TWO ENTIRE FUNCTIONS SHARING FOUR SMALL FUNCTIONS IN THE SENSE OF _(k))(β,f)=_(k))(β,g)

This paper proves a result that if two entire functions f(z) and g(z) share four small functions aj(z) (j = 1,2,3,4) in the sense of Ek)(aj, f) = Ek)(aj,g), (j = 1,2,3,4) (k ≥ 11), then there exists f(z) = g(z).     

ASYMPTOTIC NORMALITY OF KERNEL ESTIMATES OF A DENSITY FUNCTION UNDER ASSOCIATION DEPENDENCE

Let {X_n, n≥1} be a strictly stationary sequence of random variables, whichare either associated or negatively associated, f(·) be their common density. In this paper,the author shows a central limit theorem for a kernel estimate of f(·) under certain regularconditions.     

AREA FUNCTIONS ON HARDY SPACES ASSOCIATED TO SCHRDINGER OPERATORS

In this paper, the author gives a characterization of atomic Hardy spaces associated to Schrodinger operators by using area functions, and hence gets the dual spaces of atomic Hardy spaces.     

Uniform convexity of ψ-direct sums of Banach spaces

Let X and Y be Banach spaces and ψ a continuous convex function on the unit interval [0,1] satisfying certain conditions. Let X⊕_ψY be the direct sum of X and Y equipped with the associated norm with ψ. We show that X⊕_ψY is uniformly convex if and only if X,Y are uniformly convex and ψ is strictly convex. As a corollary we obtain that the ?_p,q-direct sum (not p=q=1 nor ∞), is uniformly convex if and only if X,Y are, where ?_p,q is the Lorentz sequence space. These results extend the well-known fact for the ?_p-sum . Some other examples are also presented.     

Some weighted estimates for Littlewood-Paley functions and radial multipliers

We prove some weighted estimates for certain Littlewood-Paley operators on the weighted Hardy spaces H_w^p (0<p?1) and on the weighted L^p spaces. We also prove some weighted estimates for the Bochner-Riesz operators and the spherical means.     

Boundary behavior of the Bergman kernel function on pseudoconvex domains with comparable Levi form

Let Ω be a smoothly bounded pseudoconvex domain in and let be a point of finite type. We also assume that the Levi form of bΩ is comparable in a neighborhood of z₀. Then we get a quantity which bounds from above and below the Bergman kernel function in a small constant and large constant sense.     

Some properties of an extended Wigner function

A covariant generalisation of Wigner function proposed by us some years ago is reviewed, and its remarkable and useful properties are elucidated; its being a natural solution to a relativistically covariant Liouville equation is also demonstrated.     

Extending the Balas-Yu bounds on the number of maximal independent sets in graphs to hypergraphs and lattices

A result of Balas and Yu (1989) states that the number of maximal independent sets of a graph G is at most ^p+1, where is the number of pairs of vertices in G at distance 2, and p is the cardinality of a maximum induced matching in G. In this paper, we give an analogue of this result for hypergraphs and, more generally, for subsets of vectors in the product of n lattices =₁××_n, where the notion of an induced matching in G is replaced by a certain binary tree each internal node of which is mapped into . We show that our bounds may be nearly sharp for arbitrarily large hypergraphs and lattices. As an application, we prove that the number of maximal infeasible vectors x=₁××_n for a system of polymatroid inequalities does not exceed max{Q,^logt/c(2Q,)}, where is the number of minimal feasible vectors for the system, , , and c(,) is the unique positive root of the equation 2^c(^c/log–1)=1. This bound is nearly sharp for the Boolean case ={0,1}ⁿ, and it allows for the efficient generation of all minimal feasible sets to a given system of polymatroid inequalities with quasi-polynomially bounded right-hand sides . This research was supported by the National Science Foundation (Grant IIS-0118635), and by the Office of Naval Research (Grant N00014-92-J-1375). The second and third authors are also grateful for the partial support by DIMACS, the National Science Foundation's Center for Discrete Mathematics and Theoretical Computer Science.Mathematics Subject Classification (2000):20E28, 20G40, 20C20     

Exposed points and extremal problems in H1 on a bidisc

An essential bounded function ? gives a continuous linear functional on the Hardy space H¹ on the bitorus. In this paper, we consider extremal problems on H¹ when ? is a rational function, ? is a product of one variable functions or ? = |f|/f for some outer function f in H¹ such that f(z, w) has a good property with respect to w for a.e. z. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Integrated squared error of kernel-type estimator of distribution function

Let X ₁ ,...,X _n be a random sample drawn from distribution function F(x) with density function f(x) and suppose we want to estimate X(x). It is already shown that kernel estimator of F(x) is better than usual empirical distribution function in the sense of mean integrated squared error. In this paper we derive integrated squared error of kernel estimator and compare the error with that of the empirical distribution function. It is shown that the superiority of kernel estimators is not necessarily true in the sense of integrated squared error.