Pub Date : 2023-11-17DOI: 10.1142/s021919972350058x
P. H. A. Ngoc, Le Trung Hieu, Thai Bao Tran
{"title":"On exponential stability of switched functional differential equations with average dwell-time","authors":"P. H. A. Ngoc, Le Trung Hieu, Thai Bao Tran","doi":"10.1142/s021919972350058x","DOIUrl":"https://doi.org/10.1142/s021919972350058x","url":null,"abstract":"","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"19 8","pages":""},"PeriodicalIF":1.6,"publicationDate":"2023-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139265298","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 : 2023-11-17DOI: 10.1142/s0219199723500542
Inkang Kim, Xueyuan Wan, Genkai Zhang
{"title":"Convexity of energy functions of harmonic maps homotopic to covering maps of surfaces","authors":"Inkang Kim, Xueyuan Wan, Genkai Zhang","doi":"10.1142/s0219199723500542","DOIUrl":"https://doi.org/10.1142/s0219199723500542","url":null,"abstract":"","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"9 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2023-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139263453","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 : 2023-11-10DOI: 10.1142/s0219199723500426
Elaine Crooks, Yini Du
In this paper, we present an approach to characterizing fast-reaction limits of systems with nonlinear diffusion, when there are either two reaction–diffusion equations, or one reaction–diffusion equation and one ordinary differential equation, on unbounded domains. Here, we replace the terms of the form [Formula: see text] in usual reaction–diffusion equation, which represent linear diffusion, by terms of form [Formula: see text], representing nonlinear diffusion. We prove the convergence in the fast-reaction limit [Formula: see text] that is determined by the unique solution of a certain scalar nonlinear diffusion problem.
{"title":"Fast reaction limit of reaction diffusion systems with nonlinear diffusion","authors":"Elaine Crooks, Yini Du","doi":"10.1142/s0219199723500426","DOIUrl":"https://doi.org/10.1142/s0219199723500426","url":null,"abstract":"In this paper, we present an approach to characterizing fast-reaction limits of systems with nonlinear diffusion, when there are either two reaction–diffusion equations, or one reaction–diffusion equation and one ordinary differential equation, on unbounded domains. Here, we replace the terms of the form [Formula: see text] in usual reaction–diffusion equation, which represent linear diffusion, by terms of form [Formula: see text], representing nonlinear diffusion. We prove the convergence in the fast-reaction limit [Formula: see text] that is determined by the unique solution of a certain scalar nonlinear diffusion problem.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"74 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135088359","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 : 2023-11-08DOI: 10.1142/s0219199723500517
Leonardo Biliotti, Joshua O. Windare
Let [Formula: see text] be a Kähler manifold and let [Formula: see text] be a compact connected Lie group with Lie algebra [Formula: see text] acting on [Formula: see text] and preserving [Formula: see text]. We assume that the [Formula: see text]-action extends holomorphically to an action of the complexified group [Formula: see text] and the [Formula: see text]-action on [Formula: see text] is Hamiltonian. Then there exists a [Formula: see text]-equivariant momentum map [Formula: see text]. If [Formula: see text] is a closed subgroup such that the Cartan decomposition [Formula: see text] induces a Cartan decomposition [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] is the Lie algebra of [Formula: see text], there is a corresponding gradient map [Formula: see text]. If [Formula: see text] is a [Formula: see text]-invariant compact and connected real submanifold of [Formula: see text] we may consider [Formula: see text] as a mapping [Formula: see text] Given an [Formula: see text]-invariant scalar product on [Formula: see text], we obtain a Morse like function [Formula: see text] on [Formula: see text]. We point out that, without the assumption that [Formula: see text] is a real analytic manifold, the Lojasiewicz gradient inequality holds for [Formula: see text]. Therefore, the limit of the negative gradient flow of [Formula: see text] exists and it is unique. Moreover, we prove that any [Formula: see text]-orbit collapses to a single [Formula: see text]-orbit and two critical points of [Formula: see text] which are in the same [Formula: see text]-orbit belong to the same [Formula: see text]-orbit. We also investigate convexity properties of the gradient map [Formula: see text] in the Abelian case. In particular, we study two-orbit variety [Formula: see text] and we investigate topological and cohomological properties of [Formula: see text].
{"title":"Properties of gradient maps associated with action of real reductive group","authors":"Leonardo Biliotti, Joshua O. Windare","doi":"10.1142/s0219199723500517","DOIUrl":"https://doi.org/10.1142/s0219199723500517","url":null,"abstract":"Let [Formula: see text] be a Kähler manifold and let [Formula: see text] be a compact connected Lie group with Lie algebra [Formula: see text] acting on [Formula: see text] and preserving [Formula: see text]. We assume that the [Formula: see text]-action extends holomorphically to an action of the complexified group [Formula: see text] and the [Formula: see text]-action on [Formula: see text] is Hamiltonian. Then there exists a [Formula: see text]-equivariant momentum map [Formula: see text]. If [Formula: see text] is a closed subgroup such that the Cartan decomposition [Formula: see text] induces a Cartan decomposition [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] is the Lie algebra of [Formula: see text], there is a corresponding gradient map [Formula: see text]. If [Formula: see text] is a [Formula: see text]-invariant compact and connected real submanifold of [Formula: see text] we may consider [Formula: see text] as a mapping [Formula: see text] Given an [Formula: see text]-invariant scalar product on [Formula: see text], we obtain a Morse like function [Formula: see text] on [Formula: see text]. We point out that, without the assumption that [Formula: see text] is a real analytic manifold, the Lojasiewicz gradient inequality holds for [Formula: see text]. Therefore, the limit of the negative gradient flow of [Formula: see text] exists and it is unique. Moreover, we prove that any [Formula: see text]-orbit collapses to a single [Formula: see text]-orbit and two critical points of [Formula: see text] which are in the same [Formula: see text]-orbit belong to the same [Formula: see text]-orbit. We also investigate convexity properties of the gradient map [Formula: see text] in the Abelian case. In particular, we study two-orbit variety [Formula: see text] and we investigate topological and cohomological properties of [Formula: see text].","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":" 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135292689","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 : 2023-10-13DOI: 10.1142/s0219199723500487
Gianmarco Giovannardi, Manuel Ritore
{"title":"The bernstein problem for (<i>X</i>, <i>Y</i>)-lipschitz surfaces in three-dimensional sub-finsler heisenberg groups","authors":"Gianmarco Giovannardi, Manuel Ritore","doi":"10.1142/s0219199723500487","DOIUrl":"https://doi.org/10.1142/s0219199723500487","url":null,"abstract":"","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135853587","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 : 2023-10-13DOI: 10.1142/s0219199723500499
Mathias Braun, Matteo Calisti
We prove that a globally hyperbolic smooth spacetime endowed with a $smash{mathrm{C}^1}$-Lorentzian metric whose Ricci tensor is bounded from below in all timelike directions, in a distributional sense, obeys the timelike measure-contraction property. This result includes a class of spacetimes with borderline regularity for which local existence results for the vacuum Einstein equation are known in the setting of spaces with timelike Ricci bounds in a synthetic sense. In particular, these spacetimes satisfy timelike Brunn-Minkowski, Bonnet-Myers, and Bishop-Gromov inequalities in sharp form, without any timelike nonbranching assumption. If the metric is even $smash{mathrm{C}^{1,1}}$, in fact the stronger timelike curvature-dimension condition holds. In this regularity, we also obtain uniqueness of chronological optimal couplings and chronological geodesics.
{"title":"Timelike Ricci bounds for low regularity spacetimes by optimal transport","authors":"Mathias Braun, Matteo Calisti","doi":"10.1142/s0219199723500499","DOIUrl":"https://doi.org/10.1142/s0219199723500499","url":null,"abstract":"We prove that a globally hyperbolic smooth spacetime endowed with a $smash{mathrm{C}^1}$-Lorentzian metric whose Ricci tensor is bounded from below in all timelike directions, in a distributional sense, obeys the timelike measure-contraction property. This result includes a class of spacetimes with borderline regularity for which local existence results for the vacuum Einstein equation are known in the setting of spaces with timelike Ricci bounds in a synthetic sense. In particular, these spacetimes satisfy timelike Brunn-Minkowski, Bonnet-Myers, and Bishop-Gromov inequalities in sharp form, without any timelike nonbranching assumption. If the metric is even $smash{mathrm{C}^{1,1}}$, in fact the stronger timelike curvature-dimension condition holds. In this regularity, we also obtain uniqueness of chronological optimal couplings and chronological geodesics.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135853269","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 : 2023-10-13DOI: 10.1142/s0219199723500438
Zhiqi Chen, Saiyu Wang, Hui Zhang
We consider the moment map $m:mathbb{P}V_nrightarrow text{i}mathfrak{u}(n)$ for the action of $text{GL}(n)$ on $V_n=otimes^{2}(mathbb{C}^{n})^{*}otimesmathbb{C}^{n}$, and study the functional $F_n=|m|^{2}$ restricted to the projectivizations of the algebraic varieties of all $n$-dimensional Leibniz algebras $L_n$ and all $n$-dimensional symmetric Leibniz algebras $S_n$, respectively. Firstly, we give a description of the maxima and minima of the functional $F_n: L_n rightarrow mathbb{R}$, proving that they are actually attained at the symmetric Leibniz algebras. Then, for an arbitrary critical point $[mu]$ of $F_n: S_n rightarrow mathbb{R}$, we characterize the structure of $[mu]$ by virtue of the nonnegative rationality. Finally, we classify the critical points of $F_n: S_n rightarrow mathbb{R}$ for $n=2$, $3$, respectively.
{"title":"The moment map for the variety of Leibniz algebras","authors":"Zhiqi Chen, Saiyu Wang, Hui Zhang","doi":"10.1142/s0219199723500438","DOIUrl":"https://doi.org/10.1142/s0219199723500438","url":null,"abstract":"We consider the moment map $m:mathbb{P}V_nrightarrow text{i}mathfrak{u}(n)$ for the action of $text{GL}(n)$ on $V_n=otimes^{2}(mathbb{C}^{n})^{*}otimesmathbb{C}^{n}$, and study the functional $F_n=|m|^{2}$ restricted to the projectivizations of the algebraic varieties of all $n$-dimensional Leibniz algebras $L_n$ and all $n$-dimensional symmetric Leibniz algebras $S_n$, respectively. Firstly, we give a description of the maxima and minima of the functional $F_n: L_n rightarrow mathbb{R}$, proving that they are actually attained at the symmetric Leibniz algebras. Then, for an arbitrary critical point $[mu]$ of $F_n: S_n rightarrow mathbb{R}$, we characterize the structure of $[mu]$ by virtue of the nonnegative rationality. Finally, we classify the critical points of $F_n: S_n rightarrow mathbb{R}$ for $n=2$, $3$, respectively.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"144 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135853420","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 : 2023-10-13DOI: 10.1142/s0219199723500414
Chenfeng Zhu, Dachun Yang, Wen Yuan
Let $gammainmathbb{R}setminus{0}$ and $X(mathbb{R}^n)$ be a ball Banach function space satisfying some extra mild assumptions. Assume that $Omega=mathbb{R}^n$ or $Omegasubsetmathbb{R}^n$ is an $(varepsilon,infty)$-domain for some $varepsilonin(0,1]$. In this article, the authors prove that a function $f$ belongs to the homogeneous ball Banach Sobolev space $dot{W}^{1,X}(Omega)$ if and only if $fin L_{mathrm{loc}}^1(Omega)$ and $$ sup_{lambdain(0,infty)}lambda left|left[int_{{yinOmega: |f(cdot)-f(y)|>lambda|cdot-y|^{1+frac{gamma}{p}}}} left|cdot-yright|^{gamma-n},dy right]^frac{1}{p}right|_{X(Omega)}
{"title":"Brezis–Seeger–Van Schaftingen–Yung-Type Characterization of Homogeneous Ball Banach Sobolev Spaces and Its Applications","authors":"Chenfeng Zhu, Dachun Yang, Wen Yuan","doi":"10.1142/s0219199723500414","DOIUrl":"https://doi.org/10.1142/s0219199723500414","url":null,"abstract":"Let $gammainmathbb{R}setminus{0}$ and $X(mathbb{R}^n)$ be a ball Banach function space satisfying some extra mild assumptions. Assume that $Omega=mathbb{R}^n$ or $Omegasubsetmathbb{R}^n$ is an $(varepsilon,infty)$-domain for some $varepsilonin(0,1]$. In this article, the authors prove that a function $f$ belongs to the homogeneous ball Banach Sobolev space $dot{W}^{1,X}(Omega)$ if and only if $fin L_{mathrm{loc}}^1(Omega)$ and $$ sup_{lambdain(0,infty)}lambda left|left[int_{{yinOmega: |f(cdot)-f(y)|>lambda|cdot-y|^{1+frac{gamma}{p}}}} left|cdot-yright|^{gamma-n},dy right]^frac{1}{p}right|_{X(Omega)}<infty, $$ where $pin[1,infty)$ is related to $X(mathbb{R}^n)$. This result is of wide generality and can be applied to various specific Sobolev-type function spaces, including Morrey [Bourgain--Morrey-type, weighted (or mixed-norm or variable) Lebesgue, local (or global) generalized Herz, Lorentz, and Orlicz (or Orlicz-slice)] Sobolev spaces, which is new even in all these special cases; in particular, it coincides with the well-known result of H. Brezis, A. Seeger, J. Van Schaftingen, and P.-L. Yung when $X(Omega):=L^q(mathbb{R}^n)$ with $1","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135853579","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 : 2023-10-13DOI: 10.1142/s0219199723500451
Carmelo A. Finocchiaro, Sophie Frisch, Daniel Windisch
In this work we present descriptions of prime ideals and in particular of maximal ideals in products $R = prod D_lambda$ of families $(D_lambda)_{lambda in Lambda}$ of commutative rings. We show that every maximal ideal is induced by an ultrafilter on the Boolean algebra $prod mathcal{P}(max(D_lambda))$. If every $D_lambda$ is in a certain class of rings including finite character domains and one-dimensional domains, then this leads to a characterization of the maximal ideals of $R$. If every $D_lambda$ is a Prufer domain, we depict all prime ideals of $R$. Moreover, we give an example of a (optionally non-local or local) Prufer domain such that every non-zero prime ideal is of infinite height.
本文给出了交换环族$(D_lambda)_{lambda in Lambda}$积$R = prod D_lambda$中的素理想，特别是极大理想的描述。我们证明了在布尔代数$prod mathcal{P}(max(D_lambda))$上每一个极大理想都是由一个超滤波器诱导出来的。如果每个$D_lambda$都在包含有限特征域和一维域的某一类环中，那么这将导致$R$的最大理想的表征。如果每个$D_lambda$都是一个普鲁特域，我们描述了$R$的所有素数理想。此外，我们给出了一个(可选的非局部或局部)普鲁弗域的例子，使得每个非零素数理想都是无限高的。
{"title":"Prime ideals in infinite products of commutative rings","authors":"Carmelo A. Finocchiaro, Sophie Frisch, Daniel Windisch","doi":"10.1142/s0219199723500451","DOIUrl":"https://doi.org/10.1142/s0219199723500451","url":null,"abstract":"In this work we present descriptions of prime ideals and in particular of maximal ideals in products $R = prod D_lambda$ of families $(D_lambda)_{lambda in Lambda}$ of commutative rings. We show that every maximal ideal is induced by an ultrafilter on the Boolean algebra $prod mathcal{P}(max(D_lambda))$. If every $D_lambda$ is in a certain class of rings including finite character domains and one-dimensional domains, then this leads to a characterization of the maximal ideals of $R$. If every $D_lambda$ is a Prufer domain, we depict all prime ideals of $R$. Moreover, we give an example of a (optionally non-local or local) Prufer domain such that every non-zero prime ideal is of infinite height.","PeriodicalId":50660,"journal":{"name":"Communications in Contemporary Mathematics","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135805168","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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