Sets of integers form a monoid, where the product of two sets and is defined as the set containing for all $a\in A$ and $b\in B$. We give a characterization of when a family of finite sets is a code in this monoid, that is when the sets do not satisfy any nontrivial relation. We also extend this result for some infinite sets, including all infinite rational sets.

Let be a distribution function (d.f) in the domain of attraction of an extreme value distribution $H_{\gamma }$; it is well-known that , where is the d.f of the excesses over , converges, when tends to , the end-point of , to $G_{\gamma }\left(\frac{x}{\sigma \left(u\right)}\right)$, where $G_{\gamma }$ is the d.f. of the Generalized Pareto Distribution. We provide conditions that ensure that there exists, for $\gamma >-1$, a function which verifies ${lim}_{u\to s_{+}\left(F\right)}\Lambda \left(u\right)=\gamma$ and is such that $\Delta \left(u\right)={sup}_{x\in [0,s_{+}\left(F\right)-u[}|{\overline{F}}_u\left(x\right)-{\overline{G}}_{\Lambda \left(u\right)}(x/\sigma \left(u\right))|$ converges to faster than $d\left(u\right)={sup}_{x\in [0,s_{+}\left(F\right)-u[}|{\overline{F}}_u\left(x\right)-{\overline{G}}_{\gamma }(x/\sigma \left(u\right))|$.

Let $𝒬$ be a partition of a polygonal domain of the plan into convexe quadrilaterals. Given a regular function , we construct a function πƒ ∈ (Ω), interpolating position values and derivatives of up of order 2 at vertices of $𝒬.$ On each quadrilateral $Q\in 𝒬,$ πƒ is a finite element obtained from a polynomial scheme of FVS type by adding some rational functions.

We consider an estimate of the mode of a multivariate probability density with support in ${ℝ}^d$ using a kernel estimate drawn from a sample . The estimate is defined as any in {} such that $f_n\left(x\right)={max}_{i=1,\cdots ,n}{f}_n\left(X_i\right)$. It is shown that behaves asymptotically as any maximizer ${\widehat{\theta }}_n$ of . More precisely, we prove that for any sequence ${\left(r_n\right)}_{n\ge 1}$ of positive real numbers such that $r_n\to \infty$ and $r_n^{d}logn/n\to 0$, one has $r_n\parallel {\theta }_n-{\widehat{\theta }}_n\parallel \to 0$ in probability. The asymptotic normality of follows without further work.