Related papers: Hyperbolic angles from Heegner points

Let $\calM=\Gamma\bs \calH^{(n)}$, where $\calH^{(n)}$ is a product of $n+1$ hyperbolic planes and $\Gamma\subset\PSL(2,\bbR)^{n+1}$ is an irreducible cocompact lattice. We consider closed geodesics on $\calM$ that propagate locally only in…

We study the error of the number of unimodular lattice points that fall into a dilated and centred ellipse around $0$. We first show that the study of the error, when the error is normalized by $\sqrt{t}$ with $t$ the parameter of…

We investigate number systems for the ring of integers of hyperbolic and dual numbers. We characterize all canonical number systems providing radix form for hyperbolic and dual numbers. Our approach allows us to get suitable bases by means…

Let $\omega$ be a point in the upper half plane, and let $\Gamma$ be a discrete, finite covolume subgroup of $\mathrm{PSL}_2(\mathbb{R})$. We conjecture an explicit formula for the pair correlation of the angles between geodesic rays of the…

For $\Gamma$ a cocompact or cofinite Fuchsian group, we study the lattice point problem on the Riemann surface $\Gamma\backslash\mathbb{H}$. The main asymptotic for the counting of the orbit $\Gamma z$ inside a circle of radius $r$ centered…

Recently, Haase and Ilten initiated the study of classifying algebraically hyperbolic surfaces in toric threefolds. We complete this classification for $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$, $\mathbb{P}^2 \times…

The causal structure of Einstein's evolution equations is considered. We show that in general they can be written as a first order system of balance laws for any choice of slicing or shift. We also show how certain terms in the evolution…

The continuous point symmetry algebra of the hyperbolic Ernst equation is presented. In a second step the corresponding group transformations are considered. Accordingly, the solutions of the hyperbolic Ernst equation that are invariant…

We explicate a number of notions of algebraic laminations existing in the literature, particularly in the context of an exact sequence $$1\to H\to G \to Q \to 1 $$ of hyperbolic groups. These laminations arise in different contexts:…

We study the problem of counting lattice points of a polytope that are weighted by an Ehrhart quasi-polynomial of a family of parametric polytopes. As applications one can compute integrals and maximum values of such quasi-polynomials, as…

We estimate the distribution of relatively $r$-prime lattice points in number fields $K$ with their components having a norm less than $x$. In the previous paper we obtained uniform upper bounds as $K$ runs through all number fields under…

We fully characterize the set of finite shapes with minimal perimeter on hyperbolic lattices given by regular tilings of the hyperbolic plane whose tiles are regular $p$-gons meeting at vertices of degree $q$, with $1/p+1/q<\frac{1}{2}$. In…

The construction of discrete velocity models or numerical methods for the Boltzmann equation, may lead to the necessity of computing the collision operator as a sum over lattice points. The collision operator involves an integral over a…

Let $\Lambda$ be a non-elementary convex co-compact fuchsian group which is a subgroup of an arithmetic fuchsian group. We prove that the Laplace operator of the hyperbolic surface $X=\Lambda \backslash\H$ has infinitely many resonances in…

We consider the question of how approximations satisfying Dirichlet's theorem spiral around vectors in $\mathbb{R}^d$. We give pointwise almost everywhere results (using only the Birkhoff ergodic theorem on the space of lattices). In…

The Hessian Topology is a subject with interesting relations with some classical problems of analysis and geometry. In this article we prove a conjecture on this subject stated by V.I. Arnold concerning the number of connected components of…

We investigate degenerate cross-diffusion equations with a rank-deficient diffusion matrix that are considered to model populations which move as to avoid spatial crowding and have recently been found to arise in a mean-field limit of…

We study existence, uniqueness, and distributional aspects of generalized solutions to the Cauchy problem for first-order symmetric (or Hermitian) hyperbolic systems of partial differential equations with Colombeau generalized functions as…

Let $Y$ be a projective submanifold of the total space of the inverse of a very ample line bundle $\pi:L^{-1}\rightarrow B$ over a projective manifold $B$. Any section of $L^{-1}\rightarrow B$ is isomorphic to $B$ and the Hodge numbers of…

Wigner limits are given formally as the difference between a lattice sum, associated to a positive definite quadratic form, and a corresponding multiple integral. To define these limits, which arose in work of Wigner on the energy of static…