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is-link">Go</button> </div> </div> </form> </div> </div> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> <a href="" class="pagination-previous is-invisible">Previous </a> <a href="/search/?searchtype=author&query=Ma%2C+J&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Ma%2C+J&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Ma%2C+J&start=50" class="pagination-link " aria-label="Page 2" aria-current="page">2 </a> </li> <li> <a href="/search/?searchtype=author&query=Ma%2C+J&start=100" class="pagination-link " aria-label="Page 3" aria-current="page">3 </a> </li> <li> <a href="/search/?searchtype=author&query=Ma%2C+J&start=150" class="pagination-link " aria-label="Page 4" aria-current="page">4 </a> </li> <li> <a href="/search/?searchtype=author&query=Ma%2C+J&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/2502.09498">arXiv:2502.09498</a> <span> [<a href="https://arxiv.org/pdf/2502.09498">pdf</a>, <a href="https://arxiv.org/ps/2502.09498">ps</a>, <a href="https://arxiv.org/format/2502.09498">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> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> On the monodromy and spin parity of single-cylinder origamis in the minimal stratum </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Aougab%2C+T">Tarik Aougab</a>, <a href="/search/math?searchtype=author&query=Friedman-Brown%2C+A">Adam Friedman-Brown</a>, <a href="/search/math?searchtype=author&query=Jeffreys%2C+L">Luke Jeffreys</a>, <a href="/search/math?searchtype=author&query=Ma%2C+J">Jiajie Ma</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2502.09498v1-abstract-short" style="display: inline;"> In a paper with Menasco-Nieland, the first author constructed factorially many origamis in the minimal stratum of the moduli space of translation surfaces having simultaneously a single vertical cylinder and a single horizontal cylinder. Moreover, these origamis were constructed using the minimal number of squares required for origamis in the minimal stratum. We shall call such origamis minimal… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.09498v1-abstract-full').style.display = 'inline'; document.getElementById('2502.09498v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.09498v1-abstract-full" style="display: none;"> In a paper with Menasco-Nieland, the first author constructed factorially many origamis in the minimal stratum of the moduli space of translation surfaces having simultaneously a single vertical cylinder and a single horizontal cylinder. Moreover, these origamis were constructed using the minimal number of squares required for origamis in the minimal stratum. We shall call such origamis minimal $[1,1]$-origamis. In this work, by calculating their spin parities, we determine that the odd genus origamis in this construction all lie in the odd component of the minimal stratum, while the even genus origamis are contained in both the odd and even components with an asymptotic ratio of 3:1. Noticing that the even component is missed in the odd genus case, we provide a generalisation of the odd genus construction that gives rise to factorially many minimal $[1,1]$-origamis lying in the even component. Motivated by understanding the $SL(2,\mathbb{Z})$-orbits of these origamis, we investigate their monodromy groups (a weak $SL(2,\mathbb{Z})$-invariant). We also prove that, with one exception, the monodromy group of any primitive minimal $[1,1]$-origami in the minimal stratum must be simple. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.09498v1-abstract-full').style.display = 'none'; document.getElementById('2502.09498v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">47 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/2502.02301">arXiv:2502.02301</a> <span> [<a href="https://arxiv.org/pdf/2502.02301">pdf</a>, <a href="https://arxiv.org/format/2502.02301">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 a conjecture of Pach-Spencer-T贸th for graph crossing numbers </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+K">Kaizhe Chen</a>, <a href="/search/math?searchtype=author&query=Ma%2C+J">Jie Ma</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2502.02301v1-abstract-short" style="display: inline;"> The crossing number of a graph $G$ denotes the minimum number of crossings in any planar drawing of $G$. In this short note, we confirm a long-standing conjecture posed by Pach, Spencer, and T贸th over 25 years ago, establishing an optimal lower bound on the crossing number of graphs that satisfy some monotone properties. Furthermore, we address a related open problem introduced by Pach and T贸th in… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.02301v1-abstract-full').style.display = 'inline'; document.getElementById('2502.02301v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.02301v1-abstract-full" style="display: none;"> The crossing number of a graph $G$ denotes the minimum number of crossings in any planar drawing of $G$. In this short note, we confirm a long-standing conjecture posed by Pach, Spencer, and T贸th over 25 years ago, establishing an optimal lower bound on the crossing number of graphs that satisfy some monotone properties. Furthermore, we address a related open problem introduced by Pach and T贸th in 2000, which explores the interplay between the crossing number of a graph, its degree sequence, and its bisection width. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.02301v1-abstract-full').style.display = 'none'; document.getElementById('2502.02301v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">10 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/2502.01079">arXiv:2502.01079</a> <span> [<a href="https://arxiv.org/pdf/2502.01079">pdf</a>, <a href="https://arxiv.org/ps/2502.01079">ps</a>, <a href="https://arxiv.org/format/2502.01079">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> On eigenfunctions and nodal sets of the Witten-Laplacian </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+R">Ruifeng Chen</a>, <a href="/search/math?searchtype=author&query=Mao%2C+J">Jing Mao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2502.01079v1-abstract-short" style="display: inline;"> In this paper, we successfully establish a Courant-type nodal domain theorem for both the Dirichlet eigenvalue problem and the closed eigenvalue problem of the Witten-Laplacian. Moreover, we also characterize the properties of the nodal lines of the eigenfunctions of the Witten-Laplacian on smooth Riemannian $2$-manifolds. Besides, for a Riemann surface with genus $g$, an upper bound for the multi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.01079v1-abstract-full').style.display = 'inline'; document.getElementById('2502.01079v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.01079v1-abstract-full" style="display: none;"> In this paper, we successfully establish a Courant-type nodal domain theorem for both the Dirichlet eigenvalue problem and the closed eigenvalue problem of the Witten-Laplacian. Moreover, we also characterize the properties of the nodal lines of the eigenfunctions of the Witten-Laplacian on smooth Riemannian $2$-manifolds. Besides, for a Riemann surface with genus $g$, an upper bound for the multiplicity of closed eigenvalues of the Witten-Laplacian can be provided. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.01079v1-abstract-full').style.display = 'none'; document.getElementById('2502.01079v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">11 pages. Comments are welcome</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35P15; 49Jxx; 35J15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2502.01055">arXiv:2502.01055</a> <span> [<a href="https://arxiv.org/pdf/2502.01055">pdf</a>, <a href="https://arxiv.org/format/2502.01055">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="Robotics">cs.RO</span> </div> </div> <p class="title is-5 mathjax"> On the Surprising Robustness of Sequential Convex Optimization for Contact-Implicit Motion Planning </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+Y">Yulin Li</a>, <a href="/search/math?searchtype=author&query=Han%2C+H">Haoyu Han</a>, <a href="/search/math?searchtype=author&query=Kang%2C+S">Shucheng Kang</a>, <a href="/search/math?searchtype=author&query=Ma%2C+J">Jun Ma</a>, <a href="/search/math?searchtype=author&query=Yang%2C+H">Heng Yang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2502.01055v1-abstract-short" style="display: inline;"> Contact-implicit motion planning-embedding contact sequencing as implicit complementarity constraints-holds the promise of leveraging continuous optimization to discover new contact patterns online. Nevertheless, the resulting optimization, being an instance of Mathematical Programming with Complementary Constraints, fails the classical constraint qualifications that are crucial for the convergenc… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.01055v1-abstract-full').style.display = 'inline'; document.getElementById('2502.01055v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.01055v1-abstract-full" style="display: none;"> Contact-implicit motion planning-embedding contact sequencing as implicit complementarity constraints-holds the promise of leveraging continuous optimization to discover new contact patterns online. Nevertheless, the resulting optimization, being an instance of Mathematical Programming with Complementary Constraints, fails the classical constraint qualifications that are crucial for the convergence of popular numerical solvers. We present robust contact-implicit motion planning with sequential convex programming (CRISP), a solver that departs from the usual primal-dual algorithmic framework but instead only focuses on the primal problem. CRISP solves a convex quadratic program with an adaptive trust region radius at each iteration, and its convergence is evaluated by a merit function using weighted penalty. We (i) provide sufficient conditions on CRISP's convergence to first-order stationary points of the merit function; (ii) release a high-performance C++ implementation of CRISP with a generic nonlinear programming interface; and (iii) demonstrate CRISP's surprising robustness in solving contact-implicit planning with naive initialization. In fact, CRISP solves several contact-implicit problems with all-zero initialization. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.01055v1-abstract-full').style.display = 'none'; document.getElementById('2502.01055v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2025. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2501.15416">arXiv:2501.15416</a> <span> [<a href="https://arxiv.org/pdf/2501.15416">pdf</a>, <a href="https://arxiv.org/ps/2501.15416">ps</a>, <a href="https://arxiv.org/format/2501.15416">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Periodic solutions for McKean-Vlasov SDEs under periodic distribution-dependent Lyapunov conditions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ma%2C+J">Jun Ma</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2501.15416v1-abstract-short" style="display: inline;"> In this paper, we prove the existence of periodic solutions for McKean-Vlasov SDEs under periodic distribution-dependent Lyapunov conditions, which is obtained by periodic Markov processes with state space $\mathbb R^d\times \mathcal P(\mathbb R^d)$. Here $\mathcal P(\mathbb R^d)$ denotes the space of probability measures on $\mathbb R^d$. In addition, we show the convergence to the periodic solut… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.15416v1-abstract-full').style.display = 'inline'; document.getElementById('2501.15416v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2501.15416v1-abstract-full" style="display: none;"> In this paper, we prove the existence of periodic solutions for McKean-Vlasov SDEs under periodic distribution-dependent Lyapunov conditions, which is obtained by periodic Markov processes with state space $\mathbb R^d\times \mathcal P(\mathbb R^d)$. Here $\mathcal P(\mathbb R^d)$ denotes the space of probability measures on $\mathbb R^d$. In addition, we show the convergence to the periodic solution and the continuous dependence on parameters of periodic solutions for McKean-Vlasov SDEs. Finally, we provide several examples to illustrate our theoretical results. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.15416v1-abstract-full').style.display = 'none'; document.getElementById('2501.15416v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2025. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2501.05753">arXiv:2501.05753</a> <span> [<a href="https://arxiv.org/pdf/2501.05753">pdf</a>, <a href="https://arxiv.org/format/2501.05753">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <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="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Exactly Solvable and Integrable Systems">nlin.SI</span> </div> </div> <p class="title is-5 mathjax"> Dubrovin duality and mirror symmetry for ADE resolutions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Brini%2C+A">Andrea Brini</a>, <a href="/search/math?searchtype=author&query=Ma%2C+J">Jingxiang Ma</a>, <a href="/search/math?searchtype=author&query=Strachan%2C+I+A+B">Ian A. B. Strachan</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2501.05753v1-abstract-short" style="display: inline;"> We show that, under Dubrovin's notion of ''almost'' duality, the Frobenius manifold structure on the orbit spaces of the extended affine Weyl groups of type $\mathrm{ADE}$ is dual, for suitable choices of weight markings, to the equivariant quantum cohomology of the minimal resolution of the du Val singularity of the same Dynkin type. We also provide a uniform Lie-theoretic construction of Landau-… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.05753v1-abstract-full').style.display = 'inline'; document.getElementById('2501.05753v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2501.05753v1-abstract-full" style="display: none;"> We show that, under Dubrovin's notion of ''almost'' duality, the Frobenius manifold structure on the orbit spaces of the extended affine Weyl groups of type $\mathrm{ADE}$ is dual, for suitable choices of weight markings, to the equivariant quantum cohomology of the minimal resolution of the du Val singularity of the same Dynkin type. We also provide a uniform Lie-theoretic construction of Landau-Ginzburg mirrors for the quantum cohomology of $\mathrm{ADE}$ resolutions. The mirror B-model is described by a one-dimensional LG superpotential associated to the spectral curve of the $\widehat{\mathrm{ADE}}$ affine relativistic Toda chain. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.05753v1-abstract-full').style.display = 'none'; document.getElementById('2501.05753v1-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 January, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">26 pages. Ancillary files of this arXiv submission are accessible in the directory anc/ included in the source of this manuscript. The directory contains the Wolfram Language code used to verify Proposition 3.16 for the exceptional series. A Mathematica package verifying the main conjecture in IMRN 19 (2003), 1035-1051 is also included</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.14420">arXiv:2412.14420</a> <span> [<a href="https://arxiv.org/pdf/2412.14420">pdf</a>, <a href="https://arxiv.org/ps/2412.14420">ps</a>, <a href="https://arxiv.org/format/2412.14420">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"> An inverse theorem for generalized arithmetic progression with mild multiplicative property </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Croot%2C+E">Ernie Croot</a>, <a href="/search/math?searchtype=author&query=Mao%2C+J">Junzhe Mao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.14420v1-abstract-short" style="display: inline;"> We prove a structural theorem for generalized arithmetic progressions in $\F_p$ which contain a large product set of two other progressions. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.14420v1-abstract-full" style="display: none;"> We prove a structural theorem for generalized arithmetic progressions in $\F_p$ which contain a large product set of two other progressions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.14420v1-abstract-full').style.display = 'none'; document.getElementById('2412.14420v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.12030">arXiv:2412.12030</a> <span> [<a href="https://arxiv.org/pdf/2412.12030">pdf</a>, <a href="https://arxiv.org/format/2412.12030">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"> Memory-Reduced Meta-Learning with Guaranteed Convergence </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yang%2C+H">Honglin Yang</a>, <a href="/search/math?searchtype=author&query=Ma%2C+J">Ji Ma</a>, <a href="/search/math?searchtype=author&query=Yu%2C+X">Xiao Yu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.12030v1-abstract-short" style="display: inline;"> The optimization-based meta-learning approach is gaining increased traction because of its unique ability to quickly adapt to a new task using only small amounts of data. However, existing optimization-based meta-learning approaches, such as MAML, ANIL and their variants, generally employ backpropagation for upper-level gradient estimation, which requires using historical lower-level parameters/gr… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.12030v1-abstract-full').style.display = 'inline'; document.getElementById('2412.12030v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.12030v1-abstract-full" style="display: none;"> The optimization-based meta-learning approach is gaining increased traction because of its unique ability to quickly adapt to a new task using only small amounts of data. However, existing optimization-based meta-learning approaches, such as MAML, ANIL and their variants, generally employ backpropagation for upper-level gradient estimation, which requires using historical lower-level parameters/gradients and thus increases computational and memory overhead in each iteration. In this paper, we propose a meta-learning algorithm that can avoid using historical parameters/gradients and significantly reduce memory costs in each iteration compared to existing optimization-based meta-learning approaches. In addition to memory reduction, we prove that our proposed algorithm converges sublinearly with the iteration number of upper-level optimization, and the convergence error decays sublinearly with the batch size of sampled tasks. In the specific case in terms of deterministic meta-learning, we also prove that our proposed algorithm converges to an exact solution. Moreover, we quantify that the computational complexity of the algorithm is on the order of $\mathcal{O}(蔚^{-1})$, which matches existing convergence results on meta-learning even without using any historical parameters/gradients. Experimental results on meta-learning benchmarks confirm the efficacy of our proposed algorithm. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.12030v1-abstract-full').style.display = 'none'; document.getElementById('2412.12030v1-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 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 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, 2 figures; Accepted by the 39th Annual AAAI Conference on Artificial Intelligence (AAAI)</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.04972">arXiv:2412.04972</a> <span> [<a href="https://arxiv.org/pdf/2412.04972">pdf</a>, <a href="https://arxiv.org/format/2412.04972">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"> Undecidability of polynomial inequalities in tournaments </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+H">Hao Chen</a>, <a href="/search/math?searchtype=author&query=Lin%2C+Y">Yupeng Lin</a>, <a href="/search/math?searchtype=author&query=Ma%2C+J">Jie Ma</a>, <a href="/search/math?searchtype=author&query=Wei%2C+F">Fan Wei</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.04972v2-abstract-short" style="display: inline;"> Many fundamental problems in extremal combinatorics are equivalent to proving certain polynomial inequalities in graph homomorphism densities. In 2011, a breakthrough result by Hatami and Norine showed that it is undecidable to verify polynomial inequalities in graph homomorphism densities. Recently, Blekherman, Raymond and Wei extended this result by showing that it is also undecidable to determi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.04972v2-abstract-full').style.display = 'inline'; document.getElementById('2412.04972v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.04972v2-abstract-full" style="display: none;"> Many fundamental problems in extremal combinatorics are equivalent to proving certain polynomial inequalities in graph homomorphism densities. In 2011, a breakthrough result by Hatami and Norine showed that it is undecidable to verify polynomial inequalities in graph homomorphism densities. Recently, Blekherman, Raymond and Wei extended this result by showing that it is also undecidable to determine the validity of polynomial inequalities in homomorphism densities for weighted graphs with edge weights taking real values. These two results resolved a question of Lov谩sz. In this paper, we consider the problem of determining the validity of polynomial inequalities in digraph homomorphism densities for tournaments. We prove that the answer to this problem is also undecidable. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.04972v2-abstract-full').style.display = 'none'; document.getElementById('2412.04972v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">22 pages, 3 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.01574">arXiv:2412.01574</a> <span> [<a href="https://arxiv.org/pdf/2412.01574">pdf</a>, <a href="https://arxiv.org/format/2412.01574">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Unifying AMP Algorithms for Rotationally-Invariant Models </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+S">Songbin Liu</a>, <a href="/search/math?searchtype=author&query=Ma%2C+J">Junjie Ma</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.01574v1-abstract-short" style="display: inline;"> This paper presents a unified framework for constructing Approximate Message Passing (AMP) algorithms for rotationally-invariant models. By employing a general iterative algorithm template and reducing it to long-memory Orthogonal AMP (OAMP), we systematically derive the correct Onsager terms of AMP algorithms. This approach allows us to rederive an AMP algorithm introduced by Fan and Opper et al.… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.01574v1-abstract-full').style.display = 'inline'; document.getElementById('2412.01574v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.01574v1-abstract-full" style="display: none;"> This paper presents a unified framework for constructing Approximate Message Passing (AMP) algorithms for rotationally-invariant models. By employing a general iterative algorithm template and reducing it to long-memory Orthogonal AMP (OAMP), we systematically derive the correct Onsager terms of AMP algorithms. This approach allows us to rederive an AMP algorithm introduced by Fan and Opper et al., while shedding new light on the role of free cumulants of the spectral law. The free cumulants arise naturally from a recursive centering operation, potentially of independent interest beyond the scope of AMP. To illustrate the flexibility of our framework, we introduce two novel AMP variants and apply them to estimation in spiked models. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.01574v1-abstract-full').style.display = 'none'; document.getElementById('2412.01574v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.19773">arXiv:2411.19773</a> <span> [<a href="https://arxiv.org/pdf/2411.19773">pdf</a>, <a href="https://arxiv.org/ps/2411.19773">ps</a>, <a href="https://arxiv.org/format/2411.19773">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"> Complete tripartite subgraphs of balanced tripartite graphs with large minimum degree </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+Y">Yihan Chen</a>, <a href="/search/math?searchtype=author&query=He%2C+J">Jialin He</a>, <a href="/search/math?searchtype=author&query=Lo%2C+A">Allan Lo</a>, <a href="/search/math?searchtype=author&query=Luo%2C+C">Cong Luo</a>, <a href="/search/math?searchtype=author&query=Ma%2C+J">Jie Ma</a>, <a href="/search/math?searchtype=author&query=Zhao%2C+Y">Yi 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="2411.19773v1-abstract-short" style="display: inline;"> In 1975 Bollob谩s, Erd艖s, and Szemer茅di asked what minimum degree guarantees an octahedral subgraph $K_3(2)$ in any tripartite graph $G$ with $n$ vertices in each vertex class. We show that $未(G)\geq n+2n^{\frac{5}{6}}$ suffices thus improving the bound $n+(1+o(1))n^{\frac{11}{12}}$ of Bhalkikar and Zhao obtained by following their approach. Bollob谩s, Erd艖s, and Szemer茅di conjectured that… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.19773v1-abstract-full').style.display = 'inline'; document.getElementById('2411.19773v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.19773v1-abstract-full" style="display: none;"> In 1975 Bollob谩s, Erd艖s, and Szemer茅di asked what minimum degree guarantees an octahedral subgraph $K_3(2)$ in any tripartite graph $G$ with $n$ vertices in each vertex class. We show that $未(G)\geq n+2n^{\frac{5}{6}}$ suffices thus improving the bound $n+(1+o(1))n^{\frac{11}{12}}$ of Bhalkikar and Zhao obtained by following their approach. Bollob谩s, Erd艖s, and Szemer茅di conjectured that $n+cn^{\frac{1}{2}}$ suffices and there are many $K_3(2)$-free tripartite graphs $G$ with $未(G)\geq n+cn^{\frac{1}{2}}$. We confirm this conjecture under the additional assumption that every vertex in $G$ is adjacent to at least $(1/5+\varepsilon)n$ vertices in any other vertex class. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.19773v1-abstract-full').style.display = 'none'; document.getElementById('2411.19773v1-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 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/2411.18974">arXiv:2411.18974</a> <span> [<a href="https://arxiv.org/pdf/2411.18974">pdf</a>, <a href="https://arxiv.org/format/2411.18974">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Robotics">cs.RO</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"> Synergizing Decision Making and Trajectory Planning Using Two-Stage Optimization for Autonomous Vehicles </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+W">Wenru Liu</a>, <a href="/search/math?searchtype=author&query=Liu%2C+H">Haichao Liu</a>, <a href="/search/math?searchtype=author&query=Zheng%2C+L">Lei Zheng</a>, <a href="/search/math?searchtype=author&query=Huang%2C+Z">Zhenmin Huang</a>, <a href="/search/math?searchtype=author&query=Ma%2C+J">Jun Ma</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="2411.18974v1-abstract-short" style="display: inline;"> This paper introduces a local planner that synergizes the decision making and trajectory planning modules towards autonomous driving. The decision making and trajectory planning tasks are jointly formulated as a nonlinear programming problem with an integrated objective function. However, integrating the discrete decision variables into the continuous trajectory optimization leads to a mixed-integ… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.18974v1-abstract-full').style.display = 'inline'; document.getElementById('2411.18974v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.18974v1-abstract-full" style="display: none;"> This paper introduces a local planner that synergizes the decision making and trajectory planning modules towards autonomous driving. The decision making and trajectory planning tasks are jointly formulated as a nonlinear programming problem with an integrated objective function. However, integrating the discrete decision variables into the continuous trajectory optimization leads to a mixed-integer programming (MIP) problem with inherent nonlinearity and nonconvexity. To address the challenge in solving the problem, the original problem is decomposed into two sub-stages, and a two-stage optimization (TSO) based approach is presented to ensure the coherence in outcomes for the two stages. The optimization problem in the first stage determines the optimal decision sequence that acts as an informed initialization. With the outputs from the first stage, the second stage necessitates the use of a high-fidelity vehicle model and strict enforcement of the collision avoidance constraints as part of the trajectory planning problem. We evaluate the effectiveness of our proposed planner across diverse multi-lane scenarios. The results demonstrate that the proposed planner simultaneously generates a sequence of optimal decisions and the corresponding trajectory that significantly improves driving performance in terms of driving safety and traveling efficiency as compared to alternative methods. Additionally, we implement the closed-loop simulation in CARLA, and the results showcase the effectiveness of the proposed planner to adapt to changing driving situations with high computational efficiency. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.18974v1-abstract-full').style.display = 'none'; document.getElementById('2411.18974v1-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 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/2411.15579">arXiv:2411.15579</a> <span> [<a href="https://arxiv.org/pdf/2411.15579">pdf</a>, <a href="https://arxiv.org/format/2411.15579">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"> Phase transition of degenerate Tur谩n problems in $p$-norms </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gao%2C+J">Jun Gao</a>, <a href="/search/math?searchtype=author&query=Liu%2C+X">Xizhi Liu</a>, <a href="/search/math?searchtype=author&query=Ma%2C+J">Jie Ma</a>, <a href="/search/math?searchtype=author&query=Pikhurko%2C+O">Oleg Pikhurko</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="2411.15579v2-abstract-short" style="display: inline;"> For a positive real number $p$, the $p$-norm $\left\lVert G \right\rVert_p$ of a graph $G$ is the sum of the $p$-th powers of all vertex degrees. We study the maximum $p$-norm $\mathrm{ex}_{p}(n,F)$ of $F$-free graphs on $n$ vertices, focusing on the case where $F$ is a bipartite graph. The case $p = 1$ corresponds to the classical degenerate Tur谩n problem, which has yielded numerous results indic… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.15579v2-abstract-full').style.display = 'inline'; document.getElementById('2411.15579v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.15579v2-abstract-full" style="display: none;"> For a positive real number $p$, the $p$-norm $\left\lVert G \right\rVert_p$ of a graph $G$ is the sum of the $p$-th powers of all vertex degrees. We study the maximum $p$-norm $\mathrm{ex}_{p}(n,F)$ of $F$-free graphs on $n$ vertices, focusing on the case where $F$ is a bipartite graph. The case $p = 1$ corresponds to the classical degenerate Tur谩n problem, which has yielded numerous results indicating that extremal constructions tend to exhibit certain pseudorandom properties. In contrast, results such as those by Caro--Yuster, Nikiforov, and Gerbner suggest that for large $p$, extremal constructions often display a star-like structure. It is natural to conjecture that for every bipartite graph $F$, there exists a threshold $p_F$ such that for $p< p_{F}$, the order of $\mathrm{ex}_{p}(n,F)$ is governed by pseudorandom constructions, while for $p > p_{F}$, it is governed by star-like constructions. We confirm this conjecture by determining the exact value of $p_{F}$, under a mild assumption on the growth rate of $\mathrm{ex}(n,F)$. Our results extend to $r$-uniform hypergraphs as well. We also prove a general upper bound that is tight up to a $\log n$ factor for $\mathrm{ex}_{p}(n,F)$ when $p = p_{F}$. We conjecture that this $\log n$ factor is unnecessary and prove this conjecture for several classes of well-studied bipartite graphs, including one-side degree-bounded graphs and families of short even cycles. Our proofs involve $p$-norm adaptions of fundamental tools from degenerate Tur谩n problems, including the Erd艖s--Simonovits Regularization Theorem and the Dependent Random Choice. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.15579v2-abstract-full').style.display = 'none'; document.getElementById('2411.15579v2-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 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 23 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">29 pages, we added a remark at the end of the Introduction</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.06393">arXiv:2411.06393</a> <span> [<a href="https://arxiv.org/pdf/2411.06393">pdf</a>, <a href="https://arxiv.org/format/2411.06393">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="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"> Evolution of weights on a connected finite graph </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ma%2C+J">Jicheng Ma</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yunyan Yang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.06393v1-abstract-short" style="display: inline;"> On a connected finite graph, we propose an evolution of weights including Ollivier's Ricci flow as a special case. During the evolution process, on each edge, the speed of change of weight is exactly the difference between the Wasserstein distance related to two probability measures and certain graph distance. Here the probability measure may be chosen as an $伪$-lazy one-step random walk, an $伪$-l… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.06393v1-abstract-full').style.display = 'inline'; document.getElementById('2411.06393v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.06393v1-abstract-full" style="display: none;"> On a connected finite graph, we propose an evolution of weights including Ollivier's Ricci flow as a special case. During the evolution process, on each edge, the speed of change of weight is exactly the difference between the Wasserstein distance related to two probability measures and certain graph distance. Here the probability measure may be chosen as an $伪$-lazy one-step random walk, an $伪$-lazy two-step random walk, or a general probability measure. Based on the ODE theory, we show that the initial value problem has a unique global solution. A discrete version of the above evolution is applied to the problem of community detection. Our algorithm is based on such a discrete evolution, where probability measures are chosen as $伪$-lazy one-step random walk and $伪$-lazy two-step random walk respectively. Note that the later measure has not been used in previous works [2, 16, 20, 23]. Here, as in [20], only one surgery needs to be performed after the last iteration. Moreover, our algorithm is much easier than those of [2, 16, 20], which were all based on Lin-Lu-Yau's Ricci curvature. The code is available at https://github.com/mjc191812/Evolution-of-weights-on-a-connected-finite-graph. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.06393v1-abstract-full').style.display = 'none'; document.getElementById('2411.06393v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">35 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C21; 05C85; 35R02; 68Q06 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.06126">arXiv:2411.06126</a> <span> [<a href="https://arxiv.org/pdf/2411.06126">pdf</a>, <a href="https://arxiv.org/ps/2411.06126">ps</a>, <a href="https://arxiv.org/format/2411.06126">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> On the error term concerning the number of cyclic subgroups of Z_l \times Z_m \times Z_n with lmn\leqslant x </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ma%2C+J">Jing Ma</a>, <a href="/search/math?searchtype=author&query=Li%2C+J">Jiaming Li</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+J">Jia 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="2411.06126v1-abstract-short" style="display: inline;"> Let Zn denote the additive group of residue classes modulo n. Let c(l,m,n) denote the number of cyclic subgroups of Zl *Zm *Zn. For any x > 1, we consider the asymptotic behavior of D3c(x):= \sum_{lmn\leq x} c(l,m,n), obtain an asymptotic formula by complex method, and get an upper bound for the integral mean-square of the error term in that asymptotic formula. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.06126v1-abstract-full" style="display: none;"> Let Zn denote the additive group of residue classes modulo n. Let c(l,m,n) denote the number of cyclic subgroups of Zl *Zm *Zn. For any x > 1, we consider the asymptotic behavior of D3c(x):= \sum_{lmn\leq x} c(l,m,n), obtain an asymptotic formula by complex method, and get an upper bound for the integral mean-square of the error term in that asymptotic formula. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.06126v1-abstract-full').style.display = 'none'; document.getElementById('2411.06126v1-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">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">no</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 11Axx Elementary number theory <span class="has-text-black-bis has-text-weight-semibold">ACM Class:</span> G.0 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.05492">arXiv:2411.05492</a> <span> [<a href="https://arxiv.org/pdf/2411.05492">pdf</a>, <a href="https://arxiv.org/ps/2411.05492">ps</a>, <a href="https://arxiv.org/format/2411.05492">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Signal Processing">eess.SP</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"> Covariance-Based Device Activity Detection with Massive MIMO for Near-Field Correlated Channels </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+Z">Ziyue Wang</a>, <a href="/search/math?searchtype=author&query=Li%2C+Y">Yang Li</a>, <a href="/search/math?searchtype=author&query=Liu%2C+Y">Ya-Feng Liu</a>, <a href="/search/math?searchtype=author&query=Ma%2C+J">Junjie Ma</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="2411.05492v1-abstract-short" style="display: inline;"> This paper studies the device activity detection problem in a massive multiple-input multiple-output (MIMO) system for near-field communications (NFC). In this system, active devices transmit their signature sequences to the base station (BS), which detects the active devices based on the received signal. In this paper, we model the near-field channels as correlated Rician fading channels and form… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.05492v1-abstract-full').style.display = 'inline'; document.getElementById('2411.05492v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.05492v1-abstract-full" style="display: none;"> This paper studies the device activity detection problem in a massive multiple-input multiple-output (MIMO) system for near-field communications (NFC). In this system, active devices transmit their signature sequences to the base station (BS), which detects the active devices based on the received signal. In this paper, we model the near-field channels as correlated Rician fading channels and formulate the device activity detection problem as a maximum likelihood estimation (MLE) problem. Compared to the traditional uncorrelated channel model, the correlation of channels complicates both algorithm design and theoretical analysis of the MLE problem. On the algorithmic side, we propose two computationally efficient algorithms for solving the MLE problem: an exact coordinate descent (CD) algorithm and an inexact CD algorithm. The exact CD algorithm solves the one-dimensional optimization subproblem exactly using matrix eigenvalue decomposition and polynomial root-finding. By approximating the objective function appropriately, the inexact CD algorithm solves the one-dimensional optimization subproblem inexactly with lower complexity and more robust numerical performance. Additionally, we analyze the detection performance of the MLE problem under correlated channels by comparing it with the case of uncorrelated channels. The analysis shows that when the overall number of devices $N$ is large or the signature sequence length $L$ is small, the detection performance of MLE under correlated channels tends to be better than that under uncorrelated channels. Conversely, when $N$ is small or $L$ is large, MLE performs better under uncorrelated channels than under correlated ones. Simulation results demonstrate the computational efficiency of the proposed algorithms and verify the correctness of the analysis. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.05492v1-abstract-full').style.display = 'none'; document.getElementById('2411.05492v1-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 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">15 pages, 8 figures, submitted for possible publication</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.22904">arXiv:2410.22904</a> <span> [<a href="https://arxiv.org/pdf/2410.22904">pdf</a>, <a href="https://arxiv.org/ps/2410.22904">ps</a>, <a href="https://arxiv.org/format/2410.22904">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Backward Uniqueness of Extrinsic Geometric Flow in general ambient manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+D">Dasong Li</a>, <a href="/search/math?searchtype=author&query=Ma%2C+J+M+S">John Man Shun Ma</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="2410.22904v1-abstract-short" style="display: inline;"> In this paper we prove two backward uniqueness theorems for extrinsic geometric flow of possibly non-compact hypersurfaces in general ambient complete Riemannian manifolds. These are applicable to a wide range of extrinsic geometric flow, including the mean curvature flow, inverse mean curvature flow, Gauss curvature flow and so on. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.22904v1-abstract-full" style="display: none;"> In this paper we prove two backward uniqueness theorems for extrinsic geometric flow of possibly non-compact hypersurfaces in general ambient complete Riemannian manifolds. These are applicable to a wide range of extrinsic geometric flow, including the mean curvature flow, inverse mean curvature flow, Gauss curvature flow and so on. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.22904v1-abstract-full').style.display = 'none'; document.getElementById('2410.22904v1-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">MSC Class:</span> 53C44 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.20057">arXiv:2410.20057</a> <span> [<a href="https://arxiv.org/pdf/2410.20057">pdf</a>, <a href="https://arxiv.org/format/2410.20057">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="Probability">math.PR</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"> Mechanism learning: Reverse causal inference in the presence of multiple unknown confounding through front-door causal bootstrapping </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Mao%2C+J">Jianqiao Mao</a>, <a href="/search/math?searchtype=author&query=Little%2C+M+A">Max A. Little</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="2410.20057v1-abstract-short" style="display: inline;"> A major limitation of machine learning (ML) prediction models is that they recover associational, rather than causal, predictive relationships between variables. In high-stakes automation applications of ML this is problematic, as the model often learns spurious, non-causal associations. This paper proposes mechanism learning, a simple method which uses front-door causal bootstrapping to deconfoun… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.20057v1-abstract-full').style.display = 'inline'; document.getElementById('2410.20057v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.20057v1-abstract-full" style="display: none;"> A major limitation of machine learning (ML) prediction models is that they recover associational, rather than causal, predictive relationships between variables. In high-stakes automation applications of ML this is problematic, as the model often learns spurious, non-causal associations. This paper proposes mechanism learning, a simple method which uses front-door causal bootstrapping to deconfound observational data such that any appropriate ML model is forced to learn predictive relationships between effects and their causes (reverse causal inference), despite the potential presence of multiple unknown and unmeasured confounding. Effect variables can be very high dimensional, and the predictive relationship nonlinear, as is common in ML applications. This novel method is widely applicable, the only requirement is the existence of a mechanism variable mediating the cause (prediction target) and effect (feature data), which is independent of the (unmeasured) confounding variables. We test our method on fully synthetic, semi-synthetic and real-world datasets, demonstrating that it can discover reliable, unbiased, causal ML predictors where by contrast, the same ML predictor trained naively using classical supervised learning on the original observational data, is heavily biased by spurious associations. We provide code to implement the results in the paper, online. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.20057v1-abstract-full').style.display = 'none'; document.getElementById('2410.20057v1-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, 6 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">ACM Class:</span> I.2.4; G.3 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.16409">arXiv:2410.16409</a> <span> [<a href="https://arxiv.org/pdf/2410.16409">pdf</a>, <a href="https://arxiv.org/format/2410.16409">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Integrated timetabling, vehicle scheduling, and dynamic capacity allocation of modular autonomous vehicles under demand uncertainty </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xia%2C+D">Dongyang Xia</a>, <a href="/search/math?searchtype=author&query=Ma%2C+J">Jihui Ma</a>, <a href="/search/math?searchtype=author&query=Azadeh%2C+S+S">Shadi Sharif Azadeh</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="2410.16409v1-abstract-short" style="display: inline;"> The Integrated Timetabling and Vehicle Scheduling (TTVS) problem has extensive applications in all sorts of transit networks. Recently, the emerging modular autonomous vehicles composed of modular autonomous units have made it possible to dynamically adjust on-board capacity to better match space-time imbalanced passenger flows. In this paper, we introduce an integrated framework for the TTVS prob… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.16409v1-abstract-full').style.display = 'inline'; document.getElementById('2410.16409v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.16409v1-abstract-full" style="display: none;"> The Integrated Timetabling and Vehicle Scheduling (TTVS) problem has extensive applications in all sorts of transit networks. Recently, the emerging modular autonomous vehicles composed of modular autonomous units have made it possible to dynamically adjust on-board capacity to better match space-time imbalanced passenger flows. In this paper, we introduce an integrated framework for the TTVS problem within a dynamically capacitated and modularized bus network, taking the time-varying and uncertain passenger demand patterns into account. The fixed-line modularized bus network operates units that can be (de)coupled and rerouted across different lines within the network at various times and locations to respond to the time-varying demand, providing passengers with the opportunity to make in-vehicle transfers. We formulate a stochastic programming model to jointly determine the optimal robust timetable, dynamic formations of vehicles, and cross-line circulations of these units, aiming to minimize the weighted sum of operational and passengers' costs. To obtain high-quality solutions of realistic instances, we propose a tailored integer L-shaped method coupled with valid inequalities to solve the stochastic mixed-integer programming model dynamically through a rolling-horizon optimization algorithm. An extensive computational study based on the real-world data of the Beijing bus network shows the effectiveness of the proposed approaches. Our method outperforms the two-step optimization method involving sequential decision-making for timetables and vehicle schedules. Furthermore, the computational results illustrate that our approaches are able to find timetables and vehicle schedules requiring fewer units and lower operational costs compared with using fixed-formation vehicles. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.16409v1-abstract-full').style.display = 'none'; document.getElementById('2410.16409v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 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/2410.07028">arXiv:2410.07028</a> <span> [<a href="https://arxiv.org/pdf/2410.07028">pdf</a>, <a href="https://arxiv.org/ps/2410.07028">ps</a>, <a href="https://arxiv.org/format/2410.07028">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 proof of a conjecture about cages </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Pan%2C+X">Xiang-Feng Pan</a>, <a href="/search/math?searchtype=author&query=Mao%2C+J">Jing-Zhong Mao</a>, <a href="/search/math?searchtype=author&query=Liu%2C+H">Hui-Qing Liu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.07028v1-abstract-short" style="display: inline;"> The girth of a graph is defined as the length of a shortest cycle in the graph. A $(k; g)$-cage is a graph of minimum order among all $k$-regular graphs with girth $g$. A cycle $C$ in a graph $G$ is termed nonseparating if the graph $G-V(C)$ remains connected. A conjecture, proposed in [T. Jiang, D. Mubayi. Connectivity and Separating Sets of Cages. J. Graph Theory 29(1)(1998) 35--44], posits that… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.07028v1-abstract-full').style.display = 'inline'; document.getElementById('2410.07028v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.07028v1-abstract-full" style="display: none;"> The girth of a graph is defined as the length of a shortest cycle in the graph. A $(k; g)$-cage is a graph of minimum order among all $k$-regular graphs with girth $g$. A cycle $C$ in a graph $G$ is termed nonseparating if the graph $G-V(C)$ remains connected. A conjecture, proposed in [T. Jiang, D. Mubayi. Connectivity and Separating Sets of Cages. J. Graph Theory 29(1)(1998) 35--44], posits that every cycle of length $g$ within a $(k; g)$-cage is nonseparating. While the conjecture has been proven for even $g$ in the aforementioned work, this paper presents a proof demonstrating that the conjecture holds true for odd $g$ as well. Thus, the previously mentioned conjecture was proven to be true. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.07028v1-abstract-full').style.display = 'none'; document.getElementById('2410.07028v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">This paper was published in Chinese on April 25, 2001, in Volume 14, Issue 2 of the "MATHEMATICA APPLICATA" on pages 99 to 102. This is the English version of the paper, incorporating additional details and rectifying certain typographical errors</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C40 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.06835">arXiv:2410.06835</a> <span> [<a href="https://arxiv.org/pdf/2410.06835">pdf</a>, <a href="https://arxiv.org/ps/2410.06835">ps</a>, <a href="https://arxiv.org/format/2410.06835">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> On the baroclinic instability of inviscid non-conducting Boussinesq equations with rotation in 3-D </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Mao%2C+J">Jingjing Mao</a>, <a href="/search/math?searchtype=author&query=Wang%2C+Y">Yan-Lin 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="2410.06835v1-abstract-short" style="display: inline;"> In this paper we prove the nonlinear instability of the shear flow in three dimensional inviscid non-conducting Boussinesq equations with rotation. We establish the nonlinear instability of the shear motion with respect to a general class of perturbation, by means of constructing an approximate solution containing the shear flow and an exponentially growing profile, which is deduced from the geost… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.06835v1-abstract-full').style.display = 'inline'; document.getElementById('2410.06835v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.06835v1-abstract-full" style="display: none;"> In this paper we prove the nonlinear instability of the shear flow in three dimensional inviscid non-conducting Boussinesq equations with rotation. We establish the nonlinear instability of the shear motion with respect to a general class of perturbation, by means of constructing an approximate solution containing the shear flow and an exponentially growing profile, which is deduced from the geostrophic limit model. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.06835v1-abstract-full').style.display = 'none'; document.getElementById('2410.06835v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">25 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.02765">arXiv:2409.02765</a> <span> [<a href="https://arxiv.org/pdf/2409.02765">pdf</a>, <a href="https://arxiv.org/ps/2409.02765">ps</a>, <a href="https://arxiv.org/format/2409.02765">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 codegree Tur谩n density of the 3-uniform tight cycle $C_{11}$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ma%2C+J">Jie Ma</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="2409.02765v1-abstract-short" style="display: inline;"> Piga, Sanhueza-Matamala, and Schacht recently established that the codegree Tur谩n density of 3-uniform tight cycles $C_\ell$ is $1/3$ for $\ell\in \{10, 13, 16\}$ and for all $\ell\geq 19$. In this note, we extend their proof to determine the codegree Tur谩n density of the 3-uniform tight cycle $C_{11}$, thereby completing the picture for tight cycles of length at least 10. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.02765v1-abstract-full" style="display: none;"> Piga, Sanhueza-Matamala, and Schacht recently established that the codegree Tur谩n density of 3-uniform tight cycles $C_\ell$ is $1/3$ for $\ell\in \{10, 13, 16\}$ and for all $\ell\geq 19$. In this note, we extend their proof to determine the codegree Tur谩n density of the 3-uniform tight cycle $C_{11}$, thereby completing the picture for tight cycles of length at least 10. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.02765v1-abstract-full').style.display = 'none'; document.getElementById('2409.02765v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">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">3 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2409.01654">arXiv:2409.01654</a> <span> [<a href="https://arxiv.org/pdf/2409.01654">pdf</a>, <a href="https://arxiv.org/format/2409.01654">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"> Uniquely colorable hypergraphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+X">Xizhi Liu</a>, <a href="/search/math?searchtype=author&query=Ma%2C+J">Jie Ma</a>, <a href="/search/math?searchtype=author&query=Wang%2C+T">Tianhen Wang</a>, <a href="/search/math?searchtype=author&query=Zhu%2C+T">Tianming Zhu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2409.01654v1-abstract-short" style="display: inline;"> An $r$-uniform hypergraph is uniquely $k$-colorable if there exists exactly one partition of its vertex set into $k$ parts such that every edge contains at most one vertex from each part. For integers $k \ge r \ge 2$, let $桅_{k,r}$ denote the minimum real number such that every $n$-vertex $k$-partite $r$-uniform hypergraph with positive codegree greater than $桅_{k,r} \cdot n$ and no isolated verti… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.01654v1-abstract-full').style.display = 'inline'; document.getElementById('2409.01654v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2409.01654v1-abstract-full" style="display: none;"> An $r$-uniform hypergraph is uniquely $k$-colorable if there exists exactly one partition of its vertex set into $k$ parts such that every edge contains at most one vertex from each part. For integers $k \ge r \ge 2$, let $桅_{k,r}$ denote the minimum real number such that every $n$-vertex $k$-partite $r$-uniform hypergraph with positive codegree greater than $桅_{k,r} \cdot n$ and no isolated vertices is uniquely $k$-colorable. A classic result by of Bollob谩s\cite{Bol78} established that $桅_{k,2} = \frac{3k-5}{3k-2}$ for every $k \ge 2$. We consider the uniquely colorable problem for hypergraphs. Our main result determines the precise value of $桅_{k,r}$ for all $k \ge r \ge 3$. In particular, we show that $桅_{k,r}$ exhibits a phase transition at approximately $k = \frac{4r-2}{3}$, a phenomenon not seen in the graph case. As an application of the main result, combined with a classic theorem by Frankl--F眉redi--Kalai, we derive general bounds for the analogous problem on minimum positive $i$-degrees for all $1\leq i<r$, which are tight for infinitely many cases. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2409.01654v1-abstract-full').style.display = 'none'; document.getElementById('2409.01654v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">29 pages, 10 figures, 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/2408.09435">arXiv:2408.09435</a> <span> [<a href="https://arxiv.org/pdf/2408.09435">pdf</a>, <a href="https://arxiv.org/format/2408.09435">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 modified Ricci flow on arbitrary weighted graph </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ma%2C+J">Jicheng Ma</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yunyan Yang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2408.09435v1-abstract-short" style="display: inline;"> In this paper, we propose a modified Ricci flow, as well as a quasi-normalized Ricci flow, on arbitrary weighted graph. Each of these two flows has a unique global solution. In particular, these global existence and uniqueness results do not require an exit condition proposed by Bai et al in a recent work [2]. As applications, these two Ricci flows are applied to community detection for complex ne… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.09435v1-abstract-full').style.display = 'inline'; document.getElementById('2408.09435v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.09435v1-abstract-full" style="display: none;"> In this paper, we propose a modified Ricci flow, as well as a quasi-normalized Ricci flow, on arbitrary weighted graph. Each of these two flows has a unique global solution. In particular, these global existence and uniqueness results do not require an exit condition proposed by Bai et al in a recent work [2]. As applications, these two Ricci flows are applied to community detection for complex networks, including Karate Club, American football games, Facebook, as well as artificial networks. In our algorithms, unlike in [5,15], there is no need to perform surgery at every iteration, only one surgery needs to be performed after the last iteration. From three commonly used criteria for evaluating community detection algorithms, ARI, NMI and Q, we conclude that our algorithms outperform existing algorithms, including Ollivier's Ricci flow [5], normalized Ollivier's Ricci flow and normalized Lin-Lu-Yau's Ricci flow [15]. The codes for our algorithms are available at https://github.com/mjc191812/Modified-Ricci-Flow. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.09435v1-abstract-full').style.display = 'none'; document.getElementById('2408.09435v1-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 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">15 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05C21; 05C85; 35R02; 68Q06 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2408.07949">arXiv:2408.07949</a> <span> [<a href="https://arxiv.org/pdf/2408.07949">pdf</a>, <a href="https://arxiv.org/ps/2408.07949">ps</a>, <a href="https://arxiv.org/format/2408.07949">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gao%2C+Y">Ya Gao</a>, <a href="/search/math?searchtype=author&query=Mao%2C+J">Jing Mao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2408.07949v1-abstract-short" style="display: inline;"> For a given smooth convex cone in the Euclidean $(n+1)$-space $\mathbb{R}^{n+1}$ which is centered at the origin, we investigate the evolution of strictly mean convex hypersurfaces, which are star-shaped with respect to the center of the cone and which meet the cone perpendicularly, along an inverse curvature flow with the speed equal to $\left(f(r)H\right)^{-1}$, where $f$ is a positive function… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.07949v1-abstract-full').style.display = 'inline'; document.getElementById('2408.07949v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.07949v1-abstract-full" style="display: none;"> For a given smooth convex cone in the Euclidean $(n+1)$-space $\mathbb{R}^{n+1}$ which is centered at the origin, we investigate the evolution of strictly mean convex hypersurfaces, which are star-shaped with respect to the center of the cone and which meet the cone perpendicularly, along an inverse curvature flow with the speed equal to $\left(f(r)H\right)^{-1}$, where $f$ is a positive function of the radial distance parameter $r$ and $H$ is the mean curvature of the evolving hypersurfaces. The evolution of those hypersurfaces inside the cone yields a fully nonlinear parabolic Neumann problem. Under suitable constraints on the first and the second derivatives of the radial function $f$, we can prove the long-time existence of this flow, and moreover the evolving hypersurfaces converge smoothly to a piece of the round sphere. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.07949v1-abstract-full').style.display = 'none'; document.getElementById('2408.07949v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">17 pages. Comments are welcome. arXiv admin note: substantial text overlap with arXiv:2104.08884, arXiv:2104.10600, arXiv:2106.05973</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53E10; 35K10 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2408.06020">arXiv:2408.06020</a> <span> [<a href="https://arxiv.org/pdf/2408.06020">pdf</a>, <a href="https://arxiv.org/ps/2408.06020">ps</a>, <a href="https://arxiv.org/format/2408.06020">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"> Discrepancies of perfect matchings in hypergraphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lu%2C+H">Hongliang Lu</a>, <a href="/search/math?searchtype=author&query=Ma%2C+J">Jie Ma</a>, <a href="/search/math?searchtype=author&query=Xie%2C+S">Shengjie Xie</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="2408.06020v2-abstract-short" style="display: inline;"> In this paper, we determine the minimum degree threshold of perfect matchings with high discrepancy in $r$-edge-colored $k$-uniform hypergraphs for all $k\geq 3$ and $r\geq 2$, thereby completing the investigation into discrepancies of perfect matchings that has recently attracted significant attention. Our approach identifies this discrepancy threshold with a novel family of multicolored uniform… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.06020v2-abstract-full').style.display = 'inline'; document.getElementById('2408.06020v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.06020v2-abstract-full" style="display: none;"> In this paper, we determine the minimum degree threshold of perfect matchings with high discrepancy in $r$-edge-colored $k$-uniform hypergraphs for all $k\geq 3$ and $r\geq 2$, thereby completing the investigation into discrepancies of perfect matchings that has recently attracted significant attention. Our approach identifies this discrepancy threshold with a novel family of multicolored uniform hypergraphs and reveals new phenomena not covered in previous studies. In particular, our results address a question of Balogh, Treglown and Z谩rate-Guer茅n concerning 3-uniform hypergraphs. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.06020v2-abstract-full').style.display = 'none'; document.getElementById('2408.06020v2-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 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 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">25 pages with 1 figure; added acknowledgements and a new appendix to address a missing case in the original Lemma 12</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.14905">arXiv:2407.14905</a> <span> [<a href="https://arxiv.org/pdf/2407.14905">pdf</a>, <a href="https://arxiv.org/ps/2407.14905">ps</a>, <a href="https://arxiv.org/format/2407.14905">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 multicolor Tur谩n conjecture for color-critical graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xihe Li</a>, <a href="/search/math?searchtype=author&query=Ma%2C+J">Jie Ma</a>, <a href="/search/math?searchtype=author&query=Zheng%2C+Z">Zhiheng Zheng</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.14905v1-abstract-short" style="display: inline;"> A {\it simple $k$-coloring} of a multigraph $G$ is a decomposition of the edge multiset as a disjoint sum of $k$ simple graphs which are referred as colors. A subgraph $H$ of a multigraph $G$ is called {\it multicolored} if its edges receive distinct colors in a given simple $k$-coloring of $G$. In 2004, Keevash-Saks-Sudakov-Verstra毛te introduced the {\it $k$-color Tur谩n number} $ex_k(n,H)$, which… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.14905v1-abstract-full').style.display = 'inline'; document.getElementById('2407.14905v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.14905v1-abstract-full" style="display: none;"> A {\it simple $k$-coloring} of a multigraph $G$ is a decomposition of the edge multiset as a disjoint sum of $k$ simple graphs which are referred as colors. A subgraph $H$ of a multigraph $G$ is called {\it multicolored} if its edges receive distinct colors in a given simple $k$-coloring of $G$. In 2004, Keevash-Saks-Sudakov-Verstra毛te introduced the {\it $k$-color Tur谩n number} $ex_k(n,H)$, which denotes the maximum number of edges in an $n$-vertex multigraph that has a simple $k$-coloring containing no multicolored copies of $H$. They made a conjecture for any $r\geq 3$ and $r$-color-critical graph $H$ that in the range of $k\geq \frac{r-1}{r-2}(e(H)-1)$, if $n$ is sufficiently large, then $ex_k(n, H)$ is achieved by the multigraph consisting of $k$ colors all of which are identical copies of the Tur谩n graph $T_{r-1}(n)$. In this paper, we show that this holds in the range of $k\geq 2\frac{r-1}{r}(e(H)-1)$, significantly improving earlier results. Our proof combines the stability argument of Chakraborti-Kim-Lee-Liu-Seo with a novel graph packing technique for embedding multigraphs. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.14905v1-abstract-full').style.display = 'none'; document.getElementById('2407.14905v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 July, 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/2407.10861">arXiv:2407.10861</a> <span> [<a href="https://arxiv.org/pdf/2407.10861">pdf</a>, <a href="https://arxiv.org/format/2407.10861">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"> Kohayakawa-Nagle-R{枚}dl-Schacht conjecture for subdivisions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+H">Hao Chen</a>, <a href="/search/math?searchtype=author&query=Lin%2C+Y">Yupeng Lin</a>, <a href="/search/math?searchtype=author&query=Ma%2C+J">Jie Ma</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="2407.10861v2-abstract-short" style="display: inline;"> In this paper, we study the well-known Kohayakawa-Nagle-R{枚}dl-Schacht (KNRS) conjecture, with a specific focus on graph subdivisions. The KNRS conjecture asserts that for any graph $H$, locally dense graphs contain asymptotically at least the number of copies of $H$ found in a random graph with the same edge density. We prove the following results about $k$-subdivisions of graphs (obtained by rep… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.10861v2-abstract-full').style.display = 'inline'; document.getElementById('2407.10861v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.10861v2-abstract-full" style="display: none;"> In this paper, we study the well-known Kohayakawa-Nagle-R{枚}dl-Schacht (KNRS) conjecture, with a specific focus on graph subdivisions. The KNRS conjecture asserts that for any graph $H$, locally dense graphs contain asymptotically at least the number of copies of $H$ found in a random graph with the same edge density. We prove the following results about $k$-subdivisions of graphs (obtained by replacing edges with paths of length $k+1$): (1). If $H$ satisfies the KNRS conjecture, then its $(2k-1)$-subdivision satisfies Sidorenko's conjecture, extending a prior result of Conlon, Kim, Lee and Lee; (2). If $H$ satisfies the KNRS conjecture, then its $2k$-subdivision satisfies a constant-fraction version of the KNRS conjecture; (3). If $H$ is regular and satisfies the KNRS conjecture, then its $2k$-subdivision also satisfies the KNRS conjecture. These findings imply that all balanced subdivisions of cliques satisfy the KNRS conjecture, improving upon a recent result of Brada膷, Sudakov and Wigerson. Our work provides new insights into this pivotal conjecture in extremal graph theory. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.10861v2-abstract-full').style.display = 'none'; document.getElementById('2407.10861v2-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 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 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">Add a new lemma (Lemma 3.2) from real analysis</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.18870">arXiv:2406.18870</a> <span> [<a href="https://arxiv.org/pdf/2406.18870">pdf</a>, <a href="https://arxiv.org/ps/2406.18870">ps</a>, <a href="https://arxiv.org/format/2406.18870">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"> Exact results on traces of sets </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+M">Mingze Li</a>, <a href="/search/math?searchtype=author&query=Ma%2C+J">Jie Ma</a>, <a href="/search/math?searchtype=author&query=Rong%2C+M">Mingyuan Rong</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="2406.18870v1-abstract-short" style="display: inline;"> For non-negative integers $n$, $m$, $a$ and $b$, we write $\left( n,m \right) \rightarrow \left( a,b \right)$ if for every family $\mathcal{F}\subseteq 2^{[n]}$ with $|\mathcal{F}|\geqslant m$ there is an $a$-element set $T\subseteq [n]$ such that $\left| \mathcal{F}_{\mid T} \right| \geqslant b$, where $\mathcal{F}_{\mid T}=\{ F \cap T : F \in \mathcal{F} \}$. A longstanding problem in extremal s… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.18870v1-abstract-full').style.display = 'inline'; document.getElementById('2406.18870v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.18870v1-abstract-full" style="display: none;"> For non-negative integers $n$, $m$, $a$ and $b$, we write $\left( n,m \right) \rightarrow \left( a,b \right)$ if for every family $\mathcal{F}\subseteq 2^{[n]}$ with $|\mathcal{F}|\geqslant m$ there is an $a$-element set $T\subseteq [n]$ such that $\left| \mathcal{F}_{\mid T} \right| \geqslant b$, where $\mathcal{F}_{\mid T}=\{ F \cap T : F \in \mathcal{F} \}$. A longstanding problem in extremal set theory asks to determine $m(s)=\lim_{n\rightarrow +\infty}\frac{m(n,s)}{n}$, where $m(n,s)$ denotes the maximum integer $m$ such that $\left( n,m \right) \rightarrow \left( n-1,m-s \right)$ holds for non-negatives $n$ and $s$. In this paper, we establish the exact value of $m(2^{d-1}-c)$ for all $1\leqslant c\leqslant d$ whenever $d\geqslant 50$, thereby solving an open problem posed by Piga and Sch眉lke. To be precise, we show that $$m(n,2^{d-1}-c)=\frac{2^{d}-c}{d}n \mbox{ for } 1\leq c\leq d-1 \mbox{ and } d\mid n, \mbox{ and } m(n,2^{d-1}-d)=\frac{2^{d}-d-0.5}{d}n \mbox{ for } 2d\mid n $$ holds for $d\geq 50$. Furthermore, we provide a proof that confirms a conjecture of Frankl and Watanabe from 1994, demonstrating that $m(11)=5.3$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.18870v1-abstract-full').style.display = 'none'; document.getElementById('2406.18870v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 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/2406.10959">arXiv:2406.10959</a> <span> [<a href="https://arxiv.org/pdf/2406.10959">pdf</a>, <a href="https://arxiv.org/ps/2406.10959">ps</a>, <a href="https://arxiv.org/format/2406.10959">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> </div> </div> <p class="title is-5 mathjax"> On Convergence Analysis of Policy Iteration Algorithms for Entropy-Regularized Stochastic Control Problems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ma%2C+J">Jin Ma</a>, <a href="/search/math?searchtype=author&query=Wang%2C+G">Gaozhan Wang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+J">Jianfeng Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2406.10959v3-abstract-short" style="display: inline;"> In this paper we investigate the issues regarding the convergence of the Policy Iteration Algorithm(PIA) for a class of general continuous-time entropy-regularized stochastic control problems. In particular, instead of employing sophisticated PDE estimates for the iterative PDEs involved in the PIA (see, e.g., Huang-Wang-Zhou(2023)), we shall provide a simple proof from scratch for the convergence… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.10959v3-abstract-full').style.display = 'inline'; document.getElementById('2406.10959v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.10959v3-abstract-full" style="display: none;"> In this paper we investigate the issues regarding the convergence of the Policy Iteration Algorithm(PIA) for a class of general continuous-time entropy-regularized stochastic control problems. In particular, instead of employing sophisticated PDE estimates for the iterative PDEs involved in the PIA (see, e.g., Huang-Wang-Zhou(2023)), we shall provide a simple proof from scratch for the convergence of the PIA. Our approach builds on probabilistic representation formulae for solutions of PDEs and their derivatives. Moreover, in the infinite horizon model with large discount factor and in the finite horizon model, the similar arguments lead to the exponential rate of convergence of PIA without tear. Finally, with some extra efforts we show that our approach can also be extended to the case when diffusion contains control, in the one dimensional setting but without much extra constraints on the coefficients. We believe that these results are new in the literature. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.10959v3-abstract-full').style.display = 'none'; document.getElementById('2406.10959v3-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 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 16 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">In this version, we have added results on convergence and rate of convergence for the diffusion control problem in the scalar case</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 93E35; 60H30; 35Q93 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.10047">arXiv:2406.10047</a> <span> [<a href="https://arxiv.org/pdf/2406.10047">pdf</a>, <a href="https://arxiv.org/ps/2406.10047">ps</a>, <a href="https://arxiv.org/format/2406.10047">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> </div> </div> <p class="title is-5 mathjax"> On automorphism groups of polar codes </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ma%2C+J">Jicheng Ma</a>, <a href="/search/math?searchtype=author&query=Yan%2C+G">Guiying Yan</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="2406.10047v1-abstract-short" style="display: inline;"> Over the past years, Polar codes have arisen as a highly effective class of linear codes, equipped with a decoding algorithm of low computational complexity. This family of codes share a common algebraic formalism with the well-known Reed-Muller codes, which involves monomial evaluations. As useful algebraic codes, more specifically known as decreasing monomial codes, a lot of decoding work has be… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.10047v1-abstract-full').style.display = 'inline'; document.getElementById('2406.10047v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.10047v1-abstract-full" style="display: none;"> Over the past years, Polar codes have arisen as a highly effective class of linear codes, equipped with a decoding algorithm of low computational complexity. This family of codes share a common algebraic formalism with the well-known Reed-Muller codes, which involves monomial evaluations. As useful algebraic codes, more specifically known as decreasing monomial codes, a lot of decoding work has been done on Reed-Muller codes based on their rich code automorphisms. In 2021, a new permutation group decoder, referred to as the automorphism ensemble (AE) decoder, was introduced. This decoder can be applied to Polar codes and has been shown to produce similar decoding effects. However, identifying the right set of code automorphisms that enhance decoding performance for Polar codes remains a challenging task. This paper aims to characterize the full automorphism group of Polar codes. We will prove a reduction theorem that effectively reduces the problem of determining the full automorphism group of arbitrary random Polar codes to that of a specified class of Polar codes. Besides, we give exact classification of the full automorphism groups of families of Polar codes that are constructed using the Reed-Muller codes. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.10047v1-abstract-full').style.display = 'none'; document.getElementById('2406.10047v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">14 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 05E18; 20B25; 94B05; 94B35 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.05675">arXiv:2406.05675</a> <span> [<a href="https://arxiv.org/pdf/2406.05675">pdf</a>, <a href="https://arxiv.org/ps/2406.05675">ps</a>, <a href="https://arxiv.org/format/2406.05675">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"> Finding irregular subgraphs via local adjustments </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ma%2C+J">Jie Ma</a>, <a href="/search/math?searchtype=author&query=Xie%2C+S">Shengjie Xie</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="2406.05675v1-abstract-short" style="display: inline;"> For a graph $H$, let $m(H,k)$ denote the number of vertices of degree $k$ in $H$. A conjecture of Alon and Wei states that for any $d\geq 3$, every $n$-vertex $d$-regular graph contains a spanning subgraph $H$ satisfying $|m(H,k)-\frac{n}{d+1}|\leq 2$ for every $0\leq k \leq d$. This holds easily when $d\leq 2$. An asymptotic version of this conjecture was initially established by Frieze, Gould, K… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.05675v1-abstract-full').style.display = 'inline'; document.getElementById('2406.05675v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.05675v1-abstract-full" style="display: none;"> For a graph $H$, let $m(H,k)$ denote the number of vertices of degree $k$ in $H$. A conjecture of Alon and Wei states that for any $d\geq 3$, every $n$-vertex $d$-regular graph contains a spanning subgraph $H$ satisfying $|m(H,k)-\frac{n}{d+1}|\leq 2$ for every $0\leq k \leq d$. This holds easily when $d\leq 2$. An asymptotic version of this conjecture was initially established by Frieze, Gould, Karo艅ski and Pfender, subsequently improved by Alon and Wei, and most recently enhanced by Fox, Luo and Pham, approaching its complete range. All of these approaches relied on probabilistic methods. In this paper, we provide a novel framework to study this conjecture, based on localized deterministic techniques which we call local adjustments. We prove two main results. Firstly, we show that every $n$-vertex $d$-regular graph contains a spanning subgraph $H$ satisfying $|m(H,k)-\frac{n}{d+1}|\leq 2d^2$ for all $0\leq k \leq d$, which provides the first bound independent of the value of $n$. Secondly, we confirm the case $d=3$ of the Alon-Wei Conjecture in a strong form. Both results can be generalized to multigraphs and yield efficient algorithms for finding the desired subgraphs $H$. Furthermore, we explore a generalization of the Alon-Wei Conjecture for multigraphs and its connection to the Faudree-Lehel Conjecture concerning irregularity strength. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.05675v1-abstract-full').style.display = 'none'; document.getElementById('2406.05675v1-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 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/2406.00309">arXiv:2406.00309</a> <span> [<a href="https://arxiv.org/pdf/2406.00309">pdf</a>, <a href="https://arxiv.org/ps/2406.00309">ps</a>, <a href="https://arxiv.org/format/2406.00309">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Continuous dependence for McKean-Vlasov SDEs under distribution-dependent Lyapunov conditions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ma%2C+J">Jun Ma</a>, <a href="/search/math?searchtype=author&query=Liu%2C+Z">Zhenxin Liu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2406.00309v2-abstract-short" style="display: inline;"> In this paper, we consider the continuous dependence on initial values and parameters of solutions as well as invariant measures for McKean-Vlasov SDEs under distribution-dependent Lyapunov conditions. In contrast to the classical SDEs, the solutions for McKean-Vlasov SDEs do not converge in probability although the initial values converge in probability, which is due to the mismatch of the distan… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.00309v2-abstract-full').style.display = 'inline'; document.getElementById('2406.00309v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.00309v2-abstract-full" style="display: none;"> In this paper, we consider the continuous dependence on initial values and parameters of solutions as well as invariant measures for McKean-Vlasov SDEs under distribution-dependent Lyapunov conditions. In contrast to the classical SDEs, the solutions for McKean-Vlasov SDEs do not converge in probability although the initial values converge in probability, which is due to the mismatch of the distances between measures. Finally, we give some examples to illustrate our theoretical results. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.00309v2-abstract-full').style.display = 'none'; document.getElementById('2406.00309v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 1 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">arXiv admin note: text overlap with arXiv:2309.05411</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.18081">arXiv:2405.18081</a> <span> [<a href="https://arxiv.org/pdf/2405.18081">pdf</a>, <a href="https://arxiv.org/format/2405.18081">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Information Theory">cs.IT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Optimality of Approximate Message Passing Algorithms for Spiked Matrix Models with Rotationally Invariant Noise </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Dudeja%2C+R">Rishabh Dudeja</a>, <a href="/search/math?searchtype=author&query=Liu%2C+S">Songbin Liu</a>, <a href="/search/math?searchtype=author&query=Ma%2C+J">Junjie Ma</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="2405.18081v1-abstract-short" style="display: inline;"> We study the problem of estimating a rank one signal matrix from an observed matrix generated by corrupting the signal with additive rotationally invariant noise. We develop a new class of approximate message-passing algorithms for this problem and provide a simple and concise characterization of their dynamics in the high-dimensional limit. At each iteration, these algorithms exploit prior knowle… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.18081v1-abstract-full').style.display = 'inline'; document.getElementById('2405.18081v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.18081v1-abstract-full" style="display: none;"> We study the problem of estimating a rank one signal matrix from an observed matrix generated by corrupting the signal with additive rotationally invariant noise. We develop a new class of approximate message-passing algorithms for this problem and provide a simple and concise characterization of their dynamics in the high-dimensional limit. At each iteration, these algorithms exploit prior knowledge about the noise structure by applying a non-linear matrix denoiser to the eigenvalues of the observed matrix and prior information regarding the signal structure by applying a non-linear iterate denoiser to the previous iterates generated by the algorithm. We exploit our result on the dynamics of these algorithms to derive the optimal choices for the matrix and iterate denoisers. We show that the resulting algorithm achieves the smallest possible asymptotic estimation error among a broad class of iterative algorithms under a fixed iteration budget. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.18081v1-abstract-full').style.display = 'none'; document.getElementById('2405.18081v1-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> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.12824">arXiv:2405.12824</a> <span> [<a href="https://arxiv.org/pdf/2405.12824">pdf</a>, <a href="https://arxiv.org/ps/2405.12824">ps</a>, <a href="https://arxiv.org/format/2405.12824">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1017/S0004972724000868">10.1017/S0004972724000868 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A note on the finitely generated fixed subgroup property </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lei%2C+J">Jialin Lei</a>, <a href="/search/math?searchtype=author&query=Ma%2C+J">Jiming Ma</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+Q">Qiang Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2405.12824v3-abstract-short" style="display: inline;"> We study when a group of form $G\times\mathbb{Z}^m (m\geq 1)$ has the finitely generated fixed subgroup property of automorphisms ($\rm{FGFP}_a$), by using the BNS-invariant, and provide some partial answers and non-trivial examples. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.12824v3-abstract-full" style="display: none;"> We study when a group of form $G\times\mathbb{Z}^m (m\geq 1)$ has the finitely generated fixed subgroup property of automorphisms ($\rm{FGFP}_a$), by using the BNS-invariant, and provide some partial answers and non-trivial examples. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.12824v3-abstract-full').style.display = 'none'; document.getElementById('2405.12824v3-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 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 21 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">10 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2404.07412">arXiv:2404.07412</a> <span> [<a href="https://arxiv.org/pdf/2404.07412">pdf</a>, <a href="https://arxiv.org/ps/2404.07412">ps</a>, <a href="https://arxiv.org/format/2404.07412">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="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Brock-type isoperimetric inequality for Steklov eigenvalues of the Witten-Laplacian </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Mao%2C+J">Jing Mao</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+S">Shijie 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="2404.07412v1-abstract-short" style="display: inline;"> In this paper, by imposing suitable assumptions on the weighted function, (under the constraint of fixed weighted volume) a Brock-type isoperimetric inequality for Steklov-type eigenvalues of the Witten-Laplacian on bounded domains in a Euclidean space or a hyperbolic space has been proven. This conclusion is actually an interesting extension of F. Brock's classical result about the isoperimetric… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.07412v1-abstract-full').style.display = 'inline'; document.getElementById('2404.07412v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2404.07412v1-abstract-full" style="display: none;"> In this paper, by imposing suitable assumptions on the weighted function, (under the constraint of fixed weighted volume) a Brock-type isoperimetric inequality for Steklov-type eigenvalues of the Witten-Laplacian on bounded domains in a Euclidean space or a hyperbolic space has been proven. This conclusion is actually an interesting extension of F. Brock's classical result about the isoperimetric inequality for Steklov eigenvalues of the Laplacian given in the influential paper [Z. Angew. Math. Mech. 81 (2001) 69-71]. Besides, a related open problem has also been proposed in this paper. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2404.07412v1-abstract-full').style.display = 'none'; document.getElementById('2404.07412v1-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 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">18 pages. Comments are welcome. arXiv admin note: text overlap with arXiv:2403.08070</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35P15; 49Jxx; 35J15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2403.19815">arXiv:2403.19815</a> <span> [<a href="https://arxiv.org/pdf/2403.19815">pdf</a>, <a href="https://arxiv.org/format/2403.19815">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Anisotropic capillary hypersurfaces in a wedge </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ma%2C+H">Hui Ma</a>, <a href="/search/math?searchtype=author&query=Ma%2C+J">Jiaxu Ma</a>, <a href="/search/math?searchtype=author&query=Yang%2C+M">Mingxuan Yang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2403.19815v2-abstract-short" style="display: inline;"> We investigate anisotropic capillary hypersurfaces within a wedge in Euclidean space. In this study, we generalize the Minkowski norm $F$, traditionally employed to define the anisotropic surface energy, to a gauge on the unit sphere $S^n$. This generalization helps to illuminate a significant relationship between capillary hypersurfaces and hypersurfaces with free boundary. Our main results i… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.19815v2-abstract-full').style.display = 'inline'; document.getElementById('2403.19815v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.19815v2-abstract-full" style="display: none;"> We investigate anisotropic capillary hypersurfaces within a wedge in Euclidean space. In this study, we generalize the Minkowski norm $F$, traditionally employed to define the anisotropic surface energy, to a gauge on the unit sphere $S^n$. This generalization helps to illuminate a significant relationship between capillary hypersurfaces and hypersurfaces with free boundary. Our main results include new Minkowski formulae and a Heintze-Karcher type inequality. As an application, we prove an Alexandrov-type theorem, thereby extending the known results to the anisotropic setting. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.19815v2-abstract-full').style.display = 'none'; document.getElementById('2403.19815v2-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 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> 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">22 pages, 5 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53C24; 53C42; 53C40 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2403.15662">arXiv:2403.15662</a> <span> [<a href="https://arxiv.org/pdf/2403.15662">pdf</a>, <a href="https://arxiv.org/ps/2403.15662">ps</a>, <a href="https://arxiv.org/format/2403.15662">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"> Set-Valued Stochastic Differential Equations with Unbounded Coefficients </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Almuzaini%2C+A">Atiqah Almuzaini</a>, <a href="/search/math?searchtype=author&query=Ma%2C+J">Jin Ma</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="2403.15662v1-abstract-short" style="display: inline;"> In this paper, we mainly focus on the set-valued (stochastic) analysis on the space of convex, closed, but possibly unbounded sets, and try to establish a useful theoretical framework for studying the set-valued stochastic differential equations with unbounded coefficients. The space that we will be focusing on are convex, closed sets that are "generated" by a given cone, in the sense that the Hau… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.15662v1-abstract-full').style.display = 'inline'; document.getElementById('2403.15662v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.15662v1-abstract-full" style="display: none;"> In this paper, we mainly focus on the set-valued (stochastic) analysis on the space of convex, closed, but possibly unbounded sets, and try to establish a useful theoretical framework for studying the set-valued stochastic differential equations with unbounded coefficients. The space that we will be focusing on are convex, closed sets that are "generated" by a given cone, in the sense that the Hausdorff distance of all elements to the "generating" cone is finite. Such space should in particular include the so-called "upper sets", and has many useful cases in finance, such as the well-known set-valued risk measures, as well as the solvency cone in some super-hedging problems. We shall argue that, for such a special class of unbounded sets, under some conditions, the cancellation law is still valid, eliminating a major obstacle for extending the set-valued analysis to non-compact sets. We shall establish some basic algebraic and topological properties of such spaces, and show that some standard techniques will again be valid in studying the set-valued SDEs with unbounded (drift) coefficients which, to the best of our knowledge, is new. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.15662v1-abstract-full').style.display = 'none'; document.getElementById('2403.15662v1-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 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/2403.08075">arXiv:2403.08075</a> <span> [<a href="https://arxiv.org/pdf/2403.08075">pdf</a>, <a href="https://arxiv.org/ps/2403.08075">ps</a>, <a href="https://arxiv.org/format/2403.08075">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="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Several isoperimetric inequalities of Dirichlet and Neumann eigenvalues of the Witten-Laplacian </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+R">Ruifeng Chen</a>, <a href="/search/math?searchtype=author&query=Mao%2C+J">Jing Mao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2403.08075v2-abstract-short" style="display: inline;"> In this paper, by mainly using the rearrangement technique and suitably constructing trial functions, under the constraint of fixed weighted volume, we can successfully obtain several isoperimetric inequalities for the first and the second Dirichlet eigenvalues, the first nonzero Neumann eigenvalue of the Witten-Laplacian on bounded domains in space forms. These spectral isoperimetric inequalities… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.08075v2-abstract-full').style.display = 'inline'; document.getElementById('2403.08075v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.08075v2-abstract-full" style="display: none;"> In this paper, by mainly using the rearrangement technique and suitably constructing trial functions, under the constraint of fixed weighted volume, we can successfully obtain several isoperimetric inequalities for the first and the second Dirichlet eigenvalues, the first nonzero Neumann eigenvalue of the Witten-Laplacian on bounded domains in space forms. These spectral isoperimetric inequalities extend those classical ones (i.e. the Faber-Krahn inequality, the Hong-Krahn-Szeg艖 inequality and the Szeg艖-Weinberger inequality) of the Laplacian. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.08075v2-abstract-full').style.display = 'none'; document.getElementById('2403.08075v2-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 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 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">31 pages. Comments are welcome. Several typos have been corrected to v1</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35P15; 35J10; 35J15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2403.08070">arXiv:2403.08070</a> <span> [<a href="https://arxiv.org/pdf/2403.08070">pdf</a>, <a href="https://arxiv.org/ps/2403.08070">ps</a>, <a href="https://arxiv.org/format/2403.08070">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="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> On the Ashbaugh-Benguria type conjecture about lower-order Neumann eigenvalues of the Witten-Laplacian </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+R">Ruifeng Chen</a>, <a href="/search/math?searchtype=author&query=Mao%2C+J">Jing Mao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2403.08070v2-abstract-short" style="display: inline;"> An isoperimetric inequality for lower order nonzero Neumann eigenvalues of the Witten-Laplacian on bounded domains in a Euclidean space or a hyperbolic space has been proven in this paper. About this conclusion, we would like to point out two things: It strengthens the well-known Szeg艖-Weinberger inequality for nonzero Neumann eigenvalues of the classical free membrane problem given in [J. Ratio… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.08070v2-abstract-full').style.display = 'inline'; document.getElementById('2403.08070v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.08070v2-abstract-full" style="display: none;"> An isoperimetric inequality for lower order nonzero Neumann eigenvalues of the Witten-Laplacian on bounded domains in a Euclidean space or a hyperbolic space has been proven in this paper. About this conclusion, we would like to point out two things: It strengthens the well-known Szeg艖-Weinberger inequality for nonzero Neumann eigenvalues of the classical free membrane problem given in [J. Rational Mech. Anal. 3 (1954) 343-356] and [J. Rational Mech. Anal. 5 (1956) 633-636]; Recently, Xia-Wang [Math. Ann. 385 (2023) 863-879] gave a very important progress to the celebrated conjecture of M. S. Ashbaugh and R. D. Benguria proposed in [SIAM J. Math. Anal. 24 (1993) 557-570]. It is easy to see that our conclusion here covers Xia-Wang's this progress as a special case. In this paper, we have also proposed two open problems which can be seen as a generalization of Ashbaugh-Benguria's conjecture mentioned above. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.08070v2-abstract-full').style.display = 'none'; document.getElementById('2403.08070v2-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 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 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">19 pages. Comments are welcome. Several typos have been corrected to v1. A new reference has been added</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35P15; 49Jxx; 35J15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2403.04318">arXiv:2403.04318</a> <span> [<a href="https://arxiv.org/pdf/2403.04318">pdf</a>, <a href="https://arxiv.org/ps/2403.04318">ps</a>, <a href="https://arxiv.org/format/2403.04318">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> </div> <p class="title is-5 mathjax"> A hypergraph bipartite Tur谩n problem with odd uniformity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ma%2C+J">Jie Ma</a>, <a href="/search/math?searchtype=author&query=Yang%2C+T">Tianchi Yang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2403.04318v2-abstract-short" style="display: inline;"> In this paper, we investigate the hypergraph Tur谩n number $ex(n,K^{(r)}_{s,t})$. Here, $K^{(r)}_{s,t}$ denotes the $r$-uniform hypergraph with vertex set $\left(\cup_{i\in [t]}X_i\right)\cup Y$ and edge set $\{X_i\cup \{y\}: i\in [t], y\in Y\}$, where $X_1,X_2,\cdots,X_t$ are $t$ pairwise disjoint sets of size $r-1$ and $Y$ is a set of size $s$ disjoint from each $X_i$. This study was initially ex… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.04318v2-abstract-full').style.display = 'inline'; document.getElementById('2403.04318v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.04318v2-abstract-full" style="display: none;"> In this paper, we investigate the hypergraph Tur谩n number $ex(n,K^{(r)}_{s,t})$. Here, $K^{(r)}_{s,t}$ denotes the $r$-uniform hypergraph with vertex set $\left(\cup_{i\in [t]}X_i\right)\cup Y$ and edge set $\{X_i\cup \{y\}: i\in [t], y\in Y\}$, where $X_1,X_2,\cdots,X_t$ are $t$ pairwise disjoint sets of size $r-1$ and $Y$ is a set of size $s$ disjoint from each $X_i$. This study was initially explored by Erd艖s and has since received substantial attention in research. Recent advancements by Brada膷, Gishboliner, Janzer and Sudakov have greatly contributed to a better understanding of this problem. They proved that $ex(n,K_{s,t}^{(r)})=O_{s,t}(n^{r-\frac{1}{s-1}})$ holds for any $r\geq 3$ and $s,t\geq 2$. They also provided constructions illustrating the tightness of this bound if $r\geq 4$ is {\it even} and $t\gg s\geq 2$. Furthermore, they proved that $ex(n,K_{s,t}^{(3)})=O_{s,t}(n^{3-\frac{1}{s-1}-\varepsilon_s})$ holds for $s\geq 3$ and some $蔚_s>0$. Addressing this intriguing discrepancy between the behavior of this number for $r=3$ and the even cases, Brada膷 et al. post a question of whether \begin{equation*} \mbox{$ex(n,K_{s,t}^{(r)})= O_{r,s,t}(n^{r-\frac{1}{s-1}- \varepsilon})$ holds for odd $r\geq 5$ and any $s\geq 3$.} \end{equation*} In this paper, we provide an affirmative answer to this question, utilizing novel techniques to identify regular and dense substructures. This result highlights a rare instance in hypergraph Tur谩n problems where the solution depends on the parity of the uniformity. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.04318v2-abstract-full').style.display = 'none'; document.getElementById('2403.04318v2-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 March, 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/2403.01531">arXiv:2403.01531</a> <span> [<a href="https://arxiv.org/pdf/2403.01531">pdf</a>, <a href="https://arxiv.org/format/2403.01531">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"> Complex Hyperbolic Geometry of Chain Links </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ma%2C+J">Jiming Ma</a>, <a href="/search/math?searchtype=author&query=Xie%2C+B">Baohua Xie</a>, <a href="/search/math?searchtype=author&query=Xu%2C+M">Mengmeng Xu</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2403.01531v1-abstract-short" style="display: inline;"> The complex hyperbolic triangle group $螕=螖_{4,\infty,\infty;\infty}$ acting on the complex hyperbolic plane ${\bf H}^2_{\mathbb C}$ is generated by complex reflections $I_1$, $I_2$, $I_3$ such that the product $I_2I_3$ has order four, the products $I_3I_1$, $I_1I_2$ are parabolic and the product $I_1I_3I_2I_3$ is an accidental parabolic element. Unexpectedly, the product $I_1I_2I_3I_2$ is a hidden… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.01531v1-abstract-full').style.display = 'inline'; document.getElementById('2403.01531v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.01531v1-abstract-full" style="display: none;"> The complex hyperbolic triangle group $螕=螖_{4,\infty,\infty;\infty}$ acting on the complex hyperbolic plane ${\bf H}^2_{\mathbb C}$ is generated by complex reflections $I_1$, $I_2$, $I_3$ such that the product $I_2I_3$ has order four, the products $I_3I_1$, $I_1I_2$ are parabolic and the product $I_1I_3I_2I_3$ is an accidental parabolic element. Unexpectedly, the product $I_1I_2I_3I_2$ is a hidden accidental parabolic element. We show that the 3-manifold at infinity of $螖_{4,\infty,\infty;\infty}$ is the complement of the chain link $8^4_1$ in the 3-sphere. In particular, the quartic cusped hyperbolic 3-manifold $S^3-8^4_1$ admits a spherical CR-uniformization. The proof relies on a new technique to show that the ideal boundary of the Ford domain is an infinite-genus handlebody. Motivated by this result and supported by the previous studies of various authors, we conjecture that the chain link $C_p$ is an ancestor of the 3-manifold at infinity of the critical complex hyperbolic triangle group $螖_{p,q,r;\infty}$, for $3 \leq p \leq 9$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.01531v1-abstract-full').style.display = 'none'; document.getElementById('2403.01531v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 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">30 pages, 13 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/2402.18141">arXiv:2402.18141</a> <span> [<a href="https://arxiv.org/pdf/2402.18141">pdf</a>, <a href="https://arxiv.org/ps/2402.18141">ps</a>, <a href="https://arxiv.org/format/2402.18141">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Urysohn 1-width for 4 and 5 manifolds with positive biRicci curvature </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ma%2C+J">Junyu Ma</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="2402.18141v2-abstract-short" style="display: inline;"> It was proved by Gromov-Lawson\cite{gl83} that complete three manifold with positive scalar curvature bounded below has finite Urysohn 1-width only depends on the uniform positive scalar curvature bounds. It is natural to ask the same question for the four manifolds. In this paper, we can show that closed four and five manifolds with positive biRicci curvature has finite Urysohn 1-width only depen… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.18141v2-abstract-full').style.display = 'inline'; document.getElementById('2402.18141v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.18141v2-abstract-full" style="display: none;"> It was proved by Gromov-Lawson\cite{gl83} that complete three manifold with positive scalar curvature bounded below has finite Urysohn 1-width only depends on the uniform positive scalar curvature bounds. It is natural to ask the same question for the four manifolds. In this paper, we can show that closed four and five manifolds with positive biRicci curvature has finite Urysohn 1-width only depends on the curvature bounds. During the proof we can also observe that the fundamental group of those manifolds are virtually free. This gives a quick application that $T^{2}\times S^{2}$ can't admit positive biRicci curvature. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.18141v2-abstract-full').style.display = 'none'; document.getElementById('2402.18141v2-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">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">MSC Class:</span> 53C12 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.06756">arXiv:2402.06756</a> <span> [<a href="https://arxiv.org/pdf/2402.06756">pdf</a>, <a href="https://arxiv.org/format/2402.06756">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"> Convergence of Gradient Descent with Small Initialization for Unregularized Matrix Completion </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ma%2C+J">Jianhao Ma</a>, <a href="/search/math?searchtype=author&query=Fattahi%2C+S">Salar Fattahi</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="2402.06756v1-abstract-short" style="display: inline;"> We study the problem of symmetric matrix completion, where the goal is to reconstruct a positive semidefinite matrix $\rm{X}^\star \in \mathbb{R}^{d\times d}$ of rank-$r$, parameterized by $\rm{U}\rm{U}^{\top}$, from only a subset of its observed entries. We show that the vanilla gradient descent (GD) with small initialization provably converges to the ground truth $\rm{X}^\star$ without requiring… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.06756v1-abstract-full').style.display = 'inline'; document.getElementById('2402.06756v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.06756v1-abstract-full" style="display: none;"> We study the problem of symmetric matrix completion, where the goal is to reconstruct a positive semidefinite matrix $\rm{X}^\star \in \mathbb{R}^{d\times d}$ of rank-$r$, parameterized by $\rm{U}\rm{U}^{\top}$, from only a subset of its observed entries. We show that the vanilla gradient descent (GD) with small initialization provably converges to the ground truth $\rm{X}^\star$ without requiring any explicit regularization. This convergence result holds true even in the over-parameterized scenario, where the true rank $r$ is unknown and conservatively over-estimated by a search rank $r'\gg r$. The existing results for this problem either require explicit regularization, a sufficiently accurate initial point, or exact knowledge of the true rank $r$. In the over-parameterized regime where $r'\geq r$, we show that, with $\widetilde惟(dr^9)$ observations, GD with an initial point $\|\rm{U}_0\| \leq 蔚$ converges near-linearly to an $蔚$-neighborhood of $\rm{X}^\star$. Consequently, smaller initial points result in increasingly accurate solutions. Surprisingly, neither the convergence rate nor the final accuracy depends on the over-parameterized search rank $r'$, and they are only governed by the true rank $r$. In the exactly-parameterized regime where $r'=r$, we further enhance this result by proving that GD converges at a faster rate to achieve an arbitrarily small accuracy $蔚>0$, provided the initial point satisfies $\|\rm{U}_0\| = O(1/d)$. At the crux of our method lies a novel weakly-coupled leave-one-out analysis, which allows us to establish the global convergence of GD, extending beyond what was previously possible using the classical leave-one-out analysis. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.06756v1-abstract-full').style.display = 'none'; document.getElementById('2402.06756v1-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 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/2402.02395">arXiv:2402.02395</a> <span> [<a href="https://arxiv.org/pdf/2402.02395">pdf</a>, <a href="https://arxiv.org/format/2402.02395">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Computer Vision and Pattern Recognition">cs.CV</span> </div> </div> <p class="title is-5 mathjax"> A fast and gridless ORKA algorithm for tracking moving and deforming objects </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bossmann%2C+F">Florian Bossmann</a>, <a href="/search/math?searchtype=author&query=Ma%2C+J">Jianwei Ma</a>, <a href="/search/math?searchtype=author&query=wu%2C+W">Wenze 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="2402.02395v1-abstract-short" style="display: inline;"> Identifying objects in given data is a task frequently encountered in many applications. Finding vehicles or persons in video data, tracking seismic waves in geophysical exploration data, or predicting a storm front movement from meteorological measurements are only some of the possible applications. In many cases, the object of interest changes its form or position from one measurement to another… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.02395v1-abstract-full').style.display = 'inline'; document.getElementById('2402.02395v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.02395v1-abstract-full" style="display: none;"> Identifying objects in given data is a task frequently encountered in many applications. Finding vehicles or persons in video data, tracking seismic waves in geophysical exploration data, or predicting a storm front movement from meteorological measurements are only some of the possible applications. In many cases, the object of interest changes its form or position from one measurement to another. For example, vehicles in a video may change its position or angle to the camera in each frame. Seismic waves can change its arrival time, frequency, or intensity depending on the sensor position. Storm fronts can change its form and position over time. This complicates the identification and tracking as the algorithm needs to deal with the changing object over the given measurements. In a previous work, the authors presented a new algorithm to solve this problem - Object reconstruction using K-approximation (ORKA). The algorithm can solve the problem at hand but suffers from two disadvantages. On the one hand, the reconstructed object movement is bound to a grid that depends on the data resolution. On the other hand, the complexity of the algorithm increases exponentially with the resolution. We overcome both disadvantages by introducing an iterative strategy that uses a resampling method to create multiple resolutions of the data. In each iteration the resolution is increased to reconstruct more details of the object of interest. This way, we can even go beyond the original resolution by artificially upsampling the data. We give error bounds and a complexity analysis of the new method. Furthermore, we analyze its performance in several numerical experiments as well as on real data. We also give a brief introduction on the original ORKA algorithm. Knowledge of the previous work is thus not required. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.02395v1-abstract-full').style.display = 'none'; document.getElementById('2402.02395v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 February, 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/2401.13698">arXiv:2401.13698</a> <span> [<a href="https://arxiv.org/pdf/2401.13698">pdf</a>, <a href="https://arxiv.org/format/2401.13698">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"> Finite-volume hyperbolic Coxeter 4-dimensional polytopes with 7 facets </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ma%2C+J">Jiming Ma</a>, <a href="/search/math?searchtype=author&query=Zheng%2C+F">Fangting Zheng</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2401.13698v3-abstract-short" style="display: inline;"> In this paper, we obtain a complete classification of 331 finite-volume hyperbolic Coxeter 4-dimensional polytopes with 7 facets. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2401.13698v3-abstract-full" style="display: none;"> In this paper, we obtain a complete classification of 331 finite-volume hyperbolic Coxeter 4-dimensional polytopes with 7 facets. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.13698v3-abstract-full').style.display = 'none'; document.getElementById('2401.13698v3-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 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 20 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2401.11525">arXiv:2401.11525</a> <span> [<a href="https://arxiv.org/pdf/2401.11525">pdf</a>, <a href="https://arxiv.org/ps/2401.11525">ps</a>, <a href="https://arxiv.org/format/2401.11525">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 class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1002/jgt.23211">10.1002/jgt.23211 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Weak rainbow saturation numbers of graphs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xihe Li</a>, <a href="/search/math?searchtype=author&query=Ma%2C+J">Jie Ma</a>, <a href="/search/math?searchtype=author&query=Xie%2C+T">Tianying Xie</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="2401.11525v1-abstract-short" style="display: inline;"> For a fixed graph $H$, we say that an edge-colored graph $G$ is \emph{weakly $H$-rainbow saturated} if there exists an ordering $e_1, e_2, \ldots, e_m$ of $E\left(\overline{G}\right)$ such that, for any list $c_1, c_2, \ldots, c_m$ of pairwise distinct colors from $\mathbb{N}$, the non-edges $e_i$ in color $c_i$ can be added to $G$, one at a time, so that every added edge creates a new rainbow cop… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.11525v1-abstract-full').style.display = 'inline'; document.getElementById('2401.11525v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2401.11525v1-abstract-full" style="display: none;"> For a fixed graph $H$, we say that an edge-colored graph $G$ is \emph{weakly $H$-rainbow saturated} if there exists an ordering $e_1, e_2, \ldots, e_m$ of $E\left(\overline{G}\right)$ such that, for any list $c_1, c_2, \ldots, c_m$ of pairwise distinct colors from $\mathbb{N}$, the non-edges $e_i$ in color $c_i$ can be added to $G$, one at a time, so that every added edge creates a new rainbow copy of $H$. The \emph{weak rainbow saturation number} of $H$, denoted by $rwsat(n,H)$, is the minimum number of edges in a weakly $H$-rainbow saturated graph on $n$ vertices. In this paper, we show that for any non-empty graph $H$, the limit $\lim_{n\to \infty} \frac{rwsat(n, H)}{n}$ exists. This answers a question of Behague, Johnston, Letzter, Morrison and Ogden [{\it SIAM J. Discrete Math.} (2023)]. We also provide lower and upper bounds on this limit, and in particular, we show that this limit is nonzero if and only if $H$ contains no pendant edges. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.11525v1-abstract-full').style.display = 'none'; document.getElementById('2401.11525v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 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">13 pages, 1 figure</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal of Graph Theory, 2025 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2401.06245">arXiv:2401.06245</a> <span> [<a href="https://arxiv.org/pdf/2401.06245">pdf</a>, <a href="https://arxiv.org/ps/2401.06245">ps</a>, <a href="https://arxiv.org/format/2401.06245">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Distributed Optimal Output Consensus Control of Heterogeneous Multi-Agent Systems with Safety Constraints </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ma%2C+J">Ji Ma</a>, <a href="/search/math?searchtype=author&query=Liang%2C+S">Shu Liang</a>, <a href="/search/math?searchtype=author&query=Hong%2C+Y">Yiguang 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="2401.06245v1-abstract-short" style="display: inline;"> In this paper, we develop a novel dynamic distributed optimal safe consensus protocol to simultaneously achieve safety requirements and output optimal consensus. Specifically, we construct a distributed projection optimization algorithm with an expanding constraint set in the decision-making layer, while we propose a reference tracking safety controller to ensure that each agent's output remains w… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.06245v1-abstract-full').style.display = 'inline'; document.getElementById('2401.06245v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2401.06245v1-abstract-full" style="display: none;"> In this paper, we develop a novel dynamic distributed optimal safe consensus protocol to simultaneously achieve safety requirements and output optimal consensus. Specifically, we construct a distributed projection optimization algorithm with an expanding constraint set in the decision-making layer, while we propose a reference tracking safety controller to ensure that each agent's output remains within a shrinking safety set in the control layer.We also establish the convergence and safety analysis of the closed-loop system using the small-gain theorem and time-varying control barrier function (CBF) theory, respectively. Besides, unlike previous works on distributed optimal consensus, our approach does not require prior knowledge of the local objective or gradient function and adopts a mild assumption on the dynamics of multiagent systems (MASs) by using the transmission zeros condition. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.06245v1-abstract-full').style.display = 'none'; document.getElementById('2401.06245v1-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 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, 4 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2401.00702">arXiv:2401.00702</a> <span> [<a href="https://arxiv.org/pdf/2401.00702">pdf</a>, <a href="https://arxiv.org/format/2401.00702">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"> Stability of strong viscous shock wave under periodic perturbation for 1-D isentropic Navier-Stokes system in the half space </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chang%2C+L">Lin Chang</a>, <a href="/search/math?searchtype=author&query=He%2C+L">Lin He</a>, <a href="/search/math?searchtype=author&query=Ma%2C+J">Jin Ma</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="2401.00702v1-abstract-short" style="display: inline;"> In this paper, a viscous shock wave under space-periodic perturbation of 1-D isentropic Navier-Stokes system in the half space is investigated. It is shown that if the initial periodic perturbation around the viscous shock wave is small, then the solution time asymptotically tends to a viscous shock wave with a shift partially determined by the periodic oscillations. Moreover, the strength of {the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.00702v1-abstract-full').style.display = 'inline'; document.getElementById('2401.00702v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2401.00702v1-abstract-full" style="display: none;"> In this paper, a viscous shock wave under space-periodic perturbation of 1-D isentropic Navier-Stokes system in the half space is investigated. It is shown that if the initial periodic perturbation around the viscous shock wave is small, then the solution time asymptotically tends to a viscous shock wave with a shift partially determined by the periodic oscillations. Moreover, the strength of {the} shock wave could be arbitrarily large. This result essentially improves the previous work " A. Matsumura, M. Mei, Convergence to travelling fronts of solutions of the p-system with viscosity in the presence of a boundary. Arch. Ration. Mech. Anal. 146 (1999), no. 1, 1-22." where the strength of shock wave is sufficiently small and the initial periodic oscillations vanish. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.00702v1-abstract-full').style.display = 'none'; document.getElementById('2401.00702v1-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 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">2 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2312.00240">arXiv:2312.00240</a> <span> [<a href="https://arxiv.org/pdf/2312.00240">pdf</a>, <a href="https://arxiv.org/format/2312.00240">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Commutative Algebra">math.AC</span> </div> </div> <p class="title is-5 mathjax"> Betti graphs and atomization of Puiseux monoids </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chapman%2C+S+T">Scott T. Chapman</a>, <a href="/search/math?searchtype=author&query=Jang%2C+J">Joshua Jang</a>, <a href="/search/math?searchtype=author&query=Mao%2C+J">Jason Mao</a>, <a href="/search/math?searchtype=author&query=Mao%2C+S">Skyler Mao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2312.00240v1-abstract-short" style="display: inline;"> Let $M$ be a Puiseux monoid, that is, a monoid consisting of nonnegative rationals (under addition). A nonzero element of $M$ is called an atom if its only decomposition as a sum of two elements in $M$ is the trivial decomposition (i.e., one of the summands is $0$), while a nonzero element $b \in M$ is called atomic if it can be expressed as a sum of finitely many atoms allowing repetitions: this… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.00240v1-abstract-full').style.display = 'inline'; document.getElementById('2312.00240v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2312.00240v1-abstract-full" style="display: none;"> Let $M$ be a Puiseux monoid, that is, a monoid consisting of nonnegative rationals (under addition). A nonzero element of $M$ is called an atom if its only decomposition as a sum of two elements in $M$ is the trivial decomposition (i.e., one of the summands is $0$), while a nonzero element $b \in M$ is called atomic if it can be expressed as a sum of finitely many atoms allowing repetitions: this formal sum of atoms is called an (additive) factorization of $b$. The monoid $M$ is called atomic if every nonzero element of $M$ is atomic. In this paper, we study factorizations in atomic Puiseux monoids through the lens of their associated Betti graphs. The Betti graph of $b \in M$ is the graph whose vertices are the factorizations of $b$ with edges between factorizations that share at least one atom. Betti graphs have been useful in the literature to understand several factorization invariants in the more general class of atomic monoids. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.00240v1-abstract-full').style.display = 'none'; document.getElementById('2312.00240v1-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 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">14 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 13F15; 13A05; Secondary: 20M13; 13F05 </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=Ma%2C+J&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Ma%2C+J&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Ma%2C+J&start=50" class="pagination-link " aria-label="Page 2" aria-current="page">2 </a> </li> <li> <a href="/search/?searchtype=author&query=Ma%2C+J&start=100" class="pagination-link " aria-label="Page 3" aria-current="page">3 </a> </li> <li> <a href="/search/?searchtype=author&query=Ma%2C+J&start=150" class="pagination-link " aria-label="Page 4" aria-current="page">4 </a> </li> <li> <a href="/search/?searchtype=author&query=Ma%2C+J&start=200" class="pagination-link " aria-label="Page 5" aria-current="page">5 </a> </li> <li><span 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