{"title":"On the isoperimetric Riemannian Penrose inequality","authors":"Luca Benatti, Mattia Fogagnolo, Lorenzo Mazzieri","doi":"10.1002/cpa.22239","DOIUrl":"10.1002/cpa.22239","url":null,"abstract":"<p>We prove that the Riemannian Penrose inequality holds for asymptotically flat 3-manifolds with nonnegative scalar curvature and connected horizon boundary, provided the optimal decay assumptions are met, which result in the <span></span><math>\u0000 <semantics>\u0000 <mo>ADM</mo>\u0000 <annotation>$operatorname{ADM}$</annotation>\u0000 </semantics></math> mass being a well-defined geometric invariant. Our proof builds on a novel interplay between the Hawking mass and a potential-theoretic version of it, recently introduced by Agostiniani, Oronzio, and the third named author. As a consequence, we establish the equality between <span></span><math>\u0000 <semantics>\u0000 <mo>ADM</mo>\u0000 <annotation>$operatorname{ADM}$</annotation>\u0000 </semantics></math> mass and Huisken's isoperimetric mass under the above sharp assumptions. Moreover, we establish a Riemannian Penrose inequality in terms of the isoperimetric mass on any 3-manifold with nonnegative scalar curvature, connected horizon boundary, and which supports a well-posed notion of weak inverse mean curvature flow (IMCF). In particular, such isoperimetric Riemannian Penrose inequality does not require the asymptotic flatness of the manifold. The argument is based on a new asymptotic comparison result involving Huisken's isoperimetric mass and the Hawking mass.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 5","pages":"1042-1085"},"PeriodicalIF":3.1,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22239","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142788421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yifan Chen, Ethan N. Epperly, Joel A. Tropp, Robert J. Webber
{"title":"Randomly pivoted Cholesky: Practical approximation of a kernel matrix with few entry evaluations","authors":"Yifan Chen, Ethan N. Epperly, Joel A. Tropp, Robert J. Webber","doi":"10.1002/cpa.22234","DOIUrl":"10.1002/cpa.22234","url":null,"abstract":"<p>The randomly pivoted Cholesky algorithm (<span>RPCholesky</span>) computes a factorized rank-<span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math> approximation of an <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>N</mi>\u0000 <mo>×</mo>\u0000 <mi>N</mi>\u0000 </mrow>\u0000 <annotation>$N times N$</annotation>\u0000 </semantics></math> positive-semidefinite (psd) matrix. <span>RPCholesky</span> requires only <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>k</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 <mi>N</mi>\u0000 </mrow>\u0000 <annotation>$(k + 1)N$</annotation>\u0000 </semantics></math> entry evaluations and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>O</mi>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>k</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mi>N</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathcal {O}(k^2 N)$</annotation>\u0000 </semantics></math> additional arithmetic operations, and it can be implemented with just a few lines of code. The method is particularly useful for approximating a kernel matrix. This paper offers a thorough new investigation of the empirical and theoretical behavior of this fundamental algorithm. For matrix approximation problems that arise in scientific machine learning, experiments show that <span>RPCholesky</span> matches or beats the performance of alternative algorithms. Moreover, <span>RPCholesky</span> provably returns low-rank approximations that are nearly optimal. The simplicity, effectiveness, and robustness of <span>RPCholesky</span> strongly support its use in scientific computing and machine learning applications.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 5","pages":"995-1041"},"PeriodicalIF":3.1,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22234","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142776401","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Hydrodynamic large deviations of TASEP","authors":"Jeremy Quastel, Li-Cheng Tsai","doi":"10.1002/cpa.22233","DOIUrl":"10.1002/cpa.22233","url":null,"abstract":"<p>We consider the large deviations from the hydrodynamic limit of the Totally Asymmetric Simple Exclusion Process (TASEP). This problem was studied by Jensen and Varadhan and was shown to be related to entropy production in the inviscid Burgers equation. Here we prove the full large deviation principle. Our method relies on the explicit formula of Matetski, Quastel, and Remenik for the transition probabilities of the TASEP.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 5","pages":"913-994"},"PeriodicalIF":3.1,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142690704","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the derivation of the homogeneous kinetic wave equation","authors":"Charles Collot, Pierre Germain","doi":"10.1002/cpa.22232","DOIUrl":"10.1002/cpa.22232","url":null,"abstract":"<p>The nonlinear Schrödinger equation in the weakly nonlinear regime with random Gaussian fields as initial data is considered. The problem is set on the torus in any dimension greater than two. A conjecture in statistical physics is that there exists a kinetic time scale depending on the frequency localization of the data and on the strength of the nonlinearity, on which the expectation of the squares of moduli of Fourier modes evolve according to an effective equation: the so-called kinetic wave equation. When the kinetic time for our setup is 1, we prove this conjecture up to an arbitrarily small polynomial loss. When the kinetic time is larger than 1, we obtain its validity on a more restricted time scale. The key idea of the proof is the use of Feynman interaction diagrams both in the construction of an approximate solution and in the study of its nonlinear stability. We perform a truncated series expansion in the initial data, and obtain bounds in average in various function spaces for its elements. The linearized dynamics then involves a linear Schrödinger equation with a corresponding random potential whose operator norm in Bourgain spaces we are able to estimate on average. This gives a new approach for the analysis of nonlinear wave equations out of equilibrium, and gives hope that refinements of the method could help settle the conjecture.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 4","pages":"856-909"},"PeriodicalIF":3.1,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142684236","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the stability of Runge–Kutta methods for arbitrarily large systems of ODEs","authors":"Eitan Tadmor","doi":"10.1002/cpa.22238","DOIUrl":"10.1002/cpa.22238","url":null,"abstract":"<p>We prove that Runge–Kutta (RK) methods for numerical integration of arbitrarily large systems of Ordinary Differential Equations are linearly stable. Standard stability arguments—based on spectral analysis, resolvent condition or strong stability, fail to secure the stability of RK methods for arbitrarily large systems. We explain the failure of different approaches, offer a new stability theory based on the numerical range of the underlying large matrices involved in such systems, and demonstrate its application with concrete examples of RK stability for hyperbolic methods of lines.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 4","pages":"821-855"},"PeriodicalIF":3.1,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22238","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142678569","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The \u0000 \u0000 α\u0000 $alpha$\u0000 -SQG patch problem is illposed in \u0000 \u0000 \u0000 C\u0000 \u0000 2\u0000 ,\u0000 β\u0000 \u0000 \u0000 $C^{2,beta }$\u0000 and \u0000 \u0000 \u0000 W\u0000 \u0000 2\u0000 ,\u0000 p\u0000 \u0000 \u0000 $W^{2,p}$","authors":"Alexander Kiselev, Xiaoyutao Luo","doi":"10.1002/cpa.22236","DOIUrl":"10.1002/cpa.22236","url":null,"abstract":"<p>We consider the patch problem for the <span></span><math>\u0000 <semantics>\u0000 <mi>α</mi>\u0000 <annotation>$alpha$</annotation>\u0000 </semantics></math>-(surface quasi-geostrophic) SQG system with the values <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>α</mi>\u0000 <mo>=</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$alpha =0$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>α</mi>\u0000 <mo>=</mo>\u0000 <mfrac>\u0000 <mn>1</mn>\u0000 <mn>2</mn>\u0000 </mfrac>\u0000 </mrow>\u0000 <annotation>$alpha = frac{1}{2}$</annotation>\u0000 </semantics></math> being the 2D Euler and the SQG equations respectively. It is well-known that the Euler patches are globally wellposed in non-endpoint <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>C</mi>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 <mo>,</mo>\u0000 <mi>β</mi>\u0000 </mrow>\u0000 </msup>\u0000 <annotation>$C^{k,beta }$</annotation>\u0000 </semantics></math> Hölder spaces, as well as in <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>W</mi>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mo>,</mo>\u0000 <mi>p</mi>\u0000 </mrow>\u0000 </msup>\u0000 <annotation>$W^{2,p}$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo><</mo>\u0000 <mi>p</mi>\u0000 <mo><</mo>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 <annotation>$1<p<infty$</annotation>\u0000 </semantics></math> spaces. In stark contrast to the Euler case, we prove that for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>0</mn>\u0000 <mo><</mo>\u0000 <mi>α</mi>\u0000 <mo><</mo>\u0000 <mfrac>\u0000 <mn>1</mn>\u0000 <mn>2</mn>\u0000 </mfrac>\u0000 </mrow>\u0000 <annotation>$0<alpha < frac{1}{2}$</annotation>\u0000 </semantics></math>, the <span></span><math>\u0000 <semantics>\u0000 <mi>α</mi>\u0000 <annotation>$alpha$</annotation>\u0000 </semantics></math>-SQG patch problem is strongly illposed in <i>every</i> <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>C</mi>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mo>,</mo>\u0000 <mi>β</mi>\u0000 </mrow","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"78 4","pages":"742-820"},"PeriodicalIF":3.1,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142645950","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}