Related papers: Local square mean in the hyperbolic circle problem

We consider two problems of estimation in high-dimensional Gaussian models. The first problem is that of estimating a linear functional of the means of $n$ independent $p$-dimensional Gaussian vectors, under the assumption that most of…

We consider space-time tracking optimal control problems for linear para\-bo\-lic initial boundary value problems that are given in the space-time cylinder $Q = \Omega \times (0,T)$, and that are controlled by the right-hand side…

In [FI09], Friedlander and Iwaniec studied the so-called Hyperbolic Prime Number Theorem, which asks for an infinitude of elements gamma in SL(2,Z) such that the norm squared |gamma|^2 = a^2 + b^2 + c^2 + d^2 = p, a prime. Under the…

We derive explicit bounds for the remainder term in the local Weyl law for locally hyperbolic manifolds, we also give the estimates of the derivative of this remainder. We use these to obtain explicit bounds for the C^k-norms of the…

Let $\psi$ be an $L^2$-normalized Hecke-Maass form with a large spectral parameter $\lambda>0$ on a compact arithmetic congruence hyperbolic 3-manifold $X=\Gamma\backslash\mathrm{SL}(2,\mathbb{C})/\mathrm{SU}(2)$, and let $Y$ be a totally…

We elaborate on the principle that for gapped quantum spin systems with local interaction "local perturbations [in the Hamiltonian] perturb locally [the ground state]". This principle was established in [Bachmann et al. 2012], relying on…

For any cofinite Fuchsian group $\Gamma\subset {\rm PSL}(2, \mathbb{R})$, we show that any set of $N$ points on the hyperbolic surface $\Gamma\backslash\mathbb{H}^2$ determines $\geq C_{\Gamma} \frac{N}{\log N}$ distinct distances for some…

Let $\Gamma$ be a geometrically-finite Fuchsian group acting on the upper half plane $\hh.$ Let $\E$ denote the set of elliptic fixed points of $\Gamma$ in $\hh.$ We give a lower bound on the minimal hyperbolic distance between points in…

We study the growth of the local $L^2$-norms of the unitary Eisenstein series for reductive groups over number fields, in terms of their parameters. We derive a \emph{poly-logarithmic} bound on an average, for a large class of reductive…

We investigate the distribution of orbits of a non-elementary discrete hyperbolic group acting on the n-dimensional hyperbolic space and its geometric boundary. In particular, we show that if the group $\Gamma$ admits a finite…

Let $\Gamma$ be a non-elementary Gromov-hyperbolic group, and $\partial \Gamma$ denote its Gromov boundary. We consider $\Gamma$-invariant proper $\delta$-hyperbolic, quasi-convex metric $d$ on $\Gamma$, and the associated…

Let $\Gamma\subset\mathrm{PSL}_{2}(\mathbb{R})$ be a Fuchsian subgroup of the first kind acting on the upper half-plane $\mathbb{H}$. Consider the $d_{2k}$-dimensional space of cusp forms $\mathcal{S}_{2k}^{\Gamma}$ of weight $2k$ for…

In this paper, a few dual least-squares finite element methods and their application to scalar linear hyperbolic problems are studied. The purpose is to obtain $L^2$-norm approximations on finite element spaces of the exact solutions to…

In the prototypical setting of non-Euclidean geometry, the 2-dimensional Real Hyperbolic space $\mathbb{H}^2$, we consider the Carleson's problem for the Schr\"odinger equation and improve the best known result until now by proving that the…

Upper and lower bounds are given for the maximum Euclidean curvature among faces in Bianchi's fundamental polyhedron for $PSL_2(O)$ in the upper-half space model of hyperbolic space, where $O$ is an imaginary quadratic ring of integers with…

The local behavior of the lowest order boundary element method on quasi-uniform meshes for Symm's integral equation and the stabilized hyper-singular integral equation on polygonal/polyhedral Lipschitz domains is analyzed. We prove local a…

We consider the mean curvature rigidity problem of an equatorial zone on a sphere which is symmetric about the equator with width $2w$. There are two different notions on rigidity, i.e. strong rigidity and local rigidity. We prove that for…

A classic theorem of Kazhdan and Margulis states that for any semisimple Lie group without compact factors, there is a positive lower bound on the covolume of lattices. H. C. Wang's subsequent quantitative analysis showed that the…

Analytic functions in the Hardy class $H^2$ over the upper half-plane $\mathbb{H}_+$ are uniquely determined by their values on any curve $\Gamma$ lying in the interior or on the boundary of $\mathbb{H}_+$. The goal of this paper is to…

Contrary to an assumption made by Bojanczyk, Higham, and Patel [SIAM J. Matrix Anal. Appl., 24(4):914-931, 2003], a perturbation bound for indefinite least square problems is capable of arbitrarily large overestimates for all perturbations…