Some topological properties of ω-covering sets

We prove the following theorems:1. There exists an -covering with the property s ₀.2. Under cov there exists X such that is not an -covering orX \ B is not an -covering].3. Also we characterize the property of being an -covering.     

On copies of c 0 in the bounded linear operator space

In this note we study some properties concerning certain copies of the classic Banach space c ₀ in the Banach space of all bounded linear operators between a normed space X and a Banach space Y equipped with the norm of the uniform convergence of operators.     

A Limit Result for U-Statistics of Binary Variables

Define , where is a symmetric U-type statistic, H _k() is the Hermite polynomial of degree k, and {X, X _n, n1} are independent identically distributed binary random variables with Pr(X{–1, 1}})=1. We show that according as EX=0 or EX0, respectively.     

Some remarks on the product of two C α-compact subsets

For a cardinal , we say that a subset B of a space X is C -compact in X if for every continuous function is a compact subset of . If B is a C-compact subset of a space X, then (B, X) denotes the degree of C -compactness of B in X. A space X is called -pseudocompact if X is C -compact into itself. For each cardinal , we give an example of an -pseudocompact space X such that X × X is not pseudocompact: this answers a question posed by T. Retta in Some cardinal generalizations of pseudocompactness Czechoslovak Math. J. 43 (1993), 385–390. The boundedness of the product of two bounded subsets is studied in some particular cases. A version of the classical Glicksberg's Theorem on the pseudocompactness of the product of two spaces is given in the context of boundedness. This theorem is applied to several particular cases.     

Symmetric Tensors And Geometry of {\mathbb{P}} ^N Subvarieties

This paper using a geometric approach produces vanishing and nonvanishing results concerning the spaces of twisted symmetric differentials on subvarieties , with k ≤ m. Emphasis is given to the case of k = m which is special and whose nonvanishing results on the dimensional range dim X > 2/3(N − 1) are related to the space of quadrics containing X and the variety of all tangent trisecant lines of X. The paper ends with an application showing that the twisted symmetric plurigenera, along smooth families of projective varieties X_t are not invariant even for α arbitrarily large. Received: September 2006, Revision: May 2007, Accepted: June 2007     

Some remarks on trigonometric sums

Let

From light tails to heavy tails through multiplier

Let X and Y be two independent nonnegative random variables, of which X has a distribution belonging to the class or for some γ ≥ 0 and Y is unbounded. We study how their product XY inherits the tail behavior of X. Under some mild technical assumptions we prove that the distribution of XY belongs to the class or accordingly. Hence, the multiplier Y builds a bridge between light tails and heavy tails.      

On some properties of cohomotopy-type functors

It is shown that if II ⁿ ,n\s>2, are Chogoshvili's cohomotopy functors [1, 2, 3], then

Boundary Limits for Bounded Quasiregular Mappings

In this paper we establish results on the existence of nontangential limits for weighted -harmonic functions in the weighted Sobolev space , for some q>1 and w in the Muckenhoupt A _q class, where is the unit ball in . These results generalize the ones in Sect. 3 of Koskela et al., Trans. Am. Math. Soc. 348(2), 755–766, 1996, where the weight was identically equal to one. Weighted -harmonic functions are weak solutions of the partial differential equation

Equations in finite fields with restricted solution sets. II (Algebraic equations)

Generalizing earlier results, it is shown that if are “large” subsets of a finite field F _q , then the equations a + b = cd, resp. ab + 1 = cd can be solved with . Other algebraic equations with solutions restricted to “large” subsets of F _q are also studied. The proofs are based on character sum estimates proved in Part I of the paper. Research partially supported by the Hungarian National Foundation for Scientific Research, Grants No. T 043623, T 043631 and T 049693.     

Complex Interpolation of Operators and Optimal Domains

Let X ₀ and X ₁ be two order continuous Banach function spaces on a finite measure space, (E ₀, E ₁) a Banach space interpolation pair, and \({T: X_0 + X_1 \to E_0 + E_1}\) an admissible operator between the pairs (X ₀,X ₁) and (E ₀,E ₁). If \({T_{\theta} : [X_0, X_1]_{[\theta ]} \to [E_0, E_1]_{[\theta]}}\) is the interpolated operator by the first complex method of Calderón and m ₀, m ₁ and m _θ are the vector measures coming from \({{T\vert}_{X_0}}\) and \({{T\vert}_{X_1}}\) and T _θ, respectively, then we study the relationship between the optimal domain \({L^1(m_{\theta})}\) of T _θ and the complex interpolation space \({[L^1(m_0),L^1(m_1)]_{[\theta]}}\) of the optimal domains of \({{T\vert}_{X_0}}\) and \({{T\vert}_{X_1}}\) . Then, we apply the obtained result to study interpolation of p-th power factorable and bidual (p,q)-power-concave operators.     

Precise asymptotics in the law of the iterated logarithm*

Let X₁, X₂, ... be i.i.d. random variables with EX₁ = 0 and positive, finite variance σ², and set S_n = X₁ + ... + X_n. For any α > −1, β > −1/2 and for κ_n(ε) a function of ε and n such that κ_n(ε) log log n → λ as n ↑ ∞ and , we prove that

Meromorphic approximation theorem in a Stein space

We prove the meromorphic version of the Weil–Oka approximation theorem in a reduced Stein space X and give some characterizations of meromorphically -convex open sets of X. As an application we prove that for every meromorphically -convex open set D of a reduced Stein space X with no isolated points there exists a family of holomorphic functions on X such that the normality domain of coincides with D. Mathematics Subject Classification (2000)  32E10, 32C15, 32E30, 32A19     

The lattice points of ann-dimensional tetrahedron

We show that the number of orderedm-tuples of points on the integer lattice, inside or on then-dimensional tetrahedron bounded by the hyperplanesX ₁=0,X ₂=0, ...,X _n=0 andw ₁ X ₁+w ₂ X _n+...+w _n X_n=X, with the property that, for eachj, no more thank such points have non-zerojth ordinate, is asymptotically

Interpolation by Rational Functions with Nodes on the Unit Circle

From the Erds–Turán theorem, it is known that if f is a continuous function on and L _n (f, z) denotes the unique Laurent polynomial interpolating f at the (2 n + 1)th roots of unity, then Several years later, Walsh and Sharma produced similar result but taking into consideration a function analytic in and continuous on and making use of algebraic interpolating polynomials in the roots of unity.In this paper, the above results will be generalized in two directions. On the one hand, more general rational functions than polynomials or Laurent polynomials will be used as interpolants and, on the other hand, the interpolation points will be zeros of certain para-orthogonal functions with respect to a given measure on .     

The G-Fredholm Property of the \bar{\partial} -Neumann Problem

Let G be a unimodular Lie group, X a compact manifold with boundary, and M be the total space of a principal bundle G→M→X so that M is also a strongly pseudoconvex complex manifold. In this work, we show that if G acts by holomorphic transformations in M, then the Laplacian on M has the following properties: The kernel of □ restricted to the forms Λ ^p,q with q>0 is a closed, G-invariant subspace in L ²(M,Λ ^p,q ) of finite G-dimension. Secondly, we show that if q>0, then the image of □ contains a closed, G-invariant subspace of finite G-codimension in L ²(M,Λ ^p,q ). These two properties taken together amount to saying that □ is a G-Fredholm operator. It is a corollary of the first property mentioned that the reduced L ²-Dolbeault cohomology spaces of M are finite G-dimensional for q>0. The boundary Laplacian □ _b has similar properties.      

An Extension of the Admissibility-Type Conditions for the Exponential Dichotomy of C 0-Semigroups

In the present paper we obtain a sufficient condition for the exponential dichotomy of a strongly continuous, one-parameter semigroup , in terms of the admissibility of the pair . It is already known the equivalence between the -admissibility condition and and the hyperbolicity of a C ₀-semigroup , when we assume a priori that the kernel of the dichotomic projector (denoted here by X ₂) is T(t)-invariant and is an invertible operator. We succeed to prove in this paper that the admissibility of the pair still implies the existence of an exponential dichotomy for a C ₀-semigroup even in the general case where the kernel of the dichotomic projector, X ₂, is not assumed to be T(t)-invariant.      

Remarks on Gagliardo–Nirenberg type inequality with critical Sobolev space and BMO

We consider the generalized Gagliardo–Nirenberg inequality in in the homogeneous Sobolev space with the critical differential order s = n/r, which describes the embedding such as for all q with p ≦ q < ∞, where 1 < p < ∞ and 1 < r < ∞. We establish the optimal growth rate as q → ∞ of this embedding constant. In particular, we realize the limiting end-point r = ∞ as the space of BMO in such a way that with the constant C _n depending only on n. As an application, we make it clear that the well known John–Nirenberg inequality is a consequence of our estimate. Furthermore, it is clarified that the L ^∞-bound is established by means of the BMO-norm and the logarithm of the -norm with s > n/r, which may be regarded as a generalization of the Brezis–Gallouet–Wainger inequality.