A general result on precise asymptotics for linear processes of positively associated sequences

Let {εt; t ∈ Z^＋} be a strictly stationary sequence of associated random variables with mean zeros, let 0〈Eε1^2〈∞ and σ^2=Eε1^2＋1∑j=2^∞ Eε1εj with 0〈σ^2〈∞.{aj;j∈Z^＋} is a sequence of real numbers satisfying ∑j=0^∞｜aj｜〈∞.Define a linear process Xt=∑j=0^∞ ajεt-j,t≥1,and Sn=∑t=1^n Xt,n≥1.Assume that E｜ε1｜^2＋δ′〈 for some δ′〉0 and μ（n）=O（n^-ρ） for some ρ〉0.This paper achieves a general law of precise asymptotics for {Sn}.     

Asymptotic behaviour of the phase in non-smooth obstacle scattering

In this paper, we study the asymptotic behaviour of the scattering phases(λ) of the Dirichlet Laplacian associated with obstacle , where Ω is a bounded open subset of ℝ ⁿ (n≥2) with non-smooth boundary ∂Ω and connected complement Ω _e =ℝ ⁿ . We can prove that if Ω satisfies a certain geometrical condition, then

Solvability of elliptic functional-differential equations with compressions of the arguments in weighted spaces

In the paper, the equation

Existence of solutions of certain quasilinear elliptic equations in ℝN without conditions at infinity

This paper deals with conditions for the existence of solutions of the equations

A Robin boundary problem with Hardy potential and critical nonlinearities

Let Ω be a bounded domain with a smooth C ² boundary in ℝⁿ (n ≥ 3), 0 ∈ , and υ denote the unit outward normal to ∂Ω. In this paper, we are concerned with the following class of boundary value problems:

On a class of nonlinear problems involving a p(x)-Laplace type operator

We study the boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝ ^N . Our attention is focused on two cases when , where m(x) = max{p ₁(x), p ₂(x)} for any x ∈ or m(x) < q(x) < N · m(x)/(N − m(x)) for any x ∈ . In the former case we show the existence of infinitely many weak solutions for any λ > 0. In the latter we prove that if λ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a ℤ₂-symmetric version for even functionals of the Mountain Pass Theorem and some adequate variational methods.     

A Fully Nonlinear Nonhomogeneous Neumann Problem

Let Ω be an open bounded set in ℝ^N, N≥3, with connected Lipschitz boundary ∂Ω and let a(x,ξ) be an operator of Leray–Lions type (a(⋅,∇u) is of the same type as the operator |∇u|^p−2∇u, 1<p<N). If τ is the trace operator on ∂Ω, [φ] the jump across ∂Ω of a function φ defined on both sides of ∂Ω, the normal derivative ∂/∂ν_a related to the operator a is defined in some sense as 〈a(⋅,∇u),ν〉, the inner product in ℝ^N, of the trace of a(⋅,∇u) on ∂Ω with the outward normal vector field ν on ∂Ω. If β and γ are two nondecreasing continuous real functions everywhere defined in ℝ, with β(0)=γ(0)=0, f∈L¹(ℝ^N), g∈L¹(∂Ω), we prove the existence and the uniqueness of an entropy solution u for the following problem,

Asymptotic properties of solutions of semilinear second-order elliptic equations in cylindrical domains

The equations under consideration have the following structure:

Polynomial solutions of quasi-homogeneous partial differential equations

By means of a method of analytic number theory the following theorem is proved. Letp be a quasi-homogeneous linear partial differential operator with degreem,m > 0, w.r.t a dilation given by ( a₁, …, a_n). Assume that either a₁, …, a_n are positive rational numbers or for some Then the dimension of the space of polynomial solutions of the equationp[u] = 0 on ℝⁿ must be infinite     

Localization type Berezin-Toeplitz

Let II be a bounded symmetric domain, ω ⇉ I a bounded subdomain, and let denote the weighted Bergman space of holomorphic square integrable functions on I. Let T_{λ, ω} be the Berezin-Toeplitz operator on with symbol χ_Ω and k^th eigenvalue λ _k (T _λ,Ω). We prove that for δ₁ sufficiently close to 0 and δ₂ sufficiently close to 1 the estimate

On families of faces in discrete cubes

Let ℱ be a family ofn−k-dimensional faces of the discrete cube {0,1} ⁿ such that for allF ε ℱ, F ⊄ ∪ { F′: F ≠ F′ ∈ ℱ}. It is shown that ifn≥n ₀ (k) then |ℱ| ≤ . This was conjectured by Aharoni and Holzman and is the casem=2 of a more general result on faces of {0,...,m−1}ⁿ.     

The Balian-Low theorem and regularity of Gabor systems

For any positive real numbers A, B, and d satisfying the conditions , d>2, we construct a Gabor orthonormal basis for L²(ℝ), such that the generating function g∈L²(ℝ) satisfies the condition:∫_ℝ|g(x)|²(1+|x| ^A )/log ^d (2+|x|)dx < ∞ and .     

On a problem in probability theory

For continuous random variables, we study a problem similar to that considered earlier by one of the authors for discrete random variables. Let numbers $$N > 0, E > 0, 0 \leqslant \lambda _1 \leqslant \lambda _2 \leqslant \cdots \leqslant \lambda _s $$ be given. Consider a random vector x = (x ₁, …, x _s), uniformly distributed on the set $$x_j \geqslant 0, j = 1, \ldots ,s; \sum\limits_{j = 1}^s {x_j = N} , \sum\limits_{j = 1}^s {\lambda _j x_j \leqslant E} .$$ We study the weak limit of x as s → ∞.     

Decay of solutions of the first mixed problem for a high-order parabolic equation with minor terms

In a cylindric domain D = (0, ∞) × Ω where Ω ⊂ ℝ _n+1 is an unbounded domain, the first mixed problem for a high-order parabolic equation

Some remarks on the identity between a variational integral and its relaxed functional

Summary Letf: (x, z)∈R ⁿ×Rⁿ→f(x, z)∈[0, +∞] be measurable inx and convex inz. It is proved, by an example, that even iff verifies a condition as|z| ^p≤f(x, z)≤Λ(a(x)+|z|^q) with 1<p<q,a∈L _loc ^s (R ⁿ),s>1, the functional that isL ¹(Ω)-lower semicontinuous onW ^1,1(Ω), does not agree onW ^1,1(Ω) with its relaxed functional in the topologyL ¹(Ω) given by inf

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Riassunto Siaf: (x, z)∈R ⁿ×Rⁿ→f(x, z)∈[0, +∞] misurabile inx e convessa inz. Si mostra con un esempio che anche sef verifica una condizione del tipo|z| ^p≤f(x, z)≤Λ(a(x)+|z|^q) con 1<p<q,a∈L _loc ^s (R ⁿ),s>1, il funzionale , che èL ¹(Ω)-semicontinuo inferiormente suW ^1,1(Ω), non coincide suW ^1,1(Ω) con il suo funzionale rilassato nella topologiaL ¹(Ω) definito da inf

     

Hölder Regularity for Solutions of Ultraparabolic Equations in Divergence Form

We consider the second order differential equation , where (x,t) ^N+1, 0<m ₀N, the coefficients a _i,j belong to a suitable space of vanishing mean oscillation functions VMO _L and B=(b _i,j ) is a constant real matrix. The aim of this paper is to study interior regularity for weak solutions to the above equation assuming that F _j belong to a function space of Morrey type.     

Représentation intégrale de fonctionnelles convexes sur un espace de mesures

Résumé Partant d’un résultat abstrait de représentation intègrale pour une fonctionnelle convexe faiblement s.c.i. sur [M _b (Ω)] ^d (obtenu grace à [1]); on établit un résultat de convergence pour une suite de fonctionnelles du type μ _n εM _b ⁺ (Ω),f _n : Ω×R ^d →−∞, +∞] inté Quelques exemples motivés par des applications à la mécanique sont ensuite traités.

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Riassunto Partendo da un risultato astratto di rappresentazione integrale per un funzionale convesso debolmente s.c.i. su [M _b (Ω)] ^d (ottenuto grazie a [1]), si dimostra un risultato di convergenza per una successione di funzionali del tipo con μ _n εM _b ⁺ (Ω),f _n : Ω×R ^d →−∞, +∞] integrando normale convesso. Vengono inoltre trattati alcuni esempi motivati da applicazioni alla Meccanica.

     

The <Emphasis Type="Italic">s</Emphasis>-energy of spherical designs on <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript>

This paper investigates the s-energy of (finite and infinite) well separated sequences of spherical designs on the unit sphere S ². A spherical n-design is a point set on S ² that gives rise to an equal weight cubature rule which is exact for all spherical polynomials of degree ≤n. The s-energy E _s (X) of a point set of m distinct points is the sum of the potential for all pairs of distinct points . A sequence Ξ = {X _m } of point sets X _m ⊂S ², where X _m has the cardinality card(X _m )=m, is well separated if for each pair of distinct points , where the constant λ is independent of m and X _m . For all s>0, we derive upper bounds in terms of orders of n and m(n) of the s-energy E _s (X _m(n)) for well separated sequences Ξ = {X _m(n)} of spherical n-designs X _m(n) with card(X _m(n))=m(n).      

A Hyperplane Inequality for Measures of Convex Bodies in ? n , n≤4

Let 2≤n≤4. We show that for an arbitrary measure μ with even continuous density in ℝ ⁿ and any origin-symmetric convex body K in ℝ ⁿ ,