How many intervals cover a point in Dvoretzky covering?

Consider the Dvoretzky random covering with length sequence {α/n} _n≥1 (α>0). We are interested in the setF _β of points on the circle which are covered by a numberβ logn of the firstn randomly placed intervals. It is proved among others that for a certain interval ofβ>0, the Hausdorff dimension ofF _β is equal to 1−[βlog(β/α)−(β−α)]. This implies that points on the circle are differently covered. The research was partially supported by the Zheng Ge Ru Foundation and the RGC grant of Hong Kong.     

Characterizations of BMO associated with Gauss measures via commutators of local fractional integrals

Let dγ(x) ≡ π ^−n/2 e ^{−|x| 2} dx for all x ∈ ℝ ⁿ be the Gauss measure on ℝ ⁿ . In this paper, the authors establish the characterizations of the space BMO(γ) of Mauceri and Meda via commutators of either local fractional integral operators or local fractional maximal operators. To this end, the authors first prove that such a local fractional integral operator of order β is bounded from L ^p (γ) to L ^p/(1−pβ)(γ), or from the Hardy space H ¹(γ) of Mauceri and Meda to L ^1/(1−β)(γ) or from L ^1/β (γ) to BMO(γ), where β ∈ (0, 1) and p ∈ (1, 1/β).     

Limiting process for integral functionals of a wiener process on a cylinder

We prove that integral functionals, whose integrands are bounded functions of a Wiener process on a cylinder, weakly converge to the processw ₁(τ(t)), τ(t) = β₁ t + (β₂ − β₁)mes {s:w ₂(s)≥0,s<t}, wherew ₁(t andw ₂(t) are independent one-dimensional Wiener processes, β₁ and β₂ are nonrandom values, and β₂≥β₁≥0. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 765–768, June, 1994.     

Necessary and sufficient conditions for boundedness of commutators of strongly singular integral operators with weighted Lipschitz functions

In this paper, we prove the commutator T _b generated by the strongly singular integral operator T and the function b is bounded from L ^p (w) to L ^q (w ^1−q ) if and only if b ∈ Lip _β (w), where w ∈ A ₁, 0 < β < 1, 1 < p < n/β and 1/q = 1/p − β/n. To do this, we first show a maximal function estimate for the commutator.     

UNIFORM CONVERGENCE OF CESARO MEANS OF NEGATIVE ORDER OF DOUBLE TRIGONOMETRIC FOURIER SERIES

In this paper we prove that iff ∈ C([-π,π]2) and the function f is bounded partial p-variation for some p ∈ [1, ∞), then the double trigonometric Fourier series of a function f is uniformly (C;-α,-β) summable (α β＜ 1/p,α,β＞ 0) in the sense of Pringsheim. If α β≥ 1/p, then there exists a continuous function f0 of bounded partial double trigonometric Fourier series of fo diverge over cubes.     

带振荡因子的粗双曲奇异积分算子

The singular integral operator J Ω,α, and the Marcinkiewicz integral operator (~μ)Ω,α are studied. The kernels of the operators behave like |y|-n-α(α＞0) near the origin, and contain an oscillating factor ei|y|-β(β＞0) and a distribution Ω on the unit sphere Sn-1 It is proved that, if Ω is in the Hardy space Hr (Sn-1) with 0＜r= (n-1)/(n-1 )(＞0), and satisfies certain cancellation condition,then J Ω,α and uΩ,α extend the bounded operator from Sobolev space Lpγ to Lebesgue space Lp for some p. The result improves and extends some known results.     

New approach to the numerical solution of forward-backward equations

This paper is concerned with the approximate solution of functional differential equations having the form: x′(t) = αx(t) + βx(t - 1) + γx(t + 1). We search for a solution x, defined for t ∈ [−1, k], k ∈ ℕ, which takes given values on intervals [−1, 0] and (k-1, k]. We introduce and analyse some new computational methods for the solution of this problem. Numerical results are presented and compared with the results obtained by other methods.      

Regularly varying tail of the waiting time distribution in M/G/1 retrial queue

We consider an M/G/1 retrial queue where the service time distribution has a regularly varying tail with index −β, β>1. The waiting time distribution is shown to have a regularly varying tail with index 1−β, and the pre-factor is determined explicitly. The result is obtained by comparing the waiting time in the M/G/1 retrial queue with the waiting time in the ordinary M/G/1 queue with random order service policy.     

On the average crosscap number II: Bounds for a graph

The bounds are obtained for the average crosscap number. Let G be a graph which is not a tree. It is shown that the average crosscap number of G is not less thanβ(G)-1/2β(G)-1β(G) and not larger thanβ(G). Furthermore, we also describe the structure of the graphs which attain the bounds of the average crosscap number.     

The Lambert W-functions and some of their integrals: a case study of high-precision computation

The real-valued Lambert W-functions considered here are w ₀(y) and w _− 1(y), solutions of we ^w  = y, − 1/e < y < 0, with values respectively in ( − 1,0) and ( − ∞ , − 1). A study is made of the numerical evaluation to high precision of these functions and of the integrals ò₁^￥ [-w₀(-xe^-x)]^a x^-bx\int_1^\infty [-w_0(-xe^{-x})]^\alpha x^{-\beta}\d x, α > 0, β ∈ ℝ, and ò₀¹ [-w_-1(-x e^-x)]^a x^-bx\int_0^1 [-w_{-1}(-x e^{-x})]^\alpha x^{-\beta}\d x, α > − 1, β < 1. For the latter we use known integral representations and their evaluation by nonstandard Gaussian quadrature, if α ≠ β, and explicit formulae involving the trigamma function, if α = β.     

On the error terms for representation numbers of quadratic forms

Let f be a primitive positive integral binary quadratic form of discriminant −D, and r _f (n) the number of representations of n by f up to automorphisms of f. We first improve the error term E(x) of $ \sum\limits_{n \leqq x} {r_f (n)^\beta } $ \sum\limits_{n \leqq x} {r_f (n)^\beta } for any positive integer β. Next, we give an estimate of ∫₁ ^T |E(x)|² x ^−3/2 dx when β = 1.     

Rates of uniform convergence for empirical processes of strictly stationary β-mixing sequences indexed by an unbounded class of functions

Assume that {Xn} is a strictly stationary β-mixing random sequence with the β-mixing coefficient βk = O(k-r), 0 < r ≤1. Yu (1994) obtained convergence rates of empirical processes of strictly stationary β-mixing random sequence indexed by bounded classes of functions. Here, a new truncation method is proposed and used to study the convergence for empirical processes of strictly stationary β-mixing sequences indexed by an unbounded class of functions. The research results show that if the envelope of the index class of functions is in Lp, p > 2 or p > 4, uniform convergence rates of empirical processes of strictly stationary β-mixing random sequence over the index classes can reach O((nr/(l+r)/logn)-1/2) or O((nr/(1+r)/ log n)-3/4) and that the Central Limit Theorem does not always hold for the empirical processes.``     

On αβ-sets

A closed setE⊂T is an αβ-set, where α and β are elements of infinite order inT ifE⊂E−α). ∪ (E−β). We give two constructions of “thin” αβ-sets.     

Approximation of convolutions by accompanying laws under the existence of moments of low orders

It is shown that if a one-dimensional distribution F has finite moment of order 1+β for some β, 1/2≤β≤1, then the rate of approximation of the n-fold convolution Fⁿ by accompanying laws is O(n^−1/2). Futhermore, if Eξ² = ∞ and 1/2<β<1, then the rate of approximation is o(n^−1/2). The question about the true rate of approximation of Fⁿ by infinitely divisible and accompanying laws is discussed. Bibliography: 27 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 228, 1996, pp. 135–141.     

A note on Lagrange interpolation of functions of generalized bounded variation

In this paper we solve a remained problem in [2], whether the following estimate approximation for the classf∈[-1, 1]∩BV by Lagrange interpolation based on the Jacobi abscissas: L _n ^(a,d) (f,x)−f(x)=O(1/n) holds, if α≠β α,β≥−1. The project is supported by the Natural Science Foundation of Zhejiang Province.     

Worst-case minimum rectilinear steiner trees in all dimensions

It is proved that the length of the longest possible minimum rectilinear Steiner tree ofn points in the unitd-cube is asymptotic toβ _dn^(d−1)/d , whereβ _d is a constant that depends on the dimensiond≥2. A method of Chung and Graham (1981) is generalized to dimensiond to show that 1≤β _d≤d4^(1−d)/d . In addition to replicating Chung and Graham's exact determination ofβ ₂=1, this generalization yields new bounds such as 1≤β ₃<1.191 and .     

On weighted approximation by Bernstein-Durrmeyer operators

In this paper, we consider weighted approximation by Bernstein-Durrmeyer operators in L_p[0, 1] (1≤p≤∞), where the weight function w(x)=x^α(1−x)^β,−1/p<α, β<1-1/p. We obtain the direct and converse theorems. As an important tool we use appropriate K-functionals. Supported by Zhejiang Provincial Science Foundation.     

The Fourier integral operators on hardy spaces associated with Herz spaces

In this paper, it is proved that the Fourier integral operators of order m, with −n < m ⩽ −(n−1)/2, are bounded from three kinds of Hardy spaces associated with Herz spaces to their corresponding Herz spaces.     

Best Approximations and Widths of Classes of Convolutions of Periodic Functions of High Smoothness

We consider classes of 2π-periodic functions that are represented in terms of convolutions with fixed kernels Ψ _β ⁻ whose Fourier coefficients tend to zero at exponential rate. We determine exact values of the best approximations of these classes in the uniform and integral metrics. In several cases, we determine the exact values of the Kolmogorov, Bernstein, and linear widths for these classes in the metrics of the spaces C and L. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 7, pp. 946–971, July, 2005.     

Multivariate Generalization of the Hypergeometric Function Type I Distribution

The hypergeometric function type I distribution with the pdf proportional to x ^ν−1(1−x) ^γ−1 ₂ F ₁(α,β;γ;1−x), 0<x<1 occurs as the distribution of the product of two independent beta variables. In this article, a multivariate generalization of this distribution is defined and its properties are derived.