, is introduced at Charlie’s station. The initial state of atom is $|\phi\rangle_{a}{=}\frac{1}{\sqrt{2}}(|g_{L}\rangle_{a}{+}|g_{R}\rangle_{a})$. Charlie guides photons $2$, $3$ into the cavity sequentially, and then lets them pass QWP1 and QWP2 respectively after leaving the cavity $C_{a}$. By performing the Hadamard operation on atom $a$, the quantum state after this evolution process will be $$\begin{split} \sum_{j,k{=}L,R}\frac{1}{2}[{\mp}i|j\rangle_{2}|k\rangle_{3}|g_{L}\rangle_{a}(a_{1}a_{2}|L\rangle_{1}|R\rangle_{4}{+}b_{1}b_{2}|R\rangle_{1}|L\rangle_{4})\\ {+}|j\rangle_{2}|k\rangle_{3}|g_{R}\rangle_{a}(a_{1}b_{2}|R\rangle_{1}|R\rangle_{4}{\mp}b_{1}a_{2}|L\rangle_{1}|L\rangle_{4})]. \end{split}$$ Charlie then measures the quantum states of photons and atom at his side, followed by the collapse of photonic states at Alice’s and Bob’s side to one of the corresponding components in Eq. (7). In detail, the quantum state of photons $1$, $4$ will be $|\psi\rangle_{14}{=}a_{1}a_{2}|L\rangle_{1}|R\rangle_{4}{+}b_{1}b_{2}|R\rangle_{1}|L\rangle_{4}$ or $|\psi'\rangle_{14}{=}a_{1}b_{2}|R\rangle_{1}|R\rangle_{4}{\mp}b_{1}a_{2}|L\rangle_{1}|L\rangle_{4 }$ corresponding to the measurement outcomes $|j\rangle_{2}|k\rangle_{3}|g_{L}\rangle_{a}$ or $|j\rangle_{2}|k\rangle_{3}|g_{R}\rangle_{a}$ $(j, k{=}L, R)$, respectively. Similar to atomic ECP, $|\psi'\rangle_{14}$ are maximally entangled and then the successful probability of our photonic ECP is $P{=}2a_{1}^{2}(1{-}a_{1}^{2})$. In this photonic ECP, Charlie needs to strictly control the time interval of photons $2$ and $3$ passing the low-Q cavity, in order to avoid the case that both of photons interact with the atom simultaneously. Otherwise, the ECP will fail. However, the order of photon ($2$ or $3$) interacting with atom $a$ will not affect the final results of ECP because the situation of photon $2$ and $3$ is completely equivalent. Similar to the atomic ECP, we can also generalize the photonic ECP to reconstruct multi-photon GHZ state from the partially entangled photonic GHZ-class states $|\Phi\rangle_{1}{=}a_{1}|LR{\cdots}R\rangle_{C_{1}A_{1}{\cdots}A_{N}}{+}b_{1}|RL{\cdots}L\rangle_{C_{1}A_{1}{\cdots}A_{N}}$ and $|\Phi\rangle_{2}{=}a_{2}|LR{\cdots}R\rangle_{C_{2}B_{1}{\cdots}B_{N}}{+}b_{2}|RL{\cdots}L\rangle_{C_{2}B_{1}{\cdots}B_{N}}$ with the same successful probability. It is noted that our ECPs for atomic and photonic states may be universal as the entanglement can be concentrated whenever $a_i{<}b_i$ or $a_i{>}b_i$ for every $i{=}1,2$. We briefly discuss the experimental feasibility of our protocols. Consider a $^{87}$Rb atom trapped in the fiber-based Fabry-Perot cavity [@21]. The states $|\mathrm{F}{=}2,m_{\mathrm{F}}{=}{\pm}1\rangle$ of level $5S_{1/2}$ correspond to degenerate ground states $|g_{L}\rangle$ and $|g_{R}\rangle$ respectively, the state $|\mathrm{F}'{=}3,m_{\mathrm{F}}{=}0\rangle$ of level $5P_{3/2}$ is chosen as the excited state $|e\rangle$ and the corresponding transition frequency $\omega_{0}{=}2\pi c/\lambda$ with $\lambda{=}780$nm (D$_2$ line). In Ref. [@21], the cavity length $L{=}38.6\mu$m, waist radius $w_{0}{=}3.9\mu$m and finesse $\mathcal{F}{=}37000$, which correspond to longitudinal mode number $n{=}99$, the cavity decay rate $\kappa{=}2\pi{\times}53$MHz (the relevant Q factor $Q{=}\omega_c/(2\kappa){=}3.63{\times}10^6$) and the maximal coupling strength $g_{0}{=}2\pi{\times}215$MHz. In our protocols, the atom-cavity coupling strength $g{=}g_{0}\cos(2\pi x/\lambda)$ should be matched with cavity decay rate ($g{=}\kappa/2$), which can be satisfied by adjusting the appropriate atomic longitudinal coordinate ($x{=}n\frac{\lambda}{2}{+}179$nm). Meanwhile, the input photon can be tuned to be nearly resonant with the atom-cavity system, i.e., $\omega_{p}{=}\omega_{c}{-}\kappa/2$. Therefore, based on present experiment technology in cavity QED [@13; @21], the required atom-cavity parameters can be tuned to control the reflectivity of the input photon for obtaining the desired phase shifts. In the following, we consider the possible realization of our ECPs in the context of low-Q cavity. In the experiment, the cavity Q factor and the decay rate are closely related with the cavity finesse which depends solely on the intensity transmission and loss of cavity mirror. In Ref. [@21], if the atom is located at the antinode of the cavity field ($x{=}n\frac{\lambda}{2}$), we can obtain the maximal atom-cavity coupling ($g{=}g_{0}{=}2\pi{\times}215$MHz). Consider the transmission of cavity mirror $\mathcal {T}{=}666$ppm (i.e., the cavity finesse $\mathcal{F}{=}4510$), the practical Q factor of cavity reduces to only $Q{=}4.47\times10^5$ and then the decay rate satisfies $\kappa{=}2g_{0}$. As to the ultra low-Q cavity (high decay rate of cavity $\kappa$), the condition $g{=}\kappa/2$ may also be satisfied by obtaining the large enough coupling between atom and cavity field. However, there are still some imperfections in the realistic experiment. For instance, the cavity resonance frequency may be deviated from the atomic eigenfrequency due to the tiny change of cavity length, and the coupling strength may be not strictly matched with the cavity decay rate because of the variation of atomic position in the cavity. The slight deviation of resonance ($\omega_{c}{\sim}\omega_{0}$) and mismatch of coupling strength ($g{\sim}\kappa/2$) will not change the reflection amplitudes but phase shifts $\phi(\phi_{0})$. In the case of $\omega_{c}{-}\omega_{0}{\approx}\kappa/10$, the phase shifts $\phi{\approx}2.75$ and $\phi_{0}{\approx}1.36$. The fidelity of obtaining atomic and photonic states $|\psi\rangle_{14}$ is about $F{=}\frac{1}{2}[1{-\cos2(\phi{-}\phi_{0})}]{\approx}0.955$. If the coupling strength satisfies $g{\approx}3\kappa/5$, we can obtain the phase shift $\phi{\approx}2.31$ and the fidelity of $|\psi\rangle_{14}$ $F{\approx}0.455$. Interestingly, the fidelity of quantum state $|\psi'\rangle_{14}$ is just $1$ as it is independent of the Faraday rotation angle. Thus, our atomic and photonic ECPs are immune to the experimental imperfections as discussed above. In our atomic and photonic ECPs, we have assumed that the initial condition of the entangled states satisfies $a_{1}{=}a_{2}$ and $b_{1}{=}b_{2}$. But in practice, there may be imperfections in the entanglement preparation and distribution processes, which lead the entangled states into less entangled pure or even mixed ones. Here, we consider that the entanglement preparation process is near perfect and the communication channels between Alice (Bob) and Charlie are of high quality. Then the resulting entangled states, after the entanglement preparation and distribution processes, may have a tiny deviation to the ideal ones, i.e., $a_{2}{=}a_{1}{+}ka_{1}$ with $k$ being a small constant. In this case, the fidelity of obtaining the desired state $|\psi'\rangle_{14}$ is $F(a_{1},k){=}\frac{[\sqrt{1{-}a^{2}_{1}(1{+}k)^{2}}{+}(1{+}k)\sqrt{1{-}a^{2}_{1}}]^2}{2[1{+}(1{+}k)^{2}{-}2a^{2}_{1}(1{+}k)^{2}]}$ [@9]. If we consider $a_{1}\in(0,0.7)$ and $k{=\pm}0.1$, the minimal fidelity $F{=}0.989,0.991$ for $a_{1}{=}0.7$ and $k{=\pm}0.1$, which indicates that the small deviation of coefficients, due to the effect of imperfections described above, only affects the fidelity of the result state slightly. In the following, we make comparison with the previous ECPs. Note that ECPs involving a pair of partially entangled states can be realized via entanglement swapping [@20]. The crucial step of entanglement swapping is the implementation of joint Bell-state measurement, which is also at the heart of other QIP tasks such as quantum teleportaion [@1] and dense coding [@3]. In our atomic and photonic ECPs, we have introduced low-Q cavities, three-level atom and single-photon pulse, and can implement entanglement swapping without joint Bell-state measurement, only by detecting the quantum state of atoms and photons separately. For concentration of atomic entanglement, the distinct advantage of our atomic ECP is that we only need low-Q optical cavity while the high-Q cavity is usually required in Refs. [@9; @10; @11]. In Refs. [@9; @11], the atomic state is used as the flying qubit, but it is actually suitable for acting as stationary qubit which will be feasible in experiment. In Ref. [@10], Cao *et al.* proposed atomic ECP through cavity decay which relies on two leaking photon reaching the beam splitter simultaneously, as well as the high efficiency of two photon detectors. Due to the large inefficiency of photon detector, our atomic ECP may be more efficient than Ref. [@10] as only a single-photon detector is involved for ours. Furthermore, we can use the coherent input pulse to replace the single-photon pulse as shown in Ref. [@17], and also implement atomic ECP with homodyne detection of coherent light, which can greatly relaxes the experiment requirement for photon source and reduce measurement difficulties. On the other hand, we propose to concentrate photonic entanglement via single-photon input-output process in cavity QED for the first time. With the assistance of three-level atom trapped in the low-Q cavity, two-photon and multi-photon maximally entangled states can be reconstructed with the same efficiency (successful probability) as that in Ref. [@6]. In fact, the low-Q cavity and three-level atom function as photonic phase-shift controller in photonic Faraday rotation, which is similar to the cross-kerr nonlinearity [@23]. Therefore, we can also construct photonic parity gate via photonic Faraday rotation and then implement photonic ECP as Ref. [@8]. In conclusion, we have proposed to concentrate atomic and photonic entanglement via photonic Faraday rotation. Through the single-photon input-output process in cavity QED, it is shown that the maximally entangled atomic and photonic state can be extracted from two partially entangled states. In our ECPs, we only need the low-Q cavity, three-level atom and the basic optical elements such as QWP and photon detector to complete entanglement concentration, and they may be feasible with current cavity QED and quantum optics technology. The essential idea in our ECPs is the single-photon input-output process in cavity QED, and thus it may be worth studying entanglement purification and concentration using other similar cavity QED schemes [@24] in the future. 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--- abstract: 'We study the wetting transition and the directed polymer delocalization transition on diamond hierarchical lattices. These two phase transitions with frozen disorder correspond to the critical points of quadratic renormalizations of the partition function. ( These exact renormalizations on diamond lattices can also be considered as approximate Migdal-Kadanoff renormalizations for hypercubic lattices). In terms of the rescaled partition function $z=Z/Z_{typ}$, we find that the critical point corresponds to a fixed point distribution with a power-law tail $P_c(z) \sim \Phi(\ln z)/z^{1+\mu}$ as $z \to +\infty$ ( up to some sub-leading logarithmic correction $\Phi(\ln z)$), so that all moments $z^{n}$ with $n>\mu$ diverge. For the wetting transition, the first moment diverges $\overline{z}=+\infty$ (case $0<\mu<1$), and the critical temperature is strictly below the annealed temperature $T_c<T_{ann}$. For the directed polymer case, the second moment diverges $\overline{z^2}=+\infty$ (case $1<\mu<2$), and the critical temperature is strictly below the exactly known transition temperature $T_2$ of the second moment. We then consider the correlation length exponent $\nu$ : the linearized renormalization around the fixed point distribution coincides with the transfer matrix describing a directed polymer on the Cayley tree, but the random weights determined by the fixed point distribution $P_c(z)$ are broadly distributed. This induces some changes in the travelling wave solutions with respect to the usual case of more narrow distributions.' author: - Cécile Monthus and Thomas Garel title: | Critical points of quadratic renormalizations of random variables\ and phase transitions of disordered polymer models on diamond lattices --- \#1\#2 \#1\#2 Introduction ============= Real-space renormalizations for disordered systems -------------------------------------------------- The choice to work in real-space to define renormalization procedures, which already present a great interest for pure systems [@realspaceRG], becomes the unique choice for disordered systems if one wishes to describe spatial heterogeneities. Whenever these disorder heterogeneities play a dominant role over thermal or quantum fluctuations, the most appropriate renormalizations are strong disorder renormalizations [@StrongRGreview] introduced by Ma-Dasgupta [@Ma-Dasgupta] : as shown by Fisher [@daniel], these strong disorder renormalization rules lead to asymptotic exact results if the broadness of the disorder distribution grows indefinitely at large scales. However, for disordered systems governed by finite-disorder fixed points, where disorder fluctuations remain of the same order of thermal fluctuations, one needs to use more standard real-space renormalization procedures, such as Migdal-Kadanoff block renormalizations [@MKRG]. They can be considered in two ways, either as approximate renormalization procedures on hypercubic lattices, or as exact renormalization procedures on certain hierarchical lattices [@berker; @hierarchical]. One of the most studied hierarchical lattice is the diamond lattice which is constructed recursively from a single link called here generation $n=0$ (see Figure \[figdiamond\]) : generation $n=1$ consists of $b$ branches, each branch containing $2$ bonds in series ; generation $n=2$ is obtained by applying the same transformation to each bond of the generation $n=1$. At generation $n$, the length $L_n$ between the two extreme sites $A$ and $B$ is $L_n=2^n$, the total number $B_n$ of bonds is $B_n=(2 b)^n=L_n^{d_{eff}}$ so that $d_{eff}(b)= \frac{ \ln (2b)}{\ln 2}$ represents some effective dimensionality. ![ Hierarchical construction of the diamond lattice of branching ratio $b$. []{data-label="figdiamond"}](hierarchique.eps){height="6cm"} On this diamond lattice, various disordered spin models have been studied, such as for instance the diluted Ising model [@diluted], random bond Potts model [@potts], and spin-glasses [@hierarchicalspinglass]. Disordered polymer models have also been considered, in particular the wetting on a disordered substrate [@Der_wett; @Tang_Chate] and the directed polymer model [@Der_Gri; @Coo_Der; @Tim; @roux; @kardar; @cao; @tang; @Muk_Bha; @Bou_Sil]. In this paper, we focus on these two polymer models that are described by quadratic renormalization of their partition functions as we now recall. Wetting transition with disorder on the diamond lattice ------------------------------------------------------- On the diamond lattice, the adsorption of a polymer on a disordered substrate is described by the following quadratic recursion for the partition function $Z_n$ of generation $n$ [@Der_wett] $$\begin{aligned} Z_{n+1} = Z_n^{(1)} Z_n^{(2)} + (b-1) Y_n^2 \label{zfullwetting}\end{aligned}$$ where $Y_n= b^{L_n-1}$ represents the number of walks between the two extreme points and satisfies the recursion without disorder $$\begin{aligned} Y_{n+1}=b Y_n^2 \label{ynwetting}\end{aligned}$$ and where $Z_n^{(1)}$ and $Z_n^{(2)}$ represent two independent copies of generation $n$. At generation $n=0$, the lattice reduces to a single bond with a random energy $\epsilon$, for instance drawn from the Gaussian distribution $$\begin{aligned} \rho (\epsilon) = \frac{1}{\sqrt{2\pi} } e^{- \frac{\epsilon^2}{2} } \label{gaussian}\end{aligned}$$ and thus the initial condition for the recursion of Eq. \[zfullwetting\] is simply $$\begin{aligned} Z_{n=0} = e^{- \beta \epsilon} \label{zrecursioninitial}\end{aligned}$$ The temperature only appears in this initial condition. Directed polymer on the diamond lattice --------------------------------------- The model of a directed polymer in a random medium [@Hal_Zha] can also be studied on the diamond hierarchical lattice [@Der_Gri; @Coo_Der; @Tim; @roux; @kardar; @cao; @tang; @Muk_Bha; @Bou_Sil]. The partition function $Z_n$ of the $n$-generation satisfies the exact recursion [@Coo_Der] $$\begin{aligned} Z_{n+1} = \sum_{a=1}^b Z_n^{(2a-1)} Z_n^{(2a)} \label{zrecursion}\end{aligned}$$ where $(Z_n^{(1)},...,Z_n^{(2b)})$ are $(2b)$ independent partition functions of generation $n$. At generation $n=0$, the lattice reduces to a single bond with a random energy $\epsilon$, for instance drawn from the Gaussian of Eq. \[gaussian\] and thus the initial condition for the recursion of Eq. \[zrecursion\] is again given by Eq. \[zrecursioninitial\]. Organization of the paper --------------------------- In this paper, we study the critical points of the quadratic renormalizations described above that correspond to delocalization transitions for the polymer. These two transitions are of course different in nature, since the wetting transition already exists in the pure case, whereas the directed polymer transition only exists in the presence of disorder. However, we will show below that the quadratic form of the renormalizations induce some common properties. It is thus interesting to study them along the same lines to stress their similarities and differences. The paper is organized as follows. The wetting transition on a disordered substrate is discussed in Section \[secwetting\], and studied numerically in Section \[secwettingnume\]. The directed polymer transition is discussed in Section \[secdp\], and studied numerically in Section \[secdpnume\]. In Section \[comparison\], we compare the results on the diamond lattice with respect to the same disordered polymer models defined on hypercubic lattices. Section \[conclusion\] contains the conclusion. The Appendix \[multiplicative\] contains a reminder on multiplicative stochastic processes which is used in Sections \[secwetting\] and \[secdp\]. Wetting on a disordered substrate {#secwetting} =================================== To study the wetting recursion, it is convenient to introduce the reduced partition function $z_n$ and the associated free-energy $f_n$ defined by [@Der_wett] $$\begin{aligned} z_{n} \equiv \frac{ Z_n} {Y_n} \equiv e^{- \beta f_n } \label{defzwetting}\end{aligned}$$ to rewrite the recursion of Eq. \[zfullwetting\] as $$\begin{aligned} z_{n+1} = \frac{z_n^{(1)} z_n^{(2)} + b-1}{b} \label{zrecursionwetting}\end{aligned}$$ Reminder on the pure case {#purewetting} -------------------------- In the pure case, the ratios $z_n$ defined in Eq. \[defzwetting\] are not random but take a single value $R_n$, and the recursion of Eq. \[zrecursionwetting\] reduces to a one-dimensional mapping $T$ $$\begin{aligned} R_{n+1} = \frac{R_n^2 + b-1}{b} \equiv T(R_n) \label{mappingpurewetting}\end{aligned}$$ discussed in [@Der_wett] : for $b>2$, there exists two attractive fixed points $R_{\infty}=1$ (delocalized phase) and $R_{\infty}=+\infty$ (localized phase) separated by the repulsive fixed point $R_c$ (critical point) with $$\begin{aligned} R_c = b-1 \label{rcpure}\end{aligned}$$ The critical exponents are determined by the linearization of the recurrence around the fixed point. Setting $R_n=R_c + \delta_n$, one obtains at linear order $$\begin{aligned} \delta_{n+1} \simeq \lambda \delta_{n} \ \ {\rm with \ \ } \lambda = T'(R_c) = \frac{ 2 R_c}{b} = \frac{ 2 (b-1)}{b} \label{lambdapurewetting}\end{aligned}$$ Note that this factor $\lambda=T'(R_c)>1$ describing the instability of the critical point also governs the growing of the energy $E_n$ exactly at criticality [@Muk_Bha], since the recursion for the energy $$\begin{aligned} E_{n+1} = \frac{ R_n^2 (2 E_n)}{R_n^2 + b-1 } \label{epurewetting}\end{aligned}$$ becomes at criticality $$\begin{aligned} E_{n+1}(T_c) = \frac{ 2 R_c^2 }{R_c^2 + b-1 } E_n(T_c) = \lambda E_n(T_c) \label{ecritipurewetting}\end{aligned}$$ To understand why the same factor $\lambda$ appears, one may introduce the product $U_n=R_n E_n$ that satisfies the recursion $$\begin{aligned} U_{n+1} = \frac{ 2 R_n }{b } U_n \label{upurewetting}\end{aligned}$$ It is then clear that at criticality it coincides with the recursion of the variables $\delta_n$ (Eq. \[lambdapurewetting\]). In conclusion, the variable $\delta_n$ or the energy $E_n$ at criticality grows as $\lambda^n = L_n^{1/\nu}$ in terms of the length $L_n=2^n$ with the critical exponent $$\begin{aligned} \nu = \frac{\ln 2}{ \ln \lambda} = \frac{\ln 2}{ \ln T'(R_c)}\end{aligned}$$ The specific heat exponent satisfies the hyperscaling relation $\alpha=2-\nu$. We refer to [@Der_wett] for more details. Let us now summarize the changes that the presence of frozen disorder will induce : \(i) the one-dimensional mapping of the pure case $R_{n+1}=T(R_n)$ will become the iteration of a probability distribution $Q_{n+1}(z) = {\cal F } \{ Q_n(z) \}$ \(ii) the critical value $R_c$ of the pure case will become an invariant probability distribution $Q_c(z)= {\cal F } \{ Q_c(z) \} $ \(iii) the critical exponent $\nu$ determined by the derivative $T'(R_c)$ in the pure case will be determined by the linearized iteration around the fixed point distribution $Q_c(z)$. But before concentrating on the critical point, we first describe the properties of the renormalization group (RG) flow with disorder in the limits of high and low temperatures. High-temperature RG flow ------------------------- In the high temperature phase, the variables $z_n$ defined in Eq. \[defzwetting\] flow towards $1$ or equivalently the free-energies $f_n$ decay to zero. The linearization of the recursion in this regime yields $$\begin{aligned} f_{n+1} \opsimeq \frac{f_n^{(1)} +f_n^{(2)} }{b} \label{recursionwettinghigh}\end{aligned}$$ For $b>2$, this high-temperature phase exists, the probability distribution of the free-energy converges to a Gaussian, the average and the width decays as power-laws of the length $L_n=2^n$ $$\begin{aligned} \overline{ f_n} && \propto L_n^{- \frac{\ln \frac{b}{2}}{\ln 2} } \\ \sqrt{ \overline{ f_n^2} - (\overline{ f_n} )^2} && \propto L_n^{- \omega_W'(b) } \ \ \ {\rm with } \ \ \ \omega_W'(b)= \frac{\ln \frac{b^2}{2}}{2 \ln 2} \label{wettinghighexponents}\end{aligned}$$ Low-temperature RG flow ------------------------ In the low-temperature phase, the free-energies $f_n$ of Eq. \[defzwetting\] grow extensively with the length $L_n=2^n$, and thus at large scale, the recursion is dominated by the first term in Eq. \[zrecursionwetting\] $$\begin{aligned} f_{n+1} \opsimeq f_n^{(1)} +f_n^{(2)} +... \label{recursionwettinglow}\end{aligned}$$ The probability distribution of the free-energy thus converges to a Gaussian, the average and the width grows as $$\begin{aligned} \overline{ f_n} \propto L_n \\ \sqrt{ \overline{ f_n^2} - (\overline{ f_n} )^2} \propto L_n^{1/2 } \label{wettinglowexponents}\end{aligned}$$ Analysis of the critical invariant distribution ------------------------------------------------ At criticality, to avoid the high-temperature and low-temperature described above, the free-energy $f_n$ of Eq. \[defzwetting\] should remain a random variable of order $O(1)$ with some scale-invariant probability distribution $P_c(f)$ defined on $]-\infty,0]$. Equivalently, the variable $z_n=e^{- \beta_c f_n}$ should have a scale-invariant probability distribution $Q_c(z)$ defined on $[1,+\infty[$. In the following, we derive some of their properties. ### Left-tail behavior of the free-energy distribution Let us introduce the left-tail exponent $\eta_c$ $$\begin{aligned} \ln P_c(f) \opsimeq_{f \to -\infty} - \gamma (-f)^{\eta_c} +... \label{tailscritiwett}\end{aligned}$$ where $(...)$ denote the subleading terms. In the region where $f \to -\infty$, one has effectively the low-temperature recursion $$\begin{aligned} f \opsimeq f^{(1)} +f^{(2)} +... \label{recursioncritileft}\end{aligned}$$ A saddle-point analysis shows that if $f^{(1)}$ and $f^{(2)}$ have a probability distribution with the left tail given by Eq \[tailscritiwett\], their sum $f$ has for left tail $\ln P_c(f) \opsimeq - \gamma (-f)^{\eta_c} 2^{1-\eta_c} +...$. The stability of the critical distribution thus fixes the value of the left tail exponent to $$\begin{aligned} \eta_c=1 \label{etacwett}\end{aligned}$$ So the distribution $P_c(f)$ decays exponentially $$\begin{aligned} P_c(f) \opsimeq_{f \to - \infty} e^{\gamma f} (...) \label{lefttailexpo}\end{aligned}$$ This means that the corresponding distribution $Q_c(z)$ of $z=e^{-\beta_c f}$ presents a power-law tail $$\begin{aligned} Q_c(z) \opsimeq_{z \to + \infty} \frac{ \Phi(\ln z) }{ z^{1+\mu} } \label{powerlawqc}\end{aligned}$$ with some exponent $$\begin{aligned} \mu= \frac{\gamma}{\beta_c} \label{defimu}\end{aligned}$$ and where $\Phi(\ln z)$ represents the subleading terms. ### Analysis in terms of multiplicative stochastic processes {#wettmsp} The fact that a power-law appears in the stationary distribution of some random iteration is reminiscent of multiplicative stochastic processes, whose main properties are recalled in Appendix \[multiplicative\]. For a multiplicative stochastic process $X_n$ described by Eq. \[msp\], the stationary distribution presents a power-law tail of exponent $\mu$ that can be computed in terms of the statistics of the random coefficient $a_n$ via Eq. \[mspmu\]. Here, for the quadratic renormalization of Eq. \[zrecursionwetting\], it is the process $z_n$ itself that also plays the role of the random multiplicative coefficient. As a consequence, it is instructive to analyse the recursion of Eq. \[zrecursionwetting\] along the same lines used to study multiplicative stochastic processes. The necessary stability condition of Eq. \[stabcondition\] translates here into the following condition $$\begin{aligned} \overline{ \ln \frac{z}{b} } \equiv \int_1^{+\infty} dz Q_c(z) \ln z - \ln b <0 \label{stabconditionbis}\end{aligned}$$ The condition of Eq. \[mspmu\] that ensures the stability of the power-law tail via iteration translates into the following self-consistent condition for the tail exponent $\mu$ introduced in Eq. \[powerlawqc\] $$\begin{aligned} 2 \overline{\left( \frac{z}{b} \right)^{\mu} } \equiv 2 \int_1^{+\infty} dz Q_c(z) \ \left( \frac{z}{b} \right)^{\mu} =1 \label{selfmu}\end{aligned}$$ The argument is similar to the computation of Eqs \[eqmsp\]-\[eqmspasymp\], the additional factor of $2$ coming from the fact that $z$ large corresponds to either $z^{(1)}$ large or $z^{(2)}$ large. The condition of Eq. \[selfmu\] means in particular that the subleading term $\Phi(\ln z)$ in Eq. \[powerlawqc\] should ensure the convergence at $(+ \infty)$ of the following integral $$\begin{aligned} \overline{z^{\mu} } = \int^{+\infty} dz Q_c(z) z^{\mu} \sim \int^{+\infty} \frac{dz}{z} \Phi(\ln z) \sim \int^{+\infty} dw \Phi(w) < +\infty \label{subphi}\end{aligned}$$ Since $\Phi$ is a subleading term in Eq \[powerlawqc\], it should not contain an exponential, so its decay should be a power-law $$\begin{aligned} \Phi(w) \opsimeq_{w \to \infty} \frac{1}{ w^{1+\sigma} } \ \ \ \ {\rm with } \ \ \ \sigma>0 \label{subphipower}\end{aligned}$$ Then moments of order $k \leq \mu$ are finite, whereas moments of order $k > \mu$ diverge $$\begin{aligned} \int^{+\infty} dz Q_c(z) z^{k} = + \infty \ \ {\rm for} \ \ \ \ \ \ \ k>\mu \label{momentsdv}\end{aligned}$$ In contrast with multiplicative stochastic processes where the condition of Eq. \[mspmu\] allows to compute the tail exponent $\mu$ in terms of the known statistics of the random coefficient $a_n$, we have obtained here only a self-consistent equation : the selected tail exponent $\mu$ in the region $z \to \infty$ is the exponent that satisfies the condition of Eq. \[selfmu\] that involves the whole distribution for $z \in (1,+\infty[$. However, even if we cannot explicitly compute this exponent $\mu$, we can try to locate it with respect to integer values by considering the integer moments. ### Reminder on transitions of integer moments An important property of quadratic renormalizations is that they lead to closed renormalizations for the integer moments. We now briefly recall the behavior of the first moments discussed in [@Der_wett]. The closed recursion satisfied by the first moment [@Der_wett] $$\begin{aligned} \overline{z_{n+1}} = \frac{ (\overline{z_{n}})^2+b-1 }{b} \label{recursionzav}\end{aligned}$$ coincides with the pure case equation of Eq. \[mappingpurewetting\]. Using the initial condition of Eq. \[zrecursioninitial\], the unstable fixed point of Eq. \[rcpure\] allows the define the annealed temperature via $\overline{ e^{ -\epsilon_i/T_{ann}}} = b-1$ : for $T> T_{ann}$, the averaged value $ \overline{z_{n}}$ goes to $1$, whereas for $T<T_{ann}$, the averaged value $ \overline{z_{n}}$ goes to $+\infty$. So $T_{ann}$ represents the transition of the first moment. To locate $T_c$ with respect to $T_{ann}$, we have to distinguish two possibilities \(i) if the tail exponent $\mu$ of Eq. \[powerlawqc\] satisfies $0<\mu<1$, then its first moment diverges at criticality $\overline{z_c}=+ \infty$ and we have the strict inequality $T_c<T_{ann}$. \(ii) if the tail exponent $\mu$ satisfies $\mu>1$, then its first moment is finite at criticality. The only possible finite stable value is $\overline{z_c}=b-1$ and the critical temperature then coincides with the annealed temperature $T_c=T_{ann}$. However, the analysis of the recursion for the variance leads to the conclusion that the critical temperature is strictly lower than the annealed temperature $T_c<T_{ann}$ as soon as disorder is relevant $b \geq 2+\sqrt{2} \simeq 3.414$ [@Der_wett]. In conclusion, whenever disorder is relevant at criticality, one has the strict inequality $T_c<T_{ann}$, the first moment diverges $\overline z_c = +\infty$, and the tail exponent $\mu$ of Eq. \[powerlawqc\] is smaller than $1$ $$\begin{aligned} 0<\mu<1 \label{rangewettingmu}\end{aligned}$$ Critical exponent $\nu$ {#nuwetting} ------------------------- ### Equivalence with a directed polymer on a Cayley tree In the pure case, the critical exponents are obtained from the linearization around the fixed point (see section \[purewetting\]). To follow the same strategy in the disordered case, we set $z_n=z_c+\delta_n$. At linear order, we obtain the recursion $$\begin{aligned} \delta_{n+1} \simeq \frac{z_c^{(1)}}{b } \delta_{n}^{(2)} + \frac{z_c^{(2)}}{b } \delta_{n}^{(1)} \label{epswetting}\end{aligned}$$ where $z_c^{(1,2)}$ are distributed with the critical distribution $Q_c(z)$. As in the pure case, it is also interesting to write the recursion for the energy $E_n$ $$\begin{aligned} E_{n+1} \simeq \frac{ z_c^{(1)} z_c^{(2)} (E_n^{(1)}+E_n^{(2)}) }{z_c^{(1)}z_c^{(2)} + b-1 } \label{enerwetting}\end{aligned}$$ so that the combination $U_n \equiv z_n E_n$ satisfies at criticality the same recursion as in Eq. \[epswetting\] $$\begin{aligned} U_{n+1} = \frac{z_c^{(1)}}{b } U_{n}^{(2)} + \frac{z_c^{(2)}}{b } U_{n}^{(1)} \label{unwetting}\end{aligned}$$ The recurrence of Eq \[epswetting\] coincides with the transfer matrix $$\begin{aligned} Z_{L+1}= \sum_{i=1}^K e^{- \epsilon_i} Z_L^{(i)} \label{defcayley}\end{aligned}$$ for the partition function $Z_L$ of a directed polymer on a Cayley tree of branching ratio $K=2$ with random bond energies $\epsilon_i$. [@Der_Spo; @Der_review]. The differences with Eq. \[epswetting\] we are interested in are the following : \(i) the partition function $Z_L$ in Eq \[defcayley\] is positive by definition, whereas here the random perturbation $\delta_n$ in Eq. \[epswetting\] are a priori of arbitrary sign. Eq. \[epswetting\] is thus more related to the case of a directed polymer model with complex weights studied in [@cayleycomplex]. \(ii) the weights $ e^{- \beta \epsilon_i}$ associated to the bond energies $\epsilon_i$ in Eq. \[defcayley\] are now random weights $\frac{z_c}{b}$ distributed with the fixed point distribution $Q_c(z)$ $$\begin{aligned} e^{- \beta \epsilon_i} \to \frac{z_c}{b} \label{equiwetting}\end{aligned}$$ In particular, these weights present a broad power-law tail in $1/z_c^{1+\mu}$ in contrast with the usual case where the energies $\epsilon_i$ are Gaussian. The difference (ii) turns out to be very important as we now explain. ### Tails analysis Let us consider the first iteration $$\begin{aligned} \delta_1 = \frac{z_c^{(1)}}{b } \delta_{0}^{(2)} + \frac{z_c^{(2)}}{b } \delta_{0}^{(1)}\end{aligned}$$ Suppose we start with a narrow distribution ${\cal P}_0(\delta_0)$ for the random initial perturbation $\delta_0$. The distribution ${\cal P}_1(\delta_1) $ after the first iteration will nevertheless present power-law tails inherited from the fixed point distribution $Q_c(z_c) \sim \Phi(\ln z_c) /z_c^{1+\mu}$ with $0<\mu<1$ (Eqs \[powerlawqc\] and \[rangewettingmu\]). More precisely, the tail in the region $\delta_1 \to +\infty$ is dominated by the events where $z_c^{(1)}$ is large with $\delta_{0}^{(2)} >0$ or where $z_c^{(2)}$ is large with $\delta_{0}^{(1)} >0$, and one obtains $$\begin{aligned} {\cal P}_1 (\delta_1) && \opsimeq_{\delta_1 \to +\infty} 2 \int dz_c Q_c(z_c) \int_0^{+\infty} d \delta_0 {\cal P}_0(\delta_0) \delta \left[ \delta_1- \frac{z_c}{b } \delta_{0} \right] \\ && \opsimeq_{\delta_1 \to +\infty} \frac{\Phi(\ln \delta_1)}{\delta_1^{1+\mu}} \left[ \frac{2}{b^{\mu}} \int_0^{+\infty} d \delta_0 {\cal P}_0(\delta_0)\ \delta_0^{\mu} \right]\end{aligned}$$ Similarly, the left tail reads $$\begin{aligned} {\cal P}_1 (\delta_1) && \opsimeq_{\delta_1 \to -\infty} \frac{\Phi(\ln \vert \delta_1 \vert)}{\vert \delta_1 \vert^{1+\mu}} \left[ \frac{2}{b^{\mu}} \int_{-\infty}^{0} d \delta_0 {\cal P}_0(\delta_0)\ \vert \delta_0\vert^{\mu} \right]\end{aligned}$$ It is then clear that by iteration all distributions ${\cal P}_n(\delta_n) $ will present these power-law tails $$\begin{aligned} {\cal P}_n (\delta_n) && \oppropto_{\delta_n \to \pm \infty} \frac{\Phi(\ln \vert \delta_n \vert)}{\vert \delta_n \vert^{1+\mu}} \label{defamplitude}\end{aligned}$$ Since we are looking for the Lyapunov exponent $v$ governing the typical growth of the perturbation $$\begin{aligned} \frac{\delta_{n+1}}{\delta_n} \sim e^{v } \label{defvwett}\end{aligned}$$ it is convenient to rescale the iteration of Eq. \[epswetting\] by the factor $e^{-v}$ $$\begin{aligned} y_{n+1} = e^{-v} \left[ \frac{z_c^{(1)}}{b } y_{n}^{(2)} + \frac{z_c^{(2)}}{b } y_{n}^{(1)} \right] \label{epswettingrescal}\end{aligned}$$ and to ask that the probability distribution $P_n(y)$ converges as $n \to \infty$ towards a stable distribution $P_{\infty}(y)$ presenting the tails (Eq \[defamplitude\]) $$\begin{aligned} P_{\infty} (y) && \opsimeq_{y \to \pm \infty} B^{\pm} \frac{\Phi(\ln \vert y \vert)}{\vert y \vert^{1+\mu}} \label{defamplitudey}\end{aligned}$$ Reasoning as before, a large value of $y_{n+1}$ corresponds to a large value of one of the four variables $(y_{n}^{(1)},y_{n}^{(2)},z_c^{(1)},z_c^{(2)}) $, and one obtains the following equations $$\begin{aligned} B^+ && = \left[ B^+ \frac{2 e^{-\mu v} \overline{ z_c^{\mu} } }{b^{\mu}} + \frac{2 e^{-\mu v}}{b^{\mu}} \int_0^{+\infty} dy P_{\infty}(y) \ y^{\mu} \right] \\ B^- && = \left[ B^- \frac{2 e^{-\mu v} \overline{ z_c^{\mu} } }{b^{\mu}} + \frac{2 e^{-\mu v}}{b^{\mu}} \int_{-\infty}^0 dy P_{\infty}(y) \vert y \vert^{\mu} \right]\end{aligned}$$ This shows that positive perturbations (initial distribution ${\cal P}_0(\delta_0<0)=0$) or symmetric perturbations (symmetric initial distribution ${\cal P}_0(\delta_0)={\cal P}_0(-\delta_0)$) actually lead to the same Lyapunov exponent $v$ $$\begin{aligned} e^{\mu v} = \frac{2 \overline{ z_c^{\mu} } }{b^{\mu}} + \frac{2 }{b^{\mu}} \int_0^{+\infty} dy \frac{P_{\infty}(y)}{B_+} \ y^{\mu} = 1+ \frac{2 }{b^{\mu}} \int_0^{+\infty} dy \frac{P_{\infty}(y)}{B_+} \ y^{\mu} \label{resvelocitywett}\end{aligned}$$ where we have used Eq. \[selfmu\]. The first term corresponds to the usual term for the velocity of the travelling wave approach [@Der_Spo; @Der_review], whereas the second term originates from the broad distribution of weights $(z_c/b)$. Its physical meaning is the following : in the usual case of a narrow distribution of the weights, the travelling wave approach allows to compute the velocity in terms of the weight statistics alone, because one can write a closed equation for the tail of the process [@Der_Spo; @Der_review]; in the present case, the tail of the process does not satisfy a closed equation, because the broadness of the weight distribution induces some interaction between the tail and the bulk of the process : the second term in Eq. \[resvelocitywett\] represents the influence of the bulk of the distribution $P_{\infty}(y)$ onto the tail of exponent $\mu$. The exponent $\nu$ describing the power-law growth $\delta_n \sim L_n^{1/\nu} \sim e^{vn}$ reads in terms of the Lyapunov exponent $$\begin{aligned} \nu= \frac{\ln 2}{v} \label{nuwett}\end{aligned}$$ Note that the presence of the second term in Eq. \[resvelocitywett\] is crucial to obtain a finite exponent $\nu$ : without this second term, the Lyapunov exponent $v$ would vanish ($v=0$) and the correlation length exponent would diverge ($\nu=\infty$). Numerical study of the wetting transition {#secwettingnume} =========================================== Numerical method {#wettnume} ------------------ We have performed numerical simulations with the so-called ’pool-method’ which is very much used for disordered systems on hierarchical lattices [@hierarchicalspinglass; @Coo_Der] : the idea is to represent the probability distribution $P_n(F_n)$ of the free-energy $F_n=-T \ln Z_n$ at generation $n$, by a pool of $N$ values $\{F_n^{(1)},..,F_n^{(N)} \}$. The pool at generation $(n+1)$ is then obtained as follows : each value $F_{n+1}^{(i)}$ is obtained by choosing two values at random from the pool of generation $n$ and by applying the renormalization Eq. \[zrecursionwetting\]. The results presented in this Section have been obtained for the branching ratio $b=5$, with a pool number $N=4.10^7$, with initial Gaussian energies (Eq. \[gaussian\]). The corresponding annealed temperature is $$\begin{aligned} T_{ann} = \frac{1}{\sqrt{2 \ln(b-1) } } \simeq 0.60055 \label{tannealed}\end{aligned}$$ Finally, the relation between the true free-energy $F_n=- T \ln Z_n$ and the reduced free-energy $f_n=-T \ln z_n$ used in the previous section is simply (Eq \[defzwetting\]) $$\begin{aligned} F_n= f_n-T \ln Y_n \label{grandfpetitf}\end{aligned}$$ where $Y_n=b^{L_n-1}$ does not contain any disorder. As a consequence, the two free-energy distributions have the same width $\Delta F_n = \Delta f_n$, and the same tail properties. Flow of the free-energy width $\Delta F_L$ -------------------------------------------- ![(Color online) Wetting transition : log-log plot of the width $\Delta F(L)$ of the free-energy distribution as a function of $L$, for many temperatures. []{data-label="figwettb5freewidth"}](xm56wett.mcb5.tall.eps){height="6cm"} The flow of the free-energy width $\Delta F_L$ as $L$ grows is shown on Fig. \[figwettb5freewidth\] for many temperatures. One clearly sees the two attractive fixed points on this log-log plot. For $T>T_c$, the free-energy width decays asymptotically with the exponent $\omega_W'(b)$ introduced in Eq. \[wettinghighexponents\] $$\begin{aligned} \Delta F(L) \simeq \left( \frac{L}{\xi_F^+(T)} \right)^{- \omega_W'(b=5)} \ \ { \rm with } \ \ \omega_W'(b=5)= \frac{\ln (b^2/2) }{ 2 \ln 2} = 1.8219.. \label{fwidthabovewett}\end{aligned}$$ where $\xi_F^+(T)$ is the corresponding correlation length that diverges as $T \to T_c^+$. For $T<T_c$, the free-energy width grows asymptotically with the exponent $1/2$ (see Eq. \[wettinglowexponents\] ) $$\begin{aligned} \Delta F(L) \simeq \left( \frac{L}{\xi_F^-(T)} \right)^{ 1/2} \label{fwidthbelowwett}\end{aligned}$$ where $\xi_F^-(T)$ is the corresponding correlation length that diverges as $T \to T_c^-$. The critical temperature obtained by this pool method depends of the pool, i.e. of the discrete sampling with $N$ values of the continuous probability distribution. It is expected to converge towards the thermodynamic critical temperature $T_c$ only in the limit $N \to \infty$. Nevertheless, for each given pool, the flow of free-energy width allows a very precise determination of this pool-dependent critical temperature, for instance in the case considered $0.52415 < T_c^{pool} < 0.52416$, which is significantly below the annealed temperature of Eq \[tannealed\]. Divergence of the correlation lengths $\xi_F^{\pm}(T)$ -------------------------------------------------------- ![(Color online) Wetting transition : Correlation length $\xi_F^{\pm}(T)$ as measured from the behavior of the free-energy width (Eqs \[fwidthabovewett\] and \[fwidthbelowwett\]) (a) $\ln \xi_F^{\pm}(T)$ as a function of $T$ (b) $\ln \xi_F^{\pm}(T)$ as a function of $\ln \vert T_c-T \vert$ : the asymptotic slopes are of order $\nu \sim 6.2 $ []{data-label="figb5logxifreewett"}](xmlogxi.56wettb5.tall.eps "fig:"){height="6cm"} ![(Color online) Wetting transition : Correlation length $\xi_F^{\pm}(T)$ as measured from the behavior of the free-energy width (Eqs \[fwidthabovewett\] and \[fwidthbelowwett\]) (a) $\ln \xi_F^{\pm}(T)$ as a function of $T$ (b) $\ln \xi_F^{\pm}(T)$ as a function of $\ln \vert T_c-T \vert$ : the asymptotic slopes are of order $\nu \sim 6.2 $ []{data-label="figb5logxifreewett"}](xmfitpowerlogxi.56wettb5.tall.eps "fig:"){height="6cm"} The correlation lengths $\xi_F^{\pm}(T)$ as measured from the free-energy width asymptotic behaviors above and below $T_c$ (Eqs \[fwidthabovewett\] and \[fwidthbelowwett\] ) are shown on Fig. \[figb5logxifreewett\] (a). The plot in terms of the variable $\ln \vert T_c-T \vert$ shown on Fig. \[figb5logxifreewett\] (b) indicate a power-law divergence with the same exponent $$\begin{aligned} \xi_F^{\pm}(T) \oppropto_{T \to T_c} \vert T-T_c \vert^{-\nu} \ \ { \rm with } \ \ \nu \simeq 6.2 \label{xifreenuwett}\end{aligned}$$ Histogram of the free-energy ------------------------------ ![(Color online) Wetting transition: Log-plot of the asymptotic distribution $\Pi_F$ of the rescaled free-energy $x_F= \frac{F-F_{av(L)}}{\Delta F(L)} $ in the low-temperature phase (here $T=0.25$), in the high-temperature phase ( here $T=1$) and at criticality (here $T_c^{pool}=0.524155$) []{data-label="figwettb5freehisto"}](xm52wett.mchistob5.tall.eps){height="6cm"} The asymptotic probability distribution $\Pi_F$ of the rescaled free-energy $$\begin{aligned} x_F \equiv \frac{F-F_{av(L)}}{\Delta F(L)}\end{aligned}$$ is shown on Fig \[figwettb5freehisto\] for three temperatures : \(i) the distribution is Gaussian both for $T>T_c$ and $T<T_c$ as expected from Eqs \[recursionwettinghigh\] and \[recursionwettinglow\]. \(ii) at criticality, one clearly see that a left-tail develops in the region $f \to -\infty$ with tail exponent $\eta_c =1$ in agreement with Eq \[etacwett\]. The corresponding power-law exponent of Eq \[defimu\] of the fixed-point distribution of Eq. \[powerlawqc\] is of order $$\begin{aligned} \mu \sim 0.45 \label{muvaluewett}\end{aligned}$$ The measure is not very precise because one clearly sees on Fig. \[figwettb5freehisto\] that on top of this power-law, there exists oscillations reflecting the discrete nature of the renormalization. But this value is anyway in the expected interval of Eq \[rangewettingmu\]. Flow of the energy width -------------------------- ![(Color online) Wetting transition : Flow of the widths $\Delta E(L)$ of the energy distribution as $L$ grows (a) $\ln \Delta E(L)$ as a function of $\ln L$ for many temperatures (b) Comparison of $\ln \Delta E(L)$ , $ \ln \Delta S(L)$ and $ \ln \Delta F(L)$ as a function of $\ln L$ at criticality ($T_c^{pool}=0.524155$).[]{data-label="figwettb5enerwidth"}](xm76wett.mcb5.tall.eps "fig:"){height="6cm"} ![(Color online) Wetting transition : Flow of the widths $\Delta E(L)$ of the energy distribution as $L$ grows (a) $\ln \Delta E(L)$ as a function of $\ln L$ for many temperatures (b) Comparison of $\ln \Delta E(L)$ , $ \ln \Delta S(L)$ and $ \ln \Delta F(L)$ as a function of $\ln L$ at criticality ($T_c^{pool}=0.524155$).[]{data-label="figwettb5enerwidth"}](xm567686.wettb5t0.524155.criti.eps "fig:"){height="6cm"} The flow of the energy width $\Delta E(L)$ as $L$ grows are shown on Fig. \[figwettb5enerwidth\] for many temperatures. For $T>T_c$, we find that the width decays asymptotically with the same exponent $\omega_{\infty}^{W}(b)$ as the free-energy (Eq \[fwidthabovewett\]) $$\begin{aligned} \Delta E(L) \simeq L^{- \omega_{\infty}^{W}(b)} \ \ { \rm with } \ \ \omega_{\infty}^{W}(b=5)= \frac{\ln (b^2/2) }{ 2 \ln 2} = 1.8219.. \label{wettewidthabove}\end{aligned}$$ For $T<T_c$, this width grows asymptotically with the exponent $1/2$ as the free-energy (Eq \[fwidthbelowwett\]) $$\begin{aligned} \Delta E(L) \simeq L^{\frac{1}{2}} \\ \label{wettewidthbelow}\end{aligned}$$ Exactly at criticality, the free-energy $\Delta F(L)$ width converges towards a constant, whereas the energy width grows as a power-law (see Fig \[figwettb5enerwidth\] b) $$\begin{aligned} \Delta E(L) \simeq L^{y_c} \ \ \ {\rm with } \ \ y_c \simeq 0.16 \label{wettewidthcriti}\end{aligned}$$ This exponent is in agreement with the finite-size scaling relation $y_c=1/\nu$ with $\nu \simeq 6.2$ (see Eqs \[xifreenuwett\]) Directed polymer model on diamond hierarchical lattices {#secdp} ======================================================== In this Section we study the directed polymer model, whose partition function satisfies the quadratic renormalization of Eq \[zrecursion\]. In contrast with the wetting case described above, the transition only exists in the presence of disorder. Since we are interested into the asymptotic distribution of the free-energy, it is convenient to rewrite the free-energy $F_n^{(a)}$ of a sample $(a)$ of generation $n$ as $$\begin{aligned} F_n^{(a)} \equiv \ln Z_n^{(a)} = \overline{ F_n}+ \Delta_n u_a \label{scalingfree}\end{aligned}$$ where $u_a$ is a random variable of zero mean and width unity $$\begin{aligned} \overline{u_a^2}=1 \label{unorma}\end{aligned}$$ So $\Delta_n$ represents the width $$\begin{aligned} \Delta_n= \left( \overline{ F_n^2} - (\overline{ F_n})^2 \right)^{1/2} \label{defdeltan}\end{aligned}$$ Low-temperature RG flow ------------------------ In the low-temperature phase, the width $\Delta_n$ of the free-energy distribution grows with $n$. So at large scale, the recursion is dominated by the maximal term in Eq. \[zrecursion\] $$\begin{aligned} F_{n+1} \opsimeq min_{1 \leq a \leq b} \left( F_n^{(2a-1)} + F_n^{(2a)} \right) \label{recursionDPlow}\end{aligned}$$ This effective low-temperature recursion coincides with the recursion of the energy $E_0$ of the ground state studied in [@Der_Gri; @Coo_Der]. The whole low-temperature phase is thus described by the zero-temperature fixed-point. In particular, the width of the free-energy distribution grows as $$\begin{aligned} \Delta_n \opsimeq L_n^{\omega_0(b)} \label{widthDPlow}\end{aligned}$$ where $\omega_0(b)$ is the exponent governing the width of the ground-state energy $\Delta E_0 \sim L_n^{\omega_0(b)} $ studied in [@Der_Gri]. High-temperature RG flow ------------------------- In the high-temperature phase, the width $\Delta_n$ of the free-energy distribution is expected to decay to zero. The linearization in $\Delta_n$ of the recursion of Eq. \[zrecursion\] yields $$\begin{aligned} \beta F_{n+1} && = - \ln \left[ \sum_{a=1}^b e^{- \beta \left( F_n^{(2a-1)}+ F_n^{(2a)}\right) } \right] = 2 \beta \overline{ F_n } - \ln \left[ b - \beta \Delta_n \sum_{a=1}^b (u_{2a-1}+u_{2a}) + O(\beta^2 \Delta_n^2)\right] \\ && = 2 \beta \gamma_n - \ln (b) + \frac{ \beta \Delta_n }{b} \sum_{a=1}^b (u_{2a-1}+u_{2a}) + O(\beta^2 \Delta_n^2) \label{frecursionhigh}\end{aligned}$$ The consistence with the scaling form of Eq. \[scalingfree\] at generation $(n+1)$ $$\begin{aligned} F_{n+1} = \overline{F_{n+1}} + \Delta_{n+1} u \label{scalinghighbis}\end{aligned}$$ yields $$\begin{aligned} \overline{F_{n+1}} && = 2 \overline{F_{n}} - T \ln (b) \nonumber \\ \Delta_{n+1} u && = \frac{ \Delta_n }{b} \sum_{a=1}^b (u_{2a-1}+u_{2a}) \label{recurrencehigh}\end{aligned}$$ The normalization condition of Eq. \[unorma\] yields $$\begin{aligned} \Delta_{n+1} = \sqrt{ \frac{2}{b} } \Delta_n \label{widthhigh}\end{aligned}$$ and $$\begin{aligned} u = \frac{1}{\sqrt{2b}} \sum_{a=1}^b (u_{2a-1}+u_{2a}) \label{uhigh}\end{aligned}$$ For $b>2$, this high-temperature phase exists, the probability distribution of the free-energy converges to a Gaussian. The width decays as the following power-law of the length $L_n=2^n$ $$\begin{aligned} \Delta_n \propto L_n^{- \omega_{\infty}(b)} \ \ \ {\rm with} \ \ \ \ \ \omega_{\infty}(b)=\frac{\ln \frac{b}{2}}{2 \ln 2} \label{fwidthhighdp}\end{aligned}$$ In this regime, the rescaled variable $u$ evolves according to Eq. \[uhigh\] and thus becomes Gaussian upon iteration. Analysis of the critical point ------------------------------- At criticality, to avoid the high-temperature and low-temperature described above, the width $\Delta_n$ should converge as $n \to \infty$ towards a finite value $\Delta_c$. In particular, the fluctuating part of the free-energy $$\begin{aligned} f_n^{(a)} \equiv F_n^{(a)} - \overline{ F_n} = \Delta_c u^{(a)} \label{fluctuatingfree}\end{aligned}$$ should remain a random variable of order $O(1)$, of zero mean, distributed with some scale-invariant probability distribution $P_c(f)$ defined on $]-\infty,+\infty[$. Equivalently, the variable $$\begin{aligned} z_n^{(a)} \equiv e^{- \beta_c f_n^{(a)} } = e^{- \beta_c \Delta_c u^{(a)}} \label{fluctuatingz}\end{aligned}$$ should have a scale-invariant probability distribution $Q_c(z)$ defined on $[0,+\infty[$, with $$\begin{aligned} \overline{ \ln z} = \int_0^{+\infty} dz \ln z Q_c(z)=0 \label{conditionlnz}\end{aligned}$$ The recursion for these variables $z_n$ reads $$\begin{aligned} z_{n+1} = \frac{1}{\cal B} \sum_{a=1}^b z_n^{(2a-1)} z_n^{(2a)} \label{zreducedrecursion}\end{aligned}$$ where $$\begin{aligned} {\cal B} \equiv \lim_{n \to \infty} \left( \overline{ F_{n+1} }- 2 \overline{ F_n} \right) \label{defcalB}\end{aligned}$$ should be finite ( otherwise the recursion of Eq. \[zreducedrecursion\] would not lead to a non-trivial stationary distribution $Q_c(z)$). ### Left-tail behavior of the free-energy distribution Let us introduce the left-tail exponent $\eta_c$ $$\begin{aligned} \ln P_c(f) \opsimeq_{f \to -\infty} - \gamma (-f)^{\eta_c} +... \label{tailscritidp}\end{aligned}$$ where $(...)$ denote the subleading terms. In the region where $f \to -\infty$, one has effectively the low-temperature recursion of Eq. \[recursionDPlow\] $$\begin{aligned} f \opsimeq min_{1 \leq a \leq b} \left( f^{(2a-1)} + f^{(2a)} \right) \label{recursioncritileftdp}\end{aligned}$$ As in the wetting case, a saddle-point analysis shows that the only stable left tail exponent is $$\begin{aligned} \eta_c=1 \label{etacdp}\end{aligned}$$ So the distribution $P_c(f)$ decays exponentially $$\begin{aligned} P_c(f) \opsimeq_{f \to - \infty} e^{\gamma f} (...) \label{lefttailexpodp}\end{aligned}$$ This means that the corresponding distribution $Q_c(z)$ of $z=e^{-\beta_c f}$ presents a power-law tail $$\begin{aligned} Q_c(z) \opsimeq_{z \to + \infty} \frac{ \Phi(\ln z) }{ z^{1+\mu} } \label{powerlawqcdp}\end{aligned}$$ with some exponent $$\begin{aligned} \mu= \frac{\gamma}{\beta_c} \label{defimudp}\end{aligned}$$ and where $\Phi(\ln z)$ represents the subleading terms. The first moment is fixed by the recursion of Eq \[zreducedrecursion\] $$\begin{aligned} \overline{ z} = \frac{ \cal B}{b} \label{averagez}\end{aligned}$$ As a consequence, the exponent $\mu$ of the power-law of Eq \[powerlawqcdp\] should satisfy $$\begin{aligned} \mu >1 \label{mubigger1}\end{aligned}$$ and the parameter ${\cal B}$ representing the correction to extensivity (Eq \[defcalB\]) is determined by $$\begin{aligned} { \cal B} = b \overline{ z} = b \int dz z Q_c(z) \label{calB}\end{aligned}$$ in terms of the fixed point distribution $Q_c(z)$. ### Analysis in terms of multiplicative stochastic processes {#analysis-in-terms-of-multiplicative-stochastic-processes} Again, as explained in section \[wettmsp\], it is instructive to analyse the recursion of Eq. \[zreducedrecursion\] from the point of view of multiplicative stochastic processes (see Appendix \[multiplicative\]). The condition of Eq. \[stabcondition\] translates here into the following condition using Eq. \[conditionlnz\] $$\begin{aligned} \overline{ \ln \frac{z}{{\cal B}} } = - \ln {\cal B } = - \ln ( b \overline{ z} ) <0 \label{stabconditiondp}\end{aligned}$$ The condition of Eq. \[mspmu\] translates into the following condition for the exponent $\mu$ introduced in Eq. \[powerlawqcdp\] $$\begin{aligned} 2 b \overline{\left( \frac{z}{\cal B} \right)^{\mu} } = \frac{2}{b^{\mu-1}} \ \frac{ \overline{z^{\mu} }}{ (\overline{z})^{\mu}} \equiv \frac{2}{b^{\mu-1}} \frac{ \int_1^{+\infty} dz Q_c(z) z^{\mu} }{ \left[ \int_1^{+\infty} dz Q_c(z) z \right]^{\mu} } =1 \label{selfmudp}\end{aligned}$$ The argument is similar to Eqs \[eqmsp\]-\[eqmspasymp\], the additional factor of $2b$ coming from the fact that $z$ large corresponds to one of the $z^{(i)}$ being large. This condition means in particular that the subleading term $\Phi(z)$ in Eq. \[powerlawqc\] should ensure the convergence at $(+ \infty)$ as in the wetting case (see Eqs \[subphi\] and \[momentsdv\]). ### Reminder on transitions of integer moments Let us now briefly recall the behaviors of the first moments discussed in [@Coo_Der]. From Eq \[zrecursion\], one obtains [@Coo_Der] $$\begin{aligned} \overline{ Z_{n+1} } && = b \left( \overline{Z_n} \right)^2 \\ \overline{ Z_{n+1}^2 } && = b \left( \overline{Z_n^2 } \right)^2 + b (b-1) \left( \overline{Z_n} \right)^4 \label{momentsdp}\end{aligned}$$ so that the ratio of the moments of the rescaled variable $z$ defined in Eq \[fluctuatingz\] $$\begin{aligned} r_2(n) \equiv \frac{\overline{ Z_{n}^2 }}{\left( \overline{Z_n} \right)^2} = \frac{\overline{ z_{n}^2 }}{\left( \overline{z_n} \right)^2} \label{defr2dp}\end{aligned}$$ follows the closed recursion [@Coo_Der] $$\begin{aligned} r_2(n+1) = \frac{ r_2^2(n) +b-1 }{b} \label{recr2dp}\end{aligned}$$ that actually coincides with Eq. \[mappingpurewetting\] for the pure wetting model. The repulsive fixed point $r^*=b-1$ allows to define the temperature $T_2$ via $r_2(n=0)=b-1$ [@Coo_Der]: for $T>T_2$, the ratio $r_2(n)$ flows to $1$, whereas for $T>T_2$, the ratio $r_2(n)$ flows to $(+\infty)$. Similarly, the RG flow of ratios corresponding to higher moments have been studied in [@Coo_Der], with the conclusion that for generic $b$ (more precisely $b>2.303...$) their transition temperatures are higher than $T_2$. Since we already know $\mu>1$ (Eq \[mubigger1\]), we have to distinguish two cases \(i) if the tail exponent $\mu$ satisfies $1<\mu<2$, then the second moment diverges at criticality $\overline{z^2}=+ \infty$ and we have the strict inequality $T_c<T_2$. \(ii) if the tail exponent $\mu$ satisfies $\mu>2$, then the second moment is finite at criticality. The only possible finite stable value is for the ratio $r_2$ is $r_2=b-1$. The critical temperature then coincides with the transition temperature of the second moment $T_c=T_2$. The scenario (i) is the most plausible, since the possibility (ii) would require some ’fine tuning’ in some sense : as explained in the introduction, the temperature only appears in the initial condition (Eq \[zrecursioninitial\]) of the renormalization ; any initial temperature $T>T_c$ flows towards the high temperature fixed-point, any initial temperature $T<T_c$ flows towards the low temperature fixed-point, so that $T_c$ is defined as the only initial temperature from which the critical distribution $Q_c(z)$ is accessible. The critical distribution $Q_c(z)$ has to satisfy the self-consistent equation of Eq \[selfmudp\] to be stable. If (ii) were true, the distribution $Q_c(z)$ should in addition satisfy a second completely independent condition $r_2=b-1$, which seems unlikely. In conclusion, we expect that the exponent $\mu$ introduced in Eq. \[powerlawqcdp\] satisfies $$\begin{aligned} 1<\mu<2 \label{rangeDPmu}\end{aligned}$$ This is in agreement with the numerical simulations presented below in section \[secdpnume\]. ### Right tail behavior of the free-energy distribution Let us introduce the right-tail exponent $\eta_c$ $$\begin{aligned} \ln P_c(f) \opsimeq_{f \to +\infty} - \gamma' f^{\eta_c'} +... \label{righttailDP}\end{aligned}$$ In the region where $f \to +\infty$, one has effectively the high-temperature recursion $$\begin{aligned} f \opsimeq \frac{ 1 }{b} \sum_{i=1}^{2b} f^{(i)} \label{recursionDPcritiright}\end{aligned}$$ where all free-energies are large. A saddle-point analysis with the right tail of Eq. \[righttailDP\] shows that the only stable right exponent $\eta_c'$ should satisfy $b = 2^{\eta_c'-1}$ i.e. $$\begin{aligned} \eta_c'=1+ \frac{\ln b}{\ln 2} \label{etacright}\end{aligned}$$ Critical exponent $\nu$ {#critical-exponent-nu} ------------------------- To compute the critical exponent $\nu$, we consider a small perturbation in the fluctuating part of the free-energy of Eq \[fluctuatingfree\] $$\begin{aligned} -\beta_c f_n^{(a)} \equiv -\beta_c (F_n^{(a)} - \overline{ F_n} ) = -\beta \Delta_c u^{(a)} + \delta_n^{(a)} \label{fluctuatingfreebis}\end{aligned}$$ where $\delta_n^{(a)}$ represent the random perturbations of zero mean $$\begin{aligned} \overline{ \delta_n } =0 \label{conditionzeromean}\end{aligned}$$ Equivalently, these variables $\delta_n$ represent the perturbation at linear order of the variables of Eq \[fluctuatingz\] $$\begin{aligned} z_n^{(a)} \equiv e^{- \beta_c f_n^{(a)} } = z_c^{(a)} + \delta_n^{(a)} \label{fluctuatingzbis}\end{aligned}$$ The linearization of the recursion of Eq. \[zreducedrecursion\] around the fixed point, yields $$\begin{aligned} \delta_{n+1} \simeq \sum_{i=1}^{2b} \frac{z_c^{(i)}}{ \cal B } \delta_{n}^{(i)} \label{epsDP}\end{aligned}$$ where $z_c^{(i)}$ are distributed with the critical distribution $Q_c(z)$. As in the wetting case, the recurrence of Eq \[epsDP\] coincides with the recurrence describing a directed polymer on a Cayley tree [@Der_Spo; @Der_review], with the following differences \(i) the variables $\delta_n$ are random variables of zero mean (Eq \[conditionzeromean\]), which is equivalent to a directed polymer model with random signs studied in [@cayleycomplex] \(ii) more importantly, the random weights $\frac{z_c}{b}$ are distributed with the fixed point distribution $Q_c(z)$ presenting a broad power-law tail in $1/z_c^{1+\mu}$ with $1<\mu<2$ (instead of $0<\mu<1$ for the wetting case). Reasoning as in the wetting case, any narrow symmetric distribution ${\cal P}_0(\delta_0)={\cal P}_0(-\delta_0)$ will lead to power-law tails of index $(1+\mu)$ after one iteration. The study of the evolution of these tails by iteration yields that the corresponding Lyapunov exponent $v$ will be determined by an equation similar to Eq. \[resvelocitywett\] $$\begin{aligned} e^{\mu v} = \frac{2 b \overline{ z_c^{\mu} } }{{\cal B}^{\mu}} + \frac{2 b }{{\cal B}^{\mu}} \int_0^{+\infty} dy \frac{P_{\infty}(y)}{B_+} \ y^{\mu} = 1+ \frac{2 b }{{\cal B}^{\mu}} \int_0^{+\infty} dy \frac{P_{\infty}(y)}{B_+} \ y^{\mu} \label{eqvdp}\end{aligned}$$ where we have used Eq. \[selfmudp\], in terms of the stationary distribution $P_{\infty}(y)$ of the rescaled process associated to Eq \[epsDP\] $$\begin{aligned} y_{n+1} = e^{-v} \sum_{i=1}^{2b} \frac{z_c^{(i)}}{{\cal B} } y_{n}^{(i)} \label{epsdprescal}\end{aligned}$$ The correlation length exponent then reads $\nu=\frac{\ln 2}{v}$. As in the wetting case, the presence of the second term in Eq \[eqvdp\] is crucial to obtain a positive $v$ and a finite $\nu$. Numerical study of the directed polymer transition {#secdpnume} ==================================================== As for the wetting transition (see Section \[wettnume\]), we have used the ’pool-method’ with a pool number $N=4.10^7$ to study the transition of the hierarchical lattice of branching ratio $b=5$ with initial Gaussian energies (Eq. \[gaussian\]). The exact bounds on the critical temperature are [@Coo_Der] $$\begin{aligned} T_0(b) = \frac{1}{ \left[ 2 \ln b \right]^{\frac{1}{2}}} \simeq 0.557... \leq T_c(b) \leq T_2(b) = \frac{1}{ \left[ \ln (b-1) \right]^{\frac{1}{2}}} \simeq 0.849.. \label{boundsgauss}\end{aligned}$$ In [@Coo_Der], the phase transition has been studied numerically via the specific heat and the overlap. In this paper, we characterize the transition via the statistics of free-energy and energy. As in the wetting case, this allows to locate very precisely the pool-dependent critical temperature and to measure the divergence of the correlation length $\xi(T)$ above and below $T_c$. Flow of the free-energy width ------------------------------- The flow of the free-energy width $\Delta F_L$ as $L$ grows is shown on Fig. \[figb5freewidth\] for many temperatures. One clearly sees the two attractive fixed points. For $T>T_c$, the free-energy width decays asymptotically with the exponent $\omega_{\infty}(b)$ introduced in Eq. \[fwidthhighdp\] $$\begin{aligned} \Delta F(L) \simeq \left( \frac{L}{\xi_F^+(T)} \right)^{- \omega_{\infty}(b)} \ \ { \rm with } \ \ \omega_{\infty}(b=5)= \frac{\ln (b/2) }{ 2 \ln 2} = 0.6609.. \label{fwidthabove}\end{aligned}$$ where $\xi_F^+(T)$ is the corresponding correlation length that diverges as $T \to T_c^+$. For $T<T_c$, the free-energy width grows asymptotically with the exponent $\omega_0(b)$ of the ground-state energy distribution $$\begin{aligned} \Delta F(L) \simeq \left( \frac{L}{\xi_F^-(T)} \right)^{ \omega_0(b)} \ \ { \rm with } \ \ \omega_0(b=5) \simeq 0.186... \label{fwidthbelow}\end{aligned}$$ where $\xi_F^-(T)$ is the corresponding correlation length that diverges as $T \to T_c^-$. For each given pool, the flow of free-energy width allows a very precise determination of the pool-dependent critical temperature, for instance in the case considered $0.77662 < T_c^{pool} < 0.77666$ which is significantly below the upper bound $T_2$ of Eq \[boundsgauss\]. ![(Color online) Directed polymer transition : log-log plot of the width $\Delta F(L)$ of the free-energy distribution as a function of $L$, for many temperatures. []{data-label="figb5freewidth"}](xm56.mcb5.tall.eps){height="6cm"} ![(Color online) Directed polymer transition : Correlation length $\xi_F^{\pm}(T)$ as measured from the behavior of the free-energy width (Eqs \[fwidthabove\] and \[fwidthbelow\]) (a) $\ln \xi_F^{\pm}(T)$ as a function of $T$ (b) $\ln \xi_F^{\pm}(T)$ as a function of $\ln \vert T_c-T \vert$ : the asymptotic slopes are of order $\nu \sim 3.4 $ []{data-label="figb5logxifree"}](xmlogxi.56mchistob5.tall.eps "fig:"){height="6cm"} ![(Color online) Directed polymer transition : Correlation length $\xi_F^{\pm}(T)$ as measured from the behavior of the free-energy width (Eqs \[fwidthabove\] and \[fwidthbelow\]) (a) $\ln \xi_F^{\pm}(T)$ as a function of $T$ (b) $\ln \xi_F^{\pm}(T)$ as a function of $\ln \vert T_c-T \vert$ : the asymptotic slopes are of order $\nu \sim 3.4 $ []{data-label="figb5logxifree"}](xmfitpowerlogxi.56mcb5.tall.eps "fig:"){height="6cm"} Divergence of the correlation lengths $\xi_F^{\pm}(T)$ -------------------------------------------------------- The correlation lengths $\xi_F^{\pm}(T)$ as measured from the free-energy width asymptotic behaviors above and below $T_c$ (Eqs \[fwidthabove\] and \[fwidthbelow\] ) are shown on Fig. \[figb5logxifree\] (a). The plot in terms of the variable $\ln \vert T_c-T \vert$ shown on Fig. \[figb5logxifree\] (b) indicate a power-law divergence with the same exponent $$\begin{aligned} \xi_F^{\pm}(T) \oppropto_{T \to T_c} \vert T-T_c \vert^{-\nu} \ \ { \rm with } \ \ \nu \simeq 3.4 \label{xifreenu}\end{aligned}$$ Histogram of the free-energy ------------------------------ ![(Color online) Directed polymer transition : Asymptotic distribution $\Pi_F$ of the rescaled free-energy $x_F= \frac{F-F_{av(L)}}{\Delta F(L)} $ in the low-temperature phase (here $T=0.4$), in the high-temperature phase ( here $T=2$) and at criticality (here $T_c^{pool}=0.77665$) (a) Bulk representation (b) Log-representation to see the tails. []{data-label="figb5freehisto"}](xm51.mchistob5.tall.eps "fig:"){height="6cm"} ![(Color online) Directed polymer transition : Asymptotic distribution $\Pi_F$ of the rescaled free-energy $x_F= \frac{F-F_{av(L)}}{\Delta F(L)} $ in the low-temperature phase (here $T=0.4$), in the high-temperature phase ( here $T=2$) and at criticality (here $T_c^{pool}=0.77665$) (a) Bulk representation (b) Log-representation to see the tails. []{data-label="figb5freehisto"}](xm52.mchistob5.tall.eps "fig:"){height="6cm"} The asymptotic probability distribution $\Pi_F$ of the rescaled free-energy $$\begin{aligned} x_F \equiv \frac{F-F_{av(L)}}{\Delta F(L)}\end{aligned}$$ is shown on Fig \[figb5freehisto\] for three temperatures : \(i) for $T>T_c$, it is a Gaussian in agreement with Eq \[uhigh\]. \(ii) for $T<T_c$, it coincides with the ground state energy distribution. \(iii) at criticality, one clearly see that a left-tail develops with tail exponent $\eta_c =1$ in agreement with Eq \[etacdp\]. The corresponding power-law exponent of Eq \[defimu\] of the fixed-point distribution of Eq. \[powerlawqc\] is of order $$\begin{aligned} \mu \sim 1.6 \label{muvaluedp}\end{aligned}$$ Again this measure is not precise as a consequence of the unknown logarithmic correction in Eq. \[powerlawqc\], but it is in the expected interval of Eq \[rangeDPmu\]. Flow of the energy and entropy widths --------------------------------------- ![(Color online) Directed polymer transition : Flow of the widths $\Delta E(L)$ of the energy distribution as $L$ grows (a) $\ln \Delta E(L)$ as a function of $\ln L$ for many temperatures (b) Comparison of $\ln \Delta E(L)$ , $ \ln \Delta S(L)$ and $ \ln \Delta F(L)$ as a function of $\ln L$ at criticality ($T_c^{pool}=0.77665$).[]{data-label="figb5enerwidth"}](xm76.mcb5.tall.eps "fig:"){height="6cm"} ![(Color online) Directed polymer transition : Flow of the widths $\Delta E(L)$ of the energy distribution as $L$ grows (a) $\ln \Delta E(L)$ as a function of $\ln L$ for many temperatures (b) Comparison of $\ln \Delta E(L)$ , $ \ln \Delta S(L)$ and $ \ln \Delta F(L)$ as a function of $\ln L$ at criticality ($T_c^{pool}=0.77665$).[]{data-label="figb5enerwidth"}](xm567686.mchistob5t0.77665.criti.eps "fig:"){height="6cm"} The flow of the energy width $\Delta E(L)$ as $L$ grows is shown on Fig. \[figb5enerwidth\] (a) for many temperatures ( the flow of the entropy width $\Delta S(L)$ is very similar at large scale). For $T>T_c$, we find that these widths decay asymptotically with the same exponent $\omega_{\infty}(b)$ as the free-energy (Eq \[fwidthabove\]) $$\begin{aligned} \Delta E(L) \simeq L^{- \omega_{\infty}(b)} \\ \Delta S(L) \simeq L^{- \omega_{\infty}(b)} \label{ewidthabove}\end{aligned}$$ For $T<T_c$, in agreement with the Fisher-Huse droplet scaling theory for directed polymers [@Fis_Hus], we find that these widths grow asymptotically with the exponent $1/2$ which is bigger than the free-energy exponent $\omega_0(b)$ (Eq. \[fwidthbelow\]) $$\begin{aligned} \Delta E(L) \simeq L^{\frac{1}{2}} \\ \Delta S(L) \simeq L^{\frac{1}{2}} \label{ewidthbelow}\end{aligned}$$ Exactly at criticality, the free-energy $\Delta F(L)$ width converges towards a constant, whereas the energy and entropy widths grow as power-laws (see Fig \[figb5enerwidth\] b) $$\begin{aligned} \Delta E(L) \sim L^{y_c} \sim \Delta S(L) \ \ {\rm with } \ \ y_c \sim 0.29 \label{ewidthcriti}\end{aligned}$$ This exponent is in agreement with the finite-size scaling relation $y_c=1/\nu$ with $\nu \sim 3.4$ (see Eqs \[xifreenu\]) Divergence of the correlation lengths $\xi_E^{\pm}(T)$, $\xi_S^{\pm}(T)$ -------------------------------------------------------------------------- ![(Color online) Directed polymer transition : Correlation length $\xi_E^{\pm}(T)$ (circles below and square above) and $\xi_S^{\pm}(T)$ (triangles below and diamond above) as measured from the behavior of the energy and entropy widths (a) $\ln \xi_E(T)$ and $\ln \xi_S(T)$ as a function of $T$ (b) $\ln \xi_E(T)$ and $\ln \xi_S(T)$ as a function of $\ln \vert T_c-T \vert$: the asymptotic slopes are of order $\nu \sim 3.4 $ as in Fig. \[figb5logxifree\] []{data-label="figb5logxiener"}](xmlogxicorrect.76mcb5.tall.eps "fig:"){height="6cm"} ![(Color online) Directed polymer transition : Correlation length $\xi_E^{\pm}(T)$ (circles below and square above) and $\xi_S^{\pm}(T)$ (triangles below and diamond above) as measured from the behavior of the energy and entropy widths (a) $\ln \xi_E(T)$ and $\ln \xi_S(T)$ as a function of $T$ (b) $\ln \xi_E(T)$ and $\ln \xi_S(T)$ as a function of $\ln \vert T_c-T \vert$: the asymptotic slopes are of order $\nu \sim 3.4 $ as in Fig. \[figb5logxifree\] []{data-label="figb5logxiener"}](xmfitpowerlogxicorrect.76mcb5.tall.eps "fig:"){height="6cm"} According to the Fisher-Huse droplet scaling theory of spin-glasses [@Fis_Hus], the singularities of the widths of energy and entropy as $T \to T_c^-$ is given by $(L/\xi(T))^{1/2}/(T_c-T)$. We thus define the correlation lengths $\xi_E^+(T)$ and $\xi_S^+(T)$ by $$\begin{aligned} \Delta E(L) \simeq \frac{1}{T_c-T} \left( \frac{L}{\xi_E^-(T)} \right)^{\frac{1}{2}} \\ \Delta S(L) \simeq \frac{1}{T_c-T} \left( \frac{L}{\xi_S^-(T)} \right)^{\frac{1}{2}} \label{ewidthbelowcriti}\end{aligned}$$ Similarly, for $T>T_c$, we define the corresponding correlation lengths $\xi_E^+(T)$ and $\xi_S^+(T)$ by the equations $$\begin{aligned} \Delta E(L) \simeq \frac{1}{T-T_c} \left( \frac{L}{\xi_E^+(T)} \right)^{- \omega_{\infty}(b)} \\ \Delta S(L) \simeq \frac{1}{T-T_c} \left( \frac{L}{\xi_S^+(T)} \right)^{- \omega_{\infty}(b)} \label{ewidthabovecriti}\end{aligned}$$ The correlations lengths are shown on Fig. \[figb5logxiener\] (a) The plot in terms of the variable $\ln \vert T_c-T \vert$ shown on Fig. \[figb5logxiener\] (b) indicate a power-law divergence with the same exponent as in Eq. \[xifreenu\] $$\begin{aligned} \xi_E^{\pm}(T) \oppropto_{T \to T_c} \vert T-T_c \vert^{-\nu} \ \ { \rm with } \ \ \nu \simeq 3.4 \label{xienernu}\end{aligned}$$ Histogram of the energy ------------------------- The asymptotic probability distribution $\Pi_E$ of the rescaled energy $$\begin{aligned} x_E \equiv \frac{E-E_{av(L)}}{\Delta E(L)}\end{aligned}$$ is shown for three temperatures on Fig \[figb5enerhisto\] \(i) outside criticality, both for $T>T_c$ and $T<T_c$, these distributions are Gaussian. \(ii) at criticality, the distribution is strongly non-gaussian and asymmetric, with a left-tail of tail exponent $\eta_c =1$ ![(Color online) Directed polymer transition : Asymptotic distribution $\Pi_E$ of the rescaled energy $x= \frac{E-E_{av}}{\Delta E} $ in the low-temperature phase (here $T=0.4$), in the high-temperature phase (here $T=2$) and at criticality ($T_c^{pool}=0.77665$) (a) Bulk representation (b) Log-representation to see the tails []{data-label="figb5enerhisto"}](xm71.mchistob5.tall.eps "fig:"){height="6cm"} ![(Color online) Directed polymer transition : Asymptotic distribution $\Pi_E$ of the rescaled energy $x= \frac{E-E_{av}}{\Delta E} $ in the low-temperature phase (here $T=0.4$), in the high-temperature phase (here $T=2$) and at criticality ($T_c^{pool}=0.77665$) (a) Bulk representation (b) Log-representation to see the tails []{data-label="figb5enerhisto"}](xm72.mchistob5.tall.eps "fig:"){height="6cm"} Comparison with corresponding results on hypercubic lattices {#comparison} ============================================================ Since the exact renormalizations on the diamond lattice can also be considered as approximate Migdal-Kadanoff renormalizations for hypercubic lattices, it is interesting to discuss whether the results obtained for the wetting and the directed polymer on the diamond lattice are qualitatively similar to the results for hypercubic lattices. Similarities for $T<T_c$ -------------------------- The whole low-temperature phase of disordered systems is usually characterized by the zero-temperature fixed point where disorder fluctuations dominate. For the disordered polymer models considered in this paper, the free-energy fluctuations grow as power-law of the length both for the diamond lattice and for hypercubic lattice $$\begin{aligned} \Delta F(L,T<T_c) \oppropto_{L \to \infty} L^{\omega_0} \label{zerotfixedpoint}\end{aligned}$$ where $\omega_0$ is the exponent governing the fluctuations of the ground state energy $E_0(L)$. In the wetting case, this exponent has the simple value $\omega_0^{wett}=1/2$ that reflects the normal fluctuations of the $L$ random variables defining the random adsorbing energies along the wall. In the directed polymer case, the exponent $\omega_0$ is non-trivial because the ground state configuration is the result of an optimization problem. Differences for $T>T_c$ ------------------------- The high temperature phase of disordered systems is characterized by bounded disorder fluctuations, but these fluctuations are not of the same order on diamond lattices and on hypercubic lattices. More precisely, for the disordered polymer models considered in this paper, the free-energy fluctuations decays as a power-law on the diamond lattices, whereas they remain of order $O(1)$ on hypercubic lattices $$\begin{aligned} {\rm Diamond : } \ \ \ \ \Delta F(L,T>T_c) && \oppropto_{L \to \infty} L^{-\omega_{\infty}(b)} \\ {\rm Hypercubic : } \ \ \ \ \Delta F(L,T>T_c) && \oppropto_{L \to \infty} O(1) \label{hightfixedpoint}\end{aligned}$$ This difference seems to come from the boundary conditions : (i) on the diamond lattice, the polymer is fixed at the two extreme points, but by the iterative construction of the lattice, the coordinence of these two extreme points grow with the number $n$ of generations, so that it is possible to have a very efficient averaging even near the boundaries (ii) on the hypercubic lattices, the boundary conditions are sufficient to produce free-energies fluctuations of order $O(1)$ : the fixed origin has a finite coordinence, and the fluctuations of order $O(1)$ of the random variables near this origin do not disappear as $L \to \infty$. Differences at criticality ---------------------------- On the hierarchical lattice, the free-energy fluctuations of the disordered polymer considered here are of order $O(1)$ at criticality $$\begin{aligned} {\rm Diamond : } \ \ \ \ \Delta F (L, T=T_c) \oppropto_{L \to \infty} O(1) \label{deltafdiamondcriti}\end{aligned}$$ and it is the only possibility in the presence of exact renormalizations : if the free-energy width is growing, the flow will be attracted at large scale towards the zero-temperature fixed point of Eq. \[zerotfixedpoint\], whereas if free-energy width is decaying, the flow will be attracted towards the high-temperature fixed point of Eq. \[hightfixedpoint\]. On hypercubic lattices, the free-energy fluctuations $\Delta F(L,T_c) \sim L^{\omega_c}$ at criticality are expected to be governed by a vanishing exponent $\omega_c=0$, but they are not necessarily of order $O(1)$ because logarithms cannot be excluded, and have actually been found for the directed polymer transition as we now explain. Forrest and Tang [@Fo_Ta] have conjectured from their numerical results on a growth model in the KPZ universality class and from the exact solution of another growth model that the fluctuations of the height of the interface were logarithmic at criticality. For the directed polymer model, this translates into a logarithmic behavior of the free energy fluctuations at $T_c$ $$\begin{aligned} {\rm Hypercubic : } \ \ \ \ \Delta F_{DP}^{3d} (L, T=T_c) && \oppropto_{L \to \infty} (\ln L)^{\sigma} \label{deltafregularcriti}\end{aligned}$$ where the exponent was measured to be $\sigma=1/2$ in $d=3$ [@Fo_Ta; @Ki_Br_Mo; @DP3d] Further theoretical arguments in favour of this logarithmic behavior can be found in [@Ta_Na_Fo; @Do_Ko]. So the scaling of free-energy fluctuations at criticality seem to be different on hypercubic lattices and on diamond lattices. Another related issue concerns the location of the critical temperature $T_c$ with respect to upper bound $T_2$ : \(i) on the diamond lattice, the ratio $r_2$ of Eq \[defr2dp\] is finite at $T_2$, the ratio $z=Z/Z_{ann}$ is a finite random variable at $T_c$, but the probability distribution of the corresponding partition function presents a power law tail of index $(1+\mu)$ with $1<\mu<2$ (Eq. \[rangeDPmu\]), leading to the strict inequality $T_c<T_2$ \(ii) on hypercubic lattices, the location of $T_c$ with respect to $T_2$ is still controversial. In [@DPdroplet], we have argued that $T_c=T_2$ in dimension $d=3$, because the divergence of $r_2 \sim e^{a \ln L} $ at $T_2$ is compatible with the logarithmic free-energy fluctuations of Eq. \[deltafregularcriti\], provided the rescaled distribution of free-energy involves a left-tail exponent $\eta_c>1$, as measured numerically in [@DP3d]. And in [@DP3dmultif], we have found clear numerical evidence from the statistics of inverse participation ratios that the delocalization transition takes place at $T_2$. However, other arguments are in favor of the strict inequality $T_c<T_2$ in finite dimensions : a new upper bound $T^*<T_2$ was proposed in $1+3$ [@birkner], and in [@camanes] the location of $T_c$ with respect to $T_2$ was shown to depend upon dimension and probability distribution of the bond energies. In particular for the gaussian distribution, the result $T_c<T_2$ is obtained only for $d \geq 5$ [@camanes], but not for the case $d=3$ considered in numerical simulations [@DP3d; @DP3dmultif]. For the wetting transition in $1+1$ dimension, we are not aware of results concerning the scale of free-energy fluctuations at criticality. This comparison between the diamond lattice and hypercubic lattice can be summarized as follows. The free-energy fluctuations present analogous power-law behaviors in the low-temperature phase (Eq. \[zerotfixedpoint\]) but have different behaviors in the high temperature phase (Eq \[hightfixedpoint\]). At criticality, the free-energy fluctuations could also scale differently if logarithmic contributions are present on regular lattices. Conclusion ========== In this paper, we have studied the wetting transition and the directed polymer delocalization transition on diamond hierarchical lattices. These two phase transitions with frozen disorder correspond to the critical points of quadratic renormalizations of the partition function. We have first explained why the comparison with multiplicative stochastic processes allows to understand the presence of a power-law tail in the fixed point distribution $P_c(z) \sim \Phi(z)/z^{1+\mu}$ as $z \to +\infty$ ( up to some sub-leading logarithmic function $\Phi(z)$) so that all moments $z^{n}$ with $n>\mu$ diverge. The exponent $\mu$ is in the range $0<\mu<1$ for the wetting transition ( the first moment diverges $\overline{z}=+\infty$ and the critical temperature is strictly below the annealed temperature $T_c<T_{ann}$) and is the range $1<\mu<2$ for the directed polymer transition ( the second moment diverges $\overline{z^2}=+\infty$ and the critical temperature is strictly below the transition temperature $T_2$ of the second moment.) We have then obtained that the linearized renormalization around the critical point, which determines the exponent $\nu$, coincides with the transfer matrix describing a directed polymer on the Cayley tree, where the random weights determined by the fixed point distribution $P_c(z)$ are broadly distributed. We have shown that it induces important differences with respect to the usual travelling wave solutions concerning more narrow distributions of the weights [@Der_Spo; @Der_review; @cayleycomplex], where the selected velocity only depends on the tail region. Note that travelling waves also appear in other renormalization approaches of random systems [@carpentier]. Finally, we have presented detailed numerical results on the statistics of the free-energy and of the energy as a function of temperature for the wetting and the directed polymer transition on the diamond hierarchical lattice with branching ratio $b=5$. In particular, we have shown that the measure of the free-energy width $\Delta F(L)$ yields a very clear signature of the transition and allows to measure the divergence of the correlation length $\xi^{\pm}(T)$ both below and above $T_c$ : (i) for $T<T_c$, the free-energy width is governed by the zero-temperature exponent $\omega_0$ via $\Delta F(L) \sim (L/\xi_-(T))^{\omega_0}$; (ii) for $T>T_c$, the free-energy width is governed by the high-temperature exponent $\omega_{\infty}$ via $\Delta F(L) \sim (L/\xi_+(T))^{-\omega_{\infty}}$. From the point of view of histograms, the development of a left tail with exponent $\eta_c=1$ at criticality is very clear and different from histograms with exponent $\eta>1$ outside criticality. Reminder on multiplicative stochastic processes {#multiplicative} ================================================= Multiplicative stochastic processes appears in many contexts, in particular in one-dimensional disordered systems, such as random walk in random potentials [@Kesten; @Der_Pom; @Bou] or random spin chains [@Der_Hil; @Cal] In this Appendix, we recall some useful results concerning the following recurrence of random variables $X_n$ $$\begin{aligned} X_{n+1}= a_n X_n + b_n \label{msp}\end{aligned}$$ where $(a_n, b_n)$ are positive independent random numbers. The condition to have a stationary probability distribution $P_{\infty}(X)$ is $$\begin{aligned} \overline{ \ln a} <0 \label{stabcondition}\end{aligned}$$ The most important property of $P_{\infty}(X)$ is that it presents a power-law tail $$\begin{aligned} P_{\infty}(X) \opsimeq_{X \to +\infty} \frac{C}{X^{1+\mu}} \label{powermu}\end{aligned}$$ where the exponent $\mu>0$ is determined by the condition [@Kesten; @Der_Pom; @Bou; @Der_Hil; @Cal] $$\begin{aligned} \overline{ a^{\mu}} =1 \label{mspmu}\end{aligned}$$ To understand where this condition comes from, one needs to write that $P_{\infty}(X)$ is stable via the iteration of Eq. \[msp\] $$\begin{aligned} P_{\infty}(X) = \int da {\cal P}(a) \int db \psi(b) \int dY P_{\infty}(Y) \delta \left( X- (aY+b) \right) = \int da {\cal P}(a) \int db \psi(b) \frac{P_{\infty}( \frac{X-b}{a})}{a} \label{eqmsp}\end{aligned}$$ where ${\cal P}(a)$ and $\psi(b)$ are the probability distributions of $a_n$ and $b_n$ respectively. 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--- abstract: | Metropolis nested sampling evolves a Markov chain from a current livepoint and accepts new points along the chain according to a version of the Metropolis acceptance ratio modified to satisfy the likelihood constraint, characteristic of nested sampling algorithms. The geometric nested sampling algorithm we present here is a based on the Metropolis method, but treats parameters as though they represent points on certain geometric objects, namely circles, tori and spheres. For parameters which represent points on a circle or torus, the trial distribution is ‘wrapped’ around the domain of the posterior distribution such that samples cannot be rejected automatically when evaluating the Metropolis ratio due to being outside the sampling domain. Furthermore, this enhances the mobility of the sampler. For parameters which represent coordinates on the surface of a sphere, the algorithm transforms the parameters into a Cartesian coordinate system before sampling which again makes sure no samples are automatically rejected, and provides a physically intutive way of the sampling the parameter space.\ We apply the geometric nested sampler to two types of toy model which include circular, toroidal and spherical parameters. We find that the geometric nested sampler generally outperforms <span style="font-variant:small-caps;">MultiNest</span> in both cases.\ Our implementation of the algorithm can be found at <https://github.com/SuperKam91/nested_sampling> [@javid2020geometric]. author: - | Kamran Javid$^{1,2}$[^1]\ $^{1}$Astrophysics Group, Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, UK\ $^{2}$Kavli Institute for Cosmology, Madingley Road, Cambridge CB3 0HA, UK date: 'Accepted XXX. Received YYY; in original form ZZZ' title: 'Nested sampling on non-trivial geometries' --- \[firstpage\] methods: data analysis – methods: statistical Introduction {#s:gns_intro} ============ Bayesian inference is a vital tool for any astrophysicist who wants to obtain probabilistic estimates of model parameters from a dataset, whilst incorporating *prior* information about the parameters (e.g. from another experiment). Bayesian inference can also be used to perform model comparisons by calculating values for the Bayesian *evidence*, a quantity obtained by averaging the *likelihood* function over the prior. The evidence incorporates Occam’s razor: if two models fit the data equally well, the less complex model will be preferred. For most astrophysical problems, calculating the evidence numerically is infeasible, especially for high dimensional problems. Likewise, attempting to calculate parameter probability distributions exactly is computationally impossible. Thus one usually resorts to statistical sampling to gain estimates of these quantities.\ @2004AIPC..735..395S introduced a novel sampling method referred to as nested sampling. This algorithm focuses on calculating the evidence, but also generates samples from the *posterior* probability distribution. The key computational expense associated with nested sampling is the constraint that newly generated samples must be above a certain likelihood value which increases at each iteration. Nested sampling has proved to be a very popular choice for Bayesian inference in the astrophysics community, due to its ability to calculate Bayesian evidence values for model comparison as well as produce posterior distributions (see e.g. @2018arXiv180501968J, @2019MNRAS.483.3529J, @2019MNRAS.486.2116P, and @2019MNRAS.489.3135J which all use data from the Arcminute MicroKelvin Imager @2018MNRAS.475.5677H to perform model comparison and posterior analysis using nested sampling).\ Initially, @Sivia2006 suggested satisfying this constraint by evolving a Markov chain starting at one of the pre-existing samples and evaluating an acceptance ratio based on the one used by the Metropolis algorithm [@1953JChPh..21.1087M] used in Markov Chain Monte Carlo (MCMC) sampling (see e.g. @mackay2002 for a review). A variant of the nested sampling algorithm which focused on sampling from ellipsoids which approximate the region in which the likelihood constraint is satisfied was also developed [@2006ApJ...638L..51M]. A major breakthrough in the applicability of nested sampling to highly multi-modal distributions came with the invention of clustering nested sampling algorithms (@2007MNRAS.378.1365S, @2008MNRAS.384..449F, and @2009MNRAS.398.1601F). These algorithms effectively sample from multiple ellipsoids determined by some clustering algorithm, with the aim of approximating the likelihood constraint for each mode of the distribution. More recently, the slice sampling algorithm <span style="font-variant:small-caps;">POLYCHORD</span> (@2015MNRAS.450L..61H, @2015MNRAS.453.4384H) has been introduced and is effective at navigating high dimensional spaces, due to the fact that it is not a rejection sampling algorithm. Section 4.1 of @2015MNRAS.453.4384H gives further examples of nested sampling algorithms which have different ways of satisfying the likelihood constraint. Here we introduce a nested sampling algorithm which introduces a new variant for satisfying the likelihood constraint, based on the Markov method used in @Sivia2006 (and also applied in @2008MNRAS.384..449F). Certain parameters relevant to astrophysics exhibit special properties which mean they naturally parameterise points on geometric objects such as circles, tori and spheres. The algorithm we introduce here which we refer to as the geometric nested sampler, exploits these properties to generate samples efficiently and enables mobile exploration of distributions which are defined on such geometries. In Sections \[s:gns\_bayes\] and \[s:gns\_ns\] we give more formal introductions of Bayesian inference and nested sampling respectively. In Section \[s:gns\_lhood\_const\] we give a more detailed account on the Markov chain method used in @Sivia2006 and @2008MNRAS.384..449F to satisfy the likelihood constraint, before introducing the variation of this method used in geometric nested sampling in Section \[s:gns\_geom\_ns\]. We then apply the geometric nested sampling algorithm to circular, toroidal and spherical toy models in Sections \[s:gns\_tmI\] and \[s:gns\_tmII\]. Finally, we provide conclusions of our work in Section \[s:gns\_conc\]. Bayesian inference {#s:gns_bayes} ================== For a model $\mathcal{M}$ and data ${\boldsymbol{\mathcal{D}}}$, we can obtain model parameters (also known as input or sampling parameters) ${\boldsymbol{\theta}}$ conditioned on $\mathcal{M}$ and ${\boldsymbol{\mathcal{D}}}$ using Bayes’ theorem: $$\label{e:gns_bayes} \mathrm{Pr}\left({\boldsymbol{\theta}}|{\boldsymbol{\mathcal{D}}},\mathcal{M}\right) = \frac{\mathrm{Pr}\left({\boldsymbol{\mathcal{D}}}|{\boldsymbol{\theta}},\mathcal{M}\right)\mathrm{Pr}\left({\boldsymbol{\theta}}|\mathcal{M}\right)}{\mathrm{Pr}\left({\boldsymbol{\mathcal{D}}}|\mathcal{M}\right)},$$ where $\mathrm{Pr}\left({\boldsymbol{\theta}}|{\boldsymbol{\mathcal{D}}},\mathcal{M}\right) \equiv \mathcal{P}\left({\boldsymbol{\theta}}\right)$ is the posterior distribution of the model parameter set, $\mathrm{Pr}\left({\boldsymbol{\mathcal{D}}}|{\boldsymbol{\theta}},\mathcal{M}\right) \equiv \mathcal{L}\left({\boldsymbol{\theta}}\right)$ is the likelihood function for the data, $\mathrm{Pr}\left({\boldsymbol{\theta}}|\mathcal{M}\right) \equiv \pi\left({\boldsymbol{\theta}}\right)$ is the prior probability distribution for the model parameter set, and $\mathrm{Pr}\left({\boldsymbol{\mathcal{D}}}|\mathcal{M}\right) \equiv \mathcal{Z}$ is the Bayesian evidence of the data given a model $\mathcal{M}$. The evidence can be interpreted as the factor required to normalise the posterior over the model parameter space: $$\label{e:gns_evidence} \mathcal{Z} = \int \mathcal{L}\left({\boldsymbol{\theta}}\right) \pi\left({\boldsymbol{\theta}}\right)\, \mathrm{d}{\boldsymbol{\theta}}.$$ $\mathcal{Z}$ is central to the way in which nested sampling algorithms determine samples from $\mathcal{P}\left({\boldsymbol{\theta}}\right)$, and so they are often used to calculate accurate estimates of both $\mathcal{Z}$ and $\mathcal{P}\left({\boldsymbol{\theta}}\right)$. Prior distributions {#s:gns_priors} ------------------- In general for nested sampling $\pi\left({\boldsymbol{\theta}}\right)$ can take any form as long as the distribution integrates to one and has a connected support (@2008arXiv0801.3887C, informally this means that the parts of the domain at which $\pi\left({\boldsymbol{\theta}}\right) \neq 0$ is not ‘separated’ by the parts at which $\pi\left({\boldsymbol{\theta}}\right) = 0$). For simplicity, in all examples considered in this paper (except for parameters which describe the azimuthal angle of a sphere, see below) we assume that each component of the $N$-dimensional vector ${\boldsymbol{\theta}}$ is independent of one another and that each $\pi(\theta_i)$ is a uniform probability distribution so that $$\label{e:gns_priors} \pi\left({\boldsymbol{\theta}}\right) = \prod_{i=1}^{N} \pi_{i}(\theta_{i}) = \prod_{i=1}^{N} \frac{1}{\theta_{\mathrm{max},i} - \theta_{\mathrm{min},i}},$$ where $\theta_{\mathrm{max},i}$ and $\theta_{\mathrm{min},i}$ are respectively the upper and lower bounds on $\theta_i$. Values for $\theta_{\mathrm{max},i}$ and $\theta_{\mathrm{min},i}$ used in the examples presented here will be stated in the relevant Sections where the examples are introduced (i.e. Sections \[s:gns\_tmI\] and \[s:gns\_tmII\]). In the case of parameters which resemble azimuthal angles, it is appropriate to assign a sinusoidal prior, $$\label{e:sin_prior} \pi_{i}(\theta_{i}) = \frac{1}{\theta_{\mathrm{max},i} - \theta_{\mathrm{min},i}} \sin \left(\frac{\pi \theta_i}{\theta_{\mathrm{max},i} - \theta_{\mathrm{min},i}} \right).$$ This ensures that for spherical distributions (see Section \[s:gns\_tmII\]), the prior distribution integrates to one over the surface of a sphere. Nested sampling {#s:gns_ns} =============== Nested sampling exploits the relation between the likelihood and ‘prior volume’ to transform the multidimensional integral given by equation \[e:gns\_evidence\] into a one-dimensional integral. The prior volume $X$ is defined by $\mathrm{d}X = \pi\left({\boldsymbol{\theta}}\right) \mathrm{d}{\boldsymbol{\theta}}$, thus $X$ is defined on $[0,1]$ and we can set $$\label{e:gns_ns} X(\mathcal{L}) = \int_{\mathcal{L}\left({\boldsymbol{\theta}}\right) > \mathcal{L}} \pi\left({\boldsymbol{\theta}}\right) \mathrm{d}{\boldsymbol{\theta}}.$$ The integral extends over the region(s) of the parameter space contained within the iso-likelihood contour $\mathcal{L}\left({\boldsymbol{\theta}}\right) = \mathcal{L}$ (see Figure \[f:gns\_ns\_plots\]). Assuming that the inverse of equation \[e:gns\_ns\] ($\mathcal{L}(X) \equiv X^{-1}(\mathcal{L})$) exists which is the case when $\pi$ is strictly positive, then the evidence integral can be written as (a proof of this equivalence is given in Appendix D of @kj_thesis) $$\label{e:gns_nsz} \mathcal{Z} = \int_{0}^{1} \mathcal{L}(X) \mathrm{d}X.$$ Thus, if one can evaluate $\mathcal{L}(X)$ at $n_{\rm s}$ values of $X$, the integral given by equation \[e:gns\_nsz\] can be approximated by standard quadrature methods $$\label{e:gns_nsz_sum} \mathcal{Z} \approx \sum_{i = 1}^{n_{\rm s}} \mathcal{L}_{i} (X_{i-1} - X_{i}),$$ where $$\label{e:gns_X_vals} 0 < X_{n_{\rm s}} < ... < X_{1} < X_{0} = 1.$$ Distribution of $\mathcal{Z}$ ----------------------------- As explained in @2004AIPC..735..395S, the geometric uncertainty associated with the $X_i$ leads to the idea that $\log(\mathcal{Z})$ rather than $\mathcal{Z}$ is a normally distributed variable. Assuming the latter to be normally distributed can result in distributions of $\mathcal{Z}$ with variances that suggest $\mathcal{Z}$ can take negative values, which is unphysical. This is the case with the spherical toy model considered in Section \[s:gns\_tmII\]. The mean and variance of a log-normally distributed random variable, $\mathbb{E}\left[ \log(\mathcal{Z}) \right]$ and $\mathrm{var}\left[ \log(\mathcal{Z}) \right]$, can be calculated from the moments of the non-logarithmic variables as $$\begin{gathered} \label{e:gns_lognorm_mean} \mathbb{E}\left[ \log(\mathcal{Z}) \right] = 2 \log \left( \mathbb{E}[\mathcal{Z}] \right) - \frac{1}{2} \log \left( \mathbb{E}\left[\mathcal{Z}^2\right] \right), \\ \label{e:gns_lognorm_var} \mathrm{var}\left[ \log(\mathcal{Z}) \right] = \log \left( \mathbb{E}\left[ \mathcal{Z}^2 \right] \right) - 2 \log \left( \mathbb{E}[\mathcal{Z}] \right).\end{gathered}$$ Hence our geometric nested sampling algorithm calculates the moments of the linear variables, but the final evidence estimate and its associated error are calculated using equations \[e:gns\_lognorm\_mean\] and \[e:gns\_lognorm\_var\]. Stopping criterion {#s:gns_stop_crit} ------------------ The nested sampling algorithm can be terminated based on an estimate of how precisely the evidence value has been calculated up to the current iteration. One measure of this is to look at the ratio of the current estimate of $\mathcal{Z}$ to its value plus an estimate of the ‘remaining’ evidence associated with the current livepoints. Since after iteration $n_{\rm s}$ the livepoints are uniformly distributed in the range $[0, X_{n_{\rm s}}]$, we can approximate their final contribution to the evidence as $$\label{e:ngs_final} \mathcal{Z}_{\rm f} \approx \frac{X_{n_{\rm s}}}{n_{l}} \sum_{i = 1}^{n_{l}} \mathcal{L}_i,$$ where $\mathcal{L}_i$ is the likelihood value of the $i^{\mathrm{th}}$ remaining livepoint. The stopping criterion can then be quantified as $$\label{e:ngs_stop_crit} \frac{\mathcal{Z}_{\rm f}}{\mathcal{Z}_{\rm f} + \mathcal{Z}} < \epsilon.$$ $\epsilon$ is a user defined parameter, which we set to $0.01$ in our implementation. Posterior inferences {#s:gns_post_points} -------------------- Once $\mathcal{Z}$ has been determined, posterior inferences can easily be generated using the deadpoints and final livepoints from the nested sampling process to give a total of $n_{\rm s} + n_{l}$ samples. Each such point is assigned the weight $$\label{e:ngs_post_weights} \mathcal{P}_{i} = \frac{\mathcal{L}_{i} \left(X_{i-1} - X_{i}\right)}{\mathcal{Z}}.$$ The weights (along with the corresponding values of ${\boldsymbol{\theta}}$) can be used to calculate statistics of the posterior distribution, or plot it using software such as <span style="font-variant:small-caps;">getdist</span>[^2]. Satisfying the likelihood constraint {#s:gns_lhood_const} ==================================== At each step of the nested sampling iteration, one needs to sample a new point which satisfies $\mathcal{L}_{\rm t} > \mathcal{L}_{i}$. As mentioned in the Introduction, considerable work has been put into increasing the efficiency of this process, as it is by far the most computationally expensive step of the nested sampling algorithm. We now give a review of the Metropolis likelihood sampling methodology used by @Sivia2006 and @2008MNRAS.384..449F, which forms the basis of the method used in geometric nested sampling. Metropolis likelihood sampling {#s:gns_m_lhood_const} ------------------------------ The Metropolis nested sampling method is an adaption of the algorithm used in MCMC sampling of a posterior distribution [@1953JChPh..21.1087M]. The acceptance ratio $\alpha$ for the Metropolis algorithm can be derived from the *detailed balance* relation $$\label{e:gns_db} \mathcal{P}\left({\boldsymbol{\theta}}_{j}\right) T\left({\boldsymbol{\theta}}_{j+1} | {\boldsymbol{\theta}}_{j}\right) = \mathcal{P}\left({\boldsymbol{\theta}}_{j+1}\right) T\left({\boldsymbol{\theta}}_{j} | {\boldsymbol{\theta}}_{j+1}\right),$$ where $T\left({\boldsymbol{\theta}}_{j+1} | {\boldsymbol{\theta}}_{j}\right)$ denotes the probability of transitioning from state ${\boldsymbol{\theta}}_{j}$ to state ${\boldsymbol{\theta}}_{j+1}$ in one step along a Markov chain. A derivation of $\alpha$ for the standard Metropolis algorithm and more information on the transition distribution are given in @kj_thesis. For more information on Markov processes in general we refer the reader to @mackay2002 and @robert_casella2004. The acceptance ratio for the Metropolis nested sampling algorithm takes the form $$\label{e:gns_ns_accept} \alpha = \begin{cases} \mathrm{min}\left[\pi\left({\boldsymbol{\theta}}_{\rm t}\right) / \pi\left({\boldsymbol{\theta}}_{l}\right), 1\right] \quad \mathrm{ if } \quad \mathcal{L}_{\rm t} > \mathcal{L}_{i}, \\ 0 \quad \mathrm{ otherwise.} \end{cases} $$ Here ${\boldsymbol{\theta}}_{l}$ is obtained by picking one of the current livepoints at random, and using its value of ${\boldsymbol{\theta}}$. The value for ${\boldsymbol{\theta}}_{\rm t}$ is sampled from a trial distribution $q\left({\boldsymbol{\theta}}_{\rm t} | {\boldsymbol{\theta}}_{l}\right)$. Sivia & Skilling and Feroz et al. use symmetric Gaussian distributions centred on ${\boldsymbol{\theta}}_{l}$ for $q\left({\boldsymbol{\theta}}_{\rm t} | {\boldsymbol{\theta}}_{l}\right)$. The trial point is accepted to be a new livepoint (replacing the deadpoint associated with $\mathcal{L}_{i}$) with probability $\alpha$. Note that equation \[e:gns\_ns\_accept\] implicitly assumes that the proposal distribution is symmetric in its arguments, that is $q\left({\boldsymbol{\theta}}_{\rm t} | {\boldsymbol{\theta}}_{l}\right) = q\left({\boldsymbol{\theta}}_{l} | {\boldsymbol{\theta}}_{\rm t}\right)$. In the case that the proposal distribution is asymmetric, the acceptance ratio includes an additional factor $q\left({\boldsymbol{\theta}}_{l} | {\boldsymbol{\theta}}_{\rm t}\right)q\left({\boldsymbol{\theta}}_{\rm t} | {\boldsymbol{\theta}}_{l}\right)$ (in which case the algorithm is referred to as the Metropolis-Hastings algorithm, @1970Bimka..57...97H). The fact that the Metropolis nested sampling method uses the current livepoints as a ‘starting point’ for selecting ${\boldsymbol{\theta}}_{\rm t}$, means that the autocorrelation between the livepoints is high, which in turn leads to biased sampling. This can be prevented by increasing the variance of the trial distribution used, or by requiring that multiple trial points must be accepted before the final one is accepted as a livepoint, i.e. after the first accepted trial point is found, set ${\boldsymbol{\theta}}_{l} \rightarrow {\boldsymbol{\theta}}_{\rm t}$ and use this to sample a new ${\boldsymbol{\theta}}_{\rm t}$ from $q\left({\boldsymbol{\theta}}_{\rm t} | {\boldsymbol{\theta}}_{l}\right)$. This can be repeated an arbitrary number of times, but in general more iterations leads to a lower correlation between the livepoint used at the beginning of the chain and the final accepted trial point which is added to the livepoint set.\ Sivia & Skilling suggest that at each nested sampling iteration, the number of trial points generated $n_{\rm t}$ to get a new livepoint should be $\approx 20$. In our implementation we set this number to $20 \times N$ where $N$ is the dimensionality of the parameter estimation problem. Note that $n_{\rm t}$ includes both accepted and rejected trial points. Sivia and Skilling also suggest that the acceptance rate for the trial points at each nested sampling iteration should be $\approx 50\%$. This is because a high acceptance rate usually suggests high auto-correlation between the successive trial points, whilst a low acceptance rate can suggest high correlation between the final accepted trial point and the one used to initialise the chain, as too few steps have been made between the two. In the extreme case that the acceptance rate is zero, the process of picking a new livepoint has failed, as one cannot have two livepoints corresponding to the same ${\boldsymbol{\theta}}$. The acceptance rate is affected by the variance of the trial distribution, a large variance usually results in more trial points being rejected (especially near the peaks of the posterior). Sivia & Skilling suggest updating the trial standard deviation as $$\label{e:gns_trialsig} \sigma_{\rm t} \rightarrow \begin{cases} \sigma_{\rm t} \exp(1/N_{\rm a}) \quad \mathrm{ if } \quad N_{\rm a} > N_{\rm r}, \\ \sigma_{\rm t} \exp(-1/N_{\rm r}) \quad \mathrm{ if } \quad N_{\rm a} \leq N_{\rm r}, \end{cases}$$ where $N_{\rm a}$ and $N_{\rm r}$ are the number of accepted and rejected trial points in the current nested sampling iteration respectively. Note however that we determine the variance using different methods (see Sections \[s:gns\_trialvar\] and \[s:gns\_spherevar\]). Feroz et al. incorporate the Metropolis likelihood sampling into their clustering nested sampling algorithm rather than use it in isolation. The geometric likelihood sampling we introduce in the next Section is a modified version of the Metropolis algorithm used in isolation. Geometric nested sampling {#s:gns_geom_ns} ========================= One key issue with Metropolis nested sampling is that at each nested sampling iteration, if too many trial points are rejected, then the livepoints will be highly correlated with each other after a number of nested sampling iterations. To prevent this one must sample a large number of trial points in order to increase the number of acceptances and decrease the auto-correlation of the trial point chain.\ This solution can be problematic if computing the likelihood is computationally expensive. One particular case in which the sampled point is guaranteed to be rejected, is if the point lies outside of the domain of $\mathcal{P}$ (support of $\pi$). Such a case is illustrated in Figure \[f:gns\_nonwrap\]. Of course, this can be avoided by adapting $q\left({\boldsymbol{\theta}}_{\rm t} | {\boldsymbol{\theta}}_{l}\right)$ so that it is truncated to fit the support of $\pi$, but in high dimensions this can be tedious, and inefficient in itself. Hence one desires an algorithm which does not sample outside the support of $\pi$, without having to truncate $q$. Another issue which most sampling algorithms are subject to occurs when the modes of the posterior distribution are far away from each other in ${\boldsymbol{\theta}}$ space, e.g. when they are at ‘opposite ends’ of the domain of $\pi$. In the context of nested sampling this can result in one or more of the modes not being sampled accurately, particularly in the case of low livepoint runs. Thus a sampling algorithm should be able to efficiently manoeuvre between well separated modes which lie at the ‘edges’ of $\pi$’s support. Geometric nested sampling attempts to solve these two issues by interpreting parameter values as points on geometric objects, namely on circles, tori and spheres. Wrapping the trial distribution {#s:gns_wrap} ------------------------------- A relatively straightforward way of ensuring that the trial points sampled from $q$ are in the support of $\pi$ is to ‘wrap’ $q$. This is illustrated in Figure \[f:gns\_wrap\], where we consider a one-dimensional uniform prior on $[0, 1]$ for simplicity. For any point $\theta$, there will be a non-zero probability of sampling a value of $\theta'$ from the trial distribution $q(\theta' | \theta)$ that lies outside $[0, 1]$. If the point sampled has a value of say $\theta' = -0.1$, then if we consider $q$ to be wrapped around the support this can be interpreted as sampling a point at value $\theta' = 0.9$. More generally, if $\theta'$ is outside the support of $\pi$ defined by upper and lower bounds $\theta_{\mathrm{max}}$ and $\theta_{\mathrm{min}}$ it will be transformed as $$\label{e:gns_wrap} \theta' = \begin{cases} \theta_{\mathrm{max}} - W(\theta') \quad \mathrm{ if } \quad \theta' > \theta_{\mathrm{max}}, \\ \theta_{\mathrm{min}} + W(\theta') \quad \mathrm{ if } \quad \theta' < \theta_{\mathrm{min}}, \end{cases}$$ where $$\label{e:gns_wrapmod} W(\theta) = \begin{cases} (\theta - \theta_{\mathrm{max}}) \mod (\theta_{\mathrm{max}} - \theta_{\mathrm{min}}) \quad \mathrm{ if } \quad \theta > \theta_{\mathrm{max}}, \\ (\theta_{\mathrm{min}} - \theta) \mod (\theta_{\mathrm{max}} - \theta_{\mathrm{min}}) \quad \mathrm{ if } \quad \theta < \theta_{\mathrm{min}}. \end{cases}$$ Assuming the support of $\pi$ is connected (a requirement of nested sampling, as stated in Section \[s:gns\_priors\]), then this operation will be well defined for all $\pi$ with bounded supports, of arbitrary dimension. Using this transformation does not affect the argument symmetry of $q$, thus the value of $\alpha$ given by equation \[e:gns\_ns\_accept\] still holds. Furthermore, this symmetry ensures that the detailed balance relation given by equation \[e:gns\_db\] is still satisfied. Circular parameters {#s:gns_circular} ------------------- As well as ensuring that none of the sampled trial points lie outside the support of $\pi$, the wrapped trial distribution can also improve the manoeuvrability of the sampling process, since the trial point chain can always ‘move in either direction’ without stepping outside of the support of $\pi$. This proves to be particularly useful for ‘circular parameters’. Here we define circular parameters to be those whose value at $\theta_{\mathrm{max}}$ and $\theta_{\mathrm{min}}$ correspond physically to the same point. Examples of circular parameters include angles (which are circular at e.g. zero and $2\pi$) and time periods (e.g. $00$:$00$ and $24$:$00$). Often, circular parameters have probability distributions associated with them which are also circular. An example of a circular distribution is the von Mises distribution, an example of which is shown in Figure \[f:gns\_vm\] (and defined in Section \[s:gns\_tmI\]). This particular example shows that the function’s peak(s) may be split by the wrapping, so that when plotted linearly, they appear to have to ‘half peaks’ about $\theta_{\mathrm{max}}$ and $\theta_{\mathrm{min}}$. Such half peaks would be classified as two separate peaks by clustering nested sampling algorithms. Thus in general, the number of livepoints would need to be increased to accommodate for the higher number of modes, to ensure both half peaks are sampled adequately without one cluster ‘dying out’. Furthermore, the two half peaks occur at opposite ends of the domain of a linear space, making it more difficult for a sampler to explore the regions of higher probability efficiently.\ The wrapped trial distribution resolves both of these issues, as the two half peaks in linear space are treated as one full peak as far as the sampling (and allocation of livepoints) is concerned. Consequently, the second issue of the half peaks being far away from each other is automatically eradicated.\ The wrapped trial distribution methodology can thus be applied to problems which involve sampling on non-Euclidean spaces. We apply the method to toy models with distributions defined on circles and tori in Section \[s:gns\_tmI\]. Variance of the trial distribution {#s:gns_trialvar} ---------------------------------- As with any sampling procedure which relies on a trial distribution, picking a variance for the distribution is difficult without a-priori knowledge of the posterior distribution you are sampling from. A low variance results in a lot of trial points being accepted, but a high auto correlation between these points. A high variance gives a lot of trial rejections, but when these points are accepted, their correlation with the starting point is often low. Since picking the trial variance can in itself be a mammoth task, we use a simplistic approach and take it to be $$\label{e:gns_trialvar} 0.1 \times \left\lvert \max\displaylimits_{\mathrm{livepoints}}\left(\theta_{i}\right) - \min\displaylimits_{\mathrm{livepoints}}\left(\theta_{i}\right) \right\rvert ,$$ for each component $i$ of ${\boldsymbol{\theta}}$. We use this approach to avoid the sampler from taking large steps when the livepoints are close together. However, we acknowledge that this method is far from optimal when the livepoints are compactly located at the edges of the domain of $\mathcal{P}\left({\boldsymbol{\theta}}\right)$. Non-Euclidean sampling via coordinate transformations {#s:gns_coordtrans} ----------------------------------------------------- The wrapped trial distribution introduced in Section \[s:gns\_wrap\] can in theory be used in Metropolis nested sampling to sample effectively from circular and toroidal spaces parameterised in terms of circular variables. However, it is not particularly effective at sampling from spherical spaces, since wrapping around the zenith angle (usually defined on $[0, \pi]$) would result in discontinuous jumps between the poles of the sphere. One could of course just wrap the trial distribution in the dimension representing the azimuthal angle (usually defined on $[0, 2\pi]$), rather than in both angles. However, this would re-introduce the issues stated in Section \[s:gns\_geom\_ns\], i.e. wasting samples and inefficient exploration of the parameter space. We therefore propose an alternative method for exploring spherical spaces which we incorporate in the geometric nested sampling algorithm. Spherical coordinate transformations {#s:gns_spheretrans} ------------------------------------ Assuming the surface of a unit sphere is parameterised by azimuthal angle $\phi$ on $[0, 2\pi]$ and zenith angle $\theta$ on $[0, \pi]$, then the corresponding Cartesian coordinates are $$\label{e:gns_spherecoords} \begin{split} &x = r \cos(\phi)\sin(\theta), \\ &y = r \sin(\phi)\sin(\theta), \\ &z = r \cos(\theta), \end{split}$$ with $r = 1$. Note that $\phi$ is the angle measured anti-clockwise from the positive $x$-axis in the $x$–$y$ plane and $\theta$ is the angle measured from the positive $z$-axis. Thus a trial point $\phi_{\rm t}, \theta_{\rm t}$ can be sampled as follows. Starting from a point $\phi_{l}, \theta_l$, calculate $x_l, y_l, z_l$, from which a trial point $x', y', z'$ can be sampled from $q(x', y', z' | x_l, y_l, z_l)$. We use a three-dimensional spherically symmetric Gaussian distribution for $q(x', y', z' | x_l, y_l, z_l)$. In general, the point $x', y', z'$ will not lie on the unit sphere. Nevertheless the point is implicitly projected onto it by solving the equations given by \[e:gns\_spherecoords\] simultaneously for $\phi$ and $\theta$, where we set $x = x'$, $y = y'$, $z = z'$, and $r = r'$ (see Figure \[f:gns\_sphere\]). The resulting values are $\phi_{\rm t}$ and $\theta_{\rm t}$, from which the acceptance ratio given by equation \[e:gns\_ns\_accept\] can be evaluated as normal.\ There are a few things to note about sampling the trial point in the Cartesian space. Firstly, for equation \[e:gns\_ns\_accept\] to hold we must have $q(\phi_{\rm t}, \theta_{\rm t} | \phi_l, \theta_l) = q(\phi_l, \theta_l | \phi_{\rm t}, \theta_{\rm t})$, which is equivalent to $$\label{e:gns_carttrials} \int \displaylimits_{{\boldsymbol{x}}' \in \{{\boldsymbol{x}}_{\mathrm{t}, \phi,\theta}\}} q({\boldsymbol{x}}'|{\boldsymbol{x}}) \mathrm{d}{\boldsymbol{x}}' = \int\displaylimits_{{\boldsymbol{x}} \in \{{\boldsymbol{x}}_{l,\phi,\theta}\}} q({\boldsymbol{x}}|{\boldsymbol{x}}') \mathrm{d}{\boldsymbol{x}},$$ where ${\boldsymbol{x}}' = (x',y',z')$ and ${\boldsymbol{x}} = (x,y,z)$. $\{{\boldsymbol{x}}_{\mathrm{t}, \phi,\theta}\}$ are the set of Cartesian coordinates which satisfy \[e:gns\_spherecoords\] for $\phi = \phi_{\rm t}$, $\theta = \theta_{\rm t}$, and all $r \neq 0$. Similarly $\{{\boldsymbol{x}}_{l,\phi,\theta}\}$ are the ${\boldsymbol{x}}$ which satisfy \[e:gns\_spherecoords\] for $\phi = \phi_l$ & $\theta = \theta_l$ (see Figure \[f:gns\_sphere\]). Due to the symmetry of the spherical coordinate system, these sets of vectors lie along the lines given by $(\phi_{\rm t}, \theta_{\rm t})$ and $(\phi_l, \theta_l)$ respectively. The only additional requirement for equation \[e:gns\_carttrials\] to hold is that $q(x',y',z'|x,y,z)$ is symmetric in its arguments, which it is provided that $q(a|b)$ is a symmetric function about the point $b$. As in Section \[s:gns\_wrap\], the symmetry of the trial distribution ensures that the detailed balance relation given by equation \[e:gns\_db\] is still satisfied. Sampling in Cartesian coordinates eliminates the risk of sampling points which are automatically rejected (due to being outside the support of $\pi(\phi,\theta)$) to a negligible level, since the only points in Cartesian coordinates which are ill-defined in spherical coordinates are $x=y=0$ for all $z$. How the coordinate transformation improves the manoeuvrability of the sampler relative to sampling in the original parameter space is less clear-cut. For the latter, when the variance is fixed the step sizes taken by the sampler along the surface of the sphere depend on where you start from. For example, at $\theta \approx 0$, large moves in $\phi$ will result in relatively small steps along the sphere whereas at $\theta \approx \pi/2$ such moves in $\phi$ would result in large steps along the sphere. However when sampling in a Cartesian coordinate system, for a constant variance (see below), the trial points sampled will have the same average step size in Euclidean space regardless of the starting point. Furthermore due to the symmetry of a sphere, when the sampled point ($x',y',z'$) is projected back onto the sphere (implicitly when determining $\phi_{\rm t}$ and $\theta_{\rm t}$), the variance of the steps along the sphere is still independent of the starting point.\ In either the original parameter space or the transformed space, the variance of the trial distribution can be tweaked to adjust the average step size of the sampler. Nevertheless, it seems more intuitive to the writers to perform the sampling in the space in which adjusting the variance has an effect which is independent of where you are sampling from on the sphere. A spherical distribution (namely the five-parameter Fisher-Bingham distribution, also known as the Kent distribution) is used in the toy model presented in Section \[s:gns\_tmII\]. Variance of the Cartesian trial distribution {#s:gns_spherevar} -------------------------------------------- For given variances of $\phi$ and $\theta$: $\sigma_{\phi}^2$ and $\sigma_{\theta}^2$, the variance corresponding to a function of these two variables is given by $$\label{e:gns_funcvar} \sigma_{f}^2 = \left( \frac{\partial f}{\partial \phi} \right)^2 \sigma_{\phi}^2 + \left( \frac{\partial f}{\partial \theta} \right)^2 \sigma_{\theta}^2 + 2 \frac{\partial f}{\partial \phi} \frac{\partial f}{\partial \theta} \sigma_{\phi,\theta},$$ where $\sigma_{\phi,\theta}$ is the covariance between $\phi$ and $\theta$. Hence one can calculate the corresponding variance in Cartesian coordinates, $\sigma_{x}^2$, $\sigma_{y}^2$, and $\sigma_{z}^2$ by substituting the equations given by \[e:gns\_spherecoords\] into equation \[e:gns\_funcvar\]. Using these values for $q(x',y',z'|x,y,z)$ however, leads to an asymmetric trial distribution in its arguments, since the variance is now a function of $\theta$ and $\phi$. Our entire formulation of the geometric nested sampling algorithm requires $q$ to be symmetric in order for equations \[e:gns\_ns\_accept\] and \[e:gns\_db\] to hold. Thus we set $\sigma_{x}^2 = \sigma_{y}^2 = \sigma_{z}^2 = 4 / 100$ to ensure $q$ is symmetric. Non-spherical coordinate transformations {#s:gns_nonspheretrans} ---------------------------------------- The transformation of the trial sampling problem introduced in the previous Section need not be unique to the case of a sphere. Indeed, our implementation of geometric nested sampling includes the option to transform to Cartesian coordinates from circular or toroidal parameters. This is done in the same way as described for the spherical case, but with the relations given by equation \[e:gns\_spherecoords\] replaced with the equivalent transformations for a circle or torus (see @kj_thesis).\ However, given the circular nature of the variables parameterising the points on a circle / torus, we do not think that performing coordinate transformations for these objects will give any advantages over using the wrapped trial distributions in the original parameter spaces. Hence in the applications considered in this paper, parameters which exhibit circular or toroidal properties will be sampled using the wrapped trial distribution, whilst those of a spherical nature will be sampled using the coordinate transformation methodology. The coordinate transformation methodology can be applied to arbitrary geometries; however it is important to recognise that geometries which lack symmetry will in general be much more difficult to sample from without breaking the trial distribution symmetry requirement of the Metropolis acceptance ratio. This may lead to violation of detailed balance which is a *sufficient* condition for a Markov chain to asymptotically converge to the target distribution (in this case the posterior). Applications of geometric nested sampling {#s:gns_applications} ========================================= We now apply the geometric nested sampling algorithm to models which include circular, toroidal and spherical parameters. We evaluate the algorithm’s performance by plotting the posterior samples using <span style="font-variant:small-caps;">getdist</span>. For comparison we calculate posterior samples using <span style="font-variant:small-caps;">MultiNest</span> [@2009MNRAS.398.1601F], a state of the art clustering nested sampling algorithm, effective on low dimensional problems.\ We refer to the samples / distributions obtained from the geometric nested sampler as MG (Metropolis geometric nested sampling), and those obtained from <span style="font-variant:small-caps;">MultiNest</span> as MN.\ Toy model I: circular and toroidal distributions {#s:gns_tmI} ------------------------------------------------ The first toy models considered highlight the usefulness of the wrapping of the trial distribution used by the geometric nested sampler in the case of circular or toroidal parameters. ### Circular distribution {#s:gns_tmI_c} We first consider the problem of a one-dimensional circular distribution from which we would like to sample from. The model is parameterised by one variable $\phi$, which is defined on $[0, 2\pi]$. Referring back to Section \[s:gns\_priors\] we take $\pi(\phi)$ to be uniform on $[0, 2\pi]$. For the likelihood function, we use the von Mises distribution introduced in Section \[s:gns\_circular\] and defined by $$\label{e:gns_c} \mathcal{L}_{c}\left(\phi | \mu, \sigma^2\right) = \frac{\exp(\cos(\phi - \pi - \mu)/ \sigma^2)}{2\pi I_{0}\left(\frac{1}{\sigma^2}\right)},$$ where $\mu$ and $\sigma$ are the mean and standard deviation of the distribution, and $I_{0}(x)$ is the zeroth order modified Bessel function. Here we set $\mu = 0$ so that the peak of the posterior distribution is wrapped around $[0, 2\pi]$, and appears as two half peaks. We set the variance equal to $0.25$.\ Since the problem involves the circular parameter $\phi$, the geometric nested sampling algorithm uses a wrapped trial distribution. For this low-dimensional (one-dimensional) problem both MG and MN recover the correct distribution easily, even when the algorithms are run with a low number of livepoints ($ n_{l} = 50 $). We therefore consider a more complicated model in the form of a toroidal distribution in the next Section, to which the geometric argument of applying the wrapped trial distribution methodology stands equally well. ### Toroidal distribution {#s:gns_tmI_t} The ‘usual’ three-dimensional torus ($2$-torus) can be thought of as a topological space homeomorphic to the Cartesian product of two circles. The corresponding three-dimensional toroidal distribution is therefore the product of two circular distributions. This idea can be generalised to a hypertorus also known as an $n$-torus, in which case the $n$-toroidal distribution is equal to the product of $n$ circular distributions, $$\label{e:gns_t} \mathcal{L}_{t}\left(\theta_{1},...,\theta_{n} | \mu_{1},...,\mu_{n}, \sigma_{1}^2,...,\sigma_{n}^2\right) = \prod_{i=1}^{n}\mathcal{L}_{c}\left(\theta_{i} | \mu_{i}, \sigma_{i}^2\right).$$ Here we use the same circular distribution as the one defined in Section \[s:gns\_tmI\_c\] and note that the ‘half peaks’ observed in the one-dimensional parameter space of a circular distribution become ’quarter peaks’ when observing any two-dimensional subspace of a toroidal parameter space. We thus expect this $n$-toroidal distribution to have $n (n-1) / 2 \times 4$ quarter peaks across all unique two-dimensional subspaces (see e.g. Figure \[f:gns\_l\_t\_post\]) for $n\geq 2$, which most samplers would treat as a $2n (n-1)$ mode problem, while the geometric nested sampler would interpret them as just one mode. Note that when considering the entire parameter space at once, the toroidal distribution has $2^n$ modes and so for the 6-torus this corresponds to $64$ modes, but for purposes of visualisation we focus our analysis on the algorithms’ ability to infer the two-dimensional quarter peaks. We run the MG and MN algorithms on a 6-torus toy model, which uses a uniform prior on $[0, 2\pi]$ for each $\theta_1,...,\theta_6$. For each sampler we do two runs, one with a low number of livepoints ($n_l = 50$) and one with a relatively high number ($n_l = 500$).\ Figures \[f:gns\_l\_t\_post\] and \[f:mn\_l\_t\_post\] show the posterior distributions for the low livepoint runs using the MG and MN algorithms respectively. By looking at the two-dimensional marginalised posteriors it is clear that the MG sampler does a good job at recovering all $60$ quarter peaks present in the $6$-torus model. This is expected even for a low number of livepoints, as the MG still treats the problem as unimodal, and the dimensionality of the problem is not too taxing. On the other hand the MN algorithm appears to completely miss three quarter peaks in the $95\%$ mean confidence intervals (top left corner of $\theta_1-\theta_2$, bottom right corner of $\theta_2-\theta_6$, and bottom right corner of $\theta_3-\theta_6$ posteriors) while it significantly underestimates the posterior mass (does not capture the peaks in the $68\%$ contours) on a further $14$ of the quarter peaks. For a low number of livepoints this is to be expected for any sampler which treats each of the quarter peaks as separate modes. In fact MultiNest does a surprisingly good job at not completely missing peaks, given the low number of livepoints means it can’t generate nearly as many ellipsoids as there are quarter peak modes, or more importantly the $64$ modes present when considering the parameter space as a whole.\ Figures \[f:gns\_h\_t\_post\] and \[f:mn\_h\_t\_post\] show similar plots of the posteriors but with each algorithm run with $500$ livepoints. The MG algorithm shows little improvement over its performance with $50$ livepoints, which if anything emphasises how well the algorithm performed in the low livepoint case. The MultiNest algorithm shows a marked improvement over its performance using $50$ livepoints, as it no longer completely misses any quarter peaks in the $95\%$ contours, and only two peaks are not encapsulated by the $68\%$ contours. Nevertheless the algorithm still underperforms the $50$ livepoint run with the MG algorithm. Toy model II: spherical distribution {#s:gns_tmII} ------------------------------------ We now consider a spherical toy model to illustrate the Euclidean transformation technique described in Section \[s:gns\_coordtrans\]. The geometric nested sampler uses this technique for variables which parameterise points on a sphere in terms of an azimuthal angle (usually denoted $\theta$) and polar angle ($\phi$). A natural choice for this model is the five parameter Fisher-Bingham distribution (@1982kent, also known as the Kent distribution) which is described below. ### Kent distribution {#s:gns_kent} The Kent distribution is a probability distribution defined on the surface of a three-dimensional unit sphere. It is the spherical equivalent to a two-dimensional Gaussian distribution on a linear space, and can be parameterised in terms of spherical or Cartesian coordinates. In the latter case the distribution is given by $$\label{e:gns_k} \mathcal{L}_{K}\left(\hat{{\boldsymbol{x}}}\right) = \frac{1}{c(\kappa, \beta)} \exp \left( \kappa \hat{{\boldsymbol{\gamma}}}_1 \cdot \hat{{\boldsymbol{x}}} + \beta \left[ (\hat{{\boldsymbol{\gamma}}}_2 \cdot \hat{{\boldsymbol{x}}})^2 - (\hat{{\boldsymbol{\gamma}}}_3 \cdot \hat{{\boldsymbol{x}}})^2 \right] \right),$$ (suppressing the parameters being conditioned on in $\mathcal{L}_{K}(\hat{{\boldsymbol{x}}})$ for brevity). Here $\hat{{\boldsymbol{x}}}$ is a unit vector pointing from the centre of the sphere to a point on its surface. The parameters $\kappa$ and $\beta$ describe the concentration (c.f. $1 / \sigma^2$ in the normal distribution) and ellipticity of the distribution respectively. The higher $\kappa$ ($\beta$) is, the more concentrated (elliptical) the contours of equal probability are. The vectors $\hat{{\boldsymbol{\gamma}}}_1,~\hat{{\boldsymbol{\gamma}}}_2$ and $\hat{{\boldsymbol{\gamma}}}_3$ describe the orientation of the distribution, with $\hat{{\boldsymbol{\gamma}}}_1$ pointing (from the centre of the sphere to a point on its surface) to the mean of the distribution, while $\hat{{\boldsymbol{\gamma}}}_2$ and $\hat{{\boldsymbol{\gamma}}}_3$ point in the direction of the major and minor axes of the contours of the distribution. Thus the three vectors must be orthogonal to each another. $c(\kappa, \beta)$ is a normalisation factor given by $$\label{e:gns_k_const} c(\kappa, \beta) = 2 \pi \sum_{i=0}^{i = \infty} \frac{\Gamma \left(i+ \frac{1}{2} \right)}{\Gamma \left(i + 1 \right)}\beta^{2i} \left( \frac{\kappa}{2} \right)^{-2i - \frac{1}{2}} I_{2i + \frac{1}{2}}(\kappa),$$ where $I_{\alpha}(x)$ is the $\alpha^{\mathrm{th}}$ order modified Bessel function. The model we consider here is a sum of four Kent distributions, each with $\kappa = 100$ and $\beta = 50$. Defining $\mathbfss{G}_i$ as the matrix of orthogonal column vectors $\hat{{\boldsymbol{\gamma}}}_1,~\hat{{\boldsymbol{\gamma}}}_2$ and $\hat{{\boldsymbol{\gamma}}}_3$ for the $i^{\mathrm{th}}$ Kent distribution, the spherical likelihood is given by $$\label{e:gns_s} \mathcal{L}_{s}(\hat{{\boldsymbol{x}}}) = \sum_{i = 1}^{i = 4} \mathcal{L}_{K}\left(\hat{{\boldsymbol{x}}} | \mathbfss{G}_i \right),$$ where we explicitly state the conditional dependence of $\mathcal{L}_{K}$ on $\mathbfss{G}_i$ to emphasise the matrices are different for each $\mathcal{L}_{K}$ in the summation. The $\mathbfss{G}_i$ are given by $$\label{e:gns_gammas} \begin{split} \mathbfss{G}_1 &= \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix}, \qquad \mathbfss{G}_2 = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix}, \\ \mathbfss{G}_3 &= \frac{1}{\sqrt{2}}\begin{bmatrix} 0 & -1 & 1 \\ 0 & 1 & 1 \\ \sqrt{2} & 0 & 0 \end{bmatrix}, \qquad \mathbfss{G}_4 = \frac{1}{\sqrt{2}}\begin{bmatrix} 0 & 1 & -1 \\ 0 & 1 & 1 \\ \sqrt{2} & 0 & 0 \end{bmatrix}. \end{split}$$ When combined with a uniform prior over $[0, 2\pi]$ for polar angle $\phi$ and sin prior over $[0, \pi]$ for azimuthal angle $\theta$, the resultant posterior is the one shown in Figure \[f:gns\_kentg100\] (determined by evaluating the posterior over a grid of $\phi,~\theta$). We refer to this as the ‘flower’ distribution, which is centred on ‘north’ pole ($\theta = 0$) of the sphere. Note that the eight ‘petals’ of the distribution more or less correspond to modes of the distribution when projected on a linear space (bottom plot of the Figure). ### Sampling the spherical toy model {#s:gns_kent_results} Due to the complexity of the flower distribution, we only consider runs of the MG and MN algorithms with $500$ livepoints. We first ran the algorithms on a model involving a single sphere which translates to a two-dimensional sampling space in $\phi$ and $\theta$. However, as was the case with the two-dimensional circular distribution, both algorithms recovered the posterior distribution with ease due to the low dimensionality of the problem. We therefore consider the problem of six separate spheres (in the case of the geometric nested sampler, the angles parameterising each sphere are transformed individually) each containing a petal distribution as the likelihood and a spherical prior. This results in a $12$-dimensional parameter space: $\phi_1, \theta_1,..., \phi_6, \theta_6$ which should prove significantly more challenging for the samplers.\ Figure \[f:gns\_h\_s\_post\] shows the results for the MG (black curves) and MN (red curves). Neither algorithm by any means recovers the true distribution perfectly, but for the majority of the polar angles the MG infers the eight petals of the distribution more symmetrically than MN, with the exception of $\phi_1$ and $\phi_6$ where both algorithms perform poorly. This toy model presents a problem with $6 \times 8 = 48$ modes, a modest amount, but it is the geometric nature of these modes which causes difficulties for MN. Referring back to Figure \[f:gns\_kentg100\] encapsulating the areas of prior mass is difficult for ellipsoidal samplers due to the petals being connected to each other, as it requires lots of small ellipsoids to be concatenated together to approximate the shape of the region. Combining this with the higher dimensionality of the problem of six spheres means that regions of high probability are inevitably going to be missed by the algorithm. The fact that the distribution is centred around the pole of the sphere hinders algorithms which sample in the angular parameter space further, as such an algorithm must move ‘around’ the pole to sample the different petals. The geometric nested sampler does not suffer from this issue, as since it samples from the Cartesian space, it has no concept of the pole of the sphere. Geometric nested sampling implementation {#s:gns_imp} ======================================== The implementation of the geometric nested sampler (and the vanilla Metropolis nested sampler) used in this paper, along with the toy models and the gravitational wave likelihood function can be found at <https://github.com/SuperKam91/nested_sampling>. The algorithm is written in <span style="font-variant:small-caps;">Python 2.7</span>, hence our *implementation* of the algorithm cannot match that of the state of the art nested sampling algorithms such as <span style="font-variant:small-caps;">MultiNest</span> or <span style="font-variant:small-caps;">POLYCHORD</span> [@2015MNRAS.453.4384H]. These algorithms are implemented in <span style="font-variant:small-caps;">FORTRAN 90</span>, and parallelised using a master-slave paradigm (see Section 5.4 of Handley, Hobson, & Lasenby). Nevertheless there is no reason why geometric nested sampling cannot be implemented more efficiently and parallelised using this method. Conclusions {#s:gns_conc} =========== We have presented a new nested sampling algorithm based on the Metropolis nested sampler proposed in @Sivia2006 and applied in @2008MNRAS.384..449F. Our algorithm exploits the geometric properties of certain kinds of parameters which describe points on circles, tori and spheres, to sample the parameters more efficiently in the context of nested sampling. The algorithm should be more mobile in sampling distributions defined on such geometries.\ The algorithm consists of two key sampling modes which can be summarised as follows. - For circular and toroidal problems, the trial distribution used in the sampling process is wrapped around the support of the prior distribution $\pi$ (domain of the posterior distribution $\mathcal{P}$). - This wrapping ensures that no trial points are automatically rejected when evaluating the Metropolis acceptance ratio as a consequence of the point being outside the sampling space of the model. - The wrapped trial distribution also makes the sampling more mobile at the edges of the domain of $\mathcal{P}$, meaning that circular and toroidal distributions should be easier to sample, particularly in the case of posteriors with high probability densities at these edges. - For spherical problems, parameters specifying the coordinates on a sphere are transformed to Cartesian coordinates and sampled from the corresponding Euclidean space. - This again ensures that no trial points are automatically rejected because they are outside the domain of $\mathcal{P}$. - It also enhances the mobility of the sampler, whose average step size along the surface of the sphere is not dependent on the location at which the trial distribution is centred. We applied the geometric nested sampling algorithm (MG) to three toy models, which respectively represented models on a circle, hypertorus ($n$-torus) and spheres. We compared the posterior plots with those obtained from the livepoint clustering nested sampling algorithm <span style="font-variant:small-caps;">MultiNest</span> (MN, @2009MNRAS.398.1601F). Our results can be summarised as follows. - For the circular toy model (von Mises distribution centred on the origin), the MG and MN samplers perform equally well. We attribute this to the low dimensionality of the problem. - We therefore considered a toroidal ($6$-torus) distribution which was equivalent to the product of six von Mises circular distributions. - We found that when using a low number of livepoints ($50$), the MG recovers all $60$ quarter peaks present in two-dimensional parameter subspaces very well, while the MN algorithm more or less completely missed three of these peaks, and recovered a further $14$ of them poorly. - For a high livepoint run ($500$) the MG performs similar as to what it did in the low livepoint case, while the MN shows significant improvement but still underperforms relative to the low livepoint MG run. This is in essence because the MG treats the problem as a unimodal distribution. - The first spherical toy model considered comprised of a sum of four Kent distributions (a normal distribution defined on the surface of a unit $2$-sphere) whose resultant distribution resembles a ‘flower’ centred on the pole of the sphere, which had eight ‘petals’ containing high probability mass. This two-dimensional problem in polar angle $\phi$ and azimuthal angle $\theta$ proved easy for both samplers, due to low dimensionality. - We therefore considered a problem comprised of six separate spheres, each of which had its own flower distribution and was transformed to Euclidean space independently in the case of the geometric nested sampler. This translates to a $12$-dimensional parameter estimation problem in $\phi_1, \theta_1,..., \phi_6, \theta_6$ with $6 \times 8 = 48$ petals (modes of the distribution). - We found that both algorithms failed to recover the true distributions near perfectly, but the MG did a better job at inferring the correct shape of the modes (and their symmetry with respect to each other). Acknowledgements {#acknowledgements .unnumbered} ================ The authors would like to thank Greg Willatt & David Titterington from Cavendish Astrophysics for computing assistance. They would also like to thank Dave Green for his invaluable help using LaTeX. Kamran Javid acknowledges an STFC studentship. Chopin, N., Robert, C. 2008. Properties of Nested Sampling. arXiv e-prints arXiv:0801.3887. 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--- abstract: 'We study a deterministic method for particle transport in tissue in selected medical applications. Generalized Fokker-Planck (GFP) theory [@LeaLar01] has been developed to improve the Fokker-Planck (FP) equation in cases where scattering is forward-peaked and where there is a sufficient amount of large-angle scattering. We compare grid-based numerical solutions to Fokker-Planck and Generalized Fokker-Planck (GFP) in realistic applications. Electron dose calculations in heterogeneous parts of the human body are performed. Accurate electron scattering cross sections are therefore included and their incorporation in our model is extensively described. Moreover, we solve GFP approximations of the radiative transport equation to investigate reflectance and transmittance of light in tissue. All results are compared with either Monte Carlo or discrete-ordinates transport solutions.' address: 'RWTH Aachen University, Department of Mathematics & Center for Computational Engineering Science, Schinkelstrasse 2, 52062 Aachen, Germany' author: - Edgar Olbrant - Martin Frank bibliography: - 'literature.bib' nocite: '[@*]' title: 'Application of Generalized Fokker-Planck Theory To Electron And Photon Transport In Tissue' --- Introduction ============ A difficult and important challenge in electron and photon transport is still the numerical solution of the Boltzmann transport equation (BTE) [@Gar04; @Bor98]. To contribute to this field of research, we study selected approximations of the transport equation for electrons and photons and present numerical results in different geometries. Nowadays cancer patients often undergo therapies with high energy ionizing radiation. In external radiotherapy photon beams dominate in clinical use whereas less patients receive electron therapy. Treatments are also performed by using heavy charged particles like ions or protons. These have higher costs for their particle accelerators but gain more and more importance due to first high intensity laser systems for protons [@SchPfoJae06]. To aid the recovery of patients it is important to deposit a sufficient amount of energy in the tumor. Simultaneously, the ambient healthy tissue should not be damaged. Therefore, the success of such radiation treatments strongly depends on the correct dose distribution. It is recommended that uncertainties in dose distributions should be less than 2% to get an overall desired accuracy of 3% in the delivered dose to a volume [@AAPMreport85]. Additionally, thresholds have been developed to compare dose results computed by different algorithms. A suggested tolerance in homogeneous geometries is the 2% (relative pointwise difference) or 2 mm (absolute distance to agreement) criterion. However, in heterogeneities this limit increases to 3% or 3 mm [@VenWelMij01]. Up to now many clinical dose calculation algorithms rely on pencil beam models. Originally developed for cosmic ray showers, Fermi ([@Fer40], cited by [@RosGre41]) and Eyges [@Eyg48] introduced a small-angle scattering theory which was afterwards applied to electrons. This theory was used by Hogstrom et al. to propose the pencil-beam model. Their algorithm includes experimental data, taken from dose measurements in a water phantom, to compute the central-axis dose [@HogMilAlm81]. Although it was a first clinically applicable model its accuracy deep in the irradiated material or in heterogeneities is poor. This is basically due to the small-angle approximation in Fermi-Eyges theory, $\theta \approx \tan(\theta)$. It is true that single electron collisions show small deviations. However, after multiple scattering events they accumulate to big angle changes in large penetration depths. Besides crude approximations like small deviations of particles throughout their whole path through the tissue [@LarMifFra97], geometric structures transverse to the beam direction are assumed to be infinite. Although many improvements of Fermi-Eyges theory were performed, e.g., by including additional correction factors [@Jet88; @JetBie89; @AhnSaxTre92; @ShiHog91] they still suffer from the small anlge and the homogeneity assumption. Comparisons with experimental data showed disadvantages in inhomogeneous phantoms [@MarReyWag02]. A statistical simulation method for radiation transport problems is the Monte Carlo method [@And91]. It performs direct simulations of individual particle tracks which result from a random sequence of free flights and interaction events. In this way random histories are generated. If their number is large enough macroscopic quantities can be obtained by averging over the simulated histories. Monte Carlo tools model physical processes very precisely and can handle arbitrary geometries without losing accuracy. Although they rank among the most accurate methods for predicting absorbed dose distributions, their high computation times limit their use in clinics. Due to the increase in computing power and decrease in hardware costs Monte Carlo techniques have recently become a growing field in radiotherapy [@SpeLew08]. Not only general-purpose Monte Carlo codes are now publicly available but also commercial Monte Carlo treatment planning systems [@SiaWalDAl01; @CygLocDas05]. However, they have not yet gained widespread clinical use. A different approach in the solution of radiation transport problems are deterministic calculations solving the linear BTE. In principle, its solution will give very accurate dose distributions comparable to Monte Carlo simulations. The BTE can be analytically solved only in very simplified geometries, which is insufficient for clinical applications. Additionally, numerical solutions to the BTE require deterministic methods coping with a six-dimensional phase space. Because of this and their simpler implementation, Monte Carlo techniques have so far prevailed in the medical physics community. Nevertheless, Börgers [@Bor98] argued that on certain accuracy conditions deterministic methods could compete with Monte Carlo calculations. In this paper, we describe electron and photon transport in media by solving the Generalized Fokker-Planck (GFP) approximation of the linear BTE [@LeaLar01]. For electron transport, this is an extension of the Fokker-Planck (FP) model, formulated in [@HenIzaSie06], to higher-order approximations in angle scattering. It should be stressed that this approach brings along several advantages: The BTE does not require any assumption on the geometry so that arbitrary heterogeneities are possible. Furthermore, we benefit from mathematical and physical approximation ansätze because they can be directly included in the differential equation. This avoids heuristic assumptions. In contrast to Monte Carlo simulation, deterministic solutions do not suffer from statistical noise and their resolution does not depend on the number of particles traversing a certain region. Moreover, the treatment planning problem can be formulated as a PDE-constrained optimization problem. This structure can be used to obtain additional information to speed-up the optimization [@FraHerSan09; @FraHerSch08]. This is hard to achieve by Monte Carlo techniques. Previous studies in deterministic methods for radiotherapy primarily concentrated on a combination of rigorous analytic solutions and laboratory measurements (see pencil beam models above). Without explicitly using experimental data in their model, Huizenga and Storchi [@HuiSto89] presented the phase space evolution (PSE) model for electrons and subsequently applied it to multilayered geometries [@MorHui92]. Various improvements and extension to 3D beam dose calculations were performed [@JanRieMor94; @JanKorSto97]. However, 3D dose calculations showed disadvantages in computation times [@KorAkhHei00]. Moreover, PSE models use first-order discretizations in space which cannot compete with MC techniques [@Bor98]. By contrast, our access focuses on the continuous model of the Boltzmann equation discretized with high-order schemes. First studies for deterministic dose calculations in charged particle transport came from well-known procedures for numerical solutions to the transport equation of neutrons or photons. Several approximative models to the BTE have been developed. Each of these methods has its advantages and drawbacks [@Bru02]. Multi-group methods are sometimes used to discretize the energy domain [@DatAltRay96]. This leads to a number of monoenergetic equations to be separately discretized in space and angle. Whereas discretizations in space are usually done by finite difference and finite element (FE) methods, the remaining angle domain is discretized by discrete ordinates [@Bal00; @Eds05]. Such an approach is implemented in the solver Attila [@GifHorWar06]. Recently, 3D dose calculations for real clinical test cases were performed with Attila [@VasWarDav08]. Results with similar accuracy as Monte Carlo calculations were achieved in promising computation times. Less frequent are angular FE approaches [@CopRav95] seeking to reduce ray effects. Tervo et al. extended FE discretizations to all variables in spatial, angular and energy domains [@TerKolVau99] so that no group cross sections were needed. Moreover, a detailed description of three coupled BTEs with FE discretizations of all variables was proposed in [@BomTerVau05]. In this paper, the BTE is approximated by the continuous slowing down (CSD) method [@LarMifFra97]. Scattering processes are modelled by Generalized Fokker-Planck approximations [@LeaLar01] and compared with classical Fokker-Planck [@Pom92] solutions. Leakeas and Larsen [@LeaLar01] showed that scattering kernels with a sufficient amount of large-angle scattering yield inaccurate FP results. As an extension of the work in [@LeaLar01], we perform deterministic GFP simulations and investigate their behaviour in real applications. It is well-known that electron scattering is dominantly forward-peaked. Hence, many electron transport simulations used the classical FP approximation [@HenIzaSie06; @FraHenKla07; @DucMorTik09]. However, up to now no comparisons with GFP dose computations have been done including realistic physical scattering cross sections. In our case the latter are extracted from ICRU libraries [@ICRU07]. We describe in detail how transport coefficients in the BTE can be computed from these scattering cross sections and compute GFP electron dose profiles in inhomogeneous geometries. Even more challenging for GFP theory are scattering kernels including large angle-scattering. We therefore investigate transport of photons in tissues with forward-peaked and large-angle scattering. Using test cases from [@RodKim08] the radiative transport equation for GFP approximations up to order five is solved to determine reflectance and transmittance of light in tissue. In the remaining paper you will find the following structure: A short discussion of basic electron interactions with matter is given at the beginning. We describe a model for electron transport and review crucial steps of the GFP theory. In addition, we derive transport coefficients for the GFP equations from databases for electron scattering cross sections. In Section 3, a deterministic model for light propagation together with different scattering kernels is introduced. Discretization methods used in our GFP equations are studied in Section 4. In Section 5, we compute FP, GFP and discrete-ordinates results for transmittance and reflectance of light by a slab. Moreover, we numerically compare FP, GFP and Monte Carlo solutions for 5 and 10 MeV electrons in homogeneous and heterogenous slab geometries. Section 6 gives the conclusions and outlooks. Appendix A contains explicit formulae for high-order polynomial operators. In Appendix B, we present the equations to be solved for the GFP coefficients. Deterministic Model for Electron Transport ========================================== Physical Interactions --------------------- Electron beams are nowadays a widely spread tool in cancer therapy. Typical electron beams, provided by high energy linear accelerators, range from 1 to 25 MeV. During irradiation of human tissue electrons interact with matter through several competing mechanisms: 1. *Elastic Scattering:* This is usually a non-radiative interaction between electrons and the atomic shell. Projectiles experience small deflections and lose little energy. High energy electrons can also penetrate through atomic shells and are afterwards scattered at the bare nucleus without any energy loss. With kinetic energies above 1 keV elastic scattering in water dominantly occurs in the forward direction [@LaVPim97]. 2. *Soft Inelastic e$^-$-e$^-$ Scattering:* Electrons interact with other electrons of the outer atomic shell which usually leads to excitation or ionisation of the target particle. Here binding energies are only a few eV so that projectile electrons transfer little energy and are hardly deflected. 3. *Hard Inelastic e$^-$-e$^-$ Scattering:* These collisions are determined by large transfer energies to the target electron. What ’large’ exactly means is specified in MC codes by cutoff energies. In PENELOPE [@Pen09], for example, the default value of this simulation parameter is set to 1 % of the maximum energy of all particles. As a consequence, the target electrons are ejected with larger scattering angles and higher kinetic energies (delta rays). They act as an additional source in the transport equation. 4. *Bremsstrahlung:* Caused by the electrostatic field of atoms, electrons are accelerated and hence emit bremsstrahlung photons. However, for energies below 1 MeV this phenomenon can be neglected. Bremsstrahlung photons are not mainly emitted in the forward direction. The lower their kinetic energy the more isotropic their angle distribution becomes [@Kri07]. Evidently, there are more interaction processes like ejection of Auger electrons or characteristic X-ray photons. But they are very unlikely in the energy range considered. Although inelastic collisions are decisive for the energy transfer, the radiation damage in the patient strongly depends on the spatial distribution of electrons in their passage through matter. They dominantly undergo multiple scattering events with small deviations. However, single backward scattering events also occur frequently which leads to tortuous trajectories of electrons. To a big extent such trajectories are due to elastic collisions. We therefore focus on very accurate and realistic simulation of elastic processes in our model. This is achieved by transport coefficients extracted from the ICRU 77 database [@ICRU07]. Inelastic transport coefficients are obtained in the same way. Elastic and soft inelastic events lead to small energy loss. With kinetic energies above 1 keV, electrons are assumed to lose their energy continuously [@Gar05]. Because of this, we implement the CSD approximation to model energy loss of electrons. Hence, we neglect large energy loss fluctuations caused by hard inelastic collisions. Bremsstrahlung effects are considered in our model in a very restricted way: Only energy transfer of electrons after soft bremsstrahlung collisions are simulated by means of the radiative stopping power. However, the effect of hard photon emission as well as the transport of photons are disregarded so far. The Generalized Fokker-Planck Approximation for Electrons {#GFPforElectrons} --------------------------------------------------------- Particles can be described by a six dimensional phase space $(\uline{r}, E, \uline{\Omega}) \in (V\times I \times S^2)$ with $$\begin{aligned} \text{\uline{spatial variable}} \quad & \uline{r}=(x_1,x_2,x_3)\in V \subset \mathbb{R}^3 \; \text{open and bounded} \\ \text{\uline{energy variable}} \quad & E \in I=[E_{\tx{min}}, E_{\tx{max}}] \subset \mathbb{R} \\ \text{\uline{direction variable}} \quad & \uline{\Omega}=(\Omega_1,\Omega_2,\Omega_3) \in S^2 \subset \mathbb{R}^3 \; \text{unit sphere} \\ &\, \, \, \, \, =(\sqrt{1-\mu^2}\cos(\phi),\sqrt{1-\mu^2}\sin(\phi),\mu), \q \mu=\cos(\theta).\end{aligned}$$ If particle transport takes place in an isotropic and homogeneous medium in which interaction processes are Markovian and particles do not interact with themselves, their distribution can be described by the unique solution of the time-dependent linear BTE. For practical applications in radiotherapy we are faced with issues like distributions of the deposited energy in tissue or penetration depths of the beam. For many purposes it is therefore sufficient to know the steady solution: $$\sigma_a \Psi(\ul{r}, E, \ul{\Omega}) + \ul{\Omega}\cdot\nabla\Psi(\ul{r}, E, \ul{\Omega}) = L_B \Psi(\ul{r}, E,\ul{\Omega}) \label{MBE}$$ $\text{\q with \q} \q \Psi(\uline{r}, E, \uline{\Omega})=\Psi_b(\uline{r}, E,\uline{\Omega}) \q \q \forall \uline{r} \in \partial V,\; \uline{\Omega} \cdot \uline{n}<0, \; E\in I.$\ \ $\Psi(\ul{r}, E, \ul{\Omega})$ is called [*angular flux*]{} and denotes the distribution of particles travelling in direction $\ul{\Omega}$. Boundary conditions (BC) are imposed on the angular flux depending on the incident beam. Absorption of particles is described by the [*absorption cross section*]{} $\sigma_a$. The right hand side $$L_B \Psi(\ul{r},E, \ul{\Omega}):= \int_0^{\infty} \int_{4\pi} \sigma_{s}(E',E,\ul{\Omega}\cdot\ul{\Omega}') \Psi(\ul{r}, E',\ul{\Omega}') d\Omega' dE' - \Sigma_s(E) \Psi(\ul{r},E, \ul{\Omega})$$ is known as linear [*Boltzmann Operator*]{} and describes scattering. Its integral contains the [*differential scattering cross section*]{} (DSCS) $\sigma_s(E',E,\ul\Omega\cdot\ul\Omega')$ characterising interaction mechanisms in which particles are deflected. The dot product $\ul{\Omega} \cdot \ul{\Omega}'=\cos(\theta_0)=\mu_0$ indicates that the scattering probability only depends on the scattering angle. This implies that the deflection of scattered particles is axially symmetrical around the direction of incidence $\ul{\Omega}'$. Integrating the scattering kernel $\sigma_{s}(E',E,\ul{\Omega}\cdot\ul{\Omega}')$ over all angles and energies, one gets the [*total differential scattering cross section*]{} (TDSCS) $$\Sigma_s(E) = 2\pi \int_0^{\infty} \int_{-1}^1 \sigma_s(E', E,\mu_0)d\mu_0 dE'. \label{tdscs2}$$The angle and energy integral in the Boltzmann operator is the main difficulty in its numerical solution. That is why an important aim is to develop accurate approximations. However, up to now there is no predominant method used for all types of particles. In fact, depending on specific particle properties, one has to choose an appropriate approximation. One crucial property of elastic DSCSs in water is a sharp peak in the forward direction [@ItiMas05]. To express this mathematically we define the positive [*n-th scattering transport coefficient*]{} (STC) $$\begin{aligned} \xi_n(E):=2\pi \int_0^{\infty} \int_{-1}^{1}(1-\mu_0)^n \sigma_s(E',E,\mu_0)d\mu_0 dE' \quad \tx{for all } n \geq 0\end{aligned}$$ and assume that as n increases, the coefficients $\xi_n$ fall off sufficiently fast, i.e., $$\begin{aligned} \xi_{n+1}(E) \ll \xi_n(E) \quad \tx{for all } n \geq 0 \tx{ and } E \in I. \label{fpeak}\end{aligned}$$ Additionally, elastic scattering often entails small energy loss. A first approximation is therefore to expand the scattering kernel around $\mu_0=1$ and $E=E'$. In this way Pomraning [@Pom92] showed that the already known FP operator is the lowest-order asymptotic limit of the integral operator $L_B$. In [@LeaLar01] this FP operator is derived as a first order angle approximation to $L_B$:\ $L_{FP}:=\frac{\xi_1}{2}L$ $$\begin{aligned} \tx{where} \q \q \q L_B\Psi(\uline{\Omega})=L_{FP}\Psi(\uline{\Omega})+{\cal O}(\eps) \q \tx{for } \eps \ll 1. \label{LB_FP}\end{aligned}$$ Since the spherical Laplace-Beltrami operator $$\begin{aligned} L = \left[ \frac{\partial}{\partial \mu}(1-\mu^2)\frac{\partial}{\partial\mu} + \frac{1}{1-\mu^2}\frac{\partial^2}{\partial\phi^2} \right] \quad \text{with} \; \mu=\cos(\theta)\end{aligned}$$ is differential in angle the nonlocal *integral* Boltzmann operator $L_B$ is now approximated by a local *differential* operator. The crucial point is that an integro-differential equation is transformed into a partial differential equation. Although discretizations of differential equations often lead to large linear systems their numerical effort turns out to be much lower. This is due to the local character of differential equations which bring along much sparser matrices.\ Pomraning’s resulting Fokker-Planck equation for particle transport in an isotropic medium yields: $$\begin{aligned} \sigma_a \Psi(\ul{r},E,\ul{\Omega}) + \ul{\Omega}\cdot\nabla\Psi(\ul{r},E,\ul{\Omega}) &= \frac{\xi_1(E)}{2} L \Psi(\ul{r},E,\ul{\Omega}) + \frac{\partial (S(\ul{r},E) \Psi(\ul{r},E,\ul{\Omega}))}{\partial E}.\end{aligned}$$ $S(\ul{r},\eps)$ is called stopping power defined by $$\begin{aligned} S(\ul{r},E) = \int_{0}^{\infty} \int_{4\pi} (E -E') \sigma_s(E',E,\ul{\Omega}\cdot \ul{\Omega}') \, d\Omega dE'. \label{StpPom}\end{aligned}$$ The standard Fokker-Planck approximation is a frequently used method to describe transport processes in media with *highly* forward-peaked scattering. Comparisons with real data, however, reveal that many scattering processes of interest contain a small but sufficient amount of large-angle scattering. To gain higher order asymptotic approximations to $L_B$ one could expand $L_{B}$ to a\ \ $L_{Pn}:=\sum_{m=1}^{n}a_{n,m}L^m$ with $a_{n,m} \in {\cal O}(\xi_m)={\cal O}(\eps^{m-1})$ $$\begin{aligned} \tx{with} \q L_B\Psi(\uline{\Omega})=L_{Pn}\Psi(\uline{\Omega})+{\cal O}(\eps^n) \q \tx{for all } n\geq 1. \end{aligned}$$ However, Leakeas and Larsen showed that eigenvalues of $L_{Pn}$ might become positive so that the angular flux could become infinite [@LeaLar01]. This served as motivation for them to develop asymptotically equivalent operators to $L_{Pn}$ which remain stable and preserve certain eigenvalues of $L_B$. We have summarized the details of the computation in the appendix. We end up with Generalized Fokker-Planck (GFP) equations which incorporate large-angle scattering and are therefore more accurate than the conventional Fokker-Planck equation: $$\begin{aligned} \sigma_a \Psi(\ul{r},E,\ul{\Omega}) + \ul{\Omega}\cdot\nabla\Psi(\ul{r},E,\ul{\Omega}) &= L_{GFP_n} \Psi(\ul{r},E,\ul{\Omega}) + \frac{\partial (S(\ul{r},E) \Psi(\ul{r},E,\ul{\Omega}))}{\partial E}.\end{aligned}$$ For positive coefficients $\alpha_i(E), \beta_i(E)$ and $m\in \mathbb{N}$, the GFP operators are defined by $$\begin{aligned} L_{GFP_{2m}} := \sum_{i=1}^m \alpha_i(E) L(I-\beta_i(E) L)^{-1} \quad \text{ and } \quad L_{GFP_{2m+1}} := L_{GFP_{2m}} +\alpha_{m+1}(E)L.\end{aligned}$$ Determination of Physical Quantities ------------------------------------ The fundamental part in GFP theory is based on the assumption of forward-peakedness (). Its expected accuracy strongly depends on the behaviour of transport coefficients $\xi_n(E)$ which differ for different projectiles and materials. The ICRU Report 77 provides differential cross sections for elastic and inelastic scattering of electrons and positrons for different materials and energies between 50 eV and 100 MeV [@ICRU07]. To obtain transport coefficients $\xi_n(E)$ we use these cross sections and proceed in the following way: (a) [The ELSEPA code system, distributed with the report, calculates elastic and inelastic angular differential cross sections for a fixed energy $E$ $$\begin{aligned} \sigma^{\tx{el,inel}}(E,\mu_0)=\int_0^{\infty} \sigma^{\tx{el,inel}}_s(E',E,\mu_0) dE'\end{aligned}$$ in tabulated form for discrete $\mu_0$ and $\sigma^{\tx{el,inel}}(E,\mu_0)$. For a predetermined set of energy values between 50eV and 100MeV, data for $\sigma^{\tx{el,inel}}(E,\mu_0)$ are extracted from these files. ]{} (b) [With that we calculate the $n$-th transport coefficient for a fixed energy $E$ $$\begin{aligned} \xi_n^{\tx{el,inel}}(E)=2\pi \mathcal{N} \int_{-1}^1 (1-\mu_0)^n \sigma^{\tx{el,inel}}(E,\mu_0) d\mu_0 \q \tx{with} \q \mu_0=\cos(\theta_0)\end{aligned}$$ via numerical integration of the tabulated cross sections $\sigma^{\tx{el,inel}}(E,\mu_0)$ by means of the trapezodial rule. Additionally, we multiply the result by the molecular density of the transmitted matter $$\begin{aligned} \mathcal{N}=N_A\frac{\rho}{A}.\end{aligned}$$ $N_A$ is Avogadro’s number, $\rho$ the mass density of the material and $A$ its molar mass. ]{} (c) Again, all computed results of $\xi_n^{\tx{el,inel}}(E)$ are stored and used as a look-up table. To obtain the $n$-th transport coefficient at the desired energy $E$ this tabulated data is linear interpolated. Finally, we use the following transport coefficient in our equation: $$\begin{aligned} \xi_n(E) := 2\pi\mathcal{N} \int_{-1}^1 (1-\mu_0)^n (\sigma^{\tx{el}}(E,\mu_0) +\sigma^{\tx{inel}}(E,\mu_0)) d\mu_0.\end{aligned}$$ ![Comparison of scattering transport coefficients in liquid water for energies between 50eV and 100MeV.[]{data-label="STCtot"}](pict/STCtot.png) Fig. \[STCtot\] illustrates electron transport coefficients of different order in liquid water. Except for the $0$-th, every coefficient is strictly monotonically decreasing. For $E\geq 10^{-3}$, $\xi_1$ is always bigger than $\xi_2$ but, as $E$ decreases, their deviation reduces more and more. Unfortunately, for increasing $n\geq 2$ the difference between two consecutive $\xi_n$ is so small that the assumption of forward-peakedness is not fulfilled. The inelastic transport coefficients are much smaller than elastic transport coefficients for $n \geq 1$. The higher the order of the transport coefficient the larger is the difference between elastic and inelastic ones. Our stopping power in is equivalent to the physical total stopping power as the sum of collision and radiative stopping powers. For different materials both are directly included in the files of the ICRU database. Hence, we do a linear interpolation to get the stopping power $S(E)$ at the desired energy $E$. Deterministic Model for Light Propagation ========================================= The Generalized Fokker-Planck Equation for Grey Photons ------------------------------------------------------- Many medical applications like cancer treatment or optical imaging of tumors make use of propagation of laser light in tissue. Its behaviour is determined by the solution of the steady grey radiative transport equation $$\begin{aligned} \sigma_a \Psi(\ul{r},\ul{\Omega}) + \ul{\Omega}\cdot\nabla\Psi(\ul{r}, \ul{\Omega}) = \mu_s \left[ \int_{4\pi} \sigma_s(\ul{\Omega}\cdot\ul{\Omega}')\Psi(\ul{r},\ul{\Omega}) d\Omega' -\Psi(\ul{r},\ul{\Omega}) \right] \label{RTE}\end{aligned}$$ $\text{\q with \q} \q \q \Psi(\uline{r},\uline{\Omega})=\Psi_b(\uline{r},\uline{\Omega}) \q \forall \uline{r} \in \partial V,\; \uline{\Omega} \cdot \uline{n}<0.$\ Similar to \[GFPforElectrons\] one can derive GFP approximations to the radiative transport equation in a straightforward way. The intensity $\Psi(\uline{r},\uline{\Omega})$ describes the radiation power flowing in direction $\Omega$ which is influenced by scattering and absorption coefficients $\sigma_a$ and $\mu_s$. More important is the scattering kernel $\sigma_s(\ul{\Omega}\cdot\ul{\Omega}')$. It is also characteristic for biological tissue that this kernel has a sharp peak at $\ul{\Omega}\cdot\ul{\Omega}'=1$. For its simulation mathematically simple scattering kernels with a free parameter are used. Models for Scattering Kernels ----------------------------- One often cited scattering kernel is the [*(single) Henyey-Greenstein kernel*]{} defined by $$\begin{aligned} \sigma_s^{HG}(\mu_0) &= \frac{\Sigma_s^{HG}}{2\pi} f_{HG}(\mu_0) \label{HGkernel} \\ \text{where} \q \q \q f_{HG}(\mu_0) &:= \frac{1-g^2}{2(1 -2g\mu_0 +g^2)^{3/2}} \q \tx{for} \q g\in (-1;1) \nonumber\end{aligned}$$ is the corresponding phase function and $\Sigma_s^{HG}$ the TDSCS. The single parameter $g$ determines the amount of small- and large-angle scattering. If $g\approx 1$, $\sigma_s^{HG}$ is not only strongly forward-peaked but also includes large-angle scattering. Its value depends on the irradiated tissue. For human tissue typical values for $g$ are around $0.9$ (human blood: $g=0.99$, human dermis: $g=0.81$ [@ChePraWel90]). Expanding $f_{HG}(\mu_0)$ in Legendre polynomials gives expressions for $\xi_n$ depending only on g: $$\begin{aligned} \xi_1 &= 1-g \\ \xi_2 &= \frac{4}{3} -2g +\frac{2}{3}g^2 \\ \xi_3 &= 2 -\frac{18}{5}g +2g^2 -\frac{2}{5}g^3 \\ \xi_4 &= \frac{16}{5} -\frac{32}{5}g +\frac{32}{7}g^2 -\frac{8}{5}g^3 +\frac{8}{35}g^4 \\ \xi_5 &= \frac{16}{3} -\frac{80}{7}g +\frac{200}{21}g^2 -\frac{40}{9}g^3 +\frac{8}{7}g^4 -\frac{8}{63}g^5\end{aligned}$$ This is all we need to calculate $\alpha_i$ and $\beta_i$ for GFP$_2$ to GFP$_5$. It turns out that they remain throughout positive. To control large-angle and forward-peaked scattering a linear combination of forward and backward Henyey-Greenstein phase functions was introduced [@RodKim08]. For real constants $g_1 \in (-1;0], \, g_2 \in [0;1), \, b \in [0;1]$ and the phase function $$\begin{aligned} f_{DHG}(\mu_0) &:= b f_{HG}(\mu_0,g_1) +(1-b)f_{HG}(\mu_0,g_2), \intertext{the {\it double Henyey-Greenstein} scattering kernel is defined by} \sigma_s^{DHG}(\mu_0) &= \frac{\Sigma_s^{DHG}}{2\pi} f_{DHG}(\mu_0). \label{DHGkernel}\end{aligned}$$ An indicator for the amount of forward or backward scattering is the constant $b$: Setting $b=0$ the backward scattering phase function $f_{HG}(\mu_0,g_1)$ vanishes whereas $b=1$ reduces $f_{DHG}(\mu_0)$ to the single Henyey-Greenstein phase function $f_{HG}(\mu_0,g_1)$. This provides an opportunity to adapt it to the material of interest. Analogous to above procedure for the single HG phase function one can conclude that $$\sigma_{sn} =b(g_1^n -g_2^n) +g_2^n$$ from which all STCs $\xi_n$ can be calculated. Numerics for Generalized Fokker-Planck ====================================== Discretization of Differential GFP Equations {#MonDiffGFP} -------------------------------------------- Replacing the right-hand side of the Boltzmann equation by the GFP$_2$ approximation operator yields: $$\begin{aligned} S(\ul{r},E) \frac{\partial\Psi(\ul{r}, E,\ul{\Omega})}{\partial E} + \sigma_a \Psi(\ul{r},E,\ul{\Omega}) + \ul{\Omega}\cdot\nabla\Psi(\ul{r},E,\ul{\Omega}) &= \alpha(E) L (I- \beta(E)L)^{-1} \Psi(\ul{r},E,\ul{\Omega}) \nonumber \\ &+ \Psi(\ul{r},E,\ul{\Omega})) \frac{\partial S(\ul{r},E)}{\partial E}. \label{GFP2_MBE}\end{aligned}$$ To solve we restate it by setting $$\begin{aligned} \Psi^{(0)} = \Psi(\ul{r},E,\ul{\Omega}) \q \tx{and} \q \Psi^{(1)} = (I-\beta(E) L)^{-1} \Psi^{(0)}, \nonumber\end{aligned}$$ so that it becomes $$\begin{aligned} S(\ul{r},E) \frac{\partial\Psi^{(0)}(\ul{r}, E,\ul{\Omega})}{\partial E} + \sigma_a \Psi^{(0)}(\ul{r},E,\ul{\Omega}) + \ul{\Omega}\cdot\nabla\Psi^{(0)}(\ul{r},E,\ul{\Omega}) &= \alpha(E) L \Psi^{(1)}(\ul{r},E,\ul{\Omega}) \nonumber \\ &+ \Psi^{(0)}(\ul{r},E,\ul{\Omega})) \frac{\partial S(\ul{r},E)}{\partial E} \\ (I-\beta(E) L)\Psi^{(1)}(\ul{r},E,\ul{\Omega}) &= \Psi^{(0)}(\ul{r},E,\ul{\Omega}).\end{aligned}$$ These equations form a coupled system of second-order differential equations with the angular momentum operator L. Solving this system requires differencing schemes in space and angle. Initial and boundary conditions are imposed on $\Psi^{(0)}$.\ The asymptotic GFP analysis transformed the original BTE into a new type of equation which requires an additional condition to the energy variable: $$\begin{aligned} \Psi^{(0)}(\ul{r},E_{\infty},\ul{\Omega})=0. \label{IC}\end{aligned}$$ $E_{\infty}$ denotes a large cutoff energy. In the numerical simulations, it should be bigger than the energy of all particles from the incoming beam.\ A simplified model to be studied is that of a plate which is infinitely extended in x and y directions with a thickness d in z direction. Due to symmetry reasons - [the angular flux is independent of x and y directions and]{} - [its direction of motion $\ul{\Omega}$ only depends on $\theta$.]{} That is why the initial six dimensional problem has now been reduced to a three dimensional one. Although it seems that we only describe a one dimensional object in space one should not forget the fact that our *slab* still remains three dimensional. From the mathematical point of view the symmetry of this model, however, leads to a one dimensional problem in space which decreases computational costs. In slab geometry the aforementioned system reduces to $$\begin{aligned} S(\ul{r},E)\frac{\partial\Psi^{(0)}(z,E,\mu)}{\partial E} &= \alpha(E) L_{\mu} \Psi^{(1)}(z,E,\mu) + \Psi^{(0)}(\ul{r},E,\ul{\Omega})) \frac{\partial S(\ul{r},E)}{\partial E} \nonumber \\ & - \sigma_a \Psi^{(0)}(z,E,\mu) - \D{\Psi^{(0)}(z,E,\mu)}{z}\cdot \mu \label{rhs1} \\ (I-\beta(E) L_{\mu}) & \Psi^{(1)}(z,E,\mu) = \Psi^{(0)}(z,E,\mu), \label{I-betaL_Psi1} \intertext{where the one dimensional angular momentum operator $L_{\mu}$ is defined by} L_{\mu} &:= \D{}{\mu} \left [(1-\mu^2) \D{}{\mu} \right].\end{aligned}$$ We solve this system in two steps: 1. [Obtain the solution $\Psi^{(1)}(z,E,\mu)$ to .]{} 2. [Plug $\Psi^{(1)}(z,E,\mu)$ in and solve the resulting differential equation.]{} To achieve accurate results and lower computation times a high-order scheme was implemented [@Mor85]: $$\begin{aligned} \tilde{L}_{\mu} \Psi^{(1)}(\mu_j) &= \frac{1}{w_j} \left [ D_{j+1/2} \frac{\Psi^{(1)}_{j+1} -\Psi^{(1)}_{j}}{\mu_{j+1} -\mu_j} - D_{j-1/2} \frac{\Psi^{(1)}_{j} -\Psi^{(1)}_{j-1}}{\mu_{j} -\mu_{j-1} } \right ] \label{morel} \\ D_{j+1/2} &= D_{j-1/2} -2\mu_{j}w_{j} \q \tx{with} \q D_{1/2}=0=D_{M+1/2}, \nonumber\end{aligned}$$ where $\mu_{j}$ are abscissas for a Gauss-Legendre quadrature rule with weights $w_{j}$. With knowledge of the already calculated explicit values $\Psi^{(1)}_{i,j}$ the right-hand side of reduces to a single $\Psi^{(0)}$ dependence. The resulting partial differential equation is discretized with finite differences in the z-direction. Hence, we end up with an ordinary differential equation in the energy variable $E$. Its solution is obtained by the embedded 2nd/3rd order Runge-Kutta MATLAB solver [ode23]{} solving from the initial condition in backward in energy to $E=0$. The remaining discretizations are of first order in $z$ and of second order in $\mu$. Higher order GFP equations are discretized and solved analogously. However, due to more frequent occurrence of $L_{\mu}$ and $(I-\beta(E) L_{\mu})$ the right-hand side of becomes more involved and more systems have to be solved. Numerical Results ================= Slab Geometry: HG Kernel {#slabHGSAM} ------------------------ First we neglect absorption and start with a simpler form of the GFP equation $$\begin{aligned} \ul{\Omega}\cdot\nabla\Psi(\ul{r},\ul{\Omega}) = L_{GFP_n} \Psi(\ul{r},\ul{\Omega}) \end{aligned}$$ This is solved in slab geometry with the HG scattering kernel from for selected values of the anisotropy factor g. Symmetry properties mentioned above yield the following boundary value problem (exemplarily stated for GFP$_2$): $$\begin{aligned} \mu \D{\Psi^{(0)}(z,\mu)}{z} &= \alpha L_{\mu}\Psi^{(1)}(z,\mu) \label{ICBC_1} \\ \nonumber (I-\beta L_{\mu})\Psi^{(1)}(z,\mu) &= \Psi^{(0)}(z,\mu)\end{aligned}$$ $$\begin{aligned} \ul{\text{BC}}:& \quad \Psi^{(0)}(0,\mu) =10^5 \cdot e^{-10(1-\mu)^2} \hspace{2.9cm} 1\geq \mu>0 \\ \nonumber & \quad \Psi^{(0)}(d,\mu) =0 \hspace{4.7cm} -1\leq \mu<0\end{aligned}$$ For $g=0.8$ and $g=0.95$ results were computed by time marching with an adaptive Runge-Kutta solver until a steady state was reached. Incoming photons moving in positive z-direction at $z=0$ are simulated by narrow Gaussian peaks around $\mu=1$. Corresponding graphs illustrate steady solutions and use penetration depth in cm as x- and $\int_{-1}^{1}\Psi(z,\mu,s) d\mu$ in 1/(Jcm$^2$s) as y-axis. The latter quantity is sometimes called energy density and is related to the dose. Discretization parameters in z (110 points) and in $\mu$ (64 points) direction were chosen large enough to reach convergence. Figs. \[HG08\_1\]-\[HG095\_1\] additionally show converged transport solutions generated by a discrete ordinates method (DOM) which we use as benchmark in the following. ![HG kernel in slab geometry with g=0.95. Solutions for GFP$_3$-GFP$_5$ already overlap.[]{data-label="HG095_1"}](pict/080_GFPvsTrp_M=64_N=110_s=90_lesspoints.png) ![HG kernel in slab geometry with g=0.95. Solutions for GFP$_3$-GFP$_5$ already overlap.[]{data-label="HG095_1"}](pict/095_GFPvsTrp_N=110_M=64_s=90_lesspoints.png) Each distribution forms a monotonically increasing function until it reaches a maximum and strictly decreases afterwards. There is a large difference between FP, GFP$_2$ and the other GFP approximations. However, GFP$_3$-GFP$_5$ are hardly distinguishable. For increasing values of g this discrepancy between GFP$_3$ and GFP$_5$ becomes bigger whereas results for GFP$_4$ and GFP$_5$ show throughout no distance at all. As to the DOM curve we observe that FP and GFP$_2$ give poor approximations for small penetration depths. However, GFP$_{3,4,5}$ give quite good solutions throughout the whole penetration range. Single Slab: Light Propagation in Tissue ---------------------------------------- González-Rodríguez and Kim studied light propagation in tissue including both forward-peaked and large-angle scattering [@RodKim08]. They examined several approximation methods and especially implemented GFP$_2$. We want to augment this with results up to GFP$_5$ and compare the resulting simulations with the transport solution. We focus on the following problem: $$\begin{aligned} \sigma_a \Psi(z,\mu) + \mu \D{\Psi(z,\mu)}{z} = \mu_s \left[ \int_{4\pi} \sigma_s(\ul{\Omega}\cdot\ul{\Omega}')\Psi(z,\ul{\Omega}') d\Omega' -\Psi(z,\mu) \right] \label{DE_RodKim}\end{aligned}$$ $$\begin{aligned} \ul{\text{BC}}:& \quad \Psi(z=0,\mu) =e^{-10(1-\mu)^2} \hspace{3.7cm} 1\geq \mu>0 \\ & \quad \Psi(z=d=2,\mu) =0 \hspace{4cm} -1\leq \mu<0\end{aligned}$$ It is a slab geometry with a thickness of $d=$ 2mm disregarding any time dependence. Its solution enables to compute reflectance $R(\mu)$ and transmittance $T(\mu)$ defined by $$\begin{aligned} R(\mu)&=\Psi(\mu,0) \qquad -1 \leq \mu < 0 \\ T(\mu)&=\Psi(\mu,d) \hspace{1.1cm} 1 \geq \mu > 0. \\\end{aligned}$$ ![Single HG with g=0.98: Reflectance and transmittance of liver tissue arising from a slab geometry with thickness d=2mm and conditions in . Transmittance is plotted in a semilogarithmic scale.[]{data-label="HG_RodKim"}](pict/HG_Refl_M=64_N=80_s=30.png "fig:")\ ![Single HG with g=0.98: Reflectance and transmittance of liver tissue arising from a slab geometry with thickness d=2mm and conditions in . Transmittance is plotted in a semilogarithmic scale.[]{data-label="HG_RodKim"}](pict/HG_Trm_M=64_N=80_s=30.png "fig:") ![Double HG with $g_1=0.85$, $g_2=-0.34$: GFP approximations for reflectance and transmittance of liver tissue. Transmittance is plotted in a semilogarithmic scale.[]{data-label="DHG_RodKim"}](pict/DHG_Refl_N=70_M=64_s=30.png "fig:")\ ![Double HG with $g_1=0.85$, $g_2=-0.34$: GFP approximations for reflectance and transmittance of liver tissue. Transmittance is plotted in a semilogarithmic scale.[]{data-label="DHG_RodKim"}](pict/DHG_Trm_N=70_M=64_s=30.png "fig:") A. SINGLE HENYEY-GREENSTEIN KERNEL\ \ In the first run was solved with discretizations of 64 points in angle and 80 points in space using GFP approximations with the HG DSCS $\sigma_s=\sigma_s^{HG}$. Further constants were set to $$g=0.98 \quad \sigma_a=0.01 \text{mm}^{-1} \quad \mu_s=50 \text{mm}^{-1}.$$ REFLECTANCE: Starting at $R(-1)\approx 0.23$ the reflectance slightly increases and attains its maximum at $ \mu \approx -0.5$ (). Although in this interval GFP data show discrepancies among each other their results are accurate and GFP$_3$ gives the best approximation. For $\mu>-0.5$ the transport solution DOM hunches down more than GFP functions and hence, the error increases rapidly. Surprisingly, for $\mu \gtrsim -0.25$ GFP$_3$-reflectance values are closer to DOM than those of GFP$_5$. Throughout the whole interval FP values give a very poor approximation. TRANSMITTANCE: It is almost a straight line only bending for small $\mu$. In contrast to the reflectance, a more or less constant distance to the transport solution is always sustained. To the eye, there are no differences between all GFP simulations in a wide range. Only for small $\mu$ the functions start to deviate and GFP$_3$ data give best results whereas FP is inaccurate again.\ \ B. DOUBLE HENYEY-GREENSTEIN KERNEL\ \ Taking the amount of large-angle scattering in biological tissue into account Gonzá lez-Rodríguez and Kim applied the double Henyey-Greenstein DSCS to simulate transmittance and reflectance in liver tissue. The following fit parameters were used: $$g_1=0.85 \quad g_2=-0.34 \quad b=0.86.$$ $g_1=0.85$ provides a forward-peak which is not very sharp. In addition, the combination of $g_2=-0.34$ and $b=0.86$ contains a significant amount of large-angle scattering which leads to increasing STCs: $$\xi_1=0.3166 \quad \xi_2=0.3916 \quad \xi_3=0.6058 \quad \xi_4=1.0075 \quad \xi_5=1.7388$$ In this case our fundamental assumption is not valid which could negatively affect our approximations. Moreover, it is important to emphasize that simulations for GFP$_3$-GFP$_5$ ran with some *negative* coefficients $\alpha_i$, $\beta_i$. Nevertheless, our code gave reasonable results plotted in for discretization parameters of 64 points in $\mu$ and 70 points in $z$ direction. REFLECTANCE: Fig. \[DHG\_RodKim\] shows ’bump head’ functions similarly shaped to those of the single HG kernel. The x-coordinates of their maxima are, however, shifted to the right. Moreover, for large $\mu$ different GFP approximations do not match as well as they do in . In contrast to the single HG kernel GFP$_3$ data give a poor approximation whereas GFP$_5$ is the best one among all shown here. Only for $\mu \approx 0$ GFP$_2$ is not able to match GFP$_5$. As expected, FP gives even worse results than for the single HG kernel. TRANSMITTANCE: This time our transport solution DOM is more peaked at $\mu \approx 1$. Nevertheless GFP gives more accurate results than in . A comparison between the two best approximations GFP$_3$ and GFP$_5$ yields small differences which enlarge near $\mu \approx 0$. However, the classical FP deviates from our benchmark to a big extent. Due to the contribution of large angle scattering numerical computations with the double HG kernel are more challenging and, in fact, give GFP coefficients which contradict our assumptions. Nevertheless, the GFP results plotted above approximate the transport solution very well and are much more precise than those of the FP calculations. Slab Geometry: Electron Propagation in Tissue --------------------------------------------- For dose calculations the following GFP equation is to be solved (examplarily stated for GFP$_2$): $$\begin{aligned} \sigma_a \Psi^{(0)}(z,E,\mu) + \D{\Psi^{(0)}(z,E,\mu)}{z} \cdot \mu &= \alpha L_{\mu} \Psi^{(1)}(z,E,\mu) + \frac{\partial (S(z,E) \Psi^{(0)}(z,E,\mu))}{\partial E} \nonumber \\ (I-\beta L_{\mu})\Psi^{(1)}(z,\mu,s) &= \Psi^{(0)}(z,E,\mu) \label{PomGFP2}\end{aligned}$$ $$\begin{aligned} \ul{\text{BC}}:& \quad \Psi^{(0)}(0,E,\mu) =10^5 \cdot e^{-200(1-\mu)^2} e^{-50(E_0-E)^2} \hspace{1.0cm} 1\geq \mu>0, E \in I .\\ \nonumber & \quad \Psi^{(0)}(d,E,\mu) =0 \hspace{4.7cm} -1\leq \mu<0, E \in I.\end{aligned}$$ The initial boundary value problem in describes the propagation of electrons through matter with a monoenergetic pencil beam of energy $E_0$ irradiated orthogonally to the boundary surface of the material. This is modelled by a product of two narrow Gaussian functions around $\mu=1$ and $E=E_0$. After computing the solution one can calculate the absorbed dose via $$\begin{aligned} D(\ul{r}) &= \frac{2\pi T}{\rho(\ul{r})} \int_0^\infty \int_{-1}^{1} S(\ul{r},E') \Psi^{(0)}(\ul{r},\mu,E') d\mu dE'.\end{aligned}$$ $T$ is hereby the duration of the irradiation of the patient and $\rho$ the mass density of the irradiated tissue so that $D(\ul{r})$ leads to SI unit $J/kg$ or $Gy$. Several test cases were implemented for 5 MeV and 10 MeV beams. As benchmark we used solutions of the MC code systems GEANT4 (standard physics package) [@AgoAllAma03; @AllAmaApo06] and PENELOPE [@Pen09]. However, it should be stressed that all physical models were obtained independently. The following criteria are generally employed to quantify the accuracy of a dose curve [@VenWelMij01]: 2%/2mm (pointwise difference within 2% or 2mm horizontal distance-to-agreement) in homogeneous and 3%/3mm in inhomogeneous geometries.\ A. HOMOGENEOUS GEOMETRY\ \ Characteristic electron dose profiles in a semi-infinite water phantom are shown in and . First they provide a high surface dose, increase to a maximum at a certain depth and drop off with a steep slope afterwards. Solutions for GFP$_4$ and GFP$_5$ are omitted because they overlap with GFP$_3$ in our plot. Except for GFP$_2$, computations were performed according to (32 points in $\mu$, 350 points in $z$). Due to better results we applied upwind finite difference discretizations for GFP$_2$, equidistant in $z$ (400 points) and $\mu$ (200 points). All approximations are close to each other because GFP transport coefficients $\xi_n(E)$ for water do not fall off highly enough within our energy interval. All in all, the calculated results agree well with PENELOPE and GEANT4. All dose profiles for a 5 MeV beam satisfy the 2%/2mm criterion. As we neglect bremsstrahlung the difference to MC computations becomes bigger for $10$ MeV. In fact, the largest FP and GFP$_2$ distance to PENELOPE and GEANT4 becomes 3mm at $z\approx 5$ cm and hence, they do not meet the criterion.\ ![10 MeV electron beam: normalized dose in liquid water.[]{data-label="10MeVelectron"}](pict/5MeV_FP-GFP3_M=32_N=300_epsS=001.png) ![10 MeV electron beam: normalized dose in liquid water.[]{data-label="10MeVelectron"}](pict/10MeV_FP-GFP3_M=32_N=350_epsS=001.png) B. INHOMOGENEOUS GEOMETRIES\ \ Dose calculation is more challenging in parts of the body where materials of strongly varying densities meet. Here, large dosimetric differences between experiments and predictions exist [@MarReyWag02]. As deviations of already five percent in the deposited dose may result in a 20% to 30% impact on complication rates [@AAPMreport85] it is of big importance to accurately compute the dose in such transition regions. Possible clinical applications for electron beams are for example irradiation of the chest wall or the vertebral column. To simulate dose curves on the central beam we assume that 10 MeV electrons pass three different materials: muscle (0-1.5cm), bone (1.5-3cm) and lung (3-9cm). For all results, parameters for Morel’s discretization [@Mor85] were set to 32 points in $\mu$ and 400 points in $z$. Fig. \[10MeVback\] illustrates approximations up to order three because higher order results overlap with the latter on that scale. The agreement with the MC dose profile is very satisfactory although bigger differences occur for small penetration depths. The dose differences between PENELOPE and FP exceed the 3%/3mm limit only at the boundary $z=0$. Radiotherapy gains in importance not least because surgical interventions can be avoided. Especially sensitive body areas like the brain, coated by cerebral membranes, are of big interest. Between those membranes there are many voids which means that scattering and absorption properties change abruptly. Therefore we consider an air cavity irradiated by a 10 MeV electron beam first penetrating water (0-4cm), then air (4-6cm) and later water (6-9cm) again. Similar to pure water, GFP$_2$ calculations yield better solutions for equidistant upwind discretizations (200 points in $\mu$ and 300 points in $z$). Remaining curves were obtained by Morel’s scheme ($\mu$-direction: 32 points, $z$-direction: 350 points). Except for small penetration depths all deterministic solutions are very close to each other and demonstrate good aggreement with PENELOPE (). Again FP and GFP$_2$ results show the best approximations. Larger, but still comparably small, differences between them occur in the air region. Except for the boundary value ($z=0$), the FP and GFP$_2$ curves fulfill the 3%/3mm criterion.\ ![10 MeV electron beam: normalized dose in liquid water with air cavity.[]{data-label="10MeVcavity"}](pict/10MeV_FP-GFP3_N=400_epsS=001_bound2=30.png) ![10 MeV electron beam: normalized dose in liquid water with air cavity.[]{data-label="10MeVcavity"}](pict/10MeV_airCavity_FP-GFP3_N=350_epsS=001.png) Conclusions =========== We studied practical applications of Generalized Fokker Planck approximations. Numerical examples of GFP solutions for the Henyey-Greenstein kernel in slab geometry showed more accurate approximations than FP calculations. Further test cases for reflectance and transmittance in liver tissue by means of single and doulbe HG kernels also revealed GFP$_3$- and GFP$_5$-results closest to the transport solution. For electron transport, we derived an *ab initio* model from the ICRU database. This model was compared to publicly available MC Codes (PENELOPE and GEANT4) which in turn has been benchmarked against experiments. We extracted the stopping power, elastic and inelastic cross sections from the ICRU database and transformed them to transport coefficients needed for GFP computations. Dose distributions for electron beams were performed without additional coupling to photons and positrons. We are aware that our physical model neglects important interactions like energy straggling and hard radiative events with emission of photons. They are inevitable for accurate dose calculations with high-energy electrons. However, in our energy range they are less frequent and regarded as extensions for improved models in future. And in fact, comparisons of GFP approximations with Monte Carlo calculations reveal dose profiles which agree well in both homogeneous and inhomogeneous geometries. Several tasks for further examination remain: (i) The first step towards real dose calculations from CT data is an extension to two space dimensions. As the GFP theory was derived for angular fluxes in 3D space this should only be a challenge to the numerical and programming approach. (ii) Due to a rising demand for proton therapy facilities the adaption of the GFP theory to protons is certainly an interesting subject of further study. (iii) To improve computational results it is necessary to include more physical phenomena. Especially for high energy electron beams it is inevitable to simulate the transport of bremsstrahlung quanta. Acknowledgements {#acknowledgements .unnumbered} ================ The authors would like to thank Bruno Dubroca from Université Bordeaux not only for providing his code for the discrete ordinates method but also for his support and advice. We also acknowledge support from the German Research Foundation DFG under grant KL 1105/14/2 and the German Academic Exchange Sevice DAAD Program D/07/07534. Polynomial Operators ==================== $$\begin{aligned} L_{P2} &= \left (\frac{\xi_1}{2} +\frac{\xi_2}{8} \right )L + \left (\frac{\xi_2}{16} \right )L^2 \\ L_{P3} &= \left (\frac{\xi_1}{2} +\frac{\xi_2}{8} +\frac{\xi_3}{24} \right )L + \left (\frac{\xi_2}{16} +\frac{\xi_3}{36} \right )L^2 + \left (\frac{\xi_3}{288} \right )L^3 \label{LP3} \\ % L_{P4} &= \left (\frac{\xi_1}{2} +\frac{\xi_2}{8} +\frac{\xi_3}{24} +\frac{\xi_4}{64} \right )L + \left (\frac{\xi_2}{16} +\frac{\xi_3}{36} +\frac{3\xi_4}{256} \right )L^2 \nonumber \\ & \quad + \left (\frac{\xi_3}{288} +\frac{5\xi_4}{2304} \right )L^3 + \left (\frac{\xi_4}{9216} \right )L^4 \label{LP4}\end{aligned}$$ $$\begin{aligned} L_{P5} &= \left (\frac{\xi_1}{2} +\frac{\xi_2}{8} +\frac{\xi_3}{24} +\frac{\xi_4}{64} +\frac{\xi_5}{160} \right )L + \left (\frac{\xi_2}{16} +\frac{\xi_3}{36} +\frac{3\xi_4}{256} +\frac{\xi_5}{200} \right )L^2 \nonumber \\ & \quad + \left (\frac{\xi_3}{288} +\frac{5\xi_4}{2304} +\frac{127\xi_5}{115200} \right )L^3 + \left (\frac{\xi_4}{9216} +\frac{\xi_5}{11520} \right )L^4 \nonumber \\ & \quad +\frac{\xi_5}{460800}L^5 \label{LP5}\end{aligned}$$ Derivation of Generalized Fokker-Planck Operators ================================================= If $\lambda_n^{B}$ denotes one eigenvalue of $L_{B}$ for $n\geq 0$ and $\alpha$, $\beta$ are two *positive* constants the Generalized Fokker-Planck (GFP) operator defined by $$\begin{aligned} L_{P2} + \mathcal{O}(\eps^2) = L_{GFP_2} & :=\alpha L(I-\beta L)^{-1} \nonumber \\ & = \alpha L + \alpha\beta L^2 + {\cal O}(\alpha\beta^2)\end{aligned}$$ will have to satisfy three properties to substitute $L_{P2}$ in the favoured way: 1. [Eigenvalue preservation $$\begin{aligned} -\frac{\alpha n(n+1)}{1+\beta n(n+1)} & = \lambda^{GFP_2}_n \nach{!} \lambda^B_{n} = -\sigma_{an} \quad \text{for} \quad n=1,2 \nonumber \intertext{Multiplying above equation by $(1+\beta n(n+1)) \neq 0$ and dividing by n(n+1) we conclude:} (\alpha - \beta\sigma_{an}) &= \frac{\sigma_{an}}{n(n+1)} \quad n=1,2 \nonumber \\ \Leftrightarrow \begin{bmatrix} 1 & -\sigma_{a1} \\ 1 & -\sigma_{a2} \end{bmatrix} \cdot \begin{bmatrix} \alpha \\ \beta \end{bmatrix} & = \begin{bmatrix} \sigma_{a1}/2 \\ \sigma_{a2}/6 \end{bmatrix} \end{aligned}$$ ]{} 2. [Order $\q \q {\cal O}(\alpha\beta^2) \nach{!} {\cal O}(\eps^2)$ ]{} 3. [Equivalence $$\begin{aligned} \alpha L +\alpha\beta L^2 & \nach{!} L_{P2} + {\cal O}(\eps^2) \nonumber \\ & = \left( \frac{\xi_1}{2} + \frac{\xi_2}{8} \right) L + \frac{\xi_2}{16} L^2 + {\cal O}(\eps^2). \nonumber \end{aligned}$$ ]{} $\sigma_{an}$ is a quantity which can be expressed in terms of $\xi_n$: $$\begin{aligned} \sigma_{a0} &= 0 \label{Sigma_s=xi_0} \\ \sigma_{a1} &= \xi_1 \label{sigma_a1} \\ \sigma_{a2} &= 3\xi_1 -\frac{3}{2}\xi_2 \label{sigma_a2} \\ \sigma_{a3} &= 6\xi_1 -\frac{15}{2}\xi_2 +\frac{5}{2}\xi_3 \label{sigma_a3} \\ \sigma_{a4} &= 10\xi_1 -\frac{45}{2}\xi_2 +\frac{35}{2}\xi_3 -\frac{35}{8}\xi_4 \label{sigma_a4} \\ \sigma_{a5} &= 15\xi_1 -\frac{105}{2}\xi_2 +70\xi_3 -\frac{315}{8}\xi_4 +\frac{63}{8}\xi_5. \label{sigma_a5}\end{aligned}$$ In the following item (1) is first transformed to a system of linear equations and thereafter solved for the desired GFP coefficients (here: $\alpha$ and $\beta$). However, the final equations to be solved are non-linear for GFP operators of order $n\geq3$. In this case of order $n=2$ eqs. (\[sigma\_a1\])-(\[sigma\_a2\]) yield $$\begin{aligned} \alpha = \frac{\xi_1}{2} + \frac{\xi_2}{8} \q \tx{and} \q \beta = \frac{\xi_2}{8\xi_1}.\end{aligned}$$ Going on with item (2) it is to be checked: $$\begin{aligned} \alpha \beta^2 = \left( \frac{\xi_1}{2} + \frac{\xi_2}{8} \right) \left( \frac{\xi_2}{8\xi_1} \right)^2 = \underbrace{\left( \frac{\xi_1}{2} + \frac{\xi_2}{8} \right)}_{\makebox[0pt]{$\begin{array}{c} \in {\cal O}(1) \end{array}$}} \underbrace{\left( \frac{\xi_2^2}{64\xi_1^2} \right)}_{\makebox[0pt]{$\begin{array}{c} \in {\cal O}(\eps^2) \end{array}$}} \in {\cal O}(\eps^2) \quad \surd \end{aligned}$$ As equivalence condition follows straight forward it has been shown that the operator $L_{GFP_2}$ is an ${\cal O}(\eps^2)$ approximation to $L_B$ whose first three eigenvalues agree. Now it is quite intuitive to apply a similar procedure to higher order operators. We determined explicit solutions for GFP$_2$-GFP$_5$ coefficients and performed verifications for items (1)-(3). According to GFP$_3$ all items were checked without computer support. The asymptotic behaviour of GFP operators of order four and five was, however, checked by means of a symbolic toolbox. To keep our following description short we confine ourselves to final results. $$\begin{aligned} L_{P3} + \mathcal{O}(\eps^3) = L_{GFP_3} :=& \alpha_1 L(I-\beta_1 L)^{-1} +\alpha_2 L \nonumber \\ =& (\alpha_1 +\alpha_2)L +\alpha_1\beta_1 L^2 +\alpha_1\beta_1^2 L^3 +{\cal O}(\alpha_1\beta_1^3) \label{GFP3}\end{aligned}$$ 1. [Eigenvalue preservation $$\begin{aligned} & (\alpha_1 +\alpha_2) -\sigma_{an}\beta_1 +n(n+1)\beta_1\alpha_2 = \frac{\sigma_{an}}{n(n+1)} & \quad n=1,2,3 \label{linsys3} \\ % \Rightarrow \, \alpha_1 &= \frac{\xi_2(27\xi_2^2 +5\xi_3^2 -24\xi_2\xi_3)}{8\xi_3(3\xi_2 -2\xi_3)} & \label{alpha1} \\ \beta_1 &= \frac{\xi_3}{6(3\xi_2 -2\xi_3)} & \label{beta1} \\ \alpha_2 &= \frac{\xi_1}{2} -\frac{9\xi_2^2}{8\xi_3} +\frac{3\xi_2}{8} & \label{alpha2} \end{aligned}$$ ]{} 2. Equivalence\ To guarantee that $L_{GFP_3}=L_{P3} +{\cal O}(\eps^3)$ all coefficients of $L^i$ in and must coincide. Let $\alpha_1, \beta_1, \alpha_2$ be the defining *positive* coefficients of $L_{GFP_3}$ as stated in eqs. (\[alpha1\])-(\[alpha2\]) then one can show that they satisfy I. [$\alpha_1 +\alpha_2 = \ds \frac{\xi_1}{2} +\frac{\xi_2}{8} +\frac{\xi_3}{24} +{\cal O}(\eps^3)$]{} II. [$\alpha_1\beta_1 = \ds \frac{\xi_2}{16} +\frac{\xi_3}{36} +{\cal O}(\eps^3)$]{} III. [$\alpha_1\beta_1^2 = \ds \frac{\xi_3}{288} +{\cal O}(\eps^3)$]{}. $$\begin{aligned} L_{P4} + \mathcal{O}(\eps^4) = L_{GFP_4} :=& \alpha_1 L(I-\beta_1 L)^{-1} +\alpha_2 L(I-\beta_2L)^{-1} \nonumber \\ =& (\alpha_1 +\alpha_2)L + (\alpha_1\beta_1 +\alpha_2\beta_2) L^2 +(\alpha_1\beta_1^2 +\alpha_2\beta_2^2) L^3 \nonumber \\ +& (\alpha_1\beta_1^3 +\alpha_2\beta_2^3) L^4 +{\cal O}(\alpha_1\beta_1^4) +{\cal O}(\alpha_2\beta_2^4) \label{GFP4}\end{aligned}$$ 1. [Eigenvalue preservation $$\begin{aligned} (\alpha_1 +\alpha_2) -& \sigma_{an}(\beta_1 +\beta_2) -\sigma_{an}n(n+1)\beta_1\beta_2 \nonumber \\ +& n(n+1)(\alpha_1\beta_2 +\alpha_2\beta_1) = \frac{\sigma_{an}}{n(n+1)} \quad n=1,2,3,4 \label{linsys4} \end{aligned}$$ ]{} 2. Equivalence I. [ $\alpha_1 +\alpha_2 = \ds \frac{\xi_1}{2} +\frac{\xi_2}{8} +\frac{\xi_3}{24} +\frac{\xi_4}{64} +{\cal O}(\eps^4)$ ]{} II. [ $\alpha_1\beta_1 +\alpha_2\beta_2 = \ds \frac{\xi_2}{16} +\frac{\xi_3}{36} +\frac{3\xi_4}{256} +{\cal O}(\eps^4)$ ]{} III. [ $\alpha_1\beta_1^2 +\alpha_2\beta_2^2 = \ds \frac{\xi_3}{288} +\frac{5\xi_4}{2304} +{\cal O}(\eps^4)$ ]{} IV. [ $\alpha_1\beta_1^3 +\alpha_2\beta_2^3 = \ds \frac{\xi_4}{9216} +{\cal O}(\eps^4)$ ]{} $$\begin{aligned} L_{P5} + \mathcal{O}(\eps^5) = L_{GFP_5} :=& \alpha_1 L(I-\beta_1 L)^{-1} +\alpha_2 L(I-\beta_2L)^{-1} +\alpha_3L \nonumber \\ =& (\alpha_1 +\alpha_2 +\alpha_3)L +(\alpha_1\beta_1 +\alpha_2\beta_2) L^2 +(\alpha_1\beta_1^2 +\alpha_2\beta_2^2) L^3 \nonumber \\ +& (\alpha_1\beta_1^3 +\alpha_2\beta_2^3) L^4 +(\alpha_1\beta_1^4 +\alpha_2\beta_2^4) L^5 +{\cal O}(\alpha_1\beta_1^5) +{\cal O}(\alpha_2\beta_2^5) \label{GFP5}\end{aligned}$$ 1. [Eigenvalue preservation $$\begin{aligned} (\alpha_1 +\alpha_2 +\alpha_3) -& \sigma_{an}(\beta_1 +\beta_2) -\sigma_{an}n(n+1)\beta_1\beta_2 \nonumber \\ +& n(n+1)\left[ \beta_2(\alpha_1 +\alpha_3) +\beta_1(\alpha_2 +\alpha_3) \right] \nonumber \\ +& \alpha_3\beta_1\beta_2 \left[ n(n+1) \right]^2 = \frac{\sigma_{an}}{n(n+1)} \quad n=1,2,3,4,5 \label{linsys5} \end{aligned}$$ ]{} 2. Equivalence I. [ $\alpha_1 +\alpha_2 +\alpha_3 = \ds \frac{\xi_1}{2} +\frac{\xi_2}{8} +\frac{\xi_3}{24} +\frac{\xi_4}{64} +\frac{\xi_5}{160} +{\cal O}(\eps^5)$ ]{} II. [ $\alpha_1\beta_1 +\alpha_2\beta_2 = \ds \frac{\xi_2}{16} +\frac{\xi_3}{36} +\frac{3\xi_4}{256} +\frac{\xi_5}{200} +{\cal O}(\eps^5)$ ]{} III. [ $\alpha_1\beta_1^2 +\alpha_2\beta_2^2 = \ds \frac{\xi_3}{288} +\frac{5\xi_4}{2304} +\frac{127\xi_5}{115200} +{\cal O}(\eps^5)$ ]{} IV. [ $\alpha_1\beta_1^3 +\alpha_2\beta_2^3 = \ds \frac{\xi_4}{9216} +\frac{\xi_5}{11520} +{\cal O}(\eps^5)$ ]{} V. [ $\alpha_1\beta_1^4 +\alpha_2\beta_2^4 = \ds \frac{\xi_5}{460800} +{\cal O}(\eps^5)$ ]{} All linear and nonlinear equations stated above lead to explicit solutions for $\alpha_i$ and $\beta_i$. Equations posed for GFP$_2$ and GFP$_3$ actually deliver *uniquely* determined constants whereas for higher GFP operators there is no guarantee for unique or even real valued $\alpha_i$ and $\beta_i$. Nevertheless, it is important to emphasize that the resulting values of $\alpha_i$ and $\beta_i$ must be positive. Otherwise, eigenvalues $\lambda_{n}^{GFP_k}$ of a GFP$_k$ operator could become negative. For DSCS $\sigma_s(\ul\Omega\cdot\ul\Omega')$ of different materials, and thus different $\xi_n$, this has to be checked separately.

--- abstract: | In this article we derive the explicit solution of 2-D Stokes system in exterior of the disc with no-slip condition on inner boundary and given velocity ${\mathbf{v}}_\infty$ at infinity. It turned out it is the first application of the associated Weber-Orr transform to mathematical physics in comparison to classical Weber-Orr transform which is used in many researches. From no-slip condition for velocity field we will obtain Robin-type boundary condition for vorticity. Then the initial-boundary value problem for vorticity will be solved with help of the associated Weber-Orr transform. Also the explicit formula of Biot-Savart Law in polar coordinates will be given. Primary 76D07; Secondary 33C10. title: 'Associated Weber-Orr transform, Biot-Savart Law and explicit solution of 2D Stokes system in exterior of the disc.' --- Preliminary {#preliminary .unnumbered} =========== The exterior domain is the most natural area for study the Stokes flow past an obstacles. The advantage of the planar flow of the incompressible fluid is the fact that the motion equations are reduced to only one vorticity equation. It works well in the case of Cauchy problem when the domain is the hole space ${\mathbb{R}}^2$. But in exterior domains when the fluid interacts with the object by no-slip condition, then the Dirichlet boundary condition for velocity transforms to integral relations deduced from Biot-Savart law which describes the inverse of the curl operator and has rather complicated form. In this article we will show that for cylindrical domain $B_{r_0}=\{{\mathbf{x}}\in {\mathbb{R}}^2,~|{\mathbf{x}}|\geq r_0 \},~r_0>0$ the no-slip boundary condition really can be transformed to another boundary condition for vorticity. The reasoning question is how the integral relation can be transformed to boundary ones since the Biot-Savart law doesn’t admit integration by explicit formulas. The matter of fact that this boundary condition is given in terms of Fourier coefficients $w_k(t,r)$ of vorticity function $w(t,{\mathbf{x}})$ and these coefficients itself contain integration by polar angle. Since the Biot-Savart law is the convolution of the vortex fundamental solution with velocity, then the Fourier decomposition breaks the integral relations into series of Robin-type boundary conditions for $w_k(t, r)$ of the type $$\label{int:robin_bound} r_0\frac{\partial w_k(t,r)}{\partial r}\Big|_{r=r_0} + |k| w_k(t,r_0) = 0,~k \in {\mathbb{Z}}.$$ And this relations define the no-slip boundary condition ${\mathbf{v}}(t,{\mathbf{x}})\Big|_{|{\mathbf{x}}|=r_0}=0$ in terms of vorticity function for Stokes flow with prescribed velocity ${\mathbf{v}}_\infty$ at infinity. Here we suppose that initial data satisfies boundary conditions. In this article we will find explicit formula for solution of the non-stationary Stokes system in exterior of the disc $B_{r_0}$ with given velocity value at infinity and no-slip condition at the boundary. With help of vorticity operator we reduce Stokes system to heat equation with zero Robin-type boundary condition (\[int:robin\_bound\]) and zero boundary condition at infinity. The initial boundary value problem for the heat equation in exterior of the disc with various types of boundary conditions could be solved with help of the Weber-Orr transform. In general form it can be written as $$\label{int:weberorr} W_{k,l}[f](\lambda) = \int_{r_0}^\infty R_{k,l}(\lambda,s) f(s) s {\operatorname{ds}},~k,l\in {\mathbb{R}}$$ where $$R_{k,l}(\lambda,s) = J_{k}(\lambda s)Y_{l}(\lambda r_0) - Y_{k}(\lambda s)J_{l}(\lambda r_0),$$ $J_k(r)$, $Y_k(r)$ - are the Bessel functions of the first and second type (see [@BE] [@W]). The inverse transform is defined by the formula $$\label{int:weberorrinv} W^{-1}_{k,l}[\hat f](r) = \int_{0}^\infty \frac{R_{k,l}(\lambda,r)}{{J_{l}^2(\lambda r_0) + Y_{l}^2(\lambda r_0)}} \hat f (\lambda) \lambda {\operatorname{d\lambda}}.$$ Since $J_{k}$, $Y_{k}$ satisfy Bessel equation then the terms $e^{-\lambda^2 t + ik\varphi} J_{k}(\lambda r)$,\ $e^{-\lambda^2 t + ik\varphi} Y_{k}(\lambda r)$ satisfy heat equation, and $$\label{int:explsol} w_k(t,r) = W^{-1}_{k,l} \left [ e^{-\lambda^2 t} W_{k,l} [w_k(0,r)](\lambda) \right ](t,r)$$ multiplied by $e^{ik\varphi}$ do the same. In order to satisfy the initial datum $w_k(0,r)$ at $t=0$ we need invertibility of $W_{k,l}$ and $W^{-1}_{k,l}$. When $l=k$ the above formulas are referred to classical Weber-Orr transform. Titchmarsh[@Tm][@T] proved invertibility of this transform for functions $f(r)$ : $f(r) \sqrt r \in L_1(r_0,\infty)$ with bounded variations in the following form: $$\label{int:invidentity} \frac {f(r+0) + f(r-0)}2 = W^{-1}_{k,l} \left [W_{k,l} [f(\cdot)] \right ](r).$$ For $l=k+1$ and non-negative $k\in{\mathbb{Z}}$ the above formulas define associated Weber-Orr transform $W_{k,k+1}$ with inverse transform $W^{-1}_{k,k+1}$ satisfying the same identity above. This transform applies to elasticity theory and was the subject of some investigations [@S][@KO][@N]. Let’s ask the question: what happens in the case $k>l$, $k \in {\mathbb{N}}\cup \{0\}$ and what is the applications area of this transform? In present paper we will answer on this question when $l=k-1$. In this case the above formula (\[int:invidentity\]) is incorrect. We will deduce correct formula of Weber-Orr transform in order to satisfy invertibility. And then we will present transparent application of this transform to fluid dynamics problems. So, when solving the Stokes system in exterior of the disc we will concentrate on the associated Weber-Orr transform $W_{k,k-1}$. Due to the fact that $R_{k,k-1}(\lambda,r)$ satisfies no-slip boundary condition (\[int:robin\_bound\]) for $k \in {\mathbb{N}}\cup \{0\}$, and so $w_k(t,r)$ given by (\[int:explsol\]) does the same, then the associated Weber-Orr transform $W_{k,k-1}$ will be the central point of our investigation. We can say that at this moment the only equations in which this transform finds its application - are the Stokes and Navier-Stokes systems. The case of $l=k-1$ stands out from the rest Weber-Orr transforms since the formula (\[int:invidentity\]) is no longer true and needs adding some corrections. The point is that the Robin-type condition $$\frac{\partial w_k(t,r)}{\partial r}\Big|_{r=r_0} + \alpha_k w_k(t,r_0) = 0$$ strongly depend on the sign of the $\alpha_k$, and the case $\alpha_k\geq0$ requires more thorough study. Nasim[@N] investigated invertibility of $W_{k,k-1}$ and derived additional term for inverse transform $W^{-1}_{k,k-1}$. But his formula also needs some corrections to get the invertibility relation (\[int:invidentity\]). We will find final formula of $W^{-1}_{k,k-1}$ in order to be inverse to $W_{k,k-1}$. As we will see, only in the case $k \geq 2$ the inverse transform must be supplied with additional correcting term, when for $k=0, 1$ the invertibility identity stays valid with $W^{-1}_{k,k-1}$ given by (\[int:weberorr\]). For $k \geq 2$ the correcting term involves $$\int_{r_0}^\infty s^{-|k|+1}w_k(s){\operatorname{ds}},$$ which as we will see will be equal to zero when no-slip boundary condition is satisfied. And so, finally, we can say, that for both Stokes and Navier-Stokes systems with no-slip boundary condition the formula (\[int:weberorr\]) presents actual inverse transform for $W_{k,k-1}$ and the solution of the heat equation is really provided by (\[int:explsol\]). But in the common case the formulas (\[int:weberorr\]), (\[int:weberorrinv\]) do not satisfy the invertibility property (\[int:invidentity\]). The structure of the paper is following. In the next section the statement of the initial boundary value problem is given. In section 2 we will deduce Biot-Savart law in polar coordinates for exterior of the disc. There will be considered both no-slip and slip boundary conditions. In section 3 we deduce no-slip boundary condition in terms of vorticity. And in section 4 we will find the solution of the Stokes system. The statement of the problem and the main result ================================================ Consider the Stokes system defined in exterior of the disc $B_{r_0}=\{{\mathbf{x}}\in {\mathbb{R}}^2,~|{\mathbf{x}}| > r_0 \},~r_0>0$ with given horizontal flow at infinity ${\mathbf{v}}_\infty = (v_{\infty},0) \in {\mathbb{R}}^2$: $$\begin{aligned} &&\partial_t {\mathbf{v}}- \Delta {\mathbf{v}}= \nabla p \label{maineq}\\ &&{\rm div}~{\mathbf{v}}(t,{\mathbf{x}})=0 \label{freediv}\\ &&{\mathbf{v}}(0,{\mathbf{x}})={\mathbf{v}}_0({\mathbf{x}}) \label{init}\\ &&{\mathbf{v}}(t,{\mathbf{x}})=0,~|{\mathbf{x}}|=r_0 \label{bound}\\ &&{\mathbf{v}}(t,{\mathbf{x}}) \to {\mathbf{v}}_\infty,~|{\mathbf{x}}|\to \infty. \label{boundinf}\end{aligned}$$ Here ${\mathbf{v}}(t,{\mathbf{x}})=(v_1(t,{\mathbf{x}}),v_2(t,{\mathbf{x}}))$ is the velocity field and $p(t,{\mathbf{x}})$ is the pressure. Applying to equations above the curl operator $w(t,{\mathbf{x}})=$ ${\rm curl}~{\mathbf{v}}(t,{\mathbf{x}})$ $=\partial_{{\mathbf{x}}_1}v_2 - \partial_{{\mathbf{x}}_2}v_1$ we get the system of equations for vorticity $$\begin{aligned} \frac{\partial w(t,{\mathbf{x}})}{\partial t} - \Delta w = 0, \label{maineqw} \\ w(0,{\mathbf{x}})=w_0({\mathbf{x}}) \label{initw}\\ {\operatorname{curl}}^{-1} w(t,{\mathbf{x}}) \Big|_{|{\mathbf{x}}|=r_0} = 0, \label{boundw}\\ w(t,{\mathbf{x}}) \to 0,~|{\mathbf{x}}|\to \infty \label{boundinfw}\end{aligned}$$ with initial datum $w_0({\mathbf{x}})={\rm curl}~{\mathbf{v}}_0({\mathbf{x}})$. The last system will be the main object of our study instead of (\[maineq\])-(\[boundinf\]). It is naturally assumed, that the flow at infinity is the plane-parallel ones with zero circularity: $$\label{zerocirculation} \lim_{R\to\infty}\oint_{|{\mathbf{x}}|=R} {\mathbf{v}}\cdot d\mathbf{l} = 0.$$ Further considerations will be held under zero-circulation assumption which due to Stokes formula and no-slip condition leads to zero mean vorticity: $$\begin{aligned} \lim_{R\to\infty}\oint_{|{\mathbf{x}}|=R} {\mathbf{v}}(t,{\mathbf{x}}) \cdot d\mathbf{l} = \int_{B_{r_0}} {\operatorname{curl}}{\mathbf{v}}(t,{\mathbf{x}}) {\operatorname{d\mathbf{x}}}\nonumber \\ + \oint_{|{\mathbf{x}}|=r_0} {\mathbf{v}}(t,{\mathbf{x}}) \cdot d\mathbf{l} = \int_{B_{r_0}} w(t,{\mathbf{x}}) {\operatorname{d\mathbf{x}}}= 0.\end{aligned}$$ We will use Fourier expansions in polar coordinates $r$, $\varphi$: $$\begin{aligned} &&{\mathbf{v}}(t,r,\varphi) = \sum_{k=-\infty}^\infty {\mathbf{v}}_{k}(t,r)e^{ik\varphi},\nonumber \\ &&{\mathbf{v}}_0(r,\varphi) = \sum_{k=-\infty}^\infty {\mathbf{v}}^0_{k}(r)e^{ik\varphi} \nonumber \\ &&w(t,r,\varphi) = \sum_{k=-\infty}^\infty w_k(t,r)e^{ik\varphi},\nonumber \\ &&w_0(r,\varphi) = \sum_{k=-\infty}^\infty w^0_{k}(r)e^{ik\varphi}. \nonumber\end{aligned}$$ Here ${\mathbf{v}}(t,r,\varphi) = (v_r, v_\varphi)$, ${\mathbf{v}}_k(t,r) = (v_{r,k}, v_{\varphi,k}) =\frac 1{2\pi} \int_0^{2\pi} {\mathbf{v}}(t,r,\varphi) e^{-ik\varphi} d\varphi $, ${\mathbf{v}}_0(r,\varphi) = (v^0_{r}, v^0_{\varphi})$, ${\mathbf{v}}^k_0(r) = (v^0_{r,k}, v^0_{\varphi,k})=\frac 1{2\pi} \int_0^{2\pi} {\mathbf{v}}_0(r,\varphi) e^{-ik\varphi} d\varphi$ - are the vector fields decomposed on radial and tangent components. The main result of the paper is Let vector field ${\mathbf{v}}_0({\mathbf{x}})$ satisfies (\[freediv\]), (\[bound\]), (\[boundinf\]), (\[zerocirculation\]), ${\operatorname{curl}}{\mathbf{v}}_0({\mathbf{x}})$ $ \in L_1(B_{r_0})$, and its Fourier series as well as Fourier series for vorticity converges and coefficients $w_k^0(r)$ satisfy $w_k^0(r) \sqrt r \in L_1(r_0,\infty)$, $k \in {\mathbb{Z}}$. Then the solution of (\[maineqw\])-(\[boundinfw\]) is defined via Fourier coefficients: $$w_k(t,r) = W^{-1}_{|k|,|k|-1} \left [ e^{-\lambda^2 t} W_{|k|,|k|-1} [w^0_k(\cdot)](\lambda) \right ](t,r),$$ where $W_{|k|,|k|-1}$, $W^{-1}_{|k|,|k|-1}$ are the associated Weber-Orr transforms (\[int:weberorr\]), (\[int:weberorrinv\]). Biot-Savart law in polar coordinates. ===================================== Now we study when the solenoidal velocity field ${\mathbf{v}}({\mathbf{x}})$ can be uniquely restored from its vorticity $w({\mathbf{x}})$. Consider the following system $$\begin{aligned} &&\rm{div}~ {\mathbf{v}}({\mathbf{x}}) = 0 \label{freediv2} \\ &&\rm{curl}~ {\mathbf{v}}({\mathbf{x}}) = w({\mathbf{x}}) \label{curleq} \\ &&{\mathbf{v}}({\mathbf{x}})=0,~|{\mathbf{x}}|=r_0 \label{bound2}\\ &&{\mathbf{v}}({\mathbf{x}})\to{\mathbf{v}}_\infty,~|{\mathbf{x}}|\to \infty \label{boundinf2}.\end{aligned}$$ Rewrite (\[freediv2\]),(\[curleq\]) in polar coordinates in terms of Fourier coefficients $v_{r,k}$, $v_{\varphi,k}$: $$\begin{aligned} &&{\frac {1}{r}}{\frac {\partial }{\partial r}}\left(rv_{r,k}\right)+{\frac {ik}{r}} v_{\varphi,k} = 0,\\ &&{\frac {1}{r}}{\frac {\partial }{\partial r}}\left(rv_{\varphi,k}\right)-{\frac {ik}{r}} v_{r,k} = w_k.\end{aligned}$$ The basis for solutions of homogeneous system when $w_k \equiv 0$ consists of two vectors: $$\begin{aligned} \begin{pmatrix} v^1_{r,k} \\ v^1_{\varphi,k} \end{pmatrix} = \begin{pmatrix} ir^{-k-1} \\ r^{-k-1} \end{pmatrix} , \\ \begin{pmatrix} v^2_{r,k} \\ v^2_{\varphi,k} \end{pmatrix} = \begin{pmatrix} ir^{k-1} \\ -r^{k-1} \end{pmatrix}.\end{aligned}$$ Rewrite the horizontal flow ${\mathbf{v}}_\infty = (v_\infty,0)$ in polar coordinates: $$v_\infty\begin{pmatrix} \cos\varphi \\ -\sin\varphi \end{pmatrix} = v_\infty\begin{pmatrix} e^{i\varphi}/2 + e^{-i\varphi}/2 \\ -e^{i\varphi}/2i + e^{-i\varphi}/2i \end{pmatrix}.$$ Suppose, that Fourier coefficients $w_k(s)$ of $w({\mathbf{x}})$ satisfy: $$\begin{aligned} \label{BS:cond} w_k(s) \sqrt s \in L_1(r_0,\infty), k \in {\mathbb{Z}}\setminus \{0\}, \\ w_0(s) s \in L_1(r_0,\infty). \nonumber\end{aligned}$$ Using the Fourier expansion of ${\mathbf{v}}_\infty$ the solution of the system (\[freediv2\]), (\[curleq\]), (\[boundinf2\]) will be given by the following formula: k&gt;0:&& \[BS:bck1\]\ &v\_[r,k]{}(r) = 2 \_[r\_0]{}\^r s\^[k+1]{}w\_k(s)+ 2 \_r\^s\^[-k+1]{}w\_k(s)+ 2 v\_\ &+ \_k i r\^[-k-1]{},\ &v\_[,k]{}(r) = 2 \_[r\_0]{}\^r s\^[k+1]{}w\_k(s)- 2 \_r\^s\^[-k+1]{}w\_k(s)- v\_\ & + \_k r\^[-k-1]{}, k&lt;0: &&\[BS:bck2\]\ &v\_[r,k]{} = - 2 \_[r\_0]{}\^r s\^[-k+1]{}w\_k(s)- 2 \_r\^s\^[k+1]{}w\_k(s)+ 2 v\_\ &+ \_k i r\^[k-1]{},\ &v\_[,k]{} = 2 \_[r\_0]{}\^r s\^[-k+1]{}w\_k(s)- 2 \_r\^s\^[k+1]{}w\_k(s)+ v\_\ &- \_k r\^[k-1]{}, k=0: \[BS:bck3\] &&\ &v\_[r,0]{} = r,\ &v\_[,0]{} = 1r \_[r\_0]{}\^r s w\_0(s)+ r, where $\delta_{k,1}$, $\delta_{k,-1}$ are the Kronecker deltas, $\{\alpha_k\}_{k=-\infty}^\infty$, $\theta$ will be found from the rest boundary condition (\[bound2\]). The term $\frac {\theta}r$ in the last formula describes discrete vortex motion $$\label{discretevortex} \sigma_{\theta} ({\mathbf{x}}) = \theta \frac {{\mathbf{x}}^\perp } { |{\mathbf{x}}|^2},$$ where $~{\mathbf{x}}^\perp = (-x_2,x_1)$, and $\theta$ defines its intensity. No-slip condition ----------------- For no-slip condition (\[bound2\]) from $v_{r,k}(r_0)=v_{\varphi,k}(r_0)=0$ in virtue of (\[BS:bck3\]) immediately follows $\alpha_0 = 0$, and $\theta = 0$. For $k>0$ from (\[BS:bck1\]) follows &v\_[r,k]{}(r\_0) = 2 \_[r\_0]{}\^s\^[-k+1]{}w\_k(s)+ 2 v\_+ \_k i r\^[-k-1]{} = 0,\ &v\_[,k]{}(r\_0) = - 2 \_[r\_0]{}\^s\^[-k+1]{}w\_k(s)- v\_ + \_k r\^[-k-1]{} = 0, and so $\alpha_k = 0$, $k \in {\mathbb{N}}$. In similar way from (\[BS:bck2\]) we have $\alpha_k = 0$ for negative $k$. Then if there exists the solution of (\[freediv2\]) - (\[boundinf2\]) satisfying (\[BS:cond\]), then it will be given by (\[BS:bck1\])-(\[BS:bck3\]) with $\alpha_k = 0$, $k \in {\mathbb{Z}}$, and $\theta = 0$. Under assumption of zero-circularity (\[zerocirculation\]) from Stokes’ theorem we have $$\lim_{R\to\infty}\oint_{|{\mathbf{x}}|=R} {\mathbf{v}}({\mathbf{x}}) \cdot d\mathbf{l} = \int_{B_{r_0}} w({\mathbf{x}}) {\operatorname{d\mathbf{x}}}= 2\pi \int_{r_0}^\infty w_0(s) s {\operatorname{ds}}= 0,$$ and the term in (\[BS:bck3\]) is represented as $$\frac 1r \int_{r_0}^r s w_0(s){\operatorname{ds}}= \frac 1{2r} \int_{r_0}^r s w_0(s){\operatorname{ds}}- \frac 1{2r} \int_r^\infty s w_0(s){\operatorname{ds}}.$$ Finally, the above formulas, which constitute Biot-Savart law in cylindrical domains, we rewrite for $k \in {\mathbb{Z}}$: $$\begin{aligned} \label{BiotSavar1} &&v_{r,k} = {\operatorname{sign}}(k) \frac{ir^{-|k|-1}}2 \int_{r_0}^r s^{|k|+1}w_k(s){\operatorname{ds}}\\ &&~~~~~~~~~~~~~~~~~~+ {\operatorname{sign}}(k) \frac{ir^{|k|-1}}2 \int_r^\infty s^{-|k|+1}w_k(s){\operatorname{ds}}+ \frac{\delta_{|k|,1}}2 v_\infty \nonumber \\ \label{BiotSavar2} &&v_{\varphi,k} = \frac{r^{-|k|-1}}2 \int_{r_0}^r s^{|k|+1}w_k(s){\operatorname{ds}}\nonumber \\ &&~~~~~~~~~~~~~~~~~~- \frac{r^{|k|-1}}2 \int_r^\infty s^{-|k|+1}w_k(s){\operatorname{ds}}- {\operatorname{sign}}(k) \frac{\delta_{|k|,1}}{2i} v_\infty,~~\end{aligned}$$ where ${\operatorname{sign}}(k) = \begin{cases} 1,~k > 0 \\ 0,~k=0 \\ -1,~k<0 \end{cases}$ - signum function, $\delta_{|k|,1}$ is the Kronecker delta, $v_\infty$ - the velocity of horizontal flow at infinity. Formulas (\[BiotSavar1\]), (\[BiotSavar2\]) combined with (\[bound2\]) lead to relations on vorticity ($k\in {\mathbb{Z}}$): $$\label{noslipcondintegral} \int_{r_0}^\infty s^{-|k|+1}w_k(s){\operatorname{ds}}= i v_\infty \delta_{|k|,1} {\operatorname{sign}}(k) / r_0^{|k|-1}.$$ \[Biotpolarnoslip\] Let Fourier coefficients $w_k(r)$ satisfy (\[BS:cond\]), (\[noslipcondintegral\]). Then there exists the unique solution of (\[freediv2\]) - (\[boundinf2\]) given by (\[BiotSavar1\]), (\[BiotSavar2\]) with Fourier coefficients $v_{r,k}$, $v_{\varphi,k} \in L_\infty(r_0,\infty)$. Indeed, (\[bound2\]), (\[boundinf2\]) and the inclusion $v_{r,k}$, $v_{\varphi,k} \in L_\infty(r_0,\infty)$ follows from (\[BS:cond\]), (\[noslipcondintegral\]). Formulas (\[BiotSavar1\]), (\[BiotSavar2\]) satisfy (\[freediv2\]), (\[curleq\]). Since the coefficients $\alpha_k = 0$, $k \in {\mathbb{Z}}$, and $\theta = 0$ in (\[BS:bck1\])-(\[BS:bck3\]) are uniquely defined, then the solution is also unique. Slip condition -------------- Now we will weaken (\[bound2\]) by $$\label{BS:slip} \left({\mathbf{v}}({\mathbf{x}}), {\mathbf{n}}\right) = 0,~|{\mathbf{x}}|=r_0,$$ where ${\mathbf{n}}$ is the outer normal to the boundary and find the solution of the undetermined system (\[freediv2\]), (\[curleq\]), (\[boundinf2\]) with (\[BS:slip\]). It will obtain one degree of freedom which concerns to circularity of the flow. From (\[BS:slip\]) follows $v_{r,k}(r_0)=0$, $k \in {\mathbb{Z}}$. Then from (\[BS:bck1\])-(\[BS:bck3\]) $$\begin{aligned} &\alpha_k = - \frac{{\operatorname{sign}}(k) r_0^{2|k|}}2 \int_{r_0}^\infty s^{-|k|+1}w_k(s){\operatorname{ds}}- \frac{r_0^{|k|+1}\delta_{|k|,1}}{2i} v_\infty,~k\in{\mathbb{Z}}\setminus \{0\},\\ &\alpha_k=0,~k=0.\end{aligned}$$ So, the system (\[freediv2\]), (\[curleq\]), (\[BS:slip\]), (\[boundinf2\]) is not uniquely solvable due to arbitrary $\theta \in {\mathbb{R}}$ in (\[BS:bck3\]) which describes the intensity of the circulation around the origin. The vector field ${\mathbf{v}}({\mathbf{x}})$ is restored from its vorticity $w({\mathbf{x}})$ up to circulation of ${\mathbf{v}}({\mathbf{x}})$ around boundary. In cartesian coordinates it means that to any solution of (\[freediv2\]),(\[curleq\]), (\[BS:slip\]), (\[boundinf2\]) ${\mathbf{v}}(t,{\mathbf{x}})$ we can add discrete vortex (\[discretevortex\]) and ${\mathbf{v}}(t,{\mathbf{x}}) + \sigma_{\theta} ({\mathbf{x}})$ will also be the solution of the same system with any $\theta \in {\mathbb{R}}$. In general, in exterior domains in contrast to simple connected ones, the vector field ${\mathbf{v}}({\mathbf{x}})$ is restored from its vorticity $w({\mathbf{x}})$ with one degree of freedom - circularity around boundary (for more details see [@G2]). No-slip condition for vorticity in polar coordinates. ===================================================== Now we a ready to derive no-slip boundary condition for vorticity instead of integral relations (\[noslipcondintegral\]). Initial value problem (\[maineqw\]), (\[initw\]) in polar coordinates is given by equations $$\label{omegaeqpolar} \frac{\partial w_k(t,r)}{\partial t} - \Delta_k w(t,r) = 0, ~w_k(0,r) = w^0_k(r),$$ where $$\Delta_k w_k(t,r) = \frac 1r \frac {\partial}{\partial r}\left(r \frac {\partial}{\partial r}w_k(t,r)\right) - \frac{k^2}{r^2} w_k(t,r).$$ We supply (\[omegaeqpolar\]) with Robin-type boundary condition: $$\label{noslip:robin_bound} r_0\frac{\partial w_k(t,r)}{\partial r}\Big|_{r=r_0} + |k| w_k(t,r_0) = 0,~k \in {\mathbb{Z}}.$$ The set $$\Omega = \{w_k^0(s) \in L_1(r_0,\infty,s^{-|k|+1}{\operatorname{ds}}) ~|~ \int_{r_0}^\infty s^{-|k|+1}w_k^0(s){\operatorname{ds}}= const \}$$ is invariant under the flow $e^{\Delta_k t}$ corresponding to (\[omegaeqpolar\]) iff (\[noslip:robin\_bound\]) holds. $$\begin{aligned} \frac d{dt}\int_{r_0}^\infty s^{-|k|+1}w_k(t,s){\operatorname{ds}}= \int_{r_0}^\infty s^{-|k|+1} \Delta_k w_k(t,s){\operatorname{ds}}\\ = \int_{r_0}^\infty s^{-|k|} \left ( \frac {\partial}{\partial s}\left(s \frac {\partial}{\partial s}w_k(t,s)\right) - \frac{k^2}{s} w_k(t,s) \right ) {\operatorname{ds}}\\ = - r_0^{-|k|+1}\frac{\partial w_k(t,r)}{\partial r}\Big|_{r=r_0} + \int_{r_0}^\infty s^{-|k|} \left ( |k| \frac {\partial}{\partial s}w_k(t,s) - \frac{k^2}{s} w_k(t,s) \right ) {\operatorname{ds}}\\ \\ = - r_0^{-|k|}\left(r_0 \frac{\partial w_k(t,r)}{\partial r}\Big|_{r=r_0} + |k| w_k(t,r_0) \right ) + \int_{r_0}^\infty s^{-|k|-1} \\ \times \left ( k^2 w_k(t,s) - k^2 w_k(t,s) \right ) {\operatorname{ds}}= 0\end{aligned}$$ \[thmrobincond\] Let initial vector field ${\mathbf{v}}_0({\mathbf{x}})$ satisfies(\[freediv\]), (\[bound\]), (\[boundinf\]), (\[zerocirculation\]),\ ${\operatorname{curl}}{\mathbf{v}}_0({\mathbf{x}}) \in L_1(B_{r_0})$, and its Fourier series as well as ones for vorticity converges and coefficients $w_k^0(r)$ satisfy $w_k^0(r) \sqrt r \in L_1(r_0,\infty)$, $k \in {\mathbb{Z}}$. Then coefficients $w_k(t,r)$ of (\[maineqw\])-(\[boundinfw\]) satisfy Robin-type condition (\[noslip:robin\_bound\]). From ${\operatorname{curl}}{\mathbf{v}}_0({\mathbf{x}})$ $ \in L_1(B_{r_0})$ follows $w_0^0(r) r \in L_1(r_0,\infty)$ and (\[BS:cond\]) is satisfied for $k = 0$. No-slip condition imposes the relations (\[noslipcondintegral\]), which must be satisfied for all $t>0$: $$\int_{r_0}^\infty s^{-|k|+1}w_k(t,s){\operatorname{ds}}= i v_\infty \delta_{|k|,1} {\operatorname{sign}}(k) / r_0^{|k|-1},$$ i.e. $\Omega$ must be invariant under the flow $e^{\Delta_k t}$ corresponding to (\[omegaeqpolar\]), (\[noslip:robin\_bound\]). Then the statement of the theorem is the direct result of the previous lemma. Thus, the problem (\[maineqw\])-(\[boundinfw\]) is reduced to (\[omegaeqpolar\]), (\[noslip:robin\_bound\]). Now we will prove, that, as it was mentioned in preliminary, the relation $$\int_{r_0}^\infty s^{-|k|+1}w_k(s){\operatorname{ds}}= 0,~|k| \geq 2,$$ which follows from (\[noslipcondintegral\]), will give the invertibility formula (\[int:invidentity\]) for the associated Weber-Orr transform $W_{k,k-1}$. The solution of the Heat equation with Robin-type boundary condition. Derivation of the Weber-Orr transform. ============================================================================================================ In this section we will solve the initial boundary value problem (\[omegaeqpolar\]), (\[noslip:robin\_bound\]) reducing it to elliptic system with parameter (see [@AV]). Laplace transform $$\hat \omega(\tau,{\mathbf{x}})=\int_0^\infty e^{-\tau t}w(t,{\mathbf{x}}) {\operatorname{d\mathbf{x}}}$$ reduces (\[maineqw\]), (\[initw\]) to Poison equation with parameter $\tau$ $$\label{poison} \Delta \hat \omega - \tau \hat \omega = - w_0({\mathbf{x}}).$$ Then Fourier coefficients $\hat \omega_k(\tau,r)$ satisfy differential equation $$\label{besselrhs} \frac{\partial^2 \hat \omega_k}{\partial r^2} + \frac1r \frac{\partial \hat \omega_k}{\partial r} - \Big(\frac{k^2}{r^2} + \tau \Big) \hat \omega_k = -w_k^0(r).$$ Since the above equation is invariant under the change $k$ to $-k$, then we will carry out further considerations only for non-negative integer $k$. Basis of solutions to homogeneous Bessel equation (\[besselrhs\]) when right-hand side $-w_k^0(r)$ is equal to zero consists of modified Bessel functions of the first and second kind $I_k(\sqrt{\tau}r)$, $K_k(\sqrt{\tau}r)$ (see [@BE], [@W]). Following by [@G2] we exclude exponentially growing function $I_k(\sqrt{\tau}r)$ from the solution of (\[besselrhs\]). The Wronskian of these functions is equal to: $$\label{wronsk} W\left[I_k(r),K_k (r)\right]=-r^{-1}.$$ Then $$K_k(\sqrt{\tau}r)\int_{r_0}^r w_k^0(s) I_k(\sqrt{\tau}s) s ds \nonumber \\ + I_k(\sqrt{\tau}r)\int_r^{\infty} w_k^0(s) K_k(\sqrt{\tau}s) s ds$$ will be the solution of non-homogeneous Bessel equation (\[besselrhs\]). And finally we have the formula: $$\begin{aligned} \label{directsol} \hat \omega_k(\tau,r) = h_k(\tau) K_k(\sqrt{\tau}r) + K_k(\sqrt{\tau}r)\int_{r_0}^r w_k^0(s) I_k(\sqrt{\tau}s) s ds \nonumber \\ + I_k(\sqrt{\tau}r)\int_r^{\infty} w_k^0(s) K_k(\sqrt{\tau}s) s ds ,\end{aligned}$$ where $h_k(\tau)$ will be founded from boundary condition (\[noslip:robin\_bound\]). To do this we apply the operator of boundary condition $$Z_{k,r_0}[\cdot]= r_0{\frac{\partial \cdot}{\partial r }} \Big |_{r=r_0} + k \cdot$$ to the expression $$f_k(\tau,r) := K_k(\sqrt{\tau}r)\int_{r_0}^r w_k^0(s) I_k(\sqrt{\tau}s) s ds + I_k(\sqrt{\tau}r)\int_r^{\infty} w_k^0(s) K_k(\sqrt{\tau}s) s ds.$$ Since $$\frac{\partial f_k}{\partial r}(\tau, r_0) = \sqrt \tau I_k'(\sqrt \tau r_0) \int_{r_0}^\infty w_k^0(s) K_k(\sqrt{\tau}s) s ds,$$ then $$\begin{aligned} Z_{k,r_0} [f_k(\tau,r)] = \sqrt \tau r_0 I_k'(\sqrt \tau r_0) \int_{r_0}^\infty w_k^0(s) K_k(\sqrt{\tau}s) s ds + k I_k(\sqrt{\tau}r_0) \nonumber \\ \times \int_{r_0}^\infty w_k^0(s) K_k(\sqrt{\tau}s) s ds.\end{aligned}$$ Using the differentiation relations on Bessel functions $$\begin{aligned} I_{k-1}(r)-I_{k+1}(r)={\frac {2k }{r}}I_{k }(r)\\ I_{k-1}(r)+I_{k+1}(r)=2{\frac {dI_{k }}{dr}}(r) \\\end{aligned}$$ we will have $$\sqrt \tau r_0 I_k'(\sqrt \tau r_0) + k I_k(\sqrt{\tau}r_0) = \sqrt \tau r_0 I_{k-1}(\sqrt \tau r_0).$$ Then $$\begin{aligned} Z_{k,r_0}[f_k(\tau,r)] = r_0 \sqrt \tau I_{k-1}(\sqrt \tau r_0) \int_{r_0}^\infty w_k^0(s) K_k(\sqrt{\tau}s) s ds.\end{aligned}$$ From $$\begin{aligned} K_{k-1}(r)-K_{k+1}(r)=-{\frac {2k }{r}}K_{k }(r)\\ K_{k-1}(r)+K_{k+1}(r)=-2{\frac {dK_{k }}{dr}}(r)\end{aligned}$$ follows $$\begin{aligned} Z_{k,r_0}[K_k(\sqrt \tau r)] = -r_0 \sqrt \tau K_{k-1}(\sqrt \tau r_0).\end{aligned}$$ We have got the solution of (\[besselrhs\]) with boundary (\[noslip:robin\_bound\]) for any non-negative $k \in {\mathbb{Z}}$ $$\begin{aligned} \hat \omega_k(\tau,r) = \frac{K_{k}(\sqrt{\tau}r) I_{k-1}(\sqrt \tau r_0)}{K_{k-1}(\sqrt \tau r_0)} \int_{r_0}^\infty w_k^0(s) K_{k}(\sqrt{\tau}s) s ds \nonumber \\+ K_{k}(\sqrt{\tau}r)\int_{r_0}^r w_k^0(s) I_{k}(\sqrt{\tau}s) s ds + I_{k}(\sqrt{\tau}r)\int_r^{\infty} w_k^0(s) K_{k}(\sqrt{\tau}s) s ds. \end{aligned}$$ For negative $k \in {\mathbb{Z}}$ previous formula stays valid when $k$ changes to $|k|$ in Bessel functions $K_{k}$, $K_{k-1}$, $I_{k}$, $I_{k-1}$. Finally we have $$\begin{aligned} \label{wksol} \hat \omega_k(\tau,r) = \frac{K_{|k|}(\sqrt{\tau}r) I_{|k|-1}(\sqrt \tau r_0)}{K_{|k|-1}(\sqrt \tau r_0)} \int_{r_0}^\infty w_k^0(s) K_{|k|}(\sqrt{\tau}s) s ds \nonumber \\+ K_{|k|}(\sqrt{\tau}r)\int_{r_0}^r w_k^0(s) I_{|k|}(\sqrt{\tau}s) s ds + I_{|k|}(\sqrt{\tau}r)\int_r^{\infty} w_k^0(s) K_{|k|}(\sqrt{\tau}s) s ds. \end{aligned}$$ In order to obtain the solution $w(t,x)$ of the heat equation we apply inverse Laplace transform $$w(t,r,\varphi) = \frac1{2\pi i} \int_{\Gamma_\eta}e^{\tau t}\hat \omega(\tau,r,\varphi)d\tau,$$ where $\Gamma_\eta = \{ \tau \in {\mathbb{C}}, ~\operatorname{Re}\tau = \eta \}$, $\eta$ - fixed arbitrary positive real number. Since $\sqrt \tau$ involved in (\[wksol\]) is a branching function, then using Cauchy formula we change contour of integration from $\Gamma_\eta$ to $\Gamma_{-\eta, \varepsilon}^-$ with some $\varepsilon>0$, where $$\begin{aligned} \Gamma_{-\eta, \varepsilon}^- = ( -\eta -i\infty, -\eta -i\varepsilon] \cup [-\eta - i\varepsilon, - i\varepsilon] \cup [- i\varepsilon, i\varepsilon] \nonumber \\ \cup [i\varepsilon, -\eta + i\varepsilon] \cup [-\eta + i\varepsilon, -\eta + i\infty). \nonumber\end{aligned}$$ Define oriented contour $\gamma_{\pm,\eta} = [-\eta,0] \cup [0,-\eta]$. Then, going $\varepsilon \to 0$, $\eta \to \infty$ we will have equalities: $$\begin{aligned} &w(t,r,\varphi) = \frac1{2\pi i} \sum_{k=-\infty}^\infty e^{ik\varphi} \int_{-\infty}^0 e^{\tau t}\hat \omega_k(\tau-i0,r)d\tau \nonumber \\ &+ \frac1{2\pi i} \sum_{k=-\infty}^\infty e^{ik\varphi} \int_0^{-\infty} e^{\tau t}\hat \omega_k(\tau+i0,r)d\tau + \sum_{k=-\infty}^\infty e^{ik\varphi} \operatorname{res}\limits_{\tau=0}[\hat \omega_k(\tau,r)] , \nonumber\end{aligned}$$ where $${\mathrm {res}}_{\tau=0}\,[\hat \omega_k(\tau,r)]=\lim _{{\rho \to 0}}{1 \over {2\pi i}}\int \limits _{{|\tau|=\rho }}\!\hat \omega_k(\tau,r)\,d\tau.$$ Set $\tau = -\lambda^2$. Then, we will have $$\begin{aligned} \label{wformula} w(t,r,\varphi) = \nonumber \frac1{\pi i} \sum_{k=-\infty}^\infty e^{ik\varphi} \int_0^\infty e^{-\lambda^2 t} \left ( w_k(-\lambda^2-i0,r) - w_k(-\lambda^2+i0,r) \right ) \lambda \mathrm{d}\lambda \nonumber \\ + \sum_{k=-\infty}^\infty e^{ik\varphi} \operatorname{res}\limits_{\tau=0}[\hat \omega_k(\tau,r)].~~~\end{aligned}$$ We will use the following lemma, involving Bessel function of the first kind $J_k(r)$, which was proved in [@G2]: \[besselrelationlem\] For $\lambda,r,s > 0$ modified Bessel functions $I_k,K_k$ will satisfy: $$I_k(-i\lambda r)K_k(-i\lambda s) - I_k(i\lambda r)K_k(i\lambda s) = \pi i J_k(\lambda r)J_k(\lambda s).$$ Recall the definition of Hankel functions $H_k^{(1)}$, $H_{-k}^{(2)}$ (see [@BE], [@W]): $$\begin{aligned} \label{hankelfunctions} &H_k^{(1)}(\lambda r) = J_k(\lambda r) + iY_k(\lambda r) \nonumber \\ &H_{-k}^{(2)}(\lambda r) = (-1)^k(J_{k}(\lambda r) - iY_{k}(\lambda r)).\end{aligned}$$ Then analytical continuation of $I_k,K_k$ on imaginary line is given as follows(see [@W]): $$\begin{aligned} \label{besselcontnuation} &I_k(-i\lambda r)=(-i)^k J_k(\lambda r),\nonumber \\ &I_k(i\lambda r)=i^k J_k(\lambda r), \nonumber \\ &K_k(-i\lambda r) = \frac\pi 2 i^{k+1}H_k^{(1)}(\lambda r), \nonumber \\ &K_k(i\lambda r) = -\frac\pi 2 i^{k+1}H_{-k}^{(2)}(\lambda r). \end{aligned}$$ We decompose function $\hat \omega_k(\tau,r)$ in (\[wksol\]) for $k\in{\mathbb{N}}\cup\{0\}$ (as usual for negative integer $k$ the indexes in all Bessel functions involving $k$ should be changed to $|k|$): $$\hat \omega_k(\tau,r) = G_{k,1}(\tau,r)+ G_{k,2}(\tau,r),$$ where $G_{k,1}(\tau,r)$, $G_{k,2}(\tau,r)$ are defined as $$\begin{aligned} G_{k,1}(\tau,r) &= \frac{K_k(\sqrt{\tau}r) I_{k-1}(\sqrt \tau r_0)}{K_{k-1}(\sqrt \tau r_0)} \int_{r_0}^\infty w_k^0(s) K_k(\sqrt{\tau}s) s ds, \nonumber \\ G_{k,2}(\tau,r) &= K_k(\sqrt{\tau}r)\int_{r_0}^r w_k^0(s) I_k(\sqrt{\tau}s) s ds \nonumber \\ &+ I_k(\sqrt{\tau}r)\int_r^{\infty} w_k^0(s) K_k(\sqrt{\tau}s) sds. \nonumber\end{aligned}$$ Then with help of (\[hankelfunctions\]), (\[besselcontnuation\]) we will have $$\begin{aligned} &G_{k,1}(-\lambda^2-i0,r) - G_{k,1}(-\lambda^2+i0,r) = \\ &=\int_{r_0}^\infty \Big ( \frac{K_k(-i\lambda r) I_{k-1}(-i\lambda r_0)K_k(-i\lambda s)}{K_{k-1}(-i\lambda r_0)} \\ &- \frac{K_k(i\lambda r) I_{k-1}(i\lambda r_0)K_k(i\lambda s)}{K_{k-1}(i\lambda r_0)} \Big ) w_k^0(s) s ds \nonumber \\ &=\frac{\pi i}2 \int_{r_0}^\infty \Big ( - \frac{H_k^{(1)}(\lambda r) J_{k-1}(\lambda r_0)H_k^{(1)}(\lambda s)}{H_{k-1}^{(1)}(\lambda r_0)} \\ &+ \frac{H_{-k}^{(2)}(\lambda r) J_{k-1}(\lambda r_0)H_{-k}^{(2)}(\lambda s)}{H_{-(k-1)}^{(2)}(\lambda r_0)} \Big ) w_k^0(s) s ds \nonumber \\ &=-\frac{\pi i}2 \int_{r_0}^\infty \Big ( \frac{(J_k(\lambda r)+iY_k(\lambda r)) J_{k-1}(\lambda r_0)(J_k(\lambda s)+iY_k(\lambda s))}{J_{k-1}(\lambda r_0)+iY_{k-1}(\lambda r_0)} \\ ~~~~& + \frac{(J_k(\lambda r)-iY_k(\lambda r)) J_{k-1}(\lambda r_0)(J_k(\lambda s)-iY_k(\lambda s))}{J_{k-1}(\lambda r_0)-iY_{k-1}(\lambda r_0)} \Big ) w_k^0(s) s ds. \nonumber\end{aligned}$$ From lemma \[besselrelationlem\] follows $$\begin{aligned} G_{k,2}(-\lambda^2-i0,r) - G_{k,2}(-\lambda^2+i0,r) = \pi i \int_{r_0}^\infty J_k(\lambda r) J_k(\lambda s) w_k^0(s) s ds,\end{aligned}$$ and $$\begin{aligned} &\hat \omega_k(-\lambda^2-i0,r) - \hat \omega_k(-\lambda^2+i0,r) \nonumber \\ &=G_{k,1}(-\lambda^2-i0,r) - G_{k,1}(-\lambda^2+i0,r) \nonumber \\ &+ G_{k,2}(-\lambda^2-i0,r) - G_{k,2}(-\lambda^2+i0,r) =-\frac{\pi i}2 \int_{r_0}^\infty \Big ( \\ & \frac{J_{k-1}(\lambda r_0)(J_k(\lambda r)+iY_k(\lambda r)) (J_k(\lambda s)+iY_k(\lambda s)) (J_{k-1}(\lambda r_0)-iY_{k-1}(\lambda r_0) )}{J_{k-1}(\lambda r_0)^2+Y_{k-1}(\lambda r_0)^2}\\ &+ \frac{J_{k-1}(\lambda r_0)(J_k(\lambda r)-iY_k(\lambda r)) (J_k(\lambda s)-iY_k(\lambda s))(J_{k-1}(\lambda r_0)+iY_{k-1}(\lambda r_0))}{J_{k-1}(\lambda r_0)^2+Y_{k-1}(\lambda r_0)^2} \\ &-2\frac {J_k(\lambda r) J_k(\lambda s) (J_{k-1}(\lambda r_0)^2+Y_{k-1}(\lambda r_0)^2)} {J_{k-1}(\lambda r_0)^2+Y_{k-1}(\lambda r_0)^2} \Big ) w_k^0(s) s ds.\\\end{aligned}$$ After a series of transformations we will have $$\begin{aligned} &\hat \omega_k(-\lambda^2-i0,r) - \hat \omega_k(-\lambda^2+i0,r)\\ =&\pi i \int_{r_0}^\infty \frac{ \left (J_{k-1}(\lambda r_0) Y_k(\lambda r) - Y_{k-1}(\lambda r_0) J_k(\lambda r) \right ) } {J_{k-1}(\lambda r_0)^2 + Y_{k-1}(\lambda r_0)^2} \\ &\times \left ( J_{k-1}(\lambda r_0) Y_k(\lambda s) - Y_{k-1}(\lambda r_0) J_k(\lambda s) \right ) w_k^0(s) s ds \\ &= \pi i \int_{r_0}^\infty \frac{ R_{k,k-1}(\lambda, r) R_{k,k-1}(\lambda, s) w_k^0(s) s ds} {J_{k-1}(\lambda r_0)^2 + Y_{k-1}(\lambda r_0)^2}\end{aligned}$$ Recall that we assumed $k \geq 0$. Then due to invariance of Bessel equation (\[besselrhs\]) under the change $k$ to $-k$ from (\[wformula\]) we will find the solution $$\begin{aligned} &w(t,r,\varphi) = \nonumber \sum_{k=-\infty}^\infty e^{ik\varphi} \int_0^\infty \frac{ R_{|k|,|k|-1}(\lambda, r)} {J_{|k|-1}(\lambda r_0)^2 + Y_{|k|-1}(\lambda r_0)^2} \\ & \times \left ( \int_{r_0}^\infty R_{|k|,|k|-1}(\lambda, s) w_k^0(s) s ds \right ) e^{-\lambda^2 t} \lambda \mathrm{d}\lambda + \sum_{k=-\infty}^\infty e^{ik\varphi} \operatorname{res}\limits_{\tau=0}[\hat \omega_k(\tau,r)]. \end{aligned}$$ Now we will find the residues of $\hat \omega_k(\tau,r)$ at $\tau=0$. The residues of $\hat \omega_k(\tau,r)$ defined by (\[wksol\]) at $\tau=0$ are given by $$\operatorname{res}\limits_{\tau=0}[\hat \omega_k(\tau,r)] = {\begin{cases}0,~|k|\in \{0,1\},\\ \frac {2(|k|-1) r_0^{2|k|-2}} {r^{|k|}} \int_{r_0}^\infty s^{-|k|+1} w_k(0,s) {\operatorname{ds}},~|k|\in {\mathbb{N}}\setminus \{1\}.\end{cases}}$$ Without loss of generality suppose $k \geq 0$. We will use the asymptotic form for small arguments $0 < |z| \leq \sqrt{k+1}$ [@BE]: $$\begin{aligned} I_{k }(z) & \sim {\frac {1}{\Gamma (k +1)}}\left({\frac {z}{2}}\right)^{k },\\K_{k }(z)&\sim {\begin{cases}-\ln \left({\dfrac {z}{2}}\right)-\gamma & {\text{if }}k =0,\\{\frac {\Gamma (k )}{2}}\left({\dfrac {2}{z}}\right)^{k }&{\text{if }}k >0,\end{cases}}\end{aligned}$$ where $\gamma$ is the Euler-Mascheroni constant. For $k=0$: $$G_{k,1}(\tau,r) \sim \frac {\tau r_0 \left (\ln(\sqrt \tau r /2) + \gamma\right)} {4} \int_{r_0}^\infty \left ( \ln(\sqrt \tau s /2) + \gamma \right ) w_0(0,s) s {\operatorname{ds}}.$$ For $k=1$: $$G_{k,1}(\tau,r) \sim - \frac 1 {\tau r \left (\ln(\sqrt \tau r_0/2) + \gamma \right ) } \int_{r_0}^\infty w_1(0,s) {\operatorname{ds}}.$$ In both cases $k\in \{0,1\}$ the residues are equal to zero. Then only for $k>1$ $G_{k,1}(\tau,r)$ has nonzero residue: $$G_{k,1}(\tau,r) \sim \frac {2(k-1) r_0^{2k-2}} {\tau r^k} \int_{r_0}^\infty s^{-k+1} w_k(0,s) {\operatorname{ds}}$$ and $$\operatorname{res}\limits_{\tau=0}[G_{k,1}(\tau,r)] = \frac {2(k-1) r_0^{2k-2}} {r^k} \int_{r_0}^\infty s^{-k+1} w_k(0,s) {\operatorname{ds}}.$$ Now we calculate the residues of $G_{k,2}(\tau,r)$. From $$\begin{aligned} K_{k }(\sqrt \tau r)I_{k }(\sqrt \tau s)&\sim {\begin{cases} -\ln \left({\dfrac {\sqrt \tau r}{2}}\right)-\gamma & {\text{if }}k =0,\\ \frac s{2kr}&{\text{if }}k >0\end{cases}}\end{aligned}$$ the residues of $G_{k,2}(\tau,r)$ are equal to zero. And $$\operatorname{res}\limits_{\tau=0}[\hat \omega_k(\tau,r)] = \operatorname{res}\limits_{\tau=0}[\hat G_{k,1}(\tau,r)].$$ Since (\[besselrhs\]) is invariant to the change $k$ to $-k$, the solution of the heat equation (\[maineqw\]), (\[initw\]), (\[boundinfw\]) with Robin condition (\[noslip:robin\_bound\]) is represented as $$\begin{aligned} \label{heateqdirectsol} &w(t,r,\varphi) = \nonumber \sum_{k=-\infty}^\infty e^{ik\varphi} \int_0^\infty \frac{ R_{|k|,|k|-1}(\lambda, r)} {J_{|k|-1}(\lambda r_0)^2 + Y_{|k|-1}(\lambda r_0)^2} \\ &\times \left ( \int_{r_0}^\infty R_{|k|,|k|-1}(\lambda, s) w_k(0,s) s \mathrm{d} s \right ) e^{-\lambda^2 t} \lambda \mathrm{d}\lambda \nonumber \\ &+ \sum \limits_{k=-\infty,|k| \geq 2}^\infty e^{ik\varphi} \frac {2(|k|-1) r_0^{2|k|-2}} {r^{|k|}} \int_{r_0}^\infty s^{-|k|+1} w_k(0,s) {\operatorname{ds}},\end{aligned}$$ where $$R_{k,k-1}(\lambda,s) = J_{k}(\lambda s)Y_{k-1}(\lambda r_0) - Y_{k}(\lambda s)J_{k-1}(\lambda r_0).$$ \[thmheateq\] Let $w_k^0(r) \sqrt r \in L_1(r_0,\infty)$ for $k \in {\mathbb{Z}}$. Then the solution $w(t,{\mathbf{x}})$ of (\[maineqw\]), (\[initw\]),(\[boundinfw\]), (\[noslip:robin\_bound\]) is given by (\[heateqdirectsol\]) with $w_k(0,s) = w_k^0(s)$. Using asymptotic form of the Bessel functions for large $|z|$ ([@BE],[@W]): $$\begin{aligned} J_{k }(z)={\sqrt {\frac {2}{\pi z}}}\left(\cos \left(z-{\frac {k \pi }{2}}-{\frac {\pi }{4}}\right)+e^{\left|\operatorname {Im} (z)\right|}\mathrm {O} \left(|z|^{-1}\right)\right),~\left|\arg z\right|<\pi ,\\Y_{k }(z)={\sqrt {\frac {2}{\pi z}}}\left(\sin \left(z-{\frac {k \pi }{2}}-{\frac {\pi }{4}}\right)+e^{\left|\operatorname {Im} (z)\right|}\mathrm {O} \left(|z|^{-1}\right)\right),~\left|\arg z\right|<\pi \end{aligned}$$ follows that integrals in (\[heateqdirectsol\]) are well defined and the theorem is proved. Then instead of (\[int:invidentity\]) we have the following invertibility relation of the associated Weber-Orr transform: Let $f(r) \sqrt r \in L_1(r_0,\infty)$ for $k \in {\mathbb{N}}\cup \{0\}$. Then the associated Weber-Orr transforms (\[int:weberorr\]), (\[int:weberorrinv\]) satisfy almost everywhere $$\begin{aligned} f(r) - W^{-1}_{k,k-1}\left [W_{k,k-1} [f] \right ](r) = {\begin{cases} \frac {2(k-1) r_0^{2k-2}} {r^k} \int_{r_0}^\infty s^{-k+1} f(s) {\operatorname{ds}},~k\in{\mathbb{N}}\setminus\{1\}\\ 0,~k=0,1.\end{cases}}\end{aligned}$$ Let $w(t,{\mathbf{x}})$ be the solution of (\[maineqw\]), (\[boundinfw\]), (\[noslip:robin\_bound\]) with initial datum $f(r)e^{ik\varphi}$. Then from (\[heateqdirectsol\]) $$\begin{aligned} &w(t,r,\varphi) - e^{ik\varphi} \int_0^\infty \frac{ R_{k,k-1}(\lambda, r)} {J_{k-1}(\lambda r_0)^2 + Y_{k-1}(\lambda r_0)^2} \nonumber \\ &\times \left ( \int_{r_0}^\infty R_{k,k-1}(\lambda, s) f(s) s \mathrm{d} s \right ) e^{-\lambda^2 t} \lambda \mathrm{d}\lambda \nonumber \\ &= \overline{I_{0,1}}(k) e^{ik\varphi} \frac {2(k-1) r_0^{2k-2}} {r^{k}} \int_{r_0}^\infty s^{-k+1} f(s) {\operatorname{ds}},\end{aligned}$$ where $\overline{I_{0,1}}(k) = \begin{cases} 1,~k\in{\mathbb{N}}\setminus\{1\} \\ 0,~k =0,1 \end{cases}$. Going to limit $t \to 0$ we will obtain the required relation. Finally, we have Let vector field ${\mathbf{v}}_0({\mathbf{x}})$ satisfies (\[freediv\]), (\[bound\]), (\[boundinf\]), (\[zerocirculation\]), ${\operatorname{curl}}{\mathbf{v}}_0({\mathbf{x}})$ $ \in L_1(B_{r_0})$, and its Fourier series as well as ones for vorticity converges and coefficients $w_k^0(r)$ satisfy $w_k^0(r) \sqrt r \in L_1(r_0,\infty)$, $k \in {\mathbb{Z}}$. Then the solution $w(t,{\mathbf{x}})$ of (\[maineqw\])-(\[boundinfw\]) is expressed as $$\begin{aligned} w(t,r,\varphi) = \nonumber \sum_{k=-\infty}^\infty e^{ik\varphi} \int_0^\infty \frac{ R_{|k|,|k|-1}(\lambda, r)} {J_{|k|-1}(\lambda r_0)^2 + Y_{|k|-1}(\lambda r_0)^2} \\ \times \left ( \int_{r_0}^\infty R_{|k|,|k|-1}(\lambda, s) w_k^0(s) s \mathrm{d} s \right ) e^{-\lambda^2 t} \lambda \mathrm{d}\lambda.\end{aligned}$$ From ${\operatorname{curl}}{\mathbf{v}}_0({\mathbf{x}})$ $ \in L_1(B_{r_0})$ follows $w_0^0(r) r \in L_1(r_0,\infty)$ and (\[BS:cond\]) is satisfied for $k = 0$. From $w_k^0(r) \sqrt r \in L_1(r_0,\infty)$ follows (\[BS:cond\]) for $k \neq 0$. Then from Theorems \[thmrobincond\], \[thmheateq\] we have the formula (\[heateqdirectsol\]) for $w(t,{\mathbf{x}})$. And in virtue of (\[noslipcondintegral\]) the last term in (\[heateqdirectsol\]) vanishes.  \   A.V. Gorshkov\ Lomonosov Moscow State University,\ Leninskie Gory, Moscow, 119991,\ Russian Federation\ alexey.gorshkov.msu@gmail.com\ [99]{} Bateman, H., Erdelyi, A. Higher Transcendental Functions, Vol. II. McGraw-Hill, New York (1953). Watson, G.N.: A Treatise on the Theory of Bessel Functions, Second Edition. Cambridge University Press (1995). Titchmarsh, E.C.: Weber’s integral theorem. Proc. Lond. Math. Soc. 22:2, 15-28 (1924). Titchmarsh, E.C.: Introduction to the theory of Fourier integrals. Oxford Univ. Press (1948). Sirivastav, R.P.: A pair of dual integral equations involving Bessel functions of the first and the second kind, Proc. Edin. Math. Soc. 14:2, 149-158 (1964). Krajewski, J., Olesiak, Z.: Associated Weber integral transforms of $W_{\nu-l,j}[;]$ and $W_{\nu-2,j}[;]$ types. Bull. de L’Academe Polonaise des Scs. Vol. XXX, No. 7-8 (1982). Nasim, C: Associated Weber integral transforms of arbitrary order. Ind. J. Pure & Appl. Math. 20:11, 126-1138 (1989). Gorshkov, A.V.: Boundary Stabilization of Stokes System in Exterior Domains. J. of Math. Fluid Mech. 18:4, 679-697 (2016). Agranovich, M.S., Vishik, M.I.: Elliptic problems with a parameter and parabolic problems of general type. Russian Mathematical Surveys 19:3, 53-157 (1964).

--- abstract: 'We investigate classes of quantum Heisenberg spin systems which have different coupling constants but the same energy spectrum and hence the same thermodynamical properties. To this end we define various types of isospectrality and establish conditions for their occurence. The triangle and the tetrahedron whose vertices are occupied by spins $\frac{1}{2}$ are investigated in some detail. The problem is also of practical interest since isospectrality presents an obstacle to the experimental determination of the coupling constants of small interacting spin systems such as magnetic molecules.' address: - ' Universität Osnabrück, Fachbereich Physik, Barbarastr. 7, 49069 Osnabrück, Germany' - |  Ames Laboratory and Department of Physics and Astronomy, Iowa State University\ Ames, Iowa 50011, USA author: - 'Heinz-Jürgen Schmidt and Marshall Luban' title: Continuous families of isospectral Heisenberg spin systems and the limits of inference from measurements --- Introduction ============ The measurement of the temperature-dependent magnetic susceptibility, $\chi(T)$, provides a standard essential diagnostic method for establishing the magnetic properties of a system. A careful comparison between measured data and the predictions for $\chi(T)$ as derived from a model Hamiltonian is routinely performed with the goal of establishing numerical values of model parameters, for example, the exchange constant(s) of the Heisenberg model of interacting isolated spins. The success of this technique is so firmly established that it is taken for granted that there is a one-to-one correspondence between a given form of $\chi(T)$ and the numerical values of the model parameters. Most certainly it is unnatural to contemplate that one might be able to continuously vary the parameters of a model Hamiltonian and yet generate a single, invariant form for $\chi(T)$ and similarly for other thermodynamic quantities. Yet surprisingly, there are a number of exceptional systems, where there is a continuous infinity-to-one correspondence between model Hamiltonians and measurable thermodynamic quantities. One of these exceptional cases was recently encountered [@Jun],[@Lub] in the course of attempting to determine the exchange constants of a simulating Heisenberg model from experimental susceptibility data for a specific synthetic magnetic compound [@MMBSB].\ We shall refer to systems of a continuous family having the same eigenvalue spectrum as being “isospectral". This notion is chosen in analogy to the use of “isospectrality" in other areas of physics, e. g. the occurrence of supersymmetric pairs of Hamiltonians (see, e. g. [@CKS], 7.1) or the problem of bounded domains with isospectral Laplacians (“Can one hear the shape of a drum ?" [@Kac]). The subject of isospectral spin systems is not completely novel but has been discussed in the literature only on a few occasions, e. g. [@G1], [@G2]. However, there is, to our best knowledge, no systematic account of this phenomenon, the first steps of which will be presented in this paper.\ In addition to providing a general approach to isospectrality we analyze in depth two cases of distinct Heisenberg systems where continuous variation of the exchange constants gives rise to one and the same set of temperature-dependent thermodynamic quantities. The operational conclusion for an experimentalist is quite sobering, in that for these specific systems measured data alone cannot fix the exchange constants. Comparison between theory and experiment can only place a weak constraint on a continuous family of equally acceptable choices of parameters. Although we provide some helpful insights, it is very difficult to formulate the general set of conditions to be met so as to achieve such exceptional model systems. It is reasonable to expect, that if one and the same temperature dependent thermodynamic quantity is generated by a continuous family of Hamiltonians then necessarily all members of that family share the very same eigenvalue spectrum. This is indeed the case as it is proven in section 3 for two particular thermodynamic functions.\ In short, our goal in the present work is to provide a first systematic study of Heisenberg isospectral spin systems. The remarkable advances [@Ga],[@M] in synthesis magnetochemistry, of incorporating significant numbers of interacting paramagnetic centers within individual molecules, may provide the impetus for wider studies that will yield a more comprehensive set of conditions for the occurence of isospectral spin systems.\ The layout of this paper is as follows:\ In section 2 we introduce our notation and the basic concepts of “isospectrality", “complete isospectrality", and “covariant isospectrality" for spin systems with Heisenberg Hamiltonians. Families of isospectral systems are algebraic varieties in the space $\cal J$ of coupling constants. Covariant isospectrality is implemented by a unitary representation of some Lie group, which simplifies the calculations considerably. Unfortunately this is a rare case, as we will see. Complete isospectrality means that all eigenvalues of two systems with the same magnetic quantum number are in 1:1 correspondence and equal. We do not know whether this is a strictly stronger property than plain isospectrality, except for the case of a trivial counter-example. However, we need this apparently stronger concept to derive the conclusion that completely isospectral systems share the same magnetic susceptibility function. This is done in section 3 where also the inverse problem is settled as well as the analogous question for the specific heat function. The result in short is the following: Plain isospectrality is equivalent to possessing the same specific heat function and necessary for possessing the same magnetic susceptibility function. Complete isospectrality is sufficient for possessing the same magnetic susceptibility function. In section 4 we identify the isospectral invariants which are linear or quadratic in the coupling constants. This is crucial for section 5 where we show that the triangle with spin $s=\frac{1}{2}$ is both of completely and covariantly isospectral type but that for $s>\frac{1}{2}$ isospectrality breaks down. On the other side, if the number $N$ of spin sites exceeds $3$, covariant isospectrality is no longer possible. This is proved in section 6 with the aid of MATHEMATICA$^{\circledR}$ 4.0 and some trace formulae which are explained in Appendix A. The tetrahedron ($N=4$) with $s=\frac{1}{2}$ nevertheless possesses completely isospectral families of dimension one and two, as is shown in section 7 and Appendix B. In section 8 we provide some heuristic arguments in order to explain the findings of the previous sections. We expect that isospectrality only occurs if the number of all possible bonds exceeds the number of independent eigenvalues and show that this never happens except for $s=\frac{1}{2}$ and $N=3,4,5$. A table summarizing our results and conjectures on the occurence of isospectrality for different $N$ and $s$ and concluding remarks are provided in section 9. Notations and definitions ========================= We consider spin systems with $N$ spin sites, spin quantum number $s$ and isotropic Heisenberg coupling between all sites $x$ and $y$ with coupling constants $J_{xy}$. For sake of compact notation we will write the ${N\choose 2}$ coupling constants $J_{xy}$ as the components of a vector $\vec{J}\in{\cal J}={\Bbb R}^{N\choose 2}$. Thus a specific point of ${\cal J}$ uniquely specifies the strength of the interactions between all pairs of spins and will be sometimes called a “system".\ Let $S_{x}^{(i)}, (i=1,2,3),$ denote the three components of the spin observable $\bi{S}_x$ at site $x$ and, as usual, $$\label{1} \bi{S}=\sum_x \bi{S}_x,\quad S^{(i)}=\sum_x S^{(i)}_x,\quad S^\pm=S^{(1)}\pm i S^{(2)},$$ denote the total spin vector and its various components. All linear operators occuring in this context will be identified with the corresponding $dim\times dim$-matrices, $dim=(2s+1)^N$ being the dimension of the total Hilbert space of the spin system, w. r. t. the fixed basis consisting of tensor products of eigenvectors of $S_x^{(3)}$. The Hamilton operator can then be written as $$\label{2} H_0 = \bi{J}\cdot\bi{H}=\sum_{x<y}J_{xy} H_{xy},$$ where $$\label{3} H_{xy}=\bi{S}_x\cdot\bi{S}_y = \sum_{i=1}^3 S_x^{(i)}S_y^{(i)}.$$ Here $\bi{H}$ is an ${N\choose 2}$-dimensional vector the components of which are $dim\times dim$-dimensional matrices $H_{xy}$.\ If the spin system is coupled to a constant external magnetic field $\cal H$, the total Hamilton operator will be $$\label{4} H(h)=H_0 - h S^{(3)},$$ where $h\equiv g\mu_B {\cal H}$ contains the common combination of the gyromagnetic ratio $g$ and the Bohr magneton $\mu_B$. As usual, the partition function, which yields all the standard thermodynamic functions, is defined by $$\label{5} {\cal Z}(\beta,h)\equiv \Tr (\exp({-\beta H(h))}).$$ In particular, one obtains from ${\cal Z}$ the specific heat function $$\label{5a} c(\beta)=\beta^2\frac{\partial^2}{\partial\beta^2}\ln {\cal Z}(\beta,0),$$ and the magnetic zero field susceptibility function $$\label{5b} \chi(\beta)=\frac{1}{\beta}\frac{\partial^2}{\partial h^2}\ln {\cal Z}(\beta,h)|_{h=0}.$$ Further we will need the traces of powers of $H_0$, $ t_n\equiv \Tr (H_0^n), n=0,1,2,\ldots, dim,$ and the set of traces $$\label{6} {\cal T}(H_0)=\left\{ t_n|n=0,1,2,\ldots,dim\right\}.$$ Two Hamilton operators with the same $N$ and $s$, $H_0^{(1)}=\bi{J}^{(1)}\cdot \bi{H}$ and $H_0^{(2)}=\bi{J}^{(2)}\cdot \bi{H}$ are called if they have the same eigenvalues, counted with multiplicity, or, equivalently, if they generate the same characteristic polynomial: $$\label{7} \det(H_0^{(1)}-\lambda)=\det(H_0^{(2)}-\lambda)\quad \forall \lambda\in{\Bbb R}.$$ According to the above remarks, we will also speak of “isospectral systems". Sometimes we will apply the term “isospectral" to more general pairs of operators derived from $H_0^{(i)}, i=1,2$ if there is no danger of confusion. Clearly, (\[7\]) defines an equivalence relation $\sim$ on ${\cal J}$. The coefficients of a characteristic polynomial $P(\lambda)=\det(H_0-\lambda)$ can be viewed as polynomials of the coupling constants $J_{xy}$. These polynomials assume constant values exactly on the $\sim$-equivalence classes $[\bi{J}]_\sim$, which we will call . Consequently, the isospectral classes are in ${\cal J}$, since they are defined by a finite number $k$ of polynomial equations $p_\nu(\bi{J})=0, \nu=1,\ldots,k$.[^1]\ In algebraic geometry there are various equivalent definitions of the of an algebraic variety, which are, however, too technical to be reproduced here (see, for example, [@CLS], chapter 9). For our purposes it will suffice to note that for the special case where the Jacobian matrix $\left( \frac{\partial p_\nu}{\partial J_j}\right)_{\nu=1\ldots k,j=1\ldots {N \choose 2}}$ has locally a constant rank $r$, the corresponding isospectral classes are locally differentiable manifolds of dimension $\ell={N \choose 2}-r$. This is an immediate consequence of the fibration theorem (see, for example, [@AMR], theorem 3.5.18). Especially, if $k\ge {N \choose 2}$ and the rank of the Jacobian is maximal, $r={N \choose 2}$, the isospectrality classes will only consist of a discrete point set. Note, however, that the equivalence classes will never be trivial since they at least consist of the orbits of the group of discrete symmetries of ${\cal J}$ generated by permutations of spin sites.\ We will say that the pair $(N,s)$ is of if the corresponding space of coupling constants ${\cal J}$ contains at least one isospectral equivalence class of dimension $\ell \ge 1$. The largest dimension $\ell$ of isospectral equivalence classes will be called the of the isospectral type.\ Functions $f:{\cal J} \longrightarrow {\Bbb R}$ which are constant on isospectral classes will be called .\ Moreover, we will consider a special case of isospectrality in which the equivalence classes are easily calculated. Obviously, two isospectral Hamiltonians $H_0^{(1)}$ and $H_0^{(2)}$ can be related by a unitary transformation $U$: $$\label{8} H_0^{(2)}=U^\ast H_0^{(1)} U.$$ $U$ maps the eigenvectors of $H_0^{(2)}$ onto the corresponding eigenvectors of $H_0^{(1)}$. In the case of a one–dimensional isospectral equivalence class parametrized by a coordinate $t$, $U$ can be chosen to depend smoothly on $t$. Following a closed loop, the corresponding unitary transformation need not reduce to the identity transformation, but may include some phases. We will come back to this phenomenon later. If for sufficiently many curves of isospectrally equivalent points the corresponding unitary transformations $U(t)$ can, moreover, be chosen to be a one-parameter group, the system $(N,s)$ will be called of “covariant isospectral" type. More precisely, we define: $(N,s)$ is of type iff it is of isospectral type and for any isospectrally equivalent $\vec J^{(1)},\vec J^{(2)}$ which are sufficiently close, there exists an anti-Hermitean $dim\times$dim-matrix $\Omega$, and a real ${N\choose 2}\times{N\choose 2}$-matrix $M$ and some $t_0\in{\Bbb R}$, such that for all $t\in{\Bbb R}$ $$\label{9} \exp({\Omega^\ast t})\bi{J}^{(1)}\cdot\bi{H}\exp({\Omega t}) = (\exp({-M t})\bi{J}^{(1)})\cdot\bi{H}$$ and $$\label{10} \exp({-M t_0})\bi{J}^{(1)}=\bi{J}^{(2)}.$$ It follows immediately, that the orbit $\exp({-M t})\bi{J}^{(1)}$ in ${\cal J}$–space corresponds to a family of isospectral Hamilton matrices. Since in our case all Hamilton matrices are real and symmetric, $\Omega$ can also be chosen as real and hence anti–symmetric, and thus $\exp({\Omega t})$ represents a rotation in the real Hilbert space ${\Bbb R}^{dim}$.\ The condition (\[9\]) may be replaced by its equivalent infinitesimal version: $$\label{11} \left[ \bi{H},\Omega \right] = -M^\ast \bi{H}.$$ This equation could be used to show that $M$, considered as a linear transformation in the space of ${N\choose 2}$–dimensional vectors with matrix entries endowed with the scalar product $\Tr (\bi{K}\cdot\bi{H})$, will be an antisymmetric matrix. We will give an independent proof of this fact later.\ Further we note that the set of solutions $\langle \Omega,M \rangle$ of (\[11\]) will be a Lie algebra with respect to the obvious vector and commutator operations. Hence covariant isospectral equivalence classes will be orbits of the corresponding matrix Lie groups.\ Thus far we have only discussed isospectral systems in the absence of an external magnetic field. If we include the field ${\cal H}$ and allow for a corresponding $h$-dependence of the Hamiltonian, we have to consider a slightly stronger concept of “complete isospectrality". Two Hamiltonians $H_0^{(1)}$ and $H_0^{(2)}$ (or, equivalently, two vectors $\bi{J}^{(1)}$ and $\bi{J}^{(2)}\in{\cal J}$) are called ($\bi{J}^{(1)}\approx\bi{J}^{(2)}$) iff for all $h\in{\Bbb R}\quad H^{(1)}(h)=H_0^{(1)}-h S^{(3)}$ and $H^{(2)}(h)=H_0^{(2)}-h S^{(3)}$ are isospectral. \[P0\] The following conditions are equivalent:\ ------- ---------------------------------------------------------------------- (i) $ \bi{J}^{(1)}\approx\bi{J}^{(2)}$, (ii) ${\bf P}_M H_0^{(1)}{\bf P}_M \sim {\bf P}_M H_0^{(2)}{\bf P}_M$ for all projectors ${\bf P}_M$ onto the eigenspaces of $S^{(3)}$ corresponding to the eigenvalue $M$, (iii) ${\Bbb P}_S H_0^{(1)}{\Bbb P}_S \sim {\Bbb P}_S H_0^{(2)}{\Bbb P}_S$ for all projectors ${\Bbb P}_S$ onto the eigenspaces of $\bi{S}^2{}$ corresponding to the eigenvalue $S(S+1)$. ------- ---------------------------------------------------------------------- [**Proof**]{}: (i)$\Rightarrow$ (ii). Since all $H(h), h\in{\Bbb R},$ commute, there exists a system of joint eigenprojectors ${\cal P}_{\nu,M_\nu}$ such that $$\label{11a} H(h)=\sum_{\nu,M_\nu} (\epsilon_\nu - h M_\nu){\cal P}_{\nu,M_\nu}.$$ Hence $$\label{11b} {\bf P}_M H(h){\bf P}_M = \sum_{\nu,M_\nu,M_\nu=M} (\epsilon_\nu - h M_\nu){\cal P}_{\nu,M_\nu}.$$ If $H^{(1)}(h)\sim H^{(2)}(h)$ for all $h\in{\Bbb R}$, then both systems have the same set of eigenvalues $\epsilon_\nu - h M_\nu$ with the same multiplicities $\Tr{\cal P}_{\nu,M_\nu}$. Hence, by (\[11b\]), also ${\bf P}_M H^{(1)}(h){\bf P}_M \sim {\bf P}_M H^{(2)}(h){\bf P}_M$, especially ${\bf P}_M H_0^{(1)}{\bf P}_M \sim {\bf P}_M H_0^{(2)}{\bf P}_M$ for all $M$.\ (ii)$\Rightarrow$(iii). The eigenspaces corresponding to ${\cal P}_{\nu,M_\nu}$ can be further split into eigenspaces of $\bi{S}^2$. This can be written as $$\label{11c} {\cal P}_{\nu,M_\nu} =\sum_{S=|M_\nu|}^{S_{max}}{\cal P}_{\nu,M_\nu,S}.$$ Applying the ladder operators $ S^\pm$ gives $$\label{11d} \Tr{\cal P}_{\nu,M_\nu,S} =\Tr{\cal P}_{\nu,S,S}=\Tr{\cal P}_{\nu,S} \mbox{ for all }M_\nu=-S,\ldots,S.$$ These numbers are the multiplicities of the eigenvalues of the operators ${\Bbb P}_S{\bf P}_M H_0^{(i)}{\Bbb P}_S {\bf P}_M$, which are hence isospectral for $i=1,2$. By summation over $M$ we conclude that also ${\Bbb P}_S H_0^{(i)}{\Bbb P}_S $, are isospectral for $i=1,2$.\ (iii)$\Rightarrow$(ii). This can be shown analogously by considering $$\label{} {\Bbb P}_S H_0{\Bbb P}_S =\sum_{\nu,S_\nu,S_\nu=S} \epsilon_\nu {\cal P}_{\nu,S_\nu}',$$ $$\label{} {\cal P}_{\nu,S_\nu}'=\sum_{M=-S_\nu}^{S_\nu} {\cal P}_{\nu,S_\nu,M}'$$ and $$\label{} \Tr{\cal P}_{\nu,S_\nu}'=(2 S_\nu +1) \Tr {\cal P}_{\nu,S_\nu,M}' .$$ (ii)$\Rightarrow$(i). This follows from $$\label{} H(h) = \sum_M ({\bf P}_M H_0 {\bf P}_M -h M).$$ ------------------------------------------------------------------------ Obviously, $\bi{J}^{(1)}\approx\bi{J}^{(2)}$ implies $\bi{J}^{(1)}\sim\bi{J}^{(2)}$, but not conversely:\ Take $N=3, s=\frac{1}{2}$, then $\bi{J}^{(1)}=\left(\begin{array}{c} 1\\1\\1\end{array}\right)\sim \bi{J}^{(2)}=\left(\begin{array}{c} -1\\-1\\-1\end{array}\right)$, but $\bi{J}^{(1)}\not\approx\bi{J}^{(2)}$.\ Unfortunately we do not know of less trivial counter-examples. The problem is the following:\ Being rotationally symmetric, $H_0$ commutes with $\bi{S}^2$ and $S^{(3)}$, hence each eigenspace of $H_0$ with eigenvalue $\epsilon_\nu$ is spanned by simultaneous eigenvectors of $\bi{S}^2$ and $S^{(3)}$, say, $|\nu,\lambda,\mu\rangle, \nu=0,\ldots,D, \lambda=1,\ldots,d_\nu, \mu=-S_{\nu,\lambda},\ldots,S_{\nu,\lambda}$, such that $$\label{13} \bi{S}^2 |\nu,\lambda,\mu\rangle = S_{\nu,\lambda}(S_{\nu,\lambda}+1)|\nu,\lambda,\mu\rangle, \quad S^{(3)}|\nu,\lambda,\mu\rangle = \mu|\nu,\lambda,\mu\rangle.$$ The degeneracy of the eigenvalues $\epsilon_\nu$ will be $$\label{13a} n_\nu =\sum_{\lambda=1}^{d_\nu}(2 S_{\nu,\lambda}+1).$$ We do not assume that the $S_{\nu,\lambda}$ have different values. The corresponding eigenvalues of $H(h)=H_0-hS^{(3)}$ are $$\label{14} E_{\nu,\lambda,\mu}= \epsilon_\nu - h \mu.$$ For “generic" $\bi{J}\in{\cal J}$ we expect that $H_0$ will have no further degeneracy besides that dictated by rotational symmetry (“minimal" degeneracy), i. e. we expect that $d_\nu=1$. So $\lambda$ can be skipped and $\mu=-S_\nu,\ldots,S_\nu$. In this case, “isospectrality" and “complete isospectrality" would be equivalent. So, heuristically, we may consider these two notions as having equal meaning, although we have to distinguish between them for the sake of mathematical rigour.\ For the most important case of isospectral systems which are obtained by continuously varying the coupling constants we can show the following: \[P01\] Any two systems joined by a continuous curve of isospectral systems are completely isospectral. [**Proof**]{}: Consider a curve $t\mapsto\bi{J}(t)$ and consider $$\label{11e} H_0(t)\equiv \bi{H}\cdot \bi{J}(t) = \sum_\nu \epsilon_\nu {\cal P}_\nu(t)$$ and $$\label{11f} {\Bbb P}_S H_0(t){\Bbb P}_S = \sum_\nu \epsilon_\nu {\cal P}_\nu(t) {\Bbb P}_S.$$ Since $t\mapsto{\cal P}_\nu(t)$ is continuous and $\Tr ({\cal P}_\nu(t) {\Bbb P}_S)$ assumes only non-negative integer values, the latter must be constant w. r. t. the parameter $t$. However, $\Tr ({\cal P}_\nu(t) {\Bbb P}_S)$ equals the multiplicity of the eigenvalue $\epsilon_\nu$ in (\[11f\]). Thus all ${\Bbb P}_S H_0(t){\Bbb P}_S, t\in{\Bbb R},$ are isospectral, and, by proposition \[P0\], all $H_0(t)$ are completely isospectral. ------------------------------------------------------------------------ Note that, according to this proposition, in the above counter-example the two systems cannot lie in the same connected component of an isospectral class.\ We add some definitions concerning symmetrical polynomials which will be of later use:\ The equation $$\label{15} \prod_{n=1}^{d}{(x+x_n)}=\sum_{\nu =0}^{d}{s_\nu x^{d-\nu }}$$ defines the elementary symmetrical polynomials $$\label{16} s_1=\sum_{n=1}^{d}{x_n},\quad s_2=\sum_{n<m}^{}{x_n x_m}, \quad\ldots\quad s_d=\prod_{n=1}^{d}{x_n}.$$ These also appear, up to a sign, as the coefficients of the characteristic polynomial of $H_0$ $$\label{17} p(\lambda) = \prod_{i=1}^{d}{(\lambda-E_i)},$$ written as polynomials of the $E_i$, where $d=dim$. Every other symmetric polynomial can uniquely be written as a polynomial of the $s_\nu $ (see, for example [@CLS], 7.1, Theorem 3). This holds especially for $$\label{18} t_n=\Tr (H_0^n)=\sum_{i=1}^{d}{E_i^n}.$$ Conversely, each $s_\nu$ can uniquely be written as a polynomial of the $t_n, n=1\ldots d$, (see, for example [@CLS], 7.1, Theorem 8) e. g.  $$\label{19} s_5=\frac{1}{5!}\left(t_1^5- 10 t_1^3 t_2 + 15 t_1 t_2^2 + 20 t_1^2 t_3 - 20 t_2 t_3 - 30 t_1 t_4 + 24 t_5 \right).$$ This representation is independent of the dimension $d$. Specific heat and magnetic susceptibility ========================================= As mentioned in the introduction, (completely) isospectral spin systems will give rise to the same thermodynamic functions like specific heat and magnetic susceptibility. In this section we will state this more precisely and also prove the converse, up to the subtle distinction between complete and plain isospectrality. \[L1\] Two spin systems are isospectral iff ${\cal T}(H_0^{(1)})={\cal T}(H_0^{(2)})$. [**Proof**]{}: Recall that ${\cal T}(H_0)$ was defined as the set of traces $t_n=\Tr (H_0^n), n=0,\ldots,dim$. Hence the “only if" part is obvious. From the remarks at the end of section 2 it follows that if two systems possess the same $t_n, n\in{\Bbb N}$, they share also the same values of the standard symmetric polynomials $s_\nu(\bi{E})$ and hence have the same characteristic polynomial $p(\lambda)$. ------------------------------------------------------------------------ \[P1\] Two spin systems are isospectral iff they possess the same specific heat function. [**Proof**]{}: $c(\beta )$ can be expanded into a Taylor series at $\beta =0$: $$\begin{aligned} \label{20} c(\beta ) & = & \left(-\frac{t_1^2}{t_0^2}+\frac{t_2}{t_0} \right)\beta^2\\ \nonumber & & +\left(\frac{t_1 t_2}{t_0^2}-\frac{t_1}{t_0}\left(\frac{t_1^2}{t_0^2}- \frac{t_2}{2 t_0}\right)-\frac{t_3}{2 t_0} \right)\beta^3\\ \nonumber & & +\ldots .\end{aligned}$$ This is the starting point of the so-called “moment expansion method", see e. g. [@Yos] 7.3. Obviously, each coefficient of $\beta^n$ uniquely determines $t_n$, if the other traces $t_m, m<n,$ are already known. Note that $t_0=dim, t_1=0$. Together with lemma \[L1\] this completes the proof. ------------------------------------------------------------------------ Now we consider again $H(h)=H_0-h S^{(3)}$ with eigenvectors $|\nu,\lambda,\mu\rangle$ according to the previous section. By its very definition, two completely isospectral systems share the same partition function ${\cal Z}(\beta,h)$ and any other thermodynamic function which can be derived from it. Especially, the following holds: \[P11\] Two completely isospectral systems possess the same magnetic susceptibility function. To tackle the converse problem, we consider ${\cal Z}(\beta,0)=\sum_\nu n_\nu \exp({-\beta\epsilon_\nu})$ and define the coefficients $\sigma_\nu$ implicitely by $$\label{21} \Tr \left( S^{(3)2} \exp({-\beta H_0}) \right) = \sum_\nu \sigma_\nu n_\nu \exp({-\beta\epsilon_\nu})$$ In the case of minimal degeneracy, i. e. $d_\nu=1$, we have $n_\nu= 2 S_\nu +1$ and $$\label{22} \sigma_\nu=\frac{1}{2 S_\nu +1}\sum_{\mu=-S_\nu}^{S_\nu}\mu^2 =\frac{1}{3}S_\nu (S_\nu +1).$$ In the general case, $$\begin{aligned} \label{23} \sigma_\nu & = & \frac{1}{n_\nu }\sum_{\lambda=1}^{d_\nu} \sum_{\mu=-S_{\nu,\lambda}}^{S_{\nu,\lambda}} \mu^2 \\ \nonumber &=&\frac{1}{n_\nu }\sum_{\lambda=1}^{d_\nu}\frac{1}{3}S_{\nu,\lambda} (S_{\nu,\lambda} +1)(2 S_{\nu,\lambda}+1).\end{aligned}$$ \[P2\] Two spin systems with the same susceptibility function are isospectral. [**Proof**]{}: Since $$\label{24} \chi(\beta)=\frac{\beta}{{\cal Z}(\beta,0)}\Tr \left( S^{(3)2}\exp({-\beta H_0}) \right),$$ the two systems will have the same function $$\begin{aligned} \label{25} f(\beta) & \equiv & \frac{\Tr \left( \exp({-\beta H_0}) S^{(3)2} \right) } {\Tr \left(\exp({-\beta H_0})\right)} \\ \nonumber & = & \frac{\sum_{\nu=0}^{D}{\sigma_\nu n_\nu \exp({-\beta \varepsilon_\nu})}} {\sum_{\nu=0}^{D}{n_\nu \exp({-\beta \varepsilon_\nu})}}.\end{aligned}$$ Since $\varepsilon_0 < \varepsilon _1 < \varepsilon _2 < \ldots $, the terms $\exp({-\beta \varepsilon_\nu})$ are of different orders of magnitude for $\beta \rightarrow \infty $. The first term increasingly dominates, hence $$\label{26} \lim_{\beta \to \infty }f(\beta) = \frac{\sigma _0 n_0 \exp({-\beta\varepsilon_0 })} {n_0 \exp({-\beta\varepsilon_0})} = \sigma _0.$$ If we subtract this limit from $f(\beta)$, the dominant term asymptotically becomes $$\begin{aligned} \label{27} f(\beta)-\sigma _0 & \simeq _{\beta\to\infty} & \frac{\sigma_1 n_1 \exp({-\beta\varepsilon_1 })} { n_0 \exp({-\beta\varepsilon_0 })}\\ \nonumber & = & \frac{\sigma_1 n_1}{n_0}\exp({-\beta(\varepsilon_1 -\varepsilon_0 ))}.\end{aligned}$$ In the next step we have $$\label{28} f(\beta)-\sigma_0 -\frac{\sigma_1 n_1}{n_0}\exp({-\beta(\varepsilon_1 -\varepsilon_0 )}) \simeq _{\beta\to\infty} \frac{\sigma _2 n_2}{n_0} 2\exp({-\beta(\varepsilon_2 -\varepsilon_0 )}),$$ and so on. In this way, from the behaviour of $f(\beta)$ for $\beta\to\infty$, we may extract the values\ $\sigma _0,\frac{\sigma_1 n_1}{n_0},\ldots \frac{\sigma_\nu n_\nu}{n_0},\ldots \frac{\sigma_D n_D}{n_0}$ and $\varepsilon_1 - \varepsilon_0, \varepsilon_2 -\varepsilon_0, \ldots, \varepsilon_\nu - \varepsilon_0, \ldots, \varepsilon_D - \varepsilon_0$.\ Let $t_n = \Tr (H_0^n)$ as above and $\mu_n\equiv \Tr (H_0^n S^{(3)2}), n\in{\Bbb N} $. $\mu_0 = \Tr (S^{(3)2})$ can be calculated independently of $H_0$. Since $$\label{29} \mu_0=\sum_{\nu=0}^{D}{\sigma_\nu n_\nu} = n_0 \sum_{\nu=0}^{D}{\frac{\sigma_\nu n_\nu}{n_0}} ,$$ $n_0$ and hence $\sigma _\nu n_\nu, \nu=1\ldots D,$ are also uniquely determined.\ Next we consider the Taylor expansion of $f(\beta)$ at $\beta=0$: $$\begin{aligned} \label{30} f(\beta) & = & \frac{\mu_0}{t_0}-\frac{\mu_1}{t_0}\beta + \left(-\frac{t_2 \mu_0}{2 t_0^2}+\frac{\mu_2}{2 t_0}\right)\beta^2\\ \nonumber & & +\left(\frac{t_3 \mu_0}{6 t_0^2}+\frac{t_2 \mu_1}{2 t_0^2}-\frac{\mu_3}{6 t_1}\right)\beta^3\\ \nonumber & & +\left(\mu_0\left(\frac{t_2^2}{4 t_0^3}-\frac{t_4}{24 t_0^2}\right) -\frac{t_3 \mu_1}{6 t_0^2}-\frac{t_2 \mu_2}{4 t_0^2}+\frac{\mu_4}{24 t_0}\right)\beta^4\\ \nonumber & & +\ldots\end{aligned}$$ Recall that $t_0, \mu_0$ are known. The linear term then gives $$\label{31} \mu_1=\Tr (H_0 S^{(3)2})=\sum_{\nu_0=0}^{D}{\sigma_\nu n_\nu \varepsilon_\nu}.$$ On the other side we know the l. h. s. of $$\label{32} \sum_{\nu=0}^{D}{\sigma_\nu n_\nu (\varepsilon_\nu-\varepsilon_0 )} = \mu_1 - \varepsilon_0 \mu_0,$$ hence $\varepsilon_0$ and all $\varepsilon_\nu, \nu=1\ldots D$ are also known. Similarly, $$\label{33} \sum_{\nu=0}^{D}{\sigma_\nu n_\nu (\varepsilon_\nu-\varepsilon_0 )^2} = \mu_2 - 2\varepsilon_0 \mu_1 + \varepsilon_0^2\mu_0 ,$$ hence $\mu_2$ is known and from the $\beta^2$–term in (\[30\]) also $t_2$, and so on.\ Eventually, we obtain all $t_n, \mu_n, n=2\ldots dim$ solely from $\chi(\beta)$. According to lemma \[L1\] this gives us all eigenvalues of $H_0$ with multiplicity, i. e. $n_\nu, \nu=0,\ldots D$ and the two spin systems are isospectral. ------------------------------------------------------------------------ The proof does not give complete isospectrality: If some eigenvalue $\varepsilon_\nu$ belongs to different $S_{\nu\lambda}, \lambda > 2$ then from $n_\nu=\sum_{\lambda}^{}{(2S_{\nu\lambda}+1)}$ and $\sigma_\nu = \frac{\sum_{\lambda}^{}{\frac{1}{3} S_{\nu\lambda}(S_{\nu\lambda}+1)(2S_{\nu\lambda}+1)}} {\sum_{\lambda}^{}{(2 S_{\nu\lambda}+1)}}$ the $S_{\nu,\lambda}$ cannot be uniquely determined. Some isospectral invariants =========================== Criteria for non–isospectrality could, in principle, be checked by brute force methods: Calculate the characteristic polynomial of the matrix $H_0=\bi{J}\cdot\bi{H}$, say $p(\lambda)=\sum_{\nu=0}^{dim} c_\nu \lambda^\nu$. Select ${N\choose 2}$ different coefficients $c_i, c_j,\dots$ ($\nu=dim$ being excluded since $c_{dim}=1$) and calculate the Jacobian determinant $$\label{34} Jac(\bi{J})\equiv \frac{\partial(c_i,c_j,\ldots)}{\partial(J_1,\ldots,J_{N\choose 2})}$$ preferably by using a computer algebra software. If the Jacobian nowhere vanishes, according to remarks in section 2, $(N,s)$ cannot be of isospectral type.\ In practice this method will, even for small $N$ and $s$, rapidly become extremely memory- and time-consuming. To simplify the problem one could – in the case of complete isospectrality – restrict oneself to subspaces invariant under $H_0$, for example subspaces ${\cal H}(M)$ of constant magnetic quantum number $M$.\ The space with maximal $M$, ${\cal H}(M = N s)$ is one-dimensional and is spanned by the product state $$\label{35} \varphi_0=|s,s,\ldots s\rangle$$ which is an eigenstate of $\bi{J}\cdot\bi{H}$ for all $\bi{J}\in {\cal J}$ with eigenvalue $$\label{36} E_0=s^2 J \equiv s^2 \sum_{x<y}J_{xy}.$$ This proves the first part of the following \[L2\] If $\bi{J}^{(1)}\cdot\bi{H}$ and $\bi{J}^{(2)}\cdot\bi{H}$ are completely or covariantly isospectral, then $$\label{37} \sum_{x<y}J_{xy}^{(1)} = \sum_{x<y}J_{xy}^{(2)},$$ i. e. $\bi{J}^{(1)}$ and $\bi{J}^{(2)}$ lie in the same hyperplane perpendicular to $\boldsymbol{1}\equiv (1,1,\ldots 1)$. [**Proof**]{}: If all $J_{xy}\ge 0$, then $E_0=s^2 J$ will be the maximal eigenvalue of $\bi{J}^{(2)}\cdot\bi{H}$. In fact, $\langle \varphi_0| H_{xy} \varphi_0 \rangle$ is the maximal expectation value for each $H_{xy}$. Hence $J$ is an isospectral invariant at least in the domain ${\cal J}^+\equiv \{\bi{J}|\mbox{ all } J_{xy}\ge 0\}$. Now assume covariant isospectrality and let $t\mapsto \bi{J}(t)=\exp({-M t}) \bi{J}(0)$ be an isospectral curve which will be restricted to ${\cal J}^+$. According to what has been said before, $0=\dot{J}=\frac{d}{dt}\langle \bi{J} |\boldsymbol{1}\rangle = \langle \dot{\bi{J}} |\boldsymbol{1}\rangle = \langle -M \bi{J} |\boldsymbol{1}\rangle = \langle \bi{J} |-M^\ast \boldsymbol{1}\rangle =\langle \bi{J} |M \boldsymbol{1}\rangle$ for all $\bi{J}\in{\cal J}^+$. Since ${\cal J}^+$ linearly generates ${\cal J}$ it follows that $M\boldsymbol{1}=\boldsymbol{0}$, i. e. all row sums of $M$ vanish and $J$ is an isospectral invariant on the whole space ${\cal J}$. ------------------------------------------------------------------------ Before proceding with $M=Ns-1$ we will show that also \[L4\] $||\bi{J}||^2=\sum_{x<y}J_{xy}^{2}$ is an isospectral invariant. [**Proof**]{}: Obviously, $\Tr (H_0^2)$, the sum of all eigenvalues squared, is the same for isospectral Hamiltonians. After expanding the square $(\bi{J}\cdot\bi{H})^2$ one realizes that only those products $H_{xy}H_{uv}$ have a non-zero trace where $x=u$ and $y=v$. Hence $\Tr (H_0^2)= \sum_{x<y}J_{xy}^2 \Tr (H_{xy}^2)=(\sum_{x<y}J_{xy}^2 )\cdot \frac{1}{3}s^2(s+1)^2(2s+1)^N$, see Appendix A. The actual value for $\Tr (H_{xy}^2)$ is irrelevant for the proof; what matters only is that it is independent of $x,y$. This concludes the proof. ------------------------------------------------------------------------ \ From Lemma \[L4\] it follows immediately that in the covariant isospectral case the matrix $\exp({-t M})$ leaves the Euclidean norm $||\ldots||$ invariant and thus must be an orthogonal transformation and its generator $-M$ will be antisymmetric. The triangle ($N=3$) ==================== $s=\frac{1}{2}$ --------------- Next we consider eigenvalues with eigenvectors in the subspace ${\cal H}(M=Ns-1)$, but restricted to the case $N=3$. With the abbreviations $$\label{38} J=J_{12}+J_{23}+J_{13},$$ $$\label{39} \Gamma=J_{12}J_{23}+J_{12}J_{13}+J_{23}J_{13},$$ the eigenvalues are calculated to be (see [@LL], [§]{} 62, Exercise 2 ) $$\label{40} E_0= s^2 J,\quad E_{1,2}=s(s-1)J\pm \sqrt{J^2-3 \Gamma}.$$ Hence the first three eigenvalues are constant on curves with constant $J$ and $\Gamma$. These are circles with radius $$\label{41} r=\sqrt{\frac{2}{3}J^2-2\Gamma},$$ the center of which is located on the line $J_{12}=J_{23}=J_{13}$, including the degenerate case $r=0$.\ For $s=\frac{1}{2}$ the list of eigenvalues is already exhausted: Due to rotational symmetry the value $E_0$ is $4$–fold degenerate $(S=\frac{3}{2})$ and the $E_{1,2}$ are two–fold degenerate $(S=\frac{1}{2})$. We conclude: \[P3\] The system $N=3, s=\frac{1}{2}$ is of complete isospectral type with dimension $2$. We now consider the question whether the triangle with $s=\frac{1}{2}$ is of covariant isospectral type, i. e. we seek solutions of $$\label{43} [\bi{H},\Omega]= M \bi{H},$$ $\Omega$ and $M$ being anti–symmetric. Let $T=T^3$ be the unitary left shift operator which represents a cyclic permutation of the spin sites. Then a solution of (\[43\]) is given by $$\label{44} \Omega = \frac{1}{2\sqrt{3}}(T-T^\ast), \quad M=\frac{1}{\sqrt{3}} \left( \begin{array}{ccc}0&1&-1\\-1&0&1\\1&-1&0 \end{array}\right).$$ $\Omega$ can also be written as $$\label{45} \Omega=\frac{i}{4\sqrt{3}}\boldsymbol{\sigma}_3\cdot (\boldsymbol{\sigma}_1\times\boldsymbol{\sigma}_2),$$ where the $\boldsymbol{\sigma}_i, (i=1,2,3)$ denote the Pauli-matrices. Obviously, $\Omega$ is rotationally symmetric which entails complete isospectrality. The factor $\frac{1}{\sqrt{3}}$ is chosen such that the parameter $t$ in $\exp({tM})$ will be just the angle of rotation. $T^3=T$ entails $\Omega^3=-\frac{1}{4}\Omega$, hence the exponential series of $\exp({t\Omega})$ will be actually a polynomial in $\Omega$: $$\label{46} \exp({t\Omega)}=1+2\Omega\sin\frac{t}{2}+(2\Omega)^2(1-\cos\frac{t}{2}).$$ For special values of $t$ we obtain: $$\label{47} \exp({\frac{4\pi}{3}\Omega})=T,$$ $$\label{48} \exp({4\pi\Omega})=T^3={\;\smash{\raisebox{-0.5ex}{$\!\!\stackrel{\!\mbox{1} \hspace{-0.4ex}\rule[0.0ex]{0.06ex}{1.60ex}}{ }$}}},$$ $$\label{49} \exp({2\pi\Omega})=\frac{2}{3}(T^2+T-\frac{1}{2}).$$ The last expression can be rewritten using $$\label{50} \tilde{H}\equiv \left( \begin{array}{c}1\\1\\1\end{array} \right)\cdot\bi{H}= \frac{3}{4}\left( {\Bbb P}_{\frac{3}{2}} -{\Bbb P}_{\frac{1}{2}} \right),$$ where ${\Bbb P}_{\frac{3}{2}}$ (resp. ${\Bbb P}_{\frac{1}{2}}$) denotes the projector onto the subspace $S=\frac{3}{2}$ (resp. $S=\frac{1}{2}$). The result is $$\label{51} \exp({2\pi\Omega})={\Bbb P}_{\frac{3}{2}} -{\Bbb P}_{\frac{1}{2}}.$$ This means that an eigenstate of $H_0$ with $S=\frac{1}{2}$ acquires a phase of $\pi$ after a full rotation in ${\cal J}$–space, analogous to the occurence of Berry phases for adiabatic loops in parameter space. Summarizing, we state the following proposition which, in essence, is due originally to V. G. Grachev [@G1]: \[P5\] The system $N=3, s=1/2$ is of both completely and covariantly isospectral type. A physical example ------------------ An interesting and timely application of this theory is provided by the example of the molecular magnet (CN$_3$H$_6$)$_4$Na$_2$\[H$_4$V$_6$P$_4$O$_{30}$(CH$_2$)$_3$CCH$_2$OH$_2$\]$\bullet$ 14H$_2$O which features two uncoupled systems of three $V^{4+}\quad (s =\frac{1}{2})$ ions that interact via antiferromagnetic Heisenberg exchange. It has been proposed [@MMBSB] that the Coulomb interaction between an Na ion and two of the three $V^{4+}$ ions gives rise to what is essentially an isosceles triangle, with the distances between the three vanadium ions being $3.20, 3.21,$ and $3.36 $[Å]{}. It is then quite reasonable to assume, that the three exchange constants satisfy $J_{12} = J_{13} \neq J_{23}$ . In fact, calculation of the weak field susceptibilty has yielded results that are in excellent agreement with accurate susceptibility measurements from room temperature down to $2 K$ upon assigning the values $J_{12} = J_{13}= 64.7K$ and $J_{23} = 7.5 K$ [^2] [@Jun], [@Lub]. Moreover, the calculated energy level spacings that follow from these assignments have recently been confirmed to good accuracy in a direct manner by inelastic neutron scattering [@St]. Nevertheless, as the work of this section has shown, the identical energy levels and the identical temperature dependent susceptibility emerge for the continuous choices of the three different exchange constants that lie on curves with $J = 136.9K$ and $\Gamma = 5156.6 K^2$ . $s >\frac{1}{2}$ ---------------- For $s>\frac{1}{2}$ we consider the next subspace ${\cal H}(M=3s-2)$. The characteristic polynomial $p(\lambda)=\sum_{\nu=1}^6 c_\nu \lambda^\nu$ has been calculated using MATHEMATICA$^{\circledR}$ 4.0, but is too complicated to be presented here. One particular Jacobian reads $$\label{42} \frac{\partial (c_3,c_4,c_5)}{\partial(J_{12},J_{23},J_{13})}= (3s-1)(6s-1)(1-8s+6s^2)(1-6s+15s^2).$$ This is a polynomial in $s$ which has no integer or half–integer roots. Therefore we have proved the following \[P4\] The system $N=3, s>\frac{1}{2}$ is not of complete isospectral type. Isospectrality for $N>3$ ======================== The question arises whether our result that complete isospectrality only occurs for $s=\frac{1}{2}$ also holds for $N>3$. Our method of calculating the Jacobian (\[34\]) for arbitrary $s$ will no longer work for larger $N$. However, we can prove a weaker statement, namely \[P6\] Systems with $N>3$ cannot be of covariantly isospectral type. [**Proof:**]{} For this we need a trace formula which will be explained in the Appendix A: $$\begin{aligned} \label{52} \Tr (H_0^3) & = & \sum_{x<y} (J_{xy})^3\left( -\frac{1}{6} s^2(s+1)^2(2s+1)^N \right)\\ \nonumber & + & \sum_{x<y<z}J_{xy}J_{yz}J_{xz} \left( \frac{2}{3} s^3(s+1)^3(2s+1)^N \right).\end{aligned}$$ From the isospectral invariance of $\Tr (H^3)$ we conclude that also $$\label{53} f_3\equiv \sum_{x<y} (J_{xy})^3 - 4s(s+1)\sum_{x<y<z}J_{xy}J_{yz}J_{xz}$$ will be invariant. Now we consider four different spin sites (using $N>3$) denoted by $1,2,3,4$ and consider vectors $\bi{J}\in{\cal J}$ which have vanishing components except possibly for $J_{12},J_{13},J_{23},J_{14},J_{24},J_{34}$. One-parameter isospectral curves passing through $\bi{J}$ satisfy $$\label{55} \frac{d}{dt}f_3(\bi{J}(t)) = 0.$$ Using (\[53\]) and $\frac{d}{dt}\bi{J}(t)=-M \bi{J}(t)$, (\[55\]) can be written as an equation which is linear w. r. t. the $15$ relevant matrix entries of $M$ and trilinear w. r. t. the 6 non-vanishing components of $\bi{J}$. Using MATHEMATICA$^{\circledR}$ 4.0 it is easy to show that (\[55\]) has only the trivial solution $M=0$. For example, one may randomly choose 15 vectors $\bi{J}$ (it suffices to consider components $-1,0,1$) and cast the corresponding $15 $ equations of the form (\[55\]) into matrix form. Non-trivial solutions exist only if the determinant of this matrix, which is a polynomial in $s$ of degree 30 with integer coefficients, vanishes. However, the zeros of the polynomial can be numerically computed and shown not to attain half-integer or integer values. ------------------------------------------------------------------------ \ So it seems that the concept of covariant isospectrality is of little use having only one single application for $N=3, s=1/2$. However, covariance may be restored for $N\ge 4$ if the class of admissible Hamiltonians is suitably extended, e. g. to include also Hamiltonians which are bi-quadratic in the spin observables. However, this is beyond the scope of the present article.\ Of course, our proposition \[P6\] does not exclude plain isospectrality for $N>3$. Indeed, we will show that the system $N=4, s=\frac{1}{2}$ is completely isospectral, albeit not covariantly isospectral, in the next section. The $s=1/2$ tetrahedron case ============================ In the case $N=4, s=1/2$ it is still possible to calculate the coefficients of the characteristic polynomials of $H_0$ restricted to the subspaces with $M=0, S=2,1,0$. Obviously, this is enough in order to study complete isospectrality since all eigenvalues of $H_0$ appear within these subspaces. It turns out that all coefficients can be written as functions of four fundamental invariants $I_1,I_2,I_3,I_4$. These can most conveniently be written in terms of new coordinates in ${\cal J}$, which are defined as half the sums and differences of the coupling constants of adjacent edges: $$\label{56} S_{12}\equiv \frac{1}{2}(J_{12}+J_{34}),\quad D_{12}\equiv \frac{1}{2}(J_{12}-J_{34}),$$ etc.\ \[P8\] Two spin systems with coupling constants $\bi{J}^{(1)},\bi{J}^{(2)}$ with $N=4, s=1/2$ are completely isospectral iff the following four functions assume the same values for $\bi{J}^{(1)}$ and $\bi{J}^{(2)}$: $$\label{57} I_1=D_{12}^2+D_{13}^2+D_{14}^2,$$ $$\label{58} I_2=S_{12}^2+S_{13}^2+S_{14}^2,$$ $$\label{59} I_3=S_{12}+S_{13}+S_{14},$$ $$\label{60} I_4=2 D_{12}D_{13}D_{14}+D_{12}^2 S_{12}+D_{13}^2 S_{13}+D_{14}^2 S_{14}-S_{12}S_{13}S_{14}.$$ Now let $I'$ be the functional matrix obtained by partial differentiation of $I_1,I_2,I_3,I_4$ with respect to its 6 arguments $S_{12},\cdots,D_{14}$. The rank of $I'$ assumes its maximal value of 4 iff no determinant of the 15 possible $4\times 4$ submatrices of $I'$ vanishes. We denote the subset of those points with maximal rank by ${\cal R}\subset {\cal J}$. If an isospectral class lies entirely within ${\cal R}$ it will be a 2-dimensional submanifold of the 6-dimensional space ${\cal J}$. This follows by a well-known theorem of differential geometry (see e. g. [@AMR] Theorem 3.5.4). We will call this case , the other cases .\ Although ${\cal J}$ is six-dimensional, one can visualize the 2-dimensional submanifolds in the generic case. $I_1=const.$ defines a sphere in the 3-dimensional $\bi{D}$-space with coordinates $D_{12},D_{13},D_{14}$. $I_2=const.$ and $I_3=const.$ define the intersection of a sphere and a plane, i. e. a circle in $\bi{S}$-space. For given $\bi{D}$ the last equation $I_4(\bi{D},\bi{S})=const.$ picks out a finite number (actually $\le 6$) of points in that circle. If the corresponding angles $\psi_\nu$ are drawn as different radii $r_\nu = 1+\frac{\psi_\nu}{4 \pi}$ in $\bi{D}$-space (identifying points with $r_\nu=1/2 \mbox{ and } 3/2$) we obtain a surface folded in a complicated way.\ A large number of 2-dimensional generic isospectral classes have been identified numerically. The exceptional classes are one- or zero-dimensional and will be discussed further in Appendix B. Heuristic arguments for the (non-)occurrence of isospectrality ============================================================== In the previous sections we have studied isospectrality for the cases $s=\frac{1}{2}, N=3,4$ and excluded certain other cases, e. g. complete isospectrality for $s>\frac{1}{2}$ and covariant isospectrality for $N>3$. However, we have been unable to present a complete list of criteria for the (non-)occurrence of isospectrality. What is also missing is some simple and intuitive argument why isospectrality is so rare. As a substitute for a complete theory we will, in this section, provide some heuristic arguments for the (non-)occurrence of isospectrality which also may give more insight into isospectrality than detailed proofs. We think that these arguments could be made rigorous as far as necessary conditions for isospectrality in the cases $s=\frac{1}{2}, N=3,4,5$ are involved.(See the remarks in section 2 on the dimension of isospectral classes and the fibration theorem.) However, a detailed proof would require technical issues from the theory of algebraic varieties which are beyond the scope of this article and, moreover, would appear as superfluous given that isospectrality in some of these cases has already been proven by case inspection.\ The heuristic argument goes as follows: Isospectrality will (only) occur if the number ${N \choose 2}$ of bonds between spin pairs exceeds the number $L$ of independent eigenvalues of $H_0$. In this case, typically the systems corresponding to an $n={N \choose 2}-L$-dimensional sub-variety of ${\cal J}$ will possess the same eigenvalues. The argument may even be applied if one is not aware of all relations among the eigenvalues which determine $L$. In such a case of unknown relations one would perhaps over-estimate $L$ and hence under-estimate the dimension $n$ of the isospectrality classes but, depending on the case, one could correctly predict the occurrence of isospectrality.\ When counting the number $L$ of independent eigenvalues one first has to consider the $(2S+1)$-fold degeneracy dictated by the rotational invariance of $H_0$. In the simplest example, one has to couple $N=2$ spins $s=\frac{1}{2}$, obtaining one triplet and one singlet as eigenspaces of $H_0$, symbolically $2\times 2=3+1$. Thus there are not four, but only two independent eigenvalues of $H_0$. Similarly, for $N=3$ one has $2\times 2 \times 2=4+2+2$, hence $3$ independent eigenvalues. In the latter case there are ${3\choose 2}=3$ bonds. Thus the heuristic argument does not yet explain isospectrality with $n=1$-dimensional classes. However, it is easy to find a “missing relation" among the eigenvalues which reduces the number of independent eigenvalues to $L=2$: It is just the relation $\Tr H_0 =0$ which yields a linear relation of the form $4 E_1 + 2 E_2 +2 E_3 =0$ between the eigenvalues $E_\nu$.\ For $s=\frac{1}{2}$ and arbitrary $N$ it can be shown that there are exactly ${N \choose \left\lfloor N/2 \right\rfloor}$ independent eigenvalues due to rotational degeneracy and hence, considering $\Tr H_0 =0$, $L\le {N \choose \left\lfloor N/2 \right\rfloor} -1$. In this manner we obtain the results summarized in table 1.\ \[T1\] [@llll]{} N & $ {N \choose \left\lfloor N/2 \right\rfloor} -1$ & ${N\choose 2}$ & Isospectrality expected\ $2$ & $1$ & $1$ & no\ $3$ & $2$ & $3$ & yes\ $4$ & $5$ & $6$ & yes\ $5$ & $9$ & $10$ & yes\ $6$ & $19$ &$15$ & no\ $7$ & $34$ &$21$ & no\ $\vdots$ & $\vdots$ & $\vdots$ & no\ \[T2\] [@llll]{} N &$L\le$ & ${N\choose 2}$ & Isospectrality expected\ $2$ & $3$ & $1$ & no\ $3$ & $8$ & $3$ & no\ $4$ & $24$ & $6$ & no\ $\vdots$ & $\vdots$ & $\vdots$ & no\ For large $N$ the the entries in the second column of table 1 grow asymptotically as $2^N \sqrt{\frac{2}{\pi N}}$, hence almost exponential, whereas ${N \choose 2}$ grows only quadratically. Therefore our heuristic argument will only predict isospectrality in the cases $s=\frac{1}{2}, N=3,4,5$ but not for larger $N$. For larger $s>\frac{1}{2}$ the growth of the second column will prevail from the outset, see table 2, and our argument will not even be applicable for small $N$. Of course, this does not strictly exclude isospectrality for those cases, but makes it very unlikely in our opinion.\ There is another aspect which shows up in table 1 and which we now discuss: Note that for $N=4$ the difference between the numbers in the 3rd and the 2nd column, $6-5=1$, would only explain one-dimensional isospectrality classes, whereas we have encountered two-dimensional classes in section 7. This indicates that the correct $L$ should be $4$, not $5$, and that there is a further “missing relation" between the eigenvalues of $H_0$, comparable to $\Tr H_0=0$. Indeed, as shown in the following paragraph, there holds a general property of the eigenvalues of arbitrary Heisenberg Hamiltonians $H_0$ corresponding to the distribution of the eigenvalues among the quantum numbers $S$.\ Let, as above, denote by ${\Bbb P}_S$ the projector onto the eigenspace of $\bi{S}^2$ with the eigenvalue $S(S+1)$. Then $$\label{61} \Tr (H_0 {\Bbb P}_S) = \sum_{x <y}J_{xy} \Tr(\bi{S}_x\cdot\bi{S}_y {\Bbb P}_S ).$$ Since $\bi{S}^2$ and hence all ${\Bbb P}_S$ commute with arbitrary permutations of spin sites, the last factor in (\[60\]) does not depend on $x,y$ and can be factored out: $$\begin{aligned} \label{62} \Tr (H_0 {\Bbb P}_S) & = & \left( \sum_{x<y}J_{xy}\right) \Tr(\bi{S}_1\cdot\bi{S}_2 {\Bbb P}_S )\\ & = & J \Tr(\bi{S}_1\cdot\bi{S}_2 {\Bbb P}_S ).\end{aligned}$$ Being independent of the $J_{xy}$ this factor can be calculated for any suitable $H_0$, e. g. the one with constant $J_{xy}\equiv 1$, $$\label{63} \tilde{H_0}= \frac{1}{2} \left( \bi{S}^2 -N s(s+1) \right),$$ which yields, after some computation, $$\label{64} \Tr (H_0 {\Bbb P}_S) = J \frac{1}{N(N-1)}\left( S(S+1) -N s(s+1) \right) \Tr {\Bbb P}_S .$$ Hence for all $H_0$, the vectors with the components $\left(\Tr (H_0 {\Bbb P}_S) \right)_{S=S_{min},\ldots,S_{max}}$ are proportional to the constant vector given by the r. h. s. of (\[63\]) with $J=1$. This gives a number of $S_{max}-S_{min}=\left\lfloor Ns \right\rfloor$ independent linear equations for the eigenvalues of $H_0$. Of course, $\Tr H_0 = \sum_S \Tr(H_0 {\Bbb P}_S )=0 $ is a consequence of these equations and must not be counted seperately. For $s=\frac{1}{2}, N=4,5$ we obtain $\left\lfloor Ns \right\rfloor =2 $ independent equations, which explains the two-dimensional classes for $N=4$ we found in section 7, and predicts, at least, two-dimensional classes for the case $N=5$ not yet analyzed in detail. If this case would show isospectrality classes of dimension $n>2$ one could try to explain this by invoking more complicated relations derived from the higher moments, $\Tr (H_0^k {\Bbb P}_S ), k>1$. However, we will not further pursue this question here. Conclusion ========== We summarize our results in table 3. It is in order to add some remarks on the possibility of determining the coupling constants ${\bi J}$. Our results on the limits of uniquely determining the values of ${\bi J}$ in the case of isospectrality does not mean that these values could not be determined otherwise. First, we did not consider thermodynamical functions which do not come solely from the partition function, such as correlation functions, etc. Second, we do not adhere to a positivistic attitude which would in principle deny the physical reality of unmeasurable quantities. As in other domains of physics, these parameters could also be determined with the aid of additional assumptions, e. g. based on the symmetry of the molecules and supported by chemical considerations, which although plausible have not been confirmed directly. So we think the situation is different from theories with gauge freedom. \[T3\] [@rrlll]{} N & s & Plain Isospectrality & Complete Isospectrality & Covariant Isospectrality\ $3$ & $\frac{1}{2}$ & yes & yes & yes\ $3$ & $>\frac{1}{2}$ & unlikely & no & unlikely\ $4$ & $\frac{1}{2}$ & yes & yes& no\ $4$ & $>\frac{1}{2}$ & unlikely & no & no\ $5$ & $\frac{1}{2}$ & likely & likely & no\ $>5$ & $\frac{1}{2}$ & unlikely & unlikely & no\ $>5$ & $>\frac{1}{2}$ & unlikely & no & no\ Acknowledgement {#acknowledgement .unnumbered} =============== M. L. would like to thank members of Fachbereich Physik of Universität Osnabrück for their warm hospitality during a visit when a part of this work was performed. We also acknowledge the financial support of a travel grant awarded jointly by NSF-DAAD. Finally, it is a pleasure to thank K. Bärwinkel, P. C. Canfield, V. G. Grachev, S. Jun, D. Mentrup, J. Schnack, and H. Spindler for stimulating and helpful discussions. Ames Laboratory is operated for the United States Department of Energy by Iowa State University under Contract No. W-7405-Eng-82. Trace formulae ============== Expanding the terms occuring in the trace $$\label{71} \Tr H_0^n =\Tr \left( \sum_{x<y} J_{xy} \sum_{i=1}^{3} S_x^{(i)}S_y^{(i)}\right)^n$$ and using $\Tr (A\otimes B)=(\Tr A)(\Tr B)$ one ends up with terms of the form $$\label{72} \Tr (A_1 \ldots A_\ell), A_\nu\in\{S^{(1)},S^{(2)},S^{(3)} \}.$$ Here the spin operators without a site index denote operators operating in the single site Hilbert space ${\Bbb C}^{2s+1}$, not total spin operators.\ Let $\ell_i (i=1,2,3)$ denote the number of occurences of $S^{(i)}$ in the product $A_1\ldots A_\ell$. One can easily show that $\Tr (A_1\ldots A_\ell)$ is non-zero only if all $\ell_i$ are even or all $\ell_i$ are odd. We give a list of the simplest cases, where the trace is non-zero: $$\begin{aligned} \label{73} \ell=0 & : & \Tr ({{\;\smash{\raisebox{-0.5ex}{$\!\!\stackrel{\!\mbox{1} \hspace{-0.4ex}\rule[0.0ex]{0.06ex}{1.60ex}}{ }$}}}})= 2s+1,\\ \ell=2 & : & \Tr S^{(i)2}=\sum_{m=-s}^s m^2 = \frac{1}{3}s(s+1)(2s+1),\\ \ell=3 & : & \Tr S^{(1)}S^{(2)}S^{(3)} = - \Tr S^{(3)}S^{(2)}S^{(1)}= \frac{i}{6}s(s+1)(2s+1).\end{aligned}$$ From this we obtain for $n=2$: $$\begin{aligned} \label{74} \Tr H_0^2 & = & \sum_{x<y} J_{xy}^2 \sum_{i=1}^3 \Tr \left(S_x^{(i)2}\otimes S_y^{(i)2}\otimes{{{\;\smash{\raisebox{-0.5ex}{$\!\!\stackrel{\!\mbox{1} \hspace{-0.4ex}\rule[0.0ex]{0.06ex}{1.60ex}}{ }$}}}}}_{N-2}\right)\\ & = & \left( \sum_{x<y} J_{xy}^2\right) \frac{1}{3} s^2 (s+1)^2 (2s+1)^N.\end{aligned}$$ For $n=3$ there occur two kinds of non-zero terms: $$\label{75} \Tr \left( S_x^{(1)} S_x^{(2)}S_x^{(3)}\otimes S_y^{(1)} S_y^{(2)}S_y^{(3)}\otimes{{{\;\smash{\raisebox{-0.5ex}{$\!\!\stackrel{\!\mbox{1} \hspace{-0.4ex}\rule[0.0ex]{0.06ex}{1.60ex}}{ }$}}}}}_{N-2} \right)$$ and those terms obtained by permutations of $\{ 1,2,3\}$, and $$\label{76} \Tr \left( S_x^{(i)2}\otimes S_y^{(i)2}\otimes S_z^{(i)2} \otimes{{{\;\smash{\raisebox{-0.5ex}{$\!\!\stackrel{\!\mbox{1} \hspace{-0.4ex}\rule[0.0ex]{0.06ex}{1.60ex}}{ }$}}}}}_{N-3} \right),$$ where $x,y,z$ are pairwise distinct and $i=1,2,3$. Consequently we obtain: $$\begin{aligned} \label{77} \Tr H_0^3 & = & T_1 + T_2, \\ T_1 & = & \sum_{x<y} J_{xy}^3 3!(2s+1)^{N-2} \left( \Tr S^{(1)} S^{(2)}S^{(3)} \right)^2\\ & = & - \sum_{x<y} J_{xy}^3 \frac{1}{6} s^2 (s+1)^2 (2s+1)^N,\\ T_2 & = & 3!\left( \sum_{x<y<z} J_{xy} J_{xz}J_{yz} \right) (2s+1)^{N-3} \sum_{i=1}^3\left(\Tr S^{(i)2} \right)^3\\ & = & \left( \sum_{x<y<z} J_{xy} J_{xz}J_{yz} \right) \frac{2}{3}s^3(s+1)^3(2s+1)^N.\end{aligned}$$ The exceptional cases ===================== Here we collect some properties of isospectral classes for $N=4,s=1/2$ which belong to the exceptional case. Although we did not obtain a complete classification of all isospectral classes these results may be useful for further studies.\ The subset ${\cal J}\backslash{\cal R}$, which denotes the complement of ${\cal R}$, is characterized by the vanishing of all $4\times 4$ submatrices of $I'$, hence it will be also an algebraic variety. After some computations one shows that ${\cal J}\backslash{\cal R}$ is a union ${\cal J}\backslash{\cal R}={\cal C}_1\cup{\cal C}_2\cup{\cal C}_2$ of three simpler varieties given by the following equations $$\label{78} {\cal C}_1:I_1=D_{12}^2+D_{13}^2+D_{14}^2=0,$$ $$\label{79} {\cal C}_2:I_2 = 3 I_3^2, \mbox{ i.~e.~} S_{12}=S_{13}=S_{14},$$ $$\label{80} {\cal C}_3:f=g=0,$$ where $$\label{81} f\equiv (D_{12}^2 - D_{14}^2 )D_{13}+ D_{12} D_{14}(S_{14}-S_{12}) =0,$$ $$\label{82} g\equiv (D_{13}^2 - D_{14}^2) D_{12}+ D_{13} D_{14}(S_{14}-S_{13}) =0.$$ In the first case it is clear that an isospectral class lies entirely inside ${\cal C}_1$ or outside ${\cal C}_1$, since $I_1$ is constant on this class. An analogous remark applies for ${\cal C}_2$ but the case ${\cal C}_3$ is more complicated. In the first case with $I_1=0$ the isospectral classes are $0$-dimensional, since the circles defined by $I_2=i_2$ and $I_3=i_3$ intersect the variety $I_4=-S_{12}S_{13}S_{14}=i_4$ at most at six points. In the second case with $S_{12}=S_{13}=S_{14}=\sigma$ the isospectral classes are the $1$-dimensional intersections of the sphere $I_1=i_1, i_1>0$ and the variety $I_4=2 D_{12}D_{13}D_{14}+(D_{12}^2 +D_{13}^2 +D_{14}^2)\sigma -\sigma^3=i_4$. In the third case we have shown, using the Eliminate-command of MATHEMATICA$^{\circledR}$, that $f=g=0$ implies an equation of the form $P(I_1,I_2,I_3,I_4)=0$, namely $$\begin{aligned} \label{83} P & = & -108 I_4^2+4 I_3 I_4(9 I_1+18 I_2-5 I_3^2)\nonumber\\ & & +(I_1+2 I_2-I_3^2)^2(2 I_2+4 I_2-I_3^2)\end{aligned}$$ Hence those isospectral classes with $P(I_1,I_2,I_3,I_4)\ne 0, I_1\ne 0, I_2 \ne 3 I_3^2$ lie entirely inside ${\cal R}$ and belong to the generic case. We conjecture that also the converse holds, namely that $P=0$ implies $f=g=0$ but could not prove it. By calculating the corresponding Groebner bases, it can be shown that $P$ and $f,g$ generate different ideals in the ring of polynomials in 6 variables. However, this will not exclude the possibility that the corresponding real algebraic varieties may be equal. Hence we cannot exclude further exceptional cases in the realm $P=0$ but $f\ne 0$ or $g\ne 0$. The isospectral classes in the case $f=g=0$ will be 1-dimensional: The constraint $f=g=0$ allows one to express the variables $S_{12},S_{13},S_{14}$ in terms of $I_3,D_{12},D_{13},D_{14}$. The remaining constraints $I_1=i_1,I_4=i_4$ define a family of curves obtained as intersections between spheres and tube-like surfaces. 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Note that algebraic varieties need not be differentiable manifolds since they may contain “boundaries" like the vertex of the light cone $x^2+y^2+z^2=c^2 t^2$ or “hairs" like $x=y=0$ in the variety $(x^2+y^2)z=0$. [^2]: It is usual to measure the coupling constants in units of Kelvin. The corresponding energies are obtained by multiplying with the Boltzmann constant $k_B$.

--- abstract: 'Communication services with heterogeneous performance requirements are emerging as key use cases for 5G and beyond. This paper deals with the coexistence of two service classes, i.e., critical service (CS) and non-critical service (NCS) on a grant-free channel consisting of the radio access and backhaul segments. On the radio access segment, Internet-of-Things (IoT) devices send packets to a set of non-cooperative access points (APs) using slotted ALOHA (SA). The APs then forward correctly received messages to a base station over a shared wireless backhaul segment adopting SA. The APs hence play the role of low-complexity relays that improve space diversity and reduce performance losses caused by interference on the access segment. We study first a simplified erasure channel model, which is well suited for non-terrestrial applications. Then, in order to account for terrestrial scenarios, the impact of fading is considered. Throughput and packet success rate metrics are derived, and numerical results are provided to assess the performance trade-offs between CS and NCS. Among the main conclusions, we show that orthogonal inter-service resource allocation is generally preferred for NCS devices, while non-orthogonal protocols can improve the throughput and packet success rate of CS devices for both terrestrial and non-terrestrial scenarios.' author: - '\' bibliography: - 'Biblio.bib' title: | Uncoordinated Grant-Free Radio Access via Diversity for Critical and Non-Critical IoT Services\ [^1] --- Beyond 5G, IoT, Grant-Free, Radio Access, mMTC, URLLC Introduction ============ Future generations of cellular and satellite networks will include new services with vastly different performance requirements. In recent 3GPP releases [@3gpp], a distinction is made among Ultra-Reliable and Low-Latency Communications (URLLC), with stringent delays and packet success rate requirements; enhanced Mobile Broadband (eMBB) for high throughput; and massive Machine Type Communications (mMTC) for sporadic transmissions with large spatial densities of devices [@5g_tutorial; @5Goverview; @opportunistic_coexistence]. In this paper, we focus on Internet of Things (IoT) scenarios, which are typically assumed to fall into the mMTC service category [@nbiot_mmtc]. We take a further step as compared to the mentioned 3GPP classification by considering a beyond-5G scenario characterized by the coexistence of heterogeneous IoT devices having critical or non-critical service requirements. Devices with requirements must be provided more stringent throughput and packet success rate performance guarantees than devices. The model under study also applies to the case in which each device may require alternatively or , as envisioned for the massive URLLC (mURLLC) service class in recent proposals [@saad2019vision]. In the presence of a large number of IoT devices [@mmtcsaad] requiring the transmission of small amounts of data, conventional grant-based radio access protocols can cause a significant overhead on the access network due to the large number of handshakes to be established. A potentially more efficient solution is given by *grant-free* radio access protocols. Under grant-free access, devices transmit whenever they have a packet to deliver without any prior handshake [@grant_free_popovski; @rahif_grant_free; @grant_free_cavdar]. This is typically done via some variants of the classical ALOHA random access scheme [@abramson1970aloha]. Grant-free access protocols are used by many commercial solutions in the terrestrial domain, e.g., by Sigfox [@sigfox] and LoRaWAN [@lora]; as well as in the satellite domain, using constellations of low-earth orbit satellites, e.g. Orbcomm [@orbcomm] and Myriota [@myriota]. [.5]{} [.5]{} In classical cellular IoT scenarios, orthogonal inter-service resource allocation schemes are typically used [@3gpp_nbiot]. However, due to their static nature, orthogonal schemes may cause an inefficient use of resources when traffic patterns are hard to predict as in grant-free IoT systems. To obviate this problem, dynamic spectrum access schemes have been proposed whereby devices can use idle resources allocated to other devices. Reinforcement learning or online optimization solutions can be used to derive transmission strategies that maximize the throughput [@RL_DSA_suracruse; @RL_DSA_negev; @destounis2019learn2mac]. Though promising, these solutions can fall short when devices are equipped with very limited computational capabilities and battery capacity, low memory, and when they are placed in highly dynamic environments. *Non-orthogonal* resource allocation, which allows the allocation of multiple devices to the same time-frequency resource, presents a promising alternative solution [@noma_saito; @noma_performance; @noma_challenges_potential; @ding2014performance]. Recent work has proposed to apply non-orthogonal resource allocation to heterogeneous services [@rahif_access_2018; @popovski2018slicing; @rahifuplink]. In order to mitigate interference in non-orthogonal schemes, one can leverage successive interference cancellation (SIC) [@aloha_noma], time diversity [@coded_slotted_aloha], and/or space diversity [@munari_multiple_aloha][@vladimir_cooperative_ALOHA]. As illustrated in Fig. \[fig:system\_model\], space diversity is provided by multiple Access Points (APs) that play the role of relays between the devices and the Base Station (BS). For the terrestrial networks, this topology reflects important deployments such as Cloud-RAN (C-RAN), ultra-dense networks as well as the use of Unmanned Aerial Vehicles (UAV) as flying base stations [@UAV_tutorial; @flying_BS; @flying_BS_2]. But, the space diversity model is also relevant for non-terrestrial scenarios. In particular, with the renewed interest for the deployment of Low-Earth Orbit (LEO), mega-constellations such as, Amazon Kuiper [@amazon_kuiper], SpaceX Starlink [@spacex_starlink] and OneWeb [@oneweb] can provide low-latency and high-speed broadband to unserved and under-served locations. With thousands of LEO satellites, these constellations will offer connectivity to each earth location with multiple satellites at a time. These satellites can act as relays on a larger scale than in terrestrial networks. *Main Contributions:* In this work, we study grant-free access for both and in space diversity-based models for both non-terrestrial and terrestrial applications. We analytically derive throughput and packet success rate measures for both and as a function of key parameters such as the number of APs, traffic load and frame size. The analysis accounts for orthogonal and non-orthogonal inter-service access schemes, as well as for binary erasure channels modeling non-terrestrial applications (Fig. \[fig:system\_model\](a)) and for fading channels modeling terrestrial application (Fig. \[fig:system\_model\](b)). Finally, two receiver models are considered, namely, a collision model, where packets transmitted by the same device are assumed to undergo destructive collision, and a superposition model, where packets transmitted from the same device are superposed at the receiver. The analysis sheds lights on the advantages of each access scheme with respect to each type of service in addition to giving insights on different regimes as function of the number of time and space resources available. This work was partly presented in the conference papers [@frederico2019modern] and [@rahif_space]. In [@frederico2019modern], a simplified system with a single service was considered with non-orthogonal access, erasure channels model and a simplified collision model. In [@rahif_space], the coexistence and was considered under the erasure channels model and a simpler collision model whereby transmissions are assumed to be unaffected by regardless of the NCS load. In addition, in contrast to this paper, the analysis performed in [@rahif_space] does not allow the derivation of closed-form expressions which are of practical interest. Finally, these papers do not consider the terrestrial scenario. The rest of the paper is organized as follows. In Sec. \[sec:system\_model\_performance\_metrics\] we describe the system model used and the performance metrics. In Sec. \[sec:oneclass\] we study the system in the presence of a single service while Sec. \[sec:heterogeneous\_collision\] and \[sec:heterogeneous\_superposition\] tackle the heterogeneous services case under the general collision and superposition model respectively. Finally, the heterogeneous service case is evaluated under the fading channel model in Sec. \[sec:throughput\_reliability\_fading\], conclusions and extensions are discussed in Sec. \[sec:conclusions\]. **Notation:** Throughout, we denote as $X\sim \operatorname{Bin}(n,p)$ a Binomial random variable (RV) with $n$ trials and probability of success $p$; as $X ~\sim \operatorname{Poiss}(\lambda)$ a Poisson RV with parameter $\lambda$. We also write $(X,Y)\sim f \cdot g$ for two independent RVs $X$ and $Y$ with respective probability density functions $f$ and $g$. System model and Performance metrics {#sec:system_model_performance_metrics} ==================================== System Model ------------ We first consider the system illustrated in Fig. \[fig:system\_model\], in which $L$ APs, e.g., LEO satellites, provide connectivity to IoT devices. The APs are in turn connected to a BS, e.g., a ground station, through a shared wireless backhaul channel. We assume that time over both access and backhaul channels is divided into frames and each frame contains $T$ time slots. At the beginning of each frame, a random number of IoT devices are active. The number of active IoT devices that generate and messages at the beginning of the frame follow independent Poisson distributions with average loads $\gamma_c G$ and $(1-\gamma_c) G\ \mathrm{[packet/frame]}$, respectively, for some parameter $\gamma_c \in [0,1]$ and total load $G$. Users select a time-slot uniformly at random among the $T$ time-slots in the frame and independently from each other. By the Poisson thinning property [@billingsley2008probability], the random number $N_c(t)$ of messages transmitted in a time-slot $t$ follows a Poisson distribution with average $G_c = \gamma_c G /T \ \mathrm{[packet/slot]}$, while the random number $N_{\bar{c}}(t)$ of messages transmitted in slot $t$ follows a Poisson distribution with average $G_{\Bar{c}}=(1-\gamma_c)G/T \ \mathrm{[packet/slot]}$. *Radio Access Model:* As in, e.g., [@frederico2019modern; @azimi2017content; @calderbank_erasure], we model the access links between any device and an AP as an independent interfering erasure channel with erasure probability $\epsilon_1$. In non-terrestrial applications, as represented in Fig. \[fig:system\_model\_space\], this captures the presence or absence of a line-of-sight link between the transmitter and the receiver. A packet sent by a user is independently erased at each receiver with probability $\epsilon_1$, causing no interference, or is received with full power with probability $1-\epsilon_1$. The erasure channels are independent and identically distributed (i.i.d.) across all slots and frames. Interference from messages of the same type received at an AP is assumed to cause a destructive collision. Furthermore, messages are assumed to be transmitted with a higher power than messages so as to improve their packet success rate, hence creating significant interference on messages. As a result, in each time-slot, an AP can be in three possible states: - a message is retrieved successfully if the AP receives only one (non-erased) message and no more than a number $K$ of (non-erased) messages. This implies that, due to their lower transmission power, messages generate a tolerable level of interference on messages as long as their number does not exceed the threshold $K$; - a message is retrieved successfully if the AP receives only one (non-erased) message; - no message is retrieved otherwise. *Backhaul model:* The APs share a wireless out-of-band backhaul that operates in a full-duplex mode and in an uncoordinated fashion as in [@frederico2019modern]. The lack of coordination among APs can be considered as a worst-case scenario in dense low-cost terrestrial cellular deployments [@het_networks_no_coordination] [@vladimir_cooperative_ALOHA] and as the standard solution for constellations of LEO satellites that act as relays between ground terminals and a central ground station. In fact, satellite coordination, although feasible through the use of inter-satellite links [@inter_satellite_links], may be costly in terms of on-board resources. In each time-slot $t+1$, an AP sends a message retrieved on the radio access channel in the corresponding slot $t$ over the backhaul channel to the BS. APs with no message retrieved in slot $t$ remain silent in the corresponding backhaul slot $t+1$. The link between each AP and the BS is modeled as an erasure channel with erasure probability $\epsilon_2$, and destructive collisions occur at the BS if two or more messages of the same type are received. As for the radio access case, erasure channels are i.i.d. across APs, slots and frames. In order to model interference between APs, we consider two scenarios. The first, referred to as *collision model*, assumes that multiple messages from the same device cause destructive collision. Under this model, in each time-slot, the BS’s receiver can be in three possible states: - as for radio access, a message is retrieved successfully at the BS if only one message is received from any AP, along with no more than $K$ messages; - a message is retrieved successfully if no other or message is received; - no message is retrieved at the BS otherwise. In the second model, referred to as *superposition model*, the BS is able to decode from the superposition of multiple instances of the same packet that are relayed by different APs on the same backhaul slot, assuming no collisions from other transmissions. In practice, this can be accomplished by ensuring that the time asynchronism between APs is no larger that the cyclic prefix in a multicarrier modulation implementation. Synchronization can be ensured, for example, by having a central master clock at the BS against which the local time bases of APs are synchronized [@timesynchro_patent_AP]. Overall, the BS’s receiver can be in three possible states: - a message is retrieved successfully at the BS in a given time-slot if no other message and no more than $K$ messages are received by the BS; - a message is retrieved successfully if no messages and no other messages are received in the same slot; - no message is retrieved at the BS otherwise. *Inter-service TDMA:* In addition to non-orthogonal resource allocation whereby devices from both services share the entire frame of $T$ time-slots, we also consider orthogonal resource allocation, namely *inter-service time division multiple access* (TDMA), whereby a fraction $\alpha T$ of the frame’s time-slots are reserved to devices and the remaining $(1-\alpha)T$ for devices. Inter-service contention in each allocated fraction follows a protocol as discussed above. In the following, we derive the performance metrics under the more general non-orthogonal scheme described above. The performance metrics under TDMA for each service can be directly obtained by replacing $T$ with the corresponding fraction of resources in the performance metrics equations and setting the interference from the other service to zero. Performance Metrics {#sec:performance_metrics} ------------------- We are interested in computing the throughputs $R_c$ and $R_{\bar{c}}\ [\mathrm{packet/slot}]$ and the packet success rate $\Gamma_c$ and $\Gamma_{\bar{c}}\ [\mathrm{packet/frame}]$ for and respectively. The throughput is defined as the average number of packets received correctly in any given time-slot at the BS for each type of service. The packet success rate is defined as the average probability of successful transmission of a given user given that the user is active, i.e., that it transmits a packet in a given frame. Single Service Under Collision Model {#sec:oneclass} ==================================== Performance Analysis -------------------- We start by considering the baseline case of a single service under the collision model. While this can account for either or , we consider here without loss of generality only the by setting $\gamma_c = 1$. An AP successfully retrieves a packet when *only one* of the $N_c = n_c$ transmitted packets arrives unerased, i.e. with probability $${\psU = \nTx \,(1-\perasU) \, \perasU^{\nTx-1}}, \label{eq:p_nc}$$ where the term $\epsilon_1^{n_c-1}$ is the probability that the remaining $n_c-1$ packets are erased. Removing the conditioning on $\NTx$, one can obtain the average radio access throughput as $$\begin{aligned} \mathbb{E}_{N_c}[p_{n_c}] = \sum_{\nTx=0}^{\infty} \frac{\load^\nTx \, e^{-\load}}{\nTx!} \cdot \psU = \load (1-\perasU) \, e^{-\load (1- \perasU)}, \label{eq:truSA}\end{aligned}$$ which corresponds to the throughput of a link with erasures. The overall throughput $R_c$ depends also on the backhaul channel. In particular, for a successful packet transmission, an AP must successfully decode one packet, which should then reach the BS unerased over the backhaul channel. This occurs with probability $$\psD = \psU \,(1-\perasD). \label{eq:q_nc_single_Service}$$ In addition, the packet should not collide with other packets. By virtue of the independence of erasure events, the number of incoming packets on the backhaul during a slot follows the binomial distribution $\mathrm{Bin}(\nRx,\psD)$. Recalling that collisions are regarded as destructive under the collision model, a packet is retrieved only when a single packet reaches the BS, i.e. with probability . The throughput can then be derived as $$R_c = \mathbb{E}_{N_c}[q_c] = \sum_{\nTx=0}^{\infty} \frac{\load^\nTx e^{-\load}}{\nTx!} \cdot \nRx \, \psD (1-\psD)^{\nRx-1}. \label{eq:truSum}$$ This can be computed in closed form as stated in the following proposition. ***Proposition 1:** Under the collision model, assuming $\gamma_c = 1$ (single service), the throughput $R_c$ is given as function of the number of APs $L$, channel erasure probabilities $\epsilon_1$ and $\epsilon_2$, and packet load $G_c$ as* $$R_c = \sum_{\ell=0}^{\nRx -1} (-1)^\ell \, \nRx \, {\nRx-1 \choose \ell} \left[\frac{(1-\perasU) (1-\perasD)}{\perasU}\right]^{\ell+1} e^{-\load} \cdot \ancF_{\ell+1}\!\left(\load \,\perasU^{\ell+1}\right), \label{eq:rate_single_service}$$ *where the auxiliary function $\ancF_m(x)$ is defined recursively as* $$\begin{aligned} \begin{split} \ancF_0(x) &= e^x\\ \mathrm{and}\ \ancF_m(x) & = x \sum_{\ell=0}^{m-1} {m-1 \choose \ell} \ancF_\ell (x) \quad m\geq1 . \end{split} \label{eq:ancFunc}\end{aligned}$$ ***Proof:*** Denoting $\beta = (1-\perasU) (1-\perasD)$, and recalling the definitions of probabilities $\psU$ and $\psD$ in equations and , the throughput can be written as $$\begin{aligned} \begin{split} R_c &= \sum_{\nTx=0}^{\infty} \frac{\load^\nTx e^{-\load}}{\nTx!} \cdot \nRx \, \beta \, \nTx \perasU^{\nTx-1} \left( 1 - \beta \, \nTx \perasU^{\nTx-1}\right)^{\nRx-1}\\ &\stackrel{(a)}{=}\sum_{i=0}^{\nRx-1}(-1)^i \, \nRx {\nRx-1 \choose i} \frac{\beta^{i+1} \, e^{-\load}}{\perasU^{i+1}} \sum_{\nTx=0}^{\infty} \frac{\left(\load \, \perasU^{i+1}\right)^\nTx}{\nTx!} \cdot \nTx^{i+1}, \end{split} \label{eq:truDerivation}\end{aligned}$$ where $(a)$ follows by applying Newton’s binomial expansion and after some simple yet tedious rearrangements. Let us now introduce the auxiliary function $$\ancF_{m}(x) = \sum_{\nTx=0}^{\infty} \frac{x^\nTx \,\nTx^m}{\nTx!}.$$ From the definition of Taylor’s series for the exponential function, we have $\ancF_0(x) = e^x$. Moreover, for $m\geq1$, we have $$\begin{aligned} \ancF_m(x) &= x \sum_{\nTx=0}^{\infty} \frac{x^{\nTx-1} \,\nTx^{m-1}}{(\nTx-1)!} \stackrel{(b)}{=} \,\,x \sum_{t=0}^{\infty} \frac{x^{t} \,(t+1)^{m-1}}{t!} \\ &\stackrel{(c)}{=} x \sum_{\ell=0}^{m-1} {m-1 \choose \ell}\sum_{t=0}^{\infty} \frac{x^{t} \,t^{\ell}}{t!}\\ &= x \sum_{\ell=0}^{m-1} {m-1 \choose \ell}\ancF_\ell(x),\end{aligned}$$ where equality $(b)$ applies the change of variable $t=\nTx-1$ and equality $(c)$ results from applying once more Newton’s binomial expansion to $(t+1)^{m-1}$. Plugging this result into the innermost summation within leads to the closed form expression of the throughput reported in . We now turn to the packet success rate. Define the RV $N^{\prime}_c \geq 1$ to count the number of transmitted messages given that at least one message is transmitted. This RV has the distribution $$P(N^{\prime}_c = n^{\prime}_c | N^{\prime}_c \geq 1) = (1-e^{-G_c})^{-1} \frac{(e^{-G_c}G_c^{n^{\prime}_c})}{(n^{\prime}_c!)} \label{eq:n_primec_distribution}$$ which corresponds to a normalized Poisson distribution over the interval $[1,+\infty]$. For a given value $N^{\prime}_{c} = n^{\prime}_{c}$, the probability that the packet of a given user $u$ reaches an AP given that $u$ is active is given by $p_u = (1-\epsilon_1)\epsilon_1^{n^{\prime}_c-1}$. Furthermore, the probability that the user’s packet reaches the BS is given as $$q_u = \underbrace{L p_u (1-\epsilon_2)}_{(a)} \underbrace{(1-q_{n^{\prime}_{c}})^{L-1}}_{(b)}, \label{eq:q_u_collision_single_class}$$ where $p_{n^{\prime}_{c}}$ and $q_{n^{\prime}_{c}}$ are defined in and . In , term $(a)$ is the probability that the user’s packet is received at the BS from any of the $L$ APs, while $(b)$ denotes the probability that the BS does not receive any message from the remaining $L-1$ APs. The packet success rate $\Gamma_c$, can be obtained by averaging over $N^{\prime}_c$ as $\Gamma_c = \mathbb{E}_{N^{\prime}_c}[q_u]$. This can be obtained in closed form as stated in the following proposition. ***Proposition 2:** Under the collision model, assuming $\gamma_c = 1$ (single service), the packet success rate $\Gamma_c$ is given as function of the number of APs $L$, channel erasure probabilities $\epsilon_1$ and $\epsilon_2$, and packet load $G_c$ as* $$\Gamma_c = L\beta(1-e^{-G_c})^{-1}e^{-G_c}\sum_{l=0}^{L-1} \frac{(-\beta)^l}{\epsilon_{1}^{l+1}} {L-1 \choose l}\Big[\mathbbm{1}_{l=0}(e^{\epsilon_1 G_c}-1) + \mathbbm{1}_{l>0} \mathcal{H}_l (G_c \epsilon_1^{l+1}) \Big], \label{eq:loss_single_service}$$ *where the function $\mathcal{H}_l (\cdot)$ is defined in and we have $\beta = (1-\epsilon_1)(1-\epsilon_2)$*. ***Proof:*** The proof follows using the same steps as for *Proposition 1*. Examples -------- Using expressions derived in *Proposition 1* and *Proposition 2*, in Fig. \[fig:single\_service\] we plot the throughput and packet success rate for a single service as function of the number of time-slots $T$. Increasing $T$ is seen to improve the packet success rate: an active user has a larger chance of successful transmission when more time-slots are available for random access. In contrast, there exist an optimal value of $T$ for the throughput, as the analysis of the standard ALOHA protocol. Increasing $T$ beyond this optimal value reduces the throughput owing to the larger number of idle time-slots. The asymptotic behaviors of packet success rate and throughput can be easily verified theoretically using the expressions in *Proposition 1* and *Proposition 2* by taking their limit when $G_c$ tends to zero. Heterogeneous Services Under Collision Model {#sec:heterogeneous_collision} ============================================ In this section, we extend the analysis in the previous section to derive the throughput and packet success rate of both CS and NCS under the collision model described in Sec. \[sec:system\_model\_performance\_metrics\]. Heterogeneous Services with Ideal NCS-to-CS Interference Tolerance {#sec:het_K_infty} ------------------------------------------------------------------ We start by considering the case in which messages are not affected by messages regardless of their number, i.e., we set $K \to \infty$. Under this assumption, the throughput and packet success rate expressions equals the expressions in Propositions 1 and 2. We hence focus here on the performance of , as summarized in the following proposition. ***Proposition 3:** Under the collision model with ideal -to- interference tolerance, i.e. $K \to \infty$, the throughput $R_{\bar{c}}$ and packet success rate $\Gamma_{\bar{c}}$ can be respectively written as function of the number of APs $L$, channel erasure probabilities $\epsilon_1$ and $\epsilon_2$, and and packet loads $G_c$ and $G_{\bar{c}}$ as* $$\begin{aligned} \begin{split} R_{\bar{c}} = L \sum_{i=0}^{L-1} \sum_{k=0}^{i} (-1)^i {L-1 \choose i}{i \choose k} \Bigg( \frac{\beta}{\epsilon_1} \Bigg)^{i+1} e^{-G} \mathcal{H}_{i-k}(G_c\epsilon_1^{i + 1}) \mathcal{H}_{k+1}(G_{\bar{c}}\epsilon_1^{k+1}) \label{eq:R_barc_no_K_collision} \end{split}\end{aligned}$$ $$\begin{aligned} \mathrm{and}\ \Gamma_{\bar{c}} & = \sum_{l=0}^{L-1} L \beta^{l+1} \epsilon_1^{-(l+1)} (-1)^{-l} (1-e^{-G_{\bar{c}}})^{-1} {L-1 \choose l} \bigg\{ \sum_{m=0}^{l} {l \choose m} A \cdot B \\ & + e^{-G} [\mathbbm{1}_{l=0} (e^{\epsilon_1 G_{\bar{c}}}-1) + \mathbbm{1}_{l>0} \mathcal{H}_{l}(G_{\bar{c}} \epsilon_1^{l+1})]\bigg\} , \label{eq:loss_barc_no_K_collision} \end{aligned}$$ *where* $$\begin{aligned} {1} A = e^{-G_c}(e^{G_c \epsilon_1^{l+1}}-1) \mathbbm{1}_{m=0} + e^{-G_c} \mathcal{H}_{m}(G_c \epsilon_1^{l+1}) \mathbbm{1}_{m > 0} \\ \mathrm{and}\ B = e^{-G_{\bar{c}}} \mathcal{H}_{l-m}(G_{\bar{c}} \epsilon_1^{l-m+1}) \mathbbm{1}_{l-m \neq 0} + e^{-G_{\bar{c}}}(e^{\epsilon_1G_{\bar{c}}} - 1) \mathbbm{1}_{l-m = 0}. \end{aligned}$$ ***Proof:*** The proof is detailed in Appendix A. Examples {#sec:num_results_het_coll} -------- In order to study the performance trade-offs between the two services, we start by investigating the impact of $\gamma_c$ by plotting the and throughputs versus $\gamma_c$ with $\epsilon_1=\epsilon_2=\epsilon$, $G=30\ [\mathrm{packet/frame}]$, $T=4\ [\mathrm{time\text{-}slot/frame}]$, and $L = 3$ APs. For , there is an optimal value of $\gamma_c$ that ensures an optimized load as in the standard analysis of the ALOHA protocol, discussed also in the context of Fig. \[fig:single\_service\]. In contrast, the throughput decreases as function of $\gamma_c$ due to the increasing interference from transmissions. The throughput is also seen to increase as a function of the channel erasure $\epsilon$ when $\epsilon$ is not too large. This is because a larger $\epsilon$, can reduce the interference from $\ac{CS}$ transmissions. Heterogeneous Services with Limited NCS-to-CS Interference Tolerance -------------------------------------------------------------------- We now alleviate the assumption that transmissions can withstand any level of interference by assuming that interference from at most $K$ transmissions can be tolerated without causing a collision from traffic. We derive both and performance metrics. We note that, perhaps counter-intuitively, both and performance metrics are affected by the interference tolerance parameter $K$. In fact, with a lower value of $K$, a smaller number of packets tends to reach the BS, reducing interference to transmissions. We start by detailing the performance metrics. ***Proposition 4:*** *Under the collision model with limited -to- interference tolerance, i.e. finite $K$, the throughput and packet success rate are given as a function of the number of APs $L$, channel erasure probabilities $\epsilon_1$ and $\epsilon_2$, and and packet loads $G_c$ and $G_{\bar{c}}$ as* $$R_{\bar{c}} = L \sum_{i=0}^{L-1} \sum_{k=0}^{i} (-1)^i {L-1 \choose i}{i \choose k} \Bigg( \frac{\beta}{\epsilon_1} \Bigg)^{i+1} e^{-G} \mathcal{H}_{i-k}(G_c\epsilon_1^{i + 1})\Big[ \xi_1(K, L, G_{\bar{c}}, \epsilon_1) + \xi_2(K, L, G_{\bar{c}}, \epsilon_1) \Big] \label{eq:R_barc_with_K_collision}$$ $$\mathrm{and}\ \Gamma_{\bar{c}} = \mathbb{E}_{N_c, N^{\prime}_{\bar{c}}}\bigg[ L (1-\epsilon_1)\epsilon_1^{N_{\bar{c}}^{\prime}-1} \epsilon_1^{N_c} (1-\epsilon_2)(1-q)^{L-1}\bigg], \label{eq:gamma_barc_with_K_collision}$$ *where $q = (1-\epsilon_2)(p_{N_c} \gamma_{K-1}(N_{\bar{c}}^{\prime}, \epsilon_1) + N_{\bar{c}}^{\prime}(1-\epsilon_1)\epsilon_1^{N^{\prime}_{\bar{c}}-1} \epsilon_1^{N_c})$,* $$\begin{aligned} {1} & \xi_1(K, L, G_{\bar{c}}, \epsilon_1) = \sum_{n_{\bar{c}}=0}^{K} \frac{(G_{\bar{c}} \epsilon_1^{k+1})^{n_{\bar{c}}}}{n_{\bar{c}}!} n_{\bar{c}}^{k+1} \\ & \xi_2(K, L, G_{\bar{c}}, \epsilon_1) = \sum_{n_{\bar{c}}=K+1}^{+\infty} \frac{(G_{\bar{c}} \epsilon_1^{k+1})^{n_{\bar{c}}}}{n_{\bar{c}}!} n_{\bar{c}}^{k+1} \bigg[\sum_{l=0}^{K} {n_{\bar{c}} \choose l} (1-\epsilon_1)^{l} \epsilon_1^{n_{\bar{c}}-l}\bigg]^{i-k},\end{aligned}$$ *and the expectation in is taken with respect to independent RVs $N_c$ and $N^{\prime}_{\bar{c}}$, with the latter distributed as in with the index $\bar{c}$ in lieu of $c$.* ***Proof:*** The proof is detailed in Appendix D. We now address the analysis. With finite $K$, a message is correctly received at any AP if it is the only non-erased message and no more than $K$ messages are received erasure-free at the AP. Conditioned on the number of messages $N_c=n_c$ and $N_{\bar{c}} = n_{\bar{c}}$, the probability of the first event is given by $p_{n_c}$ defined in , while the probability of the second event is given by $\gamma_K(n_{\bar{c}}, \epsilon_1)$ with $$\gamma_K(x, \epsilon) = \begin{cases} 1 & \text{if}\quad x \leq K \\ \sum_{i=0}^{K} {x \choose i} \left(1-\epsilon \right)^i \epsilon^{x - i} & \text{otherwise.} \end{cases} \label{eq:gamma_K}$$ Removing the conditioning on $N_{\bar{c}}$, the probability of the second event can be written as $$\begin{aligned} \begin{split} \sum_{\nTxNC=0}^{K} \frac{\loadNC^\nTxNC \, e^{-\loadNC}}{\nTxNC!}+ \sum_{\nTxNC=K+1}^{\infty} \frac{\loadNC^\nTxNC \, e^{-\loadNC}}{\nTxNC!}\left[ \sum_{i=0}^{K} {\nTxNC \choose i} \left(1-\perasU \right)^i \perasU^{\nTxNC - i}\right]\\ &= Q(K+1,\loadNC)+\xi(K,\loadNC), \end{split} \label{eq:gamma_K_no_conditioning}\end{aligned}$$ where the first term in is the regularized gamma function and $\xi(K,\loadNC)$ represents the second term. For the performance metrics we distinguish the following two cases depending on the number $L$ of APs. ### Small number of APs ($\nRx\leq K+1$) In this case, the effect of finite interference tolerance $K$ affects only the radio access transmission phase. In fact, in the backhaul transmission phase, if $\nRx\leq K+1$, the number of interfering transmissions on a packet at the BS cannot exceed $K$. ***Proposition 5:** Under the collision model, the throughput and packet success rate given as a function of the number of APs $\nRx\leq K+1$, channel erasure probabilities $\epsilon_1$ and $\epsilon_2$ and and packet loads $G_c$ and $G_{\bar{c}}$ as* $$\begin{aligned} \begin{split} R_c &= \sum_{\ell=0}^{\nRx -1} (-1)^\ell \, \nRx \, {\nRx-1 \choose \ell} \left[\frac{ (1-\perasU) (1-\perasD)}{\perasU}\right]^{\ell+1} e^{-\load} \,\,\ancF_{\ell+1}\left(\load \,\perasU^{\ell+1}\right)\\ &\cdot \left[\mathsf{Q}(K+1,\loadNC)+\xi(K,\loadNC,\ell)\right] \end{split} \label{eq:truDKfinite}\end{aligned}$$ $$\begin{aligned} \mathrm{and}\ \ \Gamma_c = \mathbb{E}_{N^{\prime}_c, N_{\bar{c}}}[L p_u (1-\epsilon_2)(1-q_{N^{\prime}_{c}})^{L-1}], \label{eq:reliabilityKfinite}\end{aligned}$$ *where $\xi(K,\loadNC,\ell)=\sum_{\nTxNC=K+1}^{\infty} \frac{\loadNC^\nTxNC \, e^{-\loadNC}}{\nTxNC!}\left[ \sum_{i=0}^{K} {\nTxNC \choose i} \left(1-\perasU \right)^i \perasU^{\nTxNC - i}\right]^{\ell+1}$ and $q_{n^{\prime}_c} = n^{\prime}_c (1-\epsilon_1) \epsilon_1^{n^{\prime}_{c}-1} $\ $ \gamma_K(n_{\bar{c}}, \epsilon_1) (1-\epsilon_2)$ is the probability of receiving any packet at the BS* *and the expectation in is taken with respect to independent RVs $N_{\bar{c}}$ and $N^{\prime}_c$, with the latter distributed as in* .\ ***Proof:*** The proof is provided in Appendix E.\ Comparing the throughput in with the expression for $K\rightarrow \infty$ we observe that the effect of a finite interference tolerance is measured by the multiplicative term ${\left[Q(K+1,\loadNC)+\xi(K,\loadNC,\ell)\right]}$. It can be shown that this term is always smaller than one, which is in line with the fact that a lower throughput is expected when $K$ is finite. ### Large Number of APs ($\nRx> K+1$) In this case, a successful transmission occurs in all events where a single packet and only up to $K<\nRx$ packets reach the BS. The total probability of these events given $N_c = n_c$ and $N_{\bar{c}} = n_{\bar{c}}$ can be computed as $$q_c= \sum_{\ell=0}^{K} {\nRx \choose {1,\ell,\nRx-\ell-1}}q^{\prime}_{n_c} (q_{n_{\bar{c}}})^\ell(1-q^{\prime}_{n_c}-q_{n_{\bar{c}}})^{\nRx-\ell-1}, \label{eq:truDKfiniteSum_2}$$ where $q^{\prime}_{n_c}=q_{n_{\bar{c}}}\gamma_K(n_{\bar{c}}, \epsilon_1)\,(1-\perasD)$ is the probability that a packet reaches the BS and $q_{n_{\bar{c}}}$ is the probability that a packet reaches the BS (may also be not correctly received due to a collision). Removing the conditioning on $\NTx=\nTx$ and $\NTxNC=\nTxNC$, we get the throughput as $$\begin{aligned} \begin{split} R_c &= \mathbb{E}_{\NTx, \NTxNC}\left[q_c\right], \end{split} \label{eq:truDKfiniteLGK}\end{aligned}$$ where the expectation is taken with respect to independent RVs $N_{\bar{c}}$ and $N_c$. Moving to the packet success rate, conditioned on $N_c = n_c$ and $N_{\bar{c}} = n_{\bar{c}}$, the probability $p_u$ of receiving a packet at an AP from a given user $u$ is given as in $p_u = (1-\epsilon_1)\epsilon_1^{n^\prime_c-1} \gamma_K(n_{\bar{c}}, \epsilon_1)$. The probability of receiving successfully a packet at the BS is then given as $$q_c = L p_u (1-\epsilon_2)(1-q_{n^{\prime}_c})^{L-1} \underbrace{\sum_{i=0}^{K} {L-1 \choose i } p_{\bar{c}}^{i} (1-p_{\bar{c}})^{L-1-i}}_{(a)} \label{eq:q_c_2}$$ where $q_{n^{\prime}_c} = p_{n^{\prime}_c} \gamma_K(n_{\bar{c}}, \epsilon_1) (1-\epsilon_2)$ and $p_{\bar{c}} = n_{\bar{c}}(1-\epsilon_1)\epsilon_1^{n^{\prime}_c}\epsilon_1^{n_{\bar{c}}-1} (1-\epsilon_2)$. The main difference between and the probability inside the expectation in is the multiplication by the term $(a)$ in which corresponds to the probability that a number of packets lower or equal to $K$ should be received in order to be able to recover a packet. Removing the conditioning on $\NTx=\nTx$ and $\NTxNC=\nTxNC$, we obtain the packet success rate as $$\begin{aligned} \begin{split} \Gamma_c &= \mathbb{E}_{\NTx^{\prime}, \NTxNC}\left[q_c\right], \end{split} \label{eq:lossDKfiniteLGK}\end{aligned}$$ where the expectation is taken with respect to independent RVs $N^{\prime}_{\bar{c}}$ and $N^{\prime}_c$ with the latter distributed as in . Examples {#examples-1} -------- In order to capture the effect of the number of messages $K$ on the CS, in Fig. \[fig:rate\_region\_K\] we plot the throughput region for $K=2$ and $K \to \infty$, with the latter case corresponding to the analysis in Sec. \[sec:het\_K\_infty\]. The region includes all throughput pairs that are achievable for some value of the fraction $\gamma_c$ of messages, as well as all throughput pairs that are dominated by an achievable throughput pair (i.e., for which both and throughputs are smaller than for an achievable pair). For reference, we also plot the throughput region for a conventional inter-service TDMA protocol, whereby a fraction $\alpha T$ for $\alpha \in [0,1]$ of the $T$ time-slots is allocated for messages and the remaining time-slots to messages. For TDMA, the throughput region includes all throughput pairs that are achievable for some value of $\alpha$, as well as of $\gamma_c$. A first observation from the figure is that non-orthogonal resource allocation can accommodate a significant throughput without affecting the throughput, while TDMA causes a reduction in the throughput for any increase in the throughput. This is due to the need in TDMA to allocate orthogonal time resources to messages in order to increase the corresponding throughput. However, with non-orthogonal resource allocation, the maximum throughput is generally penalized by the interference caused by the collisions from messages, while this is not the case for TDMA. In summary, TDMA is preferable when one wishes to guarantee a large throughput and the throughput requirements are loose; otherwise, non-orthogonal resource allocation outperforms TDMA in terms of throughput. Furthermore, the throughput region is generally decreased by lower value of $K$. Experiments concerning packet success rate and performance as function of the number of APs will be presented in the superposition model in the following section. Heterogeneous Services Under Superposition Model {#sec:heterogeneous_superposition} ================================================ In this section, we consider the superposition model described in Sec. \[sec:system\_model\_performance\_metrics\]. Performance Analysis -------------------- Unlike the collision model, in order to analyze the throughput and packet success rate under the superposition model, one needs to keep track of the index of the messages decoded by the APs. This is necessary to detect when multiple versions of the same message (i.e., sent by the same device) are received at the BS. Accordingly, we start by defining the RVs $B_i$ to denote the index of the message received at AP $i$ and RV $B$ for the BS at any time-slot. Accordingly, for given values $N_c=n_c$ and $N_{\bar{c}}=n_{\bar{c}}$ of transmitted messages, RVs $\{B_i\}$ can take values $$B_i = \begin{cases} 0 & \text{if no message is retrieved due to erasures or collisions}\\ 1 \leq m \leq n_c & \text{if the $m$-th \ac{CS} message is retrieved} \\ n_c + 1 \leq m \leq n_c + n_{\bar{c}} & \text{if the $(m-n_c)$-th \ac{NCS} message is retrieved.} \label{eq:Bi_superposition} \end{cases}$$ Note that we have indexed messages from $1$ to $n_c$ and messages from $n_c + 1$ to $n_c + n_{\bar{c}}$. As for the RV $B$ at the BS, it is defined as $$B = \begin{cases} c & \text{if a \ac{CS} message is retrieved} \\ \bar{c} & \text{if a \ac{NCS} message is retrieved} \\ 0 & \text{if no message is retrieved due to erasures or collisions.} \end{cases} \label{eq:B_collision}$$ Furthermore, we define as $M_m = \sum_{i=1}^{L} \mathbbm{1}_{\{B_i=m\}}$ the RVs denoting the number of APs that have message of index $m \in \{0,1, \ldots, n_c,n_c+1, \ldots , n_c + n_{\bar{c}} \}$. The joint distribution of RVs $\{M_m \}_{m=0}^{n_c+n_{\bar{c}}}$ given $N_c$ and $N_{\bar{c}}$ is multinomial and can be written as follows $$\{M_m \}_{m=0}^{n_c+n_{\bar{c}}}|N_c,N_{\bar{c}} \sim \operatorname{Multinomial}\Big(L ,\overbrace{1-p_{n_c}-p_{n_{\bar{c}}}}^{0},\overbrace{\frac{p_{n_c}}{n_c}, \ldots,\frac{p_{n_c}}{n_c}}^{n_c}, \overbrace{\frac{p_{n_{\bar{c}}}}{n_{\bar{c}}},\ldots, \frac{p_{n_{\bar{c}}}}{n_{\bar{c}}}}^{n_{\bar{c}}}\Big), \label{eq:multinomial}$$ where we used the the probabilities in and that one of the or message is received at an AP respectively in a given time-slot. The probability of retrieving a message in a given time-slot at the BS conditioned on $N_c$, $N_{\bar{c}}$ and $\{M_{m\prime} \}_{m\prime=0}^{n_c + n_{\bar{c}}}$ can be then written as $$\begin{aligned} q_c=\mathrm{Pr}[B=c | N_c=n_c , N_{\bar{c}}=n_{\bar{c}}, \{M_{m^\prime} \}_{m^\prime=0}^{n_c + n_{\bar{c}}}] = \gamma_K& \bigg(\sum_{m^\prime=n_c+1}^{n_{\bar{c}}+n_c} M_{m^{\prime}} , \epsilon_2 \bigg) \times \\ &\sum_{m=1}^{n_c} \sum_{j=1}^{M_m} {M_m \choose j} (1-\epsilon_2)^j \epsilon_2^{\sum_{ \substack{m^\prime = 0 \\ m^\prime \neq m}}^{n_c} M_{m^\prime} + M_m - j}, \label{eq:proba_c_superposition} \end{aligned}$$ where the first sum is over all possible messages, the second sum is over all combinations of APs that have the message $m$, and the third sum at the exponent is over all APs that have a message $m^{\prime} \neq m$. The throughput can be computed by averaging over all conditioning variables as $$R_c = \mathbb{E}_{N_c , N_{\bar{c}} , \{M_m\}_{m=0}^{N_c+N_{\bar{c}}}} [q_c]. \label{eq:R_c_sup}$$ In a similar manner, the conditional probability of receiving a message at the BS can be written as $$\begin{aligned} q_{\bar{c}}=\mathrm{Pr}[B = \bar{c} | & N_c=n_c , N_{\bar{c}}=n_{\bar{c}} , \{M_{m^\prime} \}_{m^\prime = 0}^{n_c + n_{\bar{c}}} ] \\ &= \sum_{m=n_c + 1}^{n_c + n_{\bar{c}}} \sum_{j=1}^{M_m} {M_m \choose j} (1-\epsilon_2)^j \epsilon_2^{\sum_{ \substack{m^{\prime\prime}=n_c + 1 \\ m^{\prime\prime} \neq m } }^{n_c + n_{\bar{c}}} M_{m^{\prime \prime}} + M_m - j + \sum_{m^\prime = 1}^{n_c} M_{m^\prime}}, \end{aligned} \label{eq:proba_cbar_one}$$ where the first sum is over all possible messages $m$; the second sum is over all possible combinations of APs that have message $m$; and the third and fourth sums at the exponent are over all APs that have a different message and a message respectively. The throughput can be then obtained by averaging over the conditioning RVs as $$R_{\bar{c}} = \mathbb{E}_{N_c, N_{\bar{c}}, \{ M_m \}_{m= 0}^{N_c + N_{\bar{c}}} } [q_{\bar{c}}]. \label{eq:R_barc_sup}$$ The packet success rate under the superposition model for and can be obtained by fixing $m$ to one and substituting $n_c$ and $n_{\bar{c}}$ by $n_c^{\prime}$ and $n_{\bar{c}}^{\prime}$ in and . Examples {#examples-2} -------- In Fig. \[fig:rate\_region\_K\], we plot the throughput region for non-orthogonal resource allocation and inter-service TDMA under the superposition model. Comparing the regions of the collision model and the superposition model, it is clear that the latter provides a larger throughput region being able to leverage transmissions of the same packets from multiple APs as compared to the collision model. This can also be seen as function of $K$ in Fig. \[fig:rate\_region\_K\]. In Fig. \[fig:rate\_L\], we explore the effect of the number of APs $L$ on the and throughputs. To capture separately the effects of the radio access and the backhaul channel erasures, we consider different values for the channel erasure probabilities $\epsilon_1$ and $\epsilon_2$. We highlight two different regimes: the first is when $\epsilon_1$ is large and $\epsilon_2$ is small, and hence larger erasures occur on the access channel; while the second covers the complementary case where $\epsilon_1$ is small and $\epsilon_2$ is large. In the first regime, increasing the number of APs is initially beneficial to both and messages in order to provide additional spatial diversity for the radio access, given the large value of $\epsilon_1$; but larger values of $L$ eventually increase the probability of collisions at the BS on the backhaul due to the low value of $\epsilon_2$. In the second regime, when $\epsilon_1=0.1$ and $\epsilon_2=0.8$ much lower throughputs are generally obtained due to the significant losses on the backhaul channel. This can be mitigated by increasing the number of APs, which increases the probability of receiving a packet at the BS. [.5]{} [.5]{} Finally, we consider the interplay between the throughputs and packet success rate levels for both non-orthogonal resource allocation and TDMA as function of the number of time-slots $T$. These are plotted in Fig. \[fig:throughput\_reliability\] for $G=15\ [\mathrm{packet/frame}]$, $\epsilon_1 = \epsilon_2 = 0.5, L=3$ APs, $\alpha=0.5$ and $\gamma_c = 0.5$. For both services, following the discussion around Fig. \[fig:single\_service\] we observe that the packet success rate level under both allocation schemes increases as function of $T$. This is because larger value of $T$ decrease chances of packet collisions. However, this not the case for the throughput, since large values of $T$ may cause some time-slots to be left unused, which penalizes the throughput. For the in Fig. \[fig:throughput\_reliability\_critical\_messages\], it is seen that non-orthogonal resource allocation outperforms TDMA in both throughput and packet success rate level due to the larger number of available resources. In contrast, Fig. \[fig:throughput\_reliability\_Noncritical \_messages\] shows that TDMA provides better throughput and packet success rate level than non-orthogonal resource allocation. The main reason for this is that the lower number of resources in TDMA is compensated by the absence of inter-service interference for messages. Throughput and packet success rate Analysis Under Fading Channels {#sec:throughput_reliability_fading} ================================================================= The binary erasure channel model discussed in the previous sections offers a tractable set-up that facilitates the analysis of the throughput and packet success rate, enabling the derivation of closed-form expressions in various cases of interest. It is also of practical interest as a simplified model for mmwave channels [@mmwave_erasurechannels] and non-terrestrial communications scenarios represented in Fig. \[fig:system\_model\]. In this section, we briefly study a more common scenario that accounts for fading channels in both radio and backhaul channels. This typically represents terrestrial scenarios as shown in Fig. \[fig:system\_model\_earth\]. More complex models that include both fading and erasures [@fading_and_erasure] can also be analyzed following the same steps presented below (see Section \[sec:conclusions\] for some details). We first detail the channel and signal models, and then we derive the throughput and packet success rate metrics. Channel and Signal Models {#sec:fading_model} ------------------------- At any time-slot $t$, the channels between each user $m$ and AP $l$ and between each AP $l$ and the BS are assumed to follow the standard Rayleigh fading model, and are denoted as $h_{l,m}(t) \sim \mathcal{CN}(0,\alpha^{2})$ and $g_{l}(t) \sim \mathcal{CN}(0,\beta^{2})$, respectively. Ensuring consistency with the erasure model, we assume that all channels are independent and that the average channel gains $\alpha^2$ and $\beta^2$ are fixed. Furthermore, as detailed below, we assume that each AP and the BS decode at most one packet in each slot. Finally, we denote the transmission rates of and messages as $r_c$ and $r_{\bar{c}}\ \mathrm{bit/s/Hz}$, respectively. Assuming that both access and backhaul channels are allocated the same amount of radio resources, the transmission rates are the same for both channels. Given the numbers $N_c(t) = n_c$ and $N_{\bar{c}} (t)= n_{\bar{c}}$ of and messages in the given time-slot $t$, the signal received at the $l$-th AP as time-slot $t$ can be written as $$y_{l}(t) = \sum_{m=1}^{n_c} h_{lm}(t)x_m (t) + \sum_{m^{\prime}=n_c+1}^{n_c + n_{\bar{c}}} h_{lm^{\prime}}(t)x_{m^{\prime}} (t) + n_l (t),$$ where $n_l (t) \sim \mathcal{CN}(0,1)$ denotes complex white Gaussian noise at the $l$-th AP. The powers of and devices are respectively denoted as $$\mathbb{E}[|x_{m}(t)|^{2} ]= P_c \label{eq:power_constraint}\ \mathrm{and}\ \mathbb{E}[|x_{m^{\prime}}(t)|^{2}] = P_{\bar{c}},$$ where we take $P_c \geq P_{\bar{c}}$ to capture the generally larger transmission power of transmissions. The Signal-to-Interference-plus-Noise Ratio (SINR) of message $m$ at AP $l$ is given as $$\mathrm{SINR}_{l,m}^{AP} = \frac{|h_{lm}(t)|^2 P_m}{1+ \sum_{\substack{m^\prime = 1 \\ m^{\prime} \neq m}}^{n_c + n_{\bar{c}}} |h_{lm^{\prime}}(t)|^2 P_{m^\prime}} \label{eq:SINR_l},$$ where $P_m = P_c$ for a message $m \in \{1,\ldots,n_c \}$ and $P_m = P_{\bar{c}}$ for a non message $m \in \{n_c + 1 , \ldots , n_c + n_{\bar{c}} \}$. Let $m_{l}^{\star}$ denote the message with the largest SINR at the $l$-th BS, i.e., $$m_{l}^{\star} = \operatorname*{argmax}_{m \in \{1,\ldots , n_c + n_{\bar{c}} \}} \mathrm{SINR}_{l,m}^{AP}. \label{eq:m_lstar}$$ The $l$-th AP only attempts to decode message $m_{l}^{\star}$. Decoding is correct if the standard Shannon capacity condition $\mathrm{SINR}_{l,m^{\star}_{l}} \geq 2^{r_{m}} -1$ is satisfied, where $r_m = r_c$ if $m_{l}^{\star} \in \{1,\ldots,n_c \}$ and $r_m = r_{\bar{c}}$ if $m_{l}^{\star} \in \{n_c,\ldots,n_c + n_{\bar{c}} \}$. Each $l$-th AP transmits the decoded message $m_l^{\star}$, if any, to the BS over the wireless backhaul channel with transmission power $P_{m^{\star}_l}^{AP} = P_c^{AP}$ if $m^{\star}_{l} \in \{1,\ldots,n_c \}$ and $P_{m^{\star}_l}^{AP} = P_{\bar{c}}^{AP}$ if $m_l^{\star} \in \{n_c+1, \ldots, n_c + n_{\bar{c}}\}$. Consequently, the signal $y_{BS}(t+1)$ received at the BS in time-slot $t+1$ can be written as the sum of messages sent by all APs as $$y_{BS}(t+1) = \sum_{l=1}^{L} g_l (t+1) x_{m_l^{\star}} (t) + n_{BS}(t+1),$$ where $n_{BS}(t) \sim \mathcal{CN}(0,1)$ denotes the white Gaussian noise at the BS. Let $\mathcal{L}_{m} = \{l : m_l^{\star} = m \}$ denote the set of indices of APs that decoded a message $m \in \mathcal{M}^{\star}$, where $\mathcal{M}^{\star}= \{ m : \exists\ l=1,\ldots,L\ \mathrm{s.t.}\ m=m^{\star}_l\}$ denotes the set of messages decoded by at least one AP in time-slot $t$. The SINR of a message $m\in \mathcal{M}^{\star}$ received at the BS can be written as $$\mathrm{SINR}_{m}^{BS} = \frac{|\sum_{l \in \mathcal{L}_{m}} g_{l}(t+1)|^2P^{AP}_{m} }{1 + \sum_{m^{\prime} \in \mathcal{M}^{\star} \setminus \{ m\}} |\sum_{l \in \mathcal{L}_{m^\prime}} g_l(t+1)|^2 P^{AP}_{m^\prime}}. \label{eq:SINR_BS}$$ In a manner similar to APs, the BS attempts decoding only of the message $m^{\star}_{BS}$ with the highest SINR, namely $$m_{BS}^{\star} = \operatorname*{argmax}_{m \in \mathcal{M}^{\star}} \mathrm{SINR}_{m}^{BS}.$$ Message $m^{\star}_{BS}$ is decoded correctly if the standard Shannon capacity condition $\mathrm{SINR}_{m^{\star}_{BS}} \geq 2^{r_{m^{\star}_{BS}}} -1$ is satisfied, where $r_{m^\star_{BS}} = r_c$ if $m^{\star}_{BS} \in \{1,\ldots,n_c \}$ and $r_{m^{\star}_{BS}} = r_{\bar{c}}$ if $m^{\star}_{BS} \in \{n_c + 1,\ldots,n_c + n_{\bar{c}} \}$. Performance Analysis -------------------- The analysis follows the same steps as in Section \[sec:heterogeneous\_superposition\], as long as one properly redefines the probabilities $p_c$ and $p_{\bar{c}}$ of decoding correctly a or a message at any given AP, as well as the probabilities $q_c$ and $q_{\bar{c}}$ of decoding correctly a or message at the BS. According to the discussion in Section \[sec:fading\_model\], the former probabilities can be respectively written as $$\begin{aligned} {1} & p_c = \mathrm{Pr}[m_l^{\star} \in \{1,\ldots,n_c \}\ \mathrm{and}\ \mathrm{SINR}^{AP}_{l,m^{\star}_l} > 2^{r_c}-1]\\ & \mathrm{and}\ p_{\bar{c}} = \mathrm{Pr}[m_l^{\star} \in \{n_c + 1,\ldots,n_c + n_{\bar{c}} \}\ \mathrm{and}\ \mathrm{SINR}^{AP}_{l,m^{\star}_l} > 2^{r_{\bar{c}}}-1], \end{aligned}$$ \[eq:p\_fading\] where $m_l^{\star}$ is defined in , while the latter probabilities can be redefined as $$\begin{aligned} {1} & q_c = \mathrm{Pr}[m_{BS}^{\star} \in \{1,\ldots,n_c \}\ \mathrm{and}\ \mathrm{SINR}^{BS}_{m^{\star}_{BS}} > 2^{r_c}-1]\\ & \mathrm{and}\ q_{\bar{c}} = \mathrm{Pr}[m_{BS}^{\star} \in \{n_c + 1,\ldots,n_c + n_{\bar{c}} \}\ \mathrm{and}\ \mathrm{SINR}^{BS}_{m^{\star}_{BS}} > 2^{r_{\bar{c}}}-1]. \end{aligned}$$ \[eq:q\_fading\] While closed-form expressions for and appear prohibitive to derive (see, e.g., [@outage_computation]), these probabilities can be easily estimated via Monte Carlo Simulations. Having computed probabilities -, the throughputs of and messages can be respectively obtained using and . The packet success rate can be computed by redefining $q_{c}$ and $q_{\bar{c}}$ to take into account a single message sent by a single user (for instance, the first one) as follows: $$\begin{aligned} {1} & q_c = \mathrm{Pr}[m_{BS}^{\star} = 1\ \mathrm{and}\ \mathrm{SINR}^{BS}_{m^{\star}_{BS}} > 2^{r_c}-1| N_c(t) \geq 1]\\ & \mathrm{and}\ q_{\bar{c}} = \mathrm{Pr}[m_{BS}^{\star}=1\ \mathrm{and}\ \mathrm{SINR}^{BS}_{m^{\star}_{BS}} > 2^{r_{\bar{c}}}-1|N_{\bar{c}}(t) \geq 1]. \end{aligned}$$ \[eq:reliability\_fading\] Examples {#examples-3} -------- We now consider the fading channels model discussed in Sec. \[sec:throughput\_reliability\_fading\] with the main aim of relating the insights obtained from the analysis of erasure channels to the more common Rayleigh fading setup. We fix $P_c = P^{AP}_c = 10$ and $T = 4\ [\mathrm{time\text{-}slot/frame}]$. In an analogy to Fig. \[fig:rate\_load\_1\], we start in Fig. \[fig:throuput\_gammac\_fading\] by investigating the throughput of and messages as function of the fraction of messages $\gamma_c$ for different values of the average channel powers $\alpha^2$ and $\beta^2$. In general, we observe similar trends as in Fig. \[fig:rate\_load\_1\]. Most notably, the throughput of messages peaks at a value of $\gamma_c$ that strikes the best balance between the combining gains due to the transmission of a message from multiple APs and the interference created by concurrent AP transmissions of different message. However, in contrast to the erasure model in which increasing the erasure rate can be advantageous, the throughput of both services improves as the average channel strengths $\alpha^2$ and $\beta^2$ are increased. This is because interference from concurrent transmissions has a more deleterious effect under the collision model assumed when considering erasures than under the SINR model. For the latter, reducing both channel strengths $\alpha^2$ and $\beta^2$ has the net effect of reducing the SINRs despite the decrease in interference power. Finally, to compare some of the design insights from the erasure model, we plot in Fig. \[fig:rate\_L\_fading\] the throughput of both services as function of the number of APs $L$. We can see that, in a manner similar to Fig.  \[fig:rate\_L\], when $\alpha^2$ is high and $\beta^2$ is low, which is akin to lower $\epsilon_1$ and high $\epsilon_2$ in Fig. \[fig:rate\_L\], the throughput of both services increases as function of $L$. This is because low values of the channel power $\beta^2$ in the backhaul channel imply that the SINR is limited by the signal power and not by interference. Therefore, increasing the space diversity via a larger $L$ can be advantageous in this regime. Furthermore, as evinced from Fig. \[fig:rate\_L\], the SINR of the backhaul channel may be limited by the level of interference and hence when $\alpha^2$ is low and $\beta^2$ is high, which is akin to high $\epsilon_1$ and low $\epsilon_2$ in Fig. \[fig:rate\_L\], increasing the number of APs beyond a given threshold reduces the throughput. Conclusions and Extensions {#sec:conclusions} ========================== This paper studies grant-free random access for coexisting and in IoT systems with shared wireless backhaul and uncoordinated APs whereby messages are transmitted with a larger power. Non-orthogonal and orthogonal inter-service resource sharing schemes based on random access are considered. From the perspective, it was found that non-orthogonal sharing is preferable to a standard inter-service TDMA protocol in terms of both throughput and packet success rate level. In contrast, this is not the case for the , since inter-service orthogonal resource allocation eliminates interference from the larger-power . The analysis is carried out under two models that assume destructive or constructive superposition of the same packet sent by multiple transmitters. Furthermore, both erasure and fading channels models are considered to capture non-terrestrial and terrestrial applications respectively. Similarities were found between these models which proves the suitability of the erasure model for such type of analysis due to its mathematical tractability properties. Through extensive numerical results, the impact of both spatial and time resources is investigated, revealing trade-offs between throughput and packet success rate for both services. Some possible extensions follow. First, it would be interesting to consider a system where both terrestrial and non-terrestrial relays coexist. In this case, several architectures could be considered and compared. For instance, the first one could correspond to the case where the is served by terrestrial access points due to low latency requirements and by satellites. Another architecture could be a hybrid architecture where both services are processed by both types of relays. Second, we assumed in this work that the APs always forward on the backhaul correctly received packets on the radio access. This might not necessarily be the case. For instance, one could assume that APs probabilistically forward received packets. An interesting direction could be the derivation of the optimal forwarding probability for each type of service. Finally, it would interesting to consider the impact of coordination among APs through the exchange of messages over capacity-limited channels. [^1]: The work of Rahif Kassab and Osvaldo Simeone has received funding from the European Research Council (ERC) under the European Union Horizon 2020 research and innovation program (grant agreement 725731).

--- abstract: | Based on the universal principle of the causality, the realization of the relativity of physics laws is discussed in this paper. First, we propose a kind of new relativity, that is the gravitational interaction survey actually may for the observation of any local region is relative. Moreover, the equivalence of such a relativity with the kinematic quantity further lead to the same mathematical form of gravity theory as Einstein’s field equation. Secondly, inertial reference frames can be naturally established resorting to “the minimal-gravity-free observer”, which is defined to apply the “law of inertia” locally. Therefore, dynamical equations would be form-invariant only to such inertial reference frames at different levels. After that, the correctness of the application of the dynamical law to any practical case has been guaranteed by use of the method of inertial reference frame.\ PACS number(s): 98.80.-k, 95.10.-a author: - 'ChiYi Chen${}^{a,b}$' - 'Kang Li${}^{c}$' title: On the Realization of the Dynamics Relativity --- INTRODUCTION ============ No matter what dynamics law is, the principle of the causality is always its soul and usually carried out by the method of inertial reference frame. For example, in classical mechanism, the dynamics equation must be established on the inertial reference frame, so to provides a guarantee for the consistence of the casualty. In the same reason, the geometry equivalence of the gravity must also be established on the background of a inertial reference frame because this is just the requirement of the principle of the causality. However, this point is not clarified in Einstein’s general relativity. In addition, the dynamics law can satisfy the principle of causality once it was adjusted by making a comparison with the experiment. Therefore, The most important property in the definition of “ inertial reference frame ” would be given by the dynamics relativity. In this paper we aim at finding a kind of reference frames, which exist universally and in all of them the causalities are unified consistently, so has provided the guarantee for the same form of the dynamics law being applied in such a kind of reference frames, then in fact this kind of reference frames are qualified to be defined as the physical “inertial reference frame”. THE SYMMETRY OF THE RELATIVITY ================================ In realistic experiences, the observation of a physical phenomenon is always localized and we must choose the different observers in different local areas. On the other hand, a most important property to a physical principle is its universality applicable. Therefore this has directly caused our physical law should be invariant for different observer, this is the so-called kinematic relativity. Therefore, as the actual carrier of any dynamical principle, the dynamical equations should keep the same mathematical form in any local area in the universe. In other words, the dynamical equations have been automatically required to be universal and its form do not depended on at least one kind of observers who stay in any local area of the universe. In theoretically, to realize this point, a natural idea is that the executed force is also relative to observers and the equivalence on the relativity of both sides of the dynamical equations should be maintained, then it is possible for the form of the dynamical equations do not change with the changes of observers. In short, to keep the symmetry of relativity is a nature way which can keep the the mathematical form of the dynamical equations invariant. Without doubt, the force executed on any particle is absolute in nature, as well as the motion of any particle is absolute in nature. But in fact the observation is always local, and our description of the motion of particles must be dependent on the observers. So the kinematical quantities of particles are just relative, this is why dynamical equations have relativity. Along this way, to realize the relativity of the dynamical equations on different observers, the counting of the executed force on the particle must also be capable being treated as an equally relative quantity to some special kinds of observers. In fact, we have never actually been able to count the absolute force for even the most special particle in the universe. THE SYMMETRY OF THE RELATIVITY AND THE GRAVITY THEORY ===================================================== It naturally brings a problem, to what kind of observers, then the kinematical quantities and the observed force in a dynamical equation can be naturally respectively defined on the equivalent(observer) relativity? To transformation between relatively accelerated observers, we have previously pointed out that if the form-invariant dynamical equations are available to some special kinds of observers, then it would be required that the influence brought by the transformation between relatively accelerated observers must be equivalently erased from the whole dynamical equation by some kinds of interaction which are executed on both the particle and the observer. Therefore, the problem that we are faced with has now turned to finding the qualified interaction which is equivalent to the kinematics of the observer. In the requirement of the equivalence of the relativity, if such a kind of interaction exists, then it must possess the following two properties: 1, The relativity induced by a relative acceleration of the observer is determined by the state of the observer, and is irrespective to the selection of the particle in the local region. That is to say, the kinematical effect induced by this kind of interaction is irrespective to the special properties of the particle. Hence it must be a kind of universal interaction at least on the local region. 2, The selection of the observer is irrespective to the distance between the observer and the particle. Hence the kinematical effect induced by this kind of interaction must be capable being counteracted in the dynamical equation by the relative acceleration for observers’ change on the large scale of the universe. Therefore, it must further be a kind of long-range interaction in the universe. By far, four kinds of interactions discovered in the physical world are concerned, only the gravitational interaction is qualified as the candidate. Furthermore, considering that the inertial mass always exactly equals to the gravitational mass, we are not able to distinguish between what part of the motion of the particle is native, and what part of the motion of the particle is induced by the gravitational interaction from the distant place of the universe. On one hand, this is just the direct reason for our investigations on particle dynamics, in reality, are always localized. On the another hand, it indicates that the gravitational interaction on the cosmological scale has actually been throughly geometrized into the global acceleration of the subsystems in the universe. For these reasons, the equivalence on the respective relativity of the kinematical quantities and the observed (or counted) gravity must further lead to the geometrization description of the gravity. In other words, as long as the equivalence between the gravitational mass and the inertial mass is presented, the geometry description must be available for the gravitational interaction. Further considering the mass-energy relation in Special Relativity and the deflection of light in gravity field, then it has been enough for us to recover the mathematical form of Einstein’s gravity theory[@kolb; @weinberg]: $$\begin{aligned} G_{\mu\nu}=-\kappa<T_{\mu\nu}>_{grav}.\end{aligned}$$ Viewed from the opposite direction, the great success of the geometrized theory of gravity has given eloquent proof of such an equivalency between the dynamical property of the gravitational interaction and the geometric language, and it also provides the guarantee for the essential equivalence of the discussed two kinds of relativity in the dynamical equations. THE SYMMETRY OF THE RELATIVITY AND THE INERTIAL REFERENCE FRAMES ================================================================ Now we turn back to the problem of finding the observers who are able to establish the dynamical equations on the equivalent relativity. As far as the gravitational interaction is concerned now, the only feasible way is that, the kinematical effect induced by “the global gravity” to a whole system must be equivalent to the global acceleration of the whole system. Then the uncertainty brought by the relativity of the kinematical quantities is capable to be erased from the whole dynamical equation. The qualified observer to above requirements are those that hold both “the global gravity” and the global acceleration of the system, we may name it as “the minimal-gravity-free observer”. According to above discussion on the realization of the dynamical relativity, then to an arbitrary particle, “the minimal-gravity-free observer” may be defined to satisfy, at the same time, the following two conditions: 1, It must be the observer who is executed only by gravitational interaction and being gravity-free in the universe. 2, the gravitational field around the observer must be completely contained within the area around the particle considered. Besides, the word “minimal” just means, to the whole system considered, such an observer is the most fundamental gravity-free one, since in the inner region of the system where may still exist the gravity-free phenomenon at another level. As an illustration, once an object as the test particle is specified, we firstly should chose a local system including this particle according to the scope of the interaction which should be counted in. Secondly, we have to find out the observer who are in the lowest-level gravity-free in this system (which would be rest on the mass center of the system if the system is in global gravity-free). Once we have find “the minimal-gravity-free observer” to the particle, it would immediately mean that we have established the frame of inertial reference. Meantime, in the requirement of the symmetry of the relativity, the external gravity executed on “the minimal-gravity-free observer” (namely “the global gravity” )should be equally counteracted in counting the force for any particle in the system. For example, in the inertial reference frame of the earth system, where the mass center of the earth is the coordinate origin where “the minimal-gravity-free observer” located, the gravitational force we are to count in should not contain the interaction originated from the cosmic dust, the galaxy, the sun and even any other planets in the solar system. That is just the one of most important differences between what we want to proposed in this Letter with the correlative issue in the Einstein’s special relativity, and it is such a subtlety which may cannot be emphasized enough for the realization of the dynamical relativity. Then, the form invariance of the dynamical equations on different observers can be realized in our method. Along this way, the statement of the “law of inertia” should be correspondingly revised as: In any inertial reference frame $with$ $a$ $minimal$-$gravity$-$free$ $observer$, every particle preserves its state of rest, or its uniform motion in a straight line, unless it is acquired a $additional$ forces $relative$ $to$ $the$ $minimal$-$gravity$-$free$ $observer$. After that, “the relative gravity” can be naturally defined in the inner region of local systems resorting to the “law of inertia” and which is just symmetric to “the relative kinematical quantities”. There is one point we should keep in mind that, in applying these dynamical equations to practical cases, the correspondence between the scope of the interaction in the counting and the choice of inertial reference frame should be one to one. For example, to ascertain the Mars’ influence upon the relative motion between the Moon and the Earth, the center of the Earth is not applicable to be “the minimal-gravity-free observer” any longer, in this case we may choose the mass center of the Sun as the origin of a new inertial reference frame instead. More importantly, at the same time, the scope of the interaction under the consideration must be spread to all the possible relative interactions in the inner region of the sun system (relativity gravity of the sun’s reference). Therefore, in our theory, the inertial reference frame is no more defined globally. The difference among the different inertial frames of different levels is the different background gravity, i.e. the different scale of the inertial space time of the physics event. Consequently, the symmetry of the relativity turns out to be a possible method to achieve a consistent scenario of the dynamical relativity. In our approach the mathematical form of the dynamical equations would not only be applicable to arbitrary particle in the inner region of a uniform gravity-free local system, but also be unchanged with the transformation between different systems being in “the global gravity” free. This is just the new principle of the relativity which is proposed. Namely, it is the principle of “minimal-gravity-free-observer” relativity. Furthermore, from the two properties of the relatively observed interaction listed above, the transformation between different minimal-gravity-free observers is a $consistent$ $relativity$, it is realizable and actually sufficient for us describe our physical universe. From the discussion below, we think that this $consistent$ $relativity$ may be more physical than the principle of general relativity in Einstein’s theory. THE DYNAMICAL RELATIVITY UNDER THE INERTIAL FORCE ================================================= On the other hand, we re-investigate the reliability of the basis of the standard cosmology—General Relativity. Firstly, we may recur to the successful experiments on the solar scale which had been made in the early period of the last century. There are two important concepts have been confirmed from these experiments: the one is that the gravitational interaction can be completely geometrized, the another is that the gravity will also lead to the photon’s redshift, which is different to the Doppler redshift in kinematical meaning. The another principle of the Einstein’s gravity theory is the general relativity, which is to say that the field equation would be form-invariant under the general coordinates transformation. However this point still has not been confirmed from the existing experiments of gravity. Einstein’s principle of general relativity based on two considerations: the one is the dynamical equivalence between the gravitational force and the inertial force, Einstein thought that his another consideration is the “Mach principle”. Considering its objective base, however, we still have not drawn out the essential connection between the two origins of the gravitational force and the inertial force. Therefore even though we are able to unify the dynamical principle of the gravity and the inertial force into the same general covariant formulism, but the contribution from the inertial force still can not be erased from the mathematical form of a general covariant gravity theory. Therefore, the complete formula of a general covariant gravity theory may be written as, $$\begin{aligned} G_{\mu\nu}=-\kappa<T_{\mu\nu}>_{grav}+\kappa<T_{\mu\nu}>_{iner}.\end{aligned}$$ The second term on the right-hand-side of the above equation in the case of Brans-Dicke gravity theory just corresponds to the contribution from the gravitational scalar. In other theories where do not claim that the inertial force originates from the external gravitational matter, then the existence of this second term would be more indispensable. In fact we do not assure how to correctly describe the inertial force in origin, so we still have to select a special coordinates reference frame in cosmology, where we do not have to consider the inertial force any more. Once such a reference frame has been naturally chosen, the mathematical form of the cosmological field equation would not be capable to be further performed by the generalized coordinates transformation. That is to say, if we apply the Einstein’s field equation to the cosmology and deal only with the gravitational energy-momentum of the ordinary matter, then the coordinates reference frame under the consideration must have been substantively chosen from a special kind of reference frames [@Rosen]. REMARKS ======= Such new relativity of the dynamical equations which we put forward, its key lies in such a fact, namely, comparing to all the objects in a gravity-free system in the universe, our direct observation of the gravitational force are always local, in fact it is also relative. The influence of the gravity originated from the outer space of the system on their relative motion is almost equally counteracted by their uniform global acceleration. A direct reason for this point is just the equality of the inertial mass and the gravitational mass. Therefore, it turns out that there actually exists a preferred coordinates system, determined by the large-scale gravity-free systems in the universe, in which the laws of the physics have their simplest form. In other words, it is just the peculiar property of gravitational interaction and its peculiar status in the universe that make the definition of “inertial reference frame” available in the practical case. This paper is just aimed to make a theoretical exploration along the realization of the dynamical relativity. Therefore, the first thing we have proposed is that the counted gravity can be treated as a relative quantity in any gravity-free system. This point is comprehended just from the viewpoint of the symmetry, we know that the observed speed of a particle is relative, the observed acceleration of the particle is also relative, so it is natural to require that the counted force should also be relative (namely the relative gravity) to keep holding the principle of relativity on the dynamical equations. The introduction of the relativity of gravity is especially important for a natural definition of the inertial reference frames. Therefore, the second thing we have proposed is that “the minimal-gravity-free observer” can naturally constitute the frames of inertial reference. After establishing the dynamical equations on this kind of observers, the symmetry of the relativity to all objects in the system is realized. Therefore, the principle of the relativity in our theory is the “minimal-gravity-free-observer” relativity. As well as the symmetry of the relativity is required by the causality in the dynamics law, the symmetrical counteracting of the relativity of dynamical quantities in different inertial reference frames is further to unify the causality in their corresponding dynamical equations. In this sense, for the reference frames with the origin of a minimal gravity-free observer have been derived out along the symmetry of the relativity, the unification of the form of the dynamics law is truly carried out in our theory. In addition, for the dynamics law concerning the principle of causation, the geometrized gravity theory should be established on the basis of the background inertial reference frame. It is to say that Einstein’s mathematical form is here maintained for gravity theory just because nothing but dynamical effects of the gravity in the inner region of a system are capable being globally geometrized on its background inertial reference frame. The principle of consistent relativity means the establishing of the geometrized equation of gravity on the basis of the inertial reference frame, namely the causality. Hence in fact it has naturally harmonized the special relativity and the geometry theory of gravity in the level of the method of inertial reference frame. [**[Acknowledgments:]{}**]{} This work has been supported in part by National Nature Science Foundation of China under grant No. 10273017. K.Li also recognize the support from the Nature Science Foundation of Zhejiang Province under the grant number M103042,102011,102028. [nn]{} N. Rosen, Phys.Rev.D(America)3, 2317(1971). Edward W.Kolb and Michael S.Turner “The Early Universe” (Addison-Wesley Publishing Company, 1990), Chap. 2. Steven. Weinberg “Gravitational and Cosmology: Principles and Applications of the General Theory of Relativity”(John Wiley [&]{} Sons, Inc, New York, 1972), Part 3.

--- address: - 'Department of Mathematics, University of California, Berkeley, CA 94720, USA' - 'Department of Mathematics, The University of Chicago, Chicago, IL 60637, USA' author: - Edward Frenkel - Dennis Gaitsgory date: November 2003 title: 'D–modules on the affine Grassmannian and representations of affine Kac-Moody algebras' --- Introduction ============ \[negative\] Let $\fg$ be a simple Lie algebra over $\BC$, and $G$ the corresponding algebraic group of adjoint type. Given an invariant inner product $\kappa$ on $\g$, let $\hat\fg_\kappa$ denote the corresponding central extension of the formal loop algebra $\fg \otimes \BC((t))$, called the affine Kac-Moody algebra $\hat\fg_\kappa$, $$0\to \BC {\mb 1}\to \hat\fg_\kappa\to \fg \otimes \BC((t))\to 0,$$ with the two-cocycle defined by the formula $$x \otimes f(t),y \otimes g(t) \mapsto -\kappa(x,y)\cdot \on{Res}_{t=0} f dg.$$ Denote by $\hat\fg_\kappa\text{--}\on{mod}$ the category of $\hat\fg_\kappa$-modules which are discrete, i.e., any vector is annihilated by the Lie subalgebra $\g \otimes t^N \BC[[t]]$ for sufficiently large $N\geq 0$, and on which ${\mb 1} \in \BC\subset \hat\fg_\kappa$ acts as the identity. We will refer to objects of these category as modules at level $\kappa$. Let $\Gr_G = G((t))/G[[t]]$ be the affine Grassmannian of $G$. For each $\kappa$ there is a category $\on{D}_\kappa(\Gr_G)\text{--}\on{mod}$ of $\kappa$-twisted right D-modules on $\Gr_G$ (see [@BD1]). We have the functor of global sections $$\Gamma: \on{D}_\kappa(\Gr_G)\text{--}\on{mod}\to \hat\fg_\kappa\text{--}\on{mod}, \qquad \F \mapsto \Gamma(\Gr_G,\F).$$ Let $\kappa_{Kil}$ be the Killing form, $\kappa_{Kil}(x,y) = \on{Tr}(\on{ad}_\fg(x)\circ \on{ad}_\fg(y))$. The level $\kappa_{crit} = -\frac{1}{2}\kappa_{Kil}$ is called critical. A level $\kappa$ is called positive (resp., negative, irrational) if $\kappa=c\cdot \kappa_{Kil}$ and $c+\frac{1}{2}\in \BQ^{>0}$ (resp., $c+\frac{1}{2}\in \BQ^{<0}$, $c\notin \BQ$). It is known that the functor of global sections cannot be exact when $\kappa$ is positive. In contrast, when $\kappa$ is negative or irrational, the functor $\Gamma$ is exact and faithful, as shown by A. Beilinson and V. Drinfeld in [@BD1], Theorem 7.15.8. This statement is a generalization for affine algebras of the famous theorem of A. Beilinson and J. Bernstein, see [@BB], that the functor of global sections from the category of $\lambda$-twisted D-modules on the flag variety $G/B$ is exact when $\lambda-\rho$ is anti-dominant and it is faithful if $\lambda-\rho$ is, moreover, regular. The purpose of this paper is to consider the functor of global sections in the case of the critical level $\kappa_{crit}$. (In what follows we will slightly abuse the notation and replace the subscript ${}_{\kappa_{crit}}$ simply by ${}_{crit}$.) Unfortunately, it appears that the approach of [@BD1] does not extend to the critical level case, so we have to use other methods to analyze it. Our main result is that the functor of global sections remains exact at the critical level: \[main\] The functor $\Gamma:\on{D}_{crit}(\Gr_G)\text{--}\on{mod}\to \hat\fg_{crit}\text{--}\on{mod}$ is exact. In other words, we obtain that for any object $\F$ of $\on{D}_{crit}(\Gr_G)\text{--}\on{mod}$ we have $\on{H}^i(\Gr_G,\F)=0$ for $i>0$. Moreover, we will show that if $\F\neq 0$, then $\on{H}^0(\Gr_G,\F)=\Gamma(\Gr_G,\F)\neq 0$. This property is sometimes referred to as “D-affineness” of $\Gr_G$. In fact, we will prove a stronger result. Namely, we note after [@BD1], that for a critically twisted D-module $\F$ on $\Gr_G$, the action of $\hat\fg_{crit}$ on $\Gamma(\Gr_G,\F)$ extends to an action of the [*renormalized enveloping algebra*]{} $U^{\ren}(\hat\fg_{crit})$ of Sect. 5.6 of [*loc.cit.*]{} Following a conjecture and suggestion of Beilinson, we show that the resulting functor from $\on{D}_{crit}(\Gr_G)\text{--}\on{mod}$ to the category of $U^{\ren}(\hat\fg_{crit})$-modules is fully-faithful. Our method of proof of [Theorem \[main\]]{} uses the chiral algebra of differential operators $\fD_{G,\kappa}$ introduced in [@AG]. Modules over $\fD_{G,\kappa}$ should be viewed as (twisted) D-modules on the loop group $G((t))$. In particular, the category of $\kappa$-twisted D-modules on $\Gr_G$ is equivalent to the subcategory in $\fD_{G,\kappa}\text{--}\on{mod}$, consisting of modules, which are integrable with respect to the [*right*]{} action of $\fg[[t]]$ (see [Theorem \[AG\]]{}). The functor of global sections on $\Gr_G$ corresponds, under this equivalence, to the functor of $\fg[[t]]$-invariants. Therefore, we need to prove that this functor of invariants is exact. This approach may be applied both when the level $\kappa$ is negative (or irrational) and critical. In the case of the negative or irrational level the argument is considerably simpler, and so we obtain a proof of the exactness of $\Gamma$, which is different from that of [@BD1] (see [Sect. \[negative level\]]{}). The argument that we use for affine Kac-Moody algebras yields also a different proof of the exactness statement from [@BB]. Let us sketch this proof. For a weight $\lambda$, let $\on{D}^\lambda(G/B)\text{--}\on{mod}$ be the category of left $\lambda$-twisted D-modules on $G/B$ (here for an integral $\lambda$, by a $\lambda$-twisted D-module on $G/B$ we understand a module over the sheaf of differential operators acting on the line bundle $G \times_B \la$). Let $\pi$ denote the natural projection $G\to G/B$, and observe that the pull-back functor (in the sense of quasicoherent sheaves) lifts to a functor $\pi^*:\on{D}^\lambda(G/B)\text{--}\on{mod}\to \on{D}(G)\text{--}\on{mod}$. Furthermore, for a D-module $\F'$ on $G$, the space of its global sections $\Gamma(G,\F')$ is naturally a bimodule over $\fg$ due to the action of $G$ on itself by left and the right translations. For $\F\in \on{D}^\lambda(G/B)\text{--}\on{mod}$ we have $$\Gamma(G/B,\F)\simeq \on{Hom}_{\fb}\left(\BC^{-\lambda},\Gamma(G,\pi^*(\F))\right),$$ where $\fb$ is the Borel subalgebra of $\fg$, $\BC^{-\lambda}$ its one-dimensional representation corresponding to weight $-\lambda$, and $\Gamma(G,\pi^*(\F))$ is a $\fb$-module via $\fb\hookrightarrow \fg$ and the right action of $\g$. But the $\fg$-module $\Gamma(G,\F')$, where $\F' = \pi^*(\F)$ (with respect the right $\g$-action), belongs to the category $\CO$. Thus, we obtain a functor $$\Gamma':\on{D}^\lambda(G/B)-\on{mod} \to \CO, \qquad \F\mapsto \Gamma(G,\pi^*(\F)),$$ and $$\Gamma(G/B,\F)\simeq \on{Hom}_{\CO}\left(M(-\lambda),\Gamma'(\F)\right),$$ where $M(-\lambda)$ is the Verma module with highest weight $-\lambda$. The functor $\Gamma'$ is exact because $G$ is affine, and it is well-known that $M(\mu)$ is a projective object of $\CO$ precisely when $\mu+\rho$ is dominant. Hence $\Gamma$ is the composition of two exact functors and, therefore, is itself exact. This reproves the Beilinson-Bernstein exactness statement. Note, however, that the methods described above do not give the non-vanishing assertion of [@BB]. The proof of the exactness result in the negative (or irrational) level case is essentially a word for word repetition of the above argument, once we are able to make sense of the category of D-modules on $G((t))$ as the category of $\fD_{G,\kappa}$-modules. The key fact that we will use will be the same: that the corresponding vacuum Weyl module $\BV_{\fg,\kappa'}$ is projective in the appropriate category $\CO$ if $\kappa'$ is positive or irrational. This argument does not work at the critical level, because in this case the corresponding Weyl module $\BV_{\fg,crit}$ is far from being projective in the category $\hat\fg_{crit}-\on{mod}$. Roughly, the picture is as follows. Modules over $\hat\fg_{crit}$ give rise to quasicoherent sheaves over the ind-scheme $\on{Spec}(\FZ_{\fg,x})$, where $\FZ_{\fg,x}$ is the center of the completed universal enveloping algebra of $\hat\fg_{crit}$ (this is the ind-scheme of $^L \fg$-opers on the punctured disc, where $^L \fg$ is the Langlands dual Lie algebra to $\g$). The ind-scheme $\on{Spec}(\FZ_{\fg,x})$ contains a closed subscheme $\on{Spec}(\fz_{\fg,x})$ (this is the scheme of $^L \fg$-opers on the disc). The module $\BV_{\fg,crit}$ is supported on $\on{Spec}(\fz_{\fg,x})$ and is projective in the category of $\hat\fg_{crit}$-modules, which are supported on $\on{Spec}(\fz_{\fg,x})$ and are $G(\hat\CO_x)$-integrable. The problem is, however, that the $\hat\fg_{crit}$-modules of the form $\Gamma\left(G((t)),\pi^*(\F)\right)$, where $\pi$ is the projection $G((t))\to G((t))/G[[t]]\simeq \Gr_G$, are never supported on $\on{Spec}(\fz_{\fg,x})$. Therefore we need to show that the functor of taking the maximal submodule of $\Gamma\left(G((t)),\pi^*(\F)\right)$, which is supported on $\on{Spec}(\fz_{\fg,x})$, is exact. We do that by showing that the action of $\hat\fg_{crit}$ on $\Gamma\left(G((t)),\pi^*(\F)\right)$ automatically extends to the action of the renormalized chiral algebra $\CA^{\ren,\tau}_\fg$, which is closely related to the renormalized enveloping algebra $U^{\ren}(\hat\fg_{crit})$, mentioned above. Consider the following analogy. Let $X$ be a smooth variety and $Y$ its smooth closed subvariety. Then we have a natural functor, denoted $i^!$, from the category of ${\mathcal O}_X$-modules, set-theoretically supported on $Y$, to the category of ${\mathcal O}_Y$-modules: this functor takes an ${\mathcal O}_X$-module $\F$ to its maximal submodule supported scheme-theoretically on $Y$. This is not an exact functor. But the corresponding functor from the category of right D-modules on $X$, also set-theoretically supported on $Y$, to the category of right D-modules on $Y$ is exact, according to a basic theorem due to Kashiwara. In our situation the role of the category of ${\mathcal O}_X$-modules is played by the category $\hat\fg_{crit}-\on{mod}$, and the role of the category of D-modules is played by the category of modules over the chiral algebra $\CA^{\ren,\tau}_\fg$. We show that the above functor of taking the maximal submodule of $\Gamma\left(G((t)),\pi^*(\F)\right)$, which is supported on $\on{Spec}(\fz_{\fg,x})$, factors through the latter category, and this allows us to prove the required exactness. Let us briefly describe how this paper is organized. In [Sect. \[negative level\]]{} we treat the negative level case. In [Sect. \[sect center\]]{} we recall some facts about commutative D-algebras and the description of the center of the Kac-Moody chiral algebra at the critical level. In [Sect. \[renormalized algebra\]]{} we discuss several versions of the renormalized universal enveloping algebra at the critical level in the setting of chiral algebras. In [Sect. \[diff op\]]{} we study the chiral algebra of differential operators $\fD_{G,\kappa}$ when $\kappa=\kappa_{crit}$. In [Sect. \[sections on Gr\]]{} we derive our main [Theorem \[main\]]{} from two other statements, Theorems \[support on tangent bundle\] and \[Kashiwara\]. In [Sect. \[proof of Kashiwara\]]{} we prove [Theorem \[Kashiwara\]]{}, generalizing Kashiwara’s theorem about D-modules supported on a subvariety. In [Sect. \[other results\]]{} we prove [Theorem \[support on tangent bundle\]]{} and describe the category of $\hat\fg_{crit}$-modules, which are supported on $\on{Spec}(\fz_{\fg,x})$ and are $G(\hat\CO_x)$-integrable. Finally, in [Sect. \[faithfulness\]]{} we prove that the functor $\Gamma$ is faithful. Our basic tool in this paper is the theory of chiral algebras. The foundational work [@BD] on this subject will soon be published (in our references we use the most recent version; a previous one is currently available on the Web). In addition, an abridged summary of the results of [@BD] that are used in this paper may be found in [@AG]. We wish to remark that all chiral algebras considered in this paper are universal in the sense that they come from quasi-conformal vertex algebras by a construction explained in [@FB], Ch. 18. Therefore all results of this paper may be easily rephrased in the language of vertex algebras. We have chosen the language of chiral algebras in order to be consistent with the language used in [@AG]. We also use some the results from [@BD1], which is still unpublished, but available on the Web. The notation in this paper mainly follows that of [@AG]. Throughout the paper, $X$ will be a fixed smooth curve; we will denote by $\CO_X$ (resp., $\omega_X$, $T_X$ and $D_X$) its structure sheaf (resp., the sheaf of differentials, the tangent sheaf and the sheaf of differential operators). We will work with D-modules on $X$, and in our notation we will not distinguish between left and right D-modules, i.e., we will denote by the same symbol a left D-module $\CM$ and the corresponding right D-module $\CM\otimes \omega_X$. The operations of tensor product, taking symmetric algebra, and restriction to a subvariety must be understood accordingly. We will denote by $\Delta$ the diagonal embedding $X\to X\times X$, and by $j$ the embedding of its complement $X\times X-\Delta(X)\to X\times X$. If $x\in X$ is a point, we will often consider D-modules supported at $x$. In this case, our notation will not distinguish between such a D-module and the underlying vector space. We will use the notation $A \underset{B}\times C$ for a fiber product of $A$ and $C$ over $B$, and the notation ${\mathcal P} \times_G V$ for the twist of a $G$-module $V$ by a $G$-torsor ${\mathcal P}$. Finally, if $\C$ is a category and $C$ is an object of $\C$, we will often write $C\in \C$. D.G. would like to express his deep gratitude to A. Beilinson for explaining to him the theory of chiral algebras, as well as for numerous conversations related to this paper. He would also like to thank S. Arkhipov, J. Bernstein for stimulating and helpful discussions. In addition, both authors would like to thank A. Beilinson for helpful remarks and suggestions and B. Feigin for valuable discussions. The research of E.F. was supported by grants from the Packard foundation and the NSF. D.G. is a long-term prize fellow of the Clay Mathematics Institute. The case of affine algebras at the negative and irrational levels {#negative level} ================================================================= In this section we will show that the functor of global sections $$\Gamma:\on{D}_{\kappa}(\Gr_G)-\text{mod}\to \hat\fg_{\kappa}-\text{mod}$$ is exact when $\kappa$ is negative or irrational. A similar result has been proved by Beilinson and Drinfeld in [@BD1], Theorem 7.15.8, by other methods. The setting of [@BD1] is slightly different: they consider twisted D-modules on the affine flag variety $\on{Fl}_G=G((t))/I$ instead of $\Gr_G=G((t))/G[[t]]$, where $I\subset G[[t]]$ is the Iwahori subgroup, i.e., the preimage of a fixed Borel subgroup $B \subset G$ under the projection $G[[t]] \to G$. Here is the precise statement of their theorem: Recall that for any affine weight $\hat{\lambda}=(\lambda,2\check h\cdot c)$ (where $\lambda$ is a weight of $\fg$, $c\in \BC$ and $\check h$ is the dual Coxeter number), we can consider the corresponding category $\on{D}_{\hat{\lambda}}(\on{Fl}_G)\text{--}\on{mod}$, of right $\hat{\lambda}$-twisted D-modules on $\on{Fl}_G$. A weight $\hat{\lambda}$ is called anti-dominant if the corresponding Verma module $M(\hat{\lambda})$ over $\hat\fg_\kappa$ (where $\kappa=c\cdot \kappa_{Kil}$) is irreducible. According to a theorem of Kac and Kazhdan (see [@KK]), this condition can be expressed combinatorially as $\langle \hat{\lambda}+\rho_{aff},\check \alpha_{aff}\rangle\notin \BZ^{> 0}$, where $\alpha_{aff}$ runs over the set of all positive affine coroots. We have: \[on flags\] If $\hat{\lambda}$ is anti-dominant, then the functor of sections $\Gamma:\on{D}_{\hat{\lambda}}(\on{Fl}_G)\text{--}\on{mod}\to \hat\fg_\kappa\text{--}\on{mod}$ is exact. [Theorem \[on flags\]]{} formally implies the exactness statement on $\Gr_G$ (i.e., [Theorem \[exactness on negative\]]{} below) only for $\kappa=c\cdot \kappa_{Kil}$ with $c$ either irrational, or $c+\frac{1}{2}< -1+\frac{1}{2\check h}$; so our exactness result is slightly sharper than that of [@BD1]. The proof of [Theorem \[exactness on negative\]]{} given below can be extended in a rather straightforward way to reprove [Theorem \[on flags\]]{}. In contrast, in the case of the critical level, it is essential that we consider D-modules on $\Gr_G$ and not on $\on{Fl}_G$; in the latter case the naive analogue of the exactness statement is not true. Finally, note that Theorem 7.15.8 of [@BD1] contains also the assertion that for $0\neq \F\in \on{D}_{\wt{\lambda}}(\on{Fl}_G)\text{--}\on{mod}$, then the space of sections $\Gamma(\Gr_G, \F)$ is non-zero, implying a similar statement for $\F\in \on{D}_{\kappa}(\Gr_G)$. In [Sect. \[faithfulness\]]{}, we will reprove this fact as well, by a different method. This proof is the same in the negative and the critical level cases. Thus, our goal in this section is to prove the following theorem: \[exactness on negative\] The functor $\Gamma:\on{D}_{\kappa}(\Gr_G)\text{--}\on{mod}\to \hat\fg_{\kappa}\text{--}\on{mod}$ is exact when $\kappa$ is negative or irrational. The starting point of our proof is the following. Recall the chiral algebra $\fD_{G,\kappa}$ (on our curve $X$), introduced in [@AG]. Let $\fD_{G,\kappa}\text{--}\on{mod}$ denote the category of chiral $\fD_{G,\kappa}$-modules concentrated at a point $x\in X$. In [@AG] it was shown that $\fD_{G,\kappa}\text{--}\on{mod}$ is a substitute for the category of twisted D-modules on the loop group $G((t))$, where $t$ is a formal coordinate on $X$ near $x$. In particular, we have the forgetful functor $$\fD_{G,\kappa}\text{--}\on{mod}\to (\hat\fg_{\kappa}\times \hat\fg_{2\kappa_{crit}-\kappa})\text{--}\on{mod},$$ where $\hat\fg_{\kappa}\text{--}\on{mod}$ (resp., $\hat\fg_{2\kappa_{crit}-\kappa}\text{--}\on{mod}$) is the category of representations of the affine algebra at the level $\kappa$ (resp., $2\kappa_{crit}-\kappa$). This functor corresponds to the action of the Lie algebra $\fg((t))$ on $G((t))$ by left and right translations. In what follows, for a module $\CM\in \fD_{G,\kappa}\text{--}\on{mod}$, we will refer to the corresponding actions of $\hat\fg_{\kappa}$ and $\hat\fg_{2\kappa_{crit}-\kappa}$ on it as “left” and “right”, respectively. Let $\wh{\CO}_x \simeq \BC[[t]]$ be the completed local ring at $x$. Consider the subalgebra $\fg(\wh{\CO}_x)\subset \hat\fg_{2\kappa_{crit}-\kappa}$. Let $\hat\fg_{2\kappa_{crit}-\kappa}\text{--}\on{mod}^{G(\wh{\CO}_x)}$ be the subcategory of $\hat\fg_{2\kappa_{crit}-\kappa}\text{--}\on{mod}$ whose objects are the $\hat\fg_{2\kappa_{crit}-\kappa}$-modules, on which the action of $\fg(\wh{\CO}_x)$ may be exponentiated to an action of the corresponding group $G(\wh{\CO}_x)$. Let $\fD_{G,\kappa}\text{--}\on{mod}^{G(\wh{\CO}_x)}$ denote the full subcategory of $\fD_{G,\kappa}\text{--}\on{mod}$ whose objects belong to $\hat\fg_{2\kappa_{crit}-\kappa}\text{--}\on{mod}^{G(\wh{\CO}_x)}$ under the right action of $\hat\fg_{2\kappa_{crit}-\kappa}\text{--}\on{mod}$. The following result has been established in [@AG]: \[AG\] There exists a canonical equivalence of categories $$\on{D}_{\kappa}(\Gr_G)\text{--}\on{mod}\simeq \fD_{G,\kappa}\text{--}\on{mod}^{G(\wh{\CO}_x)}.$$ If $\F$ is an object of $\on{D}_{\kappa}(\Gr_G)\text{--}\on{mod}$, and $\CM_\F$ the corresponding object of $\fD_{G,\kappa}\text{--}\on{mod}^{G(\wh{\CO}_x)}$, then the $\hat\fg_{\kappa}$-module $\Gamma(\Gr_G,\F)$ identifies with $(\CM_\F)^{\fg(\wh{\CO}_x)}$, the space of invariants in $\CM_\F$ with respect to the Lie subalgebra $\fg(\wh{\CO}_x)\subset \hat\fg_{2\kappa_{crit}-\kappa}$ under the right action. \[Weyl\] To prove the exactness of the functor $\Gamma:\on{D}_{\kappa}(\Gr_G)\text{--}\on{mod}\to \hat\fg_{\kappa}\text{--}\on{mod}$, for negative or irrational $\kappa$, we compose it with the tautological forgetful functor $\hat\fg_{\kappa}\text{--}\on{mod}\to \on{Vect}$. By [Theorem \[AG\]]{}, this composition can be rewritten as $$\fD_{G,\kappa}\text{--}\on{mod}^{G(\wh{\CO}_x)}\to \hat\fg_{2\kappa_{crit}-\kappa}\text{--}\on{mod}^{G(\wh{\CO}_x)} \to \on{Vect},$$ where the first arrow is the forgetful functor, and the second arrow is $\CM\mapsto \CM^{\fg(\wh{\CO}_x)}$. For an arbitrary level $\kappa'$, let $\BV_{\fg,\kappa'}$ be the vacuum Weyl module, i.e., $$\BV_{\fg,\kappa'}\simeq \on{Ind}^{\hat\fg_{\kappa'}}_{\fg(\wh{\CO}_x)\oplus \BC {\mb 1}}(\BC),$$ where $\fg(\wh{\CO}_x)$ acts on $\BC$ by zero and ${\mb 1}$ acts as the identity. Tautologically, for any $\CM\in \hat\fg_{\kappa'}\text{--}\on{mod}$, we have: $$\label{hom} \on{Hom}_{\hat\fg_{\kappa'}}(\BV_{\fg,\kappa'},\CM)\simeq \CM^{\fg(\wh{\CO}_x)}.$$ Moreover, $\BV_{\fg,\kappa'}$ is $G(\wh{\CO}_x)$-integrable, i.e., belongs to $\hat\fg_{\kappa'}\text{--}\on{mod}^{G(\wh{\CO}_x)}$. Observe that the condition that $\kappa$ is negative or irrational is equivalent to $\kappa':=2\kappa_{crit}-\kappa$ being positive or irrational. Therefore, to prove the exactness of $\Gamma$, it is enough to establish the following: \[noncrit\] If $\kappa'$ is positive or irrational, the module $\BV_{\fg,\kappa'}$ is projective in $\hat\fg_{\kappa'}\text{--}\on{mod}^{G(\wh{\CO}_x)}$. This proposition is well-known, and the proof is based on considering eigenvalues of the Segal-Sugawara operator $L_0$. We include the proof for completeness. Recall that for every non-critical value of $\kappa'$, the vector space underlying every object $\CM\in \hat\fg_{\kappa'}\text{--}\on{mod}$ carries a canonical endomorphism $L_0$ obtained via the Segal-Sugawara construction, such that the action of $\hat\fg_{\kappa'}$ commutes with $L_0$ in the following way: $$\label{comm rel} [L_0,x\otimes t^n]=-n\cdot x\otimes t^n, \qquad x\in\fg, n \in \BZ.$$ Explicitly, let $\{x^a, x_a \}$ be bases in $\g$, dual with respect to $\kappa_{Kil}$. The operator $$\label{S0} S_0 = \sum_a x^a\cdot x_a + 2\sum_a \sum_{n>0} x^a \otimes t^{-n}\cdot x_a\otimes t^n$$ is well-defined on every object of $\hat\fg_{\kappa'}\text{--}\on{mod}$, and it has the following commutation relation with elements of $\hat\fg_{\kappa'}$: $$[S_0,x \otimes t^n]=-(2c'+1)\cdot n\cdot x\otimes t^n, \qquad x\in\fg, n \in \BZ,$$ where $c'$ is such that $\kappa'=c'\cdot \kappa_{Kil}$. Therefore, for $c'\neq -\frac{1}{2}$, the operator $L_0:=\frac{1}{2c'+1}\cdot S_0$ has the required properties. For an integral dominant weight $\la$ of $\fg$, let $V^\la$ be the finite-dimensional irreducible $\fg$-module with highest weight $\la$ and $\BV^\lambda_{\fg,\kappa'}$ the corresponding Weyl module over $\hat\fg_{\ka'}$, $$\BV^\lambda_{\fg,\kappa'}= \on{Ind}^{\hat\fg_{\kappa'}}_{\fg(\wh{\CO}_x)\oplus \BC {\mb 1}}(V^\lambda),$$ where $\fg(\wh{\CO}_x)$ acts on $V^\la$ through the homomorphism $\fg(\wh{\CO}_x) \to \g$ and ${\mb 1}$ acts as the identity. Then we find from formula that $L_0$ acts on the subspace $V^\lambda \subset \BV^\lambda_{\fg,\kappa'}$ by the scalar $\frac{C_\fg(\lambda)}{2c'+1}$, where $C_{\fg}(\lambda)$ is the scalar by which the Casimir element $\sum_a x^a \cdot x_a$ of $U(\fg)$ acts on $V^\lambda$. Note that $C_{\fg}(\lambda)$ is a non-negative rational number for any dominant integral weight $\lambda$, and $C_{\fg}(\lambda)\neq 0$ if $\lambda\neq 0$. Since $\BV^\lambda_{\fg,\kappa'}$ is generated from $V^\lambda$ by the elements $x\otimes t^n\in \hat\fg_{\kappa'}$, $n<0$, we obtain that the action of $L_0$ on $\BV^\lambda_{\fg,\kappa'}$ is semi-simple. Moreover, since every object $\CM\in\hat\fg_{\kappa'}\text{--}\on{mod}^{G(\wh{\CO}_x)}$ has a filtration whose subquotients are quotients of the $\BV^\lambda_{\fg,\kappa'}$’s, the action of $L_0$ on any such $\CM$ is locally-finite. Suppose now that we have an extension $$\label{ext} 0\to \CM\to \wt{\CM}\to \BV_{\fg,\kappa'}\to 0$$ in $\hat\fg_{\kappa'}\text{--}\on{mod}^{G(\wh{\CO}_x)}$. Let $\wt{v}^0\in \wt{\CM}$ be a lift to $\wt{\CM}$ of the generating vector $v^0\in \BV_{\fg,\kappa'}$. Without loss of generality we may assume that $\wt{v}^0$ has the same generalized eigenvalue as $v^0$, i.e., $0$, with respect to the action of $L_0$. It is sufficient to show that then $\wt{v}^0$ belongs to $(\wt{\CM})^{\fg \otimes t\BC[[t]]}$. Indeed, if this is so, then $\wt{v}^0$ is annihilated by the entire Lie subalgebra $\fg(\wh{\CO}_x)$, due to the eigenvalue condition, which would mean that the extension splits. Suppose that this is not the case, i.e., that $\wt{v}^0$ is not annihilated by $\fg \otimes t\BC[[t]]$. Then we can find a sequence of elements $x_i\otimes t^{n_i} \in \fg \otimes t\BC[[t]]$, which we can assume to be homogeneous, automatically of negative degrees with respect to $L_0$, such that the vector $$w=x_1\otimes t^{n_1}\cdot ...\cdot x_k\otimes t^{n_k}\cdot \wt{v}^0\in\CM$$ is non-zero and is annihilated by $\fg \otimes t\BC[[t]]$. But then, on the one hand, the eigenvalue of $L_0$ on $w$ is $\on{deg} (x_1\otimes t^{n_1}) +...+\on{deg} (x_k\otimes t^{n_k})= -(n_1+...+n_k) \in \BZ^{<0}$, but on the other hand, it must be of the form $\frac{C_\fg(\lambda)}{c'+\frac{1}{2}}$, which is not in $\BQ^{<0}$, by our assumption. Center of the Kac-Moody chiral algebra at the critical level {#sect center} ============================================================ Let $\CA$ be a unital chiral algebra on $X$. In what follows we will work with a fixed point $x\in X$ and denote by $\CA\text{--}\on{mod}$ the category of chiral $\CA$-modules, supported at $x$. Recall that the center of $\CA$, denoted by $\fz(\CA)$, is by definition the maximal D-submodule of $\CA$ for, which the Lie-\* bracket $\fz(\CA)\boxtimes \CA\to \Delta_!(\CA)$ vanishes. It is easy to see that $\fz(\CA)$ is a commutative chiral subalgebra of $\CA$. For example, the unit $\omega_X\hookrightarrow \CA$ is always contained in $\fz(\CA)$. Let $\CA_{\fg,\kappa}$ be the chiral universal enveloping algebra of the Lie-\* algebra $L_{\fg,\kappa}:=\fg\otimes D_X\oplus \omega_X$ at the level $\kappa$ (modulo the relation equating the two embeddings of $\omega_X$). We have the basic equivalence of categories: $$\CA_{\fg,\kappa}\text{--}\on{mod}\simeq \hat\fg_{\kappa}\text{--}\on{mod}.$$ It is well-known that when $\kappa\neq \kappa_{crit}$, the inclusion $\omega_X\to \fz(\CA_{\fg,\kappa})$ is an isomorphism. Let us denote by $\fz_\fg$ the commutative chiral algebra $\fz(\CA_{\fg,crit})$. In [Theorem \[FF isom\]]{} below we will recall the description of $\fz_\fg$ obtained in [@FF; @F]. Let $\fz_{\fg,x}$ be the fiber of $\fz_\fg$ at $x$; this is a commutative algebra. We have the natural maps $$\label{center and endomorphisms} \fz_{\fg,x} \longrightarrow (\BV_{\fg,crit})^{\fg(\wh{\CO}_x)} \overset{\sim}\longleftarrow \on{End}_{\hat\fg_{crit}}(\BV_{\fg,crit}),$$ where the left arrow is obtained from the definition of the center of a chiral algebra, and the right arrow assigns to $e \in \on{End}_{\hat\fg_{crit}}(\BV_{\fg,crit})$ the vector $e \cdot v^0$, where $v^0$ is the canonical generator of $\BV_{\fg,crit}$. The resulting homomorphism of algebras $\fz_{\fg,x}\to \on{End}_{\hat\fg_{crit}}(\BV_{\fg,crit})$ is an isomorphism. In fact, for any chiral algebra $\CA$, its center $\fz(\CA)$ identifies with the D-module of endomorphisms of $\CA$ regarded as a chiral $\CA$-module. At the level of fibers, we have a map in one direction $\fz(\CA)_x\to \on{End}_{\CA\text{--}\on{mod}}(\CA_x)$. This map is an isomorphism if a certain flatness condition is satisfied. This condition is always satisfied if $\CA$ is “universal”, i.e., comes from a quasi-conformal vertex algebra, which is the case of $\CA_{\fg,crit}$. For a chiral algebra $\CA$, let $\hat\CA_x$ be the canonical topological associative algebra attached to the point $x$, see [@BD], Sect. 3.6.2. By definition, the category $\CA\text{--}\on{mod}$ endowed with the tautological forgetful functor to the category of vector spaces, is equivalent to the category of discrete continuous $\hat\CA_x$-modules, denoted $\hat\CA_x\text{--}\on{mod}$. For example, when $\CA=\CA_{\fg,\kappa}$, the corresponding algebra $\hat\CA_{\fg,\kappa,x}$ identifies with the completed universal enveloping algebra of $\hat\fg_\kappa$ modulo the relation $1={\mathbf 1}$. We denote this algebra by $U'(\hat\fg_\kappa)$. When $\CA=\CB$ is commutative, the algebra $\hat\CB_x$ is commutative as well, see [@BD], Sects. 3.6.6 and 2.4.8. In fact, $\hat\CB_x$ can be naturally represented as $\underset{\longleftarrow}{lim}\, \CB_x^i$, where $\CB^i$ are subalgebras of $\CB$, such that $\CB^i|_{X-x}\simeq \CB|_{X-x}$. In particular, we have a surjective homomorphism $\hat\CB_x\to \CB_x$; the subcategory $\CB_x\text{--}\on{mod}\subset \hat\CB_x\text{--}\on{mod}$ is the full subcategory of $\CB\text{--}\on{mod}$, whose objects are central $\CB$-modules, supported at $x\in X$. (Recall that a $\CB$-module $\CM$ is called central if the action map $j_*j^*(\CB\boxtimes \CM)\to \Delta_!(\CB)$ comes from a map $\CB\otimes\CM\to \CM$, i.e., factors through $j_*j^*(\CB\boxtimes \CM)\twoheadrightarrow \Delta_!(\CB\otimes \CM)$.) We will view $\on{Spec}(\hat\CB_x)$ as an ind-scheme $\underset{\longrightarrow}{lim}\, \on{Spec}(\CB_x^i)$; we have a closed embedding $\on{Spec}(\CB_x)\hookrightarrow \on{Spec}(\hat\CB_x)$. By taking $\CB=\fz_\fg$, we obtain a topological commutative algebra $\hat\fz_{\fg,x}$, which we will also denote by $\FZ_{\fg,x}$. The corresponding map $\on{Spec}(\fz_{\fg,x})\hookrightarrow \on{Spec}(\FZ_{\fg,x})$ will be denoted by $\imath$. For any chiral algebra $\CA$ we have a homomorphism $$\widehat{\fz(A)}_x\to Z(\hat\CA_x),$$ where $Z(\hat\CA_x)$ is the center of $\hat\CA_x$. We do not know whether this map is always an isomorphism, but can show that it is an isomorphism for $\CA=\CA_{\fg,crit}$, using the description of $\fz_{\fg}$, given by [Theorem \[FF isom\]]{}(1) below (see [@BD1], Theorem 3.7.7). In other words, $\FZ_{\fg,x}$ maps isomorphically to the center of $U'(\hat\fg_{crit})$. \[FF description\] Let us recall the explicit description of $\fz_{\fg}$ and $\FZ_{\fg,x}$ due to [@FF; @F]. Let $^L G$ be the algebraic group of adjoint type whose Lie algebra is the Langlands dual to $\g$. Denote by $\on{Op}_{^L G}({\mc D}_x)$ the affine scheme of $^L G$-[*opers*]{} on the disc ${\mc D}_x = \on{Spec} (\wh{\mc O}_x)$. These are triples $({\mc F},{\mc F}_B,\nabla)$, where ${\mc F}$ is a $^L G$–torsor over ${\mc D}_x$, ${\mc F}_B$ is its reduction to a fixed Borel subgroup $^L B \subset {} ^L G$ and $\nabla$ is a connection on ${\mc F}$ (automatically flat) such that ${\mc F}_B$ and $\nabla$ are in a special relative position (see, e.g., [@F] for details). There exists an affine $D_X$-scheme $J(\on{Op}_{^L G}(X))$ of jets of opers on $X$, whose fiber at $x \in X$ is $\on{Op}_{^L G}({\mc D}_x)$ (see [@BD1], Sect. 3.3.3), and so the corresponding sheaf of algebras of functions $\on{Fun}\left(J(\on{Op}_{^L G}(X))\right)$ on $X$ is a commutative chiral algebra. (In what follows, $\on{Fun}(\Y)$ stands for the ring of regular functions on a scheme $\Y$.) The canonical topological algebra associated to $\on{Fun}\left(J(\on{Op}_{^L G}(X))\right)$ at the point $x$ is nothing but the topological algebra of functions on the ind-affine space $\on{Op}_{^L G}({\mc D}_x^\times)$ of $^L G$-opers on the punctured disc ${\mc D}_x^\times = \on{Spec} (\wh{\mc K}_x)$, where $\wh{\mc K}_x$ is the field of fractions of $\wh{\mc O}_x$. The following was established in [@FF; @F]: \[FF isom\] [*(1)*]{} There exists a canonical isomorphism of $D_X$-algebras $$\fz_{\fg} \simeq \on{Fun}\left(J(\on{Op}_{^L G}(X))\right).$$ In particular, we have an isomorphism of commutative algebras $\fz_{\fg,x} \simeq \on{Fun}\left(\on{Op}_{^L G}({\mc D}_x)\right)$ and of commutative topological algebras $\FZ_{\fg,x} \simeq \on{Fun} \left(\on{Op}_{^L G}({\mc D}_x^\times)\right)$. [*(2)*]{} On the associated graded level, we have a commutative diagram of isomorphisms: $$\CD \on{gr} (\fz_{\fg,x}) @<<< \on{gr}\bigl(\on{Fun}\left(\on{Op}_{^L G}({\mc D}_x)\right)\bigr) \\ @VVV @VVV \\ \on{Fun}\left(\left(\fg^*\times_{\BG_m}\Gamma({\mc D}_x,\Omega_X) \right)^{G(\wh\CO_x)}\right) @<<< \on{Fun}\left((^L\fg/^L G) \times_{\BG_m}\Gamma({\mc D}_x,\Omega_X)\right), \endCD$$ where $^L\fg/^L G=\on{Spec}(\on{Fun}(^L \fg)^{^L G})$. Note that in the lower left corner of the above commutative diagram we have used the identification $\on{gr}(\BV_{\fg,crit}) \simeq \on{Sym}\left(\fg\otimes (\wh\CK_x/\wh\CO_x)\right) \simeq \on{Fun}\left(\fg^*\times_{\BG_m}\Gamma({\mc D}_x,\Omega_X) \right)$, and $$\fg^*/G\simeq \fh^*/W\simeq {}^L \fh/W\simeq {}^L \fg/^L G.$$ \[comm alg\] To proceed we need to recall some more material from [@BD] about commutative D-algebras (which, according to our conventions, we do not distinguish them from commutative chiral algebras). If $\CB$ is a commutative $D_X$-algebra, consider the $\CB$-module $\Omega^1(\CB)$ of (relative with respect to $X$) differentials on $\CB$, i.e., $\Omega^1(\CB)\simeq I_B/I^2_B$, where $I_B$ is the kernel of the product $\CB\underset{\CO_X}\otimes \CB\to \CB$. From now on we will assume that $\CB$ is finitely generated as a $D_X$-algebra; in this case $\Omega^1(\CB)$ is finitely generated as a $\CB\otimes D_X$-module. Recall that geometric points of the scheme $\on{Spec}(\CB_x)$ (resp., of the ind-scheme $\on{Spec}(\hat\CB_x)$) are the same as horizontal sections of $\on{Spec}(\CB)$ over the formal disc $\D_x$ (resp., the formal punctured disc $\D_x^\times$), see [@BD], Sect. 2.4.9. Let us explain the geometric meaning of $\Omega^1(\CB)$ in terms of these identifications. Let $z$ be a point of $\on{Spec}(\CB_x)$, corresponding to a horizontal section $\phi_z:\widehat\CO_x\to \CB_x$. Evidently, we have: $\phi_z^*(\Omega^1(\CB))_x\simeq T^*_z(\on{Spec}(\CB_x))$, where $T^*_z$ denotes the cotangent space at $z$. From the definition of $\hat\CB_x$ we obtain a map $$\label{map of cotangent} H^0_{DR}\left(\D^\times_x,\phi_z^*(\Omega^1(\CB))\right)\to T^*_z(\on{Spec}(\hat\CB_x)).$$ (Since the D-module $\phi_z^*(\Omega^1(\CB))$ on $\D_x$ is finitely generated, its de Rham cohomology over the formal and formal punctures disc makes obvious sense.) One can show that the map of is actually an isomorphism. From the short exact sequence $$\label{cotangent sequence} 0\to H^0_{DR}(\D_x,\phi_z^*(\Omega^1(\CB)))\to H^0_{DR}(\D^\times_x,\phi_z^*(\Omega^1(\CB)))\to \phi_z^*(\Omega^1(\CB))_x\to 0,$$ we obtain also an identification $$H^0_{DR}(\D_x,\phi_z^*(\Omega^1(\CB)))\simeq N^*_{z}(\CB_x),$$ where $N_z^*(\CB_x)$ denotes the conormal to $\on{Spec}(\CB_x)$ inside $\on{Spec}(\hat\CB_x)$ at the point $z$. Assume now that $\CB$ is smooth (see [@BD], Sect. 2.3.15 for the definition of smoothness). In this case $\Omega^1(\CB)$ is a finitely generated projective $\CB\otimes D_X$-module. Consider the dual of $\Omega^1(\CB)$, i.e., $$\Theta(\CB):= \on{Hom}_{\CB\otimes D_X}\left(\Omega^1(\CB),\CB\otimes D_X\right).$$ This is a central $\CB$-module, called the tangent module to $\CB$. Moreover, $\Theta(\CB)$ carries a canonical structure of Lie-\* algebroid over $\CB$ (see below). Evidently, $\Theta(\CB)$ is also projective and finitely generated as a $\CB\otimes D_X$-module. By dualizing the members of the short exact sequence , we obtain the identifications (cf. [@BD], Sect. 2.5.21): $$\begin{aligned} &H^0_{DR}(\D_x,\phi_z^*(\Theta(\CB)))\simeq T_z(\on{Spec}(\CB_x)),\,\, H^0_{DR}(\D^\times_x,\phi_z^*(\Theta(\CB)))\simeq T_z(\on{Spec}(\hat\CB_x)), \\ & \text{ \hskip3cm and } \phi_z^*(\Theta(\CB))_x\simeq N_z(\CB_x).\end{aligned}$$ The next definition will be needed in [Sect. \[sections on Gr\]]{}. Let $\CI$ denote the kernel $\hat\CB_x\to \CB_x$. The quotient $\CI/\CI^2$ is a topological module over $\CB_x$, and the normal bundle, $N(\CB_x)$, to $\on{Spec}(\CB_x)$ inside $\on{Spec}(\hat\CB_x)$ can always be defined as the group ind-scheme $\on{Spec}(\on{Sym}_{\CB_x}(\CI/\CI^2))$. Let now $\CE\subset N(\CB_x)$ be a group ind-subscheme, and let $\CE^\perp$ be its annihilator in $\CI/\CI^2$. We introduce the subcategory $\hat\CB_x\text{--}\on{mod}_\CE$ inside the category $\hat\CB_x\text{--}\on{mod}$ of all chiral $\CB$-modules supported at $x$ by imposing the following two conditions: \(1) We require that a module $\CM$, viewed as a quasicoherent sheaf on $\on{Spec}(\hat\CB_x)$, is supported on the formal neighborhood of $\on{Spec}(\CB_x)$. In particular, $\CM$ acquires a canonical increasing filtration $\CM=\underset{i\geq 1}\cup\, \CM_i$, where $\CM_i\subset \CM$ is the submodule consisting of sections annihilated by $\CI^i$. \(2) We require that the natural map $\CI/\CI^2\underset{\CB_x}\otimes \CM_{i+1}/\CM_i\to \CM_i/\CM_{i-1}$ vanish on $\CE^\perp\subset \CI/\CI^2$. Note that the category $\hat\CB_x\text{--}\on{mod}_\CE$ is in general not abelian. \[notion of algebroid\] Let us now recall the notion of Lie-\* algebroid over a commutative $D_X$ algebra $\CB$ (cf. [@BD], Sect. 2.5). Let $L$ be a central $\CB$-module. A structure of a [*Lie-\* algebroid over*]{} $\CB$ on $L$ is the data of a Lie-\* bracket $L\boxtimes L\to \Delta_!(L)$ and an action map $L\boxtimes \CB\to \Delta_!(\CB)$, which satisfy the natural compatibility conditions given in [@BD], Sect. 1.4.11 and 2.5.16. If $\CB$ is smooth, then $\Theta(\CB)$ is well-defined, and it carries a canonical structure of Lie-\* algebroid over $\CB$. It is universal in the sense that for any Lie-\* algebroid $L$, its action on $\CB$ factors through a canonical map of Lie-\* algebroids $\varpi:L\to \Theta(\CB)$, called the anchor map. Recall now that a structure on $\CB$ of chiral-Poisson (or, coisson, in the terminology of [@BD]) algebra is a Lie-\* bracket (called chiral-Poisson bracket) $\CB\boxtimes \CB\to \Delta_!(\CB)$, satisfying the Leibniz rule with respect to the multiplication on $\CB$ (cf. [@BD], Sect. 1.4.18 and 2.6.). If $\CB$ is a chiral-Poisson algebra, $\Omega^1(\CB)$ acquires a unique structure of Lie-\* algebroid, such that the de Rham differential $d:\CB\to \Omega^1(\CB)$ is a map of Lie-\* algebras, and the composition $$\CB\boxtimes \CB\overset{d\times \on{id}}\longrightarrow \Omega^1(\CB)\boxtimes \CB\to \Delta_!(\CB)$$ coincides with the chiral-Poisson bracket. Following [@BD], Sect. 2.6.6, we call a chiral-Poisson structure on $\CB$ elliptic if (a) $\CB$ is smooth, (b) the anchor map $\varpi:\Omega^1(\CB)\to \Theta(\CB)$ is injective, and (c) $\on{coker}(\varpi)$ is a projective $\CB$-module of finite rank. \[coisson\] Finally, let us recall the definition of the chiral-Poisson structure on $\fz_\fg$. Consider the flat $\BC[[\hslash]]$-family of chiral algebras $\CA_{\fg,\hslash}$, corresponding to the pairing $\kappa_\hslash=\kappa_{crit}+\hslash\cdot \kappa_0$, where $\kappa_0$ is an arbitrary fixed non-zero invariant inner product. For two sections $a,b\in \fz_\fg$, consider two arbitrary sections $a_\hslash,b_\hslash\in \CA_{\fg,\hslash}$, whose values modulo $\hslash$ are $a$ and $b$ respectively, and consider $[a_\hslash,b_\hslash]\in \Delta_!(\CA_{\fg,\hslash})$. By assumption, the last expression vanishes modulo $\hslash$. Therefore the section $\frac{1}{\hslash}[a_\hslash,b_\hslash]\in \Delta_!(\CA_{\fg,\hslash})$ is well-defined. Moreover, its value mod $\hslash$ does not depend on the choice of $a_\hslash$ and $b_\hslash$. Therefore we obtain a map $$a,b\in \fz_\fg \; \mapsto \; \frac{1}{\hslash}[a_\hslash,b_\hslash] \; \on{mod} \; \hslash \in \Delta_!(\CA_{\fg,crit}),$$ and it is easy to see that its image belongs to $\Delta_!(\fz_\fg)$. Furthermore, it is straightforward to verify that the resulting map $\fz_\fg\boxtimes \fz_\fg\to \Delta_!(\fz_\fg)$ satisfies the axioms of the chiral-Poisson bracket, see [@BD], Sect. 2.7.1. Let us now describe in terms of [Theorem \[FF isom\]]{} above the Lie-\* algebroid $\Omega^1(\fz_\fg)$, resulting from the chiral-Poisson structure on $\fz_\fg$. First, recall from [@BD], Sect. 2.4.11, that if $\CM$ is a central module over a commutative chiral algebra $\CB$, then we can form a topological module, denoted $\hat{h}{}^{\CB}_x(\CM)$ over $\CB_x$. Applying this construction to $\CB=\fz_\fg$ and $\CM=\Omega^1(\fz_\fg)$ we obtain a topological Lie-\* algebroid $\CG_{crit}:=\hat{h}{}^{\fz_\fg}_x(\Omega^1(\fz_\fg))$ (see [@BD], Sect. 2.5.18 for details). Let now $\fF_x$ be the universal $^L G$-torsor over $\on{Op}_{^L G}({\mc D}_x)$, whose fiber over a given oper $({\mc F},{\mc F}_B,\nabla)\in \on{Op}_{^L G}({\mc D}_x)$ is the $^L G$-torsor of horizontal sections of $\F$, or, equivalently, the fiber of $\F$ at $x\in \D_x$. Let us denote by $^L \CG_{\on{Op}}$ the corresponding Atiyah algebroid over $\on{Op}_{^L G}({\mc D}_x)$, which by definition consists of $^L G$-invariant vector fields on the total space of $\fF_x$. We have a short exact sequence $$0\to (^L \fg)_{\fF_x}\to {}^L \CG_{\on{Op}}\to T\left(\on{Op}_{^L G}({\mc D}_x)\right)\to 0,$$ where $T\left(\on{Op}_{^L G}({\mc D}_x)\right)$ denotes the tangent algebroid, and $(^L \fg)_{\fF_x}$, which is the kernel of the anchor map, is the twist of the adjoint representation by the $^L G$-torsor $\fF_x$. The following was established in [@BD1], Theorem 3.6.7 (see also [@BD], Sect. 2.6.8), using the fact that the isomorphism of $D_X$-algebras, given by [Theorem \[FF isom\]]{}(1), respects the chiral-Poisson structures on both sides, where $J(\on{Op}_{^L G}(X))$ acquires a chiral-Poisson structure by its realization via the Drinfeld-Sokolov reduction. \[BD descr of Gelfand-Dikii\] [*(1)*]{} The chiral-Poisson structure on $\fz_\fg$ is elliptic. [*(2)*]{} Under the isomorphism $\on{Spec}(\fz_{\fg,x}) \simeq \on{Op}_{^L G}({\mc D}_x)$, the algebroid $\CG_{crit}$ corresponds to the algebroid $^L \CG_{\on{Op}}$. The renormalized chiral algebra {#renormalized algebra} =============================== We will now refine the structure of chiral-Poisson algebra on $\fz_\fg$ and obtain a chiral version of the renormalized universal enveloping algebra at the critical level introduced in [@BD1]. First, we introduce a Lie-\* algebra $\CA^\sharp_\fg$, which fits in a short exact sequence $$0\to \CA_{\fg,crit}\to \CA^\sharp_\fg\to \fz_\fg\to 0.$$ Namely, in the family of chiral algebras $\CA_{\fg,\hslash}$ consider the following subspace $\CA^\sharp_{\fg,\hslash}$, which contains $\CA_{\fg,\hslash}$ and is contained in $\frac{1}{\hslash}\cdot \CA_{\fg,\hslash}$: $$\CA^\sharp_{\fg,\hslash} = \{ \frac{a}{\hslash} \, | \, a \in \CA_{\fg,\hslash}, a \, \on{mod} \, \hslash \in \fz_\fg \}.$$ Define $\CA^\sharp_\fg$ as $\CA^\sharp_{\fg,\hslash}/\hslash\cdot \CA_{\fg,\hslash}$. By repeating the construction of the chiral-Poisson structure on $\fz_\fg$ from [Sect. \[coisson\]]{}, we obtain a Lie-\* algebra structure on $\CA^\sharp_\fg$. Note that the composition $$\fz_\fg\boxtimes \CA^\sharp_\fg\to \CA^\sharp_\fg\boxtimes \CA^\sharp_\fg\to \Delta_!(\CA^\sharp_\fg)$$ factors as $\fz_\fg\boxtimes \CA^\sharp_\fg\to \fz_\fg\boxtimes \fz_\fg\to \Delta_!(\fz_\fg)$, where the last arrow is chiral-Poisson bracket on $\fz_\fg$. \[BD algebroid\] There exist a unique Lie-\* algebroid $\CA^{\flat}_\fg$ over $\fz_\fg$, which fits into the following commutative diagram: $$\CD 0 @>>> \CA_{\fg,crit}/\fz_\fg @>>> \CA^{\flat}_\fg @>>> \Omega^1(\fz_\fg) @>>> 0 \\ @. @AAA @AAA @AAA @. \\ 0 @>>> \CA_{\fg,crit} @>>> \CA^\sharp_\fg @>>> \fz_\fg @>>> 0. \endCD$$ In the above diagram the rows are exact, and the rightmost vertical map is the de Rham differential $\fz_\fg\to \Omega^1(\fz_\fg)$. Recall the following general construction. Let $L$ be a Lie-\* algebra acting on a commutative chiral algebra $\CB$. Then we can form a central $\CB$-module $\ol{\on{Ind}}_\CB(L):=\CB\otimes L$, which will carry a natural structure of Lie-\* algebroid over $\CB$. This is analogous to the usual construction in differential geometry, when we have a Lie-\* algebra $\mathfrak l$ acting on a manifold $\Y$ and we form the algebroid $\CO_\Y\otimes {\mathfrak l}$. By taking $\CB=\fz_\fg$ and $L=\CA^\sharp_\fg/\fz_\fg$, we thus obtain a Lie-\* algebroid $\ol{\on{Ind}}_{\fz_\fg}(\CA^\sharp_\fg/\fz_\fg)$ on $\fz_\fg$. We have a short exact sequence $$0\to \fz_\fg\otimes (\CA_{\fg,crit}/\fz_\fg)\to \ol{\on{Ind}}_{\fz_\fg}(\CA^\sharp_\fg/\fz_\fg)\to \fz_\fg\otimes \fz_\fg\to 0.$$ To obtain from $\ol{\on{Ind}}_{\fz_\fg}(\CA^\sharp_\fg/\fz_\fg)$ the desired extension $\CA^{\flat}_\fg$, we need to take the quotient by two kinds of relations. First, we must pass from $\fz_\fg\otimes (\CA_{\fg,crit}/\fz_\fg)$ to just $\CA_{\fg,crit}/\fz_\fg$, using the structure of $\fz_\fg$-module on $\CA_{\fg,crit}$. Secondly, we must impose the Leibniz rule to pass from the free $\fz_\fg$-module $\fz_\fg\otimes \fz_\fg$ to $\Omega^1(\fz_\fg)$. We will impose these two relations simultaneously. Consider the following three maps $\CA^\sharp_\fg\otimes \CA^\sharp_\fg\to \ol{\on{Ind}}_{\fz_\fg}(\CA^\sharp_\fg/\fz_\fg)$: \(1) The first map is the projection $\CA^\sharp_\fg\otimes \CA^\sharp_\fg\to \fz_\fg\otimes (\CA^\sharp_\fg/\fz_\fg)$. \(2) The second map is the projection $\CA^\sharp_\fg\otimes \CA^\sharp_\fg\to (\CA^\sharp_\fg/\fz_\fg)\otimes \fz_\fg\simeq \fz_\fg\otimes (\CA^\sharp_\fg/\fz_\fg)$. \(3) To define the third map, note that chiral bracket on $\CA_{\fg,\hslash}$, multiplied by $\hslash$, induces a map $j_*j^*(\CA^\sharp_\fg\boxtimes \CA^\sharp_\fg)\to \Delta_!(\CA^\sharp_\fg)$. Composing the latter with the projection $\Delta_!(\CA^\sharp_\fg)\to \Delta_!(\CA^\sharp_\fg/\fz_\fg)$, we obtain a map that vanishes on $\CA^\sharp_\fg\boxtimes\CA^\sharp_\fg\subset j_*j^*(\CA^\sharp_\fg\boxtimes \CA^\sharp_\fg)$, thereby giving rise to a map $\CA^\sharp_\fg\otimes \CA^\sharp_\fg\to \fz_\fg\otimes (\CA^\sharp_\fg/\fz_\fg)$. By taking the linear combination of these three maps, namely (1)–(2)–(3), we obtain a new map $\CA^\sharp_\fg\otimes \CA^\sharp_\fg\to \fz_\fg\otimes (\CA^\sharp_\fg/\fz_\fg)$. We define $\CA^{\flat}_\fg$ as the quotient of $\ol{\on{Ind}}_{\fz_\fg}(\CA^\sharp_\fg/\fz_\fg)$ by the $\fz_\fg$-module, generated by the image of the latter map. One checks in a straightforward way that the Lie-\* bracket on $\ol{\on{Ind}}_{\fz_\fg}(\CA^\sharp_\fg/\fz_\fg)$ descends to a Lie-\* bracket on $\CA^{\flat}_\fg$, so that it becomes a Lie-\* algebroid over $\fz_\fg$. Moreover, by construction, we have a short exact sequence $$0\to (\CA_{\fg,crit}/\fz_\fg)'\to \CA^{\flat}_\fg\to \Omega^1(\fz_\fg)\to 0,$$ where $(\CA_{\fg,crit}/\fz_\fg)'$ is a quotient of $\CA_{\fg,crit}/\fz_\fg$. Let us show that $\CA_{\fg,crit}/\fz_\fg\to (\CA_{\fg,crit}/\fz_\fg)'$ is an isomorphism. Observe that the Lie-\* algebra $\CA^\sharp_\fg$ acts on $\CA_{\fg,crit}$. This action gives rise to an action of the Lie-\* algebroid $\ol{\on{Ind}}_{\fz_\fg}(\CA^\sharp_\fg/\fz_\fg)$ on $\CA_{\fg,crit}$, which is compatible with the $\fz_\fg$-module structure on $\CA_{\fg,crit}$, and, moreover, it descends to an action of the algebroid $\CA^{\flat}_\fg$ on $\CA_{\fg,crit}$. The resulting Lie-\* action of $\CA_{\fg,crit}$ on $\CA_{\fg,crit}$ obtained via $$\CA_{\fg,crit}\twoheadrightarrow (\CA_{\fg,crit}/\fz_\fg)'\hookrightarrow \CA^{\flat}_\fg$$ coincides with the initial Lie-\* action of $\CA_{\fg,crit}$ on itself. By the definition of the center, the kernel of the latter action is exactly $\fz_\fg$. \[flat modules\] Let us now introduce a category of modules over $\CA^{\flat}_\fg$, which will be of interest for us. First, note that the action of $\CA^\flat_\fg$ on $\CA_{\fg,crit}$, introduced in the course of the proof of [Proposition \[BD algebroid\]]{}, is compatible in the natural sense with the chiral bracket on $\CA_{\fg,crit}$. We define $\CA_{\fg}^{\flat}\text{--}\on{mod}$ to have as objects $\CM\in\CA_{\fg,crit}\text{--}\on{mod}$, endowed with an additional action of the Lie-\* algebroid $\CA^{\flat}_\fg$ (see [@BD], Sect. 2.5.16 and 1.4.12 for the definition of the latter), such that \(a) As a chiral module over $\fz_\fg$ (via $\fz_\fg\hookrightarrow \CA_{\fg,crit}$), $\CM$ is central. \(b) The two induced Lie-\* actions $(\CA_{\fg,crit}/\fz_\fg)\boxtimes \CM\to \Delta_!(\CM)$ (one coming from the $\CA^{\flat}_\fg$-action, and the other from the $\CA_{\fg,crit}$-action and point (a) above) coincide. \(c) The chiral action of $\CA_{\fg,crit}$ and the Lie-\* action of $\CA^{\flat}_\fg$ on $\CM$ are compatible with the Lie-\* action of $\CA_{\fg}^{\flat}$ on $\CA_{\fg,crit}$. One can show that the category $\CA_{\fg}^{\flat}\text{--}\on{mod}$ is tautologically equivalent to the category of (discrete) modules over the renormalized universal enveloping algebra introduced in [@BD1], Sect. 5.6.1. For example, it is easy to see that if $\CM_\hslash$ is a flat $\BC[[\hslash]]$-family of chiral $\CA_{\fg,\hslash}$-modules such that the chiral $\fz_\fg$-module $\CM:=\CM/\hslash\CM$ is central, then this $\CM$ is naturally an object of $\CA_{\fg}^{\flat}\text{--}\on{mod}$. In addition to the notion of a Lie-\* algebroid there is also the notion of a chiral Lie algebroid over a commutative chiral algebra $\CB$, see [@BD], Sect. 3.9.6. A Lie-\* algebra $L$ is called a [*chiral Lie algebroid over*]{} $\CB$ if we are given: \(1) An action $L\boxtimes \CB\to\Delta_!(\CB)$ of $L$ as a Lie-\* algebra on the commutative chiral algebra $\CB$, \(2) A chiral action $j_*j^*(\CB\boxtimes L)\to\Delta_!(L)$, compatible with the action of $L$ on $\CB$ and the bracket on $L$. \(3) A map $\eta:\CB\to L$, compatible with both the $\CB$- and $L$-actions, such that the following conditions are satisfied: \(a) The action in (1) is $\CB$-linear, in the sense that the two natural maps $j_*j^*(\CB\boxtimes L)\boxtimes \CB\to \Delta_!(\CB)$ on $X^3$ coincide, \(b) The map $L\boxtimes \CB\overset{\on{id}\times \eta}\longrightarrow L\boxtimes L \to\Delta_!(L)$ equals the negative of $\CB\boxtimes L\hookrightarrow j_*j^*(\CB\boxtimes L)\to \Delta_!(L)$, Note that if for a chiral Lie algebroid $L$ as above, the data of $\eta$ is zero, we retrieve the notion of Lie-\* algebroid. In most examples, however, the map $\eta$ is an injection. In this case, the data of (1) is completely determined by (2) and (3), and condition (a) is superfluous. It would be interesting to find out whether there exists a chiral Lie algebroid $\CA^{\ren}_\fg$ over $\fz_\fg$, which is an extension $$0\to \CA_{\fg,crit}\to \CA^{\ren}_\fg \to \Omega^1(\fz_\fg)\to 0,$$ such that the map $\CA^\sharp_\fg\to \CA^{\flat}_\fg$ lifts to a map $\CA^\sharp_\fg\to \CA^{\ren}_\fg$. However, we do not know how to construct such an object. Instead, we will construct another chiral Lie algebroid $\CA^{\ren,d}_\fg$, which is, in some sense, a double of $\CA^{\ren}_\fg$. The construction of $\CA^{\ren,d}_\fg$ below is in terms of generators and relations. In the next section we will give a natural construction of $\CA^{\ren,d}_\fg$ via chiral differential operators on the group $G$. Consider the Lie-\* algebra $\CA^{\sharp,d}_\fg$ equal to $$(\CA^\sharp_\fg\times \CA^\sharp_\fg) \underset{\fz_\fg\times \fz_\fg}\times \fz_\fg,$$ where the map $\fz_\fg\to \fz_\fg\times \fz_\fg$ is the anti-diagonal, i.e., $(\on{id},-\on{id})$. It fits into a short exact sequence $$0\to \CA_{\fg,crit}\times \CA_{\fg,crit}\to \CA^{\sharp,d}_\fg\to \fz_\fg\to 0.$$ \[doubled algebroid\] There exist a unique chiral algebroid $\CA^{\ren,d}_\fg$ over $\fz_\fg$, which fits into the following commutative diagram with exact rows: $$\CD 0 @>>> (\CA_{\fg,crit}\times\CA_{\fg,crit})/\fz_\fg @>>> \CA^{\ren,d}_\fg @>>> \Omega^1(\fz_\fg) @>>> 0 \\ @. @AAA @AAA @AAA @. \\ 0 @>>> \CA_{\fg,crit}\times\CA_{\fg,crit} @>>> \CA^{\sharp,d}_\fg @>>> \fz_\fg @>>> 0, \endCD$$ where $\fz_\fg\hookrightarrow \CA_{\fg,crit}\times\CA_{\fg,crit}$ is the anti-diagonal embedding. Let $L$ be a Lie-\* algebra acting on a commutative chiral algebra $\CB$, as in the proof of [Proposition \[BD algebroid\]]{}. Then, following [@BD], Sect. 3.9.9, one constructs a chiral Lie algebroid $\on{Ind}_\CB(L)$, which fits into a short exact sequence $$\label{induced algebroid} 0\to \CB\to \on{Ind}_\CB(L)\to \ol{\on{Ind}}_\CB(L)\to 0.$$ Indeed, consider the D-modules $j_*j^*(\CB\boxtimes L)$ and $\Delta_!(\CB)$ on $X\times X$. We have the maps $$j_*j^*(\CB\boxtimes L) \leftarrow \CB\boxtimes L \rightarrow \Delta_!(\CB),$$ where the left arrow is the natural inclusion, and the right arrow is the negative of the Lie-\* action. Then the quotient $\bigl(j_*j^*(\CB\boxtimes L)\oplus \Delta_!(\CB)\bigr)/\CB\boxtimes L$ is supported on the diagonal, and therefore corresponds to a D-module on $X$, which is by definition our $\on{Ind}_\CB(L)$. By construction, we have the inclusions $\eta:\CB\to \on{Ind}_\CB(L)$ and $L\to \on{Ind}_\CB(L)$, and a chiral action $j_*j^*(\CB\boxtimes L)\to \Delta_!(\on{Ind}_\CB(L))$. It is a straightforward verification to show that these data extend uniquely to a Lie-\* algebra structure on $\on{Ind}_\CB(L)$ and a chiral action of $\CB$ on $\on{Ind}_\CB(L)$, which satisfy the conditions of chiral Lie algebroid. Let us view $\CA^{\sharp,d}_\fg$ as a Lie-\* algebra, which acts on $\fz_\fg$ via $\CA^{\sharp,d}_\fg\to \fz_\fg$ and the chiral-Poisson bracket on $\fz_\fg$. Consider the chiral Lie algebroid $\on{Ind}_{\fz_\fg}(\CA^{\sharp,d}_\fg)$. As in the case of $\CA^\flat_\fg$ (see the proof of [Proposition \[BD algebroid\]]{}), to obtain from $\on{Ind}_{\fz_\fg}(\CA^{\sharp,d}_\fg)$ the desired chiral algebroid $\CA^{\ren,d}_\fg$, we must take the quotient by some additional relations. The first set of relations is that we must identify the three copies of $\fz_\fg$ inside $\on{Ind}_{\fz_\fg}(\CA^{\sharp,d}_\fg)$. One copy is the image of the canonical embedding $\fz_\fg\hookrightarrow \on{Ind}_{\fz_\fg}(\CA^{\sharp,d}_\fg)$ coming from the definition of the induced algebroid. The other two copies come from $\fz_\fg\times \fz_\fg\subset \CA^{\sharp,d}_\fg\hookrightarrow \on{Ind}_{\fz_\fg}(\CA^{\sharp,d}_\fg)$. When we identify them, we obtain a new chiral algebroid over $\fz_\fg$ which we denote by $\on{Ind}'_{\fz_\fg}(\CA^{\sharp,d}_\fg)$. The second set of relations is similar to what we had in the case of $\CA^\flat_\fg$: they amount to killing the chiral $\fz_\fg$-submodule generated by the image of a certain map $\CA^{\sharp,d}_\fg\otimes \CA^{\sharp,d}_\fg\to \on{Ind}'_{\fz_\fg}(\CA^{\sharp,d}_\fg)$. To construct this map, we consider three morphisms from the D-module $j_*j^*(\CA^{\sharp,d}_\fg\boxtimes \CA^{\sharp,d}_\fg)$ to $\Delta_!(\on{Ind}_{\fz_\fg}(\CA^{\sharp,d}_\fg))$: \(1) The first map is $j_*j^*(\CA^{\sharp,d}_\fg\boxtimes \CA^{\sharp,d}_\fg)\to j_*j^*(\fz_\fg\boxtimes \CA^{\sharp,d}_\fg)\to \Delta_!(\on{Ind}_{\fz_\fg}(\CA^{\sharp,d}_\fg))$, where the first arrow comes from the natural projection $\CA^{\sharp,d}_\fg\to \fz_\fg$. \(2) The second map is obtained from the first one by interchanging the roles of the factors in $j_*j^*(\CA^{\sharp,d}_\fg\boxtimes \CA^{\sharp,d}_\fg)$. \(3) To construct the third map, note that the chiral bracket on $\CA_{\fg,\hslash}$ gives rise to a map $$\hslash\cdot(\{\cdot,\cdot\},-\{\cdot,\cdot\}): j_*j^*(\CA^{\sharp,d}_\fg\boxtimes \CA^{\sharp,d}_\fg)\to\Delta_!(\CA^{\sharp,d}_\fg),$$ and we compose it with the canonical map $\Delta_!(\CA^{\sharp,d}_\fg)\to \Delta_!(\on{Ind}_{\fz_\fg}(\CA^{\sharp,d}_\fg))$. Consider the linear combination (1)-(2)-(3) of the above three maps as a new map from $j_*j^*(\CA^{\sharp,d}_\fg\boxtimes \CA^{\sharp,d}_\fg)$ to $\Delta_!(\on{Ind}_{\fz_\fg}(\CA^{\sharp,d}_\fg))$. It is easy to see that the composition $$\CA^{\sharp,d}_\fg\boxtimes \CA^{\sharp,d}_\fg\hookrightarrow j_*j^*(\CA^{\sharp,d}_\fg\boxtimes \CA^{\sharp,d}_\fg)\to \Delta_!(\on{Ind}_{\fz_\fg}(\CA^{\sharp,d}_\fg))\to \on{Ind}'_{\fz_\fg}(\CA^{\sharp,d}_\fg)$$ vanishes. Thus, we obtain the desired map $\CA^{\sharp,d}_\fg\otimes \CA^{\sharp,d}_\fg\to \on{Ind}'_{\fz_\fg}(\CA^{\sharp,d}_\fg)$. We define $\CA^{\ren,d}_\fg$ as the quotient of $\on{Ind}'_{\fz_\fg}(\CA^{\sharp,d}_\fg)$ by the chiral $\fz_\fg$-module, generated by the image of this map. By construction, $\CA^{\ren,d}_\fg$ is a chiral $\fz_\fg$-module. One readily checks that the Lie-\* bracket on $\on{Ind}'_{\fz_\fg}(\CA^{\sharp,d}_\fg)$ descends to a Lie-\* bracket on $\CA^{\ren,d}_\fg$, such that, together with the map $\fz_\fg\to \on{Ind}'_{\fz_\fg}(\CA^{\sharp,d}_\fg)\to \CA^{\ren,d}_\fg$, these data define on $\CA^{\ren,d}_\fg$ a structure of chiral Lie algebroid over $\fz_\fg$. As in the case of $\CA^\flat_\fg$, we have a short exact sequence $$0\to ((\CA_{\fg,crit}\times \CA_{\fg,crit})/\fz_\fg)'\to \CA^{\ren,d}_\fg\to \Omega^1(\fz_\fg)\to 0,$$ where $((\CA_{\fg,crit}\times \CA_{\fg,crit})/\fz_\fg)'$ is a certain quotient of $(\CA_{\fg,crit}\times \CA_{\fg,crit})/\fz_\fg$. Let us show that $$\label{inj} (\CA_{\fg,crit}\times \CA_{\fg,crit})/\fz_\fg\to ((\CA_{\fg,crit}\times \CA_{\fg,crit})/\fz_\fg)'$$ is in fact an isomorphism. Let $\CA^{\flat,d}_\fg$ be the Lie-\* algebroid over $\fz_\fg$ equal to $\CA^{\flat,d}_\fg/\fz_\fg$. We have a surjection $$\label{inj of ker anch} (\CA_{\fg,crit}/\fz_\fg)\times (\CA_{\fg,crit}/\fz_\fg)\twoheadrightarrow \on{ker}\left(\CA^{\flat,d}_\fg\to \Omega^1_{\fz_\fg}\right).$$ As in the case of $\CA^\flat_\fg$, we show that $\CA^{\flat,d}_\fg$ acts naturally on the chiral algebra $\CA_{\fg,crit}\underset{\fz_\fg}\otimes \CA_{\fg,crit}$. This implies that the map of is an isomorphism. Thus, it remains to show that the canonical map $\fz_\fg\to \CA^{\ren,d}_\fg$ is injective. If it were not so, the ideal $\on{ker}(\fz_\fg\to \CA^{\ren,d}_\fg)$ would be stable under the chiral-Poisson bracket on $\fz_\fg$. However, this is impossible, since the above chiral-Poisson structure is elliptic by [Theorem \[BD descr of Gelfand-Dikii\]]{}(1). We remark that the isomorphism of can be alternatively deduced from [Theorem \[embedding of algebroids\]]{} below. \[categories of modules\] Let $\CA_{\fg}^{\flat,d}$ be the Lie-\* algebroid introduced in the proof of [Proposition \[doubled algebroid\]]{}. We introduce the category $\CA_{\fg}^{\flat,d}\text{--}\on{mod}$ in a way analogous to $\CA_{\fg}^\flat\text{--}\on{mod}$. Namely, the objects of $\CA_{\fg}^{\flat,d}\text{--}\on{mod}$ are modules over the chiral algebra $\CA_{\fg,crit}\underset{\fz_\fg}\otimes \CA_{\fg,crit}$ (supported at $x\in X$) equipped with an extra Lie-\* action of the Lie-\* algebroid $\CA_{\fg}^{\flat,d}$ such that the conditions, analogous to (a), (b) and (c) in the definition of $\CA_{\fg}^\flat\text{--}\on{mod}$, hold. Next, we will introduce an appropriate category of chiral modules over $\CA_{\fg}^{\ren,d}$. First, recall from [@BD], Sect. 3.9.24, the notion of chiral module over a chiral Lie algebroid. If $L$ is a chiral algebroid over a commutative $D_X$-algebra $\CB$, there exists a canonical chiral algebra $U(\CB,L)$, such the category of chiral modules over $L$ (regarded as a chiral algebroid) is equivalent to the category of chiral modules over $U(\CB,L)$ as a chiral algebra. Now let us introduce the category $\CA_{\fg}^{\ren,d}\text{--}\on{mod}$. By definition, its objects are, as before, D-modules $\CM$ on $X$ supported at $x$ equipped with \(1) An action of the chiral algebra $\CA_{\fg,crit}\underset{\fz_\fg}\otimes \CA_{\fg,crit}$, \(2) An action of the chiral algebroid $\CA_{\fg}^{\ren,d}$, such that the two induced chiral brackets $j_*j^*\left((\CA_{\fg,crit}\times\CA_{\fg,crit}/\fz_\fg) \boxtimes \CM\right)\to \Delta_!(\CM)$ coincide. Observe that one can reformulate the definition of $\CA_{\fg}^{\ren,d}\text{--}\on{mod}$ as modules (supported at $x\in X$) over a certain chiral algebra. Namely, let $U^{\ren,d}(L_{\fg,crit})$ be the quotient of the chiral algebra $U(\fz_\fg,\CA_{\fg}^{\ren,d})$ by the following relation: We have a map $U((\CA_{\fg,crit}\times\CA_{\fg,crit})/\fz_\fg)\to U(\fz_\fg,\CA_{\fg}^{\ren,d})$ coming from the embedding of Lie-\* algebras $(\CA_{\fg,crit}\times\CA_{\fg,crit})/\fz_\fg\to \CA_{\fg}^{\ren,d}$. In addition, we have a map $U((\CA_{\fg,crit}\times\CA_{\fg,crit})/\fz_\fg)\to \CA_{\fg,crit}\underset{\fz_\fg}\otimes \CA_{\fg,crit}$. We need to kill the ideal in $U(\fz_\fg,\CA_{\fg}^{\ren,d})$ generated by the image of the kernel of the latter map. We define a PBW-type filtration on $U^{\ren,d}(L_{\fg,crit})$, by setting $F^0\left(U^{\ren,d}(L_{\fg,crit})\right)$ to be the image of $\CA_{\fg,crit}\underset{\fz_\fg}\otimes \CA_{\fg,crit}$, and by requiring inductively that $F^{i+1}\left(U^{\ren,d}(L_{\fg,crit})\right)$ is the smallest $D_X$-submodule such that $$j_*j^*\left((\CA^{\ren,d}_\fg\boxtimes F^i\left(U^{\ren,d}(L_{\fg,crit})\right)\right)\to \Delta_!\left(F^{i+1}\left(U^{\ren,d}(L_{\fg,crit})\right)\right) \text{ and }$$ $$j_*j^*\left((\CA_{\fg,crit}\underset{\fz_\fg}\otimes \CA_{\fg,crit})\boxtimes F^{i+1}\left(U^{\ren,d}(L_{\fg,crit})\right)\right)\to \Delta_!\left(F^{i+1}\left(U^{\ren,d}(L_{\fg,crit})\right)\right).$$ In this case we automatically have also: $$\CA^{\ren,d}_\fg\boxtimes F^i\left(U^{\ren,d}(L_{\fg,crit})\right)\to \Delta_!\left(F^{i+1}\left(U^{\ren,d}(L_{\fg,crit})\right)\right).$$ We have a natural surjection on the associated graded level: $$\label{ass graded} (\CA_{\fg,crit}\underset{\fz_\fg}\otimes \CA_{\fg,crit}) \underset{\fz_\fg}\otimes \on{Sym}_{\fz_\fg}(\Omega^1(\fz_\fg)) \twoheadrightarrow \on{gr}\left(U^{\ren,d}(L_{\fg,crit})\right).$$ From [@BD], Theorem 3.9.12 it follows that this map is an isomorphism. \[twist by tau\] Let now $\tau$ be an automorphism of $\fz_\fg$ as a chiral-Poisson algebra. We can form the Lie-\* algebra $$\CA^{\sharp,\tau}_\fg:=(\CA^\sharp_\fg\times \CA^\sharp_\fg) \underset{\fz_\fg\times \fz_\fg}\times \fz_\fg,$$ where the map $\fz_\fg\to \fz_\fg\times \fz_\fg$ is now $(\on{id},-\tau)$. Repeating the construction of [Proposition \[doubled algebroid\]]{}, we obtain a chiral algebroid $\CA^{\ren,\tau}_\fg$, which fits in a short exact sequence $$0\to (\CA_{\fg,crit}\times\CA_{\fg,crit})/\fz_\fg\to \CA^{\ren,\tau}_\fg\to \Omega^1(\fz_\fg)\to 0,$$ where $\fz_\fg$ is embedded into $\CA_{\fg,crit}\times\CA_{\fg,crit}$ also via $(\on{id},-\tau)$. We will denote by $\CA^{\flat,\tau}_\fg$ the Lie-\* algebroid on $\fz_\fg$ equal to the quotient $\CA^{\ren,\tau}_\fg/\fz_\fg$. Finally, in a way similar to the above, we introduce the corresponding categories of modules, $\CA_{\fg}^{\flat,\tau}\text{--}\on{mod}$ and $\CA_{\fg}^{\ren,\tau}\text{--}\on{mod}$, and the chiral algebra $U^{\ren,\tau}(L_{\fg,crit})$. Note, however, that according to [@BD1] (and which something that we will have to use later), every automorphism of $\fz_\fg$, respecting the chiral-Poisson structure, comes from an outer automorphism of $\fg$. This implies that, as abstract algebroids, $\CA^{ren,\tau}_\fg$ and $\CA^{ren,d}_\fg$ are, in fact, isomorphic. Chiral differential operators at the critical level {#diff op} =================================================== Recall that $\fD_{G,\kappa}$ denotes the chiral algebra of differential operators on the group $G$ at level $\kappa$ (see [@AG]), and $\fl,\fr$ are the two embeddings $$\CA_{\fg,\kappa}\overset{\fl} \to \fD_{G,\kappa}\overset{\fr}\leftarrow \CA_{\fg,2\kappa_{crit}-\kappa}.$$ Recall that if $\CM$ is a Lie-\* module over a Lie-\* algebra $L$, the centralizer of $L$ is the maximal D-submodule $\CM'\subset \CM$ such that the Lie-\* bracket $L\boxtimes \CM'\to \Delta_!(\CM)$ vanishes. \[two centralizers\] The centralizer of $\fl(\CA_{\fg,\kappa})$ in $\fD_{G,\kappa}$ equals $\fr(\CA_{\fg,2\kappa_{crit}-\kappa})$. Conversely, the centralizer of $\fr(\CA_{\fg,2\kappa_{crit}-\kappa})$ equals $\fl(\CA_{\fg,\kappa})$. The inclusion of $\fl(\CA_{\fg,\kappa})$ into the centralizer of $\fr(\CA_{\fg,2\kappa_{crit}-\kappa})$ is just the fact that the images of $\fl$ and $\fr$ Lie-\* commute with each other. The fact that this inclusion is an equality is established as follows. Let $\hat\fg_\kappa$ be the affine Kac-Moody algebra corresponding to a point $x\in X$. The fiber $\CA_{\fg,\kappa,x}$ of $\CA_{\fg,\kappa}$ at $x$ is a $\hat\fg_\kappa$-module, equal to the vacuum module $\BV_{\fg,\kappa}$. Denote by $\fD_{G,\kappa,x}$ the fiber of $\fD_{G,\kappa}$ at $x$. This is a module over $\hat\fg_\kappa\times \hat{\fg}_{2\kappa_{crit}-\kappa}$. Recall that as $\hat\fg_\kappa$-module, $\fD_{G,\kappa,x}$ is the induced module $$\on{Ind}_{\fg(\wh{\CO}_x)\oplus \BC {\mb 1}}^{\hat{\fg}_\kappa} \left(\on{Fun}\left(G(\wh{\CO}_x)\right) \right).$$ Moreover, the commuting right action of $\fg(\wh{\CO}_x) \subset \hat{\fg}_{2\kappa_{crit}-\kappa}$ comes by transport of structure from the right action of $\fg(\wh{\CO}_x)$ on $\on{Fun}\left(G(\wh{\CO}_x)\right)$. In other words, as a right $\fg(\wh{\CO}_x)$-module, $$\on{Ind}_{\fg(\wh{\CO}_x)\oplus \BC {\mb 1}}^{\hat{\fg}_\kappa} \left(\on{Fun} \left(G(\wh{\CO}_x)\right) \right)\simeq U(\fg \otimes t^{-1}\BC[t^{-1}]) \otimes \on{Fun} \left(G(\wh{\CO}_x)\right),$$ where $\fg(\wh{\CO}_x)$ acts through the second factor and $t$ is a uniformizer in $\wh{\CO}_x$. At the level of fibers, the embedding $\fl$ is just the natural embedding $$\BV_{\fg,\kappa}\simeq \on{Ind}_{\fg(\wh{\CO}_x)\oplus \BC {\mb 1}}^{\hat{\fg}_\kappa} (\BC)\to \on{Ind}_{\fg(\wh{\CO}_x)\oplus \BC {\mb 1}}^{\hat{\fg}_\kappa} \left(\on{Fun} \left(G(\wh{\CO}_x)\right) \right)$$ corresponding to the unit $\BC\to \on{Fun} \left(G(\wh{\CO}_x)\right)$. We have to show that $\BV_{\fg,\kappa}\subset \fD_{G,\kappa,x}$ equals $\left(\fD_{G,\kappa,x}\right)^{\fg(\wh{\CO}_x)}$, for $\fg(\wh{\CO}_x)\subset \hat\fg_{2\kappa_{crit}-\kappa}$. But this immediately follows from the above description of $\fD_{G,\kappa,x}$ as a $\fg(\wh{\CO}_x)$-module. To finish the proof, observe that the roles of $\fl$ and $\fr$ in the definition of $\fD_{G,\kappa}$ are symmetric, and in particular, $\fD_{G,\kappa,x}$ is isomorphic to $\on{Ind}_{\fg(\wh{\CO}_x)\oplus \BC}^{\hat{\fg}_{2\kappa_{crit}-\kappa}} \left(\on{Fun} \left(G(\wh{\CO}_x)\right) \right)$ as a $\hat\fg(\wh{\CO}_x)\times \hat{\fg}_{2\kappa_{crit}-\kappa}$-module. Indeed, we have a map from the latter to the former, by the definition of the induction, and this is map is clearly an isomorphism at the level of associate graded spaces, by the PBW theorem. Now we specialize to $\kappa=\kappa_{crit}$. Then $\fl$ and $\fr$ are two different embeddings of $\CA_{\fg,crit}$ into $\fD_{G,crit}$. [Lemma \[two centralizers\]]{} implies the following: \[centers\] $\fl(\fz_\fg)=\fl(\CA_{\fg,crit})\cap \fr(\CA_{\fg,crit})=\fr(\fz_\fg)$. Let $\tau$ be the involution of the Dynkin diagram of $\fg$, which sends a weight $\lambda$ to $-w_0(\lambda)$. We lift $\tau$ to an outer automorphism of $\fg$, and it gives rise to a canonically defined involution of $\fz_\fg$, which we will also denote by $\tau$. \[embedding of algebroids\] The two compositions $\fz_\fg\hookrightarrow \CA_{\fg,crit}\overset{\fl}\to \fD_{G,crit}$ and $\fz_\fg\hookrightarrow \CA_{\fg,crit}\overset{\fr}\to \fD_{G,crit}$ are intertwined by the automorphism $\tau:\fz_\fg\to \fz_\fg$. We have an embedding of the chiral algebroid $\CA^{\ren,\tau}_\fg$ into $\fD_{G,crit}$ such that the maps $\fl$ and $\fr$ are the compositions of this embedding and the canonical maps $$\CA_{\fg,crit}\rightrightarrows (\CA_{\fg,crit}\times \CA_{\fg,crit})/\fz_\fg \to \CA^{\ren,\tau}_\fg.$$ This embedding extends to a homomorphism of chiral algebras $U^{\ren,\tau}(L_{\fg,crit})\to \fD_{G,crit}$. The rest of this section is devoted to the proof of this theorem. The first step will be to construct a map $$\psi:\Omega^1(\fz_\fg)\to \fD_{G,crit}/ \bigl(\fl(\CA_{\fg,crit})+\fr(\CA_{\fg,crit})\bigr).$$ Note that if $\CM$ is a central module over a commutative chiral algebra $\CB$, we have a naturally defined notion of derivation $\CB\to \CM$, which amounts to a map of $\CB$-modules $\Omega^1(\CB)\to \CM$. We take $\CB=\fz_\fg$, and $\CM$ to be the centralizer of $\fz_\fg$ in the chiral $\fz_\fg$-module $\fD_{G,crit}/\left(\fl(\CA_{\fg,crit})+\fr(\CA_{\fg,crit})\right)$. Thus, we need to construct a map $$\fz_\fg\to \fD_{G,crit}/\left(\fl(\CA_{\fg,crit})+\fr(\CA_{\fg,crit})\right),$$ whose image Lie-\* commutes with $\fz_\fg$, and which satisfies the Leibniz rule. By letting the level $\kappa$ vary in the $\BC[[\hslash]]$-family $\kappa_{\hslash}$, we obtain a flat $\BC[[\hslash]]$-family of chiral algebras $\fD_{G,\hslash}$. Note that the map $\fl$ extends to a map $\fl_\hslash:\CA_{\fg,\hslash}\to \fD_{G,\hslash}$, whereas the map $\fr$ gives rise to a map $\fr_\hslash:\CA_{\fg,-\hslash}\to \fD_{G,\hslash}$ (the negative appears due to the sign inversion in $\kappa\mapsto 2\kappa_{crit}-\kappa$). Let $a$ be an element of $\fz_\fg$, and choose elements $a'_\hslash\in \CA_{\fg,\hslash}$ and $a''_{-\hslash}\in \CA_{\fg,-\hslash}$, which map to $a$ mod $\hslash$. Consider the element $\fl_\hslash(a'_\hslash)-\fr_\hslash(a''_{-\hslash})\in \fD_{G,\hslash}$. By definition, it vanishes mod $\hslash$; hence we obtain an element $$\frac{\fl_\hslash(a'_\hslash)-\fr_\hslash(a''_{-\hslash})}{\hslash} \; \on{mod} \; \hslash\in \fD_{G,crit}$$ which is well-defined modulo $\fl(\CA_{\fg,crit})+\fr(\CA_{\fg,crit})$. This defines the required map. The fact that it is a derivation is a straightforward verification. Note that the chiral bracket on $\fD_{G,crit}$ gives rise to a well-defined Lie-\* bracket $$\fz_\fg\boxtimes \bigl(\fD_{G,crit}/\left(\fl(\CA_{\fg,crit})+\fr(\CA_{\fg,crit})\right)\bigr)\to \Delta_!\left(\fD_{G,crit}\right),$$ where $\fz_\fg$ is thought of as embedded into $\fD_{G,crit}$ via $\fz_\fg\hookrightarrow \CA_{\fg,crit}\overset{\fl}\to \fD_{G,crit}$. \[compatibility with Poisson\] The composition $$\fz_\fg\boxtimes \Omega^1(\fz_\fg)\overset{\fl\times \psi}\rightarrow \fz_\fg\boxtimes \left(\fD_{G,crit}/\left(\fl(\CA_{\fg,crit})+\fr(\CA_{\fg,crit})\right)\right)\to \Delta_!\left(\fD_{G,crit}\right)$$ factors as $\fz_\fg\boxtimes \Omega^1(\fz_\fg)\to \Delta_!(\fz_\fg)\to \Delta_!\left(\fD_{G,crit}\right)$, where the first arrow is the chiral-Poisson structure on $\fz_\fg$. A similar assertion holds for $\fz_\fg$ mapping to $\fD_{G,crit}$ via $\fr$. For two sections $a,b\in \fz_\fg$, and $a'_\hslash,a''_{-\hslash}$ as above, we have $$[\fl_\hslash(a'_\hslash)- \fr_\hslash(a''_{-\hslash}),\fl_\hslash(b_\hslash)]= [\fl_\hslash(a'_\hslash),\fl_\hslash(b_\hslash)]=\fl_\hslash([a'_\hslash,b_\hslash]),$$ because the images of $\fl_\hslash$ and $\fr_\hslash$ Lie-\* commute in $\fD_{G,\hslash}$. Hence, the assertion follows from the definition of the chiral-Poisson structure on $\fz_\fg$. \[jet construction\] Since the images of $\fz_\fg$ in $\fD_{G,crit}$ under $\fl$ and $\fr$ coincide, we obtain that there exists an automorphism $\tau'$ of $\fz_\fg$, as a commutative chiral algebra such that $\fl|_{\fz_\fg}=\fr|_{\fz_\fg}\circ \tau'$. Our goal now is to show that $\tau'=\tau$. [Lemma \[compatibility with Poisson\]]{} implies that $\tau'$ is in fact an automorphism of $\fz_\fg$ as a chiral-Poisson algebra. According to Proposition 3.5.13 and Theorem 3.6.7 of [@BD1], the chiral-Poisson structure on $\fz_\fg$ is rigid, i.e., its group of automorphisms equals the group of automorphisms of the Dynkin diagram of $\fg$. Therefore, in order to prove that $\tau'=\tau$, it suffices to show, that the two automorphisms coincide at the associate graded level. Recall that if $\C$ is a commutative $\CO_X$-algebra, $\CJ(\C)$ denotes the corresponding commutative chiral algebra, obtained by the jet construction from $\C$ (see [@BD], Sect. 2.3.2). Recall that the chiral algebras $\CA_{\fg,\kappa}$ and $\fD_{G,crit}$ are naturally filtered (see [@BD], Sect. 3.7.13 and 3.9.11), and we have: $$\on{gr}(\CA_{\fg,\kappa}) \simeq \CJ\left(\on{Sym}(\fg\otimes \omega^{\otimes -1}_X)\right) \simeq \CJ\left(\on{Fun}\left( \fg^* \times_{\BG_m} \omega_X\right) \right)$$ and $$\on{gr}(\fD_{G,\kappa})\simeq \CJ(\CO_G\otimes \CO_X)\otimes \CJ\left(\on{Sym}(\fg\otimes \omega^{\otimes -1}_X)\right)\simeq \CJ\left(\on{Fun}\left( T^*G \times_{\BG_m} \omega_X \right)\right),$$ so that the maps $\on{gr}(\fl)$ and $\on{gr}(\fr)$ come from the (moment) maps $T^*G\rightrightarrows \fg^*$ corresponding to the action of $\fg$ on $G$ by left and right translation, respectively. Moreover, $$\on{gr}(\fz_\fg)\hookrightarrow \CJ\left(\on{Sym}(\fg\otimes \omega^{\otimes -1}_X)^G\right) \simeq \CJ\left(\on{Fun}\left( \fg^*/G \times_{\BG_m} \omega_X\right) \right).$$ (This inclusion is, in fact, an equality, by [Theorem \[FF isom\]]{}(2).) Therefore, the required assertion follows from the fact that the two maps $T^*G\rightrightarrows \fg^*\to \fg^*/G$ differ by the automorphism $\tau$. To finish the proof of [Theorem \[embedding of algebroids\]]{}, we will identify $\CA^{\ren,\tau}_\fg$ with $$\Omega^1(\fz_\fg) \underset{\fD_{G,crit}/\left(\fl(\CA_{\fg,crit}) + \fr(\CA_{\fg,crit})\right)}\times \fD_{G,crit}.$$ Note that the construction of the map $\psi$ gives in fact a map $\CA^{\sharp,\tau}_\fg/\fz_\fg\to \fD_{G,crit}$. Indeed, a section of $\CA^{\sharp,\tau}_\fg$ has a form $\frac{(a'_\hslash,a''_{-\hslash})}{\hslash}$ for $a'_\hslash\in \CA_{\fg,\hslash}$, $a''_{-\hslash}\in \CA_{\fg,-\hslash}$, such that $$a'\, mod\, \hslash=-\tau(a'') \; \on{mod} \; \hslash\in \fz_\fg.$$ We associate to it a section of $\fD_{G,crit}$ equal to $\frac{\fl_\hslash(a'_\hslash)+\fr_\hslash(a''_{-\hslash})}{\hslash}$. In addition, $\fD_{G,crit}$ is obviously a chiral $\fz_\fg$-module, so we obtain a map $\on{Ind}_{\fz_\fg}(\CA^{\sharp,\tau}_\fg)\to \fD_{G,crit}$, and it is straightforward to check that the relations, defining $\CA^{\ren,\tau}_\fg$ as a quotient of $\on{Ind}_{\fz_\fg}(\CA^{\sharp,\tau}_\fg)$, hold. Finally, we obtain a homomorphism of chiral algebras $U(\fz_\fg,\CA^{\ren,\tau}_\fg)\to \fD_{G,crit}$, and it is easy to see that it annihilates the ideal defining $U^{\ren,\tau}(L_{\fg,crit})$ as a quotient of $U(\fz_\fg,\CA^{\ren,\tau}_\fg)$. The functor of global sections on the affine Grassmannian {#sections on Gr} ========================================================= Let $\fD_{G,crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}$ be the category of chiral $\fD_{G,crit}$-modules supported at the point $x\in X$, which are $G(\wh{\CO}_x)$-integrable with respect to the embedding $\fr:\CA_{\fg,crit}\to \fD_{G,crit}$. Let $\F$ be a critically twisted D-module on $\Gr_G$, and $\CM_\F$–the corresponding object of $\fD_{G,crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}$. According to [Theorem \[AG\]]{}, $$\label{relate} \Gamma(\Gr_G,\F)\simeq \on{Hom}_{\fg(\wh{\CO}_x)}(\BC,\CM_\F)\simeq \on{Hom}_{\hat\fg_{crit}}(\BV_{\fg,crit},\CM_\F),$$ where $\CM_\F$ is regarded as a $\hat\fg_{crit}$-module via $\fr$, and $\BV_{\fg,crit}\simeq \on{Ind}^{\hat\fg_{crit}}_{\fg(\wh{\CO}_x)\oplus \BC {\mb 1}}(\BC)$ is the vacuum module, i.e., the fiber $\CA_{\fg,crit,x}$ of $\CA_{\fg,crit}$ at $x$. Recall that $\hat\fg_{crit}\text{--}\on{mod}$ denotes the category of all discrete $\hat\fg_{crit}$-modules supported at $x\in X$, and let $\hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}$ be the subcategory of $G(\wh{\CO}_x)$-integrable modules. Obviously, $\BV_{\fg,crit}$ belongs to $\hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}$, but the main difficulty in the proof of [Theorem \[main\]]{} is that, in contrast to the negative or irrational level cases, $\BV_{\fg,crit}$ is not projective in this category. Let now $\hat\fg_{crit}\text{--}\on{mod}_{\reg}$ denote the subcategory of $\hat\fg_{crit}\text{--}\on{mod}$ consisting of modules, which are central (cf. [@BD], Sect. 3.3.7) with respect to the action of $\fz_\fg$. Let us denote by $\hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\reg}$ the intersection $\hat\fg_{crit}\text{--}\on{mod}_{\reg}\cap \hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}$. The module $\BV_{\fg,crit}$ belongs to $\hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\reg}$, but the modules from $\fD_{G,crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}$, regarded as objects of $\hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}$, do not belong there. The following projectivity result is essentially due to [@BD1] (see [Sect. \[other results\]]{} for the proof). \[drinfeld\] The module $\BV_{\fg,crit}$ is a projective generator of the category $\hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\reg}$. In particular, the functor $\hat\fg_{crit}-\on{mod}^{G(\wh{\CO}_x)}_{\reg} \to \on{Vect}$ given by $\CM \mapsto \on{Hom}_{\hat\fg_{crit}}(\BV_{\fg,crit},\CM)$, is exact. Consider the functors $$\begin{aligned} \sF: \hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\reg} &\to \fz_{\fg,x}\text{--}\on{mod}, \qquad {\mc M} \mapsto \Hom_{\hat\fg_{crit}}(\BV_{\fg,crit},\CM), \\ \sG: \fz_{\fg,x}\text{--}\on{mod} &\to \hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\reg}, \qquad {\mc F} \mapsto \BV_{\fg,crit} \underset{\fz_{\fg,x}}\otimes {\mc F}.\end{aligned}$$ Now [Theorem \[drinfeld\]]{} implies the following: \[Drinfeld’s equivalence of categories\] The functors $\sF$ and $\sG$ are mutually inverse equivalences of categories. By combining this theorem with [Theorem \[FF isom\]]{}, we obtain that the category $\hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\reg}$ is equivalent to the category of quasicoherent sheaves on the scheme $\on{Op}_{^L G}({\mc D}_x)$. \[functors\] Consider the functor $\imath^!:\FZ_{\fg,x}\text{--}\on{mod}\longrightarrow \fz_{\fg,x}\text{--}\on{mod}$, which takes a $\FZ_{\fg,x}$-module to its maximal submodule, scheme-theoretically supported on $\on{Spec}(\fz_{\fg,x})$, i.e., for an object $\CM\in \FZ_{\fg,x}\text{--}\on{mod}$, $\imath^!(\CM)$ consists of elements annihilated by $\on{ker}(\FZ_{\fg,x}\to \fz_{\fg,x})$. We will denote by the same symbol $\imath^!$ the corresponding functors $$\hat\fg_{crit}\text{--}\on{mod} \to \hat\fg_{crit}\text{--}\on{mod}_{\reg} \qquad \on{and} \qquad \hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)} \to \hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\reg}.$$ According to formula , the functor of global sections $\Gamma: \on{D}_{crit}(\Gr_G)\text{--}\on{mod} \to \on{Vect}$ can be viewed as a functor $\fD_{G,crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}\to \on{Vect}$ given by $\CM\mapsto \on{Hom}_{\hat\fg_{crit}}(\BV_{\fg,crit},\CM)$. Since $\BV_{\fg,crit}$ is supported on $\on{Spec}(\fz_{\fg,x})$, we obtain that this functor factors as $$\CM\mapsto \imath^!(\CM)\mapsto \on{Hom}_{\hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\reg}}(\BV_{\fg,crit},\imath^!(\CM)).$$ But according to [Theorem \[drinfeld\]]{}, the second functor is exact. Therefore [Theorem \[main\]]{} is equivalent to the following: \[critical reformulation\] The composition $$\fD_{G,crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}\to \hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)} \overset{\imath^!}\longrightarrow \hat\fg_{crit} \text{--}\on{mod}^{G(\wh{\CO}_x)}_{\reg},$$ where the first arrow is the forgetful functor corresponding to the embedding $\fr$, is exact. Let $\FZ_{\fg,x}\text{--}\on{mod}_{\nil}$ be the full subcategory of $\FZ_{\fg,x}\text{--}\on{mod}$, whose objects are those modules, which are set-theoretically supported on $\on{Spec}(\fz_{\fg,x})\subset \on{Spec}(\FZ_{\fg,x})$, i.e., modules supported on the formal neighborhood of $\on{Spec}(\fz_{\fg,x})$. Let $\hat\fg_{crit}\text{--}\on{mod}_{\nil}$, (resp., $\hat\fg_{crit}\text{--}\on{mod}_{\nil}^{G(\wh{\CO}_x)}$) denote the corresponding full subcategory of $\hat\fg_{crit}\text{--}\on{mod}$ (resp., $\hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}$). Let $\wt{\imath}{}^!:\FZ_{\fg,x}\text{--}\on{mod}\longrightarrow \FZ_{\fg,x}\text{--}\on{mod}_{\nil}$ be the functor that attaches to a $\FZ_{\fg,x}$-module its maximal submodule, which is supported on the formal neighborhood of $\on{Spec}(\fz_{\fg,x})$. In other words, for $\CM\in \FZ_{\fg,x}\text{--}\on{mod}$, $\wt{\imath}{}^!(\CM)$ consists of all sections annihilated by some power of the ideal of $\on{Spec}(\fz_{\fg,x})$ in $\on{Spec}(\FZ_{\fg,x})$. We will denote in the same way the corresponding functors $$\hat\fg_{crit}\text{--}\on{mod} \to \hat\fg_{crit}\text{--}\on{mod}_{\nil} \qquad \on{and} \qquad \hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)} \to \hat\fg_{crit}\text{--}\on{mod}_{\nil}^{G(\wh{\CO}_x)}.$$ Clearly, $\imath^!\simeq \imath^!\circ \wt{\imath}{}^!$. \[restriction to formal\] Every object $\CM\in\hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}$ can be canonically decomposed as a direct sum $\CM=\CM^{\nil}\oplus \CM^{\on{non-reg}}$, where $\wt{\imath}{}^!(\CM^{\nil})\simeq \CM^{\nil}$ and $\wt{\imath}{}^!(\CM^{\on{non-reg}})$ is supported away from $\on{Spec}(\fz_{\fg,x})\subset \on{Spec}(\FZ_{\fg,x})$. \[exactness of restr to formal\] The functor $\wt{\imath}{}^!:\hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)} \to \hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\nil}$ is exact. (of [Proposition \[restriction to formal\]]{}) For an irreducible $\fg$-module $V^\lambda$ with a dominant highest weight $\lambda \in \Lambda^+$, let $\BV^\lambda_{\fg,crit}$ be the corresponding Weyl module in $\hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}$, as defined in [Sect. \[Weyl\]]{}; in particular, $\BV_{\fg,crit}=\BV^0_{\fg,crit}$. Let $\Y^\lambda\subset \on{Spec}(\FZ_{\fg,x})$ be the closed sub ind-scheme corresponding to the annihilating ideal of $\BV^\lambda_{\fg,crit}$ in $\FZ_{\fg,x}$. In particular, $\Y^0 = \on{Spec}(\fz_{\fg,x})$. By definition, every object in the category $\hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}$ has a filtration whose successive quotients are generated by vectors, on which the subalgebra $\fg\otimes t\BC[[t]]\subset \fg(\wh{\CO}_x)$ acts trivially. In particular, such a subquotient is a quotient of $\BV^\lambda_{\fg,crit}$ for some $\lambda$. Therefore, the support in $\on{Spec}(\FZ_{\fg,x})$ of every object from $\hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}$ is contained in the union of the formal neighborhoods of $\Y^\lambda$ for $\lambda\in \Lambda^+$. \[Sugawara calculation\] For $\lambda\neq 0$, $\Y^\lambda \cap \on{Spec}(\fz_{\fg,x}) = \emptyset$. This lemma implies the proposition. Indeed, for $\CM\in \hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}$ we define $\CM^{\nil}$ to be direct summand of $\CM$ supported on the formal neighbourhood of $\Y^0$, and $\CM^{\on{non-reg}}$ to be the direct summand supported on the union of the formal neighborhoods of $\Y^\la$ with $\la \neq 0$. (of [Lemma \[Sugawara calculation\]]{}) Recall the operator $S_0$ given by formula . At the critical level this operator commutes with the action of $\hat\fg_{crit}$, i.e., it belongs to $\FZ_{\fg,x}$. But according to formula , $S_0$ acts on $V^\lambda \subset \BV^\lambda_{\fg,crit}$, and hence on the entire $\BV^\lambda_{\fg,crit}$, by the scalar $C_\fg(\la)$ equal to the value of the Casimir operator on $V^\lambda$. This scalar is zero for $\lambda=0$ and non-zero for $\lambda\neq 0$. This proves the lemma. Recall from [Sect. \[comm alg\]]{} that if $\CE$ is a group ind-subscheme of the normal bundle $N(\fz_{\fg,x})$, we can introduce the subcategory $\FZ_{\fg,x}\text{--}\on{mod}_\CE$, such that $$\fz_{\fg,x}\text{--}\on{mod}\subset \FZ_{\fg,x}\text{--}\on{mod}_\CE\subset \FZ_{\fg,x}\text{--}\on{mod}.$$ We define $\CE$ as follows. By [Theorem \[FF isom\]]{}(1), the $D_X$-algebra $\fz_\fg$ is non-canonically isomorphic to a free algebra. This implies, in particular, that the fiber $\Theta(\fz_\fg)_x$ of $\Theta(\fz_\fg)$ at $x$ is locally free of countable rank over $\fz_{\fg,x}$, and $N(\fz_{\fg,x})$ can be identified with the total space of the resulting vector bundle. Therefore, to specify a group ind-subscheme $\CE\subset N(\fz_{\fg,x})$ it would be sufficient to specify a $\fz_{\fg,x}$-submodule in $\Theta(\fz_\fg)_x$, which is locally a direct summand. Such a submodule is given by the image of the anchor map $\varpi: \Omega^1(\fz_\fg)_x\to \Theta(\fz_\fg)_x$; it is locally a direct summand, as follows from [Theorem \[BD descr of Gelfand-Dikii\]]{}(1). For $\CE$ defined in this way, let us denote by $\hat\fg_{crit}\text{--}\on{mod}_{\CE}$ the subcategory of $\hat\fg_{crit}\text{--}\on{mod}$ whose objects are the $\hat\fg_{crit}$-modules, such that the underlying chiral $\fz_\fg$-module belongs to $\FZ_{\fg,x}\text{--}\on{mod}_\CE$. In [Sect. \[other results\]]{} we will prove the following \[support on tangent bundle\] The category $\hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\nil}$ is contained in $\hat\fg_{crit}\text{--}\on{mod}_{\CE}$. In other words, this theorem says that the inclusion $$\hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\CE}:= \hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}\cap\, \hat\fg_{crit}\text{--}\on{mod}_{\CE}\subset \hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\nil}$$ is in fact an equivalence. \[Beilinson\] The following remark was suggested by A. Beilinson: Let $\on{Spec}(\FZ_{\fg,x,\on{m.f.}})$ be the smallest formal subscheme inside $\on{Spec}(\FZ_{\fg,x})$, which contains $\on{Spec}(\fz_{\fg,x})$, and which is preserved by the Poisson bracket. (The subscript “$\on{m.f.}$” stands for “monodromy free”.) Let $\FZ_{\fg,x}\text{--}\on{mod}_{\on{m.f.}}$ be the subcategory of $\FZ_{\fg,x}\text{--}\on{mod}$ consisiting of modules supported on $\on{Spec}(\FZ_{\fg,x,\on{m.f.}})$, and let $\hat\fg_{crit}\text{--}\on{mod}_{\on{m.f.}}$ be the corresponding subcategory in $\hat\fg_{crit}\text{--}\on{mod}$. Beilinson has suggested that the following strengthening of [Theorem \[support on tangent bundle\]]{} might be true: \[monodromy free\] The subcategory $\hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\nil}$ is contained in $\hat\fg_{crit}\text{--}\on{mod}_{\on{m.f.}}$. In other words, this conjecture says that any $G(\CO_x)$-integrable $\hat\fg_{crit}$-module, which is set-theoretically supported on $\on{Spec}(\fz_{\fg,x})$, is supported on the formal subscheme $\on{Spec}(\FZ_{\fg,x,\on{m.f.}})$. If we could prove this conjecture, the proof of [Theorem \[critical reformulation\]]{} would have been more elegant, since instead of the obscure condition (2) in the definition of $\FZ_{\fg,x}\text{--}\on{mod}_\CE$, we would work with a clearer geometric concept of support on a subscheme. \[some cat\] Recall now the category $\CA_{\fg}^{\ren,\tau}\text{--}\on{mod}$, introduced in [Sect. \[categories of modules\]]{} and \[twist by tau\]. We have a natural forgetful functor $\CA_{\fg}^{\ren,\tau}\text{--}\on{mod}\to \hat\fg_{crit}\text{--}\on{mod}$ coming from the “right” copy of $\CA_{\fg,crit}$ in $\CA_{\fg}^{\ren,\tau}$. Let $\CA_{\fg}^{\ren,\tau}\text{--}\on{mod}^{G(\wh{\CO}_x)}$ (resp., $\CA_{\fg}^{\ren,\tau}\text{--}\on{mod}_{\nil}$, $\CA_{\fg}^{\ren,\tau}\text{--}\on{mod}_\CE$, $\CA_{\fg}^{\ren,\tau}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\nil}$, etc.) be the preimages of the corresponding subcategories of $\hat\fg_{crit}\text{--}\on{mod}$ under the above forgetful functor. Note, that by [Theorem \[support on tangent bundle\]]{}, the inclusion $$\CA_{\fg}^{\ren,\tau}\text{--}\on{mod}^{G(\wh{\CO}_x)}_\CE\hookrightarrow \CA_{\fg}^{\ren,\tau}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\nil}$$ is in fact an equivalence. It is easy to see that the functor $\wt{\imath}{}^!:\FZ_{\fg,x}\text{--}\on{mod}\to \FZ_{\fg,x}\text{--}\on{mod}_{\nil}$ gives rise to a well-defined functor $\wt{\imath}{}^!: \CA_{\fg}^{\ren,\tau}\text{--}\on{mod}\to \CA_{\fg}^{\ren,\tau}\text{--}\on{mod}_{\nil}$. In particular, the corresponding functor $$\CA_{\fg}^{\ren,\tau}\text{--}\on{mod}^{G(\wh{\CO}_x)} \overset{\wt{\imath}^!}\longrightarrow \CA_{\fg}^{\ren,\tau}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\nil}$$ is exact, by [Corollary \[exactness of restr to formal\]]{},. Recall now the Lie-\* algebroid $\CA_{\fg}^{\flat,\tau}$ and the corresponding category $\CA_{\fg}^{\flat,\tau}\text{--}\on{mod}$ (see [Sect. \[categories of modules\]]{}, [Sect. \[twist by tau\]]{}). We claim that the functor $\imath^!:\FZ_{\fg,x}\text{--}\on{mod}\to \fz_{\fg,x}\text{--}\on{mod}$ gives rise to a functor from $\CA_{\fg}^{\ren,\tau}\text{--}\on{mod}$ to $\CA_{\fg}^{\flat,\tau}\text{--}\on{mod}$. Indeed, given an object ${\mc M}$ of the category $\CA_{\fg}^{\ren,\tau}\text{--}\on{mod}$, we consider it as a $\FZ_{\fg,x}$-module and take its maximal submodule $\imath^!(\CM)$ supported on $\on{Spec} (\fz_{\fg,x})$. We consider $\imath^!(\CM)$ as a Lie-\* module over $\CA_{\fg}^{\ren,\tau}$. But now the Lie-\* action of the diagonal $\fz_{\fg} \subset (\CA_{\fg,crit}\times\CA_{\fg,crit})/\fz_\fg \subset \CA^{\ren,\tau}_\fg$ will be zero. Therefore, the Lie-\* action of $\CA_{\fg}^{\ren,\tau}$ on $\imath^!(\CM)$ will factor through the action of the Lie-\* algebra $\CA_{\fg}^{\flat,\tau} = \CA_{\fg}^{\ren,\tau}/\fz_\fg$. Moreover, $\imath^!(\CM)$ is clearly preserved by the chiral action of $\CA_{\fg,crit}\underset{\fz_\fg}\otimes \CA_{\fg,crit}$, and these two structures make $\imath^!(\CM)$ an object of the category $\CA_{\fg}^{\flat,\tau}\text{--}\on{mod}$. By a slight abuse of notation, we denote the resulting functor $\CA_{\fg}^{\ren,\tau}\text{--}\on{mod}\to \CA_{\fg}^{\flat,\tau}\text{--}\on{mod}$ also by $\imath^!$. In the next section we will prove the following theorem, which can be regarded as a version of the Kashiwara theorem in the theory of D-modules. \[Kashiwara\] The functor $\imath^!:\CA_{\fg}^{\ren,\tau}\text{--}\on{mod}_\CE\to \CA^{\flat,\tau}_\fg\text{--}\on{mod}$ is an equivalence of categories. In particular, it is exact. If we denote by $\CA_{\fg}^{\flat,\tau}\text{--}\on{mod}^{G(\wh{\CO}_x)}$ the corresponding subcategory of $\CA_{\fg}^{\ren,\tau}$, we obtain that the functor $$\CA_{\fg}^{ren,\tau}\text{--}\on{mod}^{G(\wh{\CO}_x)}_\CE\to \CA_{\fg}^{\flat,\tau}\text{--}\on{mod}^{G(\wh{\CO}_x)}$$ is also exact (and, in fact, an equivalence). We are now able to finish the proof of [Theorem \[critical reformulation\]]{}, modulo Theorems \[support on tangent bundle\] and \[Kashiwara\]. Recall from [Theorem \[embedding of algebroids\]]{} that we have a homomorphism of chiral algebras $U^{\ren,\tau}(L_{\fg,crit})\to \fD_{G,crit}$. Hence, the forgetful functor $\fD_{G,crit}\text{--}\on{mod}\to \hat\fg_{crit}\text{--}\on{mod}$ factors as $$\fD_{G,crit}\text{--}\on{mod}\to \CA_{\fg}^{\ren,\tau}\text{--}\on{mod}\to \hat\fg_{crit}\text{--}\on{mod}.$$ We have a commutative diagram of functors $$\label{comm diag funct} \CD \CA_{\fg}^{ren,\tau}\text{--}\on{mod}^{G(\wh{\CO}_x)} @>{\imath^!}>> \CA_{\fg}^{\flat,\tau}\text{--}\on{mod}^{G(\wh{\CO}_x)} \\ @VVV @VVV \\ \hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)} @>{\imath^!}>> \hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\on{reg}}, \endCD$$ where the vertical arrows are the forgetful functors. Thus, to prove [Theorem \[critical reformulation\]]{}, it is sufficient to show that the composition $$\CA_{\fg}^{ren,\tau}\text{--}\on{mod}^{G(\wh{\CO}_x)}\overset{\wt{\imath}^!}\longrightarrow \CA_{\fg}^{ren,\tau}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\nil} \simeq \CA_{\fg}^{ren,\tau}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\CE}\overset{\imath^!}\longrightarrow \CA_{\fg}^{\flat,\tau}\text{--}\on{mod}^{G(\wh{\CO}_x)}$$ is exact. But, as we have just seen, all the above arrows are exact functors. Our plan now is as follows. In the next section we will prove [Theorem \[Kashiwara\]]{} and hence complete the proof of Theorems \[critical reformulation\] and \[main\], modulo Theorems \[drinfeld\] and \[support on tangent bundle\]. These theorems will be proved simultaneously in [Sect. \[other results\]]{}. Finally, in [Sect. \[faithfulness\]]{} we will prove that the functor of global sections, considered as a functor $\on{D}_{crit}(\Gr_G)\text{--}\on{mod}\to \CA_{\fg}^\flat\text{--}\on{mod}$, is fully faithful. Proof of [Theorem \[Kashiwara\]]{} {#proof of Kashiwara} ================================== \[Kash gen\] Let us recall the setting of the original Kashiwara theorem. Let $\X$ be a smooth variety, and $\imath:\Y\hookrightarrow \X$ an embedding of a smooth closed subvariety. Consider the category $D_\X\text{--}\on{mod}$ of right D-modules on $\X$, and its subcategory $D_\X\text{--}\on{mod}_\Y$ of right D-modules set-theoretically supported on $\Y$. Finally, consider the category $D_\Y\text{--}\on{mod}$ of D-modules on $\Y$. We have the functor $\imath^!:D_\X\text{--}\on{mod}_\Y\to \CO_\Y\text{--}\on{mod}$ which sends a D-module $\CM$ to its maximal $\CO_\X$-submodule consisting of sections annihilated by the ideal of $\Y$. Then the resulting $\CO_\X$-module naturally acquires a right action of the ring of differential operators on $\Y$, and so we obtain a functor $\imath^!:D_\X\text{--}\on{mod}_\Y\to D_\Y\text{--}\on{mod}$. Kashiwara’s theorem asserts that this functor is an equivalence of categories. Our [Theorem \[Kashiwara\]]{} should be regarded as a generalization of the above theorem, when the ring of differential operators is replaced by a certain algebroid (cf. [Sect. \[sec Beil\]]{} below). In the proof we will use the same argument as in the proof of the original Kashiwara theorem. \[functor !\] We start by constructing a functor $\imath_!: \CA^{\flat,\tau}_\fg\text{--}\on{mod}\to \CA_{\fg}^{\ren,\tau}\text{--}\on{mod}_\CE$, which will be the left adjoint of $\imath^!$. Given an object $\CN$ in $\CA^{\flat,\tau}_\fg\text{--}\on{mod}$, we regard it as a Lie-\* module over $\CA_{\fg}^{\ren,\tau}$, considered as a Lie-\* algebra. Let $\on{Ind}(\CN)$ denote the induced chiral $\CA_{\fg}^{\ren,\tau}$-module (see [@BD], Sect. 3.7.15), where $\CA_{\fg}^{\ren,\tau}$ is again considered merely as a Lie-\* algebra (and not as a chiral algebroid). Thus, $\on{Ind}(\CN)$ is a chiral module over the chiral universal enveloping algebra $U(\CA_{\fg}^{\ren,\tau})$. We have the surjections $$U(\CA_{\fg}^{\ren,\tau})\twoheadrightarrow U(\fz_\fg,\CA_{\fg}^{\ren,\tau})\twoheadrightarrow U^{\ren,\tau}(L_{\fg,crit}),$$ and we set $\imath_!(\CN)$ to be the (maximal) quotient of $\on{Ind}(\CN)$, on which the action of $U(\CA_{\fg}^{\ren,\tau})$ factors through an action of $U^{\ren,\tau}(L_{\fg,crit})$, and for which the two maps $$j_*j^*\left((\CA_{\fg,crit}\underset{\fz_\fg}\otimes \CA_{\fg,crit})\boxtimes \CN\right) \to \Delta_!\left(\imath_!(\CN)\right),$$ one coming from the inital chiral action of $\CA_{\fg,crit}\underset{\fz_\fg}\otimes \CA_{\fg,crit}$ on $\CN$, and the other from the homomorphism $\CA_{\fg,crit}\underset{\fz_\fg}\otimes \CA_{\fg,crit}\to U^{\ren,\tau}(L_{\fg,crit})$, coincide. By definition, $\imath_!(\CN)$ is an object of $\CA_{\fg}^{\ren,\tau}\text{--}\on{mod}$. It is easy to see that the functor $\CN\mapsto \imath_!(\CN):\CA_{\fg}^{\flat,\tau}\text{--}\on{mod}\to \CA_{\fg}^{\ren,\tau}\text{--}\on{mod}$ is the left adjoint to $\imath^!:\CA_{\fg}^{\ren,\tau}\text{--}\on{mod}\to \CA_{\fg}^{\flat,\tau}\text{--}\on{mod}$. One readily checks that for $\CN=\CA_{\fg,crit}\underset{\fz_\fg}\otimes \CA_{\fg,crit}$, with the action of $\CA^{\flat,\tau}_\fg$ introduced in the proof of [Theorem \[doubled algebroid\]]{}, the resulting object $\imath_!(\CN)$ is isomorphic to $U^{\ren,\tau}(L_{\fg,crit})$ itself. Let us regard $\imath_!(\CN)$ as a chiral module over $\CA_{\fg,crit}\underset{\fz_\fg}\otimes \CA_{\fg,crit}$. We have a canonical map $\CN\to \imath_!(\CN)$, and the PBW filtration on $U^{\ren,\tau}(L_{\fg,crit})$ induces an increasing filtration $F^i(\imath_!(\CN)), i\geq 1$ on $\imath_!(\CN)$ with the $F^1(\imath_!(\CN))$ term being the image of $\CN$. The terms of this filtration are stable under the chiral action of $\CA_{\fg,crit}\underset{\fz_\fg}\otimes \CA_{\fg,crit}$ and the Lie-\* action of the entire $\CA_{\fg}^{\ren,\tau}$; the action of $\fz_{\fg,x}$ on $\on{gr}(\imath_!(\CN))$ is central. The description of $\on{gr}(U^{\ren,\tau}(L_{\fg,crit}))$ given by implies that we have an isomorphism $$\label{ass graded module} \CN\underset{\fz_{\fg,x}}\otimes \on{Sym}_{\fz_{\fg,x}}(\Omega^1(\fz_{\fg,x}))\simeq\on{gr}(\imath_!(\CN)).$$ Evidently, as a module over $\FZ_{\fg,x}$, $\imath_!(\CN)$ is supported on the formal neighborhood of $\on{Spec}(\fz_{\fg,x})$. Recall the setting if [Sect. \[comm alg\]]{}; in particular, let $\CI$ be the ideal $\on{ker}(\FZ_{\fg,x}\to \fz_{\fg,x})$. For $\CN$ as above, $F^i(\imath_!(\CN))$ equals the submodule of $\imath_!(\CN)$ annihilated by $\CI^i$. Moreover, as a module over $\FZ_{\fg,x}$, $\imath_!(\CN)$ belongs to the category $\FZ_{\fg,x}\text{--}\on{mod}_\CE$. Since the action of $\fz_\fg$ on $\on{gr}(\imath_!(\CN))$ is central, we have $\CI\cdot F^{i+1}(\imath_!(\CN))\subset F^i(\imath_!(\CN))$. Therefore, by induction, every element of $F^i(\imath_!(\CN))$ is annihilated by $\CI^i$. To prove the inclusion in the other direction, consider the map $$\CI/\CI^2\underset{\fz_{\fg,x}}\otimes \left(F^{i+1}(\imath_!(\CN))/F^i(\imath_!(\CN))\right)\to \left(F^{i}(\imath_!(\CN))/F^{i-1}(\imath_!(\CN))\right),$$ and the corresponding dual map $$\label{dual map} \left(F^{i+1}(\imath_!(\CN))/F^i(\imath_!(\CN))\right)\to \Theta(\fz_\fg)_x\underset{\fz_{\fg,x}}\otimes \left(F^{i}(\imath_!(\CN))/F^{i-1}(\imath_!(\CN))\right).$$ We need to show that the map of is injective. But this follows by combining the isomorphism and [Theorem \[BD descr of Gelfand-Dikii\]]{}. Indeed, this map is obtained by tensoring with $\CN$ from $$\on{Sym}^i_{\fz_{\fg,x}}(\Omega^1(\fz_{\fg,x}))\to \on{Sym}^{i-1}_{\fz_{\fg,x}}(\Omega^1(\fz_{\fg,x})) \underset{\fz_{\fg,x}}\otimes \Omega^1(\fz_{\fg,x})\to \on{Sym}^{i-1}_{\fz_{\fg,x}}(\Omega^1(\fz_{\fg,x})) \underset{\fz_{\fg,x}}\otimes \Theta(\fz_\fg)_x.$$ This also proves the second assertion of the lemma, since $\CE$ is by definition the image of $\Omega^1(\fz_{\fg,x})$ in $\Theta(\fz_\fg)_x$. Thus, we obtain that $\CN\mapsto \imath_!(\CN)$ is a functor $\CA^{\flat,\tau}_\fg\text{--}\on{mod}\to \CA_{\fg}^{\ren,\tau}\text{--}\on{mod}_\CE$, left adjoint to $\imath^!$, and the adjunction map $\CN\mapsto \imath^!\circ \imath_!(\CN)$ is an isomorphism. To prove [Theorem \[Kashiwara\]]{}, it remains to show that for every $\CM\in \CA_{\fg}^{\ren,\tau}\text{--}\on{mod}_\CE$, the adjunction map $\imath_!\circ \imath^!(\CM)\to \CM$ is surjective. Indeed, from the fact that $\imath^!\circ \imath_!(\CN) \simeq \CN$, we know that the map $\imath^! \circ \imath_!\circ \imath^!(\CM) \to \imath^!(\CM)$ is an isomorphism, and we conclude that $\imath^!\left(\on{Ker} (\imath_!\circ \imath^!(\CM)\to \CM)\right)=0$. However, the functor $\imath^!$ is evidently faithful, by condition (1) in the definition of $\fZ_{\fg,x}\text{--}\on{mod}_\CE$. Locally on $\on{Spec}(\fz_{\fg,x})$, let us choose a basis $\xi_k$ in the space of sections of the vector bundle $\CE\subset N(\fz_{\fg,x})$. (This is possible since $\CE\simeq \Omega(\fz_\fg)_x$, and we know that $\Omega(\fz_\fg)$ is locally free over $\fz_\fg\otimes D_X$, by [Theorem \[FF isom\]]{}(1).) Let us choose functions $f_k$ in the ideal $\CI$, such that under the pairing $\langle\cdot,\cdot\rangle:\CI/\CI^2\otimes \Theta(\fz_\fg)_x\to \fz_{\fg,x}$, we have $\langle f_l,\xi_k\rangle=\delta_{k,l}$. Let $\wh{\xi}_k$ be an arbitrary element in $H^0_{DR}(\D^\times_x,\CA_{\fg}^{\ren,\tau})$, which projects to $\xi_k$ under $$H^0_{DR}(\D^\times_x,\CA_{\fg}^{\ren,\tau})\to H^0_{DR}(\D^\times_x,\Omega^1(\fz_\fg))\to (\Omega^1(\fz_\fg))_x\simeq \CE.$$ Consider the natural action of $H^0_{DR}(\D^\times_x,\CA_{\fg}^{\ren,\tau})$ on $\FZ_{\fg,x}$; we have: $$\wh{\xi}_k(f_l)=\delta_{k,l} \; \on{mod} \; \CI.$$ Let $\CM$ be an object of $\CA_{\fg}^{\ren,\tau}\text{--}\on{mod}_\CE$, and let $\CM_i$ be the canonical filtration on it, as in [Sect. \[comm alg\]]{}. We have to show that the subspace $\CM_1$ generates $\CM$ under the action of $\CA_{\fg}^{\ren,\tau}$. We will argue by induction, and assume that the subspace $\CM_{i-1}$ can be obtained from $\CM_1$ by the action of $\CA_{\fg}^{\ren,\tau}$. Consider the action of $\CI/\CI^2$ on the extension $$0\to \CM_{i-1}/\CM_{i-2}\to \CM_i/\CM_{i-2}\to \CM_i/\CM_{i-1}\to 0.$$ By definition, this action factors through $(\CI/\CI^2)/\CE^\perp$, and the span of $f_k$’s is dense in the latter quotient. If $m$ is an element of $\CM_i$, then $f_k\cdot m \in \CM_{i-1}$, and for all but finitely many indices $k$ the element $f_k\cdot m$ will belong to $\CM_{i-2}$. Therefore, the operator $\delta:=\sum_k \wh{\xi}_k\cdot f_k$ is well-defined on $\CM_i/\CM_{i-1}$. To perform the induction step, it would be enough to show that $\delta$ is surjective. We claim that $\delta$ is in fact a scalar operator that acts as multiplication by $i-1$. To prove this statement, we assume by induction, that $\delta$ acts as $j-1$ on $\CM_j/\CM_{j-1}$ for all $j<i$. Given an element $m\in \CM_i/\CM_{i-1}$, consider the finite sum $\displaystyle \sum_{k=1,...,N} \wh{\xi}_k\cdot f_k\cdot m$, which includes all the indices $k$ for which $f_k\cdot m\notin \CM_{i-2}$. We must show that for any index $l$, $$f_l\cdot \left(\sum_{k=1,...,N} \wh{\xi}_k\cdot f_k\cdot m- (i-1)\cdot m\right)\in \CM_{i-2}.$$ Without loss of generality, we can assume that the initial finite set of $k$’s included $l$, as well as the corresponding set of indices for the element $f_l\cdot m\in \CM_{i-1}/\CM_{i-2}$. Then we have $$\begin{aligned} &f_l\cdot \left(\sum_{k=1,...,N} \wh{\xi}_k\cdot f_k\cdot m- (i-1)\cdot m\right)= \left(\sum_{k=1,...,N} \wh{\xi}_k\cdot f_k\cdot f_l \cdot m -(i-2)\cdot f_l\cdot m\right)+ \\ &+\left(\sum_{l\neq k=1,...,N} \wh{\xi}_k(f_l)\cdot f_k\cdot m\right)+ \left(\wh{\xi}_l(f_l)\cdot f_l\cdot m-f_l\cdot m\right).\end{aligned}$$ In the above expression, the first term belongs to $\CM_{i-2}$, by the induction hypothesis on the action of $\delta$ on $\CM_{i-1}/\CM_{i-2}$. The second term belongs to $\CM_{i-2}$, because for $k\neq l$ we have $\wh{\xi}_k(f_l)\in \CI$, and the third term also belongs to $\CM_{i-2}$, because $\wh{\xi}_l(f_l)=1\,mod\,\, \CI$. This completes the proof of the induction step, and hence, of [Theorem \[Kashiwara\]]{}. \[sec Beil\] Recall that formal scheme $\on{Spec}(\FZ_{\fg,x,\on{m.f.}})$ introduced in [Sect. \[Beilinson\]]{}. From [Theorem \[Kashiwara\]]{} we obtain the following corollary: Every object of $\CA^{ren,\tau}_\fg\text{--}\on{mod}_\CE$ is supported on $\on{Spec}(\FZ_{\fg,x,\on{m.f.}})$. Let us write $\CM\in \CA^{ren,\tau}_\fg\text{--}\on{mod}_\CE$ as $\imath_!(\CN)$ for $\CN\in \CA^{\flat,\tau}_\fg\text{--}\on{mod}$, and consider the filtration $F^i(\imath_!(\CN))$. Of course, $F^1(\imath_!(\CN))$ is supported on $\on{Spec}(\fz_{\fg,x})\subset \on{Spec}(\FZ_{\fg,x,\on{m.f.}})$, and let us assume by induction that $F^{i-1}(\imath_!(\CN))$ is supported on $\on{Spec}(\FZ_{\fg,x,\on{m.f.}})$. However, since $F^{i-1}(\imath_!(\CN))$ is stable under the chiral action of $\CA_{\fg,crit}\underset{\fz_\fg}\otimes \CA_{\fg,crit}$, and $j_*j^*\left(\CA^{\ren,\tau}\boxtimes F^{i-1}(\imath_!(\CN))\right)$ maps surjectively onto $F^i(\imath_!(\CN))$, and taking into account that $\CA^{\ren,\tau}/\left(\CA_{\fg,crit}\times \CA_{\fg,crit}/\fz_\fg\right) \simeq \Omega^1(\fz_\fg)$, we obtain that $F^i(\imath_!(\CN))$ is also supported on $\on{Spec}(\FZ_{\fg,x,\on{m.f.}})$. Following a suggestion of Beilinson, let us note that [Theorem \[Kashiwara\]]{} can be viewed in the following general framework. (We formulate it in the finite-dimensional situation, for simplicity.) Let $\X$ and $\Y$ be as in [Sect. \[Kash gen\]]{}. Let $L_X$ be a Lie algebroid on $\X$, and let $L_\Y$ be its pull-back back to $\Y$. Let $\hat\Y$ be the formal neighbourhood of $\Y$ in $\X$, and let $\hat\Y'\supset \Y$ be the smallest ind-subscheme of $\hat\Y$, stable under the action of $L_X$. Let $L_X\text{--}\on{mod}$ be the category of all right $L_X$-modules, $L_X\text{--}\on{mod}_\Y$ its subcategory of modules supported on $\hat\Y$, and let $L_Y\text{--}\on{mod}$ be the category of right $L_Y$-modules. Let also $L_X\text{--}\on{mod}'_\Y$ be the subcategory of $L_X\text{--}\on{mod}_\Y$, consisting of modules supported on $\hat\Y'$. We have the direct image functor $\imath_!:L_Y\text{--}\on{mod}\to L_X\text{--}\on{mod}_\Y$, but it is easy to see that its image belongs in fact to $L_X\text{--}\on{mod}'_\Y$. And we have the right adjoint of $\imath_!$, denoted $\imath^!:L_X\text{--}\on{mod}_\Y\to L_Y\text{--}\on{mod}$. Assume now that $L_X$ is $\CO_\X$-flat, and that $L_Y$ is a locally direct summand in $N(\Y)$. Let $\CO_\X\text{--}\on{mod}_{L_Y}$ be the subcategory of $\CO_\X\text{--}\on{mod}$ defined as in [Sect. \[comm alg\]]{}. We have: [*(1)*]{} An object $\CM\in L_X\text{--}\on{mod}_\Y$ belongs to $L_X\text{--}\on{mod}'_\Y$ if and only if, as a $\CO_\X$-module, it belongs to $\CO_\X\text{--}\on{mod}_{L_Y}$. [*(2)*]{} The functor $\imath_!$ is an equivalence of categories $L_Y\text{--}\on{mod}\to L_X\text{--}\on{mod}'_\Y$ with the quasi-inverse given by $\imath^!$. Proof of Theorems \[support on tangent bundle\] and \[drinfeld\]. {#other results} ================================================================= Let us start by observing that we have a natural map $$\label{deform} \Omega^1(\fz_{\fg,x}) \to \on{Ext}^1_{\hat\fg_{crit}\text{--}\on{mod}}(\BV_{\fg,crit},\BV_{\fg,crit}).$$ This map can be constructed in the following general framework. Let $\CA_{\BC[\hslash]/\hslash^2}$ be a flat $\BC[\hslash]/\hslash^2$-family of chiral algebras, and set $\CA=\left(\CA_{\BC[\hslash]/\hslash^2}\right)/\hslash$. We claim that there is a canonical map $$\label{gen deform} \Omega^1(\fz(\CA)_x)\to \on{Ext}^1_{\CA\text{--}\on{mod}}(\CA_x,\CA_x).$$ Indeed, consider $(\CA_{\BC[\hslash]/\hslash^2})_x$ as an extension $$0\to \CA_x\to (\CA_{\BC[\hslash]/\hslash^2})_x\to \CA_x\to 0$$ in the category of $\CA_{\BC[\hslash]/\hslash^2}$-modules. Let ${\mathbf e}$ denote its class in $\on{Ext}^1_{\CA_{\BC[\hslash]/\hslash^2}\text{--}\on{mod}}(\CA_x,\CA_x)$. For an element $a\in \fz(\CA)_x$, viewed as an endomorphism of $\CA_x$ as a $\CA_{\BC[\hslash]/\hslash^2}$-module, we can produce two more elements of $\on{Ext}^1_{\CA_{\BC[\hslash]/\hslash^2}\text{--}\on{mod}}(\CA_x,\CA_x)$, namely, $a\cdot {\mathbf e}$ and ${\mathbf e}\cdot a$. However, it is easy to see that their difference already belongs to $\on{Ext}^1_{\CA\text{--}\on{mod}}(\CA_x,\CA_x)$. Moreover, one readily checks that the resulting map $\fz(\CA)_x\to \on{Ext}^1_{\CA\text{--}\on{mod}}(\CA_x,\CA_x)$ is a derivation, i.e., gives rise to a map in . Explicitly, for $\CA=\CA_\fg$ the map of looks as follows. Note that $$\on{Ext}^1_{\hat\fg_{crit}\text{--}\on{mod}}(\BV_{\fg,crit},\BV_{\fg,crit})\simeq \on{H}^1(\fg(\wh{\CO}_x),\BV_{\fg,crit}).$$ Given an element $da\in \Omega^1(\fz_{\fg,x})$, where $a \in \fz_{\fg,x} \subset \BV_{\fg,crit}$, consider its deformation $a_\hslash\in \BV_{\fg,\hslash}$ and define the corresponding 1-cocycle on $\fg(\wh{\CO}_x)$ by $$\label{cocycle} g\in \fg(\wh{\CO}_x)\mapsto \left. \frac{g\cdot a_\hslash}{\hslash}\right|_{\hslash=0}.$$ This gives the desired map. The next proposition states that the map of is an isomorphism. This is a particular case of the following general theorem established in [@FT]: \[FT\] We have a canonical isomorphism between $\Omega^i(\fz_{\fg,x})$ and the relative cohomology $\on{H}^i(\fg(\wh{\CO}_x),\fg,\BV_{\fg,crit})$. For $i=1$ we have $\on{H}^1(\fg(\wh{\CO}_x),\fg,\BV_{\fg,crit})\simeq \on{H}^1(\fg(\wh{\CO}_x),\BV_{\fg,crit})$, and we obtain that is indeed an isomorphism. Here we will give a different proof of this fact, using some results from [@BD1]. \[Ext\^1\] The above map $\Omega^1(\fz_{\fg,x}) \to \on{Ext}^1_{\fg_{crit}}(\BV_{\fg,crit},\BV_{\fg,crit})$ is an isomorphism. Recall the setting of [Sect. \[diff op\]]{}. Consider the short exact sequence $$0\to \BV_{\fg,crit}\overset{\fr}\longrightarrow \fD_{G,crit,x}\to \fD_{G,crit,x}/\fr(\BV_{\fg,crit})\to 0.$$ We know that $\on{H}^0(\fg(\wh{\CO}_x),\fD_{G,crit,x})\simeq \fl(\BV_{\fg,crit})$, and by a similar argument we obtain that $\on{H}^1(\fg(\wh{\CO}_x),\fD_{G,crit,x})=0$, since $\on{H}^1\left(\fg(\wh{\CO}_x),\on{Fun}\left(G(\wh{\CO}_x)\right)\right)=0$. Therefore from the long exact sequence we obtain an isomorphism $$\on{H}^1(\fg(\wh{\CO}_x),\BV_{\fg,crit})\simeq \bigl(\fD_{G,crit,x}/ \fr(\BV_{\fg,crit})\bigr)^{\fr(\fg(\wh{\CO}_x))}/\fl(\BV_{\fg,crit}).$$ By letting the point $x$ move, we obtain from the subspace $$\bigl(\fD_{G,crit,x}/ \fr(\BV_{\fg,crit})\bigr)^{\fr(\fg(\wh{\CO}_x))}/\fl(\BV_{\fg,crit}) \subset \fD_{G,crit,x}/(\fr(\BV_{\fg,crit})+\fl(\BV_{\fg,crit}))$$ a D-submodule, which we will denote by $\wt{\Omega}^1(\fz_\fg) \subset \fD_{G,crit}/(\fr(\CA_{\fg,crit})+\fl(\CA_{\fg,crit}))$. We can form the Cartesian squares $$\begin{aligned} &\wt{\CA}^\flat_\fg:= \wt{\Omega}^1(\fz_\fg)\underset{\fD_{G,crit}/(\fr(\CA_{\fg,crit}) + \fl(\CA_{\fg,crit}))} \times \fD_{G,crit}/\fr(\CA_{\fg,crit}), \\ &\wt{\CA}^{\ren,\tau}_\fg := \wt{\Omega}^1(\fz_\fg)\underset{\fD_{G,crit}/(\fr(\CA_{\fg,crit}) + \fl(\CA_{\fg,crit}))} \times \fD_{G,crit}.\end{aligned}$$ Let us show that the chiral bracket on $\fD_{G,crit}$ induces on $\wt{\Omega}^1(\fz_\fg)$ and $\wt{\CA}^\flat_\fg$ a structure of Lie-\* algebroids over $\fz_\fg$, and on $\wt{\CA}^{\ren,\tau}_\fg$ a structure of chiral algebroid. For that, let us note that $\wt{\CA}^{\ren,\tau}_\fg$ is by definition the normalizer on $\fr(\CA_{\fg,crit})$ in $\fD_{G,crit}$, i.e., the maximal D-submodule of $\fD_{G,crit}$, for which the Lie-\* bracket sends $\fr(\CA_{\fg,crit})$ to itself. Since $\fl(\CA_{\fg,crit})$ is the centralizer of $\fr(\CA_{\fg,crit})$ (by [Lemma \[two centralizers\]]{}), we obtain that $\wt{\CA}^{\ren,\tau}_\fg$ normalizes $\fl(\CA_{\fg,crit})$ as well. (By symmetry, we immediately obtain that $\wt{\CA}^{\ren,\tau}_\fg$ is in fact the entire normalizer of $\fl(\CA_{\fg,crit})$ in $\fD_{G,crit}$.) Now, [Corollary \[centers\]]{} implies that $\wt{\CA}^{\ren,\tau}_\fg$ normalizes also $\fz_\fg$. This implies the above assertion about the algebroid structures. Note also, that if $\CM$ is a chiral module over $\fD_{G,crit}$, we obtain that the Lie-\* algebroid $\wt{\CA}^\flat_\fg$ acts naturally on the subspace $\CM^{\fr(\fg(\wh{\CO}_x))}$, and $\wt{\Omega}^1(\fz_\fg)$ acts on the subspace $\CM^{\fl(\fg(\wh{\CO}_x))}\cap \CM^{\fr(\fg(\wh{\CO}_x))}$. From the construction of the map $\Omega^1(\fz_\fg)\to \fD_{G,crit}/\fr(\CA_{\fg,crit})+\fl(\CA_{\fg,crit})$ in [Sect. \[diff op\]]{}, it is clear that its image is in $\wt{\Omega}^1(\fz_\fg)$. (Of course, we are about to prove that $\Omega^1(\fz_\fg)$ is in fact isomorphic to $\wt{\Omega}^1(\fz_\fg)$.) On the level of fibers, the above map coincides with the map $$\Omega^1(\fz_\fg)\underset{\fz_\fg}\otimes \fz_{\fg,x} = \Omega^1(\fz_{\fg,x}) \to \on{H}^1(\fg(\wh{\CO}_x),\BV_{\fg,crit}),$$ given by formula . We obtain also morphisms of algebroids $\CA^\flat_\fg\to \wt{\CA}^\flat_\fg$ and $\wt{\CA}^{\ren,\tau}_\fg\to \CA^{\ren,\tau}_\fg$. Thus, we have a sequence of morphisms of algebroids $\Omega^1(\fz_\fg)\to \wt{\Omega}^1(\fz_\fg)\to \Theta(\fz_\fg)$, and we want to prove that the first arrow is an isomorphism. At the level of fibers, the second map can be viewed as follows. For an extension $$0\to \BV_{\fg,crit}\to \CM\to \BV_{\fg,crit}\to 0,$$ the action of $\CI/\CI^2\simeq N^*(\fz_{\fg,x})$ on $\CM$ defines an endomorphism of $\BV_{\fg,crit}$, i.e., an element of $\fz_{\fg,x}$. According to Proposition 6.2.4 of [@BD1], there are no non-trivial extensions of $\BV_{\fg,crit}$ by itself in the category $\hat\fg_{crit}\text{--}\on{mod}_{\reg}$. This implies that $\wt{\Omega}^1(\fz_\fg)\to \Theta(\fz_\fg)$ is an injection. Indeed, otherwise, we would obtain an extension $\CM$ as above, which belongs to $\hat\fg_{crit}\text{--}\on{mod}_{\reg}$. Thus, $\wt{\Omega}^1(\fz_\fg)$ is “squeezed” between $\Omega^1(\fz_\fg)$ and $\Theta(\fz_\fg)$. To prove that $\wt{\Omega}^1(\fz_\fg)$ in fact coincides with $\Omega^1(\fz_{\fg,x})$, we will use Theorem 5.5.3 of [@BD1]. This theorem asserts that $\Omega^1(\fz_\fg)$ coincides with the Atiyah algebroid corresponding to a certain principal $^L G$-bundle over $\on{Spec}(\fz_{\fg})$. This bundle is constructed as follows. We have an equivalence between the category $\on{Rep}({}^L G)$ of finite-dimensional representations of $^L G$ and the category of $G(\wh{\CO}_x)$-equivariant objects in $\on{D}_{crit}(\Gr_G)\text{--}\on{mod}$; for $V\in \on{Rep}({}^L G)$, let $\F^V\in \on{D}_{crit}(\Gr_G)\text{--}\on{mod}$ be the corresponding D-module. Set $$\CV_{\fz_{\fg,x}}:= \on{Hom}_{\hat\fg_{crit}}(\BV_{\fg,crit},\Gamma(\Gr_G,\F^V)).$$ Theorem 5.4.8 and Sect. 5.5.1 of [@BD1] imply that $V\mapsto \CV_{\fz_{\fg,x}}$ is a tensor functor from $\on{Rep}({}^L G)$ to the category of locally free finitely generated $\fz_{\fg,x}$-modules. By letting the point $x$ move along the curve, for every $V$ as above, we obtain a $\fz_\fg\otimes D_X$-module, denoted $\CV_{\fz_\fg}$, and the assignment $V\mapsto \CV_{\fz_\fg}$ is the sought-for principal $^L G$-bundle on $\on{Spec}(\fz_{\fg})$. The assertion of Theorem 5.5.3 of [@BD1], combined with [Theorem \[BD descr of Gelfand-Dikii\]]{}, implies that the Lie-\* algebroid $\Omega^1(\fz_\fg)$ is the universal Lie-\* algebroid over $\fz_\fg$, whose action lifts to the above $^L G$-bundle. However, as we have seen above, the Lie-\* algebroid $\wt{\Omega}^1(\fz_\fg)$ acts on every $\fz_{\fg,x}$-module of the form $\on{Hom}_{\hat\fg_{crit}}(\BV_{\fg,crit},\Gamma(\Gr_G,\F))$, for $\F\in \on{D}_{crit}(\Gr_G)\text{--}\on{mod}$. Again, globally over $X$, we obtain that $\wt{\Omega}^1(\fz_\fg)$ acts on all $\CV_{\fz_\fg}$. This implies that we have a splitting $\wt{\Omega}^1(\fz_\fg)\to \Omega^1(\fz_\fg)$, which is automatically an isomorphism. \[flatness\] Next we will prove that $\on{Ext}^i_{\hat\fg_{crit}\text{--}\on{mod}}(\BV_{\fg,crit},\BV_{\fg,crit})$ is flat as a $\fz_{\fg,x}$-module for any $i$. This statement can be formally deduced from [Theorem \[FT\]]{}, but we will give a different proof. First, we claim that the topological Lie algebroid $\hat{h}^{\fz_\fg}_x(\Omega^1(\fz_\fg))$ over $\on{Spec}(\fz_{\fg,x})$ (defined as in [@BD], Sect. 2.5.18) acts on every such $\on{Ext}^i$. Indeed, consider the Lie algebra $H^0_{DR}(\D_x,\Omega^1(\fz_\fg))$. The Lie-\* action of $\CA^\flat_{\fg}$ on $\CA_{\fg,crit}$ yields an action of $H^0_{DR}(\D_x,\Omega^1(\fz_\fg))$ on the associative algebra $\hat\CA_{\fg,crit,x}$ by outer derivations. Since the $\CA_{\fg,crit}$-action on $\BV_{\fg,crit}$ lifts to an action of $\CA^\flat_{\fg}$, we obtain that $H^0_{DR}(\D_x,\Omega^1(\fz_\fg))$ indeed acts on every $\on{Ext}^i_{\hat\fg_{crit}\text{--}\on{mod}}(\BV_{\fg,crit},\BV_{\fg,crit})$. Since $\BV_{\fg,crit}$ is a $\fz_\fg$-module, so is $\on{Ext}^i$, and the above $H^0_{DR}(\D_x,\Omega^1(\fz_\fg))$-action extends to an action of its completion $\hat{h}^{\fz_\fg}_x(\Omega^1(\fz_\fg))$. By identifying ${\mathcal K}_x$ with $\BC((t))$, we endow $\hat\fg_{crit}$ with a $\BZ$-grading by letting $t$ have degree $-1$. In this case $\BV_{\fg,crit}$ is a non-negatively graded $\hat\fg_{crit}$-module. Moreover, the terms of the standard complex computing the cohomology $\on{Ext}^i_{\hat\fg_{crit}\text{--}\on{mod}}(\BV_{\fg,crit},\BV_{\fg,crit})\simeq \on{H}^i(\fg(\wh{\CO}_x),\BV_{\fg,crit})$ are also non-negatively graded. By applying Lemma 6.2.2 of [@BD1], we conclude that the above $\on{Ext}^i$ is free over $\fz_{\fg,x}$. \[freeness\] The module $\BV_{\fg,crit}$ is flat over $\fz_{\fg,x}$. The lemma is proved by passing to the associate graded. Recall from [Theorem \[FF isom\]]{}(1) that $\BV_{\fg,crit}$ and $\fz_{\fg,x}$ are naturally filtered, and $$\begin{aligned} &\on{gr}(\BV_{\fg,crit})\simeq \on{Fun}\left(\g^*\times_{\BG_m} \Gamma({\mc D}_x,\Omega_X)\right)\simeq \CJ\left(\on{Fun}(\fg^*\times_{\BG_m}\omega_X)\right)_x \\ &\on{gr} (\fz_{\fg,x})\simeq\on{Fun}\left((\g^*/G) \times_{\BG_m} \Gamma({\mc D}_x,\Omega_X)\right) \simeq \CJ\left(\on{Fun}(\fg^*/G\times_{\BG_m}\omega_X)\right)_x.\end{aligned}$$ Now we apply Theorem A.4 of [@EF], which exactly asserts that $\CJ\left(\on{Fun}(\fg^*\times_{\BG_m}\omega_X)\right)$ is flat over $\CJ\left(\on{Fun}(\fg^*/G\times_{\BG_m}\omega_X)\right)$. Finally, we are ready to prove the following: \[all Exts\] For any $\fz_{\fg,x}$-module $\CL$, the natural map $$\on{Ext}^i_{\hat\fg_{crit}\text{--}\on{mod}}(\BV_{\fg,crit}, \BV_{\fg,crit}) \underset{\fz_{\fg,x}}\otimes \CL\to \on{Ext}^i_{\hat\fg_{crit}\text{--}\on{mod}}(\BV_{\fg,crit}, \BV_{\fg,crit}\underset{\fz_{\fg,x}}\otimes \CL)$$ is an isomorphism. From the identification $\on{Ext}^i_{\hat\fg_{crit}\text{--}\on{mod}}(\BV_{\fg,crit}, \BV_{\fg,crit}\underset{\fz_{\fg,x}}\otimes \CL)\simeq \on{H}^i(\fg(\wh{\CO}_x),\BV_{\fg,crit}\underset{\fz_{\fg,x}}\otimes \CL)$ and the standard complex, computing cohomology of the Lie algebra $\fg(\wh{\CO}_x)$, we obtain that the functor $\CL\mapsto \on{Ext}^i_{\hat\fg_{crit}\text{--}\on{mod}}(\BV_{\fg,crit}, \BV_{\fg,crit}\underset{\fz_{\fg,x}}\otimes \CL)$ commutes with direct limits. Therefore, to prove the proposition, we can suppose that $\CL$ is finitely presented. Since $\fz_{\fg,x}$ is isomorphic to a polynomial algebra (by [Theorem \[FF isom\]]{}(1)), any finitely presented module $\CL$ admits a finite resolution by projective modules: $$0\to \CP_n\to...\to \CP_1\to \CP_0\to \CL\to 0.$$ By [Lemma \[freeness\]]{}, the complex $$0\to \BV_{\fg,crit}\underset{\fz_{\fg,x}}\otimes\CP_n\to...\to \BV_{\fg,crit}\underset{\fz_{\fg,x}}\otimes\CP_1\to \BV_{\fg,crit}\underset{\fz_{\fg,x}}\otimes\CP_0\to \BV_{\fg,crit}\underset{\fz_{\fg,x}}\otimes\CL\to 0$$ is also exact. Thus, we have a spectral sequence, converging to $\on{Ext}^i_{\hat\fg_{crit}\text{--}\on{mod}}(\BV_{\fg,crit}, \BV_{\fg,crit}\underset{\fz_{\fg,x}}\otimes \CL)$, with the $E^{i,-j}_1$-term isomorphic to $\on{Ext}^i_{\hat\fg_{crit}\text{--}\on{mod}}(\BV_{\fg,crit}, \BV_{\fg,crit}\underset{\fz_{\fg,x}}\otimes \CP^j)$. Since each $\CP^j$ is projective, we evidently have $$\on{Ext}^i_{\hat\fg_{crit}\text{--}\on{mod}}(\BV_{\fg,crit}, \BV_{\fg,crit}\underset{\fz_{\fg,x}}\otimes \CP^j)\simeq \on{Ext}^i_{\hat\fg_{crit}\text{--}\on{mod}}(\BV_{\fg,crit}, \BV_{\fg,crit}) \underset{\fz_{\fg,x}}\otimes \CP^j.$$ But since all $\on{Ext}^i_{\hat\fg_{crit}\text{--}\on{mod}}(\BV_{\fg,crit}, \BV_{\fg,crit})$ are $\fz_{\fg,x}$-flat, this spectral sequence degenerates at $E_2$, implying the assertion of the proposition. \[vanishing of ext\] $\on{Ext}^1_{\hat\fg_{crit}\text{--}\on{mod}_{\reg}}(\BV_{\fg,crit}, \BV_{\fg,crit}\underset{\fz_{\fg,x}}\otimes \CL)=0$. We have a map $$\CI/\CI^2\underset{\fz_{\fg,x}}\otimes \on{Ext}^1_{\hat\fg_{crit}\text{--}\on{mod}}(\BV_{\fg,crit}, \BV_{\fg,crit}\underset{\fz_{\fg,x}}\otimes \CL)\to \on{Hom}_{\hat\fg_{crit}\text{--}\on{mod}}(\BV_{\fg,crit}, \BV_{\fg,crit}\underset{\fz_{\fg,x}}\otimes \CL),$$ and its adjoint $$\label{dmap} \on{Ext}^1_{\hat\fg_{crit}\text{--}\on{mod}}(\BV_{\fg,crit}, \BV_{\fg,crit}\underset{\fz_{\fg,x}}\otimes \CL)\to \Theta(\fz_\fg)_x\underset{\fz_{\fg,x}}\otimes \on{Hom}_{\hat\fg_{crit}\text{--}\on{mod}}(\BV_{\fg,crit}, \BV_{\fg,crit}\underset{\fz_{\fg,x}}\otimes \CL).$$ It is easy to see that $$\on{Ext}^1_{\hat\fg_{crit}\text{--}\on{mod}_{\reg}}(\BV_{\fg,crit}, \BV_{\fg,crit}\underset{\fz_{\fg,x}}\otimes \CL)\subset \on{Ext}^1_{\hat\fg_{crit}\text{--}\on{mod}}(\BV_{\fg,crit}, \BV_{\fg,crit}\underset{\fz_{\fg,x}}\otimes \CL)$$ is exactly the kernel of the latter map. However, by [Proposition \[all Exts\]]{} applied to $i=0$ and $1$, we can identify both sides in with $$\Omega^1(\fz_{\fg,x})\underset{\fz_{\fg,x}}\otimes \CL\to \Theta(\fz_{\fg,x})\underset{\fz_{\fg,x}}\otimes \CL,$$ and the latter map is injective, since $\on{coker}(\Omega^1(\fz_\fg)\to \Theta(\fz_{\fg,x}))$ is flat as a $\fz_\fg$-module, by [Theorem \[BD descr of Gelfand-Dikii\]]{}. Recall the functor $\sF:\hat\fg_{crit} \text{--} \on{mod}_{\reg}^{G(\wh{\CO}_x)}\to \fz_{\fg,x}\text{--}\on{mod}$ and its left adjoint $\sG:\fz_{\fg,x}\text{--}\on{mod}\to \hat\fg_{crit}\text{--}\on{mod}_{\reg}^{G(\wh{\CO}_x)}$ defined in [Sect. \[functors\]]{}. Note that the functor $\sF$ is faithful. Indeed, a module $\CM\in \hat\fg_{crit}\text{--}\on{mod}_{\reg}^{G(\wh{\CO}_x)}$ necessarily contains a non-zero vector, which is annihilated by $\fg\otimes t\BC[[t]]\subset \hat\fg_{crit}$. Therefore we have a non-zero map $\BV^\lambda_{\fg,crit} \to \CM$, and by [Lemma \[Sugawara calculation\]]{}, $\lambda$ must be equal to $0$. Note that the assertion of [Proposition \[all Exts\]]{} for $i=0$ implies that the adjunction morphism $$\CL\to \sF\circ \sG(\CL)$$ is an isomorphism. We claim that this, combined with [Corollary \[vanishing of ext\]]{}, formally implies [Theorem \[drinfeld\]]{}. We have to show that for $\CM\in \hat\fg_{crit}\text{--}\on{mod}_{\reg}^{G(\wh{\CO}_x)}$ the adjunction map $$\sG\circ \sF(\CM)\to \CM$$ is an isomorphism. This map is injective. Indeed, if $\CM'$ is the kernel of $\sG\circ \sF(\CM)\to \CM$, by the left exactness of $\sF$, we would obtain that $$\sF(\CM')=\on{ker}\bigl(\sF\circ \sG\circ \sF(\CM)\to \sF(\CM)\bigr)\simeq \on{ker}\bigl(\sF(\CM)\to \sF(\CM)\bigr)=0.$$ But we know that the functor $\sF$ is faithful, so $\CM'=0$. Let us prove that $\sG\circ \sF\to \on{Id}$ is surjective. Let $\CM''$ be the cokernel of $\sG\circ \sF(\CM)\to \CM$. We have the long exact sequence $$0\to \sF\circ \sG\circ \sF(\CM) \to \sF(\CM)\to \sF(\CM'')\to R^1\sF(\sG\circ \sF(\CM))\to...$$ However, [Corollary \[vanishing of ext\]]{} implies that $R^1\sF(\sG(\CL))=0$ for any $\fz_{\fg,x}$-module $\CL$. Therefore, the above portion of the long exact sequence amounts to a short exact sequence $$0\to \sF\circ \sG\circ \sF(\CM) \to \sF(\CM)\to \sF(\CM'')\to 0.$$ But the first arrow is an isomorphism, which implies that $\sF(\CM'')=0$ and hence $\CM''=0$. Thus, [Theorem \[Drinfeld’s equivalence of categories\]]{} is proved. As a corollary, we obtain the following result. Let $\sigma\in \on{Spec}(\fz_{\fg,x})$ be a $\BC$-point, and consider the subcategory $\hat\fg_{crit}\text{--}\on{mod}_{\sigma}^{G(\wh{\CO}_x)}$ of $\hat\fg_{crit}\text{--}\on{mod}_{\reg}^{G(\wh{\CO}_x)}$ whose objects are the $\hat{\fg}_{crit}$-modules with central character equal to $\sigma$. [Theorem \[Drinfeld’s equivalence of categories\]]{} implies that the category $\hat\fg_{crit}\text{--}\on{mod}_{\sigma}^{G(\wh{\CO}_x)}$ is equivalent to the category of vector spaces. In particular, the module $$\BV_{\fg,\sigma}:=\BV_{\fg,crit}\underset{\fz_{\fg,x}}\otimes \CC_\sigma$$ is irreducible. Finally, let us prove [Theorem \[support on tangent bundle\]]{}. Let $\CM$ be an object of $\hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}$, and let $\CM_i$ be the filtration as in [Sect. \[comm alg\]]{}. We need to show that the action of $\CI/\CI^2$ on $\CM_{i+1}/\CM_{i-1}$, that maps $\CM_{i+1}/\CM_i$ to $\CM_i/\CM_{i-1}$, factors through $(\CI/\CI^2)/\CE^\perp$. We claim that for any extension in $\hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}$ $$0\to \CM^1\to \CM^2\to\CM^3\to 0$$ with $\CM^1,\CM^3\in \hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\reg}$, the map $\CI/\CI^2\underset{\fz_{\fg,x}}\otimes \CM_3\to \CM_1$ factors through $(\CI/\CI^2)/\CE^\perp$. Indeed, by [Theorem \[drinfeld\]]{}, we can map surjectively onto $\CM^3$ a module of the form $\BV_{\fg,crit}\underset{\fz_{\fg,x}}\otimes \CL$, where $\CL$ is a free $\fz_{\fg,x}$-module. Hence, we can replace $\CM^3$ by $\BV_{\fg,crit}$. Again, by [Theorem \[drinfeld\]]{}, $\CM^1$ has the form $\BV_{\fg,crit}\underset{\fz_{\fg,x}}\otimes \CL^1$ for some $\fz_{\fg,x}$-module $\CL^1$. Now, our assertion follows from [Proposition \[all Exts\]]{} for $i=1$. Faithfulness ============ Recall the category $\CA_{\fg}^{\flat,\tau}\text{--}\on{mod}^{G(\wh{\CO}_x)}$ introduced in [Sect. \[some cat\]]{}. Observe that the functor $\sF:\hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\reg}\to \fz_{\fg,x}\text{--}\on{mod}$ extends to a functor $$\label{dr with tau} \CA_{\fg}^{\flat,\tau}\text{--}\on{mod}^{G(\wh{\CO}_x)}\to \CA_{\fg}^\flat\text{--}\on{mod}.$$ Moreover, [Theorem \[Drinfeld’s equivalence of categories\]]{} implies that the latter is also an equivalence of categories, with the quasi-inverse being $\CM\to \CM\underset{\fz_{\fg,x}}\otimes \BV_{\fg,crit}$. We obtain that the functor $\Gamma:\on{D}_{crit}(\Gr_G)\text{--}\on{mod}\to \hat\fg_{crit}\text{--}\on{mod}$ naturally factors as $$\on{D}_{crit}(\Gr_G)\text{--}\on{mod}\to \CA_{\fg}^\flat\text{--}\on{mod}\to \hat\fg_{crit}\text{--}\on{mod}_{\reg}\hookrightarrow \hat\fg_{crit}\text{--}\on{mod},$$ where the second arrow is the tautological forgetful functor. We will denote the resulting functor $\on{D}_{crit}(\Gr_G)\text{--}\on{mod}\to \CA_{\fg}^\flat\text{--}\on{mod}$ by $\Gamma^\flat$. [*Remark.*]{} Suppose that $\F_\hslash$ is a $\BC[[\hslash]]$-flat family of $\kappa_\hslash$-twisted D-modules on $\Gr_G$. By taking global sections, we obtain a $\BC[[\hslash]]$-family of $\hat\fg_{\kappa_\hslash}$-modules. [Theorem \[main\]]{} implies that this family is flat as well. Set $\F_0=\F/\hslash$. We obtain that the $\hat\fg_{crit}$-module $\Gamma(\Gr_G,\F_0)$ has two (a priori different) structures of object of $\CA_{\fg}^\flat\text{--}\on{mod}$: one such structure has been described above, and another is as in [Sect. \[flat modules\]]{}. However, it is easy to see that these structures in fact coincide. The main result of this section is the following \[fully faithfulness\] The above functor $\Gamma^\flat:\on{D}_{crit}(\Gr_G)\text{--}\on{mod}\to \CA_{\fg}^\flat\text{--}\on{mod}$ is fully faithful. Recall the category $\CA_{\fg}^{\ren,\tau}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\nil}$. By combining Theorems \[Kashiwara\] and \[support on tangent bundle\] we obtain: \[total equivalence\] We have the following sequence of equivalences: $$\CA_{\fg}^{\ren,\tau}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\nil} \overset{\imath^!}\longrightarrow \CA_{\fg}^{\flat,\tau}\text{--}\on{mod}^{G(\wh{\CO}_x)}\to \CA_{\fg}^\flat\text{--}\on{mod},$$ where the last functor is as in . Recall now the functor $\wt{\imath}^!:\hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}\to \hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\nil}$, and the corresponding functor $$\wt{\imath}^!:\CA_{\fg}^{\ren,\tau}\text{--}\on{mod}^{G(\wh{\CO}_x)}\to \CA_{\fg}^{\ren,\tau}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\nil}.$$ Let us denote by $\wt{\Gamma}$ the functor $\on{D}_{crit}(\Gr_G)\text{--}\on{mod}\to \CA_{\fg}^{\ren,\tau}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\nil}$ equal to the composition $$\on{D}_{crit}(\Gr_G)\text{--}\on{mod}\simeq \fD_{G,crit}\text{--}\on{mod}^{G(\wh{\CO}_x)} \to \CA_{\fg}^{\ren,\tau}\text{--}\on{mod}^{G(\wh{\CO}_x)}\overset{\wt{\imath}^!} \longrightarrow \CA_{\fg}^{\ren,\tau}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\nil}.$$ The functor $\Gamma^\flat$ is the composition of $\wt{\Gamma}$, followed by the $\CA_{\fg}^{\ren,\tau}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\nil}\to \CA_{\fg}^\flat\text{--}\on{mod}$ of [Corollary \[total equivalence\]]{}. So we are reduced to proving \[sectors\] The functor $\wt{\Gamma}:\fD_{G,crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}\to \CA_{\fg}^{\ren,\tau}\text{--}\on{mod}^{G(\wh{\CO}_x)}_{\nil}$ is fully faithful. Recall from [Proposition \[restriction to formal\]]{} that for $\CM\in \hat\fg_{crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}$, the object $\wt{\imath}^!(\CM)$ is in fact a direct summand of $\CM$, denoted $\CM^{\nil}$. Let us first analyze the decomposition $\CM\simeq \CM^{\nil}\oplus \CM^{\on{non-reg}}$ when the module $\CM$ equals $\fD_{G,crit,x}$ itself. Letting $x$ move, we obtain a direct sum decomposition of D-modules $\fD_{G,crit}\simeq \fD^{\nil}_{G,crit}\oplus \fD^{\on{non-reg}}_{G,crit}$. \[neutral component\] The homomorphism $U^{\ren,\tau}(L_{\fg,crit})\to \fD_{G,crit}$ is an isomorphism onto $\fD^{\nil}_{G,crit}$. In particular, $\fD^{\nil}_{G,crit}$ is a chiral subalgebra of $\fD_{G,crit}$. It is enough to show that the homomorphism $U^{\ren,\tau}(L_{\fg,crit})\to \fD_{G,crit}$ induces an isomorphism at the level of fibers. The fiber of $U^{\ren,\tau}(L_{\fg,crit})$, viewed as an object of $\CA_{\fg}^{\ren,\tau}\text{--}\on{mod}$, corresponds, under the equivalence of categories given by [Corollary \[total equivalence\]]{}, to $\BV_{\fg,crit}\in \CA_{\fg}^\flat\text{--}\on{mod}$. Hence it remains to show that $(\fD_{G,crit,x})^{\fg(\wh{\CO}_x)}\simeq \BV_{\fg,crit}$, but this is the content of [Lemma \[two centralizers\]]{}. This lemma implies, in particular, that for any $\CM\in \fD_{G,crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}$, the chiral action of $\fD^{\nil}_{G,crit}$ maps $\CM^{\nil}$ to $\CM^{\nil}$ and $\CM^{\on{non-reg}}$ to $\CM^{\on{non-reg}}$. \[orthogonality\] For $\CM\in \fD_{G,crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}$, the chiral action of $\fD^{\on{non-reg}}_{G,crit}$ maps $\CM^{\nil}$ to $\CM^{\on{non-reg}}$. According to [Lemma \[Sugawara calculation\]]{}, we can find an element of $\fZ_{\fg,x}$, such that its action is nilpotent on $\fD^{0}_{G,crit,x}$ and invertible on $\fD^{\on{non-reg}}_{G,crit,x}$. We can assume that this element comes from a local section $a\in \fz_\fg$. For example, $a$ can be taken to be the section corresponding to the Segal-Sugawara $S_0$ operator. Moreover, we can find a section $a'$ of $\fz_\fg\boxtimes \CO_X$ such that the $\CO$-module endomorphism of $\fD_{G,crit}$ given by $$\label{action of a'} b\mapsto (h\boxtimes \on{id})[a'\otimes b]$$ is nilpotent on $\fD^{0}_{G,crit}$, and invertible on $\fD^{\on{non-reg}}_{G,crit}$. In the above formula $a'\otimes b$ is viewed as an element of the D-module $\fD_{G,crit}\boxtimes \fD_{G,crit}$ on $X\times X$, $[\cdot,\cdot]$ denotes the chiral bracket, and $(h\boxtimes \on{id})$ denotes the De Rham projection $\Delta_!(\fD_{G,crit})\to \fD_{G,crit}$. The chiral action gives rise to a map of D-modules on $X$ $$\varphi:j_x{}_*j_x^*(\fD^{\on{non-reg}}_{G,crit})\otimes \CM^{\nil}\to i_x{}_!(\CM),$$ where $i_x$ (resp., $j_x$) is the embedding of the point $x$ (resp., of its complement). For a section $b\in j_x{}_*j_x^*(\fD^{\on{non-reg}}_{G,crit})$ and an element $m\in \CM^{\nil}$, consider the section $$a'\otimes b\otimes m\in j_{2,x}{}_*j_{2,x}^*(\fz_\fg\boxtimes\fD^{\on{non-reg}}_{G,crit})\otimes \CM^{\nil},$$ where $j_{2,x}$ is the embedding of the complement to $\Delta_X\cup X\times x$ into $X\times X$. By applying the Jacobi identity to the above section, we obtain that the action of $a'$ on the image of $\varphi$, given by the same formula as , is invertible. But this means that the subspace of $\CM$ corresponding to the D-submodule $\on{Im}(\varphi)\subset i_x{}_!(\CM)$ is supported off $\on{Spec}(\fz_{\fg,x})$. Therefore, this subspace belongs to $\CM^{\on{non-reg}}$. Let us assume for a moment that for any non-zero $\CM \in \fD_{G,crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}$, the component $\CM^{\nil}\simeq \wt\Gamma(\CM)$ is necessarily non-zero. Let us show that the functor $\wt\Gamma$ is then full. Let $\wh\fD_{G,crit,x}$ be the canonical associative algebra corresponding to the chiral algebra $\fD_{G,crit}$ and the point $x\in X$, see [Sect. \[functors\]]{}. We have a decomposition $\wh\fD_{G,crit,x}=\wh\fD^{\nil}_{G,crit,x}\oplus \wh\fD^{\on{non-reg}}_{G,crit,x}$, where the first summand is a subalgebra. For $\CM \in \fD_{G,crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}$, the action of $\wh\fD^{\nil}_{G,crit,x}$ preserves the decomposition $\CM=\CM^{\nil}\oplus \CM^{\on{non-reg}}$; moreover, by [Lemma \[orthogonality\]]{}, the action of $\wh\fD^{\on{non-reg}}_{G,crit,x}$ sends $\CM^{\nil}$ to $\CM^{\on{non-reg}}$. Observe that for $\CM\in \fD_{G,crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}$, the map $$\label{generating map} \wh\fD_{G,crit,x}\otimes \CM^{\nil}\to \CM$$ is automatically surjective. Indeed, its image is a $\fD_{G,crit}$-submodule $\CM_1\subset \CM$, which satisfies $\CM_1^{\nil}=\CM^{\nil}$. But then, for the quotient module $\CM_2:=\CM/\CM_1$, we have: $\CM_2^{\nil}=0$, which implies $\CM_2=0$. Similarly, for any element $m\in \CM$ we can always find a section $b$ of $\wh\fD_{G,crit,x}$, such that $b\cdot m$ is a non-zero element of $\CM^{\nil}$. Let now $\CM$ and $\CN$ be two objects of $\fD_{G,crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}$, and let $\phi:\CM^{\nil}\to \CN^{\nil}$ be a map in $\CA^{\ren,\tau}_\fg\text{--}\on{mod}$. By [Lemma \[neutral component\]]{}, $\phi$ is a homomorphism of $\wh\fD^{\nil}_{G,crit,x}$-modules. We have to show that this map extends uniquely to a map of $\wh\fD_{G,crit,x}$-modules $\CM\to\CN$. The uniqueness statement is clear from the surjectivity of . To prove the existence, let us suppose by contradiction that the required extension does not exist. This means that there exist elements $a_i\in \wh\fD_{G,crit,x}$, and $m_i\in \CM^{\nil}$, such that $\underset{i}\sum\, a_i\cdot m_i=0\in \CM$, but $\underset{i}\sum\, a_i\cdot \phi(m_i)=n\neq 0$ in $\CN$. Let $b\in \wh\fD_{G,crit,x}$ be an element such that $0\neq b\cdot n\in \CN^{\nil}$. Let us write $b\cdot a_i=:c_i=c'_i+c''_i$, where $c'_i\in \wh\fD^{\nil}_{G,crit,x}$, and $c''_i\in \wh\fD^{\on{non-reg}}_{G,crit,x}$. By [Lemma \[orthogonality\]]{}, we have $\underset{i}\sum\, c'_i\cdot \phi(m_i)\neq 0$. However, for the same reason, $\underset{i}\sum\, c'_i\cdot m_i=0$, which contradicts the fact that $\phi$ was a morphism of $\wh\fD^{\nil}_{G,crit,x}$-modules. Finally, let us show that $0\neq \F\in \on{D}_{crit}(\Gr_G)\text{--}\on{mod}$, implies that $\CM_\F^{\nil}\neq 0,$ where $\CM_\F$ is the corresponding object of $\fD_{G,crit}\text{--}\on{mod}^{G(\wh{\CO}_x)}$. By [Theorem \[main\]]{} and [Theorem \[total equivalence\]]{}, this is equivalent to the fact that $$0\neq \F\in \on{D}_{crit}(\Gr_G)\text{--}\on{mod}\Rightarrow \Gamma(\Gr_G,\F)\neq 0.$$ Note that the same argument works also in the negative and irrational level cases: For a congruence subgroup $K\subset G(\wh{\CO}_x)$, let $\on{D}_{\kappa}(\Gr_G)\text{--}\on{mod}^K$ be the subcategory of (strongly) $K$-equivariant D-modules, and let $\hat\fg_{\kappa}\text{--}\on{mod}^K$ be the subcategory of $K$-integrable modules. The functor $\Gamma$ of global sections evidently maps $\on{D}_{\kappa}(\Gr_G)\text{--}\on{mod}^K$ to $\hat\fg_{\kappa}\text{--}\on{mod}^K$. Recall now the setting for the Harish-Chandra action of [@BD1], Sect. 7.14. Namely, let $G((t))$ be the loop group corresponding to the point $x\in X$, and $G((t))/K$-the corresponding ind-scheme. We have the convolution functor $$\star:\on{D}_{\kappa}(G((t))/K)\text{--}\on{mod}\times \on{D}_{\kappa}(\Gr_G)\text{--}\on{mod}^K\longrightarrow D^b(\on{D}_{\kappa}(\Gr_G)\text{--}\on{mod}),$$ where $D^b(\cdot )$ stands for the bounded derived category. In addition, we have the functor $$\star:\on{D}_{\kappa}(G((t))/K)\text{--}\on{mod}\times \hat\fg_{\kappa}\text{--}\on{mod}^K\longrightarrow D^b(\hat\fg_{\kappa}\text{--}\on{mod}).$$ Moreover, the (derived) functor of global sections $$R\Gamma: D^b(\on{D}_{\kappa}(\Gr_G)\text{--}\on{mod}^K)\to D^b(\hat\fg_{\kappa}\text{--}\on{mod}^K)$$ intertwines the two actions. \[convolution\] For any non-zero object $\F\in \on{D}_{\kappa}(\Gr_G)\text{--}\on{mod}^K$, there exists a $G(\wh{\CO}_x)$-equivariant object $\F'\in \on{D}_{\kappa}(G((t))/K)\text{--}\on{mod}$, such that $\F'\star \F\in D^b(\on{D}_{\kappa}(\Gr_G)\text{--}\on{mod}^{G(\wh{\CO}_x)})$ is non-zero. We have an equivalence of categories $$\F\mapsto \F^*: \on{D}_{\kappa}(\Gr_G)\text{--}\on{mod}^K\to \on{D}_{-\kappa}(G((t))/K)\text{--}\on{mod}^{G(\wh{\CO}_x)},$$ corresponding to the involution $g\mapsto g^{-1}$ on $G((t))$. For $\F\in \on{D}_{\kappa}(\Gr_G)\text{--}\on{mod}^K$ and $\F'\in \on{D}_{\kappa}(G((t))/K)\text{--}\on{mod}^{G(\wh{\CO}_x)}$, the fiber at $1\in \Gr_G$ of the convolution $\F'\star \F$ is canonically isomorphic to $\on{H}_{DR}(G((t))/K,\F'\otimes \F^*)$. (Note that $\F'\otimes \F^*$ is an object of the derived category of usual (i.e. non-twisted) right D-modules on $G((t))/K$, therefore, global cohomology makes sense.) In particular, this global cohomology is non-zero for $\F'$ being (the direct image of) the constant D-module on a $G(\wh{\CO}_x)$-orbit $G(\wh{\CO}_x)\cdot g \subset G((t))/K$, for some $g\in G((t))/K$, such that the fiber $(\F^*)_g$ is non-zero. Using this lemma, our non-vanishing assertion reduces to the fact that for a non-zero $\F\in D^b(\on{D}_{\kappa}(\Gr_G)\text{--}\on{mod}^{G(\wh{\CO}_x)})$, the object $R\Gamma(\Gr_G,\F)$ is non-zero either. To prove it, note that since the functor $\Gamma$ is exact, we can assume that $\F$ belongs to the abelian category of D-modules. By the semi-smallness result [@BD1], Sect. 5.3.6, the convolution $\star$ is exact on $\on{D}_{\kappa}(\Gr_G)\text{--}\on{mod}^{G(\wh{\CO}_x)}$, i.e., $\on{D}_{\kappa}(\Gr_G)\text{--}\on{mod}^{G(\wh{\CO}_x)}$ acquires a structure of monoidal category. (Note that for $\kappa$ integral, the Satake equivalence identifies $\on{D}_{\kappa}(\Gr_G)\text{--}\on{mod}^{G(\wh{\CO}_x)}$ with the category of representation of the Langlands dual group $^L G$.) For an object $\F\in \on{D}_{\kappa}(\Gr_G)\text{--}\on{mod}^{G(\wh{\CO}_x)}$ (which we can assume to be finitely generated), let $\F^*\in \on{D}_{-\kappa}(\Gr_G)\text{--}\on{mod}^{G(\wh{\CO}_x)}$ be the object as in the proof of [Lemma \[convolution\]]{}, and take $\F'\in \on{D}_{\kappa}(\Gr_G)\text{--}\on{mod}^{G(\wh{\CO}_x)}$ be the Verdier dual of $\F^*$. Let $\delta_1$ be the delta-function twisted D-module, corresponding to the unit point $1\in \Gr_G$. By adjunction, we obtain a non-zero map $\delta_1\to \F'\star \F$. This map is necessarily an injection, because $\delta_1$ is irreducible in $\on{D}_{\kappa}(\Gr_G)\text{--}\on{mod}$. Hence, by the exactness of $\Gamma$, we obtain $$\BV_{\fg,\kappa}\simeq \Gamma(\Gr_G,\delta_1)\neq 0\Rightarrow \Gamma(\Gr_G,\F'\star \F)\neq 0,$$ which, in turn, implies that $\Gamma(\Gr_G,\F)\neq 0$. [199]{} S.  Arkhipov and D. Gaitsgory, [*Differential operators on the loop group via chiral algebras*]{}, IMRN 2002, no. 4, 165–210. A. Beilinson and J. Bernstein, [*Localisation de $g$-modules*]{}, C.R.Acad.Sci.Paris Ser. I Math. 292 (1981), no. 1, 15–18. A. Beilinson and V. Drinfeld, [*Chiral algebras*]{}, Preprint, available at www.math.uchicago.edu/$\sim$benzvi. A. Beilinson and V. Drinfeld, [*Quantization of Hitchin’s integrable system and Hecke eigensheaves*]{}, Preprint, available at www.math.uchicago.edu/$\sim$benzvi. D. Eisenbud and E. Frenkel, Appendix to [*Jet schemes of locally complete intersection canonical singularities*]{}, by M.Mustata, Inv. Math. [**145**]{} (2001) 397–424. E. Frenkel and D. Ben-Zvi, [*Vertex Algebras and Algebraic Curves*]{}, Mathematical Surveys and Monographs [**88**]{}, AMS, 2001. B. Feigin and E. Frenkel, [*Affine Kac-Moody algebras at the critical level and Gelfand-Dikii algebras*]{}, in [*Infinite Analysis*]{}, eds. A. Tsuchiya, T. Eguchi, M. Jimbo, Adv. Ser. in Math. Phys. [**16**]{}, 197–215, Singapore: World Scientific, 1992. E. Frenkel, [*Lectures on Wakimoto modules, opers and the center at the critical level*]{}, math.QA/0210029. E. Frenkel and K. Teleman, in preparation. V. Kac and D. Kazhdan, [*Structure of representations with highest weight of infinite-dimensional Lie algebras*]{}, Adv. in Math. [**34**]{} (1979), 97–108.

--- abstract: | \[Definition of Zeta\_ab function\] Let $d_{\alpha ,\beta }(n)=\sum\limits_{\substack{ n=kl \\ \alpha l<k\leq \beta l}}1$ be the number of ways of factoring n into two almost equal integers. For rational numbers $0<\alpha <\beta $, we consider the following Zeta function $\zeta _{\alpha ,\beta }(s)=\sum\limits_{n=1}^{\infty }\frac{d_{\alpha ,\beta }(n)}{n^{s}}$ for $\Re s>1.$ It has an analytic continuation to $\Re s>1/3.$ We get an asymptotic formula for the mean square of $\zeta _{\alpha,\beta }(s)$ in the strip $1/2<\Re s<1$. As an application, we improve an result on the distribution of primitive Pythagorean triangles. address: 'Department of Mathematics, Qingdao University, Qingdao 266071, P. R. China' author: - Kui Liu title: On the mean value of a kind of Zeta functions --- Introduction and main results ============================= All through this paper, we always suppose $s=\sigma +it$ and $x\geq 2$. Let $$d(n)=\sum \limits_{n=kl}1$$ be the classical divisor function and $$D(n)=\sum_{n\leq x}d(n)$$ be its summatory function. Dirichlet proved $$\label{Sum of d(n)} D(x)=x(\log x+2\gamma -1)+\Delta (x),$$where $\gamma =\lim\limits_{n\rightarrow \infty }\left( \sum\limits_{k=1}^{n}\frac{1}{k}-\log n\right) \approx 0.5721\cdots $ is the Euler constant and $$\Delta (x)\ll x^{\frac{1}{2}}.$$ Voronoi [@Voronoi] improved Dirichlet’s result to $$\Delta (x)\ll x^{\frac{1}{3}}\log x.$$ It is conjectured that for any $\varepsilon >0$, we have $$\Delta (x)\ll_{\varepsilon} x^{ \frac{1}{4}+\varepsilon }.$$ The best result to date is $$\Delta (x)\ll x^{\frac{131}{416}}(\log x)^\frac{26947}{8320},$$due to Huxley [@Huxley]. Let $\zeta \left( s\right) $ be the Riemann Zeta-function, then the generated function of $d(n)$ is $$\zeta ^{2}\left( s\right) =\sum\limits_{n=1}^{\infty }\frac{d\left(n\right) }{n^{s}},{\rm\ \ \ \ for\ }\sigma>1.$$ Hardy-Littlewood [@Hardy; @and; @Littlewood] considered the mean square of $\zeta^{2}\left( s\right)$ $$I_{\sigma }\left( T,\zeta^{2}\right) =\int_{T}^{2T}\left\vert \zeta\left( \sigma +it\right) \right\vert ^{4}dt,\text{\ \ \ \ for\ } 1/2<\sigma<1,$$ and proved $$\label{Fourth moments for Zeta (s)} I_{\sigma }\left( T,\zeta ^{2}\right) =\frac{\zeta ^{4}\left( 2\sigma \right) }{\zeta \left( 4\sigma \right) }T+o\left( T\right).$$ Note that their proof is based on the approximation (for example, see Section 3 of [@Ivic]) $$\label{approximation for square of Zeta(x)} \zeta ^{2}\left( s\right) =\sum_{n\leq x}\frac{d\left( n\right) }{n^{s}} +\chi ^{2}\left( s\right) \sum_{n\leq y}\frac{d\left( n\right) }{n^{1-s}} +O\left( x^{\frac{1}{2}-\sigma }\log t\right), \text{\ \ \ \ for\ } 1/2<\sigma<1,$$ where $x,y\geq 2$, $4\pi^2 xy=t^{2}$ and $$\chi \left( s\right)=\frac{\left( 2\pi \right) ^{s}}{2\Gamma \left( s\right) \cos \left( \frac{\pi s}{2}\right) }$$ is the $\Gamma $-factor in the functional equation$$\label{Functional equation of Riemann zeta-function} \zeta \left( s\right) =\chi \left( s\right) \zeta \left( 1-s\right).$$ In this paper, we focus on the following type divisor function given by $$d_{\alpha ,\beta }(n)=\sum_{\substack{ n=kl \\ \alpha l<k\leq \beta l}}1,$$ where $\alpha ,\beta $ are fixed rational numbers satisfying $0<\alpha <\beta $. Define its generated Zeta function as $$\zeta _{\alpha ,\beta }(s)=\sum_{n=1}^{\infty }\frac{d_{\alpha ,\beta }(n)}{ n^{s}},{\rm\ \ \ \ for\ }\sigma>1.$$ We prove that $\zeta _{\alpha ,\beta }(s)$ has an analytic continuation to $\sigma>1/3$ and get an asymptotic formula for the mean square of $\zeta _{\alpha ,\beta }(s)$ in the strip $1/2<\sigma<1$. \[Main theorem\] For any $\frac{1}{2}<\sigma <1$ and rational numbers $0<\alpha<\beta$, there exists a constant $\varepsilon \left(\sigma\right) >0$ such that $$\int_{T}^{2T}\left\vert \zeta _{\alpha ,\beta }(\sigma +it)\right\vert ^{2}dt=T\sum_{n=1}^{\infty }\frac{d_{\alpha ,\beta }^{2}(n)}{n^{2\sigma }}+O_{\alpha ,\beta ,\sigma }\left( T^{1-\varepsilon \left( \sigma \right) }\right) . \label{Main asymptotic formula}$$ Theorem \[Main theorem\] can be used to study the distribution of primitive Pythagorean triangles (i.e. triples $\left( a,b,c\right) $ with $a,b,c\in\mathbb{N},$ $a^{2}+b^{2}=c^{2},$ $a<b$ and $\gcd \left( a,b,c\right) =1$). Let $P(x)$ denote the number of primitive Pythagorean triangles with perimeter $a+b+c\leq x.$ D. H. Lehmer [@H.Lehmer] proved$$P\left( x\right) =\frac{\log 2}{\pi ^{2}}x+O\left( x^{1/2}\log x\right) .$$ It is difficult to reduce the exponents $1/2$ in the error term, which depends on the zero-free region of the Riemann zeta function. However, assuming the Riemann Hypothesis, it was showed in [@Liu] that, for any $\varepsilon >0,$ we have $$\label{old theorem on triangles} P\left( x\right) =\frac{\log 2}{\pi ^{2}}x+O_{\varepsilon }\left( x^{\frac{5805}{15408}+\varepsilon }\right) .$$ We improve this result by applying Theorem \[Main theorem\] and get \[Theorem 1.2\] If the Riemann Hypothesis is true, then for any $\varepsilon >0,$ we have $$P\left( x\right) =\frac{\log 2}{\pi ^{2}}x+O_{\varepsilon }\left( x^{\frac{4}{11}+\varepsilon }\right) .$$ Note that $\frac{5805}{15408}=0.3767\cdots $ and $\frac{4}{11}=0.3636\cdots .$ Main steps in the proof of Theorem \[Main theorem\] =================================================== First, Let’s recall a way to get the asymptotic formula (\[Fourth moments for Zeta (s)\]). In Chapters 3 of [@Ivic], using the functional equation (\[Functional equation of Riemann zeta-function\]), Ivic derive the Voronoi formula for the error term $\Delta \left( x\right) $ in (\[Sum of d(n)\]). Then in Chapter 4 of [@Ivic], Ivic get the approximation (\[approximation for square of Zeta(x)\]) by the Voronoi formula, from which one can obtain (\[Fourth moments for Zeta (s)\]) in a standard way. Now observing that $\zeta _{\alpha ,\beta }(s)$ is similar to $\zeta ^{2}(s),$ we can realize $\int_{T}^{2T}\left\vert \zeta _{\alpha ,\beta }(\sigma+it)\right\vert ^{2}dt $ as an analogue of $\int_{T}^{2T}\left\vert \zeta (\sigma +it)\right\vert ^{4}dt.$ Our main steps in the proof of Theorem \[Main theorem\] similar to the proof of (\[Fourth moments for Zeta (s)\]). In Section 4, we study the asymptotic property of the summatory function $$D_{\alpha ,\beta }(x)=\sum_{n\leq x}d_{\alpha ,\beta }(n). \label{Summatory function of d_ab(n)}$$In Section 5, we derive a Voronoi type formula for the error term $$\Delta _{\alpha ,\beta }(x)=D_{\alpha ,\beta }(x)-\text{Main terms.}$$In Section 6, using the asymptotic formula of $D_{\alpha ,\beta }(x)$ and the Voronoi type formula for $\Delta_{\alpha ,\beta }(x)$, we obtain the following approximation for $\zeta _{\alpha ,\beta }(s)$, which is the key for the proof of Theorem \[Main theorem\]. \[Proposition of Approximation\] The function $\zeta _{\alpha ,\beta }(s)$ can be analytically extended to the half plane $\sigma >\frac{1}{3}$ with simple poles at $s=\frac{1}{2},1.$ Moreover, suppose $ T\geq 2,$ $s=\sigma +it$ and $4\pi ^{2}xy=t^{2}$, then for any $\frac{1}{2}<\sigma <1,$ $T<t\leq 2T$ and $0<\alpha<\beta$, we have $$\label{Approximation for Zeta_ab} \zeta _{\alpha ,\beta }(s)=\sum_{n\leq x}\frac{d_{\alpha ,\beta }(n)}{n^{s}}+\chi ^{2}\left( s\right) \sum_{n\leq y}\frac{d_{\alpha ,\beta }(n)}{n^{s-1}}+E_{\alpha ,\beta }\left( s\right) ,$$ where $\chi\left( s\right)$ is given by [(\[Functional equation of Riemann zeta-function\])]{} and $E_{\alpha ,\beta }\left( s\right) $ satisfies $$\label{Meansquare for E_ab} \int_{T}^{2T}\left\vert E_{\alpha ,\beta }(\sigma +it)\right\vert ^{2}dt\ll _{\alpha ,\beta ,\sigma }\left( x^{-2\sigma }T^{2}+x^{1-\sigma }T^{\frac{1}{2}}+x^{\frac{1}{2}-\sigma }T+x^{-\sigma }T^{\frac{3}{2}}\right) \log ^{3}T.$$ From (\[Approximation for Zeta\_ab\]), we can derive Theorem \[Main theorem\] in a standard way. Hence the main work of paper is to prove Proposition \[Proposition of Approximation\]. Priliminary lemmas ================== Denote the integral part of $u$ by $[u]$. let $\psi \left( u\right)=u-[u]-\frac{1}{2}$ and $e(x)=e^{2\pi ix}$. It is well known that $\psi \left( u\right)$ has a truncated Fourier expansion (for example, see [@Heath-Brown]). \[Fourier expansion of Psi\] For any real number $H>2,$ we have $$\psi \left( u\right)=-\frac{1}{2\pi i}\sum_{1\leq \left\vert h\right\vert \leq H}\frac{1}{h}e\left( hu\right) +O\left( G\left( u,H\right) \right),$$ where $$\label{Def. of G(u,H)} G\left(u,H\right) =\min \left( 1,\frac{1}{H\left\vert \left\vert u\right\vert \right\vert }\right).$$ We will use the first derivative test (for example, see Chapter 21 of [@Pan; @and; @Pan]). \[First drivitive test\] Let $G\left( x\right) $ and $F\left(x\right) $ be a real differentiable functions such that $\frac{F^{\prime }\left( x\right) }{G\left( x\right) }$ is monotonic and $\frac{F^{\prime }\left( x\right) }{G\left( x\right) }\geq m>0$ or $\frac{F^{\prime }\left( x\right)}{G\left( x\right) }\leq -m<0.$ Then we have $$\left\vert \int_{a}^{b}G\left( x\right) e^{iF\left( x\right) }dx\right\vert \leq 4m^{-1}.$$ We will also use the following Van der Corput B-process (see [@Kuhleitner], Lemma 2.2). \[Convert formula\] Let $C_{i},i=1,\cdots ,7$ be absolute positive constants. Suppose that $g$ is a real-valued function which has four continuous derivatives on the interval $[A,B].$ Let $L$ and $W$ be real parameters not less than $1,$ such that $C_{1}L\leq B-A\leq C_{2}L,$ $$\left\vert g^{\left( j\right) }\left( \omega \right) \right\vert \leq -C_{j+2}WL^{1-j},\ \mathrm{\ \ \ \ for}\ \omega \in \lbrack A,B],\text{ }j=1,2,3,4,$$ and $$g^{\prime \prime }\left( \omega \right) \geq C_{7}WL^{-1}\ \mathrm{\ \ or\ \ }\ g^{\prime \prime }\left( \omega \right) \leq -C_{7}WL^{-1},\ \mathrm{\ \ \ \ for\ }\ \omega \in \lbrack A,B].$$ Let $\phi $ denote the inverse function of $g^{\prime }.$ Define$$\epsilon _{f}=\left\{ \begin{array}{cc} e^{\frac{\pi i}{4}}, &\ \mathrm{\ if}\ g^{\prime \prime }\left( \omega \right) >0\ \mathrm{\ \ \ \ for\ }\ \omega \in \lbrack A,B], \\ e^{-\frac{\pi i}{4}}, &\ \mathrm{if}\ g^{\prime \prime }\left( \omega \right) <0\ \mathrm{\ \ \ \ for\ }\ \omega \in \lbrack A,B] \end{array} \right.$$ and $$r\left( x\right) =\left\{ \begin{array}{cc} 0, &\ \mathrm{if}\ g^{\prime }\left( x\right) \in\mathbb{Z}, \\ \min \left( \frac{1}{\left\vert \left\vert g^{\prime }\left( x\right) \right\vert \right\vert },\sqrt{\frac{L}{W}}\right) , &\ \mathrm{else,}\ \end{array} \right.$$ with $\left\vert \left\vert \cdot \right\vert \right\vert $ denoting the distance from the nearest integer. Then it follows that $$\begin{aligned} \sum_{A< l\leq B}e\left( g\left( l\right) \right) &=&\epsilon _{f}\sideset{}{^{\prime \prime}}\sum\limits_{\min \left( g^{\prime }\left( A\right) ,g^{\prime} \left( B\right) \right) \leq k\leq \max \left( g^{\prime }\left(A\right) ,g^{\prime }\left( B\right) \right) } \frac{e\left( g\left(\phi \left( k\right) \right) -k\phi \left( k\right) \right) }{\sqrt{ \left\vert g^{\prime \prime }\left( \phi \left( k\right) \right) \right\vert}} \\ &&+O\left( r\left( A\right) +r\left( B\right) +\log \left( 2+W\right) \right) ,\end{aligned}$$ with the notation $$\sideset{}{^{\prime \prime }}\sum\limits_{a\leq m\leq b}\Phi \left( n\right) =\frac{1}{2}\left( \chi_{\mathbb{Z} }\left( a\right) \Phi \left( a\right) +\chi _{ \mathbb{Z}}\left( b\right) \Phi \left( b\right) \right) +\sum\limits_{a<m<b}\Phi \left( n\right),$$ where $\chi _{\mathbb{Z} }\left( \cdot \right) $ is the indicator function of the integers and the $O$-constant depends on the constants $C_{i},i=1,\cdots ,7.$ Asymptotic formula for the summatory function ============================================= \[Asymptotic formula for D\_ab\] Let $\alpha =\frac{p_{1}}{q_{1}}$ and $\beta =\frac{p_{2}}{q_{2}}$ with $p_{1},p_{2},q_{1},q_{2}\in\mathbb{N},$ $\gcd \left( p_{1},q_{1}\right)=1$ and $\gcd \left(q_{1},q_{2}\right) =1.$ We have $$D_{\alpha ,\beta }(x)=c_{1}x+c_{2}\sqrt{x}+\Delta _{\alpha ,\beta}(x),$$ where $$c_{1}=c_{1}\left( \alpha ,\beta \right) =\frac{\log \alpha -\log \beta }{2},\ \ \ \ c_{2}=c_{2}\left( \alpha ,\beta \right) =\frac{1}{2}\left( \sqrt{\frac{1}{p_{2}q_{2}}}-\sqrt{\frac{1}{p_{1}q_{1}}}\right) ,$$ and $$\Delta _{\alpha ,\beta }(x)=-\sum_{\sqrt{\frac{x}{\beta }}<l\leq \sqrt{\frac{x}{\alpha }}}\psi \left( \frac{x}{l}\right) +O_{\alpha ,\beta }\left( 1\right) . \label{Expression of Delta_ab}$$ It is enough to consider $d_{\alpha }(n)=\sum\limits_{\substack{n=kl\\k\leq \alpha l}}1$ and $D_{\alpha }(x)=\sum\limits_{n\leq x}d_{\alpha }(n)$. Clearly, $$\begin{aligned} D_{\alpha }(x)&=&\sum_{\substack{ kl\leq x \\ k\leq \alpha l}}1 =\sum_{l\leq x}\sum_{k\leq \min \left( x/l,\alpha l\right) }1.\end{aligned}$$ Write $$\begin{aligned} \label{Sum1+Sum2} D_{\alpha }(x)=\sum\nolimits_{1}+\sum\nolimits_{2}\ ,\end{aligned}$$ with $$\begin{aligned} \sum\nolimits_{1}=\sum_{l\leq \sqrt{\frac{x}{\alpha }}}\sum_{k\leq \alpha l}1\ \ \ \ \mathrm{and} \ \ \ \ \ \sum\nolimits_{2}=\sum_{\sqrt{ \frac{x}{\alpha }}<l\leq x}\sum_{k\leq x/l}1.\end{aligned}$$ It is easy to see that$$\begin{aligned} \label{Sum 1 in the first lemma} \sum\nolimits_{1} &=&\sum_{l\leq \sqrt{\frac{x}{\alpha }}}{\left(\alpha l-\psi \left( \alpha l\right)-1/2\right)} \nonumber\\ &=&\frac{x}{2}-\sqrt{\alpha x}\psi \left( \sqrt{\frac{x}{\alpha }}\right) -\sum_{l\leq \sqrt{\frac{x}{\alpha }}}\psi \left( \alpha l\right) -\frac{1}{2}\sqrt{\frac{x}{\alpha }}+O_{\alpha }\left( 1\right).\end{aligned}$$ Similarly, $$\begin{aligned} \label{Sum 2 in the first lemma} \sum\nolimits_{2} &=&\sum_{\sqrt{\frac{x}{\alpha }}<l\leq x}\left( x/l-\psi (x/l)-1/2\right)\nonumber \\ &=&x\sum_{\sqrt{\frac{x}{\alpha }}<l\leq x}1/l-\sum_{\sqrt{\frac{x}{\alpha }}<l\leq x}\psi (x/l)-\frac{1}{2}x+\frac{1}{2}\sqrt{\frac{x}{\alpha }}+O\left( 1\right) .\end{aligned}$$By the Euler-Maclaurin summation, we have $$\begin{aligned} \label{sum 1/l} \sum_{\sqrt{\frac{x}{\alpha }}<l\leq x}1/l &=&\frac{1}{2}\log x+\frac{1}{2}\log \alpha +\sqrt{\frac{\alpha }{x}}\psi \left( \sqrt{\frac{x}{\alpha }}\right) +O_{a}\left( \frac{1}{x}\right).\end{aligned}$$Combining (\[Sum1+Sum2\])-(\[sum 1/l\]), we get$$\begin{aligned} D_{\alpha }(x) =\frac{x}{2}\log x+\frac{\log \alpha }{2}x-\sum_{\sqrt{\frac{x}{\alpha }}<l\leq x}\psi (x/l)-\sum_{l\leq \sqrt{\frac{x}{\alpha }}}\psi \left( \alpha l\right) +O_{\alpha }\left( 1\right) .\end{aligned}$$Note that $$-\sum_{l\leq \sqrt{\frac{x}{\alpha }}}\psi \left( \alpha l\right) =-\sum_{l\leq \sqrt{\frac{q_1x}{p_1}}}\psi \left( \frac{p_1 l}{q_1} \right) =\frac{1}{2}\sqrt{\frac{x}{p_1q_1}}+O_\alpha\left( 1\right).$$Hence $$\begin{aligned} \label{D_alpha} D_{\alpha }(x) =\frac{x}{2}\log x+\frac{\log \alpha }{2}x-\sum_{\sqrt{\frac{x}{\alpha }} <l\leq x}\psi (x/l)+\frac{1}{2}\sqrt{\frac{x}{p_1q_1}}+O_{\alpha }\left( 1\right) .\end{aligned}$$ Similarly, for $d_{\beta }(n)=\sum\limits_{\substack{n=kl\\k\leq \beta l}}1$ and $D_{\beta }(x)=\sum\limits_{n\leq x}d_{\beta }(n)$, we have $$\label{D_beta} D_{\beta }(x)=\frac{x}{2}\log x+\frac{\log \beta }{2}x-\sum_{\sqrt{\frac{x}{\beta }}<l\leq x}\psi (x/l)+\frac{1}{2}\sqrt{\frac{x}{p_2q_2}}+O_{\beta }\left( 1\right).$$ Now Proposition \[Asymptotic formula for D\_ab\] follows from (\[D\_alpha\]), (\[D\_beta\]) and $$D_{\alpha ,\beta }(x)=D_{\beta }(x)-D_{\alpha }(x).$$ \[Upper bound of Delta\_ab\] We have $$D_{\alpha ,\beta }(x)=c_{1}x+c_{2}\sqrt{x}+O_{\alpha ,\beta }\left( x^{\frac{1}{3}}\right),$$ where $c_{1},\ c_{2}$ are the same as Proposition [\[Asymptotic formula for D\_ab\]]{}. This can be proved easily (even with a better upper bound for the error term) by applying Lemma \[Fourier expansion of Psi\] and exponential pairs (see [@Graham; @and; @Kolesnik]) to Proposition \[Asymptotic formula for D\_ab\]. A Voronoi type formula ====================== Define $$d_{\alpha ,\beta }\left( n,H\right) =\underset{n=hk}{\sum_{1\leq h\leq H}\sideset{}{^{\prime \prime }}\sum\limits_{\ h\alpha \leq k\leq h\beta }}1,$$ where the notation $\sideset{}{^{\prime \prime}}\sum\limits$ is the same as Lemma \[Convert formula\]. Using the Van der Corput B-process and the same argument as Section 6.2 of [@Zhai], we can derive the following Voronoi type formula for $\Delta _{\protect\alpha,\protect\beta }(x)$. \[Voronoi formula\] For any $H\geq 2$ and rational numbers $0<\alpha<\beta$, we have $$\Delta _{\alpha ,\beta }(x)=M_{\alpha ,\beta }\left( x,H\right) +E_{\alpha ,\beta }\left( x,H\right) +F_{\alpha ,\beta }\left(x,H\right) ,$$ where $$\label{M_ab(x,H)} M_{\alpha ,\beta }\left( x,H\right)=\frac{x^{\frac{1}{4}}}{\pi \sqrt{2}}\sum_{n\leq \beta H^{2}} \frac{d_{\alpha ,\beta }\left( n,H\right) }{n^{\frac{3}{4}}}\cos \left( 4\pi \sqrt{nx}-\frac{\pi }{4}\right),$$ $$E_{\alpha ,\beta }\left( x,H\right) \ll \sum_{\sqrt{\frac{x}{\alpha }}<l\leq \sqrt{\frac{x}{\beta }}}G\left( \frac{x}{l},H\right)$$ and $$\label{bound for F(x,H)} F_{\alpha ,\beta }\left( x,H\right) \ll _{\alpha ,\beta }\log H.$$ Applying Lemma \[Fourier expansion of Psi\] to (\[Expression of Delta\_ab\]), we get $$\Delta _{\alpha ,\beta }(x)=\frac{1}{2\pi i}\sum_{1\leq \left\vert h\right\vert \leq H}\frac{1}{h}\sum_{\sqrt{\frac{x}{\beta }}<l\leq \sqrt{\frac{x}{\alpha }}}e\left( \frac{hx}{l}\right) +E_{\alpha ,\beta }\left( x,H\right)+O_{\alpha,\beta }\left(1\right),$$ with $$\label{E_ab <<} E_{\alpha ,\beta }\left( x,H\right) \ll \sum_{\sqrt{\frac{x}{\alpha}}<l\leq \sqrt{\frac{x}{\beta }}}G\left( \frac{x}{l},H\right).$$ Let $$\label{Sigma over h and l} S_{\alpha,\beta}(x,H) =\frac{1}{2\pi i}\sum_{1\leq h\leq H}\frac{1}{h}\sum_{\sqrt{\frac{x}{ \beta }}<l\leq \sqrt{\frac{x}{\alpha }}}e\left( \frac{hx}{l}\right),$$ then we can write $$\label{Delta_ab=M+E+F} \Delta _{\alpha ,\beta }(x)=\frac{1}{2\pi i}\left( S_{\alpha,\beta}(x,H) -\overline{S_{\alpha,\beta}(x,H)} \right) +E_{\alpha ,\beta }\left( x,H\right) +O_{\alpha ,\beta}\left(1\right).$$ To treat the inner sum $$\sum\limits_{\sqrt{\frac{x}{\beta }}<l\leq \sqrt{\frac{x}{\alpha }}}e\left( \frac{hx}{l}\right) \text{ \ for }1\leq h\leq H$$ in (\[Sigma over h and l\]), we apply Lemma \[Convert formula\]. Let $$A=\sqrt{\frac{x}{\beta }},\ \ B=\sqrt{\frac{x}{\alpha }}\ \ {\rm{and}} \ \ g\left( l\right) =\frac{hx}{l},$$ then we have $$g^{\prime }\left( l\right) =-\frac{hx}{l^{2}},\ \ g^{\prime \prime }\left( l\right) =\frac{2hx}{l^{3}},\ \ g^{\left( 3\right) }\left( l\right) =-\frac{6hx}{l^{4}},\ \ g^{\left( 4\right) }\left( l\right) =\frac{24hx}{l^{5}},$$ $$g^{\prime }\left( B\right) =-h\alpha ,\ \ g^{\prime }\left( A\right) =-h\beta,\ \ \frac{2\alpha ^{\frac{3}{2}}h}{\sqrt{x}}<g^{\prime \prime }\left( l\right) \leq \frac{2\beta ^{\frac{3}{2}}h}{\sqrt{x}}$$ and $$\left\vert g^{\prime \prime \prime }\left( l\right) \right\vert \ll _{\alpha ,\beta }\frac{h}{x}.$$ Hence we can take $$W=1,\ \ L=\frac{\sqrt{x}}{h},\ \ \phi \left( k\right) =\sqrt{-\frac{hx}{k}},$$ $$g\left( \phi \left( k\right) \right) -k\phi \left( k\right) =2\sqrt{-hkx},\ \ {\rm{and}}\ \ g^{\prime \prime }\left( \phi \left( k\right) \right) =2\sqrt{\frac{\left( -k\right) ^{3}}{hx}}.$$ Noting $\alpha ,\beta $ are rational numbers, we have $$\label{r(A) and r(B)} r\left( A\right) ,r\left( B\right) \ll _{\alpha ,\beta }1.$$ Now for $1\leq h\leq H,$ by Lemma \[Convert formula\], we get $$\begin{aligned} \label{Sum over l e()} \sum\limits_{\sqrt{\frac{x}{\alpha }}<l\leq \sqrt{\frac{x}{\beta }}}e\left( \frac{hx}{l}\right) &=&\frac{e^{\frac{\pi i}{4}}}{\sqrt{2}} \sideset{}{^{\prime \prime }}\sum\limits_{\ -h\beta \leq k\leq -h\alpha }\frac{h^{\frac{1}{ 4}}x^{\frac{1}{4}}}{\left( -k\right) ^{\frac{3}{4}}}e\left( 2\sqrt{-hkx} \right) +O_{\alpha ,\beta }\left( 1\right) \\ &=&\frac{1}{\sqrt{2}}\sideset{}{^{\prime \prime }}\sum\limits_{\ h\alpha \leq k\leq h\beta }\frac{h^{\frac{1}{4}}x^{\frac{1}{4}}}{k^{\frac{3}{4}}}e\left( 2\sqrt{hkx}+ \frac{1}{8}\right) +O_{\alpha ,\beta }\left( 1\right) .\text{ } \notag\end{aligned}$$ Inserting (\[Sum over l e()\]) to (\[Sigma over h and l\]) gives $$\begin{aligned} S_{\alpha,\beta}(x,H) &=&\frac{1}{\sqrt{2}}\sum_{1\leq h\leq H}\frac{1}{h} \sideset{}{^{\prime \prime}}\sum\limits_{\ h\alpha \leq k\leq h\beta }\frac{h^{\frac{1}{4}} x^{\frac{1}{4}}}{k^{\frac{3}{4}}}e\left( 2\sqrt{hkx}+\frac{1}{8}\right)+O_{\alpha ,\beta}\left( \log H\right)\\ &=&\frac{x^{\frac{1}{4}}}{\sqrt{2}}\sum_{1\leq h\leq H}\sideset{}{^{\prime \prime}}\sum\limits_{\ h\alpha \leq k\leq h\beta }\frac{1}{\left( hk\right) ^{\frac{3}{4}}}e\left( 2\sqrt{hkx}+\frac{1}{8}\right) +O_{\alpha ,\beta } \left(\log H\right) \\ &=&\frac{x^{\frac{1}{4}}}{\sqrt{2}}\sum_{n\leq \beta H^{2}}\frac{d_{\alpha ,\beta }\left( n,H\right) }{n^{\frac{3}{4}}}e\left( 2\sqrt{nx}+\frac{1}{8}\right) +O_{\alpha ,\beta }\left( \log H\right) .\end{aligned}$$ Thus$$\frac{1}{2\pi i}\left( S_{\alpha,\beta}(x,H) -\overline{S_{\alpha,\beta}(x,H) }\right) =\frac{x^{\frac{1}{4}}}{ \pi \sqrt{2}}\sum_{n\leq \beta H^{2}}\frac{d_{\alpha ,\beta }\left( n,H\right) }{n^{\frac{3}{4}}}\cos \left( 4\pi \sqrt{nx}-\frac{\pi }{4} \right) +O_{\alpha ,\beta }\left( \log H\right).$$ This combining with (\[Delta\_ab=M+E+F\]) and (\[E\_ab &lt;&lt;\]) yields Lemma \[Voronoi formula\]. The bound (\[bound for F(x,H)\]) is important in the proof of Theorem \[Main theorem\]. If $\alpha,\beta$ are not rational numbers, the author can’t get the estimate (\[bound for F(x,H)\]). Because in this case the estimate (\[r(A) and r(B)\]) does not holds. Proof of Proposition \[Proposition of Approximation\] ===================================================== First, let’s show that $\zeta _{\alpha ,\beta }(s)$ can be analyticly extended to $\sigma >\frac{1}{3}.$ For $\sigma >1$ and any $N\geq 1,$ write $$\begin{aligned} \zeta _{\alpha ,\beta }(s) &=&\sum_{n\leq N}\frac{d_{\alpha ,\beta }(n)}{n^{s}} +\sum_{n>N}\frac{d_{\alpha ,\beta }(n)}{n^{s}} \\ &=&\sum_{n\leq N}\frac{d_{\alpha ,\beta}(n)}{n^{s}}+\int_{N^{+}}^{\infty }u^{-s}dD_{\alpha ,\beta}(u),\end{aligned}$$ where $D_{\alpha ,\beta }(u)$ is defined by (\[Summatory function of d\_ab(n)\]). Applying Proposition \[Asymptotic formula for D\_ab\], we get $$\begin{aligned} \zeta _{\alpha ,\beta }(s) &=&\sum_{n\leq N}\frac{d_{\alpha ,\beta }(n)}{n^{s}} +\int_{N^{+}}^{\infty }u^{-s}d\left(c_{1}u+c_{2}\sqrt{u}+\Delta_{\alpha ,\beta }(u)\right) \\ &=&\sum_{n\leq N}\frac{d_{\alpha ,\beta }(n)}{n^{s}}+c_{1}\int_{N^{+}}^{\infty }u^{-s}du +\frac{c_{2}}{2}\int_{N^{+}}^{\infty}u^{-s-1/2}du+\int_{N^{+}}^{\infty }u^{-s}d\Delta _{\alpha ,\beta}(u).\end{aligned}$$ By partial integration, we have $$\label{Zeta_ab break 1} \zeta _{\alpha ,\beta }(s)=\sum_{n\leq N}\frac{d_{\alpha ,\beta }(n)}{n^{s}} -\frac{c_{1}N^{1-s}}{1-s}-\frac{c_{2}N^{\frac{1}{2}-s}}{1-2s} +s\int_{N^{+}}^{\infty }\Delta _{\alpha ,\beta }(u)u^{-s-1}du+O\left( N^{\frac{1}{3}-\sigma }\right) .$$ From Corollary \[Upper bound of Delta\_ab\], we can see that the integral in (\[Zeta\_ab break 1\]) is absolutely convergent for $\sigma>\frac{1}{3},$ hence (\[Zeta\_ab break 1\]) gives an analytic continuation of $\zeta _{\alpha ,\beta }(s)$ for $\sigma >\frac{1}{3}.$ This proves the first assertion of Proposition \[Proposition of Approximation\]. Now suppose $\sigma >\frac{1}{3}$ and $2\leq T<t\leq 2T$. From now on, we take $N=T^{A}$ with $A>0$ being a constant, sufficiently large. Break the sum in (\[Zeta\_ab break 1\]) into $$\label{Zeta_ab break 2} \sum_{n\leq N}\frac{d_{\alpha ,\beta }(n)}{n^{s}} =\sum_{n\leq x}\frac{d_{\alpha ,\beta }(n)}{n^{s}}+\sum_{x<n\leq N}\frac{d_{\alpha ,\beta }(n)}{n^{s}}.$$ For the second sum, applying Proposition \[Asymptotic formula for D\_ab\] again, we have$$\begin{aligned} \sum_{x<n\leq N}\frac{d_{\alpha ,\beta }(n)}{n^{s}} &=&\int_{x}^{N}u^{-s}dD_{\alpha ,\beta }(u) \\ &=&\int_{x}^{N}u^{-s}d\left( c_{1}\left( \alpha ,\beta \right)u +c_{2}\left( \alpha ,\beta \right) \sqrt{u}+\Delta _{\alpha ,\beta}(u)\right) .\end{aligned}$$ By partial integration, we have $$\begin{aligned} \label{Zeta_ab break 33} \sum_{x<n\leq N}\frac{d_{\alpha ,\beta }(n)}{n^{s}} &=&c_{1}\left( \alpha ,\beta \right) \left( \frac{N^{1-s}}{1-s}-\frac{x^{1-s}}{1-s}\right)+c_{2}\left( \alpha ,\beta \right) \left( \frac{N^{1/2-s}}{1-2s}-\frac{x^{1/2-s}}{1-2s}\right)\\ &+&N^{-s}\Delta _{\alpha ,\beta }(N)-x^{-s}\Delta _{\alpha ,\beta}(x)+s\int_{x}^{N}\Delta _{\alpha ,\beta }(u)u^{-s-1}du.\notag\end{aligned}$$ Combining (\[Zeta\_ab break 1\]), (\[Zeta\_ab break 2\]) and (\[Zeta\_ab break 33\]), we get $$\label{Zeta_ab break 4} \zeta _{\alpha ,\beta }(s)=\sum_{n\leq x}\frac{d_{\alpha ,\beta }(n)}{n^{s}} +s\int_{x}^{N}\Delta _{\alpha ,\beta }(u)u^{-s-1}du+O_{\alpha ,\beta ,\sigma}\left( x^{1-\sigma }t^{-1}+x^{1/3-\sigma }\right)$$ holds for any $\sigma >\frac{1}{3}.$ Our tool to prove Proposition \[Proposition of Approximation\] is the Voronoi formula for $\Delta _{\alpha ,\beta }(x).$ Using Lemma \[Voronoi formula\], we can write $$\label{Start point. Break Zeta_ab(s)} s\int_{x}^{N}\Delta _{\alpha ,\beta }(u)u^{-s-1}du=\mathfrak{M}\left( s\right) +\mathfrak{E}\left( s\right)+\mathfrak{F}\left( s\right),$$ where $$\label{M(s)} \mathfrak{M}\left( s\right) =\mathfrak{M}_{\alpha ,\beta } \left(s,H,x,N\right) =s\int_{x}^{N}M_{\alpha ,\beta }\left( u,H\right)u^{-s-1}du,$$ $$\label{E(s)} \mathfrak{E}\left( s\right) =\mathfrak{E}_{\alpha ,\beta } \left(s,H,x,N\right) =s\int_{x}^{N}E_{\alpha ,\beta }\left( u,H\right)u^{-s-1}du,$$ and $$\label{F(s)} \mathfrak{F}\left( s\right) =\mathfrak{F}_{\alpha ,\beta }\left(s,H,x,N\right) =s\int_{x}^{N}F_{\alpha ,\beta }\left( u,H\right)u^{-s-1}du.$$ In Section 7, we will show that the upper bound of $\mathfrak{E}\left( s\right)$ is small, when $H$ is large comparing to $N$ and the mean square of $\mathfrak{F}\left( s\right)$ has an acceptable estimate; see Lemma \[Lemma Flower F(s)\] and Lemma \[Lemma Flower E(s)\], respectively. In Section 8, we will pick out the second term in (\[Approximation for Zeta\_ab\]) from $\mathfrak{M}(s)$; see Lemma \[Lemma Flower M(s)\], Combining (\[Zeta\_ab break 4\]) with Lemmas \[Lemma Flower F(s)\]-\[Approximation for Zeta\_ab\], we get Proposition \[Proposition of Approximation\]. An upper bound and a mean square estimate ========================================= To bound $\mathfrak{E}\left( s\right),$ we need the following mean value estimate for $G\left( u,H\right) $ defined by (\[Def. of G(u,H)\]). \[mean value of G(u,H)\] For any $N\geq 1$ and $H\geq 2,$ we have $$\int_{0}^{N}G\left( u,H\right) du\ll \frac{N\log H}{H}.$$ Note that $G\left( u,H\right) $ is a positive peridodic function with period $1,$ then we have $$\begin{aligned} \int_{0}^{N}G\left( u,H\right) du &\leq &\sum_{k=0}^{\left[ N\right]}\int_{k}^{k+1}G\left( u,H\right) du \\ &\ll&N\int_{0}^{1}G\left( u,H\right) du \\ &=&N\int_{-1/2}^{1/2}\min \left( 1,\frac{1}{H\left\vert \left\vert u\right\vert \right\vert }\right) du.\end{aligned}$$ Noting $\left\vert \left\vert u\right\vert \right\vert =\left\vert u\right\vert $ for $u\in \lbrack -1/2,1/2],$ we get $$\begin{aligned} \int_{0}^{N}G\left( u,H\right) du &\ll &N\int_{-1/2}^{1/2}\min \left( 1,\frac{1}{H\left\vert u\right\vert }\right) du \\ &\ll&N\int_{0}^{1/2}\min \left( 1,\frac{1}{Hu}\right) du \\ &\ll&N\int_{0}^{1/H}du+\frac{N}{H}\int_{1/H}^{1/2}\frac{1}{u}du,\end{aligned}$$ which yields $$\int_{0}^{N}G\left( u,H\right) du\ll \frac{N\log H}{H}.$$ By Lemma \[mean value of G(u,H)\], we can get \[Lemma Flower F(s)\] For any $\sigma >1/2$, we have $$\mathfrak{E}\left( s\right) \ll \frac{tx^{-\sigma -1}N^{2}\log H}{H}.$$ By (\[E(s)\]) and trivial estimates, we get $$\begin{aligned} \mathfrak{E}\left(s\right) &\ll &t\int_{x}^{N}\sum_{\sqrt{\frac{u}{\alpha}} <l\leq \sqrt{\frac{u}{\beta }}}G\left( \frac{u}{l},H\right)u^{-\sigma -1}du\\ &\ll&tx^{-\sigma -1}\sum_{\sqrt{\frac{x}{\alpha }}<l\leq \sqrt{\frac{N}{\beta }}}\int_{x}^{N}G\left( \frac{u}{l},H\right) du \\ &=&tx^{-\sigma -1}\sum_{\sqrt{\frac{x}{\alpha }}<l\leq \sqrt{\frac{N}{\beta }}}l\int_{\frac{x}{l}}^{\frac{N}{l}}G\left( u,H\right) du.\end{aligned}$$ This combining with Lemma \[mean value of G(u,H)\] yields $$\mathfrak{E}\left( s\right) \ll tx^{-\sigma -1}\sum_{l\leq \sqrt{\frac{N}{\beta}}}l\int_{0}^{N}G\left( u,H\right) du \ll \frac{tx^{-\sigma-1}N^{2}\log H}{H}.$$ Now we consider the mean square of $\mathfrak{F}\left( s\right)$. \[Lemma Flower E(s)\] For $\sigma >1/2,$ we have $$\int_{T}^{2T}\left\vert \mathfrak{F}\left( s\right) \right\vert^{2}dt\ll_{\alpha ,\beta,\sigma} x^{-2\sigma }T^{2}\log ^{2}H\log N.$$ Noting $F_{\alpha ,\beta }\left( u\right) \ll _{\alpha ,\beta }\log H$ and unfolding the square in the integral, we get $$\begin{aligned} \int_{T}^{2T}\left\vert \mathfrak{F}\left( s\right) \right\vert^{2}dt &\ll & T^{2}\int_{T}^{2T}\left\vert \int_{x}^{N}F_{\alpha,\beta }\left( u\right)u^{-s-1}du\right\vert ^{2}dt\\ &\ll_{\alpha ,\beta } &T^{2}\log ^{2}H\int_{x}^{N}\int_{x}^{N}\left(u_{1}u_{2}\right) ^{-\sigma -1}\left\vert \int_{T}^{2T}\left( \frac{u_{2}}{u_{1}}\right) ^{it}dt\right\vert du_{1}du_{2}\end{aligned}$$ Applying Lemma \[First drivitive test\] to the above integral over $t$, we have $$\begin{aligned} \int_{T}^{2T}\left\vert \mathfrak{F}\left( s\right) \right\vert^{2}dt &\ll_{\alpha ,\beta }& T^{2}\log^{2}H\int_{x}^{N}\int_{x}^{N}\left(u_{1}u_{2}\right)^{-\sigma -1} \min \left( T,\frac{1}{\left\vert \log \frac{u_{2}}{u_{1}}\right\vert }\right) du_{1}du_{2} \\ &\ll_{\alpha ,\beta } &T^{2}\log^{2}H\int_{x}^{N}\int_{u_{1}}^{N}\left(u_{1}u_{2}\right)^{-\sigma -1} \min \left( T,\frac{1}{\log \frac{u_{2}}{u_{1}}}\right) du_{1}du_{2}.\end{aligned}$$ Write this as $$\label{Meansquare of E(s)} \int_{T}^{2T}\left\vert \mathfrak{F}\left( s\right) \right\vert^{2}dt \ll _{\alpha ,\beta }\int_{1}+\int_{2}+\int_{3},$$ where $$\int_{1}=T^{3}\log ^{2}H\int_{x}^{N}u_{1}^{-\sigma -1}\int_{u_{1}}^{e^{\frac{1}{T}}u_{1}}u_{2}^{-\sigma -1}du_{2}du_{1},$$ $$\int_{2}=T^{2}\log ^{2}H\int_{x}^{N}u_{1}^{-\sigma -1}\int_{e^{\frac{1}{T}}u_{1}}^{\frac{3}{2}u_{1}}u_{2}^{-\sigma -1} \frac{1}{\log \left( \frac{u_{2}}{u_{1}}\right) }du_{2}du_{1}$$ and $$\int_{3}=T^{2}\log ^{2}H\int_{x}^{N}u_{1}^{-\sigma -1}\int_{\frac{3}{2}u_{1}}^{N}u_{2}^{-\sigma -1} \frac{1}{\log\frac{u_{2}}{u_{1}}}du_{2}du_{1}.$$ Let’s deal with $\int_{i},\ i=1,2,3$ respectively. For $\int_{1},$ we have $$\begin{aligned} \int_{1} &\ll &T^{3}\log ^{2}H\int_{x}^{N}u_{1}^{-2\sigma-2}\int_{u_{1}}^{e^{\frac{1}{T}}u_{1}}du_{2}du_{1} \\ &\ll &T^{3}\log ^{2}H\int_{x}^{N}u_{1}^{-2\sigma -2}\left( e^{\frac{1}{T}}u_{1}-u_{1}\right) du_{1} \\ &\ll &T^{3}\left( e^{\frac{1}{T}}-1\right) \log^{2}H\int_{x}^{N}u_{1}^{-2\sigma -1}du_{1},\end{aligned}$$ which yields $$\label{Meansquare of E(s) ingegral 1} \int_{1}\ll _{\sigma }x^{-2\sigma }T^{2}\log ^{2}H.$$ For $\int_{2},$ we have $$\begin{aligned} \int_{2} &=&T^{2}\log ^{2}H\int_{x}^{N}u_{1}^{-\sigma -1} \int_{e^{\frac{1}{T}}u_{1}}^{\frac{3}{2}u_{1}}u_{2}^{-\sigma -1}\frac{1}{\log \left( \frac{u_{2}}{u_{1}}\right)}du_{2}du_{1} \\ &=&T^{2}\log ^{2}H\int_{x}^{N}u_{1}^{-\sigma -1}\int_{e^{\frac{1}{T}}u_{1}}^{\frac{3}{2}u_{1}}u_{2}^{-\sigma -1}\frac{1}{\log \left(1+\frac{u_{2}-u_{1}}{u_{1}}\right) }du_{2}du_{1} \\ &\ll &T^{2}\log ^{2}H\int_{x}^{N}u_{1}^{-2\sigma -1}\int_{e^{\frac{1}{T}}u_{1}}^{\frac{3}{2}u_{1}}\frac{1}{u_{2}-u_{1}}du_{2}du_{1} \\ &\ll &T^{2}\log ^{2}H\int_{x}^{N}u_{1}^{-2\sigma -1}\log u_{1}du_{1},\end{aligned}$$ which yields $$\int_{2}\ll _{\sigma }x^{-2\sigma }T^{2}\log ^{2}H\log N. \label{Meansquare of E(s) ingegral 2}$$ For $\int_{3},$ we have $$\label{Meansquare of E(s) ingegral 3} \int_{3}\ll T^{2}\log ^{2}H\left( \int_{x}^{N}u^{-\sigma-1}du\right) ^{2}\ll _{\sigma }x^{-2\sigma }T^{2}\log ^{2}H.$$ From (\[Meansquare of E(s)\])-(\[Meansquare of E(s) ingegral 3\]), we get Lemma \[Lemma Flower E(s)\]. Picking out the second term in Proposition \[Proposition of Approximation\] =========================================================================== The second term in Proposition \[Proposition of Approximation\] is hidden in $\mathfrak{M}\left( s\right)$. In this Section, we will pick it out and prove \[Lemma Flower M(s)\] For $\sigma>1/2,$ we have $$\begin{aligned} \label{approximation for flower M} \mathfrak{M}\left( s\right)&=&\chi ^{2}\left( s\right) \sum_{n\leq x}d_{\alpha ,\beta }\left( n\right) n^{s-1}\\ \notag&&\indent+O\left( t^{-\frac{1}{2}}x^{1-\sigma }\log H+x^{1/2-\sigma }\log H+x^{\frac{1}{2}-\sigma }\log t+x^{-\sigma }t^{\frac{1}{2}}\log t\right) .\end{aligned}$$ The idea of the proof for Lemma \[Lemma Flower M(s)\] comes from Chapter 4 of [@Ivic]. By (\[M(s)\]) and (\[M\_ab(x,H)\]), we have $$\mathfrak{M}\left( s\right) \mathfrak{=}\frac{s}{\pi \sqrt{2}}\int_{x}^{N}u^{-s-\frac{3}{4}} \sum\limits_{n\leq \beta H^{2}}\frac{d_{\alpha,\beta }\left( n,H\right)}{n^{\frac{3}{4}}} \cos \left( 4\pi \sqrt{nu}-\frac{\pi }{4}\right) du.$$ Let $\eta >0$ be a fixed, sufficiently small constant. Using $\cos z=\frac{e^{iz}+e^{-iz}}{2},$ we can write $$\label{Flower M(s)} \mathfrak{M}\left( s\right) \mathfrak{=M}_{1}\left( s\right)+\mathfrak{M}_{2}\left( s\right) +\mathfrak{M}_{3}\left( s\right)+\mathfrak{M}_{4}\left( s\right)$$ with $$\mathfrak{M}_{1}\left( s\right) =\frac{s}{2\pi \sqrt{2}}\int_{x}^{N}u^{-s-\frac{3}{4}} \sum\limits_{n\leq \left( 1+\eta \right)y}\frac{d_{\alpha,\beta }\left( n,H\right) }{n^{\frac{3}{4}}}e\left( 2\sqrt{nu}-\frac{1}{8} \right) du,$$ $$\mathfrak{M}_{2}\left( s\right) =\frac{s}{2\pi \sqrt{2}}\int_{x}^{N}u^{-s-\frac{3}{4}}\sum\limits_{\left( 1+\eta \right) y<n\leq \beta H^{2}}\frac{d_{\alpha ,\beta }\left( n,H\right) }{n^{\frac{3}{4}}}e\left( 2\sqrt{nu}- \frac{1}{8}\right) du,$$ $$\mathfrak{M}_{3}\left( s\right) =\frac{s}{2\pi \sqrt{2}}\int_{x}^{N}u^{-s- \frac{3}{4}}\sum\limits_{n\leq y}\frac{d_{\alpha ,\beta }\left( n,H\right) }{ n^{\frac{3}{4}}}e\left( -2\sqrt{nu}+\frac{1}{8}\right) du$$ and $$\mathfrak{M}_{4}\left( s\right) =\frac{s}{2\pi \sqrt{2}}\int_{x}^{N}u^{-s-\frac{3}{4}}\sum\limits_{y<n\leq \beta H^{2}}\frac{d_{\alpha ,\beta }\left( n,H\right) }{n^{\frac{3}{4}}}e\left( -2\sqrt{nu}+\frac{1}{8}\right) du.$$ We will bound $\mathfrak{M}_{2}\left( s\right),$ $\mathfrak{M}_{3}\left( s\right)$ and $\mathfrak{M}_{4}\left( s\right)$ in the following Lemmas \[Flower M\_2(s)\]-\[Flower M\_4(s)\] and pick out the first term on the right side of (\[approximation for flower M\]) in Lemma \[Lemma Flower M\_1\]. From Lemmas \[Flower M\_2(s)\]-\[Lemma Flower M\_1\] and (\[Flower M(s)\]). \[Flower M\_2(s)\] For $\sigma >1/2,$ we have $$\mathfrak{M}_{2}\left( s\right) \ll t^{-\frac{1}{2}}x^{1-\sigma }\log H.$$ Write $$\begin{aligned} \mathfrak{M}_{2}\left( s\right) =\frac{s}{2\pi \sqrt{2}}\sum_{\left(1+\eta\right) y<n\leq \beta H^{2}} \frac{d_{\alpha ,\beta }\left( n;H\right) }{n^{3/4}}\int_{x}^{N}u^{-\sigma -3/4}e\left( -\frac{t}{2\pi }\log u +2\sqrt{nu}-1/8\right) du.\end{aligned}$$ In Lemma \[First drivitive test\], taking $$G\left( u\right) =u^{-\sigma -3/4}$$ and $$F_{t}\left( u\right) =-\frac{t}{2\pi }\log u+2\sqrt{nu}-1/8,$$ we have $$F_{t}^{\prime }\left( u\right) =-\frac{t}{2\pi u}+\sqrt{\frac{n}{u}}$$ and $$\frac{F_{t}^{\prime }\left( u\right) }{G\left( u\right) }=-\frac{t}{2\pi }u^{\sigma -1/4}+\sqrt{n}u^{\sigma +1/4}.$$ Since $n>\left(1+\eta\right)y,\ u>x$ and $4\pi ^{2}xy=t^{2},$ then $$\left( \frac{F_{t}^{\prime }\left( u\right) }{G\left( u\right)}\right) ^{\prime } =-\left( \sigma -1/4\right) \frac{t}{2\pi}u^{\sigma -5/4}+\left( \sigma +1/4\right) \sqrt{n}u^{\sigma-3/4}>0.$$ Thus $\frac{F^{\prime }\left( u\right) }{G\left( u\right) }$ is monotonic and $$\begin{aligned} \frac{F_{t}^{\prime }\left( u\right) }{G\left( u\right) } &=&-\left( \frac{t^{2}}{4\pi ^{2}nu}\right) ^{\frac{1}{2}}\sqrt{n}u^{\sigma +1/4}+\sqrt{n}u^{\sigma +1/4} \\ &\geq &-\left( \frac{t^{2}}{4\pi ^{2}(1+\eta)yx}\right) ^{\frac{1}{2}}\sqrt{n}x^{\sigma +1/4}+\sqrt{n}x^{\sigma +1/4} \\ &\geq &\left( 1-\frac{1}{\sqrt{1+\eta }}\right) \sqrt{n}x^{\sigma +1/4} \\ &\gg &\sqrt{n}x^{\sigma +1/4}.\end{aligned}$$ Hence Lemma \[First drivitive test\] gives $$\int_{x}^{N}u^{-\sigma -3/4}e\left( -\frac{t}{2\pi }\log u-\sqrt{nu}+1/8\right) \ll x^{-\sigma -1/4}n^{-\frac{1}{2}},$$ which yields $$\begin{aligned} \mathfrak{M}_{2}\left( s\right) &\ll &x^{-\sigma+3/4}\sum_{\left( 1+\eta \right) y<n\leq \beta H^{2}} \frac{d_{\alpha,\beta }\left(n;H\right)}{n^{5/4}} \\ &\ll &t^{-\frac{1}{2}}x^{1-\sigma }\log H.\end{aligned}$$ \[Flower M\_3(s)\] For $\sigma >1/2,$ we have $$\mathfrak{M}_{3}\left( s\right) \ll_{\sigma } \left( x^{\frac{1}{2}-\sigma }+x^{-\sigma }t^{\frac{1}{2}}\right) \log t.$$ Write $$\begin{aligned} \mathfrak{M}_{3}\left( s\right) &=&\frac{s}{2\pi \sqrt{2}}\int_{x}^{N}u^{-s-\frac{3}{4}} \sum\limits_{n\leq y}\frac{d_{\alpha ,\beta }\left( n,H\right) }{n^{\frac{3}{4}}}e\left( -2\sqrt{nu}+1/8\right) du \\ &=&-\frac{1}{2\pi \sqrt{2}}\int_{x}^{N}\left( -s+1/4\right) \sum_{n\leq y}\frac{d_{\alpha ,\beta }\left( n;H\right) }{n^{3/4}}e\left( -2\sqrt{nu}+1/8\right) u^{-s-3/4}du \\ &&+\frac{1}{8\pi \sqrt{2}}\int_{x}^{N}\sum_{n\leq y}\frac{d_{\alpha,\beta}\left( n;H\right) }{n^{3/4}}e\left( -2\sqrt{nu}+1/8\right) u^{-s-3/4}du \\ &=&-\frac{1}{2\pi \sqrt{2}}\sum_{n\leq y}\frac{d_{\alpha ,\beta}\left(n;H\right) }{n^{3/4}} \int_{x}^{N}e\left( -2\sqrt{nu}+1/8\right) du^{-s+\frac{1}{4}} \\ &&+\frac{1}{8\pi \sqrt{2}}\sum_{n\leq y}\frac{d_{\alpha ,\beta}\left( n;H\right) }{n^{3/4}}\int_{x}^{N}e\left(-2\sqrt{nu}+1/8\right) u^{-s-3/4}du.\end{aligned}$$ By partial integration, we have $$\label{Expression of flower M_3(s)} \mathfrak{M}_{3}\left( s\right) =\mathfrak{M}_{31}\left( s\right) +\mathfrak{M}_{32}\left( s\right) +\mathfrak{M}_{33}\left( s\right) +\mathfrak{M}_{34}\left( s\right)$$ where $$\mathfrak{M}_{31}\left( s\right) =-\frac{N^{-s+1/4}}{2\pi \sqrt{2}} \sum_{n\leq y}\frac{d_{\alpha ,\beta}\left( n;H\right) }{n^{3/4}}e\left( -2\sqrt{nN}+1/8\right) ,$$ $$\mathfrak{M}_{32}\left( s\right)=\frac{x^{-s+1/4}}{2\pi \sqrt{2}} \sum_{n\leq y}\frac{d_{\alpha ,\beta}\left( n;H\right) }{n^{3/4}}e\left(-2\sqrt{nx}+1/8\right),$$ $$\begin{aligned} &&\mathfrak{M}_{33}\left( s\right)=-\frac{i}{\sqrt{2}}\sum_{n\leq y}\frac{d_{\alpha ,\beta }\left(n;H\right) }{n^{1/4}} \int_{x}^{N}u^{-\sigma -1/4}e\left( -\frac{t}{2\pi }\log u-2\sqrt{nu}-1/8\right) du\end{aligned}$$ and $$\begin{aligned} &&\mathfrak{M}_{34}\left( s\right)=\frac{1}{8\pi \sqrt{2}}\sum_{n\leq y}\frac{d_{\alpha ,\beta }\left(n;H\right) }{n^{3/4}} \int_{x}^{N}e\left( -2\sqrt{nu}-1/8\right) u^{-s-3/4}du.\end{aligned}$$ Using $d_{\alpha ,\beta }\left( n;H\right) \leq d\left( n\right) $ and trivial estimates, it is easy to get $$\label{Upper bound of flower M_31,M_32} \mathfrak{M}_{31}\left( s\right) ,\mathfrak{M}_{32}\left( s\right)\ll _{\sigma }x^{\frac{1}{2}-\sigma }\log t$$ and $$\label{Upper bound of flower M_34} \mathfrak{M}_{34}\left( s\right) \ll _{\sigma }x^{-\sigma +\frac{1}{4}}y^{\frac{1}{4}}\log y\ll x^{-\sigma }t^{\frac{1}{2}}\log t.$$ Now we deal with $\mathfrak{M}_{33}\left( s\right) $. In Lemma \[First drivitive test\], let $$H\left( u\right) =1,G\left( u\right) =u^{-\sigma -1/4}\ \ \ \ {\rm and}\ \ \ \ F_{t}\left( u\right) =-\frac{t}{2\pi }\log u-2\sqrt{nu}-1/8,$$ then we have $$F_{t}^{\prime }\left( u\right) =-\frac{t}{2\pi u}-\sqrt{\frac{n}{u}}$$ and $$\frac{F_{t}^{\prime }\left( u\right) }{G\left( u\right) }=-\frac{t}{2\pi }u^{\sigma -\frac{3}{4}}-\sqrt{n}u^{\sigma -\frac{1}{4}}.$$ Obviously, $$\frac{F_{t}^{\prime }\left( u\right) }{G\left( u\right) }<-\sqrt{n}u^{\sigma -\frac{1}{4}}\leq -\sqrt{n}x^{\sigma -\frac{1}{4}}.$$ Noting $$\left( \frac{F_{t}^{\prime }\left( u\right) }{G\left( u\right) }\right) ^{\prime }=-\left( \sigma -\frac{3}{4}\right) \frac{t}{2\pi }u^{\sigma -\frac{7}{4}}-\left( \sigma -\frac{1}{4}\right) \sqrt{n}u^{\sigma -\frac{5}{4}},$$let $u_{0}=\frac{\left( 3/4-\sigma \right) t}{\left( \sigma -1/4\right) 2\pi \sqrt{n}}$ be the root of $\left( \frac{F_{t}^{\prime }\left( u\right) }{G\left( u\right) }\right) ^{\prime }=0.$ If $u_{0}\in \lbrack x,N],$ then $\frac{F_{t}^{\prime }\left( u\right) }{G\left( u\right) }$ is monotonic in $[x,u_{0}]$ and $[u_{0},N]$ respectively, otherwise$\frac{F_{t}^{\prime }\left( u\right) }{G\left( u\right) }$ is monotonic in $[x,N].$ In either case, Lemma \[First drivitive test\] is valid and gives $$\int_{x}^{N}u^{-\sigma -1/4}e\left( -\frac{t}{2\pi }\log u-2\sqrt{nu}-1/8\right) \ll n^{-\frac{1}{2}}x^{\frac{1}{4}-\sigma },$$ which yields $$\label{Upper bound of flower M_33} \mathfrak{M}_{33}\left( s\right) \ll \sum_{n\leq y}\frac{d\left( n\right) }{n^{1/4}}n^{-\frac{1}{2}}x^{\frac{1}{4}-\sigma } \ll x^{\frac{1}{4}-\sigma }y^{1/4}\log y\ll x^{-\sigma}t^{\frac{1}{2}}\log t.$$ Then Lemma \[Flower M\_3(s)\] follows from collecting (\[Expression of flower M\_3(s)\])-(\[Upper bound of flower M\_33\]). \[Flower M\_4(s)\] For $\sigma >1/2,$ we have$$\mathfrak{M}_{4}\left( s\right) \ll x^{1/2-\sigma }\log H.$$ Write $$\mathfrak{M}_{4}\left( s\right) =\frac{s}{2\pi \sqrt{2}}\sum_{y<n\leq \beta H^{2}} \frac{d_{\alpha ,\beta }\left( n;H\right) }{n^{3/4}}\int_{x}^{N}u^{-\sigma -3/4}e\left( -\frac{t}{2\pi }\log u-2\sqrt{nu}+1/8\right) du$$ In Lemma \[First drivitive test\], taking $$G\left( u\right) =u^{-\sigma -3/4}$$ and $$F_{t}\left( u\right) =-\frac{t}{2\pi }\log u-2\sqrt{nu}+1/8,$$ we have $$F_{t}^{\prime }\left( u\right) =-\frac{t}{2\pi u}-\sqrt{\frac{n}{u}}$$ and$$\frac{F_{t}^{\prime }\left( u\right) }{G\left( u\right) }=-\frac{t}{2\pi } u^{\sigma -1/4}-\sqrt{n}u^{\sigma +1/4}.$$ Thus $\frac{F^{\prime }\left( u\right) }{G\left( u\right) }$ is monotonic and $$\begin{aligned} \frac{F_{t}^{\prime }\left( u\right) }{G\left( u\right) } &=&-\frac{t}{2\pi }u^{\sigma -1/4}-\sqrt{n}u^{\sigma +1/4} \\ &<&-\sqrt{n}x^{\sigma +1/4}\end{aligned}$$ Hence Lemma \[First drivitive test\] gives $$\int_{x}^{N}u^{-\sigma -3/4}e\left( -\frac{t}{2\pi }\log u-\sqrt{nu} +1/8\right) \ll x^{-\sigma -1/4}n^{-\frac{1}{2}},$$ which yields $$\mathfrak{M}_{4}\left( s\right) \ll x^{-\sigma +3/4}\sum_{y<n\leq \beta H^{2}} \frac{d_{\alpha ,\beta }\left( n;H\right) }{n^{5/4}}\ll x^{1/2-\sigma }\log H.$$ \[Lemma Flower M\_1\] For $\sigma >1/2,$ we have $$\mathfrak{M}_{1}\left( s\right) =\chi ^{2}\left( s\right)\sum_{n\leq y}d_{\alpha ,\beta }\left( n\right) n^{s-1} +O\left(x^{1/2-\sigma }\log t\right) .$$ Similar to the the proof of Lemma \[Flower M\_3(s)\], we rewrite $\mathfrak{M}_{1}\left( s\right) $ as $$\begin{aligned} \mathfrak{M}_{1}\left( s\right) &=&\frac{s}{2\pi \sqrt{2}}\int_{x}^{N}u^{-s-\frac{3}{4}} \sum\limits_{n\leq \left( 1+\eta \right)y}\frac{d_{\alpha ,\beta }\left( n,H\right)}{n^{\frac{3}{4}}}e\left( 2\sqrt{nu}-1/8\right) du\\ &=&-\frac{1}{2\pi \sqrt{2}}\int_{x}^{N}\left( -s+1/4\right)\sum_{n\leq\left( 1+\eta \right) y} \frac{d_{\alpha ,\beta }\left( n;H\right) }{n^{3/4}}e\left( 2\sqrt{nu}-1/8\right) u^{-s-3/4}du \\ &&+\frac{1}{8\pi \sqrt{2}}\int_{x}^{N}\sum_{n\leq \left( 1+\eta \right) y} \frac{d_{\alpha ,\beta }\left( n;H\right) }{n^{3/4}}e\left( 2\sqrt{nu}-1/8\right) u^{-s-3/4}du \\ &=&-\frac{1}{2\pi \sqrt{2}}\sum_{n\leq \left( 1+\eta \right) y}\frac{d_{\alpha ,\beta }\left( n;H\right) }{n^{3/4}} \int_{x}^{N}e\left( 2\sqrt{nu}-1/8\right) du^{-s+\frac{1}{4}} \\ &&+\frac{1}{8\pi \sqrt{2}}\sum_{n\leq \left( 1+\eta \right) y}\frac{d_{\alpha ,\beta }\left( n;H\right) }{n^{3/4}} \int_{x}^{N}e\left( 2\sqrt{nu}-1/8\right) u^{-s-3/4}du.\end{aligned}$$ By partial integration, we have $$\label{Flower M_1(s)=} \mathfrak{M}_{1}\left( s\right) =\mathfrak{M}_{11}\left( s\right) +\mathfrak{M}_{12}\left( s\right) +\mathfrak{M}_{13}\left( s\right) +\mathfrak{M}_{14}\left( s\right) ,$$ where $$\mathfrak{M}_{11}\left( s\right)=-\frac{1}{i\sqrt{2}}\sum_{n\leq\left( 1+\eta \right) y}d_{\alpha ,\beta } \left( n;H\right)n^{-\frac{1}{4}}I_{n}$$ with $$I_{n}=\int_{x}^{N}u^{-\sigma -\frac{1}{4}}e\left( -\frac{t}{2\pi }\log u +2\sqrt{nu}-\frac{1}{8}\right) du,$$ $$\mathfrak{M}_{12}\left( s\right)=-\frac{1}{2\pi\sqrt{2}}\sum_{n\leq\left( 1+\eta \right) y} \frac{d_{\alpha ,\beta }\left( n;H\right) }{n^{3/4}}e\left( 2\sqrt{nN}-1/8\right) N^{-s+\frac{1}{4}},$$ $$\mathfrak{M}_{13}\left( s\right)=\frac{1}{2\pi\sqrt{2}}\sum_{n\leq \left( 1+\eta \right) y}\frac{d_{\alpha ,\beta} \left( n;H\right) }{n^{3/4}}e\left( 2\sqrt{nx}-1/8\right) x^{-s+\frac{1}{4}}$$ and $$\mathfrak{M}_{14}\left( s\right)=\frac{1}{8\pi\sqrt{2}}\sum_{n\leq \left(1+\eta \right) y} \frac{d_{\alpha ,\beta }\left( n;H\right) }{n^{3/4}}\int_{x}^{N}e\left( 2\sqrt{nu}-1/8\right) u^{-s-3/4}du.$$ Note that $\eta >0$ is a fixed, sufficiently small constant, then by $d_{\alpha ,\beta }\left( n;H\right) \leq d\left( n\right) $ and trivial estimates, we get $$\label{Flower M_1,234} \mathfrak{M}_{12}\left( s\right) ,\mathfrak{M}_{13}\left( s\right) , \mathfrak{M}_{14}\left( s\right) \ll _{\sigma}x^{\frac{1}{4}-\sigma }y^{\frac{1}{4}} \log y\ll _{\sigma}x^{-\sigma }t^{\frac{1}{2}}\log t.$$ Now only $\mathfrak{M}_{11}\left( s\right) $ is left. In Chapter 4 of [@Ivic] (Page 108-110), Ivic discussed $I_{n}$ and showed $$-\frac{1}{i\sqrt{2}}\sum_{n\leq \left( 1+\eta \right) y}d\left( n\right) n^{-\frac{1}{4}}I_{n} =\chi ^{2}\left( s\right) \sum_{n\leq y}d\left(n\right) n^{s-1}+O\left( x^{\frac{1}{2}-\sigma }\log t\right),$$ where $\chi\left( s\right)$ is given by [(\[Functional equation of Riemann zeta-function\])]{}. Replacing $d\left( n\right) $ by $d_{\alpha ,\beta }\left(n;H\right) ,$ the same argument is also valid, which gives $$\begin{aligned} \mathfrak{M}_{11}\left( s\right) &=&-\frac{1}{i\sqrt{2}}\sum_{n\leq\left(1+\eta \right) y}d_{\alpha ,\beta } \left( n;H\right) n^{-\frac{1}{4}}I_{n} \\ &=&\chi ^{2}\left( s\right) \sum_{n\leq y}d_{\alpha ,\beta }\left(n;H\right) n^{s-1}+O\left( x^{1/2-\sigma }\log t\right).\end{aligned}$$ Take $H=T^{B}$ with $B>3A>0$ being a constant, sufficiently large, then we have $$\begin{aligned} \label{Flower M_11} \mathfrak{M}_{11}\left( s\right) &=&\chi ^{2}\left( s\right) \sum_{n\leq y}\left( \underset{n=hk}{\sum_{1\leq h\leq H}\sideset{}{^{\prime \prime }} \sum\limits_{\ h\alpha \leq k\leq h\beta }}1\right)n^{s-1}+O\left( x^{1/2-\sigma }\log t\right)\\ &=&\chi ^{2}\left( s\right) \sum_{n\leq y}\left(\underset{n=hk}{\sum_{1\leq h\leq H} \sum\limits_{h\alpha \leq k\leq h\beta }}1\right) n^{s-1}+O\left(\left\vert \chi \left( s\right) \right\vert ^{2} \sum_{h\ll \sqrt{y}}h^{2\sigma -2}\right)\notag\\ \indent &&+O\left( x^{1/2-\sigma }\log T\right) \notag \\ &=&\chi ^{2}\left( s\right) \sum_{n\leq y}d_{\alpha ,\beta }\left(n\right) n^{s-1} +O\left( t^{1-2\sigma }y^{\sigma -1/2}+x^{1/2-\sigma}\log T\right)\notag \\ &=&\chi ^{2}\left( s\right) \sum_{n\leq y}d_{\alpha ,\beta }\left(n\right) n^{s-1}+O\left( x^{1/2-\sigma }\log T\right), \notag\end{aligned}$$ where we used $$\chi \left( \sigma +it\right) =\left( \frac{2\pi }{t}\right) ^{\sigma +it-\frac{1}{2}}e^{i\left( t+\frac{\pi }{4}\right) } \left( 1+O\left(t^{-1}\right) \right),\ \ \ \ \mathrm{ for }\ t\geq 2.$$ Combining (\[Flower M\_1(s)=\])-(\[Flower M\_11\]) gives Lemma \[Lemma Flower M\_1\]. Out line for the proof of Theorem \[Theorem 1.2\] ================================================= A primitive Pythagorean triangle is a triple $(a,b,c)$ of natural numbers with $a^{2}+b^{2}=c^{2},a<b$ and $\gcd (a,b,c)=1.$ Let $P(x)$ denote the number of primitive Pythagorean triangles with perimeter less than $x.$ D. H. Lehmer [@H.Lehmer] showed $$P\left( x\right) =\frac{\log 2}{\pi ^{2}}x+O\left( x^{1/2}\log x\right)$$ which was revisited by J. Lambek and L. Moser in [@Lambek]. The exponents $1/2$ in the error term can not be reduced because the current technique depends on the best zero-free regions of the Riemann zeta function, which hard to be improved. In [@Liu], the author showed if Riemann Hypothesis (RH) is true, then (\[old theorem on triangles\]) holds. Let $$r\left( n\right) =\sum_{\substack{ 2d^{2}+2dl=n \\ l<d}}1=\sum_{\substack{2dl=n \\ d<l<2d}}1$$ and $$Z\left( s\right) =\sum_{n=1}^{\infty }\frac{r\left( n\right)}{n^{s}},\text{ for }\Re s>1.$$ We can prove that $Z(s)$ has an analytic continuation to $\sigma>1/3$ and has two simple poles at $s=1,\frac{1}{2}.$ The exponent $\frac{5805}{15408}$ in (\[old theorem on triangles\]) depends on the estimate of the following type exponential sum $$\label{Exponential sum} \sum_{m\sim M}\mu \left( m\right) \sum_{n\sim N}a_{n}e\left( \frac{cx^{\frac{1}{2}}n^{\frac{1}{2}}}{m}\right)$$ with $a_{n}\ll 1$ and $c$ being a constant. Here the ranges of $M,\ N$ are determined by the smallest $\sigma $ such that $$\label{Z(s)} \int_{T}^{2T}\left\vert Z\left( \sigma +it\right) \right\vert dt\ll_{\sigma ,\varepsilon }T^{1+\varepsilon }$$ holds for any $\varepsilon >0.$ In [@Liu], the author showed $\sigma >\frac{1064}{1644}=0.6472\cdots $ is admissible. Then by estimating the exponential sum (\[Exponential sum\]) for $M\leq x^{\frac{651}{1926}},$ $N\leq x^{\frac{3798}{15408}}$, we get (\[old theorem on triangles\]). In the review of [@Liu], R. C. Baker mentioned that using the method in his paper [@Baker; @2], it is possible to prove $\sigma >\frac{3}{5}=0.6,$ which implies an improvement of (\[old theorem on triangles\]). Now by Theorem \[Main theorem\], we have (\[Z(s)\]) holds for any $\sigma>\frac{1}{2}$, which forces us to deal with exponential sum (\[Exponential sum\]) for $M,$ $N\leq $ $x^{\frac{1}{4}+\varepsilon }.$ However, the estimate in this range has been investigated carefully by R. C. Baker in [@Baker; @1], which yields Theorem \[Theorem 1.2\]. [100]{} R. C. Baker, The square-free divisor problem , *Quart. J. Math.* (Oxford)(2). 47 (1996), 133-146. R. C. Baker, Primitive lattice points in planar domains, *Acta Arith.* 142 (2010), 267-302 S. W. Graham and G. Kolesnik, *Van der Corput’s method of exponential sums,* (London Mathematical Society Lecture Note Series 126). D. R. Heath-Brown,The Piatetski-Shapiro prime theorem, *J. of Number theory*, Vol. 16(1983), 242-266. M. N. Huxley, Exponential sums and Lattices points , *Proc. London Math. Soc.* Vol. 87 (3) (2003), 591-609. G. H. Hardy and J. E. Littlewood, The approximate functional equation in the theory of the zeta-function, with applications to the divisor-problems of Dirichlet and Piltz, *Proc. London Math. Soc.* (2) 21 (1923) 39-74. A.Ivic, *The Riemann Zeta-Function,* Wiley (1985). J. Lambek and L. Moser, On the distribution of Pythagorean triangles, *Pacific J. Math.* 5 (1955), 73-83. D. H. Lehmer, A conjecture of Krishnaswami, *Bull. Amer. Math. Soc.* 54 (1948), 1185-1190 M. Kuhleitner and W. G. Nowak, The asymptotic behaviour of the mean-square of fractional part sums, *Proc. Edinb. Math. Soc.* 43 (2000), 309-323. Kui Liu, On the distribution of primitive Pythagorean triangles, *Acta Arith.* 144 (2010), 135-150. C. D. Pan and C. B. Pan, *Foundations of the Analytic Number Theory*, Science Press, Beijing, 1991 (in Chinese). G. F. Voronoi, Sur une fonction transcendante et ses applications a la sommation de quelques series. *Ann Ecole Normale,* 21: 207¨C268; 21: 459¨C534 W. G. Zhai, On the error term in Weyl¡¯s law for the Heisenberg , *Acta Arith.* 134. 3 (2008), 219-257 Acta Arith.

--- abstract: 'In this article, we provide a detailed account of a construction sketched by Kashiwara in an unpublished manuscript concerning generalized HKR isomorphisms for smooth analytic cycles whose conormal exact sequence splits. It enables us, among other applications, to solve a problem raised recently by Arinkin and Căldăraru about uniqueness of such HKR isomorphisms in the case of the diagonal injection. Using this construction, we also associate with any smooth analytic cycle endowed with an infinitesimal retraction a cycle class which is an obstruction for the cycle to be the vanishing locus of a transverse section of a holomorphic vector bundle.' address: - - | CNRS, LATP\ UMR 6632\ CMI, Université de Provence\ 39, rue Frédéric Joliot-Curie\ 13453 Marseille Cedex 13\ France. author: - Julien Grivaux bibliography: - 'biblio.bib' title: 'The Hochschild-Kostant-Rosenberg isomorphism for quantized analytic cycles' --- Introduction ============ The existence of the Hochschild-Kostant-Rosenberg (HKR) isomorphism is a fundamental result both in algebraic geometry and in homological algebra. Let us recall the statement: Let $A$ be a finitely generated regular commutative algebra over a field $k$ of characteristic zero. Then for any nonnegative integer $i$, the Hochschild homology group $\emph{HH}_i {^{\vphantom{*}} }(A)$ is isomorphic to the module $\smash{\Omega^{\,i}_{A/k}} $ of Kähler differentials of degree $i$ of $A$. The HKR isomorphism has been generalized in the context of algebraic geometry in [@SW] and [@YE]: for any smooth quasi-projective variety over a field of characteristic zero, the derived tensor product ${{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{\oo_{X \times X}}{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}$ is isomorphic in the derived category of sheaves of ${\ensuremath{\oo_{X}{^{\vphantom{*}} }}}$-modules to the direct sum of its cohomology objects, which is $\bigoplus_{i} \, \Omega_X^i[i]$. The same result also holds for smooth (or even singular) complex manifolds as shown in [@BF] and [@SH], but the proof is much more involved. We are interested here in a generalization of the analytic HKR isomorphism consisting in replacing the diagonal injection by an arbitrary closed embedding. If $(X, Y)$ is a pair of complex manifolds such that $X$ is a closed complex submanifold of $Y$, the derived tensor product ${{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}$ is not isomorphic in general to the direct sum of its cohomology objects. This fact is the main issue of [@AC], where it is proved that ${{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} $ is isomorphic to $\smash[b]{\bigoplus_{i} \,\bw{i}{} N\ee_{X/Y}[i]} $ if and only if the normal bundle of $X$ in $Y$ extends to a locally-free sheaf on the first formal neighbourhood ${\ensuremath{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{\textrm{X}{^{\vphantom{*}} }}}}}} \hspace*{-2.13ex}X}$ of $X$ in $Y$. More precisely, if $\mathcal{N}$ is such an extension, the authors construct a specific generalized HKR isomorphism, generally depending on $\mathcal{N}$, between ${\ensuremath{\oo_{X}{^{\vphantom{*}} }}}{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}$ and $\bigoplus_{i}\,\bw{i}{} N\ee_{X/Y}[i]$. Therefore, it appears clearly that it is necessary to quantize an analytic cycle (i.e. to add some additional geometric data) in order to associate with this cycle a well-defined HKR isomorphism. Quantizing the normal bundle allows to define HKR isomorphisms for the most general cycles (while dealing with smooth cycles), but the counterpart of this generality is that the space of the possible quantizations of a cycle cannot be easily handled. For instance, the following problem is raised in [@AC]: in the case of the diagonal injection, are the HKR isomorphisms associated with the quantizations given by the two canonical projections the same? More generally, the comparison of HKR isomorphisms associated with different quantizations of an analytic cycle is still an open problem. In this article, our aim is to present a different construction of HKR isomorphisms associated with pairs $(X, Y)$ of complex manifolds satisfying a more restrictive condition than the aforementioned one: the normal (or conormal) exact sequence associated with the cycle $X$ has to be holomorphically split, which means in an equivalent way that the injection of $X$ into ${\ensuremath{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{\textrm{X}{^{\vphantom{*}} }}}}}} \hspace*{-2.13ex}X}$ admits a holomorphic retraction. For any such retraction $\sigma$, the locally-free sheaf $\smash{\sigma{\ensuremath{^{\, *}}}N_{X/Y}{^{\vphantom{*}} }} $ is a quantization of $\smash{N_{X/Y}{^{\vphantom{*}} }} $ as defined above. As far as the diagonal injection is concerned, this process is carried out in [@KA] (which is reproduced in [@KS1 chap. 5]); the general case is sketched in [@KA]. Considering its importance, we provide a detailed account of the construction. In this setting, analytic cycles are quantized by retractions of their first formal injection (that is the injection into their first formal neighbourhood) so that the set of possible quantizations of an analytic cycle is an affine space whose underlying vector space is $\smash{\textrm{Hom}(\, \Omega_X^1, N\ee_{X/Y})} $. With such a quantization $\sigma$ is associated a complex $\mathcal{P}_{\sigma} {^{\vphantom{*}} }$ of coherent sheaves on ${\ensuremath{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{\textrm{X}{^{\vphantom{*}} }}}}}} \hspace*{-2.13ex}X}$ which is quasi-isomorphic to ${\ensuremath{\oo_{X}{^{\vphantom{*}} }}}$ and reduces to the first part of the Atiyah exact sequence when $X$ is a divisor in $Y$ (this is why we call $\mathcal{P}_{\sigma} {^{\vphantom{*}} }$ the Atiyah-Kashiwara complex associated with $\sigma$). The sheaves defining $\mathcal{P}_{\sigma} {^{\vphantom{*}} }$ are torsion sheaves, so that they are definitely not flat over ${\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}$. However, a remarkable fact is that $\mathcal{P}_{\sigma} {^{\vphantom{*}} }$ can be used to compute the derived tensor product $\smash[b]{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}$ and therefore to get a specific HKR isomorphism $\Gamma_{\sigma} {^{\vphantom{*}} }$ between $\smash[b]{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}$ and $\bigoplus_{i}\, \bw{i}{} N\ee_{X/Y}[i]$. It turns out that $\Gamma_{\sigma} {^{\vphantom{*}} }$ is exactly the HKR isomorphism constructed in [@AC] associated with the quantization $\smash{\sigma{\ensuremath{^{\, *}}}N_{X/Y} {^{\vphantom{*}} }} $ of $\smash{N_{X/Y}{^{\vphantom{*}} }} $. Our first result provides sufficient conditions in order that two different retractions of the first formal injection of an analytic cycle define the same HKR isomorphism: \[11\] Let $(X,Y)$ be a pair of complex manifolds such that $X$ is a closed submanifold of $Y$ and let ${\ensuremath{\overline{j}}}$ be the injection of $X$ into its first neighbourhood ${\ensuremath{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{\textrm{X}{^{\vphantom{*}} }}}}}} \hspace*{-2.13ex}X}$ in $Y$. 1. Assume that $N\ee_{X/Y}$ carries a global holomorphic connection. Then for any retractions $\sigma $ and $\sigma '$ of ${\ensuremath{\overline{j}}}$, ${\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }$ is naturally isomorphic to ${\ensuremath{\mathcal{P}}}_{\sigma '}{^{\vphantom{*}} }$. 2. Let $\sigma $ and $\sigma '$ be two retractions of ${\ensuremath{\overline{j}}}$ such that the element $\sigma '-\sigma $ in $\emph{Hom}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }(\Omega ^{1}_{X}, N\ee_{X/Y})$ is an isomorphism. Then ${\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }$ is naturally isomorphic to ${\ensuremath{\mathcal{P}}}_{\sigma '}{^{\vphantom{*}} }$. In the case of the diagonal injection, the quantizations $\operatorname{pr}_1 {^{\vphantom{*}} }$ and $\textrm{pr}_{2}$ satisfy the second condition of the theorem, which gives a positive answer to the problem mentioned above. Another important outcome of this construction is what we call the dual HKR isomorphism. To explain this notion, we consider the complex ${\mathcal{RH}om_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}}, {\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$ corresponding to Hochschild cohomology in the case of the diagonal injection. This complex is well-defined up to a unique isomorphism in the bounded derived category $D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{Y}{^{\vphantom{*}} }}})$ of sheaves of ${\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}$-modules, but not in $D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$. Indeed, the canonical isomorphism in $D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{Y}{^{\vphantom{*}} }}})$ between $\textrm{R}[\operatorname{\mathcal{H}\mathnormal{om}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }(\,*\,, {\ensuremath{\oo_{X}{^{\vphantom{*}} }}})]\, ({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$ and $\textrm{R}[\operatorname{\mathcal{H}\mathnormal{om}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}},\,*\,)]\, ({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$ is not induced in general by an isomorphism in the category $D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$. The purpose of the dual construction is to construct a specific isomorphism (the dual HKR isomorphism) between $\textrm{R}[\operatorname{\mathcal{H}\mathnormal{om}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}},\,*\,)]\, ({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$ and $\smash{\bigoplus_{i}\,\bw{i}{} N_{X/Y} {^{\vphantom{*}} }[-i]} $ in $D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$. This is achieved by replacing ${\ensuremath{\oo_{X}{^{\vphantom{*}} }}}$ by the dual complex $\smash{\operatorname{\mathcal{H}\mathnormal{om}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }(\mathcal{P}_{\sigma} {^{\vphantom{*}} }, {\ensuremath{\oo_{X}{^{\vphantom{*}} }}})}$, which is also a bounded complex of coherent sheaves on ${\ensuremath{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{\textrm{X}{^{\vphantom{*}} }}}}}} \hspace*{-2.13ex}X}$. The dual HKR isomorphism is a powerful tool, which has been used initially in [@KA] for the diagonal injection to give a functorial definition of Euler classes of coherent sheaves; it has led to a simple proof of the Grothendieck-Riemann-Roch theorem in Hodge cohomology for arbitrary proper morphisms between complex manifolds [@G]. We provide here another application: for any quantized analytic cycle $(X, \sigma)$ in a complex manifold $Y$, we construct a cohomology class $q_{\sigma} {^{\vphantom{*}} }(X)$ in $\smash[b]{\bigoplus_{i}\,\textrm{H}^{i}{_{\vphantom{i}} }(X, \bw{i}{} N\ee_{X/Y})} $ called the quantized cycle class of $(X, \sigma)$. We prove that this class provides an obstruction for $X$ to be defined as the vanishing locus of a transverse section of a holomorphic vector bundle on $Y$: \[22\] Let $(X,\sigma )$ be a quantized analytic cycle of codimension $r$ in $Y$ and assume that there exists a couple $(E,s)$ such that 1. $E$ is a holomorphic vector bundle of rank $r$ on $Y$ . 2. $s$ is a holomorphic section of $E$ vanishing exactly on $X$, and $s$ is transverse to the zero section. 3. The locally-free ${\ensuremath{\oo_{X}{^{\vphantom{*}} }}}$-modules $E{\ensuremath{\otimes}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}$ and $\sigma \ee{_{\vphantom{i}} }N_{X/Y}{^{\vphantom{*}} }$ are isomorphic. Then $q_{\sigma} {^{\vphantom{*}} }(X)=1$. For the diagonal injection, it follows from the results of [@MA], [@RA1], [@G] and [@RA2] that $q_{\,\textrm{pr}_1}(\Delta_X)$ is the Todd class of $X$. Up to the author’s knowledge, the quantized cycle class $q_{\sigma} {^{\vphantom{*}} }(X)$ has not yet appeared in the literature, and it would be interesting to compute it in purely geometrical terms. To conclude this introduction, let us discuss the link of this construction with the generalized Duflo isomorphism. The aim of this isomorphism is to understand precisely how the HKR isomorphism between the algebras $\textrm{Ext}\ee_{\oo_{X \times X}}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}}, {\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$ and $\textrm{H}\ee(X, \bw{*} \, TX)$ fails to be multiplicative. After the seminal work [@KO], the following result (conjectured in [@C1]) was proved: For any complex manifold $X$, if $\Gamma$ denotes the standard *HKR* isomorphism between $\emph{Ext}\ee_{\oo_{X \times X}}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}}, {\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$ and $\textrm{H}\ee(X, \bw{*} \, TX)$, then $(\emph{td}(X))^{-1/2} \lrcorner \, \Gamma$ is a ring isomorphism. For general cycles, the algebra $\smash[t]{\textrm{Ext}\ee_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}}, {\ensuremath{\oo_{X}{^{\vphantom{*}} }}})} $ is no longer graded commutative, and its structure is the object of current active research (see the program initiated in [@CCT1]). The quantized cycle class $\smash{q_{\sigma} {^{\vphantom{*}} }(X)} $, which generalizes the Todd class for arbitrary quantized analytic cycles, is likely to play a part in the understanding of this algebra. Let us describe more precisely the outline of this article. After a preliminary section (§ \[SectAlgExt\]), it is divided into three main parts: the local construction of HKR isomorphisms is carried out in § \[LocCx\], these results are globalized in § \[Atiyah Global\] and provide an application in § \[cycle\] where we construct and study the quantized cycle class. We now turn to the specific organization of each part. In § \[cup\], we recall some elementary constructions in exterior algebra such as contraction morphisms and Koszul complexes, mainly to fix sign conventions. In § \[DgAlg\], an abstract construction on dg-algebras is performed, the aim of which is to provide a general setting for Atiyah-Kashiwara complexes. At the beginning of § \[LocCx\], we define specific notation for the derived functors of the functor Hom and for the tensor product, since they cannot be derived as bifunctors in our setting. In § \[LocHKR\] are defined the Atiyah-Kashiwara complex (Definition \[DefUnHKR\]) together with the dual Atiyah-Kashiwara complex (Definition \[DefDeuxHKR\]); and in Propositions \[PropUnHKR\] and \[PropDeuxHKR\] we establish the corresponding local HKR isomorphisms. The proofs we give here are bound to extend naturally to a global setting. In Proposition \[PropDeuxBisHKR\], we compare in a weak sense the HKR and dual HKR isomorphisms. The argument of this proof will be used anew in the proof of Theorem \[BaProCyClThUn\] (which is Theorem \[22\] in this introduction). In § \[SectionArikinCalda\], the construction performed in § \[LocHKR\] is compared to the construction of [@AC] in the local case, and both are shown to be compatible in Proposition \[PropUnArinkinCalda\]. In § \[SousSecAKComplexes\], Proposition \[PropTroisHKR\] provides conditions to construct naturally automorphisms of Atiyah-Kashiwara complexes, and is the local version of Theorem \[11\]. The next part (§ \[Atiyah Global\]) deals with the complex analytic case. In the first section (§ \[anhkr\]), the results of and are stated in a global setting, Propositions \[PropUnAnalyticHKR\], \[HKR2\] and Theorem \[PropDeuxAnalyticHKR\] (which is Theorem \[11\] in this introduction) extending Propositions \[PropUnHKR\], \[PropDeuxHKR\] and \[PropTroisHKR\] respectively. In § \[TwistedCase\], we explain how to twist Atiyah-Kashiwara complexes by extension classes. In Proposition \[PropUnTwisted\], we prove that two Atiyah-Kashiwara complexes associated with different retractions become isomorphic after twisting by extension classes depending on the Atiyah class of the conormal bundle $\smash{N\ee_{X/Y}} $. In Theorem \[PropDeuxTwisted\] we recall (in slightly more general terms) the principal result of [@AC] and we prove in Theorem \[PropTroisTwisted\] that, when the cycle admits an infinitesimal retraction, the HKR isomorphisms of [@AC] associated with arbitrary quantizations of the normal bundle are again twisted HKR isomorphisms in our sense. In the case of the canonical quantization associated with a retraction, we obtain the compatibility of HKR isomorphisms (this globalizes proposition \[PropUnArinkinCalda\]). The aim of § \[comparison\] is to study and carefully compare twisted HKR isomorphisms (and so to compare HKR isomorphisms associated with different retractions, thanks to Proposition \[PropUnTwisted\]). We give some results in particular cases, namely when the twist are obtained by tensorization with holomorphic line bundles on ${\ensuremath{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{\textrm{X}{^{\vphantom{*}} }}}}}} \hspace*{-2.13ex}X}$ (Theorem \[ThUnCompHKR\]), and then for general extension classes when only the last but one term of each Atiyah-Kashiwara complex is twisted (Theorem \[ModuleLunaireUn\]). As a corollary, we deduce in Theorem \[ModuleLunaireDeux\] the general comparison theorem between HKR isomorphisms associated with different retractions for cycles of codimension two. We are led to propose a conjecture for the general case (Conjecture \[conj\]). The last part (§ \[cycle\]) deals with the quantized cycle class. In § \[cycle1\], using the dual HKR isomorphism, we define this cycle class and compute it in specific cases. In Theorem \[BaProCyClThUn\] (which is Theorem \[22\] in this introduction), we prove that the quantized cycle class is one when the cycle $X$ is the zero locus of a transverse section of a holomorphic vector bundle on $Y$ satisfying a compatibility condition with the retraction $\sigma$. In Theorem \[BaProCyClThDeux\], we obtain that the class $q_{\,\textrm{pr}_1}(\Delta_X)$ is the Todd class of $X$; this is equivalent to the main result of [@G]. Finally, we deal with the divisor case in theorem \[divisor\]. Preliminary constructions for § \[kash\] are carried out in § \[six\]: if $j$ denotes the injection of the cycle $X$ into $Y$, we study the right and left adjoints $j\ee$ and $j\pe$ of the direct image functor $j\ei$ operating on the corresponding derived categories. In § \[kash\], following [@KS1 chap. 5] for the diagonal injection, we establish in Theorem \[KashIsoThDeux\] that the natural isomorphism between $j\ee{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\omega _{X/Y}{^{\vphantom{*}} }$ and $j\pe{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}$ obtained using the local cycle class of $X$ in $Y$ is given via the HKR isomorphisms by contraction with the quantized cycle class $q_{\sigma} {^{\vphantom{*}} }(X)$. **Acknowledgments.** I wish to thank Pierre Schapira who has encouraged me all along, and also Damien Calaque and Richard Thomas for useful conversations and comments. Preliminary constructions {#SectAlgExt} ========================= Duality and cup-product {#cup} ----------------------- Let $r$ be a positive integer, $A$ be a commutative $k$-algebra over a field $k$ of characteristic zero, and $E$ be a free $A$-module of rank $r$. In this section, all tensor and exterior products are taken over $A$. \[DefUnSecAlgExt\] For any nonnegative integers $p$ and $q$, we denote by $\apl{W_{p,\,q}}{\bwi{p+q}{}E}{\bw{p}{E}{\ensuremath{\otimes}}\!\bw{q}{E}}$ the transpose of the cup-product map from $\bw{p}{\Ee}{\ensuremath{\otimes}}\bw{q}{\Ee}$ to ${\ensuremath{\smash[t]{\Lambda^{p+q}{_{\vphantom{i}} }\Ee}}}$ multiplied by $\dfrac{p! \, q!}{(p+q)!}$. It is possible to give another natural definition of $W_{p,q}$ as follows: for any nonnegative integer $n$, let $\mathfrak{S}_n {^{\vphantom{*}} }$ denote the symmetric group with $n$ letters, and let $\apl{\varepsilon}{\mathfrak{S}_n {^{\vphantom{*}} }}{\{-1, \,1\}}$ be the signature morphism. We define the symmetrization and antisymmetrization maps $\apl{\mathfrak{a}_n {^{\vphantom{*}} }}{{\smash{\bigotimes_{{}^{{^{\vphantom{*}} }}}^{{n}\vphantom{\big)}}}}E}{\bw{n {_{\vphantom{i}} }}{E}}$ and $\apl{\mathfrak{s}_n {^{\vphantom{*}} }}{\bw{n {_{\vphantom{i}} }}{E}}{{\smash{\bigotimes_{{}^{{^{\vphantom{*}} }}}^{{n}\vphantom{\big)}}}}E}$ by the formulae below: $$\label{EqUnPrelim} \begin{cases} \mathfrak{a}_n {^{\vphantom{*}} }\, (v_{1}{^{\vphantom{*}} }{\ensuremath{\otimes}}\dots{\ensuremath{\otimes}}v_{n}{^{\vphantom{*}} })=v_{1}{^{\vphantom{*}} }{\ensuremath{\wedge}}\dots{\ensuremath{\wedge}}v_{n}{^{\vphantom{*}} }\\ \mathfrak{s}_n {^{\vphantom{*}} }\, (v_{1}{^{\vphantom{*}} }{\ensuremath{\wedge}}\dots{\ensuremath{\wedge}}v_{n}{^{\vphantom{*}} })=\dfrac{1}{n!} \, \ds\sum_{\sigma \in \mathfrak{S}_n {^{\vphantom{*}} }} \varepsilon(\sigma)\, v_{\sigma(1)}{^{\vphantom{*}} }{\ensuremath{\otimes}}\dots{\ensuremath{\otimes}}v_{\sigma(n)}{^{\vphantom{*}} }\\ \end{cases}$$ A straightforward computation shows that $$\label{EqDeuxBisSecAlgExt} W_{p,q}{^{\vphantom{*}} }(v_1 {^{\vphantom{*}} }{\ensuremath{\wedge}}\ldots {\ensuremath{\wedge}}v_{p+q} {^{\vphantom{*}} })= \dfrac{p! \, q!}{(p+q)!} {\ensuremath{\displaystyle}}\sum_{\sigma} \varepsilon(\sigma) \, (v_{\sigma(1)}{^{\vphantom{*}} }{\ensuremath{\wedge}}\dots{\ensuremath{\wedge}}v_{\sigma(p)}{^{\vphantom{*}} }) \, {\ensuremath{\otimes}}\, (v_{\sigma(p+1)}{^{\vphantom{*}} }{\ensuremath{\wedge}}\dots{\ensuremath{\wedge}}v_{\sigma(p+q)}{^{\vphantom{*}} })$$ where $\sigma$ runs through all $(p\,$-$q)$ shuffles, which implies that $W_{p,q}=(\mathfrak{a}_{p} {\ensuremath{\otimes}}\mathfrak{a}_{q}) \circ \mathfrak{s}_{p+q}.$ \[DefUnBisSecAlgExt\] For any nonnegative integers $m$, $p$, $k$ and any $\phi $ in $\textrm{Hom}(\bw{p}{E}, \bw{k}{E})$, we define $t_{k,\,p} ^{\,m}(\phi )$ in $\textrm{Hom}(\bw{p+m}{E}, \bw{k+m}{E})$ by the composition $$\xymatrix@C=20pt{t_{k,p} ^{\,m}(\phi )\,:\,\bw{p+m}{E}\ar[rr]^-{W_{p,\,m}{^{\vphantom{*}} }}&&\bw{p}{E}{\ensuremath{\otimes}}\bw{m}{E}\ar[rr]^-{\phi\, {\ensuremath{\otimes}}\,\operatorname{id}}&&\bw{k}{E}{\ensuremath{\otimes}}\bw{m}{E}\ar[r]^-{{\ensuremath{\wedge}}}&\bw{k+m}{E.} }$$ The translation operator $\apl{t_{k,\,p}^{\,m}(\phi )}{\textrm{Hom}(\bw{p}{E}, \bw{k}{E})}{\textrm{Hom}(\bw{p+m}{E}, \bw{k+m}{E})}$ satisfies the following important property: \[LemUnBisSecAlgExt\] For any nonnegative integers $m$, $p$, $k$ such that $k\geq p$ and for any $a$ in $\bw{k-p}E$, we have $t_{k, \,p}^{\,m}(a{\ensuremath{\wedge}}\,.\,)=a{\ensuremath{\wedge}}\,.$ By (\[EqDeuxBisSecAlgExt\]), for any $e_{1}{^{\vphantom{*}} },\dots, e_{p+m}{^{\vphantom{*}} }$ in $E$, we have $$\begin{aligned} t_{k, \, p}^{\,m}(a{\ensuremath{\wedge}}\,.\,)\,(e_{1}{^{\vphantom{*}} }{\ensuremath{\wedge}}\dots{\ensuremath{\wedge}}e_{p+m}{^{\vphantom{*}} })&=\dfrac{p! \, m!}{(p+m)!} {\ensuremath{\displaystyle}}\sum_{\sigma} \varepsilon(\sigma) \, \, a {\ensuremath{\wedge}}e_{\sigma(1)}{^{\vphantom{*}} }{\ensuremath{\wedge}}\dots\wedge e_{\sigma(p)}{^{\vphantom{*}} }{\ensuremath{\wedge}}e_{\sigma(p+1)}{^{\vphantom{*}} }{\ensuremath{\wedge}}\dots\wedge e_{\sigma(p+m)}{^{\vphantom{*}} }\\ &=a{\ensuremath{\wedge}}e_{1}{^{\vphantom{*}} }{\ensuremath{\wedge}}\dots{\ensuremath{\wedge}}e_{p+m}{^{\vphantom{*}} }.\end{aligned}$$ For any positive integer $p$, any vector $v$ in $\bw{}{E}$ defines two endomorphisms $\ell_v$ and $r_v$ of $\bw{}{E}$ given by $\ell_v(x)=v {\ensuremath{\wedge}}x$ and $r_v(x)=x {\ensuremath{\wedge}}v$. The map ${}^t r_v\,$ (resp. ${}^t \ell_v\,$) is by definition the *left* (resp. *right*) contraction by $v$; it is an endomorphism of $\bw{}{\Ee}$ denoted by $\smash{\xymatrix@C=17pt{\phi\ar@{|->}[r]&v\lrcorner\, \phi}} $ (resp.$\smash{\xymatrix@C=17pt{\phi\ar@{|->}[r]&\phi \, \llcorner\, v).}} $ The left (resp. right) contraction morphism endows $\bw{}{\Ee}$ with the structure of a left (resp. right) $\bw{}{E}$-module. There are two duality isomorphisms $D^{\ell}{_{\vphantom{i}} }$ and $D^{r}{_{\vphantom{i}} }$ from $\bw{}{E} {\ensuremath{\otimes}}\textrm{det}\, E\ee$ to $\bw{}{\Ee}$ given by $$\begin{aligned} D^{\ell}(v {\ensuremath{\otimes}}\xi)=v \lrcorner \, \xi \qquad \textrm{and} \qquad D^{r}(v {\ensuremath{\otimes}}\xi)=\xi \, \llcorner \, v\end{aligned}$$ Remark that $D^{\ell}{_{\vphantom{i}} }$ (resp. $D^{r}{_{\vphantom{i}} }$) is an isomorphism of left (resp. right) $\bw{}{E}$-modules. We pause for a moment in order to discuss sign conventions concerning contraction morphisms. Let $\smash{\Delta_r=\{(i,j) \in \N^2 \, \, \textrm{such that}\, \, i+j \leq r\}} $. For any sign function $\smash{\apl{\chi}{\Delta_r {^{\vphantom{*}} }}{\Z_2 {^{\vphantom{*}} },}} $ we can consider the left (resp. right) twisted contraction morphism from $\bw{}{E} {\ensuremath{\otimes}}\bw{}{\Ee}$ (resp. $\bw{}{\Ee} {\ensuremath{\otimes}}\bw{}{E}$) to $\bw{}{\Ee}$ defined on homogeneous elements by the formula $$\begin{cases} v \, \lrcorner_{\chi} {^{\vphantom{*}} }\, \phi=\chi[\textrm{deg(}v\textrm{)}, r-\textrm{deg(}\phi\textrm{)}] \, \, v\, \lrcorner \, \phi \\ \phi \, \llcorner_{\chi} {^{\vphantom{*}} }\, v=\chi[\textrm{deg(}v\textrm{)}, r-\textrm{deg(}\phi\textrm{)}] \, \, \phi\, \llcorner \, v \end{cases}$$ A routine computation shows that the left (resp. right) twisted contraction by a sign function $\chi$ defines a left (resp. right) action of $\bw{}{E}$ on $\bw{}{\Ee}$ if and only if $\chi$ is one of the four following functions: 1. $\,\chi(p,q)=1$ 2. $\,\chi(p,q)=(-1)^p {_{\vphantom{i}} }$ 3. $\,\chi(p,q)=(-1)^{\frac{p(p+1)}{2}+pq} {_{\vphantom{i}} }$ 4. $\,\chi(p,q)=(-1)^{\frac{p(p+1)}{2}+pq+p} {_{\vphantom{i}} }$ Therefore there are *four* different sign conventions for a left (resp. right) action of $\bw{}{E}$ on $\bw{}{\Ee\!.}$ We end this section with Koszul complexes. Let $M$ be an and let $\phi $ be an $A$-linear form on $M$. The *Koszul complex* $L(M,\phi )$ is the exterior algebra $\bwi{}{A}{M}$ endowed with the differential $\delta $ of degree $-1$ given for any positive integer $p$ by $$\delta _{p}{^{\vphantom{*}} }(m_{1}{^{\vphantom{*}} }{\ensuremath{\wedge}}\dots{\ensuremath{\wedge}}m_{p}{^{\vphantom{*}} })=\sum_{i=1}^{p}\,(-1)^{i-1}{_{\vphantom{i}} }\phi (m_{i}{^{\vphantom{*}} })\,\, m_{1}{^{\vphantom{*}} }{\ensuremath{\wedge}}\dots{\ensuremath{\wedge}}m_{i-1}{^{\vphantom{*}} }{\ensuremath{\wedge}}m_{i+1}{^{\vphantom{*}} }{\ensuremath{\wedge}}\dots{\ensuremath{\wedge}}m_{p}{^{\vphantom{*}} }.$$ If $x_{1}{^{\vphantom{*}} },\dots,x_{k}{^{\vphantom{*}} }$ are elements in $A$, we recover the classical Koszul complex associated with the $x_{i}{^{\vphantom{*}} }$’s by taking $M=A^{k}{_{\vphantom{i}} }$ and $\phi (a_{1}{^{\vphantom{*}} },\dots a_{k}{^{\vphantom{*}} })=\sum_{i=1}^{k}x_{i}{^{\vphantom{*}} }a_{i}{^{\vphantom{*}} }$. Assume now that $M$ is free of finite rank $r$, and consider $M$ as the dual of $M\ee{_{\vphantom{i}} }$. In this case, $\delta $ is exactly the right contraction by $\phi $ acting on $\bwi{}{A}{M}$. Using the standard sign convention for $\textrm{Hom}$ complexes (see for instance [@KS1] Remark 1.8.11 and [@D] Remark 1.1.11), the differential $\delta \ee{_{\vphantom{i}} }$ of $L\ee{_{\vphantom{i}} }$ is given for any nonnegative integer $p$ by $$\delta \ee_{p}=(-1)^{p+1}{_{\vphantom{i}} }\,\phi \,{\ensuremath{\wedge}}.=-\,(\,.{\ensuremath{\wedge}}\phi )$$ Thus, the right duality morphism $\aplexp{D^{r}{_{\vphantom{i}} }}{\bw{}{M\ee{_{\vphantom{i}} }{\ensuremath{\otimes}}\det M}}{\bw{}{M}}{\sim}$ induces an isomorphism $$\label{EqTroisPrelim} (L\ee{_{\vphantom{i}} },\delta \ee{_{\vphantom{i}} })\simeq (L,-\delta ) {\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }\det M\ee{_{\vphantom{i}} }[-r].$$ Extensions and dg-algebras {#DgAlg} -------------------------- Let $A$ be a (non necessarily commutative) unitary algebra over a field $k$ of characteristic zero, $I$ be a $A{\ensuremath{\otimes}}A^{\textrm{op}}{_{\vphantom{i}} }$-module, and $B$ be the trivial $k$-extension of $A$ by $I$. This means that $B=I\oplus A$, endowed with the algebra structure defined by $$(i,a)\,.\,(i',a')=(i a'+a i',aa').$$ Let us take a dg-algebra $(\aaa, d\,)$ over $k$ concentrated in positive degrees, whose differential has degree $-1$, and satisfying the following compatibility condition: $$\label{EqUnDgAlg} \emph{The truncated dg-algebra}\ \smash[t]{\flgdexp{\aaa_{1}{^{\vphantom{*}} }}{\aaa_{0}{^{\vphantom{*}} }}{d_{1}{^{\vphantom{*}} }}}\ \emph{is isomorphic to}\ \smash[t]{\flgdexp{B}{A.}{\operatorname{pr}_{2}{^{\vphantom{*}} }}}$$ We denote by $|a|$ the degree of an homogeneous element $a$ in $\aaa$. \[DefUnDgAlg\] Let $\bb$ denote the graded module $\bop\nolimits_{k\ge 1}\aaa_{k}{^{\vphantom{*}} }$, where each $\aaa_{k}{^{\vphantom{*}} }$ sits in degree $k-1$. 1. For any homogeneous elements $a$ and $a'$ in $\bb$, we put $$a*a'=a\,.\,da'+(-1)^{|a|+1}{_{\vphantom{i}} }da\,.\, a'+(-1)^{|a|}{_{\vphantom{i}} }\, da\,.\,1_{B}{^{\vphantom{*}} }\,.\,da'$$ where $1_{B}{^{\vphantom{*}} }=(0, 1_{A})$ is the the unit of $B$ considered as an element of $\aaa_{1}{^{\vphantom{*}} }$. 2. We endow $\bb$ with a differential $\wh{d}$ of degree $-1$ given for any positive integer $k$ by $\wh{d}_{k}{^{\vphantom{*}} }=k\,d_{k+1}$. Remark that via the isomorphism between $\aaa_{1}{^{\vphantom{*}} }$ and $B$, the product $*$ from $\aaa_{1}{^{\vphantom{*}} }{\ensuremath{\otimes}}_{k}{^{\vphantom{*}} }\aaa_{1}{^{\vphantom{*}} }$ to $\aaa_{1}{^{\vphantom{*}} }$ is exactly the product in the algebra $B$. \[PropUnDgAlg\] For any dg-algebra satisfying *(\[EqUnDgAlg\])*, $(\bb,*,\wh{d} \,)$ is a dg-algebra. This is proved by direct computation. Let us prove for instance that $\wh{d}$ satisfies Leibniz rule, and leave to the reader the associativity of $*$. We take two homogeneous elements $b$ and $b'$ in $\bb$ of respective degrees $k$ and $k'$. Then $$\begin{aligned} \wh{d}_{k+k'}{^{\vphantom{*}} }(b*b')&=(k+k')\,d_{k+k'+1}{^{\vphantom{*}} }(b*b')=(k+k')\,db\,.\,db'\\ &=k \,(d_{k+1}{^{\vphantom{*}} }b*b')+(-1)^{k}{_{\vphantom{i}} }\, k'\,(b*d_{k'+1}{^{\vphantom{*}} }b')=\wh{d}_{k}{^{\vphantom{*}} }b*b'+(-1)^{|b|}{_{\vphantom{i}} }\, b*\wh{d}_{k'}{^{\vphantom{*}} }b'.\end{aligned}$$ It follows from this result that all the $\aaa_{k}{^{\vphantom{*}} }$’s are naturally $B {\ensuremath{\otimes}}_{k}{^{\vphantom{*}} }B^{\,\textrm{op}}$-modules. Besides, $A$ can be endowed with the structure of a $ B {\ensuremath{\otimes}}_{k}{^{\vphantom{*}} }B^{\,\textrm{op}}{_{\vphantom{i}} }$-module, and there is a natural $B {\ensuremath{\otimes}}_{k}{^{\vphantom{*}} }B^{\,\textrm{op}}{_{\vphantom{i}} }$-linear morphism $\apl{\pi }{\bb}{A}$ obtained via the composition $\smash{\xymatrix@C=17pt{\bb\ar[r]&\bb_{0}{^{\vphantom{*}} }\simeq\aaa_{1}\ar[r]^-{\smash[b]{d_{1}{^{\vphantom{*}} }} }&\aaa_{0}\simeq A.}} $ Besides, the diagram below $$\xymatrix@C=30pt@R=20pt{ \bb{\ensuremath{\otimes}}_{B}{^{\vphantom{*}} }\bb\ar[r]^-{*}\ar[d]_-{\pi {\ensuremath{\otimes}}\pi }&\bb\ar[d]^-{\pi }\\ A{\ensuremath{\otimes}}_{B}{^{\vphantom{*}} }A\ar[r]&A }$$ is commutative, the bottom line being given by $\xymatrix@C=17pt{a_{1}{^{\vphantom{*}} }{\ensuremath{\otimes}}a_{2}{^{\vphantom{*}} }\ar@{|->}[r]&a_{1}{^{\vphantom{*}} }a_{2}{^{\vphantom{*}} }.}$ The situation is more comfortable in the commutative case, i.e. when $A$ is commutative, $I$ is a $A$-module (hence a $\smash{A{\ensuremath{\otimes}}_{k}{^{\vphantom{*}} }A^{\textrm{op}}{_{\vphantom{i}} }} $-module), and $\aaa$ is graded-commutative. In that case, $\bb$ is also graded-commutative and all the $\aaa_{k}{^{\vphantom{*}} }$’s are endowed with a $B$-module structure commuting with the differential $\wh{d}$. In the main example of application, which we describe now, we assume $A$ to be commutative, and we take for $\aaa$ the exterior algebra $\bwi{}{A}{B}$ endowed with the Koszul differential given by the $A$-linear form $\apl{\operatorname{pr}_{2}{^{\vphantom{*}} }}{B}{A.}$ Thus, for any positive integer $k$, $$d_{k}(b_{1}{^{\vphantom{*}} }{\ensuremath{\wedge}}\dots{\ensuremath{\wedge}}b_{k}{^{\vphantom{*}} })=\sum_{i=1}^{k}(-1)^{i-1}\operatorname{pr}_{2}{^{\vphantom{*}} }(b_{i})\,\, b_{1}{^{\vphantom{*}} }{\ensuremath{\wedge}}\dots{\ensuremath{\wedge}}b_{i-1}{^{\vphantom{*}} }{\ensuremath{\wedge}}b_{i+1}{^{\vphantom{*}} }{\ensuremath{\wedge}}\dots{\ensuremath{\wedge}}b_{k}{^{\vphantom{*}} }.$$ Besides, via the isomorphism $$\begin{aligned} \bwi{k}{\!A}{I}\oplus\bwi{k-1}{\!A}{I}\ &\xymatrix@C=17pt{\ar[r]^-{\sim}&}\ \bwi{k}{\!A}{B}\label{EqDeuxDgAlg}\\ (\ubi\,,\,\ubj)\hphantom{\bwi{k}{aaa}{\bb}} &\xymatrix@C=17pt{\ar@{|->}[r]&}\ \ubi+1_{B}{^{\vphantom{*}} }{\ensuremath{\wedge}}{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}}\notag\end{aligned}$$ the differential $\apl{d_{k}{^{\vphantom{*}} }}{\bwi{k}{\!A}{B}}{\bwi{k-1}{\!A}{B}}$ is obtained as the composition $$\sutr{\bwi{k}{\!A}{B}}{\bwi{k-1}{\!A}{I}}{\bwi{k-1}{\!A}{B.}}$$ Thus $\aaa$ (considered as a complex of $A$-modules) is exact. Via the isomorphism (\[EqDeuxDgAlg\]), the product $*$ has the following explicit form: $$\begin{aligned} *\,:\,\,\bigl( \bwi{k+1}{\!A}{I}\oplus\bwi{k}{\!A}{I}\bigr){\ensuremath{\otimes}}_{k}{^{\vphantom{*}} }\bigl( \bwi{l+1}{\!A}{I}\oplus\bwi{l}{\!A}{I}\bigr)&\xymatrix@C=17pt{\ar[r]&}\bwi{k+l+1}{\!A}{I}\oplus\bwi{k+l}{\!A}{I}\label{EqTroisDgAlg}\\ (\ubi_{\,1}{^{\vphantom{*}} }\,,\,\,{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}}_{\,1}{^{\vphantom{*}} }){\ensuremath{\otimes}}({\ensuremath{\,\underline{i{_{\vphantom{i}} }}\,}}_{\,2}{^{\vphantom{*}} }\,,\,\,{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}}_{\,2}{^{\vphantom{*}} })\hphantom{iiiiiiiiii}&\xymatrix@C=17pt{\ar@{|->}[r]&}({\ensuremath{\,\underline{i{_{\vphantom{i}} }}\,}}_{\,1}{^{\vphantom{*}} }{\ensuremath{\wedge}}{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}}_{\,2}{^{\vphantom{*}} }+(-1)^{k}{_{\vphantom{i}} }{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}}_{\,1}{^{\vphantom{*}} }{\ensuremath{\wedge}}{\ensuremath{\,\underline{i{_{\vphantom{i}} }}\,}}_{\,2}{^{\vphantom{*}} }\,,\,\,{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}}_{\,1}{^{\vphantom{*}} }{\ensuremath{\wedge}}{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}}_{\,2}{^{\vphantom{*}} }).\notag\end{aligned}$$ In particular, the $B$-module structure on $\bwi{k}{\!A}{B}$ is given by the formula $$\label{EqQuatreDgAlg} (i,a)*({\ensuremath{\,\underline{i{_{\vphantom{i}} }}\,}}_{\,1}{^{\vphantom{*}} },{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}}_{\,1}{^{\vphantom{*}} })=(a\,{\ensuremath{\,\underline{i{_{\vphantom{i}} }}\,}}_{\,1}+i{\ensuremath{\wedge}}{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}}_{1}{^{\vphantom{*}} },\,a{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}}_{\,1}{^{\vphantom{*}} }).$$ Atiyah complexes (I) {#LocCx} ==================== In this section we assume that $A$ is a commutative algebra with unit over a field $k$ of characteristic zero, and we adopt the notations of § \[DgAlg\], except that from now on we use the *cohomological* grading for complexes, which means that all differentials are of degree $+1$. We also introduce extra notation for derived functors. Let $R$ be a commutative $k$-algebra, $M$ be an $A$-module, and assume that $A$ is a quotient of $R$. We consider the following four functors, from $\textrm{Mod}(R)$ to $\textrm{Mod}(A)$ for the three first ones and from $\textrm{Mod}(R)^{\textrm{\,op}}{_{\vphantom{i}} }$ to $\textrm{Mod}(A)$ for the last one: $$S\fl M{\ensuremath{\otimes}}_{R}{^{\vphantom{*}} }S,\quad S\fl S{\ensuremath{\otimes}}_{R}M,\quad S\fl \textrm{Hom}_{R}{^{\vphantom{*}} }(M,S),\quad S\fl \textrm{Hom}_{R}{^{\vphantom{*}} }(S,M)$$ The associated derived functors are denoted by $$S\fl M\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L},\, r}}_{}}{}_{\!R}{^{\vphantom{*}} }\,S,\quad S\fl S\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}, \, \ell}}_{}}{}_{\!R}{^{\vphantom{*}} }\,M,\quad S\fl \textrm{RHom}^{r}_{R}(M,S),\quad S\fl \textrm{RHom}^{\ell}_{R}(S,M)$$ Of course, these functors can be defined for any $M$ in $D^{-}{_{\vphantom{i}} }(A)$ for the three first ones and for any $M$ in $D^{+}(A)$ for the last one. There is a slightly subtle point behind these definitions: for any elements $M$, $N$ in $D^{-}{_{\vphantom{i}} }(A)$, there is a canonical isomorphism between $M\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L},\, r}}_{}}{}_{\!R}{^{\vphantom{*}} }\,N$ and $M\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}, \, \ell}}_{}}{}{^{\vphantom{*}} }_{\!R}\,N$ in $D^{-}{_{\vphantom{i}} }(R)$, but this isomorphism is not a priori induced by an isomorphism in $D^{-}{_{\vphantom{i}} }(A)$. The same thing happens with the isomorphism between $\textrm{RHom}^{r}_{R}(M,N)$ and $\textrm{RHom}^{\ell}_{R}(M,N)$ in $D^{+}{_{\vphantom{i}} }(R)$. HKR isomorphisms for regular ideals {#LocHKR} ----------------------------------- Let $I$ be a free $A$-module of finite rank $r$. The construction performed in § \[DgAlg\] allows to make the following definition: \[DefUnHKR\] The *Atiyah-Kashiwara* (AK) complex associated with $I$ is the complex of $B$-modules $$P\,:\,\,\xymatrix@C=35pt{0\ar[r]&\bwi{r+1}{\!A}{B}\ar[r]^-{rd_{r+1}{^{\vphantom{*}} }}&\bwi{r}{\!A}{B}\ar[r]^-{(r-1)d_{r+1}{^{\vphantom{*}} }}&\ \cdots\ \ar[r]^-{d_{2}{^{\vphantom{*}} }}&B\ar[r]&0}$$ where $B$ is in degree $0$. There is a quasi-isomorphism $\xymatrix@C=17pt{P\ar[r]^-{\sim}&A}$ in $\textrm{Mod}(B)$. As a complex of $A$-modules, $P$ splits as the direct sum of $A$ and of a null-homotopic complex. Let us now take a commutative $k$-algebra $C$ with unit as well as a regular ideal $J$ in $C$ of length $r$. If $(j_{1}{^{\vphantom{*}} },\dots,j_{r}{^{\vphantom{*}} })$ is a regular sequence defining $J$, then $J/J^{2}$ is a free $C/J$-module of rank $r$, a basis being given by the classes of the elements $j_1, \ldots, j_r$. Then, if we put $A=C/J$ and $I=J/J^{2}$, we see that $C/J^{2}$ is a $k$-extension of $A$ by $I$ via the Atiyah exact sequence $$\label{EqUnHKR} \sutrgdpt{J/J^{2}{_{\vphantom{i}} }}{C/J^{2}{_{\vphantom{i}} }}{C/J}{.}$$ If this exact sequence splits over $A$, the algebra $C/J^{2}$ is isomorphic (in a non-canonical way) to the trivial $k$-extension of $A$ by $I$, so that we can identify $C/J^{2}{_{\vphantom{i}} }$ with $B=I\oplus A$ after the choice of a splitting of (\[EqUnHKR\]). \[PropUnHKR\] Let $C$ be a commutative $k$-algebra with unit and $J$ be a regular ideal of $C$ such that the associated Atiyah sequence *(\[EqUnHKR\])* splits. If we choose an isomorphism between $C/J^{2}$ and $B$, the quasi-isomorphism $\smash{\xymatrix@C=17pt{P\ar[r]^-{\sim}&A}} $ in *Mod*$(C)$ induces isomorphisms $$\xymatrix@C=25pt{A\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L},\, r}}_{}}{}_{C}{^{\vphantom{*}} }A&A\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L},\, r}}_{}}{}_{C}{^{\vphantom{*}} }P\ar[l]_-{\sim}\ar[r]^-{\sim}&A{\ensuremath{\otimes}}_{C}{^{\vphantom{*}} }P\simeq\bop_{i=0}^{r}\bwi{i}{\!A}{I}[i]}$$ $$\xymatrix{\ar[r]^-{\sim} \emph{RHom}^{\ell}_{C}(A,A)&\emph{RHom}^{\ell}_{C} (P,A)&\emph{Hom}_{C}{^{\vphantom{*}} }(P,A)\simeq\bop_{i=0}^{r}\bwi{i}{\!A}{I\ee{_{\vphantom{i}} }}[-i]\ar[l]_-{\sim} }$$ in the bounded derived category $D^{\emph{b}}(A)$, where $I\ee{_{\vphantom{i}} }=\emph{Hom}_{\!A}{^{\vphantom{*}} }(I,A)$. For any element $c$ in $C$, we denote by ${\ensuremath{\overline{c{^{\vphantom{*}} }}}}$ the class of $c$ in $B$. We also denote by $(e_{1}{^{\vphantom{*}} },\dots,e_{r}{^{\vphantom{*}} })$ the canonical basis of $k^{r}{_{\vphantom{i}} }$. If $(j_{1}{^{\vphantom{*}} },\dots,j_{r}{^{\vphantom{*}} })$ is a regular sequence defining the ideal $J$, the Koszul complex $L=(C{\ensuremath{\otimes}}_{k}{^{\vphantom{*}} }\bw{}{k^{r}{_{\vphantom{i}} }},\delta )$ associated with $(j_{1}{^{\vphantom{*}} },\dots,j_{r}{^{\vphantom{*}} })$ is a free resolution of $A$ over $C$. For any nonnegative integer $p$, we define a map $\smash{\apl{\gamma _{-p}{^{\vphantom{*}} }}{L_{-p}{^{\vphantom{*}} }}{P_{-p}{^{\vphantom{*}} }}} $ by the formula $$\gamma _{-p}{^{\vphantom{*}} }(c{\ensuremath{\otimes}}e_{l_{1}}{^{\vphantom{*}} }{\ensuremath{\wedge}}\dots{\ensuremath{\wedge}}e_{l_{p}}{^{\vphantom{*}} })={\ensuremath{\overline{c{^{\vphantom{*}} }}}}*(1_{B}{^{\vphantom{*}} }{\ensuremath{\wedge}}{\ensuremath{\overline{j{^{\vphantom{*}} }}}}_{l_{1}}{^{\vphantom{*}} }{\ensuremath{\wedge}}\dots{\ensuremath{\wedge}}{\ensuremath{\overline{j{^{\vphantom{*}} }}}}_{l_{p}}{^{\vphantom{*}} }).$$ where the product $*$ is defined in § \[DgAlg\]. The map $\gamma _{-p}{^{\vphantom{*}} }$ is obviously $C$-linear, and if $p$ is positive, $$\begin{aligned} \wh{d}_{-p}{^{\vphantom{*}} }\circ\gamma _{-p}{^{\vphantom{*}} }(c{\ensuremath{\otimes}}e_{l_{1}}{^{\vphantom{*}} }{\ensuremath{\wedge}}\dots{\ensuremath{\wedge}}e_{l_{p}}{^{\vphantom{*}} })&= \wh{d}_{-p}{^{\vphantom{*}} }(\,{\ensuremath{\overline{c{^{\vphantom{*}} }}}}*1_{B}{^{\vphantom{*}} }{\ensuremath{\wedge}}{\ensuremath{\overline{j{^{\vphantom{*}} }}}}_{l_{1}}{^{\vphantom{*}} }{\ensuremath{\wedge}}\dots{\ensuremath{\wedge}}{\ensuremath{\overline{j{^{\vphantom{*}} }}}}_{l_{p}}{^{\vphantom{*}} })\\ &={\ensuremath{\overline{c{^{\vphantom{*}} }}}}*\wh{d}_{-p}{^{\vphantom{*}} }(1_{B}{^{\vphantom{*}} }{\ensuremath{\wedge}}{\ensuremath{\overline{j{^{\vphantom{*}} }}}}_{l_{1}}{^{\vphantom{*}} }{\ensuremath{\wedge}}\dots{\ensuremath{\wedge}}{\ensuremath{\overline{j{^{\vphantom{*}} }}}}_{l_{-p}}{^{\vphantom{*}} })\\ &=p\,{\ensuremath{\overline{c{^{\vphantom{*}} }}}}* (\, {\ensuremath{\overline{j{^{\vphantom{*}} }}}}_{l_{1}}{^{\vphantom{*}} }{\ensuremath{\wedge}}\dots{\ensuremath{\wedge}}{\ensuremath{\overline{j{^{\vphantom{*}} }}}}_{l_{p}}){^{\vphantom{*}} }\intertext{and}\\[-6ex] \gamma _{-(p-1)}{^{\vphantom{*}} }\circ\delta _{-p}{^{\vphantom{*}} }(c{\ensuremath{\otimes}}e_{l_{1}}{^{\vphantom{*}} }{\ensuremath{\wedge}}\dots{\ensuremath{\wedge}}e_{l_{p}}{^{\vphantom{*}} })&=\gamma _{-(p-1)}{^{\vphantom{*}} }\Bigl(\, \sum_{i=1}^{p}(-1)^{i-1}{_{\vphantom{i}} }c\,j_{l_{i}}{^{\vphantom{*}} }{\ensuremath{\otimes}}e_{l_{1}}{^{\vphantom{*}} }{\ensuremath{\wedge}}\dots{\ensuremath{\wedge}}e_{l_{i-1}}{^{\vphantom{*}} }{\ensuremath{\wedge}}e_{l_{i+1}}{^{\vphantom{*}} }{\ensuremath{\wedge}}\dots{\ensuremath{\wedge}}e_{l_{p}}{^{\vphantom{*}} }\Bigr)\\[-2ex] &=\sum_{i=1}^{p}(-1)^{i-1}{_{\vphantom{i}} }\,{\ensuremath{\overline{c{^{\vphantom{*}} }}}}*\bigl[ \,{\ensuremath{\overline{j}}}_{l_{i}}{^{\vphantom{*}} }*(1_{B}{^{\vphantom{*}} }{\ensuremath{\wedge}}{\ensuremath{\overline{j}}}_{l_{1}}{^{\vphantom{*}} }{\ensuremath{\wedge}}\dots{\ensuremath{\wedge}}{\ensuremath{\overline{j}}}_{l_{i-1}}{^{\vphantom{*}} }{\ensuremath{\wedge}}{\ensuremath{\overline{j}}}_{l_{i+1}}{^{\vphantom{*}} }{\ensuremath{\wedge}}\dots{\ensuremath{\wedge}}{\ensuremath{\overline{j}}}_{l_{p}}{^{\vphantom{*}} })\bigr]\\[-1ex] &=\sum_{i=1}^{p}(-1)^{i-1}{_{\vphantom{i}} }\,{\ensuremath{\overline{c{^{\vphantom{*}} }}}}*(\, {\ensuremath{\overline{j}}}_{l_{i}}{\ensuremath{\wedge}}{\ensuremath{\overline{j}}}_{l_{1}}{\ensuremath{\wedge}}\dots{\ensuremath{\wedge}}{\ensuremath{\overline{j}}}_{l_{i-1}}{\ensuremath{\wedge}}{\ensuremath{\overline{j}}}_{l_{i+1}}{\ensuremath{\wedge}}\dots{\ensuremath{\wedge}}{\ensuremath{\overline{j}}}_{l_{p}})\\ &=p\,{\ensuremath{\overline{c{^{\vphantom{*}} }}}}*(\, {\ensuremath{\overline{j}}}_{l_{1}}{\ensuremath{\wedge}}\dots{\ensuremath{\wedge}}{\ensuremath{\overline{j}}}_{l_{p}}).\end{aligned}$$ Thus $\apl{\gamma }{L}{P}$ is a morphism of complexes. Hence we get two commutative diagrams $$\xymatrix@C=25pt@R=1pt{ &A\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L},\, r}}_{}}{}_{C}{^{\vphantom{*}} }P\ar[dl]_-{\sim}\ar[r]&A{\ensuremath{\otimes}}_{C}{^{\vphantom{*}} }P\\ A\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L},\, r}}_{}}{}_{C}{^{\vphantom{*}} }A&&\\ &A\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L},\, r}}_{}}{}_{C}{^{\vphantom{*}} }L\ar[ul]^-{\sim}\ar[uu]_-{\operatorname{id}{\ensuremath{\otimes}}\gamma }\ar[r]^-{\sim}&A{\ensuremath{\otimes}}_{C}{^{\vphantom{*}} }L\ar[uu]_-{\operatorname{id}{\ensuremath{\otimes}}\gamma } }\quad \xymatrix@C=25pt@R=8pt{ &\textrm{RHom}^{\ell}_{C} (P,A)\ar[dd]^-{.\,\circ\,\gamma }_-{\sim}&\textrm{Hom}_{C}{^{\vphantom{*}} }(P,A)\ar[l]\ar[dd]^-{.\,\circ\,\gamma }\\ \textrm{and\quad RHom}\smash{^{\ell}_{C}} (A,A)\ar[ur]^-{\sim}\ar[dr]_-{\sim}&&\\ &\textrm{RHom}^{\ell}_{C} (L,A)&\textrm{Hom}_{C}{^{\vphantom{*}} }(L,A)\ar[l]_-{\sim} }\hspace*{-5pt}$$ The vertical right arrow in the first diagram is the map from $A{\ensuremath{\otimes}}_{k}{^{\vphantom{*}} }\bw{}{k^{r}{_{\vphantom{i}} }}$ to $\bwi{}{\!A}{I}$ obtained by mapping each vector $e_k$ to ${\ensuremath{\overline{j}}}_k$, hence is an isomorphism. The dual of this map over $A$ is precisely (up to sign) the vertical right arrow in the second diagram, so that it is an isomorphism too. This finishes the proof. We now recall Kashiwara’s construction of the dual HKR isomorphism. For any free $A$-module $I$ of finite rank, we denote by $\theta _{I}{^{\vphantom{*}} }$ its top exterior power. \[DefDeuxHKR\] If $I$ is a free module of rank $r$ and $P$ is the associated AK complex, the *dual *AK* complex* $Q$ is the complex of $B$-modules defined by $Q=\textrm{Hom}_{A}{^{\vphantom{*}} }(P,\theta _{I}{^{\vphantom{*}} }[r])$ with a specific sign convention: the differential of $Q$ is $(-1)^r$ times the differential of $\textrm{Hom}_{A}{^{\vphantom{*}} }(P,\theta _{I}{^{\vphantom{*}} }[r])$. To describe $Q$, notice that for every integer $p$ between $0$ and $r-1$, there is an isomorphism of $B$-modules $$\label{dualityfirst} \bwi{r-p}{\!A}{B}\simeq \textrm{Hom}_{A}{^{\vphantom{*}} }(\bwi{p+1}{\!A}{B},\theta _{I}{^{\vphantom{*}} }) \qquad \quad (\ub{u}, \ub{v}) \flba \{\, (\ub{i}, \ub{j}) \flba \ub{j}{\ensuremath{\wedge}}\ub{u}+(-1)^p \, \ub{i} {\ensuremath{\wedge}}\ub{v} \,\}.$$ Therefore the dual AK complex is isomorphic to $$Q\,:\,\xymatrix@C=25pt{0\ar[r]&\bwi{r}{\!A}{B}\ar[r]&\bwi{r-1}{\!A}{B}\ar[r]&\ \cdots\cdots\ \ar[r]&\bwi{2}{\!A}{B}\ar[r]&B\ar[r]&A\ar[r]&0}$$ where $A$ is in degree zero, and the differential is $-(p+1)\, d_{r-p}{^{\vphantom{*}} }$ on each $\bwi{r-p}{\!A}{B}$. We have a natural quasi-isomorphism $\smash{\xymatrix@C=17pt{{\ensuremath{\theta _{I}{^{\vphantom{*}} }[r]}}{^{\vphantom{*}} }\ar[r]^-{\sim}&Q.}}$ given by the map $d_{r+1}{^{\vphantom{*}} }$. Besides, as a complex of $A$-modules, $Q$ splits as the direct sum of $\theta _{I}{^{\vphantom{*}} }[r]$ and of a null-homotopic complex. The isomorphism (\[dualityfirst\]) induces another one, namely: $$\label{dualitysecond} \bwi{r-p}{\!A}{I}\simeq \textrm{Hom}_{C}{^{\vphantom{*}} }(\bwi{p+1}{\!A}{B},\theta _{I}{^{\vphantom{*}} }).$$ There is a natural product $\apl{\, \wh{*} \, }{P{\ensuremath{\otimes}}_{B}{^{\vphantom{*}} }Q}{Q}$ which is defined by the same formula as the product $\,*\,$: $$\label{EqOnzeBis} \xymatrix@C=40pt@R=2pt{ \wh{*}\,:\,\bwi{l+1}{A}{B}{\ensuremath{\otimes}}_{B}{^{\vphantom{*}} }\bwi{r-k}{A}{B}\ar[r]&\bwi{r-k+l}{A}{B}\hspace*{118pt}\\ \hspace*{10pt}(\ubi_{1}{^{\vphantom{*}} },\ubj_{1}{^{\vphantom{*}} }){\ensuremath{\otimes}}(\ubi_{2}{^{\vphantom{*}} }, \ubj_{2}{^{\vphantom{*}} })\ar@{|->}[r]& {_{\vphantom{i}} }\, (\ubi_{1}{^{\vphantom{*}} }{\ensuremath{\wedge}}\ubj_{2}{^{\vphantom{*}} }+(-1)^{l}{_{\vphantom{i}} }\ubj_{1}{^{\vphantom{*}} }{\ensuremath{\wedge}}\ubi_{2}{^{\vphantom{*}} },\ubj_{1} {^{\vphantom{*}} }{\ensuremath{\wedge}}\ubj_{2}{^{\vphantom{*}} }) }$$ A straightforward computation shows that $\wh{*}$ is indeed a morphism of complexes. \[PropDeuxHKR\] Under the hypotheses of Proposition *\[PropUnHKR\]*, the quasi-isomorphism $\smash{\xymatrix@C=17pt{\theta _{I}{^{\vphantom{*}} }[r]\ar[r]^-{\sim}&Q}} $ induces isomorphisms in $D^{\emph{b}}{_{\vphantom{i}} }(A)\!:$ $$\xymatrix@C=30pt{ \emph{RHom}^{r}_{C} (A,{\ensuremath{\theta _{I}{^{\vphantom{*}} }[r]}})\ar[r]^-{\sim}&\emph{RHom}^{r}_{C}(A,Q)&\emph{Hom}_{C}(A,Q)\simeq\bop_{i=0}^{r}\bwi{i}{\!A}{I}[i]\ar[l]_-{\sim}}$$ Since $P$ is a complex of free $A$-modules, the natural map from $\textrm{Hom}_{A}{^{\vphantom{*}} }(P,{\ensuremath{\theta _{I}{^{\vphantom{*}} }[r]}})$ to $\textrm{RHom}_{A}{^{\vphantom{*}} }(P,{\ensuremath{\theta _{I}{^{\vphantom{*}} }[r]}})$ is an isomorphism. Let us consider the following commutative diagram in $D^{\textrm{b}}(C)$: $${\xymatrix@C=25pt{\st \textrm{RHom}^{\ell}_{C}(A, \,{\ensuremath{\theta _{I}{^{\vphantom{*}} }[r]}})\ar[d]^-{\sim}\ar[r]^-{\sim}&\st\textrm{RHom}^{\ell}_{C}(P,\,{\ensuremath{\theta _{I}{^{\vphantom{*}} }[r]}})\ar[d]^-{\sim}&{\ensuremath{\scriptstyle}}\textrm{Hom}_{C}(P,\,{\ensuremath{\theta _{I}{^{\vphantom{*}} }[r]}})\ar[l]_-{\phi _{1}{^{\vphantom{*}} }}\ar[d]^-{\sim}\\ \st\textrm{RHom}_{A}(A\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L},\, r}}_{}}{}_{C}{^{\vphantom{*}} }A,\,{\ensuremath{\theta _{I}{^{\vphantom{*}} }[r]}})\ar[r]^-{\sim}&\st\textrm{RHom}_{A}(A\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L},\, r}}_{}}{}_{C}{^{\vphantom{*}} }P,\,{\ensuremath{\theta _{I}{^{\vphantom{*}} }[r]}})&{\ensuremath{\scriptstyle}}\textrm{Hom}_{A}{^{\vphantom{*}} }(A\,{\ensuremath{\otimes}}_{C}{^{\vphantom{*}} }P,\,{\ensuremath{\theta _{I}{^{\vphantom{*}} }[r]}})\ar[l]_-{\phi _{2}{^{\vphantom{*}} }}\\ \st\textrm{RHom}^{r}_{C}(A,\,\textrm{RHom}_{A}(A,\,{\ensuremath{\theta _{I}{^{\vphantom{*}} }[r]}}))\ar[u]_-{\sim}\ar[r]^-{\sim}&\st\textrm{RHom}^{r}_{C}(A,\,\textrm{RHom}_{A}(P,\,{\ensuremath{\theta _{I}{^{\vphantom{*}} }[r]}}))\ar[u]_-{\sim}&{\ensuremath{\scriptstyle}}\textrm{Hom}_{C}{^{\vphantom{*}} }(A,\,\textrm{Hom}_{A}(P,\,{\ensuremath{\theta _{I}{^{\vphantom{*}} }[r]}}))\ar[u]_-{\sim}\ar[l]\\ \st\textrm{RHom}^{r}_{C}(A,\,{\ensuremath{\theta _{I}{^{\vphantom{*}} }[r]}})\ar[u]_-{\sim}\ar[r]^-{\sim}&\st\textrm{RHom}^{r}_{C}(A,\,Q)\ar[u]_-{\sim}&{\ensuremath{\scriptstyle}}\textrm{Hom}_{C}(A,\,Q)\ar[u]_-{\sim} \ar[l]_-{\phi _{3}{^{\vphantom{*}} }}} }$$ By Proposition \[PropUnHKR\], $\phi _{1}{^{\vphantom{*}} }$ and $\phi _{2}{^{\vphantom{*}} }$ are isomorphisms. This implies that $\phi _{3}{^{\vphantom{*}} }$ is also an isomorphism. We provide now another proof of Proposition \[PropDeuxHKR\], which gives a more precise result: \[PropDeuxBisHKR\] Under the hypotheses of Proposition *\[PropUnHKR\]*, let $\phi $ be the morphism in $D^{\emph{b}}{_{\vphantom{i}} }(C)$ obtained by the composition $$\xymatrix{ \bop_{i=1}^{r}\bwi{i}{A}I[i]\simeq\emph{Hom}_{C}{^{\vphantom{*}} }(A,Q)\ar[r]&\emph{RHom}_{C}{^{\vphantom{*}} }(A,\theta _{I}{^{\vphantom{*}} }[r])&\emph{Hom}_{C}{^{\vphantom{*}} }(P,\theta _{I}{^{\vphantom{*}} }[r])\simeq\bop_{i=0}^{r}\bwi{i}{A}{I[i]}\ar[l]_-{\sim} }$$ where the last isomorphism is *(\[dualitysecond\])*. Then, as a morphism in $D^{\emph{b}}{_{\vphantom{i}} }(k)$, $\phi $ acts by multiplication by the sign $(-1)^{\frac{(r-i)(r-i-1)}{2}}{_{\vphantom{i}} }$ on each factor $\bwi{i}{A}{I[i]}$. Let $L$ be the Koszul complex associated with $(j_{1}{^{\vphantom{*}} },\dots, j_{r}{^{\vphantom{*}} })$ and $\apl{\gamma }{L}{P}$ be the quasi-isomorphism constructed in the proof of Proposition \[PropUnHKR\]. We must describe the composition $$\xymatrix@R=3pt{ \textrm{Hom}_{C}{^{\vphantom{*}} }(A,Q)\ar[r]&\textrm{Hom}_{C}{^{\vphantom{*}} }(L,Q)&\textrm{Hom}_{C}{^{\vphantom{*}} }(L,\theta _{I}{^{\vphantom{*}} }[r])\ar[l]_-{\sim}&\textrm{Hom}_{C}{^{\vphantom{*}} }(P,\theta _{I}{^{\vphantom{*}} }[r])\ar[l]^-{\circ\,\gamma} _-{\sim}\\ }$$ Let $M$ denote the free $B$-module $I{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }B$ and $\apl{\tau }{M}{B}$ be the form defined by the composition $\smash{\xymatrix@C=17pt{I{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }B\ar[r]&I{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }A=I\ar[r]&B.}} $ If we identify $M$ with $B^{r}{_{\vphantom{i}} }$ via the basis $({\ensuremath{\overline{j}}}_{1}{^{\vphantom{*}} },\dots, {\ensuremath{\overline{j}}}_{r}{^{\vphantom{*}} })$, the linear form $\tau $ is simply the composition $\smash[b]{\xymatrix@C=40pt{B^{r}{_{\vphantom{i}} }\ar[r]^-{({\ensuremath{\overline{j}}}_{1}{^{\vphantom{*}} },\dots, \,{\ensuremath{\overline{j}}}_{r}{^{\vphantom{*}} })}&B.}} $ Thus, using the notation of § \[cup\], $L{\ensuremath{\otimes}}_{C}{^{\vphantom{*}} }B$ is isomorphic to the complex $L(M,\tau )$. We denote this latter complex by $(\ti{L},\delta )$. Hence we get by (\[EqTroisPrelim\]) the chain of isomorphisms $$\textrm{Hom}_{C}{^{\vphantom{*}} }(L,Q)\simeq\textrm{Hom}_{B}{^{\vphantom{*}} }(\ti{L},Q)\simeq Q {\ensuremath{\otimes}}_{B}{^{\vphantom{*}} }\ti{L}\ee{_{\vphantom{i}} }\simeq Q {\ensuremath{\otimes}}_{B}{^{\vphantom{*}} }(\ti{L}, -\delta ){\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }\theta \ee_{I}[-r] \simeq \theta \ee_{I}[-r] {\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }(\ti{L}, -\delta ) {\ensuremath{\otimes}}_{B}{^{\vphantom{*}} }Q.$$ Let $(N,s',s'')$ denote the double complex $(\ti{L}, -\delta ) {\ensuremath{\otimes}}_{B}{^{\vphantom{*}} }Q$ and let $s$ be the total differential. To avoid cumbersome notation, we use homological grading for $N$. Then, for $0\le p,q\le r$, we have (for the definition of $W_{p,q}{^{\vphantom{*}} }$, see § \[cup\]): –$N_{p,\,q}{^{\vphantom{*}} }=\bwi{p}{A}{M}{\ensuremath{\otimes}}_{B}{^{\vphantom{*}} }\bwi{q}{A}{B}\simeq\bwi{p}{A}{I}{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }\bwi{q}{A}{B}$ –$ \xymatrix@C=10pt{s'_{p,\,q}\,:\,\bwi{p}{A}{I}{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }\bwi{q}{A}{B}\ar[r]&\bwi{p}{A}{I}{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }\bwi{q-1}{A}{I}\ar[rrrr]^-{-p\,W_{p-1,\,1}{^{\vphantom{*}} }{\ensuremath{\otimes}}\,\operatorname{id}}&&&&\bwi{p-1}{A}{I}{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }I{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }\bwi{q-1}{A}{I}}\\ \xymatrix@=40pt{&&&&\hspace*{30pt}\ar[r]^-{\operatorname{id}{\ensuremath{\otimes}}\,{\ensuremath{\wedge}}}&\bwi{p-1}{A}{I}{\ensuremath{\otimes}}_{A }{^{\vphantom{*}} }\bwi{q}{A}{I}\ar[r]&\bwi{p-1}{A}{I}{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }\bwi{q}{A}{B }} $ –$s''_{p,\,q}=\operatorname{id}{\ensuremath{\otimes}}\, [-(r-q+1)\,d_{q}{^{\vphantom{*}} }]$. Besides, an easy verification yields that for $0\le i\le r$, –The morphism $\xymatrix@C=17pt{\alpha _{i}{^{\vphantom{*}} }:\bwi{i}{A}{I[i]}\ar[r]&\textrm{Hom}_{C}{^{\vphantom{*}} }(P,\theta _{I}{^{\vphantom{*}} }[r] ) \ar[r]&\textrm{Hom}_{C}{^{\vphantom{*}} }(L,Q)\simeq\theta \ee_{I}[-r]{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }N}$ is given by the inclusion $$\xymatrix@C=30pt{\bwi{i}{A}{I} \ar[r]^-{(-1)^{r}}& \bwi{i}{A}{I}\simeq\theta \ee_{I}{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }(\bwi{i}{A}{I}{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }\bwi{r}{A}{I})\ar[r]&\theta \ee_{I}{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }(\bwi{i}{A}{I}{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }\bwi{r}{A}{B})=\theta \ee_{I}{\ensuremath{\otimes}}N_{i,\,r}{^{\vphantom{*}} }}$$ –The morphism $\xymatrix@C=17pt{\beta _{i}{^{\vphantom{*}} }:\bwi{i}{A}{I[i]}\ar[r]&\textrm{Hom}_{C}{^{\vphantom{*}} }(A,Q)\ar[r]&\textrm{Hom}_{C}{^{\vphantom{*}} }(L,Q)\simeq \theta \ee_{I}[-r]{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }\! N}$ is given by the inclusion $$\xymatrix@C=30pt{ \bwi{i}{A}{I} \ar[r]^-{(-1)^{r} {_{\vphantom{i}} }}&\bwi{i}{A}{I}\simeq\theta \ee_{I}{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }(\bwi{r}{A}{I}{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }\bwi{i}{A}{I})\ar[r]&\theta \ee_{I}{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }(\bwi{r}{A}{B}{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }\bwi{i}{A}{I})=\theta \ee_{I}{\ensuremath{\otimes}}N_{r,\,i}{^{\vphantom{*}} }}$$ Let $R$ be the subcomplex of $N$ defined by $R_{p,\,q}{^{\vphantom{*}} }=\bwi{p}{A}{I}{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }\bwi{q}{A}{I}{\ensuremath{\subseteq}}N_{p,\,q}{^{\vphantom{*}} }.$ *Claim 1.* For $r\le n\le 2r$, $\ker s_{n}{^{\vphantom{*}} }=R_{n}{^{\vphantom{*}} }$. For any $x$ in $\ker s_{n}{^{\vphantom{*}} }$, let $x_{p,\,q}{^{\vphantom{*}} }$ (with $n-r\le p,q\le r$ and $p+q=n$) denote the graded components of $x$. Then we have –$s''_{r,\,n-r}(x_{r,\, n-r}{^{\vphantom{*}} })=0$ –$s'_{i,\,n-i}(x_{i,\,n-i}{^{\vphantom{*}} })+(-1)^{i-1}{_{\vphantom{i}} }s''_{i-1,\,n-i+1}(x_{i-1,\,n-i+1}{^{\vphantom{*}} })=0$ –$s'_{n-r,\,r}(x_{n-r,\,r}{^{\vphantom{*}} })=0$ Notice that for $p,q\ge 0$, $R_{p,\,q}{^{\vphantom{*}} }\suq\ker s'_{p,\,q}$. Furthermore, if $q$ is positive, $R_{p,\,q}{^{\vphantom{*}} }=\ker s''_{p,\,q}$. Thus, if $n>r$, $x_{r,\,n-r}{^{\vphantom{*}} }$ belongs to $R_{r,\,n-r}{^{\vphantom{*}} }$ and it follows that $x_{i-1,\,n-i+1}{^{\vphantom{*}} }$ belongs to $R_{i-1,\,n-i+1}{^{\vphantom{*}} }$ for $n-r+1\le i\le r$. If $n=r$, $N_{r,\,0}{^{\vphantom{*}} }=R_{r,\,0}{^{\vphantom{*}} }$ and $s'_{r,\,0}=s''_{r,\,0}=0$. Thus $x_{r,\,0}{^{\vphantom{*}} }$ belongs to $R_{r,\,0}{^{\vphantom{*}} }$ and $s''_{r-1,\,1}(x_{r-1,\,1}{^{\vphantom{*}} })=0$. Hence $x_{r-1,\,1}{^{\vphantom{*}} }$ belongs to $R_{r-1,\,1}{^{\vphantom{*}} }$ and we argue as in the case $n>r$. This proves the claim. For any integers $p$ and $q$ such that $0\le p,q\le r$ and $p+q\ge r$, let $\smash{\apl{\pi _{p,\,q}{^{\vphantom{*}} }}{R_{p,\,q}{^{\vphantom{*}} }}{R_{p+q-r,\,r}{^{\vphantom{*}} }}} $ be defined by the composition $$\xymatrix@C=20pt{ \pi _{p,\,q}{^{\vphantom{*}} }\,:\,\bwi{p}{A}{I}{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }\bwi{q}{A}{I}\ar[rrr]^-{W_{p+q-r,\,r-q}{^{\vphantom{*}} }\,{\ensuremath{\otimes}}\,\operatorname{id}}&&&\bwi{p+q-r}{A}{I}{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }\bwi{r-q}{A}{I}{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }\bwi{q}{A}{I}\ar[rr]^-{\operatorname{id}\,{\ensuremath{\otimes}}\,{\ensuremath{\wedge}}}&&\bwi{p+q-r}{A}{I}{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }\bwi{r}{A}{I.} }$$ Then, for any integer $n$ such that $r\le n\le 2r$, we define a projector $\smash{\apl{\pi _{n}{^{\vphantom{*}} }}{R_{n}{^{\vphantom{*}} }}{R_{n-r,\,r}{^{\vphantom{*}} }}} $ by the formula $$\pi _{n}{^{\vphantom{*}} }=\sum_{p=n-r}^{r}\epsilon_{n,\,p}{^{\vphantom{*}} }\,\binom{p}{n-r}\,\pi _{p,\,n-p}{^{\vphantom{*}} }.$$ where $\epsilon_{n,p}{^{\vphantom{*}} }=(-1)^{\frac{(p+1)(p+2)}{2}-\frac{(n-r+1)(n-r+2)}{2}}{_{\vphantom{i}} }$. *Claim 2.* For $n\le r\le 2n$, $\ker \pi _{n}{^{\vphantom{*}} }=\operatorname{im}s_{n+1}{^{\vphantom{*}} }$. We begin by proving the inclusion $\operatorname{im}s_{n+1}{^{\vphantom{*}} }{\ensuremath{\subseteq}}\ker \pi _{n}{^{\vphantom{*}} }$. The module $\operatorname{im}s_{n+1}{^{\vphantom{*}} }$ is spanned by elements of the form $s'_{i,\,n+1-i}(x)+(-1)^{i}s''_{i,\,n+1-i}(x)$, with $n+1-r\le i\le r $ and $x$ in $N_{i,\,n+1-i}{^{\vphantom{*}} }$. If $y$ denotes the projection of $x$ on $\bwi{i}{A}{I}{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }\bwi{n-i}{A}{I}\vphantom{^{\ds)}}$, we have by (\[EqDeuxBisSecAlgExt\]) the identity $$(\operatorname{id}{\ensuremath{\otimes}}\,{\ensuremath{\wedge}}\,)\,(W_{n-r,\,r-n+i-1}{^{\vphantom{*}} }{\ensuremath{\otimes}}\operatorname{id})\,(\operatorname{id}{\ensuremath{\otimes}}\,{\ensuremath{\wedge}}\,)\,(W_{i-1,\,1}{\ensuremath{\otimes}}\operatorname{id})(y)=(\operatorname{id}{\ensuremath{\otimes}}\,{\ensuremath{\wedge}}\,)\,(W_{n-r,\,r-n+i}{^{\vphantom{*}} }{\ensuremath{\otimes}}\operatorname{id})(y).$$ This implies that $$\dfrac{1}{i}\,\pi _{i-1,\,n+1-i}{^{\vphantom{*}} }\,(s'_{i, \, n+1-i}(x))=\dfrac{1}{r-n+i}\,\pi _{i,\,n-i}{^{\vphantom{*}} }\,(s''_{i,\,n+1-i}(x)).$$ Hence we get $$\begin{aligned} \pi _{n}{^{\vphantom{*}} }[s'_{i,\,n+1-i}(x)+(-1)^{i}s''_{i,\,n+1-i}(x)]&=\epsilon_{n,\,i-1}{^{\vphantom{*}} }\,\binom{i-1}{n-r}\,\pi _{i-1,\,n+1-i}{^{\vphantom{*}} }(s'_{i,\,n+1-i}(x))\\ &\hspace*{-60pt}+(-1)^{i}{_{\vphantom{i}} }\epsilon_{n,\,i} {^{\vphantom{*}} }\,\binom{i}{n-r}\,\pi _{i,\,n-i}{^{\vphantom{*}} }(s''_{i,\,n+1-i}(x))\\ &\hspace*{-130pt}=\dfrac{\pi _{i}{^{\vphantom{*}} }(s''_{i,\,n+1-i}(x))}{r-n+i}\tim\left(i\,\epsilon_{n,\,i-1} {^{\vphantom{*}} }\binom{i-1}{n-r}+(-1)^{i}{_{\vphantom{i}} }\epsilon_{n,\,i}{^{\vphantom{*}} }(r-n+i)\binom{i}{n-r}\!\right)=0.\end{aligned}$$ Since we know that $\alpha _{n-r}{^{\vphantom{*}} }$ is a quasi-isomorphism in degree $-(n-r)$, by the first claim we have the equality $R_{n-r,\,r}{^{\vphantom{*}} }\oplus\operatorname{im}s_{n+1}{^{\vphantom{*}} }=\ker s_{n}{^{\vphantom{*}} }=R_{n}{^{\vphantom{*}} }$. It follows that the kernel of the projector $\pi _{n}{^{\vphantom{*}} }$ is exactly the image of $s_{n+1}{^{\vphantom{*}} }$. A quick computation shows that the map $\smash[t]{\apl{\pi _{r,\,n-r}{^{\vphantom{*}} }}{R_{r,\,n-r}{^{\vphantom{*}} }}{R_{n-r,\,r}{^{\vphantom{*}} }}} $ is the multiplication by the constant $\smash{(-1)^{n(n-r)}{_{\vphantom{i}} }\bigm/\binom{r}{n-r}}$. Thus $\pi _{n\,|\,R_{r,\,n-r}{^{\vphantom{*}} }}{^{\vphantom{*}} }=(-1)^{n(n-r)}{_{\vphantom{i}} }\epsilon_{n,\,r}\tim\operatorname{id}$. It follows from the second claim that $$\operatorname{im}\bigl[\alpha _{n-r}{^{\vphantom{*}} }- (-1)^{n(n-r)}{_{\vphantom{i}} }\epsilon_{n,\,r} \,\beta _{n-r}{^{\vphantom{*}} }\bigr]{\ensuremath{\subseteq}}\theta \ee_{I}{\ensuremath{\otimes}}\operatorname{im}s_{n+1}{^{\vphantom{*}} }.$$ This yields the result. The construction of Arinkin and Căldăraru {#SectionArikinCalda} ----------------------------------------- Let $M$ be the free $B$-module $B{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }I$ and let $\tau$ be the $B$-linear form on $M$ obtained via the composition $\smash{\sutr{B{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }I}{A{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }I=I}{B.}} $ \[DefUnAriKinCalda\] The *Arinkin–Căldăraru complex* $(K,\nu )$ associated with the pair $(I, A)$ is the tensor algebra $K=\bigoplus\limits_{i\ge 0}{\smash{\bigotimes_{{A}^{{^{\vphantom{*}} }}}^{{i}\vphantom{\big)}}}}M[i]$ endowed with a differential $\nu$ given for any positive integer $p$ by the formula $$\nu _{-p}{^{\vphantom{*}} }(m_{1}{^{\vphantom{*}} }{\ensuremath{\otimes}}\dots{\ensuremath{\otimes}}m_{p}{^{\vphantom{*}} })=\smash[t]{\dfrac{1}{p!}} \, \tau (m_{1}{^{\vphantom{*}} })\,m_{2}{^{\vphantom{*}} }{\ensuremath{\otimes}}\dots{\ensuremath{\otimes}}m_{p}{^{\vphantom{*}} }.$$ Remark that $(K,\nu )$ is a free resolution of $A$ over $B$. Indeed, for any nonnegative integer $p$, $$\label{EqUnArikinCalda} \xymatrix{{\smash{\bigotimes_{{B}^{{^{\vphantom{*}} }}}^{{\,p}\vphantom{\big)}}}}M\simeq{\smash{\bigotimes_{{A}^{{^{\vphantom{*}} }}}^{{\,p+1}\vphantom{\big)}}}}\!I\oplus {\smash{\bigotimes_{{A}^{{^{\vphantom{*}} }}}^{{\,p}\vphantom{\big)}}}}I}$$ and the map $p!\,\nu _{-p}{^{\vphantom{*}} }$ is simply the composition $\sutr{{\smash{\bigotimes_{{B}^{{^{\vphantom{*}} }}}^{{\,p}\vphantom{\big)}}}}M}{{\smash{\bigotimes_{{A}^{{^{\vphantom{*}} }}}^{{\,p}\vphantom{\big)}}}}I}{{\smash{\bigotimes_{{B}^{{^{\vphantom{*}} }}}^{{\,p-1}\vphantom{\big)}}}}\!M.}$ The $B$-module structure on ${\smash{\bigotimes_{{B}^{{^{\vphantom{*}} }}}^{{\,p}\vphantom{\big)}}}}M$ is given via the isomorphism (\[EqUnArikinCalda\]) by $$\label{EqDeuxArikinCalda} (a+i)\,.\,({\ensuremath{\,\underline{i{_{\vphantom{i}} }}\,}},\,\ubj)=(a\,\ubi+i{\ensuremath{\otimes}}\ubj,\,a\,\ubj).$$ Besides, there is a canonical sequence of $B$-modules $$\label{EqTroisArikinCalda} \sutrgdpt{{\smash{\bigotimes_{{A}^{{^{\vphantom{*}} }}}^{{\,p}\vphantom{\big)}}}}I}{{\smash{\bigotimes_{{B}^{{^{\vphantom{*}} }}}^{{\,p}\vphantom{\big)}}}}M}{{\smash{\bigotimes_{{A}^{{^{\vphantom{*}} }}}^{{\,p-1}\vphantom{\big)}}}}I}{.}$$ Let $\mathfrak{a}$ be the antisymmetrization map from ${\smash{\bigotimes_{{A}^{{^{\vphantom{*}} }}}^{{}\vphantom{\big)}}}}I$ to $\bwi{}{\!A}{I}$ defined by (\[EqUnPrelim\]). Then there exists a natural morphism $$\apl{\zeta _{-p}{^{\vphantom{*}} }\, }{{\smash{\bigotimes_{{B}^{{^{\vphantom{*}} }}}^{{\,p}\vphantom{\big)}}}}M}{\bwi{p+1}{\!A}{B}}$$ given by $\zeta _{-p}{^{\vphantom{*}} }(\ubi,\ubj)=({\mathfrak{a}}_{\,p+1}{^{\vphantom{*}} }(\ubi), {\mathfrak{a}}_{\,p}{^{\vphantom{*}} }(\ubj))$. Thanks to (\[EqQuatreDgAlg\]) and ($\ref{EqDeuxArikinCalda}$), $\zeta _{-p}{^{\vphantom{*}} }$ is $B$-linear. Besides, if $P$ is the AK complex associated with $I$, then ${\apl{\zeta }{K}{P}} $ is a morphim of complexes which is a quasi-isomorphism and commutes to the quasi-morphisms $\smash{\xymatrix@C=17pt{K\ar[r]^-{\sim}&A}}$ and $\smash{\xymatrix@C=17pt{P\ar[r]^-{\sim}&A.}}$ This construction allows to prove Arinkin–Căldăraru’s HKR theorem in the local case: \[PropUnArinkinCalda\] Under the hypotheses of Proposition *\[PropUnHKR\]*, the map obtained as the composition $$\xymatrix@C=40pt{A\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L},\, r}}_{}}{}_{C}{^{\vphantom{*}} }\,A&A\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L},\, r}}_{}}{}_{C}{^{\vphantom{*}} }\,K\ar[l]_-{\sim}\ar[r]&A\,{\ensuremath{\otimes}}_{C}{^{\vphantom{*}} }\,K \simeq\smash{\bigoplus\limits_{i\ge 0}} {\smash{\bigotimes_{{A}^{{^{\vphantom{*}} }}}^{{i}\vphantom{\big)}}}}I[i]\ar[r]^-{\smash[t]{\bop_{i=0}^{r} {\mathfrak{a}_i} {^{\vphantom{*}} }} }&\smash[t]{\bop_{i=0}^{r}} \bwi{i}{\!A}{I[i]}}$$ is an isomorphism in $D^{\emph{b}}(A)$. We prove that this morphism is exactly the HKR isomorphism appearing in Proposition \[PropUnHKR\]. This is done by looking at the commutative diagram: $$\xymatrix@C=35pt@R=1pt{ &A\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L},\, r}}_{}}{}_{C}{^{\vphantom{*}} }\,K\ar[dd]^-{\operatorname{id}{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L},\, r}}_{}}{}_{C}{^{\vphantom{*}} }\, \zeta }\ar[dl]_-{\sim}\ar[r]&A{\ensuremath{\otimes}}_{C}{^{\vphantom{*}} }K\ar[dd]^-{\operatorname{id}{\ensuremath{\otimes}}_{C}\, \zeta }\ar@<1ex>@{}[r]_-{{\ensuremath{\displaystyle}}\simeq}&\bop_{i\ge 0}{\smash{\bigotimes_{{A}^{{^{\vphantom{*}} }}}^{{i}\vphantom{\big)}}}}I[i]\ar[dd]^-{\bop_{i=0}^{r}{\mathfrak{a}_i} {^{\vphantom{*}} }}\\ A\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L},\, r}}_{}}{}_{C}{^{\vphantom{*}} }\,A&&\\ &A\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L},\, r}}_{}}{}_{C}{^{\vphantom{*}} }\,P\ar[ul]^-{\sim}\ar[r]^-{\sim}&A{\ensuremath{\otimes}}_{C} {^{\vphantom{*}} }P\ar@<1ex>@{}[r]_-{{\ensuremath{\displaystyle}}\simeq}&\bop_{i=0}^{r}\bwi{i}{\!A}I[i] }$$ Additional properties of local Atiyah complexes {#SousSecAKComplexes} ----------------------------------------------- Let $\Omega _{A/k}{^{\vphantom{*}} }$ be the module of Kähler differentials of $A$ over $k$, and put ${\ensuremath{\Omega_{A/k}^{\,i}}}=\bwi{i}{A}{}\,\Omega _{A/k}{^{\vphantom{*}} }$ for $1\le i\le r$. An $A$-connection $\nabla$ on $I$ is a $k$-linear morphism $\smash{\apl{\nabla \, }{\, I}{\,\Omega _{A/k}{^{\vphantom{*}} }{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }I}}$ satisfying Leibniz’s rule $\nabla(a\,i)=a\nabla i+da{\ensuremath{\otimes}}i$ for any $a$ in $A$ and any $i$ in $I$. In our setting, an $A$-connection on $I$ is the same thing as the datum of a $k$-vector space of rank $r$ in $I$ (corresponding to the space of flat sections of $\nabla$). Recall that the automorphism group of $B$ in the category of $k$-extensions of $A$ by $I$ is the set $\textrm{Der}_{k}{^{\vphantom{*}} }(A,I)$ of $k$-derivations of $A$ with values in $I$, which is isomorphic to $\smash[b]{\textrm{Hom}_{A}{^{\vphantom{*}} }({\ensuremath{\Omega_{A/k}^{\,}}},I)} $. For such a derivation $\chi $, we denote by $u_{\chi }{^{\vphantom{*}} }$ the associated isomorphism of $B$ given explicitly by the formula $u_{\chi }{^{\vphantom{*}} }(i,a)=(i+\chi (a),a)$. \[PropTroisHKR\] Let $\chi $ be an element of $\emph{Der}_{k}{^{\vphantom{*}} }(A,I)$ and $\wh{\chi }$ be the associated morphism in $\emph{Hom}_{A}{^{\vphantom{*}} }({\ensuremath{\Omega_{A/k}^{\,}}}, I)$. Then: 1. Every $A$-connection on $I$ induces a $u_{\chi }{^{\vphantom{*}} }$-linear isomorphism of the AK-complex $P$ (resp. of the dual AK complex $Q$) commuting with the quasi-isomorphism $\smash[t]{\xymatrix@C=17pt{P\ar[r]^-{\sim}&A}} $ (resp. $\smash[t]{\xymatrix@C=17pt{\theta_{I}{^{\vphantom{*}} }[r] \ar[r]^-{\sim}&Q\!\!}} $). 2. If $\smash[b]{\apl{\, \wh{\chi \,}}{\, {\ensuremath{\Omega_{A/k}^{\,}}}}{I}}$ is an isomorphism, there exists a canonical $u_{\chi }{^{\vphantom{*}} }$-linear isomorphism of $P$ (resp. $Q$) commuting with the quasi-isomorphism $\smash{\xymatrix@C=17pt{P\ar[r]^-{\sim}&A}}$ (resp. $\smash{\xymatrix@C=17pt{\theta_{I}{^{\vphantom{*}} }[r] \ar[r]^-{\sim}&Q\!\!}} $). \(1) For any positive integer $p$, an $A$-connection $\nabla$ on $I$ induces an $A$-connection $\bw{p}{}{\,\nabla}$ on $\bwi{p}{\!A}{I}$. Let $\apl{R_{p}{^{\vphantom{*}} }}{\bwi{p}{\!A}{I}}{\bwi{p+1}{\!A}{I}}^{\vphantom{\ds)}}$ be defined as the composition $$\xymatrix@C=30pt{\bwi{p}{\!A}{I}\ar[r]^-{\bwi{p} {\vphantom{A}}{\,\nabla}}&{\ensuremath{\Omega_{A/k}^{\,}}}{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }\bwi{p}{\!A}{I}\ar[r]^-{\chi\, {\ensuremath{\otimes}}\,\operatorname{id}}&I{\ensuremath{\otimes}}_{A}{^{\vphantom{*}} }\bwi{p}{\!A}{I}\ar[r]^-{\wedge{}{}}&\bwi{p+1}{\!A}{I.}}$$ Using the isomorphism (\[EqDeuxDgAlg\]), we define $\apl{\varphi _{-p}{^{\vphantom{*}} }}{\bwi{p+1}{\!A}{B}}{\bwi{p+1}{\!A}{B}}$ by $\varphi _{-p}{^{\vphantom{*}} }({\ensuremath{\,\underline{i{_{\vphantom{i}} }}\,}},{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}})=({\ensuremath{\,\underline{i{_{\vphantom{i}} }}\,}}+R_{p}{^{\vphantom{*}} }({\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}}),{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}}).$ Then, using (\[EqQuatreDgAlg\]), we obtain that for any $(i,a)$ in $B$, $$\begin{aligned} \varphi _{-p}{^{\vphantom{*}} }[(i,a)*({\ensuremath{\,\underline{i{_{\vphantom{i}} }}\,}},{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}})]&=\varphi _{-p}{^{\vphantom{*}} }(a{\ensuremath{\,\underline{i{_{\vphantom{i}} }}\,}}+i\wedge{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}},a{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}})\\ &=(a{\ensuremath{\,\underline{i{_{\vphantom{i}} }}\,}}+i\wedge{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}}+R_{p}{^{\vphantom{*}} }(a{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}}),a{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}})\\ &=(a{\ensuremath{\,\underline{i{_{\vphantom{i}} }}\,}}+i\wedge{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}}+aR_{p}{^{\vphantom{*}} }({\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}})+\chi (a)\wedge \ubj, a{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}})\\ &=\bigl[a ({\ensuremath{\,\underline{i{_{\vphantom{i}} }}\,}}+R_{p}{^{\vphantom{*}} }({\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}}))+(i+\chi (a))\wedge \ubj, a{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}}\bigr]\\ &=u_{\chi }{^{\vphantom{*}} }(i,a)*({\ensuremath{\,\underline{i{_{\vphantom{i}} }}\,}}+R_{p}{^{\vphantom{*}} }({\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}}),{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}})\\ &=u_{\chi }{^{\vphantom{*}} }(i,a)*\varphi _{-p}{^{\vphantom{*}} }({\ensuremath{\,\underline{i{_{\vphantom{i}} }}\,}},{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}}).\end{aligned}$$ If we take for $\apl{\varphi _{0}{^{\vphantom{*}} }}{B}{B}$ the isomorphism $u_{\chi }{^{\vphantom{*}} }$, which is of course $u_{\chi }{^{\vphantom{*}} }$-linear, the $\varphi _{-p}{^{\vphantom{*}} }$’s define the required automorphism of $P$. \(2) For any positive integer $p$, the map $\wh{\chi }$ induces an isomorphism $\smash{\apl{\bw{p}{}\, \wh{\chi }\, }{{\ensuremath{\Omega_{A/k}^{\,p}}}}{\, \bwi{p}{\!A}{I.}}}$ Then we define ${\apl{R_{p}{^{\vphantom{*}} }}{\bwi{p}{\!A}{I}}{\bwi{p+1}{\!A}{I}}} $ by ${R_{p}{^{\vphantom{*}} }=\bwi{p+1}{}{}\, \wh{\chi }\circ \mathfrak{d}_{p}{^{\vphantom{*}} }\circ(\bwi{p}{}{}\, \wh{\chi }\, )^{-1}}{_{\vphantom{i}} }$, where ${\apl{\mathfrak{d}_{p}{^{\vphantom{*}} }}{{\ensuremath{\Omega_{A/k}^{\,p}}}}{{\ensuremath{\Omega_{A/k}^{\,p+1}}}}}$ is the exterior differential. For $a$ in $A$ and ${\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}}$ in $\bwi{p}{\!A}{I}$, we have $$\begin{aligned} R_{p}{^{\vphantom{*}} }(a{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}})&=\bw{p+1}{}{}\, \wh{\chi }\,\bigl[\, a\mathfrak{d}_{p}{^{\vphantom{*}} }\bigl( (\bwi{p}{}{}\, \wh{\chi }\, )^{-1}{_{\vphantom{i}} }({\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}})\bigr)+da\wedge(\bw{p}{}{}\, \wh{\chi }\, )^{-1}{_{\vphantom{i}} }({\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}})\bigr]\\ &=aR_{p}{^{\vphantom{*}} }({\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}})+\wh{\chi }(da)\wedge\ubj\\ &=aR_{p}{^{\vphantom{*}} }({\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}})+\chi (a) \wedge\ubj\!.\end{aligned}$$ Then we argue exactly as in (i). This proposition implies as a corollary that the two local HKR isomorphisms of Proposition \[PropUnHKR\] and Proposition \[PropDeuxHKR\] are in fact independent of the splitting of the Atiyah sequence (\[EqUnHKR\]), since two different splittings yield isomorphic extensions. Atiyah complexes (II) {#Atiyah Global} ===================== In this section, we fix two connected analytic manifolds $X$ and $Y$ such that $X$ is a proper closed complex submanifold of $Y$. We introduce some notation which will be used extensively in the sequel: $r$ is the codimension of $X$ in $Y$, $\smash{\apl{j}{X}{Y}}$ is the canonical inclusion, $\smash[t]{{\ensuremath{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{\textrm{X}{^{\vphantom{*}} }}}}}} \hspace*{-2.13ex}X}}$ is the first formal [neighbourhood]{} of $X$ in $Y$, $\smash{\apl{{\ensuremath{\overline{j}}}}{X}{{\ensuremath{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{\textrm{X}{^{\vphantom{*}} }}}}}} \hspace*{-2.13ex}X}}}$ is the associated inclusion and $\bb$ is the trivial $\C_{X}{^{\vphantom{*}} }$-extension of ${{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}^{\vphantom{stupide}}$ by $N\ee_{X/Y}$. Remark that by the adjunction formula, $\smash{\textrm{det} \, N_{X/Y}{^{\vphantom{*}} }[-r]}$ is isomorphic to the relative dualizing complex $\smash{\omega_{X/Y}{^{\vphantom{*}} }}$. Although $\omega_{X/Y} {^{\vphantom{*}} }$ is an object of $D^{\textrm{b}}{_{\vphantom{i}} }(X)$, we will always consider it as the object $\det N_{X/Y}{^{\vphantom{*}} }[-r]$ in the category of complexes of sheaves of ${\ensuremath{\oo_{X}{^{\vphantom{*}} }}}$-modules. The Atiyah sequence associated with the pair $(X,Y)$ is the exact sequence $$\label{EqUnAnalyticHKR} \sutrgd{N\ee_{X/Y}}{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}}{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}$$ in $\textrm{Mod}(\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}})$, which is a sheafified version of (\[EqUnHKR\]). A *quantized analytic cycle* in a complex manifold $Y$ is a couple $(X, \sigma)$ such that: 1. $X$ is a closed complex submanifold of $Y.$ 2. $\sigma$ is a holomorphic retraction of ${\ensuremath{\overline{j}}}.$ If $(X, \sigma)$ is a quantized analytic cycle, then the Atiyah sequence (\[EqUnAnalyticHKR\]) is automatically split over ${\ensuremath{\oo_{X}{^{\vphantom{*}} }}}$. Analytic HKR isomorphisms {#anhkr} ------------------------- The constructions of § \[SectAlgExt\] can be sheafified in an obvious manner. Thus, for every positive integer $p$, $\bwi{p}{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{\bb}_{\vphantom{\ds)}}$ is naturally a sheaf of $\bb$-modules on $X$. We get in this way two AK complexes ${\ensuremath{\mathcal{P}}}$ and ${\ensuremath{\mathcal{Q}}}$ which are complexes of $\bb$-modules. If $\apl{\sigma }{{\ensuremath{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{\textrm{X}{^{\vphantom{*}} }}}}}} \hspace*{-2.13ex}X}}{X}$ is the retraction of ${\ensuremath{\overline{j}}}$ obtained from a splitting of (\[EqUnAnalyticHKR\]), then $\sigma $ induces an isomorphism $\smash{\aplexp{\psi _{\sigma }}{\bb}{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}}{\sim}}$of $\C_{X}{^{\vphantom{*}} }$-algebras. \[DefUnAnalyticHKR\] For any positive integer $p$, we put $\bwi{p}{\sigma {^{\vphantom{*}} }}{\,\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}}=\psi \ee_{\sigma }\bigl( \bwi{p}{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{\bb}\bigr)$ in $\textrm{Mod}(\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}})$, and we define the AK complexes ${\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }$ and ${\ensuremath{\mathcal{Q}}}_{\sigma }{^{\vphantom{*}} }$ by ${\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }=\psi \ee_{\sigma }{\ensuremath{\mathcal{P}}}$ and ${\ensuremath{\mathcal{Q}}}_{\sigma }{^{\vphantom{*}} }=\psi \ee_{\sigma }{\ensuremath{\mathcal{Q}}}\vphantom{\Bigl)}$. They are both complexes of $\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}$ -modules. The results of § \[SousSecAKComplexes\] can be extended in our setting. \[PropUnAnalyticHKR\] Let $(X, \sigma)$ be a quantized analytic cycle of codimension $r$ in a complex manifold $Y$. Then for any locally free sheaves $\mathcal{E}$ and $\eee'$ on $X$, the quasi-isomorphism $\smash{\xymatrix@C=17pt{{\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }{\ensuremath{\otimes}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }\mathcal{E}' \ar[r]^-{\sim}& \mathcal{E}'}}$ in $\emph{Mod}\,(\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}})$ induces isomorphisms in $D^{\emph{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$: $$\begin{aligned} &\xymatrix@C=15pt{\Gamma _{\sigma }{^{\vphantom{*}} }\,:\,{\ensuremath{\mathcal{E}}}{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L},\, r}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\mathcal{E}' &{\ensuremath{\mathcal{E}}}{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L},\, r}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }({\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }{\ensuremath{\otimes}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }\mathcal{E}') \ar[l]_-{\sim}\ar[r]^-{\sim}&{\ensuremath{\mathcal{E}}}{\ensuremath{\otimes}}_{\oo_{Y}}{^{\vphantom{*}} }({\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }{\ensuremath{\otimes}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }\mathcal{E}')\,\simeq\,\bop_{i=0}^{r}\eee\, {\ensuremath{\otimes}}\,\mathcal{E}' \,{\ensuremath{\otimes}}\, \bwi{i}{}{N\ee_{X/Y}}\,[i]}\\ &\xymatrix@C=15pt@R=0pt{{\Gamma }\ee_{\sigma }\,:\,{\mathcal{RH}om^{\,\ell\vphantom{p}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}} ({\ensuremath{\mathcal{E}}},\,\eee' )\ar[r]^-{\sim}&{\mathcal{RH}om^{\,\ell\vphantom{p}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}} ({\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }{\ensuremath{\otimes}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }{\eee},\,\eee' )&\operatorname{\mathcal{H}\mathnormal{om}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}({\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }{\ensuremath{\otimes}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }{\eee},\,\eee' ) \ar[l]_-{\sim}\\ &&\simeq\, \bop_{i=0}^{r} \operatorname{\mathcal{H}\mathnormal{om}}(\eee,\, \eee') \, {\ensuremath{\otimes}}\bwi{i}{}{N_{X/Y}{^{\vphantom{*}} }} \, [-i].}\end{aligned}$$ We refer the reader to the proof of Proposition \[PropUnHKR\]. For any locally-free sheaves $\eee$ and $\eee'$ on $X$, there are canonical isomorphisms $${\ensuremath{\mathcal{E}}}\, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L},\, r}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}} \eee' \simeq {\ensuremath{\mathcal{E}}}\, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }( {\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L},\, r}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }\eee' )\quad\textrm{and}\quad {\mathcal{RH}om^{\,\ell\vphantom{p}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}}({\ensuremath{\mathcal{E}}},\eee' ) \simeq \eee' \, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }{\mathcal{RH}om^{\,\ell\vphantom{p}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}}(\eee, {\ensuremath{\oo_{X}{^{\vphantom{*}} }}}).$$ in $D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$ which are compatible with the HKR isomorphisms of Proposition \[PropUnAnalyticHKR\]. As in the local case, we also have a dual HKR isomorphism. To state the result, we consider for any holomorphic vector bundle $\eee$ on $X$ the isomorphism $$\begin{aligned} \label{right} \operatorname{\mathcal{H}\mathnormal{om}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }({\ensuremath{\mathcal{E}}}, {\ensuremath{\mathcal{Q}}}_{\sigma }{^{\vphantom{*}} }\, {\ensuremath{\otimes}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }\omega_{X/Y} {^{\vphantom{*}} }{\ensuremath{\otimes}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }\eee')\simeq\bop_{i=0}^{r}\operatorname{\mathcal{H}\mathnormal{om}}(\eee,\, \eee') \, {\ensuremath{\otimes}}\, \bwi{i}{}{N_{X/Y}{^{\vphantom{*}} }} \, [-i]\end{aligned}$$ given by the *left* duality map $D^{\ell}{_{\vphantom{i}} }$ introduced in § \[cup\] (for this we consider the normal bundle as the dual of the conormal bundle, so that (\[right\]) is an isomorphism of left modules over the graded exterior algebra of $N\ee_{X/Y}$). \[HKR2\] Let $(X, \sigma)$ be a quantized analytic cycle of codimension $r$ in a complex manifold $Y$. Then for any locally free sheaves $\mathcal{E}$ and $\eee'$ on $X$, the quasi-isomorphism $\smash{\xymatrix@C=17pt{\eee' \ar[r]^-{\sim}&{\ensuremath{\mathcal{Q}}}_{\sigma }{^{\vphantom{*}} }\, {\ensuremath{\otimes}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }\omega_{X/Y} {^{\vphantom{*}} }{\ensuremath{\otimes}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }\eee'}}$ induces an isomorphism $$\begin{aligned} &\xymatrix@C=35pt@R=1pt{\wh\Gamma_{\sigma }\, :\, {\mathcal{RH}om^{\,r\vphantom{p}}_{\oo_{Y}}}({\ensuremath{\mathcal{E}}},\eee')\ar[r]^-{\sim}&{\mathcal{RH}om^{\,r\vphantom{p}}_{\oo_{Y}}}({\ensuremath{\mathcal{E}}},\, {\ensuremath{\mathcal{Q}}}_{\sigma }{^{\vphantom{*}} }\, {\ensuremath{\otimes}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }\omega_{X/Y} {^{\vphantom{*}} }{\ensuremath{\otimes}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }\eee')\hspace*{20pt}}\\[-1ex] &\xymatrix@C=35pt@R=1pt {\hspace*{40pt}&\operatorname{\mathcal{H}\mathnormal{om}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }({\ensuremath{\mathcal{E}}},\, {\ensuremath{\mathcal{Q}}}_{\sigma }{^{\vphantom{*}} }\, {\ensuremath{\otimes}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }\omega_{X/Y} {^{\vphantom{*}} }{\ensuremath{\otimes}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }\eee')\ar[l]_-{\sim} \simeq \, \bop_{i=0}^{r}\operatorname{\mathcal{H}\mathnormal{om}}(\eee,\, \eee') \, {\ensuremath{\otimes}}\, \bwi{i}{}{N_{X/Y}{^{\vphantom{*}} }} \, [-i] \ar[l]_-{\sim}}\end{aligned}$$ in $D^{\emph{b}}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$, the last isomorphism being given by *(\[right\])*. We refer the reader to the proof of Proposition \[PropDeuxHKR\]. The set of retractions of ${\ensuremath{\overline{j}}}$ is an affine space over $\smash[b]{\textrm{Der}_{\C_{X}}{^{\vphantom{*}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}}, N\ee_{X/Y})} $, the latter being isomorphic to $\smash[b]{\textrm{Hom}_{\oo_{X}}{^{\vphantom{*}} }(\Omega ^{1}_{X}, N\ee_{X/Y})} $. The main difference with the local situation is that the HKR isomorphism can depend a priori on $\sigma $. This problem will be discussed in § \[comparison\]. At this stage, we only give the following result, which is the global analog of Proposition \[PropTroisHKR\]: \[PropDeuxAnalyticHKR\] Let $(X,Y)$ be a pair of complex manifolds such that $X$ is a closed submanifold of $Y$. 1. Assume that $\smash[b]{N\ee_{X/Y}} $ carries a global holomorphic connection. Then for any retractions $\sigma $ and $\sigma '$ of ${\ensuremath{\overline{j}}}$, ${\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }$ (resp. ${\ensuremath{\mathcal{Q}}}_{\sigma }{^{\vphantom{*}} }$) is naturally isomorphic to ${\ensuremath{\mathcal{P}}}_{\sigma '}{^{\vphantom{*}} }$ (resp. ${\ensuremath{\mathcal{Q}}}_{\sigma '}{^{\vphantom{*}} }$) and this isomorphism commutes with the quasi-isomorphism $\smash{\xymatrix@C=17pt{{\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }\ar[r]^-{\sim}&{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}}$ (resp. $\smash{\xymatrix@C=17pt{\omega_{X/Y}^{\otimes -1}\ar[r]^-{\sim}&{\ensuremath{\mathcal{Q}}}_{\sigma }{^{\vphantom{*}} }}}\!\!$). 2. Let $\sigma $ and $\sigma '$ be two retractions of ${\ensuremath{\overline{j}}}$ such that the element $\sigma '-\sigma $ in $\emph{Hom}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }(\Omega ^{1}_{X}, N\ee_{X/Y})$ is an isomorphism. Then ${\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }$ (resp. ${\ensuremath{\mathcal{Q}}}_{\sigma }{^{\vphantom{*}} }$) is naturally isomorphic to ${\ensuremath{\mathcal{P}}}_{\sigma '}{^{\vphantom{*}} }$ (resp. ${\ensuremath{\mathcal{Q}}}_{\sigma '}{^{\vphantom{*}} }$) and this isomorphism commutes with the quasi-isomorphism $\smash{\xymatrix@C=17pt{{\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }\ar[r]^-{\sim}&{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}}$(resp. $\smash{\xymatrix@C=17pt{\omega_{X/Y}^{\otimes -1}\ar[r]^-{\sim}&{\ensuremath{\mathcal{Q}}}_{\sigma }{^{\vphantom{*}} }}}\!\!$). We refer the reader to the proof of Proposition \[PropTroisHKR\]. As a consequence, we obtain immediately: \[ThUn\] Assume that $Y=X{\ensuremath{\times}}X$, and let $\sigma _{1}{^{\vphantom{*}} }$ and $\sigma _{2}{^{\vphantom{*}} }$ be the retractions of ${\ensuremath{\overline{j}}}$ induced by the first and second projections. For any complex number $t$, we put $\sigma_{t}{^{\vphantom{*}} }=(2-t)\sigma _{1}{^{\vphantom{*}} }+(t-1)\sigma _{2}{^{\vphantom{*}} }$. Then for any $s$ and $t$ in $\C$, $\Gamma _{\sigma _{s}{^{\vphantom{*}} }}{^{\vphantom{*}} }=\Gamma _{\sigma _{t}{^{\vphantom{*}} }}{^{\vphantom{*}} }$. The map $\sigma _{1}{^{\vphantom{*}} }-\sigma _{2}{^{\vphantom{*}} }$ in $\textrm{Der}_{\C_{X}{^{\vphantom{*}} }}{^{\vphantom{*}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}},N\ee_{X/X \times X})$ is given by $$(\sigma _{1}{^{\vphantom{*}} }-\sigma _{2}{^{\vphantom{*}} })(f)=\{\xymatrix@C=17pt{\! \!(x,y)\ar@{|->}[r]&f(x)-f(y)\! \!}\}\ \textrm{modulo}\ \jj_{X}^{2}.$$ It induces an isomorphism between $\Omega ^{1}_{X}$ and $N\ee_{X/X \times X}$. Now, for any complex numbers $s$ and $t$ such that $s\neq t$, $\sigma _{t}{^{\vphantom{*}} }-\sigma _{s}{^{\vphantom{*}} }=(s-t)(\sigma _{1}{^{\vphantom{*}} }-\sigma _{2}{^{\vphantom{*}} })$ so that $\Gamma _{\sigma _{s}{^{\vphantom{*}} }}{^{\vphantom{*}} }=\Gamma _{\sigma _{t}{^{\vphantom{*}} }}{^{\vphantom{*}} }$ by Theorem \[PropDeuxAnalyticHKR\] (2). The twisted case {#TwistedCase} ---------------- For any nonnegative integer $p$, the sheaf $\bwi{p+1}{\sigma {^{\vphantom{*}} }}{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}}$ is isomorphic (as a sheaf of ${\ensuremath{\oo_{X}{^{\vphantom{*}} }}}$-modules) to $^{\vphantom{\bigl)}}\bwi{p+1}{}{N\ee_{X/Y}}\oplus\bwi{p}{}{N\ee_{X/Y}}$ via the isomorphism (\[EqDeuxDgAlg\]). Besides, any section $s$ of the sheaf $\operatorname{\mathcal{H}\mathnormal{om}}^{\vphantom{\bigl)}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}(\bwi{p}{}{N\ee_{X/Y}},\bwi{p+1}{}{N\ee_{X/Y}})$ induces a section of $\mathcal{A}ut_{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}\, }(\bwi{p+1}{\sigma {^{\vphantom{*}} }}{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}}) {}^{\vphantom{a}}$ given by $$\xymatrix@C=17pt{({\ensuremath{\,\underline{i{_{\vphantom{i}} }}\,}},{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}})\ar@{|->}[r]&({\ensuremath{\,\underline{i{_{\vphantom{i}} }}\,}}+s({\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}}),{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}})}$$ so that we have a canonical embedding of ${\operatorname{\mathcal{H}\mathnormal{om}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }(\bwi{p}{}{N\ee_{X/Y}},\bwi{p+1}{}{N\ee_{X/Y}})}$ in $\mathcal{A}ut^{\vphantom{stupide}}_{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}\, }(\bwi{p+1}{\sigma {^{\vphantom{*}} }}{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}}).$ Recall that for any vector bundles $\mathcal{E}$ and $\mathcal{F}$ on $X$ and every nonnegative integer $i$, there is a canonical isomorphism between $\textrm{Ext}^{i}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} (\mathcal{E}, \mathcal{F})$ and $\textrm{H}^{i}{_{\vphantom{i}} }(X,\operatorname{\mathcal{H}\mathnormal{om}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }(\mathcal{E}, \mathcal{F})).$ \[DefUnTwisted\] For any nonnegative integer $p$ and any $\lambda $ in $\smash[b]{\textrm{Ext}^{1}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}(\bwi{p}{}{N\ee_{X/Y}},\bwi{p+1}{}{N\ee_{X/Y}}))} $, we denote by $\smash{\bwi{p}{\sigma ,\,\lambda }\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}} $ any twisted sheaf associated with the image of the class $\lambda $ in the Č${\vphantom{\Bigl)}}$ech cohomology group $\check{\textrm{H}}^{1}{_{\vphantom{i}} }(X,\mathcal{A}ut_{\oo_{{\ensuremath{\overline{X}}}}{^{\vphantom{*}} }}{^{\vphantom{*}} }(\bwi{p+1}{\sigma {^{\vphantom{*}} }}{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}}))\vphantom{\Bigl)}$. This definition makes sense because all such twisted sheaves are isomorphic. The sheaves $\bwi{p}{\sigma, \,\lambda }{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}}$ are sheaves of $\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}$ -modules which are locally isomorphic to $\bwi{p}{\sigma {^{\vphantom{*}} }}{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}}$. They fit into exact sequences $$\label{EqUnTwisted} \sutrgdpt{\bwi{p+1}{}{N\ee_{X/Y}}}{\bwi{p}{\sigma ,\,\lambda }{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}}}{\bwi{p}{}{N\ee_{X/Y}}}{.}$$ If we fix for each integer $p$ between $0$ and $r-1$ a class $\lambda _{p}$ in $\textrm{Ext}^{1}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}(\bwi{p}{}{N\ee_{X/Y}},\bwi{p+1}{}{N\ee_{X/Y}})$, the exact sequences (\[EqUnTwisted\]) allow to define twisted AK complexes ${\ensuremath{\mathcal{P}}}_{\sigma ,\,\lambda _{0},\dots,\,\lambda _{r-1}}{^{\vphantom{*}} }$ and ${\ensuremath{\mathcal{Q}}}_{\sigma ,\,\lambda _{0},\dots,\,\lambda _{r-1}}^{\vphantom{idiot}} $ which are well-defined modulo isomorphism. Then the results of Proposition \[PropUnAnalyticHKR\] also hold in the twisted case. \[PropUnTwisted\] Let $\sigma $ be a retraction of ${\ensuremath{\overline{j}}}$, $\chi $ be in $\emph{Der}_{\C_{X}{^{\vphantom{*}} }}{^{\vphantom{*}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}}, N\ee_{X/Y})$ and $\wh{\chi }$ be the associated section of the sheaf $\smash[b]{\operatorname{\mathcal{H}\mathnormal{om}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }(\Omega ^{1}_{X}, N\ee_{X/Y})} $. For any nonnegative integer $p$, let $\lambda_{p}$ denote the image of the Atiyah class of $\vphantom{\Bigl)}\bwi{p}{}{N\ee_{X/Y}}$ by $\wh{\chi} {\ensuremath{\wedge}}\emph{id}$ in $\emph{Ext}^{1^{\vphantom{idiot}}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}(\bwi{p}{}{N\ee_{X/Y}}, \bwi{p+1}{}{N\ee_{X/Y}}).$ Then $\bwi{p+1}{\sigma +\chi \vphantom{\lambda} }\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}$ and $\bwi{p+1}{\sigma ,\,\lambda_{p}}{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}}$ are isomorphic as sheaves of $\oo{^{\vphantom{*}} }{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}$ -modules. Let $\bigl( U_{\alpha }{^{\vphantom{*}} }\bigr)_{\alpha \in J}$ be an open covering of $X$ such that $N\ee_{X/Y}$ admits a holomorphic connection $\nabla_{\alpha}$ on each $U_{\alpha }{^{\vphantom{*}} }$. For every $\alpha^{\vphantom{\bigl)}}$ in $J$, $\bwi{p+1}{\sigma +\chi }{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}}$ is isomorphic on $U_{\alpha }{^{\vphantom{*}} }$ to $\bwi{p+1}{\sigma {^{\vphantom{*}} }}{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}}$ via $$\xymatrix{\varphi _{\alpha }{^{\vphantom{*}} }:({\ensuremath{\,\underline{i{_{\vphantom{i}} }}\,}},{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}})\ar@{|->}[r]&({\ensuremath{\,\underline{i{_{\vphantom{i}} }}\,}}-(\wh{\chi }\wedge\operatorname{id})\,(\Lambda ^{p}{_{\vphantom{i}} }\,\nabla{^{\vphantom{*}} }_{\!\alpha }\,({\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}})),{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}}).}$$ Thus, for any $\alpha $, $\beta $ in $J$, if $M_{\alpha \beta }{^{\vphantom{*}} }=\Lambda ^{p}{_{\vphantom{i}} }\,\nabla{^{\vphantom{*}} }_{\!\alpha\,|\,U_{\alpha \beta } }-\Lambda ^{p}{_{\vphantom{i}} }\,\nabla{^{\vphantom{*}} }_{\!\beta\,|\,U_{\alpha \beta } }$, $$\varphi _{\beta }{^{\vphantom{*}} }\circ \varphi_{\alpha }^{-1}({\ensuremath{\,\underline{i{_{\vphantom{i}} }}\,}},{{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}}})=({\ensuremath{\,\underline{i{_{\vphantom{i}} }}\,}}+(\wh{\chi }\wedge\operatorname{id})\,(M_{\alpha \beta }({\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}})),{\ensuremath{\,\underline{j{_{\vphantom{i}} }}\,}}).$$ Since $M_{\alpha \beta }{^{\vphantom{*}} }$ is a $1$-cocycle representing the Atiyah class of the holomorphic vector bundle $\bwi{p}{}{N\ee_{X/Y}}$ in $\textrm{Ext}^{1}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}(\bwi{p}{}{N\ee_{X/Y}},\Omega ^{1}_{X} {{\vphantom{\Bigl(A}}}\otimes \bwi{p}{}{N\ee_{X/Y}}),$ we get the result. We now recall Arinkin–Căldăraru’s construction of general analytic HKR isomorphisms and make the link with twisted AK complexes and twisted HKR isomorphisms. Recall that for any locally-free sheaf $\eee$ on $X$, if $\eee$ admits a locally-free extension ${\,\ensuremath{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{\textrm{X}{^{\vphantom{*}} }}}}}} \hspace*{-1.8ex}\eee}$ on ${\ensuremath{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{\textrm{X}{^{\vphantom{*}} }}}}}} \hspace*{-2.13ex}X}$, there is an exact sequence $$\label{EqDeuxTwisted} \sutrgd{N\ee_{X/Y} {\ensuremath{\otimes}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\eee}{{\,\ensuremath{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{\textrm{X}{^{\vphantom{*}} }}}}}} \hspace*{-1.8ex}\eee}}{\eee}$$ of sheaves of $\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}\,$-modules. Thus, for any nonnegative integer $n$, if $\kk_{-n}{^{\vphantom{*}} }$ is a locally free extension of  ${\smash{\bigotimes_{{}^{{^{\vphantom{*}} }}}^{{n}\vphantom{\big)}}}} N\ee_{X/Y}$ on ${\ensuremath{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{\textrm{X}{^{\vphantom{*}} }}}}}} \hspace*{-2.13ex}X}^{\vphantom {\bigl)}}$, we have an exact sequence $$\label{EqTroisTwisted} \xymatrix@C=20pt{0\ar[r]&{\smash{\bigotimes_{{}^{{^{\vphantom{*}} }}}^{{n+1}\vphantom{\big)}}}}N\ee_{X/Y}\ar[r]^-{i_{n}{^{\vphantom{*}} }}&\kk_{-n}{^{\vphantom{*}} }\ar[r]^-{\pi _{n}{^{\vphantom{*}} }}&{\smash{\bigotimes_{{}^{{^{\vphantom{*}} }}}^{{n}\vphantom{\big)}}}}N\ee_{X/Y}\ar[r]&0.}$$ \[defUnTwisted\] If $\smash[b]{\bigl( \kk_{-n}{^{\vphantom{*}} }\bigr)_{n\ge 0}{^{\vphantom{*}} }} $ are locally free sheaves on ${\ensuremath{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{\textrm{X}{^{\vphantom{*}} }}}}}} \hspace*{-2.13ex}X}$ extending $\smash[b]{\bigl( {\smash{\bigotimes_{{}^{{^{\vphantom{*}} }}}^{{n}\vphantom{\big)}}}}N\ee_{X/Y}\bigr)_{n\ge 0}{^{\vphantom{*}} }} $, the twisted Arinkin–Căldăraru complex $(\kk,\nu )$ is the complex $\smash[b]{\bop_{n\ge 0}{^{\vphantom{*}} }\kk_{-n}{^{\vphantom{*}} }} $ endowed with the differential $\nu$ given for each positive integer $n$ by $\nu _{-n}{^{\vphantom{*}} }=\smash[b]{\frac{1}{n!}} \, i_{n-1}{^{\vphantom{*}} }\circ\pi _{n}{^{\vphantom{*}} }$. Since the sequences (\[EqTroisTwisted\]) are exact, $(\kk,\nu )$ is a free resolution of ${\ensuremath{\oo_{X}{^{\vphantom{*}} }}}$ over $\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}$. The main result about $(\kk,\nu )$ is: \[PropDeuxTwisted\] Let $(X,Y)$ be a couple of connected complex manifolds such that $X$ is a closed complex submanifold of $Y$ of codimension $r$. Then 1. The complex $\smash[b]{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L},\, r}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}$ is formal in $D^{\emph{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$ if and only if $\smash{N\ee_{X/Y}}$ can be extended to a locally-free sheaf on ${\ensuremath{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{\textrm{X}{^{\vphantom{*}} }}}}}} \hspace*{-2.13ex}X}{\vphantom{\Bigl )}}$. 2. If $\bigl( \kk_{-n}{^{\vphantom{*}} }\bigr)_{n\ge 0}{^{\vphantom{*}} }$ is a sequence of locally-free sheaves on ${\ensuremath{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{\textrm{X}{^{\vphantom{*}} }}}}}} \hspace*{-2.13ex}X}$ extending $\bigl({\smash{\bigotimes_{{}^{{^{\vphantom{*}} }}}^{{n}\vphantom{\big)}}}}N\ee_{X/Y}\bigr)_{n\ge 0}{^{\vphantom{*}} }$, then the map $$\xymatrix@C=28pt{ \Gamma _{\kk}{^{\vphantom{*}} }\, : \, {\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}& {\ensuremath{\oo_{X}{^{\vphantom{*}} }}}{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\kk\ar[l]_-{\sim}\ar[r]& {\ensuremath{\oo_{X}{^{\vphantom{*}} }}}{\ensuremath{\otimes}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }{\ensuremath{\mathcal{K}}}\, \simeq\, \bop_{i\ge 0}{\smash{\bigotimes_{{}^{{^{\vphantom{*}} }}}^{{i}\vphantom{\big)}}}}N\ee_{X/Y}\, [i]\ar[r]^-{\smash[t]{{\ensuremath{\bigoplus\limits}}\limits_{i=0}^r \mathfrak{a}_i {^{\vphantom{*}} }}}& \bop_{i=0}^{r}\bwi{i}{}{N\ee_{X/Y}\,[i]}}$$ is an isomorphism in $D^{\emph{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$. ${^{\vphantom{*}} }$ 1. If the Atiyah sequence (\[EqUnAnalyticHKR\]) splits, then any retraction $\sigma $ of ${\ensuremath{\overline{j}}}$ allows to produce an extension of $N\ee_{X/Y}$ on ${\ensuremath{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{\textrm{X}{^{\vphantom{*}} }}}}}} \hspace*{-2.13ex}X}$, namely $\sigma \ee{_{\vphantom{i}} }N\ee_{X/Y}$. 2. This theorem appears in [@AC] only when $\smash[b]{\kk_{0}{^{\vphantom{*}} }=\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}} $ and $\smash[b]{\kk_{-n}{^{\vphantom{*}} }={\smash{\bigotimes_{{\oo_{{\ensuremath{\overline{X}}}{^{\vphantom{*}} }}}^{{^{\vphantom{*}} }}}^{{n}\vphantom{\big)}}}}\kk_{1}{^{\vphantom{*}} }} $ for $n\ge 1$ (which corresponds to the untwisted case), but the proof remains unchanged $^{\vphantom{\bigl)}}$ under these slightly more general hypotheses. Assume now that (\[EqUnAnalyticHKR\]) splits, and let $\sigma $ be a retraction of ${\ensuremath{\overline{j}}}$. Then, if $\smash[b]{\bigl( \kk_{-n}{^{\vphantom{*}} }\bigr)_{n\ge 0}{^{\vphantom{*}} }} $ is a sequence of locally-free sheaves on ${\ensuremath{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{\textrm{X}{^{\vphantom{*}} }}}}}} \hspace*{-2.13ex}X}^{\vphantom{\bigl)}}$ extending $\smash[b]{\bigl({\smash{\bigotimes_{{}^{{^{\vphantom{*}} }}}^{{n}\vphantom{\big)}}}}N\ee_{X/Y}\bigr)_{n\ge 0}{^{\vphantom{*}} }} $, each exact sequence (\[EqTroisTwisted\]) defines (via $\sigma $) an extension class $\mu _{n}{^{\vphantom{*}} }$ in ${\textrm{Ext}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}^{1{\vphantom{\bigl)}}}\bigl({\smash{\bigotimes_{{}^{{^{\vphantom{*}} }}}^{{n}\vphantom{\big)}}}} N\ee_{X/Y}, {\smash{\bigotimes_{{}^{{^{\vphantom{*}} }}}^{{n+1}\vphantom{\big)}}}} N\ee_{X/Y} \bigr)}$. \[PropTroisTwisted\] Let $(X, \sigma)$ be a quantized analytic cycle of codimension $r$ in a complex manifold $Y$, let $\bigl( \kk_{-n}{^{\vphantom{*}} }\bigr)_{n\ge 0}{^{\vphantom{*}} }$ be a sequence of locally-free sheaves on ${\ensuremath{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{\textrm{X}{^{\vphantom{*}} }}}}}} \hspace*{-2.13ex}X}$ extending $\smash[t]{\bigl({\smash{\bigotimes_{{}^{{^{\vphantom{*}} }}}^{{n}\vphantom{\big)}}}}N\ee_{X/Y}\bigr)_{n\ge 0}{^{\vphantom{*}} }},$ and let $(\mu_{n})_{n \geq 0 } {^{\vphantom{*}} }$ be the associated extension classes in ${\bigl(}^{\vphantom{stupide}}\emph{Ext}\,_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}^{1}\bigl({\smash{\bigotimes_{{}^{{^{\vphantom{*}} }}}^{{n}\vphantom{\big)}}}}N\ee_{X/Y},{\smash{\bigotimes_{{}^{{^{\vphantom{*}} }}}^{{n+1 \vphantom{)}}\vphantom{\big)}}}}N\ee_{X/Y} \bigr)\bigr)_{n \geq 0}$. For every integer $n$ between $0$ and $r-1$, let $\lambda _{n}$ be any element in $\emph{Ext}\,_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}^{1} \bigl(\bwi{n}{}{N\ee_{X/Y}},\bwi{n+1}{}{N\ee_{X/Y}} \bigr)$ such that $\mu_{n}{\vphantom{\Bigr(}}{^{\vphantom{*}} }$ and $\lambda _{n}{^{\vphantom{*}} }$ map to the same extension class in $\emph{Ext}\,_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}^{1 \vphantom{)}}\bigl({\smash{\bigotimes_{{}^{{^{\vphantom{*}} }}}^{{n}\vphantom{\big)}}}}N\ee_{X/Y},\bwi{n+1}{}{N\ee_{X/Y}} \bigr)$ via the antisymmetrization morphisms. Then $\Gamma _{{\ensuremath{\mathcal{K}}}{\vphantom{\bigl)}}}=\Gamma _{\sigma ,\,\lambda _{0}{^{\vphantom{*}} },\dots,\,\lambda _{r-1}{^{\vphantom{*}} }}{^{\vphantom{*}} }$. In particular, if $\kk_{0}{^{\vphantom{*}} }=\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}$ and $\kk_{-n}{^{\vphantom{*}} }={\smash{\bigotimes_{{\oo_{{\ensuremath{\overline{X}}}{^{\vphantom{*}} }}}^{{^{\vphantom{*}} }}}^{{n}\vphantom{\big)}}}}\sigma \ee{_{\vphantom{i}} }N\ee_{X/Y}$ for $n\ge 1$, then $\Gamma _{{\ensuremath{\mathcal{K}}}{\vphantom{\bigl)}}}=\Gamma_{\sigma} {^{\vphantom{*}} }$. We start with the case $\kk_{0}{^{\vphantom{*}} }=\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}$ and $\kk_{-n}{^{\vphantom{*}} }={\smash{\bigotimes_{{\oo_{{\ensuremath{\overline{X}}}{^{\vphantom{*}} }}}^{{^{\vphantom{*}} }}}^{{n}\vphantom{\big)}}}}\sigma \ee{_{\vphantom{i}} }N\ee_{X/Y}$ for $n\ge 1$, so that all the classes $\mu_n$ vanish. Then for every nonnegative integer $n$, there exists a natural morphism $\smash[b]{\apl{\zeta _{n}{^{\vphantom{*}} }\, }{\, \kk_{-n}{^{\vphantom{*}} }}{({\ensuremath{\mathcal{P}}}_{\sigma})_{-n}{^{\vphantom{*}} }}} $ given $^{\vphantom{\bigl)}}$by the composition $$\xymatrix@C=30pt{ \zeta _{n}{^{\vphantom{*}} }\, :\, {\smash{\bigotimes_{{\oo_{{\ensuremath{\overline{X\vphantom{\vert}}}}}}^{{^{\vphantom{*}} }}}^{{n}\vphantom{\big)}}}}\sigma \ee{_{\vphantom{i}} }{N\ee_{X/Y}}\ar[r]&\bwi{n}{\oo_{{\ensuremath{\overline{X\vphantom{\vert}}}}{^{\vphantom{*}} }}}{\sigma \ee{_{\vphantom{i}} }{N\ee_{X/Y}}}=\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}{\ensuremath{\otimes}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\bwi{n}{}{{N\ee_{X/Y}}}\ar[r]&\bwi{n+1}{\sigma }{{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}}}. }$$ where the last arrow is induced by the map $x \fl \sigma^{*}(1) {\ensuremath{\wedge}}x$. Then $\smash[b]{\apl{\zeta }{\kk}{{\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }}} $ is a morphism of complexes, which is a global version of the morphism $\zeta $ constructed in § \[SectionArikinCalda\], and we can argue exactly as in Proposition \[PropUnArinkinCalda\]. In the twisted case, our hypothesis implies that there is a morphism $\smash{\apl{\zeta }{\kk}{{\ensuremath{\mathcal{P}}}_{\sigma ,\,\lambda _{0}{^{\vphantom{*}} },\dots,\,\lambda _{r-1}{^{\vphantom{*}} }}{^{\vphantom{*}} }}}$ which is locally isomorphic to the previous one. Details are left to the reader. Comparison of HKR isomorphisms {#comparison} ------------------------------ Let $(X, \sigma )$ be a quantized analytic cycle in a complex manifold $Y$, and fix two sequences of cohomology classes $\smash[b]{\ub{\lambda {_{\vphantom{i}} }}=(\lambda _{p}{^{\vphantom{*}} })_{0\le p\le r-1}{^{\vphantom{*}} }} $ and $\smash[b]{\ub{\mu {_{\vphantom{i}} }}=(\mu _{p}{^{\vphantom{*}} })_{0\le p\le r-1}{^{\vphantom{*}} }} $ such that for each $p$, $\lambda_p$ and $\mu_p$ belong to $\textrm{Ext}\,^{1 \vphantom{\bigl)}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}(\bwi{p}{}{N\ee_{X/Y}},\bwi{p+1}{}{N\ee_{X/Y}})$. If ${\ensuremath{\mathcal{P}}}_{\sigma ,\,{\ensuremath{\underline{\lambda{_{\vphantom{i}} }}}}} {\vphantom{\Bigl)}} $ and ${\ensuremath{\mathcal{P}}}_{\sigma ,\,\ubm}{\vphantom{\bigl )}}{^{\vphantom{*}} }$ are the twisted AK complexes associated $^{\vphantom{\bigl)}}$with ${\ensuremath{\underline{\lambda{_{\vphantom{i}} }}}}$ and ${\ensuremath{\underline{\mu{_{\vphantom{i}} }}}}$, the isomorphism $\xymatrix{\varphi _{{\ensuremath{\underline{\mu{_{\vphantom{i}} }}}},\,{\ensuremath{\underline{\lambda{_{\vphantom{i}} }}}}}{^{\vphantom{*}} }:\,{\ensuremath{\mathcal{P}}}_{\sigma ,\,{\ensuremath{\underline{\lambda{_{\vphantom{i}} }}}}}{^{\vphantom{*}} }\ar[r]^-{\sim}&{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}&{\ensuremath{\mathcal{P}}}_{\sigma ,\,\ub{\mu{_{\vphantom{i}} }}}{^{\vphantom{*}} }\ar[l]_-{\sim}}$ in $D^{\,\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{Y}{^{\vphantom{*}} }}})$ induces an automorphism $\Delta_{\sigma} ({\ensuremath{\underline{\mu{_{\vphantom{i}} }}}},\, \ubl)$ of $\bop\nolimits_{i=0}^{r}\bwi{i}{}{N\ee_{X/Y}}\,[i]$ in $D^{\,\textrm{b}}{_{\vphantom{i}} }(\oo_{X}{^{\vphantom{*}} })$, as shown in the following diagram: $$\xymatrix@C=30pt@C=30pt{ \bop_{j=0}^{r}\bwi{j}{}{{N\ee_{X/Y}}[j]}\ar@<1ex>@{}[r]_-{\ds=}\ar[d]^-{\sim}_-{\Delta_{\sigma} ({\ensuremath{\underline{\mu{_{\vphantom{i}} }}}},\, \ubl)}&{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}{\ensuremath{\otimes}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }{\ensuremath{\mathcal{P}}}_{\sigma ,\,{\ensuremath{\underline{\lambda{_{\vphantom{i}} }}}}}{^{\vphantom{*}} }&{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }{\ensuremath{\mathcal{P}}}_{\sigma ,\,{\ensuremath{\underline{\lambda{_{\vphantom{i}} }}}}}{^{\vphantom{*}} }\ar[l]_-{\sim}\ar[d]_-{\sim}^-{\operatorname{id}\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\varphi _{{\ensuremath{\underline{\mu{_{\vphantom{i}} }}}},\,{{\ensuremath{\underline{\lambda{_{\vphantom{i}} }}}}}}}\\ \bop_{i=0}^{r}\bwi{i}{}{{N\ee_{X/Y}}[i]}\ar@<1ex>@{}[r]_-{\ds=}&{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}{\ensuremath{\otimes}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }{\ensuremath{\mathcal{P}}}_{\sigma ,\,\ub{\mu {_{\vphantom{i}} }}}{^{\vphantom{*}} }&{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }{\ensuremath{\mathcal{P}}}_{\sigma ,\,{\ensuremath{\underline{\mu{_{\vphantom{i}} }}}}}{^{\vphantom{*}} }\ar[l]_-{\sim} }$$ If we put $\Delta_{\sigma}(\ubl)=\Delta_{\sigma}(0, \ubl)$, then $\Delta_{\sigma}(\ubm, \ubl)=\Delta_{\sigma}(\ubm)^{-1}{_{\vphantom{i}} }\circ \Delta_{\sigma}(\ubl).$ Recall that $$\textrm{Hom}_{D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})}{^{\vphantom{*}} }\Bigl[ \bop\limits_{j=0}^{r}\bwi{j}{}{{N\ee_{X/Y}}[j]},\,\bop\limits_{i=0}^{r}\bwi{i}{}{{N\ee_{X/Y}}[i]}\Bigr]=\bop_{0\le j \le i\le r}{^{\vphantom{*}} }\textrm{Ext}^{i-j}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}\bigl( \bwi{j}{}{{N\ee_{X/Y}}},\bwi{i}{}{{N\ee_{X/Y}}}\bigr).$$ Therefore, an endomorphism of $\bop_{i=0}^{r}\bwi{i}{}{{N\ee_{X/Y}}[i]}$ in the derived category $D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$ can be represented by a lower triangular $(r+1)\tim(r+1)$ matrix $\bigl( M_{i,\,j}{^{\vphantom{*}} }\bigr)_{0\le i,\,j\le r}{^{\vphantom{*}} }$ such that for $i\ge j$, the entry $M_{i,\,j}{^{\vphantom{*}} }$ is a cohomology $^{\vphantom{\bigl)}}$class in $ \textrm{Ext}^{i-j}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}(\bwi{j}{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{{N\ee_{X/Y}}}, \bwi{i}{}{{N\ee_{X/Y}}}).$ The computation of the coefficients $\Delta _{\sigma }{^{\vphantom{*}} }(\ubm,\ubl)_{i, \, j}$ seems to be a delicate problem. We solve it only in particular cases. Let us introduce some preliminary material. For any integers $i$ and $j$ such that $0\le j\le i\le r$ and any cohomology class $v$ in $\textrm{H}^{i-j}{_{\vphantom{i}} }\bigl(X,\bwi{i-j}{}{{N\ee_{X/Y}}}\bigr)$ considered as an element of $\textrm{Hom}_{D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})}{^{\vphantom{*}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}}, \,\bwi{i-j}{}{{N\ee_{X/Y}}}[i-j])$, we define a morphism $\mathfrak{l}_{i,\,j}{^{\vphantom{*}} }(v)$ from $\bwi{j}{}{{N\ee_{X/Y}}}[j^{{\vphantom{\bigl)}}}]$ to $\bwi{i}{}{{N\ee_{X/Y}}}[i]$ in $D^{\textrm{b}^{\vphantom{A}}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$ by the compostion $$\xymatrix@C=40pt{\mathfrak{l}_{i,\,j}{^{\vphantom{*}} }(v) \colon \bwi{j}{}{{N\ee_{X/Y}}}[j] \ar[r]^-{v \, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} \,\textrm{id}}&\bwi{i-j}{}{{N\ee_{X/Y}}}[i-j] \, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }\bwi{j}{}{{N\ee_{X/Y}}}[j] \ar[r]^-{\wedge}&\bwi{i}{}{{N\ee_{X/Y}}}[i]. }$$ In this way, we obtain a morphism $$\apl{\, \mathfrak{l}_{i,\,j}{^{\vphantom{*}} }\, }{\textrm{H}^{i-j}{_{\vphantom{i}} }\bigl(X,\bwi{i-j}{}{{N\ee_{X/Y}}}\bigr)}{\textrm{Ext}^{i-j}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}(\bwi{j}{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{{N\ee_{X/Y}}}, \bwi{i}{}{{N\ee_{X/Y}}}).}$$ If $\,*\,$ denotes the Yoneda product, for any integers $i,j_{\vphantom{\bigl)}},k$ such that $0 \leq k\le j\le i\le r$ and any cohomology classes $v$ and $w$ in $\textrm{H}^{i-j}{_{\vphantom{i}} }\bigl( X,\bwi{i-j}{}{{N\ee_{X/Y}}}\bigr)$ and $\textrm{H}^{j-k}{_{\vphantom{i}} }\bigl( X,\bwi{j-k}{}{{N\ee_{X/Y}}}\bigr)_{\vphantom{stupide}}$ respectively, we have $$\mathfrak{l}_{i,\,j}{^{\vphantom{*}} }(v)*\mathfrak{l}_{j,\,k}{^{\vphantom{*}} }(w)=(-1)^{(i-j)(j-k)}{_{\vphantom{i}} }\,\mathfrak{l}_{i,\,k}{^{\vphantom{*}} }(v \cup w).$$ We introduce some notation concerning Čech cohomology. Let $\mathfrak{U}=(U_{\alpha }{^{\vphantom{*}} })_{\alpha \in J}{^{\vphantom{*}} }$ be a locally finite open covering of $X$. For any bounded complex of sheaves $(\ff, d)$ on $X$, we denote by $(\mathscr{C}(\ff),\delta, d )$ the associated Čech bicomplex, which is quasi-isomorphic to $(\ff, d)$. Besides, we denote by ${\ensuremath{\wedge}}$ the wedge product on the exterior algebra $\bwi{}{}N\ee_{X/Y}$ at the level of Čech cochains. It is given by the well-known formula: $$\apl{\, {\ensuremath{\wedge}}\, }{\,\mathscr{C}^{p}{_{\vphantom{i}} }\bigl( \bwi{k}{}{{N\ee_{X/Y}}}\bigr)\times\mathscr{C}^{q}{_{\vphantom{i}} }\bigl( \bwi{l}{}{{N\ee_{X/Y}}}\bigr)}{\mathscr{C}^{p+q}{_{\vphantom{i}} }\bigl( \bwi{k+l}{}{{N\ee_{X/Y}}}\bigr)}$$ $$(\eta {\ensuremath{\wedge}}\eta')_{{\alpha'}_{0}{^{\vphantom{*}} },\dots,\,\alpha _{p+q}{^{\vphantom{*}} }}{^{\vphantom{*}} }=u_{\alpha_{0}{^{\vphantom{*}} },\dots,\,\alpha _{p}{^{\vphantom{*}} }}{^{\vphantom{*}} }{\ensuremath{\wedge}}{\eta'}_{\alpha _{p}{^{\vphantom{*}} },\dots,\,\alpha _{p+q}{^{\vphantom{*}} }}{^{\vphantom{*}} }.$$ Let $v$ be a cohomology class in $\textrm{H}^{i-j}{_{\vphantom{i}} }\bigl( X,\bwi{i-j}{}{{N\ee_{X/Y}}}\bigr)$. Since $X$ is paracompact, we can choose the covering $\mathfrak{U}$ sufficiently fine in order that $v$ be representable by a Čech cocycle $(\mathfrak{v}_{\alpha }{^{\vphantom{*}} })_{\alpha\, \in \,J^{i-j+1}}{^{\vphantom{*}} }.$ Define $\smash[b]{\apl{\, \mathfrak{q}_{i,\,j}\, {^{\vphantom{*}} }(\mathfrak{v}) \,}{\mathscr{C}\bigl( \bwi{j}{}{{N\ee_{X/Y}}[j]}\bigr)}{{\mathscr{C}\bigl( \bwi{i}{}{{N\ee_{X/Y}}[i]}\bigr)}}} $ by the formula $\mathfrak{q}_{i,\,j}{^{\vphantom{*}} }(\mathfrak{v})(\eta )=(-1)^{(i-j)\deg(\eta)}{_{\vphantom{i}} }\, \mathfrak{v}{\ensuremath{\wedge}}\eta $, where $\deg(\eta )$ denotes the degree of the Čech cochain $\eta $. Then $\mathfrak{q}_{i,\,j}{^{\vphantom{*}} }(\mathfrak{v})$ is a morphism of complexes and the diagram $$\xymatrix@C=40pt@R=30pt{ \bwi{j}{}{{N\ee_{X/Y}}[j]}\quad\ar[r]^-{\mathfrak{l}_{i,\,j}{^{\vphantom{*}} }(v)}\ar[d]_-{\sim}&\bwi{i}{}{{N\ee_{X/Y}}[i]}\quad\ar[d]^-{\sim}\\ \mathscr{C}\bigl( \bwi{j}{}{{N\ee_{X/Y}}[j]}\bigr)\ar[r]^-{\mathfrak{q}_{i,\,j}{^{\vphantom{*}} }(\mathfrak{v})}&\mathscr{C}\bigl( \bwi{i}{}{{N\ee_{X/Y}}[i]}\bigr) }$$ commutes in $D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$. \[ThUnCompHKR\] Let $(X, \sigma)$ be a quantized analytic cycle of codimension $r$ in a complex manifold $Y$. Fix two sequences $(c_{n}{^{\vphantom{*}} })_{\,0\le n\le r}{^{\vphantom{*}} }$ and $(d_{n}{^{\vphantom{*}} })_{\,0\le n\le r}{^{\vphantom{*}} }$ of cohomology classes in $\smash[t]{\emph{H}^{1_{\vphantom{)}}}{_{\vphantom{i}} }\bigl(X, N\ee_{X/Y}\bigr)}^{{^{\vphantom{*}} }}$. For any integer $p$ between $0$ and $r-1$, let $\lambda _{p}{^{\vphantom{*}} }$ and $\mu _{p}{^{\vphantom{*}} }$ be defined in $\emph{Ext}^{1}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}\bigl(\bwi{p}{}{{N\ee_{X/Y}}},\bwi{p+1}{}{{N\ee_{X/Y}}}\bigr)$ by $\lambda _{p}{^{\vphantom{*}} }=\mathfrak{l}_{p+1,\,p}{^{\vphantom{*}} }(c_{p}{^{\vphantom{*}} })$ and $\mu _{p}{^{\vphantom{*}} }=\mathfrak{l}_{p+1,\,p}{^{\vphantom{*}} }(d_{p}{^{\vphantom{*}} })$. Then for any integers $i$ and $j$ such that $0\le j\le i\le r$, we can write $\Delta _{\sigma }{^{\vphantom{*}} }(\ubm,\ubl)_{i,\,j}{^{\vphantom{*}} }=\mathfrak{l}_{i,\,j} (\zeta _{i,\,j}){^{\vphantom{*}} }$, where the classes $\zeta _{i,\,j}{^{\vphantom{*}} }$ are defined inductively by $$\begin{aligned} \zeta _{i,\,i}{^{\vphantom{*}} }&=1 &&\textrm{for} \quad 0 \leq i \leq r, \\ \zeta_{i+1,\,0}&=(-1)^i (c_0-d_i)\cup \zeta_{i,\, 0} &&\textrm{for} \quad 0 \leq i \leq r-1,\\[-1ex] \zeta _{i+1,\,j}{^{\vphantom{*}} }&=\dfrac{1}{i+1} \bigl[ j\,\zeta _{i,\,j-1}{^{\vphantom{*}} }+ (-1)^{i-j}{_{\vphantom{i}} }(c_{j}{^{\vphantom{*}} }-d_{i}{^{\vphantom{*}} })\cup\zeta _{i,\,j}\bigr] &&\textrm{for} \quad 1 \leq j \leq i \leq r-1.\end{aligned}$$ We choose a locally finite covering $\mathfrak{U}=(U_{\alpha }{^{\vphantom{*}} })_{\alpha \,\in\,J}{^{\vphantom{*}} }$ of $X$ such that for each integer $n$ between $0$ and $r-1$, the classes $c_{n}{^{\vphantom{*}} }$ and $d_{n}{^{\vphantom{*}} }$ are representable by Čech cocycles $(\mathfrak{c}_{n,\,\alpha ,\,\beta }{^{\vphantom{*}} })_{\alpha ,\,\beta \,\in\,J}{^{\vphantom{*}} }$ and $(\mathfrak{d}_{n,\,\alpha ,\,\beta }{^{\vphantom{*}} })_{\alpha ,\,\beta \,\in\,J}{^{\vphantom{*}} }$ in $\mathscr{C}^{1}{_{\vphantom{i}} }(X,{N\ee_{X/Y}})$. For any $\alpha $ in $J$, we fix isomorphisms $$\xymatrix@C=40pt@R=10pt{ \varphi _{n,\,\alpha }{^{\vphantom{*}} }\,:\,\bwi{n+1}{\sigma ,\,\lambda _{n}{^{\vphantom{*}} }}{\oo_{{\ensuremath{\overline{X\vphantom{X'}}}}\vert U_{\alpha }{^{\vphantom{*}} }}}\ar[r]^-{\sim}&\bwi{n+1}{\sigma }{\oo_{{\ensuremath{\overline{X\vphantom{X'}}}}\vert U_{\alpha }{^{\vphantom{*}} }}} \quad \textrm{and} \quad \psi _{n,\,\alpha }{^{\vphantom{*}} }\,:\,\bwi{n+1}{\sigma ,\,\mu _{n}{^{\vphantom{*}} }}{\oo_{{\ensuremath{\overline{X\vphantom{X'}}}}\vert U_{\alpha }{^{\vphantom{*}} }}}\ar[r]^-{\sim}&\bwi{n+1}{\sigma }{\oo_{{\ensuremath{\overline{X\vphantom{X'}}}}\vert U_{\alpha }{^{\vphantom{*}} }}} }$$ such that for any $\alpha $, $\beta $ in $J$, $\varphi _{n,\,\beta }{^{\vphantom{*}} }\circ \varphi _{n,\,\alpha }^{-1}$ and $\psi _{n,\,\beta }{^{\vphantom{*}} }\circ \psi _{n,\,\alpha }^{-1}$ are given via the isomorphism (\[EqDeuxDgAlg\]) by $$\varphi _{n,\,\beta }{^{\vphantom{*}} }\circ \varphi _{n,\,\alpha }^{-1}\,(\ubi,\ubj)=(\ubi+\mathfrak{c}_{n,\,\alpha ,\,\beta }{^{\vphantom{*}} }{\ensuremath{\wedge}}\ubj,\ubj)\quad\textrm{and}\quad \psi _{n,\,\beta }{^{\vphantom{*}} }\circ \psi _{n,\,\alpha }^{-1}\,(\ubi,\ubj)=(\ubi+\mathfrak{d}_{n,\,\alpha ,\,\beta }{^{\vphantom{*}} }{\ensuremath{\wedge}}\ubj,\ubj).$$ For any integers $i$, $j$ such that $0\le j\le i\le r$, we define inductively cocycles $(\eta _{i,\,j,\,\uba}{^{\vphantom{*}} })_{\uba\,\in\,J^{i-j+1}}{^{\vphantom{*}} }$ by the formulae $$\eta _{i,\,i}{^{\vphantom{*}} }=1 \quad \textrm{for} \quad 0\le i\le r, \qquad \eta _{i+1,\,0}{^{\vphantom{*}} }=(-1)^{i}(\mathfrak{c}_{0}{^{\vphantom{*}} }-\mathfrak{d}_{i}{^{\vphantom{*}} }){\ensuremath{\wedge}}\eta _{i,\,0}{^{\vphantom{*}} }\quad \textrm{for} \quad 0\le i\le r-1,$$ $$\eta _{i+1,\,j}{^{\vphantom{*}} }=\dfrac{1}{i+1}\,\bigl[ j\,\eta _{i,\,j-1}{^{\vphantom{*}} }+(-1)^{i-j}\,(\mathfrak{c}_{j}{^{\vphantom{*}} }-\mathfrak{d}_{i}{^{\vphantom{*}} }){\ensuremath{\wedge}}\eta _{i,\,j}{^{\vphantom{*}} }\bigr]\quad \textrm{for}\quad 1\le j\le i\le r-1.$$ Let $\ti{d}$ be the total differential of the Čech bicomplex $\mathscr{C}({\ensuremath{\mathcal{P}}}_{\sigma ,\,\ubm}{^{\vphantom{*}} })$. For any integer $k$, $$\mathscr{C}({\ensuremath{\mathcal{P}}}_{\sigma ,\,\ubm}{^{\vphantom{*}} })_{k}{^{\vphantom{*}} }=\bop_{l=\max(0,\,k)}^{r+k}\mathscr{C}^{\,l}{_{\vphantom{i}} }\bigl( \bwi{l+1-k}{\sigma ,\,\mu _{l-k}{^{\vphantom{*}} }}{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}}\bigr) \quad \textrm{if}\ k\ge -n \quad \textrm{and} \quad \mathscr{C}({\ensuremath{\mathcal{P}}}_{\sigma ,\,\ubm}{^{\vphantom{*}} })_{k}{^{\vphantom{*}} }=0 \quad \textrm{if}\ k<-n.$$ Besides, $\ti{d}=\delta +(-1)^{l}{_{\vphantom{i}} }\,\wh{d}_{l-k}{^{\vphantom{*}} }$ on each $\smash[b]{\mathscr{C}^{\,l}{_{\vphantom{i}} }\bigl( \bwi{l+1-k}{\sigma ,\,\mu _{l-k}{^{\vphantom{*}} }}{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}}\bigr)} $. For any nonnegative integers $n$ and $k$ such that $n+l\le k$ and any $\alpha _{0}{^{\vphantom{*}} },\dots,\,\alpha _{k}^{\vphantom{\bigl)}}$ in $J$, we define two morphisms of sheaves $$\xymatrix@C=25pt{ S_{-n,\,l,\,\uba}{^{\vphantom{*}} }\,:\,\bwi{n+1}{\sigma }{\oo_{{\ensuremath{\overline{X\vphantom{X'}}}}\vert U_{{\ensuremath{\underline{\alpha {_{\vphantom{i}} }}}}}{^{\vphantom{*}} }}}\ar[r]&\bwi{n+l+1}{\sigma }{\oo_{{\ensuremath{\overline{X\vphantom{X'}}}}\vert U_{{\ensuremath{\underline{\alpha {_{\vphantom{i}} }}}}}{^{\vphantom{*}} }}} \quad \textrm{and} \quad T_{-n,\,\lambda ,\,\uba}\,:\,\bwi{n+1}{\sigma ,\,\lambda _{n}{^{\vphantom{*}} }}{\oo_{{\ensuremath{\overline{X\vphantom{X'}}}}\vert U_{{\ensuremath{\underline{\alpha {_{\vphantom{i}} }}}}}{^{\vphantom{*}} }}}\ar[r]&\bwi{n+k+1}{\sigma ,\,\mu _{n+l{^{\vphantom{*}} }}}{\oo_{{\ensuremath{\overline{X\vphantom{X'}}}}\vert U_{{\ensuremath{\underline{\alpha {_{\vphantom{i}} }}}}}{^{\vphantom{*}} }}} }$$ by the formulae $$S_{-n,\,l,\,\uba}{^{\vphantom{*}} }\,(\ubi,\ubj)=\bigl( (-1)^{l}{_{\vphantom{i}} }\,\eta _{n+l,\,n,\,\uba}{^{\vphantom{*}} }{\ensuremath{\wedge}}\ubi,\eta _{n+l,\,n,\,\uba}{^{\vphantom{*}} }{\ensuremath{\wedge}}\ubj\bigr)\quad\textrm{and}\quad T_{-n,\,l,\,\uba}{^{\vphantom{*}} }=\psi ^{-1}_{n+l,\,\alpha _{0}{^{\vphantom{*}} }}\circ S_{-n,\,l,\,\uba}{^{\vphantom{*}} }\circ\varphi _{n,\,\alpha _{0}{^{\vphantom{*}} }}{^{\vphantom{*}} }.$$ By (\[EqQuatreDgAlg\]), $S_{-n,\,l,\,\uba}{^{\vphantom{*}} }$ and $T_{-n,\,l,\,\uba}{^{\vphantom{*}} }$ are $\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}$-linear. Since the covering $\mathfrak{U}$ is locally finite, the morphisms $\bigl( T_{-n,\,l,\,\uba}{^{\vphantom{*}} }\bigr)_{0\le l\le n-r,\ \alpha\,\in\, J^{l+1}{_{\vphantom{i}} }}{^{\vphantom{*}} }$ define a morphism $\apl{T_{-n}{^{\vphantom{*}} }}{\bwi{n+1}{\sigma ,\,\lambda _{n}{^{\vphantom{*}} }}{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}}}{\mathscr{C}\bigl( {\ensuremath{\mathcal{P}}}_{\sigma ,\,\ubm}{^{\vphantom{*}} }\bigr)_{-n}{^{\vphantom{*}} }.}$ A tedious but straightforward computation shows that the $\bigl( T_{-n}{^{\vphantom{*}} }\bigr)_{0\le n\le r}{^{\vphantom{*}} }$ define an element of $\textrm{Hom}_{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}}{^{\vphantom{*}} }\bigl( {\ensuremath{\mathcal{P}}}_{\sigma ,\ubl}{^{\vphantom{*}} },\,\mathscr{C}({\ensuremath{\mathcal{P}}}_{\sigma ,\,\ubm}{^{\vphantom{*}} })\bigr),$ so that we get the following commutative diagram in $D^{\textrm{b}}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}}){_{\vphantom{i}} }$: $$\xymatrix@C=40pt@R=30pt{ {\ensuremath{\mathcal{P}}}_{\sigma ,\,\ubl}{^{\vphantom{*}} }\ar[d]_-{\sim}\ar[r]_-{T}^-{\sim}&\mathscr{C}\bigl( {\ensuremath{\mathcal{P}}}_{\sigma ,\,\ubm}{^{\vphantom{*}} }\bigr)\ar[d]&{\ensuremath{\mathcal{P}}}_{\sigma ,\,{\ensuremath{\underline{\mu{_{\vphantom{i}} }}}}}{^{\vphantom{*}} }\ar[d]\ar[l]_-{\sim}\\ {\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\ar[r]^-{\sim} &\mathscr{C}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}}) &{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\ar[l]_-{\sim} }$$ This proves that the quasi-isomorphism ${\apl{\varphi _{\ubm,\,\ubl}{^{\vphantom{*}} }}{{\ensuremath{\mathcal{P}}}_{\sigma ,\,\ubl}{^{\vphantom{*}} }}{{\ensuremath{\mathcal{P}}}_{\sigma ,\,\ubm}{^{\vphantom{*}} }}} $ is obtained by composing the two isomorphisms of the first line. Now we have another commutative diagram, namely $$\xymatrix@C=40pt@R=20pt{ {\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\,{\ensuremath{\mathcal{P}}}_{\sigma ,\,\ubl}\ar[r]^-{\operatorname{id}{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L},\, r}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\,T}\ar[d]_-{\sim}&{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L},\, r}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\,\mathscr{C}\bigl( {\ensuremath{\mathcal{P}}}_{\sigma ,\,\ubm}\bigr)\ar[d]&{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L},\, r}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\,{\ensuremath{\mathcal{P}}}_{\sigma ,\,\ubm}\ar[l]_-{\sim}\ar[d]_-{\sim}\\ {\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\,{\ensuremath{\otimes}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\,{\ensuremath{\mathcal{P}}}_{\sigma ,\,\ubl}\ar[r]^-{\operatorname{id}{\ensuremath{\otimes}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }T}\ar@<-1.4ex>@{}^{\bigl|\!\bigl|}[d]&{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\,{\ensuremath{\otimes}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\,\mathscr{C}\bigl( {\ensuremath{\mathcal{P}}}_{\sigma ,\,\ubm}\bigr)\ar@<-1.4ex>@{}^{\bigl|\!\bigl|}[d]&{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\,{\ensuremath{\otimes}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\,{\ensuremath{\mathcal{P}}}_{\sigma ,\,\ubm}\ar[l]_-{\sim}\ar@<-1.4ex>@{}^{\bigl|\!\bigl|}[d]\\ \bop_{j=0}^{r}\bw{j}{{N\ee_{X/Y}}[j]}\ar[r]^-{\sim}&\mathscr{C}\bigl( \bop_{i=0}^{r}\bw{i}{{N\ee_{X/Y}}[i]}\bigr)&\bop_{i=0}^{r}\bw{i}{{N\ee_{X/Y}}[i]}\ar[l]_-{\sim} }$$ Thus $\Delta _{\sigma }{^{\vphantom{*}} }(\ubm,\ubl)$ is obtained by composing the two isomorphisms of the last line. The first one is explicitly given by $$\xymatrix@C=17pt{\bw{j}{{N\ee_{X/Y}}}\ar[r]&\mathscr{C}^{i-j}{_{\vphantom{i}} }(\bw{i}{{N\ee_{X/Y}}}),}\quad\xymatrix@C=17pt{ \ubi\ar@{|->}[r]&\eta _{\,i,\,j}{^{\vphantom{*}} }{\ensuremath{\wedge}}\ubi\!. }$$ Hence $\Delta _{\sigma }{^{\vphantom{*}} }(\ubm,\ubl)_{i, \, j}$ is equal to $\mathfrak{l}_{i,\,j}{^{\vphantom{*}} }\, (\zeta _{i,\,j}{^{\vphantom{*}} })$ where $\zeta _{i,\,j}{^{\vphantom{*}} }$ is the cohomology class of $\eta _{\,i,\,j}{^{\vphantom{*}} }$. This finishes the proof. \[interpretation\] The twist of the AK complex by classes in $\textrm{H}^1\bigl(X, N^*_{X/Y}\bigr)$ admits the following geometric interpretation: the existence of a retraction $\sigma$ of ${\ensuremath{\overline{j}}}$ implies that the natural sequence $$\label{Pic} \xymatrix@=20pt{0 \ar[r]&\textrm{H}^{1}(X, N\ee_{X/Y}) \ar[r]&\textrm{Pic}(\,{\ensuremath{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{\textrm{X}{^{\vphantom{*}} }}}}}} \hspace*{-2.13ex}X}) \ar[r]&\textrm{Pic}(X) \ar[r]& 0}$$ is exact. This allows to identify $\textrm{H}^1\bigl(X, N^*_{X/Y}\bigr)$ with isomorphism classes of holomorphic line bundles on ${\ensuremath{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{\textrm{X}{^{\vphantom{*}} }}}}}} \hspace*{-2.13ex}X}$ whose restriction on $X$ is trivial. Then for any integer $p$ between $0$ and $r-1$ and any class $\mu_p$ in $\textrm{H}^1\bigl(X, N^*_{X/Y}\bigr)$, if $\mathcal{L}_p$ is a line bundle on $\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}$ associated with $\mu_p$ and if $\lambda_p={_{\vphantom{i}} }\,\mathfrak{l}_{p+1, \,p}(\mu_p)$, it follows from (\[EqQuatreDgAlg\]) that $^{\vphantom{\bigl)}}\bwi{p+1}{\sigma ,\,\lambda_{p}}{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}}\simeq \bwi{p+1}{\sigma }{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}} \otimes_{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}}{^{\vphantom{*}} }\mathcal{L}\ee_p$. We compute $\Delta _{\sigma }{^{\vphantom{*}} }(\ubm,\ubl)$ in another particular case: \[ModuleLunaireUn\] Let $(X, \sigma)$ be a quantized analytic cycle of codimension $r$ in a complex manifold $Y$. For any integer $p$ between $0$ and $r-1$, let $\lambda _{p}{^{\vphantom{*}} }$ and $\mu _{p}{^{\vphantom{*}} }$ be extension classes in $\emph{Ext}^{1}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}(\bw{p} N\ee_{X/Y}, \bw{p+1} N\ee_{X/Y})$ such that $\lambda _{p}{^{\vphantom{*}} }=\mu _{p}{^{\vphantom{*}} }$ for $p\neq r-1$. Then $$\begin{cases} \Delta _{\sigma }{^{\vphantom{*}} }(\ubm,\ubl)_{i,\,i}{^{\vphantom{*}} }=1&\emph{for}\ 0\le i \le r\\ \Delta _{\sigma }{^{\vphantom{*}} }(\ubm,\ubl)_{r,\,r-1}{^{\vphantom{*}} }=\dfrac{1}{r} (\lambda _{r-1}{^{\vphantom{*}} }-\mu _{r-1}{^{\vphantom{*}} })\\ \emph{All other coefficients}\ \Delta _{\sigma }{^{\vphantom{*}} }(\ubm,\ubl)_{i,\,j}{^{\vphantom{*}} }\ \emph{vanish}. \end{cases}$$ We argue exactly as in the proof of Theorem \[ThUnCompHKR\]. For any integer $n$ between $0$ and $r-1$, we represent the extension classes $\lambda _{n}{^{\vphantom{*}} }$ and $\mu _{n}{^{\vphantom{*}} }$ by Čech cocycles $\bigl( \mathfrak{c}_{n,\,\alpha ,\,\beta}{^{\vphantom{*}} }\bigr)_{(\alpha ,\,\beta )\,\in\,J}{^{\vphantom{*}} }$ and $\bigl( \mathfrak{d}_{n,\,\alpha ,\,\beta}{^{\vphantom{*}} }\bigr)_{(\alpha ,\,\beta )\,\in\,J}{^{\vphantom{*}} }$ in $\smash[t]{\mathscr{C}^{1}{_{\vphantom{i}} }\bigl( X,\operatorname{\mathcal{H}\mathnormal{om}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }(\bw{p}N\ee_{X/Y},\bw{p+1}N\ee_{X/Y})\bigr)} $ so that $$\varphi _{n,\,\beta }{^{\vphantom{*}} }\circ\varphi _{n,\,\alpha }^{-1}(\ubi,\ubj)=(\ubi+\mathfrak{c}_{n,\,\alpha ,\,\beta }{^{\vphantom{*}} }(\ubj),\ubj)\qquad\textrm{and}\qquad \psi _{n,\,\beta }{^{\vphantom{*}} }\circ\psi _{n,\,\alpha }^{-1}(\ubi,\ubj)=(\ubi+\mathfrak{d}_{n,\,\alpha ,\,\beta }{^{\vphantom{*}} }(\ubj),\ubj).$$ Then we define the morphisms $S_{-n,\,l,\,\uba}{^{\vphantom{*}} }$ as follows: –$S_{-n,\,0,\,\alpha _{0{^{\vphantom{*}} }}}{^{\vphantom{*}} }(\ubi, \ubj)=(\ubi,\ubj)\qquad \textrm{for}\ 0\le n\le r$ –$S_{-(r-1),\,1,\,\alpha _{0}{^{\vphantom{*}} },\,\alpha _{1}{^{\vphantom{*}} }}{^{\vphantom{*}} }(\ubi,\ubj)=\dfrac{1}{r} (\mathfrak{c}_{r-1,\,\alpha _{0}{^{\vphantom{*}} },\,\alpha _{1}{^{\vphantom{*}} }}{^{\vphantom{*}} }(\ubj)-\mathfrak{d}_{r-1,\,\alpha _{0}{^{\vphantom{*}} },\,\alpha _{1}{^{\vphantom{*}} }}{^{\vphantom{*}} }(\ubj))$ –$\textrm{All other}\ S_{-n,\,\lambda ,\,\uba}{^{\vphantom{*}} }\ \textrm{vanish}$. The morphisms $T_{-n,\,l,\,\uba}{^{\vphantom{*}} }$ define a morphism of complexes from ${\ensuremath{\mathcal{P}}}_{\sigma ,\,\ubl}{^{\vphantom{*}} }$ to $\mathcal{C}({\ensuremath{\mathcal{P}}}_{\sigma ,\,\ubm}{^{\vphantom{*}} })$, and we conclude as in the proof of Theorem \[ThUnCompHKR\]. As a corollary, we obtain: \[ModuleLunaireDeux\] Let $(X, \sigma)$ be a quantized analytic cycle of codimension two in a complex manifold $Y$, $\chi $ be an element of $\emph{Der}{^{\vphantom{*}} }_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}},N\ee_{X/Y})$, $\wh{\chi }$ be the associated element in $\emph{Hom}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }(\Omega ^{1}_{X},N\ee_{X/Y})$ and $\emph{at}(N\ee_{X/Y})$ be the Atiyah class of $N\ee_{X/Y}$ in $\emph{Ext}^{1}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}(N\ee_{X/Y},\Omega ^{1}_{X}{\ensuremath{\otimes}}N\ee_{X/Y})$. If $\theta (\chi )$ denotes the class $\dfrac{1}{2} (\wh{\chi }{\ensuremath{\wedge}}\operatorname{id})(\emph{at}(N\ee_{X/Y}))$ in $\emph{Ext}^{1}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}(N\ee_{X/Y},\bw{2}N\ee_{X/Y})$, the automorphism $\Gamma {^{\vphantom{*}} }_{\sigma +\chi }\circ\Gamma ^{-1}_{\sigma }$ of ${\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\oplus N\ee_{X/Y}[1]\oplus\bw{2}N\ee_{X/Y}[2^{\vphantom{AA}}]$ is given by the $3 \times 3$ matrix $$\begin{pmatrix} 1&0&0\\0&1&0\\0&\theta (\chi )&1 \end{pmatrix}$$ In particular, $\Gamma _{\sigma + \chi}{^{\vphantom{*}} }\circ\Gamma _{\sigma }^{-1}=\operatorname{id}$ in $\emph{Aut}_{D^{\emph{b}}{_{\vphantom{i}} }(\C_{X}{^{\vphantom{*}} })} ({\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\oplus N\ee_{X/Y}[1]\oplus\bw{2}N\ee_{X/Y}[2]) $. The first part of the theorem follows directly from Theorem \[ModuleLunaireUn\] and Proposition \[PropUnTwisted\]. The second part follows from the first one. Indeed, $2\theta (\chi )$ is obtained as the composition $$\xymatrix@C=45pt{N\ee_{X/Y}\ar[r]^-{\textrm{at}(N\ee_{X/Y})}&\Omega ^{1}_{X}{\ensuremath{\otimes}}N\ee_{X/Y}[1]\ar[r]^-{\chi {\ensuremath{\otimes}}\,\operatorname{id}}&N\ee_{X/Y}{\ensuremath{\otimes}}N\ee_{X/Y}[1]\ar[r]^-{\wedge}&\bw{2}N\ee_{X/Y}[1].}$$ The class $\textrm{at}(N\ee_{X/Y})$ is obtained as the extension class of the exact sequence of $1$-jets of $N\ee_{X/Y}$: $$\xymatrix{ 0\ar[r]&\Omega ^{1}_{X}{\ensuremath{\otimes}}N\ee_{X/Y}\ar[r]&J^{1}{_{\vphantom{i}} }(N\ee_{X/Y})\ar[r]&N\ee_{X/Y}\ar[r]&0. }$$ This exact sequence splits over $\C_{X}{^{\vphantom{*}} }$, so that $\textrm{at}(N\ee_{X/Y})=0$ in $\textrm{Ext}\,^{1}_{\C_{X}{^{\vphantom{*}} }}\bigl( N\ee_{X/Y},\Omega ^{1}_{X}{\ensuremath{\otimes}}N\ee_{X/Y}\bigr).$ Thus $\theta (\chi )=0$ in $\textrm{Ext}^{1}_{\C_{X}{^{\vphantom{*}} }}\bigl( N\ee_{X/Y},\bw{2}N\ee_{X/Y}\bigr).$ We end this section by giving a conjectural expression for the matrix $\Delta _{\sigma }{^{\vphantom{*}} }(\ubm,\ubl)$. For this purpose we introduce the derived analog of the translation operator defined in § \[cup\]. For any nonnegative integers $m$, $p$ and $k$ such that $k \geq p$ and any $\phi$ in $\textrm{Ext}^{k-p}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} \bigl(\bw{p}N\ee_{X/Y}, \bw{k}N\ee_{X/Y})$ considered as an element of $\textrm{Hom}^{\vphantom{\bigl)}}_{D^{\textrm{b}}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})} (\bwi{p}{}{{N\ee_{X/Y}}}[p], \,\bwi{k}{}{{N\ee_{X/Y}}}[k])$, we define a morphism $\mathfrak{t}_{k,\,p}^{\, m}(\phi)$ in $\textrm{Hom}^{\vphantom{\bigl)}}_{D^{\textrm{b}}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})}(\bwi{p+m}{}{{N\ee_{X/Y}}}[p+m], \,\bwi{k+m}{}{{N\ee_{X/Y}}}[k+m])$ by the composition $$\xymatrix@C=12pt{{\ensuremath{\scriptstyle}}\bwi{p+m}{}{{N\ee_{X/Y}}}[p+m]\ar[rr]^-{\smash[t]{W_{p,\,m}{^{\vphantom{*}} }} }&&\st\bwi{p}{}{{N\ee_{X/Y}}}[p+m]\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\bwi{m}{}N\ee_{X/Y} \ar[rrr]^-{\smash[t]{\phi[m]\, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}\operatorname{id}} }&&&\st\bwi{k}{}{{ \!\!N\ee_{X/Y}}}[k+m]\, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\bwi{m}{}\!\!N\ee_{X/Y}\ar[r]^-{{\ensuremath{\wedge}}}&\st\bwi{k+m}{}{{N\ee_{X/Y}}}[k+m]\! }$$ The derived version of Lemma \[LemUnBisSecAlgExt\] tells us that for any class $v$ in $\textrm{H}^{k-p_{\vphantom{)}}}{_{\vphantom{i}} }\bigl(X, N\ee_{X/Y}\bigr)$, $$\mathfrak{t}_{k,\,p} ^{\,m}[\mathfrak{l}_{k, \,p} {^{\vphantom{*}} }(v)]=\mathfrak{l}_{k+m,\,p+m}(v).$$ This justifies the following conjecture: \[conj\] Let $(X, \sigma)$ be a quantized analytic cycle in a complex manifold $Y$. For any integer $p$ between $0$ and $r-1$, let $\lambda _{p}{^{\vphantom{*}} }$ and $\mu _{p}{^{\vphantom{*}} }$ be extension classes in ${}^{\vphantom{\bigl)}}\emph{Ext}^{1}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}\bigl(\bwi{p}{}{{N\ee_{X/Y}}},\bwi{p+1}{}{{N\ee_{X/Y}}}\bigr)$. For any integers $i$ and $j$ such that $0\le j\le i\le r$, we put $\Delta_{i,\,j}=\Delta _{\sigma }{^{\vphantom{*}} }(\ubm,\ubl)_{i,\,j}{^{\vphantom{*}} }$. If $\,*\,$ denotes the Yoneda product, then the coefficients ${}^{\vphantom{iiiiI \bigl(}}\Delta _{i,\,j}{^{\vphantom{*}} }$ are determined inductively by the following relations: $$\begin{array}{ll} -\ \Delta _{i,\,i}{^{\vphantom{*}} }=\operatorname{id}&\emph{for}\ 0 \leq i \leq r\\[2ex] -\ \Delta_{i+1,\,0}=(-1)^i \,\,( \mathfrak{l}_{i+1,\,i}(\lambda_0)-\mu_i) * \Delta_{i,\, 0}&\emph {for}\ 0 \leq i \leq r-1\\[1ex] -\ \Delta _{i+1,\,j}{^{\vphantom{*}} }=\dfrac{1}{i+1} \bigl[ j\, \mathfrak{t}_{i, \,j-1}^{\,1}(\Delta _{i,\,j-1}{^{\vphantom{*}} }) \,+(-1)^{i-j}{_{\vphantom{i}} }\, (\mathfrak{t}_{j+1,\, j}^{\,i} \, \lambda_{j}{^{\vphantom{*}} }-\mu_{i}{^{\vphantom{*}} }) * \Delta _{i,\,j}\bigr]&\emph{for}\ 1 \leq j \leq i \leq r-1 \end{array}$$ The cycle class of a quantized analytic cycle {#cycle} ============================================= Construction and basic properties of the cycle class {#cycle1} ---------------------------------------------------- For any complex manifolds $X$ and $Y$ such that $X$ is a closed complex submanifold of $Y$ of codimension $r$, ${\mathcal{RH}om_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}},{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}})$ is canonically isomorphic to $j\ei\omega _{X/Y}{^{\vphantom{*}} }$ in $D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{Y}{^{\vphantom{*}} }}})$. This implies that ${\mathcal{RH}om^{\,r\vphantom{p}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}},{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}})$ is concentrated in degree $r$, so that there exists an isomorphism $$\label{BaProCyClEqUn} {\mathcal{RH}om^{\,r\vphantom{p}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}},{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}})\simeq \omega _{X/Y}{^{\vphantom{*}} }$$ in $D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$ such that the composition $$\xymatrix{j\ei\omega _{X/Y}{^{\vphantom{*}} }\simeq j{\ensuremath{_{*}}}{\mathcal{RH}om^{\,r\vphantom{p}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}},{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}) \ar[r]^-{\sim} & {\mathcal{RH}om_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}},{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}) \simeq j\ei\omega _{X/Y}{^{\vphantom{*}} }}$$ is the multiplication by $(-1)^{\frac{r(r+1)}{2}}$ (the choice of this sign will become clear in the proof of Theorem \[BaProCyClThUn\] below). For any integer $i$ between $0$ and $r$, we have an isomorphism $$\label{BaProCyClEqDeux} \textrm{Hom}_{D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})}{^{\vphantom{*}} }(\omega _{X/Y}{^{\vphantom{*}} },\bw{i}{N_{X/Y}{^{\vphantom{*}} }[-i]})\simeq \textrm{H}^{r-i}{_{\vphantom{i}} }(X,\bw{r-i}{N\ee_{X/Y}})$$ obtained as follows: for any cohomology class $\alpha $ in $\textrm{H}^{r-i}{_{\vphantom{i}} }(X,\bw{r-i}{N\ee_{X/Y}})$ considered as a morphism from ${\ensuremath{\oo_{X}{^{\vphantom{*}} }}}$ to $\bw{r-i}{N\ee_{X/Y}[r-i]}$ in $D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$, we associate the morphism $$\xymatrix@C=40pt{ \omega _{X/Y}{^{\vphantom{*}} }\ar[r]^-{\alpha\, {\ensuremath{\otimes}}\,\operatorname{id}}&\bw{r-i}{N\ee_{X/Y}[r-i]}{\ensuremath{\otimes}}\omega _{X/Y}{^{\vphantom{*}} }\ar[r]^-{\smash{D^{\ell}{_{\vphantom{i}} }}}&\bw{i}{N_{X/Y}{^{\vphantom{*}} }[-i].} }$$ \[BaProCyClDefUn\] Let $(X, \sigma)$ be a quantized analytic cycle of codimension $r$ in a complex manifold $Y$. Using the isomorphism (\[BaProCyClEqUn\]), the morphism $$\xymatrix@C=25pt{ \omega _{X/Y}{^{\vphantom{*}} }\simeq {\mathcal{RH}om^{\,r\vphantom{p}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}},{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}})\ar[r]&{\mathcal{RH}om^{\,r\vphantom{p}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}},{\ensuremath{\oo_{X}{^{\vphantom{*}} }}})\ar[r]^-{\sim}_-{\wh{\Gamma }_{\sigma }{^{\vphantom{*}} }}&\bop_{i=0}^{r}\bw{i}{N_{X/Y}{^{\vphantom{*}} }[-i]} }$$ defines via (\[BaProCyClEqDeux\]) a class in $\bop_{i=0}^{r}\textrm{H}^{i}{_{\vphantom{i}} }(X,\bw{i}N\ee_{X/Y})$, which is the *quantized cycle class* $q_{\sigma} {^{\vphantom{*}} }(X)$ of $(X,\sigma )$. We now compute the quantized cycle class in specific situations. \[BaProCyClThUn\] Let $(X,\sigma )$ be a quantized analytic cycle of codimension $r$ in $Y$ and assume that there exists a couple $(E,s)$ such that 1. $E$ is a holomorphic vector bundle of rank $r$ on $Y$. 2. $s$ is a holomorphic section of $E$ vanishing exactly on $X$ and $s$ is transverse to the zero section. 3. The locally-free ${\ensuremath{\oo_{X}{^{\vphantom{*}} }}}$-modules $E{\ensuremath{\otimes}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}$ and $\sigma \ee{_{\vphantom{i}} }N_{X/Y}{^{\vphantom{*}} }$ are isomorphic.\[CondTrois\] Then $q_{\sigma} {^{\vphantom{*}} }(X)=1$. Let $s\ee{_{\vphantom{i}} }$ be the dual of $s$; it is a cosection of $E\ee{_{\vphantom{i}} }$. Since $s$ is transverse to the zero section, the Koszul complex $(\mathcal{L},\delta )=L(E\ee{_{\vphantom{i}} },s\ee{_{\vphantom{i}} })$ is a free resolution of ${\ensuremath{\oo_{X}{^{\vphantom{*}} }}}$ over ${\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}$. Using (\[EqTroisPrelim\]), the isomorphism ${\mathcal{RH}om_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}},{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}})\simeq j\ei\omega _{X/Y}{^{\vphantom{*}} }$ in $D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{Y}{^{\vphantom{*}} }}})$ is given by the chain $$\xymatrix@C=25pt{ {\mathcal{RH}om_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}},{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}})&\LL\ee{_{\vphantom{i}} }\simeq(\LL,-\delta ){\ensuremath{\otimes}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\det E[-r]\ar[l]_-{\sim} \ar[r]^-{\sim}&{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}{\ensuremath{\otimes}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\det E[-r]\simeq j\ei\omega _{X/Y}{^{\vphantom{*}} }. }$$ If $\tau$ denotes the canonical cosection of $\sigma {\ensuremath{^{\, *}}}N\ee_{X/Y}$ and $h$ is the isomorphism between $E{\ensuremath{\otimes}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}$ and $\sigma \ee{_{\vphantom{i}} }N_{X/Y}{^{\vphantom{*}} }$ given in condition (3), then $s\ee{_{\vphantom{i}} }\! \circ {}^t h$ is a cosection of $\sigma{\ensuremath{^{\, *}}}N\ee_{X/Y}$ vanishing on $X$. Hence there exists an endomorphism $F$ of the conormal bundle $N\ee_{X/Y}$ such that $s\ee{_{\vphantom{i}} }\! \circ {}^t h$ is obtained as the composition $$\xymatrix{\sigma{\ensuremath{^{\, *}}}N\ee_{X/Y} \ar[r] & N\ee_{X/Y} \ar[r]^-{F} & N\ee_{X/Y} \ar[r] & {\ensuremath{\mathcal{O}}}{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}}$$ This means that $s\ee{_{\vphantom{i}} }\! \circ {}^t h=\tau \circ \sigma{\ensuremath{^{\, *}}}(F)$. Using again that $s$ is transverse to the zero section, we get that $F$ is an isomorphism. Therefore, if we replace $h$ by $\sigma{\ensuremath{^{\, *}}}[\,{}^t F^{-1}] \circ h$, we have $s\ee{_{\vphantom{i}} }\! \circ {}^t h= \tau$. We can now construct a global quasi-isomorphism $\smash{\aplexp{\gamma }{\LL}{\mathcal{P}_{\sigma }{^{\vphantom{*}} }}{\sim}}$ (which is the global analog of the quasi-isomorphism $\gamma$ constructed in the proof of Proposition \[PropUnHKR\]) as follows: for $0\le p\le r$, $\gamma _{-p}{^{\vphantom{*}} }$ is given by the composition $$\xymatrix@C=25pt{ \bwi{p}{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{E\ee{_{\vphantom{i}} }}\ar[r]&\bwi{p}{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}}{(E\ee{_{\vphantom{i}} }{\ensuremath{\otimes}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}})}\simeq\bwi{p}{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}}{(\sigma \ee{_{\vphantom{i}} }N\ee_{X/Y})}\simeq\bw{p}{N\ee_{X/Y}{\ensuremath{\otimes}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}}\ar[r]&\bwi{p+1}{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}} }$$ where the last arrow is $\xymatrix@C=17pt{\ub{x}{\ensuremath{\otimes}}(i,a)\ar@{|->}[r]&(i\wedge\ub{x},a\ub{x}).}$ Let $\smash[t]{\apl{\Delta }{\omega _{X/Y}{^{\vphantom{*}} }}{\bop_{i=0}^{r}\bw{i}{N_{X/Y}{^{\vphantom{*}} }}[-i]}}$ be the morphism in $D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$ defining the quantized cycle class $q_{\sigma} {^{\vphantom{*}} }(X)$ (c.f. Definition \[BaProCyClDefUn\]) and let $\psi $ be the automorphism of $\smash[b]{\bop_{i=0}^{r} j\ei\bw{i}{N_{X/Y}{^{\vphantom{*}} }[-i]}}$ in $D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{Y}{^{\vphantom{*}} }}})$ such that $(-1)^{\frac{r(r+1)}{2}} \psi$ is given by the composition $$\xymatrix@C=25pt{ \bop_{i=0}^{r} j\ei\bw{i}{N_{X/Y}{^{\vphantom{*}} }[-i]}&j\ei{\mathcal{RH}om^{\,\ell\vphantom{p}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}},{\ensuremath{\oo_{X}{^{\vphantom{*}} }}})\simeq j\ei{\mathcal{RH}om^{\,r\vphantom{p}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}},{\ensuremath{\oo_{X}{^{\vphantom{*}} }}})\ar[l]^-{\Gamma \ee_{\sigma }}_-{\sim}\ar[r]^-{\sim}_-{\wh{\Gamma }_{\sigma }{^{\vphantom{*}} }}&\bop_{i=0}^{r}j\ei\bw{i}{N_{X/Y}{^{\vphantom{*}} }[-i].} }$$ Then $j\ei\Delta $ can be expressed as the chain $$\xymatrix@C=25pt{ j\ei\omega _{X/Y}{^{\vphantom{*}} }&(\LL,-\delta ){\ensuremath{\otimes}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\det E[-r]\ar[l]_-{\sim}\ar[r]&\bop_{i=0}^{r}j{\ensuremath{_{*}}}(\bw{i}{N\ee_{X/Y}[i]}{\ensuremath{\otimes}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\omega _{X/Y}{^{\vphantom{*}} })\hspace*{100pt} }$$ $$\xymatrix@C=25pt{ \hspace*{200pt}\ar[r]^-{D^{\ell}{_{\vphantom{i}} }}&\bop_{i=0}^{r}j\ei\bw{i}{N_{X/Y}{^{\vphantom{*}} }[-i]}\ar[r]^-{\sim}_-{\psi }&\bop_{i=0}^{r}j\ei\bw{i}{N_{X/Y}{^{\vphantom{*}} }[-i].} }$$ Define two morphisms $\ti{\Delta }$ and $\ti{\psi }$ in $D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$ and $D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{Y}{^{\vphantom{*}} }}})$ by the diagrams $$\xymatrix@C=70pt@R=30pt{ {\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\ar[r]^-{\ti{\Delta }}&\smash{\bop_{i=0}^{r}\bw{i}{N\ee_{X/Y}[i]}}\ar[d]^-{D^{\ell}{_{\vphantom{i}} }}_-{\sim}\\ \omega _{X/Y}{^{\vphantom{*}} }{\ensuremath{\otimes}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\det E\ee{_{\vphantom{i}} }[r]\ar[u]^-{\sim}\ar[r]^-{\Delta \,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\operatorname{id}}&\bop_{i=0}^{r}\bw{i}{N_{X/Y}{^{\vphantom{*}} }[-i]}{\ensuremath{\otimes}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\det E\ee{_{\vphantom{i}} }[r] }$$ and $$\xymatrix@C=50pt@R=30pt{ \bop_{i=0}^{r}j\ei\bw{i}{N\ee_{X/Y}[i]}\ar[r]^-{\ti{\psi }}_-{\sim}\ar[d]^-{D^{\ell}{_{\vphantom{i}} }}_-{\sim}&\smash{\bop_{i=0}^{r}j\ei\bw{i}{N\ee_{X/Y}[i]}}\ar[d]^-{\sim}\\ \bop_{i=0}^{r}j\ei\bw{i}{N_{X/Y}{^{\vphantom{*}} }[-i]}{\ensuremath{\otimes}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\det E\ee{_{\vphantom{i}} }[r]\ar[r]^-{\psi \,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\operatorname{id}}&\bop_{i=0}^{r}j\ei\bw{i}{N_{X/Y}{^{\vphantom{*}} }[-i]}{\ensuremath{\otimes}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\det E\ee{_{\vphantom{i}} }[r] }$$ Then: –For $0\le i\le r$, the $i$-th component of $\ti{\Delta }$ in $\textrm{Hom}_{D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})}{^{\vphantom{*}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}},\bw{i}{N\ee_{X/Y}}[i])$ is $q_{\sigma} {^{\vphantom{*}} }(X)_{i}{^{\vphantom{*}} }\cdot$ –The morphism $j\ei\ti{\Delta }$ is the composition of the chain of morphisms $$\xymatrix@C=25pt{ {\ensuremath{\oo_{X}{^{\vphantom{*}} }}}&(\LL,-\delta )\ar[l]_-{\sim}\ar[r]&\bop_{i=0}^{r}j\ei\bw{i}{N\ee_{X/Y}[i]}\ar[r]_-{\sim}^-{\ti{\psi }}&\bop_{i=0}^{r}j\ei\bw{i}N\ee_{X/Y}[i]. }$$ Using the quasi-isomorphism $\smash{\aplexp{\gamma }{\LL}{{\ensuremath{\mathcal{P}}}_{\sigma {^{\vphantom{*}} }},}{\sim}}$ we get that $j\ei\ti{\Delta }$ is equal to the composition $$\xymatrix@C=25pt {{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}&({\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} },-\wh{d}\,\,)\ar[l]_-{\sim}\ar[r]&\bop_{i=0}^{r}j\ei\bw{i}{N\ee_{X/Y}[i]}\ar[r]^-{\ti{\psi} }_-{\sim}&\bop ^{r}j\ei\bw{i}{N\ee_{X/Y}[i].}}$$ We now make the two following observations: –As a complex of $\C_{Y}{^{\vphantom{*}} }$-modules, ${\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }$ splits as the direct sum of ${\ensuremath{\oo_{X}{^{\vphantom{*}} }}}$ and a null-homotopic complex. –The global version of Proposition \[PropDeuxBisHKR\] shows that $\ti{\psi }$, as a morphism in the derived category $D^{\textrm{b}}{_{\vphantom{i}} }(\C_{Y}{^{\vphantom{*}} })$, acts by $(-1)^{\frac{(r-i)(r-i-1)}{2}+r(r-i)+\frac{r(r+1)}{2}}{_{\vphantom{i}} }$ on each factor $j\ei\bw{i}{N\ee_{X/Y}[i]}$. Thus, as a morphism in $D^{\textrm{b}}{_{\vphantom{i}} }(\C_{Y}{^{\vphantom{*}} })$, $j{\ensuremath{_{*}}}\ti{\Delta }$ is simply the injection ${\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\fl\bop_{i=0}^{r}j{\ensuremath{_{*}}}\bw{i}{N\ee_{X/Y}[i]}$. Hence we get $q_{\sigma} {^{\vphantom{*}} }(X)_{0}{^{\vphantom{*}} }=1$ and $q_{\sigma} {^{\vphantom{*}} }(X)_{i}{^{\vphantom{*}} }=0$ for $1\le i\le r$. As an immediate consequence, we get: \[BaProCyClCorUn\] For any quantized cycle $(X,\sigma )$, $q_{\sigma} {^{\vphantom{*}} }(X)_{0}{^{\vphantom{*}} }=1$. The class $q_{\sigma} {^{\vphantom{*}} }(X)_{0}{^{\vphantom{*}} }$ is a holomorphic function on $X$, so that it can be computed locally. Hence we can assume that $X$ is open in $\C^{n}{_{\vphantom{i}} }$ and $Y=X{\ensuremath{\times}}U$ where $U$ is open in $\C^{r}{_{\vphantom{i}} }$. If $E$ is the trivial rank $r$ bundle on $Y$ and $s$ is the section $(z_{n+1}{^{\vphantom{*}} },\dots, z_{n+r}{^{\vphantom{*}} })$, then Theorem \[BaProCyClThUn\] yields $ q_{\,\operatorname{pr}_{1}} (X)=1$. Since $N_{X/Y}{^{\vphantom{*}} }$ is trivial, $q_{\sigma} {^{\vphantom{*}} }(X)$ is independent of $\sigma $ by Proposition \[PropDeuxAnalyticHKR\] (1). This gives the result. We now turn to the case of the diagonal injection. For any complex manifold $X$, we identify the conormal bundle of $\Delta_X{^{\vphantom{*}} }$ in $X \times X{^{\vphantom{*}} }$ with $\Omega_{X}^{1}$ as follows: for any germ on holomorphic function $f$ on $X$, the local section $\textrm{pr}\ee_1(f)-\textrm{pr}\ee_2(f)$ of the conormal sheaf of the diagonal corresponds to the local section $df$ of the cotangent bundle of $X$. \[BaProCyClThDeux\] For any complex manifold $X$, $q_{\,\emph{pr}_1}(\Delta_X {^{\vphantom{*}} })$ is the Todd class of $X$. If ${\ensuremath{\mathcal{Q}}}$ is the dual Atiyah-Kashiwara complex associated with $(\Delta _{X}{^{\vphantom{*}} },\operatorname{pr}_{1}{^{\vphantom{*}} })$, the main result of [@G] is that for a specific isomorphism between ${\ensuremath{\oo_{X}{^{\vphantom{*}} }}}$ and ${\mathcal{RH}om^{\,r\vphantom{p}}_{\oo_{X \times X}}}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}}, \omega _{X}{^{\vphantom{*}} }\boxtimes {\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$, the composition $$\xymatrix@C=25pt{ {{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\simeq{\mathcal{RH}om^{\,r\vphantom{p}}_{\oo_{X \times X}{^{\vphantom{*}} }}}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}}, \omega _{X}{^{\vphantom{*}} }\boxtimes {\ensuremath{\oo_{X}{^{\vphantom{*}} }}})\ar[r]}&{{\mathcal{RH}om^{\,r\vphantom{p}}_{\oo_{X \times X}{^{\vphantom{*}} }}}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}},\omega _{X}{^{\vphantom{*}} })\simeq\operatorname{\mathcal{H}\mathnormal{om}}_{\oo_{X \times X}{^{\vphantom{*}} }}{^{\vphantom{*}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}},{\ensuremath{\mathcal{Q}}})\simeq\bop_{i=0}^{r}\Omega _{X}^{i}[i]}}$$ is the Todd class of $X$. It follows that $q_{\,\textrm{pr}_1}(\Delta_X {^{\vphantom{*}} })=\varphi \, \textrm{td}(X)$ where $\varphi$ is a nowhere zero holomorphic function on $X$. By Corollary \[BaProCyClCorUn\], $\varphi=1$. To conclude this section, we compute the quantized cycle class in the case of divisors. For any cohomology class $\delta $ in $H^{1}{_{\vphantom{i}} }(X,N\ee_{X/Y})$, we denote by $\LL_{\delta }{^{\vphantom{*}} }$ the associated line bundle on ${\ensuremath{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{\textrm{X}{^{\vphantom{*}} }}}}}} \hspace*{-2.13ex}X}$. Then, for any $\LL$ in $\textrm{Pic}({\ensuremath{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{\textrm{X}{^{\vphantom{*}} }}}}}} \hspace*{-2.13ex}X})$ such that $j\ee{_{\vphantom{i}} }\LL\simeq{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}$, there exists a unique cohomology class $\delta $ in $H^{1}{_{\vphantom{i}} }(X,N\ee_{X/Y})$ such that $\LL$ is isomorphic to $\LL_{\delta }{^{\vphantom{*}} }$ (c.f. Remark \[interpretation\]). \[divisor\] Let $(X, \sigma)$ be a quantized analytic cycle of codimension one in a complex manifold $Y$, and let $\delta $ be the cohomology class in $H^{1}{_{\vphantom{i}} }(X,N\ee_{X/Y})$ such that $\smash[b]{{\ensuremath{\overline{j}}}\ee{_{\vphantom{i}} }{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}(X){\ensuremath{\otimes}}_{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}}{^{\vphantom{*}} }\sigma \ee{_{\vphantom{i}} }N\ee_{X/Y}}$ is isomorphic to $\LL_{\delta }{^{\vphantom{*}} }$. Then $\smash{q_{\sigma} {^{\vphantom{*}} }(X)=1+\delta.}$ Let $\nn$ (resp. $\nn'$) denote the holomorphic line bundle $\sigma \ee{_{\vphantom{i}} }N_{X/Y}{^{\vphantom{*}} }$ (resp. ${\ensuremath{\overline{j}}}\ee{_{\vphantom{i}} }{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}(X)$) on ${\ensuremath{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{\textrm{X}{^{\vphantom{*}} }}}}}} \hspace*{-2.13ex}X}$. Then we have two natural exact sequences $$\xymatrix@C=17pt{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}\ar[r]^-{i}&\nn\ar[r]^-{\pi }&N{^{\vphantom{*}} }_{X/Y}\ar[r]&0}\qquad\textrm{and}\qquad \xymatrix@C=17pt{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}\ar[r]^-{i'}&\nn'\ar[r]^-{\pi' }&N{^{\vphantom{*}} }_{X/Y}\ar[r]&0}$$ If $\apl{\Delta }{N{^{\vphantom{*}} }_{X/Y}[-1]}{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\oplus N{^{\vphantom{*}} }_{X/Y}[-1]}$ is the morphism in $D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$ defining $q_{\sigma} {^{\vphantom{*}} }(X)$, then $\Delta $ is obtained as the composition of quasi-isomorphisms $$\xymatrix@C=35pt@R=30pt{ &{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\ar[r]^-{i}\ar[d]_-{-i'}&\nn\ar[d]^-{(-i'{\ensuremath{\otimes}}\,\operatorname{id},-\pi )}&{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\ar[l]_-{i}\ar[d]\ar@<1.499ex>@{}[d]_(.45){\textrm{\LARGE o}}\\ N_{X/Y}{^{\vphantom{*}} }&\nn'\ar[l]_-{-\pi '}\ar[r]_-{(\operatorname{id}{\ensuremath{\otimes}}\,i,\,0)}&[\,\nn'{\ensuremath{\otimes}}\nn\,]\oplus N_{X/Y}{^{\vphantom{*}} }\ar[d]^-{(-\pi '{\ensuremath{\otimes}}\pi ,\,0)}&N_{X/Y}{^{\vphantom{*}} }\ar[l]^-{(0,\,\operatorname{id})}\\ &&N_{X/Y}^{\,{\ensuremath{\otimes}}\,2} }$$ Let $\Delta '={\ensuremath{\overline{j}}}\ei{^{\vphantom{*}} }\Delta\, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}}{^{\vphantom{*}} }\,\nn'[1]={\ensuremath{\overline{j}}}\ei{^{\vphantom{*}} }\bigl( \Delta\, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }N\ee_{X/Y}[1]\bigr).$ If $s$, $s'$, $t$, $t'$ are the maps occurring in the two natural sequences $$\xymatrix@C=17pt{0\ar[r]&N\ee_{X/Y}\ar[r]^(0.5){s}&\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}\ar[r]^-{t}&{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\ar[r]&0}\qquad\textrm{and}\qquad \xymatrix@C=17pt{0\ar[r]&N\ee_{X/Y}\ar[r]^(0.5){s'}&\LL_{-\delta }{^{\vphantom{*}} }\ar[r]^-{t'}&{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\ar[r]&0}$$ then $\Delta '$ is the composition $$\xymatrix@C=35pt@R=30pt{&N\ee_{X/Y}\ar[r]^-{s'}\ar[d]_-{-s}&\LL_{-\delta }{^{\vphantom{*}} }\ar[d]_-{(-t',-t')}&N\ee_{X/Y}\ar[l]_(0.5){s'}\ar[d]\ar@<1.48ex>@{}[d]_(.47){\textrm{\LARGE o}}\\ {\ensuremath{\oo_{X}{^{\vphantom{*}} }}}&\oo{\ensuremath{{}_{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{ {\textrm{\Large a}}}}}}\hspace*{-1.6ex}X}}}\ar[l]^-{-t}\ar[r]_-{(t,\,0)}&{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\oplus{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}&{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\ar[l]^-{(0,\,\operatorname{id})} }$$ Thus, as a morphism in $D^{\textrm{b}}{_{\vphantom{i}} }(\C_{Y}{^{\vphantom{*}} })$, $\Delta '$ is the composition $$\xymatrix@C=35pt@R=30pt{&\LL_{-\delta }{^{\vphantom{*}} }\ar[d]_-{(-t',-t')}&N\ee_{X/Y}\ar[l]_(0.5){s'}\ar[d]\ar@<1.48ex>@{}[d]_(.47){\textrm{\LARGE o}}\\ {\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\ar[r]_(0.5){(-\operatorname{id},\,0)}&{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\oplus{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}&{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\ar[l]^-{(0,\,\operatorname{id})} }$$ Therefore, via the isomorphism $\textrm{Hom}_{D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})}{^{\vphantom{*}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}},{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\oplus N\ee_{X/Y}[1])\simeq H^{0}{_{\vphantom{i}} }(X,{\ensuremath{\oo_{X}{^{\vphantom{*}} }}})\oplus H^{1}{_{\vphantom{i}} }(X,N\ee_{X/Y})$, we have $\Delta '=1+\delta $. This yields the result. The six operations for a closed immersion {#six} ----------------------------------------- We denote by $j\ee{_{\vphantom{i}} }$ (resp. $j\pe{_{\vphantom{i}} }$) the derived pullback (resp. exceptional inverse image) induced by the closed immersion $j$. More explicitly, $$\begin{aligned} \apl{j\ee{_{\vphantom{i}} }}{D^{-}{_{\vphantom{i}} }({\ensuremath{\oo_{Y}{^{\vphantom{*}} }}})}{D^{-}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})}&&j\ee{_{\vphantom{i}} }\ff&={\ensuremath{\oo_{X}{^{\vphantom{*}} }}}{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L},\, r}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\ff\\ \apl{j\pe{_{\vphantom{i}} }}{D^{+}{_{\vphantom{i}} }({\ensuremath{\oo_{Y}{^{\vphantom{*}} }}})}{D^{+}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})}&&j\pe{_{\vphantom{i}} }\ff&={\mathcal{RH}om^{\,r\vphantom{p}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}},\ff)\end{aligned}$$ These two functors satisfy the adjunction formulae $$\begin{cases} \textrm{Hom}_{D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{Y}{^{\vphantom{*}} }}})}{^{\vphantom{*}} }(\ff,j\ei{^{\vphantom{*}} }\g)\simeq \textrm{Hom}_{D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})}(j\ee{_{\vphantom{i}} }\ff,\g)\\ \textrm{Hom}_{D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{Y}{^{\vphantom{*}} }}})}{^{\vphantom{*}} }(j\ei{^{\vphantom{*}} }\g,\ff)\simeq \textrm{Hom}_{D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})}(\g,j\pe{_{\vphantom{i}} }\ff) \end{cases}$$ as well as the projection formula $$j{\ensuremath{_{*}}}{^{\vphantom{*}} }(j\ee{_{\vphantom{i}} }{\ensuremath{\mathcal{F}}}\, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\, \g)\simeq {\ensuremath{\mathcal{F}}}\, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\, j{\ensuremath{_{*}}}{^{\vphantom{*}} }\g$$ for any $\ff$ and $\g$ in $D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{Y}{^{\vphantom{*}} }}})$ and $D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$ respectively. For any element $\ff$ in $D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$, there is a natural isomorphism $$j {\ensuremath{_{*}}}{^{\vphantom{*}} }(j {\ensuremath{^{\, *}}}{_{\vphantom{i}} }j {\ensuremath{_{*}}}{^{\vphantom{*}} }\ff) \simeq j {\ensuremath{_{*}}}{^{\vphantom{*}} }(j {\ensuremath{^{\, *}}}{_{\vphantom{i}} }j {\ensuremath{_{*}}}{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }\ff)$$ in ${D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})}{^{\vphantom{*}} }$ obtained by the chain $$j{\ensuremath{_{*}}}{^{\vphantom{*}} }(j {\ensuremath{^{\, *}}}{_{\vphantom{i}} }j {\ensuremath{_{*}}}{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }\ff) \simeq j {\ensuremath{_{*}}}{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }\, j {\ensuremath{_{*}}}{^{\vphantom{*}} }{\ensuremath{\mathcal{F}}}\simeq j {\ensuremath{_{*}}}{^{\vphantom{*}} }{\ensuremath{\mathcal{F}}}\, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\, j {\ensuremath{_{*}}}{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\simeq j{\ensuremath{_{*}}}{^{\vphantom{*}} }j {\ensuremath{^{\, *}}}{_{\vphantom{i}} }j {\ensuremath{_{*}}}{^{\vphantom{*}} }\ff$$ using the projection formula twice. It is important to notice that for general pairs $(X,Y)$ of complex analytic cycles, the objects $j {\ensuremath{^{\, *}}}{_{\vphantom{i}} }j {\ensuremath{_{*}}}{^{\vphantom{*}} }\ff$ and $j {\ensuremath{^{\, *}}}{_{\vphantom{i}} }j {\ensuremath{_{*}}}{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }\ff$ are not always isomorphic in $D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$. This can be seen as follows: assuming that $\ff$ is locally free, it is proved in [@AC §2.6] that if $j {\ensuremath{^{\, *}}}{_{\vphantom{i}} }j {\ensuremath{_{*}}}{^{\vphantom{*}} }\, \ff$ is formal in $D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$ then $\ff$ can be lifted to a locally-free sheaf on ${\ensuremath{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{\textrm{X}{^{\vphantom{*}} }}}}}} \hspace*{-2.13ex}X}$. Therefore, if $N\ee_{X/Y}$ can be lifted to a locally-free sheaf on ${\ensuremath{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{\textrm{X}{^{\vphantom{*}} }}}}}} \hspace*{-2.13ex}X}$ but $\ff$ cannot, then ${j {\ensuremath{^{\, *}}}{_{\vphantom{i}} }j {\ensuremath{_{*}}}{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }\ff}$ is formal and $j {\ensuremath{^{\, *}}}{_{\vphantom{i}} }j {\ensuremath{_{*}}}{^{\vphantom{*}} }\ff$ is not. Of course, if $j$ admits an infinitesimal retraction $\sigma$, both objects are isomorphic but the isomorphism cannot in general be chosen independent of $\sigma$. For any elements $\ff$ and $\g$ in $D^{\textrm{b}}_{\textrm{coh}}({\ensuremath{\oo_{Y}{^{\vphantom{*}} }}})$, the natural morphism $$\xymatrix@C=25pt{ {\ensuremath{\mathcal{F}}}\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}, \, \ell}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }{\mathcal{RH}om^{\,r\vphantom{p}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}},\g)\ar[r]&{\mathcal{RH}om^{\,r\vphantom{p}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}},\ff\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\g). }$$ is an isomorphism. This means that we have an isomorphism $j\ee{_{\vphantom{i}} }(\,.\,)\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }j\pe{_{\vphantom{i}} }(\,.\,)\fl j\pe{_{\vphantom{i}} }(\,.\,\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\,.\,)$ of bifunctors from $D^{\textrm{b}}_{\textrm{coh}}({\ensuremath{\oo_{Y}{^{\vphantom{*}} }}})\times D^{\textrm{b}}_{\textrm{coh}}({\ensuremath{\oo_{Y}{^{\vphantom{*}} }}})$ to $D^{\textrm{b}}_{\textrm{coh}}({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$. Kashiwara’s isomorphism {#kash} ----------------------- Let $\hh\hh_{Y}{^{\vphantom{*}} }(X)$ be the *generalized derived Hochschild complex*. It is defined by $$\label{KashIsoEqUn} \hh\hh_{Y}{^{\vphantom{*}} }(X)=j\ee{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}.$$ Then $\hh\hh_{Y}{^{\vphantom{*}} }(X)$ is a ring object in $D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$, the multiplication being given by the chain of morphisms $$\xymatrix@C=25pt{ j\ee{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\, j\ee{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\ar[r]^-{\sim}&j\ee{_{\vphantom{i}} }\bigl( j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\,j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\bigr)\ar[r]&j\ee{_{\vphantom{i}} }\bigl( j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}{\ensuremath{\otimes}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\bigr)=j\ee{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}. }$$ The object $\bop\limits_{i=0}^{r}\bw{i}N\ee_{X/Y}[i]$ is also a ring object in $D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$, with multiplication given by cup-product. \[KashIsoPropUn\] Let $(X, \sigma)$ be a quantized analytic cycle of codimension $r$ in a complex manifold $Y.$ Let $\sigma $ be a retraction of ${\ensuremath{\overline{j}}}$. Then $$\xymatrix@C=25pt{\Gamma _{\sigma }{^{\vphantom{*}} }\,:\,\hh\hh_{Y}{^{\vphantom{*}} }(X)\ar[r]^-{\sim}&\bop_{i=0}^{r}\bw{i}N\ee_{X/Y}[i]}$$ is a ring isomorphism. We consider the following commutative diagram $$\xymatrix@C=45pt@R=20pt{ \textrm{H}^{0}{_{\vphantom{i}} }(j\ee{_{\vphantom{i}} })({\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }){\ensuremath{\otimes}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\! \textrm{H}^{0}{_{\vphantom{i}} }(j\ee{_{\vphantom{i}} })({\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} })\ar[r]^-{\sim}&\textrm{H}^{0}{_{\vphantom{i}} }(j\ee{_{\vphantom{i}} })({\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }{\ensuremath{\otimes}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\!{\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} })\ar[r]^-{*}&\textrm{H}^{0}{_{\vphantom{i}} }(j\ee{_{\vphantom{i}} })({\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} })\\ j\ee{_{\vphantom{i}} }{\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }j\ee{_{\vphantom{i}} }{\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }\ar[u]^-{\sim}\ar[r]^-{\sim}\ar[d]_-{\sim}&j\ee{_{\vphantom{i}} }\bigl( {\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }{\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }\bigr)\ar[r]\ar[u]\ar[d]_-{\sim}&j\ee{_{\vphantom{i}} }{\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }\ar[u]_-{\sim}\ar[d]^-{\sim}\\ j\ee{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\, j\ee{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\ar[r]^-{\sim}&j\ee{_{\vphantom{i}} }\bigl( j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\, j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\bigr)\ar[r]&j\ee{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}$$ where $\smash[b]{\apl{*}{{\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }{\ensuremath{\otimes}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\!{\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }}{{\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }}} $ has been constructed in § \[DgAlg\]. By (\[EqTroisDgAlg\]), the composition of the two arrows of the first line is the cup-product map via the isomorphism $\smash[b]{\textrm{H}^{0}{_{\vphantom{i}} }(j\ee{_{\vphantom{i}} })({\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} })\simeq\bop\limits_{i=0}^{r}\bw{i}N\ee_{X/Y}[i]} $. This finishes the proof. \[KashIsoRemUn\] This proposition holds in a more general setting, namely when $N\ee_{X/Y}$ extends to ${\ensuremath{{\ensuremath{\overline{\hphantom{\textrm{X}}\vphantom{\textrm{X}{^{\vphantom{*}} }}}}}} \hspace*{-2.13ex}X}$ and $\Gamma _{\sigma }{^{\vphantom{*}} }$ is replaced by $\Gamma _{\mathcal{K}}{^{\vphantom{*}} }$ (where $\mathcal{K}$ is the corresponding untwisted Arinkin-Căldăraru complex). We refer the reader to [@AC] for more details. The object $j\pe{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}$ can also be naturally equipped with an action of $\hh\hh_{Y}{^{\vphantom{*}} }(X)$. This is done using the chain of morphisms $$\label{KashIsoEqDeux} j\ee{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\,j\pe{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\flexp{\sim}j\pe{_{\vphantom{i}} }\bigl( j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\,j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\bigr)\fl j\pe{_{\vphantom{i}} }\bigl( j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}{\ensuremath{\otimes}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\! j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\bigr)\flexp{\sim}j\pe{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}$$ \[KashIsoThUn\] For any quantized analytic cycle $(X, \sigma)$ of codimension $r$ in a complex manifold $Y$, the isomorphism $$\xymatrix@C=25pt@R=1pt{(\Gamma _{\sigma }{^{\vphantom{*}} }, \wh{\Gamma} _{\sigma }{^{\vphantom{*}} })\,:\,(\hh\hh_{Y}{^{\vphantom{*}} }(X), \, \,j\pe{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}) \ar[r]^-{\sim}&\bigl(\bop_{i=0}^{r}\bw{i}N\ee_{X/Y}[i], \bop_{i=0}^{r}\bw{i}N{^{\vphantom{*}} }_{X/Y}[-i]\bigr) }$$ preserves the module structure, where $\bop_{i=0}^{r}\bw{i}N\ee_{X/Y}[i]$ acts on $\bop_{i=0}^{r}\bw{i}N{^{\vphantom{*}} }_{X/Y}[-i]$ by left contraction. Let us consider the following commutative diagram, in which we use implicitly the isomorphism $j\ee{_{\vphantom{i}} }{\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }\flexp{\sim}H^{0}{_{\vphantom{i}} }(j\ee{_{\vphantom{i}} })({\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} })$: $$\xymatrix@C=35pt{ {\ensuremath{\scriptstyle}}j\ee{_{\vphantom{i}} }{\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }\,{\ensuremath{\otimes}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\,\textrm{H}^{0}{_{\vphantom{i}} }(j\pe{_{\vphantom{i}} })\,(\,{\ensuremath{\mathcal{Q}}}_{\sigma }{^{\vphantom{*}} }\, {\ensuremath{\otimes}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }\omega_{X/Y} {^{\vphantom{*}} })\ar[r]\ar[d]^-{\sim} &\st\textrm{H}^{0}{_{\vphantom{i}} }(j\pe{_{\vphantom{i}} })\,({\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }\,{\ensuremath{\otimes}}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }[\,{\ensuremath{\mathcal{Q}}}_{\sigma }{^{\vphantom{*}} }\, {\ensuremath{\otimes}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }\omega_{X/Y} {^{\vphantom{*}} }])\ar[r]^-{\operatorname{id}{\ensuremath{\otimes}}\,\wh{*}\,}\ar[d] &\st\textrm{H}^{0}{_{\vphantom{i}} }(j\pe{_{\vphantom{i}} })\,(\,{\ensuremath{\mathcal{Q}}}_{\sigma }{^{\vphantom{*}} }\, {\ensuremath{\otimes}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }\omega_{X/Y} {^{\vphantom{*}} })\ar[d]^-{\sim}\\ {\ensuremath{\scriptstyle}}j\ee{_{\vphantom{i}} }{\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}\,j\pe{_{\vphantom{i}} }(\,{\ensuremath{\mathcal{Q}}}_{\sigma }{^{\vphantom{*}} }\, {\ensuremath{\otimes}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }\omega_{X/Y} {^{\vphantom{*}} }) \ar[r]^-{\sim}\ar[d]^-{\sim} &{\ensuremath{\scriptstyle}}j\pe{_{\vphantom{i}} }({\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\,[\,{\ensuremath{\mathcal{Q}}}_{\sigma }{^{\vphantom{*}} }\, {\ensuremath{\otimes}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }\omega_{X/Y} {^{\vphantom{*}} }])\ar[r]\ar[d]^-{\sim} &{\ensuremath{\scriptstyle}}j\pe{_{\vphantom{i}} }(\,{\ensuremath{\mathcal{Q}}}_{\sigma }{^{\vphantom{*}} }\, {\ensuremath{\otimes}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }\omega_{X/Y} {^{\vphantom{*}} })\\ {\ensuremath{\scriptstyle}}j\ee{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\, j\pe{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\ar[r]^-{\sim} &{\ensuremath{\scriptstyle}}j\pe{_{\vphantom{i}} }(\,j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\, j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}})\ar[r] &{\ensuremath{\scriptstyle}}j\pe{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\ar[u]_-{\sim} }$$ where $\wh{*}$ is defined by $(\ref{EqOnzeBis})$. Now we have isomorphisms $$j\ee{_{\vphantom{i}} }{\ensuremath{\mathcal{P}}}_{\sigma }{^{\vphantom{*}} }\simeq \bop_{i=0}^{r}\bw{i}N\ee_{X/Y}[i] \qquad \textrm{and} \qquad \textrm{H}^{0}{_{\vphantom{i}} }(j\pe{_{\vphantom{i}} })\,(\,{\ensuremath{\mathcal{Q}}}_{\sigma }{^{\vphantom{*}} }\,{\ensuremath{\otimes}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\,\omega _{X/Y}{^{\vphantom{*}} }) \simeq \bop_{i=0}^{r}\bw{i}N_{X/Y}[-i],$$ the second one being given by (\[right\]). Thanks to (\[EqOnzeBis\]), a direct computation shows that the composition of the arrows in the first horizontal row of the diagram is via the above isomorphisms the left contraction morphism. This yields the result. \[KashIsoDefUn\] For any pair $(X, Y)$ of complex manifolds such that $X$ is a closed complex submanifold of $Y$, the *Kashiwara isomorphism* $\Dg$ is a specific isomorphism in the derived category $D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$ between $j\ee{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\omega _{X/Y}{^{\vphantom{*}} }$ and $j\pe{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}$ given by the chain of morphisms $$\xymatrix{{j\ee{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\omega _{X/Y}{^{\vphantom{*}} }\simeq j\ee{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }j\pe{_{\vphantom{i}} }{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}\, \ar[r]^-{\sim}}&{\, j\pe{_{\vphantom{i}} }(j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}} {^{\vphantom{*}} }{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}})\, \, \simeq\,\, j\pe{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}.}$$ \[KashIsoPropDeux\] The isomorphism $\Dg$ is an isomorphism of left $\hh\hh_{Y}{^{\vphantom{*}} }(X)$-modules. This follows directly from the commutative diagram $$\xymatrix@C=30pt{ {\ensuremath{\scriptstyle}}j\ee{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\,j\ee{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }j\pe{_{\vphantom{i}} }{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}\ar[r]^-{\sim}\ar[d]^-{\sim}_-{\operatorname{id}\, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\Dg}&{\ensuremath{\scriptstyle}}j\ee{_{\vphantom{i}} }(j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\, j\ei{\ensuremath{\oo_{X}{^{\vphantom{*}} }}})\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\, j{\ensuremath{^{\, !}}}{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}\ar[r]\ar[d]^-{\sim}&{\ensuremath{\scriptstyle}}j\ee{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\, {{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\,j\pe{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}\,\ar[d]^-{\sim}_-{\Dg}\, \\ \st j\ee{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }j\pe{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\ar[r]^-{\sim}&{\ensuremath{\scriptstyle}}j\pe{_{\vphantom{i}} }( j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{Y}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\, j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}})\ar[r]&{\ensuremath{\scriptstyle}}j\pe{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}$$ We fix an isomorphism between $j\ee{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\omega _{X/Y}{^{\vphantom{*}} }$ and $\bop_{i=0}^{r}\bw{i}{N_{X/Y}{^{\vphantom{*}} }[-i]}$ as follows: $$\label{KashIsoEqTrois} \xymatrix@C=50pt{ j\ee{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\omega _{X/Y}{^{\vphantom{*}} }\ar[r]^-{\sim}_-{\Gamma _{\sigma }{^{\vphantom{*}} }\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\operatorname{id}}&\bop_{i=0}^{r}\bw{i}{N\ee_{X/Y}[i]}{\ensuremath{\otimes}}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\omega _{X/Y}{^{\vphantom{*}} }\ar[r]^-{\sim}_-{D^{\ell}{_{\vphantom{i}} }}&\bop_{i=0}^{r}\bw{i}{N_{X/Y}{^{\vphantom{*}} }[-i].} }$$ Then the main result of this section is: \[KashIsoThDeux\] Let $(X,\sigma )$ be a quantized analytic cycle in a complex manifold $Y$, and let $M$ be the automorphism of $\bop_{i=0}^{r}\bw{i}{N_{X/Y}{^{\vphantom{*}} }[-i]}$ occurring in the diagram $$\xymatrix@C=40pt@R=20pt{ j\ee{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\omega _{X/Y}{^{\vphantom{*}} }\ar[r]^-{\sim}_-{\mathfrak{D}}\ar[d]_-{\sim}&j\pe{_{\vphantom{i}} }j\ei{^{\vphantom{*}} }{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}\ar[d]^-{\sim}\\ \bop_{i=0}^{r}\bw{i}{N_{X/Y}{^{\vphantom{*}} }[-i]}\ar[r]^-{\sim}_-{M}&\bop_{i=0}^{r}\bw{i}{N_{X/Y}{^{\vphantom{*}} }[-i]} }$$ where the left vertical isomorphism is defined by *(\[KashIsoEqTrois\])*. Then for any integers $i$, $j$ such that $0\le i\le j\le r$, the component $M_{i,\,j}{^{\vphantom{*}} }$ of $M$ is given by $$\xymatrix@C=30pt{ \bw{j}{N_{X/Y}{^{\vphantom{*}} }[-j]}\ar[rr]^-{\smash[t]{q_{\sigma} {^{\vphantom{*}} }(X)_{j-i}{^{\vphantom{*}} }\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\operatorname{id}} }&&\bw{j-i}{N\ee_{X/Y}[j-i]}\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\bw{j}{N_{X/Y}{^{\vphantom{*}} }[-j]}\ar[r]^-{\lrcorner}&\bw{i}{N_{X/Y}{^{\vphantom{*}} }[-i].} }$$ In particular, $\mathfrak{D}$ is completely determined by the quantized cycle class $q_{\sigma} {^{\vphantom{*}} }(X)$. Let $\apl{\Delta }{\omega _{X/Y}{^{\vphantom{*}} }}{\bop_{i=0}^{k}\bw{i}{N_{X/Y}{^{\vphantom{*}} }[-i]}}$ be the morphism in $D^{\textrm{b}}{_{\vphantom{i}} }({\ensuremath{\oo_{X}{^{\vphantom{*}} }}})$ defining the quantized cycle class. For any integers $i$ and $j$ such that $0\le i\le j\le r$, Propositions \[KashIsoPropUn\], \[KashIsoThUn\] and \[KashIsoPropDeux\] imply that $M_{i,\,j}{^{\vphantom{*}} }$ is given by the composition $$\xymatrix@C=35pt{ \bw{j}{N_{X/Y}{^{\vphantom{*}} }[-j]}&\bw{r-j}{N\ee_{X/Y}[r-j]}\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\omega _{X/Y}{^{\vphantom{*}} }\ar[l]_-{\sim}^-{D^{\ell}{_{\vphantom{i}} }}\ar[rr]^-{\operatorname{id}\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\Delta _{r-j+i}{^{\vphantom{*}} }}&&\hspace*{150pt} }$$ $$\xymatrix@C=35pt{ \hspace*{100pt}\bw{r-j}{N\ee_{X/Y}[r-j]\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }}\bw{r-j+i}{N_{X/Y}{^{\vphantom{*}} }[j-i-r]}\ar[r]^-{\lrcorner}&\bw{i}{N_{X/Y}{^{\vphantom{*}} }[-i]} }$$ which is exactly $$\xymatrix@C=30pt{ \bw{j}{N_{X/Y}{^{\vphantom{*}} }[-j]}&\bw{r-j}{N\ee_{X/Y}[r-j]}\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\omega _{X/Y}{^{\vphantom{*}} }\ar[l]_-{\sim}^-{D^{\ell}{_{\vphantom{i}} }}\ar[rrr]^-{(\operatorname{id}{\ensuremath{\wedge}}{\, q_{\sigma} {^{\vphantom{*}} }(X)_{j-i}{^{\vphantom{*}} }})\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }\operatorname{id}}&&&\hspace*{100pt} }$$ $$\xymatrix@C=35pt{ \hspace*{150pt}\bw{r-i}{N\ee_{X/Y}[r-i]\,{{\mathop{{\mathbin{\otimes_{\raise1.5ex\hbox to-.1em{}{}}}}}\limits^{\rm \mathbb{L}}}_{}}{}_{{\ensuremath{\oo_{X}{^{\vphantom{*}} }}}}{^{\vphantom{*}} }}\omega _{X/Y}{^{\vphantom{*}} }\ar[r]^-{\sim}_-{D^{\ell}{_{\vphantom{i}} }}&\bw{i}{N_{X/Y}{^{\vphantom{*}} }[-i].} }$$ This yields the result.

--- abstract: 'We present multiband photometry of 60 spectroscopically-confirmed supernovae (SN): 39 SN II/IIP, 19 IIn, one IIb and one that was originally classified as a IIn but later as a Ibn. Forty-six have only optical photometry, six have only near infrared (NIR) photometry and eight have both optical and NIR. The median redshift of the sample is 0.016. We also present 192 optical spectra for 47 of the 60 SN. All data are publicly available. There are 26 optical and two NIR light curves of SN II/IIP with redshifts $z > 0.01$, some of which may give rise to useful distances for cosmological applications. All photometry was obtained between 2000 and 2011 at the Fred Lawrence Whipple Observatory (FLWO), via the 1.2m and 1.3m PAIRITEL telescopes for the optical and NIR, respectively. Each SN was observed in a subset of the $u''UBVRIr''i''JHK_s$ bands. There are a total of 2932 optical and 816 NIR light curve points. Optical spectra were obtained using the FLWO 1.5m Tillinghast telescope with the FAST spectrograph and the MMT Telescope with the Blue Channel Spectrograph. Our photometry is in reasonable agreement with other samples from the literature. Comparison with Pan-STARRS shows that two-thirds of our individual star sequences have weighted-mean $V$ offsets within $\pm$0.02 mag. In comparing our standard-system SN light curves with common Carnegie Supernova Project objects using their color terms, we found that roughly three-quarters have average differences within $\pm$0.04 mag. The data from this work and the literature will provide insight into SN II explosions, help with developing methods for photometric SN classification, and contribute to their use as cosmological distance indicators.' author: - Malcolm Hicken - 'Andrew S. Friedman' - Peter Challis - Perry Berlind - Stephane Blondin - Mike Calkins - Gil Esquerdo - Thomas Matheson - Maryam Modjaz - Armin Rest - 'Robert P. Kirshner' title: Type II Supernova Light Curves and Spectra From the CA --- Introduction ============ This paper presents CfA SN II light curves and spectra, all of which are publicly available from the journal, the CfA SN Group website[^1] and The Open Supernova Catalog.[^2] It is our hope that these data will be of use to the broader SN community for use in SN II analysis and cosmological calculations. Massive stars ($M \gtrsim 8M_\odot$) end their lives as core-collapse supernovae (CCSN). Those that have retained a large portion of their hydrogen envelope are known as Type II SN, with spectra dominated by Balmer features. Those that have lost their hydrogen envelope and have no Balmer lines but do have helium features are known as Type Ib. Finally, those that have also lost much or all of their helium envelope and show no helium spectral features are known as Type Ic. For reviews of these classifications, see, for example, @galyam and the introductions of @smartt, @vandyk, @modjaz16 and @liu16. A continuum seems to exist across the CCSN classes, depending on how much of the outer layer(s) has been lost. SN II with thick hydrogen envelopes have a long plateau phase with roughly constant or slowly declining luminosity (SN IIP) whereas those with thinner envelopes decline more quickly (SN IIL) [see, for example: @anderson; @sanders]. @gall15 point out that, in addition to thinner envelopes, SN IIL need larger progenitor radii than SN IIP to give rise to their observed luminosities. There also appears to be a continuous range of decline rates joining SN IIP and IIL, giving further weight to the continuum between CCSN types. However, see @valenti16 for more on the debate over SN IIP and IIL and @morozova17 for discussion on whether there is a physical process that smoothly gives rise to a continuum between SN IIP and IIL or whether there is some specific mechanism that more abruptly gives rise to their differences. Type IIb are an intermediate class between SN II and Ib. They have hydrogen features in their early spectra but these quickly disappear as the SN ages, suggesting a thin hydrogen layer in the progenitor. @liu16 showed that there is a continuum of H$\alpha$ strengths between SN IIb and Ib. Another kind of SN II is the Type IIn, distinguished by its narrow Balmer emission lines that are believed to arise from the SN blast wave colliding with previously-ejected progenitor material from stellar winds or eruptions. SN II are of interest for several reasons. They mark the death of many massive stars and play an important role in the chemical enrichment of the Universe. SN II light curves and spectra give insight into SN II explosion mechanisms and progenitor properties. SN II also serve as accurate distance indicators. SN II are not as luminous as SN Ia [see, for example, @richardson02] but do provide an alternative and independent means of measuring cosmological distances [see, for example, @dejaeger]. As more powerful telescopes, such as the Large Synoptic Survey Telescope,[^3] observe larger numbers of SN II at both low and high redshift, SN II will become more useful cosmological tools than in the past, offering an independent source of distances for calculating the expansion history of the universe. Since most of the SN observed with large surveys will not be spectroscopically identified, it is important to understand the photometric nature of all types of SN so that sufficiently pure subsets (e.g., consisting only of SN Ia or SN IIP) can be photometrically separated and used for cosmology. The potential cosmological use of SN II was pioneered by @kirshner74 [@kirshner75] with the expanding photospheric method for measuring distances. @schmidt92 applied this method to 10 SN II to calculate their distances and a value for the Hubble constant. @schmidt94 improved their method and applied it to 18 SN II to find a Hubble constant of $H_o = 73 \pm 6(\mathrm{stat}) \pm 7(\mathrm{syst})$ km/s/Mpc, consistent within the error bars with more recent measurements such as the SN Ia and Cepheid-based value of $H_o = 73.24 \pm 1.74$ km/s/Mpc from @riess16 and the $Planck$ cosmic microwave background radiation-based value of $H_o = 67.8 \pm 0.9$ km/s/Mpc from the @planck16. More recently, several optical and NIR SN II data sets have been published while other unpublished data sets have been used for analysis or cosmological applications. We present a summary of many of them and their findings, in largely chronological order. This may also assist anyone interested in compiling SN II data from the literature. @poznanski09 combined optical light curves and spectra of 17 new SN IIP with those of 23 from the literature to find a Hubble diagram dispersion of 0.38 mag, which reduced to 0.22 mag when three $>3\sigma$ outliers were removed. @dandrea presented light curves and spectra of 15 SDSS SN IIP and combined them with others from the literature to find a Hubble diagram dispersion of 0.29 mag for the combination. NIR photometry of SN II has also been produced and analyzed. Dust extinction at NIR wavelengths is diminished relative to the optical, and there is the additional potential for smaller intrinsic dispersion in NIR light curves of SN II. @maguire10 explain that reduced extinction in the NIR gives rise to a lower error in the extinction estimate and has less of an effect on the fit between expansion velocity and NIR luminosity. They also point out that SN IIP plateau-phase spectra have far fewer lines in the NIR than the optical. They thus presume that estimates of SN IIP NIR luminosity should be less affected by variations in line strengths and line widths between different SN IIP. They examined 12 SN IIP that had spectra and both $VI$ and NIR light curves. However, only one of their SN was in the Hubble flow ($cz>3000$ km/s), suggesting the Hubble diagram dispersion they found would be larger than that of Hubble flow objects. In the optical, they found a dispersion of 0.56 mag by combining $V$-band photometry and an estimate of the expansion velocity at +50d post-explosion. In $I$, they found a dispersion of 0.5 mag. @maguire10 confirmed their hopes for the NIR by measuring a $J$ band dispersion of 0.39 mag, $\sim0.1$ mag lower than in $I$, suggesting that NIR light curves of Hubble-flow SN II would offer a similar improvement when compared to the optical. In the past year, @rodriguez16 presented preliminary results from a set of 16 SN II, showing a 0.12 mag dispersion Hubble diagram in each of the $JHK_s$ bands, better than the $BVI$ dispersions of 0.23, 0.17 and 0.19 mag, respectively, providing evidence that the smaller dispersion seen by @maguire10 for very-nearby NIR SN II light curves would also be found in the Hubble flow. More data was published by various groups in the following years. Returning to the optical, @arcavi produced 21 SN II light curves from the Caltech Core Collapse Project. @taddia published light curves and spectra of five CSP SN IIn. @faran presented light curves and spectra of 23 SN IIP from LOSS and analyzed them. @anderson provided V-band light curves and analysis for 116 SN II from the CSP and its predecessors (CT, C&T, SOIRS and CATS). They found evidence suggesting a continuum, where low-luminosity SN II have flat light curves during the plateau phase and higher-luminosity SN II have faster decline rates. @anderson measured a dispersion of 0.56 mag around the relation between the plateau-phase decline rate and peak magnitudes, suggesting that SN II can be used as pure photometric distance indicators. @galbany published all of the bands for 51 of these 116 SN II while the CSP portion is not yet published. @gutierrez presented an in-depth analysis of spectra of 123 SN II from the CSP and its predecessors, finding evidence suggesting that differences between SN IIP and IIL are related to the pre-explosion hydrogen envelope mass and do not come from different progenitor families. @rodriguez14 found a very promising dispersion of 0.12 mag by using 13 SN II in the Hubble flow that have a well-constrained shock breakout epoch. @gonzalezgaitan studied the rise times of 223 SN II light curves from SDSS and SNLS and found evidence that the early light curves of most SN II are powered by cooling of shock-heated ejecta. They also found that massive hydrogen envelopes are indeed needed to explain the plateaus of SN II. @sanders analyzed 76 multiband Pan-STARRS1 (PS1) SN II light curves and concluded that there does not appear to be two unconnected subclasses of SN II (IIP and IIL) but rather a one-parameter family likely related to the original mass of the progenitor, similar to the findings of @anderson. @sanders further confirmed that SN IIP appear to be standardizable with an intrinsic dispersion as low as $\sim$0.2 mag. @rubin published 57 R-band SN II light curves from the Palomar Transient Factory. @valenti16 presented photometry of 12 SN II and combined them with well-sampled light curves from the literature to search for correlations between the slope of the linear light-curve decay and other properties. In the past year, @dejaeger presented a Hubble diagram based on 73 SN II with a redshift range of $0.01 \leq z \leq 0.50$ from the CSP, SDSS and SNLS and found a Hubble diagram dispersion of 0.35 mag by applying the Photometric Colour Method with no redshift information, showing that SN II can be used as pure photometric distance indicators. On a smaller sample of 61 SN II and using spectroscopic information, @dejaeger measured a dispersion of 0.27 mag. Finally, @gall17 presented light curves and spectra of nine SN IIP/L in the redshift range of $0.045 \leq z \leq 0.335$ and combined them with data from the literature to update previous EPM and SCM Hubble diagrams. An interesting finding is that their three SN IIL are indistinguishable from their SN IIP in both the EPM and SCM Hubble diagrams. Larger samples are needed to confirm this but it suggests that SN IIL may be useful cosmological distance indicators as well. Given their higher luminosity than SN IIP, this could make SN IIL easier to find at higher redshifts, assuming a sufficient observing cadence is used to account for their faster decline. [llllcccr]{} SN2000eo & IIn & MCG -2-9-3 & 0.010347 & IAUC 7524 & y & - & 14\ SN2001ez & II & CGCG 329-009 & 0.012916 & IAUC 7736 & y & - & 1\ SN2001fa & IIn & NGC 673 & 0.017285 & IAUC 7737 & y & - & 5\ SN2002bx & II & IC 2461 & 0.007539 & IAUC 7864 & y & - & 8\ SN2002em & II & UGC 3430 & 0.013539 & IAUC 7955 & y & - & 0\ SN2005ay & II & NGC 3938 & 0.002699 & CBET 128 & - & y & 0\ SN2005kd & IIn & CGCG 327-013 & 0.015040 & IAUC 8630 & y & - & 0\ SN2006at & II & Anon Gal & 0.015000 & CBET 424 & y & - & 1\ SN2006be & II & IC 4582 & 0.007155 & CBET 449 & y & - & 3\ SN2006bl & II & MCG +02-40-9 & 0.032382 & CBET 462 & y & - & 1\ SN2006bo & IIn & UGC 11578 & 0.015347 & CBET 468 & y & - & 0\ SN2006bv & IIn & UGC 7848 & 0.008382 & CBET 493 & y & - & 1\ SN2006ca & II & UGC 11214 & 0.008903 & IAUC 8707 & y & - & 1\ SN2006cd & IIP & IC 1179 & 0.037116 & CBET 508 & y & - & 1\ SN2006gy & IIn & NGC 1260 & 0.019190 & CBET 644 & y & - & 2\ SN2006it & IIP & NGC 6956 & 0.015511 & CBET 660 & y & - & 0\ SN2006iw & II & Anon Gal & 0.030700 & CBET 663 & y & - & 0\ SN2006ov & II & NGC 4303 & 0.005224 & CBET 756 & y & - & 1\ SN2007Q & II & NGC 5888 & 0.029123 & CBET 821 & y & - & 2\ SN2007T & II & NGC 5828 & 0.013581 & CBET 833 & y & - & 0\ SN2007aa & II & NGC 4030 & 0.004887 & CBET 848 & y & y & 4\ SN2007ad & II & UGC 10845 & 0.027506 & CBET 854 & y & - & 1\ SN2007ah & II & UGC 2931 & 0.019170 & CBET 869 & y & - & 1\ SN2007ak & IIn & UGC 3293 & 0.015634 & CBET 875 & y & - & 0\ SN2007av & II & NGC 3279 & 0.004650 & CBET 901 & y & y & 1\ SN2007ay & II & UGC 4310 & 0.014527 & CBET 905 & y & - & 0\ SN2007bb & IIn & UGC 3627 & 0.020858 & CBET 912 & y & - & 1\ SN2007be & II & UGC 7800 & 0.012515 & CBET 917 & y & - & 1\ SN2007bf & II & UGC 9121 & 0.017769 & CBET 919 & y & - & 1\ SN2007cd & II & NGC 5174 & 0.022799 & CBET 950 & y & - & 0\ SN2007ck & II & MCG +05-43-16 & 0.026962 & CBET 970 & y & - & 1\ SN2007cm & IIn & NGC 4644 & 0.016501 & CBET 973 & y & - & 1\ SN2007ct & II & NGC 6944 & 0.014734 & CBET 988 & y & - & 1\ SN2007cu & II & UGC 10214 & 0.031358 & CBET 988 & y & - & 0\ SN2007hv & II & UGC 2858 & 0.016858 & CBET 1056 & y & - & 1\ SN2007od & II & UGC 12846 & 0.005784 & CBET 1116 & y & - & 13\ SN2007pk & IIn & NGC 579 & 0.016655 & CBET 1129 & y & - & 7\ SN2007rt & IIn & UGC 6109 & 0.022365 & CBET 1148 & y & y & 3\ SN2008B & IIn & NGC 5829 & 0.018797 & CBET 1194 & y & - & 3\ SN2008F & IIP & MCG -01-8-15 & 0.018366 & CBET 1207 & y & - & 1\ SN2008aj & IIn & MCG +06-30-34 & 0.024963 & CBET 1259 & y & - & 1\ SN2008bj & II & MCG +08-22-20 & 0.018965 & CBET 1314 & y & - & 16\ SN2008bn & II & NGC 4226 & 0.024220 & CBET 1322 & y & - & 11\ SN2008bu & II & ESO 586-G2 & 0.022115 & CBET 1341 & y & - & 0\ SN2008bx & II & Anon Gal & 0.008399 & CBET 1348 & y & - & 3\ SN2008gm & IIn & NGC 7530 & 0.011728 & CBET 1549 & y & - & 1\ SN2008if & II & MCG -01-24-10 & 0.011475 & CBET 1619 & - & y & 1\ SN2008in & II & NGC 4303 & 0.005224 & CBET 1636 & y & y & 13\ SN2008ip & IIn & NGC 4846 & 0.015124 & CBET 1641 & y & y & 11\ SN2009K & IIb & NGC 1620 & 0.011715 & CBET 1663 & - & y & 0\ SN2009N & IIP & NGC 4487 & 0.003456 & CBET 1670 & y & - & 1\ SN2009ay & II & NGC 6479 & 0.022182 & CBET 1728 & y & y & 6\ SN2009dd & II & NGC 4088 & 0.002524 & CBET 1764 & y & - & 9\ SN2009kn & IIn & MCG -03-21-6 & 0.015798 & CBET 1997 & y & - & 4\ SN2009kr & II & NGC 1832 & 0.006468 & CBET 2006 & - & y & 2\ SN2010aj & IIP & MCG -01-32-35 & 0.021185 & CBET 2201 & y & - & 7\ SN2010al & Ibn & UGC 4286 & 0.017155 & CBET 2207 & y & y & 8\ SN2010bq & IIn & UGC 10547 & 0.030988 & CBET 2241 & y & y & 5\ SN2011an & IIn & UGC 4139 & 0.016308 & CBET 2668 & - & y & 2\ SN2011ap & IIn & IC 1277 & 0.023630 & CBET 2670 & - & y & 9\ The Harvard-Smithsonian Center for Astrophysics (CfA) Supernova Group has been a source of data for nearby SN since 1993. The primary focus has been SN Ia but numerous CCSN have been observed as well. In addition to various individual SN papers, the CfA1-CfA4 data sets include a total of 345 SN Ia multiband optical light curves [@Riess99; @Jha06; @Hicken09; @Hicken12]. Optical spectroscopy of over 400 SN Ia [@Matheson08; @Blondin12] have been published. In the near-infrared, the CfAIR2 set consists of 94 SN Ia light curves [@Friedman15] with earlier versions of a subset published by @Woodvasey08. The CCSN photometry we acquired up until 2011 was processed along with the SN Ia photometry when the CfA3, CfA4 and CfAIR2 data sets were produced: 61 optical and 25 near-infrared stripped-envelope light curves were presented in @Bianco14 and 60 SN II/IIn/IIb/Ibn light curves are presented in this work. Two additional NIR light curves have been published elsewhere: SN 2010jl [@fransson14] and SN 2011dh [@marion14]. One SN IIb NIR light curve was produced after @Bianco14 and it is presented here. Additionally, optical light curves, spectra and analysis of SN IIP 2005cs and 2006bp were presented in @dessart08. @Modjaz14 presented and analyzed optical spectra of 73 stripped-envelope CCSN while CfA SN II spectra for 47 of the 60 SN from the current paper are presented here. Finally, the CfA5 data set is currently being produced and will include optical SN light curves of all types taken after the CfA4 era (which included SN discovered up to mid-2010) plus any earlier ones that were missing calibration or host-galaxy subtraction images when the CfA4 light curves were created. Spectra from the CfA5-era will also be published. Section 2 describes the data and reduction and section 3 provides select comparisons of photometry with other data sets for overlapping objects. Data Acquisition and Reduction ============================== The CfA SN II sample consists of 60 objects: 39 SN II/IIP, 19 IIn, one IIb and one that was originally classified as a IIn but later as a Ibn [2010al; see, for example, @pastorello]. Forty-six objects have only optical photometry, six have only NIR photometry and eight have both. The median redshift is 0.016. Optical spectra for 47 of the 60 SN are presented in this work. See Table \[table\_snlist\] for information on each SN’s type, host galaxy, redshift, discovery reference, optical or NIR photometry, and number of CfA optical spectra. Most of the SN were discovered as part of targeted searches. There are 54 objects in this work with optical photometry: 36 II/IIP, 17 IIn and one Ibn. Measurements were made in the $u'UBVr'i'$ bands and consist of a total 2932 light curve points. These data were acquired on the 1.2m telescope at the FLWO[^4] at the same time as the CfA3 and CfA4 SN Ia data sets. Twenty-six SN II light curves were generated as part of the light curve production process that created the CfA3 set and 28 as part of the process that created the CfA4 set. Five of the CfA3-era SN II light curves were observed on the 4Shooter camera[^5] (hereafter referred to as CfA3$_\mathrm{4SH}$), and 21 on the KeplerCam[^6] (hereafter referred to as CfA3$_\mathrm{KEP}$). The 28 CfA4-era SN II light curves (hereafter referred to as CfA4) were all observed on the KeplerCam. Since all of the optical photometry reported here was produced as part of the CfA3 and CfA4 processing campaigns, see @Hicken09 and @Hicken12 for greater details on the instruments, observations, photometry pipeline, calibration and host-galaxy subtraction used to create the CfA SN II light curves. It should be noted that the passbands shifted during part of the CfA4 observing campaign, likely due to deposits or condensation on the KeplerCam, resulting in two sets of color terms for the CfA4 calibration: period 1 and period 2, one before the shift and one after. Anyone using the CfA4-era natural system light curves should be careful to use the appropriate-period passbands, which can be found at our website.[^7] Also, the 1.2m primary mirror deteriorated during the course of the CfA4 observing, causing a sensitivity loss of about 0.6 mag in $V$. There are 14 SN with NIR photometry in this work: seven SN II, five IIn, one IIb and one Ibn. We obtained our $JHK_s$-band photometry with the robotic 1.3-m Peters Automated InfraRed Imaging TELescope (PAIRITEL) at FLWO [@bloom06]. PAIRITEL was a refurbished version of the Two Micron All Sky Survey (2MASS) North telescope using the transplanted 2MASS South camera [@skrutskie06]. It was utilized from 2005-2013 as a dedicated NIR imager for follow-up of transients, including SN of all types discovered by optical searches [@bloom06; @Bianco14; @Friedman15]. Our NIR observing strategy was described elsewhere [@Woodvasey08; @friedman12; @Bianco14; @Friedman15]. In particular, see @Friedman15 for a detailed discussion of our image reduction and photometry pipelines including mosaicking, sky subtraction, and host galaxy subtraction. Whenever possible, we used an error-weighted mean of light curves derived from multiple host galaxy template images to remove flux contamination at the SN position. Since PAIRITEL was already on the 2MASS system, the SN brightness in each field was determined with differential photometry using reference field stars from the 2MASS point source catalog [@cutri03]. There are a total of 816 NIR light curve points. Eleven of our 14 NIR light curves had host galaxy subtraction as described in [@Friedman15], while 3 objects were sufficiently isolated from the host galaxy nucleus (SN 2010bq, SN 2011an, SN 2011ap) and forced DoPHOT photometry [@schechter93] was used at the SN position without template image subtraction. The NIR light curve of SN 2005ay is presented here while its optical light curves and spectra will be presented in a separate, future paper. The optical light curve of SN 2009K was presented in @Bianco14 but the NIR light curve did not have final calibration and host galaxy images and so it is presented here. We acquired 192 optical spectra for 47 of the 60 SN in this work. Spectra were obtained using the FLWO 1.5m Tillinghast telescope with the FAST spectrograph [@Fabricant98] and the MMT Telescope with the Blue Channel Spectrograph. The FAST spectra were taken using a 3$''$ slit with an atmospheric corrector. The observations at the MMT were taken with a 1$''$ slit and a 300 line/mm grating at the parallactic slit position. All Spectra were reduced and flux calibrated by a combination of standard IRAF and custom IDL procedures [@Matheson05]. The 2-d spectra underwent flat-fielding, cosmic-ray removal and extraction into 1-d spectra. Pairs of spectroscopic standard stars were obtained to provide flux calibration (with no second order contamination) and assist in the removal of telluric features. For more detail, see @Matheson08 and @Blondin12. The journal of observations of the spectra are in Table \[table\_spectra\_journal\]. [ccccccc]{} SN2007aa & 2454154.989 & FAST & 3556-7457 & 1.46 & 1.48 & 1200\ SN2007aa & 2454156.897 & FAST & 3477-7415 & 1.47 & 1.19 & 1500\ SN2007aa & 2454169.866 & FAST & 3479-7419 & 1.47 & 1.20 & 1200\ SN2007aa & 2454185.878 & FAST & 3476-7420 & 1.47 & 1.33 & 1500\ SN2007ad & 2454155.028 & FAST & 3555-7456 & 1.46 & 1.11 & 1500\ [cccccccccccccc]{} SN2007aa & 01 & 12:00:38.638 & -01:06:55.25 & 16.213(0.014) & 2 & – & 0 & 1.238(0.058) & 2 & 0.538(0.023) & 2 & 1.043(0.023) & 2\ SN2007aa & 02 & 12:00:36.905 & -01:05:53.67 & 17.245(0.016) & 2 & – & 0 & 0.556(0.021) & 2 & 0.149(0.009) & 2 & 0.285(0.011) & 2\ SN2007aa & 03 & 12:00:35.217 & -01:03:51.48 & 17.324(0.026) & 2 & – & 0 & 0.648(0.055) & 2 & 0.197(0.037) & 2 & 0.366(0.011) & 2\ SN2007aa & 04 & 12:00:25.058 & -01:02:49.29 & 17.802(0.030) & 2 & – & 0 & 0.625(0.044) & 2 & 0.124(0.027) & 2 & 0.334(0.023) & 2\ SN2007aa & 05 & 12:00:24.144 & -01:09:55.72 & 17.359(0.014) & 2 & – & 0 & 0.533(0.032) & 2 & 0.138(0.011) & 2 & 0.276(0.009) & 2\ SN2007aa & 06 & 12:00:20.107 & -01:06:22.99 & 17.451(0.029) & 2 & – & 0 & 0.527(0.039) & 2 & 0.306(0.068) & 2 & 0.436(0.030) & 2\ SN2007aa & 07 & 12:00:19.767 & -01:05:03.02 & 15.299(0.012) & 2 & – & 0 & 0.615(0.019) & 2 & 0.161(0.018) & 2 & 0.308(0.022) & 2\ SN2007aa & 08 & 12:00:16.229 & -01:02:14.43 & 17.954(0.033) & 2 & – & 0 & 0.736(0.047) & 2 & 0.246(0.013) & 2 & 0.504(0.010) & 2\ SN2007aa & 09 & 12:00:14.168 & -01:04:24.72 & 17.185(0.016) & 2 & – & 0 & 0.636(0.017) & 2 & 0.177(0.017) & 2 & 0.325(0.009) & 2\ SN2007aa & 10 & 12:00:08.517 & -01:07:18.48 & 15.374(0.015) & 2 & – & 0 & 1.406(0.022) & 2 & 0.574(0.014) & 2 & 1.255(0.011) & 2\ SN2007aa & 11 & 12:00:07.955 & -01:04:42.70 & 16.902(0.020) & 2 & – & 0 & 1.349(0.018) & 2 & 0.555(0.012) & 2 & 1.085(0.012) & 2\ SN2007aa & 12 & 12:00:07.616 & -01:04:57.01 & 16.756(0.012) & 2 & – & 0 & 0.685(0.015) & 2 & 0.196(0.022) & 2 & 0.357(0.009) & 2\ Star Sequences, Light Curves and Spectra ---------------------------------------- In Table \[table\_compstar\] we present the standard system optical star sequences for the CfA SN II photometry and in Tables \[table\_natlightcurve\] and \[table\_stdlightcurve\] the natural system and standard system optical light curves, respectively. Note should be made of the seven CfA3-era SN II optical light curves that were well-removed from their host galaxies and did not require host-galaxy subtraction. They can be identified in Tables \[table\_natlightcurve\] and \[table\_stdlightcurve\] as those that have N$_{\mathrm{host}}$=0 host subtractions. The rest of the CfA3-era SN II light curves have N$_{\mathrm{host}}$=1. All of the CfA4-era light curves were host-subtracted and have N$_{\mathrm{host}}\geq$1. We remind the reader that the CfA4 period-1 and period-2 natural system passbands are different and so special care should be taken to use the correct period-1 or period-2 passband for the natural system CfA4-era SN II light curves for each point. In the last column of Table \[table\_natlightcurve\], KEP1 means it was taken on the KeplerCam during CfA4 period 1 and KEP2 signifies period 2. CfA4-era data before 2009 August 15 (MJD=55058) is period 1 and CfA4-era data after is period 2. The CfA4 Ia light curves [@Hicken12] usually had multiple host subtractions for each light curve data point, giving rise to multiple values and uncertainties for such a data point. The median of the multiple values, which arise from the multiple subtractions for each data point, was chosen to be the light curve value for that date. The uncertainty was created from two components: the median of the photometric pipeline uncertainties for each light curve point was added in quadrature to the standard deviation of the multiple host-subtraction light curve values for that point. However, the CfA3-era light curves did not have multiple host galaxy subtractions and thus their uncertainty consists only of the photometric pipeline uncertainty. To ensure that the CfA3-era and CfA4-era SN II light curve uncertainties are comparable, we chose to only use the median of the photometric pipeline uncertainties for the CfA4-era SN II light curves in this work (as opposed to adding it in quadrature to the standard deviation of the multiple subtraction light curve values for a given data point). In Table \[table\_nirlightcurve\] we present the PAIRITEL NIR light curves. Figures \[snplot1\] and \[snplot2\] show two examples of CfA SN II light curves: one optical (SN 2008bn) and one with both optical and NIR (SN 2008in). Figure \[spectra\] shows the spectral series of SN 2007pk, a IIn, and SN 2008bn, a IIP. The photometry and spectra are available from the journal, the CfA SN Group website[^8] and The Open Supernova Catalog.[^9] The natural system passbands mentioned in @Hicken12 are also available at our website. The period-1 passbands can be used for the CfA3 KeplerCam and the CfA4 period-1 natural system light curves while the period-2 passbands can be used for the CfA4 period-2 natural system light curves. The 4Shooter natural system passbands can be found in @Jha06 for use with the CfA3 4Shooter natural system light curves. [cccccccc]{} SN2007aa & B & 54152.35648 & 0 & 16.167 & 0.016 & CfA3 & KEP\ SN2007aa & B & 54153.32938 & 0 & 16.226 & 0.017 & CfA3 & KEP\ SN2007aa & V & 54152.35348 & 0 & 15.703 & 0.012 & CfA3 & KEP\ SN2007aa & V & 54153.32638 & 0 & 15.719 & 0.013 & CfA3 & KEP\ SN2007aa & r’ & 54152.35117 & 0 & 15.571 & 0.013 & CfA3 & KEP\ SN2007aa & r’ & 54153.32407 & 0 & 15.580 & 0.014 & CfA3 & KEP\ SN2007aa & i’ & 54152.34885 & 0 & 15.694 & 0.012 & CfA3 & KEP\ SN2007aa & i’ & 54153.32176 & 0 & 15.717 & 0.015 & CfA3 & KEP\ SN2009N & B & 54857.50559 & 5 & 16.398 & 0.022 & CfA4 & KEP1\ SN2009N & B & 54858.55471 & 5 & 16.401 & 0.024 & CfA4 & KEP1\ SN2009N & V & 54857.51825 & 9 & 16.294 & 0.020 & CfA4 & KEP1\ SN2009N & V & 54858.55843 & 5 & 16.303 & 0.023 & CfA4 & KEP1\ SN2009N & r’ & 54857.51585 & 4 & 16.272 & 0.028 & CfA4 & KEP1\ SN2009N & r’ & 54858.54485 & 5 & 16.261 & 0.016 & CfA4 & KEP1\ SN2009N & i’ & 54857.50628 & 8 & 16.192 & 0.027 & CfA4 & KEP1\ SN2009N & i’ & 54858.54786 & 5 & 16.159 & 0.015 & CfA4 & KEP1\ [cccccc]{} SN2007aa & B & 54152.35648 & 0 & 16.197 & 0.016\ SN2007aa & B & 54153.32938 & 0 & 16.259 & 0.017\ SN2007aa & B & 54158.34955 & 0 & 16.378 & 0.021\ SN2007aa & B & 54158.36546 & 0 & 16.371 & 0.022\ SN2007aa & B & 54169.35950 & 0 & 16.552 & 0.018\ SN2007aa & V & 54152.35348 & 0 & 15.694 & 0.012\ SN2007aa & V & 54153.32638 & 0 & 15.709 & 0.013\ SN2007aa & V & 54158.34654 & 0 & 15.720 & 0.013\ SN2007aa & V & 54158.36245 & 0 & 15.729 & 0.012\ SN2007aa & V & 54159.46133 & 0 & 15.704 & 0.014\ SN2007aa & r’ & 54152.35117 & 0 & 15.568 & 0.013\ SN2007aa & r’ & 54153.32407 & 0 & 15.576 & 0.014\ SN2007aa & r’ & 54158.34423 & 0 & 15.563 & 0.013\ SN2007aa & r’ & 54158.36015 & 0 & 15.572 & 0.015\ SN2007aa & r’ & 54159.45900 & 0 & 15.557 & 0.014\ SN2007aa & i’ & 54152.34885 & 0 & 15.685 & 0.012\ SN2007aa & i’ & 54153.32176 & 0 & 15.707 & 0.015\ SN2007aa & i’ & 54158.34190 & 0 & 15.647 & 0.015\ SN2007aa & i’ & 54158.35785 & 0 & 15.644 & 0.014\ SN2007aa & i’ & 54159.45671 & 0 & 15.622 & 0.015\ [ccccc]{} SN2005ay & J & 53480.15 & 14.513 & 0.011\ SN2005ay & J & 53481.21 & 14.513 & 0.027\ SN2005ay & J & 53482.17 & 14.461 & 0.022\ SN2005ay & J & 53486.18 & 14.455 & 0.013\ SN2005ay & J & 53488.17 & 14.529 & 0.044\ SN2005ay & H & 53480.15 & 14.319 & 0.034\ SN2005ay & H & 53481.21 & 14.474 & 0.034\ SN2005ay & H & 53482.17 & 14.437 & 0.027\ SN2005ay & H & 53486.18 & 14.278 & 0.018\ SN2005ay & H & 53487.17 & 14.252 & 0.022\ SN2005ay & K$_s$ & 53480.15 & 14.177 & 0.022\ SN2005ay & K$_s$ & 53481.21 & 14.208 & 0.035\ SN2005ay & K$_s$ & 53482.17 & 14.196 & 0.031\ SN2005ay & K$_s$ & 53486.18 & 14.102 & 0.015\ SN2005ay & K$_s$ & 53487.17 & 14.188 & 0.018\ Photometry Comparison With Other Samples ========================================= We compared the $BVr'i'$ CfA SN II star sequences and light curves with overlapping data from two other groups and find reasonable agreement overall, although a few objects have larger discrepancies. Comparison of CfA and Pan-STARRS1 Star Sequences ------------------------------------------------- @Scolnic15 (S15) introduced the Supercal method which provides a very useful analysis tool to estimate systematic photometric offsets among many of the SN Ia samples, including those from the CfA. Their work enables the different samples to be placed much more reliably on one photometric system. For anyone combining SN II samples from various groups and needing to put them on one system, we recommend that either the offsets from S15 be applied for photometric systems covered by S15 or that the general method be followed to calculate offsets for photometric systems not covered. To check how accurate our star sequence calibrations are, we did not apply the Supercal method but simply compared our standard system star photometry with the recently released Pan-STARRS1 (PS1) Catalog[^10], converted to the standard system. Using the coordinates for each of the stars from Table \[table\_compstar\], we searched the PS1 catalog. We required a separation of 0.5 arc seconds or less for a match. We added the offsets from Table 3 of @Scolnic15 to transform the PS1 catalog values (which have the calibration of S15) to the @Tonry12 calibration. We then applied the transformation equations of Table 6 of @Tonry12 to convert to the standard system $BVri$. Finally, we converted from SDSS[^11] $ri$ to @Smith $r'i'$, though there is minimal difference. Similar to S15, we removed PS1 stars brighter than 14.8, 14.9 and 15.1 mag in $gri$ to avoid saturated stars and used the broader of the two S15 color ranges ($0.3 < g-i < 1.0$) in order to have more stars to compare with. After applying these conditions, 50 of the 54 CfA SN II optical star sequences still had some stars in common with the PS1 Catalog. Additionally, we had a star sequence (but no light curve) for SN 2004et that we compared with PS1, bringing the number of star sequences compared with PS1 to 51. The weighted means of the CfA-minus-PS1 matched-star differences are shown in Table \[table\_cfa\_ps1\_sample\] for the CfA SN II sample as a whole, and for the separate CfA3$_\mathrm{4SH}$, CfA3$_\mathrm{KEP}$ and CfA4 matched-star subsamples. No effort was made to convert the CfA3$_\mathrm{4SH}$ $RI$ bands into $ri$ and so only $BV$ comparisons are available for CfA3$_\mathrm{4SH}$ stars. The weighted-average differences of all the stars combined for the respective $BVr'i'$ bands are -0.018, 0.001, 0.012 and -0.014 mag. [l|rrrr]{} CfA SN II stars$ - $PS1 (all matches) & -17.5 $\pm$1.4 & 1.0 $\pm$0.8 & 12.3 $\pm$1.0 & -14.1 $\pm$1.1\ CfA3$_\mathrm{4SH}$ SN II stars$ - $PS1 & -24.9 $\pm$6.9 &-0.4 $\pm$4.6 & &\ S15 CfA3$_\mathrm{4SH} - $PS1 & -34.5 $\pm$5.3 & 1.8 $\pm$3.8 & &\ CfA3$_\mathrm{KEP}$ SN II stars$ - $PS1 & -11.8 $\pm$3.6 &12.3 $\pm$2.2 & 25.1 $\pm$2.8 & 0.0 $\pm$2.8\ S15 CfA3$_\mathrm{KEP} - $PS1 & -30.9 $\pm$5.3 & 2.6 $\pm$3.6 & 12.4 $\pm$3.5 & 0.8 $\pm$3.6\ CfA4$_\mathrm{KEP}$ SN II stars$ - $PS1 & -18.2 $\pm$1.6 &-0.8 $\pm$0.9 & 10.3 $\pm$ 1.1 & -16.5 $\pm$1.2\ S15 CfA4$_1 - $PS1 & -20.1 $\pm$5.7 & 4.5 $\pm$3.6 & 4.9 $\pm$3.3 & -0.8 $\pm$3.6\ S15 CfA4$_2 - $PS1 & 5.2 $\pm$6.4 & 5.1 $\pm$3.7 & 9.0 $\pm$3.4 & -1.4 $\pm$3.6\ [lrrrrrrrr]{} SN2001ez & -0.059 $\pm$0.023 & 10 & -0.034 $\pm$0.016 & 10 & & & &\ SN2001fa & -0.028 $\pm$0.030 & 10 & 0.004 $\pm$0.021 & 10 & & & &\ SN2002bx & 0.014 $\pm$0.021 & 8 & 0.020 $\pm$0.014 & 8 & & & &\ SN2002em & -0.026 $\pm$0.008 & 11 & 0.000 $\pm$0.005 & 11 & & & &\ SN2004et & -0.007 $\pm$0.008 & 8 & 0.014 $\pm$0.005 & 8 & 0.016 $\pm$0.005 & 8 & -0.002 $\pm$0.005 & 8\ SN2005kd & -0.074 $\pm$0.008 & 9 & -0.002 $\pm$0.005 & 9 & 0.023 $\pm$0.006 & 9 & -0.009 $\pm$0.006 & 9\ SN2006at & -0.014 $\pm$0.030 & 4 & -0.011 $\pm$0.020 & 4 & 0.027 $\pm$0.027 & 4 & -0.002 $\pm$0.027 & 4\ SN2006be & 0.026 $\pm$0.026 & 5 & 0.031 $\pm$0.017 & 5 & 0.065 $\pm$0.023 & 5 & 0.030 $\pm$0.024 & 5\ SN2006bl & -0.015 $\pm$0.006 & 18 & 0.001 $\pm$0.004 & 18 & 0.015 $\pm$0.005 & 18 & -0.014 $\pm$0.005 & 18\ SN2006bo & -0.001 $\pm$0.026 & 6 & -0.019 $\pm$0.018 & 6 & -0.001 $\pm$0.023 & 6 & -0.050 $\pm$0.024 & 6\ SN2006bv & -0.023 $\pm$0.026 & 6 & -0.008 $\pm$0.018 & 6 & 0.020 $\pm$0.023 & 6 & -0.014 $\pm$0.024 & 6\ SN2006ca & -0.013 $\pm$0.050 & 1 & -0.029 $\pm$0.035 & 1 & -0.017 $\pm$0.045 & 1 & -0.052 $\pm$0.047 & 1\ SN2006cd & -0.004 $\pm$0.023 & 6 & -0.008 $\pm$0.016 & 6 & 0.026 $\pm$0.021 & 6 & -0.002 $\pm$0.021 & 6\ SN2006gy & -0.028 $\pm$0.004 & 38 & -0.012 $\pm$0.002 & 38 & -0.008 $\pm$0.003 & 38 & -0.059 $\pm$0.003 & 38\ SN2006it & -0.007 $\pm$0.005 & 27 & -0.019 $\pm$0.003 & 27 & -0.009 $\pm$0.004 & 27 & -0.050 $\pm$0.006 & 27\ SN2006iw & -0.024 $\pm$0.048 & 2 & -0.003 $\pm$0.034 & 2 & 0.016 $\pm$0.043 & 2 & 0.003 $\pm$0.045 & 2\ SN2006ov & -0.095 $\pm$0.035 & 1 & 0.023 $\pm$0.014 & 1 & 0.024 $\pm$0.017 & 1 & -0.021 $\pm$0.017 & 1\ SN2007Q & 0.066 $\pm$0.027 & 7 & 0.030 $\pm$0.018 & 7 & 0.043 $\pm$0.024 & 7 & -0.004 $\pm$0.025 & 7\ SN2007T & 0.117 $\pm$0.022 & 8 & 0.092 $\pm$0.015 & 8 & 0.088 $\pm$0.020 & 8 & 0.052 $\pm$0.020 & 8\ SN2007aa & -0.011 $\pm$0.023 & 3 & 0.009 $\pm$0.015 & 3 & 0.023 $\pm$0.017 & 3 & -0.005 $\pm$0.015 & 3\ SN2007ad & -0.035 $\pm$0.047 & 3 & -0.025 $\pm$0.033 & 3 & -0.034 $\pm$0.042 & 3 & -0.030 $\pm$0.044 & 3\ SN2007ah & -0.014 $\pm$0.008 & 11 & -0.013 $\pm$0.005 & 11 & 0.001 $\pm$0.005 & 11 & -0.038 $\pm$0.005 & 11\ SN2007av & -0.012 $\pm$0.009 & 7 & 0.021 $\pm$0.006 & 7 & 0.018 $\pm$0.011 & 7 & -0.008 $\pm$0.011 & 7\ SN2007bb & 0.138 $\pm$0.026 & 5 & 0.111 $\pm$0.018 & 5 & 0.104 $\pm$0.023 & 5 & 0.048 $\pm$0.024 & 5\ SN2007be & -0.013 $\pm$0.023 & 8 & 0.029 $\pm$0.016 & 8 & 0.046 $\pm$0.021 & 8 & 0.028 $\pm$0.021 & 8\ SN2007bf & -0.022 $\pm$0.027 & 9 & 0.015 $\pm$0.018 & 9 & 0.043 $\pm$0.024 & 9 & 0.038 $\pm$0.025 & 9\ SN2007cd & 0.003 $\pm$0.023 & 8 & 0.035 $\pm$0.016 & 8 & -0.057 $\pm$0.021 & 8 & -0.073 $\pm$0.021 & 8\ SN2007ck & 0.008 $\pm$0.010 & 12 & 0.004 $\pm$0.005 & 12 & 0.032 $\pm$0.006 & 12 & 0.010 $\pm$0.007 & 12\ SN2007cm & 0.041 $\pm$0.027 & 1 & 0.053 $\pm$0.016 & 1 & 0.078 $\pm$0.017 & 1 & 0.059 $\pm$0.017 & 1\ SN2007ct & 0.032 $\pm$0.011 & 37 & 0.038 $\pm$0.006 & 37 & 0.048 $\pm$0.008 & 37 & 0.015 $\pm$0.007 & 37\ SN2007cu & 0.022 $\pm$0.014 & 7 & 0.036 $\pm$0.007 & 7 & 0.054 $\pm$0.009 & 7 & 0.032 $\pm$0.009 & 7\ SN2007hv & -0.053 $\pm$0.006 & 23 & -0.003 $\pm$0.003 & 23 & 0.010 $\pm$0.004 & 23 & -0.042 $\pm$0.004 & 23\ SN2007od & -0.000 $\pm$0.031 & 3 & -0.007 $\pm$0.022 & 3 & 0.030 $\pm$0.028 & 3 & 0.010 $\pm$0.028 & 3\ SN2007pk & -0.034 $\pm$0.010 & 20 & -0.013 $\pm$0.005 & 20 & -0.001 $\pm$0.008 & 20 & -0.029 $\pm$0.008 & 20\ SN2007rt & -0.001 $\pm$0.011 & 6 & 0.014 $\pm$0.006 & 6 & 0.032 $\pm$0.008 & 6 & 0.011 $\pm$0.009 & 6\ SN2008aj & -0.014 $\pm$0.014 & 4 & -0.008 $\pm$0.007 & 4 & 0.010 $\pm$0.009 & 4 & -0.011 $\pm$0.009 & 4\ SN2008B & -0.008 $\pm$0.026 & 4 & 0.015 $\pm$0.014 & 4 & 0.055 $\pm$0.017 & 4 & 0.041 $\pm$0.018 & 4\ SN2008F & 0.036 $\pm$0.013 & 7 & 0.038 $\pm$0.007 & 7 & 0.051 $\pm$0.009 & 7 & 0.024 $\pm$0.009 & 7\ SN2008bj & 0.006 $\pm$0.010 & 6 & 0.028 $\pm$0.007 & 6 & 0.054 $\pm$0.008 & 6 & 0.018 $\pm$0.008 & 6\ SN2008bn & -0.026 $\pm$0.010 & 7 & 0.001 $\pm$0.006 & 7 & 0.021 $\pm$0.007 & 7 & 0.004 $\pm$0.007 & 7\ SN2008bu & -0.077 $\pm$0.006 & 17 & -0.023 $\pm$0.004 & 17 & -0.040 $\pm$0.005 & 17 & -0.042 $\pm$0.005 & 17\ SN2008bx & -0.031 $\pm$0.011 & 11 & 0.016 $\pm$0.006 & 11 & 0.033 $\pm$0.007 & 11 & 0.003 $\pm$0.007 & 11\ SN2008gm & -0.023 $\pm$0.010 & 6 & -0.008 $\pm$0.006 & 6 & 0.009 $\pm$0.007 & 6 & -0.027 $\pm$0.007 & 6\ SN2008in & -0.057 $\pm$0.013 & 4 & -0.018 $\pm$0.008 & 4 & 0.023 $\pm$0.010 & 4 & -0.019 $\pm$0.009 & 4\ SN2008ip & -0.015 $\pm$0.012 & 5 & 0.011 $\pm$0.007 & 5 & 0.034 $\pm$0.008 & 5 & 0.019 $\pm$0.008 & 5\ SN2009ay & -0.013 $\pm$0.007 & 14 & -0.002 $\pm$0.004 & 14 & 0.011 $\pm$0.005 & 14 & -0.006 $\pm$0.005 & 14\ SN2009dd & -0.024 $\pm$0.032 & 1 & 0.014 $\pm$0.027 & 1 & 0.048 $\pm$0.020 & 1 & 0.020 $\pm$0.017 & 1\ SN2009kn & -0.001 $\pm$0.005 & 42 & 0.005 $\pm$0.003 & 42 & 0.011 $\pm$0.004 & 42 & -0.005 $\pm$0.004 & 42\ SN2009N & -0.019 $\pm$0.013 & 4 & 0.001 $\pm$0.008 & 4 & 0.011 $\pm$0.010 & 4 & -0.010 $\pm$0.010 & 4\ SN2010aj & -0.016 $\pm$0.014 & 4 & 0.003 $\pm$0.008 & 4 & 0.029 $\pm$0.010 & 4 & 0.019 $\pm$0.010 & 4\ SN2010al & -0.009 $\pm$0.006 & 19 & 0.006 $\pm$0.004 & 19 & 0.025 $\pm$0.004 & 19 & 0.015 $\pm$0.004 & 19\ [lrrrrrrrr]{} SN2006be & 0.004 & 5 & 0.036 & 5 & 0.058 & 4 & 0.036 & 5\ SN2006bl & -0.016 & 13 & 0.002 & 13 & 0.015 & 13 & 0.002 & 13\ SN2006bo & -0.023 & 4 & -0.017 & 4 & -0.083 & 4 & -0.017 & 4\ SN2006it & -0.025 & 12 & -0.015 & 12 & -0.019 & 12 & -0.015 & 12\ SN2007aa & 0.016 & 10 & 0.017 & 10 & 0.043 & 10 & 0.017 & 10\ SN2007od & -0.018 & 6 & -0.015 & 6 & -0.014 & 6 & -0.015 & 6\ SN2008bu & -0.006 & 9 & 0.019 & 9 & 0.024 & 9 & 0.019 & 9\ SN2009N & 0.029 & 8 & 0.039 & 8 & 0.051 & 7 & 0.039 & 8\ [lcrr]{} SN2006bo & B & -0.027 & 3\ SN2006bo & V & -0.046 & 4\ SN2006bo & r’ & -0.033 & 4\ SN2006bo & i’ & -0.011 & 4\ SN2006be & V & 0.071 & 7\ SN2006bl & V & 0.106 & 5\ SN2006it & V & -0.038 & 1\ SN2007aa & V & -0.010 & 5\ SN2007od & V & -0.015 & 6\ SN2008bu & V & -0.033 & 1\ SN2009N & V & 0.032 & 1\ In this work, the CfA $B$ calibration is about 0.02 brighter, in the same direction found by S15 in their Table 4 (NGSL column). It should be noted that the comparisons of this work and S15 are not the same. First of all, the star sequences in S15 are different from those in this work. More importantly, S15 used observed natural-system and synthetic natural-system photometry while this work is using standard system photometry so any similarities or differences in the values between the two works’ offsets are not conclusive but merely suggestive. In this work, CfA $V$ has virtually no offset with the PS1 conversion to $V$ for the CfA3$_\mathrm{4SH}$ and CfA4 subsamples, in rough agreement with S15. However, the CfA3$_\mathrm{KEP}$ subsample is 0.012 mag fainter, about 0.01 mag fainter than the offset S15 found. CfA $r'$ is fainter than PS1 in this work, as is the case in S15, though the offset here is greater than in S15. There is excellent agreement between the CfA3$_\mathrm{KEP}$ $i'$ offsets in the two works but the CfA4 $i'$ offset in this work is 0.0165 mag brighter while there is virtually no offset in S15. With the exception of CfA3$_\mathrm{KEP}$ $V$ and CfA4 $i'$, there is reasonable directional agreement in the CfA-minus-PS1 offsets of this work and S15, in the sense of both agreeing on which calibration is fainter or brighter. On a SN-by-SN basis, Table \[table\_cfa\_ps1\_eachsn\] shows the weighted mean of the CfA-minus-PS1 star differences for each CfA SN II that had matches. 19 of the 51 SN II (37$\%$) have $V$ offsets within $\pm$0.01 mag, 34 (67%) have $V$ offsets within $\pm$0.02 mag, 42 (82%) within $\pm$0.03 mag, and 48 (94%) within $\pm$0.038 mag. These can be considered to have reasonable agreement with PS1. However, there are 3 SN with larger calibration discrepancies. SN 2007bb has a $V$ offset of 0.111 mag based on 5 matched stars, SN 2007T has a 0.092 mag offset based on 8 stars and SN 2007cm has a 0.053 mag offset based on only 1 star. Comparison of CfA and CSP Star Sequences and SN II Light Curves --------------------------------------------------------------- The CSP provided us with standard-system $BVr'i$ star sequences for their upcoming SN II light curve publication (Carlos Contreras — private communication). Table \[table\_cfa\_csp\_stars\] shows the $BVr'i'$ comparisons between 8 CfA and CSP star sequences, with the average of the differences in $V$ ranging from -0.014 to 0.035 mag, showing reasonable agreement. We also took the CSP natural system $V$ light curves for 7 objects from @anderson and the natural system $BVr'i'$ light curves for SN 2006bo from @taddia and applied color terms[^12] to convert them to the standard system and compare with the CfA standard system light curves from Table \[table\_stdlightcurve\]. Applying star-derived color terms to SN photometry to create standard system light curves is fraught with danger and inaccuracy (the user is encouraged to use the natural system light curves whenever possible) so this is intended only as a rough comparison and sanity check. Only points that are within 0.6d of each other are matched. No interpolation was used to provide for more points of comparison. Table \[table\_cfa\_csp\_lc\] shows the average of the light curve differences for the 8 SN II in common (11 comparisons in total: 8 in $V$ and 1 in each of $Br'i'$). Of the 11 comparisons, eight are within $\pm$0.04 mag while one has CfA 0.046 mag brighter, another has CfA 0.071 mag fainter and the most disparate one has CfA 0.106 mag fainter. This is comparable to the similarity that the CfA3 and CfA4 SN Ia light curves had with other groups’ light curves in @Hicken09 and @Hicken12 and suggests that the CfA SN II light curves are of reasonable accuracy. Conclusion =========== The CfA SN II light curve sample consists of 60 multiband light curves: 46 have only optical, six have only NIR and eight have both optical and NIR. We also present 192 optical spectra for 47 of the 60 SN. This work includes a total of 2932 optical and 816 NIR light curve points. There are 26 optical and two NIR light curves of SN II/IIP with redshifts $z > 0.01$, some of which may give rise to useful distances for Hubble diagrams. Select comparisons with other groups’ data show reasonable agreement in the vast majority of cases. These light curves add to a growing body of SN II data from the literature. This collective body of data (from the literature, this work and future samples) will be useful for providing greater insight into SN II explosions, developing analytic methods for photometric SN classification, and opening the path for their increasing use as cosmological distance indicators. Having another method for measuring the expansion history of the Universe, independent from SN Ia, will serve the important purpose of either confirming the results derived from SN Ia or offering insights for improving the cosmological use of SN Ia. Having an independent set of distances will enable a deeper study of SN Ia systematic errors and evolution with redshift. A larger set of SN II light curves is also useful for efforts to better photometrically identify SN types. By having a more complete picture of the range and frequency of SN II light curve properties, positive identification of SN II will be more likely, which will be useful both in building up a sample of SN II and in excluding them with confidence from samples of other types, particularly SN Ia. This will become increasingly important as large surveys provide far more SN light curves than spectroscopic identification resources can handle. 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--- abstract: 'We obtain a characteristic-free decomposition of tensor space, regarded as a module for the Brauer centralizer algebra.' address: 'Mathematics and Statistics, Loyola University Chicago, Chicago, Illinois 60626 U.S.A.' author: - 'S.R. Doty' title: 'A characteristic-free decomposition of tensor space as a Brauer algebra module' --- Introduction {#introduction .unnumbered} ============ The representation theory of a symmetric group ${\mathfrak{S}}_r$ on $r$ letters (see [@James; @Fulton]) starts with the transitive permutation modules $M^\lambda$ indexed by the partitions $\lambda$ of $r$. For any field $k$, tensor space $(k^n)^{\otimes r}$, regarded as a $k{\mathfrak{S}}_r$-module via the place permutation action, admits a direct sum decomposition into a direct sum of the $M^\lambda$. (If $n<r$ then not all of the $M^\lambda$ appear in the decomposition.) The purpose of this paper is to give a similar characteristic-free decomposition of tensor space $(k^n)^{\otimes r}$, regarded as a module for the Brauer algebra. (Characteristic 2 is excluded from some results, in order to avoid technicalities.) The main results are summarized together in Section \[sec:main\] below, for the convenience of the reader. A different characteristic-free decomposition of $(k^n)^{\otimes r}$ as a module for the Brauer algebra was previously obtained in [@HP], by working with the action defined in terms of the standard bilinear form on $k^n$. By choosing a different bilinear form, we obtain a more refined decomposition than that of [@HP], in most cases, which should give more information. Our approach is motivated by Schur–Weyl duality (see [@Schur; @Doty:SWD; @BD; @DDH; @DH]), although its full generality is not used here. All we need is the fact that the action of the Brauer algebra commutes with that of a suitable classical group. Our results provide new characteristic-free representations $N^\xi$ of the Brauer algebra, indexed by partitions $\xi$, which may be regarded as analogues of the classical transitive permutation modules for symmetric groups. It is hoped that these representations may be of some use in the study of the representation theory of Brauer algebras, especially in the non-semisimple case, where little is known. The paper is organized as follows. After summarizing the main results in Section \[sec:main\], we recall the decomposition of tensor space regarded as a module for the symmetric group in Section \[sec:Sym\], define the Brauer algebra and its action on tensors in Section \[sec:Brauer\], and prove our results in Sections \[sec:symmetric\], \[sec:symmetric-odd\], and \[sec:skew\]. Main results {#sec:main} ============ Fix a field $k$ of characteristic different from $2$. Tensor space $(k^n)^{\otimes r}$ is regarded as a module for the Brauer algebra ${\mathfrak{B}}_r(n)$ via an action (see Section \[sec:Brauer\]) defined by the nondegenerate symmetric bilinear form $(\ ,\ )$ on $k^n$ such that $(e_i,e_{j'})=\delta_{ij}$, where $e_1, \dots, e_n$ is the standard basis of $k^n$ and $j' = n+1-j$. This choice of bilinear form is important for our results. We will need the set $\Lambda(n,r)$ of $n$-part compositions of $r$, defined by $$\Lambda(n,r) = \{(\lambda_1, \dots, \lambda_n) \in {{\mathbb Z}}^n \colon 0 \le \lambda_i\ (\forall i),\ \lambda_1 + \cdots + \lambda_n = r\}.$$ For a given positive integer $l$, let sets $\Lambda_1(l,r)$ and $\Lambda_2(l,r)$ be defined as follows: $$\begin{aligned} \Lambda_1(l,r) &= \{(\xi_1, \dots, \xi_l) \in {{\mathbb Z}}^l \colon |\xi_1| + \cdots + |\xi_l| = r-s,\ 0\le s \le r \}\\ \Lambda_2(l,r) &= \{(\xi_1, \dots, \xi_l) \in {{\mathbb Z}}^l \colon |\xi_1| + \cdots + |\xi_l| = r-2s,\ 0\le 2s \le r\}.\end{aligned}$$ If $n=2l+1$, we have a surjective map $\pi \colon \Lambda(n,r) \to \Lambda_1(l,r)$ given by the rule $$\pi(\lambda_1, \dots, \lambda_n) = (\lambda_1 - \lambda_{1'}, \dots, \lambda_l - \lambda_{l'}).$$ If $n=2l$, the same rule defines a surjective map $\pi \colon \Lambda(n,r) \to \Lambda_2(l,r)$. In either case, the fibers of $\pi$ determine the desired decomposition of tensor space. See Sections \[sec:symmetric\], \[sec:symmetric-odd\] for combinatorial descriptions of the fibers, depending on the parity of $n$. The well known characteristic-free decomposition of $(k^n)^{\otimes r}$ as a $k{\mathfrak{S}}_r$-module, where ${\mathfrak{S}}_r$ is the symmetric group on $r$ letters, is given by $$(k^n)^{\otimes r} = \textstyle\bigoplus_{\lambda \in \Lambda(n,r)} M^\lambda ,$$ where $M^\lambda$ is a transitive permutation module for $k{\mathfrak{S}}_r$, realized as the $k$-span of all simple tensors $e_{i_1} \otimes \cdots \otimes e_{i_r}$ of weight $\lambda$, with ${\mathfrak{S}}_r$ acting by place permutation on the simple tensors. Given $\xi \in \Lambda_j(n,r)$ for $j=1,2$ we define $N^\xi := \bigoplus_{\lambda \in \pi^{-1}(\xi)} M^\lambda$. Then we prove the following: \[thm:1\] A characteristic-free decomposition (for characteristic $k \ne 2$) of $(k^n)^{\otimes r}$ as a ${\mathfrak{B}}_r(n)$-module is given by $$(k^n)^{\otimes r} = \textstyle\bigoplus_{\xi \in \Lambda_j(l,r)} N^\xi \qquad\quad (j=1,2),$$ where $j=1$ if $n=2l+1$ and $j=2$ if $n=2l$. This is nothing but a weight space decomposition of $(k^n)^{\otimes r}$, regarded as a module for the diagonal torus in the orthogonal group ${{\operatorname{O}}}_n(k)$, the group of matrices preserving the bilinear form $(\ ,\ )$ on $k^n$. The proof, which is given in Sections \[sec:symmetric\] and \[sec:symmetric-odd\], is almost trivial: the main idea is just the well known fact that the actions of ${{\operatorname{O}}}_n(k)$ and ${\mathfrak{B}}_r(n)$ on $k^n$ commute. The bilinear form is chosen so that the diagonal tori in ${\operatorname{GL}}_n(k)$ and ${{\operatorname{O}}}_n(k)$ are compatible upon restriction from ${\operatorname{GL}}_n(k)$ to ${{\operatorname{O}}}_n(k)$, which is what makes everything work. (It is well known that in Lie theory, our choice of defining form for the orthogonal group, or one very similar to it, is more natural than the standard defining form.) In case $n=2l$, we obtain another characteristic-free decomposition of $(k^n)^{\otimes r}$, with no restriction on the characteristic of $k$, by replacing the role of the orthogonal group in the above by the symplectic group ${{\operatorname{Sp}}}_{n}(k)$, defined as the set of matrices preserving the skew-symmetric bilinear form $(\ , \ )$ on $k^n$ given by $(e_i, e_{j'}) = \varepsilon_i \delta_{i,j}$, where $\varepsilon_j = 1$ if $j<j'$ and $-1$ otherwise. There is an action of ${\mathfrak{B}}_r(-n)$ on tensor space $(k^n)^{\otimes r}$, defined in terms of the bilinear form. \[thm:2\] If $n=2l$, a characteristic-free decomposition of $(k^n)^{\otimes r}$ as a ${\mathfrak{B}}_r(-n)$-module is given by $$(k^n)^{\otimes r} = \textstyle\bigoplus_{\xi \in \Lambda_2(l,r)} N^\xi.$$ Again, this is just a weight space decomposition for the torus of diagonal matrices in ${{\operatorname{Sp}}}_n(k)$. The proof in this case is quite similar to the even orthogonal case, and is sketched in Section \[sec:skew\]. These results provide a new family $\{ N^\xi \}$ of characteristic free representations of the Brauer algebra, indexed by $\xi \in \Lambda_1(l,r)$ or $\Lambda_2(l,r)$. Actually, we show that the hyperoctahedral group $({{\mathbb Z}}/2{{\mathbb Z}})^l \rtimes {\mathfrak{S}}_l$ acts naturally on either of the sets $\Lambda_1(l,r)$ or $\Lambda_2(l,r)$ through signed permutations of the entries of a weight, and modules $N^\xi$ indexed by weights in the same orbit are all isomorphic, so it suffices to restrict one’s attention to the modules $N^\xi$ indexed by the dominant weights $\xi$, which are partitions. So, up to isomorphism, the Brauer algebra direct summands of tensor space are indexed by the subsets $\Lambda^+_1(l,r)$ or $\Lambda^+_2(l,r)$ of partitions in $\Lambda_1(l,r)$ or $\Lambda_2(l,r)$, respectively. The modules $N^\xi$ are defined by gluing various permutation modules $M^\lambda$ together. The analysis in Sections \[sec:symmetric\] and \[sec:skew\] reveal that when $n =2l$ and $\xi = (\xi_1, \dots, \xi_l)$ is a partition of $r-2s$ into not more than $l$ parts, for $0 \le 2s \le r$, the various $\lambda$ in the fiber $\pi^{-1}(\xi)$ are precisely the weights of the form $$(\xi+\nu)\parallel \nu^* \qquad\text{for } \nu \in \Lambda(l,s)$$ where $\nu^* = (\nu_s, \dots, \nu_1)$ is the *reverse* of $\nu = (\nu_1, \dots, \nu_s)$ and where $\parallel$ denotes concatenation of finite sequences: $$(a_1, \dots, a_i) \parallel (b_1, \dots, b_j) := (a_1, \dots, a_i, b_1, \dots b_j).$$ Hence, in this case $N^\xi$ is the direct sum of $|\Lambda(l,s)|$ permutation modules. In particular, if $s=0$ there is just one permutation module in $N^\xi$. In case $n = 2l+1$ the analysis in Section \[sec:symmetric-odd\] reveals that if $\xi = (\xi_1, \dots, \xi_l)$ is a given partition of $r-s$ into not more than $l$ parts, where $0 \le s \le r$, the various $\lambda$ in the fiber $\pi^{-1}(\xi)$ are precisely the weights of the form $$(\xi+\nu) \parallel (s-2t)\parallel \nu^* \qquad \text{for }\nu \in \Lambda(l,t)$$ as $t$ varies over all possibilities in the range $0 \le 2t \le s$. Hence, in this case $N^\xi$ is the direct sum of $\sum_{0 \le 2t \le s} |\Lambda(l,t)|$ permutation modules. Symmetric group decomposition of $(k^n)^{\otimes r}$ {#sec:Sym} ==================================================== Let $k$ be an arbitrary field. Consider an $n$-dimensional vector space $k^n$ and its associated group ${\operatorname{GL}}_n(k)$ of linear automorphisms. The group acts naturally on the space, and thus also acts naturally on the $r$-fold tensor product $(k^n)^{\otimes r}$, via the ‘diagonal’ action: $$\label{eq:GL-action} g \cdot (v_1 \otimes \cdots \otimes v_r) = (g \cdot v_1) \otimes \cdots \otimes (g \cdot v_r).$$ The symmetric group ${\mathfrak{S}}_r$ also acts on the right on $(k^n)^{\otimes r}$, via the so-called ‘place permutation’ action, which satisfies $$\label{eq:Sym-action} (v_1 \otimes \cdots \otimes v_r) \cdot \pi = v_{(1)\pi^{-1}} \otimes \cdots \otimes v_{(r)\pi^{-1}}.$$ Notice that we adopt the convention that elements of ${\mathfrak{S}}_r$ act on the right of their arguments. Now it is clear from the definitions that the actions of these two groups commute: $$g \cdot \big( (v_1 \otimes \cdots \otimes v_r) \cdot \pi \big) = \big( g \cdot (v_1 \otimes \cdots \otimes v_r) \big) \cdot \pi,$$ for all $g \in {\operatorname{GL}}_n(k)$, $\pi \in {\mathfrak{S}}_r$. In order to simplify the notation, we put $V:= k^n$. We more or less follow Section 3 of [@Green]. The group ${\operatorname{GL}}_n(k)$ contains an abelian subgroup $T$ consisting of the diagonal matrices in ${\operatorname{GL}}_n(k)$, and this subgroup (being abelian) must act semisimply on $V^{\otimes r} = (k^n)^{\otimes r}$. This leads in the usual way to a ‘weight space’ decomposition $$\label{eq:wtspace} V^{\otimes r} = \textstyle \bigoplus_{\lambda\in X(T)} V^{\otimes r}_\lambda,$$ where $\lambda$ varies over the group $X(T)$ of characters $\lambda \colon T \to k^\times$, and where the weight space $V^{\otimes r}_\lambda$ is the linear span of the tensors $v=v_1 \otimes \cdots \otimes v_r$ such that $t \cdot v = \lambda(t) v$, for all $t \in T$. Clearly $T$ is isomorphic to the direct product $(k^\times)^n$ of $n$ copies of the multiplicative group $k^\times$ of the field. Let ${\varepsilon}_i \in X(T)$ be evaluation at the $i$th diagonal entry of an element of $T$. Regarding the abelian group $X(T)$ as an additive group as usual, observe that ${\varepsilon}_1, \dots, {\varepsilon}_n$ is a basis for $X(T)$, and thus the map ${{\mathbb Z}}^n \to X(T)$ given by $(\lambda_1, \dots, \lambda_n) \mapsto \sum_i \lambda_i {\varepsilon}_i$ is an isomorphism. So we identify $X(T)$ with ${{\mathbb Z}}^n$ by means of this isomorphism. The direct sum in is formally taken over $X(T)$; however, many of the summands are actually zero. It is easy to check that the weight space decomposition of $V$, regarded as a $T$-module, is given by $V = V_{{\varepsilon}_1} \oplus \cdots \oplus V_{{\varepsilon}_n}$. It follows immediately that the set of weights of $V^{\otimes r}$ is the set $$\Lambda(n,r) = \{ (\lambda_1, \dots \lambda_n) \in {{\mathbb Z}}^n \colon \lambda_i \ge 0, \lambda_1 + \cdots + \lambda_n = r \}$$ of $n$-part compositions of $r$, under the isomorphism of $X(T)$ with ${{\mathbb Z}}^n$. Thus we may write in the better form $$\label{eq:wtspaceLambda} V^{\otimes r} = \textstyle \bigoplus_{\lambda\in \Lambda(n,r)} M^\lambda$$ where we have, partly to simplify notation but also to serve tradition, put $M^\lambda := V^{\otimes r}_\lambda$. Since the actions of ${\mathfrak{S}}_r$ and ${\operatorname{GL}}_n(k)$ commute, each $M^\lambda$ is a $k{\mathfrak{S}}_r$-module, so gives a decomposition of $V^{\otimes r}$ as $k{\mathfrak{S}}_r$-modules. Let us describe the vector space $M^\lambda$ in greater detail. Let $e_1, \dots, e_n$ be the standard basis of $V = k^n$. Then $M^\lambda$, for any $\lambda \in \Lambda(n,r)$, has a basis given by the set of simple tensors $e_{i_1} \otimes \cdots \otimes e_{i_r}$ such that in the multi-index $(i_1, \dots, i_r)$ there are exactly $\lambda_1$ occurrences of $1$, $\lambda_2$ occurrences of $2$, and so forth. Evidently the action of the symmetric group ${\mathfrak{S}}_r$ permutes such simple tensors transitively, so $M^\lambda$ is in fact a transitive permutation module. The representation theory of ${\mathfrak{S}}_r$ over $k$ starts with these permutation modules (see e.g. [@James; @Fulton]) usually defined rather differently. At this point we could introduce row standard tableaux of shape $\lambda$ (or, equivalently, the “tabloids” of [@James]) to label our basis elements of $M^\lambda$, but we shall have no need of such combinatorial gadgets. The symmetric group ${\mathfrak{S}}_n$ can be identified with the Weyl group $W$ of ${\operatorname{GL}}_n(k)$. (Recall that the theory of BN-pairs (due to J. Tits) can be used to define $W$ in any ${\operatorname{GL}}_n(k)$ by a uniform method, including the case when $k$ is finite.) The group $W$ may be identified with the subgroup of permutation matrices of ${\operatorname{GL}}_n(k)$, so it acts naturally (on the left) on $V^{\otimes r}$ by restriction of the action of ${\operatorname{GL}}_n(k)$. Moreover, $W={\mathfrak{S}}_n$ acts on the set ${{\mathbb Z}}^n$ by $$w^{-1}(\lambda_1, \dots, \lambda_n) = (\lambda_{w(1)}, \dots, \lambda_{w(n)}).$$ This action stabilizes the set $\Lambda(n,r)$, so we have also an action of $W$ on $\Lambda(n,r)$. Each $W$-orbit of $\Lambda(n,r)$ contains exactly one *dominant* weight: a weight $\lambda = (\lambda_1, \dots, \lambda_n)$ such that $\lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_n$. Denote the set of dominant weights in $\Lambda(n,r)$ by $\Lambda^+(n,r)$. This set may be identified with the set of partitions of $r$ into not more than $n$ parts. The following is immediate from Proposition (3.3a) of [@Green]. \[prop:1\] For any $w \in W$, the right $k{\mathfrak{S}}_r$-modules $M^\lambda$ and $M^{w(\lambda)}$ are isomorphic. The isomorphism is given on basis elements by mapping a simple tensor $e_{i_1} \otimes \cdots \otimes e_{i_r}$ of weight $\lambda$ to the simple tensor $e_{w(i_1)} \otimes \cdots \otimes e_{w(i_r)}$ of weight $w(\lambda)$. Thus, when considering the $M^\lambda$, we may as well confine our attention to the ones labeled by dominant weights (i.e. partitions) $\lambda \in \Lambda^+(n,r)$. The Brauer algebra {#sec:Brauer} ================== A Brauer $r$-diagram (introduced in [@Brauer]) is an undirected graph with $2r$ vertices and $r$ edges, such that each vertex is the endpoint of precisely one edge. By convention, such a graph is usually drawn in a rectangle with $r$ vertices each equally spaced along the top and bottom edges of the rectangle. For example, the picture below $$\begin{tikzpicture}[scale=0.7] \draw (0,0) rectangle (8,3); \foreach \x in {0.5,1.5,...,7.5} {\fill (\x,3) circle (2pt); \fill (\x,0) circle (2pt);} \begin{scope}[very thick] \draw (0.5,3) -- (3.5,0); \draw (2.5,3) arc (180:360:0.5 and 0.25); \draw (1.5,3) arc (180:360:1.5 and 0.75); \draw (2.5,0) arc (0:180:1 and 0.5); \draw (6.5,3) -- (1.5,0); \draw (7.5,0) arc (0:180:1.5 and 0.75); \draw (5.5,3) -- (6.5,0); \draw (7.5,3) -- (5.5,0); \end{scope} \end{tikzpicture}$$ depicts a Brauer $8$-diagram. Let $k$ be an arbitrary field. Let ${\mathfrak{B}}_r(\pm n)$ be the vector space over $k$ with basis the $r$-diagrams, where we assume that $n$ is even in the negative case. Brauer defined a natural multiplication of $r$-diagrams such that ${\mathfrak{B}}_r(\pm n)$ becomes an associative algebra. In order to describe the multiplication rule, it is convenient to introduce the notations ${\tau}(d)$ and ${\beta}(d)$ for the sets of vertices along the top and bottom edges of a diagram $d$. Then the multiplication rule works as follows. Given $r$-diagrams $d_1$ and $d_2$, place $d_1$ above $d_2$ and identify the vertices in ${\beta}(d_1)$ in order with those in ${\tau}(d_2)$. The resulting graph consists of $r$ paths whose endpoints are in ${\tau}(d_1) \cup {\beta}(d_2)$, along with a certain number, say $s$, of cycles which involve only vertices in the middle row. Let $d$ be the $r$-diagram whose edges are obtained from the paths in this graph. Then the product of $d_1$ and $d_2$ in ${\mathfrak{B}}_r(\pm n)$ is given by $d_1 d_2 = (\pm n \cdot 1_k)^s d$. Now we describe a right action of ${\mathfrak{B}}_r(\pm n)$ on $V^{\otimes r}$, which depends on the defining bilinear form $(\ ,\ )$. This is the symmetric form defined in Section \[sec:main\] in the positive case and the skew-symmetric form defined in Section \[sec:main\] in the negative case. We always assume characteristic $k \ne 2$ in the symmetric case. We let $e^*_1, \dots, e^*_n$ be the basis dual to the standard basis $e_1, \dots, e_n$ of $k^n$ with respect to the bilinear form, in either case, so that $(e_i, e^*_j) = \delta_{ij}$. Given any $r$-diagram $d$, let $(d)\varphi$ be the matrix whose $(\underline{i}, \underline{j})$-entry, for $\underline{i} = (i_1, \dots, i_r)$, $\underline{j}=(j_1,\dots, j_r)$ is determined by the following procedure: 1. Label the vertices along the top edge of $d$ from left to right with $e_{i_1}, \dots, e_{i_r}$ and label the vertices along the bottom edge from left to right with $e^*_{j_1}, \dots, e^*_{j_r}$. 2. The $(\underline{i}, \underline{j})$-entry of $(d)\varphi$ is the product of the values $(u,v)$ over the edges ${\varepsilon}$ of $d$, where for each edge, $u$ and $v$ are the labels on its vertices, ordered so that a vertex in ${\tau}(d)$ precedes one in ${\beta}(d)$, and from left to right within ${\tau}(d)$ and ${\beta}(d)$. This determines the desired action: let $d$ act on $V^{\otimes r}$ as the linear endomorphism determined by the matrix $(d)\varphi$. Then $\varphi$ extends linearly to a representation $\varphi\colon {\mathfrak{B}}_r(\pm n)^{opp} \to {\operatorname{End}}_k(V^{\otimes r})$. It will be useful to have a better understanding of the action of ${\mathfrak{B}}_r(\pm n)$. For this, observe that the $r$-diagrams in which every edge connects a vertex in the top row to a vertex in the bottom row correspond to permutations in ${\mathfrak{S}}_r$, and their action on $V^{\otimes r}$ is the same as that defined by . Let us agree to call such diagrams *permutation* diagrams. Since we write maps in ${\mathfrak{S}}_r$ on the right of their arguments, the multiplication of diagrams defined above corresponds to composition of permutations, when restricted to such diagrams. Thus we have a subalgebra of ${\mathfrak{B}}_r(\pm n)$, namely the subalgebra spanned by the permutation diagrams, isomorphic to $k{\mathfrak{S}}_r$, and this subalgebra acts on $V^{\otimes r}$ via the usual place-permutation action, independently of the choice of defining bilinear form $(\ , \ )$. Now let $c_0$ be the unique $r$-diagram in which the first two vertices in ${\tau}(c_0)$ are joined by an edge, and similarly for the first two vertices in ${\beta}(c_0)$, with the $j$th vertex in ${\tau}(c_0)$ joined to the $j$th vertex in ${\beta}(c_0)$ for $j=3, \dots, r$. For instance, in case $r = 8$ the diagram $c_0$ $$\begin{tikzpicture}[scale=0.7] \draw (0,0) rectangle (8,3); \foreach \x in {0.5,1.5,...,7.5} {\fill (\x,3) circle (2pt); \fill (\x,0) circle (2pt);} \begin{scope}[very thick] \draw (0.5,3) arc (180:360:0.5 and 0.25); \draw (1.5,0) arc (0:180:0.5 and 0.25); \draw (2.5,3) -- (2.5,0); \draw (3.5,3) -- (3.5,0); \draw (4.5,3) -- (4.5,0); \draw (5.5,3) -- (5.5,0); \draw (6.5,3) -- (6.5,0); \draw (7.5,3) -- (7.5,0); \end{scope} \end{tikzpicture}$$ is the diagram pictured above. It is well known (see [@Brown]) that ${\mathfrak{B}}_r(\pm n)$ is generated by the permutation diagrams together with the diagram $c_0$. This may be argued as follows. Call an edge in a diagram $d$ *horizontal* if its endpoints both lie in ${\tau}(d)$, or both lie in ${\beta}(d)$. The number of horizontal edges in the top edge of the enclosing rectangle must equal the number in the bottom edge. Put $B_j$ equal to the span of the diagrams with exactly $2j$ horizontal edges. Then, as a vector space, ${\mathfrak{B}}_r(\pm n) = B_0 \oplus B_1 \oplus \cdots \oplus B_m$, where $m = \lfloor r/2 \rfloor$, the integer part of $r/2$. Clearly $B_0 = k{\mathfrak{S}}_r$. By acting on $c_0$ on the left or right by permutations, one can generate $B_1$. Then by picking diagrams in $B_1$ appropriately, one may obtain a diagram in $B_2$, and thus obtain all diagrams in $B_2$ by again acting by permutations on the left and right. Continuing in this way, one eventually generates all diagrams in the algebra. Thus, in order to unambiguously specify the action of the full algebra ${\mathfrak{B}}_r(\pm n)$ on $V^{\otimes r}$, we only need to see how the diagram $c_0$ acts. This depends on the choice of the defining bilinear form $(\ , \ )$, and by direct computation we see that in the symmetric case $c_0$ acts by the rule $$\textstyle (e_{i_1} \otimes \cdots \otimes e_{i_r}) \cdot c_0 = \delta_{i_1,i'_2} \sum_{j=1}^n e_j \otimes e_{j'} \otimes e_{i_3} \otimes \cdots \otimes e_{i_r}.$$ In the skew-symmetric case $c_0$ acts by the rule $$\textstyle (e_{i_1} \otimes \cdots \otimes e_{i_r}) \cdot c_0 = \delta_{i_1,i'_2} \sum_{j=1}^n {\varepsilon}_j\, e_j \otimes e_{j'} \otimes e_{i_3} \otimes \cdots \otimes e_{i_r}.$$ These actions are closely related to Weyl’s contraction operators in [@Weyl]. The ${\mathfrak{B}}_r(n)$ decomposition of $(k^n)^{\otimes r}$ in the\ symmetric case, where $n=2l$ {#sec:symmetric} ====================================================================== From now on, until further notice, we assume that the field $k$ has characteristic not $2$. This avoids technicalities pertaining to the definition of orthogonal groups over fields of characteristic $2$. We define ${{\operatorname{O}}}_n(k)$ to be the group of isometries of $V$ with respect to the symmetric form $(\ ,\ )$ given in Section \[sec:main\]. Then the action of ${\mathfrak{B}}_r(n)$ on tensor space $V^{\otimes r} = (k^n)^{\otimes r}$, defined in the preceding section, commutes with the natural action of ${{\operatorname{O}}}_n(k)$ (given by restricting the action of ${\operatorname{GL}}_n(k)$). Let ${\dot{T}}$ be the abelian subgroup of ${{\operatorname{O}}}_n(k)$ consisting of the diagonal matrices in ${{\operatorname{O}}}_n(k)$. Thus, a diagonal matrix $\text{diag}(t_1, \dots, t_n) \in {\operatorname{GL}}_n(k)$ belongs to ${\dot{T}}$ if and only if $$\label{eq:TO} t_i t_{i'} = 1 \quad\text{ for all $i = 1, \dots, n$.}$$ It will be useful to separate the consideration of the cases where $n$ is even and odd, so we assume that $n=2l$ for the remainder of this section, and consider the odd case in the next section. The description of ${\dot{T}}$ in shows in this case that ${\dot{T}}$ is isomorphic to the direct product $(k^\times)^l$ of $l = n/2$ copies of the multiplicative group $k^\times$ of the field $k$. So the character group $X({\dot{T}})$ is isomorphic to ${{\mathbb Z}}^l$, so we identify $X({\dot{T}})$ with ${{\mathbb Z}}^l$. There is a group homomorphism $\pi \colon X(T) \to X({\dot{T}})$ given by restriction: $\pi(\lambda) = \lambda_{\mid {\dot{T}}}$ for $\lambda \in X(T)$. Since ${\dot{T}}\subset T$, given a character $\xi \in X({\dot{T}})$, one can extend it to a character $\lambda \in X(T)$ such that $\lambda_{\mid {\dot{T}}} = \xi$. It follows that the map $\pi$ is surjective. In terms of the identifications $X(T) = {{\mathbb Z}}^n$ and $X({\dot{T}}) = {{\mathbb Z}}^l$, the map $\pi$ is given by the rule $$(\lambda_1, \dots, \lambda_n) \mapsto (\lambda_1 - \lambda_{1'}, \dots, \lambda_{l}-\lambda_{l'}).$$ We next consider how to characterize the image ${\Lambda_2}(l,r)$ of the set $\Lambda(n,r)$ under the map $\pi$. \[prop:2\] When $n = 2l$, the image ${\Lambda_2}(l,r)$ of the set $\Lambda(n,r)$ under $\pi$ is the set of all $\xi = (\xi_1, \dots, \xi_l) \in {{\mathbb Z}}^l$ such that $|\xi_1| + \cdots + |\xi_l|=r-2s$, where $0 \le 2s \le r$. If $\xi = \pi(\lambda)$ for $\lambda \in \Lambda(n,r)$ then $|\xi_1|+\cdots +|\xi_l|$ satisfies the condition $$|\xi_1|+\cdots +|\xi_l| = \epsilon_1(\lambda_1-\lambda_{1'}) + \cdots + \epsilon_l(\lambda_l-\lambda_{l'})$$ where for each $i=1, \dots, l$ the sign $\epsilon_i$ is defined to be $1$ if $\lambda_i \ge \lambda_{i'}$ and $-1$ otherwise. This is just a signed sum of the parts of $\lambda$, so is congruent modulo 2 to the sum of the parts of $\lambda$. Thus $|\xi_1|+\cdots +|\xi_l| = r-2s$ for some $s\in {{\mathbb Z}}$. Clearly $0 \le 2s \le r$. This proves the necessity of the condition for membership in ${\Lambda_2}(l,r)$. It remains to prove the sufficiency of the condition. Given $\xi \in {{\mathbb Z}}^l$ satisfying the condition $|\xi_1| + \cdots + |\xi_l|=r-2s$, where $0 \le 2s \le r$, we define a corresponding $\mu \in \Lambda(n,r-2s)$ as follows: put $\mu_i = \xi_i$ if $\xi_i > 0$, put $\mu_{i'} = -\xi_i$ if $\xi_i < 0$, and put all the other entries of $\mu = (\mu_1, \dots, \mu_n)$ to zero. Now pick $\nu \in \Lambda(l,s)$ arbitrarily. Then let $\lambda$ be obtained from $\mu$ and $\nu$ by adding the parts of $\nu$ in order to $(\mu_1, \dots, \mu_l)$ and by adding the parts of $\nu$ in reverse order to $(\mu_{l+1}, \dots, \mu_{2l})$, so that $$\lambda = (\mu_1+\nu_1, \dots, \mu_l+\nu_l, \mu_{l+1}+\nu_l, \dots, \mu_{2l}+\nu_1).$$ Then it easily checked that $\pi(\lambda) = \xi$. For each $\xi \in {\Lambda_2}(l,r)$, the proof of the preceding proposition reveals an algorithm for writing down the members of the fiber $\pi^{-1}(\xi)$, and in particular shows that the cardinality of the fiber is $|\Lambda(l,s)|$, where $s$ is as above. By grouping terms in the direct sum decomposition according to the fibers we obtain $$\label{eq:wtspace2ell} \textstyle (k^n)^{\otimes r} = V^{\otimes r} = \bigoplus_{\xi \in {\Lambda_2}(l,r)} \big(\bigoplus_{\lambda \in \pi^{-1}(\xi)} M^\lambda \big) = \bigoplus_{\xi \in {\Lambda_2}(l,r)} N^{\xi}$$ where we define $N^{\xi}$ for any $\xi \in {\Lambda_2}(l,r)$ by $N^{\xi}:= \bigoplus_{\lambda \in \pi^{-1}(\xi)} M^{\lambda}$. The $N^{\xi}$ are just the weight spaces under the action of the abelian group ${\dot{T}}$, so gives the weight space decomposition of tensor space as a ${\dot{T}}$-module. Since the actions of ${{\operatorname{O}}}_n(k)$ and ${\mathfrak{B}}_r(n)$ commute, it is clear that each weight space $N^{\xi}$ for $\xi\in {\Lambda_2}(l,r)$ is a right ${\mathfrak{B}}_r(n)$-module. Hence is a decomposition of tensor space $(k^n)^{\otimes r}$ as a ${\mathfrak{B}}_r(n)$-module, and we have achieved our goal in the case $n = 2l$. It remains to notice some isomorphisms existing among the ${\mathfrak{B}}_r(n)$-modules $N^{\xi}$. As we already pointed out, the Weyl group $W$ associated to ${\operatorname{GL}}_n(k)$ acts on $\{ M^\lambda \colon \lambda \in \Lambda(n,r) \}$, and the orbits are isomorphism classes. It can be expected that the Weyl group associated to ${{\operatorname{O}}}_n(k)$ similarly acts on $\{ N^{\xi} \colon \xi \in {\Lambda_2}(l,r)) \}$, and again the orbits will be isomorphism classes. The Weyl group $\dot{W}$ of ${{\operatorname{O}}}_n(k)$ is isomorphic to the semidirect product $\{\pm 1\}^l \rtimes {\mathfrak{S}}_l$, the group of signed permutations on $l$ letters. We can realize $\dot{W}$ as a subgroup of ${{\operatorname{O}}}_n(k)$, simply by taking the intersection of $W$ (the Weyl group of ${\operatorname{GL}}_n(k)$, realized as the $n \times n$ permutation matrices) with ${{\operatorname{O}}}_n(k)$. A given $w \in W$ lies within this intersection if and only if the condition $(e_{w^{-1}(i)} , e_{w^{-1}(j)}) = (e_i, e_j)$ holds for all $i,j$. Thus, $\dot{W}$ is the set of $w \in W$ such that $$\label{eq:W-criterion} \delta_{w^{-1}(i), w^{-1}(j)'} = \delta_{i,j'} \qquad \text{for all } i,j = 1, \dots, n.$$ It is easy to check by direct calculation that for any given $\sigma \in {\mathfrak{S}}_l$, if we define a corresponding $w_\sigma\in W$ such that $$w_\sigma(i) = \begin{cases} \sigma(i) & \text{if } 1 \le i \le l\\ \sigma(i') & \text{if } l+1 \le i \le 2l \end{cases}$$ then $\sigma$ satisfies the condition . Furthermore, the transposition $\tau_i$ that interchanges $i$ with $i'$ also satisfies , and thus $\dot{W}$ may be identified with the subgroup of $W$ generated by the $w_\sigma$ ($\sigma \in {\mathfrak{S}}_l$) and the $\tau_i$ ($i = 1, \dots, l$). This subgroup acts on $\Lambda(n,r)$ by restriction of the action of $W$. This induces a corresponding action of $\dot{W}$ on the set ${\Lambda_2}(l,r)$, such that $\dot{w}(\xi) = \pi( w(\lambda) )$ if $w \in W$ corresponds to $\dot{w} \in \dot{W}$ and $\xi = \pi(\lambda)$. Since $\tau_i$ sends $\xi = (\xi_1, \dots, \xi_l)$ to $(\xi_1, \dots, \xi_{i-1}, -\xi_i, \xi_{i+1}, \dots, \xi_l)$, and $w_\sigma$ sends $\xi$ to $\sigma(\xi) = (\xi_{\sigma^{-1}(1)}, \dots, \xi_{\sigma^{-1}(l)})$, it follows that $\dot{W}$ acts on the set ${\Lambda_2}(l,r)$ by signed permutations. Thus, a fundamental domain for this action is the set ${\Lambda_2}^+(l,r)$ consisting of all $\xi \in {\Lambda_2}(l,r)$ such that $\xi_1 \ge \xi_2 \ge \cdots \ge \xi_l \ge 0$. We call elements of this set *dominant* orthogonal weights. So, in other words, each orbit of ${\Lambda_2}(l,r)$ contains a unique dominant orthogonal weight. Notice that a dominant orthogonal weight is the same as a partition of not more than $l$ parts. \[prop:3\] For any $\dot{w} \in \dot{W}$, $\xi \in {\Lambda_2}(l,r)$, the right ${\mathfrak{B}}_r(n)$-modules $N^{\xi}$ and $N^{\dot{w}(\xi)}$ are isomorphic. This is similar to the proof of Proposition \[prop:1\]. The isomorphism is given on basis elements by mapping a simple tensor $e_{i_1} \otimes \cdots \otimes e_{i_r}$ of weight $\lambda \in \pi^{-1}(\xi)$ to the simple tensor $e_{w(i_1)} \otimes \cdots \otimes e_{w(i_r)}$ of weight $w(\lambda)$, where $w \in W$ corresponds to $\dot{w}$. Since $\pi(w(\lambda)) = \dot{w}(\pi(\lambda))$ and the above holds for every $\lambda \in \pi^{-1}(\xi)$, the result follows. Hence, when studying properties of the modules $N^{\xi}$, we may as well confine our attention to the ones indexed by dominant orthogonal weights; i.e., partitions. In the decomposition each summand is isomorphic to some $N^{\xi}$ for some $\xi$ such that $\xi$ is a partition of $r - 2s$ into not more than $l$ parts, for some non-negative integer $s \le r/2$. It is easy to see that all such possibilities actually occur as direct summands in . The ${\mathfrak{B}}_r(n)$ decomposition of $(k^n)^{\otimes r}$ in the\ symmetric case, where $n=2l+1$ {#sec:symmetric-odd} ====================================================================== Now we consider the case where $n = 2l+1$, still with the symmetric bilinear form. In this case, we have $(l+1)' = l+1$. Thus, if a diagonal matrix $\text{diag}(t_1, \dots, t_n) \in {\operatorname{GL}}_n(k)$ belongs to ${\dot{T}}$ then we necessarily have $t^2_{l+1} = 1$, and $t_{i'} = t_i^{-1}$ for all $i \ne l+1$. Hence, the description of ${\dot{T}}$ in shows in this case that ${\dot{T}}$ is isomorphic to the direct product $(k^\times)^l \times \{\pm 1 \}$, where by $\{ \pm 1\}$ we mean the multiplicative group of square roots of unity. So the character group $X({\dot{T}})$ is isomorphic to ${{\mathbb Z}}^l \times ({{\mathbb Z}}/2{{\mathbb Z}})$, and thus we will identify $X({\dot{T}})$ with ${{\mathbb Z}}^l \times ({{\mathbb Z}}/2{{\mathbb Z}})$. There is a group homomorphism $\pi \colon X(T) \to X({\dot{T}})$ given by restriction: $\pi(\lambda) = \lambda_{\mid {\dot{T}}}$ for $\lambda \in X(T)$. One easily checks that in this case, given a character $\xi \in X({\dot{T}})$, one can extend it to a character $\lambda \in X(T)$ such that $\lambda_{\mid {\dot{T}}} = \xi$. It follows that the map $\pi$ is surjective. In terms of the identifications $X(T) = {{\mathbb Z}}^n$ and $X({\dot{T}}) = {{\mathbb Z}}^l \times ({{\mathbb Z}}/2{{\mathbb Z}})$, the map $\pi$ is given by the rule $$(\lambda_1, \dots, \lambda_n) \mapsto (\lambda_1 - \lambda_{1'}, \dots, \lambda_{l}-\lambda_{l'}, \overline{\lambda}_{l+1})$$ where $\overline{m}$ denotes the image of an integer $m$ under the natural quotient map ${{\mathbb Z}}\to {{\mathbb Z}}/2{{\mathbb Z}}$. We next consider how to characterize the image of the set $\Lambda(n,r)$ under the map $\pi$. This case is a bit different from the even orthogonal case, because of the presence of the ${{\mathbb Z}}/2{{\mathbb Z}}$ term in the image of $\pi$. Note, however, that for any $\lambda \in \Lambda(n,r)$, the last component $\overline{\lambda}_{l+1}$ of $\pi(\lambda)$ is uniquely determined by the preceding entries in $\pi(\lambda)$, as follows. When $n = 2l+1$, suppose that $\lambda \in \Lambda(n,r)$ and put $t:= (\lambda_1-\lambda_{1'}) + \cdots + (\lambda_{l}-\lambda_{l'})$. Then $r-t \equiv \lambda_{l+1} \pmod{2}$. Since $\lambda_1 + \cdots + \lambda_n = r$, it follows by a simple calculation that $r-t = 2(\lambda_{1'} + \cdots + \lambda_{l'}) + \lambda_{l+1}$, and the result follows. Thanks to the lemma, we may as well pay attention only to the first $l$ parts of the image of some $\lambda \in \Lambda(n,r)$ under $\pi$. Let us write ${\Lambda_1}(l,r)$ for the set of all $(\xi_1, \dots, \xi_l)$ such that $(\xi_1, \dots, \xi_l, \varepsilon) \in \pi(\Lambda(n,r))$. Then we have a bijection $\pi(\Lambda(n,r)) \to {\Lambda_1}(l,r)$, given by $(\xi_1, \dots, \xi_l, \varepsilon) \mapsto (\xi_1, \dots, \xi_l)$. The inverse map is given by $(\xi_1, \dots, \xi_l) \mapsto (\xi_1, \dots, \xi_l, \varepsilon)$, where $\varepsilon$ is the mod $2$ residue of $r - \xi_1 - \cdots - \xi_l$. This leads to the following characterization of ${\Lambda_1}(l,r)$. When $n = 2l+1$, the image of the set $\Lambda(n,r)$ under the map $\pi$ may be identified with the set ${\Lambda_1}(l,r)$ consisting of all $\xi = (\xi_1, \dots, \xi_l) \in {{\mathbb Z}}^l$ such that $|\xi_1| + \cdots + |\xi_l| = r-s$, where $0 \le s \le r$. If $(\xi_1, \dots, \xi_l, {\varepsilon}) = \pi(\lambda)$ for $\lambda \in \Lambda(n,r)$ then $|\xi_1|+\cdots +|\xi_l|$ satisfies the condition $$|\xi_1|+\cdots +|\xi_l| = \epsilon_1(\lambda_1-\lambda_{1'}) + \cdots + \epsilon_l(\lambda_l-\lambda_{l'})$$ where for each $i=1, \dots, l$ the sign $\epsilon_i$ is defined to be $1$ if $\lambda_i \ge \lambda_{i'}$ and $-1$ otherwise. This is just a signed sum of the parts of $\lambda$, excluding the $(l+1)$st part $\lambda_{l+1}$. Thus $|\xi_1|+\cdots +|\xi_l| = r-s$ where $0 \le s \le r$. This proves the necessity of the condition for membership in ${\Lambda_2}(l,r)$. It remains to prove the sufficiency of the condition. Given $\xi \in {{\mathbb Z}}^l$ satisfying the condition $|\xi_1| + \cdots + |\xi_l|=r-s$, where $0 \le s \le r$, we define a corresponding $\lambda \in \Lambda(n,r-s)$ as follows: put $\lambda_i = \xi_i$ if $\xi_i > 0$, put $\lambda_{i'} = -\xi_i$ if $\xi_i < 0$, put $\lambda_{l+1}=s$, and put all the other entries of $\lambda = (\lambda_1, \dots, \lambda_n)$ to zero. Then it easily checked that $\pi(\lambda)$ identifies with $\xi$ under the correspondence $(\xi_1,\dots,\xi_l,{\varepsilon}) \to (\xi_1,\dots,\xi_l)$. For each $\xi \in {\Lambda_1}(l,r)$, the fiber $\pi^{-1}(\xi)$ may be computed as follows. Let $|\xi_1| + \cdots + |\xi_l|=r-s$, where $0 \le s \le r$, and let $\lambda$ be defined in terms of $\xi$ as in the second paragraph of the proof of the proposition. For each integer $t$ such that $0 \le 2t \le s$, let $\mu$ be the same as $\lambda$ except that $\lambda_{l+1}=s$ is replaced by $s-2t$. Then for any $\nu \in \Lambda(l,t)$ we get a member $$(\mu_1+\nu_1, \dots, \mu_l+\nu_l, s-2t, \mu_{l'}+\nu_l, \dots, \mu_{1'}+\nu_1)$$ of the fiber $\pi^{-1}(\xi)$. Thus, the fiber in this case has cardinality given by the sum $\sum_{0 \le 2t \le s} |\Lambda(l,t)|$. By grouping terms in the direct sum decomposition according to the fibers we obtain $$\label{eq:wtspace2ell+1} \textstyle (k^n)^{\otimes r} = V^{\otimes r} = \bigoplus_{\xi \in {\Lambda_1}(l,r)} \big(\bigoplus_{\lambda \in \pi^{-1}(\xi)} M^\lambda \big) = \bigoplus_{\xi \in {\Lambda_1}(l,r)} N^{\xi}$$ where we define $N^{\xi}$ for any $\xi \in {\Lambda_1}(l,r)$ by $N^{\xi}:= \bigoplus_{\lambda \in \pi^{-1}(\xi)} M^{\lambda}$. The $N^{\xi}$ are just the weight spaces under the action of the abelian group ${\dot{T}}$, so gives the weight space decomposition of tensor space as a ${\dot{T}}$-module. Since the actions of ${{\operatorname{O}}}_n(k)$ and ${\mathfrak{B}}_r(n)$ commute, it is clear that each weight space $N^{\xi}$ for $\xi\in {\Lambda_1}(l,r)$ is a right ${\mathfrak{B}}_r(n)$-module. Hence is a decomposition of tensor space $(k^n)^{\otimes r}$ as a ${\mathfrak{B}}_r(n)$-module, and we have achieved our goal in the case $n = 2l+1$. The Weyl group $\dot{W}$ of ${{\operatorname{O}}}_n(k)$ in the case $n=2l+1$ is the same as in the case $n=2l$; it is isomorphic to the semidirect product $\{\pm 1\}^l \rtimes {\mathfrak{S}}_l$, the group of signed permutations on $l$ letters. We can realize $\dot{W}$ as a subgroup of ${{\operatorname{O}}}_n(k)$ in this case as well, by taking the intersection of $W$ with ${{\operatorname{O}}}_n(k)$. A given $w \in W$ lies within this intersection if and only if the condition $(e_{w^{-1}(i)} , e_{w^{-1}(j)}) = (e_i, e_j)$ holds for all $i,j$. Thus, $\dot{W}$ is the set of $w \in W$ such that $$\label{eq:W-criterion-2} \delta_{w^{-1}(i), w^{-1}(j)'} = \delta_{i,j'} \qquad \text{for all } i,j = 1, \dots, n.$$ Thus, for $w \in W$ to belong to $\dot{W}$, it is necessary that $w^{-1}(l+1) = l+1$, or, equivalently, $w(l+1)=l+1$. Then it is easy to check by direct calculation that for any given $\sigma \in {\mathfrak{S}}_l$, if we define a corresponding $w_\sigma\in W$ such that $$w_\sigma(i) = \begin{cases} \sigma(i) & \text{if } 1 \le i \le l\\ \sigma(i') & \text{if } l+1 \le i \le 2l \end{cases}$$ then $\sigma$ satisfies the condition . Furthermore, the transposition $\tau_i$ that interchanges $i$ with $i'$ also satisfies , and thus $\dot{W}$ may be identified with the subgroup of $W$ generated by the $w_\sigma$ ($\sigma \in {\mathfrak{S}}_l$) and the $\tau_i$ ($i = 1, \dots, l$). This subgroup acts on $\Lambda(n,r)$ by restriction of the action of $W$. This induces a corresponding action of $\dot{W}$ on the set ${\Lambda_1}(l,r)$, such that $\dot{w}(\xi) = \pi( w(\lambda) )$ if $w \in W$ corresponds to $\dot{w} \in \dot{W}$ and $\xi = \pi(\lambda)$. Since $\tau_i$ sends $\xi = (\xi_1, \dots, \xi_l)$ to $(\xi_1, \dots, \xi_{i-1}, -\xi_i, \xi_{i+1}, \dots, \xi_l)$, and $w_\sigma$ sends $\xi$ to $\sigma(\xi) = (\xi_{\sigma^{-1}(1)}, \dots, \xi_{\sigma^{-1}(l)})$, it follows that $\dot{W}$ acts on the set ${\Lambda_1}(l,r)$ by signed permutations. Thus, a fundamental domain for this action is the set ${\Lambda_1}^+(l,r)$ consisting of all $\xi \in {\Lambda_1}(l,r)$ such that $\xi_1 \ge \xi_2 \ge \cdots \ge \xi_l \ge 0$. We call elements of this set dominant orthogonal weights. So, in other words, each orbit of ${\Lambda_1}(l,r)$ contains a unique dominant orthogonal weight. Notice that a dominant orthogonal weight is the same as a partition of not more than $l$ parts. For any $\dot{w} \in \dot{W}$, $\xi \in {\Lambda_1}(l,r)$, the right ${\mathfrak{B}}_r(n)$-modules $N^{\xi}$ and $N^{\dot{w}(\xi)}$ are isomorphic. The proof is similar to the proof of Proposition \[prop:3\]. Hence, when studying properties of the modules $N^{\xi}$, we may as well confine our attention to the ones indexed by dominant orthogonal weights; i.e., partitions. In the decomposition each summand is isomorphic to some $N^{\xi}$ for some $\xi$ such that $\xi$ is a partition of $r - s$ into not more than $l$ parts, for some non-negative integer $s \le r$. It is easy to see that all such possibilities actually occur as direct summands in the decomposition . The ${\mathfrak{B}}_r(-n)$ decomposition of $(k^n)^{\otimes r}$ in the skew-symmetric case, where $n=2l$ {#sec:skew} ======================================================================================================== In this case we assume throughout that $n=2l$, and let the field $k$ be arbitrary. We define ${{\operatorname{Sp}}}_n(k)$ to be the group of isometries of $V=k^n$ with respect to the skew-symmetric form $(\ ,\ )$ defined in Section \[sec:main\]. Then the action of ${\mathfrak{B}}_r(-n)$ on tensor space $V^{\otimes r} = (k^n)^{\otimes r}$, defined in Section \[sec:Brauer\], commutes with the natural action of $\\Sp_n(k)$ (given by restricting the action of ${\operatorname{GL}}_n(k)$). Let ${\dot{T}}$ be the abelian subgroup of ${{\operatorname{Sp}}}_n(k)$ consisting of the diagonal matrices in ${{\operatorname{O}}}_n(k)$. Thus, a diagonal matrix $\text{diag}(t_1, \dots, t_n) \in {\operatorname{GL}}_n(k)$ belongs to ${\dot{T}}$ if and only if $$\label{eq:TO-Sp} t_i t_{i'} = 1 \quad\text{ for all $i = 1, \dots, n$.}$$ As before, the description of ${\dot{T}}$ in shows in this case that ${\dot{T}}$ is isomorphic to $(k^\times)^l$, and $X({\dot{T}})$ is isomorphic to ${{\mathbb Z}}^l$, so we identify $X({\dot{T}})$ with ${{\mathbb Z}}^l$. The group homomorphism $\pi \colon X(T) \to X({\dot{T}})$ given by restriction is surjective, for the same reason as before. In terms of the identifications $X(T) = {{\mathbb Z}}^n$ and $X({\dot{T}}) = {{\mathbb Z}}^l$, the map $\pi$ is given by the rule $$(\lambda_1, \dots, \lambda_n) \mapsto (\lambda_1 - \lambda_{1'}, \dots, \lambda_{l}-\lambda_{l'}).$$ The image${\Lambda_2}(l,r)$ of the set $\Lambda(n,r)$ under the map $\pi$ has the same characterization as in the even symmetric case. When $n = 2l$, the image of the set $\Lambda(n,r)$ under the map $\pi$ is the set ${\Lambda_2}(l,r)$ of all $\xi = (\xi_1, \dots, \xi_l) \in {{\mathbb Z}}^l$ such that $|\xi_1| + \cdots + |\xi_l| =r-2s$, where $0 \le 2s \le r$. The proof is the same as in the even symmetric case; see the proof of Proposition \[prop:2\]. The fiber $\pi^{-1}(\xi)$ for $\xi \in {\Lambda_2}(l,r)$ has in this case the same description as in the even symmetric case; see the remarks following the proof of Proposition \[prop:2\]. By grouping terms in the direct sum decomposition according to the fibers we obtain $$\label{eq:wtspace-Sp} \textstyle (k^n)^{\otimes r} = V^{\otimes r} = \bigoplus_{\xi \in {\Lambda_2}(l,r)} \big(\bigoplus_{\lambda \in \pi^{-1}(\xi)} M^\lambda \big) = \bigoplus_{\xi \in {\Lambda_2}(l,r)} N^{\xi}$$ where we define $N^{\xi}$ for any $\xi \in {\Lambda_2}(l,r)$ by $N^{\xi}:= \bigoplus_{\lambda \in \pi^{-1}(\xi)} M^{\lambda}$. The $N^{\xi}$ are just the weight spaces under the action of the abelian group ${\dot{T}}$, so gives the weight space decomposition of tensor space as a ${\dot{T}}$-module. Since the actions of ${{\operatorname{Sp}}}_n(k)$ and ${\mathfrak{B}}_r(-n)$ commute, it is clear that each weight space $N^{\xi}$ for $\xi\in {\Lambda_2}(l,r)$ is a right ${\mathfrak{B}}_r(-n)$-module. Hence is a decomposition of tensor space $(k^n)^{\otimes r}$ as a ${\mathfrak{B}}_r(n)$-module. The Weyl group $\dot{W}$ of ${{\operatorname{Sp}}}_n(k)$ is again isomorphic to the semidirect product $\{\pm 1\}^l \rtimes {\mathfrak{S}}_l$, the group of signed permutations on $l$ letters. Again, the group $\dot{W}$ acts on the set ${\Lambda_2}(l,r)$ by signed permutations. Thus, a fundamental domain for this action is the set ${\Lambda_2}^+(l,r)$. \[prop:7\] For any $\dot{w} \in \dot{W}$, $\xi \in {\Lambda_2}(l,r)$, the right ${\mathfrak{B}}_r(n)$-modules $N^{\xi}$ and $N^{\dot{w}(\xi)}$ are isomorphic. The proof is similar to the proof of Proposition \[prop:3\]. Hence, when studying properties of the modules $N^{\xi}$, we may as well confine our attention to the ones indexed by dominant orthogonal weights; i.e., partitions. In the decomposition each summand is isomorphic to some $N^{\xi}$ for some $\xi$ such that $\xi$ is a partition of $r - 2s$ into not more than $l$ parts, for some non-negative integer $s \le r/2$. As before, it is easy to see that all such possibilities actually occur as direct summands in . [99]{} D. Benson and S. Doty, Schur–Weyl duality over finite fields, *Arch. Math.* (Basel) 93 (2009), 425–435. R. Brauer, On algebras which are connected with the semisimple continuous groups, *Ann. of Math.* (2) 38 (1937), 857–872. W. Brown, An algebra related to the orthogonal group, *Michigan Math. J.* 3 (1955), 1–22. R. Dipper, S. Doty, and J. Hu, Brauer algebras, symplectic Schur algebras and Schur-Weyl duality, *Trans. Amer. Math. Soc.* 360 (2008), 189–213 (electronic). S. Doty, Schur-Weyl duality in positive characteristic, *Representation theory*, 15–28, Contemp. Math., 478, Amer. Math. Soc., Providence, RI, 2009. S.Doty and J. Hu, Schur-Weyl duality for orthogonal groups, *Proc. Lond. Math. Soc.* (3) 98 (2009), 679–713. G. James, *The representation theory of the symmetric groups*, Lecture Notes in Mathematics, 682, Springer, Berlin, 1978. W. Fulton, *Young Tableaux, With applications to representation theory and geometry*, London Mathematical Society Student Texts, 35, Cambridge University Press, Cambridge, 1997. J.A. Green, *Polynomial Representations of ${\operatorname{GL}}_n$*, Lecture Notes in Mathematics, 830. Springer-Verlag, Berlin-New York, 1980. A.E. Henke and R. Paget, Brauer algebras with parameter $n=2$ acting on tensor space, *Algebr. Represent. Theory* 11 (2008), 545–575. I. Schur, Über die rationalen Darstellungen der allgemeinen linearen Gruppe (1927), Gesammelte Abhandlungen, Band III (German), 68–85, herausgegeben von Alfred Brauer und Hans Rohrbach, Springer-Verlag, Berlin/New York, 1973. H. Weyl, *The Classical Groups; Their Invariants and Representations*, Princeton University Press, Princeton, N.J., 1939.

--- author: - 'Ravi N. Banavar[^1] and Arjun Narayanan[^2]' bibliography: - 'vscmg.bib' - 'textbooks.bib' title: 'The Principal Fiber Bundle Structure of the Gimbal-Spacecraft System' --- Nomenclature {#nomenclature .unnumbered} ============ ---------------------------------------- --- --------------------------------------------------------------------------------------------------------------------------------------------------------------- $\beta$,$\gamma (\in {\mathbb{S}^1}) $ = The gimbal and wheel angles. ($rad$) $R_{\beta} (\in \SO3)$ = Transformation from gimbal frame $\mathcal{G}$ to the spacecraft body frame $\mathcal{B}$. ${\mathbb{I}_{s}}$ = Spacecraft inertia without the CMG gimbal and wheel inertia. ($kg.m^2$) $I_{g},\,I_{r}$ = Gimbal frame inertia, wheel inertia about own centre of mass represented in gimbal frame. ($kg.m^2$) $({\mathbb{I}_{gr}})_{\beta}$ = Combined inertia of gimbal frame and wheel in the spacecraft frame. $R_{\beta} {\mathbb{I}_{gr}}R_{\beta}^T$. ($kg.m^2$) $\tilde{I}_{(\beta)}$ = Locked inertia tensor. $X$ = State variable $\triangleq(R_{s},\beta,\gamma)\triangleq(R_{s},x)$. $\Omega_{s}$ = Angular velocity of the spacecraft in the body frame. ($rad/s$) $\mu$ = Total spatial angular momentum of the spacecraft in inertial frame. ($kg.m^2/s$) $i_{2},i_{3}$ = The vectors $[0\,1\,0]^{T}$ and $[0\,0\,1]^{T}$. $\mathcal{S}(\ ),\widehat{(\ )}$ = Mapping $\mathbb{R}^{3}\rightarrow\mathfrak{so}(3)$ such that $\mathcal{S}(\vec{a})\vec{b}\triangleq\hat{\vec{a}}\cdot\vec{b}\triangleq\vec{a}\times\vec{b}$. ---------------------------------------- --- --------------------------------------------------------------------------------------------------------------------------------------------------------------- \ Introduction ============ Spacecrafts are actuated by two principles - internal or external actuation. The actuators in the former class consist of momentum wheels (also called internal rotors) and control moment gyros (CMGs). A refinement of CMGs are the variable speed CMGs (called VSCMGs). Examples of external actuation systems include gas jet thrusters mounted on the outer body of the spacecraft. In this article we focus on spacecraft with gimbals (or the VSCMG) as the mechanism of actuation. The modelling of a spacecraft with a VSCMG has been reported in the literature [@schaub_feedback_1998; @schaub_singularity_2000]. Studies on singularity issues of this system are found in [@tsiotras_singularity_2004; @schaub_singularity_2000]. Control law synthesis and avoidance of singularities are found in [@bedrossian_thesis_1987; @margulies_aubrun_1978; @schaub_singularity_2000; @tsiotras_singularity_2004] An early study of singularity in a geometric framework is [@kurokawa_geometric_1998]. That study considers possibility of avoiding singular gimbal angle configurations using global control. The analysis proceeds by considering the inverse images of CMG system angular momenta as union of sub*“manifolds”* of the n-dimensional gimbal angle configuration space. The nature of these manifolds are examined at and near singular gimbal configurations. Spurred by the insight and creativity of J. E. Marsden [@marsden2013introduction] ,the geometric mechanics community [@bloch_nonholo_mech_ctrl; @ostrowski1999computing] has studied very many mechanical systems in the geometric framework. This framework has proved beneficial in providing insight into these systems and their structure by preserving the mechanical objects (momentum, energy) of these systems and also proving useful in control design [@bloch2001controlledlagrangian; @bloch2000controlled]. A geometric description of the VSCMG system based on variational principles is studied in [@sanyal2013vscmg]. The configuration space of the system is shown to be a principal fiber bundle and the expression for the Ehressmann connection is derived. The paper considers a general system where the rotor mass centre is offset from the gimbal axis. A stabilising control law is derived as a function of the internal momentum. Singularity analysis of this system under typical simplifying assumptions is explored in [@sanyal2015vscmg]. The system is discretised into a model which preserves the conserved quantities using variational integrators. While studying the problem of interconnected mechanical systems, the geometry of the configuration space which is a differential manifold requires attention for elegant and insightful solutions. This configuration space $Q$, is often written as the product of two manifolds. One component is the base manifold $M$, which in our context describes the configuration of the gimbals mounted inside the spacecraft. The other component of the configuration variables depicting the attitude of the spacecraft is a Lie group $G$, in this case $\SO3$ [@boothby1975introduction]. The total configuration space of the spacecraft $Q$ then naturally appears as a product $G \times M$. Such systems follow the topology of a trivial principal fiber bundle, see [@kobayashi1963foundations]. Figure \[fiber\_bundle\] shows an explanatory figure of a fiber bundle. The components of the 2-tuple $q = (x, g)$ denote the base and the group variable respectively. The projection map $\pi: Q \rta M$ maps the configuration space to the base space. With such a separation of the configuration space, locomotion is readily seen as the means by which changes in shape affect the macro position. We refer to [@kelly1995geometric; @bloch_nonholo_mech_ctrl] for a detailed explanation on the topology of locomoting systems. ![Fiber Bundle[]{data-label="fiber_bundle"}](pfb01.png) In this article we present a completely geometric approach to the modelling of the VSCMG-spacecraft system and relate it to a conventional modelling approach. Modelling in a geometric framework ================================== The configuration space of the spacecraft-gimbal system (with one CMG) is $Q = \SO3 \times {\mathbb{S}^1}\times {\mathbb{S}^1}$ and any arbitrary configuration is expressed by the 3-tuple $(R_s, \beta, \gamma)$, where the first element denotes the attitude $R_s$ of the spacecraft with respect to fixed / inertial frame, the second denotes the degree of freedom $\beta$ of the gimbal frame, the third denotes the degree of freedom $\gamma$ of the rotor about its spin axis. For the purpose of later geometrical interpretation, we club $x \deff (\beta, \gamma)$. There are three rigid bodies involved here, each having relative motion (rotation) about the other. Therefore, three frames of reference are chosen (apart from the inertial frame) - the first is the spacecraft, denoted by the subscript $s$, the second is the gimbal frame, denoted by the subscript $g$, the third is the rotor frame, denoted by the subscript $r$. The moments of inertia of the homogeneous rotor and the gimbal in their respective body frames are assumed to be $$\begin{aligned} {\mathbb{I}_{r}}= {\begin{pmatrix}J_x & 0 & 0 \\ 0 & J_x & 0 \\ 0 & 0 & J_z \end{pmatrix}} \;\;\; {\mathbb{I}_{g}}= {\begin{pmatrix}I_t & 0 & 0 \\ 0 & I_g & 0 \\ 0 & 0 & I_s \end{pmatrix}} \end{aligned}$$ Since the rotor is assumed to be homogeneous and symmetric, its inertia is represented in the gimbal frame and the combined gimbal-rotor inertia is rewritten as $$\begin{aligned} {\mathbb{I}_{gr}}= {\begin{pmatrix}(J_x + I_t)& 0 & 0 \\ 0 & (J_x + I_g) & 0 \\ 0 & 0 & (J_z + I_s) \end{pmatrix}} \end{aligned}$$ If the rotational transformation that relates the gimbal frame to the spacecraft frame is given by $R_{\beta}$, where $\beta$ denotes the gimballing angle, then the gimbal-rotor inertia reflected in the spacecraft frame is $$\begin{aligned} ({\mathbb{I}_{gr}})_\beta \deff R_{\beta} {\mathbb{I}_{gr}}R_{\beta}^T \end{aligned}$$ Here the subscript $\beta$ denotes the dependance on the gimbal angle $\beta$. Kinetic energy and a Riemannian structure ----------------------------------------- The angular velocity of the spacecraft, $\Omega_s \in {\mathbb{R}}^3$, represented as a skew-symmetric matrix $ {\hat{\Omega}_s} $, is an element of the Lie algebra $\so3$. The inner product on this Lie algebra $\so3$ is defined in terms of the standard inner product on ${\mathbb{R}}^3$ as $$\begin{aligned} {\left\langle{\hat{\Omega}_1}, {\hat{\Omega}_2}\right\rangle}_{\so3} \deff \frac{1}{2} {\left\langle{ \Omega_1}, { {\mathbb{I}_{s}}\Omega_2}\right\rangle}_{{\mathbb{R}}^3}. \end{aligned}$$ and denotes the kinetic energy of the spacecraft body. The kinetic energy of the gimbal-rotor unit is given by $$\begin{aligned} \frac{1}{2} {\left\langle{R_{\beta}^T \Omega_s + {\begin{pmatrix}0 \\ \dot{\beta} \\ \dot{\gamma}\end{pmatrix}} }, {{\mathbb{I}_{gr}}[ R_{\beta}^T \Omega_s + {\begin{pmatrix}0 \\ \dot{\beta} \\ \dot{\gamma}\end{pmatrix}} ] }\right\rangle} \end{aligned}$$ and the total kinetic energy of the entire spacecraft-gimbal system is $$\begin{aligned} \frac{1}{2} {\left\langle{ {\begin{pmatrix}\Omega_s \\ 0 \\ \dot{\beta} \\ \dot{\gamma}\end{pmatrix}} }, { {\mathbb{I}_{total}}{\begin{pmatrix}\Omega_s \\ 0 \\ \dot{\beta} \\ \dot{\gamma}\end{pmatrix}} }\right\rangle}, \end{aligned}$$ where $$\begin{aligned} {\mathbb{I}_{total}}(\beta)= {\begin{pmatrix} (R_{\beta} {\mathbb{I}_{gr}}R_{\beta}^T + {\mathbb{I}_{s}}) & R_{\beta} {\mathbb{I}_{gr}}\\ {\mathbb{I}_{gr}}R_{\beta}^T & {\mathbb{I}_{gr}}\end{pmatrix}}. \end{aligned}$$ It is to be noted that the inertia matrix is dependant on the gimbal angle, $\beta$. The kinetic energy induces a metric on the configuration space $Q$ of the system, which enables us to impart a [*Riemannian structure*]{} to the system. The Riemannian metric $\riem$ defines a smoothly varying inner product on each tangent space of $Q$. For $q = (R_s, (\beta, \gamma)) \in Q $ and $v_q = (R_s\hat{\Omega}_1, (v_\beta,v_\gamma)), w_q = (R_s\hat{\Omega}_2, (w_\beta,w_\gamma)) \in T_qQ$, the Riemannian metric is defined as $$\begin{aligned} {\left\langle{v_q}, {w_q}\right\rangle}_{\riem} & = \riem(q) (v_q, w_q) \nonumber \\ & = \riem ( R_s, (\beta, \gamma)) \left( (R_s \hat{\Omega}_1 , (v_\beta, v_\gamma)), (R_s \hat{\Omega}_2 , (w_\beta, w_\gamma))\right) \end{aligned}$$ $$\begin{aligned} = \frac{1}{2} {\left\langle{ {\begin{pmatrix}R_s \hat{\Omega}_1 \\ 0 \\ v_\beta \\ v_\gamma \end{pmatrix}} }, { {\mathbb{I}_{total}}{\begin{pmatrix}R_s \hat{\Omega}_2 \\ 0 \\ w_\beta \\ w_\gamma \end{pmatrix}} }\right\rangle} \end{aligned}$$ Here $\hat{\Omega}_1$ and $ \hat{\Omega}_2$ belong to the Lie algebra $\so3$. Note that we have used the left-invariant property of the vector field on $\SO3$, and the fact that the Riemannian metric on $\SO3$ is induced by the inner product on the Lie algebra $\so3$, wherein $$\begin{aligned} {\left\langle{v_R}, {w_R}\right\rangle}_{\SO3} \deff {\left\langle{\hat{\Omega}_1}, {\hat{\Omega}_2}\right\rangle}_{\so3} , \end{aligned}$$ where $v_R = R \hat{\Omega}_1$ and $w_R = R \hat{\Omega}_2$, the left translations by $R$ of $\hat{\Omega}_1$ and $\hat{\Omega}_2$, respectively. Group action and a principal fiber bundle ----------------------------------------- The action of the $\SO3$ group on $Q$ induces more geometric structure into the problem. Given $M \in \SO3$, the action is defined by $$\begin{aligned} \SO3 \times Q \rta Q \;\;\;\;\;\; (M , (R_s, \beta, \gamma)) \rta (MR_s, \beta, \gamma) \end{aligned}$$ and the corresponding tangent lifted action is given by $$\begin{aligned} T \SO3 \times TQ \rta TQ \;\;\; (v_{R_s}, v_{\beta}, v_{\gamma}) \rta (Mv_{R_s}, v_{\beta}, v_{\gamma}) \end{aligned}$$ This action is chosen based on the symmetry in the system; in this case, the fact that the kinetic energy of the spacecraft in a potential field free space remains unchanged under rotational transformations. The gimbal and the rotor configuration variables are viewed in a [*base space (or shape space)*]{} and the rigid body orientation is viewed as a group variable in a [*fiber space*]{}, and with a few additional requirements, the model is amenable to a principal fiber bundle description. See [@bullo_modelling_ctrl] for more details on describing mechanical systems in a fiber bundle framework. The fiber bundle structure separates the actuation and orientation variables and proves beneficial and intuitive in control design. Based on the above model description, we identify the principal fiber bundle $(Q, B, \pi, G)$, where $Q = \SO3 \times {\mathbb{S}^1}\times {\mathbb{S}^1}$, $B = {\mathbb{S}^1}\times {\mathbb{S}^1}$ and $\pi: Q \rta B$ is a bundle projection map. Under the defined group action, the kinetic energy of the total system remains invariant. Straightforward. $\Box$. We now define a few geometric quantities on this fiber bundle on the lines of [@bloch_nonholo_mech_ctrl]. The [*infinitesimal generator*]{} of the Lie algebraic element $\hat{\eta} \in \so3$ under the group action is the vector field $$\begin{aligned} \hat{\eta}_{Q}(q) = \frac{d}{dt}|_{t=0} (\exp(\hat{\eta}t) R_s, (\beta, \gamma)) = (\hat{\eta}R_s, (0, 0)) \end{aligned}$$ The [*momentum map*]{} $J : TQ \rta \so3^*$ is given by $$\begin{aligned} [J(q, v_q), \xi] = {\left\langle{ v_q }, { \hat{\xi}_Q(q) }\right\rangle}_{\riem} \end{aligned}$$ and since the kinetic energy is invariant under the action of the $\SO3$ group, we have $$\begin{aligned} {\left\langle{ v_q }, { \hat{\xi}_Q(q) }\right\rangle}_{\riem} = {\left\langle{ v_{(e,(\beta,\gamma))} }, {(R_s^T \hat{\xi} R_s,(0,0)) }\right\rangle}_{\riem} \end{aligned}$$ which yields $$\begin{aligned} [J(q, v_q), \hat{\xi}] = {\left\langle{Ad_{R_s^T}^* [( ({\mathbb{I}_{gr}})_s + {\mathbb{I}_{s}}) \Omega_s + R_{\beta} {\mathbb{I}_{gr}}{\begin{pmatrix} 0 \\ \dot{\beta} \\ \dot{\gamma} \end{pmatrix}} ] }, {\xi}\right\rangle} \end{aligned}$$ where ${\left\langle{\cdot}, {\cdot}\right\rangle}$ denotes the inner product, while $[\cdot, \cdot]$ denotes the primal-dual action of the vector spaces $\so3$ and $\so3^*.$ In the absence of external forces, the momentum map is conserved. Since the total spatial angular momentum of the system is constant, say $\mu$, the above expression yields $$\begin{aligned} \mu = Ad_{R_s^T}^* [( ({\mathbb{I}_{gr}})_s + {\mathbb{I}_{s}}) \Omega_s + R_{\beta} {\mathbb{I}_{gr}}{\begin{pmatrix} 0 \\ \dot{\beta} \\ \dot{\gamma} \end{pmatrix}} ] \end{aligned}$$ The [*locked inertia tensor*]{} at each point $q \in Q$ is the mapping $$\begin{aligned} \mathbb I (q): \so3 \rta \so3^* \end{aligned}$$ and is defined as $$\begin{aligned} [ \mathbb I(q) \eta, \xi ] = \riem (q) \left((\hat{\eta} R_s , (0,0)), (\hat{\xi} R_s , (0, 0))\right) \end{aligned}$$ The [*mechanical connection*]{} is then defined as the $\so3$-valued one-form $$\begin{aligned} \alpha: TQ \rta \so3 \;\;\;\; (q, v_q) \rta \alpha(q, v_q) = {\mathbb I(q)}^{-1} J(q,v_q) \end{aligned}$$ We now proceed to present a kinematic model and a dynamic model for the system under consideration. With the state-space as $X \deff (R_s, (\beta, \gamma)) = (R_s, x)$, where $x \deff (\beta, \gamma)$ and defining ${\tilde{\mathbb{I}}(x)}= (R_{\beta} {\mathbb{I}_{gr}}R_{\beta}^T + {\mathbb{I}_{s}}) $, the control inputs (gimbal velocity and rotor spin) at the kinematic level as $u \deff \dot{x} = (\dot{\beta}, \dot{\gamma}) = {\begin{pmatrix}u_{\beta} \\ u _{\gamma} \end{pmatrix}} $, the affine-in-the-control system model is $$\begin{aligned} \dot{X} = f(X) + g(X) u \end{aligned}$$ where the drift and control vector fields are given by $$\begin{aligned} f(X) = {\begin{pmatrix}R_s \skew{ ( {\tilde{\mathbb{I}}(x)}^{-1} (Ad_{R_s}^* \mu)) } \\ 0 \end{pmatrix}} \end{aligned}$$ $$\begin{aligned} g_{\beta}(X) = {\begin{pmatrix}- R_s \skew{ ({\tilde{\mathbb{I}}(x)}^{-1} (Ad_{R_\beta^T}^* ({\mathbb{I}_{gr}}i_2)) } \\ {\begin{pmatrix}1 \\ 0\end{pmatrix}} \end{pmatrix}} \;\;\;\;\; g_{\gamma}(X) = {\begin{pmatrix}- R_s \skew{ ({\tilde{\mathbb{I}}(x)}^{-1} (Ad_{R_\beta^T}^* ({\mathbb{I}_{gr}}i_3)) } \\ {\begin{pmatrix}0 \\ 1\end{pmatrix}} \end{pmatrix}} \end{aligned}$$ Here $\skew{\cdot} : {\mathbb{R}}^3 \rta \so3$ is given by $$\begin{aligned} \skew{(\psi_1, \psi_2, \psi_3)} \deff {\begin{pmatrix} 0 & - \psi_3 & \psi_2 \\ \psi_3 & 0 & -\psi_1 \\ - \psi_2 & \psi_1 & 0 \end{pmatrix}} \end{aligned}$$ The dynamic model ================= To arrive at the dynamic model we proceed as follows. From the expression for the total momentum $$\begin{aligned} \mu & = Ad_{R_s^T}^* [ (({\mathbb{I}_{gr}})_s + {\mathbb{I}_{s}}) \Omega_s + R_{\beta} {\mathbb{I}_{gr}}{\begin{pmatrix} 0 \\ \dot{\beta} \\ \dot{\gamma} \end{pmatrix}} ] \nonumber \\ & = R_s {\begin{pmatrix} {\tilde{\mathbb{I}}(x)}& R_{\beta} {\mathbb{I}_{gr}}\end{pmatrix}} {\begin{pmatrix}\Omega_s \\ {\begin{pmatrix} 0 \\ \dot{\beta} \\ \dot{\gamma} \end{pmatrix}} \end{pmatrix}}\end{aligned}$$ We split the momentum in to two components - one due to the gimbal-rotor unit and the other due to the rigid spacecraft. Further, we assume an internal torque $\tau_b$ (in the gimbal-rotor frame), generated by a motor, acts on the gimbal and rotor unit. We then have, due to the principle of action and reaction $$\begin{aligned} \frac{d}{dt} (R_s {\mathbb{I}_{s}}\Omega_s) = \underbrace{- R_s \tau_b}_{reaction} \;\;\;\; \;\; \frac{d}{dt} ( R_s [ R_{\beta} {\mathbb{I}_{gr}}R_{\beta}^T \Omega_s + R_{\beta} {\mathbb{I}_{gr}}\dot{x} ] ) = \underbrace{R_s \tau_b}_{action}\end{aligned}$$ The detailed computation is now shown. Differentiating $ \mu=R_{s}R_{\beta} {\mathbb{I}_{gr}}(R_{\beta}^{T}\Omega_{s}+\begin{bmatrix}0\\ \dot{\beta}\\ \dot{\gamma} \end{bmatrix}) $ with respect to time, we have $\frac{d}{dt}\mu_{cmg}=\frac{d}{dt}\left\{ R_{s}R_{\beta} {\mathbb{I}_{gr}}(R_{\beta}^{T}\Omega_{s}+\begin{bmatrix}0\\ \dot{\beta}\\ \dot{\gamma} \end{bmatrix})\right\} =\text{torque acting on cmg external to cmg}$ $\tau_{extcmg}=\left\{ \begin{array}{c} (R_{s}\hat{\Omega}_{s})R_{\beta} {\mathbb{I}_{gr}}(R_{\beta}^{T}\Omega_{s}+\begin{bmatrix}0\\ \dot{\beta}\\ \dot{\gamma} \end{bmatrix})\\ +R_{s}(R_{\beta}\hat{i}_{2}\dot{\beta}) {\mathbb{I}_{gr}}(R_{\beta}^{T}\Omega_{s}+\begin{bmatrix}0\\ \dot{\beta}\\ \dot{\gamma} \end{bmatrix})\\ +R_{s}R_{\beta} {\mathbb{I}_{gr}}(-\hat{i}_{2}R_{\beta}^{T}\dot{\beta}\Omega_{s})\\ +R_{s}R_{\beta} {\mathbb{I}_{gr}}(R_{\beta}^{T}\dot{\Omega}_{s})\\ +R_{s}R_{\beta} {\mathbb{I}_{gr}}(\begin{bmatrix}0\\ \ddot{\beta}\\ \ddot{\gamma} \end{bmatrix}) \end{array}\right\} $ $$\begin{aligned} \tau_{extcmg}={} & R_{s}\hat{\Omega}_{s}R_{\beta} {\mathbb{I}_{gr}}(R_{\beta}^{T}\Omega_{s}+\begin{bmatrix}0\\ \dot{\beta}\\ \dot{\gamma} \end{bmatrix})\\ &+R_{s}R_{\beta}\left\{ \hat{i}_{2}\dot{\beta} {\mathbb{I}_{gr}}(R_{\beta}^{T}\Omega_{s}+\begin{bmatrix}0\\ \dot{\beta}\\ \dot{\gamma} \end{bmatrix})+ {\mathbb{I}_{gr}}(-\hat{i}_{2}R_{\beta}^{T}\dot{\beta}\Omega_{s})+ {\mathbb{I}_{gr}}(\begin{bmatrix}0\\ \ddot{\beta}\\ \ddot{\gamma} \end{bmatrix})\right\}\end{aligned}$$ In the spacecraft body coordinates, $$\begin{aligned} \tau_{extcmg}^{\mathcal{B}}={}&\hat{\Omega}_{s}R_{\beta} {\mathbb{I}_{gr}}(R_{\beta}^{T}\Omega_{s}+\begin{bmatrix}0\\ \dot{\beta}\\ \dot{\gamma} \end{bmatrix})\\ &+R_{\beta}\left\{ \hat{i}_{2}\dot{\beta} {\mathbb{I}_{gr}}(R_{\beta}^{T}\Omega_{s}+\begin{bmatrix}0\\ \dot{\beta}\\ \dot{\gamma} \end{bmatrix})+ {\mathbb{I}_{gr}}(-\hat{i}_{2}R_{\beta}^{T}\dot{\beta}\Omega_{s})+ {\mathbb{I}_{gr}}(\begin{bmatrix}0\\ \ddot{\beta}\\ \ddot{\gamma} \end{bmatrix})\right\}\end{aligned}$$ Defining two vectors $$\begin{aligned} u_{1}={} &{\mathbb{I}_{gr}}(R_{\beta}^{T}\Omega_{s}+\begin{bmatrix}0\\ \dot{\beta}\\ \dot{\gamma} \end{bmatrix})\\ u_{2}={} &\left(\hat{i}_{2} {\mathbb{I}_{gr}}- {\mathbb{I}_{gr}}\hat{i}_{2}\right)R_{\beta}^{T}\Omega_{s}\dot{\beta}+\hat{i}_{2} {\mathbb{I}_{gr}}\begin{bmatrix}0\\ \dot{\beta}\\ \dot{\gamma} \end{bmatrix}\dot{\beta}+ {\mathbb{I}_{gr}}(\begin{bmatrix}0\\ \ddot{\beta}\\ \ddot{\gamma} \end{bmatrix})\end{aligned}$$ in the gimbal frame $\mathcal{G}$ such that $\tau_{extcmg}^{\mathcal{B}}=\hat{\Omega}_{s}R_{\beta}u_{1}+R_{\beta}u_{2}$ . The second component of this vector gives the torque acting on the gimbal motor and the third gives the torque acting on the wheel motor. We now simplify this expression to obtain more explicit equations, which we then compare with a standard model existing in the literature. $$[ \hat{\Omega}_s {\tilde{\mathbb{I}}(x)}\Omega_s + {\tilde{\mathbb{I}}(x)}\dot{\Omega}_s ]$$ $$\begin{aligned} = (-1)\hat{\Omega}_s R_{\beta} {\mathbb{I}_{gr}}\dot{x} - \dot{\beta} R_{\beta} {\cal U} R_{\beta}^T \Omega_s - \dot{\beta} R_{\beta} \hat{i}_2 {\mathbb{I}_{gr}}\dot{x} - R_{\beta} {\mathbb{I}_{gr}}\ddot{x}\end{aligned}$$ where ${\cal U} \deff \hat{i}_2 {\mathbb{I}_{gr}}- {\mathbb{I}_{gr}}\hat{i}_2$ is a symmetric matrix. Comparison to the Schaub-Rao-Junkins model ========================================== We now draw connections between the approach outlined in the previous sections with that of the classical CMG modeling and analysis done in the Newtonian framework in [@schaub_feedback_1998], which is cited in much of the aerospace literature. We shall refer to this paper as the SRJ paper henceforth. We first relate the notation and then establish a connection with the main equations of the SRJ paper. The two primary variables in the SRJ paper and ours are related as in table \[tab:notationcomparison\]. Variable This paper SRJ paper ---------------------------- ---------------- ----------- Gimbal angle $\beta$ $\gamma$ Rotor spin magnitude $\dot{\gamma}$ $\Omega$ Satellite angular velocity $\Omega_s$ $\omega$ : \[tab:notationcomparison\] Comparison of notation with SRJ paper The rotation matrix in the SRJ paper, relating the gimbal and spacecraft-body frame, is described in terms of three orthogonal column vectors of unit norm, $\{ \hat{g}_s, \hat{g}_{t}, \hat{g}_g \}$, where the subscripts $s, t$ and $g$ correspond to the [*spin, transverse and gimbal*]{} axes, as $$\begin{aligned} {\begin{pmatrix}| & | & | \\ \hat{g}_s & \hat{g}_{t} & \hat{g}_g \\ | & | & | \\ \end{pmatrix}}\end{aligned}$$ and further, $$\begin{aligned} {\begin{pmatrix} {\left\langle{\hat{g}_s}, {\omega}\right\rangle} \\ {\left\langle{\hat{g}_{t}}, {\omega}\right\rangle} \\ {\left\langle{\hat{g}_{g}}, {\omega}\right\rangle} \end{pmatrix}} = {\begin{pmatrix} \omega_s \\ \omega_t \\ \omega_g \end{pmatrix}}\end{aligned}$$ In our convention, the following correspondence holds: $$\begin{aligned} R_{\beta} = {\begin{pmatrix}| & | & | \\ \hat{g}_t & \hat{g}_{g} & \hat{g}_s \\ | & | & | \\ \end{pmatrix}}\end{aligned}$$ and $$\begin{aligned} R_{\beta}^T \Omega_s \longrightarrow {\begin{pmatrix} \omega_t \\ \omega_g \\ \omega_s \end{pmatrix}} $$ The SRJ equation of motion (eqn. 28) written partially in terms of our notation is $$\begin{aligned} {\tilde{\mathbb{I}}(x)}\dot{\Omega}_s + \hat{\Omega}_s {\tilde{\mathbb{I}}(x)}\Omega_s = \end{aligned}$$ $$\begin{aligned} - \hat{g}_s [ J_s (\ddot{\gamma} + \dot{\beta} \omega_t) - (J_t - J_g) \omega_t \dot{\beta} ] - \hat{g}_t [ J_s( \dot{\gamma} + \omega_s) \dot{\beta} - (J_t + J_g) \omega_s \dot{\beta} + J_s \dot{\gamma} \omega_g ] - \hat{g}_g [ J_g \ddot{\beta} - J_s \dot{\gamma} \omega_t ] $$ while the RHS of the same equation in our notation is $$- [ \hat{\Omega}_s + \dot{\beta} \hat{i}_2 ] R_{\beta} {\mathbb{I}_{gr}}\dot{x} - \dot{\beta} R_{\beta} ( \hat{i}_2 {\mathbb{I}_{gr}}- {\mathbb{I}_{gr}}\hat{i}_2 ) R_{\beta}^T \Omega_s - R_{\beta} {\mathbb{I}_{gr}}\ddot{x}$$ $$= - \hat{g}_t [ (J_z + I_s) \dot{\gamma} \dot{\beta} - (J_x + I_g) \dot{\beta} \omega_s + (J_z + I_s) \dot{\gamma} \omega_g + (J_z + I_s) - (J_x + I_t) \dot{\beta} \omega_s]$$ $$- \hat{g}_g [ (J_x + I_g) \ddot{\beta} - (J_z + I_s) \dot{\gamma} \omega_t ]$$ $$- \hat{g}_s [ (J_z + I_s) \ddot{\gamma} + (J_x + I_g) \dot{\beta} \omega_t + ((J_z + I_s) - (J_x + I_t) ) \dot{\beta} \omega_t ]$$\ The terms in the model expand as shown below. $$\begin{aligned} R_{\beta} {\mathbb{I}_{gr}}[0,\ddot{\beta},\ddot{\gamma}]^{T} & =\vec{g}_{g}(J_{x}+I_{g})\ddot{\beta}+\vec{g}_{s}(J_{z}+I_{s})\ddot{\gamma}\\ R_{\beta}\hat{i}_{2}\dot{\beta} {\mathbb{I}_{gr}}[0,\dot{\beta},\dot{\gamma}]^{T} & =\vec{g}_{t}(J_{z}+I_{s})\dot{\gamma}\dot{\beta}\\ \hat{\Omega}_{s}R_{\beta} {\mathbb{I}_{gr}}[0,\dot{\beta},\dot{\gamma}]^{T} & =\begin{array}{c} \vec{g}_{s}((J_{x}+I_{g})\dot{\beta}\omega_{t})+\vec{g}_{g}(-(J_{z}+I_{s})\dot{\gamma}\omega_{t})\\ \vec{g}\left(-(J_{x}+I_{g})\dot{\beta}\omega_{s}+(J_{z}+I_{s})\dot{\gamma}\omega_{g}\right) \end{array}\\ \hat{i}_{2} {\mathbb{I}_{gr}}- {\mathbb{I}_{gr}}\hat{i}_{2} & = \begin{bmatrix}0 & 0 & \left(J_{z}+I_{s}-(J_{x}+I_{t})\right)\\ 0 & 0 & 0\\ \left(J_{z}+I_{s}-(J_{x}+I_{t})\right) & 0 & 0 \end{bmatrix}\\ R_{\beta}(\hat{i}_{2} {\mathbb{I}_{gr}}- {\mathbb{I}_{gr}}\hat{i}_{2})R_{\beta}^{T}\Omega_{s}\dot{\beta} & =(\vec{g}_{t}\omega_{s}+\vec{g}_{s}\omega_{t})(\left\{ J_{z}+I_{s}-(J_{x}+I_{t})\right\} \dot{\beta})\end{aligned}$$ Connection form =============== We now detail the explicit computation of the connection form. The principal fiber bundle structure introduces a vertical space in the tangent space at each point on the manifold. The vertical space consists of those vectors corresponding to the infinitesimal generator vector fields at that particular point. The tangent space is then expressed as the direct sum of the vertical space and a horizontal space, where the horizontal space gets defined as the subspace orthogonal to the vertical space in the inner product induced by the Riemannian metric. A principal connection on $Q = (M,G,\pi)$ is a $\mathfrak{g}$ valued 1-form $\mathcal{A}$ on $Q$ satisfying, 1. $\mathcal{A}(\xi_{Q}(q)) = \xi$, $\forall \xi \in \mathfrak{g}$ and $q \in Q$; 2. $\mathcal{A}(\Phi_{g \ast}X_{p}) = Ad_g^* (\mathcal{A}(X_{p}))$ for all $X_{p} \in TQ$ and for all $g \in G$. This is called the equivariance of the connection. The expression for the kinetic energy is $$[\Omega^{T}\ u_{1}\ u_{2}] \begin{bmatrix} {\mathbb{I}_{s}}+ R_{\beta} {\mathbb{I}_{gr}}R_{\beta}^{T} & (J_x + I_g)\vec{g} & (J_z + I_s)\vec{s}_{\beta}\\ (J_x + I_g)\vec{g}^{T} & (J_x + I_g) & 0\\ (J_z + I_s)\vec{s}_{\beta}^{T} & 0 & (J_z + I_s) \end{bmatrix} \begin{bmatrix}\Xi\\ w_{1}\\ w_{2} \end{bmatrix}$$ Action and infinitesimal generator are as shown in other sections. So vertical space at a point $q$ is spanned by $\{(\square,0,0)|\square\in T_{q}SO(3)\}$. Locally we can represent elements in $T_{q}Q$ as $((r_{1},\ r_{2},\ r_{3}),\ \dot{\beta},\ \dot{\gamma})$ where $(r_{1},\ r_{2},\ r_{3})\in\mathbb{R}^{3}\simeq\mathfrak{so}(3)$.Then $$V_{q}Q=\mathrm{span}\{\frac{\partial}{\partial r_{1}},\ \frac{\partial}{\partial r_{2}},\ \frac{\partial}{\partial r_{3}}\}$$ To find, $H_{q}Q$, the $\mathbb{G}$ orthogonal space has to be found out. $$\left\langle \begin{bmatrix} {\mathbb{I}_{s}}+R_{\beta} {\mathbb{I}_{gr}}R_{\beta}^{T} & (J_x + I_g)\vec{g} & (J_z + I_s)\vec{s}_{\beta}\\ (J_x + I_g)\vec{g}^{T} & (J_x + I_g) & 0\\ (J_z + I_s)\vec{s}_{\beta}^{T} & 0 & (J_z + I_s) \end{bmatrix}\begin{bmatrix}\begin{pmatrix}h_{1}\\ h_{2}\\ h_{3} \end{pmatrix}\\ h_{4}\\ h_{5} \end{bmatrix},\quad\begin{bmatrix}\square\\ 0\\ 0 \end{bmatrix}\right\rangle =0$$ where $\square\in V_{q}Q$ . This implies $$({\mathbb{I}_{s}}+R_{\beta} {\mathbb{I}_{gr}}R_{\beta}^{T})\begin{pmatrix}h_{1}\\ h_{2}\\ h_{3} \end{pmatrix}+ (J_x + I_g) \vec{g}h_{4} + (J_z + I_s)\vec{s}_{\beta}h_{5}=0$$ The horizontal vector $(h_{1}\ h_{2}\ h_{3}\ h_{4}\ h_{5})^{T}$ satisfies 3 (independent) equations in 5 variables. The horizontal space is then 2 dimensional as expected since the number of shape variables is 2. If we let $h_{4}$ and $h_{5}$ be the independent variables, then $$\begin{aligned} \begin{pmatrix}h_{1}\\ h_{2}\\ h_{3} \end{pmatrix}= & ({\mathbb{I}_{s}}+R_{\beta} {\mathbb{I}_{gr}}R_{\beta}^{T})^{-1} [(J_x + I_g)\vec{g}\ (J_z + I_s)\vec{s}_{\beta}] \begin{pmatrix}h_{4}\\ h_{5} \end{pmatrix}\\ = & \tilde{I}_{\beta}^{-1} [(J_x + I_g) \vec{g}\ (J_z + I_s) \vec{s}_{\beta}] \begin{pmatrix}h_{4}\\ h_{5} \end{pmatrix}\end{aligned}$$ Now we can write any tangent vector as the sum of a vertical vector and a horizontal vector as follows $$v_{q}=\begin{pmatrix}v_{1}\\ v_{2}\\ v_{3}\\ 0\\ 0 \end{pmatrix}+\begin{bmatrix}\tilde{I}^{-1} [(J_x + I_g) \vec{g}\ (J_z + I_s) \vec{s}_{\beta}]\\ \begin{array}{cc} 1 & 0\\ 0 & 1 \end{array} \end{bmatrix}\begin{pmatrix}h_{4}\\ h_{5} \end{pmatrix}$$ The $\mathfrak{g}$ valued connection form can be then written (locally) as (here we write it as $\alpha:TQ\rightarrow\mathfrak{so}(3)\simeq\mathbb{R}^{3}$) $$\begin{bmatrix}\alpha_{11} & \alpha_{12} & \alpha_{13} & \alpha_{14} & \alpha_{15}\\ \alpha_{21} & \alpha_{22} & \alpha_{23} & \alpha_{24} & \alpha_{25}\\ \alpha_{31} & \alpha_{32} & \alpha_{33} & \alpha_{34} & \alpha_{35} \end{bmatrix}\left(\begin{pmatrix}v_{1}\\ v_{2}\\ v_{3}\\ 0\\ 0 \end{pmatrix}+\begin{bmatrix}\tilde{I}_{\beta}^{-1}[(J_x + I_g) \vec{g}\ (J_z + I_s) \vec{s}_{\beta}]\\ \begin{array}{cc} 1 & 0\\ 0 & 1 \end{array} \end{bmatrix}\begin{pmatrix}h_{4}\\ h_{5} \end{pmatrix}\right)=\begin{pmatrix}v_{1}\\ v_{2}\\ v_{3}\\ 0\\ 0 \end{pmatrix}$$ solving we get, $$\begin{bmatrix}\alpha_{11} & \alpha_{12} & \alpha_{13} & \alpha_{14} & \alpha_{15}\\ \alpha_{21} & \alpha_{22} & \alpha_{23} & \alpha_{24} & \alpha_{25}\\ \alpha_{31} & \alpha_{32} & \alpha_{33} & \alpha_{34} & \alpha_{35} \end{bmatrix}=\begin{pmatrix}\mathrm{Id}_{3\times3}\quad & -\tilde{I}_{\beta}^{-1} [(J_x + I_g) \vec{g}\ (J_z + I_s)\vec{s}_{\beta}]\end{pmatrix}$$ Conclusions =========== We present the spacecraft system with the variable speed control moment gyros (VSCMG) cast into a geometric framework based on the principal fiber bundle. The dynamics of the system are derived. A kinematic and dynamic model for the above system is presented here. The expressions for the associated geometric objects such as the kinetic energy metric, locked inertia tensor, momentum map and mechanical connection are derived. The symmetry in the system is used to find the conserved quantity and reduce the number of state variables in the system. The corresponding reconstruction equations are derived. References {#references .unnumbered} ========== [^1]: banavar@sc.iitb.ac.in [^2]: arjun\_n@sc.iitb.ac.in

--- abstract: 'Clustering objects into synthetic groups is a natural activity of any science. Astrophysics is not an exception and is now facing a deluge of data. For galaxies, the one-century old Hubble classification and the Hubble tuning fork are still largely in use, together with numerous mono- or bivariate classifications most often made by eye. However, a classification must be driven by the data, and sophisticated multivariate statistical tools are used more and more often. In this paper we review these different approaches in order to situate them in the general context of unsupervised and supervised learning. We insist on the astrophysical outcomes of these studies to show that multivariate analyses provide an obvious path toward a renewal of our classification of galaxies and are invaluable tools to investigate the physics and evolution of galaxies.' author: - bibliography: - 'RevClustClass.bib' title: Multivariate Approaches to Classification in Extragalactic Astronomy --- 2015, [Frontiers in Astronomy and Space Sciences](http://www.frontiersin.org/milky_way_and_galaxies/10.3389/fspas.2015.00003/abstract) 2, 3 Introduction ============ Astrophysics has always adopted specific strategies to classify a relatively modest amount of diversity and has much counted on the physics to define the discriminant parameters. This discipline is now facing the need for sophisticated statistical tools to tackle the astronomical number of observed and catalogued objects and the increasing number of observed properties that describe them. The debate about the usefulness of the morphological classification of galaxies is a rather old one and is still alive. Sandage [@Sandage2005], a proponent of a (morphological) classification driven by the data, noticed that the Hubble classification and the Hubble tuning fork have not yet been replaced by anything else despite the efforts of the proponents of a classification driven by the physics [e.g. @Conselice2006]. It also has been recognised to have many flaws: it is a qualitative, subjective and visual approach, difficult to use for distant galaxies, it is based solely on the visible morphological parameter while galaxies are complex and evolving systems and while we have at our disposal morphologies from X-rays to radio wavelengths, spectra, chemical compositions, stellar populations, central black hole masses, kinematics of stars and gas... However this debate may not address the right question since from a classification point of view, a classification must be driven by the data, and thus be multivariate [e.g. @Fayyad1996]. Consequently, adapted tools must be used which are not well known to astronomers in general. Nevertheless, numerous studies have been published during the last thirty years or so, especially since the beginning of the XXIst century. In this paper we would like to present these different approaches in the general context of unsupervised (clustering) and supervised (classification) learning. Clustering approaches gather objects according to their similarities either through the choice of a distance metric or using some adequate criteria for deciding to which cluster some object belongs. There is a huge class of techniques that partition the data into a pre-defined number of clusters. A well-known algorithm is the k-means [@kmeans1967; @kmeans2010]. Another family of clustering techniques uses a hierarchical representation of the pairwise distances between objects in terms of a number of parameters (variables), through a bottom-up algorithm that constructs a tree by relating the closest objects together before relating these first clustering to closest clusters or objects, and so forth until the whole sample is exhausted. The final number of groups is then chosen by cutting the tree at a fixed distance level. The branches of the tree, called a dendrogram, may or may not represent relationships between the objects. Originally, phylogenetic methods are designed to build a graph representing the evolutionary relationships between species [see reviews in @Felsenstein2003; @Makarenkov2006]. Each node of the graph indicates a transmission with modification mechanism that creates two or more species inheriting from a common ancestor. More generally, a phylogenetic approach can be viewed as an unsupervised clustering approach in which relationships are provided. As a consequence, phylogenetic techniques are particularly versatile and powerful methods for building classification trees. They can be understood in the framework of the graph theory [@semple2003]. There are two kinds of phylogenetic methods, based either on the pairwise distances (or dissimilarities) computed from the parameters describing the objects, or on these parameters themselves. The distance-based methods build the tree entirely from the distances, putting forward the global similarities between the objects. The friends-of-friends algorithm is relatively famous in astrophysics [e.g. @More2011 and references therein]. Also known as the single linkage or Nearest Neighbor algorithm, it is mathematically related to the Minimum Spanning Tree technique which looks for the simplest graph connecting the objects under study [@Gower1969; @FeigelsonBabu2012]. A more sophisticated approach used in phylogenetic studies is the Neighbor-Joining Tree technique [@NJ1987; @NJ2006]. In the parameter-based methods, the parameters are called characters which in astrocladistics correspond to the parameters associated to the physical measurements of some properties of the objects. The parameter-based methods evaluate all possible trees that can be constructed with the objects, and select the tree(s) corresponding to an optimization criterion. The process is thus based on the distribution of the parameter values. Parameter-based methods can describe a larger variety of evolutionary scenarios and are thus more general that the distance-based methods. But this is at the cost of a larger computation time which quickly becomes prohibitive. Mathematically, formal connections between parameter- and distance-based methods are developed in the case of continuous parameters [e.g. @TF09; @TF15], explaining why both kinds of methods are successfully used in phylogenetic studies. Among the parameter-based techniques, cladistics is the most famous one. Invented in the 1950’s by William Hennig [@hennig1965], its principle looks simple: two (or more) objects are related if they share a common history, that is they possess properties inherited from a common ancestor. In practice, a cladistic analysis asks for the objects under study to be described by evolutionary characters (parameters or descriptors) for which at least two states are defined: one is said to be ancestral, the other one is said to be derived. The derived state corresponds to an innovation in the evolution and is assumed to have been acquired by an unidentified ancestor. This is the transmission phase of inheritance making descendants look similar to their parents. The accidents in this process are called modifications and generate diversity. This transmission with modification process was invoked by Darwin to explain the observed hierarchical organisation of the biological diversity. Several approaches have been developed to search for the best tree representation using Maximum Likelihood, some Bayesian approaches or Maximum Parsimony. In Maximum Parsimony, one searches for the tree representation of the data with the smallest number of evolutionary steps to explain the data. But in essence, any entity, be it biological or not, evolving with a transmission with modification process can be a priori studied by Maximum Parsimony, provided evolutionary states can be defined for the characters. A more general representation of relationships are given by networks even though their interpretation is quite complex, but they can be approximated by several trees. In this review, we do not intend to present all possible techniques in both supervised and unsupervised learning. Rather, we focus on the astrophysical published studies made with the objective of discovering structures in a data set, in other words a new clustering and possibly a new classification of galaxies, beyond the traditional Hubble morphological scheme. We refer the reader to the complete review by @Ball2010 on data mining tools used in astrophysics for further information and references in particular on the separation of sources or the classification of galaxies into morphological types. Our paper is mainly devoted to unsupervised classification (clustering) and presents the phylogenetic methods which are not included in @Ball2010. In addition we insist on the astrophysical outcome and the new insights that such studies have brought to our knowledge on galaxy physics and diversity. Part of this paper is inspired from @De2013 which compares the applicability of some of the clustering techniques on the basis of Gaussian and non Gaussian astronomical data sets. Here we do not make such a comparison. The paper is organized as follows. The first section presents a frequent prerequisite to data mining, the dimension reduction (Sect. \[reddim\]). This approach has been heavily used in the extragalactic literature to identify groups in the reduced component space, the motivation being mainly for automatic classification in large data sets. The second section describes the important difference between this motivation, called (supervised) classification, and the clustering (unsupervised classification) which is the main topic of this paper (Sect. \[supervisedunsupervised\]). We also discuss shortly the concept of similarity between objects. Partitioning methods divide the sample into distinct groups. This can be made with hard or soft bounds depending on whether the membership is a probability or not [see e.g. @Andrae2010]. The $k$-Nearest Neighbor (Sect. \[NN\]), Support Vector Machine (Sect. \[SVM\]) and k-means (Sect. \[kmeans\]) methods are of the first kind. The fuzzy clustering approach (Sect. \[fuzzy\]) belongs to the soft partitioning techniques and often extends the applicability of the previous methods. The Information Bottleneck approach is able to provide both kinds of classification (Sect. \[information\]). These partitioning methods require the number of classes as an input. Some other techniques try to fit some distributions to the data set, the optimization process providing the number of groups best fitting the data. These techniques are based on mixture model (Sect. \[mixturemodels\]) and wavelet (Sect. \[wavelet\]) methods. A different category of clustering approaches establishes relationships between the objects and derive the groups from the generated graph. The first such category are the hierarchical methods (Sect. \[hierarchical\]) which build a tree based on the pairwise distances. Different cuts on the tree result in different numbers of classes. These cuts can be chosen on the basis of objective arguments but also may vary according to the goal of the analysis since the tree provides a synthetic view of the structures within the data set, instead of just the group memberships. Another kind of graphs are the networks produce by the Minimum Spanning Tree method (Sect. \[MST\]). The last kind of relationships are evolutionary relationships. This is the domain of the phylogenetic techniques, a very wide subject of bioinformatics. We here present only the Maximum Parsimony (cladistics), Neighbor-Joining Tree Estimation and Outer Planar Networks that have been applied in the context of galaxies (Sect. \[meth:phylogenetic\]). Dimension reduction approaches {#reddim} ============================== Methods ------- When the data set is large (both in terms of number of variables and number of observations) one may first apply some appropriate dimension reduction technique and then perform clustering on the reduced data set. One must keep in mind that the discriminant usefulness of distances is lost in high dimension parameter spaces since distances tend to become similar (one of the aspects of the “curse of dimensionality”). **Principal component analysis (PCA)** In this technique, given a data set of observations on correlated variables, an orthogonal transformation is performed to convert it into a set of uncorrelated variables called the principal components. The number of principal components is less than or equal to the number of original variables. This transformation is defined in such a way that the first principal component has the largest possible variance. One rule of thumb is to consider those components whose eigen values are greater than one in the reduced space. Principal components are guaranteed to be independent only if the variables are jointly normally distributed. **Independent component analysis (ICA)** Principal component analysis, Factor Analysis, Projection Pursuit are some popular methods based on linear transformation. But ICA is different because it looks for the components in the representation that are both statistically independent and non Gaussian. ICA separates statistically independent components, which are the original source data, from an observed set of data mixtures. All information in the multivariate data sets are not equally important. There is often a need for extraction of the most useful information. ICA extracts and reveals useful hidden factors from the whole data sets. ICA defines a generative model for the observed multivariate data, which is typically given as a large database of samples. Contrarily to PCA, the components are not imposed to be orthogonal. Independent Component Analysis [@Comon1994], model assumes the form $$X = AS \label{eq:ICA1}$$ where $X$ is a data matrix, $A$ is the non-singular mixing matrix, $S$ is matrix of independent components. $A^{-\vert}$ is the unmixing matrix. The main goal of ICA is to estimate the unmixing matrix $A^{-\vert}$ . It is assumed that the data variables are linear or non-linear mixtures of some latent variables and the mixing system of equation \[eq:ICA1\] can be written as $$X_i = a_{i1} S_1 + a_{i2} S_2 + ...... + a_{in} S_n , i = 1, 2, ..., n \label{ICA2}$$ The $S_i$ are mutually independent and $a_{ij}$ are the entries of the non-singular matrix $A$. Here $n$ is the number of parameters (variables). For performing ICA, the data set has to be whitened in the sense that correlations in the data have to be removed. There is no rigorous method to determine the optimum number of ICs. For instance, the number of independent components can be taken to be equal to the number of principal components with eigen values greater than 1 [@Albazzaz2004]. As most of the data sets in Astrophysics are likely to be non Gaussian, ICA can be successfully used in many situations [@Chattopadhyay2013a; @Chattopadhyay2013b]. Applications ------------ PCA technique was applied in a few papers in the 1970s and 1980s with the goal of finding the main parameters explaining the variance among galaxy samples. For instance, @Watanabe1985 used four parameters (diameter, magnitude, mean surface brightness and mean concentration index) and found that two principal components explains 97% of the total variance in their sample of all morphological types, in agreement with other studies. While @Watanabe1985 do not find differences in the two-dimension PC plane between elliptical and disk galaxies, @Whitmore1984 more explicitly looks for an objective classification of galaxies: “ The fact that there are so many different classification systems for galaxies...demonstrates that we are still searching for the fundamental properties.”. Using more parameters (up to 15) they agreed with the other studies on two components explaining most of the variance, and tentatively identify them as scale and form. They do not devise a new classification scheme, but rather identify different correlations depending on the position of the galaxies on the 2D diagram. @Chattopadhyay2006 also found two components in a PCA analysis of samples of spiral galaxies with extended rotation curves. They constructed new “fundamental planes” with these components, pinpointing the most important physical factors. They also performed a multiple stepwise regression analysis of the variation of the overall shape of the rotation curves and find that it is mainly determined by the central surface brightness, while the shape purely in the outer part of the galaxy (beyond the optical radius) is mainly determined by the size of the galactic disk. Such a regression is interesting to predict still unobserved values for some parameters, and is improved by the reduction of the dispersion in the principal component space. @Peth2015 used PCA as a simple way to reduce the dimensionality, break internal degeneracies and find the natural distributions of data in the parameter space characterizing the structures and shapes of galaxies that they study. These principal components are then used to classify the shape of galaxies through a hierarchical clustering technique (see Sect. \[hierarchical\]). Several studies [e.g. @Connolly1995] used PCA both as a dimension reduction and as a tool for classification of spectra of galaxies. Spectra are characterized by a high number of attributes (the wavelengths) that are not independent since a spectrum is made of a continuum spectrum from stars plus absorption and emission lines from the gas. PCA has in principle the power to identify the minimum number of spectra to combine in order to obtain the observed diversity. [@Connolly1995] used a variant of the PCA technique, the Karhunen-Loève transform, which allows for weighting differently some parts of the spectra. They not only find that two eigenspectra are necessary to account for most of the variance of the spectra of galaxies, but the distribution of classes in the two-parameter space is one-dimensional. They proposed a scheme of ten classes, some corresponding to the broad morphological types Sa, Sb, S0 and E, while the six others are starburst objects. Their work was intended to be used by spectral surveys to classify automatically the observations. In a similar scope of general classification of galaxies, one must mention the attempt by @Scarlata2007 to build a morphological automated classification of galaxies, the ZEST catalog, using PCA [see @Coppa2011] but the parameters used are criticized by @Andrae2010. This illustrates the importance of the selection of the parameters for a multivariate clustering or classification analysis which at some point may appear arbitrary and subjective. A special care should be brought to this initial step through the analysis of the data set itself with dedicated data mining tools. Another instance is the classification established by @Conselice2006 using a PCA analysis together with a Spearman Rank correlation test to better understand the parameters of the data set. His approach is to use the PCA on some set of parameters and then understand the physics of the principal components. So the PCA shed light on the underlying physics from which a classification scheme can be built. He finds three dimensions for this scheme, with the mass (scale), the star formation (spectral type) and the interactions/mergers (degree of dynamical disturbances). This should remind that PCA is not a clustering technique per se, it provides a new representation of the data from which a clustering may be performed. Indeed the work by @Conselice2006 proposes new relationships between the morphological classes. His scheme appears as a more physical replacement of the 2D Hubble tuning fork diagram. The Principal Component Analysis assumes a linear combination of the parameters, a rather strong assumption. @TaghizadehPopp2012 have used a non-linear PCA, the Principal Curve analysis, “which can be seen as a nonparametric extension of linear PCA. The principal curve is the curve following the location of the local mean in the multi-dimensional cloud of data points.” They obtain a drastic dimension reduction with a one-dimension parameter space (the Principal Curve) which they divide arbitrarily into 20 groups of equal densities. They compute a distance (the arc length) that ranks the galaxies so providing “a natural and objective way of ordering, partitioning and classifying the rich zoo of galaxies in the nearby universe”. @TaghizadehPopp2012 do not include luminosity nor mass in the process in order not to bias the study of the luminosity function as a function of the arc length. This is debatable but they are right in saying that it would induce a bias since these parameters will define a strong axis of variance in the PCA. Nevertheless would it be possible to classify galaxies without their mass? Could massive galaxies have the same history as less massive ones? This shows that the choice of the parameters is never so obvious, and generally related to the choice of the technique used as well. The interesting point is that they recover known trends in the physics of galaxies, but more importantly they can identify new kinds of galaxies pointing out particular physical processes and histories of galaxies. These discoveries can only be made by multivariate analyses. @Folkes1996 applied PCA on spectra of low signal-to-noise ratio mainly as a dimensionality reduction technique. The few principal components are then used to train a neural network in order to classify galaxies into the five broad morphological types. Even though this approach is efficient for big data sets, it appears limited to normal galaxies since they find that a new classification scheme must be used where unusual features are present in the spectra. The ICA analysis is still less common than PCA for the study of galaxies. At least two studies have been published, an ensemble learning for ICA [@Lu2006] and a mean field independent component analysis [@Allen2013]. In the first case, 1326 synthetic spectra have been used coming from Single Stellar Population models. They select 74 “sufficiently” different spectra from these (using an objective criterion) since the ensemble learning part converges very slowly. The ICA analysis yields six most significant components, and the 1326 spectra are fitted on these components. Each component represent a basic element behind the spectra of galaxies, and they find that each of them can be associated closely to one or a few stellar types plus some peculiar line properties. These six components are then used on real galaxy spectra to derive the stellar contents like starlight reddening, stellar velocity dispersion, stellar content, and star formation history. Even though PCA is much faster, it does not provide this important information because of the orthogonality constraint that does not allow the components to be non-negative. @Allen2013 used the mean field ICA which is a probabilistic ICA using a prior to constrain the components. They find that ten components (divided into five continuum and five emission components) are required to produce accurate reconstructions of essentially all narrow emission-line galaxies to a very high degree of accuracy. Using these ten components on a large sample of Sloan Digital Sky Survey (SDSS) galaxies, they identify the regions of parameter space that correspond to pure star formation and pure active galactic nucleus (AGN) emission-line spectra, and produce high S/N reconstructions of these spectra. In a similar fashion, @Hurley2014 applied the Non-negative Matrix Factorization technique which has been developed for blind source separation problems. Unlike PCA, this technique imposes the condition that weights and spectral components are non-negative that is also possible in the ensemble learning approach for ICA described above [@Lu2006]. This more closely resembles the physical process of emission in the mid-infrared region studied in this work, resulting in physically intuitive components. They find seven such components, including two for active galactic nucleus emission, one for star formation, and one for the rising continuums at longer wavelengths. They show that the seven components can be used to separate out different types of objects (see Sect. \[mixturemodels\]) and to separate out the emission from AGN and star formation regions and define a new star formation/AGN diagnostic which is consistent with all mid-infrared diagnostics already in use but has the advantage that it can be applied to MIR spectra with low signal-to-noise ratio or with limited spectral range. Supervised and unsupervised learning {#supervisedunsupervised} ==================================== Distances/dissimilarities ------------------------- A lot of learning techniques require a dissimilarity measure. Among them, the distances obey the well-known triangular properties and define a metric. In hierarchical clustering, the distances mainly come from a very general distance known as the Minkowski’s distance or the $p$th norm, which may be defined as follows. For two points P $= (x_1, x_2 , ....., x_n)$ and Q $= (y_1 , y_2 , ....., y_n)$ in the $n$ dimensional space, the $p$th norm is given by $$L_p = \left(\sum_{i=1}^n \vert x_i - y_i \vert^p\right)^{1/p}$$ For $p=1$, it gives the Manhattan distance (L$_1$ norm). For $p = 2$, it reduces to the Euclidean distance (L$_2$ norm). Also for $p = \infty$, the L$_p$ norm results in Chebyshev distance. In hierarchical clustering, Euclidean and Manhattan distances are mainly used. But these measures are applicable only to continuous data. For categorical or binary data other distances must be used but will not be addressed in this paper. It may be noted that the selection of the appropriate distance matrix for clustering problems completely depends on the physical situation. Supervised learning (classification) ------------------------------------ Supervised learning technique may be viewed as a mapping between a set of input variables and an output variable. This mapping is applied to predict the outputs for unseen data. The main characteristic of supervised learning is the availability of annotated training data. It supervises the learning system to instruct on the labels to associate with training examples. These labels are known as class labels in classification problems. Supervised learning induces models for the training data and these models are then used to classify other unlabeled data. Two most popular supervised learning techniques are the Nearest Neighbor (Sect. \[NN\]) and the Support Vector Machines (Sect. \[SVM\]) classifiers. Unsupervised learning (clustering) ---------------------------------- The unsupervised learning or clustering seeks some pattern in the data set by starting from the raw data with or without any distributional assumption regarding the underlying distribution. The three main categories of this kind are (i) connectivity based clustering (like hierarchical clustering, see Sect. \[hierarchical\]), (ii) centroid based clustering (like k-means, see Sect. \[kmeans\]) and (iii) density based clustering (like DBSCAN or more generally kernel density estimation). An overview of these approaches can be found in @DAbrusco2012 with many references of applications in astrophysics. Most of the methods that we present in the following are unsupervised clustering. The reason is that the multivariate analyses of galaxies essentially are either supervised approaches based mainly on dimension reduction techniques (mostly PCA, see Sect. \[reddim\]) or unsupervised methods to discover new classification schemes of galaxies which are really objective and multivariate. Nearest Neighbor {#NN} ================ {width="8cm"} **\[fig:kNN\] Figure .**[ In this k-Nearest Neighbor illustration with $k=5$, the central black square more probably belongs to the blue class. ]{} Method ------ The $k$-Nearest Neighbor (NN) algorithm is very intuitive. It starts from a training set for which we have the class labels. In order to make a prediction about a new observation, one looks at the labels of its $k$ nearest neighbors and uses a majority vote to make the prediction (Fig. \[fig:kNN\]). As the number of neighbors used in making the prediction increases, the decision boundaries become smoother, the bias increases, but the variance decreases. Applications ------------ @Ball2007 explored the k-Nearest Neighbor technique for determining photometric redshifts in petascale databases using 55 746 quasar spectra from the SDSS. The algorithm is trained on a representative sample of the data. The main result is that the comparison between the photometric and the spectroscopic redshifts shows no region of catastrophic failure where the two derived values differ a lot, contrarily to other methods used to derive photometric redshifts. Support Vector Machine {#SVM} ====================== Method ------ Support Vector Machine (SVM) aims to find the hyperplane that best separates two classes of data through an optimization method. Instead of using just a standard orthogonal basis, SVM uses many functions to describe good separating surfaces. The input data are viewed as sets of vectors, and the data points closest to the classification boundary, determined from a training sample, are the support vectors. SVM fundamentally separates two classes of objects which is probably a limitation in its use for the classification of galaxies. They use optimization methods to find surfaces that best separate categories. Their key innovation is to express the separating surfaces in terms of a vastly expanded set of basis functions. Instead of using just a standard orthogonal basis, SVMs use many basis elements. Applications ------------ SVM has been used by @HuertasCompany2008 for the morphological classification of galaxies from the COSMOS survey. The training sample is a limited sample classified visually using a 12-dimensional volume, including 5 morphological parameters, and other characteristics of galaxies such as luminosity and redshift. The objective is to be able to classify automatically the results of big surveys. However, the result seems a little bit disappointing since it can only separate between the two broad classes of early- and late-type galaxies, with an error of about 20%, even though this is better than other methods generally used. K-Means {#kmeans} ======= Method ------ The k-means algorithm [@kmeans1967; @kmeans2010] is a partitioning approach that starts with $k$ centroids, $k$ corresponding to the number of clusters given a priori. It then assigns each data point to the closest centroid and when the clusters are built, the new $k$ centroids are computed and the process iterates until convergence (Fig. \[fig:kmeans\]). The result depends very much on the initial centroids. Repeating the analysis with several initial choices is always a good idea, but consistency is not guaranteed if the data do not contain distinguishable and roughly spherical clusters. Some strategies have been devised to guess the best initial choice for the centroids [e.g. @Sugar2003; @Tajunisha2010] and many indices are available in the package $NbClust$ [@NbClust] of R [@R]. A variant called the k-medoids algorithm [@kmedoids1987; @kmedoids] chooses data points as centers (medoids) and is known to be more robust to noise and outliers. {width="8cm"} **\[fig:kmeans\] Figure .**[ A typical result of a k-means analysis in which the clusters (four here) are clearly distinguishable [from @Fraix2010]. ]{} Applications ------------ The k-means algorithm has been used in the context of stars [e.g. @Gratton2011; @Simpson2012], galaxies [e.g. @Fraix2010; @SanchezAlmeida2010; @Fraix2012] or Gamma-Ray Bursts [@ChattopadhyayGRB2007]. @SanchezAlmeida2010 performed a k-means analysis of a large number (788 677) of spectra from the SDSS. Each spectra is a collection of about 4 000 wavelengths, making the full data set very computationally demanding for a direct k-means. They thus decided to limit the spectra to a priori informative regions, reducing the number of “parameters” to 1637. Their analysis is affected by the dependence of the result on the seed. They say that estimation tools for the number of clusters could not be applied because of the sample size. Using some criteria, they end up choosing randomly one classification having 28 classes. The result looks more like a continuum distribution of spectra, and even if not shown, overlapping between classes is important. This questions the validity of the k-means approach in this case as another k-means analysis of the same sample has shown [@De2014]. Multivariate k-means analyses of smaller sample of galaxies with the aim of discovering new classes of galaxies have been performed as a complement to other clustering methods by @Fraix2010 with the four parameters of the fundamental plane, and by @Fraix2012 with six parameters selected from 23 available. In the latter case, the selection of the parameters is made through different statistical tools, in order to find a parameter subspace in which a robust clustering of the data is present. This leads to the important result that several very different clustering techniques yield compatible clusterings, giving good confidence to the result. The astrophysical implications are numerous since a new classification is established and the average properties and the correlations varies from group to group and often differ from those of the global sample. However, the interpretation benefited from the relationships between the classes established by the phylogenetic method used in these works and discussed in Sect. \[meth:phylogenetic\]. Even though the clusters are similar, the absence of these links in the k-means results is clearly missing. @Chattopadhyay2013 performed a k-means analysis of a large sample of dynamically hot stellar systems from globular clusters to giant ellipticals, in quest of the formation theory of ultra compact dwarf galaxies (UCDs), using three parameters (logarithm of stellar mass, logarithm of effective radius and stellar mass to light ratio). The number of clusters, five, is given by the optimum criterion of @Sugar2003. The classification of UCDs provides some new clues to the long discussed hypothesis that these objects may be formed either as massive globular clusters or have an origin similar to nuclei of dwarf galaxies. Fuzzy clustering {#fuzzy} ================ Methods ------- In non-fuzzy or hard clustering, data is divided into crisp clusters, where each data point belongs to exactly one cluster. In fuzzy clustering, the data points can belong to more than one cluster, and associated with each of the points are membership grades which indicate the degree to which the data points belong to the different clusters. Many algorithms exist, many of them being extension of hard clustering algorithms. One example is the fuzzy C-means which is very similar to the k-means (Sect. \[kmeans\]) but adding a weight between 0 and 1 to each point characterizing its probability to belong to a given group, and a degree of fuzziness of the groups. Applications ------------ @Coppa2011 studied the bimodality of galaxies which comes from a double peak distribution in some scatter plots, particularly in color-color diagrams. The origin of this bimodality and the relationship between the two broad classes, “red” and “blue” or “late type” and “early type”, is still not understood. Evolution is probably involved, but then what is the status of the overlapping regions called “the green valley”? To know whether this bimodal distribution is an intrinsic property of galaxies and their evolution, multivariate analysis must be used since it appears in several scatter plots. @Coppa2011 use an unsupervised fuzzy partition clustering algorithm applied on the principal components of a PCA analysis. They use eight parameters, two coming from spectra, one from photometry and five describing the morphology. They keep three principal components to perform the clustering analysis which proceeds in two steps: a modified fuzzy k-means algorithm to guess the memberships and the cluster centroids, and a second algorithm (fuzzy modification of maximum likelihood estimation) to achieve optimal fuzzy partition [see references in @Coppa2011]. They decide to identify three clusters, blue, red and green, somewhat giving up the fuzzy nature of their study. In addition, they name the clusters after previous classifications, even though “the ’early type cluster is not intended to be made up of pure passive galaxies; rather, it is composed also by bulge-dominated weakly-star forming objects.” This is a quite confusing practice especially because they discover some new kinds of objects which are invaluable for our understanding of the physics and evolution of galaxies. Bayesian approaches can also be seen as soft classification as illustrated for instance in the separation between star forming galaxies and Active Galactic Nuclei (AGNs) in @Norman2004 to avoid confusion between different kinds of objects. Information Bottleneck technique {#information} ================================ Method ------ The Information Bottleneck Method [@Tishby2000] is a simple optimization principle for a model-free extraction of the relevant part of one random variable with respect to another. The algorithm is extremely general and may be applied to different problems in analogous ways. A great advantage of this unsupervised clustering technique is that it avoids the arbitrary choice of the distance and provides a natural quality measure for the resulting classification. Using the mathematical notations of @Slonim2001 that applied this technique to galaxies, the optimal classification is given by maximizing the functional: $${\mathcal L} [p(c|g)] = I(C ; \Lambda) - \beta^{-1} I(C; G)$$ where $C$ represents the classes, $G$ the galaxy sample and $\Lambda$ the spectral wavelengths. $I(C ; \Lambda)$ and $I(C; G)$ are the mutual information between $C$;$\Lambda$ and $C;G$. $\beta^{-1}$ is the Lagrange multiplier attached to the complexity constraint. For $\beta \rightarrow 0$ the classification is non-informative, and for $\beta \rightarrow \infty$ the representation becomes arbitrarily detailed. Applications ------------ @Slonim2001 explain that by normalizing the total photon counts in each spectrum to unity, we can consider it as a conditional probability, the probability of observing a photon at a specific wavelength from a given galaxy. The ensemble of spectra can thus be seen as a conditional probability distribution function that allows to undertake the information theory-based analysis. For any desired number of classes, galaxies are classified such that the information content about the spectra is maximally preserved. The number of classes is an issue in most unsupervised clustering techniques, and the information bottleneck shares this difficulty too. @Slonim2001 note that ’the true or correct number of classes may be an ill-defined quantity for real data sets and the number should be determined by the desired resolution, or preserved information“. However one should be careful to use objective arguments based only on statistics, since the physical interpretation should come at the end to tell whether or not the result is interesting. The main results of this study is the demonstration that an objective and automated technique can yield a classification of spectra which is very physical, in the sense that it recovers results obtained more classically, but is able to discover other classes and correlations between physical parameters. An interesting point in their study is that they applied the same techniques to two samples, one observed and one simulated. The good agreement between the two clusterings shows that the models of galaxy evolution are sensible. This is a good approach to test the models by statistically comparing two populations using multivariate data sets. Mixture Models {#mixturemodels} ============== Methods ------- Most partitioning methods use a distance to define the clusters. In model-based clustering methods, each cluster can be represented by a parametric distribution, the data set being thus considered as a mixture of such model distributions [@Qiu2007]. The parameters include the mixing proportions or the prior probabilities of the clusters since the true cluster memberships of the observations are unobserved. The optimization relies on the likelihood of the weighted linear combination of the cluster distributions through the Expectation-Minimization (EM) algorithm. Clustering is done by applying the maximum posterior (Bayes) rule. The process yields a soft classification (probability of membership) and a fit to each cluster distribution. The mixture model approach also provides expected misclassification probabilities. It requires the number of clusters to be known, which can be for instance estimated with the tools developed for the k-means analysis [@Chattopadhyay2009 Sect. \[kmeans\]]. Applications ------------ @Davoodi2006 find four Gaussian distributions best fit the color distribution of 16 698 extragalactic infrared sources. They use this result to propose a classification scheme ($C_a$ to $C_d$) of galaxies that reveal a greater variety of galaxy types than usual spectral energy distribution fitting techniques that strongly depends on the quality of the template model components. Interestingly, @Davoodi2006 use their soft classification to identify outliers (rare galaxies or transient phases) by summing up the four probability density functions for each object. @Hurley2014 used the seven components they have found with a dimension reduction approach (Sect. \[reddim\]) to define a parameter space in which they apply an unsupervised Gaussian Mixture Model clustering algorithm in order to provide a classification tool. This clustering approach is a fuzzy approach since clusters describe a probability density function indicating how likely a galaxy could be found in any one of the clusters. Eight clusters are found which are consistent with previous classifications. Strangely enough, these clusters are named according to the classical classification through a majority rule. We may ask why use an unsupervised technique if one believes in an existing ”true“ hard classification? Wavelet Analysis {#wavelet} ================ Method ------ The wavelet transform is a well known signal analysis technique widely used in many research areas. Its key property is the ability to provide a multi-resolution approximation of a given input signal through a prototype function $\Psi$: $$\label{eq:wavelet} W(s,r) = \int f(t) \frac{1}{\sqrt s} \Psi\left(\frac{t - r }{s}\right) dt$$ where $s$ characterizes the scale and $r$ the translation factor. The prototype function, also called the mother wavelet, is continuous in both time and frequency and serves as the analysing window. With this definition, wavelets appear as a parametric-model decomposition of a data set using some basis functions. They could then be used for dimension reduction and/or classification [@Thuillard2001]. Shapelets are a scaled version of two-dimensional Gauss-Hermite polynomials and form a set of complete basis functions that are orthonormal on the interval $\left[-\infty,\infty\right]$. Shapelets are thus suited to decompose images. For galaxies, their use is limited to high signal-to-noise data and rather regular galaxies since they are gaussian-shaped and spherical [@Andrae2010]. The composition is an automatic and objective representation of galaxy morphologies. Other multiresolution methods have been proposed, like for instance the hierarchical Markov models extended for the multispectral astronomical image segmentation [@Collet2004]. Applications ------------ Wavelets can be used to decompose galaxy spectra into several features that can then be used to classify the spectra. In this sense they serve as a dimension reduction technique but contrarily to PCA or ICA the basic elements (features) can be chosen to be physically meaningful, representing the three components of spectra: the continuum, the emission and absorption lines [e.g. @Starck1997; @Liu2005]. @Andrae2010 review how an automatic classification of galaxy morphologies could be done using shapelets. Their goal is not to devise a new classification, since it is extremely difficult to parametrise arbitrary galaxy morphologies apart from the question that the morphology is only one property of galaxies. To address the parametrisation problem, they use shapelets and then define the distance as the angle spanned by their (normalised) coefficient vectors of the shapelets: $$d(x_i,x_j) = \arccos(x_i\cdot x_j)$$ They then use a soft (fuzzy) clustering algorithm with the similarity matrix given by: $$W_{mn} \propto \frac{\left(d(x_i,x_j)/d_{max}\right)^{\alpha} }{s}$$ with $d_{max}$ being the maximum distance between any two objects in the given data sample, and $\alpha > 0$ and $s >1$ being free parameters that tune the similarity measure. This probabilistic clustering technique uses the graph theory in which the similarity elements $W_{mn}$ are the weights of the edges. They also evaluate the impact of hard clustering methods on the estimation of the parameters characterising the classes depending on the level of overlapping. This is an important point to keep in mind in all hard (non-fuzzy) approaches to clustering, be it by hand or algorithmic. They even suggest that the processes of galaxy evolution and observations tend to invalidate hard clustering approaches. They do not go into the details of the astrophysical interpretation, but they clearly demonstrate the advantages of such sophisticated approaches for automatic morphological classification of a huge number of galaxies. However, as they rightly say, “a lot of work is still needed on the interpretation of the results.” Hierarchical Classification Methods {#hierarchical} =================================== {width="8cm"} **\[fig:dendro\] Figure .**[ An example of a dendrogram. Distance between two horizontal branches is going from left to right. The two dashed lines illustrates two cuts yielding 9 or 3 clusters.]{} Methods ------- The hierarchical classification method builds a hierarchy of clusters. Two main approaches to form the hierarchy are agglomerative and divisive. In the agglomerative approach each observation is considered as a cluster and pairs of clusters are merged as one moves up the hierarchy (see Fig. \[fig:dendro\]). The most similar objects are grouped first and those initial groups are merged ultimately into single cluster according to some proximity measure. These proximity measures are based on either similarities or dissimilarities (distances). In the divisive analysis approach all observations at first are grouped in one cluster, and splits are performed recursively as one moves down the hierarchy. Here an initial single group of objects is divided into two subgroups such that the objects in one subgroup are far from the objects in the other. These subgroups are further divided into dissimilar groups until there are as many subgroups as objects. In order to decide which clusters should be combined or where a cluster should be split, a distance matrix is required. The distances used for hierarchical clustering are mainly Euclidean and Manhattan for continuous type data. In order to find distances between clusters different linkages like single linkage, complete linkage, average linkage etc are used. Note that the nature of the final clusters totally depends upon the choices of distances and linkages. It is interesting to note that if the metric used is the single linkage, then this method is similar to the Minimum Spanning Tree technique (Sect. \[MST\]). Applications ------------ @Peth2015 applied a Ward hierarchical agglomerative clustering to classify galaxies in distinct groups using the first three principal component eigenvectors. In this kind of approach the number of groups is chosen after the analysis. @Peth2015 selected ten groups as a compromise between too many small groups which might appear as too specific, and too large ones that would smear out the true diversity of the objects. They also try to define boundaries to these groups in the PC-space by fitting a convex hull around the points within each groups in order to classify future new observed objects. However, a Nearest-Neighbor or SVM technique could be used in this purpose without the need to compute a convex hull which is a rigid boundary. It is important to recall that a classification is never definitive and would probably evolve with the inclusion of new objects, as it has been for instance the case for the S0 (lenticular) morphological class of galaxies which were not present in the original Hubble classification. One of the main results of the studies by @Peth2015 is a refined and objective classification of structures and morphologies of the galaxies in their samples. The ten groups are analyzed separately to derive their properties and their probable evolutionary status and history. Their scheme separates quenched compact galaxies from larger, smooth proto-elliptical systems, and star-forming disc-dominated clumpy galaxies from star-forming bulge-dominated asymmetric galaxies. It also reveals a higher fraction of bulge-dominated galaxies than visual classification or one based on the Sersic index. Decision trees are a practical use of hierarchical clustering. @SanchezAlmeida2012 propose a decision tree to classify galaxy spectra according to some general features that usually serves as a classification of galaxy properties. They use the decision tree on their previously ASK classes determined with the k-means technique [Sect. \[kmeans\], @SanchezAlmeida2010]. Somehow, in this way, they classify their new classes on another classification. @Suchkov2005 have applied an oblique decision tree classifier on the homogeneous multicolor imaging data base of the SDSS, the classifier being trained on subsets of objects (stars and galaxies) whose nature is precisely known via spectroscopy. Each node in the decision tree is a criterion on one parameter, defining an hyperplane parallel to one of the axis. In an oblique decision tree, the criterion is based on a (linear) combination of parameters, so the tree is no more parallel to any of the axes in the parameter space. In @Suchkov2005 the classifier is composed of ten oblique decision trees and the final decision is made by votes which yield a class probability distribution for a given object. The main result of their study is to show the ability of this approach to automatically classify objects from the photometry instead of the spectroscopy which is harder to obtain and analyse, and accurately predict the redshifts of both normal and active galaxies. This can increase considerably the samples required to analyse statistically the evolution and diversity of galaxies, their properties and their correlations. Minimum Spanning Tree {#MST} ===================== Methods ------- The Minimum Spanning Tree (MST) is mathematically related to the single linkage clustering, known to astronomers as the friends-of-friends algorithm or Nearest Neighbor algorithm [@Gower1969; @FeigelsonBabu2012]. A spanning tree is an acyclic, connected graph $G$ which is a set $(V,E)$ of vertices (nodes) and edges (branches) together with a function $w : E \rightarrow \mathbb{R}$ that assigns a weight $w(e)$ to each edge $e$ in $E$. The minimum spanning tree (Fig. \[fig:MST\]), is the spanning tree $T$ minimizing the function : $$\label{eq:MST} w(T) = \sum_{e\ \in\ T} w(e)$$ If the weights $w(e)$ are distinct, then the solution is unique. A number of algorithms have been developed to solve exactly the Minimum Spanning Tree problem. The first algorithm is attributed to @Boruvka1926. Other popular algorithms are Prim’s, Krukal’s and the Reverse-Delete algorithms that all find solutions in polynomial time. The above algorithms also work at higher dimensions in which case the Euclidean L2 or the Manhattan L1 distances are generally used. Minimum spanning trees have found applications in phylogeny, computer vision, and cytology just to name some domains. It has been used in astrophysics, and maybe very early since a large number of constellations defined by early civilizations are also shown to correlate well with a Minimum Spanning Tree [@Dry09]. {width="8cm"} **\[fig:MST\] Figure .**[ An illustration of a Minimum Spanning Tree linking ten nodes. ]{} Applications ------------ The Minimum Spanning Tree technique has been heavily used to determine the galaxy clusters in order to map the spatial distribution of the baryonic matter, a visible signature of the gravitational structure of the Universe shaped by the Dark Matter [@Barrow1985; @Bhavsar1996]. This is not an application to clustering in the sense of classification, but this is a spatial clustering. Indeed the MST approach has been strongly adapted to the particular constraint of cosmological observations: the exact position along the line of sight is only approximately given by the redshift. We do not discuss any further this application which is not in the main scope of this paper. We know of only one use of MST for galaxy classification. @Ascasibar2011 applied this technique to understand their ASK classification of SDSS spectra obtained with the k-means method [Sect. \[kmeans\], @SanchezAlmeida2010]. They find that the majority of the spectral classes are distributed along a well-defined branch going from the earliest to the latest types, with optically bright active galaxies forming an independent branch that intersects the main sequence exactly at the transition between early and late types. This description is already an interpretation of the 23 ASK classes that present a regular distribution of their spectra as already mentioned in Sect. \[kmeans\], so that the very linear structure of the MST tree is not surprising. However, the approach is interesting because this is a rather simple and objective method to obtain relationships between classes. Phylogenetic methods {#meth:phylogenetic} ==================== Basically, all galaxies share a common origin which is the gathering of baryonic matter as a self gravitating object. This baryonic matter was very primitive and has subsequently being enriched and diversified by several generations of stars and many transforming processes like galaxy interactions and mergers. There are thus obvious evolutionary relationships between different kinds of galaxies as immediately understood by Hubble when he discovered galaxies and established his famous tuning fork diagram. Taking into account the galaxy diversity of morphologies known at that time, he built a phylogenetic tree in which the relationships are due to the evolution of the stellar orbits which, he thought, should flatten with time because of the dynamical friction. Even though we now know that this process cannot be accomplished in a time shorter than the known age of our Universe, this tuning fork diagram is still used to represent galaxy diversity. Somewhat strangely enough, phylogenetic analyses of galaxy diversity has not been attempted again for a century. This is probably because the data did not allow much progress into this direction. But we now have huge multivariate databases and it seems timely to reconsider this question. We here present only a few techniques, those that have been already used on astrophysical data sets. Methods ------- Before describing some of the most important methods, let us point out that the development of phylogenetic methods has been hindered till the 2000’s by very heated discussions on the philosophical merits of the different approaches. It is only in recent years that most of the barriers between the different schools of thoughts could be overcome by a new generation of researchers. Recently a new picture of phylogenetic methods is emerging. It becomes nowadays increasingly clear that all the different approaches can be discussed within a common framework including distance- and character-based approaches, and that this theoretical framework applies both to phylogenetic trees and networks. There are two main categories of methods: the distance-based and the character-based. The “characters” are traits, descriptors, observables, parameters or properties, which can be assigned at least two states characterizing the evolutionary stage of the objects for that character. For continuous parameters, these states can be obtained through discretization. **Distance-based Approaches: Neighbor Joining Tree Estimation** \[meth:NJ\] For distance-based approaches, Neighbor-Joining is the most popular approach to construct a phylogenetic tree. The Neighbor Joining Tree Estimation [NJ, @NJ1987; @NJ2006] is based on a distance (or dissimilarity) matrix. This method is a bottom-up hierarchical clustering methods. It starts from a star tree (unresolved tree). A “corrected” distance $Q(i,j)$ between objects $i$ and $j$ from the data set of $n$ objects, is computed from the distances $d(i,j)$: $$\label{eq:NJ} Q(i,j) = (n-2)d(i,j) - \sum_{k=1}^{n}d(i,k) - \sum_{k=1}^{n}d(j,k)$$ The branches of the two objects with the lowest $Q(i,j)$ are linked together by a new node $u$ on the tree. This node replaces the pair $(i,j)$ in the subsequent iterations through the distance to any other object $k$: $$\label{eq:NJnode} d(u,k) = \frac{1}{2}\left[d(i,k)-d(i,u)\right] + \frac{1}{2}\left[d(j,k)-d(j,u)\right]$$ Neighbor-Joining minimizes a tree length, according to a criteria that can be viewed as a Balanced Minimum Evolution [@NJ2006]. For a tree metrics, Neighbor-Joining furnishes a simple algorithm to reconstruct a tree from the distance matrix. There is a large literature on how to best approximate a metrics by a tree metrics [see for instance @Fakcharoenphol2003]. Neighbor-Joining is justified if the difference between the original distance matrix and the distance matrix describing the X-tree obtained with Neighbor-Joining is not too large. **Character-based Approaches: Cladistics, Maximum Parsimony, Maximum Likelihood...** \[meth:chalclad\] {width="8cm"} **\[fig:clado\] Figure .**[ A example tree obtained with cladistics, represented here as unrooted. When a root is chosen, the tree takes the shape of hierarchical trees. ]{} Cladistics has been associated in the 80’s to the search of a maximum parsimony tree. Maximum Parsimony is a powerful approach to find tree-like arrangements of objects (Fig. \[fig:clado\]). The drawback is that the analysis must consider all possible trees before selecting the most parsimonious one. The computation complexity depends on the number of objects and character states, so that too large samples (say more than a few thousands) cannot be analyzed. The Maximum Parsimony algorithm can take uncertainties or unknowns into account by evaluating the different possibilities allowed by the range of values and selecting among them the one that provides the smallest score. In the case of unknown parameters, the most parsimonious diversification scenario provides a prediction for the unknown values. In recent years the definition of cladistics has been extended to the classification of taxa (individuals or species) defined by characters on a rooted tree. In biological applications, a phylogenetic tree describes the possible evolution of a taxon corresponding to the root. The root may either be a real taxon or be inferred from the descendant taxa. The success of a cladistics analysis much depends on the behavior of the parameters. In particular, it is sensitive to redundancies, incompatibilities, too much variability (reversals), and parallel and convergent evolutions. It is thus a very good tool for investigating whether a given set of parameters can lead to a robust and pertinent diversification scenario. If a set of characters exactly defines a phylogeny, then the phylogeny is called perfect. In practical applications, the available characters seldom define a perfect phylogeny. A supplementary measure of the deviation to a perfect phylogeny is necessary to determine how well a candidate tree fits the characters. In the standard approach to parsimony, the score $s_p$ of a tree corresponds, after labeling of the internal nodes, to the minimum number of edges $(u,v)$ with $c(u)\neq c(v)$, $c(u)$ being the character state at node $u$. The tree with the minimum score is searched for with some heuristics [@Felsenstein1984]. The maximum parsimony approach can be directly extended to continuous characters or values. To each internal node is associated a real value $f(u)$. The score s of a tree equals the sum over all edges of the absolute difference between those values: $$\label{eq:MPs} s = \sum_{e=(u,v) \epsilon E} \lvert f(u) - f(v) \rvert$$ @robinson1973 has shown that for a tree defined by continuous characters, a maximum parsimony score is reached for values of the internal nodes belonging to the set of values (or states) defined on the leaves. The main method to search for the best tree representation of data beyond Maximum Parsimony include Maximum Likelihood. We note this technique which has never been applied to astrophysics in the context of classification but may be a pertinent approach. The problem here is that an evolutionary model must be used, and naturally the result will depend significantly on it. Maximum Likelihood is used standardly in biology, and it may be possible that astrophysicists could also have well constrained physical models of the evolution of galaxies and their properties. The phylogenetic tree of Maximum Likelihood is the tree for which the observed data are most probable [@Williams2003]. Distance-based approaches are also often quite appropriate for reconstructing a phylogenetic tree from continuous characters. Distance-based approaches are fast and can be used for data exploration and for the selection of the most appropriate variables. Cladistics when applied to domain outside of biology, like in astrocladistics, refers more generally to the classification of objects by a rooted or an unrooted tree (Fig. \[fig:clado\]). In that case, the tree represents possible relationships between taxa. The search of the best tree described by a set of characters on a set of objects (or taxa in phylogeny) can be done by several different approaches. The most popular methods are the one using Maximum Parsimony or Maximum Likelihood. For continuous parameters, the software program TNT [@TNT2008] is also quite popular to reconstruct trees from characters. As an alternative, the data can be discretized through appropriate binning. As mentioned earlier, a new picture of phylogenies is emerging after the understanding that phylogenies on multistate characters reduce through a conceptually simple grouping of the characters into a phylogeny on binary characters. For binary characters, both distance- and character-based approaches are equivalent. This approach opens new perspectives as it furnishes also a bridge between character-based phylogenies and split networks or more precisely outer planar networks. **Outer planar networks** Outer planar networks permit the simultaneous representation of alternative trees with reticulations, and are thus generalizations of trees [@Huson2006]. In order to understand the connection between outer planar networks and phylogenetic trees, one has to explain succinctly what is called a split on a circular order of the taxa. A circular order on a phylogenetic tree corresponds to an indexing of the $n$ end nodes according to a circular (clockwise or anti-clockwise) scanning of the end nodes. A split on a circular order of the taxa is a partition of the objects into two disjoint sets (Fig. \[fig:split\]). {width="8cm"} **\[fig:split\] Figure .**[ A circular order for objects A to G, with their pairs of binary states, arranged according to the circular consecutive-ones condition. The two lines show two different split, one between (0,\*) and (1,\*), the other one between (\*,0) and (\*,1).]{} For multistate characters, a split can be defined after transformation of each multistate character into a binary character. For each pair of states (A,B), a subset of states containing the state ‘A’ is attributed the ‘1’ state and the complementary subset including the subset ‘B’ is given the binary state ‘0’. If the transformation can be done on each states and characters [for details see @TF15] so that each binary character fulfills the circular consecutive-ones condition, then the data can be described exactly by an outer planar network. By definition the circular consecutive-ones condition are fulfilled if for any binary state, the taxa with the ‘1’ state are consecutive on the circular order (Fig. \[fig:split\]). Splits in an outer planar network (Fig. \[fig:splitnetwork\]) furnish neighboring relationships between objects. Objects sharing a common property, as defined by splits, are consecutive in a circular order. Outer planar networks can be regarded as a generalization of phylogenetic trees. An outer planar network reduces to a phylogenetic tree if for each pair of binary characters, the so-called 4-gamete rule is fulfilled. The 4-gamete rule states that for each pair of binary characters there is at least one of the 4 possible gametes (either (1.0), (0,1), (1,1) or (0,0)) that is missing. {width="12cm"} **\[fig:splitnetwork\] Figure .**[ An example of an outer planar network showing the eight splits of the eight parameters s1 ... s8.]{} For distance-based approach, the circular consecutive-ones conditions have to be replaced by the fulfillment of the Kalmanson inequalities. For taxa indexed according to a circular order, the distances between a reference node $n$ and the path $i-j$ are gathered in the distance matrix $\left\{Y_{i,j}^{n}\right\}$ with $Y_{i,j}^{n}=\frac{1}{2}\left(d_{i,n}+d_{j,n}-d_{i,j}\right)$, $d_{i,n}$ being the pairwise distance between leave $i$ and node $n$. This distance matrix fulfils the so-called Kalmanson inequalities: $$\label{eq:Kalmanson} Y_{i,j}^{n} \geq Y_{i,k}^{n} \; , \; Y_{k,j}^{n}\geq Y_{k,i}^{n} \;\; (i\leq j\leq k)$$ @Bandelt1992 have shown that if a distance matrix $\left\{Y_{i,j}^{n}\right\}$ fulfils Kalmanson inequalities, then the distance matrix can be exactly represented by a split network or by an X-tree. The program SplitTrees4 [@Huson2006] permits to construct outer planar networks from a distance matrix. In practice, the perfect order is not known or not feasible. The difference between the perfect order and the order one obtains with a given data set is called the contradiction. The minimum contradiction analysis [@Thuillard2007; @Thuillard2008] finds the best order one can get. It is a powerful tool for ascertaining whether the parameters can lead to a tree-like arrangement of the objects [@TF09]. Using the parameters that fulfil this property, the method then performs an optimisation of the order and provides groupings with an assessment of their robustness. We believe that outer planar networks will gain importance in applications outside of biology as they furnish a real alternative to the standard classification methods. Applications ------------ @Farrah2009 have used a bayesian framework to compare and group 102 ultra-luminous infrared galaxy spectra and yield a network diagram which is used to define three groups. An evolutionary description of these galaxies is proposed from the properties of these groups. Even though their method is not a phylogenetic technique per se since the relationships are constructed after the clustering analysis, this work illustrates the potential need of phylogenetic tools in astrophysics. The use of phylogenetic approaches in astrophysics has been pioneered and pursued through the denomination of astrocladistics [@jc1; @jc2; @FCD06]. Applications have been successfully performed for galaxies [@Fraix2010; @Fraix2012], globular clusters [@FDC09; @FD15] and Gamma-Ray bursts [@Cardone2013]. The phylogenetic approaches used on galaxy samples are clearly oriented towards a multivariate and evolutionary classification of galaxies [@Fraix2010; @Fraix2012]. To this end, several statistical analyses (PCA, k-means, cladistics and minimum contradiction analysis) are used to select the set of parameters that yields a robust classification according to several clustering analyses (k-means, cladistics and Minimum Contradiction Analysis). Six parameters were so selected among the 23 available: the central velocity dispersion, the disc-to-bulge ratio, the effective surface brightness, the metallicity, and the line indices NaD and OIII. The agreement of the clustering obtained by different techniques reinforces largely the result. The cladistics tree (cladogram) is used for the interpretation since it also provides the relationships between the groups. These relationships are hypothesized as being evolutionary so that the placement of the groups on some diagrams reflects the evolution of the properties and their correlations. For instance, the famous fundamental plane is not universal at all, this 3D correlation clearly depends on the diversification level of the group: the correlation becomes tighter when the history of a galaxy is more complex. Other well-known correlations, like Mg$_b$ vs the velocity dispersion, indeed disappear within the groups but is created by the alignment of the groups in the scatter plot. This strongly suggests that these correlations (known as scaling relations) are statistical and caused by a hidden confounding factor, which is possibly the evolution [@DFB2011]. The new classification is rather easily interpreted with all the parameters available and by comparison with numerical simulations. The galaxies within a given group share a common history, that is a sequence of transforming events (collapse, interaction, harassment, merger...) that @Fraix2012 are able to identify. Outer planar or split networks have also been applied on galaxy samples [@TF09] even though it is for a theoretical illustration of an optimisation approach to fulfil as much as possible the Kalmanson inequalities (Eq. \[eq:Kalmanson\]). Nevertheless, a classification is obtained on this limited sample of 100 galaxies and with only three parameters. The main splitting character is the surface brightness (Brie) that separates the sample in two roughly equal bins. Each branch is then split into two other branches defined by the character states, “low OIII”, “high OIII” for the “low Brie” branch and “low B-R”, “high B-R” for the “high Brie” branch. The essential point here is that the split value separating “low” and “high” are not arbitrary at all, they are optimized according to an optimisation criterion aimed at obtaining the best split network or X-tree as possible. Even though the result cannot be given too much generality due to the small sample, the astrophysical outcome is informative. First, all high Brie galaxies have high OIII, but some high OIII galaxies have low Brie. This means that some low surface brightness galaxies in this sample have star formation, and some high surface brightness objects show only an OIII absorption feature Second, all high B-R galaxies have high Brie and high OIII. This means that in this sample, the red objects have a high surface brightness and some star formation. They are thus not simply ageing galaxies, but probably form stars with high metallicity. Conversely, all low OIII galaxies of the sample have a low B-R, so that blue objects do not necessarily form a lot of stars. Conclusions and perspectives ============================ In the astrophysical literature, we have found that there is a growing interest for automated classification of galaxies, which is motivated mainly by the exploding amount of available data. For this purpose, more or less sophisticated statistical analyses are recognized to be necessary. In this paper, we have reviewed the techniques used so far. We do not claim to be exhaustive, but we think we have described quite a broad range of statistical tools. Supervised learning analyses are mainly used to separate classes, morphological types or physical components in spectra of galaxies. The Principal Component Analysis is the most frequently used, due to its simplicity and efficiency, even though it is not a classification technique but rather a dimension reduction tool. Its attractiveness lies in its ability to perform automatic classification on moderately large data sets, and maybe more importantly, its ability to extract simple and important information from multivariate data. In this respect it greatly succeeds in separating spectra of galaxies, quasars and stars in large surveys. The supervised learning approaches require a classification to be established beforehand. In nearly all cases, the traditional morphological classification is the reference. It thus appears that the astronomers are keen to devise an objective way of classifying galaxies, using modern tools and multivariate data, but the classes to retrieve are devised subjectively with a visual inspection of images in the visible, hence a rather monovariate source. In the unsupervised learning analyses of the literature, the morphological classification also often serves as a reference that must be matched. However, many studies find different classes which bring new insights to the physics of galaxies and their evolution. These classes are homogeneous in the multidimensional parameter space, and not necessarily in the traditional classification scheme. Because of the number of properties to consider, the description of these new classes is more complicated, but simpler (and more pertinent) when a comparison with models and numerical simulations is performed. In addition, new kinds of objects are found which would not be possible in a multidimensional parameter space with traditional approaches. So an automatic classification of galaxies is becoming more and more crucial. The question remains of which classification is concerned. The predominance of the morphology as the most important parameter associated with the traditional classification scheme, is nearly overwhelming. Most unsupervised learning analyses yield new classifications, but this is not really exploited as such since their goal is often to propose an automatic way to retrieve the morphological classification. We think that this goal is hopeless since it hides a fundamental contradiction between the classification obtained from a traditional visual subjective and monovariate approach and the one yielded by a multivariate automatic and objective technique. The fact that obvious correlations exist between the new classifications and the traditional one is a very strong support in favor of these advanced approaches and should not obliterate the difference in the classes. The astrophysical results described in this review provide other arguments in favor of the statistical techniques, mainly because these tools can navigate more easily in a large dimensional space: - multivariate analyses are particularly interesting in the case of spectra, both for supervised and unsupervised classification. Dimension reduction is here an obvious requirement but proper unsupervised clustering is also necessary to discover new kinds of objects. - for spectra, unsupervised techniques generally do not require fitting with model spectra, so that the comparison between models and observations can really be performed in the multivariate parameter space. - more generally, the comparison between the observations, models and numerical simulations can be made by comparing the populations coming from the classifications of real and simulated galaxies, independently or together. - soft (fuzzy), tree- or network-based classifications seem more appropriate to the continuous distribution of galaxy parameters than hard clustering. - some techniques are based on the relationships between the objects and/or the classes. It is thus possible to objectively understand for instance the links between dynamically hot systems, or the place of the “green valley” galaxies with respect to “blue” and “red” ones, or the evolution of galaxies within the fundamental plane. We conclude from this review that unsupervised analyses should not be afraid to propose new classifications of galaxies. These new classifications should be compared to other such classifications, this is the only way to draw a global view of galaxy diversity and be able to classify automatically galaxies of the present and future big surveys. In addition, and probably more importantly, the physics of galaxies being intrinsically multivariate, their classification cannot be based on only one criterion. It is important to remember that there is not a unique best classification, and not a best tool. Comparison of results is a valuable task since it brings a lot of information on the nature of the data, the objects and their parameters. Also a classification is never definitive, and necessarily evolves with our knowledge and the discovery of new objects. We wish to end this review with the cluster validation question. This is an important issue in clustering and classification. In general, cross-validation and bootstrap techniques are rather easy and provide good estimates of cluster robustness. Some other validation indices are Dunn’s Validation Index [@Dunn1973], Davies-Bouldin Validity Index [@Davies1979], C Index [@Hubert1976] and Silhoutte Validation Index [@Rousseeuw1987]. Many more are given in the $clusterCrit$ package of R. In most of the clustering algorithms, the number of clusters are user specified. This is a difficult question, there are many tools (Sect. \[kmeans\]) to objectively guess the optimum number but they all have their drawbacks and limitations. Nevertheless, they should be used as much as possible to provide some hints and justifications. Disclosure/Conflict-of-Interest Statement {#disclosureconflict-of-interest-statement .unnumbered} ========================================= The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. Author Contributions {#author-contributions .unnumbered} ==================== The contributions is mainly as follows: DFB performed the review of the astrophysical literature, MT developed the theoretical aspects of phylogenetic and wavelets methods, AKC developed the other statistical tool descriptions. All three authors equally participated to the elaboration of the documents.

--- abstract: 'We announce some Python classes for numerical solution of partial differential equations, or boundary value problems of ordinary differential equations. These classes are built on routines in `numpy` and `scipy.sparse.linalg` (or `scipy.linalg` for smaller problems).' author: - 'Asif Mushtaq,  Trond Kvamsdal, K[å]{}re Olaussen,  [^1] [^2] [^3] [^4]' title: 'Python Classes for Numerical Solution of PDE’s' --- Boundary value problems, partial differential equations, sparse scipy routines. Introduction ============ Python computer language has gained increasing popularity in recent years. For good reasons: It is fast and easy to code and use for small “prototyping” tasks, since there is no need for explicit declaration of variables or a separate compilation cycle. It is freely available for most computer platforms, and comes with a huge repository of packages covering a large area of applications. Python also have features which facilitates development and encourages documentation of large well-structured program systems. Obviously, as an interpreted language native Python is not suitable for performing extended numerical computations. But very often the code for such computations reduces to calls to precompiled library routines. The `numpy` [@Walt_etal] and `scipy` [@Jones_etal; @Oliphant] packages make a large number of such routines directly available from Python. These packages are freely available for most operating systems, including [Linux]{}, [OSX]{}, and [MSWindows]{}. We here describe a process of making some of these routines even simpler to use for a field of applications, the numerical solution of partial differential equations discretized on a rectangular grid (or a subdomain of such a grid). As a simple reference problem one may consider the solution of the wave equation in the frequency domain, $$\left(-\Delta + \omega^2\right)\varphi(\bm{x}) = f(\bm{x}),$$ f.i. in a space with periodic boundary conditions. Our work is to a considerable extent motivated by a goal to solve the 3D acoustic wave equation with position dependent material properties, and its related inverse problem [@Operto_etal; @PederEliasson], to interesting accuracy in acceptable time on current (2015) high-end laptops. However, the classes used to solve this problem are designed with additional topologies, geometries, and applications in mind. These classes are `Lattice`, `LatticeFunction`, and `LatticeOperator`. A specific application from Quantum Mechanics [@AmnaKare] has been refactored to extend these classes. The Lattice class ================= This class is intended to handle the most basic properties and operations of a discretized model. We divide them into topological and geometrical aspects of the model. The most basic properties of a dicrete model are the dimensionality of space, and how we approximate a continuous space with a number of sites in each direction (referred to as its `shape`). The code snippet L1 = Lattice(shape=(2**13, )) L2 = Lattice(bC=('P', 'A')) L3 = Lattice(shape=(2**8, 2**8, 2**7)) demonstrate how three `Lattice` instances can be defined, `L1` with a one-dimensional lattice of $2^{13} = 8\,192$ sites, `L2` with (by default) a two-dimensional lattice of $2^7 \times 2^7$ sites, and `L3` with a three-dimensional lattice of $2^{8} \times 2^{8} \times 2^{7} = 8\,388\,608$ sites. In this process the instance properties `shape`, `dim`, and `size` are specified or given default values. Boundary conditions ------------------- One additional property, `bC`, specifies the default boundary conditions in all directions. These conditions specify how functions defined on a finite lattice is extended beyond its edges, as is required when applying discrete differential operators or operations like Fast Fourier Transforms (FFT). Each specific case of `bC` is a property of each function defined on the lattice. Hence it belongs to the class `LatticeFunction`, to be used and set by methods of `LatticeOperator`. However, since `bC` is often the same for all functions and operators in a given lattice model, it is convenient to provide a default property, which may be inherited by instances of `LatticeFunction` and `LatticeOperator`. The default value of `bC` is `'allP'`, for periodic boundary conditions in all directions. Otherwise, `bC` must be a list with possible entries `'P'` (for periodic extension), `'S'` (for symmetric extension), `'A'` (for antisymmetric extension), `'F'` (for extension with fixed provided values), and `'Z'` (for extension with zero values). For `'S'` and `'A'` the symmetry point is midway between two lattice points. The boundary condition can be specified differently in different directions, and (unless periodic `'P'`) differently at the two edges of a given direction (in which case the corresponding entry in `bC` must be a two-component list). Internally `bC` is either stored as `[['allP', ]]`, or as a `dim`-component list of two-component lists. The `Lattice` class is equipped with a method, `set_bC(bC='allP')`, which returns the internal representation from a variety of possible inputs. Subdomains and slices --------------------- Assume that $\phi(\bm{n})$ and $\phi_{\text{O}}(\bm{n})$ are two arrays defined on a 3-dimensional lattice, with $s_{\text{O}}$ a constant, and that we want to perform the operation $$\phi_{\text{O}}(\bm{n}) = \phi_{\text{O}}(\bm{n}) + s_{\text{O}}\,\phi(\bm{n}).$$ Python code for this operation could be the snippet for nx in range(phi.shape[0]): for ny in range(phi.shape[1]): for nz in range(phi.shape[2]): phiO[nx, ny, nz] = \ phiO[nx, ny, xz] + \ sO*phi[nx, ny, nz] This code is lengthy (hence error-prone) and runs slowly, because all `for`-loops are executed in native Python. The `numpy` code for the same operation is simply phiO += sO*phi wherein all loop operations are delegated to `numpy` (and maybe further translated to optimized `BLAS` operations).[^5] The similar operation corresponding to $$\phi_{\text{O}}(\bm{n}) = \phi_{\text{O}}(\bm{n}) + \sum_{\bm{b}} s_{\text{O}}(\bm{b})\,\phi(\bm{n}-\bm{b}),$$ where $\bm{b}$ is a non-zero integer vector, requires more care and coding, since there will be values of $\bm{n}$ for which $\bm{n} - \bm{b}$ falls outside the lattice. In such cases the expression $\phi(\bm{n}-\bm{b})$ must be related to known values of $\phi$ by use of the boundary conditions. Assume a case where `b = (3,0,-2)`, that the lattice have (much) more than 3 sites in all directions, and that the boundary conditions is given by bC = [['P','P'],['S','A'],['A','S']] We may first treat the sites $\bm{n}$ where also $\bm{n}-\bm{b}$ fall inside the lattice: phiO[3:,:,-2] += sO[3,0,-2]*phi[:-3,:,2:] Here the *slice*-notation defines a rectangular subdomain of the lattice. For instance, the slice `[3:,:,-2]` specifies the intersection of (i) all planes in the $x$-direction except the first 3, (ii) all planes in the $y$-direction, and (iii) all planes in the $z$-direction except the last 2. Note that array positions are counted from zero, with negative numbers referring to distances from the end. For a large lattice the above operation would cover most of the cases, and everything if the boundary conditions were `'Z'` in all directions. In our example there are three more regions to be included: $$\begin{aligned} 0\le n_x < 3,&\text{ and } 0 \le n_z < -2,\label{regionI}\\ 3 \le n_x \le -1,&\text{ and } -2 \le n_z \le -1,\label{regionII}\\ 0 \le n_x < 3,&\text{ and } -2 \le n_z \le -1.\label{regionIII}\end{aligned}$$ The case  can be handled by the code phiO[:3,:,:-2] += \ sO[3,0,-2]*phi[-3:,:,2:] using the periodic boundary condition in the $x$-direction. For the cases and two planes in the $z$-direction fall outside the lattice on the upper side. Due to the symmetric `'S'` boundary condition at this edge of the lattice, the function values on these planes are related to their values on the last two planes inside the lattice (counted in opposite order). This can be handled by the code phiO[3:,:,-2:] += \ sO[3,0,-2]*phi[:-3,:,:-3:-1] phiO[:3,:,-2:] += \ sO[3,0,-2]*phi[-3:,:,:-3:-1] For detailed information about indexing and slicing in `numpy`, consult the *Indexing* section of the Numpy reference manual [@NumpyReference]. However, gory details like the above are best handled by computers. The `Lattice` class provides a method, `targetNsource(b, bC=None)`, which yields all the source and target slices required for a given vector $\bm{b}$. Using this, the code snippet for cf, dT, dS in L3.targetNsource(b): phiO[dT] += cf*phi[dS] replaces all operations above. Here the coefficient `cf` is $-1$ if an odd number of antisymmetric boundary conditions are employed (otherwise $+1$). `Lattice` also provides a related method, `domain(shape, shift)`. This returns a slice `dI` pointing to a rectangular subdomain of the lattice, of shape `shape`, shifted from the origin by an integer vector `shift`. Index arrays and broadcasting ----------------------------- Each site of a `dim`-dimensional lattice is labeled by a `dim`-dimensional integer index vector $\bm{n}$. To construct an array `A` defined on all points of a 3-dimensional lattice, one could write a code snippet similar to the following defA = lambda n: \ numpy.exp(-numpy.dot(n,n)) shape = (2**8, 2**8, 2**8) A = numpy.zeros(shape) for n in numpy.ndindex(shape): A[n] = defA(numpy.array(n)) Although this code is brief and general with respect to dimensionality, it is *not a good way to do it*. Since the `for`-loop will be executed in native Python, the code will run too slow. A better way is to define three arrays `n0`, `n1`, `n2`, all of shape $(2^8, 2^8, 2^8)$, once and for all. We may then replace the code above with the snippet defA = lambda n0, n1, n2: \ exp(-n0*n0)*exp(-n1*n1)*exp(-n2*n2) A = defA(n0, n1, n2) All loops are now implicit, and will be executed by compiled `numpy` functions. Further, the memory cost of permanently storing three large arrays can be avoided by use of the *broadcasting* facility of `numpy`. Since the index array `n0` is constant in the $y$- and $z$-directions, it only contains a one-dimensional amount of information, stored in an array of shape $(2^8, 1, 1)$. Likewise, `n1` can be stored in an array of shape $(1, 2^8, 1)$, and `n2` in an array of shape $(1,1,2^8)$. All these arrays contain the same amount of data ($2^8$ linearely stored entries). But, due to their different *shape* they will act differently under f.i. algebraic operations: `n0*n0` will still produce an array of shape $(2^8,1,1)$, and similary `n1*n1` an array of shape $(1,2^8,1)$. However, the addition of these two results produces an array of shape $(2^8, 2^8, 1)$. Finally, adding `n2*n2` generates an array of the final shape $(2^8, 2^8, 2^8)$. Hence, the cost of computing and storing index arrays are modest. We have chosen *not* to include them as properties, but provide a method `narr()` which computes them when needed. This method returns a [list]{} `n` of arrays, `[n[0], n[1],..]`. Geometric properties -------------------- The discussion above maily concerns topological properties of the lattice. For most application we also need some geometric properties. In general these may be implemented by defining a `dim`-dimensional vector of arrays, $\bm{r(n)}$, specifying the position coordinates of all sites. These coordinates (which should depend monotoneously on $\bm{n}$) could also be dynamical, i.e. part of the equation system to be solved. The wide range of possibilities indicate that several versions of $\bm{r(n)}$ should be implemented, with the appropriate version chosen when a `Lattice` instance is defined. We have introduced a property `geometry`, which specifies the version to be used. So far, `geometry` can only take the value `'fixedRect'`, wherein rectangular regions of space, aligned with the lattice directions, are modelled. Such regions can be specified by a `dim`-dimensional vector $\bm{r}_E$ of edge-lengths, plus a vector $\bm{r}_0$ specifying the position of the “lower left” corner of the spatial region. For a given lattice `shape` parameter, this defines a lattice cell with a vector of sidelengths $\bm{dr}$, such that $$\text{\lstinline!dr[d] = rE[d]/shape[d]!}. \label{dr}$$ The position coordinate $\bm{r(n)}$ is then defined such that its component in the `d`-direction is $$\text{\lstinline!r[d] = r0[d] + dr[d]*(n[d]+1/2)!}. \label{r_n}$$ This implementation introduces three new properties: `r0`, by default a `dim`-dimensional tuple with entries $0$, `rE`, by default a `dim`-dimensional tuple with entries $1$, and `dr`, calculated from equation . The method `rvec()` returns a list `r` of arrays, `[r[0], r[1],...]`, calculated from equation . Lattice initialization, methods, and properties ----------------------------------------------- All currently available keyword arguments and default values for initialization of a `Lattice` instance is specified by the code snippet below: def __init__(shape=(128, 128), bC='allP', geometry='fixedRect', rE=None, r0=None): A value of `None` will invoke a default initialization process, following the rules discussed above. The current list of `Lattice` methods, with arguments, is as follows: ----------------- ------------------ `set_bC` `(bC='allP')` `domain` `(shape, shift)` `targetNsource` `(b, bC=None)` `narr` `()` `rvec` `()` ----------------- ------------------ A summary of all `Lattice` properties, with example values, is as provided in the table below. ------------ --------------------------------- `shape` `L1.shape = (8192, )` `L3.shape = (256, 256, 128)` `dim` `L1.dim = 1` `L3.dim = 3` `size` `L1.size = 8192` `L3.size = 8388608` `bC` `L1.bC = [['allP']]` `L2.bC = [['P','P'],['A','A']]` `geometry` `L1.geometry = 'fixedRect'` `r0` `L1.r0 = (0, )` `L3.r0 = (0, 0, 0)` `rE` `L1.rE = (1, )` `L3.rE = (1, 1, 1)` `dr` `L1.dr = [1.19209290e-07]` `L3.dr = [0.00390625,` `0.00390625, 0.0078125]` ------------ --------------------------------- The LatticeFunction class ========================= Space does not allow us to continue with an equally detailed discussion of all components in the `LatticeFunction` and `LatticeOperator` classes. We will instead provide examples of uses, augmented with general comments. L = Lattice(shape=(2**15, 2**15), rE=(18, 18), r0=(-9, -9)) defF = lambda r: \ numpy.exp(-r[0]**2/2)* \ numpy.exp(-r[1]**2/2) F = LatticeFunction(L, def_F=defF) t0 = time.time() F.evalFr() print (L.size, (time.time()-t0)/L.size) Here we first define a $2^{15} \times 2^{15}$ lattice model with periodic boundary conditions, and next a gaussian function centered in the middle of this lattice. The parameter `rE` is chosen large enough to make the periodic extension of this function smooth: It acquires a discontinuity in the first derivative of magnitude $18 \exp(-9^2/2) \approx 0.5\cdot 10^{-16}$ or smaller (i.e., below double precision accuracy). The gaussian function is *not* evaluated when the instance `F` is defined, only when we execute the method `F.evalFr()`. This method evaluates the function, and stores the result in the array `F.values`. The wall-clock time used to perform this computation on a 2013 MacBook Pro with 16 Gb of memory was measured to $3.26\;\text{ns}$ per point. Note that this time is mostly spent multiplying double precision numbers; only $2\times 2^{15}$ exponential function evaluations are performed. However, if the the code for `def_F` is changed to def_F = lambda r: \ numpy.exp(-(r[0]**2 + r[1]**2)/2) the execution time increases to $118\;\text{ns}$ per point. This increase is partly due to the fact that the exponential function is now evaluated $10^{30}$ times, but also because the system now has to deal with *two* very large arrays (one for the argument of the exponential function, and one for final result), and is operating very close to the limit of available memory. A more detailed analysis, for lattices of various sizes (total number of lattice points), is shown in Fig \[figure0\]. ![\[figure0\] Comparison of NumPy and C evaluation times for a gaussian defined on lattices of various sizes. The fast evaluation occur when writing the gaussian as $\exp(-x^2/2) \times \exp(-y^2/2)$, the slow evaluation when writing it as $\exp[-(x^2 + y^2)/2]$. For these cases the wall-clock and CPU times are essentially the same. As can be seen, there is little to gain in evaluation time by writing the code in a fast, compiled language like C (and a lot to lose in coding time). ](figure0) FFT and related discrete transforms ----------------------------------- We may apply a discrete Fourier transformation to the data stored in `F.values`. This is done by the function call `F.FFT()`. The transformed data is stored in the array `F.fftvalues`. The inverse transform is performed by the function call `F.iFFT()`, with the transformed data being stored in the array `F.values` (overwriting any previous data). Acctually, the method `FFT()` (or `iFFT()`) do not necessarily perform a regular (multidimensional) discrete Fourier transform `fftn` (or its inverse `ifftn`). This is but one of several related discrete transforms available in `scipy.fftpack`. Other such transforms are the discrete cosine transform `dct` (suitable for functions with symmetric boundary conditions on both sides), the discrete sine transform `dst` (suitable for functions with antisymmetric boundary conditions on both sides), the fast fourier transform `rfft` of real data, and their inverses (`idct`, `idst`, `irfft`). The rules are 1. If `bC[0][0] == 'allP'` the transform `fftn` (or `ifftn`) is used. Complex data is allowed. Otherwise, the data is assumed to be real, and an iterated sequence of transforms over all axes is executed. 2. For directions such that `bC[d][0] == 'S'` the transform `dct` (or `idct`) is performed. 3. For directions such that `bC[d][0] == 'A'` the transform `dst` (or `idst`) is performed. 4. In all other cases the transform `rfft` (or `irfft`) is performed. Note that the `bC` used here is a property of `LatticeFunction`. This may be different from the corresponding property of its lattice instance. By default they are equal. The discrete transforms above are useful because they allow (i) differential operators to be implemented as multiplication operators on the transformed functions, and (ii) accurate interpolation of lattice functions outside the lattice sites. The latter is useful for implementation of prolongations in multigrid methods. To assess to which extent this is a practical approach, we have investigated the accuracy and the time requirements of these transforms. The code snippet below illustrate how this can be done: shape = (2**14, 2**14) L = Lattice(shape=shape, bC=('P','P')) F = LatticeFunction(L) F.values = numpy.random.rand(*shape) values = numpy.copy(myF.values) t0 = time.time(); F.FFT() t1 = time.time(); F.iFFT() t2 = time.time() err=numpy.max(numpy.abs(F.values-values)) print((t1-t0)/L.size,(t2-t1)/L.size,err) The output of this code shows that the forward transform takes about $68\,\text{ns}$ per lattice point, the inverse transform about $57\,\text{ns}$, and that the maximum difference between the original and backtransformed values is $1.7\times 10^{-15}$. I.e., the cost of a one-way transform is roughly the same as 20 multiplications. The time per site increases by almost an order of magnitude for a lattice of `shape = (2**14, 2**15)`, since this is close to the limit of available memory. We have investigated the behavior above in more detail, for different choices of the `shape` and `bC` parameters, with similar results. See Fig. \[figure1\]. The crude conclusion is that the transformation times grow roughly linearly with lattice size, with a prefactor which depends only slightly on transformation type and lattice dimensionality. ![\[figure1\] Time used to perform a discrete lattice transformations of various types. Each time plotted is the *sum* of the forward and inverse transformation time. 1D lattices are plotted in red, 2D lattices in magenta, and 3D lattices in blue. Within the range of lattice sizes allowed by available memory, the theoretically expected logarithmic growth of transformation time with size is not a very distinct feature. ](figure1) LatticeFunction initialization, methods, and properties ------------------------------------------------------- All currently available keyword arguments for initialization of a `LatticeFunction` instance is specified by the argument list below: def __init__(self, lattice, def_f=None, def_F=None, def_g=None, def_G=None, bC='allP', evalf=False, evalF=False, evalg=False, evalG=False): As can be inferred from the above, there are several ways to specify a function: (i) As a function of the index arrays, $f(\bm{n})$, or as a function of the position vectors, $F(\bm{r})$. The discrete transform of the function also lives on a lattice, the *dual lattice*, whose sites can be labelled by a list of index arrays `q = [q[0], q[1],..]`. We denote the geometric version of this lattice as *reciprocal space*, wherein each site $\bm{q}$ has a reciprocal position vector $\bm{k}(\bm{q})$. Hence, the function can also be specified from its discrete transformation, as the function (iii) $g(\bm{q})$ or (iv) $G(\bm{k})$. The current list of `LatticeFunction` methods is as follows: --------------- ------------------------------------------- `qarr()` List index vectors for the dual lattice. `kvec()` List resiprocal position vectors. `evalfn()` Compute `values` from `def_f`. `evalFr()` Compute `values` from `def_F`. `evalgq()` Compute `fftvalues` from `def_g`. `evalGk()` Compute `fftvalues` from `def_G`. `FFT()` Discrete transformation of `values`. `iFFT()` Inverse transformation of `fftvalues`. `shift(frac)` Return the function translated by `frac`. `restrict()` Return the function restricted to a cruder lattice. `prolong()` Return the function prolonged to a finer lattice. --------------- ------------------------------------------- The current list of `LatticeFunction` properties is as follows: ------------- ------------------------------------------------ `lattice` Related `Lattice` instance. `def_f` Possible function definition (default `None`). `def_F` Possible function definition (default `None`). `def_g` Possible function definition (default `None`). `def_G` Possible function definition (default `None`). `bC` Boundary conditions (`lattice.bC`). `values` Array of function values. `fftvalues` Array of transformed function values. ------------- ------------------------------------------------ The LatticeOperator class ========================= Many routines in `scipy.sparse.linalg` do not require an explicit matrix representation of the operator under analysis. Only some algorithm which returns the result of applying the operator to a given vector is needed. Such algorithms can be assigned to a `LinearOperator` instance, after which it functions essentially as an explicit matrix representation. Such algorithms should not demand too much memory or computation time, but do not require any explicitly known sparse representation of the operator. F.i., any computational process involving a fixed number of multiplication, additions and fast fourier transformations will have a memory requirement which scales linearly with the lattice size, and a time requirement which (for large systems) also scales roughly linearly with lattice size. The `LinearOperator` class requires an input vector of shape `(M, )` or `(M,1)`, and an output vector of shape `(N, )`. For higher-dimensional lattices this does not match the natural construction of lattice operators, which we do not want to interfere with. We have therefore implemented a general `linOp(phi0)` method, to be used as a universal `matvec` parameter for `LinearOperator`. The currently implemented code for this is phi = phi0.reshape(self.lattice.shape) return numpy.ravel(self.varOp(phi)) This code assumes `phi0` to represent a scalar function. It will be extended to more general (vector, spinor, tensor,…) objects. The `reshape` and `ravel` operations above do not modify or move any data; they only change how the data is interpreted (the *view* of the data). The code above also call a specific method, `varOp(phi)`. However, this is just a handle which should be assigned to the operator under analysis. The latter may either be an appropriate predefined method in the `LatticeOperator` class, or a method provided from outside. Explicit matrix representations ------------------------------- It may be useful to inspect an explicit matrix representation of a given operator on a small lattice. The method `matrix(operator)` provides such a representation: L = Lattice(shape=(4, ), rE=(4,)) O = LatticeOperator(L) laplace = O.matrix(O.laplace) print (laplace) The output from this code is [[-2. 1. 0. 1.] [ 1. -2. 1. 0.] [ 0. 1. -2. 1.] [ 1. 0. 1. -2.]] which is easily verified to have the correct form for a $3$-stensil one-dimensional lattice Laplacian with periodic boundary conditions. We may redefine the lattice to have the `'Z'` boundary condition: L = Lattice(shape=(4,), bC='Z', rE=(4,)) The output now becomes: [[-2. 1. 0. 0.] [ 1. -2. 1. 0.] [ 0. 1. -2. 1.] [ 0. 0. 1. -2.]] When applied to a small two-dimensional lattice L = Lattice(shape=(2,3),bC='Z',rE=(2,3)) the output for the correponding $5$-stensil becomes [[-4. 2. 0. 0. 0. 0.] [ 2. -4. 2. 0. 0. 0.] [ 0. 2. -4. 0. 0. 0.] [ 0. 0. 0. -4. 2. 0.] [ 0. 0. 0. 2. -4. 2.] [ 0. 0. 0. 0. 2. -4.]] We have found such applications of the `matrix()` method to be quite educating, and very useful for debugging purposes. The output matrix can also be used directly as input to all the standard (dense matrix) linear algebra routines in `scipy`. Lattice sizes up to about $10^4$ can be handled in this way, sufficient for most one-dimensional systems (and useful when comparing dense and iterative methods on small higher-dimensional systems). Methods for generating sparse matrix representations will also be implemented. Example of use -------------- An example illustrating the discussion above is provided by the code snippet: L = Lattice(shape=(2**8, ), rE=(18, ), r0=(-9, )) defF = lambda r: numpy.exp(-r[0]**2/2) F = LatticeFunction(L, def_F=defF, evalF=True) O = LatticeOperator(L) O.varOp = O.laplace F2values = O.varOp(F.values) In this simple case it does not matter if `F2values` is computed by use of `O.linOp`, `O.varOp` or `O.laplace`. The result of evaluating $\Delta_L \exp(-r^2/2)$ can be compared with the exact result, $(r^2-1)\,\exp(-r^2/2)$. A good way to assess the discretization error is to compute $\max_{\bm{r}} \left| \Delta_L F(\bm{r}) - \Delta F(\bm{r})\right|$. This is plotted in Fig. \[deltaLaplace\] for a range of square lattices. ![\[deltaLaplace\] The maximum absolute difference between the numerical and exact evaluation of the Laplace operator, divided by $dr^2$, as function of the linear lattice size. This shows that the error scales like $dr^2$, as expected for these stensils. The increase in error for large linear size is probably due to numerical roundoff (because $dr^2$ becomes very small), the decrease for small linears size due to incomplete sampling of errors (too few lattice points to compare the functions where the error is maximum). ](deltalaplace) The lattice Laplace operator ---------------------------- We have used a simple implementation of the lattice Laplacian in the examples above. This is the common $(2d +1)$-stensil approximation. For periodic boundary conditions the implemention is very simple, as indicated by the code snippet below: def laplace(self, phi): Lphi = numpy.zeros_like(phi) for d in range(self.dim): Lphi += numpy.roll(phi, 1, axis=d) Lphi += numpy.roll(phi,-1, axis=d) Lphi -= 2*phi return Lphi/self.dr**2 Here the `roll`-function rotates the entries of the `phi`-array in the `d`-direction by the specified amount ($\pm 1$ for the code above). We have investigated how fast this implementation is.. The results is plotted in Fig. \[figure2\]. As expected, the evaluation times scales (essentially) linearly will lattice size, with a prefactor which increases with the complexity of the stensil. But, somewhat surprisingly, the evaluation times are not very different from the time to make back-and-forth fast fourier fourier transformations. This suggests an alternative approach, based on fast fourier transforms. The roll-process is fast, with all loop operations done in NumPy, but requires new memory for the rolled data. To avoid this we have implemented a general method, `stensOp(phi)`. The essential algorithm of this is illustrated by the snippet below: for b in numpy.ndindex(stensil.shape): cf, dT, dS = lattice.targetNsource(b) phiO[dT] += cf*stensil[b]*phi[dS] Here `stensil` is a (small) `dim`-dimensional array defining the operator in question. ![\[figure2\] The times to evaluate $-\Delta_L \phi$, for the (standard) $(2D+1)$-sensil approximation of the Laplace operator, are plotted for various lattice sizes and dimensionalites. As expected, the times increases with the complexity of the stensil. Somewhat surprisingly, the times are not significantly different from the times to perform back-and-forth fast fourier transform (or its discrete analogs), c.f. Fig. \[figure1\]. ](figure2) Acknowledgment {#acknowledgment .unnumbered} ============== We thank dr. Peder Eliasson (Research manager, SINTEF petroleum Trondheim) for an informative discussion. This work has been partially supported by the UniCQue project. [99]{} S. van der Walt, S.C. Colbert, and G. Varoquaux, *The NumPy Array: A Structure for Efficient Numerical Computation*, Computing in Science & Engineering **13**, 22–30 (2011) Eric Jones, Travis Oliphant, Pearu Peterson and others, *SciPy: Open Source Scienific Tools for Python*, `http://www.scipy.org/` (2001) Travis E. Oliphant, *Python for Scientific Computing*, Computing in Science & Engineering **9**, 90 (2007) S. Operto, J. Virieux, P. Amestoy, J-Y. L'Excellent, L. Giraud, and H.B.H. Ali, *3D finite-difference frequency-domain modeling of visco-acoustic wave propagation using a massivly parallel direct solver: A feasibility study*, Geophysics **72**, SM195–SM211 (2007) Peder Eliasson, *3D Full Waveform Modelling in Frequency Domain*, SINTEF Petroleum Research Presentation 2010–09–08. Amna Noreen and K[å]{}re Olaussen, *A Python Class for Higher-Dimensional Schr[ö]{}dinger Equations*, accepted contribution to ICCS’15, Hong Kong 18–20 March (2015) Travis E. Oliphant and others, *Numpy reference manual, indexing section*, `http://docs.scipy.org/doc/numpy/reference/ arrays.indexing.html` [^1]: Manuscript received January 22, 2015. [^2]: Asif Mushtaq is with the Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), Trondheim. e-mail: asif.mushtaq@math.ntnu.no [^3]: Trond Kvamsdal is with the Department of Mathematical Sciences, NTNU, Trondheim, N-7491, Norway. e-mail: trond.kvamsdal@math.ntnu.no [^4]: K[å]{}re Olaussen is with the Department of Physics, NTNU, Trondheim, N-7491, Norway. e-mail: Kare.Olaussen@ntnu.no [^5]: Note that the codeline `phiO = phiO + s0*phi` is *not* equivalent to `phiO += s0*phi`. In the former a new copy of `phiO` is made; this requires more memory.

--- abstract: 'In this short note we give an answer to the following question. Let $X$ be a locally compact metric space with group of isometries $G$. Let $\{g_i\}$ be a net in $G$ for which $g_ix$ converges to $y$, for some $x,y\in X$. What can we say about the convergence of $\{ g_i\}$? We show that there exist a subnet $\{g_j\}$ of $\{g_i\}$ and an isometry $f:C_x\to X$ such that $g_{j}$ converges to $f$ pointwise on $C_x$ and $f(C_x)=C_{f(x)}$, where $C_x$ and $C_y$ denote the pseudo-components of $x$ and $y$ respectively. Applying this we give short proofs of the van Dantzig–van der Waerden theorem (1928) and Gao–Kechris theorem (2003).' address: 'Fakultät für Mathematik, SFB 701, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany' author: - Antonios Manoussos title: On the action of the group of isometries on a locally compact metric space --- [^1] The main result and some applications ===================================== A few words about the notation we shall be using. In what follows, $X$ will denote a locally compact metric space with group of isometries $G$. If we endow $G$ with the topology of pointwise convergence then $G$ is a topological group [@bour2 Ch. X, §3.5 Corollary]. On $G$ there is also the topology of uniform convergence on compact subsets which is the same as the compact-open topology. In the case of a group of isometries these topologies coincide with the topology of pointwise convergence, and the natural action of $G$ on $X$ with $(g,x)\mapsto g(x)$, $g\in G$, $x\in X$, is continuous [@bour2 Ch. X, §2.4 Theorem 1 and §3.4 Corollary 1]. For $F\subset G$, let $K(F):=\{ x\in X\, |\,\,\mbox{the set}\,\, Fx\,\,\mbox{has compact closure in}\,\, X\}$. The sets $K(F)$ are clopen [@manstra Lemma 3.1]. \[lemma1\] Let $\Gamma =\{g_i\}$ be a net in $G$ and $x\in K(\Gamma)$ such that $g_ix$ converges to $y$ for some $y\in X$. Then a subnet of $\Gamma$ converges to an isometry $f:K(\Gamma)\to X$ on $K(\Gamma)$. Let $g_i|_{K(\Gamma)}$ denote the restriction of $g_i$ on $K(\Gamma)$. Arzela–Ascoli theorem implies that the set $\{ g_i|_{K(\Gamma)}:K(\Gamma)\to X \}$ has compact closure in the set of all continuous maps from $K(\Gamma)$ to $X$. Thus, there exist a subnet $\{g_j\}$ of $\{g_i\}$ and an isometry $f:K(\Gamma)\to X$ such that $g_j\to f$ on $K(\Gamma)$. In [@kechris] S. Gao and A. S. Kechris introduced the concept of pseudo-components. These are the equivalence classes $C_x$ of the following equivalence relation: $x\sim y$ if and only if x and y, as also y and x, can be connected by a finite sequence of intersecting open balls with compact closure. The pseudo-components are clopen [@kechris Proposition 5.3]. We call $X$ pseudo-connected if it has only one pseudo-component. An immediate consequence of the definitions is that $gC_x=C_{gx}$ for every $g\in G$. Another notion, that will be used in the proofs, is the radius of compactness $\rho (x)$ of $x\in X$ [@kechris]. Let $B_r(x)$ denote the open ball centered at $x$ with radius $r>0$. Then $\rho (x):=\sup \{\, r>0\, |\,\, B_r(x)\,\,\mbox{ has compact closure}\}$. If $\rho (x)=+\infty$ for some $x\in X$ then every ball has compact closure (i.e., $X$ has the Heine–Borel property), hence $\rho (x)=+\infty$ for every $x\in X$. If $\rho (x)$ is finite for some $x\in X$ then the radius of compactness is a Lipschitz map [@kechris Proposition 5.1]. Note that $\rho$ is $G$-invariant. \[lemma2\] Let $x, y\in X$ and $\{ g_i\}_I$ be a net in $G$ with $g_ix\to y$. Then there is an index $i_0\in I$ such that $C_x\subset K(F)$, where $F:=\{ g_i\,|\,i\geq i_0\}$. Since $X$ is locally compact there exists an index $i_0$ such that the set $F(x)$ has compact closure, where $F:=\{ g_i\,|\,i\geq i_0\}$. We claim that for every $z\in C_x$ the set $F(z)$ also has compact closure, hence $C_x\subset K(F)$. The strategy is to start with an open ball $B_r(x)$ with radius $r<\rho (x)$ and prove that $F(z)$ has compact closure for every $z\in B_r(x)$. Then our claim follows from the definition of $C_x$. To prove the claim take a sequence $\{g_n z\}\subset F$. Since the closure of $F(x)$ is compact we may assume, upon passing to a subsequence, that $g_nx\to w$ for some $w$ in the closure of $F(x)$. Assume that $\rho (x)$ is finite and take a positive number $\varepsilon$ such that $r+\varepsilon < \rho (x)$. Then for $n$ big enough $$d(g_nz,w)\leq d(g_nz,g_nx) + d(g_nx,w)=d(z,x)+ d(g_nx,w)<r+\varepsilon < \rho (x).$$ Recall that the radius of convergence is a continuous map, and since $g_nx\to w$ then $ \rho (x)= \rho (w)$. So, the sequence $\{g_n z\}$ is contained eventually in a ball of $w$ with compact closure, hence it has a convergence subsequence. The same also holds in the case where $\rho (x)=+\infty$. \[theorem\] Let $X$ be a locally compact metric space with group of isometries $G$ and let $\{g_i\}$ be a net in $G$ for which $g_ix$ converges to $y$, for some $x,y\in X$. Then there exist a subnet $\{g_j\}$ of $\{g_i\}$ and an isometry $f:C_x\to X$ such that $g_{j}$ converges to $f$ pointwise on $C_x$ and $f(C_x)=C_{f(x)}$ By Lemma \[lemma2\] there is an index $i_0\in I$ such that $C_x\subset K(F)$, where $F:=\{ g_i\,|\,i\geq i_0\}$. Hence, by Lemma \[lemma1\], there exists a subnet $\{ g_j\}$ of $\{ g_i\}$ which converges to an isometry $f:K(F)\to X$ on $K(F)$. Therefore, $g_{j}\to f$ on $C_x$. Let us show that $f(C_x)=C_{f(x)}$. Since $d(x,g_j^{-1}f(x))=d(g_jx,f(x))\to 0$ it follows that $g_j^{-1}f(x)\to x$. Hence, by repeating the previous procedure, there exist a subnet $\{g_k\}$ of $\{g_j\}$ and an isometry $h:C_{f(x)}\to X$ such that $g_{k}^{-1}\to h$ pointwise on $C_{f(x)}$ and $h(f(x))=x$. Note that $g_{k}x \in C_{f(x)}$ eventually for every $k$, since $g_{k}x\to f(x)$ and $C_{f(x)}$ is clopen. Therefore, $g_kC_x=C_{g_{k}x}=C_{f(x)}$. Take a point $z\in C_x$. Then, $g_{k}z\to f(z)$ and since $C_{f(x)}$ is clopen then $f(z)\in C_{f(x)}$, so $f(C_x)\subset C_{f(x)}$. By repeating the same arguments as before, it follows that $hC_{f(x)}\subset C_x$. Take now a point $w\in C_{f(x)}$. Then $h(w)\in C_x$, hence $g_{k}^{-1}(w)\in C_x$ eventually for every $k$. So, $w=g_{k}g_{k}^{-1}(w)\to f(h(w))\in f(C_x)$ from which follows that $C_{f(x)}\subset f(C_x)$. A few words about properness. A continuous action of a topological group $H$ on a topological space $Y$ is called proper (or Bourbaki proper) if the map $H\times Y\to Y\times Y \,\,\mbox{with}\,\,(g,x)\mapsto(x,gx),\,\,\mbox{for}\,\, g\in H\,\,\mbox{and}\,\, x\in Y$, is proper, i.e., it is continuous, closed and the inverse image of a singleton is a compact set [@bour1 Ch. III, §4.1 Definition 1]. In terms of nets, a continuous action is proper if and only if whenever we have two nets $\{g_i\}$ in $H$ and $\{x_i\}$ in $Y$, for which both $\{x_i\}$ and $\{g_ix_i\}$ converge, then $\{g_i\}$ has a convergent subnet. For isometric actions, it is easy to see that a continuous action is proper if and only if whenever we have a net $\{g_i\}$ in $H$ for which $\{g_ix\}$ converges for some $x\in Y$, then $\{g_i\}$ has a convergent subnet. If $H$ is locally compact and $Y$ is Hausdorff, then $H$ acts properly on $Y$ if and only if for every $x,y\in Y$ there exist neighborhoods $U$ and $V$ of $x$ and $y$, respectively, such that the set $\{ g\in H\, |\, gU\cap V\neq\emptyset\}$ has compact closure in $H$ [@bour1 Ch. III, §4.4 Proposition 7]. Observe that if $H$ acts properly on a locally compact space $Y$ then $H$ is also locally compact. A direct implication of Theorem \[theorem\] is the van Dantzig–van der Waerden theorem [@d-w]. The advantage of our proof, comparing to the proofs given in the original work of van Dantzig–van der Waerden or in [@kob-nom Theorem 4.7, pp. 46–49], is that it is considerably shorter. \[cor1\](van Dantzig–van der Waerden theorem 1928) Let $X$ be a connected locally compact metric space with group of isometries $G$. Then $G$ acts properly on $X$ and is locally compact. Another application of Theorem \[theorem\] is that we can rederive the results of Gao and Kechris in [@kechris Theorem 5.4 and Corollary 6.2]. \[cor2\](Gao–Kechris theorem 2003) Let $X$ be a locally compact metric space with finitely many pseudo-components. Then the group of isometries $G$ of $X$ is locally compact. If $X$ is pseudo-connected, then $G$ acts properly on $X$. Let $C_1, C_2, \ldots, C_n$ denote the pseudo-components of $X$ and take points $x_1\in C_1, x_2\in C_2, \ldots, x_n\in C_n$ and open balls $B_r(x_m)\subset C_m$, $m=1,2,\ldots, n$, $r>0$ such that all $B_r(x_m)$ have compact closures. We will show that the set $V:=\bigcap_{m=1}^{n}\{ g\in G\,|\, gx_m\in B_r(x_m)\}$ is an open neighborhood of the identity in $G$ with compact closure. Indeed, take a net $\{ g_i\}$ in $V$. Since each $B_r(x_m)$ has compact closure there exist a subnet $\{ g_j\}$ of $\{ g_i\}$ and points $y_1\in C_1, y_2\in C_2, \ldots, y_n\in C_n$ such that $g_jx_m\to y_m$ for every $m=1,2,\ldots, n$. Theorem \[theorem\] implies that there exist a subnet $\{ g_l\}$ of $\{ g_j\}$ and isometries $f_m: C_{m}\to X$ such that $g_l\to f_m$ on $C_{m}$ and $f_m(C_{m})=C_{m}$ for all $m$. The last implies that $\{ g_l\}$ converges to an isometry on $X$, hence $V$ has compact closure. If $X$ is pseudo-connected the proof of the statement follows directly from Theorem \[theorem\]. Note that in Corollary \[cor2\] we do not require that $X$ is separable as in [@kechris Theorem 5.4 and Corollary 6.2]. This is not a real improvement since if $X$ has countably many pseudo-components then it is separable. Indeed, we define a relation on $X$ by $x\mathcal{S} y$ if and only if there exist separable balls $B_r(x)$ and $B_l(y)$ with $y\in B_r(x)$ and $x\in B_l(y)$. Let $U(x)$ be the equivalence class of $x$ in the transitive closure of the relation $\mathcal{S}$. Then, each $U(x)$ is a separable clopen subset of $X$ [@kob-nom Lemma 3 in Appendix 2]. By construction $C_x\subset U(x)$, therefore $X$ is separable. **Acknowledgements.** We would like to thank the referee for an extremely careful reading of the manuscript. Her/his remarks and comments helped us to improve considerably the presentation of the paper. [99]{} N. Bourbaki, *Elements of Mathematics. General topology*, Part 1. Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1966. N. Bourbaki, *Elements of Mathematics. General topology*, Part 2. Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1966. D. van Dantzig and B. L. van der Waerden, *Über metrisch homogene Räume*, Abh. Math. Seminar Hamburg **6** (1928), 367–376. S. Gao and A. S. Kechris, *On the classification of Polish metric spaces up to isometry*, Mem. Amer. Math. Soc. **161** (2003), no. 766. S. Kobayashi and K. Nomizu, *Foundations of differential geometry*, Vol I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. A. Manoussos and P. Strantzalos, *On the group of isometries on a locally compact metric space*, J. Lie Theory **13** (2003), 7–12. [^1]: During this research the author was fully supported by SFB 701 “Spektrale Strukturen und Topologische Methoden in der Mathematik" at the University of Bielefeld, Germany. He would also like to express his gratitude to Professor H. Abels for his support.

--- abstract: 'Building on the mapping relations between analytic functions and periodic functions using the abstract operators $\cos(h\partial_x)$ and $\sin(h\partial_x)$, and by defining the Zeta and related functions including the Hurwitz Zeta function and the Dirichlet L-function in the form of abstract operators, we have obtained many new series expansions associated with these functions on the whole complex plane, and investigate the number theoretical properties of them, including some new rapidly converging series for $\eta(2n+1)$ and $\zeta(2n+1)$. For $n\in\mathbb{N}$, each of these series representing $\zeta(2n+1)$ converges remarkably rapidly with its general term having the order estimate: $$O(m^{-2k}\cdot k^{-2n+1})\qquad(k\rightarrow\infty;\quad m=3,4,6).$$' --- [GBK]{}[song]{} [**On Evaluation of Zeta and Related Functions by Abstract Operators**]{} Guang-Qing Bi Preliminaries ============= The Riemann Zeta function $\zeta(s)$, the Hurwitz Zeta function $\zeta(s,a)$ and the Dirichlet L-function $L(s,\chi)$ are defined usually by $$\label{00c} \zeta(s):=\sum^\infty_{n=1}\frac{1}{n^s}\qquad(\Re(s)>1),$$ $$\label{00c'} \zeta(s,a):=\sum^\infty_{n=0}\frac{1}{(n+a)^s}\quad(\Re(s)>1,a\not\in\mathbb{Z}_0^-:=\{0,-1,-2,\ldots\})$$ and for any Dirichlet character $\chi$ of modulus $q$ (There exists a positive integer $q$ such that $\chi(n)=\chi(n+q)$ for all $n$) $$\label{drlf} L(s,\chi):=\sum^\infty_{n=1}\frac{\chi(n)}{n^s}\quad(\Re(s)>1),$$ and by their meromorphic continuations for $\Re(s)\leq1$. They are known to be meromorphic. The alternating Zeta Function $\eta(s)$ is defined usually by $$\label{02} \eta(s):=\sum^\infty_{n=1}\frac{(-1)^{n-1}}{n^s}=(1-2^{1-s})\zeta(s)\quad(\Re(s)>0),$$ and by its analytic continuations for $\Re(s)\leq0$. It is a holomorphic function on the whole complex plane by analytic continuation. The concept of abstract operators is based on the analytic continuity fundamental theorem contained in a 1997 paper entitled *Applications of abstract operators to partial differential equations* (See [@bi97 p. 7]). In other words, if two abstract operators $A$ and $B$ acts on $e^{\xi{x}},\,x\in\mathbb{R}^n,\,\xi\in\mathbb{R}^n$ such that $Ae^{\xi{x}}=Be^{\xi{x}}$, then $Ag(x)=Bg(x),\,\forall{g(x)}\in{C^\infty(\Omega)},\,\Omega\subset\mathbb{R}^n$. Therefore, the abstract operators $f(\partial_x),\,\partial_x:=(\partial_{x_1},\partial_{x_2},\cdots,\partial_{x_n})$ is defined by (See [@bi97 p. 8]) $$\label{abs} f(\partial_x)e^{\xi{x}}:=f(\xi)e^{\xi{x}},\quad\forall f(\xi)\in C^\infty(\mathbb{R}^n),$$ where $f(\xi),\xi\in\mathbb{R}^n$ is called the symbols of abstract operators $f(\partial_x)$. Further, we can derive the operation rules of abstract operators, such as $$\label{e0} f(\partial_x)(e^{\xi_0x}g(x))=g(\partial_\xi)(e^{\xi{x}}f(\xi))|_{\xi=\xi_0},$$ $$\label{e1} f(\partial_x)(e^{\xi{x}}g(x))=e^{\xi{x}}f(\xi+\partial_x)g(x),$$ $$\label{e1'} e^{h\partial_x}f(x)=f(x+h),\quad h\partial_x:=h_1\partial_{x_1}+h_2\partial_{x_2}+\cdots+h_n\partial_{x_n}$$ and so on. The abstract operators in view of the analytic continuity fundamental theorem can be called the pseudo-differential operators defined on $C^\infty(\Omega)$. Conversely, the pseudo-differential operators in view of the Fourier transform can also be called the abstract operators defined on $\mathscr{S}(\mathbb{R}^n)$. The abstract operators have been applied strongly to partial differential equations (See [@bi18]-[@bi11]). In this paper, we will see that the concept of abstract operators can be applied in defining alternating Zeta function, Riemann Zeta function, Hurwitz Zeta function and Dirichlet L-function on the whole complex plane without the experience of analytic continuation process. From this, we can easily derive their asymptotic expansions and other arithmetic properties, including some new rapidly converging series for $\eta(2n+1)$ and $\zeta(2n+1)$. Let $h\partial_x=\langle{h,\partial_x}\rangle=h_1\partial_{x_1}+h_2\partial_{x_2}+\cdots+h_n\partial_{x_n}$. Then $\cos(h\partial_x)$ and $\sin(h\partial_x)$ are the abstract operators taking $\cos(hb)$ and $\sin(hb)$ as the symbols respectively, namely $$\label{1} \cos(h\partial_x)e^{bx}:=\cos(bh)e^{bx},\quad\sin(h\partial_x)e^{bx}:=\sin(bh)e^{bx}.$$ Here $bx=b_1x_1+b_2x_2+\cdots+b_nx_n,\;hb=bh=b_1h_1+b_2h_2+\cdots+b_nh_n$. Further, their operation rules can be expressed as the following three groups of operator relationships by Guang-Qing Bi [@bi97 pp. 7-9]: **Theorem 1.1.** (See [@bi97 p. 9, Theorem 3, 4, and 6]) Let $x\in\mathbb{R}^n,\;h\in\mathbb{R}^n$, $h\partial_x=\langle{h,\partial_x}\rangle=h_1\partial_{x_1}+\cdots+h_n\partial_{x_n}$. For $\cos(h\partial_x)\,\mbox{and}\,\sin(h\partial_x)$, we have $$\label{y0} \cos(h\partial_x)f(x)=\Re[f(x+ih)],\quad\sin(h\partial_x)f(x)=\Im[f(x+ih)],$$ $\forall{f(z)}\in{C}^\infty(\Omega),\,z=x+iy\in\Omega\subseteq\mathbb{C}^n$; **Theorem 1.2.** (See [@bi97 p. 9, Theorem 5]) Let $h_0\in\mathbb{R},\;x(t)\in\mathbb{R}^n,\;t\in\mathbb{R}^1,\;X\in\mathbb{R}^n,\,Y\in\mathbb{R}^n$, $Y\partial_X=\langle{Y,\partial_X}\rangle=Y_1\partial_{X_1}+\cdots+Y_n\partial_{X_n}$. Then we have where $X_j=\cos(h_0\partial_t)x_j(t),\;Y_j=\sin(h_0\partial_t)x_j(t),\;j=1,\cdots,n$. In the special case when $n=1$, (\[y3\]) can easily be restated as where $Y=\sin(h_0\partial_t)x(t),\;X=\cos(h_0\partial_t)x(t),\;t\in\mathbb{R}^1,\;h_0\in\mathbb{R}$. **Theorem 1.3.** (See [@bi97 p. 9, Theorem 7]) Let $u=g(y)$ be a monotonic function on its domain. If $y=f(bx)$ is the inverse function of $bx=g(y)$ such that $g(f(bx))=bx$, where $bx=b_1x_1+b_2x_2+\cdots+b_nx_n,\;bh=b_1h_1+b_2h_2+\cdots+b_nh_n$, then $\sin(h\partial_x)f(bx)$ (denoted by $Y$) and $\cos(h\partial_x)f(bx)$ (denoted by $X$) can be determined by the following set of equations: $$\label{y4} \left\{\begin{array}{l@{\qquad}l}\displaystyle \cos\left(Y\frac{\partial}{\partial{X}}\right)g(X)=bx,&x\in\mathbb{R}^n,\;b\in\mathbb{R}_n,\\\displaystyle \sin\left(Y\frac{\partial}{\partial{X}}\right)g(X)=bh, &h\in\mathbb{R}^n. \end{array}\right.$$ For example, by making use of (\[1\]) and (\[y4\]), we have $$\begin{aligned} \left\{\begin{array}{l@{\qquad}l}\displaystyle e^X\cos{Y} = bx, & X=\cos(h\partial_x)\ln(bx),\\\displaystyle e^X\sin{Y} = bh, & Y=\sin(h\partial_x)\ln(bx). \end{array}\right.\end{aligned}$$ By solving this set of equations, we obtain **Theorem 1.4.** Let $f(x)\in{L^2}[-c,c]$ be the sum function of the Fourier cosine series, and $g(x)\in{L^2}[-c,c]$ be that of the corresponding Fourier sine series, namely $$f(x)=\sum^\infty_{n=0}a_n\cos\frac{n\pi{x}}{c}\quad\mbox{and}\quad g(x)=\sum^\infty_{n=0}a_n\sin\frac{n\pi{x}}{c},$$ where $x\in\Omega\subseteq\mathbb{R}^1$, $c>0$ is a given real number. If $S(t)$ is the sum function of the corresponding power series $\sum^\infty_{n=0}a_nt^n$, namely $$S(t)=\sum^\infty_{n=0}a_nt^n,\quad(t\in\mathbb{R}^1,\;|t|<r,\;0<r<+\infty),$$ then for $a<x<b$ we have the following mapping relationships: Here interval $(a,b)\subset\Omega$. The endpoints $a$ and $b$ of the interval $a<x<b$ are non-analytical points (singularities) of Fourier series, which can be uniquely determined by the detailed computation of the right-hand side of (\[12\]). **Proof.** Theorem 1.4 can be proved easily by substituting $S(e^z)=\sum^\infty_{n=0}a_ne^{nz}$ into (\[12\]). In other words, (\[12\]) gives the following trigonometric summation relationships: $$\label{12ct1} \left.\cos\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)S(e^z)\right|_{z=0}=\sum^\infty_{n=0}a_n\cos\frac{n\pi{x}}{c};$$ $$\label{12ct2} \left.\sin\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)S(e^z)\right|_{z=0}=\sum^\infty_{n=0}a_n\sin\frac{n\pi{x}}{c}.$$ Here $x\in\Omega\subseteq\mathbb{R}^1$. The $\Omega$ can be uniquely determined by the detailed computation of the left-hand side of (\[12ct1\]) and (\[12ct2\]) respectively. For example, according to the proof of Lemma 2.3 in second sections of this paper, if $S(e^z)=\ln(1+e^z)$, then we have $\Omega:=\{x\in\mathbb{R}^1|\cos(\pi{x}/(2c))>0\}$ for (\[12ct1\]) and $\Omega:=\{x\in\mathbb{R}^1|\cos(\pi{x}/(2c))\neq0\}$ for (\[12ct2\]) respectively. **Theorem 1.5.** Let $S(t)$ be an arbitrary analytic function integrable in the interval $[0,1]$. Then we have $$\label{13} \left.\cos\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)\int^{e^z}_0\!\!\!S(e^z)\,de^z\right|_{z=0} = \int^1_0\!\!S(\xi)\,d\xi-\frac{\pi}{c}\int^x_0\!\!\left.\sin\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right) [S(e^z)\,e^z]\right|_{z=0}dx.$$ $$\label{14} \left.\sin\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)\int^{e^z}_0\!\!S(e^z)\,de^z\right|_{z=0}= \frac{\pi}{c}\int^x_0\!\!\left.\cos\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)[S(e^z)\,e^z]\right|_{z=0}dx.$$ **Proof.** According to the analytic continuity fundamental theorem, we only need to prove this set of formulas when $S(x)=x^n$, $n\in\mathbb{N}_0:=\{0,1,2,\cdots\}$. This is obvious. Let $S_0(t)$ be a function analytic in the neighborhood of $t=0$ and $$S_0(t)=\sum^\infty_{n=1}a_nt^n,\qquad t\in\mathbb{R}^1,\;\;|t|<r,\quad0<r<+\infty,$$ where $a_n$ are rational numbers, then the sum function $S_m(t)$ is defined as $$\label{23} S_m(t):=\underbrace{\int^t_0\frac{dt}{t}\cdots}_m\int^t_0S_0(t)\,\frac{dt}{t}=\sum^\infty_{n=1}a_n\frac{t^n}{n^m},\quad m\in\mathbb{N}_0:=\{0,1,2,\cdots\}.$$ Apparently $S_m(1)$ is the sum function of the Dirichlet series taking $m$ as the variable. According to (\[23\]), $S_m(t)$ satisfies the following recurrence relation: $$\label{24} \int^t_0S_{m-1}(t)\,\frac{dt}{t}=S_m(t),\qquad m\in\mathbb{N}:=\{1,2,\cdots\}.$$ **Theorem 1.6.** Let $m\in\mathbb{N},\,c>0$. The sum function $S_m(t)$ has the following recurrence property: $$\label{25} \left.\cos\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)S_m(e^z)\right|_{z=0}=S_m(1) -\frac{\pi}{c}\int^x_0\left.\sin\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)S_{m-1}(e^z)\right|_{z=0}dx,$$ $$\label{26} \left.\sin\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)S_m(e^z)\right|_{z=0}= \frac{\pi}{c}\int^x_0\left.\cos\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)S_{m-1}(e^z)\right|_{z=0}dx.$$ **Proof.** Taking $S(x)=S_{m-1}(x)/x$ in Theorem 1.5, then it is proved by using (\[24\]). In particular, if $S_0(t)=\ln(1+t)$, then $S_m(1)=\eta(m+1)$; if $S_0(t)=-\ln(1-t)$, then $S_m(1)=\zeta(m+1)$. Lemmas and definitions ====================== **Lemma 2.1.** For $m,r\in\mathbb{N}$, the sum function $S_m(t)$ has the following recurrence property: $$\begin{aligned} \label{30"} & & \left.\cos\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)S_{m-1}(e^z)\right|_{z=0}\nonumber\\ &=& \sum^{r-1}_{k=0}(-1)^k\frac{1}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}S_{m-2k-1}(1)\nonumber\\ & & +\,(-1)^r\left(\frac{\pi}{c}\right)^{2r-1} \underbrace{\int^x_0dx\cdots}_{2r-1}\int^x_0\left.\sin\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)S_{m-2r}(e^z)\right|_{z=0}dx.\end{aligned}$$ **Proof.** We can use the mathematical induction to prove it. According to (\[25\]) of Theorem 1.6, it is obviously tenable when $r=1$ in (\[30"\]). Now we inductively hypothesize that it is tenable when $r=K$. Using Theorem 1.6, then it is tenable when $r=K+1$, thus Lemma 2.1 is proved. Similarly, **Lemma 2.2.** For $m,r\in\mathbb{N}$, the sum function $S_m(t)$ has the following recurrence property: $$\begin{aligned} \label{30} & & \left.\cos\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)S_{m-1}(e^z)\right|_{z=0}\nonumber\\ &=& \sum^{r-1}_{k=0}(-1)^k\frac{1}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}S_{m-2k-1}(1)\nonumber\\ & & +\,(-1)^r\left(\frac{\pi}{c}\right)^{2r} \underbrace{\int^x_0dx\cdots}_{2r}\int^x_0\left.\cos\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)S_{m-2r-1}(e^z)\right|_{z=0}dx.\end{aligned}$$ **Lemma 2.3.** Let $x\in\mathbb{R}^1$ with $|x|<c$. Then $$\label{yc7.3+} \left.\sin\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)\ln(1+e^z)\right|_{z=0}=\frac{\pi{x}}{2c}.$$ $$\label{yc7.3} \left.\cos\left(\frac{\pi x}{c}\frac{\partial}{\partial z}\right)\ln(1+e^z)\right|_{z=0}=\ln\left(2\cos\frac{\pi x}{2c}\right).$$ **Proof.** By making use of (\[1\]), (\[yb1\]) and (\[7\]), we have $$\begin{aligned} & &\left.\sin\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)\ln(1+e^z)\right|_{z=0}\,=\, \left.\sin\left(Y\frac{\partial}{\partial{X}}\right)\ln{X}\right|_{z=0}=\left.\textrm{arccot}\frac{X}{Y}\right|_{z=0}\\ &=&\textrm{arccot}\frac{1+\cos(\pi{x}/c)}{\sin(\pi{x}/c)}=\textrm{arccot}\frac{\cos^2(\pi x/(2c))}{\sin(\pi x/(2c))\cos(\pi x/(2c))}.\end{aligned}$$ When $\cos(\pi{x}/(2c))\neq0$ or $|x|\neq{c}$, the above expression can be written in the form: $$\left.\sin\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)\ln(1+e^z)\right|_{z=0}= \textrm{arccot}\cot\frac{\pi{x}}{2c}=\frac{\pi{x}}{2c}\quad(|x|<c).$$ Similarly, for $\cos(\pi{x}/(2c))>0$, we have $$\begin{aligned} & & \left.\cos\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)\ln(1+e^z)\right|_{z=0}= \left.\cos\left(Y\frac{\partial}{\partial{X}}\right)\ln{X}\right|_{z=0}=\left.\frac{1}{2}\ln(X^2+Y^2)\right|_{z=0}\\ &=& \frac{1}{2}\ln\left(\left(1+\cos\frac{\pi{x}}{c}\right)^2+\sin^2\frac{\pi{x}}{c}\right)=\ln\left(2\cos\frac{\pi{x}}{2c}\right)\quad(|x|<c).\end{aligned}$$ Thus Lemma 2.3 is proved. **Lemma 2.4.** For $\Re(s)>1$ $$\label{yc2} \sum^\infty_{n=1}\frac{1}{n^s}\cos\frac{2n\pi}{3}=\frac{1}{2}(3^{1-s}-1)\zeta(s).$$ $$\label{yc4} \sum^\infty_{n=1}\frac{1}{n^s}\cos\frac{n\pi}{2}=2^{-s}(2^{1-s}-1)\zeta(s).$$ $$\label{yc1} \sum^\infty_{n=1}\frac{1}{n^s}\cos\frac{n\pi}{3}=\frac{1}{2}(6^{1-s}-3^{1-s}-2^{1-s}+1)\zeta(s).$$ **Proof.** Let $\Phi(z,s,a)$ be the Lerch transcendent. For $\Re(s)>1$, it is given by $$\label{ycz0} \Phi(z,s,a)=\sum^\infty_{n=0}\frac{z^n}{(n+a)^s}\quad(a\not\in\mathbb{Z}_0^-,\,|z|\leq1),$$ so that $\Phi(1,s,a)=\zeta(s,a)$ and $$\label{ycz1} \zeta(s,a)+\Phi(-1,s,a)=2^{1-s}\zeta(s,a/2).$$ Since $\zeta(s,1)=\zeta(s),\;\zeta(s,2)=\zeta(s)-1$ and $$\label{ycz2} \sum^{m-1}_{k=0}\zeta(s,a+k/m)=m^s\zeta(s,ma)\quad(m\in\mathbb{N}),$$ we obtain $$\label{ycz3} \Phi(-1,s,2/3)+\Phi(-1,s,4/3)=(2^{1-s}-1)(3^s-1)\zeta(s)+3^s.$$ Thus we have $$\begin{aligned} \sum^\infty_{n=1}\frac{1}{n^s}\cos\frac{n\pi}{3} &=& \cos\frac{\pi}{3}+\sum^\infty_{n=1}\frac{1}{(3n-1)^s}\cos\left(n\pi-\frac{\pi}{3}\right) \\ & & +\sum^\infty_{n=1}\frac{1}{(3n)^s}\cos(n\pi)+\sum^\infty_{n=1}\frac{1}{(3n+1)^s}\cos\left(n\pi+\frac{\pi}{3}\right)\\ &=& \frac{3^s-(\Phi(-1,s,2/3)+\Phi(-1,s,4/3))}{2\times3^s}-\frac{\eta(s)}{3^s}\\ &=& -\frac{(2^{1-s}-1)(3^s-1)\zeta(s)}{2\times3^s}-\frac{(1-2^{1-s})\zeta(s)}{3^s}\\ &=& \frac{1}{2}(6^{1-s}-3^{1-s}-2^{1-s}+1)\zeta(s)\qquad(\Re(s)>1).\end{aligned}$$ Similarly we have (\[yc2\]) and (\[yc4\]). Lemma 2.4 is proved. **Lemma 2.5.** For $\Re(s)>1$ $$\label{yc2.7} \sum^\infty_{n=1}(-1)^{n-1}\frac{1}{n^s}\cos\frac{2n\pi}{3}=-\frac{6^{1-s}-3^{1-s}-2^{1-s}+1}{2(1-2^{1-s})}\eta(s).$$ $$\label{yc4.7} \sum^\infty_{n=1}(-1)^{n-1}\frac{1}{n^s}\cos\frac{n\pi}{2}=2^{-s}(1-2^{1-s})\zeta(s)=2^{-s}\eta(s).$$ $$\label{yc1.7} \sum^\infty_{n=1}(-1)^{n-1}\frac{1}{n^s}\cos\frac{n\pi}{3}=\frac{1-3^{1-s}}{2(1-2^{1-s})}\eta(s).$$ **Proof.** It is easily seen that for $0\leq x\leq c$ $$\label{yc7.1} \sum^\infty_{n=1}(-1)^{n-1}\frac{1}{n^s}\cos\frac{n\pi(c-x)}{c}=-\sum^\infty_{n=1}\frac{1}{n^s}\cos\frac{n\pi x}{c},\quad(\Re(s)>1).$$ By applying relationships (\[yc7.1\]) in the case $x=c/3, c/2$ and $2c/3$ respectively, and by using the relation $\eta(s)=(1-2^{1-s})\zeta(s)$, we obtain Lemma 2.5 from Lemma 2.4. **Lemma 2.6.** Let $x\in\mathbb{R}^1$ with $|x|<c$. Then $$\label{yc7.2} \ln\cos\frac{\pi x}{2c}=\sum^\infty_{k=1}(-1)^k\frac{E_{2k-1}(1)}{2(2k)!}\left(\frac{\pi x}{c}\right)^{2k},$$ where $E_n(x)$ are the Euler polynomials defined by the generating functions: $$\label{sch} \frac{2e^{xz}}{e^z+1}=\sum^\infty_{n=0}E_n(x)\frac{z^n}{n!}\quad(|z|<\pi).$$ **Proof.** By (\[yc7.3\]) of Lemma 2.3, we have $$\begin{aligned} \ln\cos\frac{\pi x}{2c}&=&\left.\cos\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)\ln(1+e^z)\right|_{z=0}-\ln2\\ &=& \sum^\infty_{k=0}\frac{(-1)^k}{(2k)!}\left(\frac{\pi x}{c}\right)^{2k}\!\left.\frac{\partial^{2k}}{\partial z^{2k}}\ln(1+e^z)\right|_{z=0}-\ln2\\ &=& \sum^\infty_{k=1}\frac{(-1)^k}{(2k)!}\left(\frac{\pi x}{c}\right)^{2k}\!\left.\frac{\partial^{2k-1}}{\partial z^{2k-1}}\frac{e^z}{1+e^z}\right|_{z=0}\\ &=& \sum^\infty_{k=1}\frac{(-1)^k}{(2k)!}\left(\frac{\pi x}{c}\right)^{2k}\frac{E_{2k-1}(1)}{2}\quad(|x|<c).\end{aligned}$$ **Lemma 2.7.** For $r\in\mathbb{N}$ and $|x|\leq c\;(x\in\mathbb{R}^1)$, $$\begin{aligned} \label{d7.1} \sum^\infty_{n=1}\frac{(-1)^{n-1}}{n^{2r+1}}\cos\frac{n\pi{x}}{c} &=& \sum^r_{k=0}\frac{(-1)^k}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}\eta(2r+1-2k)\nonumber\\ & & +\sum^\infty_{k=1}(-1)^{r+k}\frac{E_{2k-1}(1)}{2(2r+2k)!}\left(\frac{\pi{x}}{c}\right)^{2r+2k},\end{aligned}$$ which is a slightly corrected version of a result proven in a significantly different way by Tsumura [@Tsu p. 388, Proposition 1 (3)]. **Proof.** In Lemma 2.2, let $m=2r+1,\;S_0(t)=\ln(1+t)$. By using Theorem 1.4 and Lemma 2.3, since $S_{2r-2k}(1)=\eta(2r+1-2k)$, we have $$\begin{aligned} \label{d7.1"} \sum^\infty_{n=1}\frac{(-1)^{n-1}}{n^{2r+1}}\cos\frac{n\pi{x}}{c} &=& \sum^{r-1}_{k=0}\frac{(-1)^k}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}\eta(2r+1-2k)\nonumber\\ & & +\,(-1)^r\left(\frac{\pi}{c}\right)^{2r}\underbrace{\int^x_0dx\cdots}_{2r}\int^x_0\ln\left(2\cos\frac{\pi x}{2c}\right)dx,\end{aligned}$$ where $|x|\leq c\;(x\in\mathbb{R}^1)$ for $r\in\mathbb{N}$. Substituting (\[yc7.2\]) into (\[d7.1"\]), since $\eta(1)=\ln2$, then it is proved. **Lemma 2.8.** For $r\in\mathbb{N}$ and $|x|\leq c\;(x\in\mathbb{R}^1)$, $$\begin{aligned} \label{44L} \sum^{\infty}_{n=1}(-1)^{n-1}\frac{1}{n^{2r}}\cos\frac{n\pi{x}}{c} =\sum^r_{k=0}(-1)^{k}\frac{1}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}\eta(2r-2k),\end{aligned}$$ which is a slightly corrected version of a result proven in a significantly different way by Tsumura [@Tsu p. 387, Proposition 1 (1)]. **Proof.** Letting $m=2r$ in (\[30"\]), if $S_0(t)=\ln(1+t)$, then according to Lemma 2.1, Theorem 1.4 and Lemma 2.3, we can obtain (on the interval $[-c,c]$ for $r\in\mathbb{N}$): $$\begin{aligned} \label{44} \sum^{\infty}_{n=1}(-1)^{n-1}\frac{1}{n^{2r}}\cos\frac{n\pi{x}}{c} &=&\sum^{r-1}_{k=0}(-1)^{k}\frac{1}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}\eta(2r-2k)\nonumber\\ & &+(-1)^r\frac{1}{2(2r)!}\left(\frac{\pi{x}}{c}\right)^{2r}.\end{aligned}$$ Since $\eta(0)=1/2$, then it is proved. Let $\partial_z^{-s},\,z\in\mathbb{R}^1$ be an abstract operators taking $\xi^{-s}$ as the symbols, namely $$\label{cxsz} \left(\frac{\partial}{\partial z}\right)^{-s}e^{\xi z}:=\xi^{-s}e^{\xi z}\quad(\xi\in\mathbb{R}^1,\,s\in\mathbb{C}).$$ **Definition 2.1.** The alternating Zeta Function $\eta(s)$ can be defined in the complete form: $$\label{yrdy} \eta(s):=\left(\frac{\partial}{\partial z}\right)^{-s}\!\!\left.\frac{e^z}{1+e^z}\right|_{z=0},\quad s\in\mathbb{C},$$ which is a holomorphic function on the whole complex plane. Similarly, $$\label{yrkz} \eta(s)=\sum^m_{n=1}\frac{(-1)^{n-1}}{n^s}+(-1)^m\left(\frac{\partial}{\partial z}\right)^{-s}\!\!\left.\frac{e^{(m+1)z}}{1+e^z}\right|_{z=0}\quad (m\in\mathbb{N}_0).$$ **Definition 2.2.** The Hurwitz Zeta function $\zeta(s,a)$ can be defined in the complete form: $$\label{hurw} \zeta(s,a):=\frac{1}{s-1}\left(\frac{\partial}{\partial z}\right)^{1-s}\!\!\left.\frac{ze^{az}}{e^z-1}\right|_{z=0}\quad(a\not\in\mathbb{Z}_0^-,\,s\in\mathbb{C}\;\mbox{and}\;s\neq1).$$ This definition is valid for all complex $s$. In the special case when $\Re(s)>1$, by applying (\[e0\]), (\[e1\]) and (\[e1’\]) to (\[hurw\]), we have $$\begin{aligned} \zeta(s,a)&=&\frac{1}{s-1}\left(\frac{\partial}{\partial z}\right)^{1-s}\!\!\left.\frac{ze^{az}}{e^z-1}\right|_{z=0} =\sum^\infty_{k=0}\left(\frac{\partial}{\partial z}\right)^{1-s}\!\!\left.\frac{ze^{(a+k)z}}{1-s}\right|_{z=0}\\ &=&\sum^\infty_{k=0}\frac{\partial}{\partial\xi}e^{(a+k)\frac{\partial}{\partial\xi}}\left.\frac{\xi^{1-s}e^{\xi{z}}}{1-s}\right|_{\xi=0,z=0}\\ &=&\sum^\infty_{k=0}\left(z+\frac{\partial}{\partial\xi}\right)e^{(a+k)\left(z+\frac{\partial}{\partial\xi}\right)}\left.\frac{\xi^{1-s}}{1-s}\right|_{\xi=0,z=0}\\ &=&\sum^\infty_{k=0}\left(z+\frac{\partial}{\partial\xi}\right)\left.\frac{e^{(a+k)z}(\xi+a+k)^{1-s}}{1-s}\right|_{\xi=0,z=0}\\ &=&\sum^\infty_{k=0}\left.\left(z\frac{e^{(a+k)z}(a+k)^{1-s}}{1-s}+e^{(a+k)z}(a+k)^{-s}\right)\right|_{z=0}\\ &=&\sum^\infty_{k=0}\frac{1}{(a+k)^s}\quad(\Re(s)>1,\,a\not\in\mathbb{Z}_0^-).\end{aligned}$$ The result of this calculation shows that our definition is reasonable. Similarly, $$\label{huzm} \zeta(s,a)=\sum^{m-1}_{n=0}\frac{1}{(n+a)^s}+\frac{1}{s-1}\left(\frac{\partial}{\partial{z}}\right)^{1-s}\!\! \left.\frac{ze^{(a+m)z}}{e^z-1}\right|_{z=0}\quad(m\in\mathbb{N}_0).$$ **Definition 2.3.** The Riemann Zeta function $\zeta(s)$ can be defined in the complete form (This definition is valid for all complex $s$): $$\label{rize} \zeta(s):=\frac{1}{s-1}\left(\frac{\partial}{\partial z}\right)^{1-s}\!\!\left.\frac{ze^z}{e^z-1}\right|_{z=0}\quad(s\in\Omega:=\{s\in\mathbb{C}|s\neq1\}).$$ **Definition 2.4.** Let $\zeta_z(s,a)$ be an analytic function defined by $$\label{rize'} \zeta_z(s,a):=\frac{1}{s-1}\left(\frac{\partial}{\partial z}\right)^{1-s}\!\!\frac{ze^{az}}{e^z-1}\quad(s\in\Omega:=\{s\in\mathbb{C}|s\neq1\},\,a\not\in\mathbb{Z}_0^-,\,z\in\mathbb{R}^1).$$ In the special case when $\Re(s)>2$ and $z\leq0$, we have $$\label{rize"} \zeta_z(s,a)=\frac{1}{1-s}\sum^\infty_{n=0}\frac{ze^{(a+n)z}}{(a+n)^{s-1}}+\sum^\infty_{n=0}\frac{e^{(a+n)z}}{(a+n)^s} \quad(\Re(s)>2,\,a\not\in\mathbb{Z}_0^-,\,z\leq0).$$ **Definition 2.5.** Let $\chi$ be a Dirichlet character modulo $q$. The Dirichlet L-function $L(s,\chi)$ can be defined in the complete form: $$\label{lfud} L(s,\chi):=\frac{1}{s-1}\left(\frac{\partial}{\partial{z}}\right)^{1-s}\sum^q_{k=1}\left.\chi(k)\frac{ze^{kz}}{e^{qz}-1}\right|_{z=0} \quad(s\in\mathbb{C}).$$ This definition is valid for all complex $s$. In the special case when $\Re(s)>1$, by applying (\[e0\]), (\[e1\]) and (\[e1’\]) to (\[lfud\]), we have $$\begin{aligned} L(s,\chi)&=&\frac{1}{s-1}\left(\frac{\partial}{\partial{z}}\right)^{1-s}\sum^q_{k=1}\left.\chi(k)\frac{ze^{kz}}{e^{qz}-1}\right|_{z=0}\\ &=&\sum^q_{k=1}\chi(k)\sum^\infty_{n=0}\left(\frac{\partial}{\partial z}\right)^{1-s}\!\!\left.\frac{ze^{(k+qn)z}}{1-s}\right|_{z=0}\\ &=&\sum^q_{k=1}\chi(k)\sum^\infty_{n=0}\frac{\partial}{\partial\xi}e^{(k+qn)\frac{\partial}{\partial\xi}}\left.\frac{\xi^{1-s}e^{\xi{z}}}{1-s}\right|_{\xi=0,z=0}\\ &=&\sum^q_{k=1}\chi(k)\sum^\infty_{n=0}\left(z+\frac{\partial}{\partial\xi}\right) e^{(k+qn)\left(z+\frac{\partial}{\partial\xi}\right)}\left.\frac{\xi^{1-s}}{1-s}\right|_{\xi=0,z=0}\\ &=&\sum^q_{k=1}\chi(k)\sum^\infty_{n=0}\left(z+\frac{\partial}{\partial\xi}\right)\left.\frac{e^{(k+qn)z}(\xi+k+qn)^{1-s}}{1-s}\right|_{\xi=0,z=0}\\ &=&\sum^q_{k=1}\chi(k)\sum^\infty_{n=0}\left.\left(z\frac{e^{(k+qn)z}(k+qn)^{1-s}}{1-s}+e^{(k+qn)z}(k+qn)^{-s}\right)\right|_{z=0}\\ &=&\sum^\infty_{n=0}\sum^q_{k=1}\frac{\chi(k)}{(k+qn)^s}=\sum^\infty_{n=1}\sum^q_{k=1}\frac{\chi(k)}{(k+q(n-1))^s}\\ &=&\sum^\infty_{n=1}\sum^{qn}_{k=1+q(n-1)}\chi(k-qn+q)\frac{1}{k^{s}}=\sum^\infty_{n=1}\frac{\chi(n)}{n^s}\quad(\Re(s)>1).\end{aligned}$$ The result of this calculation shows that our definition is reasonable. Similarly, $$\label{lfjz} L(s,\chi)=\sum^{qm}_{n=1}\frac{\chi(n)}{n^s}+\frac{1}{s-1}\left(\frac{\partial}{\partial{z}}\right)^{1-s} \sum^q_{k=1}\left.\chi(k)\frac{ze^{(k+qm)z}}{e^{qz}-1}\right|_{z=0}\quad(m\in\mathbb{N}_0).$$ **Definition 2.6.** Let $L_z(s,\chi)$ be an analytic function defined by $$\label{rild} L_z(s,\chi):=\frac{1}{s-1}\left(\frac{\partial}{\partial{z}}\right)^{1-s}\sum^q_{k=1}\chi(k)\frac{ze^{kz}}{e^{qz}-1} \quad(s\in\Omega:=\{s\in\mathbb{C}|s\neq1\},\,z\in\mathbb{R}^1).$$ In the special case when $\Re(s)>2$ and $z\leq0$, since $\chi(k+q)=\chi(k)$, we have $$\begin{aligned} \label{rild"} L_z(s,\chi) &=& \sum^q_{k=1}\chi(k)\left(\frac{1}{1-s}\sum^\infty_{n=0}\frac{ze^{(k+qn)z}}{(k+qn)^{s-1}}+\sum^\infty_{n=0}\frac{e^{(k+qn)z}}{(k+qn)^s}\right)\nonumber\\ &=& \frac{z}{1-s}\sum^\infty_{n=1}\frac{\chi(n)}{n^{s-1}}e^{nz}+\sum^\infty_{n=1}\frac{\chi(n)}{n^s}e^{nz}.\end{aligned}$$ Main results ============ Alternating Zeta function ------------------------- **Theorem 3.1.** For $r\in\mathbb{N}$ $$\begin{aligned} \label{d7.2a} \eta(2r+1) &=& \frac{2\times3^{2r}}{3^{2r+1}-1}\sum^r_{k=1}(-1)^{k-1}\frac{1}{(2k)!}\left(\frac{2\pi}{3}\right)^{2k}\eta(2r+1-2k)\nonumber\\ & & +\,\frac{(2\pi)^{2r}}{3^{2r+1}-1}\sum^\infty_{k=1}(-1)^{r+k-1}\frac{E_{2k-1}(1)}{(2r+2k)!}\left(\frac{2\pi}{3}\right)^{2k}.\end{aligned}$$ $$\begin{aligned} \label{d7.2b} \eta(2r+1) &=& \frac{2^{2r+1}}{2^{2r+1}-1}\sum^r_{k=1}(-1)^{k-1}\frac{1}{(2k)!}\left(\frac{\pi}{2}\right)^{2k}\eta(2r+1-2k)\nonumber\\ & & +\,\frac{\pi^{2r}}{2^{2r+1}-1}\sum^\infty_{k=1}(-1)^{r+k-1}\frac{E_{2k-1}(1)}{(2r+2k)!}\left(\frac{\pi}{2}\right)^{2k}.\end{aligned}$$ $$\begin{aligned} \label{d7.2c} \eta(2r+1) &=& \frac{3^{2r}(2^{2r}-1)}{3^{2r}(2^{2r-1}-1)+2^{2r-1}}\left[\sum^r_{k=1}\frac{(-1)^{k-1}}{(2k)!}\left(\frac{\pi}{3}\right)^{2k}\eta(2r+1-2k)\right.\nonumber\\ & & \left.+\,\sum^\infty_{k=1}(-1)^{r+k-1}\frac{E_{2k-1}(1)}{2(2r+2k)!}\left(\frac{\pi}{3}\right)^{2r+2k}\right].\end{aligned}$$ **Proof.** Letting $x=2c/3$ in (\[d7.1\]), and by using (\[yc2.7\]) we have (\[d7.2a\]). Similarly we have (\[d7.2b\]) and (\[d7.2c\]) by using Lemma 2.5 and Lemma 2.7. We can use $\eta(s)=(1-2^{1-s})\zeta(s)$ and (\[d7.2c\]) to deduce the following theorem, which gives $\zeta(2r+1)$ recursively in terms of $\zeta(3),\zeta(5),\ldots,\zeta(2r-1)$: **Theorem 3.2.** For $r\in\mathbb{N}$ $$\begin{aligned} \label{dy} \zeta(2r+1)&=& \frac{(-1)^r}{3^{2r}(2^{2r-1}-1)+2^{2r-1}}\left[\sum^{r-1}_{k=1}(-1)^{k-1}\frac{(2\pi)^{2r-2k}}{(2r-2k)!}3^{2k}(2^{2k}-1)\zeta(2k+1)\right.\nonumber\\ & & \left.-\frac{(2\pi)^{2r}}{(2r)!}\ln2+(2\pi)^{2r}\sum^\infty_{k=1}(-1)^{k-1}\frac{E_{2k-1}(1)}{2(2r+2k)!}\left(\frac{\pi}{3}\right)^{2k}\right],\end{aligned}$$ which provides a significantly simpler (and much more rapidly convergent) version of a main result of Dancs and He [@Danh p. 194, Theorem 3.1]: $$\begin{aligned} \label{dh} & & (1-2^{-2r})\zeta(2r+1)\nonumber\\ &=&\sum^{r-1}_{k=1}(-1)^k\frac{\pi^{2k}}{(2k+1)!}(2^{2k-2r}-1)\zeta(2r+1-2k)-\frac{(-1)^r\pi^{2r}\ln2}{(2r+1)!}\nonumber\\ & & +\,(-1)^r\pi^{2r}\sum^\infty_{k=1}(-1)^{k-1}\frac{E_{2k-1}(1)}{2(2r+2k+1)!}\pi^{2k}\quad (r\in\mathbb{N}).\end{aligned}$$ Similarly we have $$\begin{aligned} \label{dyj} \zeta(2r+1)&=& \frac{(-1)^r2^{2r+1}}{(2^{2r}-1)(2^{2r+1}-1)}\left[\sum^{r-1}_{k=1}(-1)^{k-1}\frac{\pi^{2r-2k}}{(2r-2k)!}(2^{2k}-1)\zeta(2k+1)\right.\nonumber\\ & & \left.-\frac{\pi^{2r}}{(2r)!}\ln2+\pi^{2r}\sum^\infty_{k=1}(-1)^{k-1}\frac{E_{2k-1}(1)}{2(2r+2k)!}\left(\frac{\pi}{2}\right)^{2k}\right]\quad (r\in\mathbb{N}),\end{aligned}$$ which is a slightly corrected version of a main result proven in a different way by Tsumura [@Tsu p. 383, Theorem B]. Since $E_{2k}(1)\equiv0$ for $k\in\mathbb{N}$, we can use (\[d7.1\]) and (\[44L\]) to deduce the following theorem: **Theorem 3.3.** For $m\in\mathbb{N}\setminus\{1\}$, in the interval $[-c,c]$ we have the following Fourier series expressions (\[$\cdot$\] is Gauss mark): $$\begin{aligned} \label{44"} \sum^{\infty}_{n=1}(-1)^{n-1}\frac{1}{n^m}\cos\frac{n\pi{x}}{c} &=&\sum^{[m/2]}_{k=0}(-1)^{k}\frac{1}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}\eta(m-2k)\nonumber\\ & &+\sum^\infty_{k=[m/2]+1}(-1)^k\frac{E_{2k-m}(1)}{2(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}.\end{aligned}$$ Letting $x=c/3$ in (\[44"\]), and by using (\[yc1.7\]), we obtain **Theorem 3.4.** For $m\in\mathbb{N}\setminus\{1\}$ $$\begin{aligned} \label{44"yc1.7} \eta(m) &=& \frac{3^{m-1}(2^{m-1}-1)}{3^{m-1}(2^{m-2}-1)+2^{m-2}}\left[\sum^{[m/2]}_{k=1}\frac{(-1)^{k-1}}{(2k)!}\left(\frac{\pi}{3}\right)^{2k}\eta(m-2k)\right.\nonumber\\ & & \left.+\,\sum^\infty_{k=[m/2]+1}(-1)^{k-1}\frac{E_{2k-m}(1)}{2(2k)!}\left(\frac{\pi}{3}\right)^{2k}\right],\end{aligned}$$ which gives $\eta(m)$ recursively in terms of $\eta(2),\eta(3),\eta(4),\ldots,\eta(m)$. Since $E_{2k}(1)=0$ for $k\in\mathbb{N}$, letting $m=2,3,4,\cdots$ in (\[44"yc1.7\]), we have $$\begin{aligned} \eta(2) &=& \frac{\pi^2}{12}, \\ \eta(3) &=& \frac{3\pi^2}{22}\ln2+\frac{3\pi^2}{22}\sum^\infty_{k=1}(-1)^k\frac{E_{2k-1}(1)}{(2k+2)!}\left(\frac{\pi}{3}\right)^{2k},\\ \eta(4) &=& \frac{189}{85}\left[\frac{\pi^2}{2!\cdot3^2}\eta(2)-\frac{\pi^4}{4!\cdot3^4}\frac{1}{2}\right]=\frac{7\pi^4}{720},\\ &\cdots&,\end{aligned}$$ recursively. Since $E_{2k-1}(1)\neq0$, this provides some perspective on the difficulty of evaluating $\eta(2r+1)$ as opposed to $\eta(2r)$ for $r\in\mathbb{N}$. By using (\[sch\]), when $s=-2n,\,n\in\mathbb{N}$ in (\[yrdy\]), we obtain $$\eta(-2n)=\left.\frac{\partial^{2n}}{\partial z^{2n}}\frac{e^z}{1+e^z}\right|_{z=0}=0.$$ When $s=-2n+1,\,n\in\mathbb{N}$ in (\[yrdy\]), we obtain $$\eta(-2n+1)=\left.\frac{\partial^{2n-1}}{\partial z^{2n-1}}\frac{e^z}{1+e^z}\right|_{z=0}=\frac{1}{2}E_{2n-1}(1)=(2^{2n}-1)\frac{B_{2n}}{2n}.$$ Here $B_m$ are the Bernoulli numbers defined by the generating functions: $$\label{bnls} \frac{z}{e^z-1}=\sum^\infty_{m=0}B_m\frac{z^m}{m!}\quad(|z|<2\pi).$$ Thus, we obtain the following asymptotic expansion for $\eta(s)$: **Theorem 3.5.** For $s\in\mathbb{C}$ $$\begin{aligned} \label{yr1} \eta(s)\sim\sum^m_{n=1}\frac{(-1)^{n-1}}{n^s} +\frac{(-1)^m}{2}\left[\frac{1}{m^s}+\sum^\infty_{k=1}{-s\choose{2k-1}}\frac{E_{2k-1}(1)}{m^{s+2k-1}}\right]\quad(m\rightarrow\infty).\end{aligned}$$ Let $-Li_s(-e^z)$ be an analytic function defined by $$\label{yrdy'} -Li_s(-e^z):=\left(\frac{\partial}{\partial z}\right)^{-s}\!\!\frac{e^z}{1+e^z},\quad z\in\mathbb{R}^1,\,s\in\mathbb{C}.$$ Then $\eta(s)=\left.-Li_s(-e^z)\right|_{z=0}$ and $$\left.\frac{\partial^k}{\partial z^k}(-Li_s(-e^z))\right|_{z=0}=\eta(s-k)\quad(k\in\mathbb{N}).$$ Thus for $k\in\mathbb{N}$ $$\left.\frac{\partial^{2k-1}}{\partial x^{2k-1}}\cos\left(\frac{\pi x}{c}\frac{\partial}{\partial z}\right)(-Li_s(-e^z))\right|_{z=0,x=0}=0,$$ $$\left.\frac{\partial^{2k}}{\partial x^{2k}}\cos\left(\frac{\pi x}{c}\frac{\partial}{\partial z}\right)(-Li_s(-e^z))\right|_{z=0,x=0}=(-1)^k\left(\frac{\pi}{c}\right)^{2k}\eta(s-2k).$$ Here $c>0$ is a known real constant. Therefore, we have the following Taylor expansion in the neighborhood of $x=0$: $$\label{yr2} \left.\cos\left(\frac{\pi x}{c}\frac{\partial}{\partial z}\right)(-Li_s(-e^z))\right|_{z=0}=\sum^\infty_{k=0}\frac{(-1)^k}{(2k)!}\left(\frac{\pi x}{c}\right)^{2k}\eta(s-2k)\quad (s\in\mathbb{C}).$$ Let $x\in\mathbb{R}^1,\,c'>0$ be a real constant. By using Theorem 1.4, we have ($|x|\leq c'$) $$\label{yr3} \left.\cos\left(\frac{\pi x}{c}\frac{\partial}{\partial z}\right)(-Li_s(-e^z))\right|_{z=0}=\sum^\infty_{n=1}\frac{(-1)^{n-1}}{n^s}\cos\frac{n\pi x}{c}\quad (\Re(s)>1).$$ Since $\frac{1}{2}E_n(1)=\eta(-n)$, by Theorem 3.3, we have ($x\in\mathbb{R}^1$ with $|x|\leq c$) $$\label{yr4} \sum^\infty_{n=1}\frac{(-1)^{n-1}}{n^m}\cos\frac{n\pi x}{c}=\sum^\infty_{k=0}\frac{(-1)^k}{(2k)!}\left(\frac{\pi x}{c}\right)^{2k}\eta(m-2k)\quad(m\in\mathbb{N}\setminus\{1\}).$$ Letting $s=m$ in (\[yr3\]), and by using formula (\[yr4\]), we get $c'=c$. Thus we obtain the following generalization of Theorem 3.3: **Theorem 3.6.** For $\Re(s)>1$ and $|x|\leq c\,(x\in\mathbb{R}^1)$, we have $$\label{yr5} \sum^\infty_{n=1}\frac{(-1)^{n-1}}{n^s}\cos\frac{n\pi x}{c}=\sum^\infty_{k=0}\frac{(-1)^k}{(2k)!}\left(\frac{\pi x}{c}\right)^{2k}\eta(s-2k).$$ Similarly, let $\Phi(-1,s,a)$ be an analytic function defined by $$\label{yrdy"} \Phi(-1,s,a):=\left(\frac{\partial}{\partial z}\right)^{-s}\!\!\left.\frac{e^{az}}{1+e^z}\right|_{z=0},\quad a\not\in\mathbb{Z}_0^-,\, s\in\mathbb{C}.$$ **Theorem 3.7.** For $\Re(s)>1$ and $|x|\leq c\,(x\in\mathbb{R}^1)$, we have $$\label{yr6} \sum^\infty_{n=0}\frac{(-1)^n}{(n+a)^s}\cos\frac{(n+a)\pi x}{c}=\sum^\infty_{k=0}\frac{(-1)^k}{(2k)!}\left(\frac{\pi x}{c}\right)^{2k}\Phi(-1,s-2k,a).$$ Here $a\neq0,-1,-2,\cdots$, and $\Phi(z,s,a)$ is the Lerch transcendent. In the case when $a=1$, since $\Phi(-1,s,1)=\eta(s)$, Theorem 3.7 gives Theorem 3.6. We can use Lemma 2.5 and Theorem 3.6 to deduce the following generalizations of the Theorem 3.1: **Theorem 3.8.** For $s\in\mathbb{C}$ $$\label{yrc1} \frac{3}{2}(1-3^{-s})\eta(s)=\sum^\infty_{k=1}\frac{(-1)^{k-1}}{(2k)!}\left(\frac{2\pi}{3}\right)^{2k}\eta(s-2k).$$ $$\label{yrc2} (1-2^{-s})\eta(s)=\sum^\infty_{k=1}\frac{(-1)^{k-1}}{(2k)!}\left(\frac{\pi}{2}\right)^{2k}\eta(s-2k).$$ $$\label{yrc3} \frac{1}{2}(1-2^{2-s}+3^{1-s})\zeta(s)=\sum^\infty_{k=1}\frac{(-1)^{k-1}}{(2k)!}\left(\frac{\pi}{3}\right)^{2k}\eta(s-2k).$$ Letting $a=1/2$ in (\[yr6\]), since $\Phi(-1,s,1/2)=2^s\beta(s)$, we obtain the following theorem: **Theorem 3.9.** For $\Re(s)>0$ and $|x|\leq c/2\,(x\in\mathbb{R}^1)$, we have $$\label{yr7} \sum^\infty_{n=0}\frac{(-1)^n}{(2n+1)^s}\cos\frac{(2n+1)\pi x}{c}=\sum^\infty_{k=0}\frac{(-1)^k}{(2k)!}\left(\frac{\pi x}{c}\right)^{2k}\beta(s-2k).$$ Here $\beta(s)$ is the Dirichlet Beta function, which is a holomorphic function on the whole complex plane. For $\Re(s)>0$, it is given by $$\beta(s)=\sum^\infty_{n=0}\frac{(-1)^n}{(2n+1)^s}\quad(\Re(s)>0).$$ Letting $s=-n$ in (\[yrdy"\]), and by using (\[sch\]), we obtain $$\label{sch'} \Phi(-1,-n,a)=\frac{1}{2}E_n(a)\quad(n\in\mathbb{N}),$$ where $E_n(x)$ are the Euler polynomials. In particular, $\eta(-n)=\frac{1}{2}E_n(1)$. We can use (\[yrdy"\]) and (\[sch’\]) to deduce the following asymptotic expansion for $\Phi(-1,s,a)$: **Theorem 3.10.** For $s\in\mathbb{C},\,a\not\in\mathbb{Z}_0^-$ $$\begin{aligned} \label{yr1a} \Phi(-1,s,a)\sim\sum^{m-1}_{n=0}\frac{(-1)^n}{(n+a)^s} +\frac{(-1)^m}{2}\left[\frac{1}{m^s}+\sum^\infty_{k=1}{-s\choose k}\frac{E_k(a)}{m^{s+k}}\right]\quad(m\rightarrow\infty).\end{aligned}$$ When $a=1/2$ in (\[yr1a\]), since $\Phi(-1,s,1/2)=2^s\beta(s)$, $E_k(1/2)=E_k/2^k$ and $E_{2k-1}=0$ for $k\in\mathbb{N}$, we obtain the following asymptotic expansion for $\beta(s)$: **Theorem 3.11.** For $s\in\mathbb{C}$ $$\begin{aligned} \label{yr1b} \beta(s)\sim\sum^{m-1}_{n=0}\frac{(-1)^n}{(2n+1)^s} +\frac{(-1)^m}{2}\left[\frac{1}{(2m)^s}+\sum^\infty_{k=1}{-s\choose{2k}}\frac{E_{2k}}{(2m)^{s+2k}}\right]\quad(m\rightarrow\infty).\end{aligned}$$ Here $E_n$ are the Euler numbers defined by the generating functions: $$\label{els} \frac{1}{\cosh z}=\frac{2}{e^z+e^{-z}}=\sum^\infty_{n=0}E_n\frac{z^n}{n!},\quad|z|<\frac{\pi}{2}.$$ We can use (\[yr6\]) and (\[sch’\]) to deduce the following theorem: **Theorem 3.12.** For $r\in\mathbb{N},\,a\not\in\mathbb{Z}_0^-$ and $|x|\leq c\,(x\in\mathbb{R}^1)$, we have $$\begin{aligned} \label{yr8} \sum^\infty_{n=0}\frac{(-1)^n}{(n+a)^{2r+1}}\cos\frac{(n+a)\pi{x}}{c} &=& \sum^r_{k=0}\frac{(-1)^k}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}\Phi(-1,2r+1-2k,a)\nonumber\\ & & +\sum^\infty_{k=1}(-1)^{r+k}\frac{E_{2k-1}(a)}{2(2r+2k)!}\left(\frac{\pi{x}}{c}\right)^{2r+2k};\end{aligned}$$ $$\begin{aligned} \label{yr8'} \sum^\infty_{n=0}\frac{(-1)^n}{(n+a)^{2r}}\cos\frac{(n+a)\pi{x}}{c} &=& \sum^r_{k=0}\frac{(-1)^k}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}\Phi(-1,2r-2k,a)\nonumber\\ & & +\sum^\infty_{k=1}(-1)^{r+k}\frac{E_{2k}(a)}{2(2r+2k)!}\left(\frac{\pi{x}}{c}\right)^{2r+2k}.\end{aligned}$$ When $a=1$ in (\[yr8\]) and (\[yr8’\]), since $\Phi(-1,s,1)=\eta(s)$ and $E_{2k}(1)=0$, we obtain the Lemma 2.7 and Lemma 2.8 respectively. Letting $a=1/2$ in (\[yr8\]) and (\[yr8’\]), since $\Phi(-1,s,1/2)=2^s\beta(s)$, $E_k(1/2)=E_k/2^k$ and $E_{2k-1}=0$ for $k\in\mathbb{N}$, we obtain the following results for $\beta(s)$: **Theorem 3.13.** For $r\in\mathbb{N}$ and $|x|\leq c/2\,(x\in\mathbb{R}^1)$, we have $$\label{yr8b} \sum^\infty_{n=0}\frac{(-1)^n}{(2n+1)^{2r+1}}\cos\frac{(2n+1)\pi{x}}{c} =\sum^r_{k=0}\frac{(-1)^k}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}\beta(2r+1-2k);$$ $$\begin{aligned} \label{yr8'b} \sum^\infty_{n=0}\frac{(-1)^n}{(2n+1)^{2r}}\cos\frac{(2n+1)\pi{x}}{c} &=& \sum^r_{k=0}\frac{(-1)^k}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}\beta(2r-2k)\nonumber\\ & & +\sum^\infty_{k=1}\frac{(-1)^{r+k}E_{2k}}{2(2r+2k)!}\left(\frac{\pi{x}}{c}\right)^{2r+2k}.\end{aligned}$$ When $x=c/2$, the series relations (\[yr8b\]) and (\[yr8’b\]) immediately yields $$\label{yr8d} \beta(2r+1)=\sum^r_{k=1}\frac{(-1)^{k-1}}{(2k)!}\left(\frac{\pi}{2}\right)^{2k}\beta(2r+1-2k)\quad(r\in\mathbb{N}),$$ which gives $\beta(2r+1)$ recursively in terms of $\beta(3),\beta(5),\beta(7),\ldots,\beta(2r+1)$ using the initial value $\beta(1)=\pi/4$; $$\label{yr8'd} \beta(2r)=\sum^r_{k=1}\frac{(-1)^{k-1}}{(2k)!}\left(\frac{\pi}{2}\right)^{2k}\beta(2r-2k) +\sum^\infty_{k=1}\frac{(-1)^{r+k-1}E_{2k}}{2(2r+2k)!}\left(\frac{\pi}{2}\right)^{2r+2k},$$ which gives $\beta(2r)$ recursively in terms of $\beta(2),\beta(4),\beta(6),\ldots,\beta(2r)$ using the initial value $\beta(0)=1/2$. Since $E_{2k}\neq0$ in (\[yr8’d\]), this provides some perspective on the difficulty of evaluating $\beta(2r)$ as opposed to $\beta(2r+1)$. Letting $s=-n$ in (\[yr1a\]), since $\Phi(-1,-n,a)=E_n(a)/2$ and $$\label{elxz} E_n(x+y)=\sum^n_{k=0}{n\choose{k}}E_k(x)y^{n-k}\quad(n\in\mathbb{N}_0:=\mathbb{N}\cup\{0\}),$$ we obtain the following summation formula: **Theorem 3.14.** Let $a$ be any real number. For $m\in\mathbb{N}$ and $n\in\mathbb{N}_0$ $$\label{bfh} \sum^{m-1}_{k=0}(-1)^k(k+a)^n=\frac{1}{2}(E_n(a)-(-1)^mE_n(a+m)).$$ Hurwitz Zeta function --------------------- When $s=-n,\,n\in\mathbb{N}_0$ in (\[hurw\]), we obtain $$\label{hurn} \zeta(-n,a)=-\frac{1}{n+1}\frac{\partial^{n+1}}{\partial z^{n+1}}\!\left.\frac{ze^{az}}{e^z-1}\right|_{z=0}=-\frac{1}{n+1}B_{n+1}(a),$$ where $B_n(x)$ are the Bernoulli polynomials defined by the generating functions: $$\label{bnld} \frac{ze^{xz}}{e^z-1}=\sum^\infty_{n=0}B_n(x)\frac{z^n}{n!}\quad(|z|<2\pi).$$ It is easily seen from the definition (\[hurw\]) that $$\label{djd1} \lim_{s\rightarrow1}[(s-1)\zeta(s,a)]=\lim_{s\rightarrow1}\left(\frac{\partial}{\partial z}\right)^{1-s}\!\!\left.\frac{ze^{az}}{e^z-1}\right|_{z=0} =\left.\frac{ze^{az}}{e^z-1}\right|_{z=0}=B_0(a)=1.$$ In other words, $\zeta(s,a)$ has a simple pole at $s=1$, and $B_0(a)=1$ is the residue of $\zeta(s,a)$ at the simple pole $s=1$. Similarly, $\zeta(s)$ has a simple pole at $s=1$, and $B_0(1)=1$ is the residue of $\zeta(s)$ at the simple pole $s=1$. We can use (\[huzm\]) and (\[bnld\]) to deduce the following asymptotic expansion for $\zeta(s,a)$: **Theorem 3.15.** For $a\not\in\mathbb{Z}_0^-$, $s\in\mathbb{C}$ with $s\neq1$ $$\begin{aligned} \label{hurj} \zeta(s,a)\sim\sum^{m-1}_{n=0}\frac{1}{(n+a)^s} +\frac{1}{s-1}\sum^\infty_{k=0}{1-s\choose k}\frac{B_k(a)}{m^{s+k-1}}\quad(m\rightarrow\infty).\end{aligned}$$ Since $B_0(1)=1,\,B_1(1)=1/2$, and for $k\in\mathbb{N}$, $B_{2k+1}(1)=0,\,B_{2k}(1)=B_{2k}$, when $a=1$, the Theorem 3.15 immediately yields $$\label{zezk} \zeta(s)\sim\sum^{m-1}_{n=1}\frac{1}{n^s}+\frac{m^{1-s}}{s-1}+\frac{m^{-s}}{2} +\frac{1}{s-1}\sum^\infty_{k=1}{1-s\choose2k}\frac{B_{2k}}{m^{s+2k-1}}\quad(m\rightarrow\infty),$$ which is a slightly modified version of the following asymptotic expansion for $\zeta(s)$: $$\label{zezk'} \zeta(s)\sim\sum^{N-1}_{n=1}\frac{1}{n^s}+\frac{N^{1-s}}{s-1}+\frac{N^{-s}}{2} +N^{-s}\sum^\infty_{m=1}\frac{B_{2m}s^{\overline{2m-1}}}{(2m)!N^{2m-1}}\quad(N\rightarrow\infty),$$ where $B_{2m}$ are the Bernoulli numbers and $s^{\overline{2m-1}}$ is a rising factorial. This expansion is valid for all complex $s$ and is often used to compute the Zeta function by using a large enough value of $N$. **Theorem 3.16.** Let $a$ be any real number. For $m\in\mathbb{N}$ and $n\in\mathbb{N}_0$ $$\label{bfh"} \sum^{m-1}_{k=0}(k+a)^n=\frac{1}{n+1}(B_{n+1}(a+m)-B_{n+1}(a)).$$ **Proof.** Letting $s=-n$ in (\[hurj\]), it can be proved since $\zeta(-n,a)=-\frac{1}{n+1}B_{n+1}(a)$ and $$\label{bnhs} B_n(x+y)=\sum^n_{k=0}{n\choose{k}}B_k(x)y^{n-k}\quad(n\in\mathbb{N}_0).$$ **Theorem 3.17.** For $a\not\in\mathbb{Z}_0^-$, $s\in\mathbb{C}$ with $s\neq1$ $$\begin{aligned} \label{hurj'} \zeta(s,a)\!\!&\sim&\!\!\sum^{m-1}_{n=0}\frac{1}{(n+a)^s}+\frac{(m+a)^{1-s}}{s-1}+\frac{(m+a)^{-s}}{2}\nonumber\\ & & +\,\frac{1}{s-1}\sum^\infty_{k=1}{1-s\choose2k}\frac{B_{2k}}{(m+a)^{s+2k-1}}\quad(m\rightarrow\infty).\end{aligned}$$ **Proof.** By applying (\[e1\]) and (\[bnls\]) to (\[huzm\]), we have $$\begin{aligned} \zeta(s,a)&=&\sum^{m-1}_{n=0}\frac{1}{(n+a)^s}+\frac{1}{s-1}\left(m+a+\frac{\partial}{\partial z}\right)^{1-s}\!\!\left.\frac{z}{e^z-1}\right|_{z=0}\\ &\!\!\sim\!\!&\sum^{m-1}_{n=0}\frac{1}{(n+a)^s}+\frac{(m+a)^{1-s}}{s-1}\sum^\infty_{k=0}{1-s\choose{k}}\frac{1}{(m+a)^k} \frac{\partial^k}{\partial{z}^k}\left.\frac{z}{e^z-1}\right|_{z=0}\\ &=&\sum^{m-1}_{n=0}\frac{1}{(n+a)^s}+\frac{(m+a)^{1-s}}{s-1}\sum^\infty_{k=0}{1-s\choose{k}}\frac{B_k}{(m+a)^k}\quad(m\rightarrow\infty).\end{aligned}$$ Since $B_0=1,\,B_1=-1/2$, $B_{2k+1}=0$ for $k\in\mathbb{N}$, the Theorem 3.17 is proved. Let $s^{(n)}=s^{\overline{n}}:=s(s+1)(s+2)\cdots(s+n-1)$ be the rising factorial, then $$\frac{1}{s-1}{1-s\choose2k}=\frac{1}{s-1}{s-2+2k\choose2k}=\frac{1}{s-1}\frac{(s-1)^{\overline{2k}}}{(2k)!}=\frac{s^{\overline{2k-1}}}{(2k)!}\quad(s\neq1).$$ Thus the asymptotic expansion (\[hurj’\]) can be written in the form: $$\begin{aligned} \label{hurj"} \zeta(s,a)\!\!&\sim&\!\!\sum^{m-1}_{n=0}\frac{1}{(n+a)^s}+\frac{(m+a)^{1-s}}{s-1}+\frac{(m+a)^{-s}}{2}\nonumber\\ & & +\,\sum^\infty_{k=1}\frac{B_{2k}\,s^{\overline{2k-1}}}{(2k)!\,(m+a)^{s+2k-1}}\quad(m\rightarrow\infty).\end{aligned}$$ Letting $a=1$ in (\[hurj"\]) with $m$ replaced by $N-1$, since $ \zeta(s,1)= \zeta(s)$, we obtain the asymptotic expansion (\[zezk’\]) once again. Similarly, **Theorem 3.18.** For $s\in\mathbb{C},\,a\not\in\mathbb{Z}_0^-$ $$\begin{aligned} \label{yrsj} \Phi(-1,s,a)\!\!&\sim&\!\!\sum^{m-1}_{n=0}\frac{(-1)^n}{(n+a)^s}+\frac{(-1)^m}{2(m+a)^s}\nonumber\\ & & +\,\frac{(-1)^m}{2}\sum^\infty_{k=1}\frac{E_{2k-1}(1)\,s^{\overline{2k-1}}}{(2k-1)!\,(m+a)^{s+2k-1}}\quad(m\rightarrow\infty).\end{aligned}$$ By Definition 2.2 and Definition 2.4, we have $\left.\zeta_z(s,a)\right|_{z=0}=\zeta(s,a)$ and $$\label{wfze} \left.\frac{\partial^k}{\partial z^k}\zeta_z(s,a)\right|_{z=0}=\frac{s-1-k}{s-1}\zeta(s-k,a)\quad(k\in\mathbb{N},\,a\not\in\mathbb{Z}_0^-,\,s\neq1).$$ From the differential relation (\[wfze\]), we have for $k\in\mathbb{N},\,s\neq1$ $$\left.\frac{\partial^{2k-1}}{\partial x^{2k-1}}\cos\left(\frac{\pi x}{c}\frac{\partial}{\partial z}\right)\zeta_z(s,a)\right|_{z=0,x=0}=0;$$ $$\left.\frac{\partial^{2k}}{\partial x^{2k}}\cos\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)\zeta_z(s,a)\right|_{z=0,x=0} =(-1)^k\left(\frac{\pi}{c}\right)^{2k}\frac{s-1-2k}{s-1}\zeta(s-2k,a).$$ Here $c>0$ is a given real number. Therefore, we have the following Taylor expansion in the neighborhood of $x=0$: $$\label{taze} \left.\cos\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)\zeta_z(s,a)\right|_{z=0}=\sum^\infty_{k=0}\frac{(-1)^k}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k} \frac{s-1-2k}{s-1}\zeta(s-2k,a)\quad(s\neq1).$$ On the other hand, by using (\[rize"\]) and Theorem 1.4, since $$\left.\cos\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)e^{(a+n)z}\right|_{z=0}=\cos\frac{(a+n)\pi x}{c},$$ $$\left.\cos\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)(ze^{(a+n)z})\right|_{z=0} =-\frac{\pi{x}}{c}\sin\frac{(a+n)\pi x}{c},$$ we obtain the following summation formula for $\Re(s)>2,\,x\in\Omega\subset\mathbb{R}^1$: $$\begin{aligned} \label{sjqh} & &\left.\cos\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)\zeta_z(s,a)\right|_{z=0}\nonumber\\ &=&\sum^\infty_{n=0}\frac{1}{(a+n)^s}\cos\frac{(a+n)\pi x}{c}+\frac{\pi{x}/c}{s-1}\sum^\infty_{n=0}\frac{1}{(a+n)^{s-1}}\sin\frac{(a+n)\pi x}{c}.\end{aligned}$$ By means of the abstract operators $\cos(h\partial_x)$ and $\sin(h\partial_x)$, we can prove the following Katsurada’s series representation: Let $r\in\mathbb{N}$, $x\in\mathbb{R}^1$ with $-2c\leq{x}\leq2c$ ($c>0$ be a given real number). Then $$\begin{aligned} \label{yl8} & & r\sum^\infty_{n=1}\frac{1}{n^{2r+1}}\cos\frac{n\pi{x}}{c}+\frac{\pi x}{2c}\sum^\infty_{n=1}\frac{1}{n^{2r}}\sin\frac{n\pi x}{c}\nonumber\\ &=& \sum^{r-1}_{k=0}(-1)^k\,\frac{r-k}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}\zeta(2r+1-2k)\nonumber\\ & & +\,(-1)^{r-1}\sum^\infty_{k=0}\frac{(2k)!}{(2r+2k)!}\frac{\zeta(2k)}{(2\pi)^{2k}}\left(\frac{\pi{x}}{c}\right)^{2r+2k},\end{aligned}$$ which is a slightly modified version of a result proven in a significantly different way by Katsurada [@Kats p. 81, Theorem 1]. Since $$\zeta(2k)=(-1)^{k-1}\frac{(2\pi)^{2k}}{2(2k)!}B_{2k}\quad\mbox{and}\quad\zeta(-2k+1)=-\frac{1}{2k}B_{2k}\quad(k\in\mathbb{N}),$$ we can write the last term on the right-hand side of (\[yl8\]) in the form: $$\begin{aligned} & & \sum^\infty_{k=r}(-1)^k\,\frac{r-k}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}\zeta(2r+1-2k) \\ &=& \lim_{k\rightarrow0}a_k+\sum^\infty_{k=1}(-1)^{r+k}\,\frac{-k}{(2r+2k)!}\left(\frac{\pi{x}}{c}\right)^{2r+2k}\zeta(-2k+1) \\ &=& \sum^\infty_{k=0}(-1)^{r+k}\,\frac{B_{2k}}{2(2r+2k)!}\left(\frac{\pi{x}}{c}\right)^{2r+2k} \\ &=& (-1)^{r-1}\sum^\infty_{k=0}\frac{(2k)!}{(2r+2k)!}\frac{\zeta(2k)}{(2\pi)^{2k}}\left(\frac{\pi{x}}{c}\right)^{2r+2k},\end{aligned}$$ where $$\lim_{k\rightarrow0}a_k=\lim_{k\rightarrow0}\left[(-1)^{r+k}\,\frac{-k}{(2r+2k)!}\left(\frac{\pi{x}}{c}\right)^{2r+2k}\zeta(-2k+1)\right] =(-1)^r\,\frac{B_0}{2(2r)!}\left(\frac{\pi{x}}{c}\right)^{2r}.$$ Thus the Katsurada’s formula (\[yl8\]) can be written in the form: $$\begin{aligned} \label{yl8g} & & r\sum^\infty_{n=1}\frac{1}{n^{2r+1}}\cos\frac{n\pi{x}}{c}+\frac{\pi x}{2c}\sum^\infty_{n=1}\frac{1}{n^{2r}}\sin\frac{n\pi x}{c}\nonumber\\ &=& \sum^\infty_{k=0}(-1)^k\,\frac{r-k}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}\zeta(2r+1-2k)\quad(r\in\mathbb{N},\,|x|\leq2c).\end{aligned}$$ By making use of (\[taze\]), (\[sjqh\]) and (\[yl8g\]), we obtain the following theorem: **Theorem 3.19.** Let $\Re(s)>1$. For $x\in\mathbb{R}^1$ with $|x|\leq2c$, we have $$\begin{aligned} \label{zegh} & & s\sum^\infty_{n=0}\frac{1}{(a+n)^{s+1}}\cos\frac{(a+n)\pi x}{c}+\frac{\pi{x}}{c}\sum^\infty_{n=0}\frac{1}{(a+n)^s}\sin\frac{(a+n)\pi x}{c}\nonumber\\ &=& \sum^\infty_{k=0}(-1)^k\frac{s-2k}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}\zeta(s+1-2k,a)\quad(\Re(s)>1).\end{aligned}$$ By making use of (\[hurn\]) and (\[zegh\]), we obtain the following theorem: **Theorem 3.20.** For $x\in\mathbb{R}^1$ with $|x|\leq2c$, we have $$\begin{aligned} \label{zetl} & & r\sum^\infty_{n=0}\frac{1}{(a+n)^{2r+1}}\cos\frac{(a+n)\pi{x}}{c}+\frac{\pi x}{2c}\sum^\infty_{n=0}\frac{1}{(a+n)^{2r}}\sin\frac{(a+n)\pi x}{c}\nonumber\\ &=& \sum^{r-1}_{k=0}(-1)^k\,\frac{r-k}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}\zeta(2r+1-2k,a)\nonumber\\ & & +\,\sum^\infty_{k=0}(-1)^{r+k}\frac{B_{2k}(a)}{2(2r+2k)!}\left(\frac{\pi{x}}{c}\right)^{2r+2k}\quad(r\in\mathbb{N});\end{aligned}$$ $$\begin{aligned} \label{zetl'} & & (2r-1)\sum^\infty_{n=0}\frac{1}{(a+n)^{2r}}\cos\frac{(a+n)\pi{x}}{c}+\frac{\pi{x}}{c}\sum^\infty_{n=0}\frac{1}{(a+n)^{2r-1}}\sin\frac{(a+n)\pi{x}}{c}\nonumber\\ &=& \sum^{r-1}_{k=0}(-1)^k\,\frac{2r-1-2k}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}\zeta(2r-2k,a)\nonumber\\ & & +\,\sum^\infty_{k=0}(-1)^{r+k}\frac{B_{2k+1}(a)}{(2r+2k)!}\left(\frac{\pi{x}}{c}\right)^{2r+2k}\quad(r\in\mathbb{N}\setminus\{1\}).\end{aligned}$$ Since each series in (\[zegh\]) is uniformly convergent with respect to $x$ on the closed interval $[-2c,2c]$ for $\Re(s)>2$, multiplying the both side of (\[zegh\]) by $(\pi x/c)^{s-1}$, and executing termwise differentiation in them with respect to $x$, namely $\partial_x[(\pi x/c)^{s-1}\times(\ref{zegh})]$, then the sine series term is counteracted. Thus we can obtain the following theorem: **Theorem 3.21.** Let $\Re(s)>2$. For $x\in\mathbb{R}^1$ with $|x|\leq2c$, we have $$\begin{aligned} \label{zegh'} & &\sum^\infty_{n=0}\frac{s(s-1)}{(a+n)^{s+1}}\cos\frac{(a+n)\pi{x}}{c} +\left(\frac{\pi{x}}{c}\right)^2\sum^\infty_{n=0}\frac{1}{(a+n)^{s-1}}\cos\frac{(a+n)\pi{x}}{c}\nonumber\\ &=&\sum^\infty_{k=0}(-1)^k\frac{(s-2k)(s+2k-1)}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}\zeta(s+1-2k,a)\quad(\Re(s)>2).\end{aligned}$$ By making use of (\[hurn\]) and (\[zegh’\]), we obtain the following theorem: **Theorem 3.22.** Let $r\in\mathbb{N}\setminus\{1\}$. For $x\in\mathbb{R}^1$ with $|x|\leq2c$, we have $$\begin{aligned} \label{zetl"} & & \sum^\infty_{n=0}\frac{r(2r-1)}{(a+n)^{2r+1}}\cos\frac{(a+n)\pi{x}}{c} +\frac{1}{2}\left(\frac{\pi{x}}{c}\right)^2\sum^\infty_{n=0}\frac{1}{(a+n)^{2r-1}}\cos\frac{(a+n)\pi x}{c}\nonumber\\ &=& \sum^{r-1}_{k=0}(-1)^k\,\frac{(r-k)(2r+2k-1)}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}\zeta(2r+1-2k,a)\nonumber\\ & & +\,\sum^\infty_{k=0}(-1)^{r+k}\frac{(4r+2k-1)B_{2k}(a)}{2(2r+2k)!}\left(\frac{\pi{x}}{c}\right)^{2r+2k}.\end{aligned}$$ Dirichlet L-function -------------------- When $s=-n,\,n\in\mathbb{N}_0$ in (\[lfud\]), we obtain $$\label{hurn"} L(-n,\chi)=-\frac{1}{n+1}\frac{\partial^{n+1}}{\partial z^{n+1}}\sum^q_{k=1}\left.\chi(k)\frac{ze^{kz}}{e^{qz}-1}\right|_{z=0}=-\frac{1}{n+1}B_{n+1,\chi},$$ where $B_{n,\chi}$ are the generalized Bernoulli numbers defined by the generating functions: $$\label{bnld"} \sum^q_{k=1}\chi(k)\frac{ze^{kz}}{e^{qz}-1}=\sum^\infty_{n=0}B_{n,\chi}\frac{z^n}{n!}\quad(|z|<2\pi/q).$$ It is easily seen from the definition (\[lfud\]) that $$\label{djd2} \lim_{s\rightarrow1}[(s-1)L(s,\chi)]=\sum^q_{k=1}\left.\chi(k)\frac{ze^{kz}}{e^{qz}-1}\right|_{z=0}=B_{0,\chi}.$$ Therefore, if $\chi$ is principal, then the corresponding Dirichlet L-function $L(s,\chi)$ has a simple pole at $s=1$, and $B_{0,\chi}\neq0$ is the residue of $L(s,\chi)$ at the simple pole $s=1$. By applying (\[e1\]) and (\[bnld"\]) to (\[lfjz\]), since $$\frac{1}{s-1}{1-s\choose{k}}=\frac{(-1)^k}{s-1}{s-2+k\choose{k}}=\frac{(-1)^k(s-1)^{\overline{k}}}{k!\,(s-1)}=(-1)^k\frac{s^{\overline{k-1}}}{k!}\quad(s\neq1),$$ we obtain the following asymptotic expansion for $L(s,\chi)$: **Theorem 3.23.** For $s\in\mathbb{C}$ with $s\neq1$ $$\label{lfjz'} L(s,\chi)\sim\sum^{qm}_{n=1}\frac{\chi(n)}{n^s}+\frac{(qm)^{1-s}}{s-1}B_{0,\chi} +\sum^\infty_{k=1}(-1)^k\frac{B_{k,\chi}\,s^{\overline{k-1}}}{k!\,(qm)^{s+k-1}}\quad(m\rightarrow\infty).$$ It’s easy to see that (\[lfjz’\]) can be written in the form: $$\label{lfjz"} L(s,\chi)\sim\sum^{qm}_{n=1}\frac{\chi(n)}{n^s}+\frac{(qm)^{1-s}}{s-1} \sum^\infty_{k=0}{1-s\choose{k}}\frac{B_{k,\chi}}{(qm)^k}\quad(m\rightarrow\infty).$$ Letting $s=-n,\,n\in\mathbb{N}_0$ in (\[lfjz"\]), since $L(-n,\chi)=-\frac{1}{n+1}B_{n+1,\chi}$, we obtain the following summation formula: **Theorem 3.24.** For $n\in\mathbb{N}_0,\,m\in\mathbb{N}$ $$\label{lfqh} \sum^{qm}_{k=1}\chi(k)k^n=\frac{1}{n+1}\sum^n_{k=0}{n+1\choose{k}}(qm)^{n+1-k}B_{k,\chi}.$$ When $m=1$, we obtain the following recurrent formula of generalized Bernoulli numbers: $$\label{lfqh'} \frac{1}{n+1}\sum^n_{k=0}{n+1\choose{k}}q^{n+1-k}B_{k,\chi}=\sum^q_{k=1}\chi(k)k^n\quad(n\in\mathbb{N}_0),$$ which gives $B_{n,\chi}$ recursively in terms of $B_{0,\chi},B_{1,\chi},B_{2,\chi},\ldots,B_{n,\chi}$. By Definition 2.5 and Definition 2.6, we have $\left.L_z(s,\chi)\right|_{z=0}=L(s,\chi)$ and $$\label{wfld} \left.\frac{\partial^k}{\partial z^k}L_z(s,\chi)\right|_{z=0}=\frac{s-1-k}{s-1}L(s-k,\chi)\quad(k\in\mathbb{N},\,s\neq1).$$ From the differential relation (\[wfld\]), we have for $k\in\mathbb{N},\,s\neq1$ $$\left.\frac{\partial^{2k-1}}{\partial x^{2k-1}}\cos\left(\frac{\pi x}{c}\frac{\partial}{\partial z}\right)L_z(s,\chi)\right|_{z=0,x=0}=0;$$ $$\left.\frac{\partial^{2k}}{\partial x^{2k}}\cos\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)L_z(s,\chi)\right|_{z=0,x=0} =(-1)^k\left(\frac{\pi}{c}\right)^{2k}\frac{s-1-2k}{s-1}L(s-2k,\chi).$$ Here $c>0$ is a given real number. Therefore, we have the following Taylor expansion in the neighborhood of $x=0$: $$\label{tald} \left.\cos\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)L_z(s,\chi)\right|_{z=0}=\sum^\infty_{k=0}\frac{(-1)^k}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k} \frac{s-1-2k}{s-1}L(s-2k,\chi)\quad(s\neq1).$$ Letting $s=2r+1,\,r\in\mathbb{N}$ in (\[tald\]), since $L(-n, \chi)=-\frac{1}{n+1}B_{n+1,\chi},\,n\in\mathbb{N}_0$, we have $$\begin{aligned} \label{ldcs} & & \left.\cos\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)L_z(2r+1,\chi)\right|_{z=0}\nonumber\\ &=& \frac{1}{r}\sum^{r-1}_{k=0}(-1)^k\,\frac{r-k}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}L(2r+1-2k,\chi)\nonumber\\ & & +\,\frac{1}{2r}\sum^\infty_{k=0}(-1)^{r+k}\frac{B_{2k,\chi}}{(2r+2k)!}\left(\frac{\pi{x}}{c}\right)^{2r+2k}\quad(r\in\mathbb{N}).\end{aligned}$$ By comparing (\[ldcs\]) and (\[bnld"\]), we have $|x|\leq2c/q$ in (\[ldcs\]) for $r\in\mathbb{N}$, and thus $|x|\leq2c/q$ in (\[tald\]). On the other hand, by using (\[rild"\]) and Theorem 1.4, we obtain the following summation formula for $\Re(s)>2,\,x\in\Omega\subset\mathbb{R}^1$: $$\label{sjld} \left.\cos\left(\frac{\pi{x}}{c}\frac{\partial}{\partial{z}}\right)L_z(s,\chi)\right|_{z=0} =\sum^\infty_{n=1}\frac{\chi(n)}{n^s}\cos\frac{n\pi x}{c}+\frac{\pi{x}/c}{s-1}\sum^\infty_{n=1}\frac{\chi(n)}{n^{s-1}}\sin\frac{n\pi x}{c}.$$ By making use of (\[tald\]) and (\[sjld\]), we obtain the following theorem: **Theorem 3.25.** Let $\Re(s)>1$. For $x\in\mathbb{R}^1$ with $|x|\leq2c/q$, we have $$\begin{aligned} \label{ldgh} & & s\sum^\infty_{n=1}\frac{\chi(n)}{n^{s+1}}\cos\frac{n\pi x}{c}+\frac{\pi{x}}{c}\sum^\infty_{n=1}\frac{\chi(n)}{n^s}\sin\frac{n\pi x}{c}\nonumber\\ &=& \sum^\infty_{k=0}(-1)^k\frac{s-2k}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}L(s+1-2k,\chi)\quad(\Re(s)>1).\end{aligned}$$ By making use of (\[hurn"\]) and (\[ldgh\]), we obtain the following theorem: **Theorem 3.26.** For $x\in\mathbb{R}^1$ with $|x|\leq2c/q$, we have $$\begin{aligned} \label{ldtl} & & r\sum^\infty_{n=1}\frac{\chi(n)}{n^{2r+1}}\cos\frac{n\pi{x}}{c}+\frac{\pi x}{2c}\sum^\infty_{n=1}\frac{\chi(n)}{n^{2r}}\sin\frac{n\pi x}{c}\nonumber\\ &=& \sum^{r-1}_{k=0}(-1)^k\,\frac{r-k}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}L(2r+1-2k,\chi)\nonumber\\ & & +\,\sum^\infty_{k=0}(-1)^{r+k}\frac{B_{2k,\chi}}{2(2r+2k)!}\left(\frac{\pi{x}}{c}\right)^{2r+2k}\quad(r\in\mathbb{N});\end{aligned}$$ $$\begin{aligned} \label{ldtl'} & & (2r-1)\sum^\infty_{n=1}\frac{\chi(n)}{n^{2r}}\cos\frac{n\pi{x}}{c}+\frac{\pi x}{c}\sum^\infty_{n=1}\frac{\chi(n)}{n^{2r-1}}\sin\frac{n\pi x}{c}\nonumber\\ &=& \sum^{r-1}_{k=0}(-1)^k\,\frac{2r-1-2k}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}L(2r-2k,\chi)\nonumber\\ & & +\,\sum^\infty_{k=0}(-1)^{r+k}\frac{B_{2k+1,\chi}}{(2r+2k)!}\left(\frac{\pi{x}}{c}\right)^{2r+2k}\quad(r\in\mathbb{N}\setminus\{1\}).\end{aligned}$$ Since each series in (\[ldgh\]) is uniformly convergent with respect to $x$ on the closed interval $[-2c/q,2c/q]$ for $\Re(s)>2$, multiplying the both side of (\[ldgh\]) by $(\pi x/c)^{s-1}$, and executing termwise differentiation in them with respect to $x$, namely $\partial_x[(\pi x/c)^{s-1}\times(\ref{ldgh})]$, then the sine series term is counteracted. Thus we can obtain the following theorem: **Theorem 3.27.** Let $\Re(s)>2$. For $x\in\mathbb{R}^1$ with $|x|\leq2c/q$, we have $$\begin{aligned} \label{ldgh'} & &s(s-1)\sum^\infty_{n=1}\frac{\chi(n)}{n^{s+1}}\cos\frac{n\pi{x}}{c} +\left(\frac{\pi{x}}{c}\right)^2\sum^\infty_{n=1}\frac{\chi(n)}{n^{s-1}}\cos\frac{n\pi{x}}{c}\nonumber\\ &=&\sum^\infty_{k=0}(-1)^k\frac{(s-2k)(s+2k-1)}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}L(s+1-2k,\chi).\end{aligned}$$ By making use of (\[hurn"\]) and (\[ldgh’\]), we obtain the following theorem: **Theorem 3.28.** For $x\in\mathbb{R}^1$ with $|x|\leq2c/q$, we have $$\begin{aligned} \label{ldhu} & & r(2r-1)\sum^\infty_{n=1}\frac{\chi(n)}{n^{2r+1}}\cos\frac{n\pi{x}}{c} +\frac{1}{2}\left(\frac{\pi x}{c}\right)^2\sum^\infty_{n=1}\frac{\chi(n)}{n^{2r-1}}\cos\frac{n\pi x}{c}\nonumber\\ &=& \sum^{r-1}_{k=0}(-1)^k\,\frac{(r-k)(2r+2k-1)}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}L(2r+1-2k,\chi)\nonumber\\ & & +\,\sum^\infty_{k=0}(-1)^{r+k}\frac{(4r+2k-1)B_{2k,\chi}}{2(2r+2k)!}\left(\frac{\pi{x}}{c}\right)^{2r+2k}\quad(r\in\mathbb{N}\setminus\{1\});\end{aligned}$$ $$\begin{aligned} \label{ldhu'} & & r(2r+1)\sum^\infty_{n=1}\frac{\chi(n)}{n^{2r+2}}\cos\frac{n\pi{x}}{c} +\frac{1}{2}\left(\frac{\pi x}{c}\right)^2\sum^\infty_{n=1}\frac{\chi(n)}{n^{2r}}\cos\frac{n\pi x}{c}\nonumber\\ &=& \sum^r_{k=0}(-1)^k\,\frac{(2r+1-2k)(r+k)}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}L(2r+2-2k,\chi)\nonumber\\ & & +\,\sum^\infty_{k=0}(-1)^{r+k+1}\frac{(2r+k+1)B_{2k+1,\chi}}{(2r+2k+2)!}\left(\frac{\pi{x}}{c}\right)^{2r+2k+2}\quad(r\in\mathbb{N}).\end{aligned}$$ Riemann Zeta function --------------------- In the special case of Theorem 3.22 when $a=1$, since $\zeta(s,1)=\zeta(s)$, we obtain the following series representation: **Theorem 3.29.** Let $r\in\mathbb{N}$, $x\in\mathbb{R}^1$ with $-2c\leq{x}<2c$ (or $-2c\leq x\leq2c$ for $r\in\mathbb{N}\setminus\{1\}$). Then $$\begin{aligned} \label{yl15} & & r(2r-1)\sum^\infty_{n=1}\frac{1}{n^{2r+1}}\cos\frac{n\pi{x}}{c} +\frac{1}{2}\left(\frac{\pi x}{c}\right)^2\sum^\infty_{n=1}\frac{1}{n^{2r-1}}\cos\frac{n\pi x}{c}\nonumber\\ &=& \sum^{r-1}_{k=0}(-1)^k\frac{(r-k)(2r+2k-1)}{(2k)!}\left(\frac{\pi{x}}{c}\right)^{2k}\zeta(2r+1-2k)\nonumber\\ & & +\,(-1)^{r-1}\sum^\infty_{k=0}\frac{(2k)!(4r+2k-1)}{(2r+2k)!}\frac{\zeta(2k)}{(2\pi)^{2k}}\left(\frac{\pi{x}}{c}\right)^{2r+2k}.\end{aligned}$$ Since $L(s,\chi)=\zeta(s)$ when $q=1$, we can also obtain the Theorem 3.29 from (\[ldhu\]). In particular, by making use of (\[yl15\]) and Lemma 2.4, we can obtain the following important results: **Theorem 3.30.** For $r\in\mathbb{N}\setminus\{1\}$ $$\begin{aligned} \label{yl21} \zeta(2r+1)&=& -\frac{2\pi^2(3^{2r-2}-1)}{r(2r-1)(3^{2r+1}-1)}\,\zeta(2r-1)+\frac{2\times3^{2r}}{r(2r-1)(3^{2r+1}-1)}\nonumber\\ & & \times\left[\,\sum^{r-1}_{k=1}(-1)^{k-1}\frac{(r-k)(2r+2k-1)}{(2k)!}\left(\frac{2\pi}{3}\right)^{2k}\zeta(2r+1-2k)\right.\nonumber\\ & & \left.+\,(-1)^r\left(\frac{2\pi}{3}\right)^{2r}\sum^\infty_{k=0}\frac{(2k)!(4r+2k-1)}{(2r+2k)!}\frac{\zeta(2k)}{3^{2k}}\right];\end{aligned}$$ $$\begin{aligned} \label{yl22} \zeta(2r+1)&=& -\frac{(2^{2r-1}-2)\pi^2}{r(2r-1)(2^{4r+1}+2^{2r}-1)}\,\zeta(2r-1)+\frac{2^{4r+1}}{r(2r-1)(2^{4r+1}+2^{2r}-1)}\nonumber\\ & & \times\left[\,\sum^{r-1}_{k=1}(-1)^{k-1}\frac{(r-k)(2r+2k-1)}{(2k)!}\left(\frac{\pi}{2}\right)^{2k}\zeta(2r+1-2k)\right.\nonumber\\ & & \left.+\,(-1)^r\left(\frac{\pi}{2}\right)^{2r}\sum^\infty_{k=0}\frac{(2k)!(4r+2k-1)}{(2r+2k)!}\frac{\zeta(2k)}{4^{2k}}\right];\end{aligned}$$ $$\begin{aligned} \label{yl23} \zeta(2r+1)&=& \frac{2\pi^2(6^{2r-2}-3^{2r-2}-2^{2r-2}+1)}{r(2r-1)(3^{2r}(2^{2r}+1)+2^{2r}-1)}\,\zeta(2r-1)\nonumber\\ & & +\,\frac{2\times6^{2r}}{r(2r-1)(3^{2r}(2^{2r}+1)+2^{2r}-1)}\nonumber\\ & & \times\left[\,\sum^{r-1}_{k=1}(-1)^{k-1}\frac{(r-k)(2r+2k-1)}{(2k)!}\left(\frac{\pi}{3}\right)^{2k}\zeta(2r+1-2k)\right.\nonumber\\ & & \left.+\,(-1)^r\left(\frac{\pi}{3}\right)^{2r}\sum^\infty_{k=0}\frac{(2k)!(4r+2k-1)}{(2r+2k)!}\frac{\zeta(2k)}{6^{2k}}\right].\end{aligned}$$ For example, letting $r=1,2,3,\cdots$ in (\[yl23\]), we have $$\begin{aligned} \zeta(3) &=& -\frac{\pi^2}{6}\sum^\infty_{k=0}\frac{2k+3}{(2k+1)(2k+2)}\frac{\zeta(2k)}{6^{2k}}, \\ \zeta(5) &=& \frac{8\pi^2}{87}\,\zeta(3)+\frac{\pi^4}{261}\sum^\infty_{k=0}\frac{(2k+7)\zeta(2k)}{(2k+1)(2k+2)\cdots(2k+4)6^{2k}},\\ \zeta(7) &=& \frac{3124\pi^2}{29655}\,\zeta(5)-\frac{2\pi^4}{3295}\,\zeta(3) -\frac{16\pi^6}{88965}\sum^\infty_{k=0}\frac{(2k+11)\zeta(2k)}{(2k+1)(2k+2)\cdots(2k+6)6^{2k}},\\ &\cdots&,\end{aligned}$$ recursively. Since $\zeta(2k)\rightarrow1$ as $k\rightarrow\infty$, each of these series representing $\zeta(2r+1)$ in (\[yl21\]) to (\[yl23\]) converges remarkably rapidly with its general term having the order estimate: $$O(m^{-2k}\cdot k^{-2r+1})\qquad(k\rightarrow\infty;\quad m=3,4,6;\quad r\in\mathbb{N}).$$ [9]{} G.-Q. Bi, Solutions of Cauchy problem for multiple inhomogeneous wave equation, preprint, 2018, arXiv:1806.04817v1 \[math.AP\]. G.-Q. Bi, Applications of abstract operators to partial differential equations, *Pure and Applied Mathematics* **13**(1) (1997), 7-14 (Chinese) G.-Q. Bi, Applications of abstract operator in partial differential equation ([2]{}), *Chinese Quarterly Journal of Mathematic* **14**(3) (1999), 80-87 G.-Q. Bi, Y.-K. Bi, Abstract operators and higher-order linear partial differential equation, *Chinese Quarterly Journal of Mathematics* **26** (2011), 511-515 M. Katsurada, Rapidly convergent series representations for $\zeta(2n+1)$ and their $\chi$-analogue, *ACTA ARITHMETICA* **90** (1999), 79-89 H. Tsumura, On Evaluation of the Dirichlet Series at Positive Integers by $q$-Calculation, *J. Number Theory* **48** (1994), 383-391 J. Choi, Rapidly converging series for $\zeta(2n+1)$ from Fourier series, *Abstract and Applied Analysis* Volume 2014, Article ID 457620, 9 pages G.-Q. Bi, Y.-K. Bi, New Properties of Fourier Series and Riemann Zeta Function, preprint, 2010, arXiv:1008.5046 \[math.AP; math.NT\]. Michael J. Dancs, T.-X. He, An Euler-type formula for $\zeta(2k+1)$, *J. Number Theory* **118** (2006), 192-199 Yan’an Second School, Yan’an 716000, Shaanxi, PR China *E-mail address*: guangqingbi@sohu.com

--- abstract: 'We examine the possibility of positive spectral index of the power spectrum of the primordial tensor perturbation produced during inflation in the light of the detection of the $B$-mode polarization by the BICEP2 collaboration. We find a blue tilt is in general possible when the slow-roll parameter decays rapidly. We present two known examples in which a positive spectral index for the tensor power spectrum can be obtained. We also briefly discuss other consistency tests for further studies on inflationary dynamics.' --- 1.0cm **Blue running of the\ primordial tensor spectrum** 1.0cm 0.5cm 1.2cm Introduction ============ Cosmic inflation is considered as the leading candidate to resolve otherwise finely tuned initial conditions of the hot big bang cosmology such as the horizon problem [@inflation], and at the same time to provide the origin of large scale inhomogeneities [@Mukhanov:1981xt]. Many properties of the seed perturbation, the curvature perturbation ${{\cal R}}$ produced during inflation, are dramatically constrained by the observations on the temperature fluctuations of the cosmic microwave background (CMB). From the most recent Planck data, we have very good constraints on the amplitude of the power spectrum ${{\cal P}}_{{\cal R}}$, its spectral index $n_{{\cal R}}$, and so on [@Ade:2013zuv]. The predictions of the simplest chaotic inflation model [@Linde:1983gd] are mostly consistent with the observations, but the expected tensor-to-scalar ratio $r \sim 0.15$ was in tension with the Planck constraint $r < 0.11$ [@Ade:2013uln]. Instead, $R^2$ inflation [@Starobinsky:1980te] and related theories such as the standard model Higgs inflation [@alstar_ext] have attracted a lot of attention, as $r$ is very small in these scenarios. Recently, however, the BICEP2 experiment announced the detection of the $B$-mode polarization of the CMB in the range $30 < \ell < 150$. At 1$\sigma$ confidence level, the corresponding value of $r$ reads [@Ade:2014xna] $$r = 0.20^{+0.07}_{-0.05} \, ,$$ with $r = 0$ disfavoured at 7$\sigma$. Subtracting foreground dust modifies the allowed range slightly, $r = 0.16^{+0.06}_{-0.05}$ but still $r = 0$ is strongly disfavoured at $5.9\sigma$. This is in good agreement with the prediction of simple chaotic inflation, indicating that the energy scale during inflation is as high as $10^{16}$ GeV. Thus our next task should be, with more data available soon, to examine more closely the predictions of the models that survive the current constraints. In this regard, a tantalizing feature is that the BICEP2 $BB$ auto spectrum $C_\ell^{BB}$ exhibits a bump. This may well be due to lack of more data or systematics, and may disappear after the Keck Array data analysis is completed. This feature, however, could be real as well: in that case, the power spectrum of the primordial tensor perturbation has a [*blue*]{} tilt: $n_T \sim 0.50 \pm 0.25$. The spectrum of tensor perturbation may have a small blue tilt in some string theory motivated models [@stringgas], but in the standard inflationary scenario the tilt is given by $n_T = -2\epsilon$, with $\epsilon \equiv -\dot{H}/H^2 > 0$ being the slow-roll parameter, and thus appears to be always red (see however [@Contaldi:2014zua]). In this article we examine the running of the power spectrum of tensor perturbation and explore the possibility of a positive $n_T$. Blue spectral index =================== Our starting point is to recall a more accurate formula for the spectral index $n_T$ of the tensor power spectrum ${{\cal P}}_T$. In the general slow-roll formalism [@gsr], with “general” in the sense that we generalize the standard assumption for the slow-roll approximation in such a way that the time derivatives of $\epsilon$ need not be smaller than $\epsilon$ itself, $n_T$ can be readily calculated by taking a logarithmic derivative of $k$ of ${{\cal P}}_T$ [@Gong:2004kd]. Evaluated at the moment of horizon crossing, up to second order corrections it is given by $$\label{nT-gsr} n_T = 2\frac{p'}{p} + 2\alpha \left( \frac{p'}{p} \right)' + 2 (4-\pi) \frac{p'p''}{p^2} \, ,$$ where $p = p(\log{x}) \equiv 2\pi xa/k$ with $x \equiv -k\eta$, a prime denotes a derivative with respect to $\log{x}$, and $\alpha \equiv 2-\log2-\gamma \approx 0.729637$, with $\gamma \approx 0.577216$ being the Euler-Mascheroni constant. Requiring (\[nT-gsr\]) be positive leads to the condition on the fundamental general slow-roll function $p$, $$\label{p-condition} \frac{p'}{p} \gtrsim 1 \, .$$ Once this condition is satisfied we can have a blue tensor spectral index. It is illustrative to make use of the slow-roll parameters to understand what (\[p-condition\]) means. Writing $p'/p$ in terms of the slow-roll parameters [@Gong:2004kd], we can easily find $$\label{condition2} \epsilon \sim a^{-1} \, ,$$ that is, $\epsilon$ is rapidly decaying proportional to $1/a$ or even faster. This means another widely used slow-roll parameter, $$\label{def:eta} \eta \equiv \frac{\dot\epsilon}{H\epsilon} \, ,$$ is negatively large. If this condition is satisfied, during that period $\eta$ contribution can be larger than the leading term $-2\epsilon$ and make $n_T > 0$. Of course this period cannot last too long time otherwise inflation does not terminate. We need to provide a mechanism which leads to a graceful exit, as is obvious from the $BB$ spectrum from BICEP2 where the bump seems to disappear on smaller scales. Before closing this section, we note that in [@Gong:2004kd] $n_T$ is presented in a form which can be evaluated at any convenient point of evaluation, denoted by $\star$, in such a way that $$\label{def:alphastar} \alpha \to \alpha_\star \equiv \alpha - \log \left( -k\eta_\star \right) \, ,$$ with the slow-roll parameters being evaluated at the same moment. One may doubt that the value of $n_T$ is different depending on when it is evaluated. But it is not: as can be read from the integral formula of ${{\cal P}}_T$ in [@Gong:2004kd] computed in the context of the general slow-roll formalism [@gsr], the power spectrum is independent of the evaluation point $\star$, so is its spectral index $n_T$. As we can see from (\[def:alphastar\]), when we evaluate at the moment of horizon crossing, or more precisely $-k\eta_\star=1$ which is different from horizon crossing by ${{\cal O}}(\epsilon)$, the logarithmic factor disappears and the calculation becomes most transparent. Working examples ================ Now we consider two simple examples in which the condition for a blue spectral index $n_T$ can be accomplished. We however do not attempt extensive BICEP2 data analysis or realistic model building here, which are both beyond the scope of this short article. From (\[condition2\]), we can see that typical examples include the models with very flat inflaton potential $V(\phi)$. Let us consider a canonical inflaton field evolving on a constant potential $V(\phi) = V_0$ [@usr]. We can solve trivially the equation of motion for the background inflaton field and find $\dot\phi \propto a^{-3}$, and thus the slow-roll parameters are given by $$\begin{split} \epsilon & \propto a^{-6} \, , \\ \eta & = -6 \, . \end{split}$$ Thus (\[condition2\]) is satisfied, and the tensor power spectrum during this stage exhibits a blue tilt. This picture, however, is plagued by the graceful exit problem and we should provide a mechanism that leads to the end of inflation [@usr; @Namjoo:2012aa]. Note that in this model, there are other interesting phenomenology. For example, the local non-linear parameter is given by [@Namjoo:2012aa] $${f_{\rm NL}}= \frac{5}{2} \, ,$$ which interestingly violates the consistency relation for the bispectrum of the curvature perturbation in the squeezed configuration [@ngconsistency]. There are other similar consistency relations which include tensor perturbation: for example, the three-point correlation function of two tensor perturbations $h_s(\mathbi{k})$ with polarization $s$ and one curvature perturbation in the squeezed limit reads [@ngconsistency] $$\left\langle h_s(\mathbi{k}_1)h_{s'}(\mathbi{k}_2){{\cal R}}(\mathbi{k}_3) \right\rangle \underset{k_3\to0}{\longrightarrow} (2\pi)^3 \delta^{(3)}(\mathbi{k}_1+\mathbi{k}_2+\mathbi{k}_3) P_T(k_1) P_{{\cal R}}(k_3) n_T \delta_{ss'} \, ,$$ where $P_i(k) \equiv 2\pi^2{{\cal P}}_i(k)/k^3$ with $i={{\cal R}}$ and $T$. In the other example the inflaton potential is given by [@Jain:2008dw] $$V(\phi) = \frac{1}{2}m^2\phi^2 - \frac{\sqrt{2\lambda(n-1)}}{n} m{m_{\rm Pl}}^3 \left( \frac{\phi}{{m_{\rm Pl}}} \right)^n + \frac{1}{4}\lambda{m_{\rm Pl}}^4 \left( \frac{\phi}{{m_{\rm Pl}}} \right)^{2(n-1)} \, ,$$ with $n>2$ being an integer, which can arise in certain minimal supersymmetric extensions of the standard model [@Allahverdi:2006iq]. This potential allows, depending on the combination of the three model parameters $m$, $\lambda$ and $n$, a brief period of departure from inflation sandwiched between two stages of slow-roll inflation. (\[condition2\]) is satisfied during the intermediate stage and the tensor spectrum shows a blue tilt. However, as can be seen from the explicit illustrations of Figure 2 in [@Jain:2009pm], much more prominent is the enhancement in $r$ due to the suppression of ${{\cal P}}_{{\cal R}}$, which can be as large as $r\gg1$. For similar studies due to fast-roll phase, see e.g. [@fastroll]. Discussions and conclusions =========================== In this article we have examined whether the power spectrum of the primordial tensor perturbation ${{\cal P}}_T$ may have a non-trivial running $n_T$ in such a way that it is blue in some range as suggested by the observed BICEP2 $BB$ spectrum $C_\ell^{BB}$. This is possible when the slow-roll parameter $\eta$ is negatively large. Practically, however, we can only constrain $n_T$ as a whole, not the individual contribution. This is especially the case when we consider models, for example, with feature because the CMB power spectrum has strong degeneracies [@Gong:2005jr]. An interesting but simple way to disentangle different contributions, in particular to eliminate $\epsilon$, is the running of $r$ [@Gong:2007ha]. For a few interesting cases, we find $$\frac{d\log{r}}{d\log{k}} = 1-n_{{\cal R}}+n_T = \left\{ \begin{array}{ll} \eta & \text{ for canonical single field,} \\ \eta+s & \text{ for general single field,} \\ \eta_\text{multi} + 2 \dfrac{N_{,i}N_{,j}}{G^{kl}N_{,k}N_{,l}} \dfrac{R^i{}_{ab}{}^j}{3{m_{\rm Pl}}^2} \dfrac{\dot\phi^a\dot\phi^b}{H^2} & \text{ for multi-field,} \end{array} \right.$$ where $c_s$ is the speed of sound which may be non-trivial [@heavy], $s \equiv \dot{c}_s/(Hc_s)$, $N_{,i}$ is the derivative of the $e$-fold $N$ with respect to $\phi^i$, $R^i{}_{jkl}$ is the Riemann curvature tensor constructed from the field space metric $G_{ij}$, and[^1] [@deltaN] $$\eta_\text{multi} = \frac{r}{4} - 2{m_{\rm Pl}}^2 \frac{N_{,i}N_{,j}}{G^{kl}N_{,k}N_{,l}} \frac{V^{;ij}}{V} \, ,$$ with a semicolon denoting a covariant derivative in the field space. Combined with the consistency relation for the local ${f_{\rm NL}}$, we can build a web of consistency checks and should be able to correlate one signal to another explicitly [@corr-corr]. To summarize, we have considered the general possibility $n_T > 0$ in the context of standard inflationary models. This can be accomplished if the slow-roll parameter $\epsilon$ is exponentially decaying during inflation. We have presented two simple examples where this condition is satisfied. Combined with other consistency tests including the running of $r$, we can further study various aspects of inflationary dynamics. Acknowledgements {#acknowledgements .unnumbered} ---------------- I acknowledge the Max-Planck-Gesellschaft, the Korea Ministry of Education, Science and Technology, Gyeongsangbuk-Do and Pohang City for the support of the Independent Junior Research Group at the Asia Pacific Center for Theoretical Physics. 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--- author: - Pavel Exner title: | An isoperimetric problem for leaky loops\ and related mean-chord inequalities --- [*Department of Theoretical Physics, Nuclear Physics Institute,\ Academy of Sciences, 25068 Řež near Prague, Czechia, and\ Doppler Institute, Czech Technical University, Břehov[á]{} 7,\ 11519 Prague, Czechia\ exner@ujf.cas.cz*]{} > [We consider a class of Hamiltonians in $L^2({\mathbb{R}}^2)$ with attractive interaction supported by piecewise $C^2$ smooth loops $\Gamma$ of a fixed length $L$, formally given by $-\Delta-\alpha\delta(x-\Gamma)$ with $\alpha>0$. It is shown that the ground state of this operator is locally maximized by a circular $\Gamma$. We also conjecture that this property holds globally and show that the problem is related to an interesting family of geometric inequalities concerning mean values of chords of $\Gamma$. ]{} Introduction ============ There is a small number of topics which can be regarded as trademark for mathematical physics. One of them without any doubt concerns relations between geometric properties of constraints and/or interaction and extremal values of a spectral quantity; classical examples are Faber-Krahn inequality [@Fa; @Kr] or the PPW-conjecture proved by Ashbaugh and Benguria [@AB]. A common feature of these and analogous problems is that the extremum is reached by shapes having a rotational symmetry. At the same time, the nature of the extremum may be different. While a ball *minimizes* the principal eigenvalue of the Dirichlet Laplacian among regions of a fixed volume, for non-simply connected regions like annular strips or layers considered in [@EHL; @HKK], built over a curve (surface) of a fixed length (area), the circular shape is on the contrary a *maximizer*. A natural topological way to understand this difference becomes smeared, however, when the particle is not localized by boundary conditions but by a potential, a regular or singular one. In this paper we consider such a problem associated with a class of operators in $L^2({\mathbb{R}}^2)$ which are given formally by the expression $$\label{formal} H_{\alpha,\Gamma}= -\Delta-\alpha\delta(x-\Gamma)\,,$$ where $\alpha>0$ and $\Gamma$ is a $C^2$ loop in the plane (see below for exact assumptions) having a fixed length $L>0$. A motivation to study these operators comes from the theory of *leaky quantum graphs* – see [@BT; @EI] and related papers, a bibliography can be found in [@AGHH2] – aiming at a more realistic model of quantum wire structures which would take quantum tunneling into account. Our aim is to show that the ground-state energy of $H_{\alpha,\Gamma}$ is (sharply) maximized when $\Gamma$ is a circle. We will be able to prove that this property holds *locally* conjecturing its global validity. There are several reasons why one may expect this result to be valid. On one hand, we know from [@EY] that in the limit of strong coupling, $\alpha \to\infty$, the ground-state dependence on $\Gamma$ is given in the leading order by the lowest eigenvalue of the operator $-\frac{{\mathrm{d}}^2}{{\mathrm{d}}s^2}- \frac14 \gamma(s)^2$ on $L^2([0,L])$ with periodic boundary conditions where $\gamma$ is the curvature of $\Gamma$, and the latter is easily seen to be globally sharply maximized when $\gamma$ is constant along $\Gamma$. On the other hand, by [@EN] the operator $H_{\alpha,\Gamma}$ can be approximated in the strong resolvent sense by point interaction Hamiltonians with the point interactions equidistantly spaced along $\Gamma$ and properly chosen coupling constants, and from [@Ex] we know that the ground state of such an operator is locally maximized by a regular polygon. Needless to say, neither of the above observations proves the desired result. The first one is global, but it holds only asymptotically and we do not know whether the error term will not spoil the inequality. The second argument holds for any $\alpha>0$ suggesting the local validity, but the polygons approximating the circle do not have exactly the same lengths. Our main tools in this paper are the generalized Birman-Schwinger principle in combination with the convexity of Green’s function. They allow us to reformulate the problem in a purely geometric way, in terms of *mean value of chords* of arc segments of $\Gamma$. Since such geometric inequalities are of an independent interest, we discuss them in Sec. \[geom\] separately in a broader context, including the discrete version which arose in connection with the polygon problem treated in [@Ex]. Before doing that, we will formulate in the next section the problem and state our main result, Theorem \[mainthm\], and provide the mentioned reformulation in Sec. \[bs\]. After the discussion of the inequalities we will finish the proof of Theorem \[mainthm\] and present some concluding remarks. Formulation and the main result {#main} =============================== We will assume throughout that $\Gamma:\: [0,L]\to {\mathbb{R}}^2$ is *a closed curve, $\Gamma(0)= \Gamma(L)$, parametrized by its arc length, which is $C^1$-smooth, piecewise $C^2$, and has no cusps*[^1]. Unless stated otherwise, we will mean by the curve $\Gamma$ for simplicity both the above mentioned function and its image in the plane. Furthermore, we introduce the equivalence relation: $\Gamma$ and $\Gamma'$ belong to the same equivalence class if one can be obtained from the other by a Euclidean transformation of the plane. Spectral properties of the corresponding $H_{\alpha,\Gamma}$ and $H_{\alpha,\Gamma'}$ are obviously the same, and we will usually speak about a curve $\Gamma$ having in mind the corresponding equivalence class. It is clear that the stated regularity assumptions are satisfied, in particular, by the circle, say ${\mathcal{C}}:= \{\,((L/2\pi)\cos s, (L/2\pi)\sin s):\: s\in[0,L]\,\}$, and its equivalence class. First of all we have to give a rigorous meaning to the operator (\[formal\]). Following [@BEKS; @BT] we can do that in two ways. The more general one is to consider a positive Radon measure $m$ on ${\mathbb{R}}^2$ and $\alpha>0$ such that $$\label{basiccond} (1+\alpha) \int_{{\mathbb{R}}^2} |\psi(x)|^2\, {\mathrm{d}}m(x) \le a \int_{{\mathbb{R}}^2} |\nabla\psi(x)|^2\, {\mathrm{d}}x + b \int_{{\mathbb{R}}^2} |\psi(x)|^2\, {\mathrm{d}}x$$ holds for all $\psi$ from the Schwartz space ${\mathcal{S}}({\mathbb{R}}^2)$ and some $a<1$ and $b$. The map $I_m$ defined on ${\mathcal{S}}({\mathbb{R}}^2)$ by $I_m\psi=\psi$ extends by density uniquely to $$I_m:\: W^{1,2}({\mathbb{R}}^2) \,\to\, L^2(m):= L^2({\mathbb{R}}^2,{\mathrm{d}}m) \,;$$ abusing notation we employ the same symbol for a continuous function and the corresponding equivalence classes in both $L^2({\mathbb{R}}^2)$ and $L^2(m)$. The inequality (\[basiccond\]) extends to $W^{1,2}({\mathbb{R}}^2)$ with $\psi$ replaced by $I_m\psi$ at the left-hand side. This makes it possible to introduce the following quadratic form, $$\label{Hamform} {\mathcal{E}}_{-\alpha m}(\psi,\phi) := \int_{{\mathbb{R}}^2} \overline{\nabla\psi(x)} \nabla\phi(x) \, {\mathrm{d}}x -\alpha \int_{{\mathbb{R}}^2} (I_m\bar\psi)(x) (I_m\phi)(x)\, {\mathrm{d}}m(x)\,,$$ with the domain $W^{1,2}({\mathbb{R}}^2)$; it is straightforward to see that under the condition (\[basiccond\]) it is closed and below bounded, with $C_0^{\infty}({\mathbb{R}}^2)$ as a core, and thus associated with a unique self-adjoint operator. Furthermore, (\[basiccond\]) is satisfied with any $a>0$ provided $m$ belongs to the generalized Kato class, $$\label{Kato} \lim_{\epsilon\to 0}\: \sup_{x\in{\mathbb{R}}^2}\, \int_{B(x,\epsilon)} |\ln |x\!-\!y|| \, {\mathrm{d}}m(y) = 0\,,$$ where $B(x,\epsilon)$ is the ball of radius $\epsilon$ and center $x$. Choosing now for $m$ the Dirac measure supported by the curve one can check easily that the condition (\[Kato\]) is satisfied under our assumptions about $\Gamma$, hence we may identify the above mentioned self-adjoint operator with the formal one given by (\[formal\]). The described definition applies naturally to a much wider class of perturbations than we need here. Since $\Gamma$ is supposed to be smooth, with the normal defined everywhere, we can define $H_{\alpha,\Gamma}$ alternatively through boundary conditions. Specifically, it acts as $-\Delta\psi$ on any $\psi$ from the domain consisting of functions which belong to $W^{2,2}(\mathbb{R}^2\setminus\Gamma)$, they are continuous at the curve $\Gamma$ and their normal derivatives have a jump there, $$ {\partial\psi(x)\over\partial n_+} - {\partial\psi(x)\over\partial n_-} = -\alpha \psi(x) \quad \mathrm{for}\quad x=\Gamma(s)\,,\; \forall s\in[0,L]\,.$$ It is straightforward to check that such an operator is e.s.a. and its closure can be identified with (\[formal\]) defined in the above described way [@BEKS]. The advantage of the second definition is that it has an illustrative meaning which corresponds well to the concept of a $\delta$ interaction in the cross cut of the curve. Since the curve is finite, by [@BEKS; @BT] we have $\sigma_\mathrm{ess}(H_{\alpha,\Gamma})= [0,\infty)$ while the discrete spectrum is nonempty and finite, so that $$ \epsilon_1 \equiv \epsilon_1(\alpha,\Gamma):= \inf \sigma \left(-\Delta_{\alpha,\Gamma}\right)<0\,;$$ we ask for which $\Gamma$ the principal eigenvalue is maximal. The main result of this paper is a partial answer to this question, namely: \[mainthm\] Within the specified class of curves, $\epsilon_1(\alpha,\Gamma)$ is for any fixed $\alpha>0$ and $L>0$ locally sharply maximized by a circle. While we do not give a general answer here, we suggest what it should be. \[mainconj\] The circle is a sharp global maximizer, even under weaker regularity assumptions. Birman-Schwinger reformulation {#bs} ============================== For operators associated with the quadratic form (\[Hamform\]) one can establish a generalized Birman-Schwinger principle – we refer to [@BEKS] for a detailed discussion. In particular, if $k^2$ belongs to the resolvent set of $H_{\alpha,\Gamma}$ we put $R^k_{\alpha,\Gamma} := (H_{\alpha,\Gamma}-k^2)^{-1}$. The free resolvent $R^k_0$ is defined for ${\mathrm{Im\,}}k>0$ as an integral operator in $L^2({\mathbb{R}}^2)$ with the kernel $$ G_k(x\!-\!y) = {i\over 4}\, H_0^{(1)} (k|x\!-\!y|)\,.$$ Next we have to introduce embedding operators associated with $R^k_0$. Let $\mu, \nu$ be arbitrary positive Radon measures on ${\mathbb{R}}^2$ with $\mu(x)= \nu(x) =0$ for any $x\in{\mathbb{R}}^2$. By $R^k_{\nu,\mu}$ we denote the integral operator from $L^2(\mu):=L^2({\mathbb{R}}^2,{\mathrm{d}}\mu)$ to $L^2(\nu)$ with the kernel $G_k$, in other words we suppose that $$R^k_{\nu,\mu} \phi = G_k \ast \phi\mu$$ holds $\nu$-a.e. for all $\phi\in D(R^k_{\nu,\mu}) \subset L^2(\mu)$. In our case the two measures will be the Dirac measure supported by $\Gamma$, denoted by $m$ if necessary, and the Lebesgue measure ${\mathrm{d}}x$ on ${\mathbb{R}}^2$, in different combinations. With this notation one can express the generalized BS principle as follows: \[bsprop\] (i) There is a $\kappa_0>0$ such that the operator $I-\alpha R^{i\kappa}_{m,m}$ on $L^2(m)$ has a bounded inverse for any $\kappa \ge \kappa_0$.\ \[1mm\] (ii) Let ${\mathrm{Im\,}}k>0$. Suppose that $I-\alpha R^k_{m,m}$ is invertible and the operator $$R^k := R_0^k + \alpha R^k_{{\mathrm{d}}x,m} [I-\alpha R^k_{m,m}]^{-1} R^k_{m,{\mathrm{d}}x}$$ from $L^2({\mathbb{R}}^2)$ to $L^2({\mathbb{R}}^2)$ is everywhere defined. Then $k^2$ belongs to $\rho(H_{\alpha,\Gamma})$ and $(H_{\alpha,\Gamma} -k^2)^{-1}= R^k$.\ \[1mm\] (iii) $\:\dim\ker(H_{\alpha,\Gamma}-k^2) = \dim\ker(I-\alpha R^k_{m,m})$ for any $k$ with ${\mathrm{Im\,}}k>0$.\ \[1mm\] (iv) an eigenfunction of $H_{\alpha,\Gamma}$ associated with such an eigenvalue $k^2$ can be written as $$\psi(x)= \int_0^L R^k_{{\mathrm{d}}x,m}(x,s) \phi(s)\, {\mathrm{d}}s\,,$$ where $\phi$ is the corresponding eigenfunction of $\alpha R^k_{m,m}$ with the eigenvalue one. *Proof* of (i)-(iii) is given in [@BEKS], for (iv) see [@Po]. Denoting conventionally $k=i\kappa$ with $\kappa>0$ as corresponding to the bound-state energy $-\kappa^2$, we can thus rephrase our problem as a search for solutions to the integral-operator equation $$\label{bsform} {\mathcal{R}}_{\alpha,\Gamma}^\kappa \phi= \phi\,, \quad {\mathcal{R}}_{\alpha,\Gamma}^\kappa(s,s'):= \frac{\alpha}{2\pi} K_0\big(\kappa|\Gamma(s) \!-\!\Gamma(s')|\big)\,,$$ on $L^2([0,L])$, where $K_0$ is Macdonald function. Referring again to [@BEKS] and [@Po] we find that the operator-valued function $\kappa\mapsto {\mathcal{R}}_{\alpha,\Gamma}^\kappa$ is strictly decreasing in $(0,\infty)$ and $\|{\mathcal{R}}_{\alpha,\Gamma}^\kappa\|\to 0$ as $\kappa\to\infty$. In fact the two properties can be checked also directly. The first one follows from the one-to-one correspondence of the eigenvalue branches (as functions of $\kappa$) to those of $H_{\alpha,\Gamma}$ which are obviously strictly monotonous as functions of $\alpha$; the second one in turn comes from the explicit form of the kernel together with the dominated convergence theorem. Next we use the fact that the maximum eigenvalue of ${\mathcal{R}}_{\alpha,\Gamma}^\kappa$ is simple. This conclusion results from the following considerations: the kernel of the operator is by (\[bsform\]) strictly positive, so ${\mathcal{R}}_{\alpha,\Gamma}^\kappa$ is positivity improving. It further means that for any nonzero $\phi,\chi\ge 0$ the functions ${\mathcal{R}}_{\alpha,\Gamma}^\kappa\phi,\, {\mathcal{R}}_{\alpha, \Gamma}^\kappa\chi$ are also strictly positive. Hence $(\phi, ({\mathcal{R}}_{\alpha,\Gamma}^\kappa)^2\chi)\ne 0$, and as a consequence, ${\mathcal{R}}_{\alpha,\Gamma}^\kappa$ is ergodic; then the claim follows from Thm. XIII.43 of [@RS]. In view of Proposition \[bsprop\](iii) the ground state of $H_{\alpha,\Gamma}$ is, of course, also simple. If $\Gamma$ is a circle the operator $H_{\alpha,{\mathcal{C}}}$ has a full rotational symmetry, so the corresponding eigenspace supports a one-dimensional representation of the group $O(2)$. Let us denote the ground-state eigenfunction of $H_{\alpha,{\mathcal{C}}}$ as $-\tilde\kappa_1^2\,$ (we will use tilde to distinguish quantities referring to the circle). The correspondence between the eigenfunctions given by Proposition \[bsprop\](iv) then requires that the respective eigenfunction of ${\mathcal{R}}_{\alpha,{\mathcal{C}}}^{\tilde\kappa_1}$ corresponding to the unit eigenvalue is constant; we can choose it as $\tilde \phi_1(s)= L^{-1/2}$. Then we have $$ \max \sigma({\mathcal{R}}_{\alpha,{\mathcal{C}}}^{\tilde\kappa_1}) = (\tilde\phi_1, {\mathcal{R}}_{\alpha,{\mathcal{C}}}^{\tilde\kappa_1} \tilde\phi_1) = \frac1L \int_0^L \int_0^L {\mathcal{R}}_{\alpha,{\mathcal{C}}}^{\tilde\kappa_1}(s,s') \,{\mathrm{d}}s{\mathrm{d}}s'\,,$$ and on the other hand, for the same quantity referring to a general $\Gamma$ a simple variational estimate gives $$ \max \sigma({\mathcal{R}}_{\alpha,\Gamma}^{\tilde\kappa_1}) \ge (\tilde\phi_1, {\mathcal{R}}_{\alpha,\Gamma}^{\tilde\kappa_1} \tilde\phi_1) = \frac1L \int_0^L \int_0^L {\mathcal{R}}_{\alpha,\Gamma}^{\tilde\kappa_1}(s,s') \,{\mathrm{d}}s{\mathrm{d}}s'\,.$$ Hence to check that the circle is a maximizer it sufficient to show that $$\label{Greenineq} \int_0^L \int_0^L K_0\big(\kappa|\Gamma(s) \!-\!\Gamma(s')|\big) \,{\mathrm{d}}s{\mathrm{d}}s' \ge \int_0^L \int_0^L K_0\big(\kappa|{\mathcal{C}}(s) \!-\!{\mathcal{C}}(s')|\big) \,{\mathrm{d}}s{\mathrm{d}}s'$$ holds *for all* $\kappa>0$ and $\Gamma$ of the considered class, or at least for $\Gamma$ in the vicinity of ${\mathcal{C}}$ to prove the local result in Theorem \[mainthm\]. Since the kernel is symmetric w.r.t. the two variables, we can replace the double integral by $2\int_0^L {\mathrm{d}}s\int_0^s{\mathrm{d}}s'$. By another simple change of variables we find that the above claim is equivalent to positivity of the functional $$ F_\kappa(\Gamma):= \int_0^{L/2} {\mathrm{d}}u \int_0^L {\mathrm{d}}s \bigg[ K_0\big(\kappa|\Gamma(s\!+\!u) -\Gamma(s)|\big) - K_0\big(\kappa|{\mathcal{C}}(s\!+\!u) -{\mathcal{C}}(s)|\big) \bigg]\,,$$ where the second term in the integrand is, of course, independent of $s$ being equal to $K_0\big(\frac{\kappa L}{\pi} \sin \frac{\pi u}{L}\big)$. Now we employ the (strict) convexity of $K_0$ which yields by means of the Jensen inequality the following estimate, $$ \frac 1L \,F_\kappa(\Gamma)\ge \int_0^{L/2} \left[ K_0\left( \frac{\kappa}{L} \int_0^L |\Gamma(s\!+\!u) -\Gamma(s)| {\mathrm{d}}s \right) - K_0\left(\frac{\kappa L}{\pi} \sin \frac{\pi u}{L}\right) \right]\, {\mathrm{d}}u\,,$$ where the inequality is sharp unless $\int_0^L |\Gamma(s\!+\!u) -\Gamma(s)| {\mathrm{d}}s$ is independent of $s$. Finally, we observe that $K_0$ is decreasing in $(0,\infty)$, hence it is sufficient to check the inequality $$\label{suffic} \int_0^L |\Gamma(s\!+\!u) -\Gamma(s)|\, {\mathrm{d}}s \,\le\, \frac{L^2}{\pi} \sin \frac{\pi u}{L}$$ for all $u\in(0,\frac12 L]$ and to show that is sharp unless $\Gamma$ is a circle. Mean-chord inequalities {#geom} ======================= The inequality (\[suffic\]) to which we have reduced our problem can be regarded as an element of a wider family which we are now going to describe. Let $\Gamma:\: [0,L]\to {\mathbb{R}}^2$ be again a loop in the plane; for the moment we do not specify its regularity properties. Let us consider all the arcs of $\Gamma$ having length $u\in (0,\frac12 L]$. The mentioned inequalities are the following $$\begin{aligned} C_L^p(u):&\quad\;\; \int_0^L |\Gamma(s\!+\!u) -\Gamma(s)|^p\, {\mathrm{d}}s \,\le\, \frac{L^{1+p}}{\pi^p} \sin^p \frac{\pi u}{L}\,,& \; p>0\,, \label{C+p} \\ C_L^{-p}(u):& \int_0^L |\Gamma(s\!+\!u) -\Gamma(s)|^{-p}\, {\mathrm{d}}s \,\le\, \frac{\pi^p L^{1-p}}{\sin^p \frac{\pi u}{L}}\,,& \; p>0\,. \label{C-p} \end{aligned}$$ They have also a discrete counterpart for an equilateral polygon ${\mathcal{P}}_N$ of $N$ vertices and side length $\ell>0$. Let $\{y_n\}$ be the family of its vertices, where the index values are identified modulo $N$; then we introduce $$\begin{aligned} D_{N,\ell}^p(m):&\; \sum_{n=1}^N |y_{n+m}-y_n|^p\, \le\, \frac{N\ell^p \sin^p \frac{\pi m}{N}}{\sin^p \frac{\pi}{N}} \,,& \; p>0\,, \label{D+p} \\ D_{N,\ell}^{-p}(m):& \sum_{n=1}^N |y_{n+m}-y_n|^{-p}\, \le\, \frac{N \sin^p \frac{\pi }{N}}{\ell^p \sin^p \frac{\pi m}{N}}\,,& \; p>0\,, \label{D-p} \end{aligned}$$ for any $m=1,\dots, [\frac12 N]$, where $[\cdot]$ denotes as usual the entire part. In all the cases the right-hand side corresponds, of course, to the case with maximal symmetry, i.e. to the circle and regular polygon $\tilde{\mathcal{P}}_N$, respectively. *We conjecture* that without regularity restrictions $C_L^{\pm p}(u)$ *holds for any $p\le 2$ and the same is true for* $D_{N,\ell}^{\pm p}(m)$, and furthermore, we expect the inequalities *to be sharp unless $\Gamma={\mathcal{C}}$ or ${\mathcal{P}}_N=\tilde{\mathcal{P}}_N$, respectively*. In the polygon case it is clear that the claim may not be true for $p>2$ as the example of a rhomboid shows: $D_{4,\ell}^p(2)$ is equivalent to $\sin^p\phi +\cos^p\phi \ge 1$ for $0<\phi<\pi$. We are unable at this moment to demonstrate the inequalities (\[C+p\])–(\[D-p\]) in full generality; below we will present a few particular cases. It is obvious that the inequalities have a scaling property, so without loss of generality one can assume, e.g., $L=1$ and $\ell=1$; in such a case we drop the corresponding symbol from the label. If necessary we can include also the case $p=0$ when the inequalities turn into trivial identities. \[monotone\] $C_L^p(u)\Rightarrow C_L^{p'}(u)$ and $D_{N,\ell}^p(m)\Rightarrow D_{N,\ell}^{p'}(m)$ if $p>p'>0$. [*Proof:*]{} The claim follows from the convexity of $x\mapsto x^\alpha$ in $(0,\infty)$ for $\alpha>1$, $$\begin{aligned} \frac{L^{1+p}}{\pi^p} \sin^p \frac{\pi u}{L} &\!\ge\!& \int_0^L \left(|\Gamma(s\!+\!u) -\Gamma(s)|^{p'} \right)^{p/p'}\, {\mathrm{d}}s \\ &\!\ge\!& L \left( \frac1L \int_0^L |\Gamma(s\!+\!u) -\Gamma(s)|^{p'}\, {\mathrm{d}}s \right)^{p/p'}\,. \end{aligned}$$ It is now sufficient to take both sides to the power $p'/p$; in the same way one checks the second implication. \[inverse\] $C_L^p(u)\Rightarrow C_L^{-p}(u)$ and $D_{N,\ell}^p(m)\Rightarrow D_{N,\ell}^{-p}(m)$ for any $p>0$. [*Proof:*]{} The Schwarz inequality implies $$ \int_0^L |\Gamma(s\!+\!u) -\Gamma(s)|^{-p}\, {\mathrm{d}}s \ge \frac{L^2}{\int_0^L |\Gamma(s\!+\!u) -\Gamma(s)|^p\, {\mathrm{d}}s} \ge \frac{L^2\pi^p}{L^{1+p}\sin^p \frac{\pi u}{L}}\,,$$ and similarly for the polygon case. These simple relations mean that to check the above stated conjecture one needs only to verify $C^2(u)$ and $D_N^2(m)$. We will address the continuous case in the next section, here we notice that the results of [@Ex] in combination with the last two propositions leads to the following conclusions: \[polygon1\] (a) $\;D^1_{N,\ell}(m)$ holds locally for any $N$ and $m=1,\dots, [\frac12 N]$, i.e. in a vicinity of the regular polygon, and consequently, $D^{\pm p}_{N,\ell}(m)$ holds locally for any $p\in (0,1]$.\ \[.2em\] (b) $\;D^1_{N,\ell}(2)$ holds globally for any $N$, and so does $D^{\pm p}_{N,\ell}(2)$ for each $p\in (0,1]$. Proof of Theorem \[mainthm\] {#proof} ============================ After this interlude let us return to our main problem. Notice first that our regularity hypothesis allows us to characterize $\Gamma$ by its (signed) curvature $\gamma:= \dot\Gamma_2 \ddot\Gamma_1- \dot\Gamma_1\ddot\Gamma_2$ which is by assumption a piecewise continuous function in $[0,L]$. The advantage is that $\gamma$ specifies uniquely the equivalence class related by Euclidean transformations which can be represented by $$\label{param} \Gamma(s)= \left( \int_0^s \cos\beta(s')\, {\mathrm{d}}s', \int_0^s \sin\beta(s')\, {\mathrm{d}}s' \right)\,,$$ where $\beta(s):= \int_0^s \gamma(s')\, {\mathrm{d}}s'$ is the bending angle relative to the tangent at the chosen initial point, $s=0$. To ensure that the curve is closed, the conditions $$\label{close} \int_0^L \cos\beta(s')\, {\mathrm{d}}s' = \int_0^L \sin\beta(s')\, {\mathrm{d}}s' = 0$$ must be satisfied. Using this parametrization we can rewrite the left-hand side of the inequality (\[C+p\]) in the form $$ \int_0^L \left[ \left( \int_s^{s+u} \cos\beta(s')\, {\mathrm{d}}s'\right)^2 + \left( \int_s^{s+u} \sin\beta(s')\, {\mathrm{d}}s'\right)^2 \right]^{p/2} \!{\mathrm{d}}s:=c^p_\Gamma(u)\,,$$ or equivalently $$ c^p_\Gamma(u) = \int_0^L {\mathrm{d}}s \left[ \int_s^{s+u} {\mathrm{d}}s' \int_s^{s+u} {\mathrm{d}}s'' \cos(\beta(s')-\beta(s'') \right]^{p/2}.$$ By Proposition \[monotone\] it is sufficient to check that the quantity $c^2_\Gamma(u)$ is maximized by the circle, i.e. by $\beta(s)= \frac{2\pi s}{L}$. Rearranging the integrals we get $$\begin{aligned} && c^2_\Gamma(u) = \int_0^L {\mathrm{d}}s' \int_{s'-u}^{s'+u} {\mathrm{d}}s'' \int_{\max{\{s'-u,s''-u\}}}^{\min{\{s',s''\}}} {\mathrm{d}}s\, \cos(\beta(s')-\beta(s'')) \\ && = \int_0^L {\mathrm{d}}s' \int_{s'\!-u}^{s'\!+u} \!{\mathrm{d}}s''\, [\min{\{s',s''\}}-\max{\{s'\!-u,s''\!-u\}}] \cos(\beta(s')\!-\!\beta(s'')), \end{aligned}$$ or $$ c^2_\Gamma(u)= \int_0^L {\mathrm{d}}s' \int_{s'-u}^{s'+u} {\mathrm{d}}s''\, \left[ u-|s'\!-s''|\right]\,\cos(\beta(s')-\beta(s''))\,.$$ Next we change the integration variables to $x:=s'\!-s''$ and $z:= \frac12(s'\!+s'')$, $$ c^2_\Gamma(u)= \int_{-u}^u {\mathrm{d}}x\, (u-|x|) \int_0^L {\mathrm{d}}z\, \cos\left(\beta(z+\frac12 x)-\beta(z-\frac12 x)\right)\,,$$ and since the functions involved are even w.r.t. $x$ we finally get $$\label{c^2} c^2_\Gamma(u)= 2\int_0^u {\mathrm{d}}x\, (u-x) \int_0^L {\mathrm{d}}z\, \cos\left(\int_{z-\frac12 x}^{z+\frac12 x} \gamma(s)\, {\mathrm{d}}s \right)\,.$$ As a certain analogy to Theorem \[polygon1\](b) we can prove the sought global inequality in case when the curve arcs in question are sufficiently short and/or the tangent vector direction does change too fast. \[part global\] Suppose that $\Gamma$ has no self-intersections and the inequality $\beta(z+\frac12 u)-\beta(z-\frac12 u) \le \frac12 \pi$ is valid for all $z\in[0,L]$, then $C_L^2(u)$ holds. [*Proof:*]{} We employ concavity of cosine in $(0, \frac12\pi)$ obtaining $$\begin{aligned} c^2_\Gamma(u) &\!\le\!& 2L\int_0^u {\mathrm{d}}x\, (u-x) \cos\left(\frac1L \int_0^L {\mathrm{d}}z\, \int_{z-\frac12 x}^{z+\frac12 x} \gamma(s)\, {\mathrm{d}}s \right) \\ &\!=\!& 2L\int_0^u {\mathrm{d}}x\, (u-x) \cos\left(\frac1L \int_0^L {\mathrm{d}}s\,\gamma(s) \int_{s-\frac12 x}^{s+\frac12 x} {\mathrm{d}}z \right) \\ &\!=\!& 2L\int_0^u {\mathrm{d}}x\, (u-x) \cos \frac{2\pi x}{L} = \frac{L^3}{\pi^2}\, \sin^2 \frac{\pi u}{L}\,, \end{aligned}$$ since $\int_0^L \gamma(s)\,{\mathrm{d}}s = \pm 2\pi$ for a curve without self-intersections. Moreover, the function $z\mapsto \int_{z-\frac12 x}^{z+\frac12 x} \gamma(s)\, {\mathrm{d}}s$ is constant for $x\in (0,u)\,$ *iff* $\,\gamma(\cdot)$ is constant, hence the circle corresponds to a sharp maximum. This result, however, does not help us with our main problem, because we need the inequality to be valid for all arc lengths. As indicated before, we can prove a local result which will imply Theorem \[mainthm\]. \[smalldef\] Under the regularity assumptions of Sec. \[bs\], the inequality $C^2_L(u)$ holds locally for any $L>0$ and $u\in (0,\frac12 L]$, and consequently, $C^{\pm p}_L(u)$ holds locally for any $p\in (0,2]$. [*Proof:*]{} Gentle deformations of a circle can be characterized by the curvature $$ \gamma(s)= \frac{2\pi}{L} +g(s)\,,$$ where $g$ is a piecewise continuous functions which is small in the sense that $\|g\|_\infty\ll L^{-1}$ and satisfies the condition $\int_0^L g(s)\, {\mathrm{d}}s=0$. The function in the last integral of (\[c\^2\]) can be then expanded as $$ \cos\frac{2\pi x}{L} - \sin\frac{2\pi x}{L} \int_{z-\frac12 x}^{z+\frac12 x} g(s)\, {\mathrm{d}}s - \frac12 \cos\frac{2\pi x}{L} \left( \int_{z-\frac12 x}^{z+\frac12 x} g(s)\, {\mathrm{d}}s \right)^2 + {\mathcal{O}}(g^3)\,,$$ where the error term is a shorthand for ${\mathcal{O}}(\|Lg\|_\infty^3)$. Substituting this expansion into (\[c\^2\]) we find that the term linear in $g$ vanishes, because $$ \int_0^L {\mathrm{d}}z \int_{z-\frac12 x}^{z+\frac12 x} g(s)\, {\mathrm{d}}s = \int_0^L {\mathrm{d}}s\, g(s) \int_{s-\frac12 x}^{s+\frac12 x} {\mathrm{d}}z = 0\,,$$ and thus $$\label{c^2exp} c^2_\Gamma(u)= \frac{L^3}{\pi^2} \sin^2 \frac{\pi u}{L} - I_g(u) + {\mathcal{O}}(g^3)\,,$$ where $$ I_g(u):= \int_0^u {\mathrm{d}}x\, (u-x)\, \cos\frac{2\pi x}{L} \int_0^L {\mathrm{d}}z\, \left(\int_{z-\frac12 x}^{z+\frac12 x} g(s)\, {\mathrm{d}}s \right)^2.$$ We need to show that $I_g(u)>0$ unless $g=0$ identically. Notice that for $u\le \frac14 L$ this property holds trivially. For $u\in (\frac14 L, \frac12 L]$ we use the fact that $g$ is periodic and piecewise continuous, so we can write it through its Fourier series $$ g(s) = \sum_{n=1}^\infty \left( a_n \sin\frac{2\pi ns}{L} + b_n \cos\frac{2\pi ns}{L} \right)$$ with the zero term missing, where $\sum_n(a_n^2+b_n^2)$ is finite (and small). Using $$ \int_{z-\frac12 x}^{z+\frac12 x} g(s)\, {\mathrm{d}}s = \frac{L}{\pi} \sum_{n=1}^\infty \frac1n\, \left( a_n \sin\frac{2\pi nz}{L} + b_n \cos\frac{2\pi nz}{L} \right) \sin\frac{\pi nx}{L}$$ together with the orthogonality of the Fourier basis we find $$ I_g(u)= \int_0^{u} {\mathrm{d}}x\, (u-x)\, \cos\frac{2\pi x}{L} \sum_{n=1}^\infty \frac{L^3}{2\pi^2}\, \frac{a_n^2+b_n^2}{n^2}\, \sin\frac{\pi nx}{L}\,.$$ Since the summation and integration can be obviously interchanged, we have $$\label{I_gexpand} I_g(u)= \frac{L^5}{2\pi^4}\, \sum_{n=1}^\infty \, \frac{a_n^2+b_n^2}{n^2}\, F_n\left( \frac{\pi u}{L}\right)\,,$$ where $$ F_n(v):= \int_0^{v} (v-y)\, \cos 2y\, \sin ny\: {\mathrm{d}}y\,.$$ These integrals are equal to $$\begin{aligned} F_1(v) &\!=\!& \frac{1}{18} (\,9\sin v -\sin 3v -6v)\,, \\ F_2(v) &\!=\!& \frac{1}{32} (\,4v -\sin 4v) \,, \\ F_n(v) &\!=\!& \frac{nv}{n^2-4} - \frac{\sin(n-2)v}{2(n-2)^2} - \frac{\sin(n+2)v}{2(n+2)^2} \,, \qquad n\ge 3\,. \end{aligned}$$ Using the fact that $\sin x<x$ for $x>0$ we see immediately that $F_n(v)>0$ for $v>0$ and $n\ge 2$. On the other hand, $F_1(v)$ has in the interval $(0,\frac{\pi}{2})$ a single positive maximum, at some $v> \frac{\pi}{4}$, from which it decreases to the value $F_1(\frac{\pi}{2})= \frac{1}{18}(10-3\pi)>0$. Summing up this argument, we have found that the quantity (\[I\_gexpand\]) is positive unless all the coefficients $a_n,b_n$ are zero. \[closecurve\] [One may wonder what happened with the closedness requirement (\[close\]). As the argument shows we were able to demonstrate the claim using only the weaker property that $\beta(0)=\beta(L)$. This is possible, of course, for small deformations only! As an illustration, consider $\Gamma$ in the form of an “overgrown paperclip” which satisfies the condition $\beta(0)=\beta(L)$ but not (\[close\]), i.e. a line segment with two U-turns at the ends. Making the latter short one can get $c^2_\Gamma(\frac12 L)$ arbitrarily close to $\frac13 L^3$ which is larger than $L^3/\pi^2$. ]{} Extensions and conclusions {#concl} ========================== To support our expectations that the result given in Theorem \[mainthm\] holds globally and under weaker regularity assumptions, consider a simple example. \[lens+appla\] [Let $\Gamma$ be a curve consisting of two circular segments of radius $R> \frac{L}{4\pi}$, i.e. it is given by the equations $$\label{lens} \left( x\pm R\cos\frac{L}{2R} \right)^2 + y^2 = R^2 \qquad \mathrm{for}\quad \pm x\ge 0\,.$$ For $R> \frac{L}{2\pi}$ it is “lens-shaped”, for $\frac{L}{4\pi} <R< \frac{L}{2\pi}$ “apple-shaped”; it is not smooth except in the trivial case of a circle, $R= \frac{L}{2\pi}$. The curvature of this $\Gamma$ equals $$ \gamma(s)= \frac1R + \left(\pi- \frac{L}{2R}\right) (\delta(s) +\delta(s-L/2))\,,$$ hence $$ c^2_\Gamma(u)= 2\int_0^u {\mathrm{d}}x\, (u-x) \left[ (L-2x) \cos\frac{x}{R} - 2x \cos \frac{L-2x}{2R} \right] {\mathrm{d}}x\,,$$ and evaluating the integral, we arrive at $$ c^2_\Gamma(u)= 8R^3 \left\{ \frac{L}{2R} \sin^2 \frac{u}{2R} + 4\left( \frac{u}{2R} \cos \frac{u}{2R} - \sin \frac{u}{2R} \right) \cos \frac{L}{4R}\, \cos \frac{L-2u}{4R} \right\}\,.$$ This function has for each $u\in(0,\frac12 L]$ a maximum at $R= \frac{L}{2\pi}$ and one can check directly that its value is smaller for any other $R$. In particular, in the limit $R\to \infty$ we have $c^2_\Gamma(u)\to Lu^2-\frac43 u^3$ as one can find also directly with the “lens” degenerate into a double line segment; this value is less than $\frac{L^3}{\pi^2} \sin^2 \frac{\pi u}{L}$ because $\sin^2x > x^2- \frac{4}{3\pi} x^3$ holds in $(0,\frac{\pi}{2})$. ]{} To summarize our discussion, to prove Conjecture \[mainconj\] it is sufficient to verify the inequality $C_L^p(u)$ for some $p\ge 1$ under appropriate regularity hypothesis. Naturally, one can ask also about ground-state maximizer in smaller families of curves $\Gamma$ which do not contain the circle; examples could be polygonal loops with a fixed or limited number of vertices, or various prescribed compositions of arcs belonging to specific classes, circular, elliptic, parabolic, etc. Obviously a reasonable strategy is to look first for curves as close to the circle as possible within the given class. Sometimes one expects that the answer will be the curve with maximum symmetry as in the polygon case, in other situations it may not be true. Another, and maybe more important extension of the present problem concerns a maximizer for the generalized Schrödinger operator in ${\mathbb{R}}^3$ with an attractive $\delta$ interaction supported by a closed surface of a fixed area $A$, and its generalization to closed hypersurfaces of codimension one in ${\mathbb{R}}^d,\, d>3$. In the case of $d=3$ we have a heuristic argument relying on [@Ex2; @EHL] similar to that used in the introduction which suggests that the problem is solved by the sphere provided the discrete spectrum is not empty, of course, which is a nontrivial assumption in this case – for properties of the corresponding operators see [@AGS]. The Birman-Schwinger reduction of the problem similar to that of Sec. \[bs\] can be performed again and the task is thus reduced to verification of a geometric inequality analogous to (\[C+p\]) which we can label as $C_A^{d,p}(u)$. We will discuss this problem in a following paper. Acknowledgments {#acknowledgments .unnumbered} --------------- The research has been partially supported by the ASCR Grant Agency within the project A100480501. [99]{} S. Albeverio, F. Gesztesy, R. Høegh-Krohn, H. Holden: [ *Solvable Models in Quantum Mechanics*]{}, 2nd printing with an appendix by P. Exner, AMS Chelsea Publ., Providence, R.I., 2005. J.-P. Antoine, F. Gesztesy, J. Shabani: Exactly solvable models of spere interaction in quantum mechanics, [*J. Phys. A: Math. Gen.*]{} **20**, 3627–3712 (1987). M.S. Ashbaugh, R.D. Benguria: A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions, [ *Ann. Math.*]{} [**135**]{}, 601–628 (1992). 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E.M. Harrell, P. Kröger, K. Kurata: On the placement of an obstacle or well so as to optimize the fundamental eigenvalue, [*SIAM J. Math. Anal.*]{} [**33**]{}, 240–259 (2001). E. Krahn: Über eine von Rayleigh formulierte minimal Eigenschaft des Kreises, [*Ann. Math.*]{} [**94**]{}, 97–100 (1925). A. Posilicano: Boundary triples and Weyl functions for singular perturbations of self-adjoint operator, [*Meth. Funct. Anal. Topol.*]{} [**10**]{}, 57–63 (2004). M. Reed and B. Simon: [*Methods of Modern Mathematical Physics, IV. Analysis of Operators*]{}, Academic Press, New York 1978. [^1]: There are, of course, no local cusps under the $C^1$ assumption, but we have not excluded self-intersections, so the last requirement means that the curve meets itself at such a point at a nonzero angle. In fact our main result can be pushed through under a slightly weaker regularity assumption, namely that $\dot\Gamma$ is absolutely continuous.

--- abstract: 'We show how by careful monitoring of the inspiral signal from a compact object binary in ground-based gravitational wave (GW) detectors one can test if its components are black holes or not. Here we limit ourselves to black holes (with and without spin) in General Relativity. Such objects are characterized by horizons, which absorb gravitational radiation from the orbit during their inspiral in a binary, via a phenomenon known as tidal heating. By contrast, a compact object such as a neutron star has minimal tidal heating – but has tidal deformation – and affects the phase evolution of binaries containing it in a distinctly different way. Here we identify waveform parameters that characterize the strength of tidal heating, and are zero when there is no horizon absorption. We demonstrate how by using those parameters Bayesian methods can distinguish the presence or absence of horizons in a binary. This is a particularly exciting prospect owing to several claims that these stellar-mass objects, with masses heavier than those of neutron stars, may not have a horizon but may be black hole mimickers or exotic compact objects. Perhaps more significant is the possibility that our method can be used to test the presence or absence of horizons in mass-gap binaries and, thereby, help detect the heaviest neutron star or the lightest black hole. A proper accounting of tidal heating in binary waveform models will also be critical for an unbiased measurement of characteristics of the equation of state of neutron stars in GW observations of binaries containing them – or even to probe the existence of exotic compact objects.' author: - Sayak Datta - Khun Sang Phukon - Sukanta Bose bibliography: - 'absorption.bib' title: 'Recognizing black holes in gravitational-wave observations: Telling apart impostors in mass-gap binaries' --- =1 .—In recent times, the discovery by LIGO and Virgo detectors of several compact binary mergers has ushered in the era of gravitational wave (GW) astronomy [@LIGOScientific:2018mvr]. The LIGO-Virgo collaboration also observed the binary neutron star merger GW170817 [@gw170817]. These observations provided a fillip to tests of GR in the strong-field regime [@LIGOScientific:2019fpa; @Abbott:2018lct]. Even the behavior of vacuum spacetimes and the propagation of GWs have been tested rigorously, which has resulted in stringent bounds on the mass of the graviton and violations of Lorentz invariance [@TheLIGOScientific:2016pea; @TheLIGOScientific:2016src; @Abbott:2017vtc]. Significantly, it has also become possible to test the nature of the compact objects in binaries. The components of these binary sources are definitely very compact, which normally leads to the conclusion that they are either black holes (BHs) or neutron stars (NSs). In the case of GW170817 radius measurements were made [@Abbott:2018exr] that strongly disfavor them as black holes. A similar claim may be posited for the other binary neutron star claimant GW190425 [@Abbott:2020uma]. However, for the other LIGO-Virgo binaries (which are much heavier than GW170817 or GW190425) [@LIGOScientific:2018mvr], it remains to be conclusively proven that their components are indeed black holes of GR and not, say, some exotic compact objects (ECOs) [@Yunes:2016jcc; @Cardoso:2016oxy; @Aneesh:2018hlp]. On the other hand, if binaries show up with measured masses of any of the components in the mass-gap [@Fryer_2012] then it poses the challenge of determining whether the component(s) with mass(es) in the gap are neutron stars or black holes. Either occurrence will be significant, for it will either raise the maximum known mass of a neutron star or lower the minimum known mass of a black hole. These issues make it imperative that methods be devised to discern compact objects with horizon from those without. Planck-scale modifications of black hole horizons and modification of BH structure have been proposed in several works as resolutions to the information-loss paradox [@Lunin:2001jy; @Almheiri:2012rt]. Other compact objects such as gravastars, whose interior consists of self-repulsive de Sitter spacetime surrounded by an ordinary matter shell, have also been proposed for similar reasons [@Mazur:2004fk]; likewise for boson stars, which are macroscopic objects made of scalar fields [@Liebling:2012fv]. In light of such proposals for compact objects, as alternatives to black holes, it becomes necessary to devise a strategy to tell them apart; and GWs are a new tool that can be employed for this purpose. In this paper, we do so by using GWs emitted during the inspiral phase of binary coalescences to probe the nature of the compact components. In GR, classical black holes are perfect absorbers that behave as dissipative systems [@MembraneParadigm; @Damour_viscous; @Poisson:2009di; @Cardoso:2012zn]. This property of a black hole can be attributed to its causal structure. The defining feature of a BH is the presence of its horizon, which is a null surface and a one-way membrane. Due to the presence of the horizon, a BH in a binary absorbs energy and angular momentum from the orbit. This phenomenon is called tidal heating [@Hartle:1973zz; @Hughes:2001jr; @PoissonWill]. Energy loss via tidal heating backreacts on the binary’s evolution, resulting in a shift in the phase of the GWs emitted by the system. Therefore, the absence of a horizon – or any kind of change in the near horizon structure that modifies this absorption – will leave its imprint in the phasing of GWs emitted. A careful observation thus has the potential to measure these differences in the GW phase. Several tests have been proposed to probe the black-holeness – the presence of horizon – of the compact objects in a binary. Distinguishing binary merger remnants from black holes in the post-merger phase using [*echoes*]{} has initiated rigorous modelling and search for those features in GW data [@Tsang:2019zra; @Abedi:2016hgu; @Westerweck:2017hus]. Measurement of tidal deformability (TD) [@Cardoso:2017cfl; @Sennett:2017etc] and spin-induced multipole moments [@Krishnendu:2017shb; @Datta:2019euh] from the late inspiral can also be used to test black-holeness. Absence of tidal heating (TH) is a tell-tale signature of the absence of a horizon. Its importance in identifying horizons of intermediate-mass and super-massive compact objects has been examined for the proposed space mission LISA [@Datta:2019euh; @Maselli:2017cmm; @Datta:2019epe]. In the current work, we study its usefulness for stellar mass binaries – of the type observable by ground-based GW detectors like LIGO and Virgo. The tidal heating of a black hole or any other star can be expressed in a similar mathematical form if the viscosity coefficient $(\eta)$ of a BH is identified with its mass [@Glampedakis:2013jya]. For NS $\eta^{}_{\rm NS} \sim 10^4 \left(\frac{\rho}{10^{14} \textrm{gm}- \textrm{cm}^{-3}}\right)^{5/4} \left(\frac{10^8 \textrm{K}}{T}\right)^2 \textrm{cm}^2 \textrm{s}^{-1}$, and for a BH $\eta^{}_{\rm BH} \sim 8.6 \times 10^{14} \left(\frac{M}{M_{\odot}}\right) \textrm{cm}^2 \textrm{s}^{-1}$. Since the correction in GW phase due to TH is proportional to $\eta$, for an NS that correction is 10 orders of magnitude smaller than BH [@Glampedakis:2013jya]. Also in binaries with one or both components as NS, the GW waveform has imprint of tidal [*deformability*]{} of the NS, which is finite for NS [@Abbott:2018exr; @Abbott:2020uma] but 0 for BHs in GR [@Damour:2009vw; @Binnington:2009bb] (see, however, Ref. [@Chakravarti:2018vlt] for an example of a non-GR result). That imprint can by itself help distinguish binaries with NS from those with BHs. However, as we show here, the presence (absence) of TH for BH (NS) improves the ability to discriminate between those binaries significantly. [*Effect of tidal heating on binary waveforms*]{}.— Consider a compact binary with component masses $m_i~~(i=1,2)$, total mass $m= m_1 + m_2$, and mass-ratio $q=m_2/m_1$, with $m_2 \leq m_1$. Let the dimensionless component spins be $\chi^{}_i$. Under the adiabatic approximation the orbital evolution of the binary can be quantified in the post-Newtonian formalism with reasonable accuracy, especially, when it is far from merger [@Blanchet:2013haa]. In this case the dynamics of the system is governed by energy and angular momentum loss from the orbit. Usually this dynamics has a contribution arising from taking the components as point particles (PP) and another one originating from their finite size. The latter contribution can be decomposed into two main ingredients (i) tidal deformation of each component due to the gravitational field of the other and (ii) the amount of energy absorbed by individual components from the orbit, namely, [*tidal heating*]{}. The dynamics of the system and, therefore, the emitted GW depends on all of these contributions. Hence, the Fourier transformed GW waveform can be written as $$\Tilde{h}(f) = A(f) e^{i(\Psi_{\rm PP}+\Psi_{\rm TD}+\Psi_{\rm TH})}\,,$$ where $f$ is the instantaneous GW frequency and $A(f)$ is the frequency-dependent amplitude. The phase terms – $\Psi_{\rm PP}, \Psi_{\rm TD}$ and $\Psi_{\rm TH}$ – are the contributions to the total phase arising from the point-particle approximation, tidal deformability and tidal heating, respectively. The presence or absence of a horizon can be tested by measuring the values of the tidal heating and tidal deformability terms in a binary’s phase. In the current work we show how the tidal heating term can be used for this purpose. As per current knowledge GW absorption is negligible for all matter [@Glampedakis:2013jya]. Therefore, it is reasonable to use evidence for tidal heating, in binary GW waveforms, to discern the existence of horizons [@Datta:2019euh; @Maselli:2017cmm]. Guided by this expectation, we have already intriduced the [*horizon parameter*]{} $H$ for extreme mass-ratio inspirals that LISA may observe [@Datta:2019euh]. In the current work we extend it to binaries with similarly massive components primarily to target the population of stellar-mass binaries being detected by LIGO and Virgo. For a near-equal-mass binary we define horizon parameters for each component, $(H_1, H_2)$, such that the value of $H_i$ is 1 (0) when the $i$th component has a horizon present (absent). In the case of circular orbits, the flux of energy at the horizon can be expressed as a PN expansion [@Alvi:2001mx; @Poisson:2018qqd; @Poisson:2009di; @Nagar:2011aa; @Bernuzzi:2012ku; @Chatziioannou:2016kem; @Cardoso:2012zn]. Since tidal heating is the signature of the presence of a horizon, we multiply the energy flux absorbed by each component with the corresponding $H_i$. In the case of partial absorption, one has $0<H_i<1$. Therefore, the absorbed flux can now be written as $$\begin{aligned} -\frac{dE}{dt} = &{}\frac{32}{5}\nu^2 \frac{v^{15}}{4}\sum_{i=1}^{2} H_i\left(\frac{m^{}_i}{m}\right)^3 \left( 1 + 3\chi^2_i\right)\left\{-(\hat{L}_N.\hat{S}_i)\chi^{}_i \right.\\ &\left. + 2 \left[ 1+\left( 1 - \chi^2_i\right)^{1/2}\right]\frac{m^{}_i}{m}v^3\right \}\,, \end{aligned}$$ where $\nu = {m_1 m_2}/{m^2}$ is the symmetrized mass-ratio, and $\hat{S}^{}_i$ and $\hat{L}_N$ are the unit vectors along the directions of the $i$th spin and the orbital angular momentum, respectively. We add this contribution to the PP flux in the TaylorF2 (TF2) approximation [@Buonanno:2009zt; @Damour:2000zb] in order to calculate the phase shift. That phase term is then added to the TaylorF2 GW waveform. We will call this resulting waveform HeatedTaylorF2 (HTF2) to distinguish it from TF2. We treat the $H_i$ as independent parameters whose values can be estimated from observations, thus, revealing the presence or absence of component horizons. [*New waveform parameters characterizing tidal heating*]{}.— The horizon parameters $H_1$ and $H_2$ appear in the flux, and the GW phase, in terms that also include mass and spin factors. This makes them [*degenerate*]{} with those parameters, in that it is more practical to measure the following effective observables instead of $H_{1,2}$: \[Eq.Hparams\] $$\begin{aligned} H^{}_{eff5} \equiv &{} \sum_{i=1}^{2}H^{}_i \left(\frac{m^{}_i}{m}\right)^3 \left(\hat{L}.\hat{S}^{}_i\right)\chi _i \left(3 \chi_i^2+1\right)\,,\\ H^{}_{eff8} \equiv &{} ~4 \pi H^{}_{eff5}+\sum^2_{i=1}H^{}_i \left(\frac{m_i}{m}\right)^4 \left(3 \chi_i^2+1\right)\nonumber \\ &\quad\quad\quad\quad\quad\quad\quad \times \left(\sqrt{1-\chi_i^2}+1\right)\,.\end{aligned}$$ These are analogous to the effective spin parameter $\chi_{\rm eff}$ that was introduced [@Damour:2001tu; @Racine:2008qv; @Ajith:2011ec] to characterize spinning compact binary waveforms: While the spins of the individual binary components are themselves difficult to measure (like $H_{1,2}$ here), their combined impact on the waveform phase, captured by $\chi_{\rm eff}$, lends itself to more precise measurements. Dependence of $H_{eff5}$ and $H_{eff8}$ on the spins of the components is shown in Figs. \[Heff5\_spin\] and \[Heff8\_spin\], respectively. Note that if the system is a binary black hole (BBH), as long as any one of the component spins is finite both $H_{eff5}$ and $H_{eff8}$ will be non-zero. By contrast, for the same spins a horizonless binary would have both $H_{eff5}$ and $H_{eff8}$ vanish. Therefore, it is easiest to discern between the presence and absence of horizons in binaries that have at least one component with a sufficiently large spin. On the other hand, when both component spins of a BBH tend to zero, one has $H_{eff5}$ tending to zero but $H_{eff8}$ non-zero; see the insets in Figs. \[Heff5\_spin\] and \[Heff8\_spin\]. Therefore, in the low-spin limit $H_{eff8}$ emerges as a discriminator for the presence or absence of horizons. Here the measurement is helped for small mass-ratio ($q$), which ensures large $H_{eff8}$. ![$H_{eff5}$ is plotted for a range of $\chi_{eff}$ values and for all possible values of $m_2/m$.[]{data-label="Heff5_spin"}](H_eff5_with_chieff_plot.png){width="\linewidth"} ![Analogous to Fig. \[Heff5\_spin\], but here for the parameter $H_{eff8}$.[]{data-label="Heff8_spin"}](H_eff8_with_chieff_plot.png){width="\linewidth"} It is important to note that our choice of waveforms, based on the stationary-phase approximation (SPA), is for illustrative purpose, essentially as a proof of principle that the method proposed here is promising for identifying CBCs with horizons from those without. For making such classification in real data, it will likely be important to use more accurate templates, such as those based on the EOB-NR formalism [@Husa:2015iqa; @Khan:2015jqa; @Hannam:2013oca]. We will present those results in future. Having said that, we argue that our choice of SPA-based inspiral waveforms is a reasonable one for illustrating the power of this method for the systems studied here. We deduce the GW phase involving tidal heating by using Eq. (2.7) of Ref. [@Tichy:1999pv] (see [@Isoyama:2017tbp] for the details). We find the phase shift due to the associated horizon absorption to be $$\begin{aligned} \label{eq:phase correction} \Psi^{}_{\rm TH} = &{} \frac{3}{128\nu} \left(\frac{1}{v}\right)^5 \left[- \frac{10 }{9 }v^5 H^{}_{eff5} \left(3 \log \left(v\right)+1\right) \right. \\ &- \frac{5}{168} v^7 H^{}_{eff5} \left(952 \nu +995\right) \\ &\left.+ \frac{5}{9}v^8 \left(3 \log \left(v\right)-1\right)(-4 H^{}_{eff8}+ H_{{eff5}} \psi^{}_{\text{SO}} )\right]\,, \end{aligned}$$ where at the end of the expression above we have used $$\begin{split} \psi^{}_{\text{SO}} \equiv~ &\frac{1}{6} \big[\big(-56 \nu -73 \sqrt{1-4 \nu }+73\big) \big(\hat{L}.\hat{S}^{}_1\big) \chi _1 \\ &+\big(-56 \nu + 73 \sqrt{1-4 \nu }+73 \big) \big(\hat{L}.\hat{S}^{}_2 \big) \chi _2\big]\,. \end{split}$$ Note that $H_{eff5}$ and $H_{eff8}$ arise at different PN orders in the phase. For the non-spinning case, even though $H_{eff5} = 0$ one has $H_{eff8} \neq 0$. The ranges of values spanned by these two effective horizon parameters are shown in Fig. \[Heff5\_spin\] and Fig. \[Heff8\_spin\], respectively. [*Viability of tidal heating as a horizon discriminator*]{}.— In this work, we show (a) how Bayesian inference can be applied to constrain or measure the effective horizon parameters $H_{eff5}$ and $H_{eff8}$ in data from LIGO-like ground-based detectors, and (b) probe the presence or absence of horizon via binary GW strain signals by performing Bayesian model selection on the data. To illustrate our idea imagine injecting two types of signals in aLIGO-like noisy data of a single detector – one for a BBH and another for a BNS, where both binaries have the same component masses (and, say, no spin – since large spin itself can help discriminate between NS and BH). A Bayesian way [@Veitch:2014wba; @Thrane:2018qnx; @Sivia2006] to test if these two systems can be distinguished from each other is to do the following two experiments: (1) For the BBH injection, we first compute the evidence $p({\rm H}_{1,2}=1,~\Lambda_{1,2}=0 ~|~ \rm{BBH})$ that it has a horizon by cross-correlating it with BBH waveforms (which by definition have $H_{1,2} = 1$). We next compute the evidence $p({\rm H}_{1,2}=0,~\Lambda_{1,2} \neq 0 ~|~ \rm{BBH})$ that it lacks a horizon but has NS-like tidal deformability by cross-correlating it with BNS waveforms. The ratio of the former evidence to the latter is what is defined as the Bayes factor (BF). For large values of the log of this factor (typically around 2 or more), the probability is taken to be very high that the data supports the hypothesis that the signal in it is that of a BBH. The second experiment is the same as above but with a BNS signal injection replacing the BBH one. We will present results for that experiment elsewhere. ![Horizon injections (Heated TaylorF2): We plot the log-Bayes factors for 100 HTF2 (BBH) signals injected individually in simulated aLIGO data streams. All sources are placed at 100 Mpc, and the component masses are uniformly distributed in the range $3-5M_{\odot}$. The individual spins are also varied. Half of the injections have both component spins aligned with the orbital angular momentum and the other half have them anti-aligned with the same reference. Evidence of the presence (absence) of horizon is calculated by using HTF2 (TidalTF2) waveforms as templates. The true hypothesis is taken to be HTF2 (BBH). This plot shows that all the injections are being identified correctly (with a minimum log-Bayes factor of 2.0).[]{data-label="fig:logBFLowMass"}](logBF_chi_eff_100MPC_BBH_BNS.pdf){width="\linewidth"} ![Distribution of the log-Bayes factor values for the signals shown in Fig. \[fig:logBFLowMass\].[]{data-label="fig:logBFLowMassDistribution"}](Hist_BF_duration_4s_100MPC.pdf){width="\linewidth"} ![ Distribution of log-Bayes factors estimated from a population 100 BBHs, each injected in simulated Gaussian data streams with aLIGO-ZDHP noise. Here, all the injections are randomly oriented and uniformly distributed between 100 to 250 Mpc in comoving volume. As in Fig. \[fig:logBFLowMassDistribution\], here too the log-Bayes factor is discriminating between the HTF2 model (true hypothesis) and the TidalTF2 model.[]{data-label="fig:logBFLowMassDistributionVolume"}](Hist_BF_duration_4s_100-250MPC.pdf){width="\linewidth"} [*Simulation and Results*]{}.—For the model selection test we simulated a population of 100 binaries. The binary components spins aligned or anti-aligned with the orbital angular momentum. Each component has mass $\in [3-5] M_{\odot}$ and dimensionless-spin magnitude $\in [0, 0.75]$. For model selection we constructed two families of templates, namely: (a) TaylorF2 (TF2), modified with tidal deformability contribution (TidalTF2) for representing horizonless components with nonzero tidal deformability. Here, the GW phase is devoid of any contribution from $H_{eff5}$ or $H_{eff8}$; (b) HeatedTaylorF2 (HTF2), which is TF2 but with additional phase terms arising from tidal heating, as described in Eq. (\[eq:phase correction\]). Using the aforementioned waveform models we performed simulated signal injection studies in simulated aLIGO zero-detuned high-power (ZDHP) noise power-spectral density [@design-sensitivity]. In one study the source is taken to be CBCs of black hole components. Hence, the injected waveform used is HTF2. We then used a Bayesian analysis to measure the parameters of these sources with both TidalTF2 and HTF2 templates (see the appendices), and compared their evidences for the same (horizon) injections in order to test if such an analysis has the power to identify the true signal model. The results for all the sources located at $100$Mpc are shown in Fig. \[fig:logBFLowMass\]. The lowest value for the log-Bayes factor, $\log BF$, is $\sim 2.0$, which shows that for this choice of the binaries and detector sensitivity, the analysis is able to classify the source type as black holes correctly. In a second study we distributed BBH sources uniformly in comoving volume with luminosity distance in the range $\in [100, 250]$ Mpc. As shown in Fig. \[fig:logBFLowMassDistributionVolume\], here the log-Bayes factor got spread to lower values compared to the first study, with about 20% of the sources not distinguishable from horizonless binaries. These are mostly the ones injected beyond 200 Mpc. It is important to note that even though a BH can have very high spin, with $\chi\sim 1$, a NS can not. Pulsar observations indicate that while the fastest-spinning neutron star has an observed $\chi \leq 0.4$ [@Hessels:2006ze], the fastest-spinning BNSs that will merge within a Hubble time, namely, PSR J0737– 3039A [@Burgay:2003jj] and PSR J1946+2052 [@Stovall:2018ouw], will have at most $\chi \sim 0.04 - 0.05$ when they merge. Therefore, from the measurement of the spins it is possible to get an indication about the nature of the components. However, this property is a useful discriminator mainly for high-spin objects. Our tidal-heating based method works well for both high- and low-spin objects, as can be seen in Fig. \[fig:logBFLowMass\]. There are instances in the literature where complete NSBH waveforms have been constructed by using numerical relativity (NR) simulations of BBH mergers [@Lackey:2013axa; @Kumar:2016zlj; @Pannarale:2015jka; @Nagar:2018zoe; @Chakravarti:2018uyi; @Thompson:2020nei]. Since the NR BBH part of these waveforms would have tidal heating in them, extrapolating them to earlier inspiral part of the waveform may contain imprints of it. This can result in incorrect estimates of the tidal deformability parameter and, therefore, the equation of state, of the NS. A similar problem may occur in BNS waveforms as well where NR BBH waveforms have been used as a reference but cancellation of the tidal heating part from the complete waveform has not been carried out properly. These issues will be investigated in the future and errors corrected for wherever required. [*Summary*]{}.—We have developed a method to search for and characterize tidal heating from the inspiral phase of a binary. We have defined two new parameters that capture the effect of tidal heating in the inspiral waveform. These parameters are robust enough that even partial absorption can be modeled with them – something we will pursue in detail in future. To test the presence of horizon we performed model selection using the Bayes factor. We constructed two sets of waveforms, one for binary black holes, which incorporates tidal heating but no tidal deformability, and the other for binaries of horizonless compact objects, which does not include tidal heating but has nonzero tidal deformability. Using these two types of waveforms, we showed that it will be possible to distinguish between such stellar-mass binaries in aLIGO-like detectors in most cases considered here. These results are interesting since they make it feasible to test the presence of the horizon using the data from LIGO-Virgo observations and can also possibly be utilized to identify the nature of compact objects that show up in the mass gap [@Littenberg:2015tpa; @Gupta:2019nwj]. An immediate continuation of the current work will be to construct better waveform models than TidalTF2 and HTF2 that can be used for parameter estimation and model selection of real signals in contemporaneous GW detector data. Another problem we plan to address is the challenge posed by mixed binaries (NSBH) in discerning the presence of horizons. Thirdly, future generation detectors may allow enough precision so that not only horizon parameter values of 0 and 1 can be discriminated but putative intermediate values may also be measurable, thereby, affording the possibility of probing the existence of exotic compact objects, such as gravastars, boson stars, etc. [@Datta:2020rvo; @Datta:2019epe]. [*Acknowledgments*]{}.— It is a pleasure to thank Andrea Maselli and Paolo Pani for useful discussions. We would also like to thank Richard Brito and Otto Hannuksela for carefully reading the manuscript and providing helpful inputs, and Bhaskar Biswas, Soumak Maitra and Niladri Paul for useful comments. We gratefully acknowledge the use of the IUCAA computing cluster, Sarathi, and the computational resources provided by the LIGO Laboratory (CIT) and supported by National Science Foundation Grants PHY-0757058 and PHY-0823459. SD would like to thank University Grants Commission (UGC), India, for financial support for a senior research fellowship. KSP acknowledges support of the Science and Engineering Research Board (SERB), India and the Netherlands Organisation for Scientific Research (NWO). This work was done with partial support provided by the Tata Trusts. This paper has been assigned LIGO Document Number LIGO-P2000115. Bayes factors for horizon discrimination ======================================== In GW data analysis, the detector strain data $d(t)$ can be modeled as $d(t) = n(t) + h(t, \pmb{\theta})$, where $n(t)$ and $h(t)$ denote detector’s noise and the possible GW signal with parameters $\pmb{\theta}$, respectively. Given the detector data $d(t)$ with GW signal $h(t,\pmb{\theta})$ described by a model $\mathcal{H}$, the likelihood for the data under the assumption of Gaussian and stationary noise can be written as follows [@Cutler:1994ys]: $$\label{eq:likelihood} P\left(d|\pmb{\theta}, \mathcal{H}, I \right) \propto \exp{ \left[ -\frac{1}{2} \langle d - h(\pmb{\theta})| d - h(\pmb{\theta}) \rangle \right]}\,.$$ The angular bracket in Eq. (\[eq:likelihood\]) defines a noise-weighted inner product between two real time-series $a(t),\, b(t)$, and is given as $$\langle a, b \rangle = \Re \int_0^\infty df \frac{\tilde{a}^*(f)\tilde{b}(f)}{S_n(f)},$$ where $S_n(f)$ is the one-sided power spectral density (PSD) of the detector noise. Using the inner product, one can also define the signal-to-noise ratio (SNR) $\rho$ for the template $h(t, \pmb{\theta})$ as $$\rho = \frac{\langle d|h \rangle}{\sigma},$$ where $\sigma = \sqrt{\langle h|h\rangle}$ is the template normalization. For $N$ GW detectors, we can define the network SNR $\rho_{\textrm{network}}$ as $$\rho_{\textrm{network}} = \sqrt{ \sum_i^N \rho_i^2},$$ where $\rho_i$ denotes SNR in the $i$th detector. We will assume that non-colocated detectors on the globe have uncorrelated noise; hence, the combined likelihood is $$P\left(\mathbf{d}|\pmb{\theta}, \mathcal{H}, I \right) = \prod_i^N P\left(d_i|\pmb{\theta}, \mathcal{H}, I \right)\,,$$ where $\mathbf{d}\in\{d_1, d_2,\cdots,d_N\}$ represents combined data from all $N$ detectors. Using the coherent network likelihood function, posterior probability density can be written as $$P\left(\pmb{\theta}|\mathbf{d},\mathcal{H} \right) = \frac{P\left(\mathbf{d}\pmb{\theta}, \mathcal{H}, I \right) P\left(\pmb{\theta}|\mathcal{H} \right)}{ P\left( \mathbf{d}|\mathcal{H}\right)}\,,$$ where $P\left(\pmb{\theta}|\mathcal{H} \right)$ is the prior probability density function or prior of the parameters $\pmb{\theta}$. In the denominator, $P\left( \mathbf{d}|\mathcal{H}\right)$ is the marginalized posterior probability density over all parameters $\pmb{\theta}$, and is also known as the evidence for the model $\mathcal{H}$. The evidence $P\left(d|\mathcal{H} \right)$ serves as a normalization constant of the posterior probability for $\mathcal{H}$. The evidence computed for two competing models or hypotheses can be used to determine which one is favored by the data. In this work, we compute Bayes factors for simulated signals to compare two hypotheses, namely, 1. The horizon hypothesis $\mathcal{H}_\textrm{h}$: Signal carries an imprint of horizon absorption, 2. The no-horizon hypothesis $\mathcal{H}_{\textrm{nh}}$: Signal has no imprint of horizon absorption. ![These parameter estimation corner plots show that the measurement of the new parameters introduced here, $H_{eff5}$ and $H_{eff8}$, will add little bias in the estimation of standard intrinsic parameters of compact binaries. The horizon parameters along with the other intrinsic parameters can be recovered from GW data using the HTF2 signal model. The precision of the measurement of the HTF2 model can be seen from a probability-probability (P-P) plot of the estimated parameters, such as the one shown in Fig. \[fig:p-p-plot\] for a set of injections.[]{data-label="fig:horizon_parameters"}](injection_bbh_bbh_100MPC_4s_data72_0_analysis_H1L1_dynesty.png){width="\linewidth"} ![ Inaccuracies in waveform models and detector noise can lead to offsets in the inferences about the source parameters. The P-P plot in this figure quantifies performance of our parameter estimation, that shows the correct parameter values of P% sources from a population recovered with P% confidence interval (C. I.). Ideally, a parameter should follow a one-to-one relation between fraction of sources and confidence interval, though fluctuations in the relation is expected. We injected 100 BBH signals in simulated Gaussian noise with aLIGO-ZDHP noise to generate this figure. The dark-, dim- and light-grey regions, respectively, denote the $1\sigma, 2\sigma$ and $3\sigma$ deviations from the theoretical expected diagonal line. The deviation P-P plot from the one-to-one behavior for a parameter is quantified by the $p$-value (displayed in brackets in the figure legend), obtained by employing the Kolmogorov-Smirnov test. The individual $p$-values are combined using the Fisher method to yield the combined $p$-value of 0.0825, which indicates that the analysis worked satisfactorily. []{data-label="fig:p-p-plot"}](horizon_pp_plot_sampples_100_prior_file_4s.pdf){width="\linewidth"} In Bayesian model selection, we compute the Bayes factor, $$\textrm{BF} = \frac{P\left( \mathbf{d}|\mathcal{H}_\textrm{h}\right) }{P\left( \mathbf{d}|\mathcal{H}_{\textrm{nh}}\right)}\,.$$ If the Bayes factor is greater than some preset threshold, [*i.e.*]{}, $\textrm{BF}> \textrm{BF}_{\textrm{Th}}$ then the hypothesis $\mathcal{H}_\textrm{h}$ is preferred over the other hypothesis $\mathcal{H}_\textrm{nh}$ in the data. Moreover, we use the dynamical nested sampling [@2020MNRAS.tmp..280S], as implemented in the Bilby package [@Ashton:2018jfp], to computed the posterior probability density for our simulated signals. The posteriors of the horizon parameters are shown in Fig. \[fig:horizon\_parameters\]. For computing them we limit the signal integration above to a frequency range $(20,f_{\rm ISCO})$Hz, where $f_{\rm ISCO}$ is the instantaneous GW frequency at the innermost stable circular orbit (ISCO) of the binary [@Kidder:1992zz; @Blanchet:2001id]. In practice, it may be possible to begin the signal integration at a frequency as low as 10Hz, which is what aLIGO design targets. Similarly, when waveform modeling is available to accurately incorporate tidal heating beyond the ISCO, the upper frequency cut-off will also be raised. Both these changes will improve parameter estimation as well as Bayes-factor based model discrimination. Priors ====== The distributions and ranges of parameter priors of the simulated binary waveforms used in our Bayesian model selection studies are listed in Table \[tab:priors\]. [Parameter]{} [Distribution]{} [Range]{} [Boundary condition]{} Units ------------------------------------------------------- ------------------ ------------------ ------------------------ -------------------- Chirp mass ($\mathcal{M}$) Uniform 3.0 - 4.35 – $\textrm{M}_\odot$ Mass ratio ($q$) Uniform 0.5 - 1 – – Spin of primary object $(\chi_1)$ Uniform 0 - 0.89 Reflective – Spin of secondary object $(\chi_2)$ Uniform 0 - 0.8 Reflective – Tidal deformability of primary object $(\Lambda_1)$ Uniform 0 - 5000 – Tidal deformability of secondary object $(\Lambda_2)$ Uniform 0 - 5500 – 2.5 PN horizon parameter ($H_{eff5}$) Uniform -4 - 4 – – 4 PN horizon parameter ($H_{eff8}$) Uniform -45 - 45 – – Luminosity distance ($d_L$) Uniform 10 - 500 – Mpc Right ascension (RA) Uniform $0 - 2\pi$ Periodic radian Declination (DEC) Cosinusoidal $-\pi/2 - \pi/2$ – radian Polarization angle $(\psi)$ Uniform $0 - \pi$ Periodic radian Phase at reference frequency $(\phi^{}_0)$ Uniform $0 - 2\pi$ Periodic radian Line-of-sight angle $(\theta_{JN})$ Sinusoidal $0 - \pi$ – radian

--- abstract: 'Studying the physical environments of low mass and high mass cores using dust continuum emission provides important observational constraints on theoretical models of star formation. The motivation and procedure for modeling dust continuum emission is reviewed and the results of recent surveys towards low mass and high mass star forming regions are compared.' author: - 'Yancy L. Shirley, Kaisa E. Mueller, Chadwick H. Young, Neal J. Evans' title: Submillimeter Dust Continuum Studies of Low and High Mass Star Formation --- Introduction ============ Optically thin dust emission at submm and mm wavelengths is a powerful probe of the density and temperature structure of the outer envelope of protostars. Models of the dust continuum emission constrain theoretical predictions of the structure of forming protostellar cores. The resolution of current submm and mm bolometer arrays effectively image the outer envelope on scales of $10^3$ to $10^5$ AU. The basic procedures for understanding the density and temperature structure are reviewed as well as the need for radiative transfer modeling (§2). The density and temperature structure of the envelopes of low mass and high mass star forming regions are compared (§3) and important systematic effects are discussed (§4). Dust Continuum Emission ======================= The specific intensity, at impact parameter $b$, of optically thin dust continuum emission from a spherical envelope is given by $$I_{\nu}(b) = \int_{los} B_{\nu}[T_d(s)] d\tau = \frac{4\mu m_H h \nu ^3}{c^2} \int_b^{r_o} \frac{\kappa_{\nu}(r) n(r)}{ \exp\left[ \frac{h\nu}{k T_d(r)} \right] - 1 } \;\; \frac{r dr}{\sqrt{r^2 - b^2}} \;\;,$$ where $s$ is a distance along the line-of-sight, $\mu m_H$ is the mean molecular mass of gas in grams, $r_o$ is the outer radius, $\kappa_ {\nu }(r)$ is the dust opacity in cm$^2$/gram of gas, $n(r)$ is the gas particle density in cm$^{-3}$, and $T_d(r)$ is the dust temperature distribution (see Shirley et al. 2000). Generally, this integral must be solved numerically. However, several simplifying assumptions provide an analytical solution: (1) The dust emits in the Rayleigh-Jeans limit ($T_d >> h\nu / k$); (2) The temperature and density follow single power law distributions ($T_d(r) = T_f (r / r_f)^{-q}$ and $n(r) = n_f (r / r_f)^{-p}$); (3) The dust opacity is constant along the line of sight ($\kappa_{\nu}(r) = \kappa_{\nu}$); (4) $r_o \rightarrow \infty$. With these assumptions, the specific intensity can be expressed as a power law in the impact parameter ($I_{\nu}(b) \propto b^{-m}$) with the exponent $m = (p + q) - 1$. The density power law index, $p$, is found by fitting a power law to the observed intensity distribution and assuming a temperature power law index, $q$, to find $p = m + 1 - q$. The analytical solution is not applicable at submm wavelengths (including 1.3mm) for several reasons. The dust temperature fails the Rayleigh-Jeans criterion in the outer envelopes of low mass and high mass protostars since the dust temperature drops below 20 K at large radii (i.e., only $2h\nu /k$ at 1.3mm). The temperature distribution departs from a single power law in the inner regions of the envelope where the radiative transfer becomes optically thick at UV to mid-IR wavelengths. In low mass protostars, heating from the interstellar radiation field (ISRF) becomes important in the outer envelope causing the dust temperature rise towards the outside edge of the core. The observed specific intensity profile has been convolved with a complicated beam pattern, containing multiple sidelobes, and is further modified by chopping and the detailed observing procedure (e.g., scanning). Therefore, to understand the density structure, we must model the radiative transfer to self-consistently calculate the temperature distribution, and then simulate the observing mode (beam convolution, chopping, etc.) to compare with observations. Radiative Transfer Models ========================= Two surveys of the deeply embedded phases of low and high mass star formation were recently carried out at the University of Texas: a SCUBA 850 and 450 $\micron$ survey of 39 nearby low mass star forming regions (Shirley et al. 2000, Shirley et al. 2002, Young et al. 2002) and a SHARC 350 micron survey of 51 high mass star forming regions associated with water masers (Mueller et al. 2002). Submm images are shown in Figure 1. The normalized, azimuthally averaged, intensity profiles and spectral energy distributions (SED) of 19 sources from the SCUBA survey (3 Pre-ProtoStellar cores, 7 Class 0, and 9 Class I) and 31 sources from the SHARC survey were modeled using a one dimensional radiative transfer code (Egan, Leung, & Spagna 1988) that takes into account heating from an internal source, heating from the ISRF, beam convolution, and chopping. The detailed testing of the model parameter space is discussed in Evans et al. (2001), Shirley et al. (2002), Young et al. (2002), and Mueller et al. (2002). The modeled normalized intensity profile is very sensitive to the density structure of the core while the modeled SED is sensitive to the mass and opacity ($\kappa _\nu$). Ossenkopf & Henning (1994) opcaities for coagulated dust grains with thin ice mantles fit the observed SEDs of both samples well. The temperature profile from the best fit radiative transfer models of two low mass cores (the PPC L1544 and Class 0 core B335), and a high mass core (M8E) are shown in Figure 2. Single power law temperature distributions do not fit the calculated profiles in the regions probed by SCUBA and SHARC. The temperature profile in a PPC (no internal luminosity source) drops towards the center as UV to near-IR radiation from the ISRF is attenuated. The temperature profile increases dramatically towards the center for sources with internal luminosity (B335, M8E). Heating from the ISRF strongly affects the shape of the temperature profile for low mass sources in the region of envelope probed by SCUBA (e.g., B335) and has some effect in regions probed by SHARC. Sources with internal luminosity are well fitted by a single power law density profile. The histograms of best fit power-law index, $p$, for the low mass and high mass sample are very similar (Figure 3). The average $p$ is $1.8 \pm 0.4$ for the high mass cores and $1.6 \pm 0.4$ for the low mass cores. The low mass cores may be sub-divided into Class 0 and Class I objects based on the T$_{bol}$ criterion (Class 0 typically have T$_{bol} < 70$ K, Chen et al. 1995). The average $p$ for 10 Class 0 cores is $1.7 \pm 0.30$ and is $1.6 \pm 0.4$ for 9 Class I cores. No evidence for evolution in the shape of the density profile is seen between Class 0 and Class I cores (Young et al. 2002). These results can be compared to other submm and mm surveys towards high mass regions (van der Tak et al. 2000 towards H$_2$O masers and Hatchell et al. 2000 and Beuther et al. 2002 towards UCHII regions) and low mass regions (Chandler & Richer 2000, Hogerheijde et al. 2000, Motte & André 2001, and J[ø]{}rgensen et al. 2002). Beuther et al. (2002) observed 69 high mass cores at 1.2 mm and fit broken power laws to the intensity profile. The average $p$ is $1.6 \pm 0.5$ in the inner regions ($\theta < 32^{''}$), similar to the average $p$ found by Mueller et al. There is very little overlap between high mass samples. The two sources in common agree withing uncertainties in $p$. The Motte & André (2001) survey at 1.3 mm towards low mass cores gave an average $p$ steeper by 40% for 10 sources in common. Motte & André use the analytical approximation with a single temperature power law (with q ranging from $-0.2$ to $+0.4$); however, detailed modeling has shown that the temperature profile changes from falling (positive $q$) to rising (negative $q$) within the regions of the envelope probed by these two surveys (Figure 2). J[ø]{}rgensen et al. (2002) use a one dimensional model of the radiative transfer of SCUBA-observed, low mass cores, but ignore the effects of the ISRF. For 7 sources in common, their average $p$ is flatter by 30% . If the ISRF is not included in the model, the temperature profile will continue to drop towards the outside of the core and the resulting best fit density distribution must be flatter to compensate for the colder dust grains in the outer envelope. There are significant variations between the best fit models from these surveys. The effects of the ISRF on the temperature profile can partially explain the differences and must be included in radiative transfer models (Shirley et al. 2002). The robustness of the best fit density distribution can be tested by comparing to the density derived from near-IR extinction maps. In the case of B335, a low mass Class 0 protostar, the best fit density distribution ($n(r) \sim r^{-1.8}$, Shirley et al. 2002) agrees well with the extinction profile derived from NICMOS images (Harvey et al. 2001) for radii beyond $5000$ AU. The near-IR extinction map is unable to probe regions of high extinction (A$_{\mathrm{V}}$ $>$ 30 $-$ 50 mag) due to a lack of background sources (Alves et al. 1999, Lada et al. 1999); therefore, we are unable to compare methods in the inner regions of the envelope of B335 ($r < 5000$ AU). Interferometric observations are needed at submm wavelengths to test the findings from the dust continuum models at smaller radii. Nevertheless, it is encouraging that two different methods are consistent in the outer region of the envelope. Further comparisons with NIR extinction maps are anxiously awaited. Caveats & Future Work ===================== There are many caveats and systematic effects that may affect the interpretation of the best fit density distribution. Several sources (e.g., L1544 and L1157) have asymmetric contours that cannot be modeled with a one dimensional radiative transfer code. Multi-dimensional radiative transfer codes are needed to model asymmetric cores. Outflows are observed towards many of the sources in our sample. In the near-IR extinction study of B335, it was necessary to consider clearing of material in outflow cones with a $35^{o} - 45^{o}$ opening angle, while the submm emission displays no evidence of the outflow (Figure 1). However, several cores display emission extending along the outflow direction (e.g., L1157). A 1D radiative transfer code cannot properly model the effects of the outflow. Pure envelope models without disks have been used in the models; however, disks may contribute a significant fraction of the flux at submm wavelengths within the central beam. Since a centrally normalized radial profile is used, the disk contribution may flatten the interpretation of the density profile (in an extreme limit) up to $\Delta p \sim -0.5$ (Shirley et al. 2002, Young et al. 2002). Strong constraints on the disk flux await observations by submm interferometers (SMA and ALMA). The potential importance the disk must not be ignored in future dust continuum studies. Dust continuum modeling is a powerful diagnostic of the density and temperature structure of protostellar cores. The methods of modeling will become more refined with 3D radiative transport and the inclusion of asymmetries, outflows, and disks. Studies of protostellar envelopes on scales of 10 - 10$^5$ AU will be possible with the combination of submm interferometers and single dish bolometer cameras. Alves, J., Lada, C. J., Lada, E. A. 1999, , 515, 265 Beuther, H., Schilke, P., Menten, K. M., Motte, F., Sridharan, T. K., & Wyrowski, R. 2001, , 566, 945 Chandler, C. J., & Richer, J. S. 2000, , 530, 851 Chen, H., Myers, P. C., Ladd, E. F., & Wood, D. O. S. 1995, , 445, 377 Egan, M. P., Leung, C. M., & Spagna, G. R. 1988, Comput. Phys. Comm., 48, 271 Evans, N. J., II, Rawlings, J. M. C., Shirley, Y. L., & Mundy, L. G. 2001, , 557, 193 Harvey, D.W.A., Wilner, D. J., Lada, C. J., Myers, P. C., Alves, J. F., & Chen, H. 2001, , 563, 903 Hatchell, J., Fuller, G. A., Millar, T. J., Thompson, M. A., & MacDonald. G. H. 2000, , 357, 637 Hogerheijde, M. R., & Sandell G. 2000, , 534, 880 J[ø]{}rgensen, J. K., Schöier, F. L., & van Dishoeck, E. F. 2002, , in press Lada, C. J., Alves, J., Lada, E. A. 1999, , 512, 250 Motte, F., & André, P. 2001, , 365, 440 Mueller, K. E., Shirley, Y. L., Evans, N. J., II, & Jacobson, H. R. 2002, , submitted Ossenkopf, V., & Henning, T. 1994, A & A, 291, 943 Shirley, Y. L., Evans, N. J., II, & Rawlings J. M. C., & Gregersen, E. M. 2000, , 131, 249 Shirley, Y. L., Evans, N. J., II, & Rawlings J. M. C. 2002, , 575, in press Young, C. H., Shirley, Y. L., Evans, N. J., II, & Rawlings, J. M. C. 2001, , submitted.

--- author: - | Wim Lavrijsen, Ana Tudor, Juliane Müller, Costin Iancu, Wibe de Jong\ Lawrence Berkeley National Laboratory\ *\ * bibliography: - 'minimizers.bib' - 'quantum.bib' - 'quant\_chem.bib' title: - 'Classical Optimizers for Noisy Intermediate-Scale Quantum Devices' - 'Classical Optimizers for Noisy Intermediate-Scale Quantum Devices' --- Acknowledgment {#acknowledgment .unnumbered} ============== This work was supported by the DOE under contract DE-5AC02-05CH11231, through the Office of Advanced Scientific Computing Research (ASCR) Quantum Algorithms Team and Accelerated Research in Quantum Computing programs.

--- abstract: 'An inverse population transfer of the repulsive Bose-Einstein condensate (BEC) in a weakly bound double-well trap is explored within the 3D time-dependent Gross-Pitaevskii equation. The model avoids numerous common approximations (two-mode treatment, time-space factorization, etc) and closely follows the conditions of Heidelberg experiments, thus providing a realistic description of BEC dynamics. The transfer is driven by a time-dependent shift of a barrier separating the left and right wells. It is shown that completeness and robustness of the process considerably depend on the amplitude and time profile of the shift velocity. Soft profiles provide the most robust inversion. The repulsive interaction substantially supports the transfer making it possible i) in a wide velocity interval and ii) three orders of magnitude faster than in the ideal BEC.' author: - 'V.O. Nesterenko$^1$, A.N. Novikov$^1$ and E. Suraud$^2$' title: 'Inverse population transfer of the repulsive Bose-Einstein condensate in a double-well trap: strong interaction-induced support' --- Introduction ============ The population inversion is a typical problem met in various branches of physics (ultracold gases and condensates [@Gati_07; @Torr_12_rev; @Yuk], atomic and molecular physics [@Kral], etc.). The problem is easily solvable, if it is linear and accepts an adiabatic evolution, see e.g. the Landau-Zener scenario [@Lan; @Zen]. However, if there are significant nonlinear effects or/and we need a rapid but robust transfer, the problem becomes nontrivial, e.g. for a transport of Bose-Einstein condensate (BEC) [@Nest_LP_09; @Nest_JPB_09]. The inverse population transfer of a repulsive BEC in a double-well trap is the relevant example of such a complex problem[@Nest_JPB_09]: the repulsive interaction in BEC leads to a strong nonlinearity and a limited life-time of BEC requests a rapid transfer. However, a rapid process, if not specially designed, is usually spoiled by dipole oscillations at the final state. The question is how to produce a robust (without final dipole oscillations) and rapid [*nonlinear*]{} population inversion (NLPI) in this case? Another important aspect is how does the nonlinearity affect the process? In principle, this problem is a subject of so-called shortcuts-to-adiabaticity (StA) methods which have made a significant progress over the last years, see [@Torr_12_rev] for an extensive review. However, to our knowledge, these methods have not yet been derived for our particular case: NLPI for BEC in a double-well trap. Moreover, even though some StA methods, like the optimal control theory [@oct_Gross; @oct_Brief], may be potentially implemented in this case, their protocols might be too complicated and parameter-sensitive to be realized in experiment, while we need a simple prescription with a minimal number of control parameters. We will show that such a prescription can be built using the setup of Heidelberg experiments [@Gati_07], where a time-dependent shift $x_0(t)$ of the barrier is used as a suitable control parameter driving the trap asymmetry and thus the population transfer. In the present study, the barrier shift is used to produce NLPI of BEC in a double-well trap. The calculations are performed within the three-dimensional (3D) time-dependent Gross-Pitaevskii equation [@GPE] for the total order parameter covering both left and right parts of the condensate. Our model [@Nest_JPB_12] is free from numerous approximations (two-mode treatment, time-space factorization of the order parameter, etc) widely used in investigation of BEC dynamics in a double-well trap (see e.g. [@Nest_JPB_09] and references therein) and closely follows prescriptions of the Heidelberg’s experiments [@Albiez_exp_PRL_05; @Gati_APB_06], thus providing quite realistic picture. The NLPI is explored for different magnitudes and time profiles of the transfer velocity, thus covering adiabatic and rapid scenarios. Both ideal and repulsive BECs are considered to estimate the nonlinear effect caused by the interaction between BEC atoms. As shown below, the repulsive interaction strongly favours the NLPI. This is in agreement with our previous results obtained within less involved model [@Nest_JPB_09]. The paper is organized as follows. The calculation scheme is sketched in Sec. \[sec:calc\_scheme\], the results are discussed in Sec. \[sec:results\], the summary is done in Sec. \[sec:summary\]. Calculation scheme {#sec:calc_scheme} ================== We use the 3D time-dependent Gross-Pitaevskii equation (GPE) [@GPE] $$\label{GPE} i\hbar\frac{\partial\Psi}{\partial t}({\bf r},t) = [-\frac{\hbar^2}{2m}\nabla^2 + V({\bf{r}},t) + g_0|\Psi({\bf r},t)|^2]\Psi({\bf r},t)$$ for the total order parameter $\Psi({\bf r},t)$ describing the BEC in both left and right wells of the trap. Here $g_0=4\pi\hbar^2 a_s /m$ is the interaction parameter, $a_s$ is the scattering length, and $m$ is the atomic mass. The trap potential $$\begin{aligned} \label{trap_pot} V({\bf{r}},t)&=& V_{\rm{con}}({\bf{r}})+V_{\rm{bar}}(x,t) \\ &=&\frac{m}{2}(\omega^2_x x^2+\omega^2_y y^2+\omega^2_z z^2) \nonumber \\&+& V_0 \cos^2(\pi (x-x_0(t))/q_0) \nonumber\end{aligned}$$ includes the anisotropic harmonic confinement and the barrier in $x$-direction, whose position is driven by the control parameter $x_0(t)$. Furthermore, $V_0$ is the barrier height and $q_0$ determines the barrier width. Following the Heidelberg experiment [@Albiez_exp_PRL_05; @Gati_APB_06] for measuring Josephson oscillations (JO) and macroscopic quantum self-trapping (MQST), we consider the BEC of N=1000 $^{87}$Rb atoms with $a_s=5.75$ nm. The trap frequencies are $\omega_x=2\pi\times 78$ Hz, $\omega_y=2\pi\times 66$ Hz, $\omega_z=2\pi\times 90$ Hz, i.e. $\omega_y+\omega_z=2\omega_x$. The barrier parameters are $V_0=420\times h$ Hz and $q_0=5.2 \; \mu$m. For the symmetric trap ($x_0$(t)=0), the distance between the centers of the left and right wells is $d=$4.4 $\mu$m. This setup has been previously used in our exploration of JO/MQST in weak and strong coupling regimes [@Nest_JPB_12]. The static solutions of GPE are found within the damped gradient method [@DGM] while the time evolution is computed within the time-splitting [@TSM] and fast Fourier-transformation techniques. The total order parameter $\Psi({\bf r},t)$ is determined in a 3D cartesian grid. The requirement $\int^{-\infty}_{+\infty}dr^3 |\Psi({\bf r},t)|^2 =N$ in time is directly fulfilled by using an explicit unitary propagator. Reflecting boundary conditions are used, though they have no impact on the dynamics because the harmonic confinement makes them effectless. No time-space factorization of the order parameter is used. The conservation of the total energy $E$ and complete number of atoms $N$ is perfectly controlled. The populations of the left (L) and right (R) wells are computed as $$\label{N_LR} N_{j}(t)=\int^{+\infty}_{-\infty}dr^3 |\Psi_{j}({\bf r},t)|^2,$$ with $j = L, R$, $\Psi_{L}({\bf r},t)=\Psi(x\le 0,y,z,t)$, $\Psi_{R}({\bf r},t)=\Psi(x\ge 0,y,z,t)$ and $N_L(t)+N_R(t)=N$. The normalized population imbalance is $z(t) = (N_L(t)-N_R(t))/N$. The population inversion means that BEC population characterized at the initial time t=0 by $N_L(0) > N_R(0)$ is changed during the time interval T to the inverse population $N_L(T) < N_R(T)$ where $N_L(T)=N_R(0)$ and $N_R(T)=N_L(0)$. Following the technique of [@Albiez_exp_PRL_05; @Gati_APB_06], the initial stationary asymmetric BEC state is produced by keeping the barrier right-shifted from the symmetric case ($x(0) > 0$). The value of the shift is adjusted to provide the given initial populations $N_L(0)$ and $N_R(0)$. The population inversion is generated by the time-dependent left shift of the barrier from $x(0)$ to $x(T)=-x(0)$ with the velocity $v(t)$. Thus asymmetry of the double-well trap is changed to the opposite one. The quality of the inversion is characterized by its completeness $P=-z(T)/z(0)$ (the ratio of the final and initial population imbalance) and noise $n=A_d/N$ where $A_d$ is amplitude of dipole oscillations in the final state, i.e. $A_d=\rm{max}\{N_{L,R}\}-\rm{min}\{N_{L,R}\}$ for $t>T$. Two velocity time profiles are used: i) the sharp rectangular one with the constant $v_c(t)=v^c_0$ at $0 < t < T$ and $v_c(t) =$ 0 beyond the transfer time and ii) the soft one $v_s(t)=v^s_0 \cos^2(\frac{\pi}{2}+\frac{\pi t}{T})$ with $v_s(0)=v_s(T) \sim 0$ and $v_s(T/2)=v^s_0$. For the total barrier shift $D=2 x(0)$ in the inversion process of duration T, the velocity amplitudes are $v^c_0=v^s_0=D/T$. The profile $v_c(t)$ is simple. However, it sharply changes from 0 to $v^c_0$ at t=0 and T and so is not adiabatic. The second profile $v_s(t)$ is more complicated but softer and thus closer to an adiabatic case. The transfer time T has natural lower and upper limits. It cannot be longer than the BEC lifetime ($\sim$ 3 sec). Also it cannot be too short since then the transfer would be too sharp and cause in the final state large undesirable dipole oscillations (see discussion in the next section). The same reasons determine the upper and lower limits for the transfer velocities. Results and discussion {#sec:results} ====================== Figure 1 exhibits the trap potential in x-direction, $$V_x(x,t)=\frac{m}{2}\omega^2_x x^2 + V_0 \cos^2(\pi (x-x_0(t))/q_0) ,$$ calculated for the initial t=0, intermediate t=T/2 and final t=T times of the inversion process driven by the barrier shift $x_0(t)$ with t$\in$\[0,T\]. For the same times, the BEC density in x-direction, $$\rho (x,t)=\int^{+\infty}_{-\infty} dy dz |\Psi (x,y,z,t)|^2 ,$$ obtained for an adiabatic inversion of a long duration T is shown. The ideal and repulsive BECs with N=1000 atoms are considered. Following the plots a) and d), the initial populations of the left and right wells are $N_L(0)$=800 and $N_R(0)$=200, i.e. with the initial population imbalance $z(0)$=0.6. An adiabatic evolution provides a robust population inversion to final $N_L(T)$=200, $N_R(T)$=800 and $z(T)$=-0.6. At the intermediate time t=T/2, the trap and populations are symmetric. The initial state is stationary by construction. The intermediate and final states, if obtained adiabatically, may be also treated as stationary. {width="11cm"} {width="8cm"} {width="8cm"} Upper plots of Fig. 1 show that for getting the initial $z(0)$=0.6 in the ideal BEC, a small trap asymmetry with $x_0(0)$=0.003 $\mu$m is sufficient. The overlap of the left and right parts of the condensate at the center of the trap is very small and corresponds to the case of a weak coupling. The energy difference between the ground and first excited states at the mid of the transfer (plot b)) is $\Delta E(T/2)$ = 0.005 $\hbar$ kHz. Such a tiny value confirms that the coupling and corresponding barrier penetrability are indeed small. For the repulsive BEC (bottom plots), the initial $N_L(0)$=800 and $N_R(0)$=200 are obtained at much larger asymmetry with $x_0(0)$=0.5 $\mu$m. The energy splitting $\Delta E(T/2)$ reaches 0.036 $\hbar$ kHz. This is because the repulsive interaction significantly increases the chemical potential $\mu$ of the system and thus the coupling between the left and right parts of BEC. Then, to get the initial [*stationary*]{} population imbalance $z(0)$=0.6, one should weaken the coupling by considerably increasing the asymmetry. As compared to the ideal BEC, the repulsive condensate has wider density bumps with much stronger overlap at the center of the trap. The coupling between the left and right BEC parts is not yet weak anymore. Nevertheless, the NLPI described below has occurred through tunneling. Some examples of the time evolution of the populations $N_{L,R}(t)$ in the ideal BEC (no interaction) are given in Fig. 2. The evolution is driven by the barrier shift with the rectangular $v_c(t)$ (upper plot) and soft $v_s(t)$ (bottom plots) velocitiy profiles. The total shift is $D=2 x(0)=$6 nm. For each example, the velocity amplitudes $v^c_0$ and $v^s_0$ obtained for a given transfer duration $T$ are indicated. It is seen that, at low velocities (plots a),c)) corresponding to a long duration T=1.8 $\mu$s, we get rather robust population inversion. This transfer is close to an adiabatic evolution. The final state is about stationary for $v_s(t)$ and somewhat spoiled by dipole oscillations for $v_c(t)$. The latter is caused by the sharp change of $v_c(t)$ from 0 to $v_c(0)$ and back at the beginning and end of the process. In this sense, the $v_s(t)$-transfer is much softer and thus more adiabatic. Fig. 2 also shows that the barrier-shift technique may be used not only for the population inversion (plots (a,c)) but also for production of MQST (plot (b)) and JO (plot (d)) supplementing the process. In our task, the JO/MQST come as undesirable dipole oscillations. In Fig. 3, similar examples are presented for the repulsive BEC. At first glance, the non-linear evolution resembles the linear one given in Fig. 2 except for plot d)) where the final state converges to a symmetric form with $N_L(T) \sim N_R(T) \approx 500$ or $z(T)\approx $0). Like in the linear case, a slow transfer (plots a),c)) results in a robust NLPI while a faster process (plots b),d)) spoils the final state by dipole oscillations (plot b)) or even breaks the inversion at all (plot d)). However, the nonlinearity drastically changes rates of the process. Now the robust NLPI can be produced at much shorter time (T=250 $\mu$s instead of T=1800 $\mu$s for ideal BEC) and much faster velocities ($\mu$/s instead of nm/s). So, despite the NLPI requires much stronger asymmetry and longer barrier shift (1 $\mu$m against 0.006 $\mu$m for the ideal BEC), the process becomes much faster. Namely, the velocities become about three order of magnitude higher (!), i.e. far beyond the adiabatic case. Therefore the repulsive interaction greatly favours the population inversion and this effect is indeed huge. The reason of the effect is simple. As mentioned above, the repulsive interaction significantly enhances the chemical potential $\mu$, which in turn results in a dramatic increase of the barrier penetrability. The coupling between BEC fractions becomes strong and the inversion is realized much faster. {width="8cm"} {width="8cm"} A more general information on the robustness of the population inversion is presented in Figs. 4 and 5 where the completeness $P$ and noise $n$ of the inversion are illustrated for a wide range of velocity amplitudes. The inversions for the ideal and repulsive BEC are compared. In Fig. 4, the sharp velocity profile $v_c(t)$ is used. Following the plots a) ,c) for the ideal BEC, a complete inversion ($P$=1) takes place only at a small velocity $v^c_0 <$ 0.04 $\mu$m/s. The inversion is somewhat spoiled by a weak noise $n=$ 0.02 - 0.04. The smaller the velocity, the weaker the noise. For $v^c_0 >$ 0.04 $\mu$m/s, we see a gradual destruction of the inversion, accompanied by an enhanced noise. For even larger velocities, the inversion breaks down ($P \to $ 0) and the final state is characterized by strong Rabi oscillations ($n \to$ 0.4). The latter effect is caused by the instant change of the process velocity from zero to $v^c_0$ at t=0 and back at t=T. Following Fig. 4 (b),d)), inclusion of the repulsive interaction drastically changes the results. There appears a wide plateau, $0 < v^c_0 \le$ 19 $\mu$m/s, with about complete inversion $P \approx $ 1. The repulsive interaction thus allows to get the inversion in a much wider velocity interval and, what is important, about three order of magnitude (!) faster than for the ideal BEC. Following our estimations, this is mainly caused by a considerable increase of the chemical potential $\mu$, caused by the repulsive interaction, and thus increasing the barrier penetrability. Note that the nonlinearity plays here an important but auxiliary role, which is reduced to a mechanism of rising the chemical potential. The net effect should depend on the form of the barrier. It should be strong for barriers whose penetrability increases with the excitation energy (e.g. Gaussian and $\cos^2(\pi (x-x_0(t))/q_0)$ barrier shapes) and suppressed for barriers with an energy-independent penetrability (e.g. rectangular shape). The plot Fig. 4 b) exhibits a noise (Rabi oscillations at the final state) in both inversion $v^c_0 \le$ 19 $\mu$m/s and beyond $v^c_0 \ge$ 19 $\mu$m/s regions. In the former region, the noise rises with the velocity, i.e. the faster the process, the less robust the process. At $v^c_0 \ge$ 19 $\mu$m/s, the inversion breaks down. Unlike the linear case, the transfer completeness $P$ does not tend to zero but to the negative value $P \approx $-0.7. This means that $z(0)$ and $z(T)$ have the same sign, i.e. the process results only in a modest population transfer, keeping the initial inequality $N_L > N_R$ at t=T. In Figure 5, the similar analysis is done for the softer (more adiabatic) velocity profile $v_s(t)$. The results are very similar to those in Fig. 4. The only but important difference is that, in the repulsive BEC, the inversion at $v^s_0 \le$ 20 $\mu$m/s is accompanied by much less noise as compared to the previous $v_c(t)$ case (see plot d)). So, as might be expected, the softer (and thus more adiabatic) velocity profile $v_s(t)$ provides a much better inversion than the sharp profile $v_c(t)$. The physical sense of the critical velocity $v_{\rm{crit}} \approx$ 19-20 $\mu$m/s which marks the break of inversion for both $v_c(t)$ and $v_s(t)$ regimes should be clarified. Following our analysis, $v_{\rm{crit}}$ does not corresponds to the destruction of adiabaticity (indeed we have about the same $v_{\rm{crit}}$ for less and more adiabatic profiles $v_c(t)$ and $v_s(t)$), but is rather determined by the transfer capacity defined as the multiplicative effect of the barrier penetrability $w$ and transfer duration $T$. Just these two values suffice to control the number of atoms to be transferred. Time $T$ can be enough ($v^{c,s}_0 < v_{\rm{crit}}$) or not ($v^{c,s}_0 > v_{\rm{crit}}$) for the complete inversion. For the repulsive BEC, the barrier penetrability $w$ is high and so the full inversion can be fast which explains the high $v_{\rm{crit}}$ and wide NLPI plateau in Figs. 4 b) and 5 b). In the ideal BEC, the penetrability $w$ is much weaker and thus longer times (lower velocities) are necessary for the inversion. Altogether, we see a strong support of the inversion by the repulsive interaction. The nonlinearity does not destroy but instead greatly favours the inversion, making it much faster. These findings are in accordance with our previous results for the complete transport of BEC from the left to the right well, obtained within the simplified model employing the two-mode approximation [@Nest_JPB_09]. In that study, the appearance of a wide velocity plateau due to the repulsive interaction was also observed. Note that velocity of the process has also a lower limit (e.g. caused by the finite lifetime of BEC, which is commonly a few seconds). Summary {#sec:summary} ======= The complete population inversion of the repulsive BEC in a double-well trap was investigated within the time-dependent three-dimensional Gross-Pitaevskii equation, following parameters of experiments of the Heidelberg group [@Albiez_exp_PRL_05; @Gati_APB_06]. The calculations are performed beyond usual approximations (two-mode, etc) in the description of tunneling and transport dynamics. The inversion is driven by a time-dependent barrier shift performed with different velocity regimes. As might be expected, a soft velocity profile $v(t)$ gives a more robust inversion than the sharp one. The most remarkable result is a significant support of the complete inversion by the repulsive interaction between BEC atoms. Due to the interaction (and related nonlinearity of the problem), the inversion can be produced in a wide velocity interval. Moreover, the process can be three orders of magnitude (!) faster than in the ideal BEC. Thus the transfer can be done far beyond the adiabatic requirements. These results are in accordance with our previous findings obtained within the two-mode approximation approach [@Nest_JPB_09]. The interaction effect is mainly reduced to the rise of chemical potential. Hence it should depend on the barrier form, being strong for barriers whose penetrability increases with the excitation energy and suppressed for barriers with energy-independent penetrability. Acknowledgments {#acknowledgments .unnumbered} =============== The work was partly supported by the RFBR grant 11-02-00086à and Institut Universitaire de France. We thank Prof. D. Gu´éry-Odelin for useful discussions. [99]{} R. Gati and M.K. Oberthaler, J. Phys. B: At. Mol. Opt. Phys. [**40**]{}, R61 (2007). E. 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--- abstract: 'We examine - both experimentally and numerically - a two-dimensional nonlinear driven electrical lattice with honeycomb structure. Drives are considered over a range of frequencies both outside (below and above) and inside the band of linear modes. We identify a number of discrete breathers both existing in the bulk and also (predominantly) ones arising at the domain boundaries, localized either along the arm-chair or along the zig-zag edges. The types of edge-localized breathers observed and computed emerge in distinct frequency bands near the Dirac-point frequency of the dispersion surface while driving the lattice subharmonically (in a spatially homogeneous manner). These observations/computations can represent a starting point towards the exploration of the interplay of nonlinearity and topology in an experimentally tractable system such as the honeycomb electrical lattice.' author: - 'F. Palmero $^{1}$, L. Q. English $^{2}$, J. Cuevas-Maraver $^{3}$, and P.G. Kevrekidis $^{4,5}$' title: Nonlinear edge modes in a honeycomb electrical lattice near the Dirac points --- Introduction ============ Honeycomb lattices have attracted substantial interest within the physics community in recent years, due to their inherent potential of topological surface phenomena [@bernevig]. The interplay of topology and wave dynamics (both at the linear and more recently at the nonlinear level) has had significant impact both in the realm of optics [@joann] and in that of acoustic/mechanical systems [@suss; @ma]. Nonlinearity further adds to the complexity and the wealth of this interplay, especially since in settings such as optics [@HuangNature; @Peleg; @Ablowitz1; @Ablowitz2] and atomic physics [@carr1; @carr2], it emerges spontaneously for large amplitude/density excitations. In two dimensions (2D) the foremost example of honeycomb material is, of course, graphene [@grap1; @grap2; @grap3]. One of the most intensely studied macroscopic analogues is photonic graphene, and edge-localized states were soon predicted to exist in such materials [@savin; @savin2; @ablowitz; @koh]. Furthermore, the role of nonlinearity in this context is beginning to be examined (see e.g., [@savin; @savin2]), yet there are still numerous avenues worth exploring in this context relating to the impact of nonlinearity, especially in experimentally tractable settings. In the linear regime, some pioneering experimental results have appeared in the literature in photonic graphene that show the existence of edge-localized states [@plotnik; @noh], and these were even shown to propagate in one direction only, upon breaking the time-reversal symmetry [@mikael]. Yet, we argue that the identification of unprecedented, experimentally controlled settings where nonlinear states (both bulk and edge ones) can be obtained is of value to the efforts to understand nonlinear topological structures and how they differ from their more standard, non-topological variants (as well as how such states vary from linear topological ones). In that vein, we propose as a platform worth exploring the setting of honeycomb electrical lattices. More concretely, in this paper we report on a series of findings of nonlinear waves in a 2D electrical honeycomb lattice. That intrinsic localized modes, also known as [*discrete breathers*]{} (DBs) can exist in square lattices of this kind has been shown previously [@electric1]. In fact, such modes are well-known to exist in a wide range of physical settings, summarized, e.g., in a number of reviews [@Flach; @Aubry]. Here, however, we focus on the role of the honeycomb geometry and drive the system over a wide range of frequencies both within as well as outside the band of small amplitude excitations. We find both experimentally and numerically that not only can bulk-localized modes be identified in this setting, but also edge-localized modes can be excited with a spatially homogenous, subharmonic driver. These DBs bear frequencies around that of the Dirac points. The exact DB frequency depends on the wave amplitude, as expected from a soft nonlinear system, but interestingly also on the type of edge. The relevant zig-zag edge-localized DBs are found to exist within a frequency band that is higher, in terms of frequencies (and non-overlapping) compared to the arm-chair mode band. We complement these results with a numerical stability analysis and find that these discrete breathers do not appear to derive from a continuation of linear modes, but that they come into existence via (saddle-node) bifurcation phenomena. Our findings constitute a first step towards the more systematic examination of stable bulk and edge modes in such honeycomb electrical lattices and we hope will spurt further efforts in this direction. Our presentation is structured as follows. In section II we present the mathematical model associated with the experimental setting of interest, i.e., the honeycomb lattice of LC resonators. The underlying linear modes are identified and their band is obtained for parameters associated within the experimental range in Section III. Subsequently, in section IV, we present an anthology of experimental and numerical results for similar conditions between the experiment and the numerical computation. The findings are presented for different values of the driver frequency, progressively moving from frequencies below the band to ones above the band of linear states. Finally, we summarize our findings and present our conclusions and some challenges arising towards future work in Section V. The Model ========= The experimental system investigated in this paper is a honeycomb lattice consisting of unit cells that are comprised of LC resonators, whose nonlinearity is originated by using a varactor diode instead of the standard capacitor. These nonlinear resonators are coupled together into a two-dimensional lattice via coupling inductors. Such a system was studied in a previous publication [@electric1], where it was found that stable two-dimensional ILMs/discrete breathers could be produced. That study used periodic boundary conditions exclusively, thus eliminating any lattice edges. In the present study, we have used free-ends boundary conditions, allowing a pair of both zig-zag and armchair edges. This has permitted us to investigate the dynamical interplay between lattice edges and nonlinear localized states. More concretely, our use of a varactor diode (NTE 618) introduces a specific (experimentally determined) nonlinear capacitance $C(V)$. We also use inductors of value $L_2 = 330 \mu$H, and the resulting unit cells are driven by a [periodic voltage source $\mathcal{E}(t)$ of frequency $f$]{} [(i.e., the driving is [*uniform*]{})]{} via a resistor $R = 10 $k$\Omega$. Each single unit is coupled to its three neighbors via inductors $L_1=680 \mu$H building a honeycomb lattice. Using basic circuit theory, the system can be described by the equations [@electric1; @electric2], $$\begin{aligned} \frac{d i_{n,m}}{d \tau} & =& \frac{L_2}{L_1} \left( \sum_{j,k} v_{j,k}-K_{n,m}v_{n,m} \right)-v_{n,m} \label{lattice_eq} \\ \frac{d v_{n,m}}{d \tau} & =& \frac{1}{c(v_{n,m})}\left[ i_{n,m}-i^D(v_{n,m})-\right. \nonumber \\ & & \left. \frac{v_{n,m}}{C_0\omega_0R_e}+ {\frac{1}{ C_0\omega_0 R} \frac{\mathcal{E}(\Omega\tau)}{V_d}} \right], \nonumber\end{aligned}$$ where the sum $(j,k)$ is taken over all neighbors of the $(n,m)$ node and $K_{n,m}$ is the number of neighbors of node $(n,m)$. $K_{n,m}$ is equal to three in an infinite lattice (or finite lattice with periodic boundary conditions), but in a finite lattice with free boundaries it could be either $K_{n,m}=1$ or $K_{n,m}=2$ on the edges, depending on the particular lattice node. [The varactor can be modeled as a nonlinear resistance in parallel with a nonlinear capacitance. As shown in [@electric2], the nonlinear current $I^D(V)$ is given by $$I^D(V)=-I_s \exp(-\beta V) ,$$ where $\beta=38.8$ V$^{-1}$ and $I_s=1.25 \times 10^{-14}$ A, and its capacitance $C(V)$ as $$C(V) = \begin{cases} C_v+C_1(V')+C_2 (V')^2 & \text{if } V \leq V_c, \\ C_0 e^{-\alpha V} & \text{if } V > V_c, \end{cases}$$ where $V'=(V-V_c)$, $C_0=788$ pF, $\alpha=0.456$ V$^{-1}$, $C_v=C_0 \exp(-\alpha V_c)$, $C_1=-\alpha C_v$, $C_2=100$ nF and $V_c=-0.28$ V.]{} [The following dimensionless variables were used in Eq. (\[lattice\_eq\]): $\tau=\omega_0 t$, where $\omega_0=1/\sqrt{L_2C_0}$; $\Omega=2\pi f/\omega_0$ is the dimensionless driving frequency; the dimensionless voltage [$v_{n,m}=V_{n,m}/V_d$]{}, with $V_d$ representing the voltage amplitude of the driving; $i_{n,m}=(I_v-I_2)/(C_0\omega_0 V_d)$, where $I_v$ is the full current through the unit cell and $I_2$ the current through the inductor $L_2$, both corresponding to cell $(n,m)$ and $i^D=I^D/(C_0 \omega_0 V_d)$. A phenomenological dissipation resistor, $R_l$, was included in the model to better approximate the experimental dynamics and $R_e$ is the equivalent resistance so $1/R_e=1/R+1/R_l$. In all cases, the ratio $L_2/L_1$ characterizes the strength of the effective discreteness of the system (with the uncoupled limit obtained for $L_1 \rightarrow \infty$). We should add that this is still only a simplified model of the varactor diodes, and comparison between theoretical and experimental results will not be exact. Yet, it is an important first step in the modeling effort towards understanding this setup.]{} Linear modes ============ In the linear limit ($c(v)=1$, $i_d=0$) the undriven and undamped system reduces to $$\frac{d ^2 v_{n,m}}{d \tau^2}=\frac{L_2}{L_1}\left( \sum_{j,k} v_{j,k}-K_{n,m}v_{n,m} \right)-v_{n,m}. \label{lineal}$$ Linear modes can be found as plane-wave solutions. An infinite lattice (with nearest neighbor spacing of $1$) can be generated from lattice vectors $\mathbf{e}_\pm=(1/2,\pm\sqrt{3}/2)$ (see e.g. [@Cserti]), and the dispersion relation $\omega(\mathbf{k})$, with $\mathbf{k}=(k_x,k_y)$, is given by $$\begin{aligned} \label{eq:disprel} \omega^2 & = & \frac{1}{C_0 L_2}+\frac{1}{C_0 L_1}\left[ 3 \pm \right. \\ & & \left. \sqrt{3+2 \cos(\sqrt{3} k_y)+4\cos(3k_x/2)\cos(\sqrt3k_y/2)}\right] \nonumber,\end{aligned}$$ which yields a band of frequencies between $f_\mathrm{min}=\sqrt{1/(C_0L_2)}/(2\pi)\approx 312$ kHz and $f_\mathrm{max}=\sqrt{1/(C_0L_2)+6/(C_0L_1)}/(2\pi)\approx $ 617 kHz. As shown in Fig. \[band\], the band structure corresponds to a graphene-like surface where six Dirac points exist at a frequency of $\omega_d=\sqrt{3/(C_0L_1)+1/(C_0L_2)}$, or $f_d=\omega_d/2\pi \approx 489.11$ kHz. ![Typical band structure of the infinite honeycomb lattice in the first Brioullin zone. There are six Dirac points corresponding to $(k_x,k_y)=(0,\pm 4 \pi/(3\sqrt{3})),(2\pi/3,\pm 2\pi/(3\sqrt{3}))$, $(-2\pi/3,\pm 2\pi/(3\sqrt{3}))$ and a frequency $f_d = 489.11$ kHz, cf. Eq. (\[eq:disprel\]).[]{data-label="band"}](fig1N.eps){width="40.00000%"} In a finite lattice, wave vectors $\mathbf{k}$ are quantized. However, this quantization depends on the boundary conditions and the way the lattice is tiled. Because of this, one must be very cautious with the choice of boundary conditions, the way the honeycomb is generated and the lattice size if the Dirac point is intended to be in the linear mode spectrum. For periodic boundary conditions, an explicit expression of the eigenfrequencies can be attained [@Cserti], but, for free ends boundary conditions, one must rely on the numerical solution of Eq. (\[lineal\]) for getting the linear mode spectrum. In the present study, experimental limitations restrict us to a lattice of $6\times6$ nodes, distributed as shown in Fig. \[lattice\]. The boundaries are free, as we are interested in seeking edge-localized breathers, as shown below. With this particular choice, there is a sole eigenmode oscillating with the Dirac frequency $f=f_d=489.11$ Hz. Figure \[lattice\] also shows the oscillation pattern of such eigenmode (Dirac mode), which is similar to the oscillation pattern in an infinite lattice. [We have checked that this sole Dirac mode is present when tiling this lattice to a larger one with $6N\times6M$ with $(N,M)\in\mathbb{N}$ nodes]{}. In addition, the structure of the edges of our lattice can be crucial for the formation of edge-localized breathers. According to the types of edge modes on graphene-like systems [@koh], our system should be able to support both vertical zigzag and horizontal armchair edge-localized breathers. ![Finite size ($6 \times 6$) lattice with free boundaries; shown is the Dirac mode, i.e. the linear mode corresponding to $f=f_D=489.11$ kHz. Black and white circles correspond, respectively, to a normalized amplitude in $t=0$ of $1/2$ and $-1$.[]{data-label="lattice"}](fig2.eps){width="40.00000%"} Nonlinear modes: numerical and experimental results =================================================== In this section, we will describe some numerical and experimental results on the existence of DBs when the electric lattice is driven uniformly. We have observed two kinds of such DBs, depending on whether they are localized on the lattice boundaries or elsewhere. We will call these DBs edge breathers (EBs) or bulk breathers (BBs), respectively, hereafter. The latter owe their existence to the intrinsic nonlinearity of the lattice (see e.g. [@electric1; @electric2]). In the former case, there is an interplay between nonlinearity and the nature of the coupling in the vicinity of the boundary. Driving near the lowest frequency mode -------------------------------------- Having constructed the $6\times6$ honeycomb lattice of Fig. \[lattice\], [the simplest experiment we can perform is to drive the lattice with a sinusoidal-wave profile and a frequency close the bottom of the linear modes band, as the lowest frequency mode is uniform ($\mathbf{k}=0$, i.e., the same wavevector as that of the driver)]{}. This is performed in a progression of frequencies starting from outside (under) the linear mode band and systematically increasing the frequency of the drive. When the driver frequency is near the bottom of the linear band (i.e. $f\lesssim f_\mathrm{min}$), we can generate experimentally both BBs and EBs, where the latter seems to be the most robust state between the two. Under the same conditions, in our theoretical model we find that only EBs exist. Alternatively, the use of periodic boundary conditions enables the existence of BBs for such frequencies. Fig. \[direct1\] shows a numerical bulk breather corresponding to a $6\times 6$ lattice with periodic boundaries and its Floquet multipliers spectrum (see e.g. [@Aubry] for more details on Floquet analysis for discrete breathers). Recall that the existence of the corresponding multipliers solely within the unit circle for our driven/damped system indicates its spectral stability. In that figure, we also show the experimental BB obtained in the finite size lattice with free boundary conditions. In both cases the driver amplitude was set to 2.1 V and the frequency was 278 kHz. This is a representative example of such BBs within their relevant interval of existence (see also the discussion below). ![(a) Numerical bulk breather (BB) profile in a $6\times 6$ lattice with periodic boundaries. (b) Experimental BB in the $6\times 6$ lattice with free boundary conditions. (c) Floquet multiplier spectrum corresponding to the numerical breather showing all the multipliers lying within the unit circle (and thus leading to the conclusion of spectral stability of such breathers). (d,e) Density plots corresponding to (a,b). $V_\mathrm{rms}$ in panels (a) and (b)stands for the root mean square of the voltage during a period. In both cases the [sinusoidal]{} driver amplitude was set to 2.1 V and the frequency was 278 kHz.[]{data-label="direct1"}](fig_3_a.eps "fig:"){width="23.00000%"} ![(a) Numerical bulk breather (BB) profile in a $6\times 6$ lattice with periodic boundaries. (b) Experimental BB in the $6\times 6$ lattice with free boundary conditions. (c) Floquet multiplier spectrum corresponding to the numerical breather showing all the multipliers lying within the unit circle (and thus leading to the conclusion of spectral stability of such breathers). (d,e) Density plots corresponding to (a,b). $V_\mathrm{rms}$ in panels (a) and (b)stands for the root mean square of the voltage during a period. In both cases the [sinusoidal]{} driver amplitude was set to 2.1 V and the frequency was 278 kHz.[]{data-label="direct1"}](fig_3_b.eps "fig:"){width="23.00000%"} ![(a) Numerical bulk breather (BB) profile in a $6\times 6$ lattice with periodic boundaries. (b) Experimental BB in the $6\times 6$ lattice with free boundary conditions. (c) Floquet multiplier spectrum corresponding to the numerical breather showing all the multipliers lying within the unit circle (and thus leading to the conclusion of spectral stability of such breathers). (d,e) Density plots corresponding to (a,b). $V_\mathrm{rms}$ in panels (a) and (b)stands for the root mean square of the voltage during a period. In both cases the [sinusoidal]{} driver amplitude was set to 2.1 V and the frequency was 278 kHz.[]{data-label="direct1"}](fig_3_c.eps "fig:"){width="23.00000%"} ![(a) Numerical bulk breather (BB) profile in a $6\times 6$ lattice with periodic boundaries. (b) Experimental BB in the $6\times 6$ lattice with free boundary conditions. (c) Floquet multiplier spectrum corresponding to the numerical breather showing all the multipliers lying within the unit circle (and thus leading to the conclusion of spectral stability of such breathers). (d,e) Density plots corresponding to (a,b). $V_\mathrm{rms}$ in panels (a) and (b)stands for the root mean square of the voltage during a period. In both cases the [sinusoidal]{} driver amplitude was set to 2.1 V and the frequency was 278 kHz.[]{data-label="direct1"}](fig_3_d.eps "fig:"){width="23.00000%"} Similarly, we can induce EBs which are, as indicated above, more robust than BBs. Fig. \[direct2\] shows an example of the theoretical and experimental features of an EB whose driving parameters are the same as for the BB of Fig. \[direct1\]. The existence of both kinds of solutions for the same system parameters indicates the multistability of the system, given the different branches (bulk vs. edge) of solutions. That is, the regions of existence for the different kinds of breathers substantially overlap. The EBs are found to be somewhat more stable in the following sense: as we lower either the frequency or the amplitude of the driver (starting from 278 kHz and 2.1 V), the BB will disappear first, before the EB ceases to exist. This means that there is a small window in driving parameters where only edge breathers can be stabilized. This finding, i.e. the wider range of stabilization of the EB relative to the BB, has been also experimentally observed in a chain of coupled pendula [@pend]. ![Same as figure [\[direct1\]]{} but for an edge breather (EB) in a free-boundary lattice.[]{data-label="direct2"}](fig_4_a.eps "fig:"){width="23.00000%"} ![Same as figure [\[direct1\]]{} but for an edge breather (EB) in a free-boundary lattice.[]{data-label="direct2"}](fig_4_b.eps "fig:"){width="23.00000%"} ![Same as figure [\[direct1\]]{} but for an edge breather (EB) in a free-boundary lattice.[]{data-label="direct2"}](fig_4_c.eps "fig:"){width="23.00000%"} ![Same as figure [\[direct1\]]{} but for an edge breather (EB) in a free-boundary lattice.[]{data-label="direct2"}](fig_4_d.eps "fig:"){width="23.00000%"} It should be mentioned that breathers can also be generated via subharmonic driving. In that case, breathers (which are also denoted as [*subharmonic breathers*]{}) are characterized by a core (i.e. the peak and large amplitude nodes around it) oscillating with half of the driver frequency whereas tails oscillate with the driving frequency (see [@lars2]). Unlike what is observed in the experiments (featuring both BBs and EBs), it seems that numerically only subharmonic EBs are stable for this $6\times 6$ lattice with free boundaries (the analysis of subharmonic breathers in larger lattices will be the subject of further studies). The observation of long-lived subharmonic BBs in the experiment may be due to small spatial inhomogeneities in the lattice facilitating their stabilization [@electric2]. In terms of the subharmonic EBs, a good agreement is found regarding both their existence and their dynamical robustness. Driving near the Dirac point ---------------------------- As the driver frequency is increased from around 280 kHz, we first start producing more breathers in the lattice, as expected from previous studies [@lars1]. Then, at higher frequencies, the lattice response gradually weakens. The modes at the Dirac point (with frequency 489.11 kHz) cannot be stimulated directly; this is a consequence of the fact that the Dirac mode possesses a non-zero wavevector, whereas the driver is associated with the zero wavevector. At higher driving amplitudes, however, lattice response in the vicinity of the Dirac point can be induced via subharmonic driving. Both subharmonic BBs and EBs, which are excited close to the Dirac point frequency, are similar to the ones produced via direct driving although the oscillation frequency [of the excited sites]{} is the half of that of the driver [and the breather tail]{} (see also Ref. [@lars2]). If the (sinusoidal) driver amplitude is increased to 9 V, a clear lattice subharmonic response is observed in the range $(530,695)$ kHz. When using a square-wave driving profile, this range is slightly expanded; this phenomenon, which has been reported recently for non-sinusoidal drivings, is related to the enhancement of the “mechanical” impulse transmitted to the lattice from the driver, facilitating the generation of stationary breathers in experiments, as well as in numerical computations [@impulse1]. Above the upper edge of this frequency window, the lattice response goes to zero again (at least for the uniform driving used experimentally). Then, starting at 886 kHz and using a square-wave driving at 9 V (no subharmonic response has been found for this range of frequencies using sinusoidal driving), an EB appears firstly along the armchair edge of the honeycomb lattice (also dubbed as armchair-EB). In experiments this mode persists up to a frequency of 940 kHz, corresponding to a response frequency at the breather peak of 470 kHz. Numerical simulations show similar results, as represented in Figure \[dirac1\]. ![Same as Fig. \[direct2\] but for a subharmonic armchair-EB with a [ square-wave driving profile of amplitude 9 V.]{} The driving frequency was 960 kHz in numerics and 940 kHz in experiments[]{data-label="dirac1"}](fig_5_a.eps "fig:"){width="23.00000%"} ![Same as Fig. \[direct2\] but for a subharmonic armchair-EB with a [ square-wave driving profile of amplitude 9 V.]{} The driving frequency was 960 kHz in numerics and 940 kHz in experiments[]{data-label="dirac1"}](fig_5_b.eps "fig:"){width="23.00000%"} ![Same as Fig. \[direct2\] but for a subharmonic armchair-EB with a [ square-wave driving profile of amplitude 9 V.]{} The driving frequency was 960 kHz in numerics and 940 kHz in experiments[]{data-label="dirac1"}](fig_5_c.eps "fig:"){width="23.00000%"} ![Same as Fig. \[direct2\] but for a subharmonic armchair-EB with a [ square-wave driving profile of amplitude 9 V.]{} The driving frequency was 960 kHz in numerics and 940 kHz in experiments[]{data-label="dirac1"}](fig_5_d.eps "fig:"){width="23.00000%"} At a driver frequency of 950 kHz, we witness an abrupt switch to an EB along the zig-zag edge of the lattice (also dubbed as zig-zag-EB). Such a breather, whose main features are shown in Fig. \[dirac2\], persists up to a driving frequency of 967 kHz. In general, numerics are in qualitative agreement with the experiments. The switch between these two types of edge breathers as the frequency is adiabatically increased is very reproducible. This clearly suggests the different intervals of stability of the two edge configurations, indicating which one is the system’s lower energy state for the different frequency regimes. ![Same as Fig. \[dirac1\] but for a subharmonic zig-zag-EB. The driving frequency was 1000 kHz in numerics and 960 kHz in experiments[]{data-label="dirac2"}](fig_6_a.eps "fig:"){width="23.00000%"} ![Same as Fig. \[dirac1\] but for a subharmonic zig-zag-EB. The driving frequency was 1000 kHz in numerics and 960 kHz in experiments[]{data-label="dirac2"}](fig_6_b.eps "fig:"){width="23.00000%"} ![Same as Fig. \[dirac1\] but for a subharmonic zig-zag-EB. The driving frequency was 1000 kHz in numerics and 960 kHz in experiments[]{data-label="dirac2"}](fig_6_c.eps "fig:"){width="23.00000%"} ![Same as Fig. \[dirac1\] but for a subharmonic zig-zag-EB. The driving frequency was 1000 kHz in numerics and 960 kHz in experiments[]{data-label="dirac2"}](fig_6_d_e.eps "fig:"){width="23.00000%"} ![(a) Experimental subharmonic armchair-EB profile in a $6\times 6$ lattice with free boundaries and (b) its corresponding density plot. The [square-wave]{} driving amplitude was set to 9 V and the frequency to 886 kHz. (c) Time dependence of the voltage at the two largest amplitude nodes on the top of the lattice: numerical simulations are shown by black continuous and dashed lines, whereas experimental data are shown in blue and red lines.[]{data-label="dirac3"}](fig_7_a.eps "fig:"){width="23.00000%"} ![(a) Experimental subharmonic armchair-EB profile in a $6\times 6$ lattice with free boundaries and (b) its corresponding density plot. The [square-wave]{} driving amplitude was set to 9 V and the frequency to 886 kHz. (c) Time dependence of the voltage at the two largest amplitude nodes on the top of the lattice: numerical simulations are shown by black continuous and dashed lines, whereas experimental data are shown in blue and red lines.[]{data-label="dirac3"}](fig_7_b.eps "fig:"){width="23.00000%"} ![(a) Experimental subharmonic armchair-EB profile in a $6\times 6$ lattice with free boundaries and (b) its corresponding density plot. The [square-wave]{} driving amplitude was set to 9 V and the frequency to 886 kHz. (c) Time dependence of the voltage at the two largest amplitude nodes on the top of the lattice: numerical simulations are shown by black continuous and dashed lines, whereas experimental data are shown in blue and red lines.[]{data-label="dirac3"}](fig_7_c.eps "fig:"){width="23.00000%"} Figure \[dirac3\] examines the armchair-EB more closely. Here we showcase the most nonlinear, and therefore most localized version of that mode at a driving frequency of 886 kHz. As it can be seen from the experimental pattern depicted in panel (b), there are two of such EBs at the opposite sides of the lattice. Comparing to Fig. \[lattice\], one can observe in both cases a sharp localization of the energy at the outermost node-pairs along the armchair edges. It is evident, from the detailed time-dependent oscillations pattern of panel (c), that the two largest-amplitude nodes in the top of the lattice oscillate in anti-phase, indicating that $\mathbf{k}\neq\mathbf{0}$ for this EB. Once again, numerical simulations are in good agreement with experimental results. Similar features (not shown here) are shared with zig-zag-EBs. ![Stability range versus driving frequency for different kinds of numerically calculated EBs together with the linear modes band, which is represented as a yellow region; Dirac point ($f_d$) and twice its value are depicted as black horizontal line. Green region corresponds to (direct-driven) EBs with $V_d=2.1$ V (cfr. Fig. \[direct2\]). (b) Subharmonic armchair-EBs driven by $V_d=9$ V (cfr. Fig. \[dirac1\]) is represented by the blue region, and the lighter blue region overlapping part of the linear modes band corresponds to the subharmonic oscillations of the breather peak. Similarly to the armchair-EB case, the red region and the lighter red one in the linear modes band correspond to subharmonic zig-zag-EBs.[]{data-label="cont"}](fig8N.eps){width="45.00000%"} Edge breathers around the above mentioned frequency ranges [i.e. $f\in[886-967]$ kHz]{} were not found to occur in the square lattice, thus seeming to be particular to the honeycomb geometry and its associated boundary geometry. Furthermore, it is an interesting fact that zig-zag-EBs and armchair-EBs are found to occupy such distinct frequency bands around the calculated Dirac frequency, as shown in Fig. \[cont\]. We do not get any BB with this set of boundary conditions; rather, only EBs arise. An example of a bifurcation diagram featuring a pair of saddle-node bifurcations for numerically calculated subharmonic EBs is shown in Fig. \[cont2\]. In general, we have found that this scenario of saddle-node bifurcations for the destabilization of solutions is fairly generic (results not shown here). Analyzing such features in more detail turns out to be a rather delicate task due to the existence of numerous branches in a complex bifurcation diagram. For this reason, a more exhaustive study is deferred to a future publication. ![$V_\mathrm{rms}$ on peak node of the numerically calculated subharmonic EBs (solutions in red and blue at Fig. \[cont\]), as a function of the frequency. Stable solutions are depicted as continuous lines and unstable solutions as dotted lines. The red line marks twice Dirac frequency ($2f_d$).[]{data-label="cont2"}](fig_9b.eps){width="45.00000%"} Conclusions & Future Work ========================= Naturally, the above findings constitute only a first step in the emerging rich study of localized modes and the breathing dynamics in honeycomb electrical lattices. Here, we have explored the arguably most canonical and experimentally more straightforwardly tractable case of a uniform drive of zero wavevector. At frequencies below the linear band, we have found that this drive leads to the formation of bulk, as well as edge breathers, with the latter being more robust than the former. However, our most significant finding concerns to the subharmonic drive in the vicinity of twice the frequency of the Dirac point. There, depending on the frequency interval, both armchair and zigzag edge breathers can arise, with each one appearing as the stable state in a respective interval frame. There are numerous questions that still remain worthwhile to answer. Is it possible to achieve more elaborate forms of driving so as to excite higher order states? At the same time, it appears to be relevant to develop a systematic continuation analysis, e.g., at the Hamiltonian level of the corresponding linear and also subharmonic waves and their expected role in the bifurcation diagram. Understanding whether these edge states enjoy topologically induced propagation properties (e.g. through the lattice boundary) is an important question worth considering in its own right. Furthermore, the experimental tractability of electrical lattices renders them interesting candidates for formulating additional lattices with intriguing topological properties such as, e.g., some of the artificial flat band systems recently summarized in [@flach2]. [*Acknowledgements.*]{} This material is based upon work supported by the US National Science Foundation under Grants No. PHY-1602994 and DMS-1809074 (PGK). PGK also acknowledges support from the Leverhulme Trust via a Visiting Fellowship and the Mathematical Institute of the University of Oxford for its hospitality during part of this work. [J.C.-M. was supported by MAT2016- 79866-R project (AEI/FEDER, UE). [FP visited Dickinson College with support from the VI Plan Propio of the University of Seville (VI PPITUS). 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--- abstract: 'We describe the use of explicit isogenies to translate instances of the Discrete Logarithm Problem (DLP) from Jacobians of hyperelliptic genus $3$ curves to Jacobians of non-hyperelliptic genus $3$ curves, where they are vulnerable to faster index calculus attacks. We provide explicit formulae for isogenies with kernel isomorphic to $({{\mathbb{Z}}}/2{{\mathbb{Z}}})^3$ (over an algebraic closure of the base field) for any hyperelliptic genus $3$ curve over a field of characteristic not $2$ or $3$. These isogenies are rational for a positive fraction of all hyperelliptic genus $3$ curves defined over a finite field of characteristic $p > 3$. Subject to reasonable assumptions, our constructions give an explicit and efficient reduction of instances of the DLP from hyperelliptic to non-hyperelliptic Jacobians for around $18.57\%$ of all hyperelliptic genus $3$ curves over a given finite field. We conclude with a discussion on extending these ideas to isogenies with more general kernels. A condensed version of this work appeared in the proceedings of the EUROCRYPT 2008 conference.' author: - Benjamin Smith title: Isogenies and the Discrete Logarithm Problem in Jacobians of Genus 3 Hyperelliptic Curves --- \[definition\][Algorithm]{} Introduction {#section:introduction} ============ After the great success of elliptic curves in public-key cryptography, researchers have naturally been drawn to their higher-dimensional generalizations: Jacobians of higher-genus curves. Curves of genus $1$ (elliptic curves), $2$, and $3$ are widely believed to offer the best balance of security and efficiency. This article is concerned with the security of curves of genus $3$. There are two classes of curves of genus $3$: hyperelliptic and non-hyperelliptic. Each class has a distinct geometry: the canonical morphism of a hyperelliptic curve is a double cover of a curve of genus $0$, while the canonical morphism of a non-hyperelliptic curve of genus $3$ is a birational map to a nonsingular plane quartic curve. A hyperelliptic curve cannot be isomorphic (or birational) to a non-hyperelliptic curve. From a cryptological point of view, the Discrete Logarithm Problem (DLP) in Jacobians of hyperelliptic curves of genus $3$ over ${{\mathbb{F}}}_{q}$ may be solved in ${\ensuremath{\widetilde O}}(q^{4/3})$ group operations, using the index calculus algorithm of Gaudry, Thomé, Thériault, and Diem [@GTTD]. Jacobians of non-hyperelliptic curves of genus $3$ over ${{\mathbb{F}}}_{q}$ are amenable to Diem’s index calculus algorithm [@Diem], which requires only ${\ensuremath{\widetilde O}}(q)$ group operations to solve the DLP (for comparison, Pollard/baby-step-giant-step methods require ${\ensuremath{\widetilde O}}(q^{3/2})$ group operations to solve the DLP in Jacobians of genus $3$ curves over ${{\mathbb{F}}}_{q}$). The security of non-hyperelliptic genus $3$ curves is therefore widely held to be lower than that of their hyperelliptic cousins. Our aim is to construct explicit homomorphisms to provide a means of efficiently translating instances of the DLP from Jacobians of hyperelliptic curves of genus $3$ to Jacobians of non-hyperelliptic curves, where faster index calculus is available. In the context of DLP-based cryptography, we may assume that our Jacobians are absolutely simple. In this situation, every nontrivial homomorphism of Jacobians of curves of genus $3$ is an *isogeny*: that is, a surjective homomorphism with finite kernel. To be specific, suppose we are given a hyperelliptic curve $H$ of genus $3$ over a finite field ${{\mathbb{F}}}_{q}$, together with an instance $P = [n]Q$ of the DLP in ${\ensuremath{{J}_{H}}}({{\mathbb{F}}}_{q})$; our task is to recover $n$ given $P$ and $Q$. After applying the standard Pohlig–Hellman reduction [@Pohlig--Hellman], we may assume that $P$ and $Q$ have prime order. We want to solve this DLP instance by solving an equivalent DLP instance in a non-hyperelliptic Jacobian. Suppose we have an isogeny $\phi: {\ensuremath{{J}_{H}}} \to {\ensuremath{{J}_{C}}}$, where $C$ is a non-hyperelliptic curve of genus $3$. Further, suppose that $\phi$ is explicit (that is, we have equations for $C$ and an efficient map on divisor classes representing $\phi$) and defined over ${{\mathbb{F}}}_{q}$, so it maps ${\ensuremath{{J}_{H}}}({{\mathbb{F}}}_{q})$ into ${\ensuremath{{J}_{C}}}({{\mathbb{F}}}_{q})$. Provided $\phi(Q) \not= 0$, we can recover $n$ by solving the DLP instance $\phi(P) = [n]\phi(Q)$ in ${\ensuremath{{J}_{C}}}({{\mathbb{F}}}_{q})$ with Diem’s algorithm. The approach outlined above is conceptually straightforward; the difficulty lies in computing explicit isogenies of Jacobians of genus $3$ curves. Automorphisms, integer multiplications, and Frobenius maps aside, we know of no explicit and general formulae for isogenies from Jacobians of hyperelliptic curves of genus $3$ apart from those presented below. In §\[section:kernel\] through §\[section:equations-for-the-isogeny\], we derive explicit formulae for isogenies whose kernels are generated by differences of Weierstrass points, following the construction of Donagi and Livné [@Donagi--Livne]. The key step is making Recillas’ trigonal construction [@Recillas] completely explicit. This gives us a curve $X$ of genus $3$ and an explicit isogeny ${\ensuremath{{J}_{H}}} \to {\ensuremath{{J}_{X}}}$. While $X$ may be hyperelliptic, naïve moduli space dimension arguments suggest (and experience confirms) that $X$ will be non-hyperelliptic with an overwhelming probability, and thus explicitly isomorphic to a nonsingular plane quartic curve $C$. We can therefore compute an explicit isogeny $\phi: {\ensuremath{{J}_{H}}}\to{\ensuremath{{J}_{C}}}$; if $\phi$ is defined over ${{\mathbb{F}}}_{q}$, then we can use it to reduce DLP instances. We note that the trigonal construction (and hence our formulae) does not apply in characteristics $2$ and $3$. We show in §\[section:probabilities\] that, subject to some reasonable assumptions, given a uniformly randomly chosen hyperelliptic curve $H$ of genus $3$ over a sufficiently large finite field ${{\mathbb{F}}}_{q}$ of characteristic at least $5$, our algorithms succeed in constructing an explicit isogeny defined over ${{\mathbb{F}}}_{q}$ from ${\ensuremath{{J}_{H}}}$ to a non-hyperelliptic Jacobian with probability $\approx 0.1857$. In particular, instances of the DLP can be solved in ${\ensuremath{\widetilde O}}(q)$ group operations for around $18.57\%$ of all Jacobians of hyperelliptic curves of genus $3$ over finite fields of characteristic at least $5$. We discuss more general isogenies in §\[section:other-isogenies\]. Given explicit formulae for these isogenies, we expect that most, if not all, instances of the DLP in Jacobians of hyperelliptic curves of genus $3$ over any finite field could be reduced to instances of the DLP in non-hyperelliptic Jacobians. Our results have a number of interesting implications for curve-based cryptography, at least for curves of genus $3$. First, the difficulty of the DLP in a subgroup $G$ of ${\ensuremath{{J}_{H}}}$ depends not only on the size of the subgroup $G$, but upon the existence of other rational subgroups of ${\ensuremath{{J}_{H}}}$ that can be used to form quotients. Second, the security of a given hyperelliptic genus $3$ curve depends significantly upon the factorization of its hyperelliptic polynomial. Neither of these results has any parallel in genus $1$ or $2$. The constructions of §\[section:kernel\] through §\[section:equations-for-the-isogeny\] and §\[section:other-isogenies\] require some nontrivial algebraic geometry. We have included enough mathematical detail here to enable the reader to compute examples, to justify our claim that the construction is efficient, and to support our heuristics. A Note on the Text {#a-note-on-the-text .unnumbered} ------------------ This article presents an extended version of work that appeared in the proceedings of the EUROCRYPT 2008 conference [@Smith-Eurocrypt]. The chief results are the same; we have made some (minor) changes to our notation, expanded the derivation in §\[section:equations-for-the-isogeny\], given further details and proofs throughout, and added an appendix with algorithms to compute sets of tractable subgroups. Notation and Conventions for Hyperelliptic Curves {#section:notation-and-conventions-for-hyperelliptic-curves} ================================================= We will work over ${{\mathbb{F}}}_{q}$ throughout this article,[^1] where $q$ is a power of a prime $p > 3$. We let ${\ensuremath{\mathcal{G}}}$ denote the Galois group ${{\mathrm{Gal}({{\overline{\mathbb{F}}}}_{q}/{{\mathbb{F}}}_{q})}}$, which is (topologically) generated by the $q^\mathrm{th}$ power Frobenius map. Suppose we are given a hyperelliptic curve $H$ of genus $3$ over ${{\mathbb{F}}}_{q}$. We will use both an affine model $$H: y^2 = F(x) ,$$ where $F$ is a squarefree polynomial of degree $7$ or $8$, and a weighted projective plane model $$H: w^2 = {\ensuremath{\widetilde{F}}}(u,v)$$ for $H$ (here $u$, $v$, and $w$ have weights $1$, $1$, and $4$, respectively). The coordinates of these models are related by $x = u/v$ and $y = w/v^4$. The polynomial ${\ensuremath{\widetilde{F}}}$ is squarefree of total degree $8$, with ${\ensuremath{\widetilde{F}}}(u,v) = v^8F(u/v)$ and $F(x) = {\ensuremath{\widetilde{F}}}(x,1)$. We emphasize that $F$ need not be monic. By a *randomly chosen hyperelliptic curve*, we mean the hyperelliptic curve defined by $w^2 = {\ensuremath{\widetilde{F}}}(u,v)$, where ${\ensuremath{\widetilde{F}}}$ is a uniformly randomly chosen squarefree homogenous bivariate polynomial of degree $8$ over ${{\mathbb{F}}}_{q}$. The canonical *hyperelliptic involution* $\iota$ of $H$ is defined by $(x,y) \mapsto (x,-y)$ in the affine model, $(u:v:w) \mapsto (u:v:-w)$ in the projective model, and induces the negation map $[-1]$ on ${\ensuremath{{J}_{H}}}$. The quotient $\pi: H \to H/{\ensuremath{\left\langle{\iota}\right\rangle}}\cong {\ensuremath{\mathbb{P}^{1}_{{}}}}$ sends $(u:v:w)$ to $(u:v)$ in the projective model, and $(x,y)$ to $x$ in the affine model (where it maps onto the affine patch of ${\ensuremath{\mathbb{P}^{1}_{{}}}}$ where $v \not= 0$). To compute in ${\ensuremath{{J}_{H}}}$, we fix an isomorphism from ${\ensuremath{{J}_{H}}}$ to the group of degree-$0$ divisor classes on $H$, denoted ${\ensuremath{\mathrm{Pic}^{0}(H)}}$. Recall that divisors are formal sums of points in $H({{\overline{\mathbb{F}}}}_q)$, and if $ D = \sum_{P \in H} n_P(P) $ is a divisor, then $\sum_{P\in H} n_P$ is the *degree* of $D$. We say $D$ is *principal* if $ D = \mathrm{div}(f) := \sum_{P\in H} \mathrm{ord}_P(f)(P) $ for some function $f$ on $H$, where $\mathrm{ord}_P(f)$ denotes the number of zeroes (or the negative of the number of poles) of $f$ at $P$. Since $H$ is complete, every principal divisor has degree $0$. The group ${\ensuremath{\mathrm{Pic}^{0}(H)}}$ is defined to be the group of divisors of degree $0$ modulo principal divisors; the equivalence class of a divisor $D$ is denoted by $[D]$. We let ${\ensuremath{{J}_{H}}}[l]$ denote the $l$-torsion subgroup of ${\ensuremath{{J}_{H}}}$: that is, the kernel of the multiplication-by-$l$ map. If $l$ is prime to $q$, then ${\ensuremath{{J}_{H}}}[l]({{\overline{\mathbb{F}}}}_{q})$ is isomorphic to $({{\mathbb{Z}}}/l{{\mathbb{Z}}})^6$. The Kernel of the Isogeny {#section:kernel} ========================= The eight points of $H({{\overline{\mathbb{F}}}}_{q})$ where $w = 0$ are called the *Weierstrass points* of $H$. Each Weierstrass point $W$ corresponds to a linear factor $$L_W := v(W)u - u(W)v$$ of ${\ensuremath{\widetilde{F}}}$, which is defined up to scalar multiples. If $W_1$ and $W_2$ are Weierstrass points, then $ 2(W_1) - 2(W_2) = \mathrm{div}(L_{W_1}/L_{W_2}) $, so $2[(W_1) - (W_2)] = 0$; hence $[(W_1) - (W_2)]$ represents an element of ${\ensuremath{{J}_{H}}}[2]({{\overline{\mathbb{F}}}}_{q})$. In particular, $[(W_1) - (W_2)] = [(W_2) - (W_1)]$, so the divisor class $[(W_1) - (W_2)]$ corresponds to the pair $\{W_1, W_2\}$ of Weierstrass points, and hence to the quadratic factor $L_{W_1}L_{W_2}$ of ${\ensuremath{\widetilde{F}}}$ (up to scalar multiples). \[proposition:partitions-and-tractable-subgroups\] To every ${\ensuremath{\mathcal{G}}}$-stable partition of the eight Weierstrass points of $H$ into four disjoint pairs, we may associate an ${{\mathbb{F}}}_{q}$-rational subgroup of ${\ensuremath{{J}_{H}}}[2]({{\overline{\mathbb{F}}}}_{q})$ isomorphic to $({{\mathbb{Z}}}/2{{\mathbb{Z}}})^3$. Let $ \{ \{W_1',W_1''\}, \{W_2',W_2''\}, \{W_3',W_3''\}, \{W_4',W_4''\} \} $ be a partition of the set of Weierstrass points of $H$ into four disjoint pairs. Each pair $\{W_i', W_i''\}$ corresponds to the $2$-torsion divisor class $[(W_i') - (W_i'')]$ in ${\ensuremath{{J}_{H}}}[2]({{\overline{\mathbb{F}}}}_{q})$. We associate the subgroup $S := {\ensuremath{\left\langle{[(W_i') - (W_i'') ] : 1 \le i \le 4}\right\rangle}}$ to the partition. Observe that $$\sum_{i = 1}^4 [ (W_i') - (W_i'') ] = \Big[\mathrm{div}\big(w/\prod_{i=1}^4 L_{W_i''}\big)\Big] = 0 ;$$ this is the only relation on the classes $[(W_i') - (W_i'')]$, so $S \cong ({{\mathbb{Z}}}/2{{\mathbb{Z}}})^3~$. The action of ${\ensuremath{\mathcal{G}}}$ on ${\ensuremath{{J}_{H}}}[2]({{\overline{\mathbb{F}}}}_{q})$ corresponds to its action on the Weierstrass points, so if the partition is ${\ensuremath{\mathcal{G}}}$-stable, then the subgroup $S$ is ${\ensuremath{\mathcal{G}}}$-stable. By “an ${{\mathbb{F}}}_{q}$-rational subgroup of ${\ensuremath{{J}_{H}}}[2]({{\overline{\mathbb{F}}}}_{q})$ isomorphic to $({{\mathbb{Z}}}/2{{\mathbb{Z}}})^3$”, we mean a ${\ensuremath{\mathcal{G}}}$-stable subgroup that is isomorphic to $({{\mathbb{Z}}}/2{{\mathbb{Z}}})^3$ over ${{\overline{\mathbb{F}}}}_{q}$. We emphasize that the subgroup need *not* be contained in ${\ensuremath{{J}_{H}}}({{\mathbb{F}}}_{q})$. Requiring the pairs of Weierstrass points in Proposition \[proposition:partitions-and-tractable-subgroups\] to be disjoint ensures that the associated subgroup is isotropic with respect to the $2$-Weil pairing. We will see in §\[section:other-isogenies\] that this is necessary for the quotient by the subgroup to be an isogeny of principally polarized abelian varieties, and hence for the quotient to be an isogeny of Jacobians. We call the subgroups corresponding to partitions of the Weierstrass points of $H$ as in Proposition \[proposition:partitions-and-tractable-subgroups\] *tractable subgroups*. We let ${\ensuremath{\mathcal{S}}({H})}$ denote the set of all ${{\mathbb{F}}}_{q}$-rational tractable subgroups of ${\ensuremath{{J}_{H}}}[2]({{\overline{\mathbb{F}}}}_{q})$. Not every subgroup of ${\ensuremath{{J}_{H}}}[2]({{\overline{\mathbb{F}}}}_{q})$ that is the kernel of an isogeny of Jacobians is a tractable subgroup. For example, if $W_1,\ldots,W_8$ are the Weierstrass points of $H$, then the subgroup $${\ensuremath{\left\langle{ [ (W_1) - (W_i) + (W_j) - (W_k) ] : (i,j,k) \in \{ (2,3,4), (2,5,6), (3,5,7) \} }\right\rangle}}$$ is a maximal $2$-Weil isotropic subgroup of ${\ensuremath{{J}_{H}}}({{\overline{\mathbb{F}}}}_{q})$, and hence is the kernel of an isogeny of Jacobians (see §\[section:other-isogenies\]). However, this subgroup contains no nontrivial differences of Weierstrass points, and therefore cannot be a tractable subgroup. Computing ${\ensuremath{\mathcal{S}}({H})}$ is straightforward if we identify each tractable subgroup with its corresponding partition of Weierstrass points. Recall that each pair of Weierstrass points $\{W_i',W_i''\}$ corresponds to a quadratic factor of ${\ensuremath{\widetilde{F}}}$ (up to scalar multiples). Since the pairs are disjoint, the corresponding quadratic factors are pairwise coprime, so we may take them to form a factorization of ${\ensuremath{\widetilde{F}}}$. We therefore have a correspondence of tractable subgroups, partitions of Weierstrass points into pairs, and sets of quadratic polynomials (up to scalar multiples): $$S \longleftrightarrow \big\{ \{ W_i', W_i'' \} : 1 \le i \le 4 \big\} \longleftrightarrow \big\{ F_1, F_2, F_3, F_4 \big\}, \ \text{where} \ {\ensuremath{\widetilde{F}}} = F_1F_2F_3F_4 .$$ The action of ${\ensuremath{\mathcal{G}}}$ on ${\ensuremath{{J}_{H}}}[2]({{\overline{\mathbb{F}}}}_{q})$ corresponds to its action on the set of Weierstrass points, so the action of ${\ensuremath{\mathcal{G}}}$ on a tractable subgroup $S$ corresponds to the action of ${\ensuremath{\mathcal{G}}}$ on the corresponding set $\{ F_1, F_2, F_3, F_4 \}$ (assuming the $F_i$ have been scaled appropriately). In particular, $S$ is ${{\mathbb{F}}}_{q}$-rational precisely when $\{ F_1, F_2, F_3, F_4 \}$ is fixed by ${\ensuremath{\mathcal{G}}}$. The factors $F_i$ are themselves defined over ${{\mathbb{F}}}_{q}$ precisely when the corresponding points of $S$ are ${{\mathbb{F}}}_{q}$-rational. We can use this information to compute ${\ensuremath{\mathcal{S}}({H})}$. The set of pairs of Weierstrass points contains a ${\ensuremath{\mathcal{G}}}$-orbit $ \big( \{ W_{i_1}',W_{i_1}'' \}, \ldots, \{ W_{i_n}',W_{i_n}'' \} \big) $ if and only if (possibly after exchanging some of the $W_{i_k}'$ with the $W_{i_k}''$) either both $( W_{i_1}', \ldots, W_{i_n}' )$ and $( W_{i_1}'', \ldots, W_{i_n}'' )$ are ${\ensuremath{\mathcal{G}}}$-orbits or $ ( W_{i_1}', \ldots, W_{i_n}', W_{i_1}'', \ldots, W_{i_n}'' )~$ is a ${\ensuremath{\mathcal{G}}}$-orbit. Every ${\ensuremath{\mathcal{G}}}$-orbit of Weierstrass points corresponds to an ${{\mathbb{F}}}_{q}$-irreducible factor of $F$, so the size of ${\ensuremath{\mathcal{S}}({H})}$ depends only on the factorization of $F$. A table relating the size of ${\ensuremath{\mathcal{S}}({H})}$ to the factorization of ${\ensuremath{\widetilde{F}}}$ appears in Lemma \[lemma:number-of-tractable-subgroups\] below; this will be useful for our analysis in §\[section:probabilities\]. For completeness, we have included a naïve algorithm for enumerating ${\ensuremath{\mathcal{S}}({H})}$ in Appendix \[appendix:computing-SH\]. \[lemma:number-of-tractable-subgroups\] Let $H: w^2 = {\ensuremath{\widetilde{F}}}(u,v)$ be a hyperelliptic curve of genus $3$ over ${{\mathbb{F}}}_{q}$. The cardinality of the set ${\ensuremath{\mathcal{S}}({H})}$ depends only on the degrees of the ${{\mathbb{F}}}_{q}$-irreducible factors of ${\ensuremath{\widetilde{F}}}$, and is described by the following table: Degrees of ${{\mathbb{F}}}_{q}$-irreducible factors of ${\ensuremath{\widetilde{F}}}$ $\# {\ensuremath{\mathcal{S}}({H})}$ --------------------------------------------------------------------------------------- -------------------------------------- $( 8 ), (6, 2), (6, 1, 1), (4,2,1,1)$  $1$ $( 4, 2, 2 ), (4,1,1,1,1), (3,3,2), (3,3,1,1)$  $3$ $( 4, 4 )$  $5$ $( 2, 2, 2, 1, 1 )$  $7$ $( 2, 2, 1, 1, 1, 1 )$  $9$ $( 2, 1, 1, 1, 1, 1, 1 )$  $15$ $( 2, 2, 2, 2 )$  $25$ $( 1, 1, 1, 1, 1, 1, 1, 1 )$  $105$ Other  $0$ This is a routine combinatorial exercise after noting that every ${\ensuremath{\mathcal{G}}}$-orbit of pairs of Weierstrass points corresponds to either an even-degree factor of $F$, or a pair of factors of $F$ of the same degree. The Trigonal Construction {#section:trigonal-construction} ========================= We will now briefly outline the theoretical aspects of constructing isogenies with tractable kernels. We will make the construction completely explicit in §\[section:computing-trigonal-maps\] and §\[section:equations-for-the-isogeny\]. Suppose $S = {\ensuremath{\left\langle{[(W_i') - (W_i'')] : 1 \le i \le 4}\right\rangle}}$ is a tractable subgroup. We say that a morphism $g: {\ensuremath{\mathbb{P}^{1}_{{}}}} \to {\ensuremath{\mathbb{P}^{1}_{{}}}}$ is a *trigonal map for $S$* if $g$ has degree $3$ and $g(\pi(W_i')) = g(\pi(W_i''))$ for $1 \le i \le 4$. Given a trigonal map $g$ for some tractable subgroup $S$, Recillas’ trigonal construction [@Recillas] specifies a curve $X$ of genus $3$ and a map $f: X \to {\ensuremath{\mathbb{P}^{1}_{{}}}}$ of degree $4$.[^2] The isomorphism class of $X$ depends only on $S$, and is independent of the choice of $g$ (see Recillas [@Recillas], Donagi [@Donagi Th. 2.11], and Remark \[remark:trigonal-map-descent\] below). Theorem \[theorem:isogeny-from-trigonal-construction\], due to Donagi and Livné, states that if $g$ is a trigonal map for $S$, then $S$ is the kernel of an isogeny from ${\ensuremath{{J}_{H}}}$ to ${\ensuremath{{J}_{X}}}$. \[theorem:isogeny-from-trigonal-construction\] Let $S$ be a tractable subgroup in ${\ensuremath{\mathcal{S}}({H})}$, and let $g: {\ensuremath{\mathbb{P}^{1}_{{}}}} \to {\ensuremath{\mathbb{P}^{1}_{{}}}}$ be a trigonal map for $S$. If $X$ is the curve formed from $g$ by Recillas’ trigonal construction, then there is an isogeny $\phi: {\ensuremath{{J}_{H}}} \to {\ensuremath{{J}_{X}}}$ (defined over ${{\overline{\mathbb{F}}}}_{q}$) with kernel $S$. We will give only a brief description of the geometry of $X$ here, concentrating instead on its explicit construction; we refer the reader to Recillas [@Recillas], Vakil [@Vakil], Donagi [@Donagi §2], and Birkenhake and Lange [@Birkenhake--Lange §12.7] for proofs and further detail. The isogeny of Theorem \[theorem:isogeny-from-trigonal-construction\] is analogous to the well-known Richelot isogeny in genus $2$ (see Bost and Mestre [@Bost-Mestre], and Donagi and Livné [@Donagi--Livne §4] for details), and to the explicit isogeny described by Lehavi and Ritzenthaler in [@Lehavi--Ritzenthaler] for Jacobians of non-hyperelliptic genus $3$ curves. In abstract terms, if $U$ is the subset of the codomain of $g$ above which $g\circ\pi$ is unramified, then $X$ is by definition the closure of the curve over $U$ representing the pushforward to $U$ of the sheaf of sections of $\pi: (g\circ\pi)^{-1}(U) \to g^{-1}(U)$ (in the étale topology). This means in particular that the ${{\overline{\mathbb{F}}}}_{q}$-points of $X$ over an ${{\overline{\mathbb{F}}}}_{q}$-point $P$ of $U$ represent partitions of the six ${{\overline{\mathbb{F}}}}_{q}$-points of $(g\circ\pi)^{-1}(P)$ into two sets of three exchanged by the hyperelliptic involution. The fibre product of $H$ and $X$ over ${\ensuremath{\mathbb{P}^{1}_{{}}}}$ with respect to $g\circ\pi$ and $f$ is the union of two isomorphic curves, $R$ and $R'$, which are exchanged by the involution on $H\times_{{\ensuremath{\mathbb{P}^{1}_{{}}}}}X$ induced by the hyperelliptic involution. The natural projections induce coverings $\pi_H: R \to H$ and $\pi_X: R \to X$ of degrees $2$ and $3$, respectively, so $R$ is a $(3,2)$-correspondence between $H$ and $X$. The maps $\pi_H$ and $\pi_X$ induce homomorphisms $(\pi_H)^*: {\ensuremath{{J}_{H}}} \to {\ensuremath{{J}_{R}}}$ (the *pullback*) and $(\pi_X)_*: {\ensuremath{{J}_{R}}} \to {\ensuremath{{J}_{X}}}$ (the *pushforward*). In terms of divisor classes, the pullback is defined by $$(\pi_H)^*\Big(\Big[\sum_{P \in H}n_P(P)\Big]\Big) = \Big[ \sum_{P \in H}n_P\!\!\!\!{\sum_{Q \in \pi_H^{-1}(P)}}\!\!\!\!(Q) \Big],$$ with appropriate multiplicities where $\pi_H$ ramifies; the pushforward is defined by $$(\pi_X)_*\Big(\Big[ \sum_{Q\in R}m_Q(Q) \Big]\Big) = \Big[ \sum_{Q\in R}m_Q(\pi_X(Q)) \Big] .$$ Composing $(\pi_X)_*$ with $(\pi_H)^*$, we obtain an isogeny $\phi: {\ensuremath{{J}_{H}}} \to {\ensuremath{{J}_{X}}}$ with kernel $S$. If we replace $R$ with $R'$ in the above, we obtain an isogeny isomorphic to $-\phi$. Thus, up to isomorphism, the construction of the isogeny depends only on the subgroup $S$. The curves and Jacobians described above form the commutative diagrams shown in Figure \[table:curve-diagram\]. $$\begin{psmatrix}[rowsep=36pt] &R&&&{\ensuremath{{J}_{R}}}\\ H&&X&{\ensuremath{{J}_{H}}}&&{\ensuremath{{J}_{X}}}\\ {\ensuremath{\mathbb{P}^{1}_{{}}}}\\ &{\ensuremath{\mathbb{P}^{1}_{{}}}} \ncline[nodesep=3pt]{->}{1,2}{2,1}^{\pi_H}_2 \ncline[nodesep=3pt]{->}{1,2}{2,3}^{\pi_X}_3 \ncline[nodesep=3pt]{->}{2,1}{3,1}<{\pi}>2 \ncline[nodesep=3pt]{->}{3,1}{4,2}^{3}_g \ncline[nodesep=3pt]{->}{2,3}{4,2}^{4}_f \ncline[nodesep=3pt]{->}{2,4}{2,6}_{\phi} \ncline[nodesep=3pt]{->}{2,4}{1,5}<{\pi_H^*} \ncline[nodesep=3pt]{->}{1,5}{2,6}>{(\pi_X)_*} \end{psmatrix}$$ The hyperelliptic Jacobians form a codimension-$1$ subspace ${\ensuremath{\mathcal{H}_{g}}}$ of the moduli space of $3$-dimensional principally polarized abelian varieties — which, by the theorem of Oort and Ueno [@Oort--Ueno], is also the moduli space ${\ensuremath{\mathcal{M}_{g}}}$ of Jacobians of genus $3$ curves. The Weil hypotheses imply that $\#{\ensuremath{\mathcal{H}_{g}}}({{\mathbb{F}}}_{q})/\#{\ensuremath{\mathcal{M}_{g}}}({{\mathbb{F}}}_{q}) \sim 1/q$ for sufficiently large $q$ (cf. [@Lang-Weil Theorem 1]). In particular, for cryptographically relevant sizes of $q$, the probability that a uniformly randomly chosen curve $X$ of genus $3$ over ${{\mathbb{F}}}_{q}$ should be hyperelliptic is negligible. We will suppose that the same is true for the curve $X$ constructed in Theorem \[theorem:isogeny-from-trigonal-construction\] for a uniformly randomly chosen $H$ and $S$ in ${\ensuremath{\mathcal{S}}({H})}$. This is consistent with our experimental observations, so we postulate Hypothesis \[hypothesis:hyperellipticity\]. \[hypothesis:hyperellipticity\] The probability that the curve $X$ constructed by the trigonal construction for a randomly chosen $H/{{\mathbb{F}}}_{q}$ and $S$ in ${\ensuremath{\mathcal{S}}({H})}$ is hyperelliptic is negligible for sufficiently large $q$. Computing Trigonal Maps {#section:computing-trigonal-maps} ======================= Suppose we are given a tractable subgroup $S$ of ${\ensuremath{{J}_{H}}}[2]({{\overline{\mathbb{F}}}}_{q})$, corresponding to a partition $\{ \{W_i', W_i''\}: 1 \le i \le 4 \}$ of the Weierstrass points of $H$ into pairs. The first step in the explicit trigonal construction is to compute a trigonal map $g$ for $S$. We will compute polynomials $N = x^3 + n_1x + n_0$ and $D = x^2 + d_1x + d_0$ such that the rational map $$\label{eq:rational-map-form} g: x \longmapsto t = \frac{N(x)}{D(x)} = \frac{x^3 + n_1x + n_0}{x^2 + d_1x + d_0}$$ defines a trigonal map for $S$. The derivation is an exercise in classical geometry; we include it here to demonstrate its efficiency and to justify Hypothesis \[hypothesis:trigonal-map-rationality\], which will be important in determining the expectation of success of our reduction in §\[section:probabilities\]. The reader prepared to admit the existence of efficiently computable trigonal maps in the form of  may skip the remainder of this section on first reading. By definition, $g: {\ensuremath{\mathbb{P}^{1}_{{}}}} \to {\ensuremath{\mathbb{P}^{1}_{{}}}}$ is a degree-$3$ map with $ g(\pi(W_i')) = g(\pi(W_i''))$ for $1 \le i \le 4$. We will express $g$ as a composition $ g = p\circ e$, where $e: {\ensuremath{\mathbb{P}^{1}_{{}}}} \to {\ensuremath{\mathbb{P}^{3}_{{}}}}$ is the rational normal embedding defined by $$e: (u:v) \longmapsto (u_0:u_1:u_2:u_3) = (u^3 : u^2v : uv^2 : v^3) ,$$ and $p: {\ensuremath{\mathbb{P}^{3}_{{}}}} \to {\ensuremath{\mathbb{P}^{1}_{{}}}}$ is the projection defined as follows. For each $1 \le i \le 4$, we let $L_i$ denote the line in ${\ensuremath{\mathbb{P}^{3}_{{}}}}$ passing through $e(\pi(W_i'))$ and $e(\pi(W_i''))$. There exists at least one line $L$ intersecting all four of the $L_i$ (in fact there are two, though they may coincide; we will compute them below). We take $p$ to be the projection away from $L$; then $p(e(\pi(W_i'))) = p(e(\pi(W_i'')))$ for $1 \le i \le 4$, so $g = p\circ e$ is a trigonal map for $S$. Given linear equations for $L$ in the coordinates $u_i$, we can use Gaussian elimination to compute elements $n_1$, $n_0$, $d_1$, and $d_0$ of ${{\mathbb{F}}}_{q}$ such that $$L = {\ensuremath{V\!\left({ u_0 + n_1u_2 + n_0u_3, u_1 + d_1u_2 + d_0u_3 }\right)}} .$$ The projection $p: {\ensuremath{\mathbb{P}^{3}_{{}}}} \to {\ensuremath{\mathbb{P}^{1}_{{}}}}$ away from $L$ is then defined by $$p: (u_0:u_1:u_2:u_3) \longmapsto ( u_0 + n_1u_2 + n_0u_3 : u_1 + d_1u_2 + d_0u_3 ) ,$$ so our trigonal map $g = p\circ e$ is defined by $$g: (u:v) \longmapsto ( u^3 + n_1uv^2 + n_0v^3 : u^2v + d_1uv^2 + d_0v^3 ) .$$ Therefore, if we set $N(x) := x^3 + n_1x + n_0$ and $D(x) := x^2 + d_1x + d_0$, then $g$ will be defined by the rational map $x \longmapsto t = N(x)/D(x)$. To compute equations for $L$, we will use the classical theory of *Grassmannian varieties*. The elementary Lemmas \[lemma:Grassmannian-correspondence\] and \[lemma:hyperplane\] will be stated without proof; we refer the reader to Griffiths and Harris [@Griffiths--Harris §1.5] and Harris [@Harris Lecture 6] for details. The set of lines in ${\ensuremath{\mathbb{P}^{3}_{{}}}}$ has the structure of an algebraic variety ${\mathrm{Gr}}(1,3)$, called the Grassmannian. There is a convenient model for ${\mathrm{Gr}}(1,3)$ as a quadric hypersurface in ${\ensuremath{\mathbb{P}^{5}_{{}}}}$: if $v_0,\ldots,v_5$ are coordinates on ${\ensuremath{\mathbb{P}^{5}_{{}}}}$, then we may take $${\mathrm{Gr}}(1,3) := {\ensuremath{V\!\left({ v_0v_3 + v_1v_4 + v_2v_5 }\right)}} \subset {\ensuremath{\mathbb{P}^{5}_{{}}}}.$$ \[lemma:Grassmannian-correspondence\] There is a bijection between points of ${\mathrm{Gr}}(1,3)({{\overline{\mathbb{F}}}}_{q})$ and lines in ${\ensuremath{\mathbb{P}^{3}_{{}}}}$, defined as follows. 1. The point of ${\mathrm{Gr}}(1,3)({{\overline{\mathbb{F}}}}_{q})$ corresponding to the line through $(p_0:p_1:p_2:p_3)$ and $(q_0:q_1:q_2:q_3)$ in ${\ensuremath{\mathbb{P}^{3}_{{}}}}$ has coordinates $$\left( \left|\begin{array}{@{\:}c@{\;\;}c@{\:}} p_0 & p_1 \\ q_0 & q_1 \end{array}\right|: \left|\begin{array}{@{\:}c@{\;\;}c@{\:}} p_0 & p_2 \\ q_0 & q_2 \end{array}\right|: \left|\begin{array}{@{\:}c@{\;\;}c@{\:}} p_0 & p_3 \\ q_0 & q_3 \end{array}\right|: \left|\begin{array}{@{\:}c@{\;\;}c@{\:}} p_2 & p_3 \\ q_2 & q_3 \end{array}\right|: \left|\begin{array}{@{\:}c@{\;\;}c@{\:}} p_3 & p_1 \\ q_3 & q_1 \end{array}\right|: \left|\begin{array}{@{\:}c@{\;\;}c@{\:}} p_1 & p_2 \\ q_1 & q_2 \end{array}\right| \right) .$$ 2. The line in ${\ensuremath{\mathbb{P}^{3}_{{}}}}$ corresponding to a point $(\gamma_0:\cdots:\gamma_5)$ of ${\mathrm{Gr}}(1,3)({{\overline{\mathbb{F}}}}_{q})$ is defined by $${\ensuremath{V\!\left({ \begin{array}{r@{\;}r@{\;}r@{\;}r@{\;}r@{\;}r@{\;}r} 0 u_0 & - & \gamma_3 u_1 & - & \gamma_4 u_2 & - & \gamma_5 u_3 , \\ \gamma_3 u_0 & + & 0 u_1 & - & \gamma_2 u_2 & + & \gamma_1 u_3 , \\ \gamma_4 u_0 & + & \gamma_2 u_1 & + & 0 u_2 & - & \gamma_0 u_3 , \\ \gamma_5 u_0 & - & \gamma_1 u_1 & + & \gamma_0 u_2 & + & 0 u_3 \\ \end{array} }\right)}}$$ (two of the equations will be redundant linear combinations of the others). \[lemma:hyperplane\] Let $L$ be the line in ${\ensuremath{\mathbb{P}^{3}_{{}}}}$ corresponding to a point $(\gamma_0:\cdots:\gamma_5)$ of ${\mathrm{Gr}}(1,3)({{\overline{\mathbb{F}}}}_{q})$. The points in ${\mathrm{Gr}}(1,3)({{\overline{\mathbb{F}}}}_{q})$ corresponding to lines in ${\ensuremath{\mathbb{P}^{3}_{{}}}}$ that intersect nontrivially with $L$ are precisely the points lying in the hyperplane defined by $ \sum_{i=0}^5 \gamma_iv_{i+3} = 0 $ (where the subscripts are taken modulo $6$). Suppose $S$ is represented by a set $\{F_i=a_iu^2 + b_iuv + c_iv^2 : 1\:\le\:i\:\le~4\}$ of quadratic factors of ${\ensuremath{\widetilde{F}}}$ (as in §\[section:kernel\]), with each factor $F_i$ corresponding to a pair $\{ W_i',W_i'' \}$ of Weierstrass points. Applying Lemma \[lemma:Grassmannian-correspondence\], we see that the line $L_i$ through $e(\pi(W_i'))$ and $e(\pi(W_i''))$ corresponds to the point $$(c_i^2:-c_ib_i:b_i^2-a_ic_i:a_i^2:a_ib_i:a_ic_i) $$ on ${\mathrm{Gr}}(1,3)$. If $(\gamma_0:\cdots:\gamma_5)$ in ${\mathrm{Gr}}(1,3)({{\overline{\mathbb{F}}}}_{q})$ corresponds to a candidate for $L$, then by Lemma \[lemma:hyperplane\] we have $M(\gamma_0,\ldots,\gamma_5)^T = 0$, where $$\label{eq:M-def} M = \left(\!\begin{array}{c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c@{\ \ }c} a_1^2 & a_1b_1 & a_1c_1 & c_1^2 & - c_1b_1 & (b_1^2-a_1c_1) \\ a_2^2 & a_2b_2 & a_2c_2 & c_2^2 & - c_2b_2 & (b_2^2-a_2c_2) \\ a_3^2 & a_3b_3 & a_3c_3 & c_3^2 & - c_3b_3 & (b_3^2-a_3c_3) \\ a_4^2 & a_4b_4 & a_4c_4 & c_4^2 & - c_4b_4 & (b_4^2-a_4c_4) \\ \end{array} \!\right) .$$ The kernel of $M$ is two-dimensional, corresponding to a line $\Lambda$ in ${\ensuremath{\mathbb{P}^{5}_{{}}}}$. The kernel is independent of the ordering of the $F_i$, and does not change if we replace the $F_i$ by scalar multiples; hence, $\Lambda$ depends only on the subgroup $S$. Let $\{{\ensuremath{\underline{\alpha}}},{\ensuremath{\underline{\beta}}}\}$ be a basis for $\ker M$, writing ${\ensuremath{\underline{\alpha}}} = (\alpha_0,\ldots,\alpha_5)$ and ${\ensuremath{\underline{\beta}}} = (\beta_0,\ldots,\beta_5)$. If $S$ is ${{\mathbb{F}}}_{q}$-rational, then so is $\ker M$, so we may take the $\alpha_i$ and $\beta_i$ to be in ${{\mathbb{F}}}_{q}$ (see Cartier [@Cartier §I]). We want to find a point $P_L = (\alpha_0 + \lambda \beta_0:\cdots:\alpha_5+\lambda\beta_5)$ where $\Lambda$ intersects with ${\mathrm{Gr}}(1,3)$. The points $(u_0:\ldots:u_3)$ on the line $L$ in ${\ensuremath{\mathbb{P}^{3}_{{}}}}$ corresponding to $P_L$ satisfy $ (M_{{\ensuremath{\underline{\alpha}}}} + \lambda M_{{\ensuremath{\underline{\beta}}}}) (u_0,\ldots,u_3)^T = 0 $, where $$M_{{\ensuremath{\underline{\alpha}}}} := \left(\!\begin{array}{r@{\ \ }r@{\ \ }r@{\ \ }r} 0 & -\alpha_3 & -\alpha_4 & -\alpha_5 \\ \alpha_3 & 0 & - \alpha_2 & \alpha_1 \\ \alpha_4 & \alpha_2 & 0 & - \alpha_0 \\ \alpha_5 & -\alpha_1 & \alpha_0 & 0 \\ \end{array}\!\right) \ \ \mbox{and}\quad M_{{\ensuremath{\underline{\beta}}}} := \left(\!\begin{array}{r@{\ \ }r@{\ \ }r@{\ \ }r} 0 & -\beta_3 & -\beta_4 & -\beta_5 \\ \beta_3 & 0 & - \beta_2 & \beta_1 \\ \beta_4 & \beta_2 & 0 & - \beta_0 \\ \beta_5 & -\beta_1 & \beta_0 & 0 \\ \end{array}\!\right) .$$ By part (2) of Lemma \[lemma:Grassmannian-correspondence\], the rank of $M_{{\ensuremath{\underline{\alpha}}}} + \lambda M_{{\ensuremath{\underline{\beta}}}}$ is $2$. Using the expression $$\label{eq:lambda-eqn} \det(M_{{\ensuremath{\underline{\alpha}}}} + \lambda M_{{\ensuremath{\underline{\beta}}}}) = \Big( \frac{1}{2}\big(\sum_{i=0}^6\beta_i\beta_{i+3} \big)\lambda^2 + \big(\sum_{i=0}^6 \alpha_i\beta_{i+3} \big)\lambda + \frac{1}{2}\sum_{i=0}^6 \alpha_i\alpha_{i+3} \Big)^2$$ (where the subscripts are taken modulo $6$), we see that $M_{{\ensuremath{\underline{\alpha}}}} + \lambda M_{{\ensuremath{\underline{\beta}}}}$ has rank $2$ precisely when $\det(M_{{\ensuremath{\underline{\alpha}}}} + \lambda M_{{\ensuremath{\underline{\beta}}}}) = 0$: we can therefore solve $\det(M_{{\ensuremath{\underline{\alpha}}}} + \lambda M_{{\ensuremath{\underline{\beta}}}}) = 0$ to determine a value for $\lambda$. Finally, we use Gaussian elimination to compute $n_1$, $n_0$, $d_1$, and $d_0$ in ${{\mathbb{F}}}_{q}(\lambda)$ such that $(1,0,n_1,n_0)$ and $(0,1,d_1,d_0)$ generate the rowspace of $M_{\underline{\alpha}} + \lambda M_{\underline{\beta}}$. We then take $ L = {\ensuremath{V\!\left({u_0 + n_1u_2 + n_0u_3, u_1 + d_1u_2 + d_0u_3 }\right)}} $, and compute $p$, $e$, and the trigonal map $g = p\circ e$ as above. Since $L$ is defined over ${{\mathbb{F}}}_{q}(\lambda)$, so is the projection $p$ and the trigonal map $g$. But $\lambda$ satisfies a quadratic equation with coefficients in ${{\mathbb{F}}}_{q}$, so ${{\mathbb{F}}}_{q}(\lambda)$ is at most a quadratic extension of ${{\mathbb{F}}}_{q}$. Computing the discriminant of $\det(M_{\underline{\alpha}} + \lambda M_{\underline{\beta}})$, we obtain a criterion for existence of trigonal maps over ${{\mathbb{F}}}_{q}$ for a given tractable subgroup. \[proposition:trigonal-map-rationality-criterion\] Suppose $S$ is a tractable subgroup, and let $ \{ \underline{\alpha} = (\alpha_i), \underline{\beta} = (\beta_i) \} $ be any ${{\mathbb{F}}}_{q}$-rational basis of the nullspace of the matrix $M$ defined in . There exists an ${{\mathbb{F}}}_{q}$-rational trigonal map for $S$ if and only if $$\label{eq:discriminant} \Big(\sum_{i=0}^6\alpha_i\beta_{i+3}\Big)^2 - \Big(\sum_{i=0}^6\alpha_i\alpha_{i+3}\Big) \Big(\sum_{i=0}^6\beta_i\beta_{i+3}\Big)$$ is a square in ${{\mathbb{F}}}_{q}$, where the subscripts are taken modulo $6$. From the derivation above, we see that there exists an ${{\mathbb{F}}}_{q}$-rational trigonal map for $S$ if and only if we can find a $\lambda$ in ${{\mathbb{F}}}_{q}$ such that $\det(M_{\underline{\alpha}} + \lambda M_{\underline{\beta}}) = 0$. By Equation , we can find such a $\lambda$ if and only if the quadratic polynomial $$\frac{1}{2}\big(\sum_{i=0}^6\beta_i\beta_{i+3} \big)T^2 + \big(\sum_{i=0}^6 \alpha_i\beta_{i+3} \big)T + \frac{1}{2}\sum_{i=0}^6 \alpha_i\alpha_{i+3}$$ has two roots in ${{\mathbb{F}}}_{q}$. This occurs precisely when the discriminant of this polynomial — the expression in  above — is a square in ${{\mathbb{F}}}_{q}$. Proposition \[proposition:trigonal-map-rationality-criterion\] shows that the rationality of a trigonal map for a tractable subgroup $S$ depends only upon whether an element of ${{\mathbb{F}}}_{q}$ depending only on $S$ is a square. It seems reasonable to assume that these field elements are uniformly distributed for uniformly random choices of $H$ and $S$, and indeed this is consistent with our experimental observations. Since a uniformly randomly chosen element of ${{\mathbb{F}}}_{q}$ is a square with probability $\sim 1/2$, we propose Hypothesis \[hypothesis:trigonal-map-rationality\]. \[hypothesis:trigonal-map-rationality\] The probability that there exists an ${{\mathbb{F}}}_{q}$-rational trigonal map for a subgroup $S$ uniformly randomly chosen from ${\ensuremath{\mathcal{S}}({H})}$, where $H$ is a randomly chosen hyperelliptic curve over ${{\mathbb{F}}}_{q}$, is $1/2$. Equations for the Isogeny {#section:equations-for-the-isogeny} ========================= Suppose we have a hyperelliptic curve $H$ of genus $3$, a tractable subgroup $S$ in ${\ensuremath{\mathcal{S}}({H})}$, and a trigonal map $g$ for $S$. We will now perform an explicit trigonal construction on $g$ to compute a curve $X$ and an isogeny $\phi: {\ensuremath{{J}_{H}}} \to {\ensuremath{{J}_{X}}}$ with kernel $S$. We assume that $g$ has been derived as in §\[section:computing-trigonal-maps\], and in particular that $g:{\ensuremath{\mathbb{P}^{1}_{{}}}}\to{\ensuremath{\mathbb{P}^{1}_{{}}}}$ is defined by a rational map in the form $$g: x \longmapsto t = \frac{N(x)}{D(x)} = \frac{x^3 + n_1x + n_0}{x^2 + d_1x + d_0} .$$ Observe that $g$ maps the point at infinity to the point at infinity (that is, $(1:0)$). For notational convenience, we define $$G(t,x) = x^3 + g_2(t)x^2 + g_1(t)x + g_0(t) := N(x) - tD(x) ;$$ unless otherwise noted, we will view $G(t,x)$ as an element of ${{\mathbb{F}}}_{q}[t][x]$. We have $$g_2(t) = -t, \quad g_1(t) = n_1 - d_1t, \quad \text{and}\quad g_0(t) = n_0 - d_0t .$$ We also define $f_0$, $f_1$, and $f_2$ to be the elements of ${{\mathbb{F}}}_{q}[t]$ such that $$f_0(t) + f_1(t)x + f_2(t)x^2 \equiv F(x) \pmod{ G(t,x) }.$$ Let $U$ be the subset of ${\ensuremath{\mathbb{A}^{1}_{{}}}} = {\ensuremath{\mathbb{P}^{1}_{{}}}}\setminus\{(1:0)\}$ above which $g\circ\pi$ is unramified. With the notation above, $$U = {\mathrm{Spec}}(k[t])\setminus{\ensuremath{V\!\left({ (f_1^2 - 4f_2f_0)(4g_2^3g_0 - g_2^2g_1^2 - 18g_2g_1g_0 + 4g_1^3 + 27g_0^2) }\right)}} .$$ We will derive equations for an affine model $X|_U$ of $f^{-1}(U)$ — that is, the open subset of $X$ over $U$. We will not prove here that the normalization of $X|_U$ is isomorphic to the curve $X$ specified by Recillas, but we will exhibit a bijection on geometric points. If $X$ is not hyperelliptic, then taking the canonical map of $X|_U$ into ${\ensuremath{\mathbb{P}^{2}_{{}}}}$ will give us a nonsingular plane quartic curve $C$ isomorphic to $X$. By definition, every point $P$ in $X|_U({{\overline{\mathbb{F}}}}_{q})$ corresponds to a pair of unordered triples of points in $H({{\overline{\mathbb{F}}}}_{q})$, exchanged by the hyperelliptic involution, with each triple supported on the fibre of $g\circ\pi$ over $f(P)$. To be more explicit, suppose $Q$ is a generic point of $U$. Since $g\circ\pi$ is unramified above $Q$, we may choose three preimages $P_1$, $P_2$, and $P_3$ of $Q$ such that $$(g\circ\pi)^{-1}(Q) = \{ P_1, P_2, P_3, \iota(P_1), \iota(P_2), \iota(P_3) \} .$$ Viewing unordered triples of points as effective divisors of degree $3$ (that is, as formal sums of three points), we have $$\label{eq:preimage-of-Q-on-X} f^{-1}(Q) = \left\{ \begin{array}{c} Q_1 \leftrightarrow \big\{ P_1 + P_2 + P_3 ,\ \iota(P_1) + \iota(P_2) + \iota(P_3) \big\}\!, \\ Q_2 \leftrightarrow \big\{ P_1 + \iota(P_2) + \iota(P_3) ,\ \iota(P_1) + P_2 + P_3 \big\}\!, \\ Q_3 \leftrightarrow \big\{ \iota(P_1) + P_2 + \iota(P_3) ,\ P_1 + \iota(P_2) + P_3 \big\}\!, \\ Q_4 \leftrightarrow \big\{ \iota(P_1) + \iota(P_2) + P_3 ,\ P_1 + P_2 + \iota(P_3) \big\}\!\, \end{array} \right\} .$$ Note that $P_i$ and $\iota(P_i)$ never appear in the same divisor for any $1 \le i \le 3$. There is a one-to-one correspondence between effective divisors of degree $3$ on $H$ satisfying this condition, and ideals $(a(x), y - b(x))$ where $a$ is a monic cubic polynomial and $b$ is a quadratic polynomial satisfying $b^2 \equiv F \pmod{a}$ (this is the well-known *Mumford representation* [@Mumford §IIIa]). For example, $P_1+P_2+P_3$ corresponds to the ideal $(a(x),y-b(x))$ where $a(x) = \prod_i(x - x(P_i))$ and $b$ satisfies $y(P_i)~=~b(x(P_i))$ for $1 \le i \le 3$ (with appropriate multiplicities); we may compute $b$ using the Lagrange interpolation formula. A divisor is defined over ${{\mathbb{F}}}_q$ if and only if $a$ and $b$ are defined over ${{\mathbb{F}}}_q$. The ideal $(a(x),y-b(x))$ corresponds to $P_1 + P_2 + P_3$ if and only if $(a(x),y+b(x))$ corresponds to $\iota(P_1) + \iota(P_2) + \iota(P_3)$; so each point of $X$ over $U$ corresponds to a pair $\{ (a(x),y\pm b(x)) \}$ of ideals. We will construct a curve parametrizing these pairs of ideals, and take this as a model for $X|_U$. Suppose $\{(a(x),y\pm b(x))\}$ is a pair of ideals corresponding to one of the preimages of $Q$ on $X|_U$. The product of the two ideals is equal to the principal ideal $(a(x))$; but products of ideals correspond to sums of divisors, so $(a(x))$ must cut out the divisor $P_1 + P_2 + P_3 + \iota(P_1) + \iota(P_2) + \iota(P_3)$ on $H$. This divisor is just $(g\circ\pi)^*(Q)$, which we know is cut out by $(G(t(Q),x))$; so we conclude that $a(x) = G(t(Q),x)$ for every pair of ideals $\{(a(x),y\pm b(x))\}$ corresponding to a point in $f^{-1}(Q)$. In particular, the generic point of $X|_U$ corresponds to a pair of ideals of the form $ \{ ( G(t,x), y \pm (b_0 + b_1x + b_2x^2 ) ) \}$, where $b_0$, $b_1$, and $b_2$ are algebraic functions of $t$ such that $$\label{eq:mumford-relation} (b_0 + b_1x + b_2x^2)^2 \equiv F(x) \pmod{ G(t,x) } .$$ Viewing $b_0$, $b_1$, and $b_2$ as coordinates on ${\ensuremath{\mathbb{A}^{3}_{{}}}}$ (over ${{\mathbb{F}}}_{q}$), we expand both sides of  modulo $G(t,x)$ and equate coefficients to obtain a variety ${\widetilde{X}}$ in $U\times{\ensuremath{\mathbb{A}^{3}_{{}}}}$ parametrizing ideals: $${\widetilde{X}} = {\ensuremath{V\!\left({ \widetilde{c}_0(t,b_0,b_1,b_2), \widetilde{c}_1(t,b_0,b_1,b_2), \widetilde{c}_2(t,b_0,b_1,b_2) }\right)}} ,$$ where $$\begin{array}{r@{\;=\;}l} \widetilde{c}_0(t,b_0,b_1,b_2) & g_2(t)g_0(t)b_2^2 - 2g_0(t)b_2b_1 + b_0^2 - f_0(t) , \\ \widetilde{c}_1(t,b_0,b_1,b_2) & (g_2(t)g_1(t) - g_0(t))b_2^2 - 2g_1(t)b_2b_1 + 2b_1b_0 - f_1(t) ,\quad \text{and} \\ \widetilde{c}_2(t,b_0,b_1,b_2) & (g_2(t)^2 - g_1(t))b_2^2 - 2g_2(t)b_2b_1 + 2b_2b_0 + b_1^2 - f_2(t) . \end{array}$$ The ideals in each pair $\{ (G(t,x),y\pm (b_2x^2 + b_1x + b_0)) \}$ are exchanged by the involution $ \iota_*: {\widetilde{X}} \longrightarrow {\widetilde{X}}~$ defined by $$\iota_*: (t,b_0,b_1,b_2) \longmapsto (t,-b_0,-b_1,-b_2) ;$$ the curve $X|_U$ is therefore the quotient of ${\widetilde{X}}$ by ${\ensuremath{\left\langle{\iota_*}\right\rangle}}$. To make this quotient explicit, let $m: U\times{\ensuremath{\mathbb{A}^{3}_{{}}}}\longrightarrow U\times{\ensuremath{\mathbb{A}^{6}_{{}}}}$ be the map defined by $$m : (t,b_0,b_1,b_2) \longmapsto (t,b_{00},b_{01},b_{02},b_{11},b_{12},b_{22}) = (t,b_0^2,b_0b_1,b_0b_2,b_1^2,b_1b_2,b_2^2) ;$$ observe that $$m(U\times{\ensuremath{\mathbb{A}^{3}_{{}}}}) = {\ensuremath{V\!\left({ \begin{array}{c} b_{01}^2 - b_{00}b_{11} ,\ b_{01}b_{02} - b_{00}b_{12} ,\ b_{02}^2 - b_{00}b_{22} ,\ \\ b_{02}b_{11} - b_{01}b_{12} ,\ b_{02}b_{12} - b_{01}b_{22} ,\ b_{12}^2 - b_{11}b_{22} \end{array} }\right)}} \subset U\times {\ensuremath{\mathbb{A}^{6}_{{}}}} .$$ We have $X|_U = m({\widetilde{X}})$, so $$X|_U = {\ensuremath{V\!\left({ \begin{array}{c} c_0(t,b_{00},\ldots,b_{22}), c_1(t,b_{00},\ldots,b_{22}), c_2(t,b_{00},\ldots,b_{22}), \\ b_{01}^2 - b_{00}b_{11} ,\ b_{01}b_{02} - b_{00}b_{12} ,\ b_{02}^2 - b_{00}b_{22} ,\ \\ b_{02}b_{11} - b_{01}b_{12} ,\ b_{02}b_{12} - b_{01}b_{22} ,\ b_{12}^2 - b_{11}b_{22} \end{array} }\right)}} \subset U\times{\ensuremath{\mathbb{A}^{6}_{{}}}},$$ where $c_0$, $c_1$, and $c_2$ are the polynomials defined by $$\begin{array}{l} c_0(t,b_{00},b_{01},b_{02},b_{11},b_{12},b_{22}) := g_2g_0b_{22} - 2g_0b_{12} + b_{00} - f_0 , \\ c_1(t,b_{00},b_{01},b_{02},b_{11},b_{12},b_{22}) := (g_2g_1 - g_0)b_{22} - 2g_1b_{12} + 2b_{01} - f_1 , \ \text{ and } \\ c_2(t,b_{00},b_{01},b_{02},b_{11},b_{12},b_{22}) := (g_2^2 - g_1)b_{22} - 2g_2b_{12} + 2b_{02} + b_{11} - f_2 . \end{array}$$ Observe that $X|_U$ is defined over the field of definition of $g$. It remains to derive a correspondence $R$ between $H$ and $X|_U$ inducing the isogeny $\phi$. We know that $R$ is a component of the fibre product $H\!\times_{{\ensuremath{\mathbb{P}^{1}_{{}}}}}\! X$ (with respect to ${\ensuremath{{g}\circ{\pi}}}$ and $f$). We may realise the open affine subset $H|_U\!\times_U\! X|_U$ as the subvariety ${\ensuremath{V\!\left({G(t,x)}\right)}}$ of $H|_U\!\times X|_U$; decomposing the ideal $(G(t,x))$ will therefore give us a model for $R$. \[lemma:square-root-of-s\] Let $s$ be the polynomial in ${{\mathbb{F}}}_{q}[t]$ defined by $$\label{eq:s-definition} s := \begin{array}[t]{l} f_0^3 - f_0^2f_1g_2 - 2f_0^2f_2g_1 + f_0^2f_2g_2^2 + f_0f_1^2g_1 + 3f_0f_1f_2g_0 - f_0f_1f_2g_1g_2 \\ {} - 2f_0f_2^2g_0g_2 + f_0f_2^2g_1^2 - f_1^3g_0 + f_1^2f_2g_0g_2 - f_1f_2^2g_0g_1 + f_2^3g_0^2 , \end{array}$$ and let $\alpha$ be its leading coefficient. Then $s$ has a square root in ${{\mathbb{F}}}_{q}(\sqrt{\alpha})[t]$. The polynomial $s$ is a square in ${{\mathbb{F}}}_{q}(\sqrt{\alpha})[t]$ if and only if each of its roots in ${{\overline{\mathbb{F}}}}_{q}$ occur with multiplicity $2$. In the notation of , we have $$s(t(Q)) = F(x(P_1))F(x(P_2))F(x(P_3)) ,$$ so $s(t(Q)) = 0$ if and only if $F(x(P_i)) = 0$ for some $1 \le i \le 3$ — that is, if and only if at least one of the $P_i$ is a Weierstrass point of $H$. But the trigonal map $g$ was constructed precisely so that the Weierstrass points of $H$ appear in pairs in the fibres of $g$: hence exactly two of the $P_i$ must be Weierstrass points, and so $F(x(P_1))F(x(P_2))F(x(P_3)) = 0$ and $s(t(Q)) = 0$ with multiplicity $2$. \[proposition:b22-t-equation\] Let $s$ be the polynomial of Lemma \[lemma:square-root-of-s\], and let $\delta_0$, $\delta_1$, $\delta_2$, and $\delta_4$ be the polynomials in ${{\mathbb{F}}}_{q^2}[t]$ defined by $$\begin{array}{r@{\;:=\;}l} \delta_4 & -27g_0^2 + 18g_0g_1g_2 - 4g_0g_2^3 - 4g_1^3 + g_1^2g_2^2 , \\ \delta_2 & 12f_0g_1 - 4f_0g_2^2 - 18f_1g_0 + 2f_1g_1g_2 + 12f_2g_0g_2 - 4f_2g_1^2 , \\ \delta_1 & 8\sqrt{s}, \ \text{ and } \\ \delta_0 & -4f_0f_2 + f_1^2 . \end{array}$$ On the curve $X|_U$, we have $$\label{eq:b22-equation} \big( \delta_4(t) b_{22}^2 + \delta_2(t) b_{22} + \delta_0(t) \big)^2 - \delta_1(t)^2 b_{22} = 0 .$$ Consider again the fibre of $f: X \to {\ensuremath{\mathbb{P}^{1}_{{}}}}$ over the generic point $Q = (t)$ of $U$ (as in ). If $\{ P_1 + P_2 + P_3, \iota(P_1)+\iota(P_2)+\iota(P_3) \}$ is a pair of divisors corresponding to one of the points in the fibre, then by the Lagrange interpolation formula the value of $b_{22}$ at the corresponding point of ${\widetilde{X}}$ is $$\label{eq:b22-interpolation} b_{22} = \left(\sum y(P_i)/((x(P_i) - x(P_j))(x(P_i) - x(P_k)))\right)^2 ,$$ where the sum is taken over the cyclic permutations $(i,j,k)$ of $(1,2,3)$. After interpolating for each pair of divisors in the fibre, an elementary but involved symbolic calculation shows that $b_{22}$ satisfies $$\label{eq:d-relation} \Big( \Delta b_{22}^2 - 2\big(\sum_i\Gamma_i\big) b_{22} + \frac{1}{\Delta}\Big( 2\big(\sum_i\Gamma_i^2\big) - \big(\sum_i \Gamma_i\big)^2 \Big) \Big)^2 - 64 \big(\prod_i\Gamma_i\big)b_{22} = 0 ,$$ where $$\Gamma_i := \big( f_2(t)x(P_i)^2 + f_1(t)x(P_i) + f_0(t) \big)\Delta_i = F(x(P_i))\Delta_i$$ with $$\Delta_{i} := (x(P_j) - x(P_k))^2$$ for each cyclic permutation $(i,j,k)$ of $(1,2,3)$, and where $ \Delta := \Delta_1\Delta_2\Delta_3 $. Now $\Delta$, $\sum_i \Gamma_i$, $\sum_i \Gamma_i^2$, and $\prod_i \Gamma_i$ are symmetric functions with respect to permutations of the points in the fibre $g^{-1}(Q) = g^{-1}((t))$. They are therefore polynomials in the homogeneous elementary symmetric functions $$e_1 = \sum_i x(P_i), \quad e_2 = \sum_{i<j} x(P_i)x(P_j), \quad \mbox{and}\quad e_3 = \prod_i x(P_i) ,$$ which are polynomials in $t$. Indeed, the $e_i$ are given by the coefficients of $G(t,x)$: $$e_1 = -g_2(t),\quad e_2 = g_1(t),\quad \mbox{ and }\quad e_3 = -g_0(t) .$$ Expressing $\Delta$, $\sum_i \Gamma_i$, $\sum_i \Gamma_i^2$, and $\prod_i \Gamma_i$ in terms of $f_0$, $f_1$, $f_2$, $g_0$, $g_1$, and $g_2$, and substituting the resulting expressions into , we obtain . Equation  gives us a (singular) affine plane model for $X$. We can also use  to compute a square root for $b_{22}$ on $X|_U$: we have $$b_{22} = \rho^2, \quad \text{where}\quad \rho := \frac{ \delta_4(t) b_{22}^2 + \delta_2(t) b_{22} + \delta_0(t) }{ \delta_1(t) }.$$ Returning to , we observe that $b_{22}$ is a unit on $X|_U$, since its zeroes and poles occur only at points $Q$ where $g\circ\pi$ is ramified over $f(Q)$, and these points were excluded from $U$. Since $\rho$ is the square root of $b_{22}$, it must also be a unit on $X|_U$. Given a point $(t,b_{00},\ldots,b_{22})$ of $X|_U$, the corresponding pair of divisors of degree $3$ on $H$ is cut out by the pair of ideals $$\Big\{ \Big( G(t,x), y \pm \big(\frac{b_{02}}{\rho} + \frac{b_{12}}{\rho}x + \frac{b_{22}}{\rho}x^2\big) \Big) \Big\} .$$ This is precisely the decomposition of $(G(t,x))$ that we need to compute the correspondence from $H|_U$ to $X|_U$: we have $ {\ensuremath{V\!\left({G(t,x)}\right)}} = R \cup R' $, where $$\label{eq:correspondence-eqns} R = {\ensuremath{V\!\left({ G(t,x), y - \frac{1}{\rho} (b_{02} + b_{12}x + b_{22}x^2) }\right)}}$$ and $$R' = {\ensuremath{V\!\left({ G(t,x), y + \frac{1}{\rho}(b_{02} + b_{12}x + b_{22}x^2) }\right)}} .$$ On the level of divisor classes, the isogeny $\phi: {\ensuremath{{J}_{H}}} \to {\ensuremath{{J}_{X}}}$ is made explicit by the map $$\phi = (\pi_X)_*\circ(\pi_H)^* ,$$ where $\pi_H: R \to H$ and $\pi_X: R \to X|_U$ are the natural projections defined by $ (x,y,t,b_{00},\ldots,b_{22}) \mapsto (x,y) $ and $ (x,y,t,b_{00},\ldots,b_{22}) \mapsto (t,b_{00},\ldots,b_{22}) $, respectively. In terms of ideals cutting out effective divisors, $\phi$ is realized by the map $$I_D \longmapsto \left(I_D + \Big(G(t,x), y - \frac{1}{\rho}\big(b_{02} + b_{12}x + b_{22}x^2\big)\Big)\right) \cap {{\mathbb{F}}}_{q}[s,t,b_{00},\ldots,b_{22}] .$$ Taking $R'$ in place of $R$ in the above gives an isogeny equal to $-\phi$. It remains to determine the field of definition of $\phi$. \[proposition:X-rationality-criterion\] If $S$ is a subgroup in ${\ensuremath{\mathcal{S}}({H})}$ with an ${{\mathbb{F}}}_{q}$-rational trigonal map $g$ defined over ${{\mathbb{F}}}_{q}$, and $s(t)$ is the polynomial defined in Lemma \[lemma:square-root-of-s\]. then the explicit trigonal construction on $g$ described above yields an isogeny defined over ${{\mathbb{F}}}_{q}$ if and only if the leading coefficient of $s(t)$ is a square in ${{\mathbb{F}}}_{q}$. We noted earlier that $X|_U$ is defined over the field of definition of $g$. The correspondence $R$, and hence the induced isogeny $\phi$, are both defined over the field of definition of $\rho$, which is the field of definition of $\delta_4\delta_1$, $\delta_2/\delta_1$, and $\delta_0/\delta_1$. But $\delta_4$, $\delta_4$, and $\delta_0$ are all defined over ${{\mathbb{F}}}_{q}$ (cf. Proposition \[proposition:b22-t-equation\]), while $\delta_1$ is defined over ${{\mathbb{F}}}_{q}(\sqrt{\alpha})$ where $\alpha$ is the leading coefficient of $s$ by Lemma \[lemma:square-root-of-s\]. If $\phi$ is not defined over ${{\mathbb{F}}}_{q}$, then the Jacobian ${\ensuremath{{J}_{X}}}$ is in fact a quadratic twist of the quotient ${\ensuremath{{J}_{H}}}/S$ (see §\[section:other-isogenies\]). In fact, when $\phi$ is not ${{\mathbb{F}}}_{q}$-rational, Frobenius exchanges $\rho$ and $-\rho$, hence $R$ and $R'$, and therefore $\phi$ and $-\phi$. This is a concrete realization of the Galois cohomology referred to in the proof of Proposition \[proposition:kernels\] below: the obstruction to the existence of an isomorphism from ${\ensuremath{{J}_{H}}}/S$ to ${\ensuremath{{J}_{X}}}$ over ${{\mathbb{F}}}_{q}$ is in fact the interaction of ${\ensuremath{\mathcal{G}}}$ with $[\pm 1]$ on ${\ensuremath{{J}_{X}}}$. If we assume that the leading coefficients of the polynomials $s(t)$ are uniformly distributed for randomly chosen $H$, $S$, and $g$, then the probability that $s$ is a square in ${{\mathbb{F}}}_{q}[t]$ is $1/2$. Indeed, it is easily seen that $s(t)$ is a square for $H$ if and only if it is not a square for the quadratic twist of $H$. Suppose $H: w^2 = {\ensuremath{\widetilde{F}}}(u,v)$ is a hyperelliptic curve. Let $c$ be a non-square in ${{\mathbb{F}}}_{q}$, and let $H': w^2 = c{\ensuremath{\widetilde{F}}}(u,v)$ be the quadratic twist of $H$. Suppose $S$ in ${\ensuremath{\mathcal{S}}({H})}$ is a tractable subgroup, represented by a set $\{ F_1, F_2, F_3, F_4 \}$ of quadratic factors of ${\ensuremath{\widetilde{F}}}$. The set $\{ cF_1, F_2, F_3, F_4 \}$ is a factorization of $c{\ensuremath{\widetilde{F}}}$, so it represents a tractable subgroup $S'$ in ${\ensuremath{\mathcal{S}}({H'})}$. We noted in §\[section:computing-trigonal-maps\] that scalar multiples of quadratic polynomials do not affect the construction of trigonal maps; so if $S$ has a trigonal map $g$ defined over ${{\mathbb{F}}}_{q}$, then $g$ is also a trigonal map for $S'$. Let $s$ be the polynomial computed from $g$ and $S$ in Lemma \[lemma:square-root-of-s\], and let $s'$ be the corresponding polynomial computed for $g$ and $S'$. Looking at the form of , we see that $s'(t) = c^3s(t)$. Therefore, the leading coefficient of $s'$ is a square if and only if the leading coefficient of $s$ is *not* a square. In particular, if $S$ has a trigonal map defined over ${{\mathbb{F}}}_{q}$, then so does $S'$, and we can construct an isogeny of Jacobians with kernel $S$ if and only if we cannot construct an isogeny of Jacobians with kernel $S'$. This suggests that the probability that we can compute an isogeny defined over ${{\mathbb{F}}}_{q}$ given a randomly chosen $H$ and $S$ in ${\ensuremath{\mathcal{S}}({H})}$ with a trigonal map defined over ${{\mathbb{F}}}_{q}$ is $1/2$ — since we have a $50\%$ chance of being on the “right” quadratic twist of $H$. This hypothesis is consistent with our experimental observations. \[hypothesis:rationality\] For a randomly chosen hyperelliptic curve $H$ and a uniformly randomly chosen subgroup $S$ in ${\ensuremath{\mathcal{S}}({H})}$ with a trigonal map $g$ defined over ${{\mathbb{F}}}_{q}$, the probability that we can compute an ${{\mathbb{F}}}_{q}$-rational isogeny $\phi$ with kernel $S$ is $1/2$. Computing Isogenies {#section:examples} =================== Now we will put the ideas above into practice. Suppose we are given a hyperelliptic curve $H$ of genus $3$ over ${{\mathbb{F}}}_{q}$, and a DLP instance in ${\ensuremath{{J}_{H}}}({{\mathbb{F}}}_{q})$ to solve. Our goal is to compute a nonsingular plane quartic curve $C$ and an explcit isogeny ${\ensuremath{{J}_{H}}} \to {\ensuremath{{J}_{C}}}$ defined over ${{\mathbb{F}}}_{q}$, so that we can solve our DLP instance in ${\ensuremath{{J}_{C}}}({{\mathbb{F}}}_{q})$. We begin by computing the set ${\ensuremath{\mathcal{S}}({H})}$ of ${{\mathbb{F}}}_{q}$-rational tractable subgroups of the $2$-torsion subgroup ${\ensuremath{{J}_{H}}}[2]({{\overline{\mathbb{F}}}}_{q})$ (see Appendix \[appendix:computing-SH\] below). For each $S$ in ${\ensuremath{\mathcal{S}}({H})}$, we apply Proposition \[proposition:trigonal-map-rationality-criterion\] to determine whether there exists an ${{\mathbb{F}}}_{q}$-rational trigonal map $g$ for $S$. If so, we use the formulae of §\[section:computing-trigonal-maps\] to compute $g$; if not, we move on to the next $S$. Having computed $g$, we apply Proposition \[proposition:X-rationality-criterion\] to determine whether we can compute an isogeny over ${{\mathbb{F}}}_{q}$. If so, we use the formulae of §\[section:equations-for-the-isogeny\] to compute equations for $X$ and the isogeny $\phi: {\ensuremath{{J}_{H}}} \to {\ensuremath{{J}_{X}}}$; if not, we move on to the next $S$. The formulae of §\[section:equations-for-the-isogeny\] give an affine model of $X$ in ${\ensuremath{\mathbb{A}^{1}_{{}}}}\times{\ensuremath{\mathbb{A}^{6}_{{}}}}$. In order to apply Diem’s algorithm to the DLP in ${\ensuremath{{J}_{X}}}$, we need a nonsingular plane quartic model of $X$: that is, a nonsingular curve $C \subset {\ensuremath{\mathbb{P}^{2}_{{}}}}$ isomorphic to $X$, cut out by a quartic form. Such a model exists if and only if $X$ is not hyperelliptic. To find $C$, we compute a basis $\mathcal{B} = \{\psi_1,\psi_2,\psi_3\}$ of the Riemann–Roch space of a canonical divisor of $X$. This is a routine geometrical calculation; Hess [@Hess] describes an efficient approach. In practice, the algorithms implemented in Magma [@Magma--language; @Magma] compute $\mathcal{B}$ very quickly. The three functions in $\mathcal{B}$ define a map $\psi: X \to {\ensuremath{\mathbb{P}^{2}_{{}}}}$, mapping $P$ to $(\psi_1(P):\psi_2(P):\psi_3(P))$. Up to automorphisms of ${\ensuremath{\mathbb{P}^{2}_{{}}}}$, the map $\psi$ is independent of the choice of basis $\mathcal{B}$, and depends only on $X$. If the image of $\psi$ is a conic (that is, if the $\psi_i$ satisfy a quadratic relation), then $X$ is hyperelliptic; in this situation we move on to the next $S$, since we will gain no advantage from index calculus on $X$. Otherwise, the image of $\psi$ is a nonsingular plane quartic $C$, and $\psi$ restricts to an isomorphism $\psi: X \to C$. If the procedure outlined above succeeds for some $S$ in ${\ensuremath{\mathcal{S}}({H})}$, then we have computed an explicit ${{\mathbb{F}}}_{q}$-rational isogeny $\psi_*\circ\phi: {\ensuremath{{J}_{H}}} \to {\ensuremath{{J}_{C}}}$. We can then map our DLP from ${\ensuremath{{J}_{H}}}({{\mathbb{F}}}_q)$ into ${\ensuremath{{J}_{C}}}({{\mathbb{F}}}_q)$, and solve it using Diem’s algorithm. We emphasize that the entire procedure is very fast: the curve $X$ and the isogeny can be constructed using just a few low-degree polynomial operations and some low-dimensional linear algebra (and hence the procedure is polynomial-time in $\log q$, the size of the base field). For a rough idea of the computational effort involved, given a random $H$ over a $160$-bit prime field with a tractable subgroup $S$ in ${\ensuremath{\mathcal{S}}({H})}$, a naïve implementation of our algorithms in Magma computes the trigonal map $g$, the curve $X$, the nonsingular plane quartic $C$, and the isogeny $\phi: {\ensuremath{{J}_{H}}} \to {\ensuremath{{J}_{C}}}$ in a few seconds on a 1.2GHz laptop. Since the difficulty of the construction depends only upon the difficulty of arithmetic in ${{\mathbb{F}}}_{q}$ (and *not* upon the size of the DLP subgroup of ${\ensuremath{{J}_{H}}}({{\mathbb{F}}}_{q})$), we may conclude that instances of the DLP in $160$-bit Jacobians chosen for cryptography may also be reduced to instances of the DLP in non-hyperelliptic Jacobians in very little time. We will give an example over a small field. Let $H$ be the hyperelliptic curve over ${{\mathbb{F}}}_{37}$ defined by $$H: y^2 = x^7 + 28 x^6 + 15 x^5 + 20 x^4 + 33 x^3 + 12 x^2 + 29 x + 2 .$$ Using the ideas in §\[section:kernel\], or the algorithms in Appendix \[appendix:computing-SH\], we find that ${\ensuremath{{J}_{H}}}$ has one ${{\mathbb{F}}}_{37}$-rational tractable subgroup: $${\ensuremath{\mathcal{S}}({H})} = \left\{ S \right\} \text{\quad where\quad } S = \left\{ \begin{array}{l} u^2 + \xi_1 uv + \xi_2 v^2, \ u^2 + \xi_1^{37} uv + \xi_2^{37} v^2, \\ u^2 + \xi_1^{37^2} uv + \xi_2^{37^2} v^2 , \ uv + 20v^2 \end{array} \right\} ,$$ where $\xi_1$ is an element of ${{\mathbb{F}}}_{37^3}$ satisfying $\xi_1^3 + 29\xi_1^2 + 9\xi_1 + 13 = 0$, and $\xi_2 = \xi_1^{50100}$. Applying the methods of §\[section:computing-trigonal-maps\], we compute a trigonal map $g: x \mapsto N(x)/D(x)$ for $S$, taking $$N(x) = x^3 + 16 x + 22 \quad \mbox{and}\quad D(x) = x^2 + 32 x + 18 ;$$ clearly $g$ is defined over ${{\mathbb{F}}}_{37}$. The formulae of §\[section:equations-for-the-isogeny\] give us a curve $X \subset {\ensuremath{\mathbb{A}^{1}_{{}}}}\times{\ensuremath{\mathbb{A}^{6}_{{}}}}$ of genus $3$, defined by $$X = {\ensuremath{V\!\left({ \begin{array}{l} \scriptstyle (18t^2 + 15t)b_{22} + (36t + 30)b_{12} + b_{00} + 19t^5 + 10t^4 + 12t^3 + 7t^2 + t + 30, \\ \scriptstyle (32t^2 + 2t + 15)b_{22} + (27t + 5)b_{12} + 2b_{01} + 5t^5 + 26t^4 + 15t^3 + 23t^2 + 19t + 17, \\ \scriptstyle (t^2 + 32t + 21)b_{22} + 2tb_{12} + 2b_{02} + b_{11} + 36t^5 + 29t^4 + 7t^3 + 13t^2 + 21t + 18 , \\ \scriptstyle b_{00}b_{11} - b_{01}^2, b_{00}b_{12} - b_{01}b_{02}, b_{00}b_{22} - b_{02}^2, b_{02}b_{11} - b_{01}b_{12} , b_{02}b_{12} - b_{01}b_{22} , b_{12}^2 - b_{11}b_{22} \end{array} }\right)}} .$$ The map on divisors inducing an isogeny from ${\ensuremath{{J}_{H}}}$ to ${\ensuremath{{J}_{X}}}$ with kernel $S$ is induced by the correspondence $R$ defined as in  with $$\begin{array}{r@{\;}l} G(t,x) = & x^3 - t x^2 - (32 t - 16) x -18 t + 22 , \\ \delta_0 = & 27t^{10} + 20t^9 + 33t^8 + 6t^7 + 16t^6 + 8t^5 + 9t^4 + 2t^3 + 31t^2 + 15t + 16 , \\ \delta_1 = & 35t^3 + 8t^2 + 33t + 3 , \\ \delta_2 = & 20t^7 + 18t^6 + 29t^5 + 14t^4 + 6t^3 + 20t^2 + 12t + 16 , \quad \text{and} \\ \delta_4 = & 27t^4 + 36t^3 + 13t^2 + 21t . \\ \end{array}$$ Computing the canonical morphism of $X$, we find that $X$ is non-hyperelliptic, and isomorphic to the nonsingular plane quartic curve $$C = {\ensuremath{V\!\left({ \begin{array}{c} u^4 + 26u^3v + 2u^3w + 17u^2v^2 + 9u^2vw + 20u^2w^2 + 34uv^3 + 24uv^2w \\ {} + 5uvw^2 + 36uw^3 + 19v^4 + 13v^3w + v^2w^2 + 23vw^3 + 5w^4 \end{array} }\right)}} .$$ Composing the isomorphism with the isogeny ${\ensuremath{{J}_{H}}} \to {\ensuremath{{J}_{X}}}$, we obtain an explicit isogeny $\phi: {\ensuremath{{J}_{H}}} \to {\ensuremath{{J}_{C}}}$. We can verify that ${\ensuremath{{J}_{H}}}$ and ${\ensuremath{{J}_{C}}}$ are isogenous by checking that the zeta functions of $H$ and $C$ are identical: indeed, direct calculation with Magma shows that $$Z(H;T) = Z(C;T) = \frac{ 37^3 T^6 + 4\cdot37^2 T^5 - 6\cdot37 T^4 - 240 T^3 - 6 T^2 + 4 T + 1 }{ (37T - 1)(T - 1) } .$$ Let $D = [ (10:28:1) - (14:6:1) ]$ and $D' = [ (19:28:1) - (36:13:1) ]$ be divisor classes on $H$; we have $D' = [22359]D$. Applying $\phi$, we find that $$\begin{array}{r@{\;=\;}l} \phi(D) & \left[ (7 : 18 : 1) + (34 : 34 : 1) - (18 : 22 : 1) - (15 : 33 : 1) \right] \text{\quad and}\\ \phi(D') & \left[ (7 : 23 : 1) + (6 : 13 : 1) - (13 : 15 : 1) - (7 : 18 : 1) \right] ; \end{array}$$ direct calculation verifies that $\phi(D') = [22359]\phi(D)$, as expected. Expectation of Existence of Computable Isogenies {#section:probabilities} ================================================ Our aim in this section is to estimate the proportion of genus $3$ hyperelliptic Jacobians over ${{\mathbb{F}}}_{q}$ for which the methods of this article produce an ${{\mathbb{F}}}_{q}$-rational isogeny — and thus for which the DLP may be solved using Diem’s algorithm — as $q$ tends to infinity. We will assume that if we are given a selection of ${{\mathbb{F}}}_q$-rational tractable subgroups of a given Jacobian, then the probabilities that each will yield a rational isogeny are mutually independent. This hypothesis appears to be consistent with our experimental observations. \[hypothesis:independence\] For a randomly chosen hyperelliptic curve $H$, the probabilities that we can compute an ${{\mathbb{F}}}_{q}$-rational isogeny with kernel $S$ for each $S$ in ${\ensuremath{\mathcal{S}}({H})}$ are mutually independent. \[theorem:success-probability\] Assume Hypotheses \[hypothesis:hyperellipticity\], \[hypothesis:trigonal-map-rationality\], \[hypothesis:rationality\], and \[hypothesis:independence\]. As $q$ tends to infinity, the expectation that the algorithms in this article will give a reduction of the DLP in a subgroup of ${\ensuremath{{J}_{H}}}({{\mathbb{F}}}_{q})$ for a randomly chosen hyperelliptic curve $H$ of genus $3$ over ${{\mathbb{F}}}_{q}$ to a subgroup of ${\ensuremath{{J}_{C}}}({{\mathbb{F}}}_{q})$ for some nonsingular plane quartic curve $C$ is $$\label{eq:expectation} \sum_{T \in \mathcal{T}} \Big( \big(1 - (1 - 1/4)^{s(T)}\big) / \prod_{n \in T} \big(\nu_T(n)! \cdot n^{\nu_T(n)}\big) \Big) \approx 0.1857 ,$$ where $\mathcal{T}$ denotes the set of integer partitions of $8$ and $\nu_T(n)$ denotes the multiplicity of an integer $n$ in a partition $T$, and $s(T) = \#{\ensuremath{\mathcal{S}}({H})}$, where $H$ is *any* hyperelliptic curve over ${{\mathbb{F}}}_{q}$ such that the multiset of degrees of the ${{\mathbb{F}}}_{q}$-irreducible factors of its hyperelliptic polynomial coincides with $T$. Suppose $H$ is a randomly chosen hyperelliptic curve of genus $3$ over ${{\mathbb{F}}}_{q}$. Hypotheses \[hypothesis:hyperellipticity\], \[hypothesis:trigonal-map-rationality\], and \[hypothesis:rationality\] together imply that for each $S$ in ${\ensuremath{\mathcal{S}}({H})}$, the probability that we can compute an isogeny with kernel $S$ defined over ${{\mathbb{F}}}_{q}$ is $ 1/2\cdot1/2\cdot1 = 1/4 $. Hypothesis \[hypothesis:independence\] implies that we have an equal chance of constructing an isogeny from each $S$ in ${\ensuremath{\mathcal{S}}({H})}$, so the probability that we can compute an isogeny over ${{\mathbb{F}}}_{q}$ from ${\ensuremath{{J}_{H}}}$ is $ 1 - (1 - 1/4)^{\#{\ensuremath{\mathcal{S}}({H})}} $. The expectation that we can compute an isogeny over ${{\mathbb{F}}}_{q}$ given a curve over ${{\mathbb{F}}}_{q}$ is therefore $$\label{eq:success-probability} E_q := \frac{ \sum_{{\ensuremath{\widetilde{F}}}} ( 1 - (3/4)^{\#{\ensuremath{\mathcal{S}}({H})}} ) }{ \sum_{{\ensuremath{\widetilde{F}}}} 1 },$$ where $H$ is the curve defined by $w^2 = {\ensuremath{\widetilde{F}}}(u,v)$, and ${\ensuremath{\widetilde{F}}}$ ranges over the set of all homogeneous squarefree polynomials of degree $8$ over ${{\mathbb{F}}}_{q}$. Lemma \[lemma:number-of-tractable-subgroups\] implies that $\#{\ensuremath{\mathcal{S}}({H})}$ depends only on the degrees of the ${{\mathbb{F}}}_{q}$-irreducible factors of ${\ensuremath{\widetilde{F}}}$, so the map $T \mapsto s(T)$ is well-defined. For each $T$ in $\mathcal{T}$, let $ N_q(T)~$ denote the number of homogeneous squarefree polynomials over ${{\mathbb{F}}}_{q}$ whose multiset of degrees of ${{\mathbb{F}}}_{q}$-irreducible factors coincides with $T$. We can now rewrite  as $$E_q = \frac{ \sum_{T \in \mathcal{T}}(1 - (3/4)^{s(T)})N_q(T) }{ \sum_{T\in \mathcal{T}}N_q(T) } .$$ There are $N_q(n) = \frac{1}{n}\sum_{d|n}\mu(d)q^{n/d}$ monic irreducible polynomials of degree $n$ over ${{\mathbb{F}}}_{q}$ (here $\mu$ is the Möbius function). Clearly $ N_q(T) = (q - 1)\prod_{n \in T}\binom{N_q(n)}{\nu_T(n)} $, so $$N_q(T) = \Big(\prod_{n\in T}(\nu_T(n)!\cdot n^{\nu_T(n)})\Big)^{-1}q^9 + O(q^8) ,$$ and $\sum_{T \in \mathcal{T}}N_q(T) = q^9 + O(q^8)$. Therefore, as $q$ tends to infinity, we have $$\lim_{q \to \infty} E_q = \sum_{T \in \mathcal{T}} \Big( \left(1 - (3/4)^{s(T)}\right) / \prod_{n \in T} \left(\nu_T(n)! \cdot n^{\nu_T(n)}\right) \Big) .$$ The result follows upon explicitly computing this sum, using the values for $s(T)$ listed in Lemma \[lemma:number-of-tractable-subgroups\]. Theorem \[theorem:success-probability\] gives the expectation of our ability to construct an explicit isogeny for a randomly selected hyperelliptic curve. However, looking at the table in Lemma \[lemma:number-of-tractable-subgroups\], we see that we can be sure that a particular curve has no isogenies with tractable kernels defined over ${{\mathbb{F}}}_{q}$ if we use only curves whose hyperelliptic polynomials have an irreducible factor of degree $5$ or $7$ (or a single irreducible factor of degree $3$). It may be difficult to efficiently construct a curve in this form if we are using a CM construction, for example, to ensure that the Jacobian has a large prime-order subgroup. In any case, it is interesting to note that the security of genus $3$ hyperelliptic Jacobians depends significantly upon the factorization of their hyperelliptic polynomials. This observation has no analogue for elliptic curves or Jacobians of curves of genus $2$. Of course, if $E: y^2 = F(x)$ is an elliptic curve and $F$ is completely reducible, then $\#E({{\mathbb{F}}}_{q})$ is divisible by $4$, and in particular $\#E({{\mathbb{F}}}_{q})$ cannot be prime; but this does not reduce the security of $E({{\mathbb{F}}}_{q})$ to the extent that a completely reducible hyperelliptic polynomial does for a curve of genus $3$. \[remark:trigonal-map-descent\] We noted in §\[section:trigonal-construction\] that the ${{\overline{\mathbb{F}}}}_{q}$-isomorphism class of the curve $X$ in the trigonal construction is independent of the choice of trigonal map. If there is no trigonal map defined over ${{\mathbb{F}}}_{q}$ for a given subgroup $S$ in ${\ensuremath{\mathcal{S}}({H})}$, then the methods of §\[section:computing-trigonal-maps\] construct a pair of Galois-conjugate trigonal maps $g_1$ and $g_2$ (corresponding to the roots of ) instead. Applying the trigonal construction to $g_1$ and $g_2$, we obtain curves $X_1$ and $X_2$ over ${{\mathbb{F}}}_{q^2}$. If the isomorphism between $X_1$ and $X_2$ were made explicit, then we could descend it to compute a curve $X$ over ${{\mathbb{F}}}_{q}$ in the ${{\overline{\mathbb{F}}}}_{q}$-isomorphism class of $X_1$ and $X_2$, and hence a nonsingular plane quartic $C$ over ${{\mathbb{F}}}_{q}$ and an isogeny ${\ensuremath{{J}_{H}}} \to {\ensuremath{{J}_{C}}}$. We note that the isogeny may not be defined over ${{\mathbb{F}}}_{q}$, but this approach could still allow us to replace the $1/4$ in  and  with $1/2$, raising the expectation of success in Theorem \[theorem:success-probability\] to $31.13\%$. Let $p = 1008945029102471339$. Note that $p$ is a 60-bit prime; if $H$ is a hyperelliptic curve of genus $3$ over ${{\mathbb{F}}}_{p}$ such that ${\ensuremath{{J}_{H}}}({{\mathbb{F}}}_{p})$ has a large prime-order subgroup and if Gaudry–Thomé–Thériault–Diem index calculus is the fastest algorithm for solving DLP instances in ${\ensuremath{{J}_{H}}}({{\mathbb{F}}}_{p})$, then ${\ensuremath{{J}_{H}}}$ has roughly the same security level as an elliptic curve over a 160-bit field. We generated one million random hyperelliptic curves of genus $3$ over ${{\mathbb{F}}}_{p}$ using Magma. For each curve $H$, we computed the set ${\ensuremath{\mathcal{S}}({H})}$ of tractable subgroups; then, for each $S$ in ${\ensuremath{\mathcal{S}}({H})}$ we determined whether there was an ${{\mathbb{F}}}_{p}$-rational trigonal map for $S$, and if so whether there was an ${{\mathbb{F}}}_{p}$-rational isogeny with kernel $S$. Of these curves, $502005$ (that is, $50.02\%$) had at least one rational tractable subgroup. Between them, the $10^6$ curves had $1002244$ rational tractable subgroups, of which $501629$ had a rational trigonal map (that is, $50.05\%$, which is close to the $50\%$ predicted by Hypothesis \[hypothesis:trigonal-map-rationality\]). Of these subgroups, $250560$ led to a rational isogeny (that is, $49.95\%$, which is close to the $50\%$ predicted by Hypothesis \[hypothesis:rationality\]). We found that $185814$ of the curves had at least one ${{\mathbb{F}}}_{p}$-rational isogeny, none of which had a hyperelliptic codomain (this is compatible with Hypothesis \[hypothesis:hyperellipticity\]). In particular, we could move a discrete logarithm problem for $18.58\%$ of these curves (recall that Theorem \[theorem:success-probability\] predicts a success rate of about $18.57\%$). Other Isogenies {#section:other-isogenies} =============== So far, we have concentrated on using isogenies with kernels generated by differences of Weierstrass points to move instances of the DLP from hyperelliptic to non-hyperelliptic Jacobians. More generally, we could use isogenies with other kernels. There are two important issues to consider here: the first is a theoretical restriction on the types of subgroups that can be kernels of isogenies of Jacobians, and the second is a practical restriction on the isogenies that we can currently compute. Let $H$ be a hyperelliptic curve of genus $3$. We want to characterize the subgroups $S$ of ${\ensuremath{{J}_{H}}}$ that are kernels of isogenies of Jacobians, combining standard results from the theory of abelian varieties with some special results on curves of genus $3$. For our purposes, it is enough to know that the $l$-*Weil pairing* is a nondegenerate, bilinear pairing on the $l$-torsion of an abelian variety, which can be efficiently evaluated in the case where the abelian variety is the Jacobian of a hyperelliptic curve; for further detail, we refer the reader to [@Hindry--Silverman Ex. A.7.8]. Let $A$ be an abelian variety over ${{\mathbb{F}}}_{q}$, and let $l$ be a positive integer coprime with $q$. We say a subgroup $S$ of $A[l]$ is *maximal $l$-isotropic* if 1. the $l$-Weil pairing on $A[l]$ restricts trivially to $S$, and 2. $S$ is not properly contained in any other subgroup of $A[l]$ satisfying (1). If $l$ is a prime not dividing $q$, then every maximal $l$-isotropic subgroup of ${\ensuremath{{J}_{H}}}({{\overline{\mathbb{F}}}}_{q})[l]$ is isomorphic to $({{\mathbb{Z}}}/l{{\mathbb{Z}}})^3$. The situation is more complicated when $l$ is not prime: for example, ${\ensuremath{{J}_{H}}}[2]$ is a maximal $4$-isotropic subgroup of ${\ensuremath{{J}_{H}}}[4]$, but it is isomorphic to $({{\mathbb{Z}}}/2{{\mathbb{Z}}})^6$ and not $({{\mathbb{Z}}}/4{{\mathbb{Z}}})^3$. \[proposition:kernels\] Let $H$ be a hyperelliptic curve of genus $3$ over ${{\mathbb{F}}}_{q}$ such that ${\ensuremath{{J}_{H}}}$ is absolutely simple. Let $S$ be a finite, nontrivial, ${{\mathbb{F}}}_{q}$-rational subgroup of ${\ensuremath{{J}_{H}}}({{\overline{\mathbb{F}}}}_{q})$. There exists a curve $X$ of genus $3$ over ${{\mathbb{F}}}_{q}$, and an isogeny $\phi: {\ensuremath{{J}_{H}}} \to {\ensuremath{{J}_{X}}}$ with kernel $S$, if and only if $S$ is a maximal $l$-isotropic subgroup of ${\ensuremath{{J}_{H}}}[l]$ for some positive integer $l$. The isogeny $\phi$ is defined over ${{\mathbb{F}}}_{q^2}$. The quotient ${\ensuremath{{J}_{H}}} \to {\ensuremath{{J}_{H}}}/S$ always exists as an isogeny of abelian varieties, and is defined over ${{\mathbb{F}}}_{q}$ (see Serre [@Serre-AC §III.3.12]). For the quotient to be an isogeny of Jacobians, there must be an integer $l$ such that $S$ is a maximal $l$-isotropic subgroup (see Proposition 16.8 of Milne [@Milne]): this ensures that the canonical polarization on ${\ensuremath{{J}_{H}}}$ induces a principal polarization on the quotient ${\ensuremath{{J}_{H}}}/S$. The theorem of Oort and Ueno [@Oort--Ueno] therefore guarantees that there will be an isomorphism of principally polarized abelian varieties over ${{\overline{\mathbb{F}}}}_{q}$ from ${\ensuremath{{J}_{H}}}/S$ to the Jacobian ${\ensuremath{{J}_{X}}}$ of some irreducible curve $X$ (irreducibility of $X$ follows from the fact that ${\ensuremath{{J}_{H}}}$, and hence ${\ensuremath{{J}_{H}}}/S$, is absolutely simple). Composing this isomorphism with the quotient map gives an isogeny of Jacobians from ${\ensuremath{{J}_{H}}}$ to ${\ensuremath{{J}_{X}}}$ with kernel $S$. Standard arguments from Galois cohomology (see Serre [@Serre-GC §III.1], for example) show that the isomorphism is defined over either ${{\mathbb{F}}}_{q}$ or ${{\mathbb{F}}}_{q^2}$, and it follows that the isogeny ${\ensuremath{{J}_{H}}} \to {\ensuremath{{J}_{X}}}$ must be defined over ${{\mathbb{F}}}_{q}$ or ${{\mathbb{F}}}_{q^2}$. Proposition \[proposition:kernels\] does *not* hold in higher genus: for every $g \ge 4$, there are $g$-dimensional abelian varieties that are not isomorphic to Jacobians. Indeed, this is the generic situation: for $g \ge 2$ the moduli space of $g$-dimensional abelian varieties is $g(g+1)/2$-dimensional, with the Jacobians occupying a subspace of dimension $(3g - 3)$ — which is strictly less than $g(g+1)/2$ for $g \ge 4$. We should not therefore expect an arbitrary quotient of a Jacobian to be isomorphic to a Jacobian in genus $g \ge 4$. Proposition \[proposition:kernels\] does hold in genus $1$ and $2$, and in these cases the isogenies are always defined over ${{\mathbb{F}}}_{q}$. We can expect the curve $X$ of Proposition \[proposition:kernels\] to be non-hyperelliptic. To compute an ${{\mathbb{F}}}_{q}$-rational isogeny from ${\ensuremath{{J}_{H}}}$ to a non-hyperelliptic Jacobian, therefore, the minimum requirement is an ${{\mathbb{F}}}_{q}$-rational $l$-isotropic subgroup of ${\ensuremath{{J}_{H}}}({{\overline{\mathbb{F}}}}_{q})$ isomorphic to $({{\mathbb{Z}}}/l{{\mathbb{Z}}})^3$ for some prime $l$. We emphasize that this subgroup need *not* be contained in ${\ensuremath{{J}_{H}}}({{\mathbb{F}}}_{q})$. Indeed, there may be isogenies from ${\ensuremath{{J}_{H}}}$ to non-hyperelliptic Jacobians over ${{\mathbb{F}}}_{q}$ even when ${\ensuremath{{J}_{H}}}({{\mathbb{F}}}_{q})$ has prime order (which would be the desirable situation in cryptological applications). The major obstruction to using more general isogenies to move DLP instances is the lack of general constructions for explicit isogenies in genus $3$. Apart from integer multiplications, automorphisms, Frobenius isogenies, and the construction for isogenies with tractable kernels exhibited above, we know of no constructions for explicit isogenies of general Jacobians of genus $3$ hyperelliptic curves. In particular, while we know that the curve $X$ of Proposition \[proposition:kernels\] exists, we generally have no means of computing a defining equation for it, let alone equations for a correspondence between $H$ and $X$ that would allow us to move DLP instances from ${\ensuremath{{J}_{H}}}$ to ${\ensuremath{{J}_{X}}}$. This situation stands in marked contrast to the case of isogenies of elliptic curves, which have been made completely explicit by Vélu [@Velu]. Deriving general formulae for explicit isogenies in genus $3$ (and $2$) remains a significant problem in computational number theory. Acknowledgements {#section:acknowledgements .unnumbered} ---------------- The greater part of this work was completed in the Department of Mathematics at Royal Holloway, University of London, where the author was supported by EPSRC grant EP/C014839/1. The author gratefully acknowledges Roger Oyono and Christophe Ritzenthaler for discussions which inspired this research, and Steven Galbraith and the anonymous referees for their helpful suggestions. [XX]{} Birkenhake, C., Lange, H.: Complex abelian varieties (2e), Grundlehren der mathematischen Wissenschaften 302. Springer (2004) Bosma, W., Cannon, J., Playoust, C.: The Magma computational algebra system. I. The user language. 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C. R. Acad. Sci. Paris, Séries A 273, 305–347 (1971) Appendix: Computing ${\ensuremath{\mathcal{S}}({H})}$ {#appendix:computing-SH} ===================================================== Given a hyperelliptic curve $H$ of genus $3$ over ${{\mathbb{F}}}_{q}$, we want to compute the set ${\ensuremath{\mathcal{S}}({H})}$ of ${{\mathbb{F}}}_{q}$-rational tractable subgroups of ${\ensuremath{{J}_{H}}}$. Algorithm \[algorithm:subgroup-enumeration\] splits the hyperelliptic polynomial of $H$ into Galois orbits of factors, before calling the recursive subroutine Algorithm \[algorithm:subgroup-subroutine\] to enumerate ${\ensuremath{\mathcal{S}}({H})}$. This algorithm is included only for completeness, and is not particularly efficient (we suggest some optimisations in Remark \[remark:tractable-implementation\] below.) \[algorithm:subgroup-enumeration\] Given a hyperelliptic curve $H$ of genus $3$ over ${{\mathbb{F}}}_{q}$, enumerates the set ${\ensuremath{\mathcal{S}}({H})}$ of ${{\mathbb{F}}}_{q}$-rational tractable subgroups of ${\ensuremath{{J}_{H}}}[2]({{\overline{\mathbb{F}}}}_{q})$. Each subgroup in ${\ensuremath{\mathcal{S}}({H})}$ is represented by a set of four coprime quadratic factors of ${\ensuremath{\widetilde{F}}}$. Input : The hyperelliptic polynomial ${\ensuremath{\widetilde{F}}}(u,v)$ of $H$. Output : The set ${\ensuremath{\mathcal{S}}({H})}$. Step 1 : Let $\mathcal{F}$ be the set of irreducible factors of ${\ensuremath{\widetilde{F}}}$ over its splitting field,\ scaled so that ${\ensuremath{\widetilde{F}}} = \prod_{L\in\mathcal{F}}L$, and set $\mathcal{O} := \{ \}$. Step 2 : Choose a polynomial $L$ from $\mathcal{F}$. Set $O := (L)$, set $\mathcal{F} := \mathcal{F}\setminus\{L\}$,\ and set $L_1 := L$. Step 3 : Set $L := \sigma(L)$, where $\sigma$ denotes the $q^\mathrm{th}$ power Frobenius map.\ If $L \not= L_1$, then append $L$ to $O$, set $\mathcal{F} := \mathcal{F}\setminus\{L\}$, and go to **Step 3**.\ If $L = L_1$, then set $\mathcal{O} := \mathcal{O}\cup\{ O \}$; if $\mathcal{F} \not= \emptyset$, then go to **Step 2**. Step 4 : Return the result of Algorithm \[algorithm:subgroup-subroutine\] applied to $\mathcal{O}$. \[algorithm:subgroup-subroutine\] Given a set of ${\ensuremath{\mathcal{G}}}$-orbits of coprime linear polynomials over ${{\overline{\mathbb{F}}}}_{q}$, returns the ${\ensuremath{\mathcal{G}}}$-invariant sets of coprime quadratic products of the polynomials. Input : A set $\mathcal{O}$ of disjoint sequences of distinct linear polynomials. Each sequence $O = (O_1,\ldots,O_m)$ in $\mathcal{O}$ must satisfy $O_1 = \sigma(O_{m})$ and $O_{i+1} = \sigma(O_i)$ for $1 \le i < m$, where $\sigma$ denotes the $q^\mathrm{th}$-power Frobenius map. Output : The set $\mathcal{S}$ of ${\ensuremath{\mathcal{G}}}$-stable sets of coprime quadratic polynomials such that $ \prod_{S \in \mathcal{S}}\prod_{Q \in S} Q = \prod_{O \in \mathcal{O}}\prod_{L \in O} L $. Step 1 : If $\mathcal{O}$ is empty, then return $\mathcal{S} := \{ \emptyset \}$. Step 2 : Choose a sequence $O$ from $\mathcal{O}$, and set $m := \#O$.\ If $m$ is even, then let $\mathcal{T}$ be the result of Algorithm \[algorithm:subgroup-subroutine\] applied to $\mathcal{O}\setminus\{O\}$,\ and set $ \mathcal{S} := \{ \{ O_{i}\cdot O_{(m/2)+i}: 1\le i \le m/2 \} \cup T : T \in \mathcal{T} \} $.\ If $m$ is odd, then set $\mathcal{S} := \{ \}$. Step 3 : For each $P$ in $\mathcal{O} \setminus \{ O \}$ such that $\#P = \#O = m$, Step 3i : Set $ \mathcal{U} := \left\{\!\{ O_{1+i}\cdot P_{1+((i+j)\bmod m)} : 0\le i < m \} : 0 \le j < m \right\} $. Step 3ii : Let $\mathcal{V}$ be the result of Algorithm \[algorithm:subgroup-subroutine\] applied to $\mathcal{O}\setminus\{O,P\}$. Step 3iii : Set $ \mathcal{S} := \mathcal{S} \cup \left\{ U \cup V : U \in \mathcal{U}, \ V \in \mathcal{V} \right\} $. Step 4 : Return $\mathcal{S}$. \[remark:tractable-implementation\] As we noted above, Algorithms $4$ and $5$ are not particularly efficient: for conceptual simplicity we worked over the splitting field of the hyperelliptic polynomial, and this can be extremely slow in practice. A number of simple optimizations will significantly improve the performance of this algorithm: the key is to avoid field extensions where possible, and to minimize their degree in any case. Before factoring ${\ensuremath{\widetilde{F}}}$ over its splitting field we should factor it over ${{\mathbb{F}}}_{q}$, and then work on a case-by-case basis depending on the degrees of the ${{\mathbb{F}}}_{q}$-irreducible factors. For example, if ${\ensuremath{\widetilde{F}}}$ has an odd number of odd-degree factors, then ${\ensuremath{\mathcal{S}}({H})}$ is empty by Lemma \[lemma:number-of-tractable-subgroups\], and we can simply return the empty set. If ${\ensuremath{\widetilde{F}}}$ is ${{\mathbb{F}}}_{q}$-irreducible, then it is not necessary to factor ${\ensuremath{\widetilde{F}}}$ over its splitting field (which is ${{\mathbb{F}}}_{q^8}$): there is one tractable subgroup, and it corresponds to the four quadratic factors of ${\ensuremath{\widetilde{F}}}$ that we obtain by factoring ${\ensuremath{\widetilde{F}}}$ over ${{\mathbb{F}}}_{q^4}$. Making similar modifications for the cases where ${\ensuremath{\widetilde{F}}}$ has factors of degree $6$, we can avoid working over any extensions of degree greater than $4$. If desired, we can further avoid some field extensions in the case where ${\ensuremath{\widetilde{F}}}$ has only low-degree factors. These modifications resulted in a factor-of-$50$ speedup for our experiments with $60$-bit prime fields; the unmodified Algorithms \[algorithm:subgroup-enumeration\] and \[algorithm:subgroup-subroutine\] should *not* be used in practice. [^1]: Some of the theory carries over to more general base fields: in particular, the results of §\[section:computing-trigonal-maps\] and §\[section:equations-for-the-isogeny\] are valid over fields of characteristic not $2$ or $3$. [^2]: Recillas’ original trigonal construction is defined where $\pi$ is an étale double cover; the trigonal construction we apply here is in fact the flat limit of Recillas’ construction (see [@Donagi--Livne §3] for details).

--- abstract: 'We investigate the evolution of the optical and near-infrared colour-magnitude relation in an homogeneous sample of massive clusters from $z=1$ to the present epoch. By comparing deep [*Hubble Space Telescope*]{} ACS imaging of X-ray selected MACS survey clusters at $z\sim0.5$ to the similarly selected LARCS sample at $z\sim0.1$ we find that the rest-frame $\delta(U - V)/\delta V$ slope of the colour-magnitude relation evolves with redshift which we attribute to the build up of the red sequence over time. This rest frame slope evolution is not adequately reproduced by that predicted from semi-analytic models based on the Millennium Simulation despite a prescription for the build up of the red sequence by in-falling galaxies, ‘strangulation’. We observe no strong correlation between this slope and the cluster environment at a given redshift demonstrating that the observed evolution is not due to a secondary correlation. Also presented are near-infrared UKIRT WFCAM observations of the LARCS clusters which confirm and improve on the the result from [@Stott2007a] finding that there has been a two-fold increase in faint $M_V>-20$ galaxies on the red sequence since $z=0.5$ to a significance of 5$\sigma$.' author: - | J.P. Stott$^{1, 2}$[^1], K.A. Pimbblet$^3$, A.C. Edge$^2$, G.P. Smith$^{4, 5}$, J. L. Wardlow$^2$\ \ $^1$Astrophysics Research Institute, Liverpool John Moores University, Twelve Quays House, Egerton Wharf, Birkenhead, CH41 1LD, UK\ $^2$Institute for Computational Cosmology, Department of Physics, University of Durham, South Road, Durham DH1 3LE, UK\ $^3$Department of Physics, University of Queensland, Brisbane, QLD 4072, Australia\ $^4$California Institute of Technology, Mail Code 105-24, Pasadena, CA 91125, USA\ $^5$ School of Physics $\&$ Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK\ bibliography: - 'slope.bib' title: The evolution of the red sequence slope in massive galaxy clusters --- \[firstpage\] galaxies: clusters: general – galaxies: elliptical and lenticular, cD – galaxies: evolution Introduction {#sec:introcm} ============ Galaxy clusters are important laboratories for the study of galaxy formation and evolution as they contain a concentrated population of diverse galaxies in a relatively small volume. Early workers in the field found that if a colour-magnitude diagram is plotted for members of local clusters such as Virgo and Coma, the Elliptical/S0 galaxies were found to be confined to a prominent linear feature in colour space [@vis1977]. This feature is known as the red sequence and has a very small intrinsic scatter (typically $<$0.1 mag) which has been interpreted as evidence that the passive galaxies in clusters formed coevally at high redshift [@Bower1992]. The red sequence is observed to have a slope such that the faint galaxies are bluer than the bright cluster members. The origin of the slope has been controversial due to the age metallicity degeneracy for stellar populations. [@worthey1995] showed that the sequence of colours that comprise the slope can be equally well explained by a progressive decrease in either metallicity or stellar age. The slope is now thought to be due to a mass-metallicity relation along the red sequence as this degeneracy was broken by comparing colour-magnitude simulations to observations of distant clusters [@Kodama1997]. The origin of the mass-metallicity relation is thought to be the heating of the interstellar medium (ISM) by supernovae which triggers the formation of a galactic wind when the thermal energy of the gas exceeds the binding energy. This wind ejects gas more efficiently in smaller galaxies due to their shallower potential wells, resulting in the trend of increased metallicity with mass which manifests itself as the massive metal-rich galaxies appearing progressively redder than their less massive counterparts (@carlberg1984; @arimoto1987). Within the hierarchical picture of galaxy evolution, red sequence galaxies form from the mergers of star forming disc galaxies with the largest ellipticals being the product of the merger of the largest disc systems and are thus the most metal rich [@Kauffmann1998]. The observed frame colour-magnitude relations for clusters at similar redshifts are found to have comparable slopes which have been shown to change with redshift (@Lopez1997; @Glad1998; @Lopez2004). One explanation for this change in the observed slope is that it is a result of $K$ correction and an evolution in the mass metallicity relation in cluster ellipticals (@Kodama1997). The change of slope with redshift has been used as a method to constrain cluster galaxy evolution, with comparisons of the observed slope evolution to models suggesting that the elliptical galaxies in cluster cores have been in place since at least $z=2$ [@Glad1998]. However other studies of the rest frame slope have found little or no evolution suggesting that $K$ correction is the dominant factor (@stanford98; @blakes2003; @mei2006). This non-evolution has been used as evidence to favour monolithic collapse over hierarchical methods of galaxy formation. Current research on the build up of the red sequence suggests there may indeed be an age contribution to the red sequence slope and its evolution as faint, sub-L$^{*}$ galaxies, thought to have undergone recent star formation, are found to be transforming onto the sequence over time as they fall into the dense cluster environment (@delucia2007dgr; @Stott2007a; @smith08). This is thought to be connected to the observation that there is a greater abundance of S0 galaxies in local clusters compared to their high redshift counterparts, whose progenitors may be the quenched star forming galaxies in higher redshift systems (@dres1997; @smail1998). If this transformation scenario is correct then we expect to see evidence for it in the evolution of the rest frame colour-magnitude relation. Regardless of its origin, empirical observations and models of the colour and slope of the observed red sequence can be used in combination as a template to predict the redshift of a cluster. This is found to be in good agreement with spectroscopic redshifts and can therefore be used as an effective two waveband photometric redshift. This technique is important for current and future large area photometric surveys which hope to study cluster abundance (e.g. RCS, @Glad2000; Pan-STARSS; SDSS, @SDSS; UKIDSS, @Law2007, @Swin2007). In this work we will investigate the red sequence slope evolution in observed optical ($\delta(V - I)/\delta I$) and near infrared ($\delta(J - K)/\delta K $) bands and rest frame optical ($\delta(U - V)/\delta V$) for a homogeneous sample of X-ray selected galaxy clusters in the range $z\sim$0–1. This is the most comprehensive study of the red sequence slope undertaken thus far. We compare our findings with latest synthetic slopes calculated from analysis of the semi-analytical model of [@Bower2006] based on the Millennium N-body simulation [@Springel2005]. This model includes feedback from active galactic nuclei (AGN) which quenches star formation in massive halos to match the observed break in the luminosity function seen at bright magnitudes. Another important process in this model is ‘strangulation’ which describes the stripping of a galaxy’s hot gas halo as it falls into a cluster leading to a cessation in star formation in lower mass galaxies thus providing a mechanism for the build up of passive red galaxies in the cluster [@larson1980]. Lambda Cold Dark Matter ($\Lambda$CDM) cosmology ($\Omega_{M}=$0.3, $\Omega_{Vac}=$0.7, $H_{0}=$70 km s$^{-1}$ Mpc$^{-1}$) is used throughout this work. Observations and Reduction ========================== Cluster sample {#sec:sam} -------------- To observe the evolution of the red sequence we study X-ray selected clusters in the range $z=$0–1 in both near-infrared and optical bands. This is a large sample of the most X-ray luminous clusters known (L$_{X}>10^{44} $erg s$^{-1}$, 0.1 – 2.4 keV) which, with the exception of a handful of additional archival clusters, are all sourced from the *ROentgen SATellite* (*ROSAT*) All Sky Survey, although belong to the various sub-samples named below. Such massive clusters are ideal for this study as their colour-magnitude relations are well populated. The motivation for studying an X-ray selected sample of clusters is to ensure that we are observing objects in similar high mass, high density environments. This homogeneity is key to our study as we wish to compare clusters over a range of redshifts. The main cluster samples studied in this paper, the MAssive Cluster Survey (MACS, @MACS2001; @Ebeling2007) and the Las Campanas/AAT Rich Cluster Survey (LARCS, @LARCS2001, 2006), are sufficiently X-ray luminous that they should correspond to the most extreme environments at their respective epochs. The median X-ray luminosities of the high and low redshift samples are 16.1 and 7.4$\times10^{44}$ erg s$^{-1}$, respectively, corresponding to a difference of less than a factor of 2 in the typical total mass [@popesso2005]. However, an important issue to address is that the mass of the z$\sim$0.5 progenitors of the LARCS clusters may be even lower than the MACS clusters. If we include the growth in halo mass through N-body mergers from [@Bower2006], we see that the progenitors of the LARCS clusters at $z\sim$0.5 could be up to 3 times less massive than the MACS sample (with corresponding X-ray luminosities of 2 $\times10^{44}$ erg s$^{-1}$, @popesso2005). There is no evidence for strong variations in the galaxy luminosity function between clusters spanning such a relatively modest difference in typical mass (de Propris et al. 1999). We therefore expect that any differences between the galaxy populations in these two samples will primarily reflect evolutionary differences between $z\sim$0.5 and $z\sim$0.1. The 25 clusters observed in $V$ and $I$ or $B$ and $R$ bands belong to the following surveys: MACS ($V_{F555W}$ and $V_{F814W}$, Cycle 12 GO project 9722); LARCS ($B$ and $R$) and archival [*Hubble Space Telescope*]{} ([*HST*]{}) data ($V_{F555W/F606W}$ and $V_{F814W}$). The observations were taken with the instruments: the Advanced Camera for Surveys (ACS) and the Wide Field Planetary Camera (WFPC) on [*HST*]{} and the 1m Swope telescope at Las Campanas Observatory. For a more detailed description of the data and reduction see [@Stott2007a] (MACS) and [@LARCS2001] (LARCS). The 35 clusters studied in the near-infrared ($J$ and $K$ band) belong to: The MACS survey [@MACS2001], the LARCS survey [@LARCS2001], the *ROSAT* Brightest Cluster Survey (BCS) and extended BCS  [@Ebeling2000] and the X-ray Brightest Abell Clusters Survey (XBACS, @xbacs1996). The observations were taken with: the Wide field InfraRed Camera (WIRC) instrument on the Palomar 200" Hale telescope the Infrared Spectrometer And Array Camera (ISAAC) on the Very Large Telescope (VLT) and the Wide Field CAMera (WFCAM) on United Kingdom Infrared Telescope (UKIRT). An additional low redshift data point is included for the Coma Cluster which we sourced from a combination of the Two Micron All Sky Survey (2MASS) extended and point source catalogues  [@2mass2006]. We refer the reader to [@stott2007b] for a more detailed description of the data and their reduction. We include a high redshift cluster (ClJ1226.9+3332, $z\sim0.9$) from the Wide Angle *ROSAT* Pointed Surveys (WARPS, @WARPS1997; @WARPS1998). This X-ray selected cluster was observed in the J and K bands using the UKIRT Fast-Track Imager (UFTI) camera on UKIRT [@Ellis2004]. Photometry {#sec:phot} ---------- The colour ($V - I$, $B - R$ and $J - K$) photometry extracted for the cluster members employs 9kpc diameter apertures for colours and the magnitude used is [SExtractor]{}’s ‘Best’ magnitude  [@sextractor1996]. This choice of aperture is significantly greater than the seeing conditions in which the low redshift ground-based optical and near-infrared data were taken, typically $\sim$1.0 – 1.2 arcsec, giving LARCS sample photometric apertures of $\sim$4“ whereas the higher redshift MACS sample is extracted with $\sim$1.4” apertures, considerably larger than the $\it HST$ ACS PSF. We are therefore confident that we are collecting a comparable fraction of light for galaxies at different redshifts. We observe no trend between colour and seeing for our LARCS low redshift optical data, although the seeing is consistent, and as the high redshift sample is observed with [*HST*]{} this is not a concern. We also investigate this for the $z\sim0.1 - 0.3$ clusters observed in the near-infrared wavebands for which colour is again independent of seeing, even though the conditions are more variable for our additional [*ROSAT*]{} clusters sample (0.9“ – 1.5”). The [SExtractor]{} colour photometry was run in dual mode, on PSF matched images using the [iraf psfmatch]{} package, with the 9kpc ‘red’-band ($R$, $I$, $K$) apertures used to extract the corresponding ‘blue’-band ($B$, $V$, $J$) photometry. This is to ensure the same size aperture for both bands which is important for good colour determination. Star and Galaxy separation is performed using [SExtractor]{} where detected objects with [class\_star]{}$<$0.1 and/or $J-K>$0.95 (for the near-infrared sample) are classified as galaxies with the stars removed from the analysis. We only consider galaxies within 600kpc radius of the cluster centre, to limit contamination from field galaxies. All of our observations reach a depth which allows us to see to at least 4 magnitudes fainter than the BCG and thus perform a reliable fit to the slope and probe the sub-L$^{\star}$ galaxy population where the red sequence build up is taking place. A potential problem with all aperture photometry is that the galaxies themselves may have significant internal colour gradients thought to be associated with a metallicity gradient between the outer and inner parts of the system. As stated above we are observing the same fraction of light from galaxies at all redshifts which should limit this effect unless there is a significant evolution in the colour gradient of galaxies with redshift, although this would be an interesting effect in itself. Studies of the internal colour gradients of moderate redshift cluster galaxies using high quality [*HST*]{} data have found no such evolution in mean colour gradient and redshift, with the individual cluster galaxies appearing to have random colour gradients [@tamura2000]. We therefore conclude that any variations in internal colour gradient would act randomly to make galaxies appear either redder or bluer resulting in increased scatter of the red sequence but not an evolution in its slope. \[tab:samplecm\] WFCAM data {#sec:wfcam} ---------- We obtained near-infrared $J$ and $K$ band data for 8 of the LARCS $z\sim0.1$ clusters with WFCAM on UKIRT. WFCAM consists of 4 detectors in a square, each separated by a gap comparable in size to a single detector, with a central autoguider. Each detector is a Rockwell Hawaii II 2048 $\times$ 2048 PACE HgCdTe array, with pixel size 0.4 arcsec. WFCAM single pointings or ‘footprints’ do not observe a contiguous area of sky so to create a mosaiced image four of these WFCAM footprints are tiled to make a 4 detector $\times$ 4 detector image. In addition to this tiling the WFCAM images incorporate ‘microstepping’ which is a small dither pattern in the observation. The purpose of this process is to improve the point spread function sampling of the observations as the 0.4$''$ pixel size of WFCAM is comparable to the best atmospheric seeing. The microstepping is performed by observing 4 images each offset from each other by a whole number of pixels and a 1/2 pixel in a 2 $\times$ 2 grid pattern. These images are then coadded together with the fractional offsets taken into account so that the resulting image has double the spatial sampling. Therefore, a detector image with an original resolution of 0.4 $''$/pixel and size 2048 $\times$ 2048 pixels are converted to a higher resolution 0.2 $''$/pixel 4096 $\times$ 4096 image. The observations took place during service mode on the nights of 5th December 2006 and 19th of April 2007 in 1.0“ and 1.2” seeing conditions respectively. The total integration times for the $J$ and $K$ band images are 200s each, composed of 10s exposures in a 5 point dither pattern with 2$\times$2 microstepping. To create a contiguous Mosaic of WFCAM images we run the [Terapix SWarp]{} software on the data which gives an image with a world coordinate system consistent to within 0.1" of the 2MASS and the United States Naval Observatory (USNO) star catalogue across the whole field of view. To extract the $J$ and $K$ band catalogues from the mosaiced images we use the [SExtractor]{} software in dual mode so that the $K$ band catalogue detections are used to extract the $J$ band photometry in an identical process to above. The photometry is calibrated using the 2MASS point source catalogue and we find the $J$ and $K$ band 5 sigma vega limits of these data are 19.50 and 17.75 magnitudes respectively. ### Quantifying the dwarf to giant ratio of the LARCS clusters The WFCAM instrument allows us to create contiguous images that are $\sim$0.9 degrees on the side which corresponds to 5.6Mpc at $z=0.1$. With images of this size we can study both the cluster and the surrounding field galaxy population. A number of recent papers have studied the form of the colour-magnitude relation in galaxy clusters with some reporting a dearth of faint red sequence galaxies at high redshift (e.g. @kodama2004; @delucia2004dgr; @delucia2007dgr). [@Stott2007a] demonstrate that faint red galaxy population in clusters is built up over time by analysing the ratio of faint to luminous galaxies along the red sequence in a homogenous sample of clusters at $z\sim0.1$ and $z\sim0.5$. One of the limiting factors in that study was the uncertainty in the faint end statistical field correction for the LARCS $z\sim0.1$ colour-magnitude relations. However the uncertainty in the field correction is reduced if near infrared observations are employed as it is easier to isolate the cluster red sequence. A comparison of Fig. \[fig:cmwfcam\] with the equivalent optical colour magnitude diagram, Fig. 1 [(*bottom right panel)*]{}, of [@Stott2007a] illustrates this point. By analysing the WFCAM data in an identical way to that described in [@Stott2007a] in concert with the UKIDSS Deep eXtragalactic Survey (DXS, Survey Head: Alastair Edge) field observations we can successfully quantify both the cluster and field galaxy population. The field correction is performed by dividing the colour-magnitude space of the cluster and field samples into a two-dimensional histogram and subtracting the field number counts from the cluster population [@Stott2007a]. The Poisson errors from this field correction are folded through into the final result. To determine the relative numbers of faint and luminous galaxies on the red sequence we define the red sequence dwarf to giant ratio (RDGR). This quantity is defined as the number of dwarf (faint) galaxies divided by the number of giant (luminous) galaxies on the cluster red sequence after the statistical subtraction of the field galaxy population. The dividing line between dwarfs and giants, for consistency with the [@Stott2007a] result, is given in absolute $V$ band magnitude, M$_{V}=-19.9$ and a limiting magnitude of M$_{V}=-17.75$ for the faintest dwarfs, a correction for passive evolution is applied when studying high redshift samples. We convert these limits to apparent K band magnitudes, $K\sim14.7$ and $K\sim16.8$ respectively (depending on the precise redshift), using a [@bruzcharl2003] simple stellar population with solar metallicity and a formation redshift, $z_f=5$. The weighted mean of the RDGRs of the 8 LARCS clusters from this sample is 2.59$\pm$0.24 compared to the [@Stott2007a] result of 2.93$\pm$0.45. We can now say that when compared to our $z\sim$0.5 MACS sample, where the DGR=1.33$\pm$0.06, the ratio of dwarf to giant galaxies on the cluster red sequence has increased by a factor of 1.95$\pm$0.20 in the past 5Gyr which confirms and improves the significance of the result of [@Stott2007a] from 3$\sigma$ to 5$\sigma$. This is strong evidence for a significant build up of the red sequence in massive clusters. Fig. \[fig:dgr\] is the equivalent of Fig. 2 [*(right)*]{} of [@Stott2007a] and displays the evolution in the RDGR with redshift for rich galaxy clusters. To parameterise this evolution we fit a $(1+z)^{-\beta}$ power law to the LARCS and MACS samples where $\beta=-2.1\pm0.3$ with all clusters consistent with the fit. ![The $(J - K)$ vs $K$ colour-magnitude diagram for the cluster Abell 1084 which demonstrates the ease at which near-infrared colours separate the red sequence from stellar and field galaxy contamination[]{data-label="fig:cmwfcam"}](a1084cmdiag.ps){width="50.00000%"} ![The evolution of the of the red sequence giant dwarf ratio (RDGR) with redshift comparing the MACS sample from [@Stott2007a] with the LARCS WFCAM sample. A fit of the form $(1+z)^{-\beta}$ to the LARCS and MACS samples is plotted which yields $\beta=-2.1\pm0.3$.[]{data-label="fig:dgr"}](dgr.ps){width="50.00000%"} Analysis and Results ==================== Fitting red sequence the slope {#sec:fit} ------------------------------ Figures \[fig:larcscm\] and \[fig:macscm\] are examples of the prominent red sequences we see in the rich clusters of our sample. To fit the slope of the red sequence we need to use a robust and consistent method. We therefore employ the same technique as described in [@Glad1998], an iterated 3 $\sigma$ clipped fit. The fit is performed as follows: first we set a limiting magnitude for the red sequence which corresponds to the mean of the next 2 brightest galaxies down from the BCG magnitude+3. The reason for this is that in some cases there can be a large luminosity gap ($\sim$1 mag) between the BCG and the start of the red sequence so this ensures we measure the slope without this gap. The fit is performed over 3 magnitudes to incorporate the sub-L$^{\star}$ galaxy population which show the strongest evidence for red sequence build up [@Stott2007a]. We select the region of colour magnitude space containing the red sequence and estimate an initial fit from visual inspection. The residuals about this estimate are calculated and a Gaussian is fitted to the resulting colour distribution with the slope removed. The peak of this Gaussian corresponds to the red sequence. We then perform a fit to points that are within 3 sigma of this fit. This is a two parameter linear fit of the form $y = \kappa x + c$ where $\kappa$ is the slope of the red sequence. The process is iterated until it converges to a solution. We confirm the work of [@Glad1998] that this is a robust method for reasonable choices of limiting magnitude and initial fit. It should be noted that our colour-magnitude diagrams are not field corrected as the rich clusters in our sample have well populated red sequences in contrast to the field. At the colours and magnitudes we consider this a very small contribution to the red sequence, typically 5% contamination for the MACS clusters and 10% for the LARCS clusters. We perform a statistical field correction test to a sub-sample of our low $z$ clusters and find that the slope of the sequence varies randomly by less than 1$\sigma$ than that obtained for the uncorrected sequence. We therefore feel justified in not applying this correction. For details of the statistical field correction technique used see [@Stott2007a]. {width="40.00000%"}{width="40.00000%"} {width="40.00000%"}{width="40.00000%"} {width="40.00000%"}{width="40.00000%"} {width="40.00000%"}{width="40.00000%"} {width="40.00000%"}{width="40.00000%"} {width="40.00000%"}{width="40.00000%"} {width="40.00000%"}{width="40.00000%"} {width="40.00000%"}{width="40.00000%"} {width="40.00000%"}{width="40.00000%"} {width="40.00000%"} Model slope evolution {#sec:modev} --------------------- We form a prediction of how the red sequence slope evolves, by analysing the semi-analytical model of [@Bower2006] based on the Millennium N-body simulation [@Springel2005]. We access these data online through Virgo Millennium Database. The [@Bower2006] model accounts for the recent observations that the stellar mass in bright galaxies was in place at high redshift by including feedback from active galactic nuclei. It also includes a prescription for the build of the red sequence via ‘strangulation’. This describes a process whereby galaxies falling into a cluster are stripped of their hot gas reservoir upon interaction with the intracluster medium quenching star formation [@larson1980]. We define model clusters as dark matter haloes above a threshold mass which contain bound sub-haloes (galaxies) with observational properties assigned to them by semi-analytic modelling. To allow comparison between models and observation we assume that the model cluster members are analogous to the red sequences of our observed clusters. We need to select clusters from [@Bower2006] which are comparable in mass to those we observe. By using the relation between cluster mass and X-ray luminosity [@popesso2005] we include only the simulated haloes with M$_{200}>3.4\times10^{14}$M$_{\odot}$ which corresponds to L$_{X}>10^{44}$erg s$^{-1}$. To select only the most massive systems for comparison with our observations we limit our red sequence fit to a stack of the top five ranked clusters by mass in each redshift bin. The model slope is robust to our choice of lower mass limit as we see no significant change in our slope values when using all haloes with masses greater than mean halo mass in the required redshift interval. We model the red sequence slope evolution by creating stacked colour magnitude diagrams from the [@Bower2006] model output at a distinct set of redshift intervals between $z=0$ and $z=1$. For the creation of these colour-magnitude diagrams we ensure we only study the passive red sequence galaxies by selecting galaxies with no current star formation (L(H$\alpha)=$0). The models provide no spatial information but we assume that the synthetic colours we calculate from the total magnitudes can be compared with the observed aperture magnitudes allowing for a normalisation between the model and observed slope evolution at low redshift. We calculate the model for the observed frame slope evolution and its errors by fitting to the stacked synthetic red sequence slopes at each redshift interval with the method described in §\[sec:fit\]. As with the observations, the model slope is shown to steepen with redshift. Observed slope evolution {#sec:sev} ------------------------ Fig. \[fig:evl\] displays the observed red sequence slope evolution for our near infrared sample. The slope ($\kappa_{JK}=\delta(J - K)/\delta K$) is shown to steepen with redshift. This steepening will have contributions from both $K$ correction and perhaps an age or a metallicity evolution. The contribution from $K$ correction is due to the observed J and K bands sampling increasingly bluer rest wavebands at higher redshift which affects galaxies differentially along the red sequence depending on their spectral energy distributions (@Glad1998; @LARCS2001). When the model described in §\[sec:modev\] is plotted with our complete near-infrared dataset in Fig.\[fig:evl\] we find good quantitative agreement between the two with the data having an rms scatter about the model of 0.009 which is comparable to the calculated error on the model (mean 1$\sigma$ error is 0.006). {width="60.00000%"} {width="60.00000%"} {width="60.00000%"} We now investigate the slope ($\kappa_{VI}=\delta(V - I)/\delta I$) evolution for our optical observations. As above, the simulated slope evolution plotted is calculated from semi-analytical model of [@Bower2006]. There is an additional complication as our optical data are sourced from two different filter sets so we have to account for the observed difference in red sequence slope between them. To achieve this we normalise all data to the $\delta(V - I)/\delta I$ filters by correcting for the difference between the $\delta(V - I)/\delta I$ model and the model for the $\delta(B - R)/\delta R$ filter combination (as in @Glad1998). The resultant data-points and model are plotted in Fig. \[fig:evr\]. The slope is shown to increase with redshift as above. The rms scatter about the model is 0.017 (compared to the mean 1 $\sigma$ error in the model of 0.008) so as with the near-infrared observations we find good agreement between the model and the data. ### Rest frame slope evolution {#sec:restf} To investigate the intrinsic evolution of the red sequence slope we need to study clusters observed with matched rest frame photometry. This is to quantify the proposed contribution from the build up of the red sequence without the additional effect of $K$ correction. We can observe this intrinsic slope evolution as our two main sets optical observations, the $z\sim$0.1 LARCS clusters in $(B - R)$ colour and the $z\sim$0.5 MACS clusters in $(V - I)$ colour, both correspond to the rest frame $(U - V)$ colour at their respective epochs. $(U - V)$ colour straddles the 4000Å break and therefore is a good discriminant between the red sequence and star forming cluster members/foreground galaxies. We confirm that the filters are well matched to rest frame $(U - V)$ as we find a colour term of 1.0 $\pm$0.05 between the $B - R$ at $z=$0.11 and $V - I$ at $z=$0.53 using the technique described in [@blake2006]. The evolution of this intrinsic slope ($\kappa_{UV}=\delta(U - V)/\delta V$) is plotted in Fig. \[fig:restev\]. We include an additional low redshift data-point for the Coma cluster calculated from the Sloan Digital Sky Survey (SDSS, @SDSS) $u$ and $g$ filter photometry. In this figure we can see that the intrinsic optical slope, $\kappa_{UV}$, does evolve with redshift as the intermediate $z$ MACS clusters have a steeper red sequence than their low $z$ LARCS counterparts. The weighted mean values for the LARCS and MACS $\kappa_{UV}$ are -0.053$\pm$0.004 (s.e.m) and -0.092$\pm$0.004 (s.e.m) respectively, a difference of 6.5 $\sigma$. We can therefore say that there is a real contribution to the slope evolution from factors other than $K$ correction. The fit to the data in Fig. \[fig:restev\] is of the form $(1+z)^{\beta}$ where $\beta=1.77\pm0.25$. We note that the Coma Cluster has a steeper slope than the rest of our low redshift sample, although only a $\sim$2$\sigma$ discrepancy from the fit, which we may expect as it is found to have lower than average dwarf-to-giant ratio along its red sequence suggesting it is still undergoing faint end and therefore slope evolution [@Stott2007a]. Including the Coma Cluster does not have a significant affect on the fit, $\beta=1.67\pm0.26$. The dashed line plotted on Fig. \[fig:restev\] is the rest frame $\kappa_{UV}=\delta(U - V)/\delta V$ slope calculated from the semi-analytic model of [@Bower2006], normalised to the mean value of the LARCS sample slope, which shows only a mild evolution with redshift and is therefore unable to replicate the rest frame slope change observed. From this we can conclude that the major contribution to the agreement between the observed slope evolution in $\S\ref{sec:sev}$ and that derived from the semi-analytic modelling shown in Figs \[fig:evl\] & \[fig:evr\] was the $K$ correction differential between high and low mass galaxies along the synthetic red sequence and that a significant, intrinsic, slope evolution is not predicted by the models. Evolution with other observables {#sec:xev} -------------------------------- We now investigate whether there are further trends in red sequence slope with other observable cluster properties to ensure that the result seen in §\[sec:restf\] is not due to a secondary correlation. The most obvious of these being the X-ray luminosity, which is a proxy for mass in a relaxed system. In Figs \[fig:reslxir\] and \[fig:reslxuv\] we plot the residual values of the slope about the model line in Fig. \[fig:evl\] and fit in Fig. \[fig:restev\] against $L_{X}$. From this we see no significant trend between scatter about the model and $L_{X}$ as the Pearson correlation coefficients, r , for the near-infrared and optical data are 0.17 and 0.24 respectively which is actually a weak anti-correlation between steepening negative red sequence slope and $L_{X}$. However, because of the magnitude of the errors on both the individual slope measurements and the models, we need to quantify what level of trend the errors could accommodate before becoming observable. For the near-infrared sample we find that a steepening greater than 0.010 in the slope in L$_{X}=26.6\times10^{44}$erg $s^{-1}$ can be ruled out with a 3 sigma confidence whereas for the rest frame $(U-V)$ sample we find this magnitude of change can occur in L$_{X}=12.3\times10^{44}$erg $s^{-1}$. The difference in mean L$_{X}$ between the LARCS and MACS samples is $8.5\times10^{44}$erg $s^{-1}$ for which we can rule out, with a 3 sigma confidence, increases in slope greater than 0.007, if we use the $(U-V)$ result, or 0.003 if we use the $J-K$ result. For the rest frame $\delta(U-V)/\delta V$ slope evolution in Fig. \[fig:restev\] we see that the observed change in rest frame slope of $\sim0.04$ cannot be accounted for by the difference in $L_{X}$ alone and is, we believe, a definite trend with redshift. In Fig. \[fig:ressiguv\] we also show that there is no significant trend between red sequence slope and another mass proxy, the cluster velocity dispersion, $\sigma$ (r$=0.27$), which is again a weak anti-correlation. As above we can rule out increases in slope greater than 0.010 with 3 sigma confidence between the average velocity dispersions of the MACS and LARCS samples. In addition to the X-ray luminosity and $\sigma$, we can look to the degree of BCG dominance as an indicator of the local environment within the cluster core. This parameterises the luminosity gap between the BCG and the next brightest galaxies on the red sequence and is defined as $\Delta m_{1-2,3}= (m_2+m_3)/2 -m_1$ where $m_1$ is the magnitude of the BCG and $m_2$ and $m_3$ are the magnitudes of the 2nd and 3rd brightest members respectively  [@Kim2002; @stott2007b]. The BCG may be the dominant elliptical in a cluster centre containing much smaller galaxies or it may be in a system were it is only marginally brighter than the next brightest members. In Fig. \[fig:resdomsigir\] we demonstrate that there is no strong trend between red sequence slope and the degree of BCG dominance (r$=-0.09$) and can rule out increases of slope greater than 0.01 in 0.61 magnitudes of dominance with a 3 sigma confidence. The above results suggest that different cluster environments do not have a strong effect on the near-infrared or optical red sequence slope at a given redshift, demonstrating that the result in §\[sec:sev\] is robust. Previous studies within a similar $L_{X}$ range have also seen homogeneity in other related cluster properties such as the shape of the luminosity function and the blue galaxy fraction (@deprop1999; @Wake2005). ![The residuals about the models in Fig. \[fig:evl\] plotted against X-ray luminosity. Filled circles and squares represent galaxy clusters with redshifts, $z<0.2$ and $z>0.2$ respectively, to demonstrate there is no redshift dependence. The correlation coefficient, r, for this plot is 0.17.[]{data-label="fig:reslxir"}](resnir.ps){width="50.00000%"} ![The residuals about the models in Fig. \[fig:restev\] plotted against X-ray luminosity. Filled circles and squares represent galaxy clusters with redshifts, $z<0.2$ and $z>0.2$ respectively, to demonstrate there is no redshift dependence. The correlation coefficient, r, for this plot is 0.24.[]{data-label="fig:reslxuv"}](uvreslx.ps){width="50.00000%"} ![The residuals about the models in Fig. \[fig:restev\] plotted against velocity dispersion, $\sigma$. Filled circles and squares represent galaxy clusters with redshifts, $z<0.2$ and $z>0.2$ respectively, to demonstrate there is no redshift dependence. The MACS and LARCS cluster velocity dispersions are sourced from [@Ebeling2007] and [@LARCS2006] respectively. The correlation coefficient, r, for this plot is 0.27.[]{data-label="fig:ressiguv"}](uvressig.ps){width="50.00000%"} ![The residuals about the models in Fig. \[fig:evl\] plotted against BCG dominance. Filled circles and squares represent galaxy clusters with redshifts, $z<0.2$ and $z>0.2$ respectively, to demonstrate there is no redshift dependence. The correlation coefficient, r, for this plot is -0.09.[]{data-label="fig:resdomsigir"}](resnirdom.ps){width="50.00000%"} Discussion {#sec:sum} ========== In this work we have found a significant evolution in the rest frame slope of the red sequence in rich galaxy clusters between $z\sim0.5$ and $z\sim0.1$. We propose that this intrinsic evolution is due to galaxies falling into the cluster core and transforming onto the red sequence (@delucia2007dgr; @Stott2007a). If these galaxies have undergone recent star formation in filaments (e.g. @porter2008), which has been quenched by interactions as they fall into the cluster, they will appear bluer than other passive galaxies in the cluster due to their young age. These galaxies will redden more rapidly with time than their old, luminous counterparts flattening the red sequence at progressively lower redshift. This relates to the concept of downsizing which describes the observation that star formation ceases in the largest galaxies first so that it takes place mainly in lower mass galaxies at late times [@Cowie1996]. An alternative or additional explanation is that this result is caused by a differential chemical evolution along the red sequence. A differential chemical evolution along the sequence which preferentially enriches the faint galaxies could lead to a steeper red sequence slope at high redshift. The supposition of a metallicity differential along the sequence that decreases with redshift is borne out in the semi-analytic models as the gradient at $z=$0 is -0.0029$\pm$0.0004 Z$_{\odot}$ mag$^{-1}$ and at z=1 it is -0.061$\pm$0.009 Z$_{\odot}$ mag$^{-1}$. However, this may be non-physical with the primary goal of semi-analytic model is to match the observed properties of the local Universe and not necessarily the evolution from high redshift. This result is in disagreement with the work of [@stanford98], [@blakes2003] and [@mei2006] who find no evidence for intrinsic slope evolution. There are several potential reasons for this disagreement. We first note the scatter in the values of the slope which could obscure the intrinsic evolution in previous studies without the coverage per redshift bin we have in our sample. This scatter is important when choosing a low redshift comparison cluster, some earlier works use a single cluster, usually Coma, which we have shown to have an unusually steep red sequence compared to the mean of our low $z$ sample. The use of Coma would therefore tend to bias the result towards no evolution. The choice of $(U - B)$ rest frame colour in previous studies means that they are observing a smaller dynamic range in colour than our $(U - V)$ sample. This is particularly important if the rest frame ‘$U-B$’ colours do not effectively straddle the 4000Å break (e.g. figure 4 of @mei2006) which may result in contamination of the red sequence by both foreground galaxies and star forming cluster members. We also ensure that we perform all of the red sequence slope fits using the same homogenous method rather than importing slopes from other studies which may have a systematic offset. The good agreement found between our data and slope evolution models calculated from semi-analytical model of [@Bower2006] in Figures \[fig:evl\] and \[fig:evr\] is mainly due to the treatment of $K$ correction rather than any significant slope evolution. This demonstrates that although such models include ‘strangulation’ of star-forming galaxies falling into cluster environments, they are unable to effectively reproduce the intrinsic evolution seen in our sample. This deficiency is also seen when studying the build up of the red sequence [@gilbank2008b]. We speculate that an improved prescription for a strangulation-like process in the semi-analytic models will be an important factor in recreating observations of an intrinsic slope evolution and build up of the red sequence. When looking at slope trends with other observables we see no relationship between cluster slope and X-ray luminosity, velocity dispersion or BCG degree of dominance. This suggests that there is very little variation between the red sequence slopes due to the different cluster environments demonstrating that the main result of this paper is robust to the differing global properties within our sample. This implies that searching for massive clusters using the colour magnitude relation (CMR), as employed by current and future large area optical/near-infrared surveys, is a viable method (e.g. RCS, @Glad2000; Pan-STARSS; SDSS, @SDSS; UKIDSS, @Law2007, @Swin2007). Acknowledgements ================ We thank the referee for their useful comments which have improved the clarity and conclusions of this paper. Thanks also go to Richard Bower, Philip Best, Jim Geach and Matt Hilton for useful discussions. JPS acknowledges support through a Particle Physics and Astronomy Research Council and latterly a Science and Technology Facilities Council Studentship. KAP acknowledges partial support from the Australian Research Council and partial support from a University of Queensland ResTeach Fellowship. The United Kingdom Infrared Telescope is operated by the Joint Astronomy Centre on behalf of the Science and Technology Facilities Council of the U.K. We gratefully acknowledge the allocation of UKIRT service time for our observations. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Centre/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. The Millennium Simulation databases used in this paper and the web application providing online access to them were constructed as part of the activities of the German Astrophysical Virtual Observatory. [^1]: E-mail: jps@astro.livjm.ac.uk

--- abstract: 'We present a new algorithm to sample the constrained eigenvalues of the initial shear field associated with Gaussian statistics, called the ‘peak/dip excursion-set-based’ algorithm, at positions which correspond to peaks or dips of the correlated density field. The computational procedure is based on a new formula which extends Doroshkevich’s unconditional distribution for the eigenvalues of the linear tidal field, to account for the fact that halos and voids may correspond to maxima or minima of the density field. The ability to differentiate between random positions and special points in space around which halos or voids may form (i.e. peaks/dips), encoded in the new formula and reflected in the algorithm, naturally leads to a straightforward implementation of an excursion set model for peaks and dips in Gaussian random fields – one of the key advantages of this sampling procedure. In addition, it offers novel insights into the statistical description of the cosmic web. As a first physical application, we show how the standard distributions of shear ellipticity and prolateness in triaxial models of structure formation are modified by the constraint. In particular, we provide a new expression for the conditional distribution of shape parameters given the density peak constraint, which generalizes some previous literature work. The formula has important implications for the modeling of non-spherical dark matter halo shapes, in relation to their initial shape distribution. We also test and confirm our theoretical predictions for the individual distributions of eigenvalues subjected to the extremum constraint, along with other directly related conditional probabilities. Finally, we indicate how the proposed sampling procedure naturally integrates into the standard excursion set model, potentially solving some of its well-known problems, and into the ellipsoidal collapse framework. Several other ongoing applications and extensions, towards the development of algorithms for the morphology and topology of the cosmic web, are discussed at the end.' author: - | Graziano Rossi$^{1,2,3}$[^1]\ \ $^{1}$ Korea Institute for Advanced Study, Hoegiro 85, Dongdaemun-Gu, Seoul $130-722$, Korea\ $^{2}$ CEA, Centre de Saclay, Irfu/SPP, F-91191 Gif-sur-Yvette, France\ $^{3}$ Paris Center for Cosmological Physics (PCCP) and Laboratoire APC, Université Paris 7, 75205 Paris, France date: 'Accepted 2012 November 27. Received 2012 November 7; in original form 2012 June 29' title: 'Peaks and dips in Gaussian random fields: a new algorithm for the shear eigenvalues, and the excursion set theory' --- \[firstpage\] methods: analytical – methods: statistical – galaxies: formation – galaxies: halos – cosmology: theory – large-scale structure of Universe. Introduction ============ The large-scale spatial organization of matter in clusters, filaments, sheets and voids, known as the *cosmic web* (Peebles 1980; Bardeen et al. 1986; Bond et al. 1991), is the manifestation of the anisotropic nature of gravitational collapse. This typical filamentary pattern has been confirmed by observations, for example with the 2dFGRS, SDSS and 2MASS redshift surveys of the nearby Universe (Colless et al. 2003; Tegmark et al. 2004; Huchra et al. 2005), and is routinely seen in large scale $N$-body numerical simulations – see for example the New Horizon Runs (Kim et al. 2011) or the recent Millennium-XXL (Angulo et al. 2012). Basic characteristics of the cosmic web are the spatial arrangement of galaxies and mass into elongated filaments, sheet-like walls and dense compact clusters, the existence of large near-empty void regions, and the hierarchical nature of this mass distribution (Arag[ó]{}n-Calvo et al. 2007, 2010a,b; Zhang et al. 2009). In particular, as pointed out by Bond, Kofman & Pogosyan (1996), ‘embryonic’ cosmic web is already present in the primordial density field. These key properties, along with the alignment of shape and angular momentum of objects (i.e. Argyres et al. 1986; Catelan et al. 2001; Lee & Springel 2010), are mainly due to the effects of the tidal field, associated with the gravitational potential – while the Hessian of the density field (i.e. the inertia tensor) plays a secondary role in determining the characteristic pattern of the cosmic web. Pioneering works devoted to the key role of the initial tidal field in shaping large-scale structures trace back to Doroshkevich & Zeldovich (1964), Doroshkevich (1970), Zeldovich (1970), Sunyaev & Zeldovich (1972), Icke (1973), and Doroshkevich & Shandarin (1978); their studies have contributed to reach a solid understanding of the formation and evolution of structures. Other classical works in this direction (i.e. Peebles 1980; White 1984; Bardeen et al. 1986; Kaiser 1986; Bertschinger 1987; Dubinski 1992; Bond & Myers 1996; Bond, Kofman & Pogosyan 1996; van de Weygaert & Bertschinger 1996) have considerably improved the statistical description of the cosmic web from first principles. Their impact is broader, as there is a correspondence between structures in the evolved density field and local properties of the linear tidal shear; this allows one to estimate the morphology of the cosmic web (Bond & Myers 1996; Rossi, Sheth & Tormen 2011), and is crucial in understanding the nonlinear evolution of cosmic structures (Springel et al. 2005; Shandarin et al. 2006; Pogosyan et al. 2009), the statistical properties of voids (Lee & Park 2006; Platen, van de Weygaert & Jones 2008; Lavaux & Wandelt 2010), the alignment of shape and angular momentum of halos (West 1989; Catelan et al. 2001; Faltenbacher et al. 2009; Lee & Springel 2010), and for characterizing the geometry and topology of the cosmic web (Gott et al. 1986, 1989; Park & Gott 1991; Park et al. 1992, 2005; van de Weygaert & Bond 2008; Forero-Romero et al. 2009; Aragon-Calvo et al. 2010a,b; Choi et al. 2010; Shandarin et al. 2010; van de Weygaert et al. 2011; Cautun et al. 2012; Hidding et al. 2012). In addition, the eigenvalues of the mass tidal tensor can be used to classify the large-scale environment (Shen et al. 2006; Hahn et al. 2007; Zhang et al. 2009) – hence their fundamental importance. In the context of the initial shear field, Doroshkevich (1970) provided a key contribution by deriving the joint probability distribution of an ordered set of eigenvalues in the tidal field matrix corresponding to a Gaussian potential, given the variance of the density field. Throughout the paper, we will refer to it as the *unconditional* probability distribution of shear eigenvalues. Because the initial shear field associated with Gaussian statistics plays a major role in the formation of large scale structures, considerable analytic work has been based on Doroshkevich’s unconditional formulas since their appearance, but those relations neglect the fact that halos (voids) may correspond to maxima (minima) of the underlying density field. The local extrema of such field (i.e. peaks/dips) are plausible sites for the formation of nonlinear structures (Bardeen et al. 1986), and some numerical studies have indeed reported a good correspondence between peaks in the initial conditions and halos at late times (see in particular Ludlow & Porciani 2011). Hence, their statistical properties can be used to predict the abundances and clustering of objects of various types, and in studies of triaxial formation of large-scale structures (Bardeen et al. 1986; Bond & Myers 1996). Motivated by these reasons, recently Rossi (2012) has provided a set of analytic expressions which extend the work of Doroshkevich (1970) and Bardeen et al. (1986) – and are akin in philosophy to that of van de Weygaert & Bertschinger (1996). These new relations incorporate the peak/dip constraint into the statistical description of the initial shear field, and are able to differentiate between random positions and peak/dips in the correlated density field. In essence, they allow one to express the joint probability distribution of an ordered set of eigenvalues in the initial shear field, given the fact that positions are peaks or dips of the density – and not just random spatial locations. These relations are obtained by requiring the density Hessian (i.e. the matrix of the second derivatives of the density field, associated with the curvature) to be positive/negative definite, which is the case in the vicinity of minima/maxima of the density. The correlation strength between the gravitational and density fields is quantified via a reduced parameter $\gamma$, which plays a major role in peaks theory (i.e. $\gamma$ is the same as in Bardeen et al. 1986 for Gaussian smoothing filters – see Appendix \[notation\]). Doroshkevich’s (1970) *unconditional* formulas are then naturally recovered in the absence of correlation, when $\gamma=0$. From these new *conditional* joint probabilities, it is possible to derive the individual distributions of eigenvalues subjected to the extremum constraint, along with some other related conditional probabilities; their expressions were also provided in Rossi (2012), extending the work of Lee & Shandarin (1998). The primary goal of this paper is show how the main conditional formulas presented in Rossi (2012) and reported here (Equations \[doro\_inter\_extended\] and \[doro\_eigen\]) lead to a new algorithm – called the ‘peak/dip excursion-set-based’ algorithm – to sample the constrained eigenvalues of the initial shear field associated with Gaussian statistics, at positions which correspond to peaks or dips of the correlated density field. While it is clearly possible to sample the constrained eigenvalues of the tidal field directly from the conditional probability distribution function, as done for example by Lavaux & Wandelt (2010) in the context of cosmic voids, the theoretical work carried out by Rossi (2012) allows for a much simpler procedure, part of which was previously thought not to be achievable analytically (see again the Appendix B in Lavaux & Wandelt 2010). Besides providing novel theoretical insights, the main strength of the algorithm resides in its natural inclusion into the standard excursion set framework, allowing for a generalization. As we will discuss in Section \[esmpd\], this technique potentially solves a long-standing problem of the standard excursion set theory; moreover, it is well-suited for the ellipsoidal collapse framework. The other goal of the paper is to present a first physical application of the new algorithm, related to the morphology of the cosmic web. In particular, we provide a new expression for the conditional distribution of shape parameters (i.e. ellipticity and prolateness) in the presence of the density peak constraint, which generalizes some previous literature work and combines the formalism of Bardeen et al. (1986) – based on the density field – with that of Bond & Myers (1996) – based on the shear field. The formula has important implications for the modeling of non-spherical dark matter halos and their evolved halo shapes, in relation to the initial shape distribution. In addition, we also test and confirm our theoretical predictions for the individual distributions of eigenvalues subjected to the extremum constraint, along with other directly related conditional probabilities. Finally, we illustrate how this algorithm can be readily merged into the excursion set framework (Peacock & Heavens 1990; Bond et al. 1991), and in particular to obtain an excursion set model for peaks and dips in Gaussian random fields. The key point is the ability to differentiate between random positions and peaks/dips, which is contained in Equations (\[doro\_inter\_extended\]) and (\[doro\_eigen\]) and encapsulated in the algorithm. While we leave this latter part to a dedicated forthcoming publication, we anticipate the main ideas here. We also discuss several other ongoing applications and extensions, towards the development of algorithms for classifying cosmic web structures. The layout of the paper is organized as follows. Section \[jde\] provides a short review of the key equations derived in Rossi (2012), which constitute the theoretical framework for the new algorithm described here; the main notation adopted is summarized in Appendix \[notation\], for convenience. Moving from this mathematical background, Section \[pdesa\] presents the new ‘peak/dip excursion-set-based’ algorithm (some insights derived from this part are left in Appendix \[insights\]), while Section \[cdpnt\] tests its performance against various analytic distributions of eigenvalues and related conditional probabilities, subjected to the extremum constraint. Section \[epd\] shows a physical application of the computational procedure, towards the morphology of the cosmic web. A new expression for the conditional distribution of ellipticity and prolateness in the presence of the density peak constraint is also given; in particular, the description of Bardeen et al. (1986) is combined with that of Bond & Myers (1996). Section \[esmpd\] illustrates how this new algorithm can be readily used to implement an excursion set model for peaks and dips in Gaussian random fields, and makes the connection with some previous literature. Finally, Section \[conc\] discusses several ongoing and future promising applications, which will be presented in forthcoming publications, and in particular the use of this algorithm for triaxial models of collapse and in relation to the morphology and topology of the cosmic web. Joint conditional distribution of eigenvalues in the peak/dip picture {#jde} ===================================================================== We begin by reexamining two main results derived in Rossi (2012), which constitute the key for developing a new algorithm to sample the constrained eigenvalues of the initial shear field. This part may be also regarded as a compact review of the main formulas for the constrained shear eigenvalues, which can be used directly for several applications related to the cosmic web. The notation adopted here is the same as the one introduced by Rossi (2012), with a few minor changes to make the connection with previous literature more explicit. It is summarized in Appendix \[notation\]; a reader not familiar with the notation may want to start from the appendix first. In particular, in what follows we do not adopt ‘reduced’ variables, so that the various dependencies on $\sigma$ values (i.e. $\sigma_{\rm T} \equiv \sigma_0$ for the shear, and $\sigma_{\rm H} \equiv \sigma_2$ for the density Hessian) are now shown explicitly. However, for the sake of clarity, we omit to indicate the understood dependence of these two global parameters in the left-hand side of all the formulas. Note that we prefer to write $\sigma_{\rm T}$ and $\sigma_{\rm H}$, rather than $\sigma_0$ and $\sigma_2$, for their more intuitive meaning (i.e. the label $T$ indicates that a quantity is connected to the shear field, while the label $H$ points to the Hessian of the density field). As explained in Appendix \[notation\], we also introduce the 6-dimensional vectors ${\bf T} = (T_{11},T_{22},T_{33},T_{12},T_{13},T_{23})$ and ${\bf H} = (H_{11},H_{22},H_{33},H_{12},H_{13},H_{23})$, derived from the components of their corresponding shear and density Hessian tensors. Rossi (2012) obtained the following expression for the probability of observing a tidal field ${\bf T}$ for the gravitational potential, given a curvature ${\bf H}$ for the density field: $$p ({\bf T}|{\bf H},\gamma) = {15^3 \over 16 \sqrt {5} \pi^3} {1 \over \sigma^6_{\rm T}(1-\gamma^2)^3} {\rm exp} \Big [-{3 \over 2 \sigma_{\rm T}^2(1-\gamma^2)} (2 K_1^2 - 5 K_2 ) \Big ] \label{doro_inter_extended}$$ where $$\begin{aligned} \label{K_def} K_1 &=& (T_{11} -\eta H_{11}) + (T_{22} -\eta H_{22}) + (T_{33} -\eta H_{33}) = k_1 - \eta h_1 \\ K_2 &=& (T_{11} -\eta H_{11})(T_{22} - \eta H_{22}) + (T_{11} -\eta H_{11})(T_{33} - \eta H_{33}) \nonumber \\ &+& (T_{22} -\eta H_{22})(T_{33} - \eta H_{33}) -(T_{12} - \eta H_{12})^2 -(T_{13} -\eta H_{13})^2 -(T_{23} - \eta H_{23})^2 = k_2 + \eta^2 h_2 - \eta h_1 k_1 + \eta \tau \\ \tau &=& T_{11} H_{11} + T_{22} H_{22} + T_{33} H_{33} + 2 T_{12} H_{12} + 2 T_{13} H_{13} + 2 T_{23} H_{23} \\ k_1 &=& T_{11} + T_{22} + T_{33} \\ k_2 &=& T_{11} T_{22}+ T_{11}T_{33}+ T_{22}T_{33} -T^2_{12} -T^2_{13} -T^2_{23}\\ h_1 &=& H_{11} + H_{22} + H_{33} \\ h_2 &=& H_{11} H_{22}+ H_{11} H_{33}+ H_{22} H_{33} -H^2_{12} -H^2_{13} -H^2_{23}\\ \eta &=& \gamma \sigma_{\rm T} / \sigma_{\rm H}.\end{aligned}$$ The corresponding unconditional marginal distributions $p({\bf T})$ and $p({\bf H})$ are multidimensional Gaussians, expressed using Doroshkevich’s formulas as $$p({\bf T}) = {15^3 \over 16 \sqrt{5} \pi^3} {1 \over \sigma^6_{\rm T} } {\rm exp} \Big [ {- \rm {3 \over 2 \sigma^2_{\rm T}} (2 k_1^2 - 5 k_2)} \Big ],~~ p({\bf H}) = {15^3 \over 16 \sqrt{5} \pi^3} {1 \over \sigma_{\rm H}^6} {\rm exp} \Big [{- \rm {3 \over 2 \sigma^2_{\rm H}} (2 h_1^2 - 5 h_2)} \Big ]. \label{doro_standard}$$ Equation (\[doro\_inter\_extended\]) generalizes Doroshkevich’s formulas (\[doro\_standard\]) to include the fact that halos/voids may correspond to maxima/minima of the density field. Note in particular that $p({\bf T}|{\bf H}, \gamma)$ is a multivariate Gaussian distribution with mean $b = \eta {\bf H}$ and covariance matrix $(1-\gamma^2) \sigma^2_{\rm T} {\bf \sf A}/15$, with ${\bf \sf A}$ given in Appendix \[notation\]. Clearly, one can also consider the reverse distribution $p ({\bf H}|{\bf T}, \gamma)$, the expression of which is given in Rossi (2012). This is useful in order to make the connection and extend some results derived in Bardeen et al. (1986), the subject of a forthcoming publication. It is also possible to express (\[doro\_inter\_extended\]) in terms of the constrained eigenvalues of ${\bf T}|{\bf H}$, indicated with $\zeta_{\rm i}$ ($i=1,2,3$) and ordered as $\zeta_1 \ge \zeta_2 \ge \zeta_3$. The result is: $$\label{doro_eigen} p(\zeta_1, \zeta_2, \zeta_3|\gamma) = {15^3 \over 8 \sqrt {5} \pi} {1 \over \sigma^6_{\rm T} (1-\gamma^2)^3} {\rm exp} \Big [-{3 \over 2 \sigma^2_{\rm T} (1-\gamma^2)} (2 K_1^2 - 5 K_2 ) \Big ] (\zeta_1-\zeta_2) (\zeta_1-\zeta_3) (\zeta_2-\zeta_3)$$ where in terms of constrained eigenvalues we now have: $$\begin{aligned} \label{eigen_doro_ext} K_1 &=& \zeta_1 + \zeta_2 +\zeta_3 = k_1 - \eta h_1 \\ K_2 &=& \zeta_1 \zeta_2 + \zeta_1 \zeta_3 + \zeta_2 \zeta_3 = k_2 + \eta^2 h_2 - \eta h_1 k_1 + \gamma \tau \\ \tau &=& \lambda_1 \xi_1 + \lambda_2 \xi_2 + \lambda_3 \xi_3 \\ k_1 &=& \lambda_1 + \lambda_2 + \lambda_3 \\ k_2 &=& \lambda_1 \lambda_2 + \lambda_1 \lambda_3 + \lambda_2 \lambda_3 \\ h_1 &=& \xi_1 + \xi_2 + \xi_3 \\ h_2 &=& \xi_1 \xi_2 +\xi_1 \xi_3 + \xi_2 \xi_3\\ \zeta_{\rm i} &=& \lambda_{\rm i} - \eta \xi_{\rm i}.\end{aligned}$$ The partial distributions $p(\lambda_1,\lambda_2,\lambda_3)$ and $p(\xi_1,\xi_2,\xi_3)$ are expressed by Doroshkevich’s unconditional formulas as $$p(\lambda_1,\lambda_2,\lambda_3) = {15^3 \over 8 \sqrt{5} \pi}{1 \over \sigma^6_{\rm T}} {\rm exp} \Big [{\rm - {3 \over 2 \sigma^2_{\rm T}} (2 k_1^2 - 5 k_2)} \Big ] (\lambda_1-\lambda_2) (\lambda_1-\lambda_3) (\lambda_2-\lambda_3) \label{doro_original}$$ $$p(\xi_1,\xi_2,\xi_3) = {15^3 \over 8 \sqrt{5} \pi} {1 \over \sigma^6_{\rm H}} {\rm exp} \Big [{\rm - {3 \over 2 \sigma^2_{\rm H}} (2 h_1^2 - 5 h_2)} \Big ] (\xi_1-\xi_2) (\xi_1-\xi_3) (\xi_2-\xi_3), \label{doro_original_bis}$$ where $\lambda_1, \lambda_2, \lambda_3$ are the eigenvalues of the shear tensor, while $\xi_1, \xi_2, \xi_3$ are those of the density Hessian. Similarly, one can easily obtain formulas for the reverse probability functions. Note also that $k_1=\lambda_1+ \lambda_2+\lambda_3$ is simply the overdensity $\delta_{\rm T}$ associated to the shear field, while $\xi_1 + \xi_2 + \xi_3 = h_1 \equiv \delta_{\rm H}$. Later on, we will make use of the peak height $\nu = \delta_{\rm T}/ \sigma_{\rm T}$ and of the peak curvature $x = \delta_{\rm H}/ \sigma_{\rm H}$. Equations (\[doro\_inter\_extended\]) and (\[doro\_eigen\]) allow one to develop a new algorithm to sample the constrained eigenvalues of the initial shear field, presented in the next section. It will be then straightforward to use this algorithm in order to implement an excursion set model for peaks and dips in Gaussian random fields. The peak/dip excursion-set-based algorithm {#pdesa} ========================================== While the constrained eigenvalues of the initial shear field can be sampled directly from their probability distribution function (i.e. Equations \[doro\_inter\_extended\] or \[doro\_eigen\]), the new theoretical formalism developed by Rossi (2012) leads to a much simpler algorithm, which is particularly interesting because well-suited for the ellipsoidal collapse model (Section \[epd\]), and naturally integrable in the excursion set framework (Section \[esmpd\]). In addition, the algebra leading to the new computational procedure allows one to explain analytically how the halo (void) shape distributions are altered by the inclusion of the peak (dip) constraint, and offers a variety of applications and insights on the statistical description of the cosmic web (see Section \[conc\] and Appendix \[insights\]). In what follows, we present the mathematical aspects of the ‘peak/dip excursion-set-based’ algorithm, and describe in detail the novel computational procedure. In particular, we are mainly interested in the distribution $p({\bf T}|{\bf H}>0,\gamma)$, the joint probability of observing a tidal field ${\bf T}$ for the gravitational potential with a positive density curvature ${\bf H}>0$ (i.e. at density peak locations). Clearly, $$\label{peak_dist} p({\bf T}| {\bf H}>0, \gamma) = { p({\bf T}, {\bf H}>0| \gamma) \over p({\bf H}>0)} = { \int_{{\bf H}>0} p({\bf H}) ~ p({\bf T}|{\bf H}, \gamma) ~{\rm d}{\bf H} \over \int_{{\bf H}>0} p({\bf H})~{\rm d}{\bf H}},$$ where $p({\bf T}|{\bf H}, \gamma)$ and $p({\bf H})$ are given by Equations (\[doro\_inter\_extended\]) and (\[doro\_standard\]), and the integrals are 6-dimensional. A similar expression can be written in terms of the constrained distributions of eigenvalues, using Equation (\[doro\_eigen\]) instead. In principle, one should then compute the previous integral to obtain $p({\bf T}| {\bf H}>0, \gamma)$. However, it is easier to sample $p({\bf T}|{\bf H}, \gamma)$ and impose the condition ${\bf H}>0$ directly, so that we are effectively computing $p({\bf T}|{\bf H}>0, \gamma)$. Given the nature of $p({\bf T}|{\bf H}, \gamma)$ and $p({\bf H})$ expressed by (\[doro\_inter\_extended\]) and (\[doro\_standard\]), this can be readily achieved – as the following algebra will show. In fact, because the elements of the density Hessian are drawn from a multivariate Gaussian distribution with zero mean and covariance matrix $\sigma^2_{\rm H}{\bf \sf A}/15$, where ${\bf \sf A}$ is given by Equation (\[matrix\_A\]), one can simply obtain them by generating six independent zero-mean unit-variance Gaussian random variates $y_{\rm i}$ ($i=1,6$), and then determine the various components as: $$\begin{aligned} \label{H_gauss} H_{11}&= & {\sigma_{\rm H} \over 3} \Big ( -y_1 +{2 \over \sqrt{5}} y_2 \Big ) \nonumber \\ H_{22}&= & {\sigma_{\rm H} \over 3} \Big ( - y_1 - {1 \over \sqrt{5}} y_2 - {3 \over \sqrt{15} } y_3 \Big ) \nonumber \\ H_{33}&= & {\sigma_{\rm H} \over 3} \Big ( - y_1 - {1 \over \sqrt{5}} y_2 + {3 \over \sqrt{15} } y_3 \Big ) \nonumber \\ H_{12}& = & H_{21} = {\sigma_{\rm H} \over \sqrt{15}} y_4 \nonumber \\ H_{13}&= & H_{31} = {\sigma_{\rm H} \over \sqrt{15}} y_5 \nonumber \\ H_{23}&= & H_{32} = {\sigma_{\rm H} \over \sqrt{15}} y_6. \end{aligned}$$ Similarly, the elements of the shear tensor are also drawn from a multivariate Gaussian distribution with mean zero and covariance matrix $\sigma^2_{\rm T}{\bf \sf A}/15$. Hence, if $z_{\rm i}$ ($i=1,6$) are other six independent zero-mean unit-variance Gaussian random variates, the components of the shear tensor are given by: $$\begin{aligned} \label{T_gauss} T_{11}&= & {\sigma_{\rm T} \over 3} \Big ( -z_1 +{2 \over \sqrt{5}} z_2 \Big ) \nonumber \\ T_{22}&= & {\sigma_{\rm T} \over 3} \Big ( - z_1 - {1 \over \sqrt{5}} z_2 - {3 \over \sqrt{15} } z_3 \Big ) \nonumber \\ T_{33}&= & {\sigma_{\rm T} \over 3} \Big ( - z_1 - {1 \over \sqrt{5}} z_2 + {3 \over \sqrt{15} } z_3 \Big ) \nonumber \\ T_{12}&= & T_{21} = {\sigma_{\rm T} \over \sqrt{15}} z_4 \nonumber \\ T_{13}&= & T_{31} = {\sigma_{\rm T} \over \sqrt{15}} z_5 \nonumber \\ T_{23}&= & T_{32} = {\sigma_{\rm T} \over \sqrt{15}} z_6. \end{aligned}$$ Next, consider the 6-dimensional vector ${\bf T}|{\bf H}$, made from the components of the conditional shear field. Its elements are drawn from a multivariate Gaussian distribution $p({\bf T}|{\bf H}, \gamma)$ with mean $b = \eta {\bf H}$ and covariance matrix $(1-\gamma^2) \sigma^2_{\rm T} {\bf \sf A}/15$, where the elements of the density Hessian are expressed by (\[H\_gauss\]). Therefore, this implies that the components of ${\bf T}|{\bf H}$ are distributed according to: $$\begin{aligned} \label{T_given_H_gauss} T_{11}|H_{11} &= & \eta H_{11} + {\sigma_{\rm T}\sqrt{1 - \gamma^2} \over 3} \Big ( -l_1 +{2 \over \sqrt{5}} l_2 \Big ) \equiv {\sigma_{\rm T} \over 3} \Big ( -m_1 +{2 \over \sqrt{5}} m_2 \Big ) \nonumber \\ T_{22}|H_{22} &= & \eta H_{22} +{\sigma_{\rm T} \sqrt{1-\gamma^2} \over 3} \Big ( - l_1 - {1 \over \sqrt{5}} l_2 - {3 \over \sqrt{15} } l_3 \Big ) \equiv {\sigma_{\rm T} \over 3} \Big ( - m_1 - {1 \over \sqrt{5}} m_2 - {3 \over \sqrt{15} } m_3 \Big ) \nonumber \\ T_{33}|H_{33} &= & \eta H_{33} + {\sigma_{\rm T} \sqrt{1-\gamma^2} \over 3} \Big ( - l_1 - {1 \over \sqrt{5}} l_2 + {3 \over \sqrt{15} } l_3 \Big ) \equiv {\sigma_{\rm T} \over 3} \Big ( - m_1 - {1 \over \sqrt{5}} m_2 + {3 \over \sqrt{15} } m_3 \Big ) \nonumber \\ T_{12}|H_{12} &= & T_{21}|H_{21} = \eta H_{12} +{\sigma_{\rm T} \sqrt{1-\gamma^2} \over \sqrt{15}} l_4 \equiv {\sigma_{\rm T} \over \sqrt{15}} m_4 \nonumber \\ T_{13}|H_{13} &= & T_{31}|H_{31} = \eta H_{13} +{\sigma_{\rm T} \sqrt{1-\gamma^2} \over \sqrt{15}} l_5 \equiv {\sigma_{\rm T} \over \sqrt{15}} m_5 \nonumber \\ T_{23}|H_{23} &= & T_{32}|H_{32} = \eta H_{23} + {\sigma_{\rm T} \sqrt{1-\gamma^2} \over \sqrt{15}} l_6 \equiv {\sigma_{\rm T} \over \sqrt{15}} m_6.\end{aligned}$$ In the previous expression, the various $l_{\rm i}$ ($i=1,6$) are other six independent zero-mean unit-variance Gaussian variates, while the $m_{\rm i}$ ($i=1,6$) are six independent Gaussian distributed variates with shifted mean $\gamma y_{\rm i}$ and reduced variance $(1-\gamma^2)$, i.e. $m_{\rm i} = \gamma y_{\rm i} + \sqrt{1-\gamma^2} ~ l_{\rm i} $. Equations (\[H\_gauss\]), (\[T\_gauss\]) and (\[T\_given\_H\_gauss\]) suggest a new excursion-set-based algorithm, in order to obtain the constrained eigenvalues of the matrix having components $T_{\rm \alpha}|H_{\rm \alpha}$ (see Appendix \[notation\] for the definition of the index $\alpha$) – supplemented by the condition of a positive curvature ${\bf H}$ for the density field (or negative curvature, for voids). The procedure can be summarized as follows: 1. Draw six independent zero-mean unit-variance Gaussian distributed variates $y_{\rm i}$ ($i=1,6$), and determine the components $H_{\rm \alpha}$ of the density Hessian matrix via Equation (\[H\_gauss\]). Compute the value of $\sigma_{\rm H}$ using (\[eq\_sig\]). 2. Calculate the eigenvalues $\xi_1, \xi_2, \xi_3$ of the previous Hessian matrix, and check if they are all positive (negative). If so, proceed to the next step, otherwise try again. This will guarantee the Hessian to be positive (negative) definite (i.e. the Hessian is a real-symmetric matrix), which is the condition for maxima (minima) of the field. Note that this step is clearly not required if we want to sample only $p({\bf T}|{\bf H}, \gamma)$. 3. Draw other six independent Gaussian distributed variates $l_{\rm i}$ ($i=1,6$), with mean zero and variance one, and determine the $T_{\rm \alpha}|H_{\rm \alpha}$ components via Equation (\[T\_given\_H\_gauss\]), using the previously accepted values $H_{\rm \alpha}$ from (\[H\_gauss\]) – while $\sigma_{\rm T}$ is determined via (\[eq\_sig\]). Since we require the density Hessian to be positive definite, this means that we are effectively sampling the conditional probability $p({\bf T}|{\bf H}>0, \gamma)$ – or $p({\bf T}|{\bf H}<0, \gamma)$ for a negative definite Hessian. 4. Calculate and store the constrained eigenvalues $\zeta_{\rm i}$ of the matrix having components $T_{\rm \alpha}|H_{\rm \alpha}>0$. We leave in Appendix \[insights\] some more insights on the main conditional formula (\[doro\_inter\_extended\]), which readily follow from the previous Equations (\[H\_gauss\]), (\[T\_gauss\]) and (\[T\_given\_H\_gauss\]). In the next sections, we will show a few applications of the new algorithm – with particular emphasis on the conditional distributions of shape parameters in triaxial models of structure formation (i.e. ellipticity and prolateness). Additional applications will be presented in forthcoming publications. Conditional distributions and probabilities: numerical tests {#cdpnt} ============================================================ The algorithm illustrated in the previous section allows one to test and confirm several formulas derived by Rossi (2012), regarding conditional distributions and probabilities subjected to the extremum constraint. We present here results of the comparison (i.e. theory versus numerical), where for simplicity we set $\sigma_{\rm T} = \sigma_{\rm H} \equiv 1$; this corresponds to adopt ‘reduced variables’, as done in Rossi (2012). Note that we show explicitly the dependencies on $\sigma$ values in the following expressions, although we set them to unity in the mock tests. Individual conditional distributions ------------------------------------ The individual conditional distributions of shear eigenvalues, for a given correlation strength $\gamma$ with the density field (which encapsulate the peak/dip constraint), are given by (Rossi 2012): $$\begin{aligned} \label{zeta1} p(\zeta_1|\gamma) &=& {\sqrt{5} \over 12 \pi \sigma_{\rm T}} \Big [ {20 \over (1-\gamma^2)} {\zeta_1 \over \sigma_{\rm T}} ~{\rm exp} \Big (- {9 \over 2 (1-\gamma^2) } {\zeta_1^2 \over \sigma^2_{\rm T} } \Big ) - {\sqrt{2 \pi} \over (1-\gamma^2)^{3/2}}~{\rm exp} \Big(- {5 \over 2 (1-\gamma^2)} {\zeta_1^2 \over \sigma^2_{\rm T}} \Big ) \times \\ && {\rm erfc} \Big (-{\sqrt{2} \over \sqrt{1-\gamma^2}} {\zeta_1 \over \sigma_{\rm T}} \Big ) \Big [ (1-\gamma^2)-20 {\zeta_1^2 \over \sigma^2_{\rm T}} \Big ] + {3 \sqrt{3 \pi} \over \sqrt{1-\gamma^2}}~ {\rm exp} \Big (- {15 \over 4 (1-\gamma^2)}{\zeta^2_1 \over \sigma^2_{\rm T}} \Big ) {\rm erfc} \Big ( -{\sqrt{3} \over 2 \sqrt{1-\gamma^2}}{\zeta_1 \over \sigma_{\rm T}} \Big ) \Big ] \nonumber \end{aligned}$$ $$\begin{aligned} \label{zeta2} p(\zeta_2|\gamma) &=& {\sqrt{15} \over 2 \sqrt{\pi}} {1 \over \sigma_{\rm T}\sqrt{1-\gamma^2}} {\rm exp} \Big [- {15 \over 4 (1 -\gamma^2)} {\zeta_2^2 \over \sigma^2_{\rm T} } \Big ] \end{aligned}$$ $$\begin{aligned} \label{zeta3} p(\zeta_3|\gamma) &=& -{\sqrt{5} \over 12 \pi \sigma_{\rm T}} \Big [ {20 \over (1-\gamma^2)} {\zeta_3 \over \sigma_{\rm T}} ~{\rm exp} \Big (- {9 \over 2 (1-\gamma^2) } {\zeta_3^2 \over \sigma^2_{\rm T} } \Big ) + {\sqrt{2 \pi} \over (1-\gamma^2)^{3/2}}~{\rm exp} \Big(- {5 \over 2 (1-\gamma^2)} {\zeta_3^2 \over \sigma^2_{\rm T}} \Big ) \times \\ && {\rm erfc} \Big ({\sqrt{2} \over \sqrt{1-\gamma^2}} {\zeta_3 \over \sigma_{\rm T}} \Big ) \Big [(1-\gamma^2)-20 {\zeta_3^2 \over \sigma^2_{\rm T}} \Big ] - {3 \sqrt{3 \pi} \over \sqrt{1-\gamma^2}}~ {\rm exp} \Big (- {15 \over 4 (1-\gamma^2)}{\zeta^2_3 \over \sigma^2_{\rm T}} \Big ) {\rm erfc} \Big ( {\sqrt{3} \over 2 \sqrt{1-\gamma^2}}{\zeta_3 \over \sigma_{\rm T}} \Big ) \Big ]. \nonumber \end{aligned}$$ The previous formulas show explicitly that the conditional distributions of shear eigenvalues are Doroshkevich-like expressions, with shifted mean and reduced variance. Different panels in Figure \[fig\_p\_eigen\] display plots of these distributions, contrasted with results from the algorithm presented in Section \[pdesa\], where 500,000 mock realizations are considered. In particular, note the symmetry between $p(\zeta_1|\gamma)$ and $p(\zeta_3|\gamma)$. We expect the mean values and variances of those distributions to be: $$\begin{aligned} \langle \zeta_1|\gamma \rangle &=& {3 \over \sqrt{10 \pi}}\sigma_{\rm T} \sqrt{1-\gamma^2},~~~~ \sigma^2_{\rm \zeta_1|\gamma} = {13 \pi -27 \over 30 \pi} \sigma^2_{\rm T}(1-\gamma^2)\\ \langle \zeta_2|\gamma \rangle &=& 0,~~~~~~~~~~ \sigma^2_{\rm \zeta_2|\gamma} = {2 \over 15} \sigma^2_{\rm T} (1-\gamma^2)\\ \langle \zeta_3|\gamma \rangle &\equiv& - \langle \zeta_1|\gamma\rangle = -{3 \over \sqrt{10 \pi}} \sigma_{\rm T} \sqrt{1-\gamma^2},~~ \sigma^2_{\rm \zeta_3|\gamma} \equiv \sigma^2_{\rm \zeta_1|\gamma} = {13 \pi -27 \over 30 \pi} \sigma^2_{\rm T}(1-\gamma^2).\end{aligned}$$ In the various panels, both theoretical and numerical expectations are reported and found to be in excellent agreement. For example, when $\gamma =0.50$, $\langle \zeta_1|\gamma \rangle_{\rm num}=0.4644$ while the expected theoretical value is $\langle \zeta_1|\gamma \rangle_{\rm th}=0.4635$, and for the variances $\sigma^{2, \rm num}_{\zeta_1|\gamma} = 0.1102$ where the theoretical expectation is $\sigma^{2, \rm th}_{\zeta_1|\gamma} = 0.1101$. Obviously, to see explicitly the effect of the peak constraint, one needs to make cuts in $\xi_{\rm i}$ from the partial conditional distributions $\lambda_{\rm i}|\xi_{\rm i}$, since $\zeta_{\rm i} = \lambda_{\rm i} - \eta \xi_{\rm i} $. In particular, expressions for $p(\lambda_{\rm i}|{\bf H}>0, \gamma)$ can be derived from (\[zeta1\]–\[zeta3\]) via substitution of variables and integration over ${\bf H}>0$. For example, it is straightforward to obtain an analytic formula for $p(\lambda_3|{\bf H}>0, \gamma)$. In Section \[epd\], we will return on these issues in more detail, in connection with the conditional ellipticity and prolateness for dark matter halos. ![Individual conditional probabilities $p(\zeta_1|\gamma)$ \[top\], $p(\zeta_2|\gamma)$ \[middle\], and $p(\zeta_3|\gamma)$ \[bottom\] of the initial shear field in the peak/dip picture. Solid curves are Equations (\[zeta1\]), (\[zeta2\]), (\[zeta3\]), while histograms are obtained from 500,000 mock realizations via the algorithm described in Section \[pdesa\]. Each panel displays the numerical and theoretical expectations for the mean values and variances of these distributions; their agreement is excellent.[]{data-label="fig_p_eigen"}](conditional_zeta_1_tris.ps "fig:"){width="100.00000%"} ![Individual conditional probabilities $p(\zeta_1|\gamma)$ \[top\], $p(\zeta_2|\gamma)$ \[middle\], and $p(\zeta_3|\gamma)$ \[bottom\] of the initial shear field in the peak/dip picture. Solid curves are Equations (\[zeta1\]), (\[zeta2\]), (\[zeta3\]), while histograms are obtained from 500,000 mock realizations via the algorithm described in Section \[pdesa\]. Each panel displays the numerical and theoretical expectations for the mean values and variances of these distributions; their agreement is excellent.[]{data-label="fig_p_eigen"}](conditional_zeta_2_tris.ps "fig:"){width="100.00000%"} ![Individual conditional probabilities $p(\zeta_1|\gamma)$ \[top\], $p(\zeta_2|\gamma)$ \[middle\], and $p(\zeta_3|\gamma)$ \[bottom\] of the initial shear field in the peak/dip picture. Solid curves are Equations (\[zeta1\]), (\[zeta2\]), (\[zeta3\]), while histograms are obtained from 500,000 mock realizations via the algorithm described in Section \[pdesa\]. Each panel displays the numerical and theoretical expectations for the mean values and variances of these distributions; their agreement is excellent.[]{data-label="fig_p_eigen"}](conditional_zeta_3_tris.ps "fig:"){width="100.00000%"} Another important distribution is $p(\Delta_{\rm T|H}|\zeta_3>0, \gamma)$, with $\Delta_{\rm T|H} \equiv K_1 = \zeta_1 + \zeta_2 + \zeta_3$ being the sum of the constrained eigenvalues, namely the probability distribution of $\Delta_{\rm T|H}$ confined in the regions with $\zeta_3 > 0$ (i.e. all positive eigenvalues): $$\begin{aligned} \label{K1_positive_eq} p(\Delta_{\rm T|H} |\zeta_3 >0, \gamma) &=& -{75 \sqrt{5} \over 8 \pi \sigma^2_{\rm T} } {\Delta_{\rm T|H} \over (1-\gamma^2)}~{\rm exp} \Big( - {9 \over 8} {\Delta_{\rm T|H}^2 \over \sigma^2_{\rm T}(1-\gamma^2)} \Big) + \\ &+& {25 \over 4 \sqrt{2 \pi} } {1 \over \sigma_{\rm T}\sqrt{1-\gamma^2}}~{\rm exp} \Big ( - {\Delta_{\rm T|H}^2 \over 2\sigma^2_{\rm T}(1-\gamma^2)} \Big ) \Big[{\rm erf}\Big ({\sqrt{10} \Delta_{\rm T|H} \over 4 \sigma_{\rm T}\sqrt{1-\gamma^2}} \Big )+ {\rm erf}\Big ({\sqrt{10} \Delta_{\rm T|H} \over 2 \sigma_{\rm T} \sqrt{1-\gamma^2}} \Big ) \Big]. \nonumber\end{aligned}$$ This distribution is expected to have mean value $$\langle \Delta_{\rm T|H} |\zeta_3 > 0, \gamma \rangle = {25 \sqrt{10} \over 144 \sqrt{\pi}} (3 \sqrt{6} -2)~\sigma_{\rm T} \sqrt{1-\gamma^2}\simeq 1.66~\sigma_{\rm T}\sqrt{1-\gamma^2}. \label{K1_positive_ave}$$ It is also easy to see that the maximum of $p(\Delta_{\rm T|H}|\zeta_3>0, \gamma)$ is reached when $\Delta_{\rm T|H} \simeq 1.5~\sigma_{\rm T}\sqrt{1-\gamma^2}$, and with a more elaborate calculation its variance can be estimated analytically. The result is: $$\sigma^2_{\Delta_{\rm T|H}|\zeta_3>0, \gamma}= {25 \over 4 \pi} \sigma^2_{\rm T} (1-\gamma^2)\Big[ {\rm arctan}(\sqrt{5}/2) +{\rm arctan}(\sqrt{5}) -{11 \sqrt{5} \over 54} -{125 \over 2592} (3 \sqrt{6}-2 )^2 \Big ] \simeq 0.31~\sigma^2_{\rm T} (1-\gamma^2). \label{K1_positive_sig}$$ Figure \[fig\_p\_K1\_positive\] confirms the previous relations, by contrasting Equations (\[K1\_positive\_eq\]), (\[K1\_positive\_ave\]) and (\[K1\_positive\_sig\]) with numerical results from 500,000 realizations from the algorithm presented in Section \[pdesa\]. We find good agreement, and recover correctly the expected mean values and variances of the distributions. For example, when $\gamma=0.25$, we find $\langle \Delta_{\rm T|H}|\zeta_3>0, \gamma \rangle_{\rm num}=1.6090$ while the expected theoretical value is $\langle \Delta_{\rm T|H}|\zeta_3>0, \gamma \rangle_{\rm th}=1.6040$, and for the rms values we find $\sigma^{\rm num}_{\Delta_{\rm T|H}|\zeta_3>0, \gamma} = 0.5314$ where the expected theoretical value is $\sigma^{\rm th}_{\Delta_{\rm T|H}|\zeta_3>0, \gamma} = 0.5399$. In the absence of correlations between the potential and density fields (i.e. when $\gamma=0$), all the previous expressions reduce consistently to the unconditional limit of Lee & Shandarin (1998). ![Conditional probability distribution $p(\Delta_{\rm T|H}|\zeta_3 >0, \gamma)$. Solid lines are obtained from Equation (\[K1\_positive\_eq\]), for different values of the correlation parameter $\gamma$, while histograms are drawn from 500,000 realizations via the algorithm described in Section \[pdesa\]. Note the excellent agreement between numerical results and theoretical predictions for the corresponding mean and rms values, reported in the various panels. In addition, solid vertical lines show the expected maximum value of each distribution, while dotted lines represent their corresponding mean value.[]{data-label="fig_p_K1_positive"}](conditional_Delta_zeta3_positive_bis.ps){width="100.00000%"} Distribution of peak heights ---------------------------- An important quantity which plays a major role in peaks theory (Bardeen et al. 1986) is the distribution of peak heights, namely $p(\nu |{\bf H}>0, \gamma)$, where $\nu = \delta_{\rm T}/\sigma_{\rm T}$ and the overdensity $\delta_{\rm T}$ is the trace of the shear tensor – defined in Section \[jde\]. One can easily obtain a simplified analytic expression for this distribution as follows. First, normalize Equation (\[eigen\_doro\_ext\]) by $\sigma_{\rm T}$, defining $N=\Delta_{\rm T|H} / \sigma_{\rm T}$ and $x= \delta_{\rm H}/ \sigma_{\rm H}$, so that $N = \nu - \gamma x$. In this framework, $x$ is the peak curvature of Bardeen et al. (1986) if Gaussian filters are used for computing the moments of the smoothed power spectrum (Equation \[eq\_sig\]). Since $p(N|\gamma)$ is simply a Gaussian with mean zero and variance $(1-\gamma^2)$ – see Rossi (2012) – and because of Equation (\[eigen\_doro\_ext\]), then clearly $p(\nu|x, \gamma)$ is also Gaussian, with mean $ \gamma x$ and reduced variance $(1-\gamma^2)$. Note in fact that $p(\nu,x|\gamma)$ is a bivariate Gaussian, because both $\nu$ and $x$ are normally distributed random variates. This implies that $\langle \nu|x, \gamma\rangle = \gamma x$, and clearly $\langle \nu|{\bf H}>0, \gamma\rangle = \gamma \langle x| \xi_3>0 \rangle \equiv \gamma \langle \nu | \lambda_3>0 \rangle$ (Bardeen et al. 1986) – where $\langle \nu | \lambda_3>0 \rangle = 1.6566$ according to Equation (21) of Lee & Shandarin (1998). Starting from a bivariate Gaussian with the previous expected mean value, it is then direct to derive a fairly good approximation for the distribution of peak heights, namely $$\label{eq_peak_heights_proxy} p(\nu|{\bf H}>0, \gamma) = {1 \over \sqrt{2 \pi}}~ {\rm exp} \Big [ { \-(\nu - \chi)^2 \over 2} \Big ] \Big [ 1 + {\rm erf} \Big ( { \gamma (\nu-\chi) \over \sqrt{2 (1-\gamma^2)}} \Big ) \Big ]$$ where $$\chi = { \sqrt{2} \over \sqrt{\pi}} \Big [ {25 \sqrt{5} \over 144} (3 \sqrt{6} -2) -1 \Big ] \gamma \simeq 0.86 \gamma.$$ In essence, $p(\nu|{\bf H}>0, \gamma)$ is a Gaussian with shifted mean modulated by the role of the peak curvature, and it consistently reduces to a zero-mean unit-variance Gaussian distribution when $\gamma=0$ – as expected. Additionally, the previous distribution can be shown to be equivalent to $p(\nu) [\int_0^{\infty} p(\xi_3|\gamma, \nu) {\rm d} \xi_3/p(\xi_3>0)]$, where the term in square brackets quantifies the effect of the peak constrain – and so the scale at which the difference between peaks and random positions would appear. Figure \[fig\_peak\_heights\] shows plots of $p(\nu|{\bf H}>0, \gamma)$ for different values of $\gamma$. In the various panels, vertical lines are the expected mean values; we find good agreement between numerical estimates and theoretical expectations. For example, when $\gamma=0.25$, $\langle \nu|{\bf H}>0, \gamma \rangle_{\rm num} = 0.4146$ while $\langle \nu|{\bf H}>0, \gamma \rangle_{\rm th} = 0.4142$. Solid curves in the figure are drawn from Equation (\[eq\_peak\_heights\_proxy\]), an approximation which is particularly good for lower values of the correlation strength. ![Distribution of peak heights, $p(\nu|{\bf H}>0, \gamma$), for different values of the correlation parameter $\gamma$ – as specified in the various panels. The numerical values for the mean of the distribution as a function of $\gamma$ are indicated in each panel, and also marked with vertical lines; they are in good agreement with theoretical expectations. Solid curves are drawn from Equation (\[eq\_peak\_heights\_proxy\]).[]{data-label="fig_peak_heights"}](constrained_delta_H_positive_tris.ps){width="75.00000%"} Conditional ellipticity and prolateness {#epd} ======================================= The theoretical framework outlined in Section \[jde\], along with the algebra leading to the new computational procedure presented in Section \[pdesa\], provides a natural way to include the peak/dip constraint into the initial shape distributions of halos and voids. This is a key aspect for implementing an excursion set algorithm for peaks and dips in Gaussian random fields (which will be discussed in Section \[esmpd\]), and has direct applications in the context of triaxial models of structure formation. In particular, our description combines the formalism of Bardeen et al. (1986) – based on the density field – with that of Bond & Myers (1996) – focused on the shear field. In what follows, first we introduce some useful definitions regarding shape parameters (ellipticity and prolateness); we then provide a new expression for the joint conditional distribution of halo shapes given the density peak constraint, derived from Equation (\[doro\_eigen\]), which generalizes some previous literature work. The formula has important implications for the modeling of non-spherical dark matter halos, in relation to their initial shape distribution. Finally, we briefly discuss how the shear ellipticity and prolateness (and so the initial halo shapes) are modified by the inclusion of the density peak condition. Our analysis is based on the constrained eigenvalues of the initial shear field (Equations \[doro\_eigen\]–19), and we use the new algorithm to support our description. While here we only consider the case for dark matter halos, it is straightforward to deal with voids. In a forthcoming companion publication, we will present a more detailed study on the morphology of both halos and voids in the peak/dip picture, where we also investigate the modifications induced by primordial non-Gaussianity on their shapes (see also Section \[conc\]). Shape parameters: general definitions ------------------------------------- In triaxial models of halo formation, such as the ellipsoidal collapse (Icke 1973; White & Silk 1979; Barrow & Silk 198; Kuhlman et al. 1996; Bond & Myers 1996), it is customary to characterize the shape of a region by its ellipticity and prolateness. In particular, in the ‘peak-patch’ approach of Bond & Myers (1996), the shape parameters are associated with the eigenvalues of the shear field ($\lambda_{\rm i}$ in our notation). Considering the mapping ($\lambda_1, \lambda_2, \lambda_3 \rightarrow e_{\rm T}, p_{\rm T}, \delta_{\rm T}$), where the subscript $T$ refers to the tidal tensor, we can write: $$\delta_{\rm T} \equiv k_1 = \lambda_1 + \lambda_2 + \lambda_3, \hspace{1cm} e_{\rm T} = {\lambda_1 -\lambda_3 \over 2 \delta_{\rm T}}, \hspace{1cm} p_{\rm T} = {\lambda_1 + \lambda_3 - 2 \lambda_2 \over 2 \delta_{\rm T}} \equiv e_{\rm T} -{\lambda_2 -\lambda_3 \over \delta_{\rm T}}, \label{eq_dep_T}$$ where $e_{\rm T}$ and $p_{\rm T}$ are the ‘unconditional’ shear ellipticity and prolateness. In particular, if $\delta_{\rm T} > 0$ then $e_{\rm T} > 0$ and therefore $-e_{\rm T} \le p_{\rm T} \le e_{\rm T}$. In the ‘peaks theory’ of Bardeen et al. (1986), instead, ellipticity ($e_{\rm H}$) and prolateness ($p_{\rm H}$) are associated with the eigenvalues of the density field ($\xi_{\rm i}$ in our notation). Hence, given the mapping ($\xi_1, \xi_2, \xi_3 \rightarrow e_{\rm H}, p_{\rm H}, \delta_{\rm H}$) where now the subscript $H$ refers to the Hessian of the density field, one has: $$\delta_{\rm H} \equiv h_1 = \xi_1 + \xi_2 + \xi_3, \hspace{1cm} e_{\rm H} = {\xi_1 -\xi_3 \over 2 \delta_{\rm H}}, \hspace{1cm} p_{\rm H} = {\xi_1 + \xi_3 - 2 \xi_2 \over 2 \delta_{\rm H}} \equiv e_{\rm H} -{\xi_2 -\xi_3 \over \delta_{\rm H}}. \label{eq_dep_H}$$ As in the previous case, if $\delta_{\rm H} > 0$ then $e_{\rm H} > 0$ and $-e_{\rm H} \le p_{\rm H} \le e_{\rm H}$. It is therefore natural to consider the ‘conditional’ mapping ($\zeta_1, \zeta_2, \zeta_3 \rightarrow E_{\rm T|H}, P_{\rm T|H}, \Delta_{\rm T|H}$) and define: $$\Delta_{\rm T|H} \equiv K_1 = \zeta_1 +\zeta_2+\zeta_3, \hspace{1cm} E_{\rm T|H} = {\zeta_1 - \zeta_3 \over 2 \Delta_{\rm T|H}}, \hspace{1cm} P_{\rm T|H} = {\zeta_1 + \zeta_3 - 2 \zeta_2 \over 2 \Delta_{\rm T|H} } \equiv E_{\rm T|H} - {\zeta_2 -\zeta_3 \over \Delta_{\rm T|H}}, \label{eq_dep_T_given_H}$$ where $\zeta_{\rm i}$ are the constrained shear eigenvalues – given by Equation (19). In what follows, we will refer to $E_{\rm T|H}$ and $P_{\rm T|H}$ as to the conditional shear ellipticity and prolateness, respectively; we will also relate these quantities to the unconditional expressions previously introduced, involving ($e_{\rm T},p_{\rm T}$) and ($e_{\rm H},p_{\rm H}$). Joint conditional distribution of shear ellipticity and prolateness in the peak/dip picture ------------------------------------------------------------------------------------------- Combining (\[eq\_dep\_T\]) with Doroshkevich’s formula (\[doro\_original\]), it is straightforward to derive the ‘unconditional’ distribution of $e_{\rm T}$ and $p_{\rm T}$ given $\delta_{\rm T}$, $g(e_{\rm T},p_{\rm T}|\delta_{\rm T})$, related to the gravitational potential. In particular, $$g(\delta_{\rm T}, e_{\rm T}, p_{\rm T}) = p(\delta_{\rm T}) g( e_{\rm T}, p_{\rm T}| \delta_{\rm T}) \label{gep_uncond}$$ where $p(\delta_{\rm T})$ is simply a Gaussian with zero mean and variance $\sigma_{\rm T}^2$, and $$g(e_{\rm T}, p_{\rm T}| \delta_{\rm T}) = {15^3 \over 3 \sqrt{10 \pi} } \Big ( {\delta_{\rm T} \over \sigma_{\rm T} } \Big )^5 e_{\rm T} (e^2_{\rm T} - p^2_{\rm T}) ~{\rm exp} \Big[ -{5 \over 2} \Big({\delta_{\rm T} \over \sigma_{\rm T}} \Big)^2 (3 e^2_{\rm T} + p^2_{\rm T} )\Big]. \label{gep_uncond_bis}$$ Equation (\[gep\_uncond\]) implies that the unconditional joint distribution of shear ellipticity and prolateness is independent of that of the overdensity $\delta_{\rm T}$. Similarly, inserting (\[eq\_dep\_H\]) in Doroshkevich’s formula (\[doro\_original\_bis\]), one obtains $$g(\delta_{\rm H}, e_{\rm H}, p_{\rm H}) = p(\delta_{\rm H}) g( e_{\rm H}, p_{\rm H}| \delta_{\rm H})$$ for the quantities related to the density Hessian, where $p(\delta_{\rm H})$ is a Gaussian with zero mean and variance $\sigma_{\rm H}^2$, while $$g(e_{\rm H}, p_{\rm H}| \delta_{\rm H}) = {15^3 \over 3 \sqrt{10 \pi} } \Big ( {\delta_{\rm H} \over \sigma_{\rm H} } \Big )^5 e_{\rm H} (e^2_{\rm H} - p^2_{\rm H}) ~{\rm exp} \Big[ -{5 \over 2} \Big({\delta_{\rm H} \over \sigma_{\rm H}} \Big)^2 (3 e^2_{\rm H} + p^2_{\rm H} )\Big].$$ Along the same lines, combining (\[eq\_dep\_T\_given\_H\]) with (\[doro\_eigen\]) and using the definitions introduced in the previous section, it is direct to obtain $$G( \Delta_{\rm T|H}, E_{\rm T|H}, P_{\rm T|H}|\gamma) = p(\Delta_{\rm T|H}|\gamma) G(E_{\rm T|H},P_{\rm T|H}|\Delta_{\rm T|H}, \gamma) \label{eq_epc_base}$$ where $p(\Delta_{\rm T|H}|\gamma)$ is a Gaussian distribution with mean zero and variance $\sigma^2_{\rm T} (1 -\gamma^2)$ – i.e. Equation (57) in Rossi (2012) – and $$G(E_{\rm T|H}, P_{\rm T|H}| \Delta_{\rm T|H}, \gamma) = {15^3 \over 3 \sqrt{10 \pi}} \Big ( {\Delta_{\rm T|H} \over \sigma_{\rm T}} \Big )^5 {1 \over (1 - \gamma^2)^{5/2}} E_{\rm T|H} (E_{\rm T|H}^2 - P_{\rm T|H}^2) ~{\rm exp} \Big [ {5 \over 2 (1 -\gamma^2)} \Big ( {\Delta_{\rm T|H} \over \sigma_{\rm T}}\Big )^2 (3 E_{\rm T|H}^2 + P_{\rm T|H}^2) \Big ]. \label{eq_epc}$$ The previous expression (\[eq\_epc\]) is the joint conditional distribution of shear ellipticity and prolateness, given $\Delta_{\rm T|H}$ and a correlation strength $\gamma$ with the density field. Clearly, $G(E_{\rm T|H}, P_{\rm T|H}| \Delta_{\rm T|H}, \gamma)$ is also independent of the distribution of $(\Delta_{\rm T|H}|\gamma)$. Equation (\[eq\_epc\]) generalizes the standard joint distribution of halo shape parameters to include the peak constraint, and is another main result of this paper. Note that, in the absence of correlation (i.e. when $\gamma=0$), $E_{\rm T|H} \rightarrow e_{\rm T}$, $P_{\rm T|H} \rightarrow p_{\rm T}$, $\Delta_{\rm T|H} \rightarrow \delta_{\rm T}$, so that (\[eq\_epc\]) reduces consistently to the ‘unconditional’ Doroshkevich’s limit (\[gep\_uncond\_bis\]). From this joint conditional probability function, it is possible to derive the partial conditional distributions for the shear ellipticity and prolateness given the peak constrain (which we will discuss next), and their expressions at density peak locations, when the condition ${\bf H}>0$ is satisfied. ![\[Top\] Distribution of $(\zeta_1-\zeta_3|\gamma)$, for different values of the correlation strength $\gamma$ – as indicated in the panels. Its shape is directly related to the shear conditional ellipticity in the peak/dip picture – see Equation (\[eq\_ed\_cond\]). \[Bottom\] Distribution of $(\zeta_2-\zeta_3|\gamma)$, for different values of the correlation strength $\gamma$. This distribution is instead related to the shear conditional prolateness in the peak/dip picture – see Equation (\[eq\_pd\_cond\]). The numerical values for the mean and variance of the distributions, as a function of $\gamma$, are also indicated in the figures. In both cases, solid curves are obtained via numerical integrations starting from the joint conditional distribution of eigenvalues (\[doro\_eigen\]). Histograms are drawn from 500,000 realizations, using the algorithm described in Section \[pdesa\].[]{data-label="fig_cep"}](conditional_ellipticity_paper_A.ps "fig:"){width="100.00000%"} ![\[Top\] Distribution of $(\zeta_1-\zeta_3|\gamma)$, for different values of the correlation strength $\gamma$ – as indicated in the panels. Its shape is directly related to the shear conditional ellipticity in the peak/dip picture – see Equation (\[eq\_ed\_cond\]). \[Bottom\] Distribution of $(\zeta_2-\zeta_3|\gamma)$, for different values of the correlation strength $\gamma$. This distribution is instead related to the shear conditional prolateness in the peak/dip picture – see Equation (\[eq\_pd\_cond\]). The numerical values for the mean and variance of the distributions, as a function of $\gamma$, are also indicated in the figures. In both cases, solid curves are obtained via numerical integrations starting from the joint conditional distribution of eigenvalues (\[doro\_eigen\]). Histograms are drawn from 500,000 realizations, using the algorithm described in Section \[pdesa\].[]{data-label="fig_cep"}](conditional_prolateness_paper_A.ps "fig:"){width="100.00000%"} ![Distributions $p(\zeta_1-\zeta_3|\zeta_3>0,\gamma)$ \[top\] and $p(\zeta_2-\zeta_3|\zeta_3>0,\gamma)$ \[bottom\] for different values of the correlation strength $\gamma$. In both figures, histograms are drawn from 500,000 realizations using the algorithm described in Section \[pdesa\], while solid curves are obtained via numerical integrations of the joint conditional distribution of eigenvalues (\[doro\_eigen\]). In the various panels, we also provide the numerical values for the mean and variance of the distributions, as a function of $\gamma$. In particular, their mean values at $\gamma=0$ allow one to quantify the shift in the mean of the shear ellipticity and prolateness caused by the peak constraint.[]{data-label="fig_cep_pos"}](conditional_zetas13_zeta3_positive_paper.ps "fig:"){width="100.00000%"} ![Distributions $p(\zeta_1-\zeta_3|\zeta_3>0,\gamma)$ \[top\] and $p(\zeta_2-\zeta_3|\zeta_3>0,\gamma)$ \[bottom\] for different values of the correlation strength $\gamma$. In both figures, histograms are drawn from 500,000 realizations using the algorithm described in Section \[pdesa\], while solid curves are obtained via numerical integrations of the joint conditional distribution of eigenvalues (\[doro\_eigen\]). In the various panels, we also provide the numerical values for the mean and variance of the distributions, as a function of $\gamma$. In particular, their mean values at $\gamma=0$ allow one to quantify the shift in the mean of the shear ellipticity and prolateness caused by the peak constraint.[]{data-label="fig_cep_pos"}](conditional_zetas23_zeta3_positive_paper.ps "fig:"){width="100.00000%"} Initial distributions of triaxial halo shapes at density peak locations ----------------------------------------------------------------------- We can readily express the conditional distributions of shear ellipticity and prolateness in the peak/dip picture in terms of the unconditional quantities (\[eq\_dep\_T\]) and (\[eq\_dep\_H\]). This is done simply by recalling that the constrained shear eigenvalues are given by $\zeta_{\rm i} = \lambda_{\rm i} - \eta \xi_{\rm i}$, according to Equation (19). It is then direct to obtain: $$E_{\rm T|H} \Delta_{\rm T|H} = {(\zeta_1 - \zeta_3) \over 2} \equiv {(\lambda_1 -\lambda_3) \over 2} - \eta {(\xi_1 - \xi_3) \over 2} = \delta_{\rm T} e_{\rm T} - \eta \delta_{\rm H} e_{\rm H}. \label{eq_ed_cond}$$ The previous relation implies that, for a given $\Delta_{\rm T|H}$, the conditional shear ellipticity will have its mean value shifted by the presence of the peak constraint; the entity of the shift has to be ascribed to the additional factor $\eta \delta_{\rm H} e_{\rm H}$, which is given by the density ellipticity (essentially, the latter term quantifies the role of the peak curvature). The top panel of Figure \[fig\_cep\] shows the distribution of $(\zeta_1-\zeta_3|\gamma)$ for different values of the correlation strength $\gamma$, namely the combination of constrained shear eigenvalues which controls the conditional ellipticity – according to Equation (\[eq\_ed\_cond\]). Histograms are drawn from 500,000 realizations using the algorithm described in Section \[pdesa\], while solid curves are obtained via numerical integration – starting from the joint conditional distribution of eigenvalues (\[doro\_eigen\]). In the various panels, we also provide the numerical values for the mean and variance of the distribution, as a function of $\gamma$. Similarly, $$P_{\rm T|H} \Delta_{\rm T|H} = {(\zeta_1 + \zeta_3 -2 \zeta_2)\over 2} = {(\lambda_1 + \lambda_3 -2 \lambda_2)\over 2} -\eta {(\xi_1 + \xi_3 -2 \xi_2)\over 2} = \delta_{\rm T} p_{\rm T} - \eta \delta_{\rm H} p_{\rm H} \equiv E_{\rm T|H} \Delta_{\rm T|H} - (\zeta_2 -\zeta_3). \label{eq_pd_cond}$$ This expression shows that the conditional shear prolateness can be obtained from the conditional shear ellipticity (\[eq\_ed\_cond\]) and the combination ($\zeta_2 - \zeta_3$) of the constrained shear eigenvalues. The bottom panel of Figure \[fig\_cep\] displays the distribution of $(\zeta_2-\zeta_3|\gamma)$, for different values of the correlation strength $\gamma$. Again, solid curves are derived from a numerical integration of Equation (\[doro\_eigen\]). Of particular interest are also the distributions $g(e_{\rm T}|{\bf H}>0, \gamma)$ and $g(p_{\rm T}|{\bf H}>0, \gamma)$, which will provide the initial triaxial shapes of dark matter halos at peak locations. They are related to the previous expressions, and can be readily obtained within the outlined formalism; clearly, they will reduce to the standard (or unconditional) shear ellipticity and prolateness when $\gamma=0$, in the absence of correlation. We will present a dedicated study focused on the morphology of halos and voids in the peak/dip picture, where we characterize in detail those distributions, especially in relation to the ellipsoidal collapse model. We anticipate here that several analytic and insightful results on their shapes can be derived, starting from (\[eq\_epc\]) and using (\[doro\_eigen\]), (\[eq\_ed\_cond\]) and (\[eq\_pd\_cond\]). For example, according to (\[eq\_ed\_cond\]), when ${\bf H} >0$ (which is equivalent to impose the condition $\xi_3>0$ on an ordered set of density Hessian eigenvalues), then the quantity $\langle \xi_1 -\xi_3| \xi_3 >0 \rangle$ will essentially provide by how much the mean value of the shear ellipticity has been shifted by the peak constrain. For the prolateness the situation is slightly more complicated, but according to (\[eq\_pd\_cond\]) the shift can be derived by the additional knowledge of $\langle \xi_2 -\xi_3| \xi_3 >0 \rangle $. To this end, Figure \[fig\_cep\_pos\] shows the distributions of $(\zeta_1-\zeta_3|\zeta_3>0,\gamma)$ and $(\zeta_2-\zeta_3|\zeta_3>0,\gamma)$, for different values of the correlation strength $\gamma$ (note that the quantities just discussed are their mean values when $\gamma=0$). As in the previous plot, histograms are drawn from 500,000 realizations using the algorithm described in Section \[pdesa\], while solid curves are obtained via numerical integrations – starting from the joint conditional distribution of eigenvalues (\[doro\_eigen\]). In the various panels, we also provide the numerical values for the mean and variance of the distribution, as a function of $\gamma$. As mentioned before, we will return in more detail on these distributions in a companion publication, to quantify the impact of the peak constrain on their shapes. We will also make the connection with the work of Bardeen et al. (1986) more explicit – see their Appendix C. It will be also interesting to include the role of the peculiar gravity field in our description, along the lines of van de Weygaert & Bertschinger (1996), as well as to extend the work of Desjacques (2008) on the joint statistics of the shear tensor and on the dynamical aspect of the environmental dependence, within this formalism. Excursion set approach for peaks and dips in Gaussian random fields {#esmpd} =================================================================== The ability to distinguish between random positions and peaks/dips, contained in Equations (\[doro\_inter\_extended\]) or (\[doro\_eigen\]) and achieved by the algorithm presented in Section \[pdesa\], is the key to implement an excursion set model for peaks and dips in Gaussian random fields. Indeed, the primary motivation (and one of the main strengths) of the proposed sampling procedure resides in its direct inclusion into the excursion set framework (Epstein 1983; Peacock & Heavens 1990; Bond et al. 1991; Lacey & Cole 1993). In essence, because in our formalism the peak overdensity is simply the trace of the conditional shear tensor (recall the definitions in Section \[jde\]), and since in triaxial models of collapse the initial shape parameters are just combinations of the shear eigenvalues (Bond & Myers 1996), our prescription provides a direct way to generate the distribution of initial overdensities under the conditions that they are peaks/dips (i.e. the distribution of peak heights, when ${\bf H}>0$) – along with the corresponding conditional distribution of initial shapes (see Section \[epd\]). In this respect, it is then straightforward to include this part in standard excursion set algorithms, as those used for example in Chiueh & Lee (2001), Sheth & Tormen (2002) or Sandvik et al. (2007) to compute the mass function, or in Rossi, Sheth & Tormen (2011) to describe halo shapes; the ‘peak/dip excursion-set-based’ algorithm is also useful for computing the halo bias assuming triaxial models of structure formation (i.e. ellipsoidal collapse). The only main conceptual difference is the pre-selection of peak/dip locations, instead of random positions in the field. The mass scale of the peak will then be fixed by finding the proper $\sigma_{\rm T}$ which satisfies the combination of $(\delta_{\rm T}, e_{\rm T}, p_{\rm T}|{\bf H}>0, \gamma)$, assuming some structure formation models – as for example the ellipsoidal collapse. This is particularly useful because the excursion set theory is a powerful tool for understanding various aspects of the full dynamical complexity of halo formation. Perturbations are assumed to evolve stochastically with the smoothing scale, and the problem of computing the probability of halo formation is mapped into the classical first-passage time problem in the presence of a barrier. A very elegant reformulation of this theory has been recently proposed by Maggiore & Riotto (2010a,b,c), who made several technical and conceptual improvements (i.e. no ad hoc absorbing barrier boundary conditions, account for non-markovianity induced by filtering, unambiguous mass association to a smoothed scale, etc.) by deriving the original excursion set theory from a path integral formulation – following a microscopical approach. These authors also noted that the failure of the standard excursion set approach may be related to the inadequacy of the oversimplified physical model adopted for halo formation (either spherical or ellipsoidal), and propose to treat the critical threshold for collapse as a stochastic variable which better captures some of the dynamical complexity of the halo formation phenomenon. Even so, they find that the non-markovian contributions do not alleviate the discrepancy between excursion set predictions and $N$-body simulations. In addition to the problems pointed out by Maggiore & Riotto (2010a,b,c), there is also the fact that – in its standard formulation – the excursion set approach is unable to differentiate between peaks/dips and random locations in space – i.e. all points are treated equally. However, since local extrema are plausible sites for the formation of nonlinear structures and there is a good correspondence between peaks in the initial conditions and halos at late times, it may be important to differentiate between those special positions in space. The algorithm proposed here goes in this direction, since it allows one to pre-select those special points in space around which halos form (peaks), and not just random locations – and permits to associate their corresponding initial shape distribution: in essence, the computational procedure selects a special subset, among all the possible random walks considered in the standard excursion set procedure. Therefore, the ‘peak/dip excursion-based’ algorithm can be used to study the mass function of halos and their triaxial shapes at peak/dip positions, and also the halo bias. In fact, our prescription allows one to generate the initial distributions of overdensity, ellipticity and prolateness (i.e. shape parameters) at a scale set by the variance $\sigma_{\rm T}$, with the constraint that $\delta_{\rm T}$ is a peak (i.e. the condition ${\bf H}>0$ on the Hessian). One can then just evolve this initial conditional shape distribution by solving a dynamical equation of collapse, and study the final shape distribution (as done for example in Rossi, Sheth & Tormen 2011) but now at peak/dip locations. Alternatively, one can adopt a ‘crossing threshold’, since in the excursion set approach an halo is formed when the smoothed density perturbation reaches a critical value for the first time, and the problem is reduced to a first-passage problem in the presence of a barrier (i.e. if the overdensity exceeds a critical value, the random walk stops at this scale; if not, the walk continues to smaller scales). Moreover, our numerical technique can be easily integrated in Montecarlo realizations of the trajectories obtained from a Langevin equation with colored noise (i.e. Bond et al. 1991; Robertson et al. 2009) at peak/dip locations, and even for situations where the walks are correlated – in presence of non-markovian effects, along the lines of De Simone et al. (2011a,b). In this respect, our technique is general, since any kind of filter function can be readily implemented. All these lines of research are ongoing efforts, and will be presented in several forthcoming publications. Conclusions {#conc} =========== From the joint conditional probability distribution of an ordered set of shear eigenvalues in the peak/dip picture (Rossi 2012; Section \[jde\], Equations \[doro\_inter\_extended\] and \[doro\_eigen\]), we derived a new algorithm to sample the constrained eigenvalues of the initial shear field associated with Gaussian statistics at peak/dips positions in the correlated density field. The algorithm, described in Section \[pdesa\], was then used to test and confirm several formulas presented in Rossi (2012) regarding conditional distributions and probabilities, subjected to the extremum constraint (Section \[cdpnt\]). We found excellent agreement between numerical results and theoretical predictions (Figures \[fig\_p\_eigen\] and \[fig\_p\_K1\_positive\]). In addition, we also showed how the standard distributions of shear ellipticity and prolateness in triaxial models of structure formation are modified by the constraint (Section \[epd\]; Figures \[fig\_cep\] and \[fig\_cep\_pos\]), and provided a new expression for the conditional distribution of shape parameters given the density peak requirement (Equation \[eq\_epc\]), which generalizes some previous literature work. The formula has important implications for the modeling of non-spherical dark matter halo shapes, in relation to their initial shape distribution, and is directly applicable to the ellipsoidal collapse model (Icke 1973; White & Silk 1979; Barrow & Silk 1981; Kuhlman et al. 1996; Bond & Myers 1996). In particular, our novel description is able to combine consistently, for the first time, the formalism of Bardeen et al. (1986) – based on the density field – with that of Bond & Myers (1996) – based on the shear field. Along the way, we also discussed the distribution of peak heights (see Figure \[fig\_peak\_heights\]), which plays a major role in peaks theory (Bardeen et al. 1986). While the primary motivation of this paper was to illustrate the new ‘peak/dip excursion-set-based’ algorithm, and to show a few applications focused on the morphology of the cosmic web (following up, and complementing with some more insights, the theoretical work presented in Rossi 2012), the other goal was to describe how the new sampling procedure naturally integrates into the standard excursion set framework (Epstein 1983; Peacock & Heavens 1990; Bond et al. 1991; Lacey & Cole 1993) – potentially solving some of its well-known problems. In particular, in Section \[esmpd\] we argued that the ability to distinguish between random positions and peaks/dips, encoded in the algorithm and in Equations (\[doro\_inter\_extended\]) and (\[doro\_eigen\]) derived from first principles, is indeed the key to implement a generalized excursion set model for peaks and dips in Gaussian random fields. This is the actual strength of the proposed computational procedure, since part of the failure of the original excursion set theory may be attributed to its inability to differentiate between random positions and special points (peaks) in space around which halos may form. To this end, our simple prescription can be used to study the halo mass function, halo/void shapes and bias at peak/dip density locations, in conjunction with triaxial models of structure formation. All these research lines are ongoing efforts, subjects of several forthcoming publications. The essential part is the characterization of the distribution of peak heights, and of the initial shape distribution at peak/dip locations (Equations \[doro\_inter\_extended\], \[doro\_eigen\], and \[eq\_epc\]). The algorithm presented in this paper offers also a much broader spectrum of applications. This is because, as pointed out by Rossi (2012), the fact that the eigenvalues of the Hessian matrix can be used to discriminate between different types of structures in a particle distribution is fundamental to a number of structure-finding algorithms, shape-finders algorithms, and structure reconstruction on the basis of tessellations. For example, it can be used for studying the dynamics and morphology of cosmic voids – see for example van de Weygaert & Platen (2011), Bos et al. (2012), Pan et al. (2012), and the Monge-Ampère-Kantorovitch reconstruction procedure by Lavaux & Wandelt (2010) – and in several observationally-oriented applications, in particular for developing algorithms to find and classify structures in the cosmic web or in relation to its skeleton (Sahni et al. 1998; Schaap & van de Weygaert 2000; Novikov et al. 2006; Hahn et al. 2007; Romano-Diaz & van de Weygaert 2007; Forero-Romero et al. 2009; Zhang et al. 2009; Lee & Springel 2010; Arag[ó]{}n-Calvo et al. 2010a,b; Platen et al. 2011; Cautun et al. 2012; Hidding et al. 2012). Another application is related to the work of Bond, Strauss & Cen (2010), who presented an algorithm that uses the eigenvectors of the Hessian matrix of the smoothed galaxy distribution to identify individual filamentary structures. In addition, since galaxy clusters are related to primordial density peaks, and there is a correspondence between structures in the evolved density field and local properties of the linear tidal shear, our theoretical framework provides a direct way to relate initial conditions and observables from galaxy clusters. Other intriguing connections involve topological studies of the cosmic web, the genus statistics and Minkowski functionals (Gott et al. 1986, 1989; Park et al. 1991, 2005; Matsubara 2010), and the possibility to address open questions regarding the origin of angular momentum and halo spin within this framework; this is because the dependence of the spin alignment on the morphology of the large-scale mass distribution is due to the difference in shape of the tidal fields in different environments, and most of the halo properties depend significantly on environment, and in particular on the tidal field – i.e. the environmental dependence of halo assembly time and unbound substructure fraction has its origin from the tidal field (Wang et al. 2011). It will be also interesting to explore how the new formalism proposed here can be used to study halo spin, shape and the orbital angular momentum of subhaloes relative to the LSS, in the context of the eigenvectors of the velocity shear tensor (see the recent study by Libeskind et al. 2013). In addition, the more complex question of the local expected density field alignment/orientation distribution as a function of the local field value (Bond 1987; Lee & Pen 2002; Porciani et al. 2002; Lee, Hahn & Porciani 2009; Lee 2011) can be addressed within this framework, and is the subject of future studies. On the theoretical side, we note that our algorithm is restricted to one scale (i.e. peaks and dips in the density field, as in Bardeen et al. 1986), but the extension to a multiscale *peak-patch* approach along the lines of Bond & Myers (1996) is doable and subject of ongoing work. As argued in Rossi (2012), this will allow to account for the role of the peculiar gravity field itself, an important aspect not considered in our formalism but discussed in detail in van de Weygaert & Bertschinger (1996). Including all these effects in our framework and translating them into a more elaborated algorithm is ongoing effort, and will allow us to make the connection with the multiscale analysis of the Hessian matrix of the density field by van de Weygaert & Bertschinger (1996) and Arag[ó]{}n-Calvo et al. (2007; 2010a,b). It will also allow us to incorporate the distortion effect of the peculiar gravity field in our initial distribution of halo/void shapes (Section \[epd\]). The natural extension of the peak/dip picture for the initial shear to non-Gaussian fields is also ongoing effort, along with some other broader applications in the context of the excursion set model – for example in relation to the hot and cold spots in the Cosmic Microwave Background, including the effects of $f_{\rm NL}$-type non-Gaussianity on their shapes (i.e. Rossi, Chingangbam & Park 2011) – which will be presented in forthcoming publications. Acknowledgments {#acknowledgments .unnumbered} =============== The final stage of this work was completed during the ‘APCTP-IEU Focus Program on Cosmology and Fundamental Physics III’ (June 11-22, 2012) at Postech, in Pohang, Korea. I would like to thank the organizers of the workshop, and in particular Changrim Ahn. Also, many thanks to Changbom Park for a careful reading of the manuscript, and for many interesting discussions, suggestions and encouragement. 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In particular, let $\Psi$ denote the displacement field, $\Phi$ the potential of the displacement field (i.e. the gravitational field), $F$ the source of the displacement field (i.e. the density field). Both $F$ and $\Phi$ are Gaussian random fields, the latter specified by the matter power spectrum $P(k)$, with $k$ denoting the wave number and $W(k)$ the smoothing kernel. Use $T_{\rm ij}$ for the shear tensor (its eigenvalues are $\lambda_1, \lambda_2, \lambda_3$), $H_{\rm ij}$ for the Hessian matrix of the density $F$ (with eigenvalues $\xi_1, \xi_2, \xi_3$), $J_{\rm ij}$ for the Jacobian of the displacement field, where $i,j=1,2,3$. Indicate with ${\bf q}$ the Lagrangian coordinate, with ${\bf x}$ the Eulerian coordinate, where ${\bf x} ({\bf q}) = {\bf q} + \Psi({\bf q})$. Clearly, $$J_{\rm ij}({\bf q}) = { \partial x_{\rm i} \over \partial q_{\rm j}} = \delta_{\rm ij} + T_{\rm ij}, \hspace{1cm} T_{\rm ij} = {\partial^2 \Phi \over \partial q_{\rm i} \partial q_{\rm j}}, \hspace{1cm} H_{\rm ij} = {\partial^2 F \over \partial q_{\rm i} \partial q_{\rm j}}, \hspace{1cm} F ({\bf q}) = \sum_{\rm i=1}^3 {\partial \Psi_{\rm i} \over \partial q_{\rm i}} \equiv \sum_{\rm i=1}^3 {\partial^2 \Phi \over \partial q^2_{\rm i}}.$$ The correlations between the shear and density Hessian are expressed by: $$\langle T_{\rm ij} T_{\rm kl} \rangle = { \sigma_{\rm T}^2 \over 15} ( \delta_{\rm ij} \delta_{\rm kl} + \delta_{\rm ik} \delta_{\rm jl} + \delta_{\rm il} \delta_{\rm jk})$$ $$\langle H_{\rm ij} H_{\rm kl} \rangle = { \sigma_{\rm H}^2 \over 15} ( \delta_{\rm ij} \delta_{\rm kl} + \delta_{\rm ik} \delta_{\rm jl} + \delta_{\rm il} \delta_{\rm jk})$$ $$\langle T_{\rm ij} H_{\rm kl} \rangle = { \Gamma_{\rm TH} \over 15} ( \delta_{\rm ij} \delta_{\rm kl} + \delta_{\rm ik} \delta_{\rm jl} + \delta_{\rm il} \delta_{\rm jk})$$ where $\sigma_{\rm T}^2 = S_2 \equiv \sigma_0^2$, $\sigma_{\rm H}^2 = S_6 \equiv \sigma_2^2$, $\Gamma_{\rm TH} = S_4 \equiv \sigma_1^2$, $\delta_{\rm ij}$ is the Kronecker delta, and $$S_{\rm n} = {1 \over 2 \pi^2} \int_0^{\infty} k^{\rm n}~P(k)~W^2(k) {\rm d}k$$ $$\label{eq_sig} \sigma_{\rm j}^2 = {1 \over 2 \pi^2} \int_{0}^{\infty} k^{\rm 2(j+1)}~P(k)~W^2(k)~{\rm d}k \equiv S_{\rm 2(j+1)}.$$ In the main text we prefer to use $\sigma_{\rm T}$ and $\sigma_{\rm H}$, rather than the more familiar $\sigma_0$ and $\sigma_2$, for their intuitive meaning. In particular, the subscript $T$ always indicates that a quantity is related to the shear field, while the subscript $H$ denotes a quantity linked to the Hessian of the density field. The shear and density Hessian $T_{\rm ij}$ and $H_{\rm ij}$ are real symmetric tensors, so they are specified by 6 components. Whenever necessary, we label those components with the symbols $\alpha$ or $\beta$ to indicate the various couples, where $\alpha, \beta =(1,1), (2,2), (3,3), (1,2), (1,3), (2,3)$. It is also useful to introduce the vectors ${\bf T}$ and ${\bf H}$, derived from the components of their corresponding tensors, i.e. ${\bf T} = (T_{11},T_{22},T_{33},T_{12},T_{13},T_{23})$ and ${\bf H} = (H_{11},H_{22},H_{33},H_{12},H_{13},H_{23})$. The constrained eigenvalues of the matrix having components $T_{\alpha}|H_{\alpha}$ will be indicated with $\zeta_1, \zeta_2, \zeta_3$. As shown in Rossi (2012), the covariance matrix of the joint probability distribution of $ {\bf T}$ and $ {\bf H}$ is simply $${\bf \sf V} = \begin{pmatrix} \langle T_{\alpha} T_{\alpha} \rangle & \langle T_{\alpha} H_{\beta} \rangle \\ \langle H_{\beta} T_{\alpha} \rangle & \langle H_{\beta} H_{\beta} \rangle \end{pmatrix} = {1 \over 15} \begin{pmatrix} \sigma^2_{\rm T} {\bf \sf A} & \Gamma_{\rm TH} {\bf \sf A} \\ \Gamma_{\rm TH} {\bf \sf A} & \sigma^2_{\rm H} {\bf \sf A} \end{pmatrix}$$ where $$\label{matrix_A} {\bf \sf A} = \begin{pmatrix} {\bf \sf B} & \oslash \\ \oslash & {\bf \sf I} \end{pmatrix},~ {\bf \sf B} = \begin{pmatrix} 3 & 1 & 1\\ 1 & 3&1\\ 1&1&3 \end{pmatrix}$$ with ${\bf \sf I}$ a ($3\times3$) identity matrix and $\oslash$ a ($3 \times 3$) null matrix. Finally, an important ‘spectral parameter’ often used here is the ‘reduced’ correlation: $$\gamma = \Gamma_{\rm TH}/\sigma_{\rm T} \sigma_{\rm H} = {\sigma_1^2 \over \sigma_0 \sigma_2}, \label{tilde_corr}$$ which plays a crucial role in peaks theory (i.e. Bardeen et al. 1986). If Gaussian filters are used in (\[eq\_sig\]), then our $\gamma$ is the same as the one introduced in Bardeen et al. (1986) – specified by their Equation (4.6a). Note that in the main text we also define $\eta = \gamma \sigma_{\rm T} / \sigma_{\rm H}$; if one adopts reduced variables (i.e. $T_{\rm \alpha}$ and $H_{\rm \alpha}$ normalized by their corresponding rms values $\sigma_{\rm T}$ and $\sigma_{\rm H}$), clearly $\eta \equiv \gamma$. Invariants from the conditional formulas {#insights} ======================================== The algebra presented in Section \[pdesa\] allows one to gain more insights into the joint conditional distribution of eigenvalues in the peak/dip picture (Equation \[doro\_inter\_extended\]). For simplicity, in what follows we consider ‘reduced’ variables, so that the various components of $\bf T$ and $\bf H$ are now normalized by their corresponding rms values ($\sigma_{\rm T}$ and $\sigma_{\rm H}$, respectively). With some abuse of notation, we omit the tilde symbol (used instead in Rossi 2012) to distinguish between normalized and unnormalized quantities. It is then possible to characterize and study the properties of the first few elementary symmetric functions of degree $n$ for the density Hessian, the shear tensor, and the conditional shear tensor – along the lines of Weyl (1948), Doroshkevich (1970), Sheth & Tormen (2002) and Desjacques (2008). In particular, it is direct to note that, using Equation (\[H\_gauss\]) and considering six independent Gaussian random variates $y_{\rm i}$ ($i=1,6$) represented by the six-dimensional vector ${\bf y}$, one obtains that the first two classical invariants $$\begin{aligned} h_1 &=& H_{11} +H_{22} +H_{33} = - y_1\\ h_3^2 &=& h_1^2 - 3 h_2 = {1 \over 5} (y_2^2 + y_3^2 +y_4^2+y_5^2+y_6^2)\end{aligned}$$ are independent. This fact implies that $$p({\bf H}) = {e^{-h_1^2/2} \over \sqrt{2 \pi}} {15^3 \over 8 \sqrt{10} \pi^{5/2}} e^{-5 h_3^2/2} \equiv p(h_1) p(h_3),$$ hence $p({\bf H})$ is the product of two independent distributions, where in particular $p(h_1)$ is a one-dimensional Gaussian with mean zero and unity variance. Note also that $p({\bf H}) {\rm d} {\bf H} = p(\bf y) {\rm d}{\bf y}$, where $p ({\bf y}) \equiv \prod_{\rm i=1}^6 g_{\rm i}$ is simply the product of six independent one-dimensional zero mean unit variance Gaussians $g_{\rm i}$. Similarly, for the shear tensor ${\bf T}$, one obtains that $$\begin{aligned} k_1 &=& T_{11} +T_{22} +T_{33} = - z_1\\ k_3^2 &=& k_1^2 - 3 k_2 = {1 \over 5} (z_2^2 + z_3^2 +z_4^2+z_5^2+z_6^2)\end{aligned}$$ are also independent, with $z_{\rm i}$ ($i=1,6$) other six Gaussian random variates represented by the six-dimensional vector ${\bf z}$. Therefore, $$p({\bf T}) = {e^{-k_1^2/2} \over \sqrt{2 \pi}} {15^3 \over 8 \sqrt{10} \pi^{5/2}} e^{-5 k_3^2/2} \equiv p(k_1) p(k_3)$$ with $p(k_1)$ a one-dimensional Gaussian distribution. Note again that $p({\bf T}) {\rm d} {\bf T} = p({\bf z}) {\rm d}{\bf z}$, where $p ({\bf z}) \equiv \prod_{\rm i=1}^6 g_{\rm i}$ is the product of six independent one-dimensional zero mean unit variance Gaussians $g_{\rm i}$. Following the previous logic, one would naturally expect that the quantities $K_1$ and $K_3^2 = K_1^2 - 3 K_2 $, defined in the main text (see Equation \[K\_def\]), should also be independent. Indeed, it is direct to obtain that (Ravi Sheth, private communication): $$\begin{aligned} K_1 &=& = -( m_1 - \gamma y_1 ) = - \sqrt{1-\gamma^2}~l_1 \\ K_3^2 &=& K_1^2 - 3 K_2 \nonumber \\ &=& {1 \over 5}[ (m_2-\gamma y_2)^2+(m_3-\gamma y_3)^2+(m_4-\gamma y_4)^2+(m_5-\gamma y_5)^2+(m_6-\gamma y_6)^2 ] \nonumber \\ &= & {(1-\gamma^2) \over 5}~(l_2^2 +l_3^2 +l_4^2 +l_5^2 +l_6^2), \end{aligned}$$ where $l_{\rm i}$ ($i=1,6$) are other six independent Gaussian distributed variates with mean zero and unity variance, while $m_{\rm i}$ ($i=1,6$) are six Gaussian distributed variates with shifted mean $\gamma y_{\rm i}$ and reduced variance $(1-\gamma^2)$, i.e. $m_{\rm i} = \gamma y_{\rm i} + \sqrt{1-\gamma^2} ~ l_{\rm i} $. Hence, the joint conditional distribution of eigenvalues in the peak/dip picture (Equation \[doro\_inter\_extended\]) can be written as the product of two independent distributions as follows: $$p ({\bf T}|{\bf H},\gamma) = {e^{-K_1^2/[2(1-\gamma^2)] } \over \sqrt{2 \pi (1-\gamma^2)}} {15^3 \over 8 \sqrt {10} \pi^{5/2}} { e^{-5 K_3^2/[2(1-\gamma^2)] } \over (1-\gamma^2)^{5/2}} \equiv p(K_1|\gamma) p(K_3|\gamma). \label{doro_inter_extended_more}$$ This latter expression clearly shows that the distribution of constrained $K_1 \equiv \Delta_{\rm T|H}$ is independent of the distribution of the constrained angular momentum $K_3^2$, where in particular $p(K_1|\gamma)$ is a Gaussian with zero mean and variance given by $(1-\gamma^2)$, while $p(K_3)$ is a chi-square distribution with five degrees of freedom. Once again, note that $p({\bf T}|{\bf H},\gamma) {\rm d} ({\bf T}|{\bf H}) = p({\bf m}|{\bf y},\gamma) {\rm d}({\bf m}|{\bf y})$, where now $$p ({\bf m}|{\bf y},\gamma) \equiv \prod_{\rm i=1}^6 { e^{(m_{\rm i} -\gamma y_{\rm i})^2 /2 (1-\gamma^2)} \over \sqrt{2 \pi (1-\gamma^2)}} \equiv \prod_{\rm i=1}^6 t_{\rm i};$$ namely, $p({\bf m}|{\bf y},\gamma)$ is now the product of six independent one-dimensional Gaussians $t_{\rm i}$ with shifted mean $\gamma y_{\rm i}$ and reduced variance $(1-\gamma^2)$ – represented by the six-dimensional vector ${\bf m}|{\bf y}$. \[lastpage\] [^1]: E-mail: graziano.rossi@cea.fr