Search | arXiv e-print repository

<!DOCTYPE html> <html lang="en"> <head> <meta charset="utf-8"/> <meta name="viewport" content="width=device-width, initial-scale=1"/>  <link rel="apple-touch-icon" sizes="180x180" href="https://static.arxiv.org/static/base/1.0.0a5/images/icons/apple-touch-icon.png"> <link rel="icon" type="image/png" sizes="32x32" href="https://static.arxiv.org/static/base/1.0.0a5/images/icons/favicon-32x32.png"> <link rel="icon" type="image/png" sizes="16x16" href="https://static.arxiv.org/static/base/1.0.0a5/images/icons/favicon-16x16.png"> <link rel="manifest" href="https://static.arxiv.org/static/base/1.0.0a5/images/icons/site.webmanifest"> <link rel="mask-icon" href="https://static.arxiv.org/static/base/1.0.0a5/images/icons/safari-pinned-tab.svg" color="#b31b1b"> <link rel="shortcut icon" href="https://static.arxiv.org/static/base/1.0.0a5/images/icons/favicon.ico"> <meta name="msapplication-TileColor" content="#b31b1b"> <meta name="msapplication-config" content="images/icons/browserconfig.xml"> <meta name="theme-color" content="#b31b1b">  <title>Search | arXiv e-print repository</title> <script defer src="https://static.arxiv.org/static/base/1.0.0a5/fontawesome-free-5.11.2-web/js/all.js"></script> <link rel="stylesheet" href="https://static.arxiv.org/static/base/1.0.0a5/css/arxivstyle.css" /> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ messageStyle: "none", extensions: ["tex2jax.js"], jax: ["input/TeX", "output/HTML-CSS"], tex2jax: { inlineMath: [ ['$','$'], ["\$","\$"] ], displayMath: [ ['$$','$$'], ["\\[","\\]"] ], processEscapes: true, ignoreClass: '.*', processClass: 'mathjax.*' }, TeX: { extensions: ["AMSmath.js", "AMSsymbols.js", "noErrors.js"], noErrors: { inlineDelimiters: ["$","$"], multiLine: false, style: { "font-size": "normal", "border": "" } } }, "HTML-CSS": { availableFonts: ["TeX"] } }); </script> <script src='//static.arxiv.org/MathJax-2.7.3/MathJax.js'></script> <script src="https://static.arxiv.org/static/base/1.0.0a5/js/notification.js"></script> <link rel="stylesheet" href="https://static.arxiv.org/static/search/0.5.6/css/bulma-tooltip.min.css" /> <link rel="stylesheet" href="https://static.arxiv.org/static/search/0.5.6/css/search.css" /> <script src="https://code.jquery.com/jquery-3.2.1.slim.min.js" integrity="sha256-k2WSCIexGzOj3Euiig+TlR8gA0EmPjuc79OEeY5L45g=" crossorigin="anonymous"></script> <script src="https://static.arxiv.org/static/search/0.5.6/js/fieldset.js"></script> <style> radio#cf-customfield_11400 { display: none; } </style> </head> <body> <header><a href="#main-container" class="is-sr-only">Skip to main content</a>  <div class="attribution level is-marginless" role="banner"> <div class="level-left"> <a class="level-item" href="https://cornell.edu/"><img src="https://static.arxiv.org/static/base/1.0.0a5/images/cornell-reduced-white-SMALL.svg" alt="Cornell University" width="200" aria-label="logo" /></a> </div> <div class="level-right is-marginless"><p class="sponsors level-item is-marginless"><span id="support-ack-url">We gratefully acknowledge support from<br /> the Simons Foundation, <a href="https://info.arxiv.org/about/ourmembers.html">member institutions</a>, and all contributors. <a href="https://info.arxiv.org/about/donate.html">Donate</a></span></p></div> </div>  <div class="identity level is-marginless"> <div class="level-left"> <div class="level-item"> <a class="arxiv" href="https://arxiv.org/" aria-label="arxiv-logo"> <img src="https://static.arxiv.org/static/base/1.0.0a5/images/arxiv-logo-one-color-white.svg" aria-label="logo" alt="arxiv logo" width="85" style="width:85px;"/> </a> </div> </div> <div class="search-block level-right"> <form class="level-item mini-search" method="GET" action="https://arxiv.org/search"> <div class="field has-addons"> <div class="control"> <input class="input is-small" type="text" name="query" placeholder="Search..." aria-label="Search term or terms" /> <p class="help"><a href="https://info.arxiv.org/help">Help</a> | <a href="https://arxiv.org/search/advanced">Advanced Search</a></p> </div> <div class="control"> <div class="select is-small"> <select name="searchtype" aria-label="Field to search"> <option value="all" selected="selected">All fields</option> <option value="title">Title</option> <option value="author">Author</option> <option value="abstract">Abstract</option> <option value="comments">Comments</option> <option value="journal_ref">Journal reference</option> <option value="acm_class">ACM classification</option> <option value="msc_class">MSC classification</option> <option value="report_num">Report number</option> <option value="paper_id">arXiv identifier</option> <option value="doi">DOI</option> <option value="orcid">ORCID</option> <option value="author_id">arXiv author ID</option> <option value="help">Help pages</option> <option value="full_text">Full text</option> </select> </div> </div> <input type="hidden" name="source" value="header"> <button class="button is-small is-cul-darker">Search</button> </div> </form> </div> </div>  <div class="container"> <div class="user-tools is-size-7 has-text-right has-text-weight-bold" role="navigation" aria-label="User menu"> <a href="https://arxiv.org/login">Login</a> </div> </div> </header> <main class="container" id="main-container"> <div class="level is-marginless"> <div class="level-left"> <h1 class="title is-clearfix"> Showing 1–50 of 416 results for author: <span class="mathjax">Wang, M</span> </h1> </div> <div class="level-right is-hidden-mobile">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> <div class="content"> <form method="GET" action="/search/math" aria-role="search"> Searching in archive <strong>math</strong>. <a href="/search/?searchtype=author&query=Wang%2C+M">Search in all archives.</a> <div class="field has-addons-tablet"> <div class="control is-expanded"> <label for="query" class="hidden-label">Search term or terms</label> <input class="input is-medium" id="query" name="query" placeholder="Search term..." type="text" value="Wang, M"> </div> <div class="select control is-medium"> <label class="is-hidden" for="searchtype">Field</label> <select class="is-medium" id="searchtype" name="searchtype"><option value="all">All fields</option><option value="title">Title</option><option selected value="author">Author(s)</option><option value="abstract">Abstract</option><option value="comments">Comments</option><option value="journal_ref">Journal reference</option><option value="acm_class">ACM classification</option><option value="msc_class">MSC classification</option><option value="report_num">Report number</option><option value="paper_id">arXiv identifier</option><option value="doi">DOI</option><option value="orcid">ORCID</option><option value="license">License (URI)</option><option value="author_id">arXiv author ID</option><option value="help">Help pages</option><option value="full_text">Full text</option></select> </div> <div class="control"> <button class="button is-link is-medium">Search</button> </div> </div> <div class="field"> <div class="control is-size-7"> <label class="radio"> <input checked id="abstracts-0" name="abstracts" type="radio" value="show"> Show abstracts </label> <label class="radio"> <input id="abstracts-1" name="abstracts" type="radio" value="hide"> Hide abstracts </label> </div> </div> <div class="is-clearfix" style="height: 2.5em"> <div class="is-pulled-right"> <a href="/search/advanced?terms-0-term=Wang%2C+M&terms-0-field=author&size=50&order=-announced_date_first">Advanced Search</a> </div> </div> <input type="hidden" name="order" value="-announced_date_first"> <input type="hidden" name="size" value="50"> </form> <div class="level breathe-horizontal"> <div class="level-left"> <form method="GET" action="/search/"> <div style="display: none;"> <select id="searchtype" name="searchtype"><option value="all">All fields</option><option value="title">Title</option><option selected value="author">Author(s)</option><option value="abstract">Abstract</option><option value="comments">Comments</option><option value="journal_ref">Journal reference</option><option value="acm_class">ACM classification</option><option value="msc_class">MSC classification</option><option value="report_num">Report number</option><option value="paper_id">arXiv identifier</option><option value="doi">DOI</option><option value="orcid">ORCID</option><option value="license">License (URI)</option><option value="author_id">arXiv author ID</option><option value="help">Help pages</option><option value="full_text">Full text</option></select> <input id="query" name="query" type="text" value="Wang, M"> <ul id="abstracts"><li><input checked id="abstracts-0" name="abstracts" type="radio" value="show"> <label for="abstracts-0">Show abstracts</label></li><li><input id="abstracts-1" name="abstracts" type="radio" value="hide"> <label for="abstracts-1">Hide abstracts</label></li></ul> </div> <div class="box field is-grouped is-grouped-multiline level-item"> <div class="control"> <span class="select is-small"> <select id="size" name="size"><option value="25">25</option><option selected value="50">50</option><option value="100">100</option><option value="200">200</option></select> </span> <label for="size">results per page</label>. </div> <div class="control"> <label for="order">Sort results by</label> <span class="select is-small"> <select id="order" name="order"><option selected value="-announced_date_first">Announcement date (newest first)</option><option value="announced_date_first">Announcement date (oldest first)</option><option value="-submitted_date">Submission date (newest first)</option><option value="submitted_date">Submission date (oldest first)</option><option value="">Relevance</option></select> </span> </div> <div class="control"> <button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> <a href="" class="pagination-previous is-invisible">Previous </a> <a href="/search/?searchtype=author&query=Wang%2C+M&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Wang%2C+M&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Wang%2C+M&start=50" class="pagination-link " aria-label="Page 2" aria-current="page">2 </a> </li> <li> <a href="/search/?searchtype=author&query=Wang%2C+M&start=100" class="pagination-link " aria-label="Page 3" aria-current="page">3 </a> </li> <li> <a href="/search/?searchtype=author&query=Wang%2C+M&start=150" class="pagination-link " aria-label="Page 4" aria-current="page">4 </a> </li> <li> <a href="/search/?searchtype=author&query=Wang%2C+M&start=200" class="pagination-link " aria-label="Page 5" aria-current="page">5 </a> </li> <li><span class="pagination-ellipsis">…</span></li> </ul> </nav> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.11573">arXiv:2411.11573</a> <span> [<a href="https://arxiv.org/pdf/2411.11573">pdf</a>, <a href="https://arxiv.org/ps/2411.11573">ps</a>, <a href="https://arxiv.org/format/2411.11573">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"> Observability inequality, log-type Hausdorff content and heat equations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Huang%2C+S">Shanlin Huang</a>, <a href="/search/math?searchtype=author&query=Wang%2C+G">Gengsheng Wang</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Ming 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="2411.11573v1-abstract-short" style="display: inline;"> This paper studies observability inequalities for heat equations on both bounded domains and the whole space $\mathbb{R}^d$. The observation sets are measured by log-type Hausdorff contents, which are induced by certain log-type gauge functions closely related to the heat kernel. On a bounded domain, we derive the observability inequality for observation sets of positive log-type Hausdorff content… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.11573v1-abstract-full').style.display = 'inline'; document.getElementById('2411.11573v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.11573v1-abstract-full" style="display: none;"> This paper studies observability inequalities for heat equations on both bounded domains and the whole space $\mathbb{R}^d$. The observation sets are measured by log-type Hausdorff contents, which are induced by certain log-type gauge functions closely related to the heat kernel. On a bounded domain, we derive the observability inequality for observation sets of positive log-type Hausdorff content. Notably, the aforementioned inequality holds not only for all sets with Hausdorff dimension $s$ for any $s\in (d-1,d]$, but also for certain sets of Hausdorff dimension $d-1$. On the whole space $\mathbb{R}^d$, we establish the observability inequality for observation sets that are thick at the scale of the log-type Hausdorff content. Furthermore, we prove that for the 1-dimensional heat equation on an interval, the Hausdorff content we have chosen is an optimal scale for the observability inequality. To obtain these observability inequalities, we use the adapted Lebeau-Robiano strategy from \cite{Duyckaerts2012resolvent}. For this purpose, we prove the following results at scale of the log-type Hausdorff content, the former being derived from the latter: We establish a spectral inequality/a Logvinenko-Sereda uncertainty principle; we set up a quantitative propagation of smallness of analytic functions; we build up a Remez' inequality; and more fundamentally, we provide an upper bound for the log-type Hausdorff content of a set where a monic polynomial is small, based on an estimate in Lubinsky \cite{Lubinsky1997small}, which is ultimately traced back to the classical Cartan Lemma. In addition, we set up a capacity-based slicing lemma (related to the log-type gauge functions) and establish a quantitative relationship between Hausdorff contents and capacities. These tools are crucial in the studies of the aforementioned propagation of smallness in high-dimensional situations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.11573v1-abstract-full').style.display = 'none'; document.getElementById('2411.11573v1-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> <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">56pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.10830">arXiv:2411.10830</a> <span> [<a href="https://arxiv.org/pdf/2411.10830">pdf</a>, <a href="https://arxiv.org/format/2411.10830">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="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> One-Layer Transformer Provably Learns One-Nearest Neighbor In Context </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+Z">Zihao Li</a>, <a href="/search/math?searchtype=author&query=Cao%2C+Y">Yuan Cao</a>, <a href="/search/math?searchtype=author&query=Gao%2C+C">Cheng Gao</a>, <a href="/search/math?searchtype=author&query=He%2C+Y">Yihan He</a>, <a href="/search/math?searchtype=author&query=Liu%2C+H">Han Liu</a>, <a href="/search/math?searchtype=author&query=Klusowski%2C+J+M">Jason M. Klusowski</a>, <a href="/search/math?searchtype=author&query=Fan%2C+J">Jianqing Fan</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Mengdi 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="2411.10830v1-abstract-short" style="display: inline;"> Transformers have achieved great success in recent years. Interestingly, transformers have shown particularly strong in-context learning capability -- even without fine-tuning, they are still able to solve unseen tasks well purely based on task-specific prompts. In this paper, we study the capability of one-layer transformers in learning one of the most classical nonparametric estimators, the one-… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.10830v1-abstract-full').style.display = 'inline'; document.getElementById('2411.10830v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.10830v1-abstract-full" style="display: none;"> Transformers have achieved great success in recent years. Interestingly, transformers have shown particularly strong in-context learning capability -- even without fine-tuning, they are still able to solve unseen tasks well purely based on task-specific prompts. In this paper, we study the capability of one-layer transformers in learning one of the most classical nonparametric estimators, the one-nearest neighbor prediction rule. Under a theoretical framework where the prompt contains a sequence of labeled training data and unlabeled test data, we show that, although the loss function is nonconvex when trained with gradient descent, a single softmax attention layer can successfully learn to behave like a one-nearest neighbor classifier. Our result gives a concrete example of how transformers can be trained to implement nonparametric machine learning algorithms, and sheds light on the role of softmax attention in transformer models. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.10830v1-abstract-full').style.display = 'none'; document.getElementById('2411.10830v1-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 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/2411.02918">arXiv:2411.02918</a> <span> [<a href="https://arxiv.org/pdf/2411.02918">pdf</a>, <a href="https://arxiv.org/format/2411.02918">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 minimum number of maximal dissociation sets in unicyclic graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Zhang%2C+J">Junxia Zhang</a>, <a href="/search/math?searchtype=author&query=Ren%2C+X">Xiangyu Ren</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Maoqun 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="2411.02918v1-abstract-short" style="display: inline;"> A subset of vertices in a graph $G$ is considered a maximal dissociation set if it induces a subgraph with vertex degree at most 1 and it is not contained within any other dissociation sets. In this paper, it is shown that for $n\geq 3$, every unicyclic graph contains a minimum of $\lfloor n/2\rfloor+2$ maximal dissociation sets. We also show the graphs that attain this minimum bound. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.02918v1-abstract-full" style="display: none;"> A subset of vertices in a graph $G$ is considered a maximal dissociation set if it induces a subgraph with vertex degree at most 1 and it is not contained within any other dissociation sets. In this paper, it is shown that for $n\geq 3$, every unicyclic graph contains a minimum of $\lfloor n/2\rfloor+2$ maximal dissociation sets. We also show the graphs that attain this minimum bound. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.02918v1-abstract-full').style.display = 'none'; document.getElementById('2411.02918v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">13 pages, 7 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.23610">arXiv:2410.23610</a> <span> [<a href="https://arxiv.org/pdf/2410.23610">pdf</a>, <a href="https://arxiv.org/format/2410.23610">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> </div> <p class="title is-5 mathjax"> Global Convergence in Training Large-Scale Transformers </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gao%2C+C">Cheng Gao</a>, <a href="/search/math?searchtype=author&query=Cao%2C+Y">Yuan Cao</a>, <a href="/search/math?searchtype=author&query=Li%2C+Z">Zihao Li</a>, <a href="/search/math?searchtype=author&query=He%2C+Y">Yihan He</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Mengdi Wang</a>, <a href="/search/math?searchtype=author&query=Liu%2C+H">Han Liu</a>, <a href="/search/math?searchtype=author&query=Klusowski%2C+J+M">Jason Matthew Klusowski</a>, <a href="/search/math?searchtype=author&query=Fan%2C+J">Jianqing 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="2410.23610v1-abstract-short" style="display: inline;"> Despite the widespread success of Transformers across various domains, their optimization guarantees in large-scale model settings are not well-understood. This paper rigorously analyzes the convergence properties of gradient flow in training Transformers with weight decay regularization. First, we construct the mean-field limit of large-scale Transformers, showing that as the model width and dept… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.23610v1-abstract-full').style.display = 'inline'; document.getElementById('2410.23610v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.23610v1-abstract-full" style="display: none;"> Despite the widespread success of Transformers across various domains, their optimization guarantees in large-scale model settings are not well-understood. This paper rigorously analyzes the convergence properties of gradient flow in training Transformers with weight decay regularization. First, we construct the mean-field limit of large-scale Transformers, showing that as the model width and depth go to infinity, gradient flow converges to the Wasserstein gradient flow, which is represented by a partial differential equation. Then, we demonstrate that the gradient flow reaches a global minimum consistent with the PDE solution when the weight decay regularization parameter is sufficiently small. Our analysis is based on a series of novel mean-field techniques that adapt to Transformers. Compared with existing tools for deep networks (Lu et al., 2020) that demand homogeneity and global Lipschitz smoothness, we utilize a refined analysis assuming only $\textit{partial homogeneity}$ and $\textit{local Lipschitz smoothness}$. These new techniques may be of independent interest. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.23610v1-abstract-full').style.display = 'none'; document.getElementById('2410.23610v1-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 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">to be published in 38th Conference on Neural Information Processing Systems (NeurIPS 2024)</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35Q93 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.22759">arXiv:2410.22759</a> <span> [<a href="https://arxiv.org/pdf/2410.22759">pdf</a>, <a href="https://arxiv.org/format/2410.22759">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"> Mapped Hermite Functions and their applications to two-dimensional weakly singular Fredholm-Hammerstein integral equations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+M">Min Wang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Z">Zhimin 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="2410.22759v1-abstract-short" style="display: inline;"> The Fredholm-Hammerstein integral equations (FHIEs) with weakly singular kernels exhibit multi-point singularity at the endpoints or boundaries. The dense discretized matrices result in high computational complexity when employing numerical methods. To address this, we propose a novel class of mapped Hermite functions, which are constructed by applying a mapping to Hermite polynomials.We establish… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.22759v1-abstract-full').style.display = 'inline'; document.getElementById('2410.22759v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.22759v1-abstract-full" style="display: none;"> The Fredholm-Hammerstein integral equations (FHIEs) with weakly singular kernels exhibit multi-point singularity at the endpoints or boundaries. The dense discretized matrices result in high computational complexity when employing numerical methods. To address this, we propose a novel class of mapped Hermite functions, which are constructed by applying a mapping to Hermite polynomials.We establish fundamental approximation theory for the orthogonal functions. We propose MHFs-spectral collocation method and MHFs-smoothing transformation method to solve the two-point weakly singular FHIEs, respectively. Error analysis and numerical results demonstrate that our methods, based on the new orthogonal functions, are particularly effective for handling problems with weak singularities at two endpoints, yielding exponential convergence rate. We position this work as the first to directly study the mapped spectral method for multi-point singularity problems, to the best of our knowledge. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.22759v1-abstract-full').style.display = 'none'; document.getElementById('2410.22759v1-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 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/2410.19330">arXiv:2410.19330</a> <span> [<a href="https://arxiv.org/pdf/2410.19330">pdf</a>, <a href="https://arxiv.org/ps/2410.19330">ps</a>, <a href="https://arxiv.org/format/2410.19330">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"> Moments of Gamma type and three-parametric Mittag-Leffler function </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+M">Min 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="2410.19330v1-abstract-short" style="display: inline;"> We study a class of positive random variables having moments of Gamma type, whose density can be expressed by the three-parametric Mittag-Leffler functions. We give some necessary conditions and some sufficient conditions for their existence. As a corollary, we give some conditions for non-negativity of the three-parametric Mittag-Leffler functions. As an application, we study the infinite divisib… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.19330v1-abstract-full').style.display = 'inline'; document.getElementById('2410.19330v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.19330v1-abstract-full" style="display: none;"> We study a class of positive random variables having moments of Gamma type, whose density can be expressed by the three-parametric Mittag-Leffler functions. We give some necessary conditions and some sufficient conditions for their existence. As a corollary, we give some conditions for non-negativity of the three-parametric Mittag-Leffler functions. As an application, we study the infinite divisibility of the powers of half $\a$-Cauchy variable. In addition, we find that a random variable $\X$ having moment of Gamma type if and only if $\log \X$ is quasi infinitely divisible. From this perspective, we can solve many Hausdorff moment problems of sequences of factorial ratios. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.19330v1-abstract-full').style.display = 'none'; document.getElementById('2410.19330v1-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 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">12 pages, comments welcome</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 33E12; 60E10; 60E07; 60E05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.11474">arXiv:2410.11474</a> <span> [<a href="https://arxiv.org/pdf/2410.11474">pdf</a>, <a href="https://arxiv.org/format/2410.11474">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> <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"> How Transformers Implement Induction Heads: Approximation and Optimization Analysis </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+M">Mingze Wang</a>, <a href="/search/math?searchtype=author&query=Yu%2C+R">Ruoxi Yu</a>, <a href="/search/math?searchtype=author&query=E%2C+W">Weinan E</a>, <a href="/search/math?searchtype=author&query=Wu%2C+L">Lei Wu</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="2410.11474v2-abstract-short" style="display: inline;"> Transformers have demonstrated exceptional in-context learning capabilities, yet the theoretical understanding of the underlying mechanisms remain limited. A recent work (Elhage et al., 2021) identified a "rich" in-context mechanism known as induction head, contrasting with "lazy" $n$-gram models that overlook long-range dependencies. In this work, we provide both approximation and optimization an… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.11474v2-abstract-full').style.display = 'inline'; document.getElementById('2410.11474v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.11474v2-abstract-full" style="display: none;"> Transformers have demonstrated exceptional in-context learning capabilities, yet the theoretical understanding of the underlying mechanisms remain limited. A recent work (Elhage et al., 2021) identified a "rich" in-context mechanism known as induction head, contrasting with "lazy" $n$-gram models that overlook long-range dependencies. In this work, we provide both approximation and optimization analyses of how transformers implement induction heads. In the approximation analysis, we formalize both standard and generalized induction head mechanisms, and examine how transformers can efficiently implement them, with an emphasis on the distinct role of each transformer submodule. For the optimization analysis, we study the training dynamics on a synthetic mixed target, composed of a 4-gram and an in-context 2-gram component. This setting enables us to precisely characterize the entire training process and uncover an {\em abrupt transition} from lazy (4-gram) to rich (induction head) mechanisms as training progresses. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.11474v2-abstract-full').style.display = 'none'; document.getElementById('2410.11474v2-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 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">39 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.05253">arXiv:2410.05253</a> <span> [<a href="https://arxiv.org/pdf/2410.05253">pdf</a>, <a href="https://arxiv.org/format/2410.05253">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"> Multicontinuum splitting scheme for multiscale flow problems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Efendiev%2C+Y">Yalchin Efendiev</a>, <a href="/search/math?searchtype=author&query=Leung%2C+W+T">Wing Tat Leung</a>, <a href="/search/math?searchtype=author&query=Shan%2C+B">Buzheng Shan</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Min 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="2410.05253v1-abstract-short" style="display: inline;"> In this paper, we propose multicontinuum splitting schemes for multiscale problems, focusing on a parabolic equation with a high-contrast coefficient. Using the framework of multicontinuum homogenization, we introduce spatially smooth macroscopic variables and decompose the multicontinuum solution space into two components to effectively separate the dynamics at different speeds (or the effects of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.05253v1-abstract-full').style.display = 'inline'; document.getElementById('2410.05253v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.05253v1-abstract-full" style="display: none;"> In this paper, we propose multicontinuum splitting schemes for multiscale problems, focusing on a parabolic equation with a high-contrast coefficient. Using the framework of multicontinuum homogenization, we introduce spatially smooth macroscopic variables and decompose the multicontinuum solution space into two components to effectively separate the dynamics at different speeds (or the effects of contrast in high-contrast cases). By treating the component containing fast dynamics (or dependent on the contrast) implicitly and the component containing slow dynamics (or independent of the contrast) explicitly, we construct partially explicit time discretization schemes, which can reduce computational cost. The derived stability conditions are contrast-independent, provided the continua are chosen appropriately. Additionally, we discuss possible methods to obtain an optimized decomposition of the solution space, which relaxes the stability conditions while enhancing computational efficiency. A Rayleigh quotient problem in tensor form is formulated, and simplifications are achieved under certain assumptions. Finally, we present numerical results for various coefficient fields and different continua to validate our proposed approach. It can be observed that the multicontinuum splitting schemes enjoy high accuracy and efficiency. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.05253v1-abstract-full').style.display = 'none'; document.getElementById('2410.05253v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.16101">arXiv:2409.16101</a> <span> [<a href="https://arxiv.org/pdf/2409.16101">pdf</a>, <a href="https://arxiv.org/ps/2409.16101">ps</a>, <a href="https://arxiv.org/format/2409.16101">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"> Spreading dynamics of a Fisher-KPP nonlocal diffusion model with a free boundary </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+L">Lei Li</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Mingxin 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="2409.16101v1-abstract-short" style="display: inline;"> This paper concerns the spreading speed and asymptotical behaviors, which was left as an open problem in \cite{LLW22}, of a Fisher-KPP nonlocal diffusion model with a free boundary. Using a new lower solution, we get the exact finite spreading speed of free boundary that is proved to be the asymptotical spreading speed of solution component $u$. Moreover, for the algebraic decay kernels, we derive… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.16101v1-abstract-full').style.display = 'inline'; document.getElementById('2409.16101v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.16101v1-abstract-full" style="display: none;"> This paper concerns the spreading speed and asymptotical behaviors, which was left as an open problem in \cite{LLW22}, of a Fisher-KPP nonlocal diffusion model with a free boundary. Using a new lower solution, we get the exact finite spreading speed of free boundary that is proved to be the asymptotical spreading speed of solution component $u$. Moreover, for the algebraic decay kernels, we derive the rates of accelerated spreading and the corresponding asymptotical behaviors of solution component $u$. Especially, for the level set $E^位(t)$ with $位\in(0,u^*)$, we find an interesting propagation phenomenon different from the double free boundary problem. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.16101v1-abstract-full').style.display = 'none'; document.getElementById('2409.16101v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 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/2409.07027">arXiv:2409.07027</a> <span> [<a href="https://arxiv.org/pdf/2409.07027">pdf</a>, <a href="https://arxiv.org/ps/2409.07027">ps</a>, <a href="https://arxiv.org/format/2409.07027">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"> Log-type ultra-analyticity of elliptic equations with gradient terms </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Dong%2C+H">Hongjie Dong</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Ming 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="2409.07027v1-abstract-short" style="display: inline;"> It is well known that every solution of an elliptic equation is analytic if its coefficients are analytic. However, less is known about the ultra-analyticity of such solutions. This work addresses the problem of elliptic equations with lower-order terms, where the coefficients are entire functions of exponential type. We prove that every solution satisfies a quantitative logarithmic ultra-analytic… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.07027v1-abstract-full').style.display = 'inline'; document.getElementById('2409.07027v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.07027v1-abstract-full" style="display: none;"> It is well known that every solution of an elliptic equation is analytic if its coefficients are analytic. However, less is known about the ultra-analyticity of such solutions. This work addresses the problem of elliptic equations with lower-order terms, where the coefficients are entire functions of exponential type. We prove that every solution satisfies a quantitative logarithmic ultra-analytic bound and demonstrate that this bound is sharp. The results suggest that the ultra-analyticity of solutions to elliptic equations cannot be expected to achieve the same level of ultra-analyticity as the coefficients. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.07027v1-abstract-full').style.display = 'none'; document.getElementById('2409.07027v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> <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">24 pages, submitted</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35J15; 26E05; 35A20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.04991">arXiv:2409.04991</a> <span> [<a href="https://arxiv.org/pdf/2409.04991">pdf</a>, <a href="https://arxiv.org/ps/2409.04991">ps</a>, <a href="https://arxiv.org/format/2409.04991">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"> Error estimates of the Euler's method for stochastic differential equations with multiplicative noise via relative entropy </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+L">Lei Li</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Mengchao Wang</a>, <a href="/search/math?searchtype=author&query=Wang%2C+Y">Yuliang 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="2409.04991v2-abstract-short" style="display: inline;"> We investigate the sharp error estimate of the density under the relative entropy (or Kullback-Leibler divergence) for the traditional Euler-Maruyama discretization of stochastic differential equations (SDEs) with multiplicative noise. The foundation of the proof is the estimates of the derivatives for the logarithmic numerical density. The key technique is to adopt the Malliavin calculus to get t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.04991v2-abstract-full').style.display = 'inline'; document.getElementById('2409.04991v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.04991v2-abstract-full" style="display: none;"> We investigate the sharp error estimate of the density under the relative entropy (or Kullback-Leibler divergence) for the traditional Euler-Maruyama discretization of stochastic differential equations (SDEs) with multiplicative noise. The foundation of the proof is the estimates of the derivatives for the logarithmic numerical density. The key technique is to adopt the Malliavin calculus to get the expressions of the derivatives of the logarithmic Green's function and to obtain an estimate for the inverse Malliavin matrix. The estimate of relative entropy then naturally gives sharp error bounds under total variation distance and Wasserstein distances. Compared to the usual weak error estimate for SDEs, such estimate can give an error bound for a family of test functions instead of one test function. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.04991v2-abstract-full').style.display = 'none'; document.getElementById('2409.04991v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 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/2409.03967">arXiv:2409.03967</a> <span> [<a href="https://arxiv.org/pdf/2409.03967">pdf</a>, <a href="https://arxiv.org/format/2409.03967">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> </div> <p class="title is-5 mathjax"> Covers of surfaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Biringer%2C+I">Ian Biringer</a>, <a href="/search/math?searchtype=author&query=Chandran%2C+Y">Yassin Chandran</a>, <a href="/search/math?searchtype=author&query=Cremaschi%2C+T">Tommaso Cremaschi</a>, <a href="/search/math?searchtype=author&query=Tao%2C+J">Jing Tao</a>, <a href="/search/math?searchtype=author&query=Vlamis%2C+N+G">Nicholas G. Vlamis</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Mujie Wang</a>, <a href="/search/math?searchtype=author&query=Whitfield%2C+B">Brandis Whitfield</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="2409.03967v2-abstract-short" style="display: inline;"> We study the homeomorphism types of certain covers of (always orientable) surfaces, usually of infinite-type. We show that every surface with non-abelian fundamental group is covered by every noncompact surface, we identify the universal abelian covers and the $\mathbb{Z}/n\mathbb{Z}$-homology covers of surfaces, and we show that non-locally finite characteristic covers of surfaces have four possi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.03967v2-abstract-full').style.display = 'inline'; document.getElementById('2409.03967v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.03967v2-abstract-full" style="display: none;"> We study the homeomorphism types of certain covers of (always orientable) surfaces, usually of infinite-type. We show that every surface with non-abelian fundamental group is covered by every noncompact surface, we identify the universal abelian covers and the $\mathbb{Z}/n\mathbb{Z}$-homology covers of surfaces, and we show that non-locally finite characteristic covers of surfaces have four possible homeomorphism types. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.03967v2-abstract-full').style.display = 'none'; document.getElementById('2409.03967v2-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">v1</span> submitted 5 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">added a reference</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.02947">arXiv:2409.02947</a> <span> [<a href="https://arxiv.org/pdf/2409.02947">pdf</a>, <a href="https://arxiv.org/ps/2409.02947">ps</a>, <a href="https://arxiv.org/format/2409.02947">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Mathematics">math.GM</span> </div> </div> <p class="title is-5 mathjax"> Metric dimensions of bicyclic graphs with potential applications in Supply Chain Logistics </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+M">Muwen Wang</a>, <a href="/search/math?searchtype=author&query=Haidar%2C+G">Ghulam Haidar</a>, <a href="/search/math?searchtype=author&query=Yousafzai%2C+F">Faisal Yousafzai</a>, <a href="/search/math?searchtype=author&query=Khan%2C+M+U+I">Murad Ul Islam Khan</a>, <a href="/search/math?searchtype=author&query=Sikandar%2C+W">Waseem Sikandar</a>, <a href="/search/math?searchtype=author&query=Khan%2C+A+U+I">Asad Ul Islam Khan</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="2409.02947v1-abstract-short" style="display: inline;"> Metric dimensions and metric basis are graph invariants studied for their use in locating and indexing nodes in a graph. It was recently established that for bicyclic graph of type-III ($螛$-graphs), the metric dimension is $3$ only, when all paths have equal lengths, or when one of the outside path has a length $2$ more than the other two paths. In this article, we refute this claim and show that… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.02947v1-abstract-full').style.display = 'inline'; document.getElementById('2409.02947v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.02947v1-abstract-full" style="display: none;"> Metric dimensions and metric basis are graph invariants studied for their use in locating and indexing nodes in a graph. It was recently established that for bicyclic graph of type-III ($螛$-graphs), the metric dimension is $3$ only, when all paths have equal lengths, or when one of the outside path has a length $2$ more than the other two paths. In this article, we refute this claim and show that the case where the middle path is $2$ vertices more than the other two paths, also has metric dimension $3$. We also determine the metric dimension for other values of $p,q,r$ which were omitted in the recent research due to the constraint $p \leq q \leq r$. We also propose a graph-based technique to transform an agricultural supply chain logistics problem into a mathematical model, by using metric basis and metric dimensions. We provide a theoretical groundwork which can be used to model and solve these problems using machine learning algorithms. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.02947v1-abstract-full').style.display = 'none'; document.getElementById('2409.02947v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">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> 05C12; 05C90 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2408.13726">arXiv:2408.13726</a> <span> [<a href="https://arxiv.org/pdf/2408.13726">pdf</a>, <a href="https://arxiv.org/ps/2408.13726">ps</a>, <a href="https://arxiv.org/format/2408.13726">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"> Weighted norm inequalities of various square functions and Volterra integral operators on the unit ball </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Pang%2C+C">Changbao Pang</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Maofa Wang</a>, <a href="/search/math?searchtype=author&query=Xu%2C+B">Bang Xu</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+H">Hao 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="2408.13726v2-abstract-short" style="display: inline;"> In this paper, we investigate various square functions on the unit complex ball. We prove the weighted inequalities of the Lusin area integral associated with Poisson integral in terms of $A_p$ weights for all $1<p<\infty$; this gives an affirmative answer to an open question raised by Segovia and Wheeden. In addition, we get an equivalent characterization of weighted Hardy spaces by means of the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.13726v2-abstract-full').style.display = 'inline'; document.getElementById('2408.13726v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.13726v2-abstract-full" style="display: none;"> In this paper, we investigate various square functions on the unit complex ball. We prove the weighted inequalities of the Lusin area integral associated with Poisson integral in terms of $A_p$ weights for all $1<p<\infty$; this gives an affirmative answer to an open question raised by Segovia and Wheeden. In addition, we get an equivalent characterization of weighted Hardy spaces by means of the Lusin area integral in the context of holomorphic functions. We also obtain the weighted inequalities for Volterra integral operators. The key ingredients of our proof involve complex analysis, Calder贸n-Zygmund theory, the local mean oscillation technique and sparse domination. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.13726v2-abstract-full').style.display = 'none'; document.getElementById('2408.13726v2-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 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 25 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">43 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2408.13537">arXiv:2408.13537</a> <span> [<a href="https://arxiv.org/pdf/2408.13537">pdf</a>, <a href="https://arxiv.org/ps/2408.13537">ps</a>, <a href="https://arxiv.org/format/2408.13537">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> </div> </div> <p class="title is-5 mathjax"> Fock projections on vector-valued $L^p$-spaces with matrix weights </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+J">Jiale Chen</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Maofa 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="2408.13537v1-abstract-short" style="display: inline;"> In this paper, we characterize the $d\times d$ matrix weights $W$ on $\mathbb{C}^n$ such that the Fock projection $P_伪$ is bounded on the vector-valued spaces $L^p_{伪,W}(\mathbb{C}^n;\mathbb{C}^d)$ induced by $W$. It is proved that for $1\leq p<\infty$, the Fock projection $P_伪$ is bounded on $L^p_{伪,W}(\mathbb{C}^n;\mathbb{C}^d)$ if and only if $W$ satisfies a restricted $\mathbf{A}_p$-condition.… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.13537v1-abstract-full').style.display = 'inline'; document.getElementById('2408.13537v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.13537v1-abstract-full" style="display: none;"> In this paper, we characterize the $d\times d$ matrix weights $W$ on $\mathbb{C}^n$ such that the Fock projection $P_伪$ is bounded on the vector-valued spaces $L^p_{伪,W}(\mathbb{C}^n;\mathbb{C}^d)$ induced by $W$. It is proved that for $1\leq p<\infty$, the Fock projection $P_伪$ is bounded on $L^p_{伪,W}(\mathbb{C}^n;\mathbb{C}^d)$ if and only if $W$ satisfies a restricted $\mathbf{A}_p$-condition. In particular, when $p=1$, our result establishes a strong (1,1) type estimate for the Fock projections, which is quite different with the case of Calder贸n--Zygmund operators. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.13537v1-abstract-full').style.display = 'none'; document.getElementById('2408.13537v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">19 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2408.05224">arXiv:2408.05224</a> <span> [<a href="https://arxiv.org/pdf/2408.05224">pdf</a>, <a href="https://arxiv.org/ps/2408.05224">ps</a>, <a href="https://arxiv.org/format/2408.05224">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Biomolecules">q-bio.BM</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"> Optimal Strategy for Stabilizing Protein Folding Intermediates </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+M">Mengshou Wang</a>, <a href="/search/math?searchtype=author&query=Pengb%2C+L">Liangrong Pengb</a>, <a href="/search/math?searchtype=author&query=Jia%2C+B">Baoguo Jia</a>, <a href="/search/math?searchtype=author&query=Hong%2C+L">Liu Hong</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="2408.05224v1-abstract-short" style="display: inline;"> To manipulate the protein population at certain functional state through chemical stabilizers is crucial for protein-related studies. It not only plays a key role in protein structure analysis and protein folding kinetics, but also affects protein functionality to a large extent and thus has wide applications in medicine, food industry, etc. However, due to concerns about side effects or financial… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.05224v1-abstract-full').style.display = 'inline'; document.getElementById('2408.05224v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.05224v1-abstract-full" style="display: none;"> To manipulate the protein population at certain functional state through chemical stabilizers is crucial for protein-related studies. It not only plays a key role in protein structure analysis and protein folding kinetics, but also affects protein functionality to a large extent and thus has wide applications in medicine, food industry, etc. However, due to concerns about side effects or financial costs of stabilizers, identifying optimal strategies for enhancing protein stability with a minimal amount of stabilizers is of great importance. Here we prove that either for the fixed terminal time (including both finite and infinite cases) or the free one, the optimal control strategy for stabilizing the folding intermediates with a linear strategy for stabilizer addition belongs to the class of Bang-Bang controls. The corresponding optimal switching time is derived analytically, whose phase diagram with respect to several key parameters is explored in detail. The Bang-Bang control will be broken when nonlinear strategies for stabilizer addition are adopted. Our current study on optimal strategies for protein stabilizers not only offers deep insights into the general picture of protein folding kinetics, but also provides valuable theoretical guidance on treatments for protein-related diseases in medicine. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.05224v1-abstract-full').style.display = 'none'; document.getElementById('2408.05224v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">19 pages, 5 figures, 2 tables</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 34Hxx; 92Cxx </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.16925">arXiv:2407.16925</a> <span> [<a href="https://arxiv.org/pdf/2407.16925">pdf</a>, <a href="https://arxiv.org/ps/2407.16925">ps</a>, <a href="https://arxiv.org/format/2407.16925">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"> Randomized dual singular value decomposition </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+M">Mengyu Wang</a>, <a href="/search/math?searchtype=author&query=Zhou%2C+J">Jingchun Zhou</a>, <a href="/search/math?searchtype=author&query=Li%2C+H">Hanyu 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="2407.16925v1-abstract-short" style="display: inline;"> We first propose a concise singular value decomposition of dual matrices. Then, the randomized version of the decomposition is presented. It can significantly reduce the computational cost while maintaining the similar accuracy. We analyze the theoretical properties and illuminate the numerical performance of the randomized algorithm. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.16925v1-abstract-full" style="display: none;"> We first propose a concise singular value decomposition of dual matrices. Then, the randomized version of the decomposition is presented. It can significantly reduce the computational cost while maintaining the similar accuracy. We analyze the theoretical properties and illuminate the numerical performance of the randomized algorithm. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.16925v1-abstract-full').style.display = 'none'; document.getElementById('2407.16925v1-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 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.16134">arXiv:2407.16134</a> <span> [<a href="https://arxiv.org/pdf/2407.16134">pdf</a>, <a href="https://arxiv.org/format/2407.16134">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="Statistics Theory">math.ST</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"> Diffusion Transformer Captures Spatial-Temporal Dependencies: A Theory for Gaussian Process Data </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+H">Hengyu Fu</a>, <a href="/search/math?searchtype=author&query=Dou%2C+Z">Zehao Dou</a>, <a href="/search/math?searchtype=author&query=Guo%2C+J">Jiawei Guo</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Mengdi Wang</a>, <a href="/search/math?searchtype=author&query=Chen%2C+M">Minshuo Chen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.16134v1-abstract-short" style="display: inline;"> Diffusion Transformer, the backbone of Sora for video generation, successfully scales the capacity of diffusion models, pioneering new avenues for high-fidelity sequential data generation. Unlike static data such as images, sequential data consists of consecutive data frames indexed by time, exhibiting rich spatial and temporal dependencies. These dependencies represent the underlying dynamic mode… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.16134v1-abstract-full').style.display = 'inline'; document.getElementById('2407.16134v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.16134v1-abstract-full" style="display: none;"> Diffusion Transformer, the backbone of Sora for video generation, successfully scales the capacity of diffusion models, pioneering new avenues for high-fidelity sequential data generation. Unlike static data such as images, sequential data consists of consecutive data frames indexed by time, exhibiting rich spatial and temporal dependencies. These dependencies represent the underlying dynamic model and are critical to validate the generated data. In this paper, we make the first theoretical step towards bridging diffusion transformers for capturing spatial-temporal dependencies. Specifically, we establish score approximation and distribution estimation guarantees of diffusion transformers for learning Gaussian process data with covariance functions of various decay patterns. We highlight how the spatial-temporal dependencies are captured and affect learning efficiency. Our study proposes a novel transformer approximation theory, where the transformer acts to unroll an algorithm. We support our theoretical results by numerical experiments, providing strong evidence that spatial-temporal dependencies are captured within attention layers, aligning with our approximation theory. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.16134v1-abstract-full').style.display = 'none'; document.getElementById('2407.16134v1-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 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">52 pages, 8 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.15664">arXiv:2407.15664</a> <span> [<a href="https://arxiv.org/pdf/2407.15664">pdf</a>, <a href="https://arxiv.org/ps/2407.15664">ps</a>, <a href="https://arxiv.org/format/2407.15664">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> <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 new properties of the beta function and Ramanujan R-function </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yang%2C+Z">Zhen-Hang Yang</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Miao-Kun Wang</a>, <a href="/search/math?searchtype=author&query=Zhao%2C+T">Tie-Hong Zhao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.15664v1-abstract-short" style="display: inline;"> In this paper, the power series and hypergeometric series representations of the beta and Ramanujan functions \begin{equation*} \mathcal{B}\left( x\right) =\frac{螕\left( x\right)^{2}}{螕\left( 2x\right) }\text{ and }\mathcal{R}\left( x\right) =-2蠄\left( x\right) -2纬\end{equation*} are presented, which yield higher order monotonicity results related to $ \mathcal{B}(x)$ and $\mathcal{R}(x)$; the dec… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.15664v1-abstract-full').style.display = 'inline'; document.getElementById('2407.15664v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.15664v1-abstract-full" style="display: none;"> In this paper, the power series and hypergeometric series representations of the beta and Ramanujan functions \begin{equation*} \mathcal{B}\left( x\right) =\frac{螕\left( x\right)^{2}}{螕\left( 2x\right) }\text{ and }\mathcal{R}\left( x\right) =-2蠄\left( x\right) -2纬\end{equation*} are presented, which yield higher order monotonicity results related to $ \mathcal{B}(x)$ and $\mathcal{R}(x)$; the decreasing property of the functions $\mathcal{R}\left( x\right) /\mathcal{B}\left( x\right) $ and $[ \mathcal{B}(x) -\mathcal{R}(x)] /x^{2}$ on $\left( 0,\infty \right)$ are proved. Moreover, a conjecture put forward by Qiu et al. in [17] is proved to be true. As applications, several inequalities and identities are deduced. These results obtained in this paper may be helpful for the study of certain special functions. Finally, an interesting infinite series similar to Riemann zeta functions is observed initially. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.15664v1-abstract-full').style.display = 'none'; document.getElementById('2407.15664v1-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 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">18 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 33B15; 33C05; 11M06; 30B10; 26A48 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.15361">arXiv:2407.15361</a> <span> [<a href="https://arxiv.org/pdf/2407.15361">pdf</a>, <a href="https://arxiv.org/ps/2407.15361">ps</a>, <a href="https://arxiv.org/format/2407.15361">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 vector-host epidemic model with spatial structure and seasonality </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+M">Mingxin Wang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Q">Qianying Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.15361v1-abstract-short" style="display: inline;"> Recently, Li and Zhao [5] (Bull. Math. Biol., 83(5), 43, 25 pp (2021)) proposed and studied a periodic reaction-diffusion model of Zika virus with seasonality and spatial heterogeneous structure in host and vector populations. They found the basic reproduction ratio R0, which is a threshold parameter. In this short paper we shall use the upper and lower solutions method to study the model of [5] w… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.15361v1-abstract-full').style.display = 'inline'; document.getElementById('2407.15361v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.15361v1-abstract-full" style="display: none;"> Recently, Li and Zhao [5] (Bull. Math. Biol., 83(5), 43, 25 pp (2021)) proposed and studied a periodic reaction-diffusion model of Zika virus with seasonality and spatial heterogeneous structure in host and vector populations. They found the basic reproduction ratio R0, which is a threshold parameter. In this short paper we shall use the upper and lower solutions method to study the model of [5] with Neumann boundary conditions replaced by general boundary conditions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.15361v1-abstract-full').style.display = 'none'; document.getElementById('2407.15361v1-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 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.11702">arXiv:2407.11702</a> <span> [<a href="https://arxiv.org/pdf/2407.11702">pdf</a>, <a href="https://arxiv.org/ps/2407.11702">ps</a>, <a href="https://arxiv.org/format/2407.11702">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"> Dynamics for a diffusive epidemic model with a free boundary: sharp asymptotic profile </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xueping Li</a>, <a href="/search/math?searchtype=author&query=Li%2C+L">Lei Li</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Mingxin Wang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.11702v1-abstract-short" style="display: inline;"> This paper concerns the sharp asymptotic profiles of the solution of a diffusive epidemic model with one free boundary and one fixed boundary which is subject to the homogeneous Dirichlet boundary condition and Neumann boundary condition, respectively. The longtime behaviors has been proved to be governed by a spreading-vanishing dichotomy in \cite{LL}, and when spreading happens, the spreading sp… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.11702v1-abstract-full').style.display = 'inline'; document.getElementById('2407.11702v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.11702v1-abstract-full" style="display: none;"> This paper concerns the sharp asymptotic profiles of the solution of a diffusive epidemic model with one free boundary and one fixed boundary which is subject to the homogeneous Dirichlet boundary condition and Neumann boundary condition, respectively. The longtime behaviors has been proved to be governed by a spreading-vanishing dichotomy in \cite{LL}, and when spreading happens, the spreading speed is determined in \cite{LLW}. In this paper, by constructing some subtle upper and lower solutions, as well as employing some detailed analysis, we improve the results in \cite{LLW} and obtain the sharp asymptotic spreading profiles, which show the homogeneous Dirichlet boundary condition and Neumann boundary condition imposed at the fixed boundary $x=0$ lead to the same asymptotic behaviors of $h(t)$ and $(u,v)$ near the spreading front $h(t)$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.11702v1-abstract-full').style.display = 'none'; document.getElementById('2407.11702v1-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 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.08928">arXiv:2407.08928</a> <span> [<a href="https://arxiv.org/pdf/2407.08928">pdf</a>, <a href="https://arxiv.org/ps/2407.08928">ps</a>, <a href="https://arxiv.org/format/2407.08928">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"> Dynamics for a diffusive epidemic model with a free boundary: spreading speed </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xueping Li</a>, <a href="/search/math?searchtype=author&query=Li%2C+L">Lei Li</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Mingxin Wang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.08928v1-abstract-short" style="display: inline;"> We study the spreading speed of a diffusive epidemic model proposed by Li et al. \cite{LL}, where the Stefan boundary condition is imposed at the right boundary, and the left boundary is subject to the homogeneous Dirichlet and Neumann condition, respectively. A spreading-vanishing dichotomy and some sharp criteria were obtained in \cite{LL}. In this paper, when spreading happens, we not only obta… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.08928v1-abstract-full').style.display = 'inline'; document.getElementById('2407.08928v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.08928v1-abstract-full" style="display: none;"> We study the spreading speed of a diffusive epidemic model proposed by Li et al. \cite{LL}, where the Stefan boundary condition is imposed at the right boundary, and the left boundary is subject to the homogeneous Dirichlet and Neumann condition, respectively. A spreading-vanishing dichotomy and some sharp criteria were obtained in \cite{LL}. In this paper, when spreading happens, we not only obtain the exact spreading speed of the spreading front described by the right boundary, but derive some sharp estimates on the asymptotical behavior of solution component $(u,v)$. Our arguments depend crucially on some detailed understandings for a corresponding semi-wave problem and a steady state problem. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.08928v1-abstract-full').style.display = 'none'; document.getElementById('2407.08928v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.08667">arXiv:2407.08667</a> <span> [<a href="https://arxiv.org/pdf/2407.08667">pdf</a>, <a href="https://arxiv.org/ps/2407.08667">ps</a>, <a href="https://arxiv.org/format/2407.08667">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Factorization algebras from topological-holomorphic field theories </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+M">Minghao Wang</a>, <a href="/search/math?searchtype=author&query=Williams%2C+B+R">Brian R. Williams</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.08667v2-abstract-short" style="display: inline;"> Topological field theories and holomorphic field theories naturally appear in both mathematics and physics. However, there exist intriguing hybrid theories that are topological in some directions and holomorphic in others, such as twists of supersymmetric field theories or Costello's 4-dimensional Chern-Simons theory. In this paper, we rigorously prove the ultraviolet (UV) finiteness for such hybr… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.08667v2-abstract-full').style.display = 'inline'; document.getElementById('2407.08667v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.08667v2-abstract-full" style="display: none;"> Topological field theories and holomorphic field theories naturally appear in both mathematics and physics. However, there exist intriguing hybrid theories that are topological in some directions and holomorphic in others, such as twists of supersymmetric field theories or Costello's 4-dimensional Chern-Simons theory. In this paper, we rigorously prove the ultraviolet (UV) finiteness for such hybrid theories on flat space, and present two significant vanishing results regarding anomalies. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.08667v2-abstract-full').style.display = 'none'; document.getElementById('2407.08667v2-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 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">51 pages, no figures. An error in the proof has been fixed. Corrected several typos. Comments 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/2407.02022">arXiv:2407.02022</a> <span> [<a href="https://arxiv.org/pdf/2407.02022">pdf</a>, <a href="https://arxiv.org/ps/2407.02022">ps</a>, <a href="https://arxiv.org/format/2407.02022">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Smooth deformation limit of Moishezon manifolds is Moishezon </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+M">Mu-lin Li</a>, <a href="/search/math?searchtype=author&query=Rao%2C+S">Sheng Rao</a>, <a href="/search/math?searchtype=author&query=Wang%2C+K">Kai Wang</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Meng-jiao Wang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.02022v1-abstract-short" style="display: inline;"> We prove the conjecture that the deformation limit of Moishezon manifolds under a smooth deformation over a unit disk in $\mathbb{C}$ is Moishezon. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.02022v1-abstract-full" style="display: none;"> We prove the conjecture that the deformation limit of Moishezon manifolds under a smooth deformation over a unit disk in $\mathbb{C}$ is Moishezon. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.02022v1-abstract-full').style.display = 'none'; document.getElementById('2407.02022v1-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 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">All 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/2406.17631">arXiv:2406.17631</a> <span> [<a href="https://arxiv.org/pdf/2406.17631">pdf</a>, <a href="https://arxiv.org/ps/2406.17631">ps</a>, <a href="https://arxiv.org/format/2406.17631">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"> Tight Toughness and Isolated Toughness for $\{K_2,C_n\}$-factor critical avoidable graph </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Guan%2C+X">Xiaxia Guan</a>, <a href="/search/math?searchtype=author&query=Ma%2C+H">Hongxia Ma</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Maoqun Wang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2406.17631v1-abstract-short" style="display: inline;"> A spannning subgraph $F$ of $G$ is a $\{K_2,C_n\}$-factor if each component of $F$ is either $K_{2}$ or $C_{n}$. A graph $G$ is called a $(\{K_2,C_n\},n)$-factor critical avoidable graph if $G-X-e$ has a $\{K_2,C_n\}$-factor for any $S\subseteq V(G)$ with $|X|=n$ and $e\in E(G-X)$. In this paper, we first obtain a sufficient condition with regard to isolated toughness of a graph $G$ such that $G$… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.17631v1-abstract-full').style.display = 'inline'; document.getElementById('2406.17631v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.17631v1-abstract-full" style="display: none;"> A spannning subgraph $F$ of $G$ is a $\{K_2,C_n\}$-factor if each component of $F$ is either $K_{2}$ or $C_{n}$. A graph $G$ is called a $(\{K_2,C_n\},n)$-factor critical avoidable graph if $G-X-e$ has a $\{K_2,C_n\}$-factor for any $S\subseteq V(G)$ with $|X|=n$ and $e\in E(G-X)$. In this paper, we first obtain a sufficient condition with regard to isolated toughness of a graph $G$ such that $G$ is $\{K_2,C_{n}\}$-factor critical avoidable. In addition, we give a sufficient condition with regard to tight toughness and isolated toughness of a graph $G$ such that $G$ is $\{K_2,C_{2i+1}|i \geqslant 2\}$-factor critical avoidable respectively. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.17631v1-abstract-full').style.display = 'none'; document.getElementById('2406.17631v1-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 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.11407">arXiv:2406.11407</a> <span> [<a href="https://arxiv.org/pdf/2406.11407">pdf</a>, <a href="https://arxiv.org/ps/2406.11407">ps</a>, <a href="https://arxiv.org/format/2406.11407">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"> Note on a vector-host epidemic model with spatial structure </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+M">Mingxin Wang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2406.11407v4-abstract-short" style="display: inline;"> Magal, Webb and Wu [Nonlinearity 31, 5589-5614 (2018)] studied the model describing outbreak of Zika in Rio De Janerio, and provided a complete analysis of dynamical properties for the solutions. In this note we first use a very simple approach to prove their results, and then investigate the modified version of the model concerned in their paper, with Neumann boundary condition replaced by Dirich… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.11407v4-abstract-full').style.display = 'inline'; document.getElementById('2406.11407v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.11407v4-abstract-full" style="display: none;"> Magal, Webb and Wu [Nonlinearity 31, 5589-5614 (2018)] studied the model describing outbreak of Zika in Rio De Janerio, and provided a complete analysis of dynamical properties for the solutions. In this note we first use a very simple approach to prove their results, and then investigate the modified version of the model concerned in their paper, with Neumann boundary condition replaced by Dirichlet boundary condition. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.11407v4-abstract-full').style.display = 'none'; document.getElementById('2406.11407v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.04880">arXiv:2406.04880</a> <span> [<a href="https://arxiv.org/pdf/2406.04880">pdf</a>, <a href="https://arxiv.org/ps/2406.04880">ps</a>, <a href="https://arxiv.org/format/2406.04880">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"> The free boundary problem of an epidemic model with nonlocal diffusions and nonlocal reactions: spreading-vanishing dichotomy </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xueping Li</a>, <a href="/search/math?searchtype=author&query=Li%2C+L">Lei Li</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Mingxin Wang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2406.04880v2-abstract-short" style="display: inline;"> This paper concerns the free boundary problem of an epidemic model. The spatial movements of the infectious agents and the infective humans are approximated by nonlocal diffusion operators. Especially, both the growth rate of the agents and the infective rate of humans are represented by nonlocal reaction terms. Thus our model has four integral terms which bring some diffculties for the study of t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.04880v2-abstract-full').style.display = 'inline'; document.getElementById('2406.04880v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.04880v2-abstract-full" style="display: none;"> This paper concerns the free boundary problem of an epidemic model. The spatial movements of the infectious agents and the infective humans are approximated by nonlocal diffusion operators. Especially, both the growth rate of the agents and the infective rate of humans are represented by nonlocal reaction terms. Thus our model has four integral terms which bring some diffculties for the study of the corresponding principal eigenvalue problem. Firstly, using some elementray analysis instead of Krein-Rutman theorem and the variational characteristic, we obtain the existence and asymptotic behaviors of principal eigenvalue. Then a spreading-vanishing dichotomy is proved to hold, and the criteria for spreading and vanishing are derived. Lastly, comparing our results with those in the existing works, we discuss the effect of nonlocal reaction term on spreading and vanishing, finding that the more nonlocal reaction terms a model has, the harder spreading happens. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.04880v2-abstract-full').style.display = 'none'; document.getElementById('2406.04880v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 7 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.20763">arXiv:2405.20763</a> <span> [<a href="https://arxiv.org/pdf/2405.20763">pdf</a>, <a href="https://arxiv.org/format/2405.20763">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> <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"> Improving Generalization and Convergence by Enhancing Implicit Regularization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+M">Mingze Wang</a>, <a href="/search/math?searchtype=author&query=Wang%2C+J">Jinbo Wang</a>, <a href="/search/math?searchtype=author&query=He%2C+H">Haotian He</a>, <a href="/search/math?searchtype=author&query=Wang%2C+Z">Zilin Wang</a>, <a href="/search/math?searchtype=author&query=Huang%2C+G">Guanhua Huang</a>, <a href="/search/math?searchtype=author&query=Xiong%2C+F">Feiyu Xiong</a>, <a href="/search/math?searchtype=author&query=Li%2C+Z">Zhiyu Li</a>, <a href="/search/math?searchtype=author&query=E%2C+W">Weinan E</a>, <a href="/search/math?searchtype=author&query=Wu%2C+L">Lei Wu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2405.20763v4-abstract-short" style="display: inline;"> In this work, we propose an Implicit Regularization Enhancement (IRE) framework to accelerate the discovery of flat solutions in deep learning, thereby improving generalization and convergence. Specifically, IRE decouples the dynamics of flat and sharp directions, which boosts the sharpness reduction along flat directions while maintaining the training stability in sharp directions. We show that I… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.20763v4-abstract-full').style.display = 'inline'; document.getElementById('2405.20763v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.20763v4-abstract-full" style="display: none;"> In this work, we propose an Implicit Regularization Enhancement (IRE) framework to accelerate the discovery of flat solutions in deep learning, thereby improving generalization and convergence. Specifically, IRE decouples the dynamics of flat and sharp directions, which boosts the sharpness reduction along flat directions while maintaining the training stability in sharp directions. We show that IRE can be practically incorporated with {\em generic base optimizers} without introducing significant computational overload. Experiments show that IRE consistently improves the generalization performance for image classification tasks across a variety of benchmark datasets (CIFAR-10/100, ImageNet) and models (ResNets and ViTs). Surprisingly, IRE also achieves a $2\times$ {\em speed-up} compared to AdamW in the pre-training of Llama models (of sizes ranging from 60M to 229M) on datasets including Wikitext-103, Minipile, and Openwebtext. Moreover, we provide theoretical guarantees, showing that IRE can substantially accelerate the convergence towards flat minima in Sharpness-aware Minimization (SAM). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.20763v4-abstract-full').style.display = 'none'; document.getElementById('2405.20763v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 31 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 31 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">44 pages, accepted by NeurIPS 2024</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.18173">arXiv:2405.18173</a> <span> [<a href="https://arxiv.org/pdf/2405.18173">pdf</a>, <a href="https://arxiv.org/ps/2405.18173">ps</a>, <a href="https://arxiv.org/format/2405.18173">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"> Life span of solutions to a semilinear parabolic equation on locally finite graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Hu%2C+Y">Yuanyang Hu</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Mingxin 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="2405.18173v1-abstract-short" style="display: inline;"> Let $G=(V,E)$ be a locally finite connected graph. We develop the first eigenvalue method on $G$ introduced in 1963 by Kaplan \cite{Kaplan} on Euclidean space, the discrete Phragm茅n-Lindel枚f principle of parabolic equations and upper and lower solutions method on $G$. Using these methods, we establish the estimates and asymptotic behaviour of the life span of solutions to a semilinear heat equatio… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.18173v1-abstract-full').style.display = 'inline'; document.getElementById('2405.18173v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.18173v1-abstract-full" style="display: none;"> Let $G=(V,E)$ be a locally finite connected graph. We develop the first eigenvalue method on $G$ introduced in 1963 by Kaplan \cite{Kaplan} on Euclidean space, the discrete Phragm茅n-Lindel枚f principle of parabolic equations and upper and lower solutions method on $G$. Using these methods, we establish the estimates and asymptotic behaviour of the life span of solutions to a semilinear heat equation with initial data $位蠄(x)$ for different scales of $位$ on $G$ under some different conditions. Our results are different from the continuous case, which is related to the structure of the graph $G$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.18173v1-abstract-full').style.display = 'none'; document.getElementById('2405.18173v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">34 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/2405.14134">arXiv:2405.14134</a> <span> [<a href="https://arxiv.org/pdf/2405.14134">pdf</a>, <a href="https://arxiv.org/ps/2405.14134">ps</a>, <a href="https://arxiv.org/format/2405.14134">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="Classical Analysis and ODEs">math.CA</span> </div> </div> <p class="title is-5 mathjax"> Sharp convergence rate on Schr枚dinger type operators </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+M">Meng Wang</a>, <a href="/search/math?searchtype=author&query=Zhao%2C+S">Shuijiang Zhao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2405.14134v1-abstract-short" style="display: inline;"> For Schr枚dinger type operators in one dimension, we consider the relationship between the convergence rate and the regularity for initial data. By establishing the associated frequency-localized maximal estimates, we prove sharp results up to the endpoints. The optimal range for the wave operator in all dimensions is also obtained. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.14134v1-abstract-full" style="display: none;"> For Schr枚dinger type operators in one dimension, we consider the relationship between the convergence rate and the regularity for initial data. By establishing the associated frequency-localized maximal estimates, we prove sharp results up to the endpoints. The optimal range for the wave operator in all dimensions is also obtained. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.14134v1-abstract-full').style.display = 'none'; document.getElementById('2405.14134v1-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 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">13 pages, 2 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 42B25 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.13418">arXiv:2405.13418</a> <span> [<a href="https://arxiv.org/pdf/2405.13418">pdf</a>, <a href="https://arxiv.org/ps/2405.13418">ps</a>, <a href="https://arxiv.org/format/2405.13418">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"> Dynamics of a nonlinear infection viral propagation model with one fixed boundary and one free boundary </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+M">Mingxin 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="2405.13418v1-abstract-short" style="display: inline;"> In this paper we study a nonlinear infection viral propagation model with diffusion, in which, the left boundary is fixed and with homogeneous Dirichlet boundary conditions, while the right boundary is free. We find that the habitat always expands to the half line $[0, \infty)$, and that the virus and infected cells always die out when the {\it Basic Reproduction Number} $\mathcal{R}_0\le 1$, whil… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.13418v1-abstract-full').style.display = 'inline'; document.getElementById('2405.13418v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.13418v1-abstract-full" style="display: none;"> In this paper we study a nonlinear infection viral propagation model with diffusion, in which, the left boundary is fixed and with homogeneous Dirichlet boundary conditions, while the right boundary is free. We find that the habitat always expands to the half line $[0, \infty)$, and that the virus and infected cells always die out when the {\it Basic Reproduction Number} $\mathcal{R}_0\le 1$, while the virus and infected cells have persistence properties when $\mathcal{R}_0>1$. To obtain the persistence properties of virus and infected cells when $\mathcal{R}_0>1$, the most work of this paper focuses on the existence and uniqueness of positive equilibrium solutions for subsystems and the existence of positive equilibrium solutions for the entire system. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.13418v1-abstract-full').style.display = 'none'; document.getElementById('2405.13418v1-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 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.12190">arXiv:2405.12190</a> <span> [<a href="https://arxiv.org/pdf/2405.12190">pdf</a>, <a href="https://arxiv.org/format/2405.12190">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"> Quantitative asymptotics for polynomial patterns in the primes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Matthiesen%2C+L">Lilian Matthiesen</a>, <a href="/search/math?searchtype=author&query=Ter%C3%A4v%C3%A4inen%2C+J">Joni Ter盲v盲inen</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Mengdi 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="2405.12190v2-abstract-short" style="display: inline;"> We prove quantitative estimates for averages of the von Mangoldt and M枚bius functions along polynomial progressions $n+P_1(m),\ldots, n+P_k(m)$ for a large class of polynomials $P_i$. The error terms obtained save an arbitrary power of logarithm, matching the classical Siegel--Walfisz error term. These results give the first quantitative bounds for the Tao--Ziegler polynomial patterns in the prime… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.12190v2-abstract-full').style.display = 'inline'; document.getElementById('2405.12190v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.12190v2-abstract-full" style="display: none;"> We prove quantitative estimates for averages of the von Mangoldt and M枚bius functions along polynomial progressions $n+P_1(m),\ldots, n+P_k(m)$ for a large class of polynomials $P_i$. The error terms obtained save an arbitrary power of logarithm, matching the classical Siegel--Walfisz error term. These results give the first quantitative bounds for the Tao--Ziegler polynomial patterns in the primes result. The proofs are based on a quantitative generalised von Neumann theorem of Peluse, a recent result of Leng on strong bounds for the Gowers uniformity of the primes, and analysis of a ``Siegel model'' for the von Mangoldt function along polynomial progressions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.12190v2-abstract-full').style.display = 'none'; document.getElementById('2405.12190v2-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 20 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">27 pages; small change in abstract</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 11B30; 11N32 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.11871">arXiv:2405.11871</a> <span> [<a href="https://arxiv.org/pdf/2405.11871">pdf</a>, <a href="https://arxiv.org/ps/2405.11871">ps</a>, <a href="https://arxiv.org/format/2405.11871">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"> Qualitative properties of solutions to nonlocal infectious SIR epidemic models </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bao%2C+H">Hanxiang Bao</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Mingxin Wang</a>, <a href="/search/math?searchtype=author&query=Yao%2C+S">Shaowen Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2405.11871v1-abstract-short" style="display: inline;"> This paper studies qualitative properties of solutions of nonlocal infectious SIR epidemic models (1.3)-(1.5), with the homogeneous Neumann boundary conditions, Dirichlet boundary conditions and free boundary, respectively. We first use the upper and lower solutions method and the Lyapunov function method to prove the global asymptotically stabilities of the disease-free equilibrium and the unique… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.11871v1-abstract-full').style.display = 'inline'; document.getElementById('2405.11871v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.11871v1-abstract-full" style="display: none;"> This paper studies qualitative properties of solutions of nonlocal infectious SIR epidemic models (1.3)-(1.5), with the homogeneous Neumann boundary conditions, Dirichlet boundary conditions and free boundary, respectively. We first use the upper and lower solutions method and the Lyapunov function method to prove the global asymptotically stabilities of the disease-free equilibrium and the unique positive equilibrium of (1.3). Then we use the theory of topological degree in cones to study the positive equilibrium solutions of (1.4), including the necessary and sufficient conditions for the existence, and the uniqueness in some special case. At last, for the free boundary problem (1.5), we study the longtime behaviors of solutions and criteria for spreading and vanishing. The highlights are to overcome failures of the Lyapunov functional method and comparison principle, and difficulties in the maximum principle and Hopf boundary lemma of boundary value problems caused by nonlocal terms. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.11871v1-abstract-full').style.display = 'none'; document.getElementById('2405.11871v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.11851">arXiv:2405.11851</a> <span> [<a href="https://arxiv.org/pdf/2405.11851">pdf</a>, <a href="https://arxiv.org/ps/2405.11851">ps</a>, <a href="https://arxiv.org/format/2405.11851">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"> Lower classes and Chung's LILs of the fractional integrated generalized fractional Brownian motion </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lyu%2C+M">Mengjie Lyu</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Min Wang</a>, <a href="/search/math?searchtype=author&query=Wang%2C+R">Ran 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="2405.11851v1-abstract-short" style="display: inline;"> Let $\{X(t)\}_{t\geqslant0}$ be the generalized fractional Brownian motion introduced by Pang and Taqqu (2019): \begin{align*} \{X(t)\}_{t\ge0}\overset{d}{=}&\left\{ \int_{\mathbb R} \left((t-u)_+^伪-(-u)_+^伪 \right) |u|^{-纬/2} B(du) \right\}_{t\ge0}, \end{align*} where $ 纬\in [0,1), \ \ 伪\in \left(-\frac12+\frac纬{2}, \ \frac12+\frac纬{2} \right)$ are constants. For any $胃>0$, let \begin{ali… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.11851v1-abstract-full').style.display = 'inline'; document.getElementById('2405.11851v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.11851v1-abstract-full" style="display: none;"> Let $\{X(t)\}_{t\geqslant0}$ be the generalized fractional Brownian motion introduced by Pang and Taqqu (2019): \begin{align*} \{X(t)\}_{t\ge0}\overset{d}{=}&\left\{ \int_{\mathbb R} \left((t-u)_+^伪-(-u)_+^伪 \right) |u|^{-纬/2} B(du) \right\}_{t\ge0}, \end{align*} where $ 纬\in [0,1), \ \ 伪\in \left(-\frac12+\frac纬{2}, \ \frac12+\frac纬{2} \right)$ are constants. For any $胃>0$, let \begin{align*} Y(t)=\frac{1}{螕(胃)}\int_0^t (t-u)^{胃-1} X(u)du, \quad t\ge 0. \end{align*} Building upon the arguments of Talagrand (1996), we give integral criteria for the lower classes of $Y$ at $t=0$ and at infinity, respectively. As a consequence, we derive its Chung-type laws of the iterated logarithm. In the proofs, the small ball probability estimates play important roles. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.11851v1-abstract-full').style.display = 'none'; document.getElementById('2405.11851v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> <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 papges, comments welcome</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 60G15; 60G17; 60G18; 60G22 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.04915">arXiv:2405.04915</a> <span> [<a href="https://arxiv.org/pdf/2405.04915">pdf</a>, <a href="https://arxiv.org/ps/2405.04915">ps</a>, <a href="https://arxiv.org/format/2405.04915">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 spiders $S(4m+2,\,2m,\,1)$ are $e$-positivite </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Tang%2C+D+Q+B">Davion Q. B. Tang</a>, <a href="/search/math?searchtype=author&query=Wang%2C+D+G+L">David G. L. Wang</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M+M+Y">Monica M. Y. 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="2405.04915v1-abstract-short" style="display: inline;"> We establish the $e$-positivity of spider graphs of the form $S(4m+2,\, 2m,\, 1)$, which was conjectured by Aliniaeifard, Wang and van Willigenburg. A key to our proof is the $e_I$-expansion formula of the chromatic symmetric function of paths due to Shareshian and Wachs, where the symbol~$I$ indicates integer compositions rather than partitions. Following the strategy of the divide-and-conquer te… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.04915v1-abstract-full').style.display = 'inline'; document.getElementById('2405.04915v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.04915v1-abstract-full" style="display: none;"> We establish the $e$-positivity of spider graphs of the form $S(4m+2,\, 2m,\, 1)$, which was conjectured by Aliniaeifard, Wang and van Willigenburg. A key to our proof is the $e_I$-expansion formula of the chromatic symmetric function of paths due to Shareshian and Wachs, where the symbol~$I$ indicates integer compositions rather than partitions. Following the strategy of the divide-and-conquer technique, we pick out one or two positive $e_J$-terms for each negative $e_I$-term in an $e$-expression for the spiders, where $J$ are selected to be distinct compositions obtained by rearranging the parts of $I$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.04915v1-abstract-full').style.display = 'none'; document.getElementById('2405.04915v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.04268">arXiv:2405.04268</a> <span> [<a href="https://arxiv.org/pdf/2405.04268">pdf</a>, <a href="https://arxiv.org/ps/2405.04268">ps</a>, <a href="https://arxiv.org/format/2405.04268">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"> Dynamics of an epidemic model with nonlocal di?usion and a free boundary </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+L">Lei Li</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Mingxin 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="2405.04268v1-abstract-short" style="display: inline;"> An epidemic model, where the dispersal is approximated by nonlocal diffusion operator and spatial domain has one ?xed boundary and one free boundary, is considered in this paper. Firstly, using some elementary analysis instead of variational characterization, we show the existence and asymptotic behaviors of the principal eigenvalue of a cooperative system which can be used to characterize more ep… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.04268v1-abstract-full').style.display = 'inline'; document.getElementById('2405.04268v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.04268v1-abstract-full" style="display: none;"> An epidemic model, where the dispersal is approximated by nonlocal diffusion operator and spatial domain has one ?xed boundary and one free boundary, is considered in this paper. Firstly, using some elementary analysis instead of variational characterization, we show the existence and asymptotic behaviors of the principal eigenvalue of a cooperative system which can be used to characterize more epidemic models, not just ours. Then we study the existence, uniqueness and stability of a related steady state problem. Finally, we obtain a rather complete understanding for long time behaviors, spreading-vanishing dichotomy, criteria for spreading and vanishing, and spreading speed. Particularly, we prove that the asymptotic spreading speed of solution component (u; v) is equal to the spreading speed of free boundary which is ?nite if and only if a threshold condition holds for kernel functions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.04268v1-abstract-full').style.display = 'none'; document.getElementById('2405.04268v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.02588">arXiv:2405.02588</a> <span> [<a href="https://arxiv.org/pdf/2405.02588">pdf</a>, <a href="https://arxiv.org/ps/2405.02588">ps</a>, <a href="https://arxiv.org/format/2405.02588">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="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Inexact Adaptive Cubic Regularization Algorithms on Riemannian Manifolds and Application </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+Z+Y">Z. Y. Li</a>, <a href="/search/math?searchtype=author&query=Wang%2C+X+M">X. M. 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="2405.02588v1-abstract-short" style="display: inline;"> The adaptive cubic regularization algorithm employing the inexact gradient and Hessian is proposed on general Riemannian manifolds, together with the iteration complexity to get an approximate second-order optimality under certain assumptions on accuracies about the inexact gradient and Hessian. The algorithm extends the inexact adaptive cubic regularization algorithm under true gradient in [Math.… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.02588v1-abstract-full').style.display = 'inline'; document.getElementById('2405.02588v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.02588v1-abstract-full" style="display: none;"> The adaptive cubic regularization algorithm employing the inexact gradient and Hessian is proposed on general Riemannian manifolds, together with the iteration complexity to get an approximate second-order optimality under certain assumptions on accuracies about the inexact gradient and Hessian. The algorithm extends the inexact adaptive cubic regularization algorithm under true gradient in [Math. Program., 184(1-2): 35-70, 2020] to more general cases even in Euclidean settings. As an application, the algorithm is applied to solve the joint diagonalization problem on the Stiefel manifold. Numerical experiments illustrate that the algorithm performs better than the inexact trust-region algorithm in [Advances of the neural information processing systems, 31, 2018]. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.02588v1-abstract-full').style.display = 'none'; document.getElementById('2405.02588v1-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 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">15 pages, 1 table</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53C20(Primary); 53C22(Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.07771">arXiv:2404.07771</a> <span> [<a href="https://arxiv.org/pdf/2404.07771">pdf</a>, <a href="https://arxiv.org/format/2404.07771">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="Statistics Theory">math.ST</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"> An Overview of Diffusion Models: Applications, Guided Generation, Statistical Rates and Optimization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+M">Minshuo Chen</a>, <a href="/search/math?searchtype=author&query=Mei%2C+S">Song Mei</a>, <a href="/search/math?searchtype=author&query=Fan%2C+J">Jianqing Fan</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Mengdi Wang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2404.07771v1-abstract-short" style="display: inline;"> Diffusion models, a powerful and universal generative AI technology, have achieved tremendous success in computer vision, audio, reinforcement learning, and computational biology. In these applications, diffusion models provide flexible high-dimensional data modeling, and act as a sampler for generating new samples under active guidance towards task-desired properties. Despite the significant empi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.07771v1-abstract-full').style.display = 'inline'; document.getElementById('2404.07771v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.07771v1-abstract-full" style="display: none;"> Diffusion models, a powerful and universal generative AI technology, have achieved tremendous success in computer vision, audio, reinforcement learning, and computational biology. In these applications, diffusion models provide flexible high-dimensional data modeling, and act as a sampler for generating new samples under active guidance towards task-desired properties. Despite the significant empirical success, theory of diffusion models is very limited, potentially slowing down principled methodological innovations for further harnessing and improving diffusion models. In this paper, we review emerging applications of diffusion models, understanding their sample generation under various controls. Next, we overview the existing theories of diffusion models, covering their statistical properties and sampling capabilities. We adopt a progressive routine, beginning with unconditional diffusion models and connecting to conditional counterparts. Further, we review a new avenue in high-dimensional structured optimization through conditional diffusion models, where searching for solutions is reformulated as a conditional sampling problem and solved by diffusion models. Lastly, we discuss future directions about diffusion models. The purpose of this paper is to provide a well-rounded theoretical exposure for stimulating forward-looking theories and methods of diffusion models. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.07771v1-abstract-full').style.display = 'none'; document.getElementById('2404.07771v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2403.13219">arXiv:2403.13219</a> <span> [<a href="https://arxiv.org/pdf/2403.13219">pdf</a>, <a href="https://arxiv.org/format/2403.13219">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"> Diffusion Model for Data-Driven Black-Box Optimization </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+Z">Zihao Li</a>, <a href="/search/math?searchtype=author&query=Yuan%2C+H">Hui Yuan</a>, <a href="/search/math?searchtype=author&query=Huang%2C+K">Kaixuan Huang</a>, <a href="/search/math?searchtype=author&query=Ni%2C+C">Chengzhuo Ni</a>, <a href="/search/math?searchtype=author&query=Ye%2C+Y">Yinyu Ye</a>, <a href="/search/math?searchtype=author&query=Chen%2C+M">Minshuo Chen</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Mengdi 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="2403.13219v1-abstract-short" style="display: inline;"> Generative AI has redefined artificial intelligence, enabling the creation of innovative content and customized solutions that drive business practices into a new era of efficiency and creativity. In this paper, we focus on diffusion models, a powerful generative AI technology, and investigate their potential for black-box optimization over complex structured variables. Consider the practical scen… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.13219v1-abstract-full').style.display = 'inline'; document.getElementById('2403.13219v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.13219v1-abstract-full" style="display: none;"> Generative AI has redefined artificial intelligence, enabling the creation of innovative content and customized solutions that drive business practices into a new era of efficiency and creativity. In this paper, we focus on diffusion models, a powerful generative AI technology, and investigate their potential for black-box optimization over complex structured variables. Consider the practical scenario where one wants to optimize some structured design in a high-dimensional space, based on massive unlabeled data (representing design variables) and a small labeled dataset. We study two practical types of labels: 1) noisy measurements of a real-valued reward function and 2) human preference based on pairwise comparisons. The goal is to generate new designs that are near-optimal and preserve the designed latent structures. Our proposed method reformulates the design optimization problem into a conditional sampling problem, which allows us to leverage the power of diffusion models for modeling complex distributions. In particular, we propose a reward-directed conditional diffusion model, to be trained on the mixed data, for sampling a near-optimal solution conditioned on high predicted rewards. Theoretically, we establish sub-optimality error bounds for the generated designs. The sub-optimality gap nearly matches the optimal guarantee in off-policy bandits, demonstrating the efficiency of reward-directed diffusion models for black-box optimization. Moreover, when the data admits a low-dimensional latent subspace structure, our model efficiently generates high-fidelity designs that closely respect the latent structure. We provide empirical experiments validating our model in decision-making and content-creation tasks. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.13219v1-abstract-full').style.display = 'none'; document.getElementById('2403.13219v1-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 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">arXiv admin note: substantial text overlap with arXiv:2307.07055</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2403.11968">arXiv:2403.11968</a> <span> [<a href="https://arxiv.org/pdf/2403.11968">pdf</a>, <a href="https://arxiv.org/format/2403.11968">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="Statistics Theory">math.ST</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"> Unveil Conditional Diffusion Models with Classifier-free Guidance: A Sharp Statistical Theory </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fu%2C+H">Hengyu Fu</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Z">Zhuoran Yang</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Mengdi Wang</a>, <a href="/search/math?searchtype=author&query=Chen%2C+M">Minshuo Chen</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2403.11968v1-abstract-short" style="display: inline;"> Conditional diffusion models serve as the foundation of modern image synthesis and find extensive application in fields like computational biology and reinforcement learning. In these applications, conditional diffusion models incorporate various conditional information, such as prompt input, to guide the sample generation towards desired properties. Despite the empirical success, theory of condit… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.11968v1-abstract-full').style.display = 'inline'; document.getElementById('2403.11968v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.11968v1-abstract-full" style="display: none;"> Conditional diffusion models serve as the foundation of modern image synthesis and find extensive application in fields like computational biology and reinforcement learning. In these applications, conditional diffusion models incorporate various conditional information, such as prompt input, to guide the sample generation towards desired properties. Despite the empirical success, theory of conditional diffusion models is largely missing. This paper bridges this gap by presenting a sharp statistical theory of distribution estimation using conditional diffusion models. Our analysis yields a sample complexity bound that adapts to the smoothness of the data distribution and matches the minimax lower bound. The key to our theoretical development lies in an approximation result for the conditional score function, which relies on a novel diffused Taylor approximation technique. Moreover, we demonstrate the utility of our statistical theory in elucidating the performance of conditional diffusion models across diverse applications, including model-based transition kernel estimation in reinforcement learning, solving inverse problems, and reward conditioned sample generation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.11968v1-abstract-full').style.display = 'none'; document.getElementById('2403.11968v1-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 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">92 pages, 5 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2403.04361">arXiv:2403.04361</a> <span> [<a href="https://arxiv.org/pdf/2403.04361">pdf</a>, <a href="https://arxiv.org/format/2403.04361">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> </div> <p class="title is-5 mathjax"> Subsampling for Big Data Linear Models with Measurement Errors </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ju%2C+J">Jiangshan Ju</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Mingqiu Wang</a>, <a href="/search/math?searchtype=author&query=Zhao%2C+S">Shengli Zhao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2403.04361v2-abstract-short" style="display: inline;"> Subsampling algorithms for various parametric regression models with massive data have been extensively investigated in recent years. However, all existing studies on subsampling heavily rely on clean massive data. In practical applications, the observed covariates may suffer from inaccuracies due to measurement errors. To address the challenge of large datasets with measurement errors, this study… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.04361v2-abstract-full').style.display = 'inline'; document.getElementById('2403.04361v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.04361v2-abstract-full" style="display: none;"> Subsampling algorithms for various parametric regression models with massive data have been extensively investigated in recent years. However, all existing studies on subsampling heavily rely on clean massive data. In practical applications, the observed covariates may suffer from inaccuracies due to measurement errors. To address the challenge of large datasets with measurement errors, this study explores two subsampling algorithms based on the corrected likelihood approach: the optimal subsampling algorithm utilizing inverse probability weighting and the perturbation subsampling algorithm employing random weighting assuming a perfectly known distribution. Theoretical properties for both algorithms are provided. Numerical simulations and two real-world examples demonstrate the effectiveness of these proposed methods compared to other uncorrected algorithms. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.04361v2-abstract-full').style.display = 'none'; document.getElementById('2403.04361v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 7 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2403.04275">arXiv:2403.04275</a> <span> [<a href="https://arxiv.org/pdf/2403.04275">pdf</a>, <a href="https://arxiv.org/ps/2403.04275">ps</a>, <a href="https://arxiv.org/format/2403.04275">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="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> The small mass limit for a McKean-Vlasov equation with state-dependent friction </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Shi%2C+C">Chungang Shi</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Mengmeng Wang</a>, <a href="/search/math?searchtype=author&query=Lv%2C+Y">Yan Lv</a>, <a href="/search/math?searchtype=author&query=Wang%2C+W">Wei 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="2403.04275v1-abstract-short" style="display: inline;"> The small mass limit is derived for a McKean-Vlasov equation with state-dependent friction in $d$-dimensional space. By applying the averaging approach to a non-autonomous slow-fast system with the microscopic and macroscopic scales, the convergence in distribution is obtained. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.04275v1-abstract-full" style="display: none;"> The small mass limit is derived for a McKean-Vlasov equation with state-dependent friction in $d$-dimensional space. By applying the averaging approach to a non-autonomous slow-fast system with the microscopic and macroscopic scales, the convergence in distribution is obtained. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.04275v1-abstract-full').style.display = 'none'; document.getElementById('2403.04275v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2403.04236">arXiv:2403.04236</a> <span> [<a href="https://arxiv.org/pdf/2403.04236">pdf</a>, <a href="https://arxiv.org/ps/2403.04236">ps</a>, <a href="https://arxiv.org/format/2403.04236">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="Econometrics">econ.EM</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="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Regularized DeepIV with Model Selection </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+Z">Zihao Li</a>, <a href="/search/math?searchtype=author&query=Lan%2C+H">Hui Lan</a>, <a href="/search/math?searchtype=author&query=Syrgkanis%2C+V">Vasilis Syrgkanis</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Mengdi Wang</a>, <a href="/search/math?searchtype=author&query=Uehara%2C+M">Masatoshi Uehara</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2403.04236v1-abstract-short" style="display: inline;"> In this paper, we study nonparametric estimation of instrumental variable (IV) regressions. While recent advancements in machine learning have introduced flexible methods for IV estimation, they often encounter one or more of the following limitations: (1) restricting the IV regression to be uniquely identified; (2) requiring minimax computation oracle, which is highly unstable in practice; (3) ab… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.04236v1-abstract-full').style.display = 'inline'; document.getElementById('2403.04236v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.04236v1-abstract-full" style="display: none;"> In this paper, we study nonparametric estimation of instrumental variable (IV) regressions. While recent advancements in machine learning have introduced flexible methods for IV estimation, they often encounter one or more of the following limitations: (1) restricting the IV regression to be uniquely identified; (2) requiring minimax computation oracle, which is highly unstable in practice; (3) absence of model selection procedure. In this paper, we present the first method and analysis that can avoid all three limitations, while still enabling general function approximation. Specifically, we propose a minimax-oracle-free method called Regularized DeepIV (RDIV) regression that can converge to the least-norm IV solution. Our method consists of two stages: first, we learn the conditional distribution of covariates, and by utilizing the learned distribution, we learn the estimator by minimizing a Tikhonov-regularized loss function. We further show that our method allows model selection procedures that can achieve the oracle rates in the misspecified regime. When extended to an iterative estimator, our method matches the current state-of-the-art convergence rate. Our method is a Tikhonov regularized variant of the popular DeepIV method with a non-parametric MLE first-stage estimator, and our results provide the first rigorous guarantees for this empirically used method, showcasing the importance of regularization which was absent from the original work. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.04236v1-abstract-full').style.display = 'none'; document.getElementById('2403.04236v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.18342">arXiv:2402.18342</a> <span> [<a href="https://arxiv.org/pdf/2402.18342">pdf</a>, <a href="https://arxiv.org/ps/2402.18342">ps</a>, <a href="https://arxiv.org/format/2402.18342">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> Local Fourier uniformity of higher divisor functions on average </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+M">Mengdi 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="2402.18342v2-abstract-short" style="display: inline;"> Let $蟿_k$ be the $k$-fold divisor function. By constructing an approximant of $蟿_k$, denoted as $蟿_k^*$, which is a normalized truncation of the $k$-fold divisor function, we prove that when $\exp\left(C\log^{1/2}X(\log\log X)^{1/2}\right)\leq H\leq X$ and $C>0$ is sufficiently large, the following estimate holds for almost all $x\in[X,2X]$: \[ \Big|\sum_{x<n\leq x+H}(蟿_k(n)-蟿_k^*(n)) e(伪_dn^d+\cd… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.18342v2-abstract-full').style.display = 'inline'; document.getElementById('2402.18342v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.18342v2-abstract-full" style="display: none;"> Let $蟿_k$ be the $k$-fold divisor function. By constructing an approximant of $蟿_k$, denoted as $蟿_k^*$, which is a normalized truncation of the $k$-fold divisor function, we prove that when $\exp\left(C\log^{1/2}X(\log\log X)^{1/2}\right)\leq H\leq X$ and $C>0$ is sufficiently large, the following estimate holds for almost all $x\in[X,2X]$: \[ \Big|\sum_{x<n\leq x+H}(蟿_k(n)-蟿_k^*(n)) e(伪_dn^d+\cdots+伪_1n)\Big|=o(H\log^{k-1}X), \] where $伪_1, \dots, 伪_d\in \mathbb{R}$ are arbitrary frequencies. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.18342v2-abstract-full').style.display = 'none'; document.getElementById('2402.18342v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Version 2: corrected some minor typos</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.10810">arXiv:2402.10810</a> <span> [<a href="https://arxiv.org/pdf/2402.10810">pdf</a>, <a href="https://arxiv.org/ps/2402.10810">ps</a>, <a href="https://arxiv.org/format/2402.10810">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> <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"> Double Duality: Variational Primal-Dual Policy Optimization for Constrained Reinforcement Learning </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+Z">Zihao Li</a>, <a href="/search/math?searchtype=author&query=Liu%2C+B">Boyi Liu</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Z">Zhuoran Yang</a>, <a href="/search/math?searchtype=author&query=Wang%2C+Z">Zhaoran Wang</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Mengdi 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="2402.10810v1-abstract-short" style="display: inline;"> We study the Constrained Convex Markov Decision Process (MDP), where the goal is to minimize a convex functional of the visitation measure, subject to a convex constraint. Designing algorithms for a constrained convex MDP faces several challenges, including (1) handling the large state space, (2) managing the exploration/exploitation tradeoff, and (3) solving the constrained optimization where the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.10810v1-abstract-full').style.display = 'inline'; document.getElementById('2402.10810v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.10810v1-abstract-full" style="display: none;"> We study the Constrained Convex Markov Decision Process (MDP), where the goal is to minimize a convex functional of the visitation measure, subject to a convex constraint. Designing algorithms for a constrained convex MDP faces several challenges, including (1) handling the large state space, (2) managing the exploration/exploitation tradeoff, and (3) solving the constrained optimization where the objective and the constraint are both nonlinear functions of the visitation measure. In this work, we present a model-based algorithm, Variational Primal-Dual Policy Optimization (VPDPO), in which Lagrangian and Fenchel duality are implemented to reformulate the original constrained problem into an unconstrained primal-dual optimization. Moreover, the primal variables are updated by model-based value iteration following the principle of Optimism in the Face of Uncertainty (OFU), while the dual variables are updated by gradient ascent. Moreover, by embedding the visitation measure into a finite-dimensional space, we can handle large state spaces by incorporating function approximation. Two notable examples are (1) Kernelized Nonlinear Regulators and (2) Low-rank MDPs. We prove that with an optimistic planning oracle, our algorithm achieves sublinear regret and constraint violation in both cases and can attain the globally optimal policy of the original constrained problem. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.10810v1-abstract-full').style.display = 'none'; document.getElementById('2402.10810v1-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 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.07193">arXiv:2402.07193</a> <span> [<a href="https://arxiv.org/pdf/2402.07193">pdf</a>, <a href="https://arxiv.org/format/2402.07193">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> <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"> Parameter Symmetry and Noise Equilibrium of Stochastic Gradient Descent </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ziyin%2C+L">Liu Ziyin</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Mingze Wang</a>, <a href="/search/math?searchtype=author&query=Li%2C+H">Hongchao Li</a>, <a href="/search/math?searchtype=author&query=Wu%2C+L">Lei Wu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2402.07193v3-abstract-short" style="display: inline;"> Symmetries are prevalent in deep learning and can significantly influence the learning dynamics of neural networks. In this paper, we examine how exponential symmetries -- a broad subclass of continuous symmetries present in the model architecture or loss function -- interplay with stochastic gradient descent (SGD). We first prove that gradient noise creates a systematic motion (a ``Noether flow")… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.07193v3-abstract-full').style.display = 'inline'; document.getElementById('2402.07193v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.07193v3-abstract-full" style="display: none;"> Symmetries are prevalent in deep learning and can significantly influence the learning dynamics of neural networks. In this paper, we examine how exponential symmetries -- a broad subclass of continuous symmetries present in the model architecture or loss function -- interplay with stochastic gradient descent (SGD). We first prove that gradient noise creates a systematic motion (a ``Noether flow") of the parameters $胃$ along the degenerate direction to a unique initialization-independent fixed point $胃^*$. These points are referred to as the {\it noise equilibria} because, at these points, noise contributions from different directions are balanced and aligned. Then, we show that the balance and alignment of gradient noise can serve as a novel alternative mechanism for explaining important phenomena such as progressive sharpening/flattening and representation formation within neural networks and have practical implications for understanding techniques like representation normalization and warmup. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.07193v3-abstract-full').style.display = 'none'; document.getElementById('2402.07193v3-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">v1</span> submitted 11 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">NeurIPS camera ready</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.00590">arXiv:2402.00590</a> <span> [<a href="https://arxiv.org/pdf/2402.00590">pdf</a>, <a href="https://arxiv.org/format/2402.00590">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> On the connected coalition number </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Guan%2C+X">Xiaxia Guan</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Maoqun 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="2402.00590v1-abstract-short" style="display: inline;"> For a graph $G=(V,E)$, a pair of vertex disjoint sets $A_{1}$ and $A_{2}$ form a connected coalition of $G$, if $A_{1}\cup A_{2}$ is a connected dominating set, but neither $A_{1}$ nor $A_{2}$ is a connected dominating set. A connected coalition partition of $G$ is a partition $桅$ of $V(G)$ such that each set in $桅$ either consists of only a singe vertex with the degree $|V(G)|-1$, or forms a conn… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.00590v1-abstract-full').style.display = 'inline'; document.getElementById('2402.00590v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.00590v1-abstract-full" style="display: none;"> For a graph $G=(V,E)$, a pair of vertex disjoint sets $A_{1}$ and $A_{2}$ form a connected coalition of $G$, if $A_{1}\cup A_{2}$ is a connected dominating set, but neither $A_{1}$ nor $A_{2}$ is a connected dominating set. A connected coalition partition of $G$ is a partition $桅$ of $V(G)$ such that each set in $桅$ either consists of only a singe vertex with the degree $|V(G)|-1$, or forms a connected coalition of $G$ with another set in $桅$. The connected coalition number of $G$, denoted by $CC(G)$, is the largest possible size of a connected coalition partition of $G$. In this paper, we characterize graphs that satisfy $CC(G)=2$. Moreover, we obtain the connected coalition number for unicycle graphs and for the corona product and join of two graphs. Finally, we give a lower bound on the connected coalition number of the Cartesian product and the lexicographic product of two graphs. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.00590v1-abstract-full').style.display = 'none'; document.getElementById('2402.00590v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.00314">arXiv:2402.00314</a> <span> [<a href="https://arxiv.org/pdf/2402.00314">pdf</a>, <a href="https://arxiv.org/ps/2402.00314">ps</a>, <a href="https://arxiv.org/format/2402.00314">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> </div> </div> <p class="title is-5 mathjax"> Littlewood-type theorems for random Dirichlet series </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+J">Jiale Chen</a>, <a href="/search/math?searchtype=author&query=Guo%2C+X">Xin Guo</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Maofa 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="2402.00314v1-abstract-short" style="display: inline;"> In this paper, we completely give the solution of the problem of Littlewood-type randomization in the Hardy and Bergman spaces of Dirichlet series. The Littlewood-type theorem for Bergman spaces of Dirichlet series is very different from the corresponding version for Hardy spaces of Dirichlet series; but also exhibits various pathological phenomena compared with the setting of analytic Bergman spa… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.00314v1-abstract-full').style.display = 'inline'; document.getElementById('2402.00314v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.00314v1-abstract-full" style="display: none;"> In this paper, we completely give the solution of the problem of Littlewood-type randomization in the Hardy and Bergman spaces of Dirichlet series. The Littlewood-type theorem for Bergman spaces of Dirichlet series is very different from the corresponding version for Hardy spaces of Dirichlet series; but also exhibits various pathological phenomena compared with the setting of analytic Bergman spaces over the unit disk, due to the fact that Dirichlet series behave as power series of infinitely many variables. A description for the inclusion between some mixed norm spaces of Dirichlet series plays an essential role in our investigation. Finally, as another application of the inclusion, we completely characterize the superposition operators between Bergman spaces of Dirichlet series. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.00314v1-abstract-full').style.display = 'none'; document.getElementById('2402.00314v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 31 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 30B50; 30H20; 46E15; 47H30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2401.08113">arXiv:2401.08113</a> <span> [<a href="https://arxiv.org/pdf/2401.08113">pdf</a>, <a href="https://arxiv.org/format/2401.08113">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> </div> <p class="title is-5 mathjax"> Feynman Graph Integrals on $\mathbb{C}^d$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+M">Minghao 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="2401.08113v2-abstract-short" style="display: inline;"> We introduce a type of graph integrals which are holomorphic analogs of configuration space integrals. We prove their (ultraviolet) finiteness by considering a compactification of the moduli space of graphs with metrics, and study their failure to be holomorphic. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2401.08113v2-abstract-full" style="display: none;"> We introduce a type of graph integrals which are holomorphic analogs of configuration space integrals. We prove their (ultraviolet) finiteness by considering a compactification of the moduli space of graphs with metrics, and study their failure to be holomorphic. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.08113v2-abstract-full').style.display = 'none'; document.getElementById('2401.08113v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 31 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">33 pages, 1 figure</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2401.04459">arXiv:2401.04459</a> <span> [<a href="https://arxiv.org/pdf/2401.04459">pdf</a>, <a href="https://arxiv.org/ps/2401.04459">ps</a>, <a href="https://arxiv.org/format/2401.04459">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"> The intermediate disorder regime for stable directed polymer in Poisson environment </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+M">Min 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="2401.04459v1-abstract-short" style="display: inline;"> We consider the stable directed polymer in Poisson random environment in dimension 1+1, under the intermediate disorder regime. We show that, under a diffusive scaling involving different parameters of the system, the normalized point-to-point partition function of the polymer converges in law to the solution of the stochastic heat equation with fractional Laplacian and Gaussian multiplicative noi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.04459v1-abstract-full').style.display = 'inline'; document.getElementById('2401.04459v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2401.04459v1-abstract-full" style="display: none;"> We consider the stable directed polymer in Poisson random environment in dimension 1+1, under the intermediate disorder regime. We show that, under a diffusive scaling involving different parameters of the system, the normalized point-to-point partition function of the polymer converges in law to the solution of the stochastic heat equation with fractional Laplacian and Gaussian multiplicative noise. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.04459v1-abstract-full').style.display = 'none'; document.getElementById('2401.04459v1-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 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">15 pages, comments welcome</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 60K37; 60K35; 82D60 </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=Wang%2C+M&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Wang%2C+M&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Wang%2C+M&start=50" class="pagination-link " aria-label="Page 2" aria-current="page">2 </a> </li> <li> <a href="/search/?searchtype=author&query=Wang%2C+M&start=100" class="pagination-link " aria-label="Page 3" aria-current="page">3 </a> </li> <li> <a href="/search/?searchtype=author&query=Wang%2C+M&start=150" class="pagination-link " aria-label="Page 4" aria-current="page">4 </a> </li> <li> <a href="/search/?searchtype=author&query=Wang%2C+M&start=200" class="pagination-link " aria-label="Page 5" aria-current="page">5 </a> </li> <li><span class="pagination-ellipsis">…</span></li> </ul> </nav> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns is-desktop" role="navigation" aria-label="Secondary">  <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/about">About</a></li> <li><a href="https://info.arxiv.org/help">Help</a></li> </ul> </div> <div class="column"> <ul class="nav-spaced"> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>contact arXiv</title><desc>Click here to contact arXiv</desc><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"/></svg> <a href="https://info.arxiv.org/help/contact.html"> Contact</a> </li> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>subscribe to arXiv mailings</title><desc>Click here to subscribe</desc><path d="M476 3.2L12.5 270.6c-18.1 10.4-15.8 35.6 2.2 43.2L121 358.4l287.3-253.2c5.5-4.9 13.3 2.6 8.6 8.3L176 407v80.5c0 23.6 28.5 32.9 42.5 15.8L282 426l124.6 52.2c14.2 6 30.4-2.9 33-18.2l72-432C515 7.8 493.3-6.8 476 3.2z"/></svg> <a href="https://info.arxiv.org/help/subscribe"> Subscribe</a> </li> </ul> </div> </div> </div>   <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/help/license/index.html">Copyright</a></li> <li><a href="https://info.arxiv.org/help/policies/privacy_policy.html">Privacy Policy</a></li> </ul> </div> <div class="column sorry-app-links"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/help/web_accessibility.html">Web Accessibility Assistance</a></li> <li> <p class="help"> <a class="a11y-main-link" href="https://status.arxiv.org" target="_blank">arXiv Operational Status <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 256 512" class="icon filter-dark_grey" role="presentation"><path d="M224.3 273l-136 136c-9.4 9.4-24.6 9.4-33.9 0l-22.6-22.6c-9.4-9.4-9.4-24.6 0-33.9l96.4-96.4-96.4-96.4c-9.4-9.4-9.4-24.6 0-33.9L54.3 103c9.4-9.4 24.6-9.4 33.9 0l136 136c9.5 9.4 9.5 24.6.1 34z"/></svg></a><br> Get status notifications via <a class="is-link" href="https://subscribe.sorryapp.com/24846f03/email/new" target="_blank"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"/></svg>email</a> or <a class="is-link" href="https://subscribe.sorryapp.com/24846f03/slack/new" target="_blank"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512" class="icon filter-black" role="presentation"><path d="M94.12 315.1c0 25.9-21.16 47.06-47.06 47.06S0 341 0 315.1c0-25.9 21.16-47.06 47.06-47.06h47.06v47.06zm23.72 0c0-25.9 21.16-47.06 47.06-47.06s47.06 21.16 47.06 47.06v117.84c0 25.9-21.16 47.06-47.06 47.06s-47.06-21.16-47.06-47.06V315.1zm47.06-188.98c-25.9 0-47.06-21.16-47.06-47.06S139 32 164.9 32s47.06 21.16 47.06 47.06v47.06H164.9zm0 23.72c25.9 0 47.06 21.16 47.06 47.06s-21.16 47.06-47.06 47.06H47.06C21.16 243.96 0 222.8 0 196.9s21.16-47.06 47.06-47.06H164.9zm188.98 47.06c0-25.9 21.16-47.06 47.06-47.06 25.9 0 47.06 21.16 47.06 47.06s-21.16 47.06-47.06 47.06h-47.06V196.9zm-23.72 0c0 25.9-21.16 47.06-47.06 47.06-25.9 0-47.06-21.16-47.06-47.06V79.06c0-25.9 21.16-47.06 47.06-47.06 25.9 0 47.06 21.16 47.06 47.06V196.9zM283.1 385.88c25.9 0 47.06 21.16 47.06 47.06 0 25.9-21.16 47.06-47.06 47.06-25.9 0-47.06-21.16-47.06-47.06v-47.06h47.06zm0-23.72c-25.9 0-47.06-21.16-47.06-47.06 0-25.9 21.16-47.06 47.06-47.06h117.84c25.9 0 47.06 21.16 47.06 47.06 0 25.9-21.16 47.06-47.06 47.06H283.1z"/></svg>slack</a> </p> </li> </ul> </div> </div> </div>  </div> </footer> <script src="https://static.arxiv.org/static/base/1.0.0a5/js/member_acknowledgement.js"></script> </body> </html>

CINXE.COM

Search | arXiv e-print repository