A counterexample to the local–global principle of linear dependence for Abelian varieties

Let A be an Abelian variety defined over a number field k. Let P be a point in $A(k)$ and let X be a subgroup of $A(k)$. Gajda and Kowalski asked in 2002 whether it is true that the point P belongs to X if and only if the point $(P\mathrm{mod}\mathfrak{p})$ belongs to $(X\mathrm{mod}\mathfrak{p})$ for all but finitely many primes $\mathfrak{p}$ of k. We provide a counterexample.     

Tail of a linear diffusion with Markov switching

Let Y be a Ornstein–Uhlenbeck diffusion governed by a stationary and ergodic Markov jump process X, i.e. $\mathrm{d}{Y}_t=a(X_t)Y_t\mathrm{d}t+\sigma (X_t)\mathrm{d}{W}_t$, $Y_0=y_0$. Ergodicity conditions for Y have been obtained. Here we investigate the tail property of the stationary distribution of this model. A characterization of the only two possible cases is established: light tail or polynomial tail. Our method is based on discretizations and renewal theory. To cite this article: B. de Saporta, J.-F. Yao, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Estimation par ondelettes dans les systèmes dynamiques

Let $\{X_t,t\in \mathbb{Z}\}$ be a stochastic process valued on $[0,1]$ where $X_{t+1}=\phi (X_t)$, φ a piecewise expanding map, preserving measure μ with density f. We give consistent estimates of the invariant measure f and the map φ with the linear wavelets method using the base defined by Cohen, DeVore and Daubechies. We obtain the optimal rate of convergence of the MISE. To cite this article: M.-L. Vanharen, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On functors preserving skeletal maps and skeletally generated compacta

A map $f:X\to Y$ between topological spaces is skeletal if the preimage $f^{?1}(A)$ of each nowhere dense subset $A?Y$ is nowhere dense in X. We prove that a normal functor $F:\mathbf{Comp}\to \mathbf{Comp}$ is skeletal (which means that F preserves skeletal epimorphisms) if and only if for any open surjective map $f:X\to Y$ between metrizable zero-dimensional compacta with two-element non-degeneracy set $N^f=\{x\in X:|f^{?1}(f(x))|>1\}$ the map $Ff:FX\to FY$ is skeletal. This characterization implies that each open normal functor is skeletal. The converse is not true even for normal functors of finite degree. The other main result of the paper says that each normal functor $F:\mathbf{Comp}\to \mathbf{Comp}$ preserves the class of skeletally generated compacta. This contrasts with the known ??epin?s result saying that a normal functor is open if and only if it preserves the class of openly generated compacta.     

The w-integral closure of integral domains

Let D be an integral domain with quotient field K, $\overline{D}$ the integral closure of D, X an indeterminate over D, and $N_v=\{f\in D[X]|{(A_f)}_v=D\}$. Let w be the 1-operation on D defined by $I_w=\{x\in K|\text{there is a finitely generated ideal}A\text{such that}{A}^{\mathrm{-1}}=D\text{and}xA\subseteq I\}$, and let $D^w=\{u\in K|u{I}_w\subseteq I_w\text{for some nonzero finitely generated ideal}I\text{of}D\}$. Then $D^w$, called the w-integral closure of D, is an integrally closed overring of D. In this paper, we show that $D^w=\overline{D}{[X]}_{N_v}\cap K$ and $D^w{[X]}_{N_v}=\overline{D}{[X]}_{N_v}$. Using this result, we give several w-integral closure analogs of the integral closure. We also study the w-integral closure of UMT-domains and strong Mori domains.     

Equivariant Nielsen root theory for G-maps

Let X be a compact Hausdorff space, Y be a connected topological manifold, $f:X\to Y$ be a map between closed manifolds and $a\in Y$. The vanishing of the Nielsen root number $N(f;a)$ implies that f is homotopic to a root free map h, i.e., $h～f$ and $h^{?1}(a)=?$. In this paper, we prove an equivariant analog of this result for G-maps between G-spaces where G is a finite group.