Communications on Pure and Applied Mathematics最新文献

{"title":"Conformal covariance of connection probabilities and fields in 2D critical percolation","authors":"Federico Camia","doi":"10.1002/cpa.22171","DOIUrl":"10.1002/cpa.22171","url":null,"abstract":"<p>Fitting percolation into the conformal field theory framework requires showing that connection probabilities have a conformally invariant scaling limit. For critical site percolation on the triangular lattice, we prove that the probability that <i>n</i> vertices belong to the same open cluster has a well-defined scaling limit for every <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>≥</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$n ge 2$</annotation>\u0000 </semantics></math>. Moreover, the limiting functions <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>P</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>x</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <mtext>…</mtext>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>x</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$P_n(x_1,ldots ,x_n)$</annotation>\u0000 </semantics></math> transform covariantly under Möbius transformations of the plane as well as under local conformal maps, that is, they behave like correlation functions of primary operators in conformal field theory. In particular, they are invariant under translations, rotations and inversions, and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>P</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>s</mi>\u0000 <msub>\u0000 <mi>x</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <mtext>…</mtext>\u0000 <mo>,</mo>\u0000 <mi>s</mi>\u0000 <msub>\u0000 <mi>x</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>=</mo>\u0000 <msup>\u0000 <mi>s</mi>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>5</mn>\u0000 <mi>n</mi>\u0000 <mo>/</mo>\u0000 <mn>48</mn>\u0000 </mrow>\u0000 </msup>\u0000 <msub>\u0000 <mi>P</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>x</mi>\u0000 <mn>1</mn>\u0000 </m","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 3","pages":"2138-2176"},"PeriodicalIF":3.0,"publicationDate":"2023-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493387","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":"High-dimensional limit theorems for SGD: Effective dynamics and critical scaling","authors":"Gérard Ben Arous,&nbsp;Reza Gheissari,&nbsp;Aukosh Jagannath","doi":"10.1002/cpa.22169","DOIUrl":"10.1002/cpa.22169","url":null,"abstract":"<p>We study the scaling limits of stochastic gradient descent (SGD) with constant step-size in the high-dimensional regime. We prove limit theorems for the trajectories of summary statistics (i.e., finite-dimensional functions) of SGD as the dimension goes to infinity. Our approach allows one to choose the summary statistics that are tracked, the initialization, and the step-size. It yields both ballistic (ODE) and diffusive (SDE) limits, with the limit depending dramatically on the former choices. We show a critical scaling regime for the step-size, below which the effective ballistic dynamics matches gradient flow for the population loss, but at which, a new correction term appears which changes the phase diagram. About the fixed points of this effective dynamics, the corresponding diffusive limits can be quite complex and even degenerate. We demonstrate our approach on popular examples including estimation for spiked matrix and tensor models and classification via two-layer networks for binary and XOR-type Gaussian mixture models. These examples exhibit surprising phenomena including multimodal timescales to convergence as well as convergence to sub-optimal solutions with probability bounded away from zero from random (e.g., Gaussian) initializations. At the same time, we demonstrate the benefit of overparametrization by showing that the latter probability goes to zero as the second layer width grows.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 3","pages":"2030-2080"},"PeriodicalIF":3.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22169","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493383","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":"Sine-kernel determinant on two large intervals","authors":"Benjamin Fahs,&nbsp;Igor Krasovsky","doi":"10.1002/cpa.22147","DOIUrl":"10.1002/cpa.22147","url":null,"abstract":"<p>We consider the probability of two large gaps (intervals without eigenvalues) in the bulk scaling limit of the Gaussian Unitary Ensemble of random matrices. We determine the multiplicative constant in the asymptotics. We also provide the full explicit asymptotics (up to decreasing terms) for the transition between one and two large gaps.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 3","pages":"1958-2029"},"PeriodicalIF":3.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22147","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71493382","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":"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}