Diophantine bounds on the concentration function

Bounds on the concentration function in terms of Diophantine approximation

We demonstrate a simple analytic argument that may be used to bound the Lévy concentration function of a sum of independent random variables. The main application is a version of a recent inequality due to Rudelson and Vershynin, and its multidimensional generalisa-tion. Des bornes pour la fonction de concentration enmatì ere d'approximation Diophantienne. Nous montrons un simple raison-nement ...

A Generalization of a Theorem of Bumby on Quartic Diophantine Equations

Given a parametrized family of cubic models of elliptic curves Et over Q, it is a notoriously difficult problem to find absolute bounds for the number of integral points on Et (and, indeed, in many cases, it is unlikely such bounds even exist). Perhaps somewhat surprisingly, the situation is often radically different for quartic models. In a series of classical papers, Ljunggren (see e.g. [5] a...

Bounds on the restrained Roman domination number of a graph

A {em Roman dominating function} on a graph $G$ is a function$f:V(G)rightarrow {0,1,2}$ satisfying the condition that everyvertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex$v$ for which $f(v) =2$. {color{blue}A {em restrained Roman dominating}function} $f$ is a {color{blue} Roman dominating function if the vertices with label 0 inducea subgraph with no isolated vertex.} The wei...

On Solving Linear Diophantine Systems Using Generalized Rosser's Algorithm

An algorithm for solving a diophantine equation with lower and upper bounds on the variables

We develop an algorithm for solving a diophantine equation with lower and upper bounds on the variables. The algorithm is based on lattice basis reduction, and rst nds short vectors satisfying the diophantine equation. The next step is to branch on linear combinations of these vectors, which either yields a vector that satis es the bound constraints or provides a proof that no such vector exist...