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.

Let $ \Gamma_w $ be a non-cofinite Hecke triangle group with cusp width $ w>2 $ and let $ \varrho\colon\Gamma_w\to 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 $ \exp\left( C_{\varepsilon} \vert s\vert^{\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 any geometrically finite surface $ X = \widetilde{\Gamma}\backslash \mathbb{H}^2 $ whose fundamental group $ \widetilde{\Gamma} $ is a finite index, torsion-free subgroup of $ \Gamma_w $.

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.

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 $ \lambda $. 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 $ \lambda $ and on the time step. Finally, we did some numerical simulations which show the behavior of the solution.

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.

We consider a discrete dynamical system on a pseudo-Riemannian manifold and we determine the concept of a hyperbolic set for it. We insert a condition in the definition of a hyperbolic set which implies to the unique decomposition of a part of tangent space (at each point of this set) to two unstable and stable subspaces with exponentially increasing and exponentially decreasing dynamics on them. We prove the continuity of this decomposition via the metric created by a torsion-free pseudo-Riemannian connection. We present a global attractor for a diffeomorphism on an open submanifold of the hyperbolic space $H^{2}(1)$ which is not a hyperbolic set for it.

We describe the (complex) quaternionic geometry encoded by the embeddings of the Riemann sphere with nonnegative normal bundles.

We present a condition for towers of fiber bundles which implies that the\nfundamental group of the total space has a nilpotent subgroup of finite index\nwhose torsion is contained in its center. Moreover, the index of the subgroup\ncan be bounded in terms of the fibers of the tower.\n Our result is motivated by the conjecture that every almost nonnegatively\ncurved closed m-dimensional manifold M admits a finite cover M' for which the\nnumber of leafs is bounded in terms of m such that the torsion of the\nfundamental group of M' lies in its center.\n

We provide a coherent picture of our efforts thus far in extending real\nalgebra and its links to the theory of quadratic forms over ordered fields in\nthe noncommutative direction, using hermitian forms and "ordered" algebras with\ninvolution.\n