Electronic Research Announcements

 2019 , Volume 26

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Cluster algebras with Grassmann variables
Valentin Ovsienko and MichaeL Shapiro
2019, 26: 1-15 doi: 10.3934/era.2019.26.001 +[Abstract](1738) +[HTML](614) +[PDF](390.98KB)
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.

Orthogonal powers and Möbius conjecture for smooth time changes of horocycle flows
Livio Flaminio and Giovanni Forni
2019, 26: 16-23 doi: 10.3934/era.2019.26.002 +[Abstract](840) +[HTML](486) +[PDF](335.03KB)
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 Möbius orthogonality conjecture, already known for horocycle flows by the work of J. Bourgain, P. Sarnak and T. Ziegler.

Fractal Weyl bounds and Hecke triangle groups
Frédéric Naud, Anke Pohl and Louis Soares
2019, 26: 24-35 doi: 10.3934/era.2019.26.003 +[Abstract](453) +[HTML](179) +[PDF](368.48KB)
Abstract:

Let \begin{document}$ \Gamma_w $\end{document} be a non-cofinite Hecke triangle group with cusp width \begin{document}$ w>2 $\end{document} and let \begin{document}$ \varrho\colon\Gamma_w\to U(V) $\end{document} be a finite-dimensional unitary representation of \begin{document}$ \Gamma_w $\end{document}. In this note we announce a new fractal upper bound for the Selberg zeta function of \begin{document}$ \Gamma_w $\end{document} twisted by \begin{document}$ \varrho $\end{document}. In strips parallel to the imaginary axis and bounded away from the real axis, the Selberg zeta function is bounded by \begin{document}$ \exp\left( C_{\varepsilon} \vert s\vert^{\delta + \varepsilon} \right) $\end{document}, where \begin{document}$ \delta = \delta_w $\end{document} denotes the Hausdorff dimension of the limit set of \begin{document}$ \Gamma_w. $\end{document} This bound implies fractal Weyl bounds on the resonances of the Laplacian for any geometrically finite surface \begin{document}$ X = \widetilde{\Gamma}\backslash \mathbb{H}^2 $\end{document} whose fundamental group \begin{document}$ \widetilde{\Gamma} $\end{document} is a finite index, torsion-free subgroup of \begin{document}$ \Gamma_w $\end{document}.

On higher-order anisotropic perturbed Caginalp phase field systems
Clesh Deseskel Elion Ekohela and Daniel Moukoko
2019, 26: 36-53 doi: 10.3934/era.2019.26.004 +[Abstract](352) +[HTML](170) +[PDF](385.18KB)
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.

Finite difference scheme for 2D parabolic problem modelling electrostatic Micro-Electromechanical Systems
Hawraa Alsayed, Hussein Fakih, Alain Miranville and Ali Wehbe
2019, 26: 54-71 doi: 10.3934/era.2019.26.005 +[Abstract](424) +[HTML](206) +[PDF](710.52KB)
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.

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