Comptes Rendus de l'Académie des Sciences - Series I - Mathematics最新文献

We consider the Brownian bridge of length T on a symmetric space of the noncompact type. We prove that this process, properly rescaled, converges when T→+∞ to a process whose generalized radial part is the bridge of the Euclidean Brownian motion in the Weyl chamber killed at the boundary.

For a sequence of independent identically distributed random variables, the normalized step-processes associated are weakly convergent to the Wiener process. We strengthen for the functional distributions the convergence for the variation distance for a class of functionals.

We introduce the operators the Weyl symbols of which are automorphic distributions on $\text{R}{}^2$ and we study their symbolic calculus.

A non-oscillating Paley–Wiener function is a real entire function f of exponential type belonging to $\text{L}{}_2\text{(}\text{R}\text{)}$ and such that each derivative f⁽ⁿ⁾, $\text{n=0,1,2,…}$, has only a finite number of real zeros. We show that the class of such functions is non-empty and contains functions of arbitrarily fast decay on $\text{R}$ allowed by the convergence of the logarithmic integral. We also give a close to the best possible asymptotic (as n→∞) estimate of the size of the smallest interval containing all real zeros of n-th derivative of a function f of the class.

Une fonction entière réelle du type exponentiel appartenant à $\text{L}{}^2\text{(}\text{R}\text{)}$ est une fonction non oscillante de Paley–Wiener si chacune de ses dérivées f⁽ⁿ⁾, $\text{n=0,1,2,…}$, possède un nombre fini de zéros. Nous montrons que la classe de ces fonctions n'est pas vide. De plus, elle contient des fonctions qui décroissent arbitrairement vite, à condition que la vitesse de décroissance n'interdise pas la convergence de l'intégrale logarithmique. Nous établissons aussi une estimation de la longueur de l'intervalle minimal contenant tous les zéros réels de la n-ième dérivée d'une fonction de cette classe. D'un certain point de vue, cette estimation est optimale.

We characterize the strong limiting behavior of the kernel density estimates of the mode using law of the iterated logarithm type results.

For a finite subset $\text{X⊂}\text{R}{}^{\mathrm{n}}$ of unit vectors, G_X denotes the group generated by reflections r_x fixing hyperplanes orthogonal to x∈X. When X is a spherical t-design and π^(k)_har the representation of G_X in the harmonic polynomials in n variables of degree k, the spectrum of the Markov operator $\text{1}\text{|X|}\text{∑}{}_{\mathrm{x\in X}}\text{π}{}^{\mathrm{(k)}}{}_{\mathrm{<mtext>har</mtext>}}\text{(r}{}_{\mathrm{x}}\text{)}$ is analyzed. If k is small enough, this operator is a scalar multiple of the identity. When X is the spherical 11-design of short vectors in a Leech lattice, it is shown that the infinite group G_X contains the finite Conway group Co and is a quotient of a remarkable Coxeter group.

We obtain some conditions under which solutions for semidiscretizations of semilinear parabolic equations extinct in a finite time and estimate their extinction time. A similar study is also undertaken for full discretizations.

This paper proposes a hypothesis testing procedure for nonparametric regression models based on regression splines. We assume that the sample is a part of stationary sequence which satisfy a mild mixing property. The approach yields tests of monotonicity and convexity.

When a source term is present, the constants are no longer solutions to quasilinear equations. Some numerical techniques based on this property need to be generalized. A stationary profile scheme uses an approximate solution made of stationary solutions in each cell. We propose such a scheme, for which we prove some results of stability and convergence towards the entropy solution. The numerical tests show that this scheme is well adapted to the simulation of well balanced states, and often far better than the usual splitting methods.

This work is concerned with the indirect boundary observability of a coupled system of two wave equations. We show that for a sufficiently large time, the observation of the trace of the normal derivative of the first component of the solution on a part of the boundary allows us to get back a weakened energy of the initial data, this if the coupling parameter is sufficiently small, but non vanishing. This result leads to a new uniqueness theorem and also to an indirect exact controllability result.