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name="searchtype"><option value="all">All fields</option><option value="title">Title</option><option selected value="author">Author(s)</option><option value="abstract">Abstract</option><option value="comments">Comments</option><option value="journal_ref">Journal reference</option><option value="acm_class">ACM classification</option><option value="msc_class">MSC classification</option><option value="report_num">Report number</option><option value="paper_id">arXiv identifier</option><option value="doi">DOI</option><option value="orcid">ORCID</option><option value="license">License (URI)</option><option value="author_id">arXiv author ID</option><option value="help">Help pages</option><option value="full_text">Full text</option></select> <input id="query" name="query" type="text" value="Yao, X"> <ul id="abstracts"><li><input checked id="abstracts-0" name="abstracts" type="radio" value="show"> <label for="abstracts-0">Show abstracts</label></li><li><input id="abstracts-1" name="abstracts" 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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=Yao%2C+X&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Yao%2C+X&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Yao%2C+X&start=50" class="pagination-link " aria-label="Page 2" aria-current="page">2 </a> </li> </ul> </nav> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2503.13815">arXiv:2503.13815</a> <span> [<a href="https://arxiv.org/pdf/2503.13815">pdf</a>, <a href="https://arxiv.org/ps/2503.13815">ps</a>, <a href="https://arxiv.org/format/2503.13815">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> </div> <p class="title is-5 mathjax"> On the topology of manifolds with positive intermediate curvature </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Mazurowski%2C+L">Liam Mazurowski</a>, <a href="/search/math?searchtype=author&query=Wang%2C+T">Tongrui Wang</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xuan Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2503.13815v1-abstract-short" style="display: inline;"> We formulate a conjecture relating the topology of a manifold's universal cover with the existence of metrics with positive $m$-intermediate curvature. We prove the result for manifolds of dimension $n\in\{3,4,5\}$ and for most choices of $m$ when $n=6$. As a corollary, we show that a closed, aspherical 6-manifold cannot admit a metric with positive $4$-intermediate curvature. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2503.13815v1-abstract-full" style="display: none;"> We formulate a conjecture relating the topology of a manifold's universal cover with the existence of metrics with positive $m$-intermediate curvature. We prove the result for manifolds of dimension $n\in\{3,4,5\}$ and for most choices of $m$ when $n=6$. As a corollary, we show that a closed, aspherical 6-manifold cannot admit a metric with positive $4$-intermediate curvature. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.13815v1-abstract-full').style.display = 'none'; document.getElementById('2503.13815v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 March, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 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">29 pages, comments are welcome!</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2503.00004">arXiv:2503.00004</a> <span> [<a href="https://arxiv.org/pdf/2503.00004">pdf</a>, <a href="https://arxiv.org/format/2503.00004">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"> Congruences modulo arbitrary powers of $5$ and $7$ for Andrews and Paule's partition diamonds with $(n+1)$ copies of $n$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Du%2C+J+Q+D">Julia Q. D. Du</a>, <a href="/search/math?searchtype=author&query=Yao%2C+O+X+M">Olivia X. M. Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2503.00004v1-abstract-short" style="display: inline;"> Recently, Andrews and Paule introduced a partition function $PDN1(N)$ which denotes the number of partition diamonds with $(n+1)$ copies of $n$ where summing the parts at the links gives $N$. They also presented the generating function for $PDN1(n)$ and proved several congruences modulo 5,7,25,49 for $PDN1(n)$. At the end of their paper, Andrews and Paule asked for determining infinite families of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.00004v1-abstract-full').style.display = 'inline'; document.getElementById('2503.00004v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2503.00004v1-abstract-full" style="display: none;"> Recently, Andrews and Paule introduced a partition function $PDN1(N)$ which denotes the number of partition diamonds with $(n+1)$ copies of $n$ where summing the parts at the links gives $N$. They also presented the generating function for $PDN1(n)$ and proved several congruences modulo 5,7,25,49 for $PDN1(n)$. At the end of their paper, Andrews and Paule asked for determining infinite families of congruences similar to Ramanujan's classical $ p(5^kn +d_k) \equiv 0 \pmod {5^k}$, where $24d_k\equiv 1 \pmod {5^k}$ and $k\geq 1$. In this paper, we give an answer of Andrews and Paule's open problem by proving three congruences modulo arbitrary powers of $5$ for $PDN1(n)$. In addition, we prove two congruences modulo arbitrary powers of $7$ for $PDN1(n)$, which are analogous to Watson's congruences for $p(n)$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.00004v1-abstract-full').style.display = 'none'; document.getElementById('2503.00004v1-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 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2025. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2502.09091">arXiv:2502.09091</a> <span> [<a href="https://arxiv.org/pdf/2502.09091">pdf</a>, <a href="https://arxiv.org/ps/2502.09091">ps</a>, <a href="https://arxiv.org/format/2502.09091">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> </div> </div> <p class="title is-5 mathjax"> On uniqueness of functions in the extended Selberg class with moving targets </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+J">Jun Wang</a>, <a href="/search/math?searchtype=author&query=Wang%2C+Q">Qiongyan Wang</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiao Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2502.09091v1-abstract-short" style="display: inline;"> We study the question of when two functions L_1,L_2 in the extended Selberg class are identical in terms of the zeros of L_i-h(i=1,2). Here, the meromorphic function h is called moving target. With the assumption on the growth order of h, we prove that L_1\equiv L_2 if L_1-h and L_2-h have the same zeros counting multiplicities. Moreover, we also construct some examples to show that the assumption… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.09091v1-abstract-full').style.display = 'inline'; document.getElementById('2502.09091v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.09091v1-abstract-full" style="display: none;"> We study the question of when two functions L_1,L_2 in the extended Selberg class are identical in terms of the zeros of L_i-h(i=1,2). Here, the meromorphic function h is called moving target. With the assumption on the growth order of h, we prove that L_1\equiv L_2 if L_1-h and L_2-h have the same zeros counting multiplicities. Moreover, we also construct some examples to show that the assumption is necessary. Compared with the known methods in the literature of this area, we developed a new strategy which is based on the transcendental directions first proposed in the study of distribution of Julia set in complex dynamical system. This may be of independent interest. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.09091v1-abstract-full').style.display = 'none'; document.getElementById('2502.09091v1-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">14pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 11M36; 30D20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2501.12347">arXiv:2501.12347</a> <span> [<a href="https://arxiv.org/pdf/2501.12347">pdf</a>, <a href="https://arxiv.org/format/2501.12347">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Euclidean Domains with Nearly Maximal Yamabe Quotient </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Mazurowski%2C+L">Liam Mazurowski</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xuan Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2501.12347v1-abstract-short" style="display: inline;"> Let $惟$ be a smooth, bounded domain in $\mathbb R^3$ with connected boundary. It follows from work of Escobar that the Yamabe quotient of $惟$ is at most the Yamabe quotient of a ball, and equality holds if and only if $惟$ is a ball. We show that if equality almost holds then the following things are true: (i)$惟$ is diffeomorphic to a ball; (ii) There is a small number $蔚> 0$ such that… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.12347v1-abstract-full').style.display = 'inline'; document.getElementById('2501.12347v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2501.12347v1-abstract-full" style="display: none;"> Let $惟$ be a smooth, bounded domain in $\mathbb R^3$ with connected boundary. It follows from work of Escobar that the Yamabe quotient of $惟$ is at most the Yamabe quotient of a ball, and equality holds if and only if $惟$ is a ball. We show that if equality almost holds then the following things are true: (i)$惟$ is diffeomorphic to a ball; (ii) There is a small number $蔚> 0$ such that $B(x,r) \subset 惟\subset B(x,r(1+蔚))$; (iii) After suitable scaling, $惟$ is Gromov-Hausdorff close to the unit ball when considered as a metric space with its induced length metric. We also give a qualitative comparison between $Q$ and the coefficient of quasi-conformality studied in the theory of quasi-conformal maps. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2501.12347v1-abstract-full').style.display = 'none'; document.getElementById('2501.12347v1-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, 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">24 pages, 8 figures, comments welcome!</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.09061">arXiv:2412.09061</a> <span> [<a href="https://arxiv.org/pdf/2412.09061">pdf</a>, <a href="https://arxiv.org/ps/2412.09061">ps</a>, <a href="https://arxiv.org/format/2412.09061">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"> Decay estimates for beam equations with potentials on the line </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+S">Shuangshuang Chen</a>, <a href="/search/math?searchtype=author&query=Wan%2C+Z">Zijun Wan</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiaohua Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.09061v1-abstract-short" style="display: inline;"> This paper is devoted to the time decay estimates for the following beam equation with a potential on the line: $$ \partial_t^2 u + \left( 螖^2 + m^2 + V(x) \right) u = 0, \ \ u(0, x) = f(x),\quad \partial_t u(0, x) = g(x), $$ where $V$ is a real-valued decaying potential on $\mathbb{R}$, and $m \in \mathbb{R}$. Let $H = 螖^2 + V$ and $P_{ac}(H)$ denote the projection onto the absolutely continu… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.09061v1-abstract-full').style.display = 'inline'; document.getElementById('2412.09061v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.09061v1-abstract-full" style="display: none;"> This paper is devoted to the time decay estimates for the following beam equation with a potential on the line: $$ \partial_t^2 u + \left( 螖^2 + m^2 + V(x) \right) u = 0, \ \ u(0, x) = f(x),\quad \partial_t u(0, x) = g(x), $$ where $V$ is a real-valued decaying potential on $\mathbb{R}$, and $m \in \mathbb{R}$. Let $H = 螖^2 + V$ and $P_{ac}(H)$ denote the projection onto the absolutely continuous spectrum of $H$. Then for $m = 0$, we establish the following decay estimates of the solution operators: $$ \left\|\cos (t \sqrt{H}) P_{ac}(H)\right\|_{L^1 \rightarrow L^{\infty}} + \left\|\frac{\sin (t \sqrt{H})}{t \sqrt{H}} P_{ac}(H)\right\|_{L^1 \rightarrow L^{\infty}} \lesssim |t|^{-\frac{1}{2}}. $$ But for $m \neq 0$, the solutions have different time decay estimates from the case where $m=0$. Specifically, the $L^1$-$L^\infty$ estimates of $\cos (t \sqrt{H + m^2})$ and $\frac{\sin (t \sqrt{H + m^2})}{\sqrt{H + m^2}}$ are bounded by $O(|t|^{-\frac{1}{4}})$ in the low-energy part and $O(|t|^{-\frac{1}{2}})$ in the high-energy part. It is noteworthy that all these results remain consistent with the free cases (i.e., $V = 0$) whatever zero is a regular point or a resonance of $H$. As consequences, we establish the corresponding Strichartz estimates, which are fundamental to study nonlinear problems of beam equations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.09061v1-abstract-full').style.display = 'none'; document.getElementById('2412.09061v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> <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</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.21238">arXiv:2410.21238</a> <span> [<a href="https://arxiv.org/pdf/2410.21238">pdf</a>, <a href="https://arxiv.org/ps/2410.21238">ps</a>, <a href="https://arxiv.org/format/2410.21238">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"> A Note on Scalar curvature comparison rigidity for compact domains </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yao%2C+X">Xuan Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.21238v1-abstract-short" style="display: inline;"> We prove a generalization of Gromov's conjecture on scalar curvature rigidity of convex polytopes to arbitrary convex Riemannian polytope type domains via harmonic spinors on convex domians with boundary condition constructed by Brendle. In particular, we prove a rigidity results on comparison of scalar curvature and scaled mean curvature on the boundary for any convex domain in Euclidean space, w… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.21238v1-abstract-full').style.display = 'inline'; document.getElementById('2410.21238v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.21238v1-abstract-full" style="display: none;"> We prove a generalization of Gromov's conjecture on scalar curvature rigidity of convex polytopes to arbitrary convex Riemannian polytope type domains via harmonic spinors on convex domians with boundary condition constructed by Brendle. In particular, we prove a rigidity results on comparison of scalar curvature and scaled mean curvature on the boundary for any convex domain in Euclidean space, which is a parallel of Shi-Tam's results. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.21238v1-abstract-full').style.display = 'none'; document.getElementById('2410.21238v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">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">10 pages, comments are welcome!</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.20548">arXiv:2410.20548</a> <span> [<a href="https://arxiv.org/pdf/2410.20548">pdf</a>, <a href="https://arxiv.org/ps/2410.20548">ps</a>, <a href="https://arxiv.org/format/2410.20548">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"> Scalar curvature comparison and rigidity of $3$-dimensional weakly convex domains </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ko%2C+D">Dongyeong Ko</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xuan Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.20548v1-abstract-short" style="display: inline;"> For a compact Riemannian $3$-manifold $(M^{3}, g)$ with mean convex boundary which is diffeomorphic to a weakly convex compact domain in $\mathbb{R}^{3}$, we prove that if scalar curvature is nonnegative and the scaled mean curvature comparison $H^{2}g \ge H_{0}^{2} g_{Eucl}$ holds then $(M,g)$ is flat. Our result is a smooth analog of Gromov's dihedral rigidity conjecture and an effective version… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.20548v1-abstract-full').style.display = 'inline'; document.getElementById('2410.20548v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.20548v1-abstract-full" style="display: none;"> For a compact Riemannian $3$-manifold $(M^{3}, g)$ with mean convex boundary which is diffeomorphic to a weakly convex compact domain in $\mathbb{R}^{3}$, we prove that if scalar curvature is nonnegative and the scaled mean curvature comparison $H^{2}g \ge H_{0}^{2} g_{Eucl}$ holds then $(M,g)$ is flat. Our result is a smooth analog of Gromov's dihedral rigidity conjecture and an effective version of extremality results on weakly convex balls in $\mathbb R^3$. More generally, we prove the comparison and rigidity theorem for several classes of manifold with corners. Our proof uses capillary minimal surfaces with prescribed contact angle together with the construction of foliation with nonnegative mean curvature and with prescribed contact angles. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.20548v1-abstract-full').style.display = 'none'; document.getElementById('2410.20548v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">20 pages, comments are welcome!</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.09626">arXiv:2410.09626</a> <span> [<a href="https://arxiv.org/pdf/2410.09626">pdf</a>, <a href="https://arxiv.org/ps/2410.09626">ps</a>, <a href="https://arxiv.org/format/2410.09626">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Classical Analysis and ODEs">math.CA</span> </div> </div> <p class="title is-5 mathjax"> Mass, Conformal Capacity, and the Volumetric Penrose Inequality </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Mazurowski%2C+L">Liam Mazurowski</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xuan Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.09626v1-abstract-short" style="display: inline;"> Let $惟$ be a smooth, bounded subset of $\mathbb{R}^3$ diffeomorphic to a ball. Consider $M = \mathbb{R}^3 \setminus 惟$ equipped with an asymptotically flat metric $g = f^4 g_{\text{euc}}$, where $f\to 1$ at infinity. Assume that $g$ has non-negative scalar curvature and that $危= \partial M$ is a minimal 2-sphere in the $g$ metric. We prove a sharp inequality relating the ADM mass of $M$ with the c… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.09626v1-abstract-full').style.display = 'inline'; document.getElementById('2410.09626v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.09626v1-abstract-full" style="display: none;"> Let $惟$ be a smooth, bounded subset of $\mathbb{R}^3$ diffeomorphic to a ball. Consider $M = \mathbb{R}^3 \setminus 惟$ equipped with an asymptotically flat metric $g = f^4 g_{\text{euc}}$, where $f\to 1$ at infinity. Assume that $g$ has non-negative scalar curvature and that $危= \partial M$ is a minimal 2-sphere in the $g$ metric. We prove a sharp inequality relating the ADM mass of $M$ with the conformal capacity of $惟$. As a corollary, we deduce a sharp lower bound for the ADM mass of $M$ in terms of the Euclidean volume of $惟$. We also prove a stability type result for this ``volumetric Penrose inequality.'' The proofs are based on a monotonicity formula holding along the level sets of a 3-harmonic function. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.09626v1-abstract-full').style.display = 'none'; document.getElementById('2410.09626v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">21 pages, comments are welcome!</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.19737">arXiv:2406.19737</a> <span> [<a href="https://arxiv.org/pdf/2406.19737">pdf</a>, <a href="https://arxiv.org/ps/2406.19737">ps</a>, <a href="https://arxiv.org/format/2406.19737">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> </div> </div> <p class="title is-5 mathjax"> K枚nigs maps and commutants of composition operators on the Hardy-Hilbert space of Dirichlet series </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bayart%2C+F">Fr茅d茅ric Bayart</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xingxing Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2406.19737v1-abstract-short" style="display: inline;"> Let $\varphi$ be a holomorphic map which is a symbol of a bounded composition operator $C_\varphi$ acting on the Hardy-Hilbert space of Dirichlet series. We find a K枚nigs map for $\varphi$. We then deduce several applications on $C_\varphi$ (e.g. on its spectrum, on its dynamical properties). In particular, we study for a large class of symbols $\varphi$ if the associated composition operator has… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.19737v1-abstract-full').style.display = 'inline'; document.getElementById('2406.19737v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.19737v1-abstract-full" style="display: none;"> Let $\varphi$ be a holomorphic map which is a symbol of a bounded composition operator $C_\varphi$ acting on the Hardy-Hilbert space of Dirichlet series. We find a K枚nigs map for $\varphi$. We then deduce several applications on $C_\varphi$ (e.g. on its spectrum, on its dynamical properties). In particular, we study for a large class of symbols $\varphi$ if the associated composition operator has a minimal commutant. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.19737v1-abstract-full').style.display = 'none'; document.getElementById('2406.19737v1-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 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/2406.12584">arXiv:2406.12584</a> <span> [<a href="https://arxiv.org/pdf/2406.12584">pdf</a>, <a href="https://arxiv.org/ps/2406.12584">ps</a>, <a href="https://arxiv.org/format/2406.12584">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> </div> <p class="title is-5 mathjax"> Minimal surfaces with low genus in lens spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+X">Xingzhe Li</a>, <a href="/search/math?searchtype=author&query=Wang%2C+T">Tongrui Wang</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xuan Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2406.12584v1-abstract-short" style="display: inline;"> Given a Riemannian $\mathbb{RP}^3$ with a bumpy metric or a metric of positive Ricci curvature, we show that there either exist four distinct minimal real projective planes, or exist one minimal real projective plane together with two distinct minimal $2$-spheres. Our proof is based on a variant multiplicity one theorem for the Simon-Smith min-max theory under certain equivariant settings. In part… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.12584v1-abstract-full').style.display = 'inline'; document.getElementById('2406.12584v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.12584v1-abstract-full" style="display: none;"> Given a Riemannian $\mathbb{RP}^3$ with a bumpy metric or a metric of positive Ricci curvature, we show that there either exist four distinct minimal real projective planes, or exist one minimal real projective plane together with two distinct minimal $2$-spheres. Our proof is based on a variant multiplicity one theorem for the Simon-Smith min-max theory under certain equivariant settings. In particular, we show under the positive Ricci assumption that $\mathbb{RP}^3$ contains at least four distinct minimal real projective planes and four distinct minimal tori. Additionally, the number of minimal tori can be improved to five for a generic positive Ricci metric on $\mathbb{RP}^3$ by the degree method. Moreover, using the same strategy, we show that in the lens space $L(4m,2m\pm 1)$, $m\geq 1$, with a bumpy metric or a metric of positive Ricci curvature, there either exist $N(m)$ numbers of distinct minimal Klein bottles, or exist one minimal Klein bottle and three distinct minimal $2$-spheres, where $N(1)=4$, $N(m)=2$ for $m\geq 2$, and the first case happens under the positive Ricci assumption. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.12584v1-abstract-full').style.display = 'none'; document.getElementById('2406.12584v1-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 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">34 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> 53A10; 53C42 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.00815">arXiv:2402.00815</a> <span> [<a href="https://arxiv.org/pdf/2402.00815">pdf</a>, <a href="https://arxiv.org/ps/2402.00815">ps</a>, <a href="https://arxiv.org/format/2402.00815">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 the stability of the Yamabe invariant of $S^3$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Mazurowski%2C+L">Liam Mazurowski</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xuan Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2402.00815v1-abstract-short" style="display: inline;"> Let $g$ be a complete, asymptotically flat metric on $\mathbb{R}^3$ with vanishing scalar curvature. Moreover, assume that $(\mathbb{R}^3,g)$ supports a nearly Euclidean $L^2$ Sobolev inequality. We prove that $(\mathbb{R}^3,g)$ must be close to Euclidean space with respect to the $d_p$-distance defined by Lee-Naber-Neumayer. We then discuss some consequences for the stability of the Yamabe invari… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.00815v1-abstract-full').style.display = 'inline'; document.getElementById('2402.00815v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.00815v1-abstract-full" style="display: none;"> Let $g$ be a complete, asymptotically flat metric on $\mathbb{R}^3$ with vanishing scalar curvature. Moreover, assume that $(\mathbb{R}^3,g)$ supports a nearly Euclidean $L^2$ Sobolev inequality. We prove that $(\mathbb{R}^3,g)$ must be close to Euclidean space with respect to the $d_p$-distance defined by Lee-Naber-Neumayer. We then discuss some consequences for the stability of the Yamabe invariant of $S^3$. More precisely, we show that if such a manifold $(\mathbb{R}^3,g)$ carries a suitably normalized, positive solution to $螖_g w + 位w^5 = 0$ then $w$ must be close, in a certain sense, to a conformal factor that transforms Euclidean space into a round sphere. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.00815v1-abstract-full').style.display = 'none'; document.getElementById('2402.00815v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2024. </p> <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, comments are welcome!</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53C21 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2311.06768">arXiv:2311.06768</a> <span> [<a href="https://arxiv.org/pdf/2311.06768">pdf</a>, <a href="https://arxiv.org/ps/2311.06768">ps</a>, <a href="https://arxiv.org/format/2311.06768">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Classical Analysis and ODEs">math.CA</span> </div> </div> <p class="title is-5 mathjax"> Counterexamples and weak (1,1) estimates of wave operators for fourth-order Schr枚dinger operators in dimension three </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Mizutani%2C+H">Haruya Mizutani</a>, <a href="/search/math?searchtype=author&query=Wan%2C+Z">Zijun Wan</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiaohua Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2311.06768v3-abstract-short" style="display: inline;"> This paper is dedicated to investigating the $L^p$-bounds of wave operators $W_\pm(H,螖^2)$ associated with fourth-order Schr枚dinger operators $H=螖^2+V$ on $\mathbb{R}^3$. We consider that real potentials satisfy $|V(x)|\lesssim \langle x\rangle^{-渭}$ for some $渭>0$. A recent work by Goldberg and Green \cite{GoGr21} has demonstrated that wave operators $W_\pm(H,螖^2)$ are bounded on… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.06768v3-abstract-full').style.display = 'inline'; document.getElementById('2311.06768v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2311.06768v3-abstract-full" style="display: none;"> This paper is dedicated to investigating the $L^p$-bounds of wave operators $W_\pm(H,螖^2)$ associated with fourth-order Schr枚dinger operators $H=螖^2+V$ on $\mathbb{R}^3$. We consider that real potentials satisfy $|V(x)|\lesssim \langle x\rangle^{-渭}$ for some $渭>0$. A recent work by Goldberg and Green \cite{GoGr21} has demonstrated that wave operators $W_\pm(H,螖^2)$ are bounded on $L^p(\mathbb{R}^3)$ for all $1<p<\infty$ under the condition that $渭>9$, and zero is a regular point of $H$. In this paper, we aim to further establish endpoint estimates for $W_\pm(H,螖^2)$ in two significant ways. First, we provide counterexamples that illustrate the unboundedness of $W_\pm(H,螖^2)$ on the endpoint spaces $L^1(\mathbb{R}^3)$ and $L^\infty(\mathbb{R}^3)$, even for non-zero compactly supported potentials $V$. Second, we establish weak (1,1) estimates for the wave operators $W_\pm(H,螖^2)$ and their dual operators $W_\pm(H,螖^2)^*$ in the case where zero is a regular point and $渭>11$. These estimates depend critically on the singular integral theory of Calder贸n-Zygmund on a homogeneous space $(X,d蠅)$ with a doubling measure $d蠅$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.06768v3-abstract-full').style.display = 'none'; document.getElementById('2311.06768v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">29 pages. This a final version in Journal of Spectral Theory,2024</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2311.06763">arXiv:2311.06763</a> <span> [<a href="https://arxiv.org/pdf/2311.06763">pdf</a>, <a href="https://arxiv.org/ps/2311.06763">ps</a>, <a href="https://arxiv.org/format/2311.06763">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Classical Analysis and ODEs">math.CA</span> </div> </div> <p class="title is-5 mathjax"> $L^p$-boundedness of wave operators for fourth order Schr枚dinger operators with zero resonances on $\mathbb{R}^3$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Mizutani%2C+H">Haruya Mizutani</a>, <a href="/search/math?searchtype=author&query=Wan%2C+Z">Zijun Wan</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiaohua Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2311.06763v3-abstract-short" style="display: inline;"> Let $H = 螖^2 + V$ be the fourth-order Schr枚dinger operator on $\mathbb{R}^3$ with a real-valued fast-decaying potential $V$. If zero is neither a resonance nor an eigenvalue of $H$, then it was recently shown that the wave operators $W_\pm(H, 螖^2)$ are bounded on $L^p(\mathbb{R}^3)$ for all $1 < p < \infty$ and unbounded at the endpoints $p=1$ and $p=\infty$. This paper is to further establish t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.06763v3-abstract-full').style.display = 'inline'; document.getElementById('2311.06763v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2311.06763v3-abstract-full" style="display: none;"> Let $H = 螖^2 + V$ be the fourth-order Schr枚dinger operator on $\mathbb{R}^3$ with a real-valued fast-decaying potential $V$. If zero is neither a resonance nor an eigenvalue of $H$, then it was recently shown that the wave operators $W_\pm(H, 螖^2)$ are bounded on $L^p(\mathbb{R}^3)$ for all $1 < p < \infty$ and unbounded at the endpoints $p=1$ and $p=\infty$. This paper is to further establish the $L^p$-boundedness of $W_\pm(H, 螖^2)$ that exhibit all types of singularities at the zero energy threshold. We first prove that $W_\pm(H, 螖^2)$ are bounded on $L^p(\mathbb{R}^3)$ for all $1 < p < \infty$ in the first kind resonance case, and then proceed to establish for the second kind resonance case that they are bounded on $L^p(\mathbb{R}^3)$ for all $1 < p < 3$, but not if $3 \le p \le \infty$. In the third kind resonance case, we also show that $W_\pm(H, 螖^2)$ are bounded on $L^p(\mathbb{R}^3)$ for all $1<p<3$ and generically unbounded on $L^p(\R^3)$ for any $3\le p\le\infty$. Moreover, it is also shown that $W_\pm(H, 螖^2)$ are bounded on $L^p(\R^3)$ for all $3\le p<\infty$ if in addition $H$ has the zero eigenvalue, but no $p$-wave zero resonances and all zero eigenfunctions are orthogonal to $x_ix_jx_kV$ in $L^2(\R^3)$ for all $i,j,k=1,2,3$ with $x=(x_1,x_2,x_3)\in \R^3$. These results describe precisely the validity of the $L^p$-boundedness of $W_\pm(H, 螖^2)$ in $\mathbb{R}^3$ for all types of singularities at the zero energy threshold with some exceptions for the endpoint cases $p=1,\infty$. As an application, $L^p$-$L^q$ decay estimates are also derived for the fourth-order Schr枚dinger equations and Beam equations with zero resonance singularities. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.06763v3-abstract-full').style.display = 'none'; document.getElementById('2311.06763v3-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">v1</span> submitted 12 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">45 pages, it is the third version</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2309.13865">arXiv:2309.13865</a> <span> [<a href="https://arxiv.org/pdf/2309.13865">pdf</a>, <a href="https://arxiv.org/ps/2309.13865">ps</a>, <a href="https://arxiv.org/format/2309.13865">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> </div> <p class="title is-5 mathjax"> Generalized $S^1$-stability theorem </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+T">Tongrui Wang</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xuan Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2309.13865v1-abstract-short" style="display: inline;"> We use the equivariant $渭$-bubbles technique to prove that for any compact manifold $M^n$ with non-empty boundary, $n\in\{3,5,6\}$, the Yamabe invariant of $M^n$ is positive if and only if the Yamabe invariant of $M^n\times S^1$ is positive. This generalized the $S^1$-stability conjecture of Rosenberg to compact manifolds with boundary. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2309.13865v1-abstract-full" style="display: none;"> We use the equivariant $渭$-bubbles technique to prove that for any compact manifold $M^n$ with non-empty boundary, $n\in\{3,5,6\}$, the Yamabe invariant of $M^n$ is positive if and only if the Yamabe invariant of $M^n\times S^1$ is positive. This generalized the $S^1$-stability conjecture of Rosenberg to compact manifolds with boundary. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.13865v1-abstract-full').style.display = 'none'; document.getElementById('2309.13865v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 September, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">14 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> 53C21; 53C23 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2309.13861">arXiv:2309.13861</a> <span> [<a href="https://arxiv.org/pdf/2309.13861">pdf</a>, <a href="https://arxiv.org/format/2309.13861">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"> Improved Hebey-Vaugon conjecture on equivariant Yamabe invariants in dimension 3 </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+T">Tongrui Wang</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xuan Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2309.13861v1-abstract-short" style="display: inline;"> Consider a closed connected $3$-manifold $M$ acted diffeomorphically on by a compact Lie group $G$ with at least one orbit of finite cardinality. We show an upper bound for the $G$-equivariant Yamabe invariant $蟽_G(M)$ under certain topological assumptions, which improved a conjecture of Hebey-Vaugon. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2309.13861v1-abstract-full" style="display: none;"> Consider a closed connected $3$-manifold $M$ acted diffeomorphically on by a compact Lie group $G$ with at least one orbit of finite cardinality. We show an upper bound for the $G$-equivariant Yamabe invariant $蟽_G(M)$ under certain topological assumptions, which improved a conjecture of Hebey-Vaugon. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.13861v1-abstract-full').style.display = 'none'; document.getElementById('2309.13861v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 September, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">10 pages, 2 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/2307.16428">arXiv:2307.16428</a> <span> [<a href="https://arxiv.org/pdf/2307.16428">pdf</a>, <a href="https://arxiv.org/ps/2307.16428">ps</a>, <a href="https://arxiv.org/format/2307.16428">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"> Decay estimates for Beam equations with potentials in dimension three </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chen%2C+M">Miao Chen</a>, <a href="/search/math?searchtype=author&query=Li%2C+P">Ping Li</a>, <a href="/search/math?searchtype=author&query=Soffer%2C+A">Avy Soffer</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiaohua Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2307.16428v3-abstract-short" style="display: inline;"> This paper is devoted to studying time decay estimates of the solution for Beam equation (higher order type wave equation) with a potential $$u_{t t}+\big(螖^2+V\big)u=0, \,\ u(0, x)=f(x),\ u_{t}(0, x)=g(x)$$ in dimension three, where $V$ is a real-valued and decaying potential on $\R^3$. Assume that zero is a regular point of $H:= 螖^2+V $, we first prove the following optimal time decay estimate… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.16428v3-abstract-full').style.display = 'inline'; document.getElementById('2307.16428v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2307.16428v3-abstract-full" style="display: none;"> This paper is devoted to studying time decay estimates of the solution for Beam equation (higher order type wave equation) with a potential $$u_{t t}+\big(螖^2+V\big)u=0, \,\ u(0, x)=f(x),\ u_{t}(0, x)=g(x)$$ in dimension three, where $V$ is a real-valued and decaying potential on $\R^3$. Assume that zero is a regular point of $H:= 螖^2+V $, we first prove the following optimal time decay estimates of the solution operators \begin{equation*} \big\|\cos (t\sqrt{H})P_{ac}(H)\big\|_{L^{1} \rightarrow L^{\infty}} \lesssim|t|^{-\frac{3}{2}}\ \ \hbox{and} \ \ \Big\|\frac{\sin(t\sqrt{H})}{\sqrt{H}} P_{a c}(H)\Big\|_{L^{1} \rightarrow L^{\infty}} \lesssim|t|^{-\frac{1}{2}}. \end{equation*} Moreover, if zero is a resonance of $H$, then time decay of the solution operators above also are considered. It is noticed that the first kind resonance does not effect the decay rates for the propagator operators $\cos(t\sqrt{H})$ and $\frac{\sin(t\sqrt{H})}{\sqrt{H}}$, but their decay will be dramatically changed for the second and third resonance types. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.16428v3-abstract-full').style.display = 'none'; document.getElementById('2307.16428v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 31 July, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 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">43 Pages. This is a final version. To appear in JFA,2024</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2305.19784">arXiv:2305.19784</a> <span> [<a href="https://arxiv.org/pdf/2305.19784">pdf</a>, <a href="https://arxiv.org/ps/2305.19784">ps</a>, <a href="https://arxiv.org/format/2305.19784">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Classical Analysis and ODEs">math.CA</span> </div> </div> <p class="title is-5 mathjax"> Monotone Quantities for $p$-Harmonic functions and the Sharp $p$-Penrose inequality </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Mazurowski%2C+L">Liam Mazurowski</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xuan Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2305.19784v4-abstract-short" style="display: inline;"> Consider a complete asymptotically flat 3-manifold $M$ with non-negative scalar curvature and non-empty minimal boundary $危$. Fix a number $1 < p < 3$. We derive monotone quantities for $p$-harmonic functions on $M$ which become constant on Schwarzschild. These monotonicity formulas imply a sharp mass-capacity estimate relating the ADM mass of $M$ with the $p$-capacity of $危$ in $M$, which was fir… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.19784v4-abstract-full').style.display = 'inline'; document.getElementById('2305.19784v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2305.19784v4-abstract-full" style="display: none;"> Consider a complete asymptotically flat 3-manifold $M$ with non-negative scalar curvature and non-empty minimal boundary $危$. Fix a number $1 < p < 3$. We derive monotone quantities for $p$-harmonic functions on $M$ which become constant on Schwarzschild. These monotonicity formulas imply a sharp mass-capacity estimate relating the ADM mass of $M$ with the $p$-capacity of $危$ in $M$, which was first proved by Xiao using weak inverse mean curvature flow. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.19784v4-abstract-full').style.display = 'none'; document.getElementById('2305.19784v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 31 May, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 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">22 pages, comments are welcome!</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2305.00854">arXiv:2305.00854</a> <span> [<a href="https://arxiv.org/pdf/2305.00854">pdf</a>, <a href="https://arxiv.org/ps/2305.00854">ps</a>, <a href="https://arxiv.org/format/2305.00854">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> </div> <p class="title is-5 mathjax"> The Yamabe Invariant of $\mathbb{RP}^3$ via Harmonic Functions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Mazurowski%2C+L">Liam Mazurowski</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xuan Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2305.00854v1-abstract-short" style="display: inline;"> We use harmonic functions to give a new proof of a result of Bray and Neves on the Yamabe invariant of $\mathbb{RP}^3$ </span> <span class="abstract-full has-text-grey-dark mathjax" id="2305.00854v1-abstract-full" style="display: none;"> We use harmonic functions to give a new proof of a result of Bray and Neves on the Yamabe invariant of $\mathbb{RP}^3$ <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.00854v1-abstract-full').style.display = 'none'; document.getElementById('2305.00854v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 May, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 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">8 pages, comments welcome!</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2303.02767">arXiv:2303.02767</a> <span> [<a href="https://arxiv.org/pdf/2303.02767">pdf</a>, <a href="https://arxiv.org/ps/2303.02767">ps</a>, <a href="https://arxiv.org/format/2303.02767">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"> Difference independence of the Euler gamma function </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+Q">Qiongyan Wang</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiao Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2303.02767v1-abstract-short" style="display: inline;"> In this paper, we established a sharp version of the difference analogue of the celebrated H枚lder's theorem concerning the differential independence of the Euler gamma function $螕$. More precisely, if $P$ is a polynomial of $n+1$ variables in $\mathbb{C}[X, Y_0,\dots, Y_{n-1}]$ such that \begin{equation*} P(s, 螕(s+a_0), \dots, 螕(s+a_{n-1}))\equiv 0 \end{equation*} for some… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.02767v1-abstract-full').style.display = 'inline'; document.getElementById('2303.02767v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2303.02767v1-abstract-full" style="display: none;"> In this paper, we established a sharp version of the difference analogue of the celebrated H枚lder's theorem concerning the differential independence of the Euler gamma function $螕$. More precisely, if $P$ is a polynomial of $n+1$ variables in $\mathbb{C}[X, Y_0,\dots, Y_{n-1}]$ such that \begin{equation*} P(s, 螕(s+a_0), \dots, 螕(s+a_{n-1}))\equiv 0 \end{equation*} for some $(a_0, \dots, a_{n-1})\in \mathbb{C}^{n}$ and $a_i-a_j\notin \mathbb{Z}$ for any $0\leq i<j\leq n-1$, then we have $$P\equiv 0.$$ Our result complements a classical result of algebraic differential independence of the Euler gamma function proved by H枚lder in 1886, and also a result of algebraic difference independence of the Riemann zeta function proved by Chiang and Feng in 2006. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.02767v1-abstract-full').style.display = 'none'; document.getElementById('2303.02767v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 March, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 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">8 Pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 11M06; 39A05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2302.11445">arXiv:2302.11445</a> <span> [<a href="https://arxiv.org/pdf/2302.11445">pdf</a>, <a href="https://arxiv.org/ps/2302.11445">ps</a>, <a href="https://arxiv.org/format/2302.11445">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> </div> <p class="title is-5 mathjax"> On the Yamabe invariant of certain compact manifolds with boundary </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yao%2C+X">Xuan Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2302.11445v2-abstract-short" style="display: inline;"> We generalize Kobayashi's connected-sum inequality to the $位$-Yamabe invariants. As an application, we calculate the $位$-Yamabe invariants of $\#m_1\mathbb{RP}^n\# m_2(\mathbb{RP}^{n-1}\times S^1)\#lH^n\#kS_+^n$, for any $位\in [0,1]$, $n\geq 3$, provided $k+l\geq 1$. As a corollary, we prove that $\mathbb{RP}^n$ minus finitely many disjoint $n$-balls have the same $位$-Yamabe invariants as the he… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2302.11445v2-abstract-full').style.display = 'inline'; document.getElementById('2302.11445v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2302.11445v2-abstract-full" style="display: none;"> We generalize Kobayashi's connected-sum inequality to the $位$-Yamabe invariants. As an application, we calculate the $位$-Yamabe invariants of $\#m_1\mathbb{RP}^n\# m_2(\mathbb{RP}^{n-1}\times S^1)\#lH^n\#kS_+^n$, for any $位\in [0,1]$, $n\geq 3$, provided $k+l\geq 1$. As a corollary, we prove that $\mathbb{RP}^n$ minus finitely many disjoint $n$-balls have the same $位$-Yamabe invariants as the hemi-sphere, which forms an interesting contrast with the famous Bray-Neves results on the Yamabe invariants of $\mathbb{RP}^3$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2302.11445v2-abstract-full').style.display = 'none'; document.getElementById('2302.11445v2-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 March, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 February, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 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">22 pages, comments welcome!</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53C18 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2301.12610">arXiv:2301.12610</a> <span> [<a href="https://arxiv.org/pdf/2301.12610">pdf</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> To Define the Core Entropy for All Polynomials Having a Connected Julia Set </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+J">Jun Luo</a>, <a href="/search/math?searchtype=author&query=Tan%2C+B">Bo Tan</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yi Yang</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiao-Ting Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2301.12610v3-abstract-short" style="display: inline;"> The classical core entropy for a post critically finite (PCF) polynomial f with deg(f)>1 is defined to be the topological entropy of f restricted to its Hubbard tree. We fully generalize this notion by a new quantity, called the (general) core entropy, which is well defined whenever f has a connected Julia set. If f is PCF, the core entropy equals the classical version. If two polynomials are J-eq… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2301.12610v3-abstract-full').style.display = 'inline'; document.getElementById('2301.12610v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2301.12610v3-abstract-full" style="display: none;"> The classical core entropy for a post critically finite (PCF) polynomial f with deg(f)>1 is defined to be the topological entropy of f restricted to its Hubbard tree. We fully generalize this notion by a new quantity, called the (general) core entropy, which is well defined whenever f has a connected Julia set. If f is PCF, the core entropy equals the classical version. If two polynomials are J-equivalent they share the same core entropy. If f is renormalizable there is a direct connection between the core entropy of f and that corresponding to the small Julia set. We also analyze the map that sends every parameter c in the Mandelbrot set to the core entropy of the polynomial z^2+c. In particular, the level set of this entropy map at log2 is of full harmonic measure and the Mandelbrot set is locally connected at each point in this level set. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2301.12610v3-abstract-full').style.display = 'none'; document.getElementById('2301.12610v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 29 January, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 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">36 pages, 5 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 54D05; 54H20; 37F45; 37E99 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2209.03773">arXiv:2209.03773</a> <span> [<a href="https://arxiv.org/pdf/2209.03773">pdf</a>, <a href="https://arxiv.org/ps/2209.03773">ps</a>, <a href="https://arxiv.org/format/2209.03773">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> </div> <p class="title is-5 mathjax"> A model for planar compacta and rational Julia sets </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+J">Jun Luo</a>, <a href="/search/math?searchtype=author&query=Yang%2C+Y">Yi Yang</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiaoting Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2209.03773v2-abstract-short" style="display: inline;"> A Peano compactum is a compact metric space having locally connected components such that at most finitely many of them are of diameter greater than any fixed number C>0. Given a compactum K in the extended complex plane, it is known that there is a finest upper semi-continuous decomposition of K into subcontinua such that the resulting quotient space is a Peano compactum. We call this decompositi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2209.03773v2-abstract-full').style.display = 'inline'; document.getElementById('2209.03773v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2209.03773v2-abstract-full" style="display: none;"> A Peano compactum is a compact metric space having locally connected components such that at most finitely many of them are of diameter greater than any fixed number C>0. Given a compactum K in the extended complex plane, it is known that there is a finest upper semi-continuous decomposition of K into subcontinua such that the resulting quotient space is a Peano compactum. We call this decomposition the core decomposition of K with Peano quotient and its elements atoms of K. We show that for any branched covering f of the extended complex plane onto itself and for any atom d of K, the preimage of d under f has finitely many components each of which is an atom of the preimage of K under f. Since rational functions are branched coverings, our result extends earlier ones that are restricted to more limited cases, requiring that f be a polynomial and K completely invariant under f. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2209.03773v2-abstract-full').style.display = 'none'; document.getElementById('2209.03773v2-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 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 September, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">20 pages, 13 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 2020: 37B45; 37F10; 54D05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2205.07555">arXiv:2205.07555</a> <span> [<a href="https://arxiv.org/pdf/2205.07555">pdf</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Peridynamic modeling for impact failure of wet concrete considering the influence of saturation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wu%2C+L">Liwei Wu</a>, <a href="/search/math?searchtype=author&query=Huang%2C+D">Dan Huang</a>, <a href="/search/math?searchtype=author&query=Ma%2C+Q">Qipeng Ma</a>, <a href="/search/math?searchtype=author&query=Li%2C+Z">Zhiyuan Li</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xuehao Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2205.07555v2-abstract-short" style="display: inline;"> In this paper, a modified intermediately homogenized peridynamic (IH-PD) model for analyzing impact failure of wet concrete has been presented under the configuration of ordinary state-based peridynamic theory. The meso-structural properties of concrete are linked to the macroscopic mechanical behavior in the IH-PD model, where the heterogeneity of concrete is taken into account, and the calculati… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2205.07555v2-abstract-full').style.display = 'inline'; document.getElementById('2205.07555v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2205.07555v2-abstract-full" style="display: none;"> In this paper, a modified intermediately homogenized peridynamic (IH-PD) model for analyzing impact failure of wet concrete has been presented under the configuration of ordinary state-based peridynamic theory. The meso-structural properties of concrete are linked to the macroscopic mechanical behavior in the IH-PD model, where the heterogeneity of concrete is taken into account, and the calculation cost does not increase. Simultaneously, the porosity of concrete is considered, which is implemented by deleting the bond between two material points, as well as the influence of porosity on the mechanical properties of concrete. Moreover, the effective bulk and shear modulus of cement mortar in wet concrete (saturated and unsaturated concrete) are calculated respectively. The dynamic model for wet concrete is described from three aspects: strength, dynamic increase factor, and equation of state. Validation of the proposed model is established through analyzing some benchmark tests and comparing with the corresponding experiment and other available numerical results. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2205.07555v2-abstract-full').style.display = 'none'; document.getElementById('2205.07555v2-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 May, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 16 May, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2204.06753">arXiv:2204.06753</a> <span> [<a href="https://arxiv.org/pdf/2204.06753">pdf</a>, <a href="https://arxiv.org/ps/2204.06753">ps</a>, <a href="https://arxiv.org/format/2204.06753">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Rationality of meromorphic functions between real algebraic sets in the plane </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ng%2C+T">Tuen-Wai Ng</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiao Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2204.06753v1-abstract-short" style="display: inline;"> We study one variable meromorphic functions mapping a planar real algebraic set $A$ to another real algebraic set in the complex plane. By using the theory of Schwarz reflection functions, we show that for certain $A$, these meromorphic functions must be rational. In particular, when $A$ is the standard unit circle, we obtain an one dimensional analog of Poincar茅(1907), Tanaka(1962) and Alexander(… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2204.06753v1-abstract-full').style.display = 'inline'; document.getElementById('2204.06753v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2204.06753v1-abstract-full" style="display: none;"> We study one variable meromorphic functions mapping a planar real algebraic set $A$ to another real algebraic set in the complex plane. By using the theory of Schwarz reflection functions, we show that for certain $A$, these meromorphic functions must be rational. In particular, when $A$ is the standard unit circle, we obtain an one dimensional analog of Poincar茅(1907), Tanaka(1962) and Alexander(1974)'s rationality results for $2m-1$ dimensional sphere in $\mathbb{C}^m$ when $m\ge 2$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2204.06753v1-abstract-full').style.display = 'none'; document.getElementById('2204.06753v1-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 April, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">To appear in Proc. AMS</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 30C99; 30D99 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2201.04758">arXiv:2201.04758</a> <span> [<a href="https://arxiv.org/pdf/2201.04758">pdf</a>, <a href="https://arxiv.org/ps/2201.04758">ps</a>, <a href="https://arxiv.org/format/2201.04758">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Classical Analysis and ODEs">math.CA</span> </div> </div> <p class="title is-5 mathjax"> $L^p$-boundedness of wave operators for bi-Schr枚dinger operators on the line </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Mizutani%2C+H">Haruya Mizutani</a>, <a href="/search/math?searchtype=author&query=Wan%2C+Z">Zijun Wan</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiaohua Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2201.04758v3-abstract-short" style="display: inline;"> This paper is devoted to establishing several types of $L^p$-boundedness of wave operators $W_\pm=W_\pm(H, 螖^2)$ associated with the bi-Schr枚dinger operators $H=螖^{2}+V(x)$ on the line $\mathbb{R}$. Given suitable decay potentials $V$, we firstly prove that the wave and dual wave operators are bounded on $L^p(\mathbb{R})$ for all $1<p<\infty$:… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2201.04758v3-abstract-full').style.display = 'inline'; document.getElementById('2201.04758v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2201.04758v3-abstract-full" style="display: none;"> This paper is devoted to establishing several types of $L^p$-boundedness of wave operators $W_\pm=W_\pm(H, 螖^2)$ associated with the bi-Schr枚dinger operators $H=螖^{2}+V(x)$ on the line $\mathbb{R}$. Given suitable decay potentials $V$, we firstly prove that the wave and dual wave operators are bounded on $L^p(\mathbb{R})$ for all $1<p<\infty$: $$ \|W_\pm f\|_{L^p(\mathbb{R})}+\|W_\pm^* f\|_{L^p(\mathbb{R})}\lesssim \|f\|_{L^p(\mathbb{R})},$$ which are further extended to the $L^p$-boundedness on the weighted spaces $L^p(\mathbb{R},w)$ with general even $A_p$-weights $w$ and to the boundedness on the Sobolev spaces $W^{s,p}(\mathbb{R})$. For the limiting case, we prove that $W_\pm$ are bounded from $L^1(\R)$ to $L^{1,\infty}(\R)$ as well as bounded from the Hardy space $\H^1(\R)$ to $L^1(\R)$. These results especially hold whatever the zero energy is a regular point or a resonance of $H$. We also obtain that $W_\pm$ are bounded from $L^\infty(\R)$ to $\BMO(\R)$ if zero is a regular point or a first kind resonance of $H$. Next, we show that $W_\pm$ are neither bounded on $L^1(\mathbb{R})$ nor on $L^\infty(\mathbb{R})$ even if zero is a regular point of $H$. Moreover, if zero is a second kind resonance of $H$, then $W_\pm$ are shown to be even not bounded from $L^\infty(\R)$ to $\BMO(\R)$ in general. In particular, we remark that our results give a complete picture of the validity of $L^p$-boundedness of the wave operators for all $1\le p\le \infty$ in the regular case. Finally, as applications, we deduce the $L^p$-$L^q$ decay estimates for the propagator $e^{-itH}P_{\mathrm{ac}}(H)$ with pairs $(1/p,1/q)$ belonging to a certain region of $\mathbb{R}^2$, as well as establish the H枚rmander-type $L^p$-boundedness theorem for the spectral multiplier $f(H)$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2201.04758v3-abstract-full').style.display = 'none'; document.getElementById('2201.04758v3-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 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 January, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2022. </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">57 pages. This is a final version. To appear in Adv. Math., 2024</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2112.12995">arXiv:2112.12995</a> <span> [<a href="https://arxiv.org/pdf/2112.12995">pdf</a>, <a href="https://arxiv.org/format/2112.12995">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="K-Theory and Homology">math.KT</span> </div> </div> <p class="title is-5 mathjax"> The K-homology of 2-dimensional Crystallographic Groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Wang%2C+H">Hang Wang</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiufeng Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2112.12995v2-abstract-short" style="display: inline;"> In this paper we compute the topological K-homology of 2-dimensional crystal groups. Our method focuses on the fixed point of group action and simplifies the calculation of the K-homology of universal space. The result also verifies the Baum-Connes Conjecture of 2-dimensional crystal groups. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2112.12995v2-abstract-full" style="display: none;"> In this paper we compute the topological K-homology of 2-dimensional crystal groups. Our method focuses on the fixed point of group action and simplifies the calculation of the K-homology of universal space. The result also verifies the Baum-Connes Conjecture of 2-dimensional crystal groups. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2112.12995v2-abstract-full').style.display = 'none'; document.getElementById('2112.12995v2-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 March, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 24 December, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2110.07154">arXiv:2110.07154</a> <span> [<a href="https://arxiv.org/pdf/2110.07154">pdf</a>, <a href="https://arxiv.org/ps/2110.07154">ps</a>, <a href="https://arxiv.org/format/2110.07154">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> Decay estimates for fourth-order Schr枚dinger operators in dimension two </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+P">Ping Li</a>, <a href="/search/math?searchtype=author&query=Soffer%2C+A">Avy Soffer</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiaohua Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2110.07154v3-abstract-short" style="display: inline;"> In this paper we study the decay estimates of the fourth order Schr枚dinger operator $H=螖^{2}+V(x)$ on $\mathbb{R}^2$ with a bounded decaying potential $V(x)$. We first deduce the asymptotic expansions of resolvent of $H$ near the zero threshold in the presence of resonances or eigenvalue, and then use them to establish the $L^1-L^\infty$ decay estimates of $e^{-itH}$generated by the fourth order S… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2110.07154v3-abstract-full').style.display = 'inline'; document.getElementById('2110.07154v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2110.07154v3-abstract-full" style="display: none;"> In this paper we study the decay estimates of the fourth order Schr枚dinger operator $H=螖^{2}+V(x)$ on $\mathbb{R}^2$ with a bounded decaying potential $V(x)$. We first deduce the asymptotic expansions of resolvent of $H$ near the zero threshold in the presence of resonances or eigenvalue, and then use them to establish the $L^1-L^\infty$ decay estimates of $e^{-itH}$generated by the fourth order Schr枚dinger operator $H$. Our methods used in the decay estimates depend on Littlewood-Paley decomposition and oscillatory integral theory. Moreover, we classify these zero resonances as the distributional solutions of $H蠁=0$ in suitable weighted spaces. Due to the degeneracy of $螖^{2}$ at zero threshold and the lower even dimension (i.e. $n=2$), we remark that the asymptotic expansions of resolvent $R_V(位^4)$ and the classifications of resonances are more involved than Schr枚dinger operator $-螖+V$ in dimension two. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2110.07154v3-abstract-full').style.display = 'none'; document.getElementById('2110.07154v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 31 July, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 October, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2021. </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">62 Pages. This is a final version which was published in JFA 2023</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2109.09581">arXiv:2109.09581</a> <span> [<a href="https://arxiv.org/pdf/2109.09581">pdf</a>, <a href="https://arxiv.org/ps/2109.09581">ps</a>, <a href="https://arxiv.org/format/2109.09581">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> </div> </div> <p class="title is-5 mathjax"> Topological structure of the space of composition operators on the Hardy space of Dirichlet series </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bayart%2C+F">Fr茅d茅ric Bayart</a>, <a href="/search/math?searchtype=author&query=Wang%2C+M">Maofa Wang</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xingxing Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2109.09581v1-abstract-short" style="display: inline;"> The aim of this paper is to study when two composition operators on the Hilbert space of Dirichlet series with square summable coefficients belong to the same component or when their difference is compact. As a corollary we show that if a linear combination of composition operators with polynomial symbols of degree at most 2 is compact, then each composition operator is compact. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2109.09581v1-abstract-full" style="display: none;"> The aim of this paper is to study when two composition operators on the Hilbert space of Dirichlet series with square summable coefficients belong to the same component or when their difference is compact. As a corollary we show that if a linear combination of composition operators with polynomial symbols of degree at most 2 is compact, then each composition operator is compact. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2109.09581v1-abstract-full').style.display = 'none'; document.getElementById('2109.09581v1-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 September, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2109.01497">arXiv:2109.01497</a> <span> [<a href="https://arxiv.org/pdf/2109.01497">pdf</a>, <a href="https://arxiv.org/ps/2109.01497">ps</a>, <a href="https://arxiv.org/format/2109.01497">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 for the multi-dimensional Borg--Levinson theorem of the biharmonic operator </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+P">Peijun Li</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiaohua Yao</a>, <a href="/search/math?searchtype=author&query=Zhao%2C+Y">Yue 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="2109.01497v2-abstract-short" style="display: inline;"> In this paper, we prove a conditional H枚lder stability estimate for the inverse spectral problem of the biharmonic operator. The proof employs the resolvent estimate and a Weyl-type law for the biharmonic operator which were obtained by the authors in \cite{LYZ}. This work extends nontrivially the result in \cite{stefanov} from the second order Schr枚dinger operator to the fourth order biharmonic o… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2109.01497v2-abstract-full').style.display = 'inline'; document.getElementById('2109.01497v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2109.01497v2-abstract-full" style="display: none;"> In this paper, we prove a conditional H枚lder stability estimate for the inverse spectral problem of the biharmonic operator. The proof employs the resolvent estimate and a Weyl-type law for the biharmonic operator which were obtained by the authors in \cite{LYZ}. This work extends nontrivially the result in \cite{stefanov} from the second order Schr枚dinger operator to the fourth order biharmonic operator. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2109.01497v2-abstract-full').style.display = 'none'; document.getElementById('2109.01497v2-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 January, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 September, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2021. </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">A detailed comparison is made with reference [8]; The structures of the proofs are improved; The presentation of the work including the introduction is modified; More relevant references are added</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2108.13257">arXiv:2108.13257</a> <span> [<a href="https://arxiv.org/pdf/2108.13257">pdf</a>, <a href="https://arxiv.org/format/2108.13257">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</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="Dynamical Systems">math.DS</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00220-022-04417-7">10.1007/s00220-022-04417-7 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> The spectrum of period-doubling Hamiltonian </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Liu%2C+Q">Qinghui Liu</a>, <a href="/search/math?searchtype=author&query=Qu%2C+Y">Yanhui Qu</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiao Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2108.13257v2-abstract-short" style="display: inline;"> In this paper, we show the following: the Hausdorff dimension of the spectrum of period-doubling Hamiltonian is bigger than $\log 伪/\log 4$, where $伪$ is the Golden number; there exists a dense uncountable subset of the spectrum such that for each energy in this set, the related trace orbit is unbounded, which is in contrast with a recent result of Carvalho (Nonlinearity 33, 2020); we give a compl… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2108.13257v2-abstract-full').style.display = 'inline'; document.getElementById('2108.13257v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2108.13257v2-abstract-full" style="display: none;"> In this paper, we show the following: the Hausdorff dimension of the spectrum of period-doubling Hamiltonian is bigger than $\log 伪/\log 4$, where $伪$ is the Golden number; there exists a dense uncountable subset of the spectrum such that for each energy in this set, the related trace orbit is unbounded, which is in contrast with a recent result of Carvalho (Nonlinearity 33, 2020); we give a complete characterization for the structure of gaps and the gap labelling of the spectrum. All of these results are consequences of an intrinsic coding of the spectrum we construct in this paper. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2108.13257v2-abstract-full').style.display = 'none'; document.getElementById('2108.13257v2-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 October, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 30 August, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2021. </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">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/2107.11019">arXiv:2107.11019</a> <span> [<a href="https://arxiv.org/pdf/2107.11019">pdf</a>, <a href="https://arxiv.org/ps/2107.11019">ps</a>, <a href="https://arxiv.org/format/2107.11019">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Artificial Intelligence">cs.AI</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Neural and Evolutionary Computing">cs.NE</span> </div> </div> <p class="title is-5 mathjax"> Generating Large-scale Dynamic Optimization Problem Instances Using the Generalized Moving Peaks Benchmark </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Omidvar%2C+M+N">Mohammad Nabi Omidvar</a>, <a href="/search/math?searchtype=author&query=Yazdani%2C+D">Danial Yazdani</a>, <a href="/search/math?searchtype=author&query=Branke%2C+J">Juergen Branke</a>, <a href="/search/math?searchtype=author&query=Li%2C+X">Xiaodong Li</a>, <a href="/search/math?searchtype=author&query=Yang%2C+S">Shengxiang Yang</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xin Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2107.11019v1-abstract-short" style="display: inline;"> This document describes the generalized moving peaks benchmark (GMPB) and how it can be used to generate problem instances for continuous large-scale dynamic optimization problems. It presents a set of 15 benchmark problems, the relevant source code, and a performance indicator, designed for comparative studies and competitions in large-scale dynamic optimization. Although its primary purpose is t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2107.11019v1-abstract-full').style.display = 'inline'; document.getElementById('2107.11019v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2107.11019v1-abstract-full" style="display: none;"> This document describes the generalized moving peaks benchmark (GMPB) and how it can be used to generate problem instances for continuous large-scale dynamic optimization problems. It presents a set of 15 benchmark problems, the relevant source code, and a performance indicator, designed for comparative studies and competitions in large-scale dynamic optimization. Although its primary purpose is to provide a coherent basis for running competitions, its generality allows the interested reader to use this document as a guide to design customized problem instances to investigate issues beyond the scope of the presented benchmark suite. To this end, we explain the modular structure of the GMPB and how its constituents can be assembled to form problem instances with a variety of controllable characteristics ranging from unimodal to highly multimodal, symmetric to highly asymmetric, smooth to highly irregular, and various degrees of variable interaction and ill-conditioning. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2107.11019v1-abstract-full').style.display = 'none'; document.getElementById('2107.11019v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 July, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2021. </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:2106.06174</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2106.15966">arXiv:2106.15966</a> <span> [<a href="https://arxiv.org/pdf/2106.15966">pdf</a>, <a href="https://arxiv.org/ps/2106.15966">ps</a>, <a href="https://arxiv.org/format/2106.15966">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> Decay estimates for bi-Schr枚dinger operators in dimension one </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Soffer%2C+A">Avy Soffer</a>, <a href="/search/math?searchtype=author&query=Wu%2C+Z">Zhao Wu</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiaohua Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2106.15966v4-abstract-short" style="display: inline;"> This paper is devoted to study the time decay estimates for bi-Schr枚dinger operators $H=螖^{2}+V(x)$ in dimension one with decaying potentials $V(x)$. We first deduce the asymptotic expansions of resolvent of $H$ at zero energy threshold without/with the presence of resonances, and then characterize these resonance spaces corresponding to different types of zero resonance in suitable weighted space… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2106.15966v4-abstract-full').style.display = 'inline'; document.getElementById('2106.15966v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2106.15966v4-abstract-full" style="display: none;"> This paper is devoted to study the time decay estimates for bi-Schr枚dinger operators $H=螖^{2}+V(x)$ in dimension one with decaying potentials $V(x)$. We first deduce the asymptotic expansions of resolvent of $H$ at zero energy threshold without/with the presence of resonances, and then characterize these resonance spaces corresponding to different types of zero resonance in suitable weighted spaces $L_s^2({\mathbf{R}})$. Next we use them to establish the sharp $L^1-L^\infty$ decay estimates of Schr枚dinger groups $e^{-itH}$ generated by bi-Schr枚dinger operators also with zero resonances. As a consequence, Strichartz estimates are obtained for the solution of fourth-order Schr枚dinger equations with potentials for initial data in $L^2({\mathbf{R}})$. In particular, it should be emphasized that the presence of zero resonances does not change the optimal time decay rate of $e^{-itH}$ in dimension one, except at requiring faster decay rate of the potential. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2106.15966v4-abstract-full').style.display = 'none'; document.getElementById('2106.15966v4-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 December, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 30 June, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">49 pages. This is a final version which will be published in Annales Henri Poincare</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2106.10597">arXiv:2106.10597</a> <span> [<a href="https://arxiv.org/pdf/2106.10597">pdf</a>, <a href="https://arxiv.org/ps/2106.10597">ps</a>, <a href="https://arxiv.org/format/2106.10597">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"> Direct and inverse problems for the Schr枚dinger operator in a three-dimensional planar waveguide </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+P">Peijun Li</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiaohua Yao</a>, <a href="/search/math?searchtype=author&query=Zhao%2C+Y">Yue 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="2106.10597v1-abstract-short" style="display: inline;"> In this paper, we study the meromorphic continuation of the resolvent for the Schr枚dinger operator in a three-dimensional planar waveguide. We prove the existence of a resonance-free region and an upper bound for the resolvent. As an application, the direct source problem is shown to have a unique solution under some appropriate assumptions. Moreover, an increasing stability is achieved for the in… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2106.10597v1-abstract-full').style.display = 'inline'; document.getElementById('2106.10597v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2106.10597v1-abstract-full" style="display: none;"> In this paper, we study the meromorphic continuation of the resolvent for the Schr枚dinger operator in a three-dimensional planar waveguide. We prove the existence of a resonance-free region and an upper bound for the resolvent. As an application, the direct source problem is shown to have a unique solution under some appropriate assumptions. Moreover, an increasing stability is achieved for the inverse source problem of the Schr枚dinger operator in the waveguide by using limited aperture Dirichlet data only at multiple frequencies. The stability estimate consists of the Lipschitz type data discrepancy and the high frequency tail of the source function, where the latter decreases as the upper bound of the frequency increases. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2106.10597v1-abstract-full').style.display = 'none'; document.getElementById('2106.10597v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 June, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2021. </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:2102.04631</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2106.07323">arXiv:2106.07323</a> <span> [<a href="https://arxiv.org/pdf/2106.07323">pdf</a>, <a href="https://arxiv.org/ps/2106.07323">ps</a>, <a href="https://arxiv.org/format/2106.07323">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="Neural and Evolutionary Computing">cs.NE</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/10.1109/TCYB.2022.3179378">10.1109/TCYB.2022.3179378 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Gridless Evolutionary Approach for Line Spectral Estimation with Unknown Model Order </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Yan%2C+B">Bai Yan</a>, <a href="/search/math?searchtype=author&query=Zhao%2C+Q">Qi Zhao</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+J">Jin Zhang</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+J+A">J. Andrew Zhang</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xin Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2106.07323v1-abstract-short" style="display: inline;"> Gridless methods show great superiority in line spectral estimation. These methods need to solve an atomic $l_0$ norm (i.e., the continuous analog of $l_0$ norm) minimization problem to estimate frequencies and model order. Since this problem is NP-hard to compute, relaxations of atomic $l_0$ norm, such as nuclear norm and reweighted atomic norm, have been employed for promoting sparsity. However,… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2106.07323v1-abstract-full').style.display = 'inline'; document.getElementById('2106.07323v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2106.07323v1-abstract-full" style="display: none;"> Gridless methods show great superiority in line spectral estimation. These methods need to solve an atomic $l_0$ norm (i.e., the continuous analog of $l_0$ norm) minimization problem to estimate frequencies and model order. Since this problem is NP-hard to compute, relaxations of atomic $l_0$ norm, such as nuclear norm and reweighted atomic norm, have been employed for promoting sparsity. However, the relaxations give rise to a resolution limit, subsequently leading to biased model order and convergence error. To overcome the above shortcomings of relaxation, we propose a novel idea of simultaneously estimating the frequencies and model order by means of the atomic $l_0$ norm. To accomplish this idea, we build a multiobjective optimization model. The measurment error and the atomic $l_0$ norm are taken as the two optimization objectives. The proposed model directly exploits the model order via the atomic $l_0$ norm, thus breaking the resolution limit. We further design a variable-length evolutionary algorithm to solve the proposed model, which includes two innovations. One is a variable-length coding and search strategy. It flexibly codes and interactively searches diverse solutions with different model orders. These solutions act as steppingstones that help fully exploring the variable and open-ended frequency search space and provide extensive potentials towards the optima. Another innovation is a model order pruning mechanism, which heuristically prunes less contributive frequencies within the solutions, thus significantly enhancing convergence and diversity. Simulation results confirm the superiority of our approach in both frequency estimation and model order selection. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2106.07323v1-abstract-full').style.display = 'none'; document.getElementById('2106.07323v1-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, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2021. </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 work has been submitted to the IEEE for possible publication</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> IEEE Transactions on Cybernetics, 2022, early access </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2103.00461">arXiv:2103.00461</a> <span> [<a href="https://arxiv.org/pdf/2103.00461">pdf</a>, <a href="https://arxiv.org/ps/2103.00461">ps</a>, <a href="https://arxiv.org/format/2103.00461">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 for an inverse source problem of the damped biharmonic plate equation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+P">Peijun Li</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiaohua Yao</a>, <a href="/search/math?searchtype=author&query=Zhao%2C+Y">Yue 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="2103.00461v1-abstract-short" style="display: inline;"> This paper is concerned with the stability of the inverse source problem for the damped biharmonic plate equation in three dimensions. The stability estimate consists of the Lipschitz type data discrepancy and the high frequency tail of the source function, where the latter decreases as the upper bound of the frequency increases. The stability also shows exponential dependence on the constant damp… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2103.00461v1-abstract-full').style.display = 'inline'; document.getElementById('2103.00461v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2103.00461v1-abstract-full" style="display: none;"> This paper is concerned with the stability of the inverse source problem for the damped biharmonic plate equation in three dimensions. The stability estimate consists of the Lipschitz type data discrepancy and the high frequency tail of the source function, where the latter decreases as the upper bound of the frequency increases. The stability also shows exponential dependence on the constant damping coefficient. The analysis employs Carleman estimates and time decay estimates for the damped plate wave equation to obtain an exact observability bound and depends on the study of the resonance-free region and an upper bound of the resolvent of the biharmonic operator with respect to the complex wavenumber. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2103.00461v1-abstract-full').style.display = 'none'; document.getElementById('2103.00461v1-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 February, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2102.07324">arXiv:2102.07324</a> <span> [<a href="https://arxiv.org/pdf/2102.07324">pdf</a>, <a href="https://arxiv.org/ps/2102.07324">ps</a>, <a href="https://arxiv.org/format/2102.07324">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> </div> <div 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/10.1088/1361-6544/ac355d">10.1088/1361-6544/ac355d <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Multifractal analysis in non-uniformly hyperbolic interval maps </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ma%2C+G">Guan-Zhong Ma</a>, <a href="/search/math?searchtype=author&query=Shen%2C+W">Wen-Qiang Shen</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiao Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2102.07324v1-abstract-short" style="display: inline;"> In this paper, we study the Hausdorff dimension of the generalized intrinsic level set with respect to the given ergodic meausre in a class of non-uniformly hyperbolic interval maps with finitely many branches. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2102.07324v1-abstract-full" style="display: none;"> In this paper, we study the Hausdorff dimension of the generalized intrinsic level set with respect to the given ergodic meausre in a class of non-uniformly hyperbolic interval maps with finitely many branches. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2102.07324v1-abstract-full').style.display = 'none'; document.getElementById('2102.07324v1-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 February, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 37B40; 28A80 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2102.04631">arXiv:2102.04631</a> <span> [<a href="https://arxiv.org/pdf/2102.04631">pdf</a>, <a href="https://arxiv.org/ps/2102.04631">ps</a>, <a href="https://arxiv.org/format/2102.04631">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 for an inverse source problem of the biharmonic operator </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+P">Peijun Li</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiaohua Yao</a>, <a href="/search/math?searchtype=author&query=Zhao%2C+Y">Yue 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="2102.04631v1-abstract-short" style="display: inline;"> In this paper, we study for the first time the stability of the inverse source problem for the biharmonic operator with a compactly supported potential in $\mathbb R^3$. Firstly, to connect the boundary data with the unknown source, we shall consider an eigenvalue problem for the bi-Schr$\ddot{\rm o}$dinger operator $螖^2 + V(x)$ on a ball which contains the support of the potential $V$. We prove a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2102.04631v1-abstract-full').style.display = 'inline'; document.getElementById('2102.04631v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2102.04631v1-abstract-full" style="display: none;"> In this paper, we study for the first time the stability of the inverse source problem for the biharmonic operator with a compactly supported potential in $\mathbb R^3$. Firstly, to connect the boundary data with the unknown source, we shall consider an eigenvalue problem for the bi-Schr$\ddot{\rm o}$dinger operator $螖^2 + V(x)$ on a ball which contains the support of the potential $V$. We prove a Weyl-type law for the upper bounds of spherical normal derivatives of both the eigenfunctions $蠁$ and their Laplacian $螖蠁$ corresponding to the bi-Schr$\ddot{\rm o}$dinger operator. This type of upper bounds was proved by Hassell and Tao for the Schr$\ddot{\rm o}$dinger operator. Secondly, we investigate the meromorphic continuation of the resolvent of the bi-Schr$\ddot{\rm o}$dinger operator and prove the existence of a resonance-free region and an estimate of $L^2_{\rm comp} - L^2_{\rm loc}$ type for the resolvent. As an application, we prove a bound of the analytic continuation of the data from the given data to the higher frequency data. Finally, we derive the stability estimate which consists of the Lipschitz type data discrepancy and the high frequency tail of the source function, where the latter decreases as the upper bound of the frequency increases. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2102.04631v1-abstract-full').style.display = 'none'; document.getElementById('2102.04631v1-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 February, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2012.14351">arXiv:2012.14351</a> <span> [<a href="https://arxiv.org/pdf/2012.14351">pdf</a>, <a href="https://arxiv.org/ps/2012.14351">ps</a>, <a href="https://arxiv.org/format/2012.14351">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"> Global existences and asymptotic behavior for semilinear heat equation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Soffer%2C+A">Avy Soffer</a>, <a href="/search/math?searchtype=author&query=Wu%2C+Y">Yifei Wu</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiaohua Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2012.14351v1-abstract-short" style="display: inline;"> In this paper, we consider the global Cauchy problem for the $L^2$-critical semilinear heat equations $ \partial_t h=螖h\pm |h|^{\frac4d}h, $ with $h(0,x)=h_0$, where $h$ is an unknown real function defined on $ \R^+\times\R^d$. In most of the studies on this subject, the initial data $h_0$ belongs to Lebesgue spaces $L^p(\R^d)$ for some $p\ge 2$ or to subcritical Sobolev space $H^{s}(\R^d)$ with… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2012.14351v1-abstract-full').style.display = 'inline'; document.getElementById('2012.14351v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2012.14351v1-abstract-full" style="display: none;"> In this paper, we consider the global Cauchy problem for the $L^2$-critical semilinear heat equations $ \partial_t h=螖h\pm |h|^{\frac4d}h, $ with $h(0,x)=h_0$, where $h$ is an unknown real function defined on $ \R^+\times\R^d$. In most of the studies on this subject, the initial data $h_0$ belongs to Lebesgue spaces $L^p(\R^d)$ for some $p\ge 2$ or to subcritical Sobolev space $H^{s}(\R^d)$ with $s>0$. {\it First,} we prove that there exists some positive constant $纬_0$ depending on $d$, such that the Cauchy problem is locally and globally well-posed for any initial data $h_0$ which is radial, supported away from the origin and in the negative Sobolev space $\dot H^{-纬_0}(\R^d)$. In particular, it leads to local and global existences of the solutions to Cauchy problem considered above for the initial data in a proper subspace of $L^p(\R^d)$ with some $p<2$. {\it Secondly,} the sharp asymptotic behavior of the solutions ( i.e. $L^2$-decay estimates ) as $t\to +\infty$ are obtained with arbitrary large initial data $h_0\in \dot H^{-纬_0}(\R^d)$ in the defocusing case and in the focusing case with suitably small initial data $h_0$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2012.14351v1-abstract-full').style.display = 'none'; document.getElementById('2012.14351v1-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, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2020. </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">Revised and expanded version of arXiv:1903.08316. It also includes new section on large time asymptotics</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35K05; 35B40; 35B65 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2012.07453">arXiv:2012.07453</a> <span> [<a href="https://arxiv.org/pdf/2012.07453">pdf</a>, <a href="https://arxiv.org/ps/2012.07453">ps</a>, <a href="https://arxiv.org/format/2012.07453">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> </div> </div> <p class="title is-5 mathjax"> Inequalities Concerning Maximum Modulus and Zeros of Random Entire Functions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+H">Hui Li</a>, <a href="/search/math?searchtype=author&query=Wang%2C+J">Jun Wang</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiao Yao</a>, <a href="/search/math?searchtype=author&query=Ye%2C+Z">Zhuan Ye</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="2012.07453v1-abstract-short" style="display: inline;"> Let $f_蠅(z)=\sum\limits_{j=0}^{\infty}蠂_j(蠅) a_j z^j$ be a random entire function, where $蠂_j(蠅)$ are independent and identically distributed random variables defined on a probability space $(惟, \mathcal{F}, 渭)$. In this paper, we first define a family of random entire functions, which includes Gaussian, Rademacher, Steinhaus entire functions. Then, we prove that, for almost all functions in the f… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2012.07453v1-abstract-full').style.display = 'inline'; document.getElementById('2012.07453v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2012.07453v1-abstract-full" style="display: none;"> Let $f_蠅(z)=\sum\limits_{j=0}^{\infty}蠂_j(蠅) a_j z^j$ be a random entire function, where $蠂_j(蠅)$ are independent and identically distributed random variables defined on a probability space $(惟, \mathcal{F}, 渭)$. In this paper, we first define a family of random entire functions, which includes Gaussian, Rademacher, Steinhaus entire functions. Then, we prove that, for almost all functions in the family and for any constant $C>1$, there exist a constant $r_0=r_0(蠅)$ and a set $E\subset [e, \infty)$ of finite logarithmic measure such that, for $r>r_0$ and $r\notin E$, $$ |\log M(r, f)- N(r,0, f_蠅)|\le (C/A)^{\frac1{B}}\log^{\frac1{B}}\log M(r,f) +\log\log M(r, f), \qquad a.s. $$ where $A, B$ are constants, $M(r, f)$ is the maximum modulus, and $N(r, 0, f)$ is the weighted counting-zero function of $f$. As a by-product of our main results, we prove Nevanlinna's second main theorem for random entire functions. Thus, the characteristic function of almost all functions in the family is bounded above by a weighed counting function, rather than by two weighted counting functions in the classical Nevanlinna theory. For instance, we show that, for almost all Gaussian entire functions $f_蠅$ and for any $蔚>0$, there is $r_0$ such that, for $r>r_0$, $$ T(r, f) \le N(r,0, f_蠅)+(\frac12+蔚) \log T(r, f). $$ <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2012.07453v1-abstract-full').style.display = 'none'; document.getElementById('2012.07453v1-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 December, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">18 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 30B20; 30D15; 60G99 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2005.11600">arXiv:2005.11600</a> <span> [<a href="https://arxiv.org/pdf/2005.11600">pdf</a>, <a href="https://arxiv.org/format/2005.11600">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="Neural and Evolutionary Computing">cs.NE</span> </div> </div> <p class="title is-5 mathjax"> Knee Point Identification Based on Trade-Off Utility </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Li%2C+K">Ke Li</a>, <a href="/search/math?searchtype=author&query=Nie%2C+H">Haifeng Nie</a>, <a href="/search/math?searchtype=author&query=Gao%2C+H">Huifu Gao</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xin Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2005.11600v1-abstract-short" style="display: inline;"> Knee points, characterised as their smallest trade-off loss at all objectives, are attractive to decision makers in multi-criterion decision-making. In contrast, other Pareto-optimal solutions are less attractive since a small improvement on one objective can lead to a significant degradation on at least one of the other objectives. In this paper, we propose a simple and effective knee point ident… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2005.11600v1-abstract-full').style.display = 'inline'; document.getElementById('2005.11600v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2005.11600v1-abstract-full" style="display: none;"> Knee points, characterised as their smallest trade-off loss at all objectives, are attractive to decision makers in multi-criterion decision-making. In contrast, other Pareto-optimal solutions are less attractive since a small improvement on one objective can lead to a significant degradation on at least one of the other objectives. In this paper, we propose a simple and effective knee point identification method based on trade-off utility, dubbed KPITU, to help decision makers identify knee points from a given set of trade-off solutions. The basic idea of KPITU is to sequentially validate whether a solution is a knee point or not by comparing its trade-off utility with others within its neighbourhood. In particular, a solution is a knee point if and only if it has the best trade-off utility among its neighbours. Moreover, we implement a GPU version of KPITU that carries out the knee point identification in a parallel manner. This GPU version reduces the worst-case complexity from quadratic to linear. To validate the effectiveness of KPITU, we compare its performance with five state-of-the-art knee point identification methods on 134 test problem instances. Empirical results fully demonstrate the outstanding performance of KPITU especially on problems with many local knee points. At the end, we further validate the usefulness of KPITU for guiding EMO algorithms to search for knee points on the fly during the evolutionary process. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2005.11600v1-abstract-full').style.display = 'none'; document.getElementById('2005.11600v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 May, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2020. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2004.10115">arXiv:2004.10115</a> <span> [<a href="https://arxiv.org/pdf/2004.10115">pdf</a>, <a href="https://arxiv.org/ps/2004.10115">ps</a>, <a href="https://arxiv.org/format/2004.10115">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"> Global Kato smoothing and Strichartz estimates for higher-order Schr枚dinger operators with rough decay potentials </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Mizutani%2C+H">Haruya Mizutani</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiaohua Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2004.10115v2-abstract-short" style="display: inline;"> Let $ H = (-螖)^m + V $ be a higher-order elliptic operator on $ L^2(\mathbb{R}^n) $, where $ V $ is a general bounded decaying potential. This paper focuses on the global Kato smoothing and Strichartz estimates for solutions to Schr枚dinger-type equation associated with $ H $. In particular, we first establish sharp global Kato smoothing estimates for $ e^{itH} $, based on uniform resolve… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2004.10115v2-abstract-full').style.display = 'inline'; document.getElementById('2004.10115v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2004.10115v2-abstract-full" style="display: none;"> Let $ H = (-螖)^m + V $ be a higher-order elliptic operator on $ L^2(\mathbb{R}^n) $, where $ V $ is a general bounded decaying potential. This paper focuses on the global Kato smoothing and Strichartz estimates for solutions to Schr枚dinger-type equation associated with $ H $. In particular, we first establish sharp global Kato smoothing estimates for $ e^{itH} $, based on uniform resolvent estimates of Kato-Yajima type for the absolutely continuous part of $ H $. As a consequence, we also obtain optimal local decay estimates. Using these local decay estimates, we then prove the full set of Strichartz estimates, including the endpoint case. Notably, we derive Strichartz estimates with sharp smoothing effects for higher-order cases with rough potentials, which are applicable to the study of nonlinear higher-order Schr枚dinger equations. Finally, we introduce new uniform Sobolev estimates of the Kenig-Ruiz-Sogge type, incorporating an additional derivative term, which are crucial for establishing the sharp Kato smoothing estimates. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2004.10115v2-abstract-full').style.display = 'none'; document.getElementById('2004.10115v2-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">v1</span> submitted 21 April, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2020. </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. To appear in Rev. Math. Phys., 2024</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2002.12737">arXiv:2002.12737</a> <span> [<a href="https://arxiv.org/pdf/2002.12737">pdf</a>, <a href="https://arxiv.org/ps/2002.12737">ps</a>, <a href="https://arxiv.org/format/2002.12737">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Metric Geometry">math.MG</span> </div> </div> <p class="title is-5 mathjax"> A Generalized $蟺_2$-Diffeomorphism Finiteness Theorem </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Rong%2C+X">Xiaochun Rong</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xuchao Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2002.12737v1-abstract-short" style="display: inline;"> The $蟺_2$-diffeomorphism finiteness result (\cite{FR1,2}, \cite{PT}) asserts that the diffeomorphic types of compact $n$-manifolds $M$ with vanishing first and second homotopy groups can be bounded above in terms of $n$, and upper bounds on the absolute value of sectional curvature and diameter of $M$. In this paper, we will generalize this $蟺_2$-diffeomorphism finiteness by removing the condition… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.12737v1-abstract-full').style.display = 'inline'; document.getElementById('2002.12737v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2002.12737v1-abstract-full" style="display: none;"> The $蟺_2$-diffeomorphism finiteness result (\cite{FR1,2}, \cite{PT}) asserts that the diffeomorphic types of compact $n$-manifolds $M$ with vanishing first and second homotopy groups can be bounded above in terms of $n$, and upper bounds on the absolute value of sectional curvature and diameter of $M$. In this paper, we will generalize this $蟺_2$-diffeomorphism finiteness by removing the condition that $蟺_1(M)=0$ and asserting the diffeomorphism finiteness on the Riemannian universal cover of $M$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.12737v1-abstract-full').style.display = 'none'; document.getElementById('2002.12737v1-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 February, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2020. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2002.08644">arXiv:2002.08644</a> <span> [<a href="https://arxiv.org/pdf/2002.08644">pdf</a>, <a href="https://arxiv.org/ps/2002.08644">ps</a>, <a href="https://arxiv.org/format/2002.08644">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> </div> </div> <p class="title is-5 mathjax"> On continuous extension of conformal homeomorphisms of infinitely connected planar domains </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Luo%2C+J">Jun Luo</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiao-Ting Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2002.08644v2-abstract-short" style="display: inline;"> We consider conformal homeomorphisms $\varphi$ of generalized Jordan domains $U$ onto planar domains $惟$ %, possibly {\bf infinitely connected}, that satisfy both of the next two conditions: (1) at most countably many boundary components of $惟$ are non-degenerate and their diameters have a finite sum; (2) either the degenerate boundary components of $惟$ or those of $U$ form a set of sigma-finite l… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.08644v2-abstract-full').style.display = 'inline'; document.getElementById('2002.08644v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2002.08644v2-abstract-full" style="display: none;"> We consider conformal homeomorphisms $\varphi$ of generalized Jordan domains $U$ onto planar domains $惟$ %, possibly {\bf infinitely connected}, that satisfy both of the next two conditions: (1) at most countably many boundary components of $惟$ are non-degenerate and their diameters have a finite sum; (2) either the degenerate boundary components of $惟$ or those of $U$ form a set of sigma-finite linear measure. We prove that $\varphi$ continuously extends to the closure of $U$ if and only if every boundary component of $惟$ is locally connected. This generalizes the Carath茅odory's Continuity Theorem and leads us to a new generalization of the well known Osgood-Taylor-Carath茅odory Theorem. There are three issues that are noteworthy. Firstly, none of the above conditions (1) and (2) can be removed. Secondly, %no further requirements concerning $U$ or $惟$ are needed. So our results remain valid for non-cofat domains and do not follow from the extension results, of a similar nature, that are obtained in very recent studies on the conformal rigidity of circle domains. Finally, when $\varphi$ does extend continuously to the closure of $U$, the boundary of $惟$ is a Peano compactum. Therefore, we also show that the following properties are equivalent for any planar domain $惟$: (1) The boundary of $惟$ is a Peano compactum. (2) $惟$ has Property S. (3) Every point on the boundary of $惟$ is locally accessible. (4) Every point on the boundary of $惟$ is locally sequentially accessible. (5) $惟$ is finitely connected at the boundary. (6) The completion of $惟$ under the Mazurkiewicz distance is compact. \noindent This provides new generalizations of earlier partial results that are restricted to special cases, when additional assumptions on the topology of $U$ or its boundary are required. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.08644v2-abstract-full').style.display = 'none'; document.getElementById('2002.08644v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 March, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 20 February, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">40 pages, 6 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary 30D40; 54C20; Secondary 54F15 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Trans Amer Math Soc, 2022 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2002.08638">arXiv:2002.08638</a> <span> [<a href="https://arxiv.org/pdf/2002.08638">pdf</a>, <a href="https://arxiv.org/ps/2002.08638">ps</a>, <a href="https://arxiv.org/format/2002.08638">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Topology">math.GN</span> </div> </div> <p class="title is-5 mathjax"> On Lambda Function and a Quantification of Torhorst Theorem </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Feng%2C+L">Li Feng</a>, <a href="/search/math?searchtype=author&query=Luo%2C+J">Jun Luo</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiao-Ting Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2002.08638v2-abstract-short" style="display: inline;"> To any compact $K\subset\hat{\mathbb{C}}$ we associate a map $位_K: \hat{\mathbb{C}}\rightarrow\mathbb{N}\cup\{\infty\}$ -- the lambda function of $K$ -- such that a planar continuum $K$ is locally connected if and only if $螞_K(x)\equiv0$. We establish basic methods of determining the lambda function $位_K$ for specific compacta $K\subset\hat{\mathbb{C}}$, including a gluing lemma for lambda functio… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.08638v2-abstract-full').style.display = 'inline'; document.getElementById('2002.08638v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2002.08638v2-abstract-full" style="display: none;"> To any compact $K\subset\hat{\mathbb{C}}$ we associate a map $位_K: \hat{\mathbb{C}}\rightarrow\mathbb{N}\cup\{\infty\}$ -- the lambda function of $K$ -- such that a planar continuum $K$ is locally connected if and only if $螞_K(x)\equiv0$. We establish basic methods of determining the lambda function $位_K$ for specific compacta $K\subset\hat{\mathbb{C}}$, including a gluing lemma for lambda functions and some inequalities. One of these inequalities comes from an interplay between the topological difficulty of a planar compactum $K$ and that of a sub-compactum $L\subset K$, lying on the boundary of a component of $\hat{\mathbb{C}}\setminus K$. It generalizes and quantifies the result of Torhorst Theorem, a fundamental result from plane topology. We also find three conditions under which this inequality is actually an equality. Under one of these conditions, this equality provides a quantitative version for Whyburn's Theorem, which is a partial converse to Torhorst Theorem. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.08638v2-abstract-full').style.display = 'none'; document.getElementById('2002.08638v2-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 April, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 20 February, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary 30D40; Secondary 30C35 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2002.02163">arXiv:2002.02163</a> <span> [<a href="https://arxiv.org/pdf/2002.02163">pdf</a>, <a href="https://arxiv.org/ps/2002.02163">ps</a>, <a href="https://arxiv.org/format/2002.02163">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 class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00220-021-04229-1">10.1007/s00220-021-04229-1 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Kato smoothing, Strichartz and uniform Sobolev estimates for fractional operators with sharp Hardy potentials </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Mizutani%2C+H">Haruya Mizutani</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiaohua Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2002.02163v6-abstract-short" style="display: inline;"> Let $0<蟽<n/2$ and $H=(-螖)^蟽+V(x)$ be Schr枚dinger type operators on $\mathbb R^n$ with a class of scaling-critical potentials $V(x)$, which include the Hardy potential $a|x|^{-2蟽}$ with a sharp coupling constant $a\ge -C_{蟽,n}$ ($C_{蟽,n}$ is the best constant of Hardy's inequality of order $蟽$). In the present paper we consider several sharp global estimates for the resolvent and the solution to th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.02163v6-abstract-full').style.display = 'inline'; document.getElementById('2002.02163v6-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2002.02163v6-abstract-full" style="display: none;"> Let $0<蟽<n/2$ and $H=(-螖)^蟽+V(x)$ be Schr枚dinger type operators on $\mathbb R^n$ with a class of scaling-critical potentials $V(x)$, which include the Hardy potential $a|x|^{-2蟽}$ with a sharp coupling constant $a\ge -C_{蟽,n}$ ($C_{蟽,n}$ is the best constant of Hardy's inequality of order $蟽$). In the present paper we consider several sharp global estimates for the resolvent and the solution to the time-dependent Schr枚dinger equation associated with $H$. In the case of the subcritical coupling constant $a>-C_{蟽,n}$, we first prove {\it uniform resolvent estimates} of Kato--Yajima type for all $0<蟽<n/2$, which turn out to be equivalent to {\it Kato smoothing estimates} for the Cauchy problem. We then establish {\it Strichartz estimates} for $蟽>1/2$ and {\it uniform Sobolev estimates} of Kenig--Ruiz--Sogge type for $蟽\ge n/(n+1)$. These extend the same properties for the Schr枚dinger operator with the inverse-square potential to the higher-order and fractional cases. Moreover, we also obtain {\it improved Strichartz estimates with a gain of regularities} for general initial data if $1<蟽<n/2$ and for radially symmetric data if $n/(2n-1)<蟽\le1$, which extends the corresponding results for the free evolution to the case with Hardy potentials. These arguments can be further applied to a large class of higher-order inhomogeneous elliptic operators and even to certain long-range metric perturbations of the Laplace operator. Finally, in the critical coupling constant case (i.e. $a=-C_{蟽,n}$), we show that the same results as in the subcritical case still hold for functions orthogonal to radial functions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.02163v6-abstract-full').style.display = 'none'; document.getElementById('2002.02163v6-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 September, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 February, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">44 pages, 1 figure; revised version</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1904.12275">arXiv:1904.12275</a> <span> [<a href="https://arxiv.org/pdf/1904.12275">pdf</a>, <a href="https://arxiv.org/ps/1904.12275">ps</a>, <a href="https://arxiv.org/format/1904.12275">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> </div> </div> <p class="title is-5 mathjax"> Decay estimates for higher order elliptic operators </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Feng%2C+H">Hongliang Feng</a>, <a href="/search/math?searchtype=author&query=Soffer%2C+A">Avy Soffer</a>, <a href="/search/math?searchtype=author&query=Wu%2C+Z">Zhao Wu</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiaohua Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1904.12275v2-abstract-short" style="display: inline;"> This paper is mainly devoted to study time decay estimates of the higher-order Schr枚dinger type operator $H=(-螖)^{m}+V(x)$ in $\mathbf{R}^{n}$ for $n>2m$ and $m\in\mathbf{N}$. For certain decay potentials $V(x)$, we first derive the asymptotic expansions of resolvent $R_{V}(z)$ near zero threshold with the presence of zero resonance or zero eigenvalue, as well identify the resonance space for each… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1904.12275v2-abstract-full').style.display = 'inline'; document.getElementById('1904.12275v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1904.12275v2-abstract-full" style="display: none;"> This paper is mainly devoted to study time decay estimates of the higher-order Schr枚dinger type operator $H=(-螖)^{m}+V(x)$ in $\mathbf{R}^{n}$ for $n>2m$ and $m\in\mathbf{N}$. For certain decay potentials $V(x)$, we first derive the asymptotic expansions of resolvent $R_{V}(z)$ near zero threshold with the presence of zero resonance or zero eigenvalue, as well identify the resonance space for each kind of zero resonance which displays different effects on time decay rate. Then we establish Kato-Jensen type estimates and local decay estimates for higher order Schr枚dinger propagator $e^{-itH}$ in the presence of zero resonance or zero eigenvalue. As a consequence, the endpoint Strichartz estimate and $L^{p}$-decay estimates can also be obtained. Finally, by a virial argument, a criterion on the absence of positive embedded eigenvalues is given for $(-螖)^{m}+V(x)$ with a repulsive potential. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1904.12275v2-abstract-full').style.display = 'none'; document.getElementById('1904.12275v2-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 September, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 April, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2019. </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">55 pages. Welcome any comments! To appear in Transactions of the American Mathematical Society</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1903.08316">arXiv:1903.08316</a> <span> [<a href="https://arxiv.org/pdf/1903.08316">pdf</a>, <a href="https://arxiv.org/ps/1903.08316">ps</a>, <a href="https://arxiv.org/format/1903.08316">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> Global rough solution for $L^2$-critical semilinear heat equation in the negative Sobolev space </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Soffer%2C+A">Avy Soffer</a>, <a href="/search/math?searchtype=author&query=Wu%2C+Y">Yifei Wu</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiaohua Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1903.08316v1-abstract-short" style="display: inline;"> In this paper, we consider the Cauchy global problem for the $L^2$-critical semilinear heat equations $\partial_t h=螖h\pm |h|^{\frac4d}h, $ with $h(0,x)=h_0$, where $h$ is an unknown real function defined on $ \R^+\times\R^d$. In most of the studies on this subject, the initial data $h_0$ belongs to Lebesgue spaces $L^p(\R^d)$ for some $p\ge 2$ or to subcritical Sobolev space $H^{s}(\R^d)$ with… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1903.08316v1-abstract-full').style.display = 'inline'; document.getElementById('1903.08316v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1903.08316v1-abstract-full" style="display: none;"> In this paper, we consider the Cauchy global problem for the $L^2$-critical semilinear heat equations $\partial_t h=螖h\pm |h|^{\frac4d}h, $ with $h(0,x)=h_0$, where $h$ is an unknown real function defined on $ \R^+\times\R^d$. In most of the studies on this subject, the initial data $h_0$ belongs to Lebesgue spaces $L^p(\R^d)$ for some $p\ge 2$ or to subcritical Sobolev space $H^{s}(\R^d)$ with $s>0$. We here prove that there exists some positive constant $\varepsilon_0$ depending on $d$, such that the Cauchy problem is locally and globally well-posed for any initial data $h_0$ which is radial, supported away from origin and in the negative Sobolev space $\dot H^{-\varepsilon_0}(\R^d)$ including $L^p(\R^d)$ with certain $p<2$ as subspace. Furthermore, unconditional uniqueness, and $L^2$-estimate both as time $t\to0$ and $t\to +\infty$ were considered. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1903.08316v1-abstract-full').style.display = 'none'; document.getElementById('1903.08316v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 March, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">16 pages. Any 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/1902.10973">arXiv:1902.10973</a> <span> [<a href="https://arxiv.org/pdf/1902.10973">pdf</a>, <a href="https://arxiv.org/ps/1902.10973">ps</a>, <a href="https://arxiv.org/format/1902.10973">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Metric Geometry">math.MG</span> </div> <div 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/10.1142/S0219199721500486">10.1142/S0219199721500486 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Margulis lemma and Hurewicz fibration Theorem on Alexandrov spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Xu%2C+S">Shicheng Xu</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xuchao Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1902.10973v2-abstract-short" style="display: inline;"> We prove the generalized Margulis lemma with a uniform index bound on an Alexandrov $n$-space $X$ with curvature bounded below, i.e., small loops at $p\in X$ generate a subgroup of the fundamental group of unit ball $B_1(p)$ that contains a nilpotent subgroup of index $\le w(n)$, where $w(n)$ is a constant depending only on the dimension $n$. The proof is based on the main ideas of V.~Kapovitch, A… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1902.10973v2-abstract-full').style.display = 'inline'; document.getElementById('1902.10973v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1902.10973v2-abstract-full" style="display: none;"> We prove the generalized Margulis lemma with a uniform index bound on an Alexandrov $n$-space $X$ with curvature bounded below, i.e., small loops at $p\in X$ generate a subgroup of the fundamental group of unit ball $B_1(p)$ that contains a nilpotent subgroup of index $\le w(n)$, where $w(n)$ is a constant depending only on the dimension $n$. The proof is based on the main ideas of V.~Kapovitch, A.~Petrunin, and W.~Tuschmann, and the following results: (1) We prove that any regular almost Lipschitz submersion constructed by Yamaguchi on a collapsed Alexandrov space with curvature bounded below is a Hurewicz fibration. We also prove that such fibration is uniquely determined up to a homotopy equivalence. (2) We give a detailed proof on the gradient push, improving the universal pushing time bound given by V.~Kapovitch, A.~Petrunin, and W.~Tuschmann, and justifying in a specific way that the gradient push between regular points can always keep away from extremal subsets. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1902.10973v2-abstract-full').style.display = 'none'; document.getElementById('1902.10973v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 31 March, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 February, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 53C23 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1812.00223">arXiv:1812.00223</a> <span> [<a href="https://arxiv.org/pdf/1812.00223">pdf</a>, <a href="https://arxiv.org/ps/1812.00223">ps</a>, <a href="https://arxiv.org/format/1812.00223">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"> Time Asymptotic expansions of solution for fourth-order Schr枚dinger equation with zero resonance or eigenvalue </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Feng%2C+H">Hongliang Feng</a>, <a href="/search/math?searchtype=author&query=Wu%2C+Z">Zhao Wu</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiaohua Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1812.00223v2-abstract-short" style="display: inline;"> In this paper, we first deduce the asymptotic expansions of resolvent of $H=(-螖)^2+V$ with the presence of resonance or eigenvalue at the degenerate zero threshold for $d\geq5$. In particular, we identify these resonance spaces for full kinds of zero resonances. As a consequence, we then establish the {\it time asymptotic expansions} and {\it Kato-Jensen estimates} for the solution of fourth-order… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1812.00223v2-abstract-full').style.display = 'inline'; document.getElementById('1812.00223v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1812.00223v2-abstract-full" style="display: none;"> In this paper, we first deduce the asymptotic expansions of resolvent of $H=(-螖)^2+V$ with the presence of resonance or eigenvalue at the degenerate zero threshold for $d\geq5$. In particular, we identify these resonance spaces for full kinds of zero resonances. As a consequence, we then establish the {\it time asymptotic expansions} and {\it Kato-Jensen estimates} for the solution of fourth-order Schr枚dinger equation under the presence of zero resonance or eigenvalue. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1812.00223v2-abstract-full').style.display = 'none'; document.getElementById('1812.00223v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 March, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 1 December, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2018. </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. This is an updated version, where more backgrounds and comments are added</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1811.07578">arXiv:1811.07578</a> <span> [<a href="https://arxiv.org/pdf/1811.07578">pdf</a>, <a href="https://arxiv.org/ps/1811.07578">ps</a>, <a href="https://arxiv.org/format/1811.07578">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Dynamics of the focusing 3D cubic NLS with slowly decaying potential </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Guo%2C+Q">Qing Guo</a>, <a href="/search/math?searchtype=author&query=Wang%2C+H">Hua Wang</a>, <a href="/search/math?searchtype=author&query=Yao%2C+X">Xiaohua Yao</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1811.07578v3-abstract-short" style="display: inline;"> In this paper, we consider a 3d cubic focusing nonlinear Schr枚dinger equation (NLS) with slowing decaying potentials. Adopting the variational method of Ibrahim-Masmoudi-Nakanishi \cite{IMN}, we obtain a condition for scattering. It is actually sharp in some sense since the solution will blow up if it's false. The proof of blow-up part relies on the method of Du-Wu-Zhang \cite{DWZ} </span> <span class="abstract-full has-text-grey-dark mathjax" id="1811.07578v3-abstract-full" style="display: none;"> In this paper, we consider a 3d cubic focusing nonlinear Schr枚dinger equation (NLS) with slowing decaying potentials. Adopting the variational method of Ibrahim-Masmoudi-Nakanishi \cite{IMN}, we obtain a condition for scattering. It is actually sharp in some sense since the solution will blow up if it's false. The proof of blow-up part relies on the method of Du-Wu-Zhang \cite{DWZ} <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1811.07578v3-abstract-full').style.display = 'none'; document.getElementById('1811.07578v3-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 December, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 November, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2018. </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, This is a new version where the proof of Lemma 2.1 was added</span> </p> </li> </ol> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> <a href="" 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