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is-link">Go</button> </div> </div> </form> </div> </div> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> <a href="" class="pagination-previous is-invisible">Previous </a> <a href="/search/?searchtype=author&query=Chen%2C+L&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Chen%2C+L&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Chen%2C+L&start=50" class="pagination-link " aria-label="Page 2" aria-current="page">2 </a> </li> <li> <a href="/search/?searchtype=author&query=Chen%2C+L&start=100" class="pagination-link " aria-label="Page 3" aria-current="page">3 </a> </li> <li> <a href="/search/?searchtype=author&query=Chen%2C+L&start=150" class="pagination-link " aria-label="Page 4" aria-current="page">4 </a> </li> <li> <a href="/search/?searchtype=author&query=Chen%2C+L&start=200" class="pagination-link " aria-label="Page 5" aria-current="page">5 </a> </li> <li><span class="pagination-ellipsis">…</span></li> </ul> </nav> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.13765">arXiv:2411.13765</a> <span> [<a href="https://arxiv.org/pdf/2411.13765">pdf</a>, <a href="https://arxiv.org/ps/2411.13765">ps</a>, <a href="https://arxiv.org/format/2411.13765">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Schr枚dinger Bridge Problem for Jump Diffusions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zlotchevski%2C+A">Andrei Zlotchevski</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">Linan Chen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.13765v1-abstract-short" style="display: inline;"> The Schr枚dinger bridge problem (SBP) seeks to find the measure $\hat{\mathbf{P}}$ on a certain path space which interpolates between state-space distributions $蟻_0$ at time $0$ and $蟻_T$ at time $T$ while minimizing the KL divergence (relative entropy) to a reference path measure $\mathbf{R}$. In this work, we tackle the SBP in the case when $\mathbf{R}$ is the path measure of a jump diffusion. Un… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.13765v1-abstract-full').style.display = 'inline'; document.getElementById('2411.13765v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.13765v1-abstract-full" style="display: none;"> The Schr枚dinger bridge problem (SBP) seeks to find the measure $\hat{\mathbf{P}}$ on a certain path space which interpolates between state-space distributions $蟻_0$ at time $0$ and $蟻_T$ at time $T$ while minimizing the KL divergence (relative entropy) to a reference path measure $\mathbf{R}$. In this work, we tackle the SBP in the case when $\mathbf{R}$ is the path measure of a jump diffusion. Under mild assumptions, with both the operator theory approach and the stochastic calculus techniques, we establish an $h$-transform theory for jump diffusions and devise an approximation method to achieve the jump-diffusion SBP solution $\hat{\mathbf{P}}$ as the strong-convergence limit of a sequence of harmonic $h$-transforms. To the best of our knowledge, these results are novel in the study of SBP. Moreover, the $h$-transform framework and the approximation method developed in this work are robust and applicable to a relatively general class of jump diffusions. In addition, we examine the SBP of particular types of jump diffusions under additional regularity conditions and extend the existing results on the SBP from the diffusion case to the jump-diffusion setting. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.13765v1-abstract-full').style.display = 'none'; document.getElementById('2411.13765v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35Q93; 45K05; 60H10; 60H20; 60H30; 94A17 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.13285">arXiv:2411.13285</a> <span> [<a href="https://arxiv.org/pdf/2411.13285">pdf</a>, <a href="https://arxiv.org/ps/2411.13285">ps</a>, <a href="https://arxiv.org/format/2411.13285">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/S0550-3213(01)00405-9">10.1016/S0550-3213(01)00405-9 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the $L_{\mathrm{YJ}}(尉, 畏, X)$ constant for the Bana艣-Fr膮czek space </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+Y">Yuxin Wang</a>, <a href="/search/math?searchtype=author&query=Liu%2C+Q">Qi Liu</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">Linhui Chen</a>, <a href="/search/math?searchtype=author&query=Tan%2C+X">Xiewei Tan</a>, <a href="/search/math?searchtype=author&query=Sarfraz%2C+M">Muhammad Sarfraz</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.13285v1-abstract-short" style="display: inline;"> In this paper, for any $位\geq 1, R_位^2$ is the Bana艣-Fr膮czek space. The exact value of $L_{\mathrm{YJ}}(尉, 畏, X)$ for this space will be calculated. Specifically, $L_{\mathrm{YJ}}\left(尉, 畏, R_位^2\right)=1+\frac{2 尉畏}{尉^2+畏^2}\left(1-\frac{1}{位^2}\right)$ is the result thereafter through meticilous computation. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.13285v1-abstract-full" style="display: none;"> In this paper, for any $位\geq 1, R_位^2$ is the Bana艣-Fr膮czek space. The exact value of $L_{\mathrm{YJ}}(尉, 畏, X)$ for this space will be calculated. Specifically, $L_{\mathrm{YJ}}\left(尉, 畏, R_位^2\right)=1+\frac{2 尉畏}{尉^2+畏^2}\left(1-\frac{1}{位^2}\right)$ is the result thereafter through meticilous computation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.13285v1-abstract-full').style.display = 'none'; document.getElementById('2411.13285v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">for associated mpeg file, see http://myhost.domain/file.mpg</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> Report-no: EFI-94-11 <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 46B20 <span class="has-text-black-bis has-text-weight-semibold">ACM Class:</span> F.2.2; I.2.7 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J.Hasty Results 1 (2008) 1-9; Erratum: J.Hasty Results 2 (2008) 1-2 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.12840">arXiv:2411.12840</a> <span> [<a href="https://arxiv.org/pdf/2411.12840">pdf</a>, <a href="https://arxiv.org/format/2411.12840">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Logic in Computer Science">cs.LO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Category Theory">math.CT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> The Aldous--Hoover Theorem in Categorical Probability </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+L">Leihao Chen</a>, <a href="/search/math?searchtype=author&query=Fritz%2C+T">Tobias Fritz</a>, <a href="/search/math?searchtype=author&query=Gonda%2C+T">Tom谩拧 Gonda</a>, <a href="/search/math?searchtype=author&query=Klingler%2C+A">Andreas Klingler</a>, <a href="/search/math?searchtype=author&query=Lorenzin%2C+A">Antonio Lorenzin</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.12840v1-abstract-short" style="display: inline;"> The Aldous-Hoover Theorem concerns an infinite matrix of random variables whose distribution is invariant under finite permutations of rows and columns. It states that, up to equality in distribution, each random variable in the matrix can be expressed as a function only depending on four key variables: one common to the entire matrix, one that encodes information about its row, one that encodes i… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.12840v1-abstract-full').style.display = 'inline'; document.getElementById('2411.12840v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.12840v1-abstract-full" style="display: none;"> The Aldous-Hoover Theorem concerns an infinite matrix of random variables whose distribution is invariant under finite permutations of rows and columns. It states that, up to equality in distribution, each random variable in the matrix can be expressed as a function only depending on four key variables: one common to the entire matrix, one that encodes information about its row, one that encodes information about its column, and a fourth one specific to the matrix entry. We state and prove the theorem within a category-theoretic approach to probability, namely the theory of Markov categories. This makes the proof more transparent and intuitive when compared to measure-theoretic ones. A key role is played by a newly identified categorical property, the Cauchy--Schwarz axiom, which also facilitates a new synthetic de Finetti Theorem. We further provide a variant of our proof using the ordered Markov property and the d-separation criterion, both generalized from Bayesian networks to Markov categories. We expect that this approach will facilitate a systematic development of more complex results in the future, such as categorical approaches to hierarchical exchangeability. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.12840v1-abstract-full').style.display = 'none'; document.getElementById('2411.12840v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">39 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.06476">arXiv:2411.06476</a> <span> [<a href="https://arxiv.org/pdf/2411.06476">pdf</a>, <a href="https://arxiv.org/format/2411.06476">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Eigen-componentwise convergence of SGD for quadratic programming </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+L">Lehan Chen</a>, <a href="/search/math?searchtype=author&query=Nakatsukasa%2C+Y">Yuji Nakatsukasa</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.06476v1-abstract-short" style="display: inline;"> Stochastic gradient descent (SGD) is a workhorse algorithm for solving large-scale optimization problems in data science and machine learning. Understanding the convergence of SGD is hence of fundamental importance. In this work we examine the SGD convergence (with various step sizes) when applied to unconstrained convex quadratic programming (essentially least-squares (LS) problems), and in parti… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.06476v1-abstract-full').style.display = 'inline'; document.getElementById('2411.06476v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.06476v1-abstract-full" style="display: none;"> Stochastic gradient descent (SGD) is a workhorse algorithm for solving large-scale optimization problems in data science and machine learning. Understanding the convergence of SGD is hence of fundamental importance. In this work we examine the SGD convergence (with various step sizes) when applied to unconstrained convex quadratic programming (essentially least-squares (LS) problems), and in particular analyze the error components respect to the eigenvectors of the Hessian. The main message is that the convergence depends largely on the corresponding eigenvalues (singular values of the coefficient matrix in the LS context), namely the components for the large singular values converge faster in the initial phase. We then show there is a phase transition in the convergence where the convergence speed of the components, especially those corresponding to the larger singular values, will decrease. Finally, we show that the convergence of the overall error (in the solution) tends to decay as more iterations are run, that is, the initial convergence is faster than the asymptote. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.06476v1-abstract-full').style.display = 'none'; document.getElementById('2411.06476v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.23865">arXiv:2410.23865</a> <span> [<a href="https://arxiv.org/pdf/2410.23865">pdf</a>, <a href="https://arxiv.org/format/2410.23865">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> A Primal Staggered Discontinuous Galerkin Method on Polytopal Meshes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+L">L. Chen</a>, <a href="/search/math?searchtype=author&query=Huang%2C+X">X. Huang</a>, <a href="/search/math?searchtype=author&query=Park%2C+E">E. Park</a>, <a href="/search/math?searchtype=author&query=Wang%2C+R">R. Wang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.23865v1-abstract-short" style="display: inline;"> This paper introduces a novel staggered discontinuous Galerkin (SDG) method tailored for solving elliptic equations on polytopal meshes. Our approach utilizes a primal-dual grid framework to ensure local conservation of fluxes, significantly improving stability and accuracy. The method is hybridizable and reduces the degrees of freedom compared to existing approaches. It also bridges connections t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.23865v1-abstract-full').style.display = 'inline'; document.getElementById('2410.23865v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.23865v1-abstract-full" style="display: none;"> This paper introduces a novel staggered discontinuous Galerkin (SDG) method tailored for solving elliptic equations on polytopal meshes. Our approach utilizes a primal-dual grid framework to ensure local conservation of fluxes, significantly improving stability and accuracy. The method is hybridizable and reduces the degrees of freedom compared to existing approaches. It also bridges connections to other numerical methods on polytopal meshes. Numerical experiments validate the method's optimal convergence rates and computational efficiency. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.23865v1-abstract-full').style.display = 'none'; document.getElementById('2410.23865v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 31 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.20408">arXiv:2410.20408</a> <span> [<a href="https://arxiv.org/pdf/2410.20408">pdf</a>, <a href="https://arxiv.org/format/2410.20408">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Tangential-Normal Decompositions of Finite Element Differential Forms </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+L">Long Chen</a>, <a href="/search/math?searchtype=author&query=Huang%2C+X">Xuehai Huang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.20408v1-abstract-short" style="display: inline;"> The paper introduces a novel tangential-normal ($t$-$n$) decomposition for finite element differential forms. Its main contribution is the development of a $t$-$n$ basis where the degrees of freedom and shape functions are explicitly dual to each other. This duality simplifies the assembly of stiffness matrices and enhances the efficiency of interpolation and numerical integration in finite elemen… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.20408v1-abstract-full').style.display = 'inline'; document.getElementById('2410.20408v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.20408v1-abstract-full" style="display: none;"> The paper introduces a novel tangential-normal ($t$-$n$) decomposition for finite element differential forms. Its main contribution is the development of a $t$-$n$ basis where the degrees of freedom and shape functions are explicitly dual to each other. This duality simplifies the assembly of stiffness matrices and enhances the efficiency of interpolation and numerical integration in finite element methods. Additionally, the well-documented Lagrange element basis can be used to expedite implementation. This paper focuses on the full polynomial spaces, excluding trimmed polynomial differential forms, and provides a new perspective on constructing finite element differential forms. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.20408v1-abstract-full').style.display = 'none'; document.getElementById('2410.20408v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">21 pages, 3 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 58A10; 58J10; 65N30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.09568">arXiv:2410.09568</a> <span> [<a href="https://arxiv.org/pdf/2410.09568">pdf</a>, <a href="https://arxiv.org/format/2410.09568">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Computational Complexity">cs.CC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Cryptography and Security">cs.CR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> Second-Order Min-Max Optimization with Lazy Hessians </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+L">Lesi Chen</a>, <a href="/search/math?searchtype=author&query=Liu%2C+C">Chengchang Liu</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+J">Jingzhao Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.09568v1-abstract-short" style="display: inline;"> This paper studies second-order methods for convex-concave minimax optimization. Monteiro and Svaiter (2012) proposed a method to solve the problem with an optimal iteration complexity of $\mathcal{O}(蔚^{-3/2})$ to find an $蔚$-saddle point. However, it is unclear whether the computational complexity, $\mathcal{O}((N+ d^2) d 蔚^{-2/3})$, can be improved. In the above, we follow Doikov et al. (2023)… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.09568v1-abstract-full').style.display = 'inline'; document.getElementById('2410.09568v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.09568v1-abstract-full" style="display: none;"> This paper studies second-order methods for convex-concave minimax optimization. Monteiro and Svaiter (2012) proposed a method to solve the problem with an optimal iteration complexity of $\mathcal{O}(蔚^{-3/2})$ to find an $蔚$-saddle point. However, it is unclear whether the computational complexity, $\mathcal{O}((N+ d^2) d 蔚^{-2/3})$, can be improved. In the above, we follow Doikov et al. (2023) and assume the complexity of obtaining a first-order oracle as $N$ and the complexity of obtaining a second-order oracle as $dN$. In this paper, we show that the computation cost can be reduced by reusing Hessian across iterations. Our methods take the overall computational complexity of $ \tilde{\mathcal{O}}( (N+d^2)(d+ d^{2/3}蔚^{-2/3}))$, which improves those of previous methods by a factor of $d^{1/3}$. Furthermore, we generalize our method to strongly-convex-strongly-concave minimax problems and establish the complexity of $\tilde{\mathcal{O}}((N+d^2) (d + d^{2/3} 魏^{2/3}) )$ when the condition number of the problem is $魏$, enjoying a similar speedup upon the state-of-the-art method. Numerical experiments on both real and synthetic datasets also verify the efficiency of our method. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.09568v1-abstract-full').style.display = 'none'; document.getElementById('2410.09568v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.07581">arXiv:2410.07581</a> <span> [<a href="https://arxiv.org/pdf/2410.07581">pdf</a>, <a href="https://arxiv.org/ps/2410.07581">ps</a>, <a href="https://arxiv.org/format/2410.07581">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Clocks are $e$-positive </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+L">L. Chen</a>, <a href="/search/math?searchtype=author&query=He%2C+Y+T">Y. T. He</a>, <a href="/search/math?searchtype=author&query=Wang%2C+D+G+L">David G. L. Wang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.07581v1-abstract-short" style="display: inline;"> Along with his confirmation of the $e$-positivity of all cycle-chord graphs $胃_{ab1}$, the third author conjectured the $e$-positivity of all theta graphs $胃_{abc}$. In this paper, we establish the $e$-positivity of all clock graphs $胃_{ab2}$ by using the composition method. The key idea is to investigate the fibers of certain partial reversal transformation on compositions with all parts at least… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.07581v1-abstract-full').style.display = 'inline'; document.getElementById('2410.07581v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.07581v1-abstract-full" style="display: none;"> Along with his confirmation of the $e$-positivity of all cycle-chord graphs $胃_{ab1}$, the third author conjectured the $e$-positivity of all theta graphs $胃_{abc}$. In this paper, we establish the $e$-positivity of all clock graphs $胃_{ab2}$ by using the composition method. The key idea is to investigate the fibers of certain partial reversal transformation on compositions with all parts at least $2$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.07581v1-abstract-full').style.display = 'none'; document.getElementById('2410.07581v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">15 pages, 4 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05E05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.05048">arXiv:2410.05048</a> <span> [<a href="https://arxiv.org/pdf/2410.05048">pdf</a>, <a href="https://arxiv.org/format/2410.05048">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Focal surfaces of lightcone framed surfaces in the Lorentz-Minkowski 3-space </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xu%2C+C">Chang Xu</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">Liang Chen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.05048v2-abstract-short" style="display: inline;"> In this paper, we consider the differential geometry properties of focal surfaces of lightcone framed surfaces in Lorentz-Minkowski 3-space. In general, a mixed type surface is a connected regular surface with non-empty spacelike and timelike point sets. In order to investigate the mixed type surface with singular points, we introduce the lightcone framed surface. First, we give the lightcone fram… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.05048v2-abstract-full').style.display = 'inline'; document.getElementById('2410.05048v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.05048v2-abstract-full" style="display: none;"> In this paper, we consider the differential geometry properties of focal surfaces of lightcone framed surfaces in Lorentz-Minkowski 3-space. In general, a mixed type surface is a connected regular surface with non-empty spacelike and timelike point sets. In order to investigate the mixed type surface with singular points, we introduce the lightcone framed surface. First, we give the lightcone frame, by which we define the lightcone framed surface. Next, we consider the differential geometry properties of lightcone framed surfaces by using the lightcone frame. At last, we give the definition of focal surfaces of lightcone framed surfaces and investigate the differential geometry properties of focal surfaces. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.05048v2-abstract-full').style.display = 'none'; document.getElementById('2410.05048v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 7 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.04888">arXiv:2410.04888</a> <span> [<a href="https://arxiv.org/pdf/2410.04888">pdf</a>, <a href="https://arxiv.org/format/2410.04888">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> </div> <p class="title is-5 mathjax"> Focal surfaces and evolutes of framed curves in hyperbolic 3-space from the viewpoint of Legendrian duality </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yu%2C+H">Haibo Yu</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">Liang Chen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.04888v1-abstract-short" style="display: inline;"> A hyperbolic framed curve is a smooth curve with a moving frame in hyperbolic 3-space. It may have singularities. By using this moving frame, we can investigate the differential geometry properties of curves, even at singular points. In fact, we consider the focal surfaces and evolutes of hyperbolic framed curves by using Legendrian dualities which developed by Chen and Izumiya. The focal surfaces… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.04888v1-abstract-full').style.display = 'inline'; document.getElementById('2410.04888v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.04888v1-abstract-full" style="display: none;"> A hyperbolic framed curve is a smooth curve with a moving frame in hyperbolic 3-space. It may have singularities. By using this moving frame, we can investigate the differential geometry properties of curves, even at singular points. In fact, we consider the focal surfaces and evolutes of hyperbolic framed curves by using Legendrian dualities which developed by Chen and Izumiya. The focal surfaces are the dual surfaces of tangent indicatrix of original curves. Moreover, classifications of singularities of the serval dual surfaces are shown. By this, we give the relationship among focal surfaces, evolutes and dual surfaces of evolutes. Finally, we study duality relations of singularities between focal surfaces and dual surfaces of evolutes. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.04888v1-abstract-full').style.display = 'none'; document.getElementById('2410.04888v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">23 pages, 5 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.03112">arXiv:2410.03112</a> <span> [<a href="https://arxiv.org/pdf/2410.03112">pdf</a>, <a href="https://arxiv.org/format/2410.03112">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Learning to Select Cutting Planes in Mixed Integer Linear Programming Solving </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+X">Xuefeng Zhang</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">Liangyu Chen</a>, <a href="/search/math?searchtype=author&query=Zeng%2C+Z">Zhenbing Zeng</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.03112v1-abstract-short" style="display: inline;"> Cutting planes (cuts) are crucial for solving Mixed Integer Linear Programming (MILP) problems. Advanced MILP solvers typically rely on manually designed heuristic algorithms for cut selection, which require much expert experience and cannot be generalized for different scales of MILP problems. Therefore, learning-based methods for cut selection are considered a promising direction. State-of-the-a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.03112v1-abstract-full').style.display = 'inline'; document.getElementById('2410.03112v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.03112v1-abstract-full" style="display: none;"> Cutting planes (cuts) are crucial for solving Mixed Integer Linear Programming (MILP) problems. Advanced MILP solvers typically rely on manually designed heuristic algorithms for cut selection, which require much expert experience and cannot be generalized for different scales of MILP problems. Therefore, learning-based methods for cut selection are considered a promising direction. State-of-the-art learning-based methods formulate cut selection as a sequence-to-sequence problem, easily handled by sequence models. However, the existing sequence models need help with the following issues: (1) the model only captures cut information while neglecting the Linear Programming (LP) relaxation; (2) the sequence model utilizes positional information of the input sequence, which may influence cut selection. To address these challenges, we design a novel learning model HGTSM for better select cuts. We encode MILP problem state as a heterogeneous tripartite graph, utilizing heterogeneous graph networks to fully capture the underlying structure of MILP problems. Simultaneously, we propose a novel sequence model whose architecture is tailored to handle inputs in different orders. Experimental results demonstrate that our model outperforms heuristic methods and learning-based baselines on multiple challenging MILP datasets. Additionally, the model exhibits stability and the ability to generalize to different types of problems. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.03112v1-abstract-full').style.display = 'none'; document.getElementById('2410.03112v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.19674">arXiv:2409.19674</a> <span> [<a href="https://arxiv.org/pdf/2409.19674">pdf</a>, <a href="https://arxiv.org/format/2409.19674">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Alternating Maximization Algorithm for Mismatch Capacity with Oblivious Relaying </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xinwei Li</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">Lingyi Chen</a>, <a href="/search/math?searchtype=author&query=Wu%2C+S">Shitong Wu</a>, <a href="/search/math?searchtype=author&query=Wu%2C+H">Huihui Wu</a>, <a href="/search/math?searchtype=author&query=Wu%2C+H">Hao Wu</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+W">Wenyi Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2409.19674v4-abstract-short" style="display: inline;"> Reliable communication over a discrete memoryless channel with the help of a relay has aroused interest due to its widespread applications in practical scenarios. By considering the system with a mismatched decoder, previous works have provided optimization models to evaluate the mismatch capacity in these scenarios. The proposed models, however, are difficult due to the complicated structure of t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.19674v4-abstract-full').style.display = 'inline'; document.getElementById('2409.19674v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.19674v4-abstract-full" style="display: none;"> Reliable communication over a discrete memoryless channel with the help of a relay has aroused interest due to its widespread applications in practical scenarios. By considering the system with a mismatched decoder, previous works have provided optimization models to evaluate the mismatch capacity in these scenarios. The proposed models, however, are difficult due to the complicated structure of the mismatched decoding problem with the information flows in hops given by the relay. Existing methods, such as the grid search, become impractical as they involve finding all roots of a nonlinear system, with the growing size of the alphabet. To address this problem, we reformulate the max-min optimization model as a consistent maximization form, by considering the dual form of the inner minimization problem and the Lagrangian with a fixed multiplier. Based on the proposed formulation, an alternating maximization framework is designed, which provides the closed-form solution with simple iterations in each step by introducing a suitable variable transformation. The effectiveness of the proposed approach is demonstrated by the simulations over practical scenarios, including Quaternary and Gaussian channels. Moreover, the simulation results of the transitional probability also shed light on the promising application attribute to the quantizer design in the relay node. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.19674v4-abstract-full').style.display = 'none'; document.getElementById('2409.19674v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 29 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.09580">arXiv:2409.09580</a> <span> [<a href="https://arxiv.org/pdf/2409.09580">pdf</a>, <a href="https://arxiv.org/ps/2409.09580">ps</a>, <a href="https://arxiv.org/format/2409.09580">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> K-theoretic Gromov-Witten invariants of line degrees on flag varieties </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Buch%2C+A+S">Anders S. Buch</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">Linda Chen</a>, <a href="/search/math?searchtype=author&query=Xu%2C+W">Weihong Xu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2409.09580v1-abstract-short" style="display: inline;"> A homology class $d \in H_2(X)$ of a complex flag variety $X = G/P$ is called a line degree if the moduli space $\overline{M}_{0,0}(X,d)$ of 0-pointed stable maps to $X$ of degree $d$ is also a flag variety $G/P'$. We prove a quantum equals classical formula stating that any $n$-pointed (equivariant, K-theoretic, genus zero) Gromov-Witten invariant of line degree on $X$ is equal to a classical int… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.09580v1-abstract-full').style.display = 'inline'; document.getElementById('2409.09580v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.09580v1-abstract-full" style="display: none;"> A homology class $d \in H_2(X)$ of a complex flag variety $X = G/P$ is called a line degree if the moduli space $\overline{M}_{0,0}(X,d)$ of 0-pointed stable maps to $X$ of degree $d$ is also a flag variety $G/P'$. We prove a quantum equals classical formula stating that any $n$-pointed (equivariant, K-theoretic, genus zero) Gromov-Witten invariant of line degree on $X$ is equal to a classical intersection number computed on the flag variety $G/P'$. We also prove an $n$-pointed analogue of the Peterson comparison formula stating that these invariants coincide with Gromov-Witten invariants of the variety of complete flags $G/B$. Our formulas make it straightforward to compute the big quantum K-theory ring of $X$ modulo degrees larger than line degrees. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.09580v1-abstract-full').style.display = 'none'; document.getElementById('2409.09580v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">To appear in the proceedings of the conference GLSM@30 (Stony Brook, May 2023)</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14N35 (Primary) 19E08; 14N15; 14M15 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.08670">arXiv:2409.08670</a> <span> [<a href="https://arxiv.org/pdf/2409.08670">pdf</a>, <a href="https://arxiv.org/ps/2409.08670">ps</a>, <a href="https://arxiv.org/format/2409.08670">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Proof of the geometric Langlands conjecture IV: ambidexterity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Arinkin%2C+D">D. Arinkin</a>, <a href="/search/math?searchtype=author&query=Beraldo%2C+D">D. Beraldo</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">L. Chen</a>, <a href="/search/math?searchtype=author&query=Faergeman%2C+J">J. Faergeman</a>, <a href="/search/math?searchtype=author&query=Gaitsgory%2C+D">D. Gaitsgory</a>, <a href="/search/math?searchtype=author&query=Lin%2C+K">K. Lin</a>, <a href="/search/math?searchtype=author&query=Raskin%2C+S">S. Raskin</a>, <a href="/search/math?searchtype=author&query=Rozenblyum%2C+N">N. Rozenblyum</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2409.08670v1-abstract-short" style="display: inline;"> This paper performs the following steps toward the proof of GLC in the de Rham setting: (i) We deduce GLC for G=GL_n; (ii) We prove that the Langlands functor L_G constructed in [GLC1], when restricted to the cuspidal category, is ambidextrous; (iii) We reduce GLC to the study of a certain classical vector bundle with connection on the stack of irreducible local systems; (iv) We prove that… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.08670v1-abstract-full').style.display = 'inline'; document.getElementById('2409.08670v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.08670v1-abstract-full" style="display: none;"> This paper performs the following steps toward the proof of GLC in the de Rham setting: (i) We deduce GLC for G=GL_n; (ii) We prove that the Langlands functor L_G constructed in [GLC1], when restricted to the cuspidal category, is ambidextrous; (iii) We reduce GLC to the study of a certain classical vector bundle with connection on the stack of irreducible local systems; (iv) We prove that GLC is equivalent to the contractibility of the space of generic oper structures on irreducible local systems; (v) Using [BKS], we deduce GLC for classical groups. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.08670v1-abstract-full').style.display = 'none'; document.getElementById('2409.08670v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.07051">arXiv:2409.07051</a> <span> [<a href="https://arxiv.org/pdf/2409.07051">pdf</a>, <a href="https://arxiv.org/ps/2409.07051">ps</a>, <a href="https://arxiv.org/format/2409.07051">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Proof of the geometric Langlands conjecture III: compatibility with parabolic induction </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Campbell%2C+J">Justin Campbell</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">Lin Chen</a>, <a href="/search/math?searchtype=author&query=Faergeman%2C+J">Joakim Faergeman</a>, <a href="/search/math?searchtype=author&query=Gaitsgory%2C+D">Dennis Gaitsgory</a>, <a href="/search/math?searchtype=author&query=Lin%2C+K">Kevin Lin</a>, <a href="/search/math?searchtype=author&query=Raskin%2C+S">Sam Raskin</a>, <a href="/search/math?searchtype=author&query=Rozenblyum%2C+N">Nick Rozenblyum</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2409.07051v1-abstract-short" style="display: inline;"> We establish the compatibility of the Langlands functor with the operations of Eisenstein series constant term, and deduce that the Langlands functor induces an equivalence on Eisenstein-generated subcategories. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.07051v1-abstract-full" style="display: none;"> We establish the compatibility of the Langlands functor with the operations of Eisenstein series constant term, and deduce that the Langlands functor induces an equivalence on Eisenstein-generated subcategories. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.07051v1-abstract-full').style.display = 'none'; document.getElementById('2409.07051v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.06530">arXiv:2409.06530</a> <span> [<a href="https://arxiv.org/pdf/2409.06530">pdf</a>, <a href="https://arxiv.org/format/2409.06530">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Functionally Constrained Algorithm Solves Convex Simple Bilevel Problems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+H">Huaqing Zhang</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">Lesi Chen</a>, <a href="/search/math?searchtype=author&query=Xu%2C+J">Jing Xu</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+J">Jingzhao Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2409.06530v2-abstract-short" style="display: inline;"> This paper studies simple bilevel problems, where a convex upper-level function is minimized over the optimal solutions of a convex lower-level problem. We first show the fundamental difficulty of simple bilevel problems, that the approximate optimal value of such problems is not obtainable by first-order zero-respecting algorithms. Then we follow recent works to pursue the weak approximate soluti… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.06530v2-abstract-full').style.display = 'inline'; document.getElementById('2409.06530v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.06530v2-abstract-full" style="display: none;"> This paper studies simple bilevel problems, where a convex upper-level function is minimized over the optimal solutions of a convex lower-level problem. We first show the fundamental difficulty of simple bilevel problems, that the approximate optimal value of such problems is not obtainable by first-order zero-respecting algorithms. Then we follow recent works to pursue the weak approximate solutions. For this goal, we propose novel near-optimal methods for smooth and nonsmooth problems by reformulating them into functionally constrained problems. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.06530v2-abstract-full').style.display = 'none'; document.getElementById('2409.06530v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">to be published in NeurIPS 2024</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2408.12074">arXiv:2408.12074</a> <span> [<a href="https://arxiv.org/pdf/2408.12074">pdf</a>, <a href="https://arxiv.org/ps/2408.12074">ps</a>, <a href="https://arxiv.org/format/2408.12074">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> Vertex-primitive s-arc-transitive digraphs of symplectic groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+L">Lei Chen</a>, <a href="/search/math?searchtype=author&query=Giudici%2C+M">Michael Giudici</a>, <a href="/search/math?searchtype=author&query=Praeger%2C+C+E">Cheryl E. Praeger</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2408.12074v1-abstract-short" style="display: inline;"> A digraph is $s$-arc-transitive if its automorphism group is transitive on directed paths with $s$ edges, that is, on $s$-arcs. Although infinite families of finite $s$-arc transitive digraphs of arbitrary valency were constructed by the third author in 1989, existence of a vertex-primitive $2$-arc-transitive digraph was not known until an infinite family was constructed by the second author with… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.12074v1-abstract-full').style.display = 'inline'; document.getElementById('2408.12074v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.12074v1-abstract-full" style="display: none;"> A digraph is $s$-arc-transitive if its automorphism group is transitive on directed paths with $s$ edges, that is, on $s$-arcs. Although infinite families of finite $s$-arc transitive digraphs of arbitrary valency were constructed by the third author in 1989, existence of a vertex-primitive $2$-arc-transitive digraph was not known until an infinite family was constructed by the second author with Li and Xia in 2017. This led to a conjecture by the second author and Xia in 2018 that, for a finite vertex-primitive $s$-arc-transitive digraph, $s$ is at most $2$, together with their proof that it is sufficient to prove the conjecture for digraphs with an almost simple group of automorphisms. This paper confirms the conjecture for finite symplectic groups. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.12074v1-abstract-full').style.display = 'none'; document.getElementById('2408.12074v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2408.08232">arXiv:2408.08232</a> <span> [<a href="https://arxiv.org/pdf/2408.08232">pdf</a>, <a href="https://arxiv.org/ps/2408.08232">ps</a>, <a href="https://arxiv.org/format/2408.08232">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Equivalent Characterizations of the Aubin Property for Nonlinear Semidefinite Programming </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+L">Liang Chen</a>, <a href="/search/math?searchtype=author&query=Chen%2C+R">Ruoning Chen</a>, <a href="/search/math?searchtype=author&query=Sun%2C+D">Defeng Sun</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+L">Liping Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2408.08232v1-abstract-short" style="display: inline;"> In this paper, we study the Aubin property of the Karush-Kuhn-Tucker solution mapping for the nonlinear semidefinite programming (NLSDP) problem at a locally optimal solution. In the literature, it is known that the Aubin property implies the constraint nondegeneracy by Fusek [SIAM J. Optim. 23 (2013), pp. 1041-1061] and the second-order sufficient condition by Ding et al. [SIAM J. Optim. 27 (2017… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.08232v1-abstract-full').style.display = 'inline'; document.getElementById('2408.08232v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.08232v1-abstract-full" style="display: none;"> In this paper, we study the Aubin property of the Karush-Kuhn-Tucker solution mapping for the nonlinear semidefinite programming (NLSDP) problem at a locally optimal solution. In the literature, it is known that the Aubin property implies the constraint nondegeneracy by Fusek [SIAM J. Optim. 23 (2013), pp. 1041-1061] and the second-order sufficient condition by Ding et al. [SIAM J. Optim. 27 (2017), pp. 67-90]. Based on the Mordukhovich criterion, here we further prove that the strong second-order sufficient condition is also necessary for the Aubin property to hold. Consequently, several equivalent conditions including the strong regularity are established for NLSDP's Aubin property. Together with the recent progress made by Chen et al. on the equivalence between the Aubin property and the strong regularity for nonlinear second-order cone programming [arXiv:2406.13798v1 (2024)], this paper constitutes a significant step forward in characterizing the Aubin property for general non-polyhedral $C^2$-cone reducible constrained optimization problems. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.08232v1-abstract-full').style.display = 'none'; document.getElementById('2408.08232v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 49J53; 90C22; 90C31; 90C46 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2408.05778">arXiv:2408.05778</a> <span> [<a href="https://arxiv.org/pdf/2408.05778">pdf</a>, <a href="https://arxiv.org/format/2408.05778">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Pareto Front Shape-Agnostic Pareto Set Learning in Multi-Objective Optimization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ye%2C+R">Rongguang Ye</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">Longcan Chen</a>, <a href="/search/math?searchtype=author&query=Kou%2C+W">Wei-Bin Kou</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+J">Jinyuan Zhang</a>, <a href="/search/math?searchtype=author&query=Ishibuchi%2C+H">Hisao Ishibuchi</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2408.05778v1-abstract-short" style="display: inline;"> Pareto set learning (PSL) is an emerging approach for acquiring the complete Pareto set of a multi-objective optimization problem. Existing methods primarily rely on the mapping of preference vectors in the objective space to Pareto optimal solutions in the decision space. However, the sampling of preference vectors theoretically requires prior knowledge of the Pareto front shape to ensure high pe… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.05778v1-abstract-full').style.display = 'inline'; document.getElementById('2408.05778v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.05778v1-abstract-full" style="display: none;"> Pareto set learning (PSL) is an emerging approach for acquiring the complete Pareto set of a multi-objective optimization problem. Existing methods primarily rely on the mapping of preference vectors in the objective space to Pareto optimal solutions in the decision space. However, the sampling of preference vectors theoretically requires prior knowledge of the Pareto front shape to ensure high performance of the PSL methods. Designing a sampling strategy of preference vectors is difficult since the Pareto front shape cannot be known in advance. To make Pareto set learning work effectively in any Pareto front shape, we propose a Pareto front shape-agnostic Pareto Set Learning (GPSL) that does not require the prior information about the Pareto front. The fundamental concept behind GPSL is to treat the learning of the Pareto set as a distribution transformation problem. Specifically, GPSL can transform an arbitrary distribution into the Pareto set distribution. We demonstrate that training a neural network by maximizing hypervolume enables the process of distribution transformation. Our proposed method can handle any shape of the Pareto front and learn the Pareto set without requiring prior knowledge. Experimental results show the high performance of our proposed method on diverse test problems compared with recent Pareto set learning algorithms. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.05778v1-abstract-full').style.display = 'none'; document.getElementById('2408.05778v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">7 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> IEEE International Conference on Systems, Man, and Cybernetics (IEEE SMC 2024) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.17742">arXiv:2407.17742</a> <span> [<a href="https://arxiv.org/pdf/2407.17742">pdf</a>, <a href="https://arxiv.org/format/2407.17742">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> A high-order, high-efficiency adaptive time filter algorithm for shale reservoir model based on coupled fluid flow with porous media flow </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+J">Jian Li</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">Lele Chen</a>, <a href="/search/math?searchtype=author&query=Qin%2C+Y">Yi Qin</a>, <a href="/search/math?searchtype=author&query=Chen%2C+Z">Zhangxin Chen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.17742v1-abstract-short" style="display: inline;"> In this paper, a third-order time adaptive algorithm with less computation, low complexity is provided for shale reservoir model based on coupled fluid flow with porous media flow. The algorithm combines the three-step linear time filters method for simple post-processing and the second-order backward differential formula (BDF2), is third-order accurate and provides, at no extra computational comp… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.17742v1-abstract-full').style.display = 'inline'; document.getElementById('2407.17742v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.17742v1-abstract-full" style="display: none;"> In this paper, a third-order time adaptive algorithm with less computation, low complexity is provided for shale reservoir model based on coupled fluid flow with porous media flow. The algorithm combines the three-step linear time filters method for simple post-processing and the second-order backward differential formula (BDF2), is third-order accurate and provides, at no extra computational complexity. At the same time, the time filter method can also be used to damp non-physical oscillations inherent in the BDF2 method, ensuring stability. We proves the variable time stepsize second-order backward differential formula plus time filter (BDF2-TF) algorithm's stability and the convergence properties of the fluid velocity u and hydraulic head $蠁$ in the $L^2$ norm with an order of $O(k_{n+1}^3 + h^3)$. In the experiments, the adaptive algorithm automatically adjusts the time step in response to the varying characteristics of different models, ensuring that errors are maintained within acceptable limits. This algorithm addresses the issue that high-order algorithms may select inappropriate time steps, resulting in instability or reduced precision of the numerical solution, thereby enhancing calculation accuracy and efficiency. We perform three-dimensional numerical experiments to verify the BDF2-TF algorithm's effectiveness, stability, and third-order convergence. Simultaneously, a simplified model is employed to simulate the process of shale oil extraction from reservoirs, further demonstrating the algorithm's practical applicability. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.17742v1-abstract-full').style.display = 'none'; document.getElementById('2407.17742v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.14238">arXiv:2407.14238</a> <span> [<a href="https://arxiv.org/pdf/2407.14238">pdf</a>, <a href="https://arxiv.org/ps/2407.14238">ps</a>, <a href="https://arxiv.org/format/2407.14238">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="History and Philosophy of Physics">physics.hist-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Logic">math.LO</span> </div> </div> <p class="title is-5 mathjax"> Univalence and Ontic Structuralism </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+L">Lu Chen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.14238v1-abstract-short" style="display: inline;"> The persistent challenge of formulating ontic structuralism in a rigorous manner, which prioritizes structures over the entities they contain, calls for a transformation of traditional logical frameworks. I argue that Univalent Foundations (UF), which feature the axiom that all isomorphic structures are identical, offer such a foundation and are more attractive than other proposed structuralist fr… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.14238v1-abstract-full').style.display = 'inline'; document.getElementById('2407.14238v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.14238v1-abstract-full" style="display: none;"> The persistent challenge of formulating ontic structuralism in a rigorous manner, which prioritizes structures over the entities they contain, calls for a transformation of traditional logical frameworks. I argue that Univalent Foundations (UF), which feature the axiom that all isomorphic structures are identical, offer such a foundation and are more attractive than other proposed structuralist frameworks. Furthermore, I delve into the significance in the case of the hole argument and, very briefly, the nature of symmetries. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.14238v1-abstract-full').style.display = 'none'; document.getElementById('2407.14238v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Foundation of Physics 54, 38 (2024) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.14215">arXiv:2407.14215</a> <span> [<a href="https://arxiv.org/pdf/2407.14215">pdf</a>, <a href="https://arxiv.org/ps/2407.14215">ps</a>, <a href="https://arxiv.org/format/2407.14215">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Implementable Semismooth* Newton Methods for Generalized Equations are G-Semismooth Newton Methods </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+L">Liang Chen</a>, <a href="/search/math?searchtype=author&query=Sun%2C+D">Defeng Sun</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+W">Wangyongquan Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.14215v1-abstract-short" style="display: inline;"> Semismooth* Newton methods have been proposed in recent years targeting multi-valued inclusion problems and have been successfully implemented to deal with several concrete generalized equations. In this paper, we show that these executable implementations are exactly the applications of G-semismooth Newton methods for solving nonsmooth equations localized from these generalized equations. This ne… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.14215v1-abstract-full').style.display = 'inline'; document.getElementById('2407.14215v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.14215v1-abstract-full" style="display: none;"> Semismooth* Newton methods have been proposed in recent years targeting multi-valued inclusion problems and have been successfully implemented to deal with several concrete generalized equations. In this paper, we show that these executable implementations are exactly the applications of G-semismooth Newton methods for solving nonsmooth equations localized from these generalized equations. This new understanding expands the breadth of G-semismooth Newton methods in theory, and more importantly, facilitates the design and implementation of practical Newton-type algorithms for solving generalized equations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.14215v1-abstract-full').style.display = 'none'; document.getElementById('2407.14215v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 49J52; 49J53; 90C31; 90C33; 49M15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.12936">arXiv:2407.12936</a> <span> [<a href="https://arxiv.org/pdf/2407.12936">pdf</a>, <a href="https://arxiv.org/ps/2407.12936">ps</a>, <a href="https://arxiv.org/format/2407.12936">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Mean-Field Control for Diffusion Aggregation system with Coulomb Interaction </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+L">Li Chen</a>, <a href="/search/math?searchtype=author&query=Wang%2C+Y">Yucheng Wang</a>, <a href="/search/math?searchtype=author&query=Wang%2C+Z">Zhao Wang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.12936v2-abstract-short" style="display: inline;"> The mean-field control problem for a multi-dimensional diffusion-aggregation system with Coulomb interaction (the so called parabolic elliptic Keller-Segel system) is considered. The existence of optimal control is proved through the $螕$-convergence of the corresponding control problem of the interacting particle system. There are three building blocks in the whole argument. Firstly, for the optim… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.12936v2-abstract-full').style.display = 'inline'; document.getElementById('2407.12936v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.12936v2-abstract-full" style="display: none;"> The mean-field control problem for a multi-dimensional diffusion-aggregation system with Coulomb interaction (the so called parabolic elliptic Keller-Segel system) is considered. The existence of optimal control is proved through the $螕$-convergence of the corresponding control problem of the interacting particle system. There are three building blocks in the whole argument. Firstly, for the optimal control problem on the particle level, instead of using classical method for stochastic system, we study directly the control problem of high-dimensional parabolic equation, i.e. the Liouville equation of it. Secondly, we obtain a strong propagation of chaos result for the interacting particle system by combining the convergence in probability and relative entropy method. Due to this strong mean field limit result, we avoid giving compact support requirement for control functions, which has been often used in the literature. Thirdly, because of strong aggregation effect, additional difficulties arise from control function in obtaining the well-posedness theory of the diffusion-aggregation equation, so that the known method cannot be directly applied. Instead, we use a combination of local existence result and bootstrap argument to obtain the global solution in the sub-critical regime. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.12936v2-abstract-full').style.display = 'none'; document.getElementById('2407.12936v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.07911">arXiv:2407.07911</a> <span> [<a href="https://arxiv.org/pdf/2407.07911">pdf</a>, <a href="https://arxiv.org/ps/2407.07911">ps</a>, <a href="https://arxiv.org/format/2407.07911">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> </div> </div> <p class="title is-5 mathjax"> Linear algebra of quadratic forms and polynomial identity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+L">Li Chen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.07911v2-abstract-short" style="display: inline;"> Let $S_1=\{p_1,p_2,\cdots, p_l\}\subset\cc[z_1,z_2\cdots,z_n]$ be a set of quadratic forms such that $p_i=q_i^2$ where $\{q_i\}_{i=1}^l$ are linear forms. For $1\leq k\leq l$, let $S_k=\{p_{i_1}p_{i_2}\cdots p_{i_k}|1\leq i_1<i_2<\cdots<i_k\leq l\}$ be the set of $k$-products of distinct polynomials in $S_1$. We show somehow unexpectedly that linear independence of $S_1$ is equivalent to that of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.07911v2-abstract-full').style.display = 'inline'; document.getElementById('2407.07911v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.07911v2-abstract-full" style="display: none;"> Let $S_1=\{p_1,p_2,\cdots, p_l\}\subset\cc[z_1,z_2\cdots,z_n]$ be a set of quadratic forms such that $p_i=q_i^2$ where $\{q_i\}_{i=1}^l$ are linear forms. For $1\leq k\leq l$, let $S_k=\{p_{i_1}p_{i_2}\cdots p_{i_k}|1\leq i_1<i_2<\cdots<i_k\leq l\}$ be the set of $k$-products of distinct polynomials in $S_1$. We show somehow unexpectedly that linear independence of $S_1$ is equivalent to that of $S_k$ for $k=2$ and $3$ under certain rank conditions. Among technical treatments in the proof, of independent conceptual interest is a novel polynomial identity which elegantly incorporates quadratic forms and matrix determinants. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.07911v2-abstract-full').style.display = 'none'; document.getElementById('2407.07911v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 2 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.06664">arXiv:2407.06664</a> <span> [<a href="https://arxiv.org/pdf/2407.06664">pdf</a>, <a href="https://arxiv.org/format/2407.06664">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> PDEformer-1: A Foundation Model for One-Dimensional Partial Differential Equations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ye%2C+Z">Zhanhong Ye</a>, <a href="/search/math?searchtype=author&query=Huang%2C+X">Xiang Huang</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">Leheng Chen</a>, <a href="/search/math?searchtype=author&query=Liu%2C+Z">Zining Liu</a>, <a href="/search/math?searchtype=author&query=Wu%2C+B">Bingyang Wu</a>, <a href="/search/math?searchtype=author&query=Liu%2C+H">Hongsheng Liu</a>, <a href="/search/math?searchtype=author&query=Wang%2C+Z">Zidong Wang</a>, <a href="/search/math?searchtype=author&query=Dong%2C+B">Bin Dong</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.06664v1-abstract-short" style="display: inline;"> This paper introduces PDEformer-1, a versatile neural solver capable of simultaneously addressing various partial differential equations (PDEs). With the PDE represented as a computational graph, we facilitate the seamless integration of symbolic and numeric information inherent in a PDE. A graph Transformer and an implicit neural representation (INR) are employed subsequently to generate mesh-fre… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.06664v1-abstract-full').style.display = 'inline'; document.getElementById('2407.06664v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.06664v1-abstract-full" style="display: none;"> This paper introduces PDEformer-1, a versatile neural solver capable of simultaneously addressing various partial differential equations (PDEs). With the PDE represented as a computational graph, we facilitate the seamless integration of symbolic and numeric information inherent in a PDE. A graph Transformer and an implicit neural representation (INR) are employed subsequently to generate mesh-free predicted solutions. We generated a dataset with up to three million samples involving diverse one-dimensional PDEs to pretrain our model. Compared with baseline models trained specifically on benchmark datasets, our pretrained model achieves comparable accuracy via zero-shot inference, and the advantage expands after finetuning. For PDEs new or unseen in the pretraining stage, our model can adapt quickly by finetuning on a relatively small set of examples from the target equation. Additionally, PDEformer-1 demonstrates promising results in the inverse problem of PDE scalar coefficient recovery and coefficient field recovery. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.06664v1-abstract-full').style.display = 'none'; document.getElementById('2407.06664v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.05286">arXiv:2407.05286</a> <span> [<a href="https://arxiv.org/pdf/2407.05286">pdf</a>, <a href="https://arxiv.org/format/2407.05286">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Stability and Generalization for Stochastic Recursive Momentum-based Algorithms for (Strongly-)Convex One to $K$-Level Stochastic Optimizations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Pan%2C+X">Xiaokang Pan</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xingyu Li</a>, <a href="/search/math?searchtype=author&query=Liu%2C+J">Jin Liu</a>, <a href="/search/math?searchtype=author&query=Sun%2C+T">Tao Sun</a>, <a href="/search/math?searchtype=author&query=Sun%2C+K">Kai Sun</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">Lixing Chen</a>, <a href="/search/math?searchtype=author&query=Qu%2C+Z">Zhe Qu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.05286v1-abstract-short" style="display: inline;"> STOchastic Recursive Momentum (STORM)-based algorithms have been widely developed to solve one to $K$-level ($K \geq 3$) stochastic optimization problems. Specifically, they use estimators to mitigate the biased gradient issue and achieve near-optimal convergence results. However, there is relatively little work on understanding their generalization performance, particularly evident during the tra… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.05286v1-abstract-full').style.display = 'inline'; document.getElementById('2407.05286v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.05286v1-abstract-full" style="display: none;"> STOchastic Recursive Momentum (STORM)-based algorithms have been widely developed to solve one to $K$-level ($K \geq 3$) stochastic optimization problems. Specifically, they use estimators to mitigate the biased gradient issue and achieve near-optimal convergence results. However, there is relatively little work on understanding their generalization performance, particularly evident during the transition from one to $K$-level optimization contexts. This paper provides a comprehensive generalization analysis of three representative STORM-based algorithms: STORM, COVER, and SVMR, for one, two, and $K$-level stochastic optimizations under both convex and strongly convex settings based on algorithmic stability. Firstly, we define stability for $K$-level optimizations and link it to generalization. Then, we detail the stability results for three prominent STORM-based algorithms. Finally, we derive their excess risk bounds by balancing stability results with optimization errors. Our theoretical results provide strong evidence to complete STORM-based algorithms: (1) Each estimator may decrease their stability due to variance with its estimation target. (2) Every additional level might escalate the generalization error, influenced by the stability and the variance between its cumulative stochastic gradient and the true gradient. (3) Increasing the batch size for the initial computation of estimators presents a favorable trade-off, enhancing the generalization performance. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.05286v1-abstract-full').style.display = 'none'; document.getElementById('2407.05286v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.16627">arXiv:2406.16627</a> <span> [<a href="https://arxiv.org/pdf/2406.16627">pdf</a>, <a href="https://arxiv.org/format/2406.16627">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> A Random Integration Algorithm for High-dimensional Function Spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+L">Liang Chen</a>, <a href="/search/math?searchtype=author&query=Xu%2C+M">Minqiang Xu</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+H">Haizhang Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2406.16627v1-abstract-short" style="display: inline;"> We introduce a novel random integration algorithm that boasts both high convergence order and polynomial tractability for functions characterized by sparse frequencies or rapidly decaying Fourier coefficients. Specifically, for integration in periodic isotropic Sobolev space and the isotropic Sobolev space with compact support, our approach attains a nearly optimal root mean square error (RMSE) bo… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.16627v1-abstract-full').style.display = 'inline'; document.getElementById('2406.16627v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.16627v1-abstract-full" style="display: none;"> We introduce a novel random integration algorithm that boasts both high convergence order and polynomial tractability for functions characterized by sparse frequencies or rapidly decaying Fourier coefficients. Specifically, for integration in periodic isotropic Sobolev space and the isotropic Sobolev space with compact support, our approach attains a nearly optimal root mean square error (RMSE) bound. In contrast to previous nearly optimal algorithms, our method exhibits polynomial tractability, ensuring that the number of samples does not scale exponentially with increasing dimensions. Our integration algorithm also enjoys nearly optimal bound for weighted Korobov space. Furthermore, the algorithm can be applied without the need for prior knowledge of weights, distinguishing it from the component-by-component algorithm. For integration in the Wiener algebra, the sample complexity of our algorithm is independent of the decay rate of Fourier coefficients. The effectiveness of the integration is confirmed through numerical experiments. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.16627v1-abstract-full').style.display = 'none'; document.getElementById('2406.16627v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.13798">arXiv:2406.13798</a> <span> [<a href="https://arxiv.org/pdf/2406.13798">pdf</a>, <a href="https://arxiv.org/ps/2406.13798">ps</a>, <a href="https://arxiv.org/format/2406.13798">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Aubin Property and the Strong Regularity Are Equivalent for Nonlinear Second-Order Cone Programming </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+L">Liang Chen</a>, <a href="/search/math?searchtype=author&query=Chen%2C+R">Ruoning Chen</a>, <a href="/search/math?searchtype=author&query=Sun%2C+D">Defeng Sun</a>, <a href="/search/math?searchtype=author&query=Zhu%2C+J">Junyuan Zhu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2406.13798v2-abstract-short" style="display: inline;"> This paper solves a fundamental open problem in variational analysis on the equivalence between the Aubin property and the strong regularity for nonlinear second-order cone programming (SOCP) at a locally optimal solution. We achieve this by introducing a reduction approach to the Aubin property characterized by the Mordukhovich criterion and a lemma of alternative choices on cones to replace the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.13798v2-abstract-full').style.display = 'inline'; document.getElementById('2406.13798v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.13798v2-abstract-full" style="display: none;"> This paper solves a fundamental open problem in variational analysis on the equivalence between the Aubin property and the strong regularity for nonlinear second-order cone programming (SOCP) at a locally optimal solution. We achieve this by introducing a reduction approach to the Aubin property characterized by the Mordukhovich criterion and a lemma of alternative choices on cones to replace the S-lemma used in Outrata and Ram铆rez [SIAM J. Optim. 21 (2011) 789-823] and Opazo, Outrata, and Ram铆rez [SIAM J. Optim. 27 (2017) 2141-2151], where the same SOCP was considered under the strict complementarity condition except for possibly only one block of constraints. As a byproduct, we also offer a new approach to the well-known result of Dontchev and Rockafellar [SIAM J. Optim. 6 (1996) 1087-1105] on the equivalence of the two concepts in conventional nonlinear programming. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.13798v2-abstract-full').style.display = 'none'; document.getElementById('2406.13798v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 90C; 90C31; 90C46 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.09772">arXiv:2406.09772</a> <span> [<a href="https://arxiv.org/pdf/2406.09772">pdf</a>, <a href="https://arxiv.org/format/2406.09772">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Accelerated Over-Relaxation Heavy-Ball Methods with Provable Acceleration and Global Convergence </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wei%2C+J">Jingrong Wei</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">Long Chen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2406.09772v1-abstract-short" style="display: inline;"> The heavy-ball momentum method has gained widespread popularity for accelerating gradient descent by incorporating a momentum term. Recent studies have conclusively shown that the heavy-ball method cannot achieve an accelerated convergence rate for general smooth strongly convex optimization problems. This work introduces the Accelerated Over-Relaxation Heavy-Ball (AOR-HB) method, a novel approach… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.09772v1-abstract-full').style.display = 'inline'; document.getElementById('2406.09772v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.09772v1-abstract-full" style="display: none;"> The heavy-ball momentum method has gained widespread popularity for accelerating gradient descent by incorporating a momentum term. Recent studies have conclusively shown that the heavy-ball method cannot achieve an accelerated convergence rate for general smooth strongly convex optimization problems. This work introduces the Accelerated Over-Relaxation Heavy-Ball (AOR-HB) method, a novel approach that represents the first heavy-ball method to demonstrate provable global and accelerated convergence for smooth strongly convex optimization. The key innovation of the AOR-HB method lies in the application of an over-relaxation technique to the gradient term. This novel approach enables the method to be applied to min-max problems and meet optimal lower complexity bounds. This breakthrough addresses a long-standing theoretical gap in heavy-ball momentum methods and paves the way for developing accelerated methods that transcend the boundaries of convex optimization to non-convex optimization. Numerical experiments validate the effectiveness of the proposed algorithms, with their performance matching that of other leading first-order optimization methods. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.09772v1-abstract-full').style.display = 'none'; document.getElementById('2406.09772v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.09449">arXiv:2406.09449</a> <span> [<a href="https://arxiv.org/pdf/2406.09449">pdf</a>, <a href="https://arxiv.org/ps/2406.09449">ps</a>, <a href="https://arxiv.org/format/2406.09449">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Smooth solutions to the Christoffel problem in $\mathbb{H}^{n+1}$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+L">Li Chen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2406.09449v1-abstract-short" style="display: inline;"> The famous Christoffel problem is possibly the oldest problem of prescribed curvatures for convex hypersurfaces in Euclidean space. Recently, this problem has been naturally formulated in the context of uniformly $h$-convex hypersurfaces in hyperbolic space by Espinar-G谩lvez-Mira. Surprisingly, Espinar-G谩lvez-Mira find that the Christoffel problem in hyperbolic space is essentially equivalent to… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.09449v1-abstract-full').style.display = 'inline'; document.getElementById('2406.09449v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.09449v1-abstract-full" style="display: none;"> The famous Christoffel problem is possibly the oldest problem of prescribed curvatures for convex hypersurfaces in Euclidean space. Recently, this problem has been naturally formulated in the context of uniformly $h$-convex hypersurfaces in hyperbolic space by Espinar-G谩lvez-Mira. Surprisingly, Espinar-G谩lvez-Mira find that the Christoffel problem in hyperbolic space is essentially equivalent to the Nirenberg-Kazdan-Warner problem on prescribing scalar curvature on $\mathbb{S}^n$. This equivalence opens a new door to study the Nirenberg-Kazdan-Warner problem. In this paper, we establish a existence of solutions to the Christoffel problem in hyperbolic space by proving a full rank theorem. As a corollary, a existence of solutions to the Nirenberg-Kazdan-Warner problem follows. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.09449v1-abstract-full').style.display = 'none'; document.getElementById('2406.09449v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">22 pages. arXiv admin note: substantial text overlap with arXiv:2302.01604</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.08306">arXiv:2406.08306</a> <span> [<a href="https://arxiv.org/pdf/2406.08306">pdf</a>, <a href="https://arxiv.org/ps/2406.08306">ps</a>, <a href="https://arxiv.org/format/2406.08306">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> 2-dimensional Ricci limit spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+L">Lina Chen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2406.08306v1-abstract-short" style="display: inline;"> In this note, we will show that if a measured Gromov-Hausdorff limit space of a sequence of Riemannian manifolds with lower Ricci curvature bound has dense 2-regular set, then it is homeomorphic to a 2-dimensional manifold in an open full measure set. This result gives a positive answer to an open problem in [Naber, Open problem 3.4] in dimension 2 and for dimension larger than 2 there are counter… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.08306v1-abstract-full').style.display = 'inline'; document.getElementById('2406.08306v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.08306v1-abstract-full" style="display: none;"> In this note, we will show that if a measured Gromov-Hausdorff limit space of a sequence of Riemannian manifolds with lower Ricci curvature bound has dense 2-regular set, then it is homeomorphic to a 2-dimensional manifold in an open full measure set. This result gives a positive answer to an open problem in [Naber, Open problem 3.4] in dimension 2 and for dimension larger than 2 there are counterexamples by [HNW, Zhou]. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.08306v1-abstract-full').style.display = 'none'; document.getElementById('2406.08306v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.07655">arXiv:2406.07655</a> <span> [<a href="https://arxiv.org/pdf/2406.07655">pdf</a>, <a href="https://arxiv.org/ps/2406.07655">ps</a>, <a href="https://arxiv.org/format/2406.07655">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Analogues of Alder-Type Partition Inequalities for Fixed Perimeter Partitions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+L">Ling Chen</a>, <a href="/search/math?searchtype=author&query=Hernandez%2C+I">Isabelle Hernandez</a>, <a href="/search/math?searchtype=author&query=Shields%2C+Z">Zain Shields</a>, <a href="/search/math?searchtype=author&query=Swisher%2C+H">Holly Swisher</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2406.07655v1-abstract-short" style="display: inline;"> In a 2016 paper, Straub proved an analogue to Euler's partition identity for partitions with fixed perimeter. Later, Fu and Tang provided a refinement and generalization of Straub's analogue to $d$-distinct partitions as well as a result related to the first Rogers-Ramanujan identity. Motivated by Alder-type partition identities and their generalizations, we build on work of Fu and Tang to establi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.07655v1-abstract-full').style.display = 'inline'; document.getElementById('2406.07655v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.07655v1-abstract-full" style="display: none;"> In a 2016 paper, Straub proved an analogue to Euler's partition identity for partitions with fixed perimeter. Later, Fu and Tang provided a refinement and generalization of Straub's analogue to $d$-distinct partitions as well as a result related to the first Rogers-Ramanujan identity. Motivated by Alder-type partition identities and their generalizations, we build on work of Fu and Tang to establish generalized Alder-type partition inequalities in a fixed perimeter setting, and notably, a reverse Alder-type inequality. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.07655v1-abstract-full').style.display = 'none'; document.getElementById('2406.07655v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">13 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.07552">arXiv:2406.07552</a> <span> [<a href="https://arxiv.org/pdf/2406.07552">pdf</a>, <a href="https://arxiv.org/ps/2406.07552">ps</a>, <a href="https://arxiv.org/format/2406.07552">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Rings and Algebras">math.RA</span> </div> </div> <p class="title is-5 mathjax"> Cohomology of a restricted Lie algebra with a restricted derivation in characteristic 2 </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Mao%2C+D">Dan Mao</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">Liangyun Chen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2406.07552v1-abstract-short" style="display: inline;"> This paper mainly studies the ResLieDer pair in characteristic 2, that is, a restricted Lie algebra with a restricted derivation. We define the restricted representation of a ResLieDer pair and the corresponding cohomology complex. We show that a ResLieDer pair is rigid if the second cohomology group is trivial and a deformation of order $n$ is extensible if and only if its obstruction class is tr… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.07552v1-abstract-full').style.display = 'inline'; document.getElementById('2406.07552v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.07552v1-abstract-full" style="display: none;"> This paper mainly studies the ResLieDer pair in characteristic 2, that is, a restricted Lie algebra with a restricted derivation. We define the restricted representation of a ResLieDer pair and the corresponding cohomology complex. We show that a ResLieDer pair is rigid if the second cohomology group is trivial and a deformation of order $n$ is extensible if and only if its obstruction class is trivial. Moreover, we prove that the central extensions of a ResLieDer pair are classified by the second cohomology group. Finally, we show that a pair of restricted derivations is extensible if and only if its obstruction class is trivial. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.07552v1-abstract-full').style.display = 'none'; document.getElementById('2406.07552v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">26 page</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.00743">arXiv:2406.00743</a> <span> [<a href="https://arxiv.org/pdf/2406.00743">pdf</a>, <a href="https://arxiv.org/ps/2406.00743">ps</a>, <a href="https://arxiv.org/format/2406.00743">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Quantization property of n-Laplacian mean field equation and sharp Moser-Onofri inequality </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+L">Lu Chen</a>, <a href="/search/math?searchtype=author&query=Lu%2C+G">Guozhen Lu</a>, <a href="/search/math?searchtype=author&query=Wang%2C+B">Bohan Wang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2406.00743v1-abstract-short" style="display: inline;"> In this paper, we are concerned with the following $n$-Laplacian mean field equation \[ \left\{ {\begin{array}{*{20}{c}} { - 螖_n u = 位e^u} & {\rm in} \ \ 惟, \\ {\ \ \ \ u = 0} &\ {\rm on}\ \partial 惟, \end{array}} \right. \] \[\] where $惟$ is a smooth bounded domain of $\mathbb{R}^n \ (n\geq 2)$ and $- 螖_n u =-{\rm div}(|\nabla u|^{n-2}\nabla u)$. We first establish the quantization property of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.00743v1-abstract-full').style.display = 'inline'; document.getElementById('2406.00743v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.00743v1-abstract-full" style="display: none;"> In this paper, we are concerned with the following $n$-Laplacian mean field equation \[ \left\{ {\begin{array}{*{20}{c}} { - 螖_n u = 位e^u} & {\rm in} \ \ 惟, \\ {\ \ \ \ u = 0} &\ {\rm on}\ \partial 惟, \end{array}} \right. \] \[\] where $惟$ is a smooth bounded domain of $\mathbb{R}^n \ (n\geq 2)$ and $- 螖_n u =-{\rm div}(|\nabla u|^{n-2}\nabla u)$. We first establish the quantization property of solutions to the above $n$-Laplacian mean field equation. As an application, combining the Pohozaev identity and the capacity estimate, we obtain the sharp constant $C(n)$ of the Moser-Onofri inequality in the $n$-dimensional unit ball $B^n:=B^n(0,1)$, $$\mathop {\inf }\limits_{u \in W_0^{1,n}(B^n)}\frac{1}{ n C_n}\int_{B^n} | \nabla u|^n dx- \ln \int_{B^n} {e^u} dx\geq C(n), $$ which extends the result of Caglioti-Lions-Marchioro-Pulvirenti in \cite{Caglioti} to the case of $n$-dimensional ball. Here $C_n=(\frac{n^2}{n-1})^{n-1} 蠅_{n-1}$ and $蠅_{n-1}$ is the surface measure of $B^n$. For the Moser-Onofri inequality in a general bounded domain of $\mathbb{R}^n$, we apply the technique of $n$-harmonic transplantation to give the optimal concentration level of the Moser-Onofri inequality and obtain the criterion for the existence and non-existence of extremals for the Moser-Onofri inequality. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.00743v1-abstract-full').style.display = 'none'; document.getElementById('2406.00743v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.19637">arXiv:2405.19637</a> <span> [<a href="https://arxiv.org/pdf/2405.19637">pdf</a>, <a href="https://arxiv.org/format/2405.19637">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> </div> <p class="title is-5 mathjax"> Inference in semiparametric formation models for directed networks </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Qu%2C+L">Lianqiang Qu</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">Lu Chen</a>, <a href="/search/math?searchtype=author&query=Yan%2C+T">Ting Yan</a>, <a href="/search/math?searchtype=author&query=Chen%2C+Y">Yuguo Chen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2405.19637v1-abstract-short" style="display: inline;"> We propose a semiparametric model for dyadic link formations in directed networks. The model contains a set of degree parameters that measure different effects of popularity or outgoingness across nodes, a regression parameter vector that reflects the homophily effect resulting from the nodal attributes or pairwise covariates associated with edges, and a set of latent random noises with unknown di… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.19637v1-abstract-full').style.display = 'inline'; document.getElementById('2405.19637v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.19637v1-abstract-full" style="display: none;"> We propose a semiparametric model for dyadic link formations in directed networks. The model contains a set of degree parameters that measure different effects of popularity or outgoingness across nodes, a regression parameter vector that reflects the homophily effect resulting from the nodal attributes or pairwise covariates associated with edges, and a set of latent random noises with unknown distributions. Our interest lies in inferring the unknown degree parameters and homophily parameters. The dimension of the degree parameters increases with the number of nodes. Under the high-dimensional regime, we develop a kernel-based least squares approach to estimate the unknown parameters. The major advantage of our estimator is that it does not encounter the incidental parameter problem for the homophily parameters. We prove consistency of all the resulting estimators of the degree parameters and homophily parameters. We establish high-dimensional central limit theorems for the proposed estimators and provide several applications of our general theory, including testing the existence of degree heterogeneity, testing sparse signals and recovering the support. Simulation studies and a real data application are conducted to illustrate the finite sample performance of the proposed methods. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.19637v1-abstract-full').style.display = 'none'; document.getElementById('2405.19637v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">28 pages, 3 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.17727">arXiv:2405.17727</a> <span> [<a href="https://arxiv.org/pdf/2405.17727">pdf</a>, <a href="https://arxiv.org/ps/2405.17727">ps</a>, <a href="https://arxiv.org/format/2405.17727">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Optimal stability of Hardy-Littlewood-Sobolev and Sobolev inequalities of arbitrary orders with dimension-dependent constants </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+L">Lu Chen</a>, <a href="/search/math?searchtype=author&query=Lu%2C+G">Guozhen Lu</a>, <a href="/search/math?searchtype=author&query=Tang%2C+H">Hanli Tang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2405.17727v2-abstract-short" style="display: inline;"> Recently, Dolbeault-Esteban-Figalli-Frank-Loss [20] established the optimal stability of the first-order Sobolev inequality with dimension-dependent constant. Subsequently, Chen-Lu-Tang [18] obtained the optimal stability for the fractional Sobolev inequality of order $s$ when $0<s<1$. However, the optimal stability question for Sobolev inequality of order $s$ when $1<s<\frac{n}{2}$ is still unsol… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.17727v2-abstract-full').style.display = 'inline'; document.getElementById('2405.17727v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.17727v2-abstract-full" style="display: none;"> Recently, Dolbeault-Esteban-Figalli-Frank-Loss [20] established the optimal stability of the first-order Sobolev inequality with dimension-dependent constant. Subsequently, Chen-Lu-Tang [18] obtained the optimal stability for the fractional Sobolev inequality of order $s$ when $0<s<1$. However, the optimal stability question for Sobolev inequality of order $s$ when $1<s<\frac{n}{2}$ is still unsolved. Furthermore, the optimal stability for the HLS inequality still remains open. The purpose of this paper is to solve these problems. Our strategy is to first establish the optimal stability for the HLS inequality. The main difficulty lies in establishing the optimal local stability of HLS inequality when $1<s<\frac{n}{2}$. The proof of the local stability when $0<s\leq 1$ using duality does not work for $1<s<n/2$. Thus, we directly establish the optimal local stability for the HLS inequality. The loss of the Hilbert structure of the distance appearing in the stability of the HLS inequality brings much challenge to establishing the desired stability. To achieve our goal, we develop a new strategy based on the $H^{-s}-$decomposition instead of $L^{\frac{2n}{n+2s}}-$decomposition to obtain the local stability of the HLS inequality with $L^{\frac{2n}{n+2s}}-$distance. However, this kind of ``new local stability" adds new difficulties to deduce the global stability from the local stability using the rearrangement flow because of the non-uniqueness and non-continuity of $\|r\|_{\frac{2n}{n+2s}}$ for the rearrangement flow. We establish the norm comparison theorem for $\|r\|_{\frac{2n}{n+2s}}$ and ``new continuity" theorem for the rearrangement flow to overcome this difficulty (see Lemma 3.1, Lemma 3.3 and Lemma 3.5). We also deduce the optimal stability of the Sobolev inequality of order s when $1\le s<\frac{n}{2}$ as an important application. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.17727v2-abstract-full').style.display = 'none'; document.getElementById('2405.17727v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 27 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Its presentation is improved and polished. Some corollaries are added</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.15128">arXiv:2405.15128</a> <span> [<a href="https://arxiv.org/pdf/2405.15128">pdf</a>, <a href="https://arxiv.org/ps/2405.15128">ps</a>, <a href="https://arxiv.org/format/2405.15128">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Fluctuations around the mean-field limit for attractive Riesz potentials in the moderate regime </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+L">Li Chen</a>, <a href="/search/math?searchtype=author&query=Holzinger%2C+A">Alexandra Holzinger</a>, <a href="/search/math?searchtype=author&query=J%C3%BCngel%2C+A">Ansgar J眉ngel</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2405.15128v1-abstract-short" style="display: inline;"> A central limit theorem is shown for moderately interacting particles in the whole space. The interaction potential approximates singular attractive or repulsive potentials of sub-Coulomb type. It is proved that the fluctuations become asymptotically Gaussians in the limit of infinitely many particles. The methodology is inspired by the classical work of Oelschl盲ger on fluctuations for the porous-… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.15128v1-abstract-full').style.display = 'inline'; document.getElementById('2405.15128v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.15128v1-abstract-full" style="display: none;"> A central limit theorem is shown for moderately interacting particles in the whole space. The interaction potential approximates singular attractive or repulsive potentials of sub-Coulomb type. It is proved that the fluctuations become asymptotically Gaussians in the limit of infinitely many particles. The methodology is inspired by the classical work of Oelschl盲ger on fluctuations for the porous-medium equation. The novelty in this work is that we can allow for attractive potentials in the moderate regime and still obtain asymptotic Gaussian fluctuations. The key element of the proof is the mean-square convergence in expectation for smoothed empirical measures associated to moderately interacting $N$-particle systems with rate $N^{-1/2-\varepsilon}$ for some $\varepsilon>0$. To allow for attractive potentials, the proof uses a quantitative mean-field convergence in probability with any algebraic rate and a law-of-large-numbers estimate as well as a systematic separation of the terms to be estimated in a mean-field part and a law-of-large-numbers part. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.15128v1-abstract-full').style.display = 'none'; document.getElementById('2405.15128v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35Q70; 35Q92; 60J70; 82C22 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.12134">arXiv:2405.12134</a> <span> [<a href="https://arxiv.org/pdf/2405.12134">pdf</a>, <a href="https://arxiv.org/format/2405.12134">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Two-dimensional signal-dependent parabolic-elliptic Keller-Segel system and its means field derivation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bol%2C+L">Lukas Bol</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">Li Chen</a>, <a href="/search/math?searchtype=author&query=Li%2C+Y">Yue Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2405.12134v1-abstract-short" style="display: inline;"> In this paper, the well-posedness of two-dimensional signal-dependent Keller-Segel system and its mean field derivation from a interacting particle system on the whole space are investigated. The signal dependence effect is reflected by the fact that the diffusion coefficient in the particle system depends nonlinearly on the interactions between the individuals. Therefore, the mathematical challen… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.12134v1-abstract-full').style.display = 'inline'; document.getElementById('2405.12134v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.12134v1-abstract-full" style="display: none;"> In this paper, the well-posedness of two-dimensional signal-dependent Keller-Segel system and its mean field derivation from a interacting particle system on the whole space are investigated. The signal dependence effect is reflected by the fact that the diffusion coefficient in the particle system depends nonlinearly on the interactions between the individuals. Therefore, the mathematical challenge in studying the well-posedness of this system lies in the possible degeneracy and the aggregation effect when the concentration of signal becomes unbounded. The well-established method on bounded domain, to obtain the appropriate estimates for the signal concentration, is invalid for the whole space case. Motivated by the entropy minimization method and Onofri's inequality, which has been successfully applied for parabolic-parabolic Keller-Segel system, we establish a complete entropy estimate benefited from linear diffusion term, which plays important role in obtaining the Lp estimates for the solution. Furthermore, the upper bound for the concentration of signal is obtained. Based on estimates we obtained for the density of cells, the rigorous mean-field derivation is proved by introducing an intermediate particle system with a mollified interaction potential with logarithmic scaling. By using this mollification, we obtain the convergence of the particle trajectories in expectation, which implies the weak propagation of chaos. Additionally, under a regularity assumption of the cell-density, we derive the strong L1 convergence for the propagation of chaos by using relative entropy method. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.12134v1-abstract-full').style.display = 'none'; document.getElementById('2405.12134v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.07537">arXiv:2405.07537</a> <span> [<a href="https://arxiv.org/pdf/2405.07537">pdf</a>, <a href="https://arxiv.org/format/2405.07537">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Statistical Rounding Error Analysis for the Computation of Random Vectors and Matrices </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fang%2C+Y">Yiming Fang</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">Li Chen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2405.07537v2-abstract-short" style="display: inline;"> The conventional rounding error analysis provides worst-case bounds with an associated failure probability and ignores the statistical property of the rounding errors. In this paper, we develop a new statistical rounding error analysis for random vectors and matrices computation. By assuming the relative errors are independent random variables, we derive the approximate closed-form expressions for… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.07537v2-abstract-full').style.display = 'inline'; document.getElementById('2405.07537v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.07537v2-abstract-full" style="display: none;"> The conventional rounding error analysis provides worst-case bounds with an associated failure probability and ignores the statistical property of the rounding errors. In this paper, we develop a new statistical rounding error analysis for random vectors and matrices computation. By assuming the relative errors are independent random variables, we derive the approximate closed-form expressions for the expectation and variance of the rounding errors in various key computations for vectors and random matrices. Numerical experiments validate the accuracy of our derivations and demonstrate that our analytical expressions are generally at least two orders of magnitude tighter than alternative worst-case bounds, exemplified through the inner products. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.07537v2-abstract-full').style.display = 'none'; document.getElementById('2405.07537v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 13 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">33 pages, 7 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.03648">arXiv:2405.03648</a> <span> [<a href="https://arxiv.org/pdf/2405.03648">pdf</a>, <a href="https://arxiv.org/ps/2405.03648">ps</a>, <a href="https://arxiv.org/format/2405.03648">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Proof of the geometric Langlands conjecture II: Kac-Moody localization and the FLE </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Arinkin%2C+D">D. Arinkin</a>, <a href="/search/math?searchtype=author&query=Beraldo%2C+D">D. Beraldo</a>, <a href="/search/math?searchtype=author&query=Campbell%2C+J">J. Campbell</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">L. Chen</a>, <a href="/search/math?searchtype=author&query=Faergeman%2C+J">J. Faergeman</a>, <a href="/search/math?searchtype=author&query=Gaitsgory%2C+D">D. Gaitsgory</a>, <a href="/search/math?searchtype=author&query=Lin%2C+K">K. Lin</a>, <a href="/search/math?searchtype=author&query=Raskin%2C+S">S. Raskin</a>, <a href="/search/math?searchtype=author&query=Rozenblyum%2C+N">N. Rozenblyum</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2405.03648v3-abstract-short" style="display: inline;"> This paper is the second in a series of five that together prove the geometric Langlands conjecture. Our goals are two-fold: (1) Formulate and prove the Fundamental Local Equivalence (FLE) at the critical level; (2) Study the interaction between Kac-Moody localization and the global geometric Langlands functor of ref. [GLC1]. This paper contains an extensive Appendix, whose primary goals are… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.03648v3-abstract-full').style.display = 'inline'; document.getElementById('2405.03648v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.03648v3-abstract-full" style="display: none;"> This paper is the second in a series of five that together prove the geometric Langlands conjecture. Our goals are two-fold: (1) Formulate and prove the Fundamental Local Equivalence (FLE) at the critical level; (2) Study the interaction between Kac-Moody localization and the global geometric Langlands functor of ref. [GLC1]. This paper contains an extensive Appendix, whose primary goals are: (a) Development the theory of ind-coherent sheaves in infinite type; (b)Development of the formalism of factorization categories. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.03648v3-abstract-full').style.display = 'none'; document.getElementById('2405.03648v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.00545">arXiv:2405.00545</a> <span> [<a href="https://arxiv.org/pdf/2405.00545">pdf</a>, <a href="https://arxiv.org/format/2405.00545">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> A Double Maximization Approach for Optimizing the LM Rate of Mismatched Decoding </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+L">Lingyi Chen</a>, <a href="/search/math?searchtype=author&query=Wu%2C+S">Shitong Wu</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xinwei Li</a>, <a href="/search/math?searchtype=author&query=Wu%2C+H">Huihui Wu</a>, <a href="/search/math?searchtype=author&query=Wu%2C+H">Hao Wu</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+W">Wenyi Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2405.00545v1-abstract-short" style="display: inline;"> An approach is established for maximizing the Lower bound on the Mismatch capacity (hereafter abbreviated as LM rate), a key performance bound in mismatched decoding, by optimizing the channel input probability distribution. Under a fixed channel input probability distribution, the computation of the corresponding LM rate is a convex optimization problem. When optimizing the channel input probabil… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.00545v1-abstract-full').style.display = 'inline'; document.getElementById('2405.00545v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.00545v1-abstract-full" style="display: none;"> An approach is established for maximizing the Lower bound on the Mismatch capacity (hereafter abbreviated as LM rate), a key performance bound in mismatched decoding, by optimizing the channel input probability distribution. Under a fixed channel input probability distribution, the computation of the corresponding LM rate is a convex optimization problem. When optimizing the channel input probability distribution, however, the corresponding optimization problem adopts a max-min formulation, which is generally non-convex and is intractable with standard approaches. To solve this problem, a novel dual form of the LM rate is proposed, thereby transforming the max-min formulation into an equivalent double maximization formulation. This new formulation leads to a maximization problem setup wherein each individual optimization direction is convex. Consequently, an alternating maximization algorithm is established to solve the resultant maximization problem setup. Each step of the algorithm only involves a closed-form iteration, which is efficiently implemented with standard optimization procedures. Numerical experiments show the proposed approach for optimizing the LM rate leads to noticeable rate gains. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.00545v1-abstract-full').style.display = 'none'; document.getElementById('2405.00545v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.19308">arXiv:2404.19308</a> <span> [<a href="https://arxiv.org/pdf/2404.19308">pdf</a>, <a href="https://arxiv.org/format/2404.19308">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Physics">quant-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1002/andp.202200289">10.1002/andp.202200289 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A characterization of entangled two-qubit states via partial-transpose-moments </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+L">Lin Zhang</a>, <a href="/search/math?searchtype=author&query=Zhao%2C+M">Ming-Jing Zhao</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">Lin Chen</a>, <a href="/search/math?searchtype=author&query=Xiang%2C+H">Hua Xiang</a>, <a href="/search/math?searchtype=author&query=Shen%2C+Y">Yi Shen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2404.19308v1-abstract-short" style="display: inline;"> Although quantum entanglement is an important resource, its characterization is quite challenging. The partial transposition is a common method to detect bipartite entanglement. In this paper, the authors study the partial-transpose(PT)-moments of two-qubit states,and completely describe the whole region, composed of the second and third PT-moments, for all two-qubit states. Furthermore, they dete… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.19308v1-abstract-full').style.display = 'inline'; document.getElementById('2404.19308v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.19308v1-abstract-full" style="display: none;"> Although quantum entanglement is an important resource, its characterization is quite challenging. The partial transposition is a common method to detect bipartite entanglement. In this paper, the authors study the partial-transpose(PT)-moments of two-qubit states,and completely describe the whole region, composed of the second and third PT-moments, for all two-qubit states. Furthermore, they determine the accurate region corresponding to all entangled two-qubit states. The states corresponding to those boundary points of the whole region, and to the border lines between separable and entangled states are analyzed. As an application, they characterize the entangled region of PT-moments for the two families of Werner states and Bell-diagonal states. The relations between entanglement and the pairs of PT-moments are revealed from these typical examples. They also numerically plot the whole region of possible PT-moments for all two-qubit X-states, and find that this region is almost the same as the whole region of PT-moments for all two-qubit states. Moreover, they extend their results to detect the entanglement of multiqubit states. By utilizing the PT-moment-based method to characterize the entanglement of the multiqubit states mixed by the GHZ and W states, they propose an operational way of verifying the genuine entanglement in such states. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.19308v1-abstract-full').style.display = 'none'; document.getElementById('2404.19308v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">31 pages, LaTeX, 9 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Annalen der Physik 534, 2200289 (2022) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.12795">arXiv:2404.12795</a> <span> [<a href="https://arxiv.org/pdf/2404.12795">pdf</a>, <a href="https://arxiv.org/format/2404.12795">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Stability for a class of three-tori with small negative scalar curvature </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bryden%2C+E">Edward Bryden</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">Lizhi Chen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2404.12795v2-abstract-short" style="display: inline;"> We define a flexible class of Riemmanian metrics on the three-torus. Then, using Stern's inequality relating scalar curvature to harmonic one-forms, we show that any sequence of metrics in this family whose negative part of the scalar curvature tends to zero in $L^2$ norm has a subsequence which converges to some flat metric on the three-torus in the sense of Dong-Song. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.12795v2-abstract-full" style="display: none;"> We define a flexible class of Riemmanian metrics on the three-torus. Then, using Stern's inequality relating scalar curvature to harmonic one-forms, we show that any sequence of metrics in this family whose negative part of the scalar curvature tends to zero in $L^2$ norm has a subsequence which converges to some flat metric on the three-torus in the sense of Dong-Song. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.12795v2-abstract-full').style.display = 'none'; document.getElementById('2404.12795v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">In the second version, the abstract is updated; some typos are corrected</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.11628">arXiv:2404.11628</a> <span> [<a href="https://arxiv.org/pdf/2404.11628">pdf</a>, <a href="https://arxiv.org/ps/2404.11628">ps</a>, <a href="https://arxiv.org/format/2404.11628">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Classification of positive solutions of critical anisotropic Sobolev equation without the finite volume constraint </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+L">Lu Chen</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yabo Yang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2404.11628v3-abstract-short" style="display: inline;"> In this paper, we classify all positive solutions of the critical anisotropic Sobolev equation \begin{equation*} -螖^{H}_{p}u = u^{p^{*}-1}, \ \ x\in \mathbb{R}^n \end{equation*} without the finite volume constraint for $n \geq 2$ and $\frac{(n+1)}{3} \leq p < n$, where $p^{*} = \frac{np}{n-p}$ denotes the critical Sobolev exponent and $-螖^{H}_{p}=-div(H^{p-1}(\cdot)\nabla H(\cdot))$ denotes the an… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.11628v3-abstract-full').style.display = 'inline'; document.getElementById('2404.11628v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.11628v3-abstract-full" style="display: none;"> In this paper, we classify all positive solutions of the critical anisotropic Sobolev equation \begin{equation*} -螖^{H}_{p}u = u^{p^{*}-1}, \ \ x\in \mathbb{R}^n \end{equation*} without the finite volume constraint for $n \geq 2$ and $\frac{(n+1)}{3} \leq p < n$, where $p^{*} = \frac{np}{n-p}$ denotes the critical Sobolev exponent and $-螖^{H}_{p}=-div(H^{p-1}(\cdot)\nabla H(\cdot))$ denotes the anisotropic $p$-Laplace operator. This result removes the finite volume assumption on the classification of critical anisotropic $p$-Laplace equation which was obtained by Ciraolo-Figalli-Roncoroni in the literature \cite{CFR}. The method is based on constructing suitable vector fields integral inequality and using Newton's type inequality. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.11628v3-abstract-full').style.display = 'none'; document.getElementById('2404.11628v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.08541">arXiv:2404.08541</a> <span> [<a href="https://arxiv.org/pdf/2404.08541">pdf</a>, <a href="https://arxiv.org/ps/2404.08541">ps</a>, <a href="https://arxiv.org/format/2404.08541">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Existence of monotone Morse flow lines of the expander functional </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bernstein%2C+J">Jacob Bernstein</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">Letian Chen</a>, <a href="/search/math?searchtype=author&query=Wang%2C+L">Lu Wang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2404.08541v1-abstract-short" style="display: inline;"> Given a smooth asymptotically conical self-expander that is strictly unstable we construct a (singular) Morse flow line of the expander functional that connects it to a stable self-expander. This flow is monotone in a suitable sense and has small singular set. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.08541v1-abstract-full" style="display: none;"> Given a smooth asymptotically conical self-expander that is strictly unstable we construct a (singular) Morse flow line of the expander functional that connects it to a stable self-expander. This flow is monotone in a suitable sense and has small singular set. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.08541v1-abstract-full').style.display = 'none'; document.getElementById('2404.08541v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">46 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53E10; 49Q20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.02476">arXiv:2404.02476</a> <span> [<a href="https://arxiv.org/pdf/2404.02476">pdf</a>, <a href="https://arxiv.org/format/2404.02476">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Artificial Intelligence">cs.AI</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> Deep Reinforcement Learning for Traveling Purchaser Problems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yuan%2C+H">Haofeng Yuan</a>, <a href="/search/math?searchtype=author&query=Zhu%2C+R">Rongping Zhu</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wanlu Yang</a>, <a href="/search/math?searchtype=author&query=Song%2C+S">Shiji Song</a>, <a href="/search/math?searchtype=author&query=You%2C+K">Keyou You</a>, <a href="/search/math?searchtype=author&query=Fan%2C+W">Wei Fan</a>, <a href="/search/math?searchtype=author&query=Chen%2C+C+L+P">C. L. Philip Chen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2404.02476v5-abstract-short" style="display: inline;"> The traveling purchaser problem (TPP) is an important combinatorial optimization problem with broad applications. Due to the coupling between routing and purchasing, existing works on TPPs commonly address route construction and purchase planning simultaneously, which, however, leads to exact methods with high computational cost and heuristics with sophisticated design but limited performance. In… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.02476v5-abstract-full').style.display = 'inline'; document.getElementById('2404.02476v5-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.02476v5-abstract-full" style="display: none;"> The traveling purchaser problem (TPP) is an important combinatorial optimization problem with broad applications. Due to the coupling between routing and purchasing, existing works on TPPs commonly address route construction and purchase planning simultaneously, which, however, leads to exact methods with high computational cost and heuristics with sophisticated design but limited performance. In sharp contrast, we propose a novel approach based on deep reinforcement learning (DRL), which addresses route construction and purchase planning separately, while evaluating and optimizing the solution from a global perspective. The key components of our approach include a bipartite graph representation for TPPs to capture the market-product relations, and a policy network that extracts information from the bipartite graph and uses it to sequentially construct the route. One significant benefit of our framework is that we can efficiently construct the route using the policy network, and once the route is determined, the associated purchasing plan can be easily derived through linear programming, while, leveraging DRL, we can train the policy network to optimize the global solution objective. Furthermore, by introducing a meta-learning strategy, the policy network can be trained stably on large-sized TPP instances, and generalize well across instances of varying sizes and distributions, even to much larger instances that are never seen during training. Experiments on various synthetic TPP instances and the TPPLIB benchmark demonstrate that our DRL-based approach can significantly outperform well-established TPP heuristics, reducing the optimality gap by 40%-90%, and also showing an advantage in runtime, especially on large-sized instances. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.02476v5-abstract-full').style.display = 'none'; document.getElementById('2404.02476v5-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.00438">arXiv:2404.00438</a> <span> [<a href="https://arxiv.org/pdf/2404.00438">pdf</a>, <a href="https://arxiv.org/format/2404.00438">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Distributed, Parallel, and Cluster Computing">cs.DC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Artificial Intelligence">cs.AI</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Communication Efficient Distributed Training with Distributed Lion </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+B">Bo Liu</a>, <a href="/search/math?searchtype=author&query=Wu%2C+L">Lemeng Wu</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">Lizhang Chen</a>, <a href="/search/math?searchtype=author&query=Liang%2C+K">Kaizhao Liang</a>, <a href="/search/math?searchtype=author&query=Zhu%2C+J">Jiaxu Zhu</a>, <a href="/search/math?searchtype=author&query=Liang%2C+C">Chen Liang</a>, <a href="/search/math?searchtype=author&query=Krishnamoorthi%2C+R">Raghuraman Krishnamoorthi</a>, <a href="/search/math?searchtype=author&query=Liu%2C+Q">Qiang Liu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2404.00438v1-abstract-short" style="display: inline;"> The Lion optimizer has been a promising competitor with the AdamW for training large AI models, with advantages on memory, computation, and sample efficiency. In this paper, we introduce Distributed Lion, an innovative adaptation of Lion for distributed training environments. Leveraging the sign operator in Lion, our Distributed Lion only requires communicating binary or lower-precision vectors be… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.00438v1-abstract-full').style.display = 'inline'; document.getElementById('2404.00438v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.00438v1-abstract-full" style="display: none;"> The Lion optimizer has been a promising competitor with the AdamW for training large AI models, with advantages on memory, computation, and sample efficiency. In this paper, we introduce Distributed Lion, an innovative adaptation of Lion for distributed training environments. Leveraging the sign operator in Lion, our Distributed Lion only requires communicating binary or lower-precision vectors between workers to the center server, significantly reducing the communication cost. Our theoretical analysis confirms Distributed Lion's convergence properties. Empirical results demonstrate its robustness across a range of tasks, worker counts, and batch sizes, on both vision and language problems. Notably, Distributed Lion attains comparable performance to standard Lion or AdamW optimizers applied on aggregated gradients, but with significantly reduced communication bandwidth. This feature is particularly advantageous for training large models. In addition, we also demonstrate that Distributed Lion presents a more favorable performance-bandwidth balance compared to existing efficient distributed methods such as deep gradient compression and ternary gradients. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.00438v1-abstract-full').style.display = 'none'; document.getElementById('2404.00438v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">22 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.00078">arXiv:2404.00078</a> <span> [<a href="https://arxiv.org/pdf/2404.00078">pdf</a>, <a href="https://arxiv.org/format/2404.00078">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> Irreversible and dissipative systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Beck%2C+J">J. Beck</a>, <a href="/search/math?searchtype=author&query=Chen%2C+W+W+L">W. W. L. Chen</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Y. Yang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2404.00078v2-abstract-short" style="display: inline;"> We study some new dynamical systems where the corresponding piecewise linear flow is neither time reversible nor measure preserving. We create a dissipative system by starting with a finite polysquare translation surface, and then modifying it by including a one-sided barrier on a common vertical edge of two adjacent atomic squares, in the form of a union of finitely many intervals. The line flow… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.00078v2-abstract-full').style.display = 'inline'; document.getElementById('2404.00078v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.00078v2-abstract-full" style="display: none;"> We study some new dynamical systems where the corresponding piecewise linear flow is neither time reversible nor measure preserving. We create a dissipative system by starting with a finite polysquare translation surface, and then modifying it by including a one-sided barrier on a common vertical edge of two adjacent atomic squares, in the form of a union of finitely many intervals. The line flow in this system partitions the system into a transient set and a recurrent set. We are interested in the geometry of these two sets. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.00078v2-abstract-full').style.display = 'none'; document.getElementById('2404.00078v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">44 pages, 39 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37E35; 11K38 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.00077">arXiv:2404.00077</a> <span> [<a href="https://arxiv.org/pdf/2404.00077">pdf</a>, <a href="https://arxiv.org/format/2404.00077">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> A note on the Kronecker--Weyl equidistribution theorem </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Beck%2C+J">J. Beck</a>, <a href="/search/math?searchtype=author&query=Chen%2C+W+W+L">W. W. L. Chen</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Y. Yang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2404.00077v2-abstract-short" style="display: inline;"> We study the relationship between the discrete and the continuous versions of the Kronecker--Weyl equidistribution theorem, as well as their possible extension to manifolds in higher dimensions. We also investigate a way to deduce in some limited way uniformity results in higher dimension from results in lower dimension. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.00077v2-abstract-full" style="display: none;"> We study the relationship between the discrete and the continuous versions of the Kronecker--Weyl equidistribution theorem, as well as their possible extension to manifolds in higher dimensions. We also investigate a way to deduce in some limited way uniformity results in higher dimension from results in lower dimension. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.00077v2-abstract-full').style.display = 'none'; document.getElementById('2404.00077v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">12 pages, 1 figure</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37E35; 11K38 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2403.19960">arXiv:2403.19960</a> <span> [<a href="https://arxiv.org/pdf/2403.19960">pdf</a>, <a href="https://arxiv.org/format/2403.19960">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> A note on density of geodesics </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Beck%2C+J">J. Beck</a>, <a href="/search/math?searchtype=author&query=Chen%2C+W+W+L">W. W. L. Chen</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Y. Yang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2403.19960v1-abstract-short" style="display: inline;"> We extend the famous result of Katok and Zemlyakov on the density of half-infinite geodesics on finite flat rational surfaces to half-infinite geodesics on a finite polycube translation $3$-manifold. We also extend this original result to establish a weak uniformity statement. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.19960v1-abstract-full" style="display: none;"> We extend the famous result of Katok and Zemlyakov on the density of half-infinite geodesics on finite flat rational surfaces to half-infinite geodesics on a finite polycube translation $3$-manifold. We also extend this original result to establish a weak uniformity statement. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.19960v1-abstract-full').style.display = 'none'; document.getElementById('2403.19960v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">10 pages, 3 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37E35; 11K38 </p> </li> </ol> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> <a href="" class="pagination-previous 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