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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=Yang%2C+W&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Yang%2C+W&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Yang%2C+W&start=50" class="pagination-link " aria-label="Page 2" aria-current="page">2 </a> </li> <li> <a href="/search/?searchtype=author&query=Yang%2C+W&start=100" class="pagination-link " aria-label="Page 3" aria-current="page">3 </a> </li> <li> <a href="/search/?searchtype=author&query=Yang%2C+W&start=150" class="pagination-link " aria-label="Page 4" aria-current="page">4 </a> </li> <li> <a href="/search/?searchtype=author&query=Yang%2C+W&start=200" class="pagination-link " aria-label="Page 5" aria-current="page">5 </a> </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/2502.02039">arXiv:2502.02039</a> <span> [<a href="https://arxiv.org/pdf/2502.02039">pdf</a>, <a href="https://arxiv.org/ps/2502.02039">ps</a>, <a href="https://arxiv.org/format/2502.02039">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Operator Algebras">math.OA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</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"> Boundary actions of Bass-Serre Trees and the applications to $C^*$-algebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ma%2C+X">Xin Ma</a>, <a href="/search/math?searchtype=author&query=Wang%2C+D">Daxun Wang</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wenyuan 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="2502.02039v1-abstract-short" style="display: inline;"> In this paper, we study Bass-Serre theory from the perspectives of $C^*$-algebras and topological dynamics. In particular, we investigate the actions of fundamental groups of graphs of groups on their Bass-Serre trees and the associated boundaries, through which, we identify new families of $C^*$-simple groups including certain tubular groups and fundamental groups of certain graphs of groups with… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.02039v1-abstract-full').style.display = 'inline'; document.getElementById('2502.02039v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.02039v1-abstract-full" style="display: none;"> In this paper, we study Bass-Serre theory from the perspectives of $C^*$-algebras and topological dynamics. In particular, we investigate the actions of fundamental groups of graphs of groups on their Bass-Serre trees and the associated boundaries, through which, we identify new families of $C^*$-simple groups including certain tubular groups and fundamental groups of certain graphs of groups with one vertex group acylindrically hyperbolic. In addition, we obtain characterizations of $C^*$-simplicity for Generalized Baumslag-Solitar (GBS) groups and outer automorphism groups $\operatorname{Out}(BS(p, q))$ of Baumslag-Solitar groups. These $C^*$-simple groups also provide new examples of highly transitive groups. Moreover, we demonstrate that natural boundary actions of these $C^*$-simple fundamental groups of graphs of groups give rise to the new purely infinite crossed product $C^*$-algebras. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.02039v1-abstract-full').style.display = 'none'; document.getElementById('2502.02039v1-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> 4 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2025. </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">This paper, along with another forthcoming paper, will supersede arXiv:2202.03374. Consequently, arXiv:2202.03374 is not intended for publication</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2501.09516">arXiv:2501.09516</a> <span> [<a href="https://arxiv.org/pdf/2501.09516">pdf</a>, <a href="https://arxiv.org/format/2501.09516">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"> Proximal Quasi-Newton Method for Composite Optimization over the Stiefel Manifold </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+Q">Qinsi Wang</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W+H">Wei Hong 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="2501.09516v1-abstract-short" style="display: inline;"> In this paper, we consider the composite optimization problems over the Stiefel manifold. A successful method to solve this class of problems is the proximal gradient method proposed by Chen et al. Motivated by the proximal Newton-type techniques in the Euclidean space, we present a Riemannian proximal quasi-Newton method, named ManPQN, to solve the composite optimization problems. The global conv… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.09516v1-abstract-full').style.display = 'inline'; document.getElementById('2501.09516v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2501.09516v1-abstract-full" style="display: none;"> In this paper, we consider the composite optimization problems over the Stiefel manifold. A successful method to solve this class of problems is the proximal gradient method proposed by Chen et al. Motivated by the proximal Newton-type techniques in the Euclidean space, we present a Riemannian proximal quasi-Newton method, named ManPQN, to solve the composite optimization problems. The global convergence of the ManPQN method is proved and iteration complexity for obtaining an $蔚$-stationary point is analyzed. Under some mild conditions, we also establish the local linear convergence result of the ManPQN method. Numerical results are encouraging, which shows that the proximal quasi-Newton technique can be used to accelerate the proximal gradient method. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.09516v1-abstract-full').style.display = 'none'; document.getElementById('2501.09516v1-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> 16 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2025. </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">37 pages, 12 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 90C53 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2501.02746">arXiv:2501.02746</a> <span> [<a href="https://arxiv.org/pdf/2501.02746">pdf</a>, <a href="https://arxiv.org/ps/2501.02746">ps</a>, <a href="https://arxiv.org/format/2501.02746">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Signal Processing">eess.SP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</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"> A Large-dimensional Analysis of ESPRIT DoA Estimation: Inconsistency and a Correction via RMT </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+Z">Zhengyu Wang</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wei Yang</a>, <a href="/search/math?searchtype=author&query=Mai%2C+X">Xiaoyi Mai</a>, <a href="/search/math?searchtype=author&query=Ling%2C+Z">Zenan Ling</a>, <a href="/search/math?searchtype=author&query=Liao%2C+Z">Zhenyu Liao</a>, <a href="/search/math?searchtype=author&query=Qiu%2C+R+C">Robert C. Qiu</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="2501.02746v1-abstract-short" style="display: inline;"> In this paper, we perform asymptotic analyses of the widely used ESPRIT direction-of-arrival (DoA) estimator for large arrays, where the array size $N$ and the number of snapshots $T$ grow to infinity at the same pace. In this large-dimensional regime, the sample covariance matrix (SCM) is known to be a poor eigenspectral estimator of the population covariance. We show that the classical ESPRIT al… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.02746v1-abstract-full').style.display = 'inline'; document.getElementById('2501.02746v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2501.02746v1-abstract-full" style="display: none;"> In this paper, we perform asymptotic analyses of the widely used ESPRIT direction-of-arrival (DoA) estimator for large arrays, where the array size $N$ and the number of snapshots $T$ grow to infinity at the same pace. In this large-dimensional regime, the sample covariance matrix (SCM) is known to be a poor eigenspectral estimator of the population covariance. We show that the classical ESPRIT algorithm, that relies on the SCM, and as a consequence of the large-dimensional inconsistency of the SCM, produces inconsistent DoA estimates as $N,T \to \infty$ with $N/T \to c \in (0,\infty)$, for both widely- and closely-spaced DoAs. Leveraging tools from random matrix theory (RMT), we propose an improved G-ESPRIT method and prove its consistency in the same large-dimensional setting. From a technical perspective, we derive a novel bound on the eigenvalue differences between two potentially non-Hermitian random matrices, which may be of independent interest. Numerical simulations are provided to corroborate our theoretical findings. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.02746v1-abstract-full').style.display = 'none'; document.getElementById('2501.02746v1-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> 5 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2025. </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">25 pages, 8 figures. Part of this work was presented at the IEEE 32nd European Signal Processing Conference (EUSIPCO 2024), Lyon, France, under the title "Inconsistency of ESPRIT DoA Estimation for Large Arrays and a Correction via RMT."</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.13576">arXiv:2412.13576</a> <span> [<a href="https://arxiv.org/pdf/2412.13576">pdf</a>, <a href="https://arxiv.org/format/2412.13576">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"> GPU-based Graver Basis Extraction for Nonlinear Integer Optimization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+W">Wenbo Liu</a>, <a href="/search/math?searchtype=author&query=Wang%2C+A">Akang Wang</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wenguo 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="2412.13576v1-abstract-short" style="display: inline;"> Nonlinear integer programs involve optimizing nonlinear objectives with variables restricted to integer values, and have widespread applications in areas such as resource allocation and portfolio selection. One approach to solving these problems is the augmentation procedure, which iteratively refines a feasible solution by identifying augmenting steps from the Graver Basis--a set of test directio… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.13576v1-abstract-full').style.display = 'inline'; document.getElementById('2412.13576v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.13576v1-abstract-full" style="display: none;"> Nonlinear integer programs involve optimizing nonlinear objectives with variables restricted to integer values, and have widespread applications in areas such as resource allocation and portfolio selection. One approach to solving these problems is the augmentation procedure, which iteratively refines a feasible solution by identifying augmenting steps from the Graver Basis--a set of test directions. While this method guarantees termination in polynomially many steps, computing the Graver Basis exactly is known to be $\mathcal{NP}$-hard. To address this computational challenge, we propose Multi-start Augmentation via Parallel Extraction (MAPLE), a GPU-based heuristic designed to efficiently approximate the Graver Basis. MAPLE extracts test directions by optimizing non-convex continuous problems, leveraging first-order methods to enable parallelizable implementation. The resulting set of directions is then used in multiple augmentations, each seeking to improve the solution's optimality. The proposed approach has three notable characteristics: (i) independence from general-purpose solvers, while ensuring guaranteed feasibility of solutions; (ii) high computational efficiency, achieved through GPU-based parallelization; (iii) flexibility in handling instances with shared constraint matrices but varying objectives and right-hand sides. Empirical evaluations on QPLIB benchmark instances demonstrate that MAPLE delivers performance comparable to state-of-the-art solvers in terms of solution quality, while achieving significant gains in computational efficiency. These results highlight MAPLE's potential as an effective heuristic for solving nonlinear integer programs in practical applications. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.13576v1-abstract-full').style.display = 'none'; document.getElementById('2412.13576v1-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 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.09767">arXiv:2412.09767</a> <span> [<a href="https://arxiv.org/pdf/2412.09767">pdf</a>, <a href="https://arxiv.org/ps/2412.09767">ps</a>, <a href="https://arxiv.org/format/2412.09767">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> </div> </div> <p class="title is-5 mathjax"> Generalized Fiber Contraction Mapping Principle </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luna%2C+A">Alexandro Luna</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Weiran 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="2412.09767v1-abstract-short" style="display: inline;"> We prove a generalized non-stationary version of the fiber contraction mapping theorem. It was originally used in [HirschPugh70] to prove that the stable foliation of a $C^2$ Anosov diffeomorphism of a surface is $C^1$. Our generalized principle is used in [Luna24], where an analogous regularity result for stable foliations of non-stationary systems is proved. The result is stated in a general set… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.09767v1-abstract-full').style.display = 'inline'; document.getElementById('2412.09767v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.09767v1-abstract-full" style="display: none;"> We prove a generalized non-stationary version of the fiber contraction mapping theorem. It was originally used in [HirschPugh70] to prove that the stable foliation of a $C^2$ Anosov diffeomorphism of a surface is $C^1$. Our generalized principle is used in [Luna24], where an analogous regularity result for stable foliations of non-stationary systems is proved. The result is stated in a general setting so that it may be used in future dynamical results in the random and non-stationary settings, especially for graph transform arguments. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.09767v1-abstract-full').style.display = 'none'; document.getElementById('2412.09767v1-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 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.08206">arXiv:2412.08206</a> <span> [<a href="https://arxiv.org/pdf/2412.08206">pdf</a>, <a href="https://arxiv.org/format/2412.08206">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"> Mixed-Integer Linear Optimization via Learning-Based Two-Layer Large Neighborhood Search </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+W">Wenbo Liu</a>, <a href="/search/math?searchtype=author&query=Wang%2C+A">Akang Wang</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wenguo Yang</a>, <a href="/search/math?searchtype=author&query=Shi%2C+Q">Qingjiang Shi</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="2412.08206v1-abstract-short" style="display: inline;"> Mixed-integer linear programs (MILPs) are extensively used to model practical problems such as planning and scheduling. A prominent method for solving MILPs is large neighborhood search (LNS), which iteratively seeks improved solutions within specific neighborhoods. Recent advancements have integrated machine learning techniques into LNS to guide the construction of these neighborhoods effectively… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.08206v1-abstract-full').style.display = 'inline'; document.getElementById('2412.08206v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.08206v1-abstract-full" style="display: none;"> Mixed-integer linear programs (MILPs) are extensively used to model practical problems such as planning and scheduling. A prominent method for solving MILPs is large neighborhood search (LNS), which iteratively seeks improved solutions within specific neighborhoods. Recent advancements have integrated machine learning techniques into LNS to guide the construction of these neighborhoods effectively. However, for large-scale MILPs, the search step in LNS becomes a computational bottleneck, relying on off-the-shelf solvers to optimize auxiliary MILPs of substantial size. To address this challenge, we introduce a two-layer LNS (TLNS) approach that employs LNS to solve both the original MILP and its auxiliary MILPs, necessitating the optimization of only small-sized MILPs using off-the-shelf solvers. Additionally, we incorporate a lightweight graph transformer model to inform neighborhood design. We conduct extensive computational experiments using public benchmarks. The results indicate that our learning-based TLNS approach achieves remarkable performance gains--up to 66% and 96% over LNS and state-of-the-art MILP solvers, respectively. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.08206v1-abstract-full').style.display = 'none'; document.getElementById('2412.08206v1-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 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.13805">arXiv:2411.13805</a> <span> [<a href="https://arxiv.org/pdf/2411.13805">pdf</a>, <a href="https://arxiv.org/format/2411.13805">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"> On Representing Convex Quadratically Constrained Quadratic Programs via Graph Neural Networks </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wu%2C+C">Chenyang Wu</a>, <a href="/search/math?searchtype=author&query=Chen%2C+Q">Qian Chen</a>, <a href="/search/math?searchtype=author&query=Wang%2C+A">Akang Wang</a>, <a href="/search/math?searchtype=author&query=Ding%2C+T">Tian Ding</a>, <a href="/search/math?searchtype=author&query=Sun%2C+R">Ruoyu Sun</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wenguo Yang</a>, <a href="/search/math?searchtype=author&query=Shi%2C+Q">Qingjiang Shi</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="2411.13805v3-abstract-short" style="display: inline;"> Convex quadratically constrained quadratic programs (QCQPs) involve finding a solution within a convex feasible region defined by quadratic constraints while minimizing a convex quadratic objective function. These problems arise in various industrial applications, including power systems and signal processing. Traditional methods for solving convex QCQPs primarily rely on matrix factorization, whi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.13805v3-abstract-full').style.display = 'inline'; document.getElementById('2411.13805v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.13805v3-abstract-full" style="display: none;"> Convex quadratically constrained quadratic programs (QCQPs) involve finding a solution within a convex feasible region defined by quadratic constraints while minimizing a convex quadratic objective function. These problems arise in various industrial applications, including power systems and signal processing. Traditional methods for solving convex QCQPs primarily rely on matrix factorization, which quickly becomes computationally prohibitive as the problem size increases. Recently, graph neural networks (GNNs) have gained attention for their potential in representing and solving various optimization problems such as linear programs and linearly constrained quadratic programs. In this work, we investigate the representation power of GNNs in the context of QCQP tasks. Specifically, we propose a new tripartite graph representation for general convex QCQPs and properly associate it with message-passing GNNs. We demonstrate that there exist GNNs capable of reliably representing key properties of convex QCQPs, including feasibility, optimal value, and optimal solution. Our result deepens the understanding of the connection between QCQPs and GNNs, paving the way for future machine learning approaches to efficiently solve QCQPs. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.13805v3-abstract-full').style.display = 'none'; document.getElementById('2411.13805v3-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> 6 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 20 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/2411.03666">arXiv:2411.03666</a> <span> [<a href="https://arxiv.org/pdf/2411.03666">pdf</a>, <a href="https://arxiv.org/ps/2411.03666">ps</a>, <a href="https://arxiv.org/format/2411.03666">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"> Isolation partitions in graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+G">Gang Zhang</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Weiling Yang</a>, <a href="/search/math?searchtype=author&query=Jin%2C+X">Xian'an Jin</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="2411.03666v1-abstract-short" style="display: inline;"> Let $G$ be a graph and $k \geq 3$ an integer. A subset $D \subseteq V(G)$ is a $k$-clique (resp., cycle) isolating set of $G$ if $G-N[D]$ contains no $k$-clique (resp., cycle). In this paper, we prove that every connected graph with maximum degree at most $k$, except $k$-clique, can be partitioned into $k+1$ disjoint $k$-clique isolating sets, and that every connected claw-free subcubic graph, exc… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.03666v1-abstract-full').style.display = 'inline'; document.getElementById('2411.03666v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.03666v1-abstract-full" style="display: none;"> Let $G$ be a graph and $k \geq 3$ an integer. A subset $D \subseteq V(G)$ is a $k$-clique (resp., cycle) isolating set of $G$ if $G-N[D]$ contains no $k$-clique (resp., cycle). In this paper, we prove that every connected graph with maximum degree at most $k$, except $k$-clique, can be partitioned into $k+1$ disjoint $k$-clique isolating sets, and that every connected claw-free subcubic graph, except 3-cycle, can be partitioned into four disjoint cycle isolating sets. As a consequence of the first result, every $k$-regular graph can be partitioned into $k+1$ disjoint $k$-clique isolating sets. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.03666v1-abstract-full').style.display = 'none'; document.getElementById('2411.03666v1-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> 5 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">14 pages, 5 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C69; 05C15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.16340">arXiv:2410.16340</a> <span> [<a href="https://arxiv.org/pdf/2410.16340">pdf</a>, <a href="https://arxiv.org/format/2410.16340">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">stat.ML</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="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Limit Theorems for Stochastic Gradient Descent with Infinite Variance </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Blanchet%2C+J">Jose Blanchet</a>, <a href="/search/math?searchtype=author&query=Mijatovi%C4%87%2C+A">Aleksandar Mijatovi膰</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wenhao 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="2410.16340v3-abstract-short" style="display: inline;"> Stochastic gradient descent is a classic algorithm that has gained great popularity especially in the last decades as the most common approach for training models in machine learning. While the algorithm has been well-studied when stochastic gradients are assumed to have a finite variance, there is significantly less research addressing its theoretical properties in the case of infinite variance g… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.16340v3-abstract-full').style.display = 'inline'; document.getElementById('2410.16340v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.16340v3-abstract-full" style="display: none;"> Stochastic gradient descent is a classic algorithm that has gained great popularity especially in the last decades as the most common approach for training models in machine learning. While the algorithm has been well-studied when stochastic gradients are assumed to have a finite variance, there is significantly less research addressing its theoretical properties in the case of infinite variance gradients. In this paper, we establish the asymptotic behavior of stochastic gradient descent in the context of infinite variance stochastic gradients, assuming that the stochastic gradient is regular varying with index $伪\in(1,2)$. The closest result in this context was established in 1969 , in the one-dimensional case and assuming that stochastic gradients belong to a more restrictive class of distributions. We extend it to the multidimensional case, covering a broader class of infinite variance distributions. As we show, the asymptotic distribution of the stochastic gradient descent algorithm can be characterized as the stationary distribution of a suitably defined Ornstein-Uhlenbeck process driven by an appropriate stable L茅vy process. Additionally, we explore the applications of these results in linear regression and logistic regression models. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.16340v3-abstract-full').style.display = 'none'; document.getElementById('2410.16340v3-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> 5 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 21 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/2410.09572">arXiv:2410.09572</a> <span> [<a href="https://arxiv.org/pdf/2410.09572">pdf</a>, <a href="https://arxiv.org/format/2410.09572">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"> Boundary spike-layer solutions of the singular Keller-Segel system: existence, profiles and stability </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Carrillo%2C+J+A">Jose A. Carrillo</a>, <a href="/search/math?searchtype=author&query=Li%2C+J">Jingyu Li</a>, <a href="/search/math?searchtype=author&query=Wang%2C+Z">Zhi-An Wang</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wen 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="2410.09572v1-abstract-short" style="display: inline;"> This paper is concerned with the boundary-layer solutions of the singular Keller-Segel model proposed by Keller-Segel (1971) in a multi-dimensional domain, where the zero-flux boundary condition is imposed to the cell while inhomogeneous Dirichlet boundary condition to the nutrient. The steady-state problem of the Keller-Segel system is reduced to a scalar Dirichlet nonlocal elliptic problem with… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.09572v1-abstract-full').style.display = 'inline'; document.getElementById('2410.09572v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.09572v1-abstract-full" style="display: none;"> This paper is concerned with the boundary-layer solutions of the singular Keller-Segel model proposed by Keller-Segel (1971) in a multi-dimensional domain, where the zero-flux boundary condition is imposed to the cell while inhomogeneous Dirichlet boundary condition to the nutrient. The steady-state problem of the Keller-Segel system is reduced to a scalar Dirichlet nonlocal elliptic problem with singularity. Studying this nonlocal problem, we obtain the unique steady-state solution which possesses a boundary spike-layer profile as nutrient diffusion coefficient $\varepsilon>0$ tends to zero. When the domain is radially symmetric, we find the explicit expansion for the slope of boundary-layer profiles at the boundary and boundary-layer thickness in terms of the radius as $\varepsilon>0$ is small, which pinpoints how the boundary curvature affects the boundary-layer profile and thickness. Furthermore, we establish the nonlinear exponential stability of the boundary-layer steady-state solution for the radially symmetric domain. The main challenge encountered in the analysis is that the singularity will arise when the nutrient diffusion coefficient $\varepsilon>0$ is small for both stationary and time-dependent problems. By relegating the nonlocal steady-state problem to local problems and performing a delicate analysis using the barrier method and Fermi coordinates, we can obtain refined estimates for the solution of local steady-state problem near the boundary. This strategy finally helps us to find the asymptotic profile of the solution to the nonlocal problem as $\varepsilon \to 0$ so that the singularity is accurately captured and hence properly handled to achieve our results. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.09572v1-abstract-full').style.display = 'none'; document.getElementById('2410.09572v1-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> <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">8 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35K57; 35Q92; 92D25 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.00796">arXiv:2410.00796</a> <span> [<a href="https://arxiv.org/pdf/2410.00796">pdf</a>, <a href="https://arxiv.org/format/2410.00796">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Systems and Control">eess.SY</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> </div> </div> <p class="title is-5 mathjax"> Fast and Reliable $N-k$ Contingency Screening with Input-Convex Neural Networks </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Christianson%2C+N">Nicolas Christianson</a>, <a href="/search/math?searchtype=author&query=Cui%2C+W">Wenqi Cui</a>, <a href="/search/math?searchtype=author&query=Low%2C+S">Steven Low</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Weiwei Yang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+B">Baosen 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="2410.00796v1-abstract-short" style="display: inline;"> Power system operators must ensure that dispatch decisions remain feasible in case of grid outages or contingencies to prevent cascading failures and ensure reliable operation. However, checking the feasibility of all $N - k$ contingencies -- every possible simultaneous failure of $k$ grid components -- is computationally intractable for even small $k$, requiring system operators to resort to heur… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.00796v1-abstract-full').style.display = 'inline'; document.getElementById('2410.00796v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.00796v1-abstract-full" style="display: none;"> Power system operators must ensure that dispatch decisions remain feasible in case of grid outages or contingencies to prevent cascading failures and ensure reliable operation. However, checking the feasibility of all $N - k$ contingencies -- every possible simultaneous failure of $k$ grid components -- is computationally intractable for even small $k$, requiring system operators to resort to heuristic screening methods. Because of the increase in uncertainty and changes in system behaviors, heuristic lists might not include all relevant contingencies, generating false negatives in which unsafe scenarios are misclassified as safe. In this work, we propose to use input-convex neural networks (ICNNs) for contingency screening. We show that ICNN reliability can be determined by solving a convex optimization problem, and by scaling model weights using this problem as a differentiable optimization layer during training, we can learn an ICNN classifier that is both data-driven and has provably guaranteed reliability. Namely, our method can ensure a zero false negative rate. We empirically validate this methodology in a case study on the IEEE 39-bus test network, observing that it yields substantial (10-20x) speedups while having excellent classification accuracy. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.00796v1-abstract-full').style.display = 'none'; document.getElementById('2410.00796v1-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">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">11 pages, 4 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/2409.10268">arXiv:2409.10268</a> <span> [<a href="https://arxiv.org/pdf/2409.10268">pdf</a>, <a href="https://arxiv.org/ps/2409.10268">ps</a>, <a href="https://arxiv.org/format/2409.10268">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</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"> Growth tightness of quotients by confined subgroups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ding%2C+L">Lihuang Ding</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wenyuan 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="2409.10268v1-abstract-short" style="display: inline;"> In this paper, we establish the growth tightness of the quotient by confined subgroups in groups admitting the statistically convex-cocompact action with contracting elements. The result is sharp in the sense that the actions could not be relaxed with purely exponential growth. Applications to uniformly recurrent subgroups are discussed. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.10268v1-abstract-full" style="display: none;"> In this paper, we establish the growth tightness of the quotient by confined subgroups in groups admitting the statistically convex-cocompact action with contracting elements. The result is sharp in the sense that the actions could not be relaxed with purely exponential growth. Applications to uniformly recurrent subgroups are discussed. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.10268v1-abstract-full').style.display = 'none'; document.getElementById('2409.10268v1-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> 16 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">19 pages, 3 figures, Appendix by Lihuang Ding and Kairui Liu</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.05366">arXiv:2409.05366</a> <span> [<a href="https://arxiv.org/pdf/2409.05366">pdf</a>, <a href="https://arxiv.org/ps/2409.05366">ps</a>, <a href="https://arxiv.org/format/2409.05366">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"> Asymptotic symmetry of solutions for reaction-diffusion equations via elliptic geometry </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+B">Baiyu Liu</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wenlong 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="2409.05366v1-abstract-short" style="display: inline;"> In this paper, we investigate the asymptotic symmetry and monotonicity of positive solutions to a reaction-diffusion equation in the unit ball, utilizing techniques from elliptic geometry. Firstly, we discuss the properties of solutions in the elliptic space. Then, we establish crucial principles, including the asymptotic narrow region principle.Finally, we employ the method of moving planes to de… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.05366v1-abstract-full').style.display = 'inline'; document.getElementById('2409.05366v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.05366v1-abstract-full" style="display: none;"> In this paper, we investigate the asymptotic symmetry and monotonicity of positive solutions to a reaction-diffusion equation in the unit ball, utilizing techniques from elliptic geometry. Firstly, we discuss the properties of solutions in the elliptic space. Then, we establish crucial principles, including the asymptotic narrow region principle.Finally, we employ the method of moving planes to demonstrate the asymptotic symmetry of the solutions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.05366v1-abstract-full').style.display = 'none'; document.getElementById('2409.05366v1-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 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">MSC Class:</span> 35R11; 35B07 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.04664">arXiv:2409.04664</a> <span> [<a href="https://arxiv.org/pdf/2409.04664">pdf</a>, <a href="https://arxiv.org/ps/2409.04664">ps</a>, <a href="https://arxiv.org/format/2409.04664">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> Non degeneracy of blow-up solutions of non-quantized singular Liouville-type equations and the convexity of the mean field entropy of the Onsager vortex model with singular sources </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bartolucci%2C+D">Daniele Bartolucci</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wen Yang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+L">Lei 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="2409.04664v2-abstract-short" style="display: inline;"> We establish the non-degeneracy of bubbling solutions for singular mean field equations when the blow-up points are either regular or involve non-quantized singular sources. This extends the results from Bartolucci-Jevnikar-Lee-Yang \cite{bart-5}, which focused on regular blow-up points. As a consequence, we establish the strict convexity of the Entropy in the large energy limit for a specific cla… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.04664v2-abstract-full').style.display = 'inline'; document.getElementById('2409.04664v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.04664v2-abstract-full" style="display: none;"> We establish the non-degeneracy of bubbling solutions for singular mean field equations when the blow-up points are either regular or involve non-quantized singular sources. This extends the results from Bartolucci-Jevnikar-Lee-Yang \cite{bart-5}, which focused on regular blow-up points. As a consequence, we establish the strict convexity of the Entropy in the large energy limit for a specific class of two-dimensional domains in the Onsager mean field vortex model with singular sources. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.04664v2-abstract-full').style.display = 'none'; document.getElementById('2409.04664v2-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> 5 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 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">74 pages. Some typos and misprints are corrected. arXiv admin note: text overlap with arXiv:2401.12057</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35J60; 82C40; 53C21 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.04464">arXiv:2409.04464</a> <span> [<a href="https://arxiv.org/pdf/2409.04464">pdf</a>, <a href="https://arxiv.org/format/2409.04464">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computation and Language">cs.CL</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> </div> </div> <p class="title is-5 mathjax"> Leveraging Large Language Models for Solving Rare MIP Challenges </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+T">Teng Wang</a>, <a href="/search/math?searchtype=author&query=Yu%2C+W">Wing-Yin Yu</a>, <a href="/search/math?searchtype=author&query=She%2C+R">Ruifeng She</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wenhan Yang</a>, <a href="/search/math?searchtype=author&query=Chen%2C+T">Taijie Chen</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+J">Jianping 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="2409.04464v2-abstract-short" style="display: inline;"> Mixed Integer Programming (MIP) has been extensively applied in areas requiring mathematical solvers to address complex instances within tight time constraints. However, as the problem scale increases, the complexity of model formulation and finding feasible solutions escalates significantly. In contrast, the model-building cost for end-to-end models, such as large language models (LLMs), remains… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.04464v2-abstract-full').style.display = 'inline'; document.getElementById('2409.04464v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.04464v2-abstract-full" style="display: none;"> Mixed Integer Programming (MIP) has been extensively applied in areas requiring mathematical solvers to address complex instances within tight time constraints. However, as the problem scale increases, the complexity of model formulation and finding feasible solutions escalates significantly. In contrast, the model-building cost for end-to-end models, such as large language models (LLMs), remains largely unaffected by problem scale due to their pattern recognition capabilities. While LLMs, like GPT-4, without fine-tuning, can handle some traditional medium-scale MIP problems, they struggle with uncommon or highly specialized MIP scenarios. Fine-tuning LLMs can yield some feasible solutions for medium-scale MIP instances, but these models typically fail to explore diverse solutions when constrained by a low and constant temperature, limiting their performance. In this paper, we propose and evaluate a recursively dynamic temperature method integrated with a chain-of-thought approach. Our findings show that starting with a high temperature and gradually lowering it leads to better feasible solutions compared to other dynamic temperature strategies. Additionally, by comparing results generated by the LLM with those from Gurobi, we demonstrate that the LLM can produce solutions that complement traditional solvers by accelerating the pruning process and improving overall efficiency. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.04464v2-abstract-full').style.display = 'none'; document.getElementById('2409.04464v2-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 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 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/2408.14924">arXiv:2408.14924</a> <span> [<a href="https://arxiv.org/pdf/2408.14924">pdf</a>, <a href="https://arxiv.org/format/2408.14924">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Exactly Solvable and Integrable Systems">nlin.SI</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Toda lattice and Riemann type minimal surfaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gui%2C+C">Changfeng Gui</a>, <a href="/search/math?searchtype=author&query=Liu%2C+Y">Yong Liu</a>, <a href="/search/math?searchtype=author&query=Wang%2C+J">Jun Wang</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wen 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="2408.14924v1-abstract-short" style="display: inline;"> Toda lattice and minimal surfaces are related to each other through Allen-Cahn equation. In view of the structure of the solutions of the Toda lattice, we find new balancing configuration using techniques of integrable systems. This allows us to construct new singly periodic minimal surfaces. The genus of these minimal surfaces equals $j(j+1)/2-1$. They are natural generalization of the Riemann mi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.14924v1-abstract-full').style.display = 'inline'; document.getElementById('2408.14924v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.14924v1-abstract-full" style="display: none;"> Toda lattice and minimal surfaces are related to each other through Allen-Cahn equation. In view of the structure of the solutions of the Toda lattice, we find new balancing configuration using techniques of integrable systems. This allows us to construct new singly periodic minimal surfaces. The genus of these minimal surfaces equals $j(j+1)/2-1$. They are natural generalization of the Riemann minimal surfaces, which have genus zero. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.14924v1-abstract-full').style.display = 'none'; document.getElementById('2408.14924v1-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 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">16 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/2408.05547">arXiv:2408.05547</a> <span> [<a href="https://arxiv.org/pdf/2408.05547">pdf</a>, <a href="https://arxiv.org/format/2408.05547">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"> Triangle-free Graphs with Large Minimum Common Degree </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+J">Jian Wang</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Weihua Yang</a>, <a href="/search/math?searchtype=author&query=Zhao%2C+F">Fan Zhao</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="2408.05547v1-abstract-short" style="display: inline;"> Let $G$ be a graph. For $x\in V(G)$, let $N(x)=\{y\in V(G)\colon xy\in E(G)\}$. The minimum common degree of $G$, denoted by $未_{2}(G)$, is defined as the minimum of $|N(x)\cap N(y)|$ over all non-edges $xy$ of $G$. In 1982, H盲ggkvist showed that every triangle-free graph with minimum degree greater than $\lfloor\frac{3n}{8}\rfloor$ is homomorphic to a cycle of length 5. In this paper, we prove th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.05547v1-abstract-full').style.display = 'inline'; document.getElementById('2408.05547v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.05547v1-abstract-full" style="display: none;"> Let $G$ be a graph. For $x\in V(G)$, let $N(x)=\{y\in V(G)\colon xy\in E(G)\}$. The minimum common degree of $G$, denoted by $未_{2}(G)$, is defined as the minimum of $|N(x)\cap N(y)|$ over all non-edges $xy$ of $G$. In 1982, H盲ggkvist showed that every triangle-free graph with minimum degree greater than $\lfloor\frac{3n}{8}\rfloor$ is homomorphic to a cycle of length 5. In this paper, we prove that every triangle-free graph with minimum common degree greater than $\lfloor\frac{n}{8}\rfloor$ is homomorphic to a cycle of length 5, which implies H盲ggkvist's result. The balanced blow-up of the M枚bius ladder graph shows that it is best possible. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.05547v1-abstract-full').style.display = 'none'; document.getElementById('2408.05547v1-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">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">11 pages, 9 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/2407.21092">arXiv:2407.21092</a> <span> [<a href="https://arxiv.org/pdf/2407.21092">pdf</a>, <a href="https://arxiv.org/ps/2407.21092">ps</a>, <a href="https://arxiv.org/format/2407.21092">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computation and Language">cs.CL</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistical Mechanics">cond-mat.stat-mech</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Entropy, Thermodynamics and the Geometrization of the Language Model </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yang%2C+W">Wenzhe 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="2407.21092v1-abstract-short" style="display: inline;"> In this paper, we discuss how pure mathematics and theoretical physics can be applied to the study of language models. Using set theory and analysis, we formulate mathematically rigorous definitions of language models, and introduce the concept of the moduli space of distributions for a language model. We formulate a generalized distributional hypothesis using functional analysis and topology. We… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.21092v1-abstract-full').style.display = 'inline'; document.getElementById('2407.21092v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.21092v1-abstract-full" style="display: none;"> In this paper, we discuss how pure mathematics and theoretical physics can be applied to the study of language models. Using set theory and analysis, we formulate mathematically rigorous definitions of language models, and introduce the concept of the moduli space of distributions for a language model. We formulate a generalized distributional hypothesis using functional analysis and topology. We define the entropy function associated with a language model and show how it allows us to understand many interesting phenomena in languages. We argue that the zero points of the entropy function and the points where the entropy is close to 0 are the key obstacles for an LLM to approximate an intelligent language model, which explains why good LLMs need billions of parameters. Using the entropy function, we formulate a conjecture about AGI. Then, we show how thermodynamics gives us an immediate interpretation to language models. In particular we will define the concepts of partition function, internal energy and free energy for a language model, which offer insights into how language models work. Based on these results, we introduce a general concept of the geometrization of language models and define what is called the Boltzmann manifold. While the current LLMs are the special cases of the Boltzmann manifold. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.21092v1-abstract-full').style.display = 'none'; document.getElementById('2407.21092v1-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 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">Comments:</span> <span class="has-text-grey-dark mathjax">18 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 68T01 <span class="has-text-black-bis has-text-weight-semibold">ACM Class:</span> I.2.7 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.09678">arXiv:2407.09678</a> <span> [<a href="https://arxiv.org/pdf/2407.09678">pdf</a>, <a href="https://arxiv.org/format/2407.09678">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> </div> </div> <p class="title is-5 mathjax"> Q statistics in data depth: fundamental theory revisited and variants </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gao%2C+M">Min Gao</a>, <a href="/search/math?searchtype=author&query=Chen%2C+Y">Yiting Chen</a>, <a href="/search/math?searchtype=author&query=Shi%2C+X">Xiaoping Shi</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wenzhi 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="2407.09678v1-abstract-short" style="display: inline;"> Recently, data depth has been widely used to rank multivariate data. The study of the depth-based $Q$ statistic, originally proposed by Liu and Singh (1993), has become increasingly popular when it can be used as a quality index to differentiate between two samples. Based on the existing theoretical foundations, more and more variants have been developed for increasing power in the two sample test… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.09678v1-abstract-full').style.display = 'inline'; document.getElementById('2407.09678v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.09678v1-abstract-full" style="display: none;"> Recently, data depth has been widely used to rank multivariate data. The study of the depth-based $Q$ statistic, originally proposed by Liu and Singh (1993), has become increasingly popular when it can be used as a quality index to differentiate between two samples. Based on the existing theoretical foundations, more and more variants have been developed for increasing power in the two sample test. However, the asymptotic expansion of the $Q$ statistic in the important foundation work of Zuo and He (2006) currently has an optimal rate $m^{-3/4}$ slower than the target $m^{-1}$, leading to limitations in higher-order expansions for developing more powerful tests. We revisit the existing assumptions and add two new plausible assumptions to obtain the target rate by applying a new proof method based on the Hoeffding decomposition and the Cox-Reid expansion. The aim of this paper is to rekindle interest in asymptotic data depth theory, to place Q-statistical inference on a firmer theoretical basis, to show its variants in current research, to open the door to the development of new theories for further variants requiring higher-order expansions, and to explore more of its potential applications. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.09678v1-abstract-full').style.display = 'none'; document.getElementById('2407.09678v1-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 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/2406.15263">arXiv:2406.15263</a> <span> [<a href="https://arxiv.org/pdf/2406.15263">pdf</a>, <a href="https://arxiv.org/ps/2406.15263">ps</a>, <a href="https://arxiv.org/format/2406.15263">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Logic">math.LO</span> </div> </div> <p class="title is-5 mathjax"> On Stability and Existence of Models in Abstract Elementary Classes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Mazari-Armida%2C+M">Marcos Mazari-Armida</a>, <a href="/search/math?searchtype=author&query=Vasey%2C+S">Sebastien Vasey</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wentao 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="2406.15263v2-abstract-short" style="display: inline;"> For an abstract elementary class $\mathbf{K}$ and a cardinal $位\geq LS(\mathbf{K})$, we prove under mild cardinal arithmetic assumptions, categoricity in two succesive cardinals, almost stability for $位^+$-minimal types and continuity of splitting in $位$, that stability in $位$ is equivalent to the existence of a model in $位^{++}$. The forward direction holds without any cardinal or categoricity as… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.15263v2-abstract-full').style.display = 'inline'; document.getElementById('2406.15263v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.15263v2-abstract-full" style="display: none;"> For an abstract elementary class $\mathbf{K}$ and a cardinal $位\geq LS(\mathbf{K})$, we prove under mild cardinal arithmetic assumptions, categoricity in two succesive cardinals, almost stability for $位^+$-minimal types and continuity of splitting in $位$, that stability in $位$ is equivalent to the existence of a model in $位^{++}$. The forward direction holds without any cardinal or categoricity assumptions, this result improves both [Vas18b, 12.1] and [MaYa24, 3.14]. Moreover, we prove a categoricity theorem for abstract elementary classes with weak amalgamation and tameness under mild structural assumptions in $位$. A key feature of this result is that we do not assume amalgamation or arbitrarily large models. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.15263v2-abstract-full').style.display = 'none'; document.getElementById('2406.15263v2-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> 4 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 21 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> Primary: 03C48. Secondary: 03C45; 03C52; 03C55 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.03787">arXiv:2406.03787</a> <span> [<a href="https://arxiv.org/pdf/2406.03787">pdf</a>, <a href="https://arxiv.org/format/2406.03787">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> </div> </div> <p class="title is-5 mathjax"> Projection-Free Variance Reduction Methods for Stochastic Constrained Multi-Level Compositional Optimization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jiang%2C+W">Wei Jiang</a>, <a href="/search/math?searchtype=author&query=Yang%2C+S">Sifan Yang</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wenhao Yang</a>, <a href="/search/math?searchtype=author&query=Wang%2C+Y">Yibo Wang</a>, <a href="/search/math?searchtype=author&query=Wan%2C+Y">Yuanyu Wan</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+L">Lijun 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="2406.03787v1-abstract-short" style="display: inline;"> This paper investigates projection-free algorithms for stochastic constrained multi-level optimization. In this context, the objective function is a nested composition of several smooth functions, and the decision set is closed and convex. Existing projection-free algorithms for solving this problem suffer from two limitations: 1) they solely focus on the gradient mapping criterion and fail to mat… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.03787v1-abstract-full').style.display = 'inline'; document.getElementById('2406.03787v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.03787v1-abstract-full" style="display: none;"> This paper investigates projection-free algorithms for stochastic constrained multi-level optimization. In this context, the objective function is a nested composition of several smooth functions, and the decision set is closed and convex. Existing projection-free algorithms for solving this problem suffer from two limitations: 1) they solely focus on the gradient mapping criterion and fail to match the optimal sample complexities in unconstrained settings; 2) their analysis is exclusively applicable to non-convex functions, without considering convex and strongly convex objectives. To address these issues, we introduce novel projection-free variance reduction algorithms and analyze their complexities under different criteria. For gradient mapping, our complexities improve existing results and match the optimal rates for unconstrained problems. For the widely-used Frank-Wolfe gap criterion, we provide theoretical guarantees that align with those for single-level problems. Additionally, by using a stage-wise adaptation, we further obtain complexities for convex and strongly convex functions. Finally, numerical experiments on different tasks demonstrate the effectiveness of our methods. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.03787v1-abstract-full').style.display = 'none'; document.getElementById('2406.03787v1-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> 6 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/2406.00489">arXiv:2406.00489</a> <span> [<a href="https://arxiv.org/pdf/2406.00489">pdf</a>, <a href="https://arxiv.org/format/2406.00489">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"> Efficient Sign-Based Optimization: Accelerating Convergence via Variance Reduction </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jiang%2C+W">Wei Jiang</a>, <a href="/search/math?searchtype=author&query=Yang%2C+S">Sifan Yang</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wenhao Yang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+L">Lijun 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="2406.00489v3-abstract-short" style="display: inline;"> Sign stochastic gradient descent (signSGD) is a communication-efficient method that transmits only the sign of stochastic gradients for parameter updating. Existing literature has demonstrated that signSGD can achieve a convergence rate of $\mathcal{O}(d^{1/2}T^{-1/4})$, where $d$ represents the dimension and $T$ is the iteration number. In this paper, we improve this convergence rate to… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.00489v3-abstract-full').style.display = 'inline'; document.getElementById('2406.00489v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.00489v3-abstract-full" style="display: none;"> Sign stochastic gradient descent (signSGD) is a communication-efficient method that transmits only the sign of stochastic gradients for parameter updating. Existing literature has demonstrated that signSGD can achieve a convergence rate of $\mathcal{O}(d^{1/2}T^{-1/4})$, where $d$ represents the dimension and $T$ is the iteration number. In this paper, we improve this convergence rate to $\mathcal{O}(d^{1/2}T^{-1/3})$ by introducing the Sign-based Stochastic Variance Reduction (SSVR) method, which employs variance reduction estimators to track gradients and leverages their signs to update. For finite-sum problems, our method can be further enhanced to achieve a convergence rate of $\mathcal{O}(m^{1/4}d^{1/2}T^{-1/2})$, where $m$ denotes the number of component functions. Furthermore, we investigate the heterogeneous majority vote in distributed settings and introduce two novel algorithms that attain improved convergence rates of $\mathcal{O}(d^{1/2}T^{-1/2} + dn^{-1/2})$ and $\mathcal{O}(d^{1/4}T^{-1/4})$ respectively, outperforming the previous results of $\mathcal{O}(dT^{-1/4} + dn^{-1/2})$ and $\mathcal{O}(d^{3/8}T^{-1/8})$, where $n$ represents the number of nodes. Numerical experiments across different tasks validate the effectiveness of our proposed methods. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.00489v3-abstract-full').style.display = 'none'; document.getElementById('2406.00489v3-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 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 1 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/2405.19705">arXiv:2405.19705</a> <span> [<a href="https://arxiv.org/pdf/2405.19705">pdf</a>, <a href="https://arxiv.org/ps/2405.19705">ps</a>, <a href="https://arxiv.org/format/2405.19705">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"> Universal Online Convex Optimization with $1$ Projection per Round </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yang%2C+W">Wenhao Yang</a>, <a href="/search/math?searchtype=author&query=Wang%2C+Y">Yibo Wang</a>, <a href="/search/math?searchtype=author&query=Zhao%2C+P">Peng Zhao</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+L">Lijun 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="2405.19705v1-abstract-short" style="display: inline;"> To address the uncertainty in function types, recent progress in online convex optimization (OCO) has spurred the development of universal algorithms that simultaneously attain minimax rates for multiple types of convex functions. However, for a $T$-round online problem, state-of-the-art methods typically conduct $O(\log T)$ projections onto the domain in each round, a process potentially time-con… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.19705v1-abstract-full').style.display = 'inline'; document.getElementById('2405.19705v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.19705v1-abstract-full" style="display: none;"> To address the uncertainty in function types, recent progress in online convex optimization (OCO) has spurred the development of universal algorithms that simultaneously attain minimax rates for multiple types of convex functions. However, for a $T$-round online problem, state-of-the-art methods typically conduct $O(\log T)$ projections onto the domain in each round, a process potentially time-consuming with complicated feasible sets. In this paper, inspired by the black-box reduction of Cutkosky and Orabona (2018), we employ a surrogate loss defined over simpler domains to develop universal OCO algorithms that only require $1$ projection. Embracing the framework of prediction with expert advice, we maintain a set of experts for each type of functions and aggregate their predictions via a meta-algorithm. The crux of our approach lies in a uniquely designed expert-loss for strongly convex functions, stemming from an innovative decomposition of the regret into the meta-regret and the expert-regret. Our analysis sheds new light on the surrogate loss, facilitating a rigorous examination of the discrepancy between the regret of the original loss and that of the surrogate loss, and carefully controlling meta-regret under the strong convexity condition. In this way, with only $1$ projection per round, we establish optimal regret bounds for general convex, exponentially concave, and strongly convex functions simultaneously. Furthermore, we enhance the expert-loss to exploit the smoothness property, and demonstrate that our algorithm can attain small-loss regret for multiple types of convex and smooth functions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.19705v1-abstract-full').style.display = 'none'; document.getElementById('2405.19705v1-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 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/2405.17688">arXiv:2405.17688</a> <span> [<a href="https://arxiv.org/pdf/2405.17688">pdf</a>, <a href="https://arxiv.org/format/2405.17688">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="Hardware Architecture">cs.AR</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/10.4230/LIPIcs.TQC.2024.1">10.4230/LIPIcs.TQC.2024.1 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Multi-qubit Lattice Surgery Scheduling </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Silva%2C+A">Allyson Silva</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+X">Xiangyi Zhang</a>, <a href="/search/math?searchtype=author&query=Webb%2C+Z">Zak Webb</a>, <a href="/search/math?searchtype=author&query=Kramer%2C+M">Mia Kramer</a>, <a href="/search/math?searchtype=author&query=Yang%2C+C+W">Chan Woo Yang</a>, <a href="/search/math?searchtype=author&query=Liu%2C+X">Xiao Liu</a>, <a href="/search/math?searchtype=author&query=Lemieux%2C+J">Jessica Lemieux</a>, <a href="/search/math?searchtype=author&query=Chen%2C+K">Ka-Wai Chen</a>, <a href="/search/math?searchtype=author&query=Scherer%2C+A">Artur Scherer</a>, <a href="/search/math?searchtype=author&query=Ronagh%2C+P">Pooya Ronagh</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="2405.17688v2-abstract-short" style="display: inline;"> Fault-tolerant quantum computation using two-dimensional topological quantum error correcting codes can benefit from multi-qubit long-range operations. By using simple commutation rules, a quantum circuit can be transpiled into a sequence of solely non-Clifford multi-qubit gates. Prior work on fault-tolerant compilation avoids optimal scheduling of such gates since they reduce the parallelizabilit… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.17688v2-abstract-full').style.display = 'inline'; document.getElementById('2405.17688v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.17688v2-abstract-full" style="display: none;"> Fault-tolerant quantum computation using two-dimensional topological quantum error correcting codes can benefit from multi-qubit long-range operations. By using simple commutation rules, a quantum circuit can be transpiled into a sequence of solely non-Clifford multi-qubit gates. Prior work on fault-tolerant compilation avoids optimal scheduling of such gates since they reduce the parallelizability of the circuit. We observe that the reduced parallelization potential is outweighed by the significant reduction in the number of gates. We therefore devise a method for scheduling multi-qubit lattice surgery using an earliest-available-first policy, solving the associated forest packing problem using a representation of the multi-qubit gates as Steiner trees. Our extensive testing on random and application-inspired circuits demonstrates the method's scalability and performance. We show that the transpilation significantly reduces the circuit length on the set of circuits tested, and that the resulting circuit of multi-qubit gates has a further reduction in the expected circuit execution time compared to serial execution. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.17688v2-abstract-full').style.display = 'none'; document.getElementById('2405.17688v2-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 June, 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">23 pages, 7 figures, 4 tables</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> 19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024). LIPIcs, Volume 310, pp. 1:1-1:22, 2024 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.17191">arXiv:2405.17191</a> <span> [<a href="https://arxiv.org/pdf/2405.17191">pdf</a>, <a href="https://arxiv.org/format/2405.17191">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computer Vision and Pattern Recognition">cs.CV</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"> MCGAN: Enhancing GAN Training with Regression-Based Generator Loss </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xiao%2C+B">Baoren Xiao</a>, <a href="/search/math?searchtype=author&query=Ni%2C+H">Hao Ni</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Weixin 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="2405.17191v2-abstract-short" style="display: inline;"> Generative adversarial networks (GANs) have emerged as a powerful tool for generating high-fidelity data. However, the main bottleneck of existing approaches is the lack of supervision on the generator training, which often results in undamped oscillation and unsatisfactory performance. To address this issue, we propose an algorithm called Monte Carlo GAN (MCGAN). This approach, utilizing an innov… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.17191v2-abstract-full').style.display = 'inline'; document.getElementById('2405.17191v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.17191v2-abstract-full" style="display: none;"> Generative adversarial networks (GANs) have emerged as a powerful tool for generating high-fidelity data. However, the main bottleneck of existing approaches is the lack of supervision on the generator training, which often results in undamped oscillation and unsatisfactory performance. To address this issue, we propose an algorithm called Monte Carlo GAN (MCGAN). This approach, utilizing an innovative generative loss function, termly the regression loss, reformulates the generator training as a regression task and enables the generator training by minimizing the mean squared error between the discriminator's output of real data and the expected discriminator of fake data. We demonstrate the desirable analytic properties of the regression loss, including discriminability and optimality, and show that our method requires a weaker condition on the discriminator for effective generator training. These properties justify the strength of this approach to improve the training stability while retaining the optimality of GAN by leveraging strong supervision of the regression loss. Extensive experiments on diverse datasets, including image data (CIFAR-10/100, FFHQ256, ImageNet, and LSUN Bedroom), time series data (VAR and stock data) and video data, are conducted to demonstrate the flexibility and effectiveness of our proposed MCGAN. Numerical results show that the proposed MCGAN is versatile in enhancing a variety of backbone GAN models and achieves consistent and significant improvement in terms of quality, accuracy, training stability, and learned latent space. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.17191v2-abstract-full').style.display = 'none'; document.getElementById('2405.17191v2-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 December, 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> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.09070">arXiv:2405.09070</a> <span> [<a href="https://arxiv.org/pdf/2405.09070">pdf</a>, <a href="https://arxiv.org/ps/2405.09070">ps</a>, <a href="https://arxiv.org/format/2405.09070">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</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"> Confined subgroups in groups with contracting elements </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Choi%2C+I">Inhyeok Choi</a>, <a href="/search/math?searchtype=author&query=Gekhtman%2C+I">Ilya Gekhtman</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wenyuan Yang</a>, <a href="/search/math?searchtype=author&query=Zheng%2C+T">Tianyi Zheng</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="2405.09070v1-abstract-short" style="display: inline;"> In this paper, we study the growth of confined subgroups through boundary actions of groups with contracting elements. We establish that the growth rate of a confined subgroup is strictly greater than half of the ambient growth rate in groups with purely exponential growth. Along the way, several results are obtained on the Hopf decomposition for boundary actions of subgroups with respect to confo… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.09070v1-abstract-full').style.display = 'inline'; document.getElementById('2405.09070v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.09070v1-abstract-full" style="display: none;"> In this paper, we study the growth of confined subgroups through boundary actions of groups with contracting elements. We establish that the growth rate of a confined subgroup is strictly greater than half of the ambient growth rate in groups with purely exponential growth. Along the way, several results are obtained on the Hopf decomposition for boundary actions of subgroups with respect to conformal measures. In particular, we prove that confined subgroups are conservative, and examples of subgroups with nontrivial Hopf decomposition are constructed. We show a connection between Hopf decomposition and quotient growth and provide a dichotomy on quotient growth of Schreier graphs for subgroups in hyperbolic groups. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.09070v1-abstract-full').style.display = 'none'; document.getElementById('2405.09070v1-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">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">58 pages, 13 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20F65; 20F67; 20F69 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.04954">arXiv:2405.04954</a> <span> [<a href="https://arxiv.org/pdf/2405.04954">pdf</a>, <a href="https://arxiv.org/ps/2405.04954">ps</a>, <a href="https://arxiv.org/format/2405.04954">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"> On vector parking functions and q-analogue </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yang%2C+W">Wenkai 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="2405.04954v1-abstract-short" style="display: inline;"> In 2000, it was demonstrated that the set of $x$-parking functions of length $n$, where $x$=($a,b,...,b$) $\in \mathbbm{N}^n$, is equivalent to the set of rooted multicolored forests on [$n$]=\{1,...,$n$\}. In 2020, Yue Cai and Catherine H. Yan systematically investigated the properties of rational parking functions. Subsequently, a series of Context-free grammars possessing the requisite property… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.04954v1-abstract-full').style.display = 'inline'; document.getElementById('2405.04954v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.04954v1-abstract-full" style="display: none;"> In 2000, it was demonstrated that the set of $x$-parking functions of length $n$, where $x$=($a,b,...,b$) $\in \mathbbm{N}^n$, is equivalent to the set of rooted multicolored forests on [$n$]=\{1,...,$n$\}. In 2020, Yue Cai and Catherine H. Yan systematically investigated the properties of rational parking functions. Subsequently, a series of Context-free grammars possessing the requisite property were introduced by William Y.C. Chen and Harold R.L. Yang in 2021. %An Abelian-type identity is derived from a comparable methodology and grammatical framework. %Leveraging a comparable methodology and grammatical framework, an Abelian-type identity is derived herein. In this paper, I discuss generalized parking functions in terms of grammars. The primary result is to obtain the q-analogue about the number of '1's in certain vector parking functions with the assistance of grammars. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.04954v1-abstract-full').style.display = 'none'; document.getElementById('2405.04954v1-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> 8 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> 05A15; 05A19; 05A30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.18762">arXiv:2404.18762</a> <span> [<a href="https://arxiv.org/pdf/2404.18762">pdf</a>, <a href="https://arxiv.org/ps/2404.18762">ps</a>, <a href="https://arxiv.org/format/2404.18762">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> Genericity of sublinearly Morse directions in general metric spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Qing%2C+Y">Yulan Qing</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wenyuan 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="2404.18762v1-abstract-short" style="display: inline;"> In this paper, we show that for any proper statistically convexcocompact actions on proper metric spaces, the sublinearly Morse boundary has full Patterson-Sullivan measure in the horofundction boundary. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.18762v1-abstract-full" style="display: none;"> In this paper, we show that for any proper statistically convexcocompact actions on proper metric spaces, the sublinearly Morse boundary has full Patterson-Sullivan measure in the horofundction boundary. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.18762v1-abstract-full').style.display = 'none'; document.getElementById('2404.18762v1-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 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">23 pages, 5 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20F65; 20F67 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.18117">arXiv:2404.18117</a> <span> [<a href="https://arxiv.org/pdf/2404.18117">pdf</a>, <a href="https://arxiv.org/ps/2404.18117">ps</a>, <a href="https://arxiv.org/format/2404.18117">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Symbolic Computation">cs.SC</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 Basis-preserving Algorithm for Computing the Bezout Matrix of Newton Polynomials </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yang%2C+J">Jing Yang</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wei 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="2404.18117v1-abstract-short" style="display: inline;"> This paper tackles the problem of constructing Bezout matrices for Newton polynomials in a basis-preserving approach that operates directly with the given Newton basis, thus avoiding the need for transformation from Newton basis to monomial basis. This approach significantly reduces the computational cost and also mitigates numerical instability caused by basis transformation. For this purpose, we… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.18117v1-abstract-full').style.display = 'inline'; document.getElementById('2404.18117v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.18117v1-abstract-full" style="display: none;"> This paper tackles the problem of constructing Bezout matrices for Newton polynomials in a basis-preserving approach that operates directly with the given Newton basis, thus avoiding the need for transformation from Newton basis to monomial basis. This approach significantly reduces the computational cost and also mitigates numerical instability caused by basis transformation. For this purpose, we investigate the internal structure of Bezout matrices in Newton basis and design a basis-preserving algorithm that generates the Bezout matrix in the specified basis used to formulate the input polynomials. Furthermore, we show an application of the proposed algorithm on constructing confederate resultant matrices for Newton polynomials. Experimental results demonstrate that the proposed methods perform superior to the basis-transformation-based ones. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.18117v1-abstract-full').style.display = 'none'; document.getElementById('2404.18117v1-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 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/2404.11112">arXiv:2404.11112</a> <span> [<a href="https://arxiv.org/pdf/2404.11112">pdf</a>, <a href="https://arxiv.org/format/2404.11112">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 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/10.1007/s10589-024-00595-3">10.1007/s10589-024-00595-3 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> An Adaptive Regularized Proximal Newton-Type Methods for Composite Optimization over the Stiefel Manifold </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+Q">Qinsi Wang</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W+H">Wei Hong 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="2404.11112v3-abstract-short" style="display: inline;"> Recently, the proximal Newton-type method and its variants have been generalized to solve composite optimization problems over the Stiefel manifold whose objective function is the summation of a smooth function and a nonsmooth function. In this paper, we propose an adaptive quadratically regularized proximal quasi-Newton method, named ARPQN, to solve this class of problems. Under some mild assumpt… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.11112v3-abstract-full').style.display = 'inline'; document.getElementById('2404.11112v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.11112v3-abstract-full" style="display: none;"> Recently, the proximal Newton-type method and its variants have been generalized to solve composite optimization problems over the Stiefel manifold whose objective function is the summation of a smooth function and a nonsmooth function. In this paper, we propose an adaptive quadratically regularized proximal quasi-Newton method, named ARPQN, to solve this class of problems. Under some mild assumptions, the global convergence, the local linear convergence rate and the iteration complexity of ARPQN are established. Numerical experiments and comparisons with other state-of-the-art methods indicate that ARPQN is very promising. We also propose an adaptive quadratically regularized proximal Newton method, named ARPN. It is shown the ARPN method has a local superlinear convergence rate under certain reasonable assumptions, which demonstrates attractive convergence properties of regularized proximal Newton methods. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.11112v3-abstract-full').style.display = 'none'; document.getElementById('2404.11112v3-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 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 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">38 pages, 6 figures. Comput Optim Appl (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/2404.07486">arXiv:2404.07486</a> <span> [<a href="https://arxiv.org/pdf/2404.07486">pdf</a>, <a href="https://arxiv.org/format/2404.07486">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"> Extremal triangle-free graphs with chromatic number at least four </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ren%2C+S">Sijie Ren</a>, <a href="/search/math?searchtype=author&query=Wang%2C+J">Jian Wang</a>, <a href="/search/math?searchtype=author&query=Wang%2C+S">Shipeng Wang</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Weihua 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="2404.07486v1-abstract-short" style="display: inline;"> Let $G$ be an $n$-vertex triangle-free graph. The celebrated Mantel's theorem showed that $e(G)\leq \lfloor\frac{n^2}{4}\rfloor$. In 1962, Erd艖s (together with Gallai), and independently Andr谩sfai, proved that if $G$ is non-bipartite then $e(G)\leq \lfloor\frac{(n-1)^2}{4}\rfloor+1$. In this paper, we extend this result and show that if $G$ has chromatic number at least four and $n\geq 150$, then… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.07486v1-abstract-full').style.display = 'inline'; document.getElementById('2404.07486v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.07486v1-abstract-full" style="display: none;"> Let $G$ be an $n$-vertex triangle-free graph. The celebrated Mantel's theorem showed that $e(G)\leq \lfloor\frac{n^2}{4}\rfloor$. In 1962, Erd艖s (together with Gallai), and independently Andr谩sfai, proved that if $G$ is non-bipartite then $e(G)\leq \lfloor\frac{(n-1)^2}{4}\rfloor+1$. In this paper, we extend this result and show that if $G$ has chromatic number at least four and $n\geq 150$, then $e(G)\leq \lfloor\frac{(n-3)^2}{4}\rfloor+5$. The blow-up of Gr枚tzsch graph shows that this bound is best possible. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.07486v1-abstract-full').style.display = 'none'; document.getElementById('2404.07486v1-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 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">14 pages, 4 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/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/2402.10165">arXiv:2402.10165</a> <span> [<a href="https://arxiv.org/pdf/2402.10165">pdf</a>, <a href="https://arxiv.org/ps/2402.10165">ps</a>, <a href="https://arxiv.org/format/2402.10165">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</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"> Marked length spectrum rigidity in groups with contracting elements </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wan%2C+R">Renxing Wan</a>, <a href="/search/math?searchtype=author&query=Xu%2C+X">Xiaoyu Xu</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wenyuan 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="2402.10165v2-abstract-short" style="display: inline;"> This paper presents a study of the well-known marked length spectrum rigidity problem in the coarse-geometric setting. For any two (possibly non-proper) group actions $G\curvearrowright X_1$ and $G\curvearrowright X_2$ with contracting property, we prove that if the two actions have the same marked length spectrum, then the orbit map $Go_1\to Go_2$ must be a rough isometry. In the special case of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.10165v2-abstract-full').style.display = 'inline'; document.getElementById('2402.10165v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.10165v2-abstract-full" style="display: none;"> This paper presents a study of the well-known marked length spectrum rigidity problem in the coarse-geometric setting. For any two (possibly non-proper) group actions $G\curvearrowright X_1$ and $G\curvearrowright X_2$ with contracting property, we prove that if the two actions have the same marked length spectrum, then the orbit map $Go_1\to Go_2$ must be a rough isometry. In the special case of cusp-uniform actions, the rough isometry can be extended to the entire space. This generalizes the existing results in hyperbolic groups and relatively hyperbolic groups. In addition, we prove a finer marked length spectrum rigidity from confined subgroups and further, geometrically dense subgroups. Our proof is based on the Extension Lemma and uses purely elementary metric geometry. This study produces new results and recovers existing ones for many more interesting groups through a unified and elementary approach. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.10165v2-abstract-full').style.display = 'none'; document.getElementById('2402.10165v2-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 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 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">45 pages; updated information about reference [41]</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 20F65 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2401.12057">arXiv:2401.12057</a> <span> [<a href="https://arxiv.org/pdf/2401.12057">pdf</a>, <a href="https://arxiv.org/ps/2401.12057">ps</a>, <a href="https://arxiv.org/format/2401.12057">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"> Asymptotic Analysis and Uniqueness of blowup solutions of non-quantized singular mean field equations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bartolucci%2C+D">Daniele Bartolucci</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wen Yang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+L">Lei 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="2401.12057v3-abstract-short" style="display: inline;"> For singular mean field equations defined on a compact Riemann surface, we prove the uniqueness of bubbling solutions as far as blowup points are either regular points or non-quantized singular sources. In particular the uniqueness result covers the most general case extending or improving all previous works of Bartolucci-Jevnikar-Lee-Yang \cite{bart-4,bart-4-2} and Wu-Zhang \cite{wu-zhang-ccm}. F… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.12057v3-abstract-full').style.display = 'inline'; document.getElementById('2401.12057v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2401.12057v3-abstract-full" style="display: none;"> For singular mean field equations defined on a compact Riemann surface, we prove the uniqueness of bubbling solutions as far as blowup points are either regular points or non-quantized singular sources. In particular the uniqueness result covers the most general case extending or improving all previous works of Bartolucci-Jevnikar-Lee-Yang \cite{bart-4,bart-4-2} and Wu-Zhang \cite{wu-zhang-ccm}. For example, unlike previous results, we drop the assumption of singular sources being critical points of a suitably defined Kirchoff-Routh type functional. Our argument is based on refined estimates, robust and flexible enough to be applied to a wide range of problems requiring a delicate blowup analysis. In particular we come up with several new estimates of independent interest about the concentration phenomenon for Liouville-type equations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.12057v3-abstract-full').style.display = 'none'; document.getElementById('2401.12057v3-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 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 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">86 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35J60; 53C21 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2312.06690">arXiv:2312.06690</a> <span> [<a href="https://arxiv.org/pdf/2312.06690">pdf</a>, <a href="https://arxiv.org/ps/2312.06690">ps</a>, <a href="https://arxiv.org/format/2312.06690">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="Mathematical Finance">q-fin.MF</span> </div> </div> <p class="title is-5 mathjax"> Backward Stochastic Differential Equations in Financial Mathematics </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yang%2C+W">Weiye 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="2312.06690v1-abstract-short" style="display: inline;"> A backward stochastic differential equation (BSDE) is an SDE of the form $-dY_t = f(t,Y_t,Z_t)dt - Z_t^*dW_t;\ Y_T = 尉$. The subject of BSDEs has seen extensive attention since their introduction in the linear case by Bismut (1973) and in the general case by Pardoux and Peng (1990). In contrast with deterministic differential equations, it is not enough to simply reverse the direction of time and… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.06690v1-abstract-full').style.display = 'inline'; document.getElementById('2312.06690v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2312.06690v1-abstract-full" style="display: none;"> A backward stochastic differential equation (BSDE) is an SDE of the form $-dY_t = f(t,Y_t,Z_t)dt - Z_t^*dW_t;\ Y_T = 尉$. The subject of BSDEs has seen extensive attention since their introduction in the linear case by Bismut (1973) and in the general case by Pardoux and Peng (1990). In contrast with deterministic differential equations, it is not enough to simply reverse the direction of time and treat the terminal condition as an initial condition, as we would then run into problems with adaptedness. Intuitively, our "knowledge" at time $t$ consists only of what has happened at all times $s \in [0,t]$, and we cannot reverse the direction of time whilst keeping this true. The layout of this essay is as follows: In Section 1 we introduce BSDEs and go over the basic results of BSDE theory, including two major theorems: the existence and uniqueness of solutions and the comparison theorem. We also introduce linear BSDEs and the notion of supersolutions of a BSDE. In Section 2 we set up the financial framework in which we will price European contingent claims, and prove a result about the fair price of such claims in a dynamically complete market. In Section 3 we extend the theory of linear BSDEs to include concave BSDEs, and apply this to pricing claims in more complicated market models. In Section 4 we take a look at utility maximisation problems, and see how utilising BSDE theory allows for a relatively simple and neat solution in certain cases. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.06690v1-abstract-full').style.display = 'none'; document.getElementById('2312.06690v1-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 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2023. </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. Essay submitted for Part III of the Cambridge University Mathematical Tripos</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 60H10 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2312.01063">arXiv:2312.01063</a> <span> [<a href="https://arxiv.org/pdf/2312.01063">pdf</a>, <a href="https://arxiv.org/ps/2312.01063">ps</a>, <a href="https://arxiv.org/format/2312.01063">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"> Lump type solutions: Backlund transformation and spectral properties </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+Y">Yong Liu</a>, <a href="/search/math?searchtype=author&query=Wei%2C+J">Jun-cheng Wei</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wen 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="2312.01063v1-abstract-short" style="display: inline;"> There are various different ways to obtain traveling waves of lump type for the KP equation. We propose a general and simple approach to derive them via a Backlund transformation. This enables us to establish an explicit connection between those low energy solutions and high energy ones. Based on this construction, spectral analysis of the degree $6$ solutions is then carried out in details. The a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.01063v1-abstract-full').style.display = 'inline'; document.getElementById('2312.01063v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2312.01063v1-abstract-full" style="display: none;"> There are various different ways to obtain traveling waves of lump type for the KP equation. We propose a general and simple approach to derive them via a Backlund transformation. This enables us to establish an explicit connection between those low energy solutions and high energy ones. Based on this construction, spectral analysis of the degree $6$ solutions is then carried out in details. The analysis of higher energy ones can be done in an inductive way. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.01063v1-abstract-full').style.display = 'none'; document.getElementById('2312.01063v1-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 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2023. </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">25 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/2311.08634">arXiv:2311.08634</a> <span> [<a href="https://arxiv.org/pdf/2311.08634">pdf</a>, <a href="https://arxiv.org/ps/2311.08634">ps</a>, <a href="https://arxiv.org/format/2311.08634">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"> On the minimum degree of minimally $ t $-tough, claw-free graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ma%2C+H">Hui Ma</a>, <a href="/search/math?searchtype=author&query=Hu%2C+X">Xiaomin Hu</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Weihua 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="2311.08634v1-abstract-short" style="display: inline;"> A graph $ G $ is minimally $ t $-tough if the toughness of $ G $ is $ t $ and deletion of any edge from $ G $ decreases its toughness. Katona et al. conjectured that the minimum degree of any minimally $ t $-tough graph is $ \lceil 2t\rceil $ and proved that the minimum degree of minimally $ \frac{1}2 $-tough and $ 1 $-tough, claw-free graphs is 1 and 2, respectively. We have show that every minim… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.08634v1-abstract-full').style.display = 'inline'; document.getElementById('2311.08634v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2311.08634v1-abstract-full" style="display: none;"> A graph $ G $ is minimally $ t $-tough if the toughness of $ G $ is $ t $ and deletion of any edge from $ G $ decreases its toughness. Katona et al. conjectured that the minimum degree of any minimally $ t $-tough graph is $ \lceil 2t\rceil $ and proved that the minimum degree of minimally $ \frac{1}2 $-tough and $ 1 $-tough, claw-free graphs is 1 and 2, respectively. We have show that every minimally $ 3/2 $-tough, claw-free graph has a vertex of degree of $ 3 $. In this paper, we give an upper bound on the minimum degree of minimally $t$-tough, claw-free graphs for $ t\geq 2 $. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.08634v1-abstract-full').style.display = 'none'; document.getElementById('2311.08634v1-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 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2311.05505">arXiv:2311.05505</a> <span> [<a href="https://arxiv.org/pdf/2311.05505">pdf</a>, <a href="https://arxiv.org/format/2311.05505">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"> On regular 2-path Hamiltonian graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xia Li</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Weihua Yang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+B">Bo Zhang</a>, <a href="/search/math?searchtype=author&query=Zhao%2C+S">Shuang Zhao</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="2311.05505v1-abstract-short" style="display: inline;"> Kronk introduced the $l$-path hamiltonianicity of graphs in 1969. A graph is $l$-path Hamiltonian if every path of length not exceeding $l$ is contained in a Hamiltonian cycle. We have shown that if $P=uvz$ is a 2-path of a 2-connected, $k$-regular graph on at most $2k$ vertices and $G - V(P)$ is connected, then there must exist a Hamiltonian cycle in $G$ that contains the 2-path $P$. In this pape… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.05505v1-abstract-full').style.display = 'inline'; document.getElementById('2311.05505v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2311.05505v1-abstract-full" style="display: none;"> Kronk introduced the $l$-path hamiltonianicity of graphs in 1969. A graph is $l$-path Hamiltonian if every path of length not exceeding $l$ is contained in a Hamiltonian cycle. We have shown that if $P=uvz$ is a 2-path of a 2-connected, $k$-regular graph on at most $2k$ vertices and $G - V(P)$ is connected, then there must exist a Hamiltonian cycle in $G$ that contains the 2-path $P$. In this paper, we characterize a class of graphs that illustrate the sharpness of the bound $2k$. Additionally, we show that by excluding the class of graphs, both 2-connected, $k$-regular graphs on at most $2k + 1$ vertices and 3-connected, $k$-regular graphs on at most $3k-6$ vertices satisfy that there is a Hamiltonian cycle containing the 2-path $P$ if $G\setminus V(P)$ is connected. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.05505v1-abstract-full').style.display = 'none'; document.getElementById('2311.05505v1-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 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2023. </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">20. arXiv admin note: text overlap with arXiv:2203.04345</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2310.19254">arXiv:2310.19254</a> <span> [<a href="https://arxiv.org/pdf/2310.19254">pdf</a>, <a href="https://arxiv.org/ps/2310.19254">ps</a>, <a href="https://arxiv.org/format/2310.19254">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"> Hierarchical control for the semilinear parabolic equations with interior degeneracy </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gao%2C+H">Hang Gao</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wei Yang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+M">Muming 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="2310.19254v1-abstract-short" style="display: inline;"> This paper concerns with the hierarchical control of the semilinear parabolic equations with interior degeneracy. By a Stackelberg-Nash strategy, we consider the linear and semilinear system with one leader and two followers. First, for any given leader, we analyze a Nash equilibrium corresponding to a bi-objective optimal control problem. The existence and uniqueness of the Nash equilibrium is pr… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.19254v1-abstract-full').style.display = 'inline'; document.getElementById('2310.19254v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2310.19254v1-abstract-full" style="display: none;"> This paper concerns with the hierarchical control of the semilinear parabolic equations with interior degeneracy. By a Stackelberg-Nash strategy, we consider the linear and semilinear system with one leader and two followers. First, for any given leader, we analyze a Nash equilibrium corresponding to a bi-objective optimal control problem. The existence and uniqueness of the Nash equilibrium is proved, and its characterization is given. Then, we find a leader satisfying the null controllability problem. The key is to establish a new Carleman estimate for a coupled degenerate parabolic system with interior degeneracy. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.19254v1-abstract-full').style.display = 'none'; document.getElementById('2310.19254v1-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 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2310.14474">arXiv:2310.14474</a> <span> [<a href="https://arxiv.org/pdf/2310.14474">pdf</a>, <a href="https://arxiv.org/ps/2310.14474">ps</a>, <a href="https://arxiv.org/format/2310.14474">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Logic">math.LO</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/10.1017/jsl.2024.32">10.1017/jsl.2024.32 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Building models in small cardinals in local abstract elementary classes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Mazari-Armida%2C+M">Marcos Mazari-Armida</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wentao 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="2310.14474v3-abstract-short" style="display: inline;"> There are many results in the literature where superstablity-like independence notions, without any categoricity assumptions, have been used to show the existence of larger models. In this paper we show that \emph{stability} is enough to construct larger models for small cardinals assuming a mild locality condition for Galois types. $\mathbf{Theorem.}$ Suppose $位<2^{\aleph_0}$. Let $\mathbf{K}$… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.14474v3-abstract-full').style.display = 'inline'; document.getElementById('2310.14474v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2310.14474v3-abstract-full" style="display: none;"> There are many results in the literature where superstablity-like independence notions, without any categoricity assumptions, have been used to show the existence of larger models. In this paper we show that \emph{stability} is enough to construct larger models for small cardinals assuming a mild locality condition for Galois types. $\mathbf{Theorem.}$ Suppose $位<2^{\aleph_0}$. Let $\mathbf{K}$ be an abstract elementary class with $位\geq LS(\mathbf{K})$. Assume $\mathbf{K}$ has amalgamation in $位$, no maximal model in $位$, and is stable in $位$. If $\mathbf{K}$ is $(<位^+, 位)$-local, then $\mathbf{K}$ has a model of cardinality $位^{++}$. The set theoretic assumption that $位<2^{\aleph_0}$ and model theoretic assumption of stability in $位$ can be weakened to the model theoretic assumptions that $|\mathbf{S}^{na}(M)|< 2^{\aleph_0}$ for every $M \in \mathbf{K}_位$ and stability for $位$-algebraic types in $位$. This is a significant improvement of Theorem 0.1., as the result holds on some unstable abstract elementary classes. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.14474v3-abstract-full').style.display = 'none'; document.getElementById('2310.14474v3-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 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 03C48. Secondary: 03C45; 03C52; 03C55 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2310.12846">arXiv:2310.12846</a> <span> [<a href="https://arxiv.org/pdf/2310.12846">pdf</a>, <a href="https://arxiv.org/format/2310.12846">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="Artificial Intelligence">cs.AI</span> </div> </div> <p class="title is-5 mathjax"> Physical Information Neural Networks for Solving High-index Differential-algebraic Equation Systems Based on Radau Methods </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+J">Jiasheng Chen</a>, <a href="/search/math?searchtype=author&query=Tang%2C+J">Juan Tang</a>, <a href="/search/math?searchtype=author&query=Yan%2C+M">Ming Yan</a>, <a href="/search/math?searchtype=author&query=Lai%2C+S">Shuai Lai</a>, <a href="/search/math?searchtype=author&query=Liang%2C+K">Kun Liang</a>, <a href="/search/math?searchtype=author&query=Lu%2C+J">Jianguang Lu</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wenqiang 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="2310.12846v1-abstract-short" style="display: inline;"> As is well known, differential algebraic equations (DAEs), which are able to describe dynamic changes and underlying constraints, have been widely applied in engineering fields such as fluid dynamics, multi-body dynamics, mechanical systems and control theory. In practical physical modeling within these domains, the systems often generate high-index DAEs. Classical implicit numerical methods typic… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.12846v1-abstract-full').style.display = 'inline'; document.getElementById('2310.12846v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2310.12846v1-abstract-full" style="display: none;"> As is well known, differential algebraic equations (DAEs), which are able to describe dynamic changes and underlying constraints, have been widely applied in engineering fields such as fluid dynamics, multi-body dynamics, mechanical systems and control theory. In practical physical modeling within these domains, the systems often generate high-index DAEs. Classical implicit numerical methods typically result in varying order reduction of numerical accuracy when solving high-index systems.~Recently, the physics-informed neural network (PINN) has gained attention for solving DAE systems. However, it faces challenges like the inability to directly solve high-index systems, lower predictive accuracy, and weaker generalization capabilities. In this paper, we propose a PINN computational framework, combined Radau IIA numerical method with a neural network structure via the attention mechanisms, to directly solve high-index DAEs. Furthermore, we employ a domain decomposition strategy to enhance solution accuracy. We conduct numerical experiments with two classical high-index systems as illustrative examples, investigating how different orders of the Radau IIA method affect the accuracy of neural network solutions. The experimental results demonstrate that the PINN based on a 5th-order Radau IIA method achieves the highest level of system accuracy. Specifically, the absolute errors for all differential variables remains as low as $10^{-6}$, and the absolute errors for algebraic variables is maintained at $10^{-5}$, surpassing the results found in existing literature. Therefore, our method exhibits excellent computational accuracy and strong generalization capabilities, providing a feasible approach for the high-precision solution of larger-scale DAEs with higher indices or challenging high-dimensional partial differential algebraic equation systems. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.12846v1-abstract-full').style.display = 'none'; document.getElementById('2310.12846v1-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 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2310.12283">arXiv:2310.12283</a> <span> [<a href="https://arxiv.org/pdf/2310.12283">pdf</a>, <a href="https://arxiv.org/format/2310.12283">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"> A characterization of 4-connected graphs with no $K_{3,3}+v$-minor </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wei%2C+L">Linsong Wei</a>, <a href="/search/math?searchtype=author&query=Xu%2C+Y">Yuqi Xu</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Weihua Yang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Y">Yunxia 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="2310.12283v2-abstract-short" style="display: inline;"> Among graphs with 13 edges, there are exactly three internally 4-connected graphs which are $Oct^{+}$, cube+e and $ K_{3,3} +v$. A complete characterization of all 4-connected graphs with no $Oct^{+}$-minor is given in [John Maharry, An excluded minor theorem for the octahedron plus an edge, Journal of Graph Theory 57(2) (2008) 124-130]. Let $K_{3,3}+v$ denote the graph obtained by adding a new ve… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.12283v2-abstract-full').style.display = 'inline'; document.getElementById('2310.12283v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2310.12283v2-abstract-full" style="display: none;"> Among graphs with 13 edges, there are exactly three internally 4-connected graphs which are $Oct^{+}$, cube+e and $ K_{3,3} +v$. A complete characterization of all 4-connected graphs with no $Oct^{+}$-minor is given in [John Maharry, An excluded minor theorem for the octahedron plus an edge, Journal of Graph Theory 57(2) (2008) 124-130]. Let $K_{3,3}+v$ denote the graph obtained by adding a new vertex $v$ to $K_{3,3}$ and joining $v$ to the four vertices of a 4-cycle. In this paper, we determine all 4-connected graphs that do not contain $K_{3,3}+v$ as a minor. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.12283v2-abstract-full').style.display = 'none'; document.getElementById('2310.12283v2-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 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2023. </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">25</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2310.04659">arXiv:2310.04659</a> <span> [<a href="https://arxiv.org/pdf/2310.04659">pdf</a>, <a href="https://arxiv.org/ps/2310.04659">ps</a>, <a href="https://arxiv.org/format/2310.04659">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"> Convolution formulas for multivariate arithmetic Tutte polynomials </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ma%2C+T">Tianlong Ma</a>, <a href="/search/math?searchtype=author&query=Jin%2C+X">Xian'an Jin</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Weiling 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="2310.04659v1-abstract-short" style="display: inline;"> The multivariate arithmetic Tutte polynomial of arithmetic matroids is a generalization of the multivariate Tutte polynomial of matroids. In this note, we give the convolution formulas for the multivariate arithmetic Tutte polynomial of the product of two arithmetic matroids. In particular, the convolution formulas for the multivariate arithmetic Tutte polynomial of an arithmetic matroid are obtai… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.04659v1-abstract-full').style.display = 'inline'; document.getElementById('2310.04659v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2310.04659v1-abstract-full" style="display: none;"> The multivariate arithmetic Tutte polynomial of arithmetic matroids is a generalization of the multivariate Tutte polynomial of matroids. In this note, we give the convolution formulas for the multivariate arithmetic Tutte polynomial of the product of two arithmetic matroids. In particular, the convolution formulas for the multivariate arithmetic Tutte polynomial of an arithmetic matroid are obtained. Applying our results, several known convolution formulas including [5, Theorem 10.9 and Corollary 10.10] and [1, Theorems 1 and 4] are proved by a purely combinatorial proof. The proofs presented here are significantly shorter than the previous ones. In addition, we obtain a convolution formula for the characteristic polynomial of an arithmetic matroid. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2310.04659v1-abstract-full').style.display = 'none'; document.getElementById('2310.04659v1-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> 6 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2309.17250">arXiv:2309.17250</a> <span> [<a href="https://arxiv.org/pdf/2309.17250">pdf</a>, <a href="https://arxiv.org/ps/2309.17250">ps</a>, <a href="https://arxiv.org/format/2309.17250">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="Metric Geometry">math.MG</span> </div> </div> <p class="title is-5 mathjax"> Liouville theorems for ancient solutions of subexponential growth to the heat equation on graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Hua%2C+B">Bobo Hua</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wenhao 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="2309.17250v1-abstract-short" style="display: inline;"> Mosconi proved Liouville theorems for ancient solutions of subexponential growth to the heat equation on a manifold with Ricci curvature bounded below. We extend these results to graphs with bounded geometry: for a graph with bounded geometry, any nonnegative ancient solution of subexponential growth in space and time to the heat equation is stationary, and thus is a harmonic solution. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2309.17250v1-abstract-full" style="display: none;"> Mosconi proved Liouville theorems for ancient solutions of subexponential growth to the heat equation on a manifold with Ricci curvature bounded below. We extend these results to graphs with bounded geometry: for a graph with bounded geometry, any nonnegative ancient solution of subexponential growth in space and time to the heat equation is stationary, and thus is a harmonic solution. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.17250v1-abstract-full').style.display = 'none'; document.getElementById('2309.17250v1-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 September, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2023. </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</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2309.08183">arXiv:2309.08183</a> <span> [<a href="https://arxiv.org/pdf/2309.08183">pdf</a>, <a href="https://arxiv.org/ps/2309.08183">ps</a>, <a href="https://arxiv.org/format/2309.08183">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> </div> </div> <p class="title is-5 mathjax"> Spectral Properties and Weak Detection in Stochastic Block Models </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Han%2C+Y">Yoochan Han</a>, <a href="/search/math?searchtype=author&query=Lee%2C+J+O">Ji Oon Lee</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wooseok 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="2309.08183v1-abstract-short" style="display: inline;"> We consider the spectral properties of balanced stochastic block models of which the average degree grows slower than the number of nodes (sparse regime) or proportional to it (dense regime). For both regimes, we prove a phase transition of the extreme eigenvalues of SBM at the Kesten--Stigum threshold. We also prove the central limit theorem for the linear spectral statistics for both regimes. We… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.08183v1-abstract-full').style.display = 'inline'; document.getElementById('2309.08183v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2309.08183v1-abstract-full" style="display: none;"> We consider the spectral properties of balanced stochastic block models of which the average degree grows slower than the number of nodes (sparse regime) or proportional to it (dense regime). For both regimes, we prove a phase transition of the extreme eigenvalues of SBM at the Kesten--Stigum threshold. We also prove the central limit theorem for the linear spectral statistics for both regimes. We propose a hypothesis test for determining the presence of communities of the graph, based on the central limit theorem for the linear spectral statistics. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.08183v1-abstract-full').style.display = 'none'; document.getElementById('2309.08183v1-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 September, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2023. </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">63 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/2309.01048">arXiv:2309.01048</a> <span> [<a href="https://arxiv.org/pdf/2309.01048">pdf</a>, <a href="https://arxiv.org/ps/2309.01048">ps</a>, <a href="https://arxiv.org/format/2309.01048">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> Uniqueness of lump solutions of KP-I equation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+Y">Yong Liu</a>, <a href="/search/math?searchtype=author&query=Wei%2C+J">Jun-cheng Wei</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wen 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="2309.01048v2-abstract-short" style="display: inline;"> The KP-I equation has family of solutions which decay to zero at space infinity. One of these solutions is the classical lump solution. This is a traveling wave, and the KP-I equation in this case reduces to the Boussinesq equation. In this paper we classify the lump type solutions of the Boussinesq equation. Using a robust inverse scattering transform developed by Bilman-Miller, we show that the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.01048v2-abstract-full').style.display = 'inline'; document.getElementById('2309.01048v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2309.01048v2-abstract-full" style="display: none;"> The KP-I equation has family of solutions which decay to zero at space infinity. One of these solutions is the classical lump solution. This is a traveling wave, and the KP-I equation in this case reduces to the Boussinesq equation. In this paper we classify the lump type solutions of the Boussinesq equation. Using a robust inverse scattering transform developed by Bilman-Miller, we show that the lump type solutions are rational and their tau function has to be a polynomial of degree $k(k+1)$. In particular, this implies that the lump solution is the unique ground state of the KP-I equation (as conjectured by Klein and Saut in \cite{Klein0}). Our result generalizes a theorem by Airault-McKean-Moser on the classification of rational solutions for the KdV equation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.01048v2-abstract-full').style.display = 'none'; document.getElementById('2309.01048v2-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 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 2 September, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2308.12126">arXiv:2308.12126</a> <span> [<a href="https://arxiv.org/pdf/2308.12126">pdf</a>, <a href="https://arxiv.org/format/2308.12126">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> </div> </div> <p class="title is-5 mathjax"> An Accelerated Block Proximal Framework with Adaptive Momentum for Nonconvex and Nonsmooth Optimization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yang%2C+W">Weifeng Yang</a>, <a href="/search/math?searchtype=author&query=Min%2C+W">Wenwen Min</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="2308.12126v2-abstract-short" style="display: inline;"> We propose an accelerated block proximal linear framework with adaptive momentum (ABPL$^+$) for nonconvex and nonsmooth optimization. We analyze the potential causes of the extrapolation step failing in some algorithms, and resolve this issue by enhancing the comparison process that evaluates the trade-off between the proximal gradient step and the linear extrapolation step in our algorithm. Furth… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2308.12126v2-abstract-full').style.display = 'inline'; document.getElementById('2308.12126v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2308.12126v2-abstract-full" style="display: none;"> We propose an accelerated block proximal linear framework with adaptive momentum (ABPL$^+$) for nonconvex and nonsmooth optimization. We analyze the potential causes of the extrapolation step failing in some algorithms, and resolve this issue by enhancing the comparison process that evaluates the trade-off between the proximal gradient step and the linear extrapolation step in our algorithm. Furthermore, we extends our algorithm to any scenario involving updating block variables with positive integers, allowing each cycle to randomly shuffle the update order of the variable blocks. Additionally, under mild assumptions, we prove that ABPL$^+$ can monotonically decrease the function value without strictly restricting the extrapolation parameters and step size, demonstrates the viability and effectiveness of updating these blocks in a random order, and we also more obviously and intuitively demonstrate that the derivative set of the sequence generated by our algorithm is a critical point set. Moreover, we demonstrate the global convergence as well as the linear and sublinear convergence rates of our algorithm by utilizing the Kurdyka-Lojasiewicz (K艁) condition. To enhance the effectiveness and flexibility of our algorithm, we also expand the study to the imprecise version of our algorithm and construct an adaptive extrapolation parameter strategy, which improving its overall performance. We apply our algorithm to multiple non-negative matrix factorization with the $\ell_0$ norm, nonnegative tensor decomposition with the $\ell_0$ norm, and perform extensive numerical experiments to validate its effectiveness and efficiency. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2308.12126v2-abstract-full').style.display = 'none'; document.getElementById('2308.12126v2-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 August, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 23 August, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2308.10520">arXiv:2308.10520</a> <span> [<a href="https://arxiv.org/pdf/2308.10520">pdf</a>, <a href="https://arxiv.org/ps/2308.10520">ps</a>, <a href="https://arxiv.org/format/2308.10520">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"> Smooth Subsonic and Transonic Flows with Nonzero Angular Velocity and Vorticity to steady Euler-Poisson system in a Concentric Cylinder </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Weng%2C+S">Shangkun Weng</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wengang Yang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+N">Na 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="2308.10520v2-abstract-short" style="display: inline;"> In this paper, both smooth subsonic and transonic flows to steady Euler-Poisson system in a concentric cylinder are studied. We first establish the existence of cylindrically symmetric smooth subsonic and transonic flows to steady Euler-Poisson system in a concentric cylinder. On one hand, we investigate the structural stability of smooth cylindrically symmetric subsonic flows under three-dimensio… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2308.10520v2-abstract-full').style.display = 'inline'; document.getElementById('2308.10520v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2308.10520v2-abstract-full" style="display: none;"> In this paper, both smooth subsonic and transonic flows to steady Euler-Poisson system in a concentric cylinder are studied. We first establish the existence of cylindrically symmetric smooth subsonic and transonic flows to steady Euler-Poisson system in a concentric cylinder. On one hand, we investigate the structural stability of smooth cylindrically symmetric subsonic flows under three-dimensional perturbations on the inner and outer cylinders. On the other hand, the structural stability of smooth transonic flows under the axi-symmetric perturbations are examined. There is no any restrictions on the background subsonic and transonic solutions. A deformation-curl-Poisson decomposition to the steady Euler-Poisson system is utilized in our work to deal with the hyperbolic-elliptic mixed structure in subsonic region. It should be emphasized that there is a special structure of the steady Euler-Poisson system which yields a priori estimates and uniqueness of a second order elliptic system for the velocity potential and the electrostatic potential. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2308.10520v2-abstract-full').style.display = 'none'; document.getElementById('2308.10520v2-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, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 21 August, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2308.06991">arXiv:2308.06991</a> <span> [<a href="https://arxiv.org/pdf/2308.06991">pdf</a>, <a href="https://arxiv.org/format/2308.06991">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 convex dual problem for the rational minimax approximation and Lawson's iteration </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+L">Lei-Hong Zhang</a>, <a href="/search/math?searchtype=author&query=Yang%2C+L">Linyi Yang</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W+H">Wei Hong Yang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Y">Ya-Nan 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="2308.06991v2-abstract-short" style="display: inline;"> Computing the discrete rational minimax approximation in the complex plane is challenging. Apart from Ruttan's sufficient condition, there are few other sufficient conditions for global optimality. The state-of-the-art rational approximation algorithms, such as the adaptive Antoulas-Anderson (AAA), AAA-Lawson, and the rational Krylov fitting (RKFIT) method, perform highly efficiently, but the comp… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2308.06991v2-abstract-full').style.display = 'inline'; document.getElementById('2308.06991v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2308.06991v2-abstract-full" style="display: none;"> Computing the discrete rational minimax approximation in the complex plane is challenging. Apart from Ruttan's sufficient condition, there are few other sufficient conditions for global optimality. The state-of-the-art rational approximation algorithms, such as the adaptive Antoulas-Anderson (AAA), AAA-Lawson, and the rational Krylov fitting (RKFIT) method, perform highly efficiently, but the computed rational approximations may not be minimax solutions. In this paper, we propose a convex programming approach, the solution of which is guaranteed to be the rational minimax approximation under Ruttan's sufficient condition. Furthermore, we present a new version of Lawson's iteration for solving this convex programming problem. The computed solution can be easily verified as the rational minimax approximation. Our numerical experiments demonstrate that this updated version of Lawson's iteration generally converges monotonically with respect to the objective function of the convex optimization. It is an effective competitive approach for computing the rational minimax approximation, compared to the highly efficient AAA, AAA-Lawson, and the stabilized Sanathanan-Koerner iteration. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2308.06991v2-abstract-full').style.display = 'none'; document.getElementById('2308.06991v2-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 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 August, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2023. </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">40 pages, 11 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 41A50; 41A20; 65D15; 33F05; 90C46 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2307.10662">arXiv:2307.10662</a> <span> [<a href="https://arxiv.org/pdf/2307.10662">pdf</a>, <a href="https://arxiv.org/ps/2307.10662">ps</a>, <a href="https://arxiv.org/format/2307.10662">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</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 growth of the Green function for random walks and Poincar{茅} series </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Dussaule%2C+M">Matthieu Dussaule</a>, <a href="/search/math?searchtype=author&query=Yang%2C+W">Wenyuan Yang</a>, <a href="/search/math?searchtype=author&query=Wang%2C+L">Longmin 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="2307.10662v1-abstract-short" style="display: inline;"> Given a probability measure $渭$ on a finitely generated group $螕$, the Green function $G(x,y|r)$ encodes many properties of the random walk associated with $渭$. Finding asymptotics of $G(x,y|r)$ as $y$ goes to infinity is a common thread in probability theory and is usually referred as renewal theory in literature. Endowing $螕$ with a word distance, we denote by $H_r(n)$ the sum of the Green funct… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.10662v1-abstract-full').style.display = 'inline'; document.getElementById('2307.10662v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2307.10662v1-abstract-full" style="display: none;"> Given a probability measure $渭$ on a finitely generated group $螕$, the Green function $G(x,y|r)$ encodes many properties of the random walk associated with $渭$. Finding asymptotics of $G(x,y|r)$ as $y$ goes to infinity is a common thread in probability theory and is usually referred as renewal theory in literature. Endowing $螕$ with a word distance, we denote by $H_r(n)$ the sum of the Green function $G(e,x|r)$ along the sphere of radius $n$. This quantity appears naturally when studying asymptotic properties of branching random walks driven by $渭$ on $螕$ and the behavior of $H_r(n)$ as $n$ goes to infinity is intimately related to renewal theory. Our motivation in this paper is to construct various examples of particular behaviors for $H_r(n)$. First, our main result exhibits a class of relatively hyperbolic groups with convergent Poincar{茅} series generated by $H_r(n)$, which answers some questions raised in a previous paper of the authors. Along the way, we investigate the behavior of $H_r(n)$ for several classes of finitely generated groups, including abelian groups, certain nilpotent groups, lamplighter groups, and Cartesian products of free groups. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.10662v1-abstract-full').style.display = 'none'; document.getElementById('2307.10662v1-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 July, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2023. </p> </li> </ol> <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=Yang%2C+W&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a 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