Pub Date : 2022-09-09DOI: 10.1142/s0219199722500626
M. Winkler
{"title":"Arbitrarily fast grow-up rates in quasilinear Keller-Segel systems","authors":"M. Winkler","doi":"10.1142/s0219199722500626","DOIUrl":"https://doi.org/10.1142/s0219199722500626","url":null,"abstract":"","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2022-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46782105","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-29DOI: 10.1142/S0219199723500062
Florian Besau, Steven Hoehner
A new intrinsic volume metric is introduced for the class of convex bodies in $mathbb{R}^n$. As an application, an inequality is proved for the asymptotic best approximation of the Euclidean unit ball by arbitrarily positioned polytopes with a restricted number of vertices under this metric. This result improves the best known estimate, and shows that dropping the restriction that the polytope is contained in the ball or vice versa improves the estimate by at least a factor of dimension. The same phenomenon has already been observed in the special cases of volume, surface area and mean width approximation of the ball.
{"title":"An intrinsic volume metric for the class of convex bodies in ℝn","authors":"Florian Besau, Steven Hoehner","doi":"10.1142/S0219199723500062","DOIUrl":"https://doi.org/10.1142/S0219199723500062","url":null,"abstract":"A new intrinsic volume metric is introduced for the class of convex bodies in $mathbb{R}^n$. As an application, an inequality is proved for the asymptotic best approximation of the Euclidean unit ball by arbitrarily positioned polytopes with a restricted number of vertices under this metric. This result improves the best known estimate, and shows that dropping the restriction that the polytope is contained in the ball or vice versa improves the estimate by at least a factor of dimension. The same phenomenon has already been observed in the special cases of volume, surface area and mean width approximation of the ball.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2022-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47565165","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-23DOI: 10.1142/S021919972250081X
Carl Johan Peter Johansson, Riccardo Tione
In this paper we study the rank-one convex hull of a differential inclusion associated to entropy solutions of a hyperbolic system of conservation laws. This was introduced in Section 7 of [Kirchheim, M"uller, v{S}ver'ak, 2003] and many of its properties have already been shown in [Lorent, Peng, 2019]-[Lorent, Peng, 2020]. In particular, in [Lorent, Peng 2020] it is shown that the differential inclusion does not contain any $T_4$ configurations. Here we continue that study by showing that the differential inclusion does not contain $T_5$ configurations.
{"title":"T5 Configurations and Hyperbolic Systems","authors":"Carl Johan Peter Johansson, Riccardo Tione","doi":"10.1142/S021919972250081X","DOIUrl":"https://doi.org/10.1142/S021919972250081X","url":null,"abstract":"In this paper we study the rank-one convex hull of a differential inclusion associated to entropy solutions of a hyperbolic system of conservation laws. This was introduced in Section 7 of [Kirchheim, M\"uller, v{S}ver'ak, 2003] and many of its properties have already been shown in [Lorent, Peng, 2019]-[Lorent, Peng, 2020]. In particular, in [Lorent, Peng 2020] it is shown that the differential inclusion does not contain any $T_4$ configurations. Here we continue that study by showing that the differential inclusion does not contain $T_5$ configurations.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2022-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45533187","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-18DOI: 10.1142/s0219199722500614
C. O. Alves, Renan J. S. Isneri, P. Montecchiari
{"title":"Existence of Heteroclinic and Saddle type solutions for a class of quasilinear problems in whole ℝ2","authors":"C. O. Alves, Renan J. S. Isneri, P. Montecchiari","doi":"10.1142/s0219199722500614","DOIUrl":"https://doi.org/10.1142/s0219199722500614","url":null,"abstract":"","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2022-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42612341","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-18DOI: 10.1142/s0219199722500602
Benniao Li, W. Long, Aliang Xia
{"title":"Multiple positive and sign-changing solutions for a class of kirchhoff equations","authors":"Benniao Li, W. Long, Aliang Xia","doi":"10.1142/s0219199722500602","DOIUrl":"https://doi.org/10.1142/s0219199722500602","url":null,"abstract":"","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2022-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44640992","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-17DOI: 10.1142/s0219199723500359
Jordan Serres
We control the behavior of the Poincar{'e} constant along the Polchinski renormalization flow using a dynamic version of $Gamma$-calculus. We also treat the case of higher order eigenvalues. Our method generalizes a method introduced by B. Klartag and E. Putterman to analyze the evolution of log-concave distributions along the heat flow. Furthermore, we apply it to general $Phi$ 4-measures and discuss the interpretation in terms of transport maps.
{"title":"Behavior of the poincare constant along the polchinski renormalization flow","authors":"Jordan Serres","doi":"10.1142/s0219199723500359","DOIUrl":"https://doi.org/10.1142/s0219199723500359","url":null,"abstract":"We control the behavior of the Poincar{'e} constant along the Polchinski renormalization flow using a dynamic version of $Gamma$-calculus. We also treat the case of higher order eigenvalues. Our method generalizes a method introduced by B. Klartag and E. Putterman to analyze the evolution of log-concave distributions along the heat flow. Furthermore, we apply it to general $Phi$ 4-measures and discuss the interpretation in terms of transport maps.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2022-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45526630","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-20DOI: 10.1142/s0219199722500444
J. Bona, F. Weissler
{"title":"Blowup and ILL-Posedness for the Complex, Periodic KDV Equation","authors":"J. Bona, F. Weissler","doi":"10.1142/s0219199722500444","DOIUrl":"https://doi.org/10.1142/s0219199722500444","url":null,"abstract":"","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2022-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44899448","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-11DOI: 10.1142/s0219199723500372
L. Brasco, Mar'ia de Mar Gonz'alez, M. Ispizua
Motivated by the connection between the first eigenvalue of the Dirichlet-Laplacian and the torsional rigidity, the aim of this paper is to find a physically coherent and mathematically interesting new concept for boundary torsional rigidity, closely related to the Steklov eigenvalue. From a variational point of view, such a new object corresponds to the sharp constant for the trace embedding of $W^{1,2}(Omega)$ into $L^1(partialOmega)$. We obtain various equivalent variational formulations, present some properties of the state function and obtain some sharp geometric estimates, both for planar simply connected sets and for convex sets in any dimension.
{"title":"A steklov version of the torsional rigidity","authors":"L. Brasco, Mar'ia de Mar Gonz'alez, M. Ispizua","doi":"10.1142/s0219199723500372","DOIUrl":"https://doi.org/10.1142/s0219199723500372","url":null,"abstract":"Motivated by the connection between the first eigenvalue of the Dirichlet-Laplacian and the torsional rigidity, the aim of this paper is to find a physically coherent and mathematically interesting new concept for boundary torsional rigidity, closely related to the Steklov eigenvalue. From a variational point of view, such a new object corresponds to the sharp constant for the trace embedding of $W^{1,2}(Omega)$ into $L^1(partialOmega)$. We obtain various equivalent variational formulations, present some properties of the state function and obtain some sharp geometric estimates, both for planar simply connected sets and for convex sets in any dimension.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2022-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43596438","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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