1. Introduction. In [Ch] we gave some large sieve type inequalities involving elements of harmonic analysis in Riemann surfaces and compact Riemannian manifolds. In this paper we present some of their applications. Our results are related to the hyperbolic circle problem, which is a generalization of the classical circle problem. The latter can be formulated as counting the images of a point in...

We describe a statistical analysis of the eye tracker measurements in a database with 15 observers viewing 1003 images under free-viewing conditions. In contrast to the common approach of investigating the properties of the fixation points we analyze the properties of the transition phases between fixations. We introduce hyperbolic geometry as a tool to measure the step length between consecuti...

Contents 1. Hyperbolic geometry 1 2. Fuchsian groups and hyperbolic surfaces 4 3. Spectrum and resolvent 11 4. Spectral theory: finite-area case 16 5. Spectral theory: infinite-area case 20 6. Selberg trace formula 23 7. Arithmetic surfaces 33 References 44 1. Hyperbolic geometry In complex analysis, we learn that the upper half-plane H = {Im z > 0} has a large group of conformal automorphisms,...

We consider dynamical systems on compact manifolds, which are local diffeomorphisms outside an exceptional set (a compact submanifold). We are interested in analyzing the relation between the integrability (with respect to Lebesgue measure) of the first hyperbolic time map and the existence of positive frequency of hyperbolic times. We show that some (strong) integrability of the first hyperbol...

Horosphere is surface in hyperbolic space that is isometric to the Euclidean plane. In order to correctly visualize hyperbolic space we embed flat computer screen as horosphere and investigate geometry of central projection of hyperbolic space onto horosphere. We also discuss realization of hyperbolic isometries. Corresponding algorithms are implemented in Mathematica package L3toHorospere. We ...

In this paper, we consider hyperbolic rotation (G0), hyperbolic translation (G1) and horocyclic rotation (G2) groups in H3, which is called Minkowski model of hyperbolic space. Then, we investigate extrinsic differential geometry of invariant surfaces under subgroups of G0 in H3. Also, we give explicit parametrization of this invariant surfaces with respect to constant hyperbolic curvature of p...

We consider both hyperbolic sets and partially hyperbolic sets attracting a set of points with positive volume in a Riemannian manifold. We obtain several results on the topological structure of such sets for diffeomorphisms whose differentiability is bigger than one. We show in particular that there are no partially hyperbolic horseshoes with positive volume for such diffeomorphisms. We also g...

In this note we show that there are algebraic families of hyperbolic, Fermat-Waring type hypersurfaces in P of degree 4(n− 1)2, for all dimensions n 2. Moreover, there are hyperbolic Fermat-Waring hypersurfaces in P of degree 4n2−2n +1 possessing complete hyperbolic, hyperbolically embedded complements. Many examples have been given of hyperbolic hypersurfaces in P3 (e.g., see [ShZa] and the li...

We propose and analyse a new finite element method for convection diffusion problems based on the combination of a mixed method for the elliptic and a discontinuous Galerkin method for the hyperbolic part of the problem. The two methods are made compatible via hybridization and the combination of both is appropriate for the solution of intermediate convection-diffusion problems. By construction...

Dispersive and Strichartz estimates for solutions to general strictly hyperbolic partial differential equations with constant coefficients are considered. The global time decay estimates of Lp−Lq norms of propagators are obtained, and it is shown how the time decay rates depend on the geometry of the problem. The frequency space is separated in several zones each giving a certain decay rate. Ge...