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<|MaskedSetence|> Unlike the existing randomized low tubal rank approximation methods, the proposed algorithm finds an optimal tubal rank and the corresponding low tubal rank approximation automatically given a data tensor and an approximation error bound. <|MaskedSetence|> This algorithm can be generalized to higher order tensors according to the paper [46]. This will be our future work and we are planning to use it to develop fast tensor completion algorithms similar to the strategy done in [52]. A detailed theoretical analysis of the proposed algorithm needs to be investigated. <|MaskedSetence|> | **A**: In this paper, we proposed a new randomized fixed-precision algorithm for fast computation of tensor SVD (t-SVD).
**B**: Simulations on synthetic and real-world data-sets confirmed that the proposed algorithm is efficient and applicable.
**C**: This is also our future work.
VIII Acknowledgements.
| ACB | ABC | ABC | ABC | Selection 2 |
<|MaskedSetence|> <|MaskedSetence|> The matter is that if the group G𝐺Gitalic_G is large enough, then we get a lot of prohibited pairs from the very beginning. Therefore, every time an orbit is taken, a lot of other orbits become removed immediately. This leads to the fact that branches of the algorithm in which we choose to take an orbit several times end very quickly. <|MaskedSetence|> (The number of choices in favor of removing an orbit is unimportant.) In the case (4.1) the algorithm never goes beyond level 5555, and spends most of its time at levels 1111 and 2222.. | **A**:
Remark 4.3.
At first glance, it may seem that our algorithm will branch in an uncontrolled way.
**B**: However, this does not happen.
**C**: We conveniently say that our algorithm is on level k𝑘kitalic_k if k−1𝑘1k-1italic_k - 1 choices in favor of taking an orbit have been made so far.
| ABC | BCA | ABC | ABC | Selection 1 |
This is, of course, not a novel observation, and the disconnection between evaluating raters’ reliability and whether the best applicants were selected was noted earlier [[, e.g.,]]kraemer1991we, nelson1991process, mayo2006peering. <|MaskedSetence|> We extend the aforementioned approach to settings where a proportion of the best candidates is selected and show that, under the assumption of a normally distributed latent variable, the expected classification accuracy can be directly obtained from IRR and the proportion of selected candidates. Subsequently, the selection procedures can be characterized as binary classification and evaluated via well-known and interpretable metrics such as sensitivity or false positive/negative rates. <|MaskedSetence|> In fact, connecting reliability to binary classification recalls classical models that evaluate selection procedures based on validity [[, e.g.,]]taylor1939relationship, cronbach1957psychological.
While our approach is related to univariate classification under measurement error, this use case differs. Compared to typical classification tasks, which aim to separate subjects into different categories [73], we assume the existence of a single continuous latent trait (or a composite score based on a multidimensional assessment) measured by the ratings. Then, we aim to select the best applicants defined by the latent trait and evaluate the (miss)classification probabilities due to the measurement error contained in the observed ratings. Ideally, the validity of the observed indicators (and their combination into the overall assessment) would be evaluated directly, thus answering the question of how well the results of selection procedures predict applicants’ success. However, such a “gold standard” measure of success needed for directly assessing the validity of selection procedure is often not available ([86, 100, 111], also see [78] for alternatives); either because the success measure is located too far in the future, or it is difficult to agree on the success measure itself (see [85, 89] for suggestion to use bibliometric measures in grant reviews, and [76, 90] for a subsequent critique). Consequently, we are often left to assess the reliability of the selection procedures, as reliability limits the usefulness even of completely valid indicators [113]. <|MaskedSetence|> | **A**: Furthermore, the binary classification framework allows researchers and stakeholders to evaluate and improve selection procedures while incorporating the costs of incorrect decisions, increasing the number of raters, or modifying the rating procedure.
**B**: Therefore, our approach results in the lower bound on the corresponding error probabilities as it does not account for additional miss-classifications due to (a lack of) validity.
.
**C**: In cases where applicant selection is based on a fixed threshold (i.e., pass/fail tests), the expected classification accuracy can be estimated by methods outlined by [107, 108] and [79] [[, also see]]lee2010classification, livingston1995estimating, hanson1990investigation.
| ACB | CAB | CAB | CAB | Selection 3 |
<|MaskedSetence|> Treewidth plays a fundamental role in the design of exact and approximation algorithms on planar graphs (and more generally, H𝐻Hitalic_H-minor-free graphs) [18, 3, 28].
The main property of such graphs is that they enjoy the bounded local treewidth property. In other words, any planar graph of a small diameter has a small treewidth. A natural research direction is to extend such methods to intersection graphs of geometric objects [24, 34]. However, even for very “simple” objects like unit disks, the corresponding intersection graphs do not have locally bounded treewidth. <|MaskedSetence|> <|MaskedSetence|> Galby, Munaro, and Yang [25] use Theorem 1.1 for obtaining polynomial-time approximation schemes for several packing problems
on geometric graphs. It is interesting to note that algorithms on geometric graphs often require geometric representation of a graph. Sometimes, like for unit disk graphs, finding such a representation is a challenging computational task [29].. | **A**: On the other hand, in many scenarios, the treewidth-based methods on such graphs could be replaced by tree decompositions of bounded independence number.
**B**:
Theorem 1.1 appears to be a handy tool in the subarea of computational geometry concerning optimization problems on geometric graphs.
**C**: In particular, de Berg, Bodlaender, Kisfaludi-Bak, Marx, and van der Zanden
use tree decompositions whose bags are covered by a small number of cliques, and thus of small independence number, to design subexponential-time algorithms on geometric graph classes [17].
| BAC | ACB | BAC | BAC | Selection 4 |
<|MaskedSetence|> If the lattice is complete, we say that X𝑋Xitalic_X
is a Banach lattice. Note that this implies obviously that for any x∈X𝑥𝑋x\in Xitalic_x ∈ italic_X
the elements x𝑥xitalic_x and |x|𝑥\left|x\right|| italic_x | have the same norm. We denote
by X+={x∈X,x≥0}.subscript𝑋formulae-sequence𝑥𝑋𝑥0X_{+}=\left\{x\in X,x\geq 0\right\}.italic_X start_POSTSUBSCRIPT + end_POSTSUBSCRIPT = { italic_x ∈ italic_X , italic_x ≥ 0 } . <|MaskedSetence|> <|MaskedSetence|> | **A**: say that X𝑋Xitalic_X is a Banach lattice.
**B**: An element x𝑥xitalic_x of X𝑋Xitalic_X is a
positive if x∈X+.𝑥subscript𝑋x\in X_{+}.italic_x ∈ italic_X start_POSTSUBSCRIPT + end_POSTSUBSCRIPT .
**C**: For x𝑥xitalic_x ∈Xabsent𝑋\in X∈ italic_X let x+:=sup{x,0},x−:=sup{−x,0}formulae-sequenceassignsuperscript𝑥supremum𝑥0assignsuperscript𝑥supremum𝑥0x^{+}:=\sup\left\{x,0\right\},x^{-}:=\sup\left\{-x,0\right\}italic_x start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT := roman_sup { italic_x , 0 } , italic_x start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT := roman_sup { - italic_x , 0 } be the positive part, the.
| BAC | ABC | ABC | ABC | Selection 3 |
As discussed above, the upper bound in Theorem 1.3 relates to Widmer’s work on imprimitive Galois groups. <|MaskedSetence|> We say that G𝐺Gitalic_G is transitive (n1,n2)subscript𝑛1subscript𝑛2(n_{1},n_{2})( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )-imprimitive if G𝐺Gitalic_G is transitive and there is a partition of n𝑛nitalic_n into n1subscript𝑛1n_{1}italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT blocks ΛisubscriptΛ𝑖\Lambda_{i}roman_Λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT such that for all g∈G𝑔𝐺g\in Gitalic_g ∈ italic_G and 1≤i≤n1𝑖𝑛1\leq i\leq n1 ≤ italic_i ≤ italic_n there exists 1≤j≤n1𝑗𝑛1\leq j\leq n1 ≤ italic_j ≤ italic_n such that gΛi=Λj𝑔subscriptΛ𝑖subscriptΛ𝑗g\Lambda_{i}=\Lambda_{j}italic_g roman_Λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = roman_Λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT. In particular, |Λi|=n2subscriptΛ𝑖subscript𝑛2|\Lambda_{i}|=n_{2}| roman_Λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | = italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT for all i𝑖iitalic_i. <|MaskedSetence|> <|MaskedSetence|> | **A**: We prove
Theorem 1.6..
**B**: For example, the wreath product C2≀Sn/2≀subscript𝐶2subscript𝑆𝑛2C_{2}\wr S_{n/2}italic_C start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≀ italic_S start_POSTSUBSCRIPT italic_n / 2 end_POSTSUBSCRIPT with the imprimitive action is a transitive (2,n/2)2𝑛2(2,n/2)( 2 , italic_n / 2 )-imprimitive.
**C**: In fact, Theorem 1.3 follows from an improvement of Widmer’s general result: Assume n=n1n2𝑛subscript𝑛1subscript𝑛2n=n_{1}n_{2}italic_n = italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, n1,n2>1subscript𝑛1subscript𝑛21n_{1},n_{2}>1italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT > 1 and let G𝐺Gitalic_G be a subgroup of Snsubscript𝑆𝑛S_{n}italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT.
| CBA | CBA | CAB | CBA | Selection 2 |
The contributions of the paper are mainly in three aspects. Firstly, we propose an AS strategy to reduce the scale of the hdr lasso problem, based on checking the KKT conditions of the problem. We also prove the convergence of the AS algorithm. Secondly, taking into account of the nonsmoothness of the loss function and the L1subscript𝐿1L_{1}italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT regularizer in the hdr lasso problem, we apply PPA for solving the subproblem. Finally, for each subproblem of PPA, we use the semismooth Newton’s based augmented Lagrangian method. <|MaskedSetence|> <|MaskedSetence|> 2, we propose our AS strategy and prove the convergence. In Sect. 3, we propose PPA to solve the subproblems in AS and illustrate the convergence of PPA. In Sect. 4, we use the augmented Lagrange method (ALM) to solve each subproblem in PPA. In Sect. <|MaskedSetence|> Numerical results on different datasets are presented in Sect. 6, which verify the efficiency of our proposed AS-PPA for the hdr lasso problem. Final conclusions are given in Sect. 7.
. | **A**: 5, we apply the semismooth Newton’s method (SSN) to solve the subproblems in ALM.
**B**: Numerical results demonstrate that the proposed adaptive sieving strategy is very efficient in reducing the scale of the hdr lasso problem and the resulting AS-PPA significantly outperforms other methods.
The organization of the paper is as follows.
**C**: In Sect.
| ABC | BCA | BCA | BCA | Selection 2 |
1.3. <|MaskedSetence|> Later, by correcting the drift term of the Dean-Kawasaki equation, the authors of [AvR10] and [KvR19] constructed a solution. Cornalba, Shardlow and Zimmer [CSZ19, CSZ20] derived a suitably regularized Dean-Kawasaki model of wave equation type in one dimension, which corresponds to second-order Langevin dynamics. In the case of local interaction, Fehrman and Gess [FG24] obtained the well-posedness of functional-valued solutions of the Dean-Kawasaki equation with correlated noise. <|MaskedSetence|> <|MaskedSetence|> | **A**: Furthermore, Clini and Fehrman [CF23a] expanded this research by developing a central limit theorem for the nonlinear Dean-Kawasaki equation with correlated noise.
.
**B**: Building on this framework in [FG24], the authors also addressed small noise large deviations in [FG23].
**C**: Comments on the literature
The existence of solutions to corrected Dean-Kawasaki equations with smooth interacting kernel has been proved by von Renesse and Sturm [vRS09] by Dirichlet forms techniques, where the nonlocal interacting term is replaced by a nonlinear operator.
| CBA | CBA | CBA | CBA | Selection 1 |
<|MaskedSetence|> Now we start to prove this case. Similar to Case-2, we also have ℓ(Γ)≥min{w0,sys(Xg−2,1)}≻1ℓΓsubscript𝑤0syssubscript𝑋𝑔21succeeds1\ell(\Gamma)\geq\min\{w_{0},\mathop{\rm sys}(X_{g-2,1})\}\succ 1roman_ℓ ( roman_Γ ) ≥ roman_min { italic_w start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , roman_sys ( italic_X start_POSTSUBSCRIPT italic_g - 2 , 1 end_POSTSUBSCRIPT ) } ≻ 1. <|MaskedSetence|> <|MaskedSetence|> This gives that
. | **A**: Clearly ∂(A∩(Xg−2,1∖T))⊂(∂A)∪(∂T)=Γ∪(∂T)𝐴subscript𝑋𝑔21𝑇𝐴𝑇Γ𝑇\partial(A\cap(X_{g-2,1}\setminus T))\subset(\partial A)\cup(\partial T)=%
\Gamma\cup(\partial T)∂ ( italic_A ∩ ( italic_X start_POSTSUBSCRIPT italic_g - 2 , 1 end_POSTSUBSCRIPT ∖ italic_T ) ) ⊂ ( ∂ italic_A ) ∪ ( ∂ italic_T ) = roman_Γ ∪ ( ∂ italic_T ).
**B**:
where γ~⊂P~~𝛾~𝑃\tilde{\gamma}\subset\tilde{P}over~ start_ARG italic_γ end_ARG ⊂ over~ start_ARG italic_P end_ARG is the simple geodesic arc with the same endpoints and homotopy type as γ⊂P𝛾𝑃\gamma\subset Pitalic_γ ⊂ italic_P.
**C**: Recall that for the half collar T𝑇Titalic_T, we always have Area(T)≤2πArea𝑇2𝜋\mathop{\rm Area}(T)\leq 2\piroman_Area ( italic_T ) ≤ 2 italic_π and ℓ(∂T)≍1asymptotically-equalsℓ𝑇1\ell(\partial T)\asymp 1roman_ℓ ( ∂ italic_T ) ≍ 1.
| BCA | BCA | BCA | ACB | Selection 2 |
<|MaskedSetence|> <|MaskedSetence|> Indeed, since lim|x|→∞ui=0subscript→𝑥subscript𝑢𝑖0\lim_{|x|\rightarrow\infty}u_{i}=0roman_lim start_POSTSUBSCRIPT | italic_x | → ∞ end_POSTSUBSCRIPT italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 0, for any ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0, there esists sufficiently large R𝑅Ritalic_R such that x0∈BR(0)subscript𝑥0subscript𝐵𝑅0x_{0}\in B_{R}(0)italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_B start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ( 0 ) and u1≥u2−ϵsubscript𝑢1subscript𝑢2italic-ϵu_{1}\geq u_{2}-\epsilonitalic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - italic_ϵ on ∂BR(0)subscript𝐵𝑅0\partial B_{R}(0)∂ italic_B start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ( 0 ). <|MaskedSetence|> Let ϵitalic-ϵ\epsilonitalic_ϵ go to 00, we have u1(x0)≥u2(x0)subscript𝑢1subscript𝑥0subscript𝑢2subscript𝑥0u_{1}(x_{0})\geq u_{2}(x_{0})italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ≥ italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) and thus u1≥u2subscript𝑢1subscript𝑢2u_{1}\geq u_{2}italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT in ΩcsuperscriptΩ𝑐\Omega^{c}roman_Ω start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT. Similarly, we can prove u2≥u1subscript𝑢2subscript𝑢1u_{2}\geq u_{1}italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≥ italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT in ΩcsuperscriptΩ𝑐\Omega^{c}roman_Ω start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT. Then we have u1=u2subscript𝑢1subscript𝑢2u_{1}=u_{2}italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT in ΩcsuperscriptΩ𝑐\Omega^{c}roman_Ω start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT and thus prove the uniqueness part.
Case 2: k>n2𝑘𝑛2k>\frac{n}{2}italic_k > divide start_ARG italic_n end_ARG start_ARG 2 end_ARG. | **A**: For any x0∈Ωcsubscript𝑥0superscriptΩ𝑐x_{0}\in\Omega^{c}italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ roman_Ω start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT, we want to prove u1(x0)≥u2(x0)subscript𝑢1subscript𝑥0subscript𝑢2subscript𝑥0u_{1}(x_{0})\geq u_{2}(x_{0})italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ≥ italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ).
**B**: Note we also have u1=u2=0subscript𝑢1subscript𝑢20u_{1}=u_{2}=0italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0 on ∂ΩΩ\partial\Omega∂ roman_Ω, by comparison theorem in ΩRsubscriptΩ𝑅\Omega_{R}roman_Ω start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT, we then have u1≥u2−ϵsubscript𝑢1subscript𝑢2italic-ϵu_{1}\geq u_{2}-\epsilonitalic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - italic_ϵ in ΩRsubscriptΩ𝑅\Omega_{R}roman_Ω start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT.
**C**: Let u1,u2subscript𝑢1subscript𝑢2u_{1},u_{2}italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT be solutions of the k𝑘kitalic_k-Hessian equation.
| CAB | ACB | CAB | CAB | Selection 1 |
We thank D. Goluskin, J. <|MaskedSetence|> Chernyavsky for many discussions about this work and for offering comments on an early draft. This work also benefited from conversations with A. <|MaskedSetence|> Chernyshenko, and G. Valmorbida. <|MaskedSetence|> | **A**: Wynn, S.
**B**: We gratefully acknowledge the hospitality and support of the Banff International Research Station; this paper was finished during the Focussed Research Group “Studying PDE dynamics via optimization with integral inequality constraints” (http://www.birs.ca/events/2022/focussed-research-groups/22frg243).
Open-access statement.
**C**: Bramburger and A.
| CAB | CAB | CAB | BAC | Selection 1 |
<|MaskedSetence|> e. x∈Ω,formulae-sequence𝜑∇𝑢𝑢for a. <|MaskedSetence|> e. }x\in\Omega,italic_φ = divide start_ARG | ∇ italic_u | end_ARG start_ARG italic_u end_ARG for a. e. <|MaskedSetence|> t∈(0,1)𝑡01t\in(0,1)italic_t ∈ ( 0 , 1 ). Thus, by the rigidity of the isoperimetric inequality, we get that Utsubscript𝑈𝑡U_{t}italic_U start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT coincides with a ball up to a ℋn−1superscriptℋ𝑛1\mathcal{H}^{n-1}caligraphic_H start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT-negligible set for a.e. t∈(0,1)𝑡01t\in(0,1)italic_t ∈ ( 0 , 1 ). In particular, {u>0}=⋃tUt𝑢0subscript𝑡subscript𝑈𝑡\Set{u>0}=\bigcup_{t}U_{t}{ start_ARG italic_u > 0 end_ARG } = ⋃ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT and {u=1}=⋂tUt𝑢1subscript𝑡subscript𝑈𝑡\set{u=1}=\bigcap_{t}U_{t}{ start_ARG italic_u = 1 end_ARG } = ⋂ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT coincide with two balls up to a ℋn−1superscriptℋ𝑛1\mathcal{H}^{n-1}caligraphic_H start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT-negligible set.. | **A**: italic_x ∈ roman_Ω ,
so that, by section 2, we have equality in (2.16) for a.e.
**B**: e. 𝑥Ω\varphi=\frac{\lvert\nabla u\rvert}{u}\qquad\text{for a.
**C**: φ=|∇u|ufor a.
| CBA | CBA | ABC | CBA | Selection 4 |
<|MaskedSetence|> Here a Reeb orbit is called “hyperbolic” when its linearized return map has real eigenvalues. The differential on the chain complex counts J𝐽Jitalic_J-holomorphic curves in ℝ×Yℝ𝑌{\mathbb{R}}\times Yblackboard_R × italic_Y with ECH index 1111 for a generic almost complex structure J𝐽Jitalic_J on ℝ×Yℝ𝑌{\mathbb{R}}\times Yblackboard_R × italic_Y. See [14] for a detailed exposition.
Taubes [24] has shown that if Y𝑌Yitalic_Y is connected777If Y𝑌Yitalic_Y is disconnected, then the ECH of (Y,λ)𝑌𝜆(Y,\lambda)( italic_Y , italic_λ ) is the tensor product of the ECH of its components. It follows from Taubes’s isomorphism applied to each component that the ECH of (Y,λ)𝑌𝜆(Y,\lambda)( italic_Y , italic_λ ) is still an invariant of (Y,ξ)𝑌𝜉(Y,\xi)( italic_Y , italic_ξ )., then there is a canonical isomorphism between ECH(Y,λ)𝐸𝐶𝐻𝑌𝜆ECH(Y,\lambda)italic_E italic_C italic_H ( italic_Y , italic_λ ) and a version of Seiberg-Witten Floer homology as defined by Kronheimer-Mrowka [19]. <|MaskedSetence|> However certain additional structure on it, such as a direct sum decomposition ECH(Y,ξ)=⊕Γ∈H1(Y)ECH(Y,ξ,Γ)𝐸𝐶𝐻𝑌𝜉subscriptdirect-sumΓsubscript𝐻1𝑌𝐸𝐶𝐻𝑌𝜉ΓECH(Y,\xi)=\oplus_{\Gamma\in H_{1}(Y)}ECH(Y,\xi,\Gamma)italic_E italic_C italic_H ( italic_Y , italic_ξ ) = ⊕ start_POSTSUBSCRIPT roman_Γ ∈ italic_H start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_Y ) end_POSTSUBSCRIPT italic_E italic_C italic_H ( italic_Y , italic_ξ , roman_Γ ), depends also on ξ𝜉\xiitalic_ξ. <|MaskedSetence|> | **A**: on the contact structure ξ=Ker(λ)𝜉Ker𝜆\xi=\operatorname{Ker}(\lambda)italic_ξ = roman_Ker ( italic_λ ), and so can be denoted by ECH(Y,ξ)𝐸𝐶𝐻𝑌𝜉ECH(Y,\xi)italic_E italic_C italic_H ( italic_Y , italic_ξ ).
.
**B**: It follows from this isomorphism that ECH(Y,λ)𝐸𝐶𝐻𝑌𝜆ECH(Y,\lambda)italic_E italic_C italic_H ( italic_Y , italic_λ ) depends only888In fact, at this level of description, ECH𝐸𝐶𝐻ECHitalic_E italic_C italic_H depends only on Y𝑌Yitalic_Y.
**C**: If the contact form λ𝜆\lambdaitalic_λ is nondegenerate, then the embedded contact homology of (Y,λ)𝑌𝜆(Y,\lambda)( italic_Y , italic_λ ), denoted by ECH(Y,λ)𝐸𝐶𝐻𝑌𝜆ECH(Y,\lambda)italic_E italic_C italic_H ( italic_Y , italic_λ ), is the homology of a chain complex over ℤ/2ℤ2{\mathbb{Z}}/2blackboard_Z / 2 freely generated by orbit sets α={(αi,mi)}𝛼subscript𝛼𝑖subscript𝑚𝑖\alpha=\{(\alpha_{i},m_{i})\}italic_α = { ( italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) } which are “admissible”, meaning that mi=1subscript𝑚𝑖1m_{i}=1italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1 whenever αisubscript𝛼𝑖\alpha_{i}italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is hyperbolic.
| CBA | CBA | ABC | CBA | Selection 1 |
<|MaskedSetence|> In Section 2, we introduce the MV-SDE and associated notation, motivate MC methods to estimate expectations associated with its solution and set forth the problem to be solved. <|MaskedSetence|> Next, we state the optimal importance sampling control for the decoupled MV-SDE derived using stochastic optimal control and introduce the DLMC estimator with importance sampling from (Ben Rached et al., 2023) in Section 4. <|MaskedSetence|> We combine the multilevel DLMC estimator with the proposed importance sampling scheme and develop an adaptive multilevel DLMC algorithm that feasibly estimates rare-event quantities associated with MV-SDEs. Finally, we apply the proposed methods to the Kuramoto model from statistical physics in Section 6 and numerically verify all assumptions in this work and the derived complexity rates for the multilevel DLMC estimator for two observables.. | **A**: Then, we introduce the novel multilevel DLMC estimator in Section 5, develop an antithetic sampler for it, and derive new complexity results for the estimator.
**B**: In Section 3, we introduce the decoupling approach for MV-SDEs (dos Reis et al., 2023) and formulate a DLMC estimator.
**C**:
The remainder of this paper is structured as follows.
| CBA | CBA | CBA | ABC | Selection 3 |
<|MaskedSetence|> In section 3, we provide an overview of the paper’s main results and state the main theorems. In section 4, we review related literature and existing results on computing Brascamp–Lieb constants. Section 5 provides formal proofs of the paper’s findings. <|MaskedSetence|> <|MaskedSetence|> | **A**: We conclude with a general discussion of the paper’s results and limitations, as well as avenues for future investigation, in section 7.
.
**B**: In section 6, we provide a discussion on alternative Picard iterations that could be use to compute Brascamp–Lieb constants.
**C**: 1.3 Outline
The paper is structured as follows: In section 2 we introduce basic background and notation, including Thompson geometry on the space of positive definite matrices and the class of Brascamp–Lieb inequalities.
| ABC | CBA | CBA | CBA | Selection 2 |
Our proposed method, therefore, has broad applicability across numerous survey sampling scenarios where sampling weights are known.
Second, we have introduced adaptive estimators that asymptotically attain the semiparametric efficiency bound, rendering them asymptotically optimal within the extensive class of Regular Asymptotically Linear (RAL) estimators for the parameters under consideration. <|MaskedSetence|> <|MaskedSetence|> Utilizing the debiased/double machine learning method [6], this innovation is applied to our semiparametric adaptive estimators, enhancing their robustness. Consequently, our methodology stands out not only for its efficiency, but also for its robustness, thanks to the nonparametric estimation of the weight model.
Moreover, we have rigorously established the large-sample properties of the adaptive efficient estimator, including
n𝑛\sqrt{n}square-root start_ARG italic_n end_ARG-consistency and asymptotic normality. <|MaskedSetence|> | **A**: This approach marks a departure from traditional descriptive inference by treating weights as random variables in analytic inference, thereby leveraging the weight model to improve the efficiency of parameter estimation.
Third, to mitigate bias arising from potential misspecification of weight models, we have developed a nonparametric estimation of the weight models.
**B**: As a result, this foundation enables statistical inference using confidence intervals, providing a scientific framework for assessing the precision and reliability of estimates derived from large samples..
**C**: By innovatively incorporating models on the sampling weights, our estimators utilize these weights more effectively, enhancing the estimation accuracy of various parameters.
| CAB | BCA | CAB | CAB | Selection 1 |
<|MaskedSetence|> The Fano configuration is flag-transitive so this is unique. <|MaskedSetence|> By the corollary, the minimum number of monochromatic triangles given a line 2222-coloring should be greater than or equal to 4444. Our GAP code confirms this and establishes that it is exactly 4444.
We here also give a couple of particular examples of configurations with maximally 5 mutually intersecting lines. <|MaskedSetence|> | **A**: It is then clear that there are two disjoint sets of six mutually intersecting lines - one each from the original configurations.
**B**: Take a connected sum of two Fano configurations.
**C**: The first has minimally no monochromatic triangles given a line 2222-coloring and the second has minimally one monochromatic triangle given a line 2222-coloring..
| BAC | BAC | BAC | BAC | Selection 1 |
<|MaskedSetence|> <|MaskedSetence|> For this purpose, we use the statistical test developed by Chevyrev and Oberhauser (2022) which precisely allows to check whether two samples of stochastic processes paths come from the same distribution. It relies on the notions of signature and maximum mean distance which are presented in this article. Our contribution is to study this test from a numerical point of view in settings that are relevant for applications. <|MaskedSetence|> To this end, we apply the test to three stochastic processes in continuous time, namely the fractional Brownian motion (fBm), the Black-Scholes dynamics (BSd) and the rough Heston model, and two time series models, namely a regime-switching AR(1)𝐴𝑅1AR(1)italic_A italic_R ( 1 ) process and a random walk with i.i.d. Gamma increments. The test is also applied to a two-dimensional process combining the price in the rough Heston model and a regime-switching AR(1)𝐴𝑅1AR(1)italic_A italic_R ( 1 ) process. The numerical experiments have highlighted the need to configure the test specifically for each stochastic process to achieve a good statistical power. In particular, the path representation (original paths, log-paths, realized volatility or log-returns), the path transformation (lead-lag, time lead-lag or cumulative lead-lag), the truncation order, the signature type (signature or log-signature) and the rescaling are key ingredients to be adjusted for each model. For example, the test achieves statistical powers that are close to one in the following settings which illustrate three different risk factors (stock volatility, stock price and inflation respectively):
•. | **A**:
We propose a new approach for the validation of real-world economic scenarios motivated by insurance applications.
**B**: More specifically, we start by measuring the statistical power of the test on synthetic data under two practical constraints: first, the marginal one-year distributions of the compared samples are equal or very close so that point-in-time validation methods are unable to distinguish the two samples and second, one sample is assumed to be of small size (below 50) while the other is of larger size (1000).
**C**: This approach relies on the formulation of the problem of validating real-world economic scenarios as a two-sample hypothesis testing problem where the first sample consists of historical paths, the second sample consists of simulated paths of a given real-world stochastic model and the null hypothesis is that the two samples come from the same distribution.
| ACB | CBA | ACB | ACB | Selection 3 |
<|MaskedSetence|> However, for the viscous and resistive MHD system (1.1), the uniqueness for solutions in CtLx2subscript𝐶𝑡subscriptsuperscript𝐿2𝑥C_{t}L^{2}_{x}italic_C start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT is still unsloved. Theorem 1.2 solves this problem and extends the non-uniqueness results of the ideal MHD system to the viscous and resistive MHD system.
Compared with the ideal MHD system, the dissipative effect prevents the nonlinear term from balancing the stress error (R̊q,M̊q)subscript̊𝑅𝑞subscript̊𝑀𝑞(\mathring{R}_{q},\mathring{M}_{q})( over̊ start_ARG italic_R end_ARG start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT , over̊ start_ARG italic_M end_ARG start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ) as doing in [4]. This
leads to the major difficulty in convex integral iteration in CtLx2subscript𝐶𝑡subscriptsuperscript𝐿2𝑥C_{t}L^{2}_{x}italic_C start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT. A nature choice is using 3D box type flows instead of the Mikado flows in convex integral iteration. <|MaskedSetence|> Inspired by [10, 4, 8], we construct “temporal flows” and “Inverse traveling wave flows” to eliminate these extra errors, which help us construct a weak solution by combining with the principal flows. <|MaskedSetence|> | **A**: However, these 3D box type flows do not have enough freedom on the oscillation directions in the velocity and magnetic flows, which will give rise to additional errors in the oscillation terms.
**B**: For the ideal MHD system (1.4) with non-trivial magnetic fields, Beekie-Buckmaster-Vicol in [4] proved the non-uniqueness for the weak solutions in CtLx2subscript𝐶𝑡subscriptsuperscript𝐿2𝑥C_{t}L^{2}_{x}italic_C start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT.
**C**: Moreover, we construct the so-called “Initial flows” and “Helicity flows” to achieve.
| BAC | CAB | BAC | BAC | Selection 1 |
Acknowledgments.
We thank B. Berndtsson, R. Berman, S. <|MaskedSetence|> Jonsson, Y. Odaka, G. <|MaskedSetence|> Xia, C. Xu, X. Zhou, X. Zhu for suggesting simplifications and other improvements to the presentation of the paper. The first named author was partially supported by an Alfred P. Sloan Fellowship and National Science Foundation grant DMS–1846942. <|MaskedSetence|> | **A**: Tian, M.
**B**: The second named author is supported by NSFC grant 12101052 and the.
**C**: Boucksom, M.
| CAB | BAC | CAB | CAB | Selection 4 |
Similar to the enumeration Simion and Schmidt [9] did in 1985 and Mansour [7] in 2020, Bonichon and Morel [3] started the enumeration of d𝑑ditalic_d-permutations avoiding small patterns and made many conjectures regarding the enumeration of 3333-permutations avoiding sets of two patterns. We present two main classes of results regarding the enumeration of 3333-permutation avoiding small patterns. <|MaskedSetence|> Further, we derive a recurrence relation for 3333-permutations avoiding 132132132132 and 213213213213, whose sequence we added to the OEIS database [10], and Bonichon and Morel did not provide any conjecture. <|MaskedSetence|> In Section 2, we introduce preliminary definitions and notation. <|MaskedSetence|> In addition, we prove a recurrence relation for an avoidance class whose sequence we added to the OEIS database [10], completing our enumeration. In Section 4, we extend our enumeration to 3333-permutations avoiding three patterns of size 3 and prove recurrence relations for their avoidance classes. We conclude with open problems in Section 5.. | **A**: We first completely enumerate 3333-permutations avoiding classes of two patterns of size 3333 and prove their respective recurrence relations, solving the conjectures presented by Bonichon and Morel [3].
**B**: We then further initiate and completely enumerate 3333-permutations avoiding classes of three patterns of size 3333, similar to Simion and Schmidt’s results in 1985 [9].
This paper is organized as follows.
**C**: In Section 3, we completely enumerate sequences of 3333-permutations avoiding two patterns of size 3 and prove four conjectures of Bonichon and Morel [3].
| ABC | ABC | ABC | ABC | Selection 1 |
Online learning methods enable model updates incrementally from sequential data, offering greater efficiency and scalability than traditional batch learning. Regularization technique is widely used in online convex optimization problems [40]. Online Mirror Descent, an extension of Mirror Descent [41], utilizes a gradient update rule in the dual space, leading to improved bounds. <|MaskedSetence|> Follow-the-Regularized-Leader [43, 44] is stable extension of Follow-the-Leader [45, 46] by adding a strong convex regularization term to the objective function to achieve a sublinear regret bound.
In this work, we present an innovative methodology that combines changepoints detection with PINNs to address changes and instabilities in the dynamics of PDEs. This approach marks the first exploration into simultaneously detecting changepoints and estimating unknown parameters within PDE dynamics based on observed data. <|MaskedSetence|> <|MaskedSetence|> (ii) We propose an online learning technique aimed at optimizing the weights within the loss function during training. By adaptively adjusting these weights, our method not only enhances the model’s estimation accuracy but also increases its robustness against the instabilities associated with rapid parameter variations. (iii) We present several theoretical results to show that our re-weighting approach minimizes the training loss function with a regularizer and demonstrates that the regret is upper bounded. The theoretical results also indicate that the weight update method does not alter the neural network’s optimization objective on average.. | **A**: This approach not only identifies the timing of changes but also facilitates the estimation of unknown system parameters.
**B**: We have three main contributions: (i) We introduce a novel strategy that leverages PINNs alongside the Total Variation method for detecting changepoints within PDE dynamics.
**C**: Adaptive subgradient method [42] dynamically adjusts regularization term based on its current subgradient.
| CBA | CBA | CBA | BAC | Selection 2 |
<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Let us define on the C((xs))𝐶subscript𝑥𝑠C(\!(x_{s})\!)italic_C ( ( italic_x start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) )–space Eφsubscript𝐸𝜑E_{\varphi}italic_E start_POSTSUBSCRIPT italic_φ end_POSTSUBSCRIPT a connection by ∇φrs:=∇φ−p(φ)idEφassignsuperscriptsubscript∇𝜑rssubscript∇𝜑p𝜑subscriptidsubscript𝐸𝜑{\nabla}_{\varphi}^{\mathrm{rs}}:=\nabla_{\varphi}-\mathrm{p}(\varphi)\mathrm{%
id}_{E_{\varphi}}∇ start_POSTSUBSCRIPT italic_φ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_rs end_POSTSUPERSCRIPT := ∇ start_POSTSUBSCRIPT italic_φ end_POSTSUBSCRIPT - roman_p ( italic_φ ) roman_id start_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_φ end_POSTSUBSCRIPT end_POSTSUBSCRIPT. Consider the Levelt-Jordan decomposition ∇=S∇+N∇∇subscript𝑆∇subscript𝑁∇\nabla=S_{\nabla}+N_{\nabla}∇ = italic_S start_POSTSUBSCRIPT ∇ end_POSTSUBSCRIPT + italic_N start_POSTSUBSCRIPT ∇ end_POSTSUBSCRIPT of ∇∇\nabla∇ and denote by Sφ=S∇|Eφsubscript𝑆𝜑evaluated-atsubscript𝑆∇subscript𝐸𝜑S_{\varphi}=S_{\nabla}|_{E_{\varphi}}italic_S start_POSTSUBSCRIPT italic_φ end_POSTSUBSCRIPT = italic_S start_POSTSUBSCRIPT ∇ end_POSTSUBSCRIPT | start_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_φ end_POSTSUBSCRIPT end_POSTSUBSCRIPT. According to [Lev75, Section 6a], there exists a C((xs))𝐶subscript𝑥𝑠C(\!(x_{s})\!)italic_C ( ( italic_x start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) )–basis (𝐞φ)subscript𝐞𝜑(\mathbf{e}_{\varphi})( bold_e start_POSTSUBSCRIPT italic_φ end_POSTSUBSCRIPT ) of Eφsubscript𝐸𝜑E_{\varphi}italic_E start_POSTSUBSCRIPT italic_φ end_POSTSUBSCRIPT such that Mat(Sφ,𝐞φ)=(cφ+p(φ))IrφMatsubscript𝑆𝜑subscript𝐞𝜑subscript𝑐𝜑p𝜑subscriptIsubscript𝑟𝜑\mathrm{Mat}(S_{\varphi},\mathbf{e}_{\varphi})=(c_{\varphi}+\mathrm{p}(\varphi%. | **A**: Proof.
Existence.
**B**: For each φ∈Φ𝜑Φ\varphi\in\Phiitalic_φ ∈ roman_Φ, let cφ∈Csubscript𝑐𝜑𝐶c_{\varphi}\in Citalic_c start_POSTSUBSCRIPT italic_φ end_POSTSUBSCRIPT ∈ italic_C such that cφ+p(φ)subscript𝑐𝜑p𝜑c_{\varphi}+\mathrm{p}(\varphi)italic_c start_POSTSUBSCRIPT italic_φ end_POSTSUBSCRIPT + roman_p ( italic_φ ) is a representative of φ𝜑\varphiitalic_φ.
**C**: According to (3), the connection (E,∇)𝐸∇(E,\nabla)( italic_E , ∇ ) admits a TLJ-decomposition (E(s),∇)=⨁φ∈Φ(Eφ,∇φ)subscript𝐸𝑠∇subscriptdirect-sum𝜑Φsubscript𝐸𝜑subscript∇𝜑(E_{(s)},\nabla)=\bigoplus\limits_{\varphi\in\Phi}(E_{\varphi},\nabla_{\varphi})( italic_E start_POSTSUBSCRIPT ( italic_s ) end_POSTSUBSCRIPT , ∇ ) = ⨁ start_POSTSUBSCRIPT italic_φ ∈ roman_Φ end_POSTSUBSCRIPT ( italic_E start_POSTSUBSCRIPT italic_φ end_POSTSUBSCRIPT , ∇ start_POSTSUBSCRIPT italic_φ end_POSTSUBSCRIPT )
over C((xs))𝐶subscript𝑥𝑠C(\!(x_{s})\!)italic_C ( ( italic_x start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) ).
| ACB | ACB | ACB | CBA | Selection 1 |
and let π:𝒳→ℂA:𝜋→𝒳superscriptℂ𝐴\pi\colon{\cal X}\to\operatorname{\mathbb{C}}^{A}italic_π : caligraphic_X → blackboard_C start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT be the projection to ℂAsuperscriptℂ𝐴\operatorname{\mathbb{C}}^{A}blackboard_C start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT. <|MaskedSetence|> Denote by ℋnsuperscriptℋ𝑛\operatorname{\mathcal{H}}^{n}caligraphic_H start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT the n𝑛nitalic_n-th cohomology group of the relative de Rham complex (Ω𝒳/ℂA∙,∇x),superscriptsubscriptΩ𝒳superscriptℂ𝐴∙subscript∇𝑥(\Omega_{{\cal X}/\operatorname{\mathbb{C}}^{A}}^{\bullet},\nabla_{x}),( roman_Ω start_POSTSUBSCRIPT caligraphic_X / blackboard_C start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∙ end_POSTSUPERSCRIPT , ∇ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ) , where Ω𝒳/ℂAksuperscriptsubscriptΩ𝒳superscriptℂ𝐴𝑘\Omega_{{\cal X}/\operatorname{\mathbb{C}}^{A}}^{k}roman_Ω start_POSTSUBSCRIPT caligraphic_X / blackboard_C start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT is the sheaf of relative differential k𝑘kitalic_k-forms. <|MaskedSetence|> The differential ∇x=dx+dlogx(fsxν)subscript∇𝑥subscriptd𝑥subscriptdlog𝑥superscript𝑓𝑠superscript𝑥𝜈\nabla_{x}=\mathrm{d}_{x}+\operatorname{dlog}_{x}(f^{s}x^{\nu})∇ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT = roman_d start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT + roman_dlog start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_f start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT ) only takes derivatives with respect to x.𝑥x.italic_x . <|MaskedSetence|> | **A**: For every c*∈ℂA,superscript𝑐superscriptℂ𝐴c^{*}\in\operatorname{\mathbb{C}}^{A},italic_c start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ∈ blackboard_C start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT , there is an evaluation map.
**B**: This sheaf is locally defined by its sections
∑|I|=ks(x,c)dxI,subscript𝐼𝑘𝑠𝑥𝑐dsuperscript𝑥𝐼\sum_{|I|=k}s(x,c)\,\mathrm{d}x^{I},∑ start_POSTSUBSCRIPT | italic_I | = italic_k end_POSTSUBSCRIPT italic_s ( italic_x , italic_c ) roman_d italic_x start_POSTSUPERSCRIPT italic_I end_POSTSUPERSCRIPT , where s(x,c)𝑠𝑥𝑐s(x,c)italic_s ( italic_x , italic_c ) are sections of the structure sheaf 𝒪𝒳.subscript𝒪𝒳\mathcal{O}_{\cal X}.caligraphic_O start_POSTSUBSCRIPT caligraphic_X end_POSTSUBSCRIPT .
**C**: For c∈ℂA,𝑐superscriptℂ𝐴c\in\operatorname{\mathbb{C}}^{A},italic_c ∈ blackboard_C start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT , the fiber of π𝜋\piitalic_π is X(c).𝑋𝑐X(c).italic_X ( italic_c ) .
Note that Hn(X(c),ω(c))superscript𝐻𝑛𝑋𝑐𝜔𝑐H^{n}(X(c),\omega(c))italic_H start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( italic_X ( italic_c ) , italic_ω ( italic_c ) ) depends rationally on the ci.subscript𝑐𝑖c_{i}.italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT .
| CAB | CBA | CBA | CBA | Selection 3 |
As a consequence, many methods have been developed over the years that use insights from deterministic problems to make the Bayesian problem more tractable, or to approximate the latter. The result are methods that lie somewhere on the spectrum between the deterministic and Bayesian end points; for example, one could try to approximate p(q|z)𝑝conditional𝑞𝑧p(q|z)italic_p ( italic_q | italic_z ) by a Gaussian in return for substantial computational savings. <|MaskedSetence|> <|MaskedSetence|> In between, a range of hybrid methods can be positioned. <|MaskedSetence|> | **A**: The extremes of the spectrum are occupied by PDE-constrained optimization (used for deterministic optimization problems: computationally efficient but statistically not very insightful) and MCMC solvers for Bayesian inference (statistically accurate but computationally very expensive).
**B**: Figure 1 provides a notional characterization of how different well-known methods could be positioned on a spectrum that takes into account statistical accuracy and computational efficiency.
**C**: The comparison is of course not straightforward but conceptual; for example, some information can be computed online or offline, and some methods require sequential computations whereas others can do things simultaneously in parallel.
Nevertheless, the characterization provides a general perspective of utilizing information in inverse problems.
.
| BAC | BAC | CAB | BAC | Selection 2 |
<|MaskedSetence|> Let σ:{0,…,2m−1}→{0,…,2m−1}:𝜎→0…2𝑚10…2𝑚1\sigma:\{0,\ldots,2m-1\}\rightarrow\{0,\ldots,2m-1\}italic_σ : { 0 , … , 2 italic_m - 1 } → { 0 , … , 2 italic_m - 1 } be the involution defined by σ(i)≠i𝜎𝑖𝑖\sigma(i)\neq iitalic_σ ( italic_i ) ≠ italic_i and h¯i=h¯σ(i)subscript¯ℎ𝑖subscript¯ℎ𝜎𝑖\bar{h}_{i}=\bar{h}_{\sigma(i)}over¯ start_ARG italic_h end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = over¯ start_ARG italic_h end_ARG start_POSTSUBSCRIPT italic_σ ( italic_i ) end_POSTSUBSCRIPT for all 0≤i≤2m−10𝑖2𝑚10\leq i\leq 2m-10 ≤ italic_i ≤ 2 italic_m - 1. Since h0=h2m−1=1Hsubscriptℎ0subscriptℎ2𝑚1subscript1𝐻h_{0}=h_{2m-1}=1_{H}italic_h start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_h start_POSTSUBSCRIPT 2 italic_m - 1 end_POSTSUBSCRIPT = 1 start_POSTSUBSCRIPT italic_H end_POSTSUBSCRIPT, one has σ(0)≠1,2m−1𝜎012𝑚1\sigma(0)\neq 1,2m-1italic_σ ( 0 ) ≠ 1 , 2 italic_m - 1. By Lemma 12, there exists permutations μi:ℤ2n→ℤ2n:subscript𝜇𝑖→subscriptℤsuperscript2𝑛subscriptℤsuperscript2𝑛\mu_{i}:\mathbb{Z}_{2^{n}}\rightarrow\mathbb{Z}_{2^{n}}italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : blackboard_Z start_POSTSUBSCRIPT 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT → blackboard_Z start_POSTSUBSCRIPT 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT, i≥0𝑖0i\geq 0italic_i ≥ 0, such that conditions (i)-(ii) in Lemma 12 hold. <|MaskedSetence|> <|MaskedSetence|> | **A**: We now define the sequence 𝐠:g0,…,g2n+1m−1:𝐠subscript𝑔0…subscript𝑔superscript2𝑛1𝑚1{\bf g}:g_{0},\ldots,g_{2^{n+1}m-1}bold_g : italic_g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_g start_POSTSUBSCRIPT 2 start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT italic_m - 1 end_POSTSUBSCRIPT, in G=H×ℤ2n𝐺𝐻subscriptℤsuperscript2𝑛G=H\times\mathbb{Z}_{2^{n}}italic_G = italic_H × blackboard_Z start_POSTSUBSCRIPT 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT, and the sequence 𝐭:t0,…,t2n+1m−1:𝐭subscript𝑡0…subscript𝑡superscript2𝑛1𝑚1{\bf t}:t_{0},\ldots,t_{2^{n+1}m-1}bold_t : italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_t start_POSTSUBSCRIPT 2 start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT italic_m - 1 end_POSTSUBSCRIPT, in ℤ2nsubscriptℤsuperscript2𝑛\mathbb{Z}_{2^{n}}blackboard_Z start_POSTSUBSCRIPT 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT end_POSTSUBSCRIPT, by letting g0=1Gsubscript𝑔0subscript1𝐺g_{0}=1_{G}italic_g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 1 start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT, t0=0subscript𝑡00t_{0}=0italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 0, and
.
**B**: Given j≥0𝑗0j\geq 0italic_j ≥ 0 with j=i(mod2m)𝑗annotated𝑖pmod2𝑚j=i\pmod{2m}italic_j = italic_i start_MODIFIER ( roman_mod start_ARG 2 italic_m end_ARG ) end_MODIFIER and i∈{0,…,2m−1}𝑖0…2𝑚1i\in\{0,\ldots,2m-1\}italic_i ∈ { 0 , … , 2 italic_m - 1 }, we let hj=hisubscriptℎ𝑗subscriptℎ𝑖h_{j}=h_{i}italic_h start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT.
**C**:
Proof.
Let |H|=m𝐻𝑚|H|=m| italic_H | = italic_m and 𝐡:h0,…,h2m−1:𝐡subscriptℎ0…subscriptℎ2𝑚1{\bf h}:h_{0},\ldots,h_{2m-1}bold_h : italic_h start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_h start_POSTSUBSCRIPT 2 italic_m - 1 end_POSTSUBSCRIPT, be a cyclic D-sequence in H𝐻Hitalic_H.
| CBA | ABC | CBA | CBA | Selection 3 |
<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> We actually prove a version of Theorem 1.4 for general FAD-systems, see Theorem LABEL:thmerrorterm. This, for example, encompasses the orbit asymptotics for ergodic (not necessarily hyperbolic) endomorphisms of real tori (Waddington). Dynamical analogues of the Prime Number Theorem abound, following work of Sinai Sinai and Margulis Margulis on counting geodesics on manifolds with negative curvature (interpreted as a problem in dynamics for the geodesic flow, which in this case satisfies ‘Axiom A’); see e.g. Parry and Policott PP, or PoSh for a result including an error term.. | **A**: When applied to the Frobenius endomorphism of a Jacobian of a curve over a finite field, it relates to Weil’s theorem about the growth of class numbers of function fields under extensions of the ground field (see Example LABEL:exam:Weilstheorem), and to the analogue of the Prime Polynomial Theorem in arbitrary global function fields.
**B**: For the additive group and the Frobenius endomorphism, it specialises to the Prime Polynomial Theorem, which is essentially due to Gauß (§342–347 of the drafted eighth chapter ‘Disquisitiones generales de congruentiis’ of the Disquisitiones Arithmeticae, cf. Frei).
**C**:
In fact, for an algebraic group the analogue of the topological entropy conjecture of Shub holds in the following strong sense: the difference between the entropy (defined, as above, as the logarithmic growth rate of the number of fixed points) and the logarithm of the spectral radius (largest absolute value of eigenvalues of σ𝜎\sigmaitalic_σ acting on the total cohomology) is precisely the unipotent entropy.
Theorem 1.4 subsumes various results from the literature.
| CBA | CBA | BAC | CBA | Selection 1 |
Various other notions of rank, such as border rank and cactus rank, appear in the study of higher secant varieties and are closely related to the rank.
The cactus rank is the minimal length of an apolar subscheme to F𝐹Fitalic_F, while the border rank is the minimal r𝑟ritalic_r such that F𝐹Fitalic_F is a limit of forms of rank r𝑟ritalic_r. For an extensive description and usage of the classical concept of apolarity, we refer to (Iarrobino, Kanev, 1999) and (Ranestad, Schreyer, 2000) and the references therein, which go back to the late XIX century with A. Clebsch, J. Lüroth, T. <|MaskedSetence|> Scorza and to the beginning of the XX century with E. Lasker, F. H. S. Macaulay, J. J. <|MaskedSetence|> Terracini and E. K. <|MaskedSetence|> | **A**: Wakeford.
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**B**: Reye, G.
**C**: Sylvester, A.
| BCA | ABC | BCA | BCA | Selection 4 |
<|MaskedSetence|> This follows from the
Campbell-Baker-Hausdorff formula, since for vectors A∈logG𝐴𝐺A\in\log Gitalic_A ∈ roman_log italic_G, and X∈logN𝑋𝑁X\in\log Nitalic_X ∈ roman_log italic_N, we have exp(A)exp(X)exp(A−1)=exp(rX)=exp(X+[A,X]+c2[A,[A,X]]+…)𝐴𝑋superscript𝐴1𝑟𝑋𝑋𝐴𝑋subscript𝑐2𝐴𝐴𝑋…\exp\left(A\right)\exp\left(X\right)\exp\left(A^{-1}\right)=\exp\left(rX\right%
)=\exp\left(X+\left[A,X\right]+c_{2}\left[A,\left[A,X\right]\right]+...\right)roman_exp ( italic_A ) roman_exp ( italic_X ) roman_exp ( italic_A start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) = roman_exp ( italic_r italic_X ) = roman_exp ( italic_X + [ italic_A , italic_X ] + italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT [ italic_A , [ italic_A , italic_X ] ] + … ). <|MaskedSetence|> Because G𝐺Gitalic_G is nilpotent, {X,[A,X],[A,[A,X]],…,ad(A)k−1(X)}𝑋𝐴𝑋𝐴𝐴𝑋…adsuperscript𝐴𝑘1𝑋\left\{X,\left[A,X\right],\left[A,\left[A,X\right]\right],...,\mathrm{ad}\left%
(A\right)^{k-1}\left(X\right)\right\}{ italic_X , [ italic_A , italic_X ] , [ italic_A , [ italic_A , italic_X ] ] , … , roman_ad ( italic_A ) start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ( italic_X ) } is linearly independent. Since rX=X+[A,X]+c2[A,[A,X]]+…𝑟𝑋𝑋𝐴𝑋subscript𝑐2𝐴𝐴𝑋…rX=X+\left[A,X\right]+c_{2}\left[A,\left[A,X\right]\right]+...italic_r italic_X = italic_X + [ italic_A , italic_X ] + italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT [ italic_A , [ italic_A , italic_X ] ] + …, we have [A,X]=0𝐴𝑋0\left[A,X\right]=0[ italic_A , italic_X ] = 0. <|MaskedSetence|> | **A**:
By normality, N⊂Z(G)𝑁𝑍𝐺N\subset Z\left(G\right)italic_N ⊂ italic_Z ( italic_G ).
**B**: Let ad(A)k(X)adsuperscript𝐴𝑘𝑋\mathrm{ad}\left(A\right)^{k}\left(X\right)roman_ad ( italic_A ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_X ) be the first zero element of the sequence (X,[A,X],[A,[A,X]],…)𝑋𝐴𝑋𝐴𝐴𝑋…\left(X,\left[A,X\right],\left[A,\left[A,X\right]\right],...\right)( italic_X , [ italic_A , italic_X ] , [ italic_A , [ italic_A , italic_X ] ] , … ).
**C**: Note that since π(N)=1𝜋𝑁1\pi\left(N\right)=1italic_π ( italic_N ) = 1, if m(π,Uε)≠0𝑚𝜋subscript𝑈𝜀0m\left(\pi,U_{\varepsilon}\right)\neq 0italic_m ( italic_π , italic_U start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT ) ≠ 0, Uε(n)f=fsubscript𝑈𝜀𝑛𝑓𝑓U_{\varepsilon}\left(n\right)f=fitalic_U start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT ( italic_n ) italic_f = italic_f for.
| ABC | ABC | ABC | CAB | Selection 3 |
In order to describe module D(𝒜)𝐷𝒜D(\mathcal{A})italic_D ( caligraphic_A ) of any arrangement in general, one has an unique tool up to now: Gröbner basis (see [2]). <|MaskedSetence|> Therefore, it is very difficult to control combinatorial data as well as freeness. <|MaskedSetence|> In Section 2 we give some results that they will lead us to a system of equations describing combinatorial data of D(𝒜)𝐷𝒜D(\mathcal{A})italic_D ( caligraphic_A ); to recognize the essence of combinatorial determined property lies entirely in the non-homogeneous parts. <|MaskedSetence|> Consequently, when it is free, derivations θisubscript𝜃𝑖\theta_{i}italic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT of a basis of D(𝒜)𝐷𝒜D(\mathcal{A})italic_D ( caligraphic_A ) satisfy that sufficient condition. They yields a proof for the Terao’s conjecture in the last corollary.
. | **A**: In Section 3 we present a combinatorial structure of generators, verify the example of Ziegler and give a sufficient condition on combinatorial determined property as a generalization.
**B**: In this paper, we introduce a new approach to the problems without using Gröbner basis: to investigate its system of equations instead of module D(𝒜)𝐷𝒜D(\mathcal{A})italic_D ( caligraphic_A ).
The organization of this is as follows.
**C**: It is easy to use this tool for computer but there are many computations which would be extremely intractable to do by hand.
| BCA | CBA | CBA | CBA | Selection 3 |
<|MaskedSetence|> [7, 6] to calculate the local crossing number of each of the 3315 order types of 8 points. Our calculations verified that lcr¯(K8)≥4¯lcrsubscript𝐾84\operatorname{\overline{lcr}}(K_{8})\geq 4start_OPFUNCTION over¯ start_ARG roman_lcr end_ARG end_OPFUNCTION ( italic_K start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT ) ≥ 4 for all of them, with equality achieved by 39 order types. For n=14𝑛14n=14italic_n = 14, Aichholzer [5] extended his data base from n=11𝑛11n=11italic_n = 11 to n=14𝑛14n=14italic_n = 14 for this specific problem. <|MaskedSetence|> Thus lcr¯(K14)=15¯lcrsubscript𝐾1415\operatorname{\overline{lcr}}(K_{14})=15start_OPFUNCTION over¯ start_ARG roman_lcr end_ARG end_OPFUNCTION ( italic_K start_POSTSUBSCRIPT 14 end_POSTSUBSCRIPT ) = 15.
Figure 5: The construction for n∈{11,14,17}𝑛111417n\in\{11,14,17\}italic_n ∈ { 11 , 14 , 17 }. <|MaskedSetence|> | **A**: The 11-point set consists of the black points, the 14-point set consists of the black or white points, and the 17-point set consists of all of the points shown.
.
**B**: He confirmed our results for n<14𝑛14n<14italic_n < 14 and verified our earlier conjecture [1] that there are no geometric sets of 14 points where every edge is crossed at most 14 times.
**C**: Finally, for n=8𝑛8n=8italic_n = 8, we used the data base by Aichholzer et al.
| CBA | CBA | CBA | CAB | Selection 3 |
<|MaskedSetence|> We show that any eventually positive system is also positive with respect to some cone, which however may be hard to characterize explicitly or it may be hard to use. We also derive straightforward formulas to compute invariant cones and Lyapunov functions for eventually positive systems under some additional assumptions on the system matrix. <|MaskedSetence|> <|MaskedSetence|> In fact, it was shown that a subset of scaled diagonally dominant matrices admit diagonal Lyapunov functions [13], which can be computed by linear programming/algebra [14]. These results were then extended to block-partitioned matrices in [15]. Eventually positive systems can be considered as a complimentary extension of positive systems with a different set of retained properties.. | **A**: We then consider positive input-output systems, which are characterized by eventually positive trajectories of the states and positive outputs.
**B**: We extend some of the properties of positive systems to this case such as: computation of induced norms, properties of energy functions and discuss implications for model reduction.
It is worth mentioning that some of the strong properties of positive systems exploit scaled diagonal dominance of the system matrices rather than nonnegativity of the trajectories.
**C**:
We first consider dynamical eventually positive systems, which are studied in [12] from the linear algebra point of view and we offer a control-theoretic one.
| BCA | CAB | CAB | CAB | Selection 2 |
<|MaskedSetence|> Two notable applications are combinatorics and learning theory. Two representative results are the concentration of the chromatic number of Erdos-Rényi graphs [1], and the fact that stable algorithms have good generalization performance [2]. Namely, if the output of a learning algorithm does not change too much when a single training example is modified, then it performs well on an unseen example. <|MaskedSetence|> <|MaskedSetence|> | **A**: An overview of how McDiarmid’s inequality can be applied to several problems in information theory is found in [13].
**B**: For instance, [8] applies McDiarmid’s inequality to solve the problem of channel resolvability.
.
**C**: 2 Related work
The strength of McDiarmid’s inequality lies in its applicability (see [10] for an extensive survey): set 𝒳𝒳{\cal X}caligraphic_X may be completely arbitrary, and, even when f𝑓fitalic_f is involved, it is usually easy to check that the bounded differences assumption holds.
| CAB | CAB | BAC | CAB | Selection 4 |
<|MaskedSetence|> It is a great pleasure to thank the Institut für Algebraische Geometrie for generous fundings and excellent working conditions. <|MaskedSetence|> The first-named author would like to thank Davide Cesare Veniani for sharing his insights on some of the examples. The second author would like to express his gratitude to Łukasz Sienkiewicz for pointing out Vinberg’s paper [16] and his construction, and to Stefan Müller-Stach for useful comments. <|MaskedSetence|> | **A**:
Acknowledgement.
The present paper has grown out from discussions the authors had during the time when Piotr Pokora was visiting Roberto Laface at the Leibniz Universität Hannover.
**B**: At last, we would like to warmly thank the anonymous referee for his/her very useful comments and suggestions which allowed to improve our results.
.
**C**: The authors would like to express their gratitude to Klaus Hulek and Matthias Schütt for stimulating discussions about the topic, Igor Dolgachev for pointing out Example 3.8, and Bert van Geemen for sharing his insights on the subject.
| ACB | ACB | ABC | ACB | Selection 2 |
Therefore our main contribution in this paper is to establish the rate of convergence of resolvents operators, eigenvalues and equilibria for the problem (4) to be able to apply the results of [5] to the continuity of attractors. In addition, for the particular model considered, our work improves the works [1, 7] and [8] (in what concerns the continuity of attractors), where continuity of attractors was proved without any rate. <|MaskedSetence|> In the Section 2 we make the study of the elliptic problem in order to find a rate of attraction for the resolvent operators. <|MaskedSetence|> <|MaskedSetence|> | **A**: The current paper is the first to consider the rate of convergence of attractors for parabolic problems with localized large diffusion.
This paper is organized as follows.
**B**: In Section 4 we obtain the rate of convergence of invariant manifolds and in the Section 5 we reduce the system to finite dimensions and we finally obtain a rate of convergence of attractors..
**C**: In the Section 3 we exhibit a rate of attraction for the eigenvalues and equilibrium points.
| ACB | ACB | CAB | ACB | Selection 2 |
One way to visualize and study k𝑘kitalic_k-page book drawings is by using what we called the convex model. In this model, a given k𝑘kitalic_k-page book drawing of a graph G𝐺Gitalic_G is drawn on the plane as an edge-colored convex drawing as follows: The spine is now a circle C𝐶Citalic_C, or more generally, a simple convex curve. <|MaskedSetence|> <|MaskedSetence|> Using this model, the problem of determining νk(Kn)subscript𝜈𝑘subscript𝐾𝑛\nu_{k}(K_{n})italic_ν start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_K start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) is equivalent to finding the minimum number of monochromatic crossings in a k𝑘kitalic_k-edge coloring of a convex drawing of Knsubscript𝐾𝑛K_{n}italic_K start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT. The subgraph (subdrawing) induced by each of the colors is known as a convex or outerplanar graph (drawing). <|MaskedSetence|> Since we are interested in crossings, it is often convenient to disregard the sides of the underlying polygon as edges. We denote by Dnsubscript𝐷𝑛D_{n}italic_D start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT the complete convex graph minus all the edges corresponding to the sides of the underlying polygon. Let eℓ(n)subscript𝑒ℓ𝑛e_{\ell}(n)italic_e start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( italic_n ) be the maximum number of edges over all convex subgraphs of Dnsubscript𝐷𝑛D_{n}italic_D start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT in which each edge is crossed at most ℓℓ\ellroman_ℓ times. Such graphs are said to have local crossing number at most ℓℓ\ellroman_ℓ. For example, planar graphs have local crossing number 0, the graphs in Figure 1(b) have local crossing number 1, and those in 1(e) have local crossing number 4. (Graphs with local crossing number at most ℓℓ\ellroman_ℓ are called ℓℓ\ellroman_ℓ-planar graphs [17]. If they are also convex, then they are called outer ℓℓ\ellroman_ℓ-planar graphs [8, 12, 3].) Local crossing numbers of convex graphs were studied by Kainen [11, 12]. The problem of maximizing the number of edges over convex graphs satisfying certain crossing conditions was studied by Brass, Károlyi, and Valtr [5]. Functions equivalent to eℓ(n)subscript𝑒ℓ𝑛e_{\ell}(n)italic_e start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( italic_n ) for general drawings of graphs in the plane were studied by Pach and Tóth [15]; Pach, Radoic̆ić, Tardos, and Tóth [14]; and Ackerman [2].
In Section 2.1, we prove the following theorem that relates the functions eℓ(n)subscript𝑒ℓ𝑛e_{\ell}(n)italic_e start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( italic_n ) to the k𝑘kitalic_k-page book crossing numbers νk(Kn)subscript𝜈𝑘subscript𝐾𝑛\nu_{k}(K_{n})italic_ν start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_K start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ). Theorem 3 is used to prove Conjecture 1 for 2k<n≤3k2𝑘𝑛3𝑘2k<n\leq 3k2 italic_k < italic_n ≤ 3 italic_k, the asymptotic bound of Theorem 2, and the lower bound improvements in Table 2.
. | **A**: The vertices of G𝐺Gitalic_G are placed on C𝐶Citalic_C, typically forming the set of vertices of a convex polygon inscribed in C𝐶Citalic_C.
**B**: We denote by Gnsubscript𝐺𝑛G_{n}italic_G start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT the complete convex graph, that is, any convex drawing of Knsubscript𝐾𝑛K_{n}italic_K start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT.
**C**: The edges are diagonals or sides (straight line segments) of the polygon that are k𝑘kitalic_k-colored in such a way that two edges get the same color if and only if they originally were on the same page.
| ACB | ACB | ACB | BAC | Selection 3 |
In this paper we focus on the case of a finite-dimensional normed vector space X𝑋Xitalic_X with a polyhedral norm, meaning that the unit ball of the norm is a convex polytope containing the origin in its interior. <|MaskedSetence|> An advantage of polyhedral norms is that in this case all horofunctions are Busemann points, and that, by Walsh’s result in [Wal07], the set of horofunctions can be determined explicitly. To see the underlying topology of the set of horofunctions, the second author of this paper used Walsh’s description to characterize converging sequences in the horofunction compactification of finite-dimensional vector spaces with polyhedral norms in her diploma thesis [Sch14]. <|MaskedSetence|> <|MaskedSetence|> | **A**: This connection was also noticed by Kapovich and Leeb [KL18], who posed the following question:
.
**B**: This result is also given as Theorem 3.11 in this paper and shows a deep connection between the horofunction compactification and the shape of the dual unit ball, a convex polytope in the dual space X∗superscript𝑋X^{*}italic_X start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT.
**C**: For the same setting, Karlsson, Metz and Noskov [KMN06] use the blow-up-and-shift-technique to characterize the horofunction compactification, whereas Ciobotaru, Kramer and Schwer [CKS23] use ultrapowers.
| ACB | CBA | CBA | CBA | Selection 4 |
<|MaskedSetence|> The circuit C1(C,v)subscript𝐶1𝐶𝑣C_{1}(C,v)italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_C , italic_v ) includes the passage hv1,v,hv4superscriptsubscriptℎ𝑣1𝑣superscriptsubscriptℎ𝑣4h_{v}^{1},v,h_{v}^{4}italic_h start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT , italic_v , italic_h start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT and no other passage through v𝑣vitalic_v. If ϕC(v)=P(v)subscriptitalic-ϕ𝐶𝑣𝑃𝑣\phi_{C}(v)=P(v)italic_ϕ start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_v ) = italic_P ( italic_v ) then the initial half-edge of evsubscript𝑒𝑣e_{v}italic_e start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT is hv1¯=hv2¯¯superscriptsubscriptℎ𝑣1¯superscriptsubscriptℎ𝑣2\overline{h_{v}^{1}}=\overline{h_{v}^{2}}over¯ start_ARG italic_h start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT end_ARG = over¯ start_ARG italic_h start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG, and the terminal half-edge is hv3¯=hv4¯¯superscriptsubscriptℎ𝑣3¯superscriptsubscriptℎ𝑣4\overline{h_{v}^{3}}=\overline{h_{v}^{4}}over¯ start_ARG italic_h start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG = over¯ start_ARG italic_h start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG, so C1(C,v)¯¯subscript𝐶1𝐶𝑣\overline{C_{1}(C,v)}over¯ start_ARG italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_C , italic_v ) end_ARG traverses evsubscript𝑒𝑣e_{v}italic_e start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT in the
positive direction. If χC(v)=P(v)subscript𝜒𝐶𝑣𝑃𝑣\chi_{C}(v)=P(v)italic_χ start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_v ) = italic_P ( italic_v ) then the initial half-edge of evsubscript𝑒𝑣e_{v}italic_e start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT is hv1¯=hv4¯¯superscriptsubscriptℎ𝑣1¯superscriptsubscriptℎ𝑣4\overline{h_{v}^{1}}=\overline{h_{v}^{4}}over¯ start_ARG italic_h start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT end_ARG = over¯ start_ARG italic_h start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG, and the terminal
half-edge is hv2¯=hv3¯¯superscriptsubscriptℎ𝑣2¯superscriptsubscriptℎ𝑣3\overline{h_{v}^{2}}=\overline{h_{v}^{3}}over¯ start_ARG italic_h start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG = over¯ start_ARG italic_h start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG, so C1(C,v)¯¯subscript𝐶1𝐶𝑣\overline{C_{1}(C,v)}over¯ start_ARG italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_C , italic_v ) end_ARG does not traverse evsubscript𝑒𝑣e_{v}italic_e start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT. <|MaskedSetence|> <|MaskedSetence|> | **A**: Suppose v∈V(F)𝑣𝑉𝐹v\in V(F)italic_v ∈ italic_V ( italic_F ).
**B**: We have the following..
**C**: If ψC(v)=P(v)subscript𝜓𝐶𝑣𝑃𝑣\psi_{C}(v)=P(v)italic_ψ start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_v ) = italic_P ( italic_v ) then the initial half-edge of evsubscript𝑒𝑣e_{v}italic_e start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT is hv1¯=hv3¯¯superscriptsubscriptℎ𝑣1¯superscriptsubscriptℎ𝑣3\overline{h_{v}^{1}}=\overline{h_{v}^{3}}over¯ start_ARG italic_h start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT end_ARG = over¯ start_ARG italic_h start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG,
and the terminal half-edge is hv2¯=hv4¯¯superscriptsubscriptℎ𝑣2¯superscriptsubscriptℎ𝑣4\overline{h_{v}^{2}}=\overline{h_{v}^{4}}over¯ start_ARG italic_h start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG = over¯ start_ARG italic_h start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG, so
C1(C,v)¯¯subscript𝐶1𝐶𝑣\overline{C_{1}(C,v)}over¯ start_ARG italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_C , italic_v ) end_ARG traverses evsubscript𝑒𝑣e_{v}italic_e start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT in the positive direction.
| ABC | ACB | ACB | ACB | Selection 3 |
<|MaskedSetence|> As (𝒳,𝒴)𝒳𝒴(\mathcal{X},\mathcal{Y})( caligraphic_X , caligraphic_Y ) is complete, there is an exact sequence of complexes 0→Y′→X′→A→0→0superscript𝑌′→superscript𝑋′→𝐴→00\rightarrow Y^{\prime}\rightarrow X^{\prime}\rightarrow A\rightarrow 00 → italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT → italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT → italic_A → 0 with X′∈𝒳superscript𝑋′𝒳X^{\prime}\in\mathcal{X}italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ caligraphic_X and Y′∈𝒴superscript𝑌′𝒴Y^{\prime}\in\mathcal{Y}italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ caligraphic_Y. <|MaskedSetence|> As X→A→𝑋𝐴X\rightarrow Aitalic_X → italic_A factors through X′superscript𝑋′X^{\prime}italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT (even in C(𝒜)𝐶𝒜C(\mathcal{A})italic_C ( caligraphic_A )), it follows from condition (2) in [K, Lemma 4.7.1] that J𝐽Jitalic_J is fully faithful. <|MaskedSetence|> Given
any complex M𝑀Mitalic_M take exact sequence of complexes 0→Y→X→M→0→0𝑌→𝑋→𝑀→00\rightarrow Y\rightarrow X\rightarrow M\rightarrow 00 → italic_Y → italic_X → italic_M → 0
with X∈𝒳𝑋𝒳X\in\mathcal{X}italic_X ∈ caligraphic_X and Y∈𝒴𝑌𝒴Y\in\mathcal{Y}italic_Y ∈ caligraphic_Y. Since Y𝑌Yitalic_Y is, in particular, acyclic, the morphism
X→M→𝑋𝑀X\rightarrow Mitalic_X → italic_M is an isomorphism in D(𝒜)𝐷𝒜D(\mathcal{A})italic_D ( caligraphic_A ), so X∈K(𝒳)𝑋𝐾𝒳X\in K(\mathcal{X})italic_X ∈ italic_K ( caligraphic_X ) with J(X)≃Msimilar-to-or-equals𝐽𝑋𝑀J(X)\simeq Mitalic_J ( italic_X ) ≃ italic_M in D(𝒜)𝐷𝒜D(\mathcal{A})italic_D ( caligraphic_A ).
. | **A**: As A𝐴Aitalic_A and Y′superscript𝑌′Y^{\prime}italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT are acyclic (by assumption on 𝒴𝒴\mathcal{Y}caligraphic_Y), so is X′superscript𝑋′X^{\prime}italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and hence X′∈Kac(𝒳)superscript𝑋′subscript𝐾ac𝒳X^{\prime}\in K_{{\rm{ac}}}(\mathcal{X})italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_K start_POSTSUBSCRIPT roman_ac end_POSTSUBSCRIPT ( caligraphic_X ).
**B**: and A𝐴Aitalic_A is acyclic.
**C**: It remains to see that J𝐽Jitalic_J is essentially surjective.
| BAC | BAC | BAC | ABC | Selection 3 |
From the introduction: “This paper is concerned with probabilistic aspects of the expansion of points in n𝑛nitalic_n-dimensional Euclidean space. The expansions we consider need not converge although previous work has required convergence……\ldots… In 1869 Jacobi presented an extension of the continued fraction to two dimensions. <|MaskedSetence|> In 1964 Schweiger began an examination of the measure theoretic properties of Jacobi’s algorithm. <|MaskedSetence|> However Schweiger has ……\ldots… published some results which also concern general F𝐹Fitalic_F-expansions for n𝑛nitalic_n-dimensions. The class of algorithms he considers does not include the Jacobi algorithm and is a natural generalization of Rényi. <|MaskedSetence|> We also include a central limit theorem and a law of the iterated logarithm.”
. | **A**: It was this work which motivated our paper.
**B**: Our results generalize most of Schweiger’s work and have the Jacobi algorithm as an example.
**C**: Perron extended Jacobi’s work to n𝑛nitalic_n-dimensions.
| CAB | CBA | CAB | CAB | Selection 4 |
It is indicated Chirvasitu[6] for the standard definition of the Shilov Boundary. It generalizes classical models as the hyperquadrics (see [5],[9],[10],[11],[12],[13],[19],[24]). <|MaskedSetence|> We make computations using formal power series. <|MaskedSetence|> It is an alternative to the Method of Cartan applied by Kim-Zaitsev[16],[17]. It detects an analogue of the geometrical rank introduced by Huang[10],[11]. <|MaskedSetence|> The main result is the following:
. | **A**: This new invariant is defined by several matrices having identical rank.
**B**: It has a fundamental importance in the study of Holomorphic Isometries in Complex Analysis and Complex Geometry (see [23],[27],[28]) and their properties.
In this, we use the language of matrices in order to establish a normal form type construction (see [2],[3],[29],[30]) for Formal (Holomorphic) Embeddings between Shilov Boundaries of Bounded Symmetric Domains of First Type (see [16],[17],[18],[25]).
**C**: In particular, we use suitale linear changes of coordinates in order to implement procedures from Baouendi-Huang[1], Chern-Moser[4], Hamada[9], Huang[10],[11], Huang-Ji[12] and Kim-Zaitsev[16],[17].
| BCA | BCA | BCA | BCA | Selection 4 |
<|MaskedSetence|> <|MaskedSetence|> Hence, a point s0subscript𝑠0s_{0}italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is a pole of the function ν(s)𝜈𝑠\nu(s)italic_ν ( italic_s ) if and only if lims→s0Γ(s¯)cos(πs¯2)=0subscript→𝑠subscript𝑠0Γ¯𝑠𝜋¯𝑠20\lim_{s\to s_{0}}\,\Gamma(\bar{s})\,\cos\left(\frac{\pi\bar{s}}{2}\right)=0roman_lim start_POSTSUBSCRIPT italic_s → italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT roman_Γ ( over¯ start_ARG italic_s end_ARG ) roman_cos ( divide start_ARG italic_π over¯ start_ARG italic_s end_ARG end_ARG start_ARG 2 end_ARG ) = 0. Such poles occur either when Γ(s¯)Γ¯𝑠\Gamma(\bar{s})roman_Γ ( over¯ start_ARG italic_s end_ARG ) or cos(πs¯2)𝜋¯𝑠2\cos\left(\frac{\pi\bar{s}}{2}\right)roman_cos ( divide start_ARG italic_π over¯ start_ARG italic_s end_ARG end_ARG start_ARG 2 end_ARG ) are equal to zero. The term cos(πs¯2)𝜋¯𝑠2\cos\left(\frac{\pi\bar{s}}{2}\right)roman_cos ( divide start_ARG italic_π over¯ start_ARG italic_s end_ARG end_ARG start_ARG 2 end_ARG ) is equal to zero at the points s=±1,±3,±5,±7,…𝑠plus-or-minus1plus-or-minus3plus-or-minus5plus-or-minus7…s=\pm 1,\pm 3,\pm 5,\pm 7,...italic_s = ± 1 , ± 3 , ± 5 , ± 7 , … The Euler form of the Gamma function is expressed as Γ(s)=1s∏n=1∞()s1+snΓ𝑠1𝑠superscriptsubscriptproduct𝑛1superscript𝑠1𝑠𝑛\Gamma(s)=\frac{1}{s}\,\prod_{n=1}^{\infty}\frac{\left(\right)^{s}}{1+\frac{s}%
{n}}roman_Γ ( italic_s ) = divide start_ARG 1 end_ARG start_ARG italic_s end_ARG ∏ start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT divide start_ARG ( ) start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT end_ARG start_ARG 1 + divide start_ARG italic_s end_ARG start_ARG italic_n end_ARG end_ARG. <|MaskedSetence|> However, the Gamma function tends to infinity at the points s=0,−1,−2,−3,…,−n𝑠0123…𝑛s=0,-1,-2,-3,...,-nitalic_s = 0 , - 1 , - 2 , - 3 , … , - italic_n where n∈ℕ𝑛ℕn\in\mathbb{N}italic_n ∈ blackboard_N, hence the points s=1,3,5,7,…𝑠1357…s=1,3,5,7,...italic_s = 1 , 3 , 5 , 7 , … can be determined to be poles of ν(s)𝜈𝑠\nu(s)italic_ν ( italic_s ) with certainty. The points s=−1,−3,−5,−7,…𝑠1357…s=-1,-3,-5,-7,...italic_s = - 1 , - 3 , - 5 , - 7 , … are poles of ν(s)𝜈𝑠\nu(s)italic_ν ( italic_s ) only if the limit of Γ(s)cos(πs2)Γ𝑠𝜋𝑠2\Gamma(s)\,\cos\left(\frac{\pi s}{2}\right)roman_Γ ( italic_s ) roman_cos ( divide start_ARG italic_π italic_s end_ARG start_ARG 2 end_ARG ) is equal to zero when s𝑠sitalic_s tends to either of these points. The Taylor series of cos(πs2)𝜋𝑠2\cos\left(\frac{\pi s}{2}\right)roman_cos ( divide start_ARG italic_π italic_s end_ARG start_ARG 2 end_ARG ) in −m𝑚-m- italic_m where m𝑚mitalic_m is an odd integer is as follows: cos(πs2)=π2sin(πm2)(s+m)−(π2)3sin(πm2)(s+m)3+…𝜋𝑠2𝜋2𝜋𝑚2𝑠𝑚superscript𝜋23𝜋𝑚2superscript𝑠𝑚3…\cos\left(\frac{\pi s}{2}\right)=\frac{\pi}{2}\sin\left(\frac{\pi m}{2}\right)%. | **A**:
Proof
We use the expression of the reciprocal of ν(s)𝜈𝑠\nu(s)italic_ν ( italic_s ) introduced in proposition no 4.
**B**: We have shown in proof of proposition no 5, that the factor [u2−v2u2+v2+i(−2uvu2+v2)]delimited-[]superscript𝑢2superscript𝑣2superscript𝑢2superscript𝑣2𝑖2𝑢𝑣superscript𝑢2superscript𝑣2\left[\frac{u^{2}-v^{2}}{u^{2}+v^{2}}+i\,\left(\frac{-2uv}{u^{2}+v^{2}}\right)\right][ divide start_ARG italic_u start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_u start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + italic_i ( divide start_ARG - 2 italic_u italic_v end_ARG start_ARG italic_u start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) ] cannot take the value zero and is bounded.
**C**: As ∀n∈ℕ∗for-all𝑛superscriptℕ\forall n\in\,\mathbb{N}^{*}∀ italic_n ∈ blackboard_N start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT the term 1+1n11𝑛1+\frac{1}{n}1 + divide start_ARG 1 end_ARG start_ARG italic_n end_ARG is not equal to zero, we can say that the Gamma function is never equal to zero.
| ABC | ABC | ABC | CAB | Selection 3 |
<|MaskedSetence|> <|MaskedSetence|> Suppose that there exists a face (1-connected region in the complement of the embedded graph) of degree m≥2𝑚2m\geq 2italic_m ≥ 2. Add one vertex (different from the leafs) to each connected component in the boundary of the face and draw a polygon, joining these vertices. Contract this polygon to a point.
This contracting morphism deforms the first graph by defining a new tree, which is locally star-like. <|MaskedSetence|> This enters a singularisation process, where a new m𝑚mitalic_m-fold point arises in the topological model and also on the corresponding perturbed harmonic polynomial.
. | **A**:
Consider a topological model, being an embedded forest.
**B**: We call this operation a complete contracting Whitehead move.
**C**: The graph has an even number of leaves and vertices are of even valency.
| CAB | ACB | ACB | ACB | Selection 2 |
<|MaskedSetence|> <|MaskedSetence|> It introduces all concepts in order to describe the controlled problem. Section 3 gives the characterization of the associated value function as a PDE in the viscosity sense. Section 4 shows the viscosity properties, following and adapting the arguments of [3] in our context. <|MaskedSetence|> Section 6 gives a numerical scheme in order to solve numerically the controlled problem in practice. Finally, section 7 provides a case study of issuing CAT bonds in a optimal way, in the context of Hurricanes in Florida.
. | **A**: Section 5 provides a sufficient condition for a comparison principle for the PDE satisfied by the value function.
**B**:
The paper is organized as follows.
**C**: Section 2 presents the framework.
| BCA | BCA | BCA | BAC | Selection 1 |
obtained from cyclic codes over the ring 𝔽2+v𝔽2subscript𝔽2𝑣subscript𝔽2\mathbb{F}_{2}+v\mathbb{F}_{2}blackboard_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_v blackboard_F start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, v2=vsuperscript𝑣2𝑣v^{2}=vitalic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_v. <|MaskedSetence|> In sari2 , Sarı and Siap extended the results obtained
in qian1 over a class of nonchain rings 𝔽p[v]⟨vp−v⟩subscript𝔽𝑝delimited-[]𝑣delimited-⟨⟩superscript𝑣𝑝𝑣\frac{{{\mathbb{F}_{p}}\left[v\right]}}{{\left\langle{v^{p}-v}\right\rangle}}divide start_ARG blackboard_F start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT [ italic_v ] end_ARG start_ARG ⟨ italic_v start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT - italic_v ⟩ end_ARG. <|MaskedSetence|> obtained new
nonbinary quantum codes from the codes over the ring 𝔽p[u]⟨u3−u⟩subscript𝔽𝑝delimited-[]𝑢delimited-⟨⟩superscript𝑢3𝑢\frac{{{\mathbb{F}_{p}}\left[u\right]}}{{\left\langle{u^{3}-u}\right\rangle}}divide start_ARG blackboard_F start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT [ italic_u ] end_ARG start_ARG ⟨ italic_u start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT - italic_u ⟩ end_ARG. In tang , Tang et al. <|MaskedSetence|> | **A**: In ashraf , the authors considered quantum codes as the Gray
images of the cyclic codes over the ring 𝔽3+v𝔽3subscript𝔽3𝑣subscript𝔽3\mathbb{F}_{3}+v\mathbb{F}_{3}blackboard_F start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_v blackboard_F start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, v3=vsuperscript𝑣3𝑣v^{3}=vitalic_v start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT = italic_v.
**B**: utilized (1+u)1𝑢\left(1+u\right)( 1 + italic_u )-constacyclic codes over the ring 𝔽2m+u𝔽2msubscript𝔽superscript2𝑚𝑢subscript𝔽superscript2𝑚{\mathbb{F}_{{2^{m}}}}+u{\mathbb{F}_{{2^{m}}}}blackboard_F start_POSTSUBSCRIPT 2 start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT end_POSTSUBSCRIPT + italic_u blackboard_F start_POSTSUBSCRIPT 2 start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT end_POSTSUBSCRIPT to construct quantum codes.
.
**C**: More recently, in bag , Bag et al.
| ACB | ACB | ACB | CAB | Selection 1 |
<|MaskedSetence|> In Section 2 we collect preliminaries for nonpositively curved spaces and the groups of isometries. The concept of transverse space is introduced. In Section 3 we prove an asymptotic Margulis Lemma, which deals with the case when the transverse space is bounded. <|MaskedSetence|> In Section 5 we show that the fundamental group of each end as stated in Theorem 1.2 has vanishing algebraic entropy. The proof of Theorem 1.2 is completed in Section 6. <|MaskedSetence|> Finally in the Appendix we construct a complete manifold of finite volume with bounded nonpositive curvature whose universal cover is a visibility manifold but not a Gromov hyperbolic space.
. | **A**:
The paper is organized as follows.
**B**: In Section 7 we prove Corollary 1.9, 1.10 and 1.11.
**C**: A converse to the main result of Section 3 is shown in Section 4.
| BAC | ACB | ACB | ACB | Selection 3 |
<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Indeed, the two approaches mentioned above are not applicable as they rely strongly on the smoothing effect of the semigroup. In this work we tackle this problem and develop a technique that allows one to establish optimal weak convergence rates for hyperbolic SPDEs with multiplicative noise. A special case of our main result is presented in the following theorem.
. | **A**: One is based on regularity results for the corresponding Kolmogorov equation and Malliavin calculus; see, e.g., [BD18, Deb11].
**B**: The other is based on more elementary regularity results of the Kolmogorov equation and the mild Itô formula; see, e.g., [CJK19, HJK16, JK21].
No successful approach for proving optimal weak convergence rates has been developed yet for temporal discretisations of hyperbolic SPDEs with multiplicative noise.
**C**: Roughly speaking, there are two successful approaches to obtain optimal weak convergence rates for parabolic SPDEs with multiplicative noise.
| CAB | CAB | CAB | CAB | Selection 3 |
set π.α:=(απ(1),…,απ(r))formulae-sequence𝜋assign𝛼subscript𝛼𝜋1…subscript𝛼𝜋𝑟\pi.\alpha:=(\alpha_{\pi(1)},\dots,\alpha_{\pi(r)})italic_π . italic_α := ( italic_α start_POSTSUBSCRIPT italic_π ( 1 ) end_POSTSUBSCRIPT , … , italic_α start_POSTSUBSCRIPT italic_π ( italic_r ) end_POSTSUBSCRIPT ).
We have the following pair of commutative diagrams.
(Rα,k)𝔖αsuperscriptsubscript𝑅𝛼𝑘subscript𝔖𝛼(R_{\alpha,k})^{{\mathfrak{S}}_{\alpha}}( italic_R start_POSTSUBSCRIPT italic_α , italic_k end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT fraktur_S start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_POSTSUPERSCRIPT(Rπ.α)𝔖π.αsuperscriptsubscript𝑅formulae-sequence𝜋𝛼subscript𝔖formulae-sequence𝜋𝛼(R_{\pi.\alpha})^{{\mathfrak{S}}_{\pi.\alpha}}( italic_R start_POSTSUBSCRIPT italic_π . italic_α end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT fraktur_S start_POSTSUBSCRIPT italic_π . <|MaskedSetence|> italic_α end_POSTSUBSCRIPT ; blackboard_Z )(Sα,k)𝔖αsuperscriptsubscript𝑆𝛼𝑘subscript𝔖𝛼(S_{\alpha,k})^{{\mathfrak{S}}_{\alpha}}( italic_S start_POSTSUBSCRIPT italic_α , italic_k end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT fraktur_S start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_POSTSUPERSCRIPT(Sπ.α)𝔖π.αsuperscriptsubscript𝑆formulae-sequence𝜋𝛼subscript𝔖formulae-sequence𝜋𝛼(S_{\pi.\alpha})^{{\mathfrak{S}}_{\pi.\alpha}}( italic_S start_POSTSUBSCRIPT italic_π . <|MaskedSetence|> italic_α end_POSTSUBSCRIPT end_POSTSUPERSCRIPTH∙(Xα,k;ℚ)superscript𝐻∙subscript𝑋𝛼𝑘ℚH^{\bullet}(X_{\alpha,k};{\mathbb{Q}})italic_H start_POSTSUPERSCRIPT ∙ end_POSTSUPERSCRIPT ( italic_X start_POSTSUBSCRIPT italic_α , italic_k end_POSTSUBSCRIPT ; blackboard_Q )H∙(Xπ.α;ℚ)superscript𝐻∙subscript𝑋formulae-sequence𝜋𝛼ℚH^{\bullet}(X_{\pi.\alpha};{\mathbb{Q}})italic_H start_POSTSUPERSCRIPT ∙ end_POSTSUPERSCRIPT ( italic_X start_POSTSUBSCRIPT italic_π . <|MaskedSetence|> | **A**: italic_α end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT fraktur_S start_POSTSUBSCRIPT italic_π .
**B**: italic_α end_POSTSUBSCRIPT ; blackboard_Q )
The vertical equalities come from Theorem 1.4
(and Equation (5.16))..
**C**: italic_α end_POSTSUBSCRIPT end_POSTSUPERSCRIPTH∙(Xα,k;ℤ)superscript𝐻∙subscript𝑋𝛼𝑘ℤH^{\bullet}(X_{\alpha,k};{\mathbb{Z}})italic_H start_POSTSUPERSCRIPT ∙ end_POSTSUPERSCRIPT ( italic_X start_POSTSUBSCRIPT italic_α , italic_k end_POSTSUBSCRIPT ; blackboard_Z )H∙(Xπ.α;ℤ)superscript𝐻∙subscript𝑋formulae-sequence𝜋𝛼ℤH^{\bullet}(X_{\pi.\alpha};{\mathbb{Z}})italic_H start_POSTSUPERSCRIPT ∙ end_POSTSUPERSCRIPT ( italic_X start_POSTSUBSCRIPT italic_π .
| CAB | CAB | BAC | CAB | Selection 4 |
<|MaskedSetence|> <|MaskedSetence|> α1∈H1(M)subscript𝛼1superscript𝐻1𝑀\alpha_{1}\in H^{1}(M)italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( italic_M )),
fbsubscript𝑓𝑏f_{b}italic_f start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT is constant and fbκ=0.subscript𝑓𝑏𝜅0f_{b}\kappa=0.italic_f start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_κ = 0 . Recall that α=α1+α2𝛼subscript𝛼1subscript𝛼2\alpha=\alpha_{1}+\alpha_{2}italic_α = italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT with α2=fbχℱ.subscript𝛼2subscript𝑓𝑏subscript𝜒ℱ\alpha_{2}=f_{b}\chi_{\mathcal{F}}.italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_f start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_χ start_POSTSUBSCRIPT caligraphic_F end_POSTSUBSCRIPT . In the case where H1(M)={0}superscript𝐻1𝑀0H^{1}(M)=\{0\}italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( italic_M ) = { 0 } and
α𝛼\alphaitalic_α is ΔasubscriptΔ𝑎\Delta_{a}roman_Δ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT-harmonic 1111-form, then α1=0subscript𝛼10\alpha_{1}=0italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0 and
α=fbχℱ=(constant)χℱ𝛼subscript𝑓𝑏subscript𝜒ℱconstantsubscript𝜒ℱ\alpha=f_{b}\chi_{\mathcal{F}}={\rm(constant)}\,\chi_{\mathcal{F}}italic_α = italic_f start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_χ start_POSTSUBSCRIPT caligraphic_F end_POSTSUBSCRIPT = ( roman_constant ) italic_χ start_POSTSUBSCRIPT caligraphic_F end_POSTSUBSCRIPT. But this constant cannot be zero in view of Remark 8.2. <|MaskedSetence|> | **A**: Hence.
**B**:
which is non-negative.
**C**: Then ⟨Δaα,α⟩=0subscriptΔ𝑎𝛼𝛼0\left\langle\Delta_{a}\alpha,\alpha\right\rangle=0⟨ roman_Δ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_α , italic_α ⟩ = 0 if and only if α1subscript𝛼1\alpha_{1}italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is harmonic (i.e.
| BCA | BCA | BCA | CBA | Selection 1 |
Our idea goes similarly as in [7]. We know from Section 4 that if D>D∗𝐷subscript𝐷D>D_{*}italic_D > italic_D start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT, then f𝟎subscript𝑓0f_{{\mathbf{0}}}italic_f start_POSTSUBSCRIPT bold_0 end_POSTSUBSCRIPT is the unique mimizer of the free energy ℱℱ\mathcal{F}caligraphic_F and we can also prove the convergence. We study the time deriviative of ℱℱ\mathcal{F}caligraphic_F, which is the Fisher information ℐℐ\mathcal{I}caligraphic_I. Around f𝟎subscript𝑓0f_{{\mathbf{0}}}italic_f start_POSTSUBSCRIPT bold_0 end_POSTSUBSCRIPT, we consider the quadratic forms associated with the expansion of ℱℱ\mathcal{F}caligraphic_F and ℐℐ\mathcal{I}caligraphic_I, and we prove a coercivity result relating the two quadratic forms Q1subscript𝑄1Q_{1}italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and Q2subscript𝑄2Q_{2}italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, which is based on a Poincaré inequality, see Section 3. <|MaskedSetence|> For the equation (1.1), we consider the linearzied equation with the linear operator ℒℒ\mathcal{L}caligraphic_L. <|MaskedSetence|> The coercivity constant 𝒫Dsubscript𝒫𝐷\mathcal{P}_{D}caligraphic_P start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT is the spectral gap of ℒℒ\mathcal{L}caligraphic_L. <|MaskedSetence|> Moreover, the sharp exponential rate is just 2𝒫D2subscript𝒫𝐷2\mathcal{P}_{D}2 caligraphic_P start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT. Details can be found in Section 5.
One can change probably the potential ϕα(v)subscriptitalic-ϕ𝛼𝑣\phi_{\alpha}(v)italic_ϕ start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( italic_v ) to a polynomial of higher order in v𝑣vitalic_v (of course it should also take negative values of ϕ(α)=0italic-ϕ𝛼0\phi(\alpha)=0italic_ϕ ( italic_α ) = 0), and similar results on the phase transitions and large time asymptotic behaviour when the intensity of noise D𝐷Ditalic_D is large enough, but the phase diagram will be more complicated.. | **A**: The goal of studying the quadratic forms is to prove the result of large time asymptotic behaviour.
**B**: After choosing an adopted Hilbert space with scalar product ⟨,⟩\langle,\rangle⟨ , ⟩, such that ⟨g,g⟩=Q1[g]𝑔𝑔subscript𝑄1delimited-[]𝑔\langle g,g\rangle=Q_{1}[g]⟨ italic_g , italic_g ⟩ = italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT [ italic_g ] and ⟨g,ℒg⟩=−Q2[g]𝑔ℒ𝑔subscript𝑄2delimited-[]𝑔\langle g,\mathcal{L}g\rangle=-Q_{2}[g]⟨ italic_g , caligraphic_L italic_g ⟩ = - italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT [ italic_g ].
**C**: For the non-linear part of (1.1), we can prove that the difference with the nonlinear expansion is small enough compared to ⟨g,g⟩𝑔𝑔\langle g,g\rangle⟨ italic_g , italic_g ⟩ and ⟨g,ℒg⟩𝑔ℒ𝑔\langle g,\mathcal{L}g\rangle⟨ italic_g , caligraphic_L italic_g ⟩, so the result of large time asymptotics is proved by using Grönwall’s inequality.
| CAB | ABC | ABC | ABC | Selection 4 |
<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Section 3 is dedicated to two-parameter generalized relative entropy and its properties. We discuss the information geometric aspects of entropy in section 4. Then we conclude the article comparing similar properties of Shannon, Tsallis and two-parameter generalized entropy.. | **A**: In section 2, we define the joint entropy and the conditional entropy to present a number of properties of two-parameter generalized entropy as well as the chain rule.
**B**: To the best of our knowledge, this article develops these properties for two-parameter generalized entropy first time in literature.
This article is distributed as follows.
**C**:
The similar properties for the Tsallis entropy and divergence are investigated in detail [23], [24], [25].
| CBA | CBA | CBA | CBA | Selection 3 |
<|MaskedSetence|> Since faμ∘fbμ=fabμsuperscriptsubscript𝑓𝑎𝜇superscriptsubscript𝑓𝑏𝜇superscriptsubscript𝑓𝑎𝑏𝜇f_{a}^{\mu}\circ f_{b}^{\mu}=f_{ab}^{\mu}italic_f start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT ∘ italic_f start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT = italic_f start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT, σ𝜎\sigmaitalic_σ is a homomorphism of groups. Let a∈Kerσ𝑎𝐾𝑒𝑟𝜎a\in Ker\;\sigmaitalic_a ∈ italic_K italic_e italic_r italic_σ.Then faμ=Ieμsuperscriptsubscript𝑓𝑎𝜇superscriptsubscript𝐼𝑒𝜇f_{a}^{\mu}=I_{e}^{\mu}italic_f start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT = italic_I start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT i.e.,
μ(x−1a−1y)=μ(x−1y)𝜇superscript𝑥1superscript𝑎1𝑦𝜇superscript𝑥1𝑦\mu(x^{-1}a^{-1}y)=\mu(x^{-1}y)italic_μ ( italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_y ) = italic_μ ( italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_y ) ∀(x,y)∈G×Gfor-all𝑥𝑦𝐺𝐺\forall(x,y)\in G\times G∀ ( italic_x , italic_y ) ∈ italic_G × italic_G. <|MaskedSetence|> This holds only when x−1a−1x=esuperscript𝑥1superscript𝑎1𝑥𝑒x^{-1}a^{-1}x=eitalic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x = italic_e. <|MaskedSetence|> For any faμ∈AFμ(G)superscriptsubscript𝑓𝑎𝜇superscriptsubscript𝐴𝐹𝜇𝐺f_{a}^{\mu}\in A_{F}^{\mu}(G)italic_f start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT ∈ italic_A start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT ( italic_G ), σ(a)=faμ𝜎𝑎superscriptsubscript𝑓𝑎𝜇\sigma(a)=f_{a}^{\mu}italic_σ ( italic_a ) = italic_f start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT. Thus σ𝜎\sigmaitalic_σ is an isomorphism and hence G≅AFμ(G)𝐺superscriptsubscript𝐴𝐹𝜇𝐺G\cong A_{F}^{\mu}(G)italic_G ≅ italic_A start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT ( italic_G ).
∎. | **A**: σ(a)=faμ𝜎𝑎superscriptsubscript𝑓𝑎𝜇\sigma(a)=f_{a}^{\mu}italic_σ ( italic_a ) = italic_f start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT.
**B**: Eventually, a=e𝑎𝑒a=eitalic_a = italic_e and σ𝜎\sigmaitalic_σ is a monomorphism of groups.
**C**: In particular, for y=x𝑦𝑥y=xitalic_y = italic_x, μ(x−1a−1x)=μ(e)=1𝜇superscript𝑥1superscript𝑎1𝑥𝜇𝑒1\mu(x^{-1}a^{-1}x)=\mu(e)=1italic_μ ( italic_x start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_x ) = italic_μ ( italic_e ) = 1.
| ACB | ACB | ACB | ACB | Selection 1 |
<|MaskedSetence|> Applying Corollary 7 to a wider range of bipartite and multipartite graphs, ordering the partite sets so n1≤n2≤⋯≤nmsubscript𝑛1subscript𝑛2⋯subscript𝑛𝑚n_{1}\leq n_{2}\leq\cdots\leq n_{m}italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≤ ⋯ ≤ italic_n start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT, we get the results shown in Table 1. For comparison, an upper bound on the DP-chromatic number is provided in the table (this upper bound is sharp for large enough m𝑚mitalic_m, and is also a sharp bound on the list chromatic number). <|MaskedSetence|> Specifically, if we consider complete bipartite graphs, it is well-known that for each n∈ℕ𝑛ℕn\in\mathbb{N}italic_n ∈ blackboard_N, there is a mn∈ℕsubscript𝑚𝑛ℕm_{n}\in\mathbb{N}italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ blackboard_N such that χDP(Kn,mn)=n+1subscript𝜒𝐷𝑃subscript𝐾𝑛subscript𝑚𝑛𝑛1\chi_{{}_{DP}}(K_{n,m_{n}})=n+1italic_χ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_D italic_P end_FLOATSUBSCRIPT end_POSTSUBSCRIPT ( italic_K start_POSTSUBSCRIPT italic_n , italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) = italic_n + 1 (see e.g., [10]). <|MaskedSetence|> | **A**: Note that the difference between the DP-chromatic number and our bound on the fractional DP-chromatic number can be made arbitrarily large.
**B**: However, for pn∗subscriptsuperscript𝑝𝑛p^{*}_{n}italic_p start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, the unique solution in (0,1)01(0,1)( 0 , 1 ) to p=(1−p)n𝑝superscript1𝑝𝑛p=(1-p)^{n}italic_p = ( 1 - italic_p ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT, it can be verified that
limn→∞((n+1)−1/pn∗)=∞.subscript→𝑛𝑛11subscriptsuperscript𝑝𝑛\lim_{n\to\infty}((n+1)-1/p^{*}_{n})=\infty.roman_lim start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT ( ( italic_n + 1 ) - 1 / italic_p start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = ∞ ..
**C**: It follows from Theorem 1 and the fact that χ(K1,m)=2𝜒subscript𝐾1𝑚2\chi(K_{1,m})=2italic_χ ( italic_K start_POSTSUBSCRIPT 1 , italic_m end_POSTSUBSCRIPT ) = 2 for each m∈ℕ𝑚ℕm\in\mathbb{N}italic_m ∈ blackboard_N that χDP∗(K1,m)=2subscriptsuperscript𝜒𝐷𝑃subscript𝐾1𝑚2\chi^{*}_{DP}(K_{1,m})=2italic_χ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_D italic_P end_POSTSUBSCRIPT ( italic_K start_POSTSUBSCRIPT 1 , italic_m end_POSTSUBSCRIPT ) = 2.
| CAB | CAB | BCA | CAB | Selection 2 |
<|MaskedSetence|> Standard example are associative and coassociative bialgebras and Lie bialgebras, which are central in various topics of algebraic topology, representation theory and mathematical physics [17, 18, 5, 26, 27, 38, 74, 75]. <|MaskedSetence|> <|MaskedSetence|> Props also appear naturally in topology, for example the Frobenius bialgebra structure on the cohomology of compact oriented manifolds coming from Poincaré duality, and the involutive Lie bialgebra structure on the equivariant homology of loop spaces on manifolds, which lies at the heart of string topology ([14],[15]) and are also central in symplectic field theory and Lagrangian Floer theory by the work of Cielebak-Fukaya-Latsheev [16]. Props also provide a concise way to encode various field theories such as topological quantum field theories and conformal field theories, and have recently proven to be the kind of algebraic structure underlying the topological recursion phenomenom, as unraveled by Kontsevich and Soibelman in [61] (see [9] for connections with mathematical physics and algebraic geometry).
A meaningful idea to understand the behavior of these various structures and, accordingly, to get more information about the mathematical objects on which they act, is to organize all the possible deformations of a given structure into a single geometric object which encapsulates not only the deformations but also an equivalence relation between these deformations. That is, to define a formal moduli problem. Such ideas goes back to the pioneering work of Kodaira-Spencer in geometry and the work of Gerstenhaber on associative algebras and Hochschild cohomology. In the eighties supported by Deligne and Drinfeld, a groundbreaking principle emerged, asserting that any formal moduli problem corresponds to a certain differential graded Lie algebra which parametrizes algebraically the corresponding deformation theory. The deformations correspond to special elements of this Lie algebra called the Maurer-Cartan elements, and equivalences of deformations are determined by a quotient under the action of a gauge group.. | **A**: Here the formalism of props, which actually goes back to [70], is the convenient unifying framework to handle such structures.
**B**:
However, algebraic structure governed by operations with several inputs and several outputs also appear naturally in a variety of topics related to the same fields of mathematics.
**C**: Props plays a crucial role in the deformation quantization process for Lie bialgebras, as shown by Etingof-Kazdhan ([26], [27]), and more generally in the theory of quantization functors [23, 76].
| BAC | BAC | ACB | BAC | Selection 4 |
<|MaskedSetence|> In Section 2 we give the background on potential functions with bulk deformations. <|MaskedSetence|> In Section 4 and 5 we construct three types of Floer theories with cylinder corrections and show some geometric properties of these theories. <|MaskedSetence|> The second and third models are complexes generated by Hamiltonian chords and intersection points respectively, which will be used to study the intersection behavior of our Lagrangians under Hamiltonian perturbations. Once the equivalences between the three models is established, we apply them, in Section 6, to obtain estimates of displacement energy and prove Theorem 1.4.
. | **A**: The first model is a disk model with cylinder corrections, which gives us a deformed potential function to do concrete computations.
**B**:
The outline of this article is as follows.
**C**: In Section 3 we review the symplectic sum and cut method and prove Theorem 1.6 and Theorem 1.7.
| BCA | BCA | BCA | ABC | Selection 3 |
Here, the Hilbert space is ℋ=L2(Ω)ℋsuperscript𝐿2Ω\mathscr{H}=L^{2}(\Omega)script_H = italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ), and H=−(ℏ2/2m)∇2+V𝐻superscriptPlanck-constant-over-2-pi22𝑚superscript∇2𝑉H=-(\hbar^{2}/2m)\nabla^{2}+Vitalic_H = - ( roman_ℏ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2 italic_m ) ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_V on the following domain D(H)𝐷𝐻D(H)italic_D ( italic_H ). <|MaskedSetence|> <|MaskedSetence|> By the Sobolev imbedding theorem [1, p. 85], functions f∈Hk(ℝd)𝑓superscript𝐻𝑘superscriptℝ𝑑f\in H^{k}(\mathbb{R}^{d})italic_f ∈ italic_H start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) for k≥1𝑘1k\geq 1italic_k ≥ 1 possess a “trace” on any affine hyperplane P⊂ℝd𝑃superscriptℝ𝑑P\subset\mathbb{R}^{d}italic_P ⊂ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT, i.e., an unambiguous restriction to P𝑃Pitalic_P that lies in L2(P,dd−1𝒙)superscript𝐿2𝑃superscript𝑑𝑑1𝒙L^{2}(P,d^{d-1}\boldsymbol{x})italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_P , italic_d start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT bold_italic_x ). <|MaskedSetence|> Since ∂ΩΩ\partial\Omega∂ roman_Ω is assumed to consist of finitely many C1superscript𝐶1C^{1}italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT surfaces, a trace in L2(∂Ω,dd−1𝒙)superscript𝐿2Ωsuperscript𝑑𝑑1𝒙L^{2}(\partial\Omega,d^{d-1}\boldsymbol{x})italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( ∂ roman_Ω , italic_d start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT bold_italic_x ) exists for ψ∈H1(Ω)𝜓superscript𝐻1Ω\psi\in H^{1}(\Omega)italic_ψ ∈ italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) (and thus also for ψ∈Hk(Ω)𝜓superscript𝐻𝑘Ω\psi\in H^{k}(\Omega)italic_ψ ∈ italic_H start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( roman_Ω ) with k>1𝑘1k>1italic_k > 1). Thus, for ψ∈H2(Ω)𝜓superscript𝐻2Ω\psi\in H^{2}(\Omega)italic_ψ ∈ italic_H start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ), both ψ𝜓\psiitalic_ψ and ∇ψ∇𝜓\nabla\psi∇ italic_ψ (whose d𝑑ditalic_d components lie in H1(Ω)superscript𝐻1ΩH^{1}(\Omega)italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω )) can be evaluated on ∂ΩΩ\partial\Omega∂ roman_Ω, and (2) is a meaningful condition that defines a linear subspace of H2(Ω)superscript𝐻2ΩH^{2}(\Omega)italic_H start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ); this subspace is D(H)𝐷𝐻D(H)italic_D ( italic_H ).
. | **A**: By the Stein extension theorem [1, p. 146, 154], every f∈Hk(Ω)𝑓superscript𝐻𝑘Ωf\in H^{k}(\Omega)italic_f ∈ italic_H start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( roman_Ω ) possesses an extension in Hk(ℝd)superscript𝐻𝑘superscriptℝ𝑑H^{k}(\mathbb{R}^{d})italic_H start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ).
**B**: Since, by Rademacher’s theorem, every Lipschitz function is differentiable almost everywhere, a surface area measure dd−1𝒙superscript𝑑𝑑1𝒙d^{d-1}\boldsymbol{x}italic_d start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT bold_italic_x and a Hilbert space L2(∂Ω,dd−1𝒙)superscript𝐿2Ωsuperscript𝑑𝑑1𝒙L^{2}(\partial\Omega,d^{d-1}\boldsymbol{x})italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( ∂ roman_Ω , italic_d start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT bold_italic_x ) are uniquely defined on ∂ΩΩ\partial\Omega∂ roman_Ω, and 𝒏(𝒙)𝒏𝒙\boldsymbol{n}(\boldsymbol{x})bold_italic_n ( bold_italic_x ) is defined almost everywhere on ∂ΩΩ\partial\Omega∂ roman_Ω.
**C**: For d>1𝑑1d>1italic_d > 1, ∂ΩΩ\partial\Omega∂ roman_Ω consists not necessarily of hyperplanes but C1superscript𝐶1C^{1}italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT surfaces, and a suitable version of the Sobolev imbedding theorem [1, p. 164] provides a trace of f∈H1(ℝd)𝑓superscript𝐻1superscriptℝ𝑑f\in H^{1}(\mathbb{R}^{d})italic_f ∈ italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) or ψ∈H1(Ω)𝜓superscript𝐻1Ω\psi\in H^{1}(\Omega)italic_ψ ∈ italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) also in this case.
| BAC | ABC | ABC | ABC | Selection 4 |
<|MaskedSetence|> <|MaskedSetence|> To construct this, consider an edge of the cubical complex; transversal to it, draw a (d−1)𝑑1(d-1)( italic_d - 1 )-dimensional disk. <|MaskedSetence|> Repeating this for every edge yields the desired hypersurface. We refer to [Fun99] for more on these notions.
Note that the union of these normal hypersurfaces form precisely the codimension one-skeleton of the dual cell decomposition. Notice further that every self-intersection of that hypersurface is normal crossing; the intersections are transversal.. | **A**: Indeed, Funar [Fun99] provided a conjecture for a complete characterization for the cubical case that revealed the problem’s depth
depth: Every cubical manifold of dimension d𝑑ditalic_d has associated an immersed normal hypersurface.
**B**: Hence, there are at least two classes of cubical 2-spheres that can never be connected by cubical Pachner moves.
**C**: Continue drawing through adjacent edges, that is, edges that are in a common square, but have no vertex in common.
| BAC | BAC | BAC | BCA | Selection 1 |
<|MaskedSetence|> Thus, ab𝑎𝑏abitalic_a italic_b is a nonzero element of
𝕃𝕃\mathbb{L}blackboard_L. Hence, ord(ab)ord𝑎𝑏\operatorname*{ord}\left(ab\right)roman_ord ( italic_a italic_b ) is defined as
the smallest β∈𝕍𝛽𝕍\beta\in\mathbb{V}italic_β ∈ blackboard_V such that [tβ](ab)≠0delimited-[]subscript𝑡𝛽𝑎𝑏0\left[t_{\beta}\right]\left(ab\right)\neq 0[ italic_t start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT ] ( italic_a italic_b ) ≠ 0. But we already know that orda+ordbord𝑎ord𝑏\operatorname*{ord}a+\operatorname*{ord}broman_ord italic_a + roman_ord italic_b is the smallest such β𝛽\betaitalic_β. <|MaskedSetence|> <|MaskedSetence|> | **A**: This proves Lemma.
**B**:
From [torda+ordb](ab)≠0delimited-[]subscript𝑡ord𝑎ord𝑏𝑎𝑏0\left[t_{\operatorname*{ord}a+\operatorname*{ord}b}\right]\left(ab\right)\neq 0[ italic_t start_POSTSUBSCRIPT roman_ord italic_a + roman_ord italic_b end_POSTSUBSCRIPT ] ( italic_a italic_b ) ≠ 0, we obtain ab≠0𝑎𝑏0ab\neq 0italic_a italic_b ≠ 0.
**C**: Comparing these two
results, we conclude that ord(ab)=orda+ordbord𝑎𝑏ord𝑎ord𝑏\operatorname*{ord}\left(ab\right)=\operatorname*{ord}a+\operatorname*{ord}broman_ord ( italic_a italic_b ) = roman_ord italic_a + roman_ord italic_b.
| BCA | BCA | ABC | BCA | Selection 4 |
We examine the action of the transfer operator associated with a class of random dynamical systems. We establish appropriate anisotropic spaces by extending a spectral gap from a subshift of finite type to the transfer operator of the skew-product map. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> However, our approach differs; we obtain regularity by examining the behavior of the transfer operator in the neighbourhood of the constructed fixed point. Specifically, we achieve uniqueness and regularity as a result of convergence to the equilibrium and the presence of a spectral gap.
. | **A**: Following the general approach outlined in [19] and [20], we generalize the techniques and apply them to the system under study.
**B**: Commonly, these approaches achieve regularity through compact inclusion arguments.
**C**: In essence, we achieve a fixed point of the transfer operator of the skew-product within a specific anisotropic space by lifting to the RDS a known spectral gap from the base system.
This type of study is often carried out using the Ionescu-Tulcea and Marinescu Theorem by constructing a pair of suitable function spaces—a stronger space and an auxiliary weaker space—such that the action of the Perron-Frobenius operator on the stronger space exhibits a spectral gap (see [6], [35], [11], [17], and [40] for introductory texts).
| ACB | CBA | ACB | ACB | Selection 4 |
<|MaskedSetence|> Section 3 and Section 4 follow the same organisation: in Subsection 3.1 (resp. Subsection 4.1) we define rigorously the particle system, prove technical lemmas on the kernel and state the global well-posedness of the PDE for Burgers (resp. local well-posedness for Keller-Segel). Then in Subsection 3.2 (resp. <|MaskedSetence|> <|MaskedSetence|> | **A**: Subsection 4.2), we state and prove the convergence theorems.
**B**: Finally in the Appendix, we state and prove a bound on the stochastic convolution integral that appears in the expression of the mollified empirical measure of the Burgers particle system.
.
**C**:
In Section 2, we list the notations used throughout the paper and in Subsection 2.2 we recall or prove some useful lemmas that hold in Bessel spaces both in the torus and in the Euclidean space.
| CAB | CAB | CAB | ABC | Selection 3 |
F)\\
id.&\mbox{ on }(c-id.)(F)^{\perp},\end{cases}\quad\widetilde{c}^{unip}=\begin{%
cases}id.&\mbox{ on }(c-id.)(F)\\
\widetilde{c}&\mbox{ on }(c-id.)(F)^{\perp}.\end{cases}over~ start_ARG italic_c end_ARG start_POSTSUPERSCRIPT italic_s italic_s end_POSTSUPERSCRIPT = { start_ROW start_CELL over~ start_ARG italic_c end_ARG end_CELL start_CELL on ( italic_c - italic_i italic_d . <|MaskedSetence|> end_CELL start_CELL on ( italic_c - italic_i italic_d . ) ( italic_F ) start_POSTSUPERSCRIPT ⟂ end_POSTSUPERSCRIPT , end_CELL end_ROW over~ start_ARG italic_c end_ARG start_POSTSUPERSCRIPT italic_u italic_n italic_i italic_p end_POSTSUPERSCRIPT = { start_ROW start_CELL italic_i italic_d . <|MaskedSetence|> ) ( italic_F ) end_CELL end_ROW start_ROW start_CELL over~ start_ARG italic_c end_ARG end_CELL start_CELL on ( italic_c - italic_i italic_d . ) ( italic_F ) start_POSTSUPERSCRIPT ⟂ end_POSTSUPERSCRIPT . <|MaskedSetence|> | **A**: ) ( italic_F ) end_CELL end_ROW start_ROW start_CELL italic_i italic_d .
**B**: end_CELL start_CELL on ( italic_c - italic_i italic_d .
**C**: end_CELL end_ROW
.
| ABC | ABC | ABC | ACB | Selection 1 |
The terms l𝑙litalic_l and log\logroman_log in estimates above in fact represent a loss in the rates of convergence. A result without these terms is still an open problem which can be called an optimal resolvent rate problem. <|MaskedSetence|> In addition, constructing explicit estimates that give rise to the rate of convergence of attractors can be hard-working as we can see in [10], where an elliptic problem involving a divergente operator with localized large diffusion has been studied. <|MaskedSetence|> Also, the unstable manifold attracts exponentially. Thus, we hope that the rate of convergence of attractors will be exponential. It can be called an optimal exponential rate problem. <|MaskedSetence|> | **A**: We already know that in the situation of a family of parabolic problems whose asymptotic behavior is dictated by a system of ordinary differential equations, it is possible to obtain the optimal resolvent rate, see [9].
**B**: It is interesting to observe that, for a class of problems that generate gradient semigroups, the global attractor is given by a union of unstable manifolds of hyperbolic equilibrium points.
**C**: Both optimal resolvent and exponential rates, as far as we know, are open problems.
.
| ABC | CAB | ABC | ABC | Selection 1 |
<|MaskedSetence|> Moreover, for the set of 2-arrows, there are two products: the ‘horizontal’ and the ‘vertical’ products, which form a nonassociative group and groupoid respectively. <|MaskedSetence|> <|MaskedSetence|> These two coproducts also satisfy the interchange law. As a coherent Hopf 2-algebra, the coherence condition will be described by a coassociator, which satisfies the ‘3-cocycle’ condition.
. | **A**: Therefore, by applying the idea of ‘2-arrow’ quantization, a coherent Hopf 2-algebra could consist of two Hopf coquasigroups, which correspond to the ‘quantum 1-arrows’ and ‘quantum 2-arrows’.
**B**:
Recall that, for a classical coherent 2-group, all the 1-arrows and 2-arrows are weakly invertible.
**C**: Moreover, for the ‘quantum 2-arrows’, it is on the one hand, a Hopf coquasigroup corresponds to the ‘horizontal’ coproduct; on the other hand, a Hopf algebroid corresponds to the ‘vertical’ coproduct.
| BAC | BAC | ABC | BAC | Selection 4 |
In this paper, we take an information theoretic approach to study lossless compression of graphs with vertex labels.
We assume the graph is generated by some random graph model and investigate lossless compression schemes that achieve the theoretical limit, i.e., the entropy of the graph, asymptotically as the number of vertices goes to infinity. When the underlying distribution/statistics of the random graph model is known, optimal lossless compression can be achieved by methods like Huffman coding. However, in most real-world applications, the exact distribution is usually hard to obtain and the data we are given is a single realization of this distribution. This motivates us to consider the framework of universal compression, in which we assume the underlying distribution belongs to a known family of distributions and require that the encoder and the decoder should not be a function of the underlying distribution. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> | **A**: For this paper, we focus on the family of stochastic block models, which are widely used random graph models that capture the clustering effect in social networks.
**B**: The goal of universal compression is to design a single compression scheme that universally achieves the optimal theoretical limit, for every distribution in the family, without knowing which distribution generates the data.
**C**: Our goal is to develop a universal graph compression scheme for a family of stochastic block models with as wide a range of parameters as possible.
.
| BAC | BAC | CAB | BAC | Selection 4 |
<|MaskedSetence|> The arrival rate is a key quantity controlling defect formation.
When the arrival rate is slow, the bilayer interface can grow in size to accommodate the new mass. <|MaskedSetence|> At moderate rates, a pearling bifurcation can be triggered, the onset of which is well understood within the context of the FCH gradient flow, [12]. The pearling can be transient, subsiding as the dilute suspension of amphiphilic material is consumed. The pearling can also be lead to the formation of end-cap type defects, essentially micelles that remain connected to the underlying structure from which they emerged. The endcaps form most readily at points of high curvature of the bilayer interface. <|MaskedSetence|> | **A**:
Intuitively, both a high density of dispersed diblock polymers or a high energy associated to an isolated diblock molecule correspond to a high rate of
absorption of the dispersed polymers onto the bilayer interface.
**B**: The stem of the endcap can grow,.
**C**: The growth process is adiabatic and has been studied rigorously, [6], deriving a motion against curvature, regularized by a higher order Willmore term that includes surface diffusion.
If the rate of arrival increases beyond a critical threshold, then defects, such as pearling, endcaps, and loop formation are observed.
| BAC | ACB | ACB | ACB | Selection 2 |
which implies that (D^ii−Djj)Qij=0subscript^𝐷𝑖𝑖subscript𝐷𝑗𝑗subscript𝑄𝑖𝑗0(\hat{D}_{ii}-D_{jj})Q_{ij}=0( over^ start_ARG italic_D end_ARG start_POSTSUBSCRIPT italic_i italic_i end_POSTSUBSCRIPT - italic_D start_POSTSUBSCRIPT italic_j italic_j end_POSTSUBSCRIPT ) italic_Q start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = 0. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Since D11,…,Dnnsubscript𝐷11…subscript𝐷𝑛𝑛D_{11},\dots,D_{nn}italic_D start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT , … , italic_D start_POSTSUBSCRIPT italic_n italic_n end_POSTSUBSCRIPT are distinct, Qij=0subscript𝑄𝑖𝑗0Q_{ij}=0italic_Q start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = 0 unless π(i)=j𝜋𝑖𝑗\pi(i)=jitalic_π ( italic_i ) = italic_j, in which case Qij=1subscript𝑄𝑖𝑗1Q_{ij}=1italic_Q start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = 1 due to the orthogonality. This completes the proof.
∎
. | **A**: In other words, there exists a permutation π𝜋\piitalic_π on {1,…,n}1…𝑛\{1,\dots,n\}{ 1 , … , italic_n } such that D^ii=Dπ(i)π(i)subscript^𝐷𝑖𝑖subscript𝐷𝜋𝑖𝜋𝑖\hat{D}_{ii}=D_{\pi(i)\pi(i)}over^ start_ARG italic_D end_ARG start_POSTSUBSCRIPT italic_i italic_i end_POSTSUBSCRIPT = italic_D start_POSTSUBSCRIPT italic_π ( italic_i ) italic_π ( italic_i ) end_POSTSUBSCRIPT.
**B**: Therefore, we have (Dπ(i)π(i)−Djj)Qij=0subscript𝐷𝜋𝑖𝜋𝑖subscript𝐷𝑗𝑗subscript𝑄𝑖𝑗0(D_{\pi(i)\pi(i)}-D_{jj})Q_{ij}=0( italic_D start_POSTSUBSCRIPT italic_π ( italic_i ) italic_π ( italic_i ) end_POSTSUBSCRIPT - italic_D start_POSTSUBSCRIPT italic_j italic_j end_POSTSUBSCRIPT ) italic_Q start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = 0 for any i,j=1,…,nformulae-sequence𝑖𝑗1…𝑛i,j=1,\dots,nitalic_i , italic_j = 1 , … , italic_n.
**C**: Since eigenvalues are preserved by conjugation, we know that the set of numbers on the diagonal of D^^𝐷\hat{D}over^ start_ARG italic_D end_ARG must be the same as that of D𝐷Ditalic_D.
| CAB | ACB | CAB | CAB | Selection 1 |
Secondly, based on the theoretical results, we propose an algorithm for finding deep NN approximations of stable manifolds. One of the crucial aspects of this algorithm is a composite loss function that incorporates the maximum error, the mean error of the NN from the exact stable manifold on the sample set, and the error between the derivative of the NN at the origin and the stabilizing solution of the Riccati equation (as shown in equation (4.1) below).
Another crucial issue is adaptive data generation by solving the characteristic Hamiltonian systems which is inspired by [27]. Specifically, we solve two-point boundary value problems (BVPs) locally near the equilibrium and extend the local solutions using initial value problems (IVPs) for the characteristic Hamiltonian system. We randomly choose a number of samples along each trajectory, and adaptively select additional samples near points with large errors from the previous round of training. Our approach is causality-free and does not depend on discretizing the space, making it suitable for high-dimensional problems. <|MaskedSetence|> See e.g. [22, 24, 23, 39, 7, 13].
Our approach differs from those focused on solving the HJB equations, e.g., [37, 27]. Our method is based on the stable manifold, an intrinsic geometric property of the HJB equation. <|MaskedSetence|> There are few theoretical results on this topic in the literature. <|MaskedSetence|> Moreover, our method is different from that in [28], which devises certain architectures for approximate NN to stabilize the system. It is worth noting that in [36, 35, 5], the algorithms are based on an iterative procedure in a small neighborhood of the equilibrium, which is difficult to estimate the accuracy and is time-consuming to generate trajectories.. | **A**: Causality-free algorithms have been successful in various applications.
**B**: With this framework, we can ensure the stability of the closed loop from the controller generated by the trained NN satisfying certain accuracy.
**C**: In empirical algorithms, the ‘equilibrium’ of the closed loop system from the NN may become unstable or disappear as time goes to infinity, as shown in [28].
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Several noticeable distributed algorithms have been proposed, e.g., [17, 4, 18, 19, 20, 21]. <|MaskedSetence|> Nevertheless, various techniques, including stochastic gradient descent[17], perturbations[18], proximal methods[19], polynomial filtering[22], and successive convex approximation[4, 21] enable agents to iteratively converge to stationary or locally optimal points of nonconvex problems.
Zeroth-order Distributed Optimization
is motivated by the concern that black-box procedures or resource limitations may inhibit access to the gradients of objective functions. To address this issue, the key idea is to utilize zeroth-order information (i.e., function evaluations) to construct randomized gradient estimates. <|MaskedSetence|> <|MaskedSetence|> | **A**: Their convergence rates match those of their first-order counterparts..
**B**: Distributed zeroth-order algorithms [23, 24] perform iterative updates based on these estimates.
**C**: Their algorithmic frameworks share similarities with those for convex problems.
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<|MaskedSetence|> Brakke’s expanding holes lemma
In this section, we discuss Brakke’s expanding holes lemma [6, Lemma 6.5], which is a key tool towards the proof of Theorem 3.5. <|MaskedSetence|> The lemma is valid for Brakke flow of any codimension,
and we will state it and prove it in such generality. Hence, in this section k𝑘kitalic_k will be a fixed integer in {1,…,n}1…𝑛\{1,\ldots,n\}{ 1 , … , italic_n }, and T𝑇Titalic_T will be a plane in 𝐆(n+1,k)𝐆𝑛1𝑘\mathbf{G}(n+1,k)bold_G ( italic_n + 1 , italic_k ). <|MaskedSetence|> | **A**: Before stating the lemma, we will need some preliminary notation.
.
**B**: 5.
**C**: Given its importance in the following arguments, and for the reader’s convenience, we provide a detailed proof.
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Comparing various definitions of height pairings is a highly nontrivial issue, which is solved only in a few cases: for example, as far as I know those defined by Bloch and Beilinson in [8] and [5] have still not been checked to agree. In [44], Schneider compares an l𝑙litalic_l-adic height pairing (loc. cit., p. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> | **A**: 298) with the Néron-Tate height by comparing each to an intermediate Yoneda pairing (loc.
**B**: cit., p.
**C**: 502)
(3)
.
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<|MaskedSetence|> [16], [34]); trdeg𝑡𝑟𝑑𝑒𝑔trdegitalic_t italic_r italic_d italic_e italic_g will denote the usual transcendence degree of commutative fields. <|MaskedSetence|> <|MaskedSetence|> Lets fix a root system ΣΣ\Sigmaroman_Σ, with a fixed basis of simple roots ΔΔ\Deltaroman_Δ, n𝑛nitalic_n in total, and N𝑁Nitalic_N positive roots in total; the Weyl group will be denoted by W𝑊Witalic_W. Consider the Lie ring 𝔤ℤ,Σsubscript𝔤ℤΣ\mathfrak{g}_{\mathbb{Z},\Sigma}fraktur_g start_POSTSUBSCRIPT blackboard_Z , roman_Σ end_POSTSUBSCRIPT obtained by Chevalley basis in the corresponding complex Lie algebra (cf. [21, 25.2].)
. | **A**: Let Tdeg𝑇𝑑𝑒𝑔Tdegitalic_T italic_d italic_e italic_g denote the Gelfand-Kirillov transcendence degree of a 𝗄𝗄\mathsf{k}sansserif_k algebra (cf.
**B**: Over a field 𝗄𝗄\mathsf{k}sansserif_k of characteristic 0, Tdeg𝔻n,l(𝗄)=2n+l𝑇𝑑𝑒𝑔subscript𝔻𝑛𝑙𝗄2𝑛𝑙Tdeg\,\mathbb{D}_{n,l}(\mathsf{k})=2n+litalic_T italic_d italic_e italic_g blackboard_D start_POSTSUBSCRIPT italic_n , italic_l end_POSTSUBSCRIPT ( sansserif_k ) = 2 italic_n + italic_l, as computed in [16, Thm.
**C**: 2].
We need to recall some facts of finite dimensional semisimple Lie algebras and their enveloping algebras.
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The Crofton formula is closely related to Buffon’s problem and the literature on the development and application of this formula in the study of differential geometry is vast. <|MaskedSetence|> <|MaskedSetence|> A theorem of Besicovitch [7, Theorem 6.13] implies that the intersection of a dropped needle with a fractal that belongs to a certain large class of planar fractals is a zero probability event [10]. The Favard length of a subset E𝐸Eitalic_E of the unit square is proportional to the probability that a needle dropped in the square intersects E𝐸Eitalic_E. <|MaskedSetence|> Although a survey of the literature is beyond the scope of this work, it is important for perspective to at least recognize some appearances of ideas that are related to Buffon’s problem in both the fractal and differential geometric settings.
. | **A**: Buffon’s problem may be extended to the setting of certain planar fractals.
**B**: It appears that Peres and Solomyak [10] were the first to study the problem of estimating the decay of the Favard length for a class of self similar sets that includes K2superscript𝐾2K^{2}italic_K start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, the Cartesian product of the middle half Cantor set K𝐾Kitalic_K with itself.
**C**: Calegari’s lively article [5] is an engaging starting point for a reader who is interested in further study.
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denote a category of modules over A𝐴Aitalic_A. <|MaskedSetence|> <|MaskedSetence|> For an algebra A𝐴Aitalic_A, we use Aesuperscript𝐴𝑒A^{e}italic_A start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT for the
enveloping algebra A⊗Aoptensor-product𝐴superscript𝐴𝑜𝑝A\otimes A^{op}italic_A ⊗ italic_A start_POSTSUPERSCRIPT italic_o italic_p end_POSTSUPERSCRIPT. <|MaskedSetence|> For (bi)modules, left actions are denoted by ▷▷\triangleright▷ and right. | **A**: We extend the notation for A𝐴Aitalic_A-modules by
using S−1Xsuperscript𝑆1𝑋S^{-1}Xitalic_S start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_X for X⊗AS−1Asubscripttensor-product𝐴𝑋superscript𝑆1𝐴X\otimes_{A}S^{-1}Aitalic_X ⊗ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_S start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_A.
**B**: For an Ore subset S𝑆Sitalic_S of an algebra A𝐴Aitalic_A, we use
S−1Asuperscript𝑆1𝐴S^{-1}Aitalic_S start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_A for the localization of A𝐴Aitalic_A at S𝑆Sitalic_S.
**C**: With this definition at hand, a module in Mod-AeMod-superscript𝐴𝑒\text{{\bf Mod}-}{A^{e}}bold_Mod - italic_A start_POSTSUPERSCRIPT italic_e end_POSTSUPERSCRIPT
is a bimodule over A𝐴Aitalic_A.
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Taibleson introduced the idea of a pseudo-differential operator in the context of a local field [25]. Saloff-Coste studied such operators [23] and generalized them to the setting of local groups [24]. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Section 3 introduces the state space of the primitive process and some properties of the primitive process and its refinements. It establishes in Theorem 3.1 upper bounds for low order moments of the primitive process. Section 4 introduces a family of scaled processes in ℚpsubscriptℚ𝑝\mathds{Q}_{p}blackboard_Q start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT and uses Theorem 3.1 to determine uniform upper bounds for the scaled processes that imply that the family of measures associated to the scaled processes is uniformly tight. Uniform tightness of the measures and the convergence of the finite dimensional distributions of the scaled processes imply the main result of the paper, Theorem 4.1.. | **A**: The seminal works of Vladimirov and Volovich on p𝑝pitalic_p-adic quantum systems [30, 31] and Vladimirov’s works in which he studied the Vladimirov operator and computed its spectrum [28, 29] dramatically increased interest in analogs of diffusion equations in the p𝑝pitalic_p-adic setting.
**B**: Following the work of Kochubei [17] and Albeverio and Karwowski [1], Varadarajan introduced the idea of a Brownian motion in a state space that is a finite dimensional vector space over a division ring which is finite dimensional over a local field of arbitrary characteristic [26].
**C**: The current paper takes the perspective of Varadarajan as its starting point, but restricts to the setting of p𝑝pitalic_p-adic state spaces.
Section 2 presents some necessary background for the current work and introduces to a general readership a framework for studying scaling limits in both the real and p𝑝pitalic_p-adic settings.
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The present work was initiated by discussions with Martin Bossert and Vince Poor at the IEEE International Symposium on Information Theory 2019 in Paris. Holger Boche would like to thank Martin Bossert and Vince Poor for valuable remarks on the importance of zero error capacity for other areas of information theory.
The authors acknowledge the financial support by the Federal Ministry of Education and Research
of Germany (BMBF) in the programme of “Souverän. Digital. <|MaskedSetence|> <|MaskedSetence|> H. <|MaskedSetence|> Deppe acknowledge the financial support
from the BMBF quantum programme QuaPhySI under Grant
16KIS1598K, QUIET under Grant 16KISQ093, and the QC-. | **A**: Joint project 6G-life, project identification number: 16KISK002.
**B**: Boche and C.
**C**: Vernetzt.”.
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7.2. A coarse adjacency graph
In principle, it seems reasonable to expect that the strategy above could be imitated in the coarse setting. <|MaskedSetence|> <|MaskedSetence|> The homological approach has the advantage that it gives the adjacency graph a more homological interpretation. <|MaskedSetence|> | **A**: We will see that the adjacency graph is, in a certain sense, equivalent to a canonical homology class known as the Jordan cycle..
**B**: Instead we replace the point-set topology arguments with entirely homological arguments.
**C**: But to perform this conversion would require a fair amount of machinery.
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1.5 Paper outline and technical notation
The rest of this paper is organized as follows. Section 2 focuses on one-sample mean testing and formally develops our intuition. In particular we show that the proposed method for one-sample mean testing has the dimension-agnostic property, which is in contrast to the existing method using a U-statistic. <|MaskedSetence|> We end Section 2 by describing dimension-agnostic confidence sets for a mean vector. Section 3 provides similar results in the context of one-sample covariance testing. In Section 4, we identify general conditions under which a sample-splitting analogue of a degenerate U-statistic has a Gaussian limiting distribution. We end with a discussion and directions for future work in Section 5. Additional results are in the appendices. Appendix A discusses the approach based on multiple sample-splitting, while Appendix B presents a general strategy for studying the asymptotic power of the proposed method. <|MaskedSetence|> In Appendix D, we construct dimension-agnostic confidence sets for mean vectors by inverting dimension-agnostic p𝑝pitalic_p-values. Appendix E illustrates our main results using Gaussian MMD and studies minimax power against nonparametric alternatives. <|MaskedSetence|> | **A**: We support our theoretical findings with simulations in Appendix F, and all proofs are provided in Appendix G..
**B**: Appendix C provides an example that demonstrates the non-triviality of the asymptotic normality under sample splitting.
**C**: We also illustrate that the proposed method possesses good power properties against dense or sparse alternatives.
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<|MaskedSetence|> First one could consider the picture in the relative p𝑝pitalic_p-adic Hodge theory, for instance established in [KL15] and [KL16], or the multidimensional program of [CKZ18] and [PZ19]. <|MaskedSetence|> The applications from [Ked04] to the rigid cohomology and isocrystals have already considered some sort of multidimensional construction as in [Ked06],[Ked07],[Ked08],[Ked11],[Ked09],[Ked07]. However, one could still only get information deduced from the monodromy theorem or the slope filtration theorem essentially from the one mentioned above. The theory of partial differential equations have been regarded as non-trivial and significant generalization of the story relevant above. <|MaskedSetence|> Here let us do some summarization in a uniform way on the known pictures which one might want to follow being local or global, being étale or non-étale:. | **A**: To generalize to the higher dimensional situation, we have many ways to do so which are encoded in the following projects.
**B**: To be more precise, these are reflected in the following work: [Xiao11], [Xiao12] and [KX10], [Ked10], [Ked11], [Ked13].
That being all said, we will only focus on mixed-characteristic situation in this paper.
**C**: Also we would like to mention some relativization programs in [BC1], [Ked10], [Liu1] where some attempt to generalize the slope filtration theorem and the local monodromy theorem to the relative Frobenius situation had been successful.
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Here we introduce a generalization of oriented hypergraphs in which the coefficient of a vertex–edge incidence is an element of the complex unit circle. We call them complex unit hypergraphs. <|MaskedSetence|> In Section 2, we give the basic definitions on complex unit hypergraphs and their associated operators. <|MaskedSetence|> <|MaskedSetence|> | **A**: In Section 3, we investigate the first properties of the spectra and in Section 4 we discuss hypergraph transformations and their effect on the eigenvalues.
**B**: Finally, in Section 5, we provide several bounds for the smallest and largest eigenvalues of each operator.
.
**C**: We also define their associated adjacency, Kirchhoff Laplacian and normalized Laplacian matrices, as operators that have entries in the complex field.
The paper is structured as follows.
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<|MaskedSetence|> In Section 2, we present the BFD scheme and prove this BFD scheme is a nodal-based DG method. <|MaskedSetence|> <|MaskedSetence|> In A, we analyze the BFD scheme in the eigenvectors space, find the optimal free parameter, and highlight the benefits of using the post-processing filters. B presents the post-processing filters and their implementations.
. | **A**: This method is then generalized for the Dirichlet boundary conditions and multi-dimensional Heat equation.
**B**:
This paper is constructed as follows.
**C**: The stability of this DG method has been proven.
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Micro-fire experimental data
We consider the fire spread data from micro-fire experiments in [Thompson et al., 2020]. <|MaskedSetence|> <|MaskedSetence|> This figure demonstrates the effectiveness of anisotropic smoothing, where the boundary is implicitly detected and not smoothed over. There are clear issues in the two regions that are separated by the fire boundary, where optimal bandwidth selection methods for isotropic local constant–such as least-squares cross-validation [Li and Racine, 2007] or AIC cross-validation [Hurvich et al., 1998]–do not perform optimally for anisotropic smoothing as they are designed for isotropic smoothing. <|MaskedSetence|> While the result is satisfactory in smoothing the areas between boundaries, the method needs improvement through an asymptotically optimal bandwidth selection procedure.. | **A**: Anisotropic smoothing is applied to fire spread measurements to preserve edges and smooth between the boundaries of fuel, burning, and burnt out regions.
**B**: Figure 11 shows the anisotropic local constant estimator applied to the red channel of the RGB fire spread images.
**C**: Figure 12 shows the resulting fire image from smoothing each of the three RGB channels independently.
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One may consider a generalization of the usual multi-class logistic regression by allowing the sample data to belong to all classes, albeit with varying probabilities. We call this label smoothing. We then ask if the MLE exists. We address this question in this work, and answer affirmatively. Moreover, in contrast to the previous works, we do not impose a requirement of data separability or the full rank of the data matrix. Given that an MLE exists, one typically seeks to find it by using a numerical optimization method. In the case of small datasets, optimizers with a quadratic convergence rate such as Newton-Raphson are typically used. When datasets are very large, as is often the case in many modern datasets, or in the machine learning community, optimizers which are linear in convergence rate are used, an example being gradient descent. This provides motivation for our study of the optimization of the MLE problem using gradient descent as the optimizer. Prior studies (Freund et al. [2018], Nacson et al. <|MaskedSetence|> <|MaskedSetence|> We note that according to the results in Albert and Anderson [1984], Silvapulle [1981], data separability and binary classification imply that the MLE does not exist-therefore these cases are not relevant to our scenario. <|MaskedSetence|> | **A**: To address the convergence rate we investigate spectral properties of the Hessian of the MLE and as a consequence we provide the convergence rate in terms of a desired contraction rate.
2 Notation and setup.
**B**: [2019b], Nacson et al.
**C**: [2019a],Ji and Telgarsky [2019]) on the convergence of gradient descent for logistic regression assume data separability and binary classification.
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Let S=Specℤ[−5]𝑆Specℤdelimited-[]5S={\text{Spec}}\,\mathbb{Z}[\sqrt{-5}]italic_S = Spec blackboard_Z [ square-root start_ARG - 5 end_ARG ] and let ℒ⊆𝒪Sℒsubscript𝒪𝑆\mathcal{L}\subseteq\mathcal{O}_{S}caligraphic_L ⊆ caligraphic_O start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT be the line bundle given by the non-principal ideal I=(2,1+−5)𝐼215I=(2,1+\sqrt{-5})italic_I = ( 2 , 1 + square-root start_ARG - 5 end_ARG ). <|MaskedSetence|> That is, X𝑋Xitalic_X is the spectrum of the ℤ[−5]ℤdelimited-[]5\mathbb{Z}[\sqrt{-5}]blackboard_Z [ square-root start_ARG - 5 end_ARG ]-algebra ℤ[−5][x,y]/(x3−(−2+−5),xy−(1+−5),y2−2x)ℤdelimited-[]5𝑥𝑦superscript𝑥325𝑥𝑦15superscript𝑦22𝑥\mathbb{Z}[\sqrt{-5}][x,y]/(x^{3}-(-2+\sqrt{-5}),xy-(1+\sqrt{-5}),y^{2}-2x)blackboard_Z [ square-root start_ARG - 5 end_ARG ] [ italic_x , italic_y ] / ( italic_x start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT - ( - 2 + square-root start_ARG - 5 end_ARG ) , italic_x italic_y - ( 1 + square-root start_ARG - 5 end_ARG ) , italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 2 italic_x ), where x𝑥xitalic_x has weight 1 and y𝑦yitalic_y has weight 2. <|MaskedSetence|> <|MaskedSetence|> That is π∗(ℰ⊗ℰ′)≅ℒ≇𝒪Ssubscript𝜋tensor-productℰsuperscriptℰ′ℒnot-approximately-equalssubscript𝒪𝑆\pi_{*}(\mathcal{E}\otimes\mathcal{E}^{\prime})\cong\mathcal{L}\not\cong%
\mathcal{O}_{S}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( caligraphic_E ⊗ caligraphic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ≅ caligraphic_L ≇ caligraphic_O start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT.. | **A**: If we let 𝒳=[X/𝝁3]𝒳delimited-[]𝑋subscript𝝁3\mathscr{X}=[X/\boldsymbol{\mu}_{3}]script_X = [ italic_X / bold_italic_μ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ] and ℰ=ℰ′=𝒪𝒳[1]ℰsuperscriptℰ′subscript𝒪𝒳delimited-[]1\mathcal{E}=\mathcal{E}^{\prime}=\mathcal{O}_{\mathscr{X}}[1]caligraphic_E = caligraphic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = caligraphic_O start_POSTSUBSCRIPT script_X end_POSTSUBSCRIPT [ 1 ], we get π∗(ℰ)≅π∗(ℰ′)≅𝒪Ssubscript𝜋ℰsubscript𝜋superscriptℰ′subscript𝒪𝑆\pi_{*}(\mathcal{E})\cong\pi_{*}(\mathcal{E}^{\prime})\cong\mathcal{O}_{S}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( caligraphic_E ) ≅ italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( caligraphic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ≅ caligraphic_O start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT since the degree-zero part of 𝒪𝒳[1]subscript𝒪𝒳delimited-[]1\mathcal{O}_{\mathscr{X}}[1]caligraphic_O start_POSTSUBSCRIPT script_X end_POSTSUBSCRIPT [ 1 ] is the degree-one part of 𝒪𝒳subscript𝒪𝒳\mathcal{O}_{\mathscr{X}}caligraphic_O start_POSTSUBSCRIPT script_X end_POSTSUBSCRIPT, which is 𝒪Ssubscript𝒪𝑆\mathcal{O}_{S}caligraphic_O start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT.
**B**: On the other hand, ℰ⊗ℰ′=𝒪𝒳[1]⊗2≅𝒪𝒳[2]tensor-productℰsuperscriptℰ′subscript𝒪𝒳superscriptdelimited-[]1tensor-productabsent2subscript𝒪𝒳delimited-[]2\mathcal{E}\otimes\mathcal{E}^{\prime}=\mathcal{O}_{\mathscr{X}}[1]^{\otimes 2%
}\cong\mathcal{O}_{\mathscr{X}}[2]caligraphic_E ⊗ caligraphic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = caligraphic_O start_POSTSUBSCRIPT script_X end_POSTSUBSCRIPT [ 1 ] start_POSTSUPERSCRIPT ⊗ 2 end_POSTSUPERSCRIPT ≅ caligraphic_O start_POSTSUBSCRIPT script_X end_POSTSUBSCRIPT [ 2 ] which has ℒℒ\mathcal{L}caligraphic_L is degree zero.
**C**: Let 𝒪Xsubscript𝒪𝑋\mathcal{O}_{X}caligraphic_O start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT be the locally free ℤ/3ℤℤ3ℤ\mathbb{Z}/3\mathbb{Z}blackboard_Z / 3 blackboard_Z-graded 𝒪Ssubscript𝒪𝑆\mathcal{O}_{S}caligraphic_O start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT-module 𝒪S⊕𝒪S⊕ℒdirect-sumsubscript𝒪𝑆subscript𝒪𝑆ℒ\mathcal{O}_{S}\oplus\mathcal{O}_{S}\oplus\mathcal{L}caligraphic_O start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ⊕ caligraphic_O start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ⊕ caligraphic_L and define s1,1=1+−5,s1,2=(1+−5)/2formulae-sequencesubscript𝑠1115subscript𝑠12152s_{1,1}=1+\sqrt{-5},s_{1,2}=(1+\sqrt{-5})/2italic_s start_POSTSUBSCRIPT 1 , 1 end_POSTSUBSCRIPT = 1 + square-root start_ARG - 5 end_ARG , italic_s start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT = ( 1 + square-root start_ARG - 5 end_ARG ) / 2, and s2,2=1/2subscript𝑠2212s_{2,2}=1/2italic_s start_POSTSUBSCRIPT 2 , 2 end_POSTSUBSCRIPT = 1 / 2.
| CAB | CAB | CAB | BCA | Selection 3 |
Proof.
Note that our definition of equivariant spectral flow are based on the approach of Phillips [23], which is another approach to that of [3]. This coincides with the context that Getzler is working on, cf. <|MaskedSetence|> Hence, the equivariant notion in Definition 3.2 extends to Getzler’s setting. <|MaskedSetence|> <|MaskedSetence|> | **A**: [15, Definition 2.3].
**B**: In the following we only highlight the relevant changes to Getzler’s setting without reciting all details.
**C**: For h∈H,ℎ𝐻h\in H,italic_h ∈ italic_H , consider the associated equivariant one-form
.
| BAC | ABC | ABC | ABC | Selection 2 |
The aim of this paper is to give a complete classification of the real and strongly real unipotent elements in a classical simple Lie group. In the literature, the investigations of the reality problem have been done mostly from the viewpoint of linear algebra and the theory of algebraic groups. <|MaskedSetence|> <|MaskedSetence|> Even though there is a complete classification for simple Lie groups, we have not seen any work where general structure theory of Lie groups has been used to tackle the problem. This work is an attempt to fill this gap in the literature and initiate an investigation of reality using the adjoint representations. For this, we introduce an infinitesimal notion of reality, viz. AdG-real given below. <|MaskedSetence|> | **A**: In some geometric cases, e.g.
**B**: real rank one, local geometry of the transformation has also seen a role.
**C**: This infinitesimal reality not only helps us to classify the unipotent classes, but it may be a problem of independent interest to investigate these classes in their own rights.
.
| ABC | CAB | ABC | ABC | Selection 1 |
<|MaskedSetence|> Then the proof of [23, Lem. <|MaskedSetence|> The basic idea of the proof
is not complicated and goes back to Dehn [9]: if there are
too many edges (i.e. 𝔢(P)𝔢𝑃\mathfrak{e}(P)fraktur_e ( italic_P ) is large) then one can find a string
of letters in the reduced word of w𝑤witalic_w (e.g. <|MaskedSetence|> | **A**: is a boundary reduced tiled surface.
**B**: aba−1b−1c𝑎𝑏superscript𝑎1superscript𝑏1𝑐aba^{-1}b^{-1}citalic_a italic_b italic_a start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_c) that.
**C**: 5.18]
contains the result stated in the lemma.
| CAB | ACB | ACB | ACB | Selection 4 |
3.3. Floer’s excision theorem
Note that the proofs of Theorem 2.13 and Theorem 2.24 (c.f. [3, 42]) both involve Floer’s excision theorem in an essential way. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> | **A**: In this subsection, we follow Kronheimer and Mrowka’s idea in [36, Section 3] to prove an excision theorem for Heegaard Floer theory.
**B**: The proof in [36, Section 3] depends essentially on the TQFT properties and Axiom (A1), so it works for a general TQFT satisfying Axiom (A1).
**C**: Though for Heegaard Floer theory, we need to modify the proof to fit the settings of multi-basepoints 3-manifolds and ribbon graph cobordisms.
.
| BCA | ABC | ABC | ABC | Selection 3 |
Holger Boche thanks Martin Bossert for discussions and questions on the theory of the channel reliability function and on questions about the trustworthiness of numerical simulations on digital computers of the channel reliability function. Holger Boche also thanks Vince Poor and Martin Bossert for discussions at ISIT 2019 in Paris. These discussions initiated the research work whose results are presented in this paper. The authors acknowledge the financial support by the Federal Ministry of Education and Research
of Germany (BMBF) in the programme of “Souverän. Digital. <|MaskedSetence|> Joint project 6G-life, project identification number: 16KISK002. H. <|MaskedSetence|> Deppe acknowledge the financial support
from the BMBF quantum programme QuaPhySI under Grant
16KIS1598K, QUIET under Grant 16KISQ093, and the QC-
CamNetz Project under Grant 16KISQ077. <|MaskedSetence|> | **A**: Vernetzt.”.
**B**: Boche and C.
**C**: They were also sup-.
| ABC | ABC | ABC | ABC | Selection 1 |
<|MaskedSetence|> Pushforward operations and stability
In this section, we study pushforward operations for general groups. <|MaskedSetence|> In §2.2 we define the notion of stability and prove a better classification result for stable pushforward operations. <|MaskedSetence|> In §2.4 we prove a technical result that will be useful in §3. Throughout, we always use singular cohomology with rational coefficients.. | **A**: The behaviour with respect to the multiplicative structure is studied in §2.3.
**B**:
2.
**C**: These are defined in §2.1 and we also prove an elementary classification result there.
| BCA | BCA | ACB | BCA | Selection 4 |
5 Computing the generating series of newly defined invariants
We define and compute many new invariants using the formula derived in the previous section. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> We also argue that the appropriate generalization of the Nekrasov genus is obtained in terms of the real-oriented Witten genus due to the connection of DT4 invariants to real geometry. For the above invariants, we additionally study their symmetries and their relation to lower-dimensional geometries.
. | **A**: These include tautological series, Nekrasov genera and virtual Verlinde numbers.
**B**: Unlike the surface case, the definition of DT4 Verlinde invariants requires an additional twist by a square-root line bundle that we explain in §5.3.
**C**: The tautological series and the Verlinde series were inspired by their analogs on surfaces that we recalled in the introduction.
| ACB | ACB | ACB | ACB | Selection 2 |
(3.48)
Therefore, we have m1=−vL(ω1)subscript𝑚1subscript𝑣𝐿subscript𝜔1m_{1}=-v_{L}(\omega_{1})italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = - italic_v start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ). <|MaskedSetence|> <|MaskedSetence|> Hence we have ma=m1subscript𝑚𝑎subscript𝑚1m_{a}=m_{1}italic_m start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. By (3.28) we get mj=jmasubscript𝑚𝑗𝑗subscript𝑚𝑎m_{j}=jm_{a}italic_m start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_j italic_m start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT for all 1≤j≤n1𝑗𝑛1\leq j\leq n1 ≤ italic_j ≤ italic_n. <|MaskedSetence|> | **A**: Thus we have m1=pm1−(p−1)masubscript𝑚1𝑝subscript𝑚1𝑝1subscript𝑚𝑎m_{1}=pm_{1}-(p-1)m_{a}italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_p italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - ( italic_p - 1 ) italic_m start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT by Lemma 2.2 and Proposition 2.5(a).
**B**: Hence we have sn=−vL(dcn)=−vL(dcn′′)=mnsubscript𝑠𝑛subscript𝑣𝐿𝑑subscript𝑐𝑛subscript𝑣𝐿𝑑subscriptsuperscript𝑐′′𝑛subscript𝑚𝑛s_{n}=-v_{L}(dc_{n})=-v_{L}(dc^{\prime\prime}_{n})=m_{n}italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = - italic_v start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_d italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = - italic_v start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ( italic_d italic_c start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT..
**C**: By definition, we have ω1=db1subscript𝜔1𝑑subscript𝑏1\omega_{1}=db_{1}italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_d italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT.
| CAB | CAB | ABC | CAB | Selection 4 |
<|MaskedSetence|> It is of course a profound mystery how it achieved so much while its most significant axiom led to such a short contradiction. But that achievement is enormous: Frege provided an absolutely rigorous foundation for arithmetic, and was well on his way to providing a foundation for analysis. <|MaskedSetence|> Still less rigorous are typical logic articles today, not to mention proofs (more properly proof sketches) where there is serious disagreement over whether they actually constitute a proof. <|MaskedSetence|> | **A**: Frege’s original conception retains interest for Platonists even after the paradox.
**B**: Compare Frege’s boast:
Wenn etwa jemand etwas fehlerhaft finden sollte, muss er genau angeben können, wo der Fehler seiner Meinung nach steckt: in den Grundgesetzen, in den Definitionen, in den Regeln oder ihrer Anwendung an einer bestimmten Stelle.
.
**C**: Even Whitehead & Russell’s Principia Mathematica (according to [Gödel 1944] p. 448) was substantially less rigorous.
| ACB | BAC | ACB | ACB | Selection 1 |
<|MaskedSetence|> <|MaskedSetence|> The finite (non–coorperative) game considered in [52] is essentially static. In most dynamic models, information accumulates over time and players in a finite game naturally make a mixed strategy contingent on information available at the time a mixed strategy is made, so the accumulated information becomes imbedded in mixed strategies and payoff functions of players. Just motivated by the work of [42] on the conditional mean–variance frontier and that of [11, 20, 41, 21] on the conditional risk measures, in this paper we consider the finite game under the conditional framework (or simply, the finite conditional game). <|MaskedSetence|> | **A**: As an interesting application of Theorem 1.12 we obtain the conditional version of the classical existence theorem of Nash equilibrium points, namely Theorem 1.19 below, which is a natural generalization of Theorem 1 of [52].
To describe the finite conditional game, we first introduce the following standard terminologies and notation..
**B**: See [52] for details.
**C**:
The classical Nash theorem on the existence of equilibrium points consider n𝑛nitalic_n–fold affine (or, linear) functions defined on n𝑛nitalic_n–fold Cartesian product of simplexes, where n𝑛nitalic_n stands for the number of players.
| CBA | CBA | ABC | CBA | Selection 4 |
In the last decade, rigorous guarantees on the behavior of SON clustering have been studied by several authors, including Zhu et al. (2014); Tan and Witten (2015); Chiquet et al. (2017); Panahi et al. (2017); Radchenko and Mukherjee (2017); Jiang et al. <|MaskedSetence|> <|MaskedSetence|> Most of these works aim at the identification of sufficient conditions for SON clustering to succeed in separating clusters. Our main goal here, stated precisely in Theorem 1.1, is rather to present a seemingly simple clustering problem in which the SON clustering algorithm will typically fail. This requires us to establish necessary and sufficient conditions for the success of SON clustering, which we present in Subsection 1.3. <|MaskedSetence|> | **A**: (2021); Nguyen and Mamitsuka (Preprint, 2021).
**B**: We anticipate that these conditions will be useful in future studies of sum-of-norms clustering, and thus are interesting results in their own right.
.
**C**: (2020); Chi and Steinerberger (2019); Jiang and Vavasis (Preprint, 2020); Sun et al.
| BAC | CAB | CAB | CAB | Selection 3 |
<|MaskedSetence|> See Section 2.4 for an outline of the proof.
As discussed above, for actions by cocompact lattices, theorem D is essentially contained in [BFH]; in this case, one first produces an A𝐴Aitalic_A-invariant Borel probability measure with positive exponents by a fairly soft argument in [BFH, Proposition 4.6]. Improving the measure to one that projects to Haar is more difficult and occupies much of [BFH]. <|MaskedSetence|> As the arguments in this paper use in an essential way all arguments from [BFH] and many of those from [BFH-SLnZ], the reader may find it easier to read those papers first. <|MaskedSetence|> | **A**: An expository account of some of the arguments from [BFH] with more detailed background may be found in the lecture notes by Brown [Brown]; see also the expository account of many ideas from [BFH] in [MR4093195].
.
**B**: In the context of actions of non-uniform lattices, the arguments that construct an A𝐴Aitalic_A-invariant measure with positive exponents is as difficult as finding one that projects to Haar.
**C**:
The remainder of this paper is devoted to establishing theorem D.
| CBA | CBA | CBA | BCA | Selection 1 |
<|MaskedSetence|> As we have said, the case n=1𝑛1n=1italic_n = 1 does not occur. <|MaskedSetence|> When n=3𝑛3n=3italic_n = 3 a point (x1,x2,x3)subscript𝑥1subscript𝑥2subscript𝑥3(x_{1},x_{2},x_{3})( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT )
lies in S𝑆Sitalic_S if and only if the closed convex hull in ℂℂ\mathbb{C}blackboard_C having these
points as vertices contains 00. To use Theorem 2.3, we regard the points xisubscript𝑥𝑖x_{i}italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT as
vectors in the plane ℂℂ\mathbb{C}blackboard_C. <|MaskedSetence|> | **A**: value lower than n𝑛nitalic_n.
**B**: For each i𝑖iitalic_i,.
**C**: When n=2𝑛2n=2italic_n = 2 the
set S𝑆Sitalic_S is a copy of S1superscript𝑆1S^{1}italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT and hence is polyhedral.
| ACB | BCA | ACB | ACB | Selection 1 |
<|MaskedSetence|> Baldwin, Steven Sivek, John B. Etnyre, David Shea Vela-Vick, Ian Zemke, Linsheng Wang, Minghao Miao, and Yu Zhao for valuable discussions. The authors are grateful to Sudipta Ghosh and Ian Zemke for pointing out the proof of Lemma 4.20. The authors are also grateful to Yi Xie for pointing out the applications about detection results. <|MaskedSetence|> The second author would like to thank his supervisor Jacob Rasmussen for patient guidance and helpful comments and thank his parents for support and constant encouragement. <|MaskedSetence|> | **A**: The authors would like to thank Ciprian Manolescu, John A.
**B**: The second author is also grateful to Yi Liu for inviting him to BICMR, Peking University.
.
**C**: The authors are grateful to Hongjian Yan for comments on Remark 1.10.
| ACB | ABC | ACB | ACB | Selection 4 |
This paper aims to study sample size conditions of the method essentially proposed in Rebane and Pearl, (1987) for the recovery of polytree structures by applying the Chow-Liu algorithm to pairwise sample correlations in the case of Gaussian linear structure equation models (SEM). <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> In particular, conditions on the sample size for undirected tree structure learning via the Chow-Liu algorithm have been studied for both Ising and Gaussian models (Bresler and Karzand,, 2020; Tavassolipour et al.,, 2018; Nikolakakis et al.,, 2019), and the analyses usually rely crucially on the so-called “correlation decay” property over the true undirected tree. The correlation decay properties can usually be explicitly quantified by the pairwise population correlations corresponding to the edges of the underlying true tree. Based on this result and some perturbation results of pairwise sample correlations to their population counterparts, sufficient conditions on the sample size for undirected tree recovery with the Chow-Liu algorithm can be straightforwardly obtained.
. | **A**: Our sufficient and necessary conditions match in order in a broad regime of model parameters, and thereby characterize the difficulty of these two tasks in polytree learning.
A relevant line of research is structure learning for tree-structured undirected graphical models, including both discrete cases (Heinemann and Globerson,, 2014; Bresler and Karzand,, 2020; Netrapalli et al.,, 2010; Anandkumar et al., 2012b, ; Anandkumar et al., 2012a, ) and Gaussian cases (Tan et al.,, 2010; Tavassolipour et al.,, 2018; Nikolakakis et al.,, 2019; Katiyar et al.,, 2019).
**B**: On the other hand, we will also establish the necessary conditions on the sample sizes for these two tasks through information-theoretic lower bounds.
**C**: We establish sufficient conditions on the sample sizes for consistent recovery of both the skeleton and equivalence class for the underlying polytree structure.
| CBA | CBA | ACB | CBA | Selection 2 |
<|MaskedSetence|> Later, in [11, 12], two of the authors of this paper, inspired by the ideas of [31] (construction of an approximate solution, control of the higher order terms via Feynmann diagramms) estimated the error in Bourgain spaces instead of Strichartz spaces and were able reach the kinetic timescale up to arbitrarily small polynomial loss. At the same time, a similar result was obtained independently by Deng and Hani [14]. Recently, Deng and Hani [15] reached the kinetic timescale for the cubic NLS, which provides the first full derivation of the homogeneous (KWE) for (NLS).
In many situations of physical interest, the leading nonlinear term is quadratic: for instance, this is the case for long-wave perturbations of the acoustic type (which can exist in most media), or interaction of three-wave packets in media with a decay dispersion law. These models have extremely wide applications, ranging from solid state physics to hydrodynamics, plasma physics etc. <|MaskedSetence|> <|MaskedSetence|> | **A**: In the absence of noise, the result of [39] is conditional.
.
**B**: Recently, under the assumption of multiplicative noise, Staffilani and Tran [39] reached the kinetic timescale for the Zakharov-Kuznetsov (ZK) equation.
**C**: For the cubic NLS, the derivation of the homogeneous kinetic wave equation for random data out of statistical equilibrium was first addressed in [10] using Strichartz estimates to control the error term.
| CBA | CBA | CBA | CBA | Selection 1 |
<|MaskedSetence|> <|MaskedSetence|> Later, Conner and Thuswaldner [6] gave criteria for a self-affine tile to be a closed 3333-dimensional ball and Deng et al. [9] dealt with self-affine tiles of a special form and showed that they are 3333-dimensional balls. Kamae et al. [18] investigated a particular class of n𝑛nitalic_n-dimensional self-affine tiles. Recently, Thuswaldner and Zhang [35] studied a natural class of 3333-dimensional self-affine tiles and proved that their boundary is homeomorphic to a 2222-sphere. <|MaskedSetence|> Indeed, we want to explore if these tiles are indeed homeomorphic to a 3333-dimensional ball, which means that we have to exclude pathologies like the Alexander horned sphere which is known to occur in the context of self-affine tiles (see [6, Section 8.2]).
. | **A**: The systematic topological study of the 3333-dimensional case was initiated some years ago when Bandt [3] considered the combinatorial topology of some 3333-dimensional self-affine tiles.
**B**: It is this class of tiles that we are interested in.
**C**:
The present paper is devoted to the topology of 3333-dimensional self-affine tiles.
| CAB | CAB | ABC | CAB | Selection 4 |