On Permutation Binomials over Finite Fields

Let $\mathbb{F}_{q}$ be the finite field of characteristic $p$ containing $q = p^{r}$ elements and $f(x)=ax^{n} + x^{m}$ a binomial with coefficients in this field. If some conditions on the gcd of $n-m$ an $q-1$ are satisfied then this polynomial does not permute the elements of the field. We prove in particular that if $f(x) = ax^{n} + x^{m}$ permutes $\mathbb{F}_{p}$, where $n>m>0$ and $a \in {\mathbb{F}_{p}}^{*}$, then $p -1 \leq (d -1)d$, where $d = {{gcd}}(n-m,p-1)$, and that this bound of $p$ in term of $d$ only, is sharp. We show as well how to obtain in certain cases a permutation binomial over a subfield of $\mathbb{F}_{q}$ from a permutation binomial over $\mathbb{F}_{q}$

On stochastic differential equations driven by the renormalized square of the Gaussian white noise

We investigate the properties of the Wick square of Gaussian white noises through a new method to perform non linear operations on Hida distributions. This method lays in between the Wick product interpretation and the usual definition of nonlinear functions. We prove on Ito-type formula and solve stochastic differential equations driven by the renormalized square of the Gaussian white noise. Our approach works with standard assumptions on the coefficients of the equations, Lipschitz continuity and linear growth condition, and produces existence and uniqueness results in the space where the noise lives. The linear case is studied in details and positivity of the solution is proved.Comment: 23 page

An unexpected application of minimization theory to module decompositions

The aim of this work is to show how we can decompose a module (if decomposable) into an indecomposable module with the help of the minimization process.Comment: 15-02-200

Verification of Timed Automata Using Rewrite Rules and Strategies

ELAN is a powerful language and environment for specifying and prototyping deduction systems in a language based on rewrite rules controlled by strategies. Timed automata is a class of continuous real-time models of reactive systems for which efficient model-checking algorithms have been devised. In this paper, we show that these algorithms can very easily be prototyped in the ELAN system. This paper argues through this example that rewriting based systems relying on rules and strategies are a good framework to prototype, study and test rather efficiently symbolic model-checking algorithms, i.e. algorithms which involve combination of graph exploration rules, deduction rules, constraint solving techniques and decision procedures