Comparaison des deux composantes d''un subordinateur bivarié, puis étude de l''enveloppe supérieure d''un processus de Lévy

Soit (Y,Z) un subordinateur bivarié. Nous donnons une condition suffisante pour que Y_t/Z_t converge vers zéro quand t tend vers 0 ou +∞. Ceci généralise partiellement des résultats de Bertoin et de Kesten–Erickson.Soit X un processus de Lévy et S_t=sup{X_s: st}. Soit f une fonction sous-additive. En appliquant le résultat précédent au subordinateur bivarié d'échelle, nous donnons des conditions nécéssaires et suffisantes pour que et égalent 0 ou +∞.Let (Y,Z) be a bivariate subordinator. Generalizing theorems of Bertoin and Kesten–Erickson, we give a sufficient condition for Y_t/Z_t to converge to 0 when t tends either to 0 or +∞.Let X be a Lévy process. Denote by S_t=sup{X_s: st} and let f be any sub-additive function. Applying our first result to the bivariate ladder process, we give necessary and sufficient conditions for and to be either 0 or +∞.     

On the Domain of Divergence of Hermite–Fejér Interpolating Polynomials

Recently, the study of the behavior of the Hermite–Fejér interpolants in the complex plane was initiated by L. Brutman and I. Gopengauz (1999, Constr. Approx.15, 611–617). It was shown that, for a broad class of interpolatory matrices on [−1, 1], the sequence of polynomials induced by Hermite–Fejér interpolation to f(z)≡z diverges everywhere in the complex plane outside the interval of interpolation [−1, 1]. In this note we amplify this result and prove that the divergence phenomenon takes place without any restriction on the interpolatory matrices.     

Principe de Hartogs dans les variétés CR

Let M be a CR manifold. The main results of this paper are the following:

When M is real analytic, a semi-global Hartogs extension phenomenon occurs for real analytic CR functions if and only if M is nowhere strictly pseudoconvex and .

When M is a standard manifold, the Hartogs–Bochner extension phenomenon occurs for non-CR-confined domains if and only if M is nowhere strictly pseudoconvex and dim_CRM2.

If M is a smooth submanifold of foliated by complex curves, a semi-global Hartogs–Bochner extension phenomenon occurs for smooth non-CR-confined domains if and only if dim_CRM2.

If M is a real analytic nowhere strictly pseudoconvex manifold and if Ω is a sufficiently small domain in M, a hyperfunction which is real analytic in a neighborhood of bΩ and CR in a neighborhood of is in fact real analytic on Ω.

An efficient fixed-parameter algorithm for 3-Hitting Set

Given a collection C of subsets of size three of a finite set S and a positive integer k, the 3-Hitting Set problem is to determine a subset S′S with |S′|k, so that S′ contains at least one element from each subset in C. The problem is NP-complete, and is motivated, for example, by applications in computational biology. Improving previous work, we give an O(2.270^k+n) time algorithm for 3-Hitting Set, which is efficient for small values of k, a typical occurrence in some applications. For d-Hitting Set we present an O(c^k+n) time algorithm with c=d−1+O(d⁻¹).     

DIRICHLET PROBLEMS FOR STATIONARY VON NEUMANN-LANDAU WAVE EQUATIONS

In this article, we are concerned with the Dirichlet problem of the stationary von Neumann-Landau wave equation：
{（-△x＋△y）φ（x,y）=0,x,y∈Ω
φ｜δΩxδΩ=f
where Ω is a bounded domain in R^n. By introducing anti-inner product spaces, we show the existence and uniqueness of the generalized solution for the above Dirichlet problem by functional-analytic methods.     

On Generalized Hermite–Fejér Interpolation of Lagrange Type on the Chebyshev Nodes

For fC[−1, 1], let H_m, n(f, x) denote the (0, 1, …,anbsp;m) Hermite–Fejér (HF) interpolation polynomial of f based on the Chebyshev nodes. That is, H_m, n(f, x) is the polynomial of least degree which interpolates f(x) and has its first m derivatives vanish at each of the zeros of the nth Chebyshev polynomial of the first kind. In this paper a precise pointwise estimate for the approximation error |H_2m, n(f, x)−f(x)| is developed, and an equiconvergence result for Lagrange and (0, 1, …, 2m) HF interpolation on the Chebyshev nodes is obtained. This equiconvergence result is then used to show that a rational interpolatory process, obtained by combining the divergent Lagrange and (0, 1, …, 2m) HF interpolation methods on the Chebyshev nodes, is convergent for all fC[−1, 1].     

On the structure of biharmonic functions satisfying the clamped plate conditions on a right angle

Let u(r,θ) be biharmonic and bounded in the circular sector ¦θ¦ < π/4, 0 < r < ρ (ρ > 1) and vanish together with δu/δθ when ¦θ¦ = π/4. We consider the transform û(p,θ) = ∝₀¹r^{p − 1}u(r,θ)dr. We show that for any fixed θ₀ u(p,θ₀) is meromorphic with no real poles and cannot be entire unless u(r, θ₀) ≡ 0. It follows then from a theorem of Doetsch that u(r, θ₀) either vanishes identically or oscillates as r → 0.     

Maximum principle for elliptic operators and applications

Let f be a signed function defined on some bounded domain Ω. We give sufficient conditions ensuring the positivity of u, solution of the following equation: −Δu=f in Ω, u|_∂Ω=0.     

On Chebyshev–Markov Rational Functions over Several Intervals

Chebyshev–Markov rational functions are the solutions of the following extremal problem

Full-size image

withKbeing a compact subset of andω_n(x) being a fixed real polynomial of degree less thann, positive onK. A parametric representation of Chebyshev–Markov rational functions is found forK=[b₁, b₂]…[b_2p−1, b_2p], −∞<b₁b₂<…<b_2p−1b_2p<+∞ in terms of Schottky–Burnside automorphic functions.     

On the Divergence Phenomenon in Hermite–Fejér Interpolation

Generalizing results of L. Brutman and I. Gopengauz (1999, Constr. Approx.15, 611–617), we show that for any nonconstant entire function f and any interpolation scheme on [−1, 1], the associated Hermite–Fejér interpolating polynomials diverge on any infinite subset of \[−1, 1]. Moreover, it turns out that even for the locally uniform convergence on the open interval ]−1, 1[ it is necessary that the interpolation scheme converges to the arcsine distribution.     

Estimation de Yule–Walker d'un CAR(p) observé à temps discret

Let (X(lδ), l=0,n) be a discrete observation at mesh δ>0 of X, a CAR(p). Classical Yule–Walker estimation are biased and must be corrected. Resultant estimators converge if T=nδ→+∞, are asymptotically normal with rate , and efficient. The diffusion coefficient is also estimated, with rate .     

On sharp Burkholder–Rosenthal-type inequalities for infinite-degree U-statistics

In this paper, we present a method that allows one to obtain a number of sharp inequalities for expectations of functions of infinite-degree U-statistics. Using the approach, we prove, in particular, the following result: Let D be the class of functions f :R₊→R₊ such that the function f(x+z)−f(x) is concave in xR₊ for all zR₊. Then the following estimate holds: for all fD and all U-statistics ∑_1i1<<ilnY_i1,…,il(X_i1,…,X_il) with nonnegative kernels Y_i1,…,il :R^l→R₊, 1i_kn; i_r≠i_s, r≠s; k,r,s=1,…,l; l=0,…,m, in independent r.v.'s X₁,…,X_n. Similar inequality holds for sums of decoupled U-statistics. The class D is quite wide and includes all nonnegative twice differentiable functions f such that the function f″(x) is nonincreasing in x>0, and, in particular, the power functions f(x)=x^t, 1<t2; the power functions multiplied by logarithm f(x)= (x+x₀)^t ln(x+x₀), 1<t<2, x₀max(e^{(3t2−6t+2)/(t(t−1)(2−t))},1); and the entropy-type functions f(x)=(x+x₀)ln(x+x₀), x₀1. As an application of the results, we determine the best constants in Burkholder–Rosenthal-type inequalities for sums of U-statistics and prove new decoupling inequalities for those objects. The results obtained in the paper are, to our knowledge, the first known results on the best constants in sharp moment estimates for U-statistics of a general type.     

A generalization of a theorem on biseparating maps

In J. Math. Anal. Appl. 12 (1995) 258–265, Araujo et al. proved that for any linear biseparating map  from C(X) onto C(Y), where X and Y are completely regular, there exist ω in C(Y) and an homeomorphism h from the realcompactification vX of X onto vY, such that

The compact version of this result was proved before by Jarosz in Bull. Canad. Math. Soc. 33 (1990) 139–144. In Contemp. Math., Vol. 253, 2000, pp. 125–144, Henriksen and Smith asked to what extent the result above can be generalized to a larger class of algebras. In the present paper, we give an answer to that question as follows. Let A and B be uniformly closed Φ-algebras. We first prove that every order bounded linear biseparating map from A onto B is automatically a weighted isomorphism, that is, there exist ω in B and a lattice and algebra isomorphism ψ between A and B such that

(a)=ωψ(a) for all aA.

We then assume that every universally σ-complete projection band in A is essentially one-dimensional. Under this extra condition and according to a result from Mem. Amer. Math. Soc. 143 (2000) 679 by Abramovich and Kitover, any linear biseparating map from A onto B is automatically order bounded and, by the above, a weighted isomorphism. It turns out that, indeed, the latter result is a generalization of the aforementioned theorem by Araujo et al. since we also prove that every universally σ-complete projection band in the uniformly closed Φ-algebra C(X) is essentially one-dimensional.     

Borne sur la torsion dans les variétés abéliennes de type CM

Let A be an abelian variety of dimension g1 defined over a number field K. We study the size of the torsion group A(F)_tors where F/K is a finite extension and more precisely we study the best possible exponent γ in the inequality Card(A(F)_tors)[F:K]^γ when F is any finite extension of K. In the CM case we give an exact formula for the exponent γ in terms of the characters of the Mumford–Tate group—a torus in this case—and discuss briefly the general case.Finally we give an application of the main result in direction of a generalisation of the Manin–Mumford conjecture.     

Attractors of the reaction-diffusion systems with rapidly oscillating coefficients and their homogenization

The following reaction-diffusion system in spatially non-homogeneous almost-periodic media is considered in a bounded domain : (1)∂_tu=Au−f(u)+g, u|_∂Ω=0. Here u=(u¹,…,u^k) is an unknown vector-valued function, f is a given nonlinear interaction function and the second order elliptic operator A has the following structure: where a_ij^l(y) are given almost-periodic functions. We prove that, under natural assumptions on the nonlinear term f(u), the longtime behavior of solutions of (1) can be described in terms of the global attractor of the associated dynamical system and that the attractors  , 0<<₀1, converge to the attractor of the homogenized problem (1) as →0. Moreover, in the particular case of periodic media, we give explicit estimates for the distance between the non-homogenized and the homogenized attractors in terms of the parameter .     

Asymptotic properties of Cramér-Smirnov statistics—A new approach

Let Y₁,…, Y_n be independent identically distributed random variables with distribution function F(x, θ), θ = (θ′₁, θ′₂), where θ_i (i = 1, 2) is a vector of p_i components, p = p₁ + p₂ and for θI, an open interval in ^p, F(x, θ) is continuous. In the present paper the author shows that the asymptotic distribution of modified Cramér-Smirnov statistic under H_n: θ₁ = θ₁₀ + n^−1/2γ, θ₂ unspecified, where γ is a given vector independent of n, is the distribution of a sum of weighted noncentral χ₁² variables whose weights are eigenvalues of a covariance function of a Gaussian process and noncentrality parameters are Fourier coefficients of the mean function of the Gaussian process. Further, the author exploits the special form of the covariance function by using perturbation theory to obtain the noncentrality parameters and the weights. The technique is applicable to other goodness-of-fit statistics such as U² [G. S. Watson, Biometrika 48 (1961), 109–114].     

Extending analyticK-subanalytic functions

Letg:U→ℝ (U open in ℝⁿ) be an analytic and K-subanalytic (i. e. definable in ℝ _an ^K , whereK, the field of exponents, is any subfield ofℝ) function. Then the set of points, denoted Σ, whereg does not admit an analytic extension is K-subanalytic andg can be extended analytically to a neighbourhood of Ū\∑. Partially supported by the European RTN Network RAAG (contract no. HPRN-CT-00271)     

Quadratic Hermite–Padé polynomials associated with the exponential function

The asymptotic behavior of quadratic Hermite–Padé polynomials associated with the exponential function is studied for n→∞. These polynomials are defined by the relation

(*)

p_n(z)+q_n(z)e^z+r_n(z)e^2z=O(z³ⁿ⁺²) as z→0,

where O(·) denotes Landau's symbol. In the investigation analytic expressions are proved for the asymptotics of the polynomials, for the asymptotics of the remainder term in (*), and also for the arcs on which the zeros of the polynomials and of the remainder term cluster if the independent variable z is rescaled in an appropriate way. The asymptotic expressions are defined with the help of an algebraic function of third degree and its associated Riemann surface. Among other possible applications, the results form the basis for the investigation of the convergence of quadratic Hermite–Padé approximants, which will be done in a follow-up paper.     

On metrizability of compactoid sets in non-archimedean locally convex spaces

In 2003, N. De Grande-De Kimpe, J. Kąkol and C. Perez-Garcia using t-frames and some machinery concerning tensor products proved that compactoid sets in non-archimedean (LM)-spaces (i.e. the inductive limits of a sequence of non-archimedean metrizable locally convex spaces) are metrizable. In this paper we show a similar result for a large class of non-archimedean locally convex space with a £-base, i.e. a decreasing base (U_α)_αN^N of neighbourhoods of zero. This extends the first mentioned result since every non-archimedean (LM)-space has a £-base. We also prove that compactoid sets in non-archimedean (DF)-spaces are metrizable.