Communications on Pure and Applied Mathematics最新文献

{"title":"Directed mean curvature flow in noisy environment","authors":"Andris Gerasimovičs,&nbsp;Martin Hairer,&nbsp;Konstantin Matetski","doi":"10.1002/cpa.22158","DOIUrl":"10.1002/cpa.22158","url":null,"abstract":"<p>We consider the directed mean curvature flow on the plane in a weak Gaussian random environment. We prove that, when started from a sufficiently flat initial condition, a rescaled and recentred solution converges to the Cole–Hopf solution of the KPZ equation. This result follows from the analysis of a more general system of nonlinear SPDEs driven by inhomogeneous noises, using the theory of regularity structures. However, due to inhomogeneity of the noise, the “black box” result developed in the series of works cannot be applied directly and requires significant extension to infinite-dimensional regularity structures. Analysis of this general system of SPDEs gives two more interesting results. First, we prove that the solution of the quenched KPZ equation with a very strong force also converges to the Cole–Hopf solution of the KPZ equation. Second, we show that a properly rescaled and renormalised quenched Edwards–Wilkinson model in any dimension converges to the stochastic heat equation.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 3","pages":"1850-1939"},"PeriodicalIF":3.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22158","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493381","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":"Almost monotonicity formula for H-minimal Legendrian surfaces in the Heisenberg group","authors":"Tristan Rivière","doi":"10.1002/cpa.22179","DOIUrl":"10.1002/cpa.22179","url":null,"abstract":"<p>We prove an almost monotonicity formula for H-minimal Legendrian Surfaces (also called <i>contact stationary Legendrian immersions</i> or <i>Hamiltonian stationary immersions</i>) in the Heisenberg Group <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>H</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>${mathbb {H}}^2$</annotation>\u0000 </semantics></math>. From this formula we deduce a Bernstein-Liouville type theorem for H-minimal Legendrian Surfaces. We also present some possible range of applications of this formula.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 3","pages":"1940-1957"},"PeriodicalIF":3.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22179","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493390","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 forbidden region for random zeros: Appearance of quadrature domains","authors":"Alon Nishry,&nbsp;Aron Wennman","doi":"10.1002/cpa.22142","DOIUrl":"10.1002/cpa.22142","url":null,"abstract":"<p>Our main discovery is a surprising interplay between quadrature domains on the one hand, and the zeros of the Gaussian Entire Function (GEF) on the other. Specifically, consider the GEF conditioned on the rare <i>hole event</i> that there are no zeros in a given large Jordan domain. We show that in the natural scaling limit, a quadrature domain enclosing the hole emerges as a <i>forbidden region</i>, where the zero density vanishes. Moreover, we give a description of the class of holes for which the forbidden region is a disk. The connecting link between random zeros and potential theory is supplied by a constrained extremal problem for the Zeitouni-Zelditch functional. To solve this problem, we recast it in terms of a seemingly novel obstacle problem, where the solution is forced to be harmonic inside the hole.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 3","pages":"1766-1849"},"PeriodicalIF":3.0,"publicationDate":"2023-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22142","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493380","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":"Optimal regularity for supercritical parabolic obstacle problems","authors":"Xavier Ros-Oton,&nbsp;Clara Torres-Latorre","doi":"10.1002/cpa.22166","DOIUrl":"10.1002/cpa.22166","url":null,"abstract":"&lt;p&gt;We study the obstacle problem for parabolic operators of the type &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;∂&lt;/mi&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$partial _t + L$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, where &lt;i&gt;L&lt;/i&gt; is an elliptic integro-differential operator of order 2&lt;i&gt;s&lt;/i&gt;, such as &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mi&gt;Δ&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$(-Delta )^s$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, in the supercritical regime &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mfrac&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mfrac&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$s in (0,frac{1}{2})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. The best result in this context was due to Caffarelli and Figalli, who established the &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;annotation&gt;$C^{1,s}_x$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; regularity of solutions for the case &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mi&gt;Δ&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mi&gt;s&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$L = (-Delta )^s$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, the same regularity as in the elliptic setting.&lt;/p&gt;&lt;p&gt;Here we prove for the first time that solutions are actually &lt;i&gt;more&lt;/i&gt; regular than in the elliptic case. More precisely, we show that they are &lt;i&gt;C&lt;/i&gt;&lt;sup&gt;1, 1&lt;/sup&gt; in space and time, and that this is optimal. We also deduce the &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 ","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 3","pages":"1724-1765"},"PeriodicalIF":3.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22166","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135243489","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":"Spectrum of random d-regular graphs up to the edge","authors":"Jiaoyang Huang,&nbsp;Horng-Tzer Yau","doi":"10.1002/cpa.22176","DOIUrl":"10.1002/cpa.22176","url":null,"abstract":"<p>Consider the normalized adjacency matrices of random <i>d</i>-regular graphs on <i>N</i> vertices with fixed degree <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>⩾</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$dgeqslant 3$</annotation>\u0000 </semantics></math>. We prove that, with probability <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>−</mo>\u0000 <msup>\u0000 <mi>N</mi>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 <mo>+</mo>\u0000 <mi>ε</mi>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$1-N^{-1+varepsilon }$</annotation>\u0000 </semantics></math> for any <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ε</mi>\u0000 <mo>&gt;</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$varepsilon &gt;0$</annotation>\u0000 </semantics></math>, the following two properties hold as <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>N</mi>\u0000 <mo>→</mo>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 <annotation>$N rightarrow infty$</annotation>\u0000 </semantics></math> provided that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>⩾</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$dgeqslant 3$</annotation>\u0000 </semantics></math>: (i) The eigenvalues are close to the classical eigenvalue locations given by the Kesten–McKay distribution. In particular, the extremal eigenvalues are concentrated with polynomial error bound in <i>N</i>, that is, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>λ</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <mrow>\u0000 <mo>|</mo>\u0000 <msub>\u0000 <mi>λ</mi>\u0000 <mi>N</mi>\u0000 </msub>\u0000 <mo>|</mo>\u0000 </mrow>\u0000 <mo>⩽</mo>\u0000 <mn>2</mn>\u0000 <mo>+</mo>\u0000 <msup>\u0000 <mi>N</mi>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mi>c</mi>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$lambda _2, |lambda _N|leqslant 2+N^{-c}$</annotation>\u0000 </semantics></math>. (ii) All eigenvectors of random <i>d</i>-regular graphs are completely delocalized.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 3","pages":"1635-1723"},"PeriodicalIF":3.0,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135387299","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":"Discrete honeycombs, rational edges, and edge states","authors":"Charles L. Fefferman,&nbsp;Sonia Fliss,&nbsp;Michael I. Weinstein","doi":"10.1002/cpa.22141","DOIUrl":"10.1002/cpa.22141","url":null,"abstract":"<p>Consider the tight binding model of graphene, sharply terminated along an edge <b>l</b> parallel to a direction of translational symmetry of the underlying period lattice. We classify such edges <b>l</b> into those of “zigzag type” and those of “armchair type”, generalizing the classical zigzag and armchair edges. We prove that zero energy / flat band edge states arise for edges of zigzag type, but never for those of armchair type. We exhibit explicit formulas for flat band edge states when they exist. We produce strong evidence for the existence of dispersive (non flat) edge state curves of nonzero energy for most <b>l</b>.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 3","pages":"1575-1634"},"PeriodicalIF":3.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136010814","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":"Erratum for “Global Identifiability of Differential Models”","authors":"Hoon Hong,&nbsp;Alexey Ovchinnikov,&nbsp;Gleb Pogudin,&nbsp;Chee Yap","doi":"10.1002/cpa.22163","DOIUrl":"10.1002/cpa.22163","url":null,"abstract":"<p>We are grateful to Peter Thompson for pointing out an error in [<span>1</span>, Lemma 3.5, p. 1848]. The original proof worked only under the assumption that <math>\u0000 <semantics>\u0000 <mover>\u0000 <mi>θ</mi>\u0000 <mo>̂</mo>\u0000 </mover>\u0000 <annotation>$hat{theta }$</annotation>\u0000 </semantics></math> is a vector of constants. However, some of the components of <math>\u0000 <semantics>\u0000 <mover>\u0000 <mi>θ</mi>\u0000 <mo>̂</mo>\u0000 </mover>\u0000 <annotation>$hat{bm{theta }}$</annotation>\u0000 </semantics></math> could be the states of the dynamic under consideration, and the lemma was used in such a setup (i.e., with <math>\u0000 <semantics>\u0000 <mover>\u0000 <mi>θ</mi>\u0000 <mo>̂</mo>\u0000 </mover>\u0000 <annotation>$hat{bm{theta }}$</annotation>\u0000 </semantics></math> involving states) later in [<span>1</span>, Proposition 3.4].</p><p>We give a more explicit version of the statement and provide a correct proof. The desired statement will be deduced from the following:\u0000\u0000 </p><p>\u0000 </p><p>The following corollary is equivalent to [<span>1</span>, Lemma 3.5, p. 1848] but explicitly highlights that some of the entries of <math>\u0000 <semantics>\u0000 <mover>\u0000 <mi>θ</mi>\u0000 <mo>̂</mo>\u0000 </mover>\u0000 <annotation>$hat{bm{theta }}$</annotation>\u0000 </semantics></math> may be initial conditions, not only system parameters.\u0000\u0000 </p><p>\u0000 </p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 3","pages":"2177-2179"},"PeriodicalIF":3.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22163","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138504551","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":"An upper Minkowski dimension estimate for the interior singular set of area minimizing currents","authors":"Anna Skorobogatova","doi":"10.1002/cpa.22165","DOIUrl":"10.1002/cpa.22165","url":null,"abstract":"<p>We show that for an area minimizing <i>m</i>-dimensional integral current <i>T</i> of codimension at least two inside a sufficiently regular Riemannian manifold, the upper Minkowski dimension of the interior singular set is at most <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 <mo>−</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$m-2$</annotation>\u0000 </semantics></math>. This provides a strengthening of the existing <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>m</mi>\u0000 <mo>−</mo>\u0000 <mn>2</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(m-2)$</annotation>\u0000 </semantics></math>-dimensional Hausdorff dimension bound due to Almgren and De Lellis &amp; Spadaro. As a by-product of the proof, we establish an improvement on the persistence of singularities along the sequence of center manifolds taken to approximate <i>T</i> along blow-up scales.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 2","pages":"1509-1572"},"PeriodicalIF":3.0,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135153622","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":"Logarithmic cotangent bundles, Chern-Mather classes, and the Huh-Sturmfels involution conjecture","authors":"Laurenţiu G. Maxim,&nbsp;Jose Israel Rodriguez,&nbsp;Botong Wang,&nbsp;Lei Wu","doi":"10.1002/cpa.22156","DOIUrl":"10.1002/cpa.22156","url":null,"abstract":"<p>Using compactifications in the logarithmic cotangent bundle, we obtain a formula for the Chern classes of the pushforward of Lagrangian cycles under an open embedding with normal crossing complement. This generalizes earlier results of Aluffi and Wu-Zhou. The first application of our formula is a geometric description of Chern-Mather classes of an arbitrary very affine variety, generalizing earlier results of Huh which held under the smooth and schön assumptions. As the second application, we prove an involution formula relating sectional maximum likelihood (ML) degrees and ML bidegrees, which was conjectured by Huh and Sturmfels in 2013.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 2","pages":"1486-1508"},"PeriodicalIF":3.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22156","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135437037","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":"Global minimizers of a large class of anisotropic attractive-repulsive interaction energies in 2D","authors":"José A. Carrillo,&nbsp;Ruiwen Shu","doi":"10.1002/cpa.22162","DOIUrl":"10.1002/cpa.22162","url":null,"abstract":"<p>We study a large family of Riesz-type singular interaction potentials with anisotropy in two dimensions. Their associated global energy minimizers are given by explicit formulas whose supports are determined by ellipses under certain assumptions. More precisely, by parameterizing the strength of the anisotropic part we characterize the sharp range in which these explicit ellipse-supported configurations are the global minimizers based on linear convexity arguments. Moreover, for certain anisotropic parts, we prove that for large values of the parameter the global minimizer is only given by vertically concentrated measures corresponding to one dimensional minimizers. We also show that these ellipse-supported configurations generically do not collapse to a vertically concentrated measure at the critical value for convexity, leading to an interesting gap of the parameters in between. In this intermediate range, we conclude by infinitesimal concavity that any superlevel set of any local minimizer in a suitable sense does not have interior points. Furthermore, for certain anisotropic parts, their support cannot contain any vertical segment for a restricted range of parameters, and moreover the global minimizers are expected to exhibit a zigzag behavior. All these results hold for the limiting case of the logarithmic repulsive potential, extending and generalizing previous results in the literature. Various examples of anisotropic parts leading to even more complex behavior are numerically explored.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 2","pages":"1353-1404"},"PeriodicalIF":3.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22162","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135488330","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}