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class="pagination-link " aria-label="Page 5" aria-current="page">5 </a> </li> <li><span class="pagination-ellipsis">…</span></li> </ul> </nav> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.16398">arXiv:2411.16398</a> <span> [<a href="https://arxiv.org/pdf/2411.16398">pdf</a>, <a href="https://arxiv.org/format/2411.16398">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"> Large Deviations of Cover Time of Tori in Dimensions $d\geq 3$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xinyi Li</a>, <a href="/search/math?searchtype=author&query=Shi%2C+J">Jialu Shi</a>, <a href="/search/math?searchtype=author&query=Xu%2C+Q">Qiheng Xu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.16398v1-abstract-short" style="display: inline;"> We consider large deviations of the cover time of the discrete torus $(\mathbb{Z}/N\mathbb{Z})^d$, $d \geq 3$ by simple random walk. We prove a lower bound on the probability that the cover time is smaller than $纬\in (0,1)$ times its expected value, with exponents matching the upper bound from [Goodman-den Hollander, Probab. Theory Related Fields (2014)] and [Comets-Gallesco-Popov-Vachkovskaia, El… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.16398v1-abstract-full').style.display = 'inline'; document.getElementById('2411.16398v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.16398v1-abstract-full" style="display: none;"> We consider large deviations of the cover time of the discrete torus $(\mathbb{Z}/N\mathbb{Z})^d$, $d \geq 3$ by simple random walk. We prove a lower bound on the probability that the cover time is smaller than $纬\in (0,1)$ times its expected value, with exponents matching the upper bound from [Goodman-den Hollander, Probab. Theory Related Fields (2014)] and [Comets-Gallesco-Popov-Vachkovskaia, Electron. J. Probab. (2013)]. Moreover, we derive sharp asymptotics for $纬\in (\frac{d+2}{2d},1)$. The strong coupling of the random walk on the torus and random interlacements developed in a recent work [Pr茅vost-Rodriguez-Sousi, arXiv:2309.03192] serves as an important ingredient in the proofs. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.16398v1-abstract-full').style.display = 'none'; document.getElementById('2411.16398v1-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> 25 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">34 pages, 2 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C81; 60F10; 60G70 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.14921">arXiv:2411.14921</a> <span> [<a href="https://arxiv.org/pdf/2411.14921">pdf</a>, <a href="https://arxiv.org/format/2411.14921">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"> A boundary Harnack principle and its application to analyticity of 3D Brownian intersection exponents </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gao%2C+Y">Yifan Gao</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xinyi Li</a>, <a href="/search/math?searchtype=author&query=Li%2C+Y">Yifan Li</a>, <a href="/search/math?searchtype=author&query=Liu%2C+R">Runsheng Liu</a>, <a href="/search/math?searchtype=author&query=Liu%2C+X">Xiangyi Liu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.14921v1-abstract-short" style="display: inline;"> We show that a domain in $\mathbb{R}^3$ with the trace of a 3D Brownian motion removed almost surely satisfies the boundary Harnack principle (BHP). Then, we use it to prove that the intersection exponents for 3D Brownian motion are analytic. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.14921v1-abstract-full" style="display: none;"> We show that a domain in $\mathbb{R}^3$ with the trace of a 3D Brownian motion removed almost surely satisfies the boundary Harnack principle (BHP). Then, we use it to prove that the intersection exponents for 3D Brownian motion are analytic. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.14921v1-abstract-full').style.display = 'none'; document.getElementById('2411.14921v1-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> 22 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">49 pages, 5 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 60J65 (Primary) 31B25; 31B05 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.14828">arXiv:2411.14828</a> <span> [<a href="https://arxiv.org/pdf/2411.14828">pdf</a>, <a href="https://arxiv.org/ps/2411.14828">ps</a>, <a href="https://arxiv.org/format/2411.14828">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"> Continuous and discrete-time accelerated methods for an inequality constrained convex optimization problem </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+J">Juan Liu</a>, <a href="/search/math?searchtype=author&query=Huang%2C+N">Nan-Jing Huang</a>, <a href="/search/math?searchtype=author&query=Long%2C+X">Xian-Jun Long</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xue-song Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.14828v1-abstract-short" style="display: inline;"> This paper is devoted to the study of acceleration methods for an inequality constrained convex optimization problem by using Lyapunov functions. We first approximate such a problem as an unconstrained optimization problem by employing the logarithmic barrier function. Using the Hamiltonian principle, we propose a continuous-time dynamical system associated with a Bregman Lagrangian for solving th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.14828v1-abstract-full').style.display = 'inline'; document.getElementById('2411.14828v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.14828v1-abstract-full" style="display: none;"> This paper is devoted to the study of acceleration methods for an inequality constrained convex optimization problem by using Lyapunov functions. We first approximate such a problem as an unconstrained optimization problem by employing the logarithmic barrier function. Using the Hamiltonian principle, we propose a continuous-time dynamical system associated with a Bregman Lagrangian for solving the unconstrained optimization problem. Under certain conditions, we demonstrate that this continuous-time dynamical system exponentially converges to the optimal solution of the inequality constrained convex optimization problem. Moreover, we derive several discrete-time algorithms from this continuous-time framework and obtain their optimal convergence rates. Finally, we present numerical experiments to validate the effectiveness of the proposed algorithms. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.14828v1-abstract-full').style.display = 'none'; document.getElementById('2411.14828v1-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> 22 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/2411.13868">arXiv:2411.13868</a> <span> [<a href="https://arxiv.org/pdf/2411.13868">pdf</a>, <a href="https://arxiv.org/format/2411.13868">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="Computation and Language">cs.CL</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Robust Detection of Watermarks for Large Language Models Under Human Edits </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xiang Li</a>, <a href="/search/math?searchtype=author&query=Ruan%2C+F">Feng Ruan</a>, <a href="/search/math?searchtype=author&query=Wang%2C+H">Huiyuan Wang</a>, <a href="/search/math?searchtype=author&query=Long%2C+Q">Qi Long</a>, <a href="/search/math?searchtype=author&query=Su%2C+W+J">Weijie J. Su</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="2411.13868v1-abstract-short" style="display: inline;"> Watermarking has offered an effective approach to distinguishing text generated by large language models (LLMs) from human-written text. However, the pervasive presence of human edits on LLM-generated text dilutes watermark signals, thereby significantly degrading detection performance of existing methods. In this paper, by modeling human edits through mixture model detection, we introduce a new m… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.13868v1-abstract-full').style.display = 'inline'; document.getElementById('2411.13868v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.13868v1-abstract-full" style="display: none;"> Watermarking has offered an effective approach to distinguishing text generated by large language models (LLMs) from human-written text. However, the pervasive presence of human edits on LLM-generated text dilutes watermark signals, thereby significantly degrading detection performance of existing methods. In this paper, by modeling human edits through mixture model detection, we introduce a new method in the form of a truncated goodness-of-fit test for detecting watermarked text under human edits, which we refer to as Tr-GoF. We prove that the Tr-GoF test achieves optimality in robust detection of the Gumbel-max watermark in a certain asymptotic regime of substantial text modifications and vanishing watermark signals. Importantly, Tr-GoF achieves this optimality \textit{adaptively} as it does not require precise knowledge of human edit levels or probabilistic specifications of the LLMs, in contrast to the optimal but impractical (Neyman--Pearson) likelihood ratio test. Moreover, we establish that the Tr-GoF test attains the highest detection efficiency rate in a certain regime of moderate text modifications. In stark contrast, we show that sum-based detection rules, as employed by existing methods, fail to achieve optimal robustness in both regimes because the additive nature of their statistics is less resilient to edit-induced noise. Finally, we demonstrate the competitive and sometimes superior empirical performance of the Tr-GoF test on both synthetic data and open-source LLMs in the OPT and LLaMA families. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.13868v1-abstract-full').style.display = 'none'; document.getElementById('2411.13868v1-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 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/2411.11737">arXiv:2411.11737</a> <span> [<a href="https://arxiv.org/pdf/2411.11737">pdf</a>, <a href="https://arxiv.org/ps/2411.11737">ps</a>, <a href="https://arxiv.org/format/2411.11737">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> </div> <p class="title is-5 mathjax"> Randomization-based Z-estimation for evaluating average and individual treatment effects </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Qu%2C+T">Tianyi Qu</a>, <a href="/search/math?searchtype=author&query=Du%2C+J">Jiangchuan Du</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xinran Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.11737v1-abstract-short" style="display: inline;"> Randomized experiments have been the gold standard for drawing causal inference. The conventional model-based approach has been one of the most popular ways for analyzing treatment effects from randomized experiments, which is often carried through inference for certain model parameters. In this paper, we provide a systematic investigation of model-based analyses for treatment effects under the ra… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.11737v1-abstract-full').style.display = 'inline'; document.getElementById('2411.11737v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.11737v1-abstract-full" style="display: none;"> Randomized experiments have been the gold standard for drawing causal inference. The conventional model-based approach has been one of the most popular ways for analyzing treatment effects from randomized experiments, which is often carried through inference for certain model parameters. In this paper, we provide a systematic investigation of model-based analyses for treatment effects under the randomization-based inference framework. This framework does not impose any distributional assumptions on the outcomes, covariates and their dependence, and utilizes only randomization as the "reasoned basis". We first derive the asymptotic theory for Z-estimation in completely randomized experiments, and propose sandwich-type conservative covariance estimation. We then apply the developed theory to analyze both average and individual treatment effects in randomized experiments. For the average treatment effect, we consider three estimation strategies: model-based, model-imputed, and model-assisted, where the first two can be sensitive to model misspecification or require specific ways for parameter estimation. The model-assisted approach is robust to arbitrary model misspecification and always provides consistent average treatment effect estimation. We propose optimal ways to conduct model-assisted estimation using generally nonlinear least squares for parameter estimation. For the individual treatment effects, we propose to directly model the relationship between individual effects and covariates, and discuss the model's identifiability, inference and interpretation allowing model misspecification. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.11737v1-abstract-full').style.display = 'none'; document.getElementById('2411.11737v1-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 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/2411.11215">arXiv:2411.11215</a> <span> [<a href="https://arxiv.org/pdf/2411.11215">pdf</a>, <a href="https://arxiv.org/ps/2411.11215">ps</a>, <a href="https://arxiv.org/format/2411.11215">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Hypergeometric $\ell$-adic sheaves for reductive groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+L">Lei Fu</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xuanyou Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.11215v1-abstract-short" style="display: inline;"> We define the hypergeometric exponential sum associated to a finite family of representations of a reductive group over a finite field. We introduce the hypergeometric $\ell$-adic sheaf to describe the behavior of the hypergeometric exponential sum. It is a perverse sheaf, and it is the counterpart in characteristic $p$ of the $A$-hypergeometric $\mathcal D$-module introduced by Kapranov. Using th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.11215v1-abstract-full').style.display = 'inline'; document.getElementById('2411.11215v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.11215v1-abstract-full" style="display: none;"> We define the hypergeometric exponential sum associated to a finite family of representations of a reductive group over a finite field. We introduce the hypergeometric $\ell$-adic sheaf to describe the behavior of the hypergeometric exponential sum. It is a perverse sheaf, and it is the counterpart in characteristic $p$ of the $A$-hypergeometric $\mathcal D$-module introduced by Kapranov. Using the theory of the Fourier transform for vector bundles over a general base developed by Wang, we are able to study the hypergeometric $\ell$-adic sheaf via the hypergeometric $\mathcal D$-module. We apply our results to the estimation of the hypergeometric exponential sum. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.11215v1-abstract-full').style.display = 'none'; document.getElementById('2411.11215v1-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> 17 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Secondary 14M27; 11L07. Primary 14F10; 14F20; Secondary 14M27; 11L07. Primary 14F10; 14F20; Secondary 14M27; 11L07 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.11115">arXiv:2411.11115</a> <span> [<a href="https://arxiv.org/pdf/2411.11115">pdf</a>, <a href="https://arxiv.org/ps/2411.11115">ps</a>, <a href="https://arxiv.org/format/2411.11115">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"> Numerical integrations of stochastic contact Hamiltonian systems via stochastic contact Hamilton-Jacobi equation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhan%2C+Q">Qingyi Zhan</a>, <a href="/search/math?searchtype=author&query=Duan%2C+J">Jinqiao Duan</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiaofan Li</a>, <a href="/search/math?searchtype=author&query=Wang%2C+L">Lijin 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="2411.11115v1-abstract-short" style="display: inline;"> Stochastic contact Hamiltonian systems are a class of important mathematical models, which can describe the dissipative properties with odd dimensions in the stochastic environment. In this article, we investigate the numerical dynamics of the stochastic contact Hamiltonian systems via structure-preserving methods. The contact structure-preserving schemes are constructed by the stochastic contact… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.11115v1-abstract-full').style.display = 'inline'; document.getElementById('2411.11115v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.11115v1-abstract-full" style="display: none;"> Stochastic contact Hamiltonian systems are a class of important mathematical models, which can describe the dissipative properties with odd dimensions in the stochastic environment. In this article, we investigate the numerical dynamics of the stochastic contact Hamiltonian systems via structure-preserving methods. The contact structure-preserving schemes are constructed by the stochastic contact Hamilton-Jacobi equation. A general numerical approximation method of the stochastic contact Hamilton-Jacobi equation is devised, and the convergent order theorem is provided, too. Numerical tests are shown to confirm the theoretical results and the usability of proposed approach. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.11115v1-abstract-full').style.display = 'none'; document.getElementById('2411.11115v1-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> 17 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">27 pages, 11 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37C50; 65C30; 65P20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.08311">arXiv:2411.08311</a> <span> [<a href="https://arxiv.org/pdf/2411.08311">pdf</a>, <a href="https://arxiv.org/ps/2411.08311">ps</a>, <a href="https://arxiv.org/format/2411.08311">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link 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="Mathematical Physics">math-ph</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"> A generalization of the martingale property of entropy production in stochastic systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xiangting Li</a>, <a href="/search/math?searchtype=author&query=Chou%2C+T">Tom Chou</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="2411.08311v1-abstract-short" style="display: inline;"> By decoupling forward and backward stochastic trajectories, we develop a family of martingales and work theorems for the same stochastic process. We achieve this by introducing an alternative work theorem derivation that uses tools from stochastic calculus instead of path integrals. Our derivation applies to both overdamped and underdamped Langevin dynamics and generalizes work theorems so that th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.08311v1-abstract-full').style.display = 'inline'; document.getElementById('2411.08311v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.08311v1-abstract-full" style="display: none;"> By decoupling forward and backward stochastic trajectories, we develop a family of martingales and work theorems for the same stochastic process. We achieve this by introducing an alternative work theorem derivation that uses tools from stochastic calculus instead of path integrals. Our derivation applies to both overdamped and underdamped Langevin dynamics and generalizes work theorems so that they connect new quantities in stochastic processes, potentially revealing new applications in dissipative systems. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.08311v1-abstract-full').style.display = 'none'; document.getElementById('2411.08311v1-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 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 60H30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.07271">arXiv:2411.07271</a> <span> [<a href="https://arxiv.org/pdf/2411.07271">pdf</a>, <a href="https://arxiv.org/format/2411.07271">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="Artificial Intelligence">cs.AI</span> <span class="tag is-small is-grey 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="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Multi-hop Upstream Preemptive Traffic Signal Control with Deep Reinforcement Learning </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xiaocan Li</a>, <a href="/search/math?searchtype=author&query=Wang%2C+X">Xiaoyu Wang</a>, <a href="/search/math?searchtype=author&query=Smirnov%2C+I">Ilia Smirnov</a>, <a href="/search/math?searchtype=author&query=Sanner%2C+S">Scott Sanner</a>, <a href="/search/math?searchtype=author&query=Abdulhai%2C+B">Baher Abdulhai</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="2411.07271v1-abstract-short" style="display: inline;"> Traffic signal control is crucial for managing congestion in urban networks. Existing myopic pressure-based control methods focus only on immediate upstream links, leading to suboptimal green time allocation and increased network delays. Effective signal control, however, inherently requires a broader spatial scope, as traffic conditions further upstream can significantly impact traffic at the cur… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.07271v1-abstract-full').style.display = 'inline'; document.getElementById('2411.07271v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.07271v1-abstract-full" style="display: none;"> Traffic signal control is crucial for managing congestion in urban networks. Existing myopic pressure-based control methods focus only on immediate upstream links, leading to suboptimal green time allocation and increased network delays. Effective signal control, however, inherently requires a broader spatial scope, as traffic conditions further upstream can significantly impact traffic at the current location. This paper introduces a novel concept based on the Markov chain theory, namely multi-hop upstream pressure, that generalizes the conventional pressure to account for traffic conditions beyond the immediate upstream links. This farsighted and compact metric informs the deep reinforcement learning agent to preemptively clear the present queues, guiding the agent to optimize signal timings with a broader spatial awareness. Simulations on synthetic and realistic (Toronto) scenarios demonstrate controllers utilizing multi-hop upstream pressure significantly reduce overall network delay by prioritizing traffic movements based on a broader understanding of upstream congestion. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.07271v1-abstract-full').style.display = 'none'; document.getElementById('2411.07271v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <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">5 tables, 12 figures. arXiv admin note: text overlap with arXiv:2409.00753</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.06656">arXiv:2411.06656</a> <span> [<a href="https://arxiv.org/pdf/2411.06656">pdf</a>, <a href="https://arxiv.org/ps/2411.06656">ps</a>, <a href="https://arxiv.org/format/2411.06656">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> On moments of the error term of the multivariable k-th divisor functions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Guo%2C+Z">Zhen Guo</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xin Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.06656v1-abstract-short" style="display: inline;"> Suppose $k\geqslant3$ is an integer. Let $蟿_k(n)$ be the number of ways $n$ can be written as a product of $k$ fixed factors. For any fixed integer $r\geqslant2$, we have the asymptotic formula \begin{equation*} \sum_{n_1,\cdots,n_r\leqslant x}蟿_k(n_1 \cdots n_r)=x^r\sum_{\ell=0}^{r(k-1)}d_{r,k,\ell}(\log x)^{\ell}+O(x^{r-1+伪_k+\varepsilon}), \end{equation*} where $d_{r,k,\ell}$ and $0<伪_k<1$ ar… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.06656v1-abstract-full').style.display = 'inline'; document.getElementById('2411.06656v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.06656v1-abstract-full" style="display: none;"> Suppose $k\geqslant3$ is an integer. Let $蟿_k(n)$ be the number of ways $n$ can be written as a product of $k$ fixed factors. For any fixed integer $r\geqslant2$, we have the asymptotic formula \begin{equation*} \sum_{n_1,\cdots,n_r\leqslant x}蟿_k(n_1 \cdots n_r)=x^r\sum_{\ell=0}^{r(k-1)}d_{r,k,\ell}(\log x)^{\ell}+O(x^{r-1+伪_k+\varepsilon}), \end{equation*} where $d_{r,k,\ell}$ and $0<伪_k<1$ are computable constants. In this paper we study the mean square of $螖_{r,k}(x)$ and give upper bounds for $k\geqslant4$ and an asymptotic formula for the mean square of $螖_{r,3}(x)$. We also get an upper bound for the third power moment of $螖_{r,3}(x)$. Moreover, we study the first power moment of $螖_{r,3}(x)$ and then give a result for the sign changes of it. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.06656v1-abstract-full').style.display = 'none'; document.getElementById('2411.06656v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">22 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.06494">arXiv:2411.06494</a> <span> [<a href="https://arxiv.org/pdf/2411.06494">pdf</a>, <a href="https://arxiv.org/ps/2411.06494">ps</a>, <a href="https://arxiv.org/format/2411.06494">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"> On hydrostatic limit of Beris-Edwards system in a thin strip </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=De+Anna%2C+F">Francesco De Anna</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xingyu Li</a>, <a href="/search/math?searchtype=author&query=Paicu%2C+M">Marius Paicu</a>, <a href="/search/math?searchtype=author&query=Zarnescu%2C+A">Arghir Zarnescu</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="2411.06494v2-abstract-short" style="display: inline;"> In this paper we consider the 3D co-rotational Beris-Edwards system modeling the hydrodynamic motion of nematic liquid crystals in a thin strip. The system contains the incompressible Navier-Stokes, coupled with a parabolic system for matrix-valued functions, the $Q$-tensors. We show that under a suitable scaling, corresponding, in the Navier-Stokes part, to the hydrostatic scaling, one obtains… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.06494v2-abstract-full').style.display = 'inline'; document.getElementById('2411.06494v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.06494v2-abstract-full" style="display: none;"> In this paper we consider the 3D co-rotational Beris-Edwards system modeling the hydrodynamic motion of nematic liquid crystals in a thin strip. The system contains the incompressible Navier-Stokes, coupled with a parabolic system for matrix-valued functions, the $Q$-tensors. We show that under a suitable scaling, corresponding, in the Navier-Stokes part, to the hydrostatic scaling, one obtains in the limit a partly decoupled system. For the fluid part we obtain the Prandtl system while for the $Q$-tensors we obtain a non-standard system, involving fluids components and a non-standard combination of partly dissipative equations and algebraic constraints. We prove the convergence of the rescaled system and the well-posedness of the limit in Sobolev spaces. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.06494v2-abstract-full').style.display = 'none'; document.getElementById('2411.06494v2-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, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.03972">arXiv:2411.03972</a> <span> [<a href="https://arxiv.org/pdf/2411.03972">pdf</a>, <a href="https://arxiv.org/format/2411.03972">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="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Toward end-to-end quantum simulation for protein dynamics </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+Z">Zhenning Liu</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiantao Li</a>, <a href="/search/math?searchtype=author&query=Wang%2C+C">Chunhao Wang</a>, <a href="/search/math?searchtype=author&query=Liu%2C+J">Jin-Peng Liu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.03972v1-abstract-short" style="display: inline;"> Modeling and simulating the protein folding process overall remains a grand challenge in computational biology. We systematically investigate end-to-end quantum algorithms for simulating various protein dynamics with effects, such as mechanical forces or stochastic noises. We offer efficient quantum simulation algorithms to produce quantum encoding of the final states, history states, or density m… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.03972v1-abstract-full').style.display = 'inline'; document.getElementById('2411.03972v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.03972v1-abstract-full" style="display: none;"> Modeling and simulating the protein folding process overall remains a grand challenge in computational biology. We systematically investigate end-to-end quantum algorithms for simulating various protein dynamics with effects, such as mechanical forces or stochastic noises. We offer efficient quantum simulation algorithms to produce quantum encoding of the final states, history states, or density matrices of inhomogeneous or stochastic harmonic oscillator models. For the read-in setting, we design (i) efficient quantum algorithms for initial state preparation, utilizing counter-based random number generator and rejection sampling, and (ii) depth-efficient approaches for molecular structure loading. Both are particularly important in handling large protein molecules. For the read-out setting, our algorithms estimate various classical observables, such as energy, low vibration modes, density of states, correlation of displacement, and optimal control of molecular dynamics. We also show classical numerical experiments focused on estimating the density of states and applying the optimal control to facilitate conformation changes to verify our arguments on potential quantum speedups. Overall, our study demonstrates that the quantum simulation of protein dynamics can be a solid end-to-end application in the era of early or fully fault-tolerant quantum computing. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.03972v1-abstract-full').style.display = 'none'; document.getElementById('2411.03972v1-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 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">61 pages, 10 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/2411.03599">arXiv:2411.03599</a> <span> [<a href="https://arxiv.org/pdf/2411.03599">pdf</a>, <a href="https://arxiv.org/format/2411.03599">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="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Structure-preserving quantum algorithms for linear and nonlinear Hamiltonian systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wu%2C+H">Hsuan-Cheng Wu</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiantao Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.03599v1-abstract-short" style="display: inline;"> Hamiltonian systems of ordinary and partial differential equations are fundamental across modern science and engineering, appearing in models that span virtually all physical scales. A critical property for the robustness and stability of computational methods in such systems is the symplectic structure, which preserves geometric properties like phase-space volume over time and energy conservation… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.03599v1-abstract-full').style.display = 'inline'; document.getElementById('2411.03599v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.03599v1-abstract-full" style="display: none;"> Hamiltonian systems of ordinary and partial differential equations are fundamental across modern science and engineering, appearing in models that span virtually all physical scales. A critical property for the robustness and stability of computational methods in such systems is the symplectic structure, which preserves geometric properties like phase-space volume over time and energy conservation over an extended period. In this paper, we present quantum algorithms that incorporate symplectic integrators, ensuring the preservation of this key structure. We demonstrate how these algorithms maintain the symplectic properties for both linear and nonlinear Hamiltonian systems. Additionally, we provide a comprehensive theoretical analysis of the computational complexity, showing that our approach offers both accuracy and improved efficiency over classical algorithms. These results highlight the potential application of quantum algorithms for solving large-scale Hamiltonian systems while preserving essential physical properties. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.03599v1-abstract-full').style.display = 'none'; document.getElementById('2411.03599v1-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> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.01911">arXiv:2411.01911</a> <span> [<a href="https://arxiv.org/pdf/2411.01911">pdf</a>, <a href="https://arxiv.org/ps/2411.01911">ps</a>, <a href="https://arxiv.org/format/2411.01911">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> </div> </div> <p class="title is-5 mathjax"> Contraction property on complex hyperbolic ball </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xiaoshan Li</a>, <a href="/search/math?searchtype=author&query=Su%2C+G">Guicong Su</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="2411.01911v1-abstract-short" style="display: inline;"> We prove two isoperimetric inequalities on complex hyperbolic ball. As an application, we prove a contraction property for the holomorphic functions in Hardy and weighted Bergman spaces on the unit ball, thus extending the results of Kulikov (GAFA(2022)) to higher dimensional space. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.01911v1-abstract-full" style="display: none;"> We prove two isoperimetric inequalities on complex hyperbolic ball. As an application, we prove a contraction property for the holomorphic functions in Hardy and weighted Bergman spaces on the unit ball, thus extending the results of Kulikov (GAFA(2022)) to higher dimensional space. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.01911v1-abstract-full').style.display = 'none'; document.getElementById('2411.01911v1-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 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/2411.01509">arXiv:2411.01509</a> <span> [<a href="https://arxiv.org/pdf/2411.01509">pdf</a>, <a href="https://arxiv.org/ps/2411.01509">ps</a>, <a href="https://arxiv.org/format/2411.01509">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> </div> </div> <p class="title is-5 mathjax"> On existence questions for the functional equations $f^8+g^8+h^8=1$ and $f^6+g^6+h^6=1$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xiao-Min Li</a>, <a href="/search/math?searchtype=author&query=Yi%2C+H">Hong-Xun Yi</a>, <a href="/search/math?searchtype=author&query=Korhonen%2C+R">Risto Korhonen</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="2411.01509v3-abstract-short" style="display: inline;"> In 1985, W.K.Hayman (Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber, 1984(1985), 1-13.) proved that there do not exist non-constant meromorphic functions $f,$ $g$ and $h$ satisfying the functional equation $f^n+g^n+h^n=1$ for $n\geq 9.$ We prove that there do not exist non-constant meromorphic solutions $f,$ $g,$ $h$ satisfying the functional equation $f^8+g^8+h^8=1.$ In 1971, N. Toda (T么hoku Mat… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.01509v3-abstract-full').style.display = 'inline'; document.getElementById('2411.01509v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.01509v3-abstract-full" style="display: none;"> In 1985, W.K.Hayman (Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber, 1984(1985), 1-13.) proved that there do not exist non-constant meromorphic functions $f,$ $g$ and $h$ satisfying the functional equation $f^n+g^n+h^n=1$ for $n\geq 9.$ We prove that there do not exist non-constant meromorphic solutions $f,$ $g,$ $h$ satisfying the functional equation $f^8+g^8+h^8=1.$ In 1971, N. Toda (T么hoku Math. J. 23(1971), no. 2, 289-299.) proved that there do not exist non-constant entire functions $f,$ $g,$ $h$ satisfying $f^n+g^n+h^n=1$ for $n\geq 7.$ We prove that there do not exist non-constant entire functions $f,$ $g,$ $h$ satisfying the functional equation $f^6+g^6+h^6=1.$ Our results answer questions of G. G. Gundersen. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.01509v3-abstract-full').style.display = 'none'; document.getElementById('2411.01509v3-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 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 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">40 pages. arXiv admin note: text overlap with arXiv:math/9310226 by other authors</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 30D30; 30D35 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.20369">arXiv:2410.20369</a> <span> [<a href="https://arxiv.org/pdf/2410.20369">pdf</a>, <a href="https://arxiv.org/ps/2410.20369">ps</a>, <a href="https://arxiv.org/format/2410.20369">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"> Langevin deformation for R茅nyi entropy on Wasserstein space over Riemannian manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lei%2C+R">Rong Lei</a>, <a href="/search/math?searchtype=author&query=Li%2C+S">Songzi Li</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiang-Dong Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.20369v1-abstract-short" style="display: inline;"> We introduce the Langevin deformation for the R茅nyi entropy on the $L^2$-Wasserstein space over $\mathbb{R}^n$ or a Riemannian manifold, which interpolates between the porous medium equation and the Benamou-Brenier geodesic flow on the $L^2$-Wasserstein space and can be regarded as the compressible Euler equations for isentropic gas with damping. We prove the $W$-entropy-information formulae and t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.20369v1-abstract-full').style.display = 'inline'; document.getElementById('2410.20369v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.20369v1-abstract-full" style="display: none;"> We introduce the Langevin deformation for the R茅nyi entropy on the $L^2$-Wasserstein space over $\mathbb{R}^n$ or a Riemannian manifold, which interpolates between the porous medium equation and the Benamou-Brenier geodesic flow on the $L^2$-Wasserstein space and can be regarded as the compressible Euler equations for isentropic gas with damping. We prove the $W$-entropy-information formulae and the the rigidity theorems for the Langevin deformation for the R茅nyi entropy on the Wasserstein space over complete Riemannian manifolds with non-negative Ricci curvature or CD$(0, m)$-condition. Moreover, we prove the monotonicity of the Hamiltonian and the convexity of the Lagrangian along the Langevin deformation of flows. Finally, we prove the convergence of the Langevin deformation for the R茅nyi entropy as $c\rightarrow 0$ and $c\rightarrow \infty$ respectively. Our results are new even in the case of Euclidean spaces and compact or complete Riemannian manifolds with non-negative Ricci curvature. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.20369v1-abstract-full').style.display = 'none'; document.getElementById('2410.20369v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.19669">arXiv:2410.19669</a> <span> [<a href="https://arxiv.org/pdf/2410.19669">pdf</a>, <a href="https://arxiv.org/ps/2410.19669">ps</a>, <a href="https://arxiv.org/format/2410.19669">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"> Three types of the minimal excludant size of an overpartition </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=He%2C+T+Y">Thomas Y. He</a>, <a href="/search/math?searchtype=author&query=Huang%2C+C+S">C. S. Huang</a>, <a href="/search/math?searchtype=author&query=Li%2C+H+X">H. X. Li</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+X">X. 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="2410.19669v2-abstract-short" style="display: inline;"> Recently, Andrews and Newman studied the minimal excludant of a partition, which is defined as the smallest positive integer that is not a part of a partition. In this article, we consider the minimal excludant size of an overpartition, which is an overpartition analogue of the minimal excludant of a partition. We define three types of overpartition related to the minimal excludant size. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.19669v2-abstract-full" style="display: none;"> Recently, Andrews and Newman studied the minimal excludant of a partition, which is defined as the smallest positive integer that is not a part of a partition. In this article, we consider the minimal excludant size of an overpartition, which is an overpartition analogue of the minimal excludant of a partition. We define three types of overpartition related to the minimal excludant size. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.19669v2-abstract-full').style.display = 'none'; document.getElementById('2410.19669v2-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">v1</span> submitted 25 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/2410.15506">arXiv:2410.15506</a> <span> [<a href="https://arxiv.org/pdf/2410.15506">pdf</a>, <a href="https://arxiv.org/ps/2410.15506">ps</a>, <a href="https://arxiv.org/format/2410.15506">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Improved Explicit Near-Optimal Codes in the High-Noise Regimes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xin Li</a>, <a href="/search/math?searchtype=author&query=Mao%2C+S">Songtao Mao</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="2410.15506v2-abstract-short" style="display: inline;"> We study uniquely decodable codes and list decodable codes in the high-noise regime, specifically codes that are uniquely decodable from $\frac{1-\varepsilon}{2}$ fraction of errors and list decodable from $1-\varepsilon$ fraction of errors. We present several improved explicit constructions that achieve near-optimal rates, as well as efficient or even linear-time decoding algorithms. Our contribu… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.15506v2-abstract-full').style.display = 'inline'; document.getElementById('2410.15506v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.15506v2-abstract-full" style="display: none;"> We study uniquely decodable codes and list decodable codes in the high-noise regime, specifically codes that are uniquely decodable from $\frac{1-\varepsilon}{2}$ fraction of errors and list decodable from $1-\varepsilon$ fraction of errors. We present several improved explicit constructions that achieve near-optimal rates, as well as efficient or even linear-time decoding algorithms. Our contributions are as follows. 1. Explicit Near-Optimal Linear Time Uniquely Decodable Codes: We construct a family of explicit $\mathbb{F}_2$-linear codes with rate $惟(\varepsilon)$ and alphabet size $2^{\mathrm{poly} \log(1/\varepsilon)}$, that are capable of correcting $e$ errors and $s$ erasures whenever $2e + s < (1 - \varepsilon)n$ in linear-time. 2. Explicit Near-Optimal List Decodable Codes: We construct a family of explicit list decodable codes with rate $惟(\varepsilon)$ and alphabet size $2^{\mathrm{poly} \log(1/\varepsilon)}$, that are capable of list decoding from $1-\varepsilon$ fraction of errors with a list size $L = \exp\exp\exp(\log^{\ast}n)$ in polynomial time. 3. List Decodable Code with Near-Optimal List Size: We construct a family of explicit list decodable codes with an optimal list size of $O(1/\varepsilon)$, albeit with a suboptimal rate of $O(\varepsilon^2)$, capable of list decoding from $1-\varepsilon$ fraction of errors in polynomial time. Furthermore, we introduce a new combinatorial object called multi-set disperser, and use it to give a family of list decodable codes with near-optimal rate $\frac{\varepsilon}{\log^2(1/\varepsilon)}$ and list size $\frac{\log^2(1/\varepsilon)}{\varepsilon}$, that can be constructed in probabilistic polynomial time and decoded in deterministic polynomial time. We also introduce new decoding algorithms that may prove valuable for other graph-based codes. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.15506v2-abstract-full').style.display = 'none'; document.getElementById('2410.15506v2-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 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 20 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">28 pages. To appear in SODA 2025</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.15358">arXiv:2410.15358</a> <span> [<a href="https://arxiv.org/pdf/2410.15358">pdf</a>, <a href="https://arxiv.org/ps/2410.15358">ps</a>, <a href="https://arxiv.org/format/2410.15358">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="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> A New Adaptive Balanced Augmented Lagrangian Method with Application to ISAC Beamforming Design </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wu%2C+J">Jiageng Wu</a>, <a href="/search/math?searchtype=author&query=Jiang%2C+B">Bo Jiang</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xinxin Li</a>, <a href="/search/math?searchtype=author&query=Liu%2C+Y">Ya-Feng Liu</a>, <a href="/search/math?searchtype=author&query=Yuan%2C+J">Jianhua Yuan</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="2410.15358v1-abstract-short" style="display: inline;"> In this paper, we consider a class of convex programming problems with linear equality constraints, which finds broad applications in machine learning and signal processing. We propose a new adaptive balanced augmented Lagrangian (ABAL) method for solving these problems. The proposed ABAL method adaptively selects the stepsize parameter and enjoys a low per-iteration complexity, involving only the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.15358v1-abstract-full').style.display = 'inline'; document.getElementById('2410.15358v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.15358v1-abstract-full" style="display: none;"> In this paper, we consider a class of convex programming problems with linear equality constraints, which finds broad applications in machine learning and signal processing. We propose a new adaptive balanced augmented Lagrangian (ABAL) method for solving these problems. The proposed ABAL method adaptively selects the stepsize parameter and enjoys a low per-iteration complexity, involving only the computation of a proximal mapping of the objective function and the solution of a linear equation. These features make the proposed method well-suited to large-scale problems. We then custom-apply the ABAL method to solve the ISAC beamforming design problem, which is formulated as a nonlinear semidefinite program in a previous work. This customized application requires careful exploitation of the problem's special structure such as the property that all of its signal-to-interference-and-noise-ratio (SINR) constraints hold with equality at the solution and an efficient computation of the proximal mapping of the objective function. Simulation results demonstrate the efficiency of the proposed ABAL method. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.15358v1-abstract-full').style.display = 'none'; document.getElementById('2410.15358v1-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 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">7 pages, 1 table</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.14899">arXiv:2410.14899</a> <span> [<a href="https://arxiv.org/pdf/2410.14899">pdf</a>, <a href="https://arxiv.org/format/2410.14899">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"> Out-of-distribution Robust Optimization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Cai%2C+Z">Zhongze Cai</a>, <a href="/search/math?searchtype=author&query=Jiang%2C+H">Hansheng Jiang</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiaocheng Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.14899v2-abstract-short" style="display: inline;"> In this paper, we consider the contextual robust optimization problem under an out-of-distribution setting. The contextual robust optimization problem considers a risk-sensitive objective function for an optimization problem with the presence of a context vector (also known as covariates or side information) capturing related information. While the existing works mainly consider the in-distributio… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.14899v2-abstract-full').style.display = 'inline'; document.getElementById('2410.14899v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.14899v2-abstract-full" style="display: none;"> In this paper, we consider the contextual robust optimization problem under an out-of-distribution setting. The contextual robust optimization problem considers a risk-sensitive objective function for an optimization problem with the presence of a context vector (also known as covariates or side information) capturing related information. While the existing works mainly consider the in-distribution setting, and the resultant robustness achieved is in an out-of-sample sense, our paper studies an out-of-distribution setting where there can be a difference between the test environment and the training environment where the data are collected. We propose methods that handle this out-of-distribution setting, and the key relies on a density ratio estimation for the distribution shift. We show that additional structures such as covariate shift and label shift are not only helpful in defending distribution shift but also necessary in avoiding non-trivial solutions compared to other principled methods such as distributionally robust optimization. We also illustrate how the covariates can be useful in this procedure. Numerical experiments generate more intuitions and demonstrate that the proposed methods can help avoid over-conservative solutions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.14899v2-abstract-full').style.display = 'none'; document.getElementById('2410.14899v2-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> 22 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 18 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/2410.11349">arXiv:2410.11349</a> <span> [<a href="https://arxiv.org/pdf/2410.11349">pdf</a>, <a href="https://arxiv.org/ps/2410.11349">ps</a>, <a href="https://arxiv.org/format/2410.11349">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"> Inequalities for independent random vectors under sublinear expectations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xiaojuan Li</a>, <a href="/search/math?searchtype=author&query=Hu%2C+M">Mingshang Hu</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="2410.11349v1-abstract-short" style="display: inline;"> In this paper, by using the representation theorem for sublinear expectations, we give a simple proof to obtain two inequalities about the sample mean for independent random vectors under sublinear expectations. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.11349v1-abstract-full" style="display: none;"> In this paper, by using the representation theorem for sublinear expectations, we give a simple proof to obtain two inequalities about the sample mean for independent random vectors under sublinear expectations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.11349v1-abstract-full').style.display = 'none'; document.getElementById('2410.11349v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">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">6 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 60E05; 60E15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.11279">arXiv:2410.11279</a> <span> [<a href="https://arxiv.org/pdf/2410.11279">pdf</a>, <a href="https://arxiv.org/format/2410.11279">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="Artificial Intelligence">cs.AI</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"> Advancing the Understanding of Fixed Point Iterations in Deep Neural Networks: A Detailed Analytical Study </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ke%2C+Y">Yekun Ke</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiaoyu Li</a>, <a href="/search/math?searchtype=author&query=Liang%2C+Y">Yingyu Liang</a>, <a href="/search/math?searchtype=author&query=Shi%2C+Z">Zhenmei Shi</a>, <a href="/search/math?searchtype=author&query=Song%2C+Z">Zhao Song</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="2410.11279v1-abstract-short" style="display: inline;"> Recent empirical studies have identified fixed point iteration phenomena in deep neural networks, where the hidden state tends to stabilize after several layers, showing minimal change in subsequent layers. This observation has spurred the development of practical methodologies, such as accelerating inference by bypassing certain layers once the hidden state stabilizes, selectively fine-tuning lay… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.11279v1-abstract-full').style.display = 'inline'; document.getElementById('2410.11279v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.11279v1-abstract-full" style="display: none;"> Recent empirical studies have identified fixed point iteration phenomena in deep neural networks, where the hidden state tends to stabilize after several layers, showing minimal change in subsequent layers. This observation has spurred the development of practical methodologies, such as accelerating inference by bypassing certain layers once the hidden state stabilizes, selectively fine-tuning layers to modify the iteration process, and implementing loops of specific layers to maintain fixed point iterations. Despite these advancements, the understanding of fixed point iterations remains superficial, particularly in high-dimensional spaces, due to the inadequacy of current analytical tools. In this study, we conduct a detailed analysis of fixed point iterations in a vector-valued function modeled by neural networks. We establish a sufficient condition for the existence of multiple fixed points of looped neural networks based on varying input regions. Additionally, we expand our examination to include a robust version of fixed point iterations. To demonstrate the effectiveness and insights provided by our approach, we provide case studies that looped neural networks may exist $2^d$ number of robust fixed points under exponentiation or polynomial activation functions, where $d$ is the feature dimension. Furthermore, our preliminary empirical results support our theoretical findings. Our methodology enriches the toolkit available for analyzing fixed point iterations of deep neural networks and may enhance our comprehension of neural network mechanisms. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.11279v1-abstract-full').style.display = 'none'; document.getElementById('2410.11279v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">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/2410.09990">arXiv:2410.09990</a> <span> [<a href="https://arxiv.org/pdf/2410.09990">pdf</a>, <a href="https://arxiv.org/format/2410.09990">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"> Global convergence of gradient descent for phase retrieval </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fougereux%2C+T">Th茅odore Fougereux</a>, <a href="/search/math?searchtype=author&query=Josz%2C+C">C茅dric Josz</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiaopeng Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.09990v2-abstract-short" style="display: inline;"> We propose a tensor-based criterion for benign landscape in phase retrieval and establish boundedness of gradient trajectories. This implies that gradient descent will converge to a global minimum for almost every initial point. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.09990v2-abstract-full" style="display: none;"> We propose a tensor-based criterion for benign landscape in phase retrieval and establish boundedness of gradient trajectories. This implies that gradient descent will converge to a global minimum for almost every initial point. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.09990v2-abstract-full').style.display = 'none'; document.getElementById('2410.09990v2-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 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 13 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">MSC Class:</span> 90C26 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.09482">arXiv:2410.09482</a> <span> [<a href="https://arxiv.org/pdf/2410.09482">pdf</a>, <a href="https://arxiv.org/ps/2410.09482">ps</a>, <a href="https://arxiv.org/format/2410.09482">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"> A quantitative symmetry result for $p$-Laplace equations with discontinuous nonlinearities </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ciraolo%2C+G">Giulio Ciraolo</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiaoliang Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.09482v1-abstract-short" style="display: inline;"> In this paper, we study positive solutions $u$ of the homogeneous Dirichlet problem for the $p$-Laplace equation $-螖_p \,u=f(u)$ in a bounded domain $惟\subset\mathbb{R}^N$, where $N\ge 2$, $1<p<+\infty$ and $f$ is a discontinuous function. We address the quantitative stability of a Gidas-Ni-Nirenberg type symmetry result for $u$, which was established by Lions and Serra when $惟$ is a ball. By expl… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.09482v1-abstract-full').style.display = 'inline'; document.getElementById('2410.09482v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.09482v1-abstract-full" style="display: none;"> In this paper, we study positive solutions $u$ of the homogeneous Dirichlet problem for the $p$-Laplace equation $-螖_p \,u=f(u)$ in a bounded domain $惟\subset\mathbb{R}^N$, where $N\ge 2$, $1<p<+\infty$ and $f$ is a discontinuous function. We address the quantitative stability of a Gidas-Ni-Nirenberg type symmetry result for $u$, which was established by Lions and Serra when $惟$ is a ball. By exploiting a quantitative version of the P贸lya-Szeg枚 principle, we prove that the deviation of $u$ from its Schwarz symmetrization can be estimated in terms of the isoperimetric deficit of $惟$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.09482v1-abstract-full').style.display = 'none'; document.getElementById('2410.09482v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.07721">arXiv:2410.07721</a> <span> [<a href="https://arxiv.org/pdf/2410.07721">pdf</a>, <a href="https://arxiv.org/ps/2410.07721">ps</a>, <a href="https://arxiv.org/format/2410.07721">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"> The maximum spectral radius of $胃_{1,3,3}$-free graphs with given size </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gao%2C+J">Jing Gao</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xueliang Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.07721v1-abstract-short" style="display: inline;"> A graph $G$ is said to be $F$-free if it does not contain $F$ as a subgraph. A theta graph, say $胃_{l_1,l_2,l_3}$, is the graph obtained by connecting two distinct vertices with three internally disjoint paths of length $l_1, l_2, l_3$, where $l_1\leq l_2\leq l_3$ and $l_2\geq2$. Recently, Li, Zhao and Zou [arXiv:2409.15918v1] characterized the $胃_{1,p,q}$-free graph of size $m$ having the largest… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.07721v1-abstract-full').style.display = 'inline'; document.getElementById('2410.07721v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.07721v1-abstract-full" style="display: none;"> A graph $G$ is said to be $F$-free if it does not contain $F$ as a subgraph. A theta graph, say $胃_{l_1,l_2,l_3}$, is the graph obtained by connecting two distinct vertices with three internally disjoint paths of length $l_1, l_2, l_3$, where $l_1\leq l_2\leq l_3$ and $l_2\geq2$. Recently, Li, Zhao and Zou [arXiv:2409.15918v1] characterized the $胃_{1,p,q}$-free graph of size $m$ having the largest spectral radius, where $q\geq p\geq3$ and $p+q\geq2k+1\geq7$, and proposed a problem on characterizing the graphs with the maximum spectral radius among $胃_{1,3,3}$-free graphs. In this paper, we consider this problem and determine the maximum spectral radius of $胃_{1,3,3}$-free graphs with size $m$ and characterize the extremal graph. Up to now, all the graphs in $\mathcal{G}(m,胃_{1,p,q})$ which have the largest spectral radius have been determined, where $q\geq p\geq 2$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.07721v1-abstract-full').style.display = 'none'; document.getElementById('2410.07721v1-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 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">14 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C35; 05C50 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.07454">arXiv:2410.07454</a> <span> [<a href="https://arxiv.org/pdf/2410.07454">pdf</a>, <a href="https://arxiv.org/format/2410.07454">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</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"> Representation-Enhanced Neural Knowledge Integration with Application to Large-Scale Medical Ontology Learning </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+S">Suqi Liu</a>, <a href="/search/math?searchtype=author&query=Cai%2C+T">Tianxi Cai</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiaoou Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.07454v1-abstract-short" style="display: inline;"> A large-scale knowledge graph enhances reproducibility in biomedical data discovery by providing a standardized, integrated framework that ensures consistent interpretation across diverse datasets. It improves generalizability by connecting data from various sources, enabling broader applicability of findings across different populations and conditions. Generating reliable knowledge graph, leverag… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.07454v1-abstract-full').style.display = 'inline'; document.getElementById('2410.07454v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.07454v1-abstract-full" style="display: none;"> A large-scale knowledge graph enhances reproducibility in biomedical data discovery by providing a standardized, integrated framework that ensures consistent interpretation across diverse datasets. It improves generalizability by connecting data from various sources, enabling broader applicability of findings across different populations and conditions. Generating reliable knowledge graph, leveraging multi-source information from existing literature, however, is challenging especially with a large number of node sizes and heterogeneous relations. In this paper, we propose a general theoretically guaranteed statistical framework, called RENKI, to enable simultaneous learning of multiple relation types. RENKI generalizes various network models widely used in statistics and computer science. The proposed framework incorporates representation learning output into initial entity embedding of a neural network that approximates the score function for the knowledge graph and continuously trains the model to fit observed facts. We prove nonasymptotic bounds for in-sample and out-of-sample weighted MSEs in relation to the pseudo-dimension of the knowledge graph function class. Additionally, we provide pseudo-dimensions for score functions based on multilayer neural networks with ReLU activation function, in the scenarios when the embedding parameters either fixed or trainable. Finally, we complement our theoretical results with numerical studies and apply the method to learn a comprehensive medical knowledge graph combining a pretrained language model representation with knowledge graph links observed in several medical ontologies. The experiments justify our theoretical findings and demonstrate the effect of weighting in the presence of heterogeneous relations and the benefit of incorporating representation learning in nonparametric models. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.07454v1-abstract-full').style.display = 'none'; document.getElementById('2410.07454v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.04458">arXiv:2410.04458</a> <span> [<a href="https://arxiv.org/pdf/2410.04458">pdf</a>, <a href="https://arxiv.org/ps/2410.04458">ps</a>, <a href="https://arxiv.org/format/2410.04458">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"> A Comprehensive Framework for Analyzing the Convergence of Adam: Bridging the Gap with SGD </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jin%2C+R">Ruinan Jin</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiao Li</a>, <a href="/search/math?searchtype=author&query=Yu%2C+Y">Yaoliang Yu</a>, <a href="/search/math?searchtype=author&query=Wang%2C+B">Baoxiang Wang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.04458v2-abstract-short" style="display: inline;"> Adaptive Moment Estimation (Adam) is a cornerstone optimization algorithm in deep learning, widely recognized for its flexibility with adaptive learning rates and efficiency in handling large-scale data. However, despite its practical success, the theoretical understanding of Adam's convergence has been constrained by stringent assumptions, such as almost surely bounded stochastic gradients or uni… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.04458v2-abstract-full').style.display = 'inline'; document.getElementById('2410.04458v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.04458v2-abstract-full" style="display: none;"> Adaptive Moment Estimation (Adam) is a cornerstone optimization algorithm in deep learning, widely recognized for its flexibility with adaptive learning rates and efficiency in handling large-scale data. However, despite its practical success, the theoretical understanding of Adam's convergence has been constrained by stringent assumptions, such as almost surely bounded stochastic gradients or uniformly bounded gradients, which are more restrictive than those typically required for analyzing stochastic gradient descent (SGD). In this paper, we introduce a novel and comprehensive framework for analyzing the convergence properties of Adam. This framework offers a versatile approach to establishing Adam's convergence. Specifically, we prove that Adam achieves asymptotic (last iterate sense) convergence in both the almost sure sense and the $L_1$ sense under the relaxed assumptions typically used for SGD, namely $L$-smoothness and the ABC inequality. Meanwhile, under the same assumptions, we show that Adam attains non-asymptotic sample complexity bounds similar to those of SGD. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.04458v2-abstract-full').style.display = 'none'; document.getElementById('2410.04458v2-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, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 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/2410.02209">arXiv:2410.02209</a> <span> [<a href="https://arxiv.org/pdf/2410.02209">pdf</a>, <a href="https://arxiv.org/ps/2410.02209">ps</a>, <a href="https://arxiv.org/format/2410.02209">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> Some three-weight linear codes and their complete weight enumerators and weight hierarchies </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xiumei Li</a>, <a href="/search/math?searchtype=author&query=Chen%2C+Z">Zongxi Chen</a>, <a href="/search/math?searchtype=author&query=Li%2C+F">Fei Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.02209v1-abstract-short" style="display: inline;"> Linear codes with a few weights can be applied to secrete sharing, authentication codes, association schemes and strongly regular graphs. For an odd prime power $q$, we construct a class of three-weight $\F_q$-linear codes from quadratic functions via a bivariate construction and then determine the complete weight enumerators and weight hierarchies of these linear codes completely. This paper gene… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.02209v1-abstract-full').style.display = 'inline'; document.getElementById('2410.02209v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.02209v1-abstract-full" style="display: none;"> Linear codes with a few weights can be applied to secrete sharing, authentication codes, association schemes and strongly regular graphs. For an odd prime power $q$, we construct a class of three-weight $\F_q$-linear codes from quadratic functions via a bivariate construction and then determine the complete weight enumerators and weight hierarchies of these linear codes completely. This paper generalizes some results in Li et al. (2022) and Hu et al. (2024). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.02209v1-abstract-full').style.display = 'none'; document.getElementById('2410.02209v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">28</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 94B05; 11T71 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.01749">arXiv:2410.01749</a> <span> [<a href="https://arxiv.org/pdf/2410.01749">pdf</a>, <a href="https://arxiv.org/ps/2410.01749">ps</a>, <a href="https://arxiv.org/format/2410.01749">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"> Fully Coupled Nonlinear FBS$螖$Es: Solvability and LQ Control Insights </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Niu%2C+Z">Zhipeng Niu</a>, <a href="/search/math?searchtype=author&query=Meng%2C+Q">Qingxin Meng</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xun Li</a>, <a href="/search/math?searchtype=author&query=Tang%2C+M">Maoning Tang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.01749v1-abstract-short" style="display: inline;"> This paper explores a class of fully coupled nonlinear forward-backward stochastic difference equations (FBS$螖$Es). Building on insights from linear quadratic optimal control problems, we introduce a more relaxed framework of domination-monotonicity conditions specifically designed for discrete systems. Utilizing these conditions, we apply the method of continuation to demonstrate the unique solva… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.01749v1-abstract-full').style.display = 'inline'; document.getElementById('2410.01749v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.01749v1-abstract-full" style="display: none;"> This paper explores a class of fully coupled nonlinear forward-backward stochastic difference equations (FBS$螖$Es). Building on insights from linear quadratic optimal control problems, we introduce a more relaxed framework of domination-monotonicity conditions specifically designed for discrete systems. Utilizing these conditions, we apply the method of continuation to demonstrate the unique solvability of the fully coupled FBS$螖$Es and derive a set of solution estimates. Moreover, our results have considerable implications for various related linear quadratic (LQ) problems, particularly where stochastic Hamiltonian systems are aligned with the FBS$螖$Es meeting these introduced domination-monotonicity conditions. As a result, solving the associated stochastic Hamiltonian systems allows us to derive explicit expressions for the unique optimal controls. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.01749v1-abstract-full').style.display = 'none'; document.getElementById('2410.01749v1-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 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">arXiv admin note: text overlap with arXiv:2310.13195</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.01741">arXiv:2410.01741</a> <span> [<a href="https://arxiv.org/pdf/2410.01741">pdf</a>, <a href="https://arxiv.org/ps/2410.01741">ps</a>, <a href="https://arxiv.org/format/2410.01741">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"> Discrete-Time LQ Stochastic Two-Person Nonzero-Sum Difference Games with Random Coefficients:~Open-Loop Nash Equilibrium </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wu%2C+Y">Yiwei Wu</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xun Li</a>, <a href="/search/math?searchtype=author&query=Meng%2C+Q">Qingxin Meng</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="2410.01741v1-abstract-short" style="display: inline;"> This paper presents a pioneering investigation into discrete-time two-person nonzero-sum linear quadratic (LQ) stochastic games characterized by random coefficients. We derive necessary and sufficient conditions for the existence of open-loop Nash equilibria using convex variational calculus. To obtain explicit expressions for the Nash equilibria, we introduce fully coupled forward-backward stocha… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.01741v1-abstract-full').style.display = 'inline'; document.getElementById('2410.01741v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.01741v1-abstract-full" style="display: none;"> This paper presents a pioneering investigation into discrete-time two-person nonzero-sum linear quadratic (LQ) stochastic games characterized by random coefficients. We derive necessary and sufficient conditions for the existence of open-loop Nash equilibria using convex variational calculus. To obtain explicit expressions for the Nash equilibria, we introduce fully coupled forward-backward stochastic difference equations (stochastic Hamiltonian systems), which provide a dual characterization of these Nash equilibria. Additionally, we develop non-symmetric stochastic Riccati equations that decouple the stochastic Hamiltonian system for each player, enabling the derivation of closed-loop feedback forms for open-loop Nash equilibrium strategies. A notable aspect of this research is the complete randomness of the coefficients, which results in the corresponding Riccati equations becoming fully nonlinear higher-order backward stochastic difference equations. It distinguishes our nonzero-sum difference game from the deterministic case, where the Riccati equations reduce to algebraic forms. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.01741v1-abstract-full').style.display = 'none'; document.getElementById('2410.01741v1-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 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/2410.01543">arXiv:2410.01543</a> <span> [<a href="https://arxiv.org/pdf/2410.01543">pdf</a>, <a href="https://arxiv.org/ps/2410.01543">ps</a>, <a href="https://arxiv.org/format/2410.01543">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"> Weighted $L^p~(p\geq1)$ solutions of random time horizon BSDEs with stochastic monotonicity generators </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xinying Li</a>, <a href="/search/math?searchtype=author&query=Fan%2C+S">Shengjun Fan</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="2410.01543v1-abstract-short" style="display: inline;"> In this paper, we are concerned with a multidimensional backward stochastic differential equation (BSDE) with a general random terminal time $蟿$, which may take values in $[0,+\infty]$. Firstly, we establish an existence and uniqueness result for a weighted $L^p~(p>1)$ solution of the preceding BSDE with generator $g$ satisfying a stochastic monotonicity condition with general growth in the first… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.01543v1-abstract-full').style.display = 'inline'; document.getElementById('2410.01543v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.01543v1-abstract-full" style="display: none;"> In this paper, we are concerned with a multidimensional backward stochastic differential equation (BSDE) with a general random terminal time $蟿$, which may take values in $[0,+\infty]$. Firstly, we establish an existence and uniqueness result for a weighted $L^p~(p>1)$ solution of the preceding BSDE with generator $g$ satisfying a stochastic monotonicity condition with general growth in the first unknown variable $y$ and a stochastic Lipschitz continuity condition in the second unknown variable $z$. Then, we derive an existence and uniqueness result for a weighted $L^1$ solution of the preceding BSDE under an additional stochastic sub-linear growth condition in $z$. These results generalize the corresponding ones obtained in \cite{Li2024} to the $L^p~(p\geq 1)$ solution case. Finally, the corresponding comparison theorems for the weighted $L^p~(p\geq1)$ solutions are also put forward and verified in the one-dimensional setting. In particular, we develop new ideas and systematical techniques in order to establish the above results. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.01543v1-abstract-full').style.display = 'none'; document.getElementById('2410.01543v1-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 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">29 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/2410.00936">arXiv:2410.00936</a> <span> [<a href="https://arxiv.org/pdf/2410.00936">pdf</a>, <a href="https://arxiv.org/ps/2410.00936">ps</a>, <a href="https://arxiv.org/format/2410.00936">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> On high power moments of the error term of the Dirichlet divisor function over primes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Guo%2C+Z">Zhen Guo</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xin Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.00936v1-abstract-short" style="display: inline;"> Let $3\leqslant k\leqslant9$ be a fixed integer, $p$ be a prime and $d(n)$ denote the Dirichlet divisor function. We use $螖(x)$ to denote the error term in the asymptotic formula of the summatory function of $d(n)$. The aim of this paper is to study the $k$-th power moments of $螖(p)$, namely $\sum_{p\leqslant x}螖^k(p)$, and we give an asymptotic formula. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.00936v1-abstract-full" style="display: none;"> Let $3\leqslant k\leqslant9$ be a fixed integer, $p$ be a prime and $d(n)$ denote the Dirichlet divisor function. We use $螖(x)$ to denote the error term in the asymptotic formula of the summatory function of $d(n)$. The aim of this paper is to study the $k$-th power moments of $螖(p)$, namely $\sum_{p\leqslant x}螖^k(p)$, and we give an asymptotic formula. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.00936v1-abstract-full').style.display = 'none'; document.getElementById('2410.00936v1-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 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">15 pages. arXiv admin note: substantial text overlap with arXiv:2410.00329</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.00329">arXiv:2410.00329</a> <span> [<a href="https://arxiv.org/pdf/2410.00329">pdf</a>, <a href="https://arxiv.org/ps/2410.00329">ps</a>, <a href="https://arxiv.org/format/2410.00329">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> On a sum of the error term of the Dirichlet divisor function over primes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Guo%2C+Z">Zhen Guo</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xin Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.00329v1-abstract-short" style="display: inline;"> Let $d(n)$ be the Dirichlet divisor function and $螖(x)$ denote the error term of the sum $\sum_{n\leqslant x}d(n)$ for a large real variable $x$. In this paper we focus on the sum $\sum_{p\leqslant x}螖^2(p)$, where $p$ runs over primes. We prove that there exists an asymptotic formula. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.00329v1-abstract-full" style="display: none;"> Let $d(n)$ be the Dirichlet divisor function and $螖(x)$ denote the error term of the sum $\sum_{n\leqslant x}d(n)$ for a large real variable $x$. In this paper we focus on the sum $\sum_{p\leqslant x}螖^2(p)$, where $p$ runs over primes. We prove that there exists an asymptotic formula. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.00329v1-abstract-full').style.display = 'none'; document.getElementById('2410.00329v1-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 September, 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">13 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.19674">arXiv:2409.19674</a> <span> [<a href="https://arxiv.org/pdf/2409.19674">pdf</a>, <a href="https://arxiv.org/format/2409.19674">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Alternating Maximization Algorithm for Mismatch Capacity with Oblivious Relaying </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xinwei Li</a>, <a href="/search/math?searchtype=author&query=Chen%2C+L">Lingyi Chen</a>, <a href="/search/math?searchtype=author&query=Wu%2C+S">Shitong Wu</a>, <a href="/search/math?searchtype=author&query=Wu%2C+H">Huihui Wu</a>, <a href="/search/math?searchtype=author&query=Wu%2C+H">Hao Wu</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+W">Wenyi Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2409.19674v4-abstract-short" style="display: inline;"> Reliable communication over a discrete memoryless channel with the help of a relay has aroused interest due to its widespread applications in practical scenarios. By considering the system with a mismatched decoder, previous works have provided optimization models to evaluate the mismatch capacity in these scenarios. The proposed models, however, are difficult due to the complicated structure of t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.19674v4-abstract-full').style.display = 'inline'; document.getElementById('2409.19674v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.19674v4-abstract-full" style="display: none;"> Reliable communication over a discrete memoryless channel with the help of a relay has aroused interest due to its widespread applications in practical scenarios. By considering the system with a mismatched decoder, previous works have provided optimization models to evaluate the mismatch capacity in these scenarios. The proposed models, however, are difficult due to the complicated structure of the mismatched decoding problem with the information flows in hops given by the relay. Existing methods, such as the grid search, become impractical as they involve finding all roots of a nonlinear system, with the growing size of the alphabet. To address this problem, we reformulate the max-min optimization model as a consistent maximization form, by considering the dual form of the inner minimization problem and the Lagrangian with a fixed multiplier. Based on the proposed formulation, an alternating maximization framework is designed, which provides the closed-form solution with simple iterations in each step by introducing a suitable variable transformation. The effectiveness of the proposed approach is demonstrated by the simulations over practical scenarios, including Quaternary and Gaussian channels. Moreover, the simulation results of the transitional probability also shed light on the promising application attribute to the quantizer design in the relay node. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.19674v4-abstract-full').style.display = 'none'; document.getElementById('2409.19674v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 29 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.18914">arXiv:2409.18914</a> <span> [<a href="https://arxiv.org/pdf/2409.18914">pdf</a>, <a href="https://arxiv.org/ps/2409.18914">ps</a>, <a href="https://arxiv.org/format/2409.18914">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"> Amenable Metric Mean Dimension and Amenable Mean Hausdorff Dimension of Product Sets and Metric Varying </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xianqiang Li</a>, <a href="/search/math?searchtype=author&query=Luo%2C+X">Xiaofang Luo</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="2409.18914v1-abstract-short" style="display: inline;"> Metric mean dimension and mean Hausdorff dimension depend on metrics. In this paper, we investigate the continuity of the metric mean dimension and mean Hausdorff dimension concerning the metrics for amenable group actions, which extends recent results by Muentes, Becker, Baraviera et al.. Moreover, we give proof of the product formulas for the mean Hausdorff dimension and the metric mean dimensio… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.18914v1-abstract-full').style.display = 'inline'; document.getElementById('2409.18914v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.18914v1-abstract-full" style="display: none;"> Metric mean dimension and mean Hausdorff dimension depend on metrics. In this paper, we investigate the continuity of the metric mean dimension and mean Hausdorff dimension concerning the metrics for amenable group actions, which extends recent results by Muentes, Becker, Baraviera et al.. Moreover, we give proof of the product formulas for the mean Hausdorff dimension and the metric mean dimension for amenable group actions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.18914v1-abstract-full').style.display = 'none'; document.getElementById('2409.18914v1-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 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.18140">arXiv:2409.18140</a> <span> [<a href="https://arxiv.org/pdf/2409.18140">pdf</a>, <a href="https://arxiv.org/ps/2409.18140">ps</a>, <a href="https://arxiv.org/format/2409.18140">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"> Globally conservative weak solutions for a class of two-component nonlinear dispersive wave equations beyond wave breaking </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhou%2C+Y">Yonghui Zhou</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiaowan Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2409.18140v1-abstract-short" style="display: inline;"> In this paper, we prove that the existence of globally conservative weak solutions for a class of two-component nonlinear dispersive wave equations beyond wave breaking. We first introduce a new set of independent and dependent variables in connection with smooth solutions, and transform the system into an equivalent semi-linear system. We then establish the global existence of solutions for the s… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.18140v1-abstract-full').style.display = 'inline'; document.getElementById('2409.18140v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.18140v1-abstract-full" style="display: none;"> In this paper, we prove that the existence of globally conservative weak solutions for a class of two-component nonlinear dispersive wave equations beyond wave breaking. We first introduce a new set of independent and dependent variables in connection with smooth solutions, and transform the system into an equivalent semi-linear system. We then establish the global existence of solutions for the semi-linear system via the standard theory of ordinary differential equations. Finally, by the inverse transformation method, we prove the existence of the globally conservative weak solution for the original system. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.18140v1-abstract-full').style.display = 'none'; document.getElementById('2409.18140v1-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, 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">arXiv admin note: substantial text overlap with arXiv:2303.08640</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.17900">arXiv:2409.17900</a> <span> [<a href="https://arxiv.org/pdf/2409.17900">pdf</a>, <a href="https://arxiv.org/ps/2409.17900">ps</a>, <a href="https://arxiv.org/format/2409.17900">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"> Sharp asymptotics of disconnection time of large cylinders by simple and biased random walks </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xinyi Li</a>, <a href="/search/math?searchtype=author&query=Liu%2C+Y">Yu Liu</a>, <a href="/search/math?searchtype=author&query=Wang%2C+Y">Yuanzheng 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="2409.17900v1-abstract-short" style="display: inline;"> We investigate the asymptotic disconnection time of a large discrete cylinder $(\mathbb{Z}/N\mathbb{Z})^{d}\times \mathbb{Z}$, $d\geq 2$, by simple and biased random walks. For simple random walk, we derive a sharp asymptotic lower bound that matches the upper bound from [Sznitman, Ann. Probab., 2009]. For biased walks, we obtain bounds that asymptotically match in the principal order when the bia… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.17900v1-abstract-full').style.display = 'inline'; document.getElementById('2409.17900v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.17900v1-abstract-full" style="display: none;"> We investigate the asymptotic disconnection time of a large discrete cylinder $(\mathbb{Z}/N\mathbb{Z})^{d}\times \mathbb{Z}$, $d\geq 2$, by simple and biased random walks. For simple random walk, we derive a sharp asymptotic lower bound that matches the upper bound from [Sznitman, Ann. Probab., 2009]. For biased walks, we obtain bounds that asymptotically match in the principal order when the bias is not too strong, which greatly improves non-matching bounds from [Windisch, Ann. Appl. Probab., 2008]. As a crucial tool in the proof, we also obtain a "very strong" coupling between the trace of random walk on the cylinder and random interlacements, which is of independent interest. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.17900v1-abstract-full').style.display = 'none'; document.getElementById('2409.17900v1-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> 26 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">88 pages, 6 tables of symbols</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 60G50; 60K35; 82C41 (Primary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.16924">arXiv:2409.16924</a> <span> [<a href="https://arxiv.org/pdf/2409.16924">pdf</a>, <a href="https://arxiv.org/ps/2409.16924">ps</a>, <a href="https://arxiv.org/format/2409.16924">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"> Weak Closed-loop Solvability of Linear Quadratic Stochastic Optimal Control Problems with Partial Information </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xun Li</a>, <a href="/search/math?searchtype=author&query=Wang%2C+G">Guangchen Wang</a>, <a href="/search/math?searchtype=author&query=Xiong%2C+J">Jie Xiong</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+H">Heng 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="2409.16924v1-abstract-short" style="display: inline;"> This paper investigates a linear quadratic stochastic optimal control (LQSOC) problem with partial information. Firstly, by introducing two Riccati equations and a backward stochastic differential equation (BSDE), we solve this LQSOC problem under standard positive semidefinite assumptions. Secondly, by means of a perturbation approach, we study open-loop solvability of this problem when the weigh… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.16924v1-abstract-full').style.display = 'inline'; document.getElementById('2409.16924v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.16924v1-abstract-full" style="display: none;"> This paper investigates a linear quadratic stochastic optimal control (LQSOC) problem with partial information. Firstly, by introducing two Riccati equations and a backward stochastic differential equation (BSDE), we solve this LQSOC problem under standard positive semidefinite assumptions. Secondly, by means of a perturbation approach, we study open-loop solvability of this problem when the weighting matrices in the cost functional are indefinite. Thirdly, we investigate weak closed-loop solvability of this problem and prove the equivalence between open-loop and weak closed-loop solvabilities. Finally, we give an example to illustrate the way for obtaining a weak closed-loop optimal strategy. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.16924v1-abstract-full').style.display = 'none'; document.getElementById('2409.16924v1-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> 25 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.16716">arXiv:2409.16716</a> <span> [<a href="https://arxiv.org/pdf/2409.16716">pdf</a>, <a href="https://arxiv.org/ps/2409.16716">ps</a>, <a href="https://arxiv.org/format/2409.16716">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"> Simultaneously reconstructing potentials and internal sources for fractional Schr枚dinger equations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xinyan Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2409.16716v1-abstract-short" style="display: inline;"> The inverse problems about fractional Calder贸n problem and fractional Schr枚dinger equations are of interest in the study of mathematics. In this paper, we propose the inverse problem to simultaneously reconstruct potentials and sources for fractional Schr枚dinger equations with internal source terms. We show the uniqueness for reconstructing the two terms under measurements from two different nonho… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.16716v1-abstract-full').style.display = 'inline'; document.getElementById('2409.16716v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.16716v1-abstract-full" style="display: none;"> The inverse problems about fractional Calder贸n problem and fractional Schr枚dinger equations are of interest in the study of mathematics. In this paper, we propose the inverse problem to simultaneously reconstruct potentials and sources for fractional Schr枚dinger equations with internal source terms. We show the uniqueness for reconstructing the two terms under measurements from two different nonhomogeneous boundary conditions. By introducing the variational Tikhonov regularization functional, numerical method based on conjugate gradient method(CGM) is provided to realize this inverse problem. Numerical experiments are given to gauge the performance of the numerical method. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.16716v1-abstract-full').style.display = 'none'; document.getElementById('2409.16716v1-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> 25 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">8 pages and 2 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/2409.16711">arXiv:2409.16711</a> <span> [<a href="https://arxiv.org/pdf/2409.16711">pdf</a>, <a href="https://arxiv.org/ps/2409.16711">ps</a>, <a href="https://arxiv.org/format/2409.16711">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 numerical method for reconstructing the potential in fractional Calder贸n problem with a single measurement </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xinyan Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2409.16711v1-abstract-short" style="display: inline;"> In this paper, we develop a numerical method for determining the potential in one and two dimensional fractional Calder贸n problems with a single measurement. Finite difference scheme is employed to discretize the fractional Laplacian, and the parameter reconstruction is formulated into a variational problem based on Tikhonov regularization to obtain a stable and accurate solution. Conjugate gradie… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.16711v1-abstract-full').style.display = 'inline'; document.getElementById('2409.16711v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.16711v1-abstract-full" style="display: none;"> In this paper, we develop a numerical method for determining the potential in one and two dimensional fractional Calder贸n problems with a single measurement. Finite difference scheme is employed to discretize the fractional Laplacian, and the parameter reconstruction is formulated into a variational problem based on Tikhonov regularization to obtain a stable and accurate solution. Conjugate gradient method is utilized to solve the variational problem. Moreover, we also provide a suggestion to choose the regularization parameter. Numerical experiments are performed to illustrate the efficiency and effectiveness of the developed method and verify the theoretical results. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.16711v1-abstract-full').style.display = 'none'; document.getElementById('2409.16711v1-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> 25 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">18 pages and 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/2409.14492">arXiv:2409.14492</a> <span> [<a href="https://arxiv.org/pdf/2409.14492">pdf</a>, <a href="https://arxiv.org/ps/2409.14492">ps</a>, <a href="https://arxiv.org/format/2409.14492">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Classical Analysis and ODEs">math.CA</span> </div> </div> <p class="title is-5 mathjax"> Completely regular growth solutions to linear differential equations with exponential polynomials coefficients </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xing-Yu Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2409.14492v1-abstract-short" style="display: inline;"> Consider the linear differential equation $$ f^{(n)}+A_{n-1}f^{(n-1)}+\cdots+A_{0}f=0 $$ where the coefficients $A_j,j=0,\ldots,n-1,$ are exponential polynomials. It is known that every solution is entire. This paper will show that all transcendental solutions of finite growth order are of completely regular growth. This problem was raised in Heittokangas et al.[8, p.33], which involves an e… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.14492v1-abstract-full').style.display = 'inline'; document.getElementById('2409.14492v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.14492v1-abstract-full" style="display: none;"> Consider the linear differential equation $$ f^{(n)}+A_{n-1}f^{(n-1)}+\cdots+A_{0}f=0 $$ where the coefficients $A_j,j=0,\ldots,n-1,$ are exponential polynomials. It is known that every solution is entire. This paper will show that all transcendental solutions of finite growth order are of completely regular growth. This problem was raised in Heittokangas et al.[8, p.33], which involves an extensive question about Gol'dberg-Ostrovski菒's Problem [5, p.300]. Moreover, we define functions in a generalized class concluding exponential polynomial functions, which are also of completely regular growth. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.14492v1-abstract-full').style.display = 'none'; document.getElementById('2409.14492v1-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> 22 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.10987">arXiv:2409.10987</a> <span> [<a href="https://arxiv.org/pdf/2409.10987">pdf</a>, <a href="https://arxiv.org/ps/2409.10987">ps</a>, <a href="https://arxiv.org/format/2409.10987">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"> Relationship between stochastic maximum principle and dynamic programming principle under convex expectation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xiaojuan Li</a>, <a href="/search/math?searchtype=author&query=Hu%2C+M">Mingshang Hu</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="2409.10987v1-abstract-short" style="display: inline;"> In this paper, we study the relationship between maximum principle (MP) and dynamic programming principle (DPP) for forward-backward control system under consistent convex expectation dominated by G-expectation. Under the smooth assumptions for the value function, we get the relationship between MP and DPP under a reference probability by establishing a useful estimate. If the value function is no… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.10987v1-abstract-full').style.display = 'inline'; document.getElementById('2409.10987v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.10987v1-abstract-full" style="display: none;"> In this paper, we study the relationship between maximum principle (MP) and dynamic programming principle (DPP) for forward-backward control system under consistent convex expectation dominated by G-expectation. Under the smooth assumptions for the value function, we get the relationship between MP and DPP under a reference probability by establishing a useful estimate. If the value function is not smooth, then we obtain the first-order sub-jet and super-jet of the value function at any t. However, the processing method in this case is much more difficult than that when t equals 0. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.10987v1-abstract-full').style.display = 'none'; document.getElementById('2409.10987v1-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> 17 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</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 93E20; 60H10; 35K15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.10391">arXiv:2409.10391</a> <span> [<a href="https://arxiv.org/pdf/2409.10391">pdf</a>, <a href="https://arxiv.org/ps/2409.10391">ps</a>, <a href="https://arxiv.org/format/2409.10391">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Existence of embedded minimal tori in three-spheres with positive Ricci curvature </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xingzhe Li</a>, <a href="/search/math?searchtype=author&query=Wang%2C+Z">Zhichao 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="2409.10391v1-abstract-short" style="display: inline;"> In this paper, we prove the strong Morse inequalities for the area functional in the space of embedded tori and spheres in the three sphere. As a consequence, we prove that in the three dimensional sphere with positive Ricci curvature, there exist at least 4 distinct embedded minimal tori. Suppose in addition that the metric is bumpy, then the three-sphere contains at least 9 distinct embedded min… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.10391v1-abstract-full').style.display = 'inline'; document.getElementById('2409.10391v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.10391v1-abstract-full" style="display: none;"> In this paper, we prove the strong Morse inequalities for the area functional in the space of embedded tori and spheres in the three sphere. As a consequence, we prove that in the three dimensional sphere with positive Ricci curvature, there exist at least 4 distinct embedded minimal tori. Suppose in addition that the metric is bumpy, then the three-sphere contains at least 9 distinct embedded minimal tori. The proof relies on a multiplicity one theorem for the Simon-Smith min-max theory proved by the second author and X. Zhou. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.10391v1-abstract-full').style.display = 'none'; document.getElementById('2409.10391v1-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">28 pages; comments are welcome</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.09762">arXiv:2409.09762</a> <span> [<a href="https://arxiv.org/pdf/2409.09762">pdf</a>, <a href="https://arxiv.org/ps/2409.09762">ps</a>, <a href="https://arxiv.org/format/2409.09762">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"> A new blowup criterion for strong solutions of a coupled periodic Camassa-Holm system </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhou%2C+Y">Yonghui Zhou</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiaowan Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2409.09762v1-abstract-short" style="display: inline;"> This paper is concerned with the wave breaking phenomena for a coupled periodic Camassa-Holm system. We establish a new blowup criterion for strong solutions by the method of characteristic and convolution estimates, and also give the existence interval of the blowup point. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.09762v1-abstract-full" style="display: none;"> This paper is concerned with the wave breaking phenomena for a coupled periodic Camassa-Holm system. We establish a new blowup criterion for strong solutions by the method of characteristic and convolution estimates, and also give the existence interval of the blowup point. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.09762v1-abstract-full').style.display = 'none'; document.getElementById('2409.09762v1-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, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.01973">arXiv:2409.01973</a> <span> [<a href="https://arxiv.org/pdf/2409.01973">pdf</a>, <a href="https://arxiv.org/ps/2409.01973">ps</a>, <a href="https://arxiv.org/format/2409.01973">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"> Open-loop and closed-loop solvabilities for zero-sum stochastic linear quadratic differential games of Markovian regime switching system </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wu%2C+F">Fan Wu</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xun Li</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+X">Xin 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="2409.01973v1-abstract-short" style="display: inline;"> This paper investigates a zero-sum stochastic linear quadratic (SLQ, for short) differential games with Markovian jumps. Both open-loop and closed-loop solvabilities are studied by employing a new "decomposition method", which can decompose the open-loop and closed-loop solvability problems of zero-sum SLQ differential game into two coupled SLQ control problems for solving. Moreover, we construct… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.01973v1-abstract-full').style.display = 'inline'; document.getElementById('2409.01973v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.01973v1-abstract-full" style="display: none;"> This paper investigates a zero-sum stochastic linear quadratic (SLQ, for short) differential games with Markovian jumps. Both open-loop and closed-loop solvabilities are studied by employing a new "decomposition method", which can decompose the open-loop and closed-loop solvability problems of zero-sum SLQ differential game into two coupled SLQ control problems for solving. Moreover, we construct the open-loop saddle point and its closed-loop representation under the uniform convexity-concavity condition based on the solution of a system of constrained coupled differential Riccati equations (CDREs, for short), whose solvability is also provided by adopting the dimension extension technique and the continuation method. At the end of this paper, we provide a concrete example and give its open-loop saddle based on the obtained results. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.01973v1-abstract-full').style.display = 'none'; document.getElementById('2409.01973v1-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 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> 93E03; 93E20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.01518">arXiv:2409.01518</a> <span> [<a href="https://arxiv.org/pdf/2409.01518">pdf</a>, <a href="https://arxiv.org/format/2409.01518">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"> Modular Vehicle Routing Problem: Applications in Logistics </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhou%2C+H">Hang Zhou</a>, <a href="/search/math?searchtype=author&query=Li%2C+Y">Yang Li</a>, <a href="/search/math?searchtype=author&query=Ma%2C+C">Chengyuan Ma</a>, <a href="/search/math?searchtype=author&query=Long%2C+K">Keke Long</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiaopeng Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2409.01518v1-abstract-short" style="display: inline;"> Recent studies and industry advancements indicate that modular vehicles (MVs) have the potential to enhance transportation systems through their ability to dock and split en route. Although various applications of MVs have been explored across different domains, their use in logistics remains relatively underexplored. This study examines the application of MVs in cargo delivery to reduce costs. We… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.01518v1-abstract-full').style.display = 'inline'; document.getElementById('2409.01518v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.01518v1-abstract-full" style="display: none;"> Recent studies and industry advancements indicate that modular vehicles (MVs) have the potential to enhance transportation systems through their ability to dock and split en route. Although various applications of MVs have been explored across different domains, their use in logistics remains relatively underexplored. This study examines the application of MVs in cargo delivery to reduce costs. We model the delivery problem for MVs as a variant of the Vehicle Routing Problem, referred to as the Modular Vehicle Routing Problem (MVRP). In the MVRP, MVs can either serve customers independently or dock with other MVs to form a platoon, thereby reducing the average cost per unit. To tackle this problem, we first developed a Mixed Integer Linear Programming model, solvable by commercial optimization solvers. Given the problem's computational complexity, we also designed a Tabu Search (TS) algorithm with specialized neighborhood operators tailored for the MVRP. To escape local optima, multi-start and shaking strategies were incorporated into the TS algorithm. Additionally, we explored potential applications in logistics through various MVRP variants. The results of the numerical experiments indicate that the proposed algorithm successfully identifies all optimal solutions found by the MILP model in small-size benchmark instances, while also demonstrating good convergence speed in large-size benchmark instances. Comparative experiments show that the MVRP approach can reduce costs by approximately 5\% compared to traditional delivery methods. The code and data used in this study will be made available upon the acceptance of this paper. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.01518v1-abstract-full').style.display = 'none'; document.getElementById('2409.01518v1-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 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.01457">arXiv:2409.01457</a> <span> [<a href="https://arxiv.org/pdf/2409.01457">pdf</a>, <a href="https://arxiv.org/ps/2409.01457">ps</a>, <a href="https://arxiv.org/format/2409.01457">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> The sixth moment of Dirichlet L-functions at the central point </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chandee%2C+V">Vorrapan Chandee</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiannan Li</a>, <a href="/search/math?searchtype=author&query=Matom%C3%A4ki%2C+K">Kaisa Matom盲ki</a>, <a href="/search/math?searchtype=author&query=Radziwi%C5%82%C5%82%2C+M">Maksym Radziwi艂艂</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="2409.01457v1-abstract-short" style="display: inline;"> In 1970, Huxley obtained a sharp upper bound for the sixth moment of Dirichlet $L$-functions at the central point, averaged over primitive characters $蠂$ modulo $q$ and all moduli $q \leq Q$. In 2007, as an application of their ``asymptotic large sieve'', Conrey, Iwaniec and Soundararajan showed that when an additional short $t$-averaging is introduced into the problem, an asymptotic can be obtain… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.01457v1-abstract-full').style.display = 'inline'; document.getElementById('2409.01457v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.01457v1-abstract-full" style="display: none;"> In 1970, Huxley obtained a sharp upper bound for the sixth moment of Dirichlet $L$-functions at the central point, averaged over primitive characters $蠂$ modulo $q$ and all moduli $q \leq Q$. In 2007, as an application of their ``asymptotic large sieve'', Conrey, Iwaniec and Soundararajan showed that when an additional short $t$-averaging is introduced into the problem, an asymptotic can be obtained. In this paper we show that this extraneous averaging can be removed, and we thus obtain an asymptotic for the original moment problem considered by Huxley. The main new difficulty in our work is the appearance of certain challenging ``unbalanced'' sums that arise as soon as the $t$-aspect averaging is removed. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.01457v1-abstract-full').style.display = 'none'; document.getElementById('2409.01457v1-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 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">49 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 11M06; Secondary: 11M26 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.01361">arXiv:2409.01361</a> <span> [<a href="https://arxiv.org/pdf/2409.01361">pdf</a>, <a href="https://arxiv.org/format/2409.01361">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"> Conformal measures of (anti)holomorphic correspondences </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Hemmingsson%2C+N">Nils Hemmingsson</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiaoran Li</a>, <a href="/search/math?searchtype=author&query=Li%2C+Z">Zhiqiang Li</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2409.01361v1-abstract-short" style="display: inline;"> In this paper, we study the existence and properties of conformal measures on limit sets of (anti)holomorphic correspondences. We show that if the critical exponent satisfies $1\leq 未_{\operatorname{crit}}(x) <+\infty,$ the correspondence $F$ is (relatively) hyperbolic on the limit set $螞_+(x)$, and $螞_+(x)$ is minimal, then $螞_+(x)$ admits a non-atomic conformal measure for $F$ and the Hausdorff… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.01361v1-abstract-full').style.display = 'inline'; document.getElementById('2409.01361v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.01361v1-abstract-full" style="display: none;"> In this paper, we study the existence and properties of conformal measures on limit sets of (anti)holomorphic correspondences. We show that if the critical exponent satisfies $1\leq 未_{\operatorname{crit}}(x) <+\infty,$ the correspondence $F$ is (relatively) hyperbolic on the limit set $螞_+(x)$, and $螞_+(x)$ is minimal, then $螞_+(x)$ admits a non-atomic conformal measure for $F$ and the Hausdorff dimension of $螞_+(x)$ is strictly less than 2. As a special case, this shows that for a parameter $a$ in the interior of a hyperbolic component of the modular Mandelbrot set, the limit set of the Bullett--Penrose correspondence $F_a$ has a non-atomic conformal measure and its Hausdorff dimension is strictly less than 2. The same results hold for the LLMM correspondences, under some extra assumptions on its defining function $f$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.01361v1-abstract-full').style.display = 'none'; document.getElementById('2409.01361v1-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 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">37 pages, 2 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/2409.00995">arXiv:2409.00995</a> <span> [<a href="https://arxiv.org/pdf/2409.00995">pdf</a>, <a href="https://arxiv.org/ps/2409.00995">ps</a>, <a href="https://arxiv.org/format/2409.00995">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"> Favorite Sites for Simple Random Walk in Two and More Dimensions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Hao%2C+C">Chenxu Hao</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xinyi Li</a>, <a href="/search/math?searchtype=author&query=Okada%2C+I">Izumi Okada</a>, <a href="/search/math?searchtype=author&query=Zheng%2C+Y">Yushu 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="2409.00995v1-abstract-short" style="display: inline;"> On the trace of a discrete-time simple random walk on $\mathbb{Z}^d$ for $d\geq 2$, we consider the evolution of favorite sites, i.e., sites that achieve the maximal local time at a certain time. For $d=2$, we show that almost surely three favorite sites occur simultaneously infinitely often and eventually there is no simultaneous occurrence of four favorite sites. For $d\geq 3$, we derive sharp a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.00995v1-abstract-full').style.display = 'inline'; document.getElementById('2409.00995v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.00995v1-abstract-full" style="display: none;"> On the trace of a discrete-time simple random walk on $\mathbb{Z}^d$ for $d\geq 2$, we consider the evolution of favorite sites, i.e., sites that achieve the maximal local time at a certain time. For $d=2$, we show that almost surely three favorite sites occur simultaneously infinitely often and eventually there is no simultaneous occurrence of four favorite sites. For $d\geq 3$, we derive sharp asymptotics of the number of favorite sites. This answers an open question of Erd艖s and R茅v茅sz (1987), which was brought up again in Dembo (2005). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.00995v1-abstract-full').style.display = 'none'; document.getElementById('2409.00995v1-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 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">46 pages, 3 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 60F15; 60J55 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2408.17030">arXiv:2408.17030</a> <span> [<a href="https://arxiv.org/pdf/2408.17030">pdf</a>, <a href="https://arxiv.org/ps/2408.17030">ps</a>, <a href="https://arxiv.org/format/2408.17030">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"> Zero-sum stochastic linear-quadratic Stackelberg differential games of Markovian regime-switching system </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wu%2C+F">Fan Wu</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xun Li</a>, <a href="/search/math?searchtype=author&query=Xiong%2C+J">Jie Xiong</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+X">Xin Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2408.17030v1-abstract-short" style="display: inline;"> This paper investigates a zero-sum stochastic linear-quadratic (SLQ, for short) Stackelberg differential game problem, where the coefficients of the state equation and the weighting matrices in the performance functional are regulated by a Markov chain. By utilizing the findings in \citet{Zhang.X.2021_ILQM}, we directly present the feedback representation to the rational reaction of the follower.… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.17030v1-abstract-full').style.display = 'inline'; document.getElementById('2408.17030v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.17030v1-abstract-full" style="display: none;"> This paper investigates a zero-sum stochastic linear-quadratic (SLQ, for short) Stackelberg differential game problem, where the coefficients of the state equation and the weighting matrices in the performance functional are regulated by a Markov chain. By utilizing the findings in \citet{Zhang.X.2021_ILQM}, we directly present the feedback representation to the rational reaction of the follower. For the leader's problem, we derive the optimality system through the variational method and study its unique solvability from the Hilbert space point of view. We construct the explicit optimal control for the leader based on the solution to coupled differential Riccati equations (CDREs, for short) and obtain the solvability of CDREs under the one-dimensional framework. Finally, we provide two concrete examples to illustrate the results developed in this paper. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.17030v1-abstract-full').style.display = 'none'; document.getElementById('2408.17030v1-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 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 91A15; 49N10; 93E20 </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=Li%2C+X&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> 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