Related papers: Lattice counting problem
We examine the moments of the number of lattice points in a fixed ball of volume $V$ for lattices in Euclidean space which are modules over the ring of integers of a number field $K$. In particular, denoting by $\omega_K$ the number of…
An advection--diffusion-limited dissolution model of an object being eroded by a two-dimensional potential flow is presented. By taking advantage of the conformal invariance of the model, a numerical method is introduced that tracks the…
For a large class of expanding maps of the interval, we prove that partial sums of Lipschitz observables satisfy an almost sure central limit theorem (ASCLT). In fact, we provide a speed of convergence in the Kantorovich metric. Maxima of…
We extend to Lipschitz continuous functionals either of the true paths or of the Euler scheme with decreasing step of a wide class of Brownian ergodic diffusions, the Central Limit Theorems formally established for their marginal empirical…
This article performs a unified convergence analysis of a variety of numerical methods for a model of the miscible displacement of one incompressible fluid by another through a porous medium. The unified analysis is enabled through the…
In this paper, we extend the central limit theorem of the occupation time of the voter model on the lattice $\mathbb{Z}^d$ given in \cite{Cox1983} to the sample path case for $d\geq 3$. The proof of our main result utilizes the resolvent…
Self-normalized processes arise naturally in statistical applications. Being unit free, they are not affected by scale changes. Moreover, self-normalization often eliminates or weakens moment assumptions. In this paper we present several…
Expectation Propagation is a very popular algorithm for variational inference, but comes with few theoretical guarantees. In this article, we prove that the approximation errors made by EP can be bounded. Our bounds have an asymptotic…
We consider the Dirichlet problem for elliptic systems with periodically distributed inclusions whose conduction parameter exhibits a significant contrast compared to the background media. We develop a unified method to quantify the…
The evaluation of the interaction between objects arranged on a lattice requires the computation of lattice sums. A scenario frequently encountered are systems governed by the Helmholtz equation in the context of electromagnetic scattering…
Consider a stationary Poisson process $\eta$ in the $d$-dimensional Euclidean or hyperbolic space and construct a random graph with vertex set $\eta$ as follows. First, each point $x\in\eta$ is connected by an edge to its nearest neighbour,…
This paper introduces the order-theoretic concept of lattices along with the concept of consistent quantification where lattice elements are mapped to real numbers in such a way that preserves some aspect of the order-theoretic structure.…
The analyses of interior penalty discontinuous Galerkin methods of any order k for solving elliptic and parabolic problems with Dirac line sources are presented. For the steady state case, we prove convergence of the method by deriving a…
The Ehrhart polynomial and Ehrhart series count lattice points in integer dilations of a lattice polytope. We introduce and study a $q$-deformation of the Ehrhart series, based on the notions of harmonic spaces and Macaulay's inverse…
The $B$-meson decay constant can be measured on the lattice using a $1/m_b$ expansion. To relate the physical quantity to Monte Carlo data one has to know the renormalization coefficient, $Z$, between the lattice operators and their…
In this paper we consider a branching particle system consisting of particles moving according to the Ornstein-Uhlenbeck process in $\Rd$ and undergoing a binary, supercritical branching with a constant rate $\lambda>0$. This system is…
In this paper, we study the pointwise convergence of centain continuous-time polynomial ergodic averages. Our approach is based on the topological models of measurable flows. One of the main results of this paper is as follows: Let $a\in…
Let $\| \cdot \|$ be the euclidean norm on ${\bf R}^n$ and $\gamma_n$ the (standard) Gaussian measure on ${\bf R}^n$ with density $(2 \pi )^{-n/2} e^{- \| x\|^2 /2}$. Let $\vartheta$ ($ \simeq 1.3489795$) be defined by $\gamma_1 ([ -…
For random combinatorial optimization problems, there has been much progress in establishing laws of large numbers and computing limiting constants for the optimal value of various problems. However, there has not been as much success in…
Zeckendorf's Theorem states that any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers. We consider higher-dimensional lattice analogues, where a legal decomposition of a number $n$ is a collection of…