Related papers: Local square mean in the hyperbolic circle problem

We calculate the region of the MSSM parameter space (i.e. $M_{1/2}$, $m_{0}$, $\mu$, \ldots) compatible with a correct electroweak breaking and a realistic top-quark mass. To do so we have included {\em all} the one-loop corrections to the…

In this paper, we introduce the notion of "geodesic cover" for Fuchsian groups, which summons copies of fundamental polygons in the hyperbolic plane to cover pairs of representatives realizing distances in the corresponding hyperbolic…

For $\Gamma$ a cofinite Kleinian group acting on $\mathbb{H}^3$, we study the Prime Geodesic Theorem on $M=\Gamma \backslash \mathbb{H}^3$, which asks about the asymptotic behaviour of lengths of primitive closed geodesics (prime geodesics)…

A quantitative regularity theory is developed for weak solutions to the parabolic system $$ \partial_t u-\mathrm{div}\,{\boldsymbol{\mathsf A}}(x,t,Du)=0 \quad\text{in }E_T\subset \mathbb{R}^N\times\mathbb{R}, $$ which features the…

Quantum estimation theory provides optimal observations for various estimation problems for unknown parameters in the state of the system under investigation. However, the theory has been developed under the assumption that every observable…

This paper mainly discuss the regularity behavior of the hyperbolic magnetic Schroedinger equation with singular coefficients near the origin. We apply the techniques from the microlocal analysis to explore the upper bound of loss of…

For the Hamiltonian operator H = -{\Delta}+V(x) of the Schr\"odinger Equation with a repulsive potential, the problem of local decay is considered. It is analyzed by a direct method, based on a new, L^2 bounded, propagation observable. The…

Let $u$ be a nonnegative, local, weak solution to the porous medium equation for $m\ge2$ in a space-time cylinder $\Omega_T$. Fix a point $(x_o,t_o)\in\Omega_T$: if the average \[…

This is the main paper in a sequence in which we give a complete proof of the bounded $L^2$ curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the…

It is shown that a construction of Z. Zhang and Y. Xiao on open subsets of Ptolemaic spaces yields, when the subset has boundary containing at least two points, metrics that are Gromov hyperbolic with parameter $\log 2$ and strongly…

For the Srzednicki-W\'ojcik equation, the planar nonautonomous ODE parameterized by $\kappa \in \mathbb{R}$, $$ z'=\overline{z}(1+ |z|^2 \exp(i \kappa t)), \qquad z(t) \in \mathbb{C} $$ using averaging we show how the region of hyperbolicty…

Consider a closed Riemannian $n$-manifold $M$ admitting a negatively curved Riemannian metric. We show that for every Riemannian metric on $M$ of sufficiently small volume, there is a point in the universal cover of $M$ such that the volume…

Let $\Gamma$ be a cofinite Fuchsian group acting on hyperbolic two-space $\HH.$ Let $M=\Gamma \setminus \HH $ be the corresponding quotient space. For $\gamma,$ a closed geodesic of $M$, let $l(\gamma)$ denote its length. The prime geodesic…

We study the fine distribution of lattice points lying on expanding circles in the hyperbolic plane $\mathbb{H}$. The angles of lattice points arising from the orbit of the modular group $PSL_{2}(\mathbb{Z})$, and lying on hyperbolic…

Hilbert-Arnold (HA) problem, motivated by Hilbert 16-th problem, is to prove that for a generic k-parameter family of smooth vector fields {\dot x=v(x,\eps)}_{\eps\in B^k} on the 2-dimensional sphere S^2 has uniformly bounded number of…

Hardy conjectured that the error term arising from approximating the number of lattice points lying in a radius-$R$ disc by its area is $O(R^{1/2+o(1)})$. One source of support for this conjecture is a folklore heuristic that uses i.i.d.…

We establish the local H\"older regularity of the spatial gradient of bounded weak solutions $u\colon E_T\to\R^k$ to the non-linear system of parabolic type \begin{equation*} \partial_tu-\Div\Big(…

In this paper, we prove a local limit theorem for the chi-square distribution with $r > 0$ degrees of freedom and noncentrality parameter $\lambda \geq 0$. We use it to develop refined normal approximations for the survival function. Our…

We study the random connection model on hyperbolic space $\mathbb{H}^d$ in dimension $d=2,3$. Vertices of the spatial random graph are given as a Poisson point process with intensity $\lambda>0$. Upon variation of $\lambda$ there is a…

We determine the mean number of 2-torsion elements in class groups of cubic orders, when such orders are enumerated by discriminant. Specifically, we prove that when isomorphism classes of totally real (resp., complex) cubic orders are…