In this note, we investigate the convergence of a U-statistic of order two having stationary ergodic data. We will find sufficient conditions for the almost sure and L 1 convergence and present some counter-examples showing that the U-statistic itself might fail to converge: centering is needed as well as finiteness of sup j≥2 𝔼[|h(X 1 ,X j )|].
{"title":"Some remarks on the ergodic theorem for U-statistics","authors":"Herold G. Dehling, Davide Giraudo, Dalibor Volny","doi":"10.5802/crmath.494","DOIUrl":"https://doi.org/10.5802/crmath.494","url":null,"abstract":"In this note, we investigate the convergence of a U-statistic of order two having stationary ergodic data. We will find sufficient conditions for the almost sure and L 1 convergence and present some counter-examples showing that the U-statistic itself might fail to converge: centering is needed as well as finiteness of sup j≥2 𝔼[|h(X 1 ,X j )|].","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"83 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135087584","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The optimal transport map between the standard Gaussian measure and an α-strongly log-concave probability measure is α -1/2 -Lipschitz, as first observed in a celebrated theorem of Caffarelli. In this paper, we apply two classical covariance inequalities (the Brascamp–Lieb and Cramér–Rao inequalities) to prove a sharp bound on the Lipschitz constant of the map that arises from entropically regularized optimal transport. In the limit as the regularization tends to zero, we obtain an elegant and short proof of Caffarelli’s original result. We also extend Caffarelli’s theorem to the setting in which the Hessians of the log-densities of the measures are bounded by arbitrary positive definite commuting matrices.
{"title":"An entropic generalization of Caffarelli’s contraction theorem via covariance inequalities","authors":"Sinho Chewi, Aram-Alexandre Pooladian","doi":"10.5802/crmath.486","DOIUrl":"https://doi.org/10.5802/crmath.486","url":null,"abstract":"The optimal transport map between the standard Gaussian measure and an α-strongly log-concave probability measure is α -1/2 -Lipschitz, as first observed in a celebrated theorem of Caffarelli. In this paper, we apply two classical covariance inequalities (the Brascamp–Lieb and Cramér–Rao inequalities) to prove a sharp bound on the Lipschitz constant of the map that arises from entropically regularized optimal transport. In the limit as the regularization tends to zero, we obtain an elegant and short proof of Caffarelli’s original result. We also extend Caffarelli’s theorem to the setting in which the Hessians of the log-densities of the measures are bounded by arbitrary positive definite commuting matrices.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"121 51","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135136437","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The last decade has seen an abundance of congruences for b ℓ (n), the number of ℓ-regular partitions of n. Notably absent are congruences modulo 4 for b 3 (n). In this paper, we introduce Ramanujan type congruences modulo 4 for b 3 (2n) involving some primes p congruent to 11,13,17,19,23 modulo 24.
{"title":"Congruences modulo 4 for the number of 3-regular partitions","authors":"Cristina Ballantine, Mircea Merca","doi":"10.5802/crmath.512","DOIUrl":"https://doi.org/10.5802/crmath.512","url":null,"abstract":"The last decade has seen an abundance of congruences for b ℓ (n), the number of ℓ-regular partitions of n. Notably absent are congruences modulo 4 for b 3 (n). In this paper, we introduce Ramanujan type congruences modulo 4 for b 3 (2n) involving some primes p congruent to 11,13,17,19,23 modulo 24.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"121 8","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135137875","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce an analogue of the classical Markov equation that involves dual numbers a+αε with ε 2 =0. This equation characterizes the “shadow Markov numbers” recently considered by one of us. We show that this equation is characterized by invariance by cluster algebra mutations.
{"title":"A shadow Markov equation","authors":"Nathan Bonin, Valentin Ovsienko","doi":"10.5802/crmath.496","DOIUrl":"https://doi.org/10.5802/crmath.496","url":null,"abstract":"We introduce an analogue of the classical Markov equation that involves dual numbers a+αε with ε 2 =0. This equation characterizes the “shadow Markov numbers” recently considered by one of us. We show that this equation is characterized by invariance by cluster algebra mutations.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":" 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135141663","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The function space H s (0,T), s<1/2, allows for functions with jump discontinuities and is thus attractive for treating optimal control problems with discrete-valued control functions. We show that while arbitrary chattering controls are impossible, there exist feasible controls in H s (0,T) that have countably jump discontinuities with jump height one in each of countably many pairwise disjoint intervals. However, under mild assumptions, we show that certain types of jump discontinuities cannot be optimal. The derivation of meaningful optimality conditions via a direct variational argument using simple feasible perturbations remains a major challenge; as illustrated by an example.
函数空间H s (0,T)， s<1/2，允许具有跳跃不连续的函数，因此对于处理具有离散值控制函数的最优控制问题具有吸引力。我们证明了任意抖振控制是不可能的，但在H (0,T)中存在可行的控制，该控制具有可计数的跳跃不连续，在可计数的多个两两不相交区间中的每个区间中跳跃高度为1。然而，在温和的假设下，我们证明某些类型的跳跃不连续不能是最优的。利用简单可行扰动通过直接变分参数推导有意义的最优性条件仍然是一个主要挑战;用一个例子来说明。
{"title":"On Binary Optimal Control in H s (0,T), s<1/2","authors":"Paul Manns, Thomas M. Surowiec","doi":"10.5802/crmath.507","DOIUrl":"https://doi.org/10.5802/crmath.507","url":null,"abstract":"The function space H s (0,T), s<1/2, allows for functions with jump discontinuities and is thus attractive for treating optimal control problems with discrete-valued control functions. We show that while arbitrary chattering controls are impossible, there exist feasible controls in H s (0,T) that have countably jump discontinuities with jump height one in each of countably many pairwise disjoint intervals. However, under mild assumptions, we show that certain types of jump discontinuities cannot be optimal. The derivation of meaningful optimality conditions via a direct variational argument using simple feasible perturbations remains a major challenge; as illustrated by an example.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":" 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135141317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Erik Burman, Guillaume Delay, Alexandre Ern, Lauri Oksanen
This paper studies the unique continuation problem for the heat equation. We prove a so-called conditional stability estimate for the solution. We are interested in local estimates that are Hölder stable with the weakest possible norms of data on the right-hand side. Such an estimate is useful for the convergence analysis of computational methods dealing with data assimilation. We focus on the case of a known solution at initial time and in some subdomain but that is unknown on the boundary. To the best of our knowledge, this situation has not yet been studied in the literature.
{"title":"A stability estimate for data assimilation subject to the heat equation with initial datum","authors":"Erik Burman, Guillaume Delay, Alexandre Ern, Lauri Oksanen","doi":"10.5802/crmath.506","DOIUrl":"https://doi.org/10.5802/crmath.506","url":null,"abstract":"This paper studies the unique continuation problem for the heat equation. We prove a so-called conditional stability estimate for the solution. We are interested in local estimates that are Hölder stable with the weakest possible norms of data on the right-hand side. Such an estimate is useful for the convergence analysis of computational methods dealing with data assimilation. We focus on the case of a known solution at initial time and in some subdomain but that is unknown on the boundary. To the best of our knowledge, this situation has not yet been studied in the literature.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"83 18","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135087581","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show the local null-controllability of a fluid-structure interaction system coupling a viscous incompressible fluid with a damped beam located on a part of its boundary. The controls act on arbitrary small parts of the fluid domain and of the beam domain. In order to show the result, we first use a change of variables and a linearization to reduce the problem to the null-controllability of a Stokes-beam system in a cylindrical domain. We obtain this property by combining Carleman inequalities for the heat equation, for the damped beam equation and for the Laplace equation with high-frequency estimates. Then, the result on the nonlinear system is obtained by a fixed-point argument.
{"title":"Controllability of a fluid-structure interaction system coupling the Navier–Stokes system and a damped beam equation","authors":"Rémi Buffe, Takéo Takahashi","doi":"10.5802/crmath.509","DOIUrl":"https://doi.org/10.5802/crmath.509","url":null,"abstract":"We show the local null-controllability of a fluid-structure interaction system coupling a viscous incompressible fluid with a damped beam located on a part of its boundary. The controls act on arbitrary small parts of the fluid domain and of the beam domain. In order to show the result, we first use a change of variables and a linearization to reduce the problem to the null-controllability of a Stokes-beam system in a cylindrical domain. We obtain this property by combining Carleman inequalities for the heat equation, for the damped beam equation and for the Laplace equation with high-frequency estimates. Then, the result on the nonlinear system is obtained by a fixed-point argument.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"83 16","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135087582","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For any ring we propose the construction of a cover which increases the finitistic dimension on one side and decreases the finitistic dimension to zero on the opposite side. This complements recent work of Cummings.
{"title":"On the symmetry of the finitistic dimension","authors":"Henning Krause","doi":"10.5802/crmath.481","DOIUrl":"https://doi.org/10.5802/crmath.481","url":null,"abstract":"For any ring we propose the construction of a cover which increases the finitistic dimension on one side and decreases the finitistic dimension to zero on the opposite side. This complements recent work of Cummings.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"117 42","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135136711","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study the relative homological dimension based on the class of projectively coresolved Gorenstein flat modules (PGF-modules), that were introduced by Saroch and Stovicek in [26]. The resulting PGF-dimension of modules has several properties in common with the Gorenstein projective dimension, the relative homological theory based on the class of Gorenstein projective modules. In particular, there is a hereditary Hovey triple in the category of modules of finite PGF-dimension, whose associated homotopy category is triangulated equivalent to the stable category of PGF-modules. Studying the finiteness of the PGF global dimension reveals a connection between classical homological invariants of left and right modules over the ring, that leads to generalizations of certain results by Jensen [24], Gedrich and Gruenberg [17] that were originally proved in the realm of commutative Noetherian rings.
{"title":"Homological dimension based on a class of Gorenstein flat modules","authors":"Georgios Dalezios, Ioannis Emmanouil","doi":"10.5802/crmath.480","DOIUrl":"https://doi.org/10.5802/crmath.480","url":null,"abstract":"In this paper, we study the relative homological dimension based on the class of projectively coresolved Gorenstein flat modules (PGF-modules), that were introduced by Saroch and Stovicek in [26]. The resulting PGF-dimension of modules has several properties in common with the Gorenstein projective dimension, the relative homological theory based on the class of Gorenstein projective modules. In particular, there is a hereditary Hovey triple in the category of modules of finite PGF-dimension, whose associated homotopy category is triangulated equivalent to the stable category of PGF-modules. Studying the finiteness of the PGF global dimension reveals a connection between classical homological invariants of left and right modules over the ring, that leads to generalizations of certain results by Jensen [24], Gedrich and Gruenberg [17] that were originally proved in the realm of commutative Noetherian rings.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":"122 30","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135138342","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Our main result associates a family of congruences with each suitable system of nilpotent subgroups of a finite group. Using this result, we complete and correct the proof of a theorem of Hirsch concerning the class number of a finite group of odd order.
{"title":"Congruences associated with families of nilpotent subgroups and a theorem of Hirsch","authors":"Stefanos Aivazidis, Thomas Müller","doi":"10.5802/crmath.514","DOIUrl":"https://doi.org/10.5802/crmath.514","url":null,"abstract":"Our main result associates a family of congruences with each suitable system of nilpotent subgroups of a finite group. Using this result, we complete and correct the proof of a theorem of Hirsch concerning the class number of a finite group of odd order.","PeriodicalId":10620,"journal":{"name":"Comptes Rendus Mathematique","volume":" 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135141919","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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