Our aim in this paper is to study the existence and uniqueness of solution for hyperbolic relaxations of higher-order anisotropic Caginalp phase field systems with homogeous Dirichlet boundary conditions with regular potentials.
{"title":"On higher-order anisotropic perturbed Caginalp phase field systems","authors":"Clesh Deseskel Elion Ekohela, D. Moukoko","doi":"10.3934/ERA.2019.26.004","DOIUrl":"https://doi.org/10.3934/ERA.2019.26.004","url":null,"abstract":"Our aim in this paper is to study the existence and uniqueness of solution for hyperbolic relaxations of higher-order anisotropic Caginalp phase field systems with homogeous Dirichlet boundary conditions with regular potentials.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79810753","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Hawraa Alsayed, Hussein Fakih, A. Miranville, A. Wehbe
This paper is dedicated to study the fully discretized semi implicit and implicit schemes of a 2D parabolic semi linear problem modeling MEMS devices. Starting with the analysis of the semi-implicit scheme, we proved the existence of the discrete solution which converges under certain conditions on the voltage begin{document}$ lambda $end{document} . On the other hand, we consider a fully implicit scheme, we proved the existence of the discrete solution, which also converges to the stationary solution under certain conditions on the voltage begin{document}$ lambda $end{document} and on the time step. Finally, we did some numerical simulations which show the behavior of the solution.
This paper is dedicated to study the fully discretized semi implicit and implicit schemes of a 2D parabolic semi linear problem modeling MEMS devices. Starting with the analysis of the semi-implicit scheme, we proved the existence of the discrete solution which converges under certain conditions on the voltage begin{document}$ lambda $end{document} . On the other hand, we consider a fully implicit scheme, we proved the existence of the discrete solution, which also converges to the stationary solution under certain conditions on the voltage begin{document}$ lambda $end{document} and on the time step. Finally, we did some numerical simulations which show the behavior of the solution.
{"title":"Finite difference scheme for 2D parabolic problem modelling electrostatic Micro-Electromechanical Systems","authors":"Hawraa Alsayed, Hussein Fakih, A. Miranville, A. Wehbe","doi":"10.3934/ERA.2019.26.005","DOIUrl":"https://doi.org/10.3934/ERA.2019.26.005","url":null,"abstract":"This paper is dedicated to study the fully discretized semi implicit and implicit schemes of a 2D parabolic semi linear problem modeling MEMS devices. Starting with the analysis of the semi-implicit scheme, we proved the existence of the discrete solution which converges under certain conditions on the voltage begin{document}$ lambda $end{document} . On the other hand, we consider a fully implicit scheme, we proved the existence of the discrete solution, which also converges to the stationary solution under certain conditions on the voltage begin{document}$ lambda $end{document} and on the time step. Finally, we did some numerical simulations which show the behavior of the solution.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85330608","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We derive, from the work of M. Ratner on joinings of time-changes of horocycle flows and from the result of the authors on its cohomology, the property of orthogonality of powers for non-trivial smooth time-changes of horocycle flows on compact quotients. Such a property is known to imply P. Sarnak's Mobius orthogonality conjecture, already known for horocycle flows by the work of J. Bourgain, P. Sarnak and T. Ziegler.
{"title":"Orthogonal powers and Möbius conjecture for smooth time changes of horocycle flows","authors":"L. Flaminio, G. Forni","doi":"10.3934/ERA.2019.26.002","DOIUrl":"https://doi.org/10.3934/ERA.2019.26.002","url":null,"abstract":"We derive, from the work of M. Ratner on joinings of time-changes of horocycle flows and from the result of the authors on its cohomology, the property of orthogonality of powers for non-trivial smooth time-changes of horocycle flows on compact quotients. Such a property is known to imply P. Sarnak's Mobius orthogonality conjecture, already known for horocycle flows by the work of J. Bourgain, P. Sarnak and T. Ziegler.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86541287","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $Gamma_{w}$ be a non-cofinite Hecke triangle group with cusp width $w>2$ and let $varrhocolonGamma_wto U(V)$ be a finite-dimensional unitary representation of $Gamma_w$. In this note we announce a new fractal upper bound for the Selberg zeta function of $Gamma_{w}$ twisted by $varrho$. In strips parallel to the imaginary axis and bounded away from the real axis, the Selberg zeta function is bounded by $expleft( C_{varepsilon} vert svert^{delta + varepsilon} right)$, where $delta = delta_{w}$ denotes the Hausdorff dimension of the limit set of $Gamma_{w}$. This bound implies fractal Weyl bounds on the resonances of the Laplacian for all geometrically finite surfaces $X=widetilde{Gamma}backslashmathbb{H}$ where $widetilde{Gamma}$ is a finite index, torsion-free subgroup of $Gamma_w$.
{"title":"Fractal Weyl bounds and Hecke triangle groups","authors":"Fr'ed'eric Naud, A. Pohl, Louis Soares","doi":"10.3934/ERA.2019.26.003","DOIUrl":"https://doi.org/10.3934/ERA.2019.26.003","url":null,"abstract":"Let $Gamma_{w}$ be a non-cofinite Hecke triangle group with cusp width $w>2$ and let $varrhocolonGamma_wto U(V)$ be a finite-dimensional unitary representation of $Gamma_w$. In this note we announce a new fractal upper bound for the Selberg zeta function of $Gamma_{w}$ twisted by $varrho$. In strips parallel to the imaginary axis and bounded away from the real axis, the Selberg zeta function is bounded by $expleft( C_{varepsilon} vert svert^{delta + varepsilon} right)$, where $delta = delta_{w}$ denotes the Hausdorff dimension of the limit set of $Gamma_{w}$. This bound implies fractal Weyl bounds on the resonances of the Laplacian for all geometrically finite surfaces $X=widetilde{Gamma}backslashmathbb{H}$ where $widetilde{Gamma}$ is a finite index, torsion-free subgroup of $Gamma_w$.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89596400","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We develop a version of cluster algebra extending the ring of Laurent polynomials by adding Grassmann variables. These algebras can be described in terms of "extended quivers," which are oriented hypergraphs. We describe mutations of such objects and define a corresponding commutative superalgebra. Our construction includes the notion of weighted quivers that has already appeared in different contexts. This paper is a step towards understanding the notion of cluster superalgebra.
{"title":"Cluster algebras with Grassmann variables","authors":"V. Ovsienko, M. Shapiro","doi":"10.3934/ERA.2019.26.001","DOIUrl":"https://doi.org/10.3934/ERA.2019.26.001","url":null,"abstract":"We develop a version of cluster algebra extending the ring of Laurent polynomials by adding Grassmann variables. These algebras can be described in terms of \"extended quivers,\" which are oriented hypergraphs. We describe mutations of such objects and define a corresponding commutative superalgebra. Our construction includes the notion of weighted quivers that has already appeared in different contexts. This paper is a step towards understanding the notion of cluster superalgebra.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87493893","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce a new construction of matrix wreath products of algebras that is similar to the construction of wreath products of groups introduced by L. Kaloujnine and M. Krasner [ 17 ]. We then illustrate its usefulness by proving embedding theorems into finitely generated algebras and constructing nil algebras with prescribed Gelfand-Kirillov dimension.
{"title":"On matrix wreath products of algebras","authors":"A. Alahmadi, H. Alsulami, S. Jain, E. Zelmanov","doi":"10.3934/ERA.2017.24.009","DOIUrl":"https://doi.org/10.3934/ERA.2017.24.009","url":null,"abstract":"We introduce a new construction of matrix wreath products of algebras that is similar to the construction of wreath products of groups introduced by L. Kaloujnine and M. Krasner [ 17 ]. We then illustrate its usefulness by proving embedding theorems into finitely generated algebras and constructing nil algebras with prescribed Gelfand-Kirillov dimension.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41623969","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a condition for towers of fiber bundles which implies that the fundamental group of the total space has a nilpotent subgroup of finite index whose torsion is contained in its center. Moreover, the index of the subgroup can be bounded in terms of the fibers of the tower. Our result is motivated by the conjecture that every almost nonnegatively curved closed begin{document}$ m $end{document} -dimensional manifold begin{document}$ M $end{document} admits a finite cover begin{document}$ tilde M $end{document} for which the number of leafs is bounded in terms of begin{document}$ m $end{document} such that the torsion of the fundamental group begin{document}$ π_1 tilde M $end{document} lies in its center.
{"title":"On the torsion in the center conjecture","authors":"V. Kapovitch, A. Petrunin, Wilderich Tuschmann","doi":"10.3934/era.2018.25.004","DOIUrl":"https://doi.org/10.3934/era.2018.25.004","url":null,"abstract":"We present a condition for towers of fiber bundles which implies that the fundamental group of the total space has a nilpotent subgroup of finite index whose torsion is contained in its center. Moreover, the index of the subgroup can be bounded in terms of the fibers of the tower. Our result is motivated by the conjecture that every almost nonnegatively curved closed begin{document}$ m $end{document} -dimensional manifold begin{document}$ M $end{document} admits a finite cover begin{document}$ tilde M $end{document} for which the number of leafs is bounded in terms of begin{document}$ m $end{document} such that the torsion of the fundamental group begin{document}$ π_1 tilde M $end{document} lies in its center.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43925956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Hajek [ 3 ] showed that a dynamical system on a Tychonoff space with paracompact orbit space is parallelizable if and only if its corresponding bundle is a locally trivial fiber bundle with fiber begin{document}$mathbb{R}$end{document} . The present paper provides an enhancement for this classical theorem by omitting all topological hypotheses.
{"title":"A note on parallelizable dynamical systems","authors":"J. Souza, T. A. Pacifico, Hélio V. M. Tozatti","doi":"10.3934/ERA.2017.24.007","DOIUrl":"https://doi.org/10.3934/ERA.2017.24.007","url":null,"abstract":"Hajek [ 3 ] showed that a dynamical system on a Tychonoff space with paracompact orbit space is parallelizable if and only if its corresponding bundle is a locally trivial fiber bundle with fiber begin{document}$mathbb{R}$end{document} . The present paper provides an enhancement for this classical theorem by omitting all topological hypotheses.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47813337","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We provide necessary conditions for the refined version of the Brascamp-Lieb inequality where the input functions are allowed to belong to Lorentz spaces, thereby establishing the sharpness of the range of Lorentz exponents in the subcritical case. Using similar considerations, some sharp refinements of the Strichartz estimates for the kinetic transport equation are established.
{"title":"Sharpness of the Brascamp–Lieb inequality in Lorentz spaces","authors":"N. Bez, Sanghyuk Lee, Shohei Nakamura, Y. Sawano","doi":"10.3934/ERA.2017.24.006","DOIUrl":"https://doi.org/10.3934/ERA.2017.24.006","url":null,"abstract":"We provide necessary conditions for the refined version of the Brascamp-Lieb inequality where the input functions are allowed to belong to Lorentz spaces, thereby establishing the sharpness of the range of Lorentz exponents in the subcritical case. Using similar considerations, some sharp refinements of the Strichartz estimates for the kinetic transport equation are established.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43087648","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study the Dirichlet boundary value problem of a class of nonlinear parabolic equations. By a priori estimates, difference and variation techniques, we establish the existence and uniqueness of weak solutions of this problem.
{"title":"Existence and uniqueness of weak solutions for a class of nonlinear parabolic equations","authors":"Peiying Chen","doi":"10.3934/ERA.2017.24.005","DOIUrl":"https://doi.org/10.3934/ERA.2017.24.005","url":null,"abstract":"In this paper, we study the Dirichlet boundary value problem of a class of nonlinear parabolic equations. By a priori estimates, difference and variation techniques, we establish the existence and uniqueness of weak solutions of this problem.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47494394","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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