Finite-Term Relations for Planar Orthogonal Polynomials

We prove by elementary means that, if the Bergman orthogonal polynomials of a bounded simply-connected planar domain, with sufficiently regular boundary, satisfy a finite-term relation, then the domain is algebraic and characterized by the fact that Dirichlet’s problem with boundary polynomial data has a polynomial solution. This, and an additional compactness assumption, is known to imply that the domain is an ellipse. In particular, we show that if the Bergman orthogonal polynomials satisfy a three-term relation then the domain is an ellipse. This completes an inquiry started forty years ago by Peter Duren. To Peter Duren on the occasion of his seventieth birthday The first author was partially supported by the National Science Foundation Grant DMS- 0350911. Received: October 15, 2006. Revised: January 22, 2007.     

Gleason's problem for harmonic Bergman and Bloch functions on half-spaces

On the setting of the half-spaceR ^n–1×R ₊, we investigate Gleason's problem for harmonic Bergman and Bloch functions. We prove that Gleason's problem for the harmonicL ^p -Bergman space is solvable if and only ifp>n. We also prove that Gleason's problem for the harmonic (little) Bloch space is solvable.     

On Generalized Integral Means and Euler Type Vector Fields

Formulas for the Euler vector fields, the Neumann derivatives, and the Euler as well as Dirichlet product are derived. Extensions to a Riemann domain of the Gauss operator, the Gauss’ lemma and the related jump formulas are given, and the Gauss–Helmholtz representation with ramifications proved. Examples of elementary solutions to certain modified Laplace operators, applications to pseudospherical harmonics, and characterizations of pseudoradial, pseudospherical, nearly holomorphic, and holomorphic functions, are obtained, and constancy criterion for locally Lipschitz, semiharmonic, respectively, weakly holomorphic functions are given.     

D-Dimensional hypercubes and the Euler and MacNeish conjectures

We give a construction for large sets of mutually orthogonal hypercubes of dimensionald given sets of mutually orthogonal latin squares and hypercubes of lower dimension. We also considerd>-2 dimensional versions of the Euler and MacNeish conjectures as well as discussing applications to improved constructions of (t, m, s)-nets, useful in pseudorandom number generation and quasi-Monte-Carlo methods of numerical integration.This author would like to thank the National Security Agency for partial support under grant agreement # MDA 904-92-H-3044.     

Symmetric balanced ternary designs with ϱ1 = 1 or 2

Summary We prove the following two non-existence theorems for symmetric balanced ternary designs. If ₁ = 1 and 0 (mod 4) then eitherV = + 1 or 4₂ – + 1 is a square and (4₂ – + 1) divides ² – 1. If ₁ = 2 thenV = ((m + 1)/2) ² + 2,K = (m ² + 7)/4 and = ((m – 1)/2)² + 1 wherem 3 (mod 4). An example belonging to the latter series withV = 18 is constructed.     

Borel and Julia directions of meromorphic Schröder functions II

Let R(w) be a non-inear rational function and s be a complex constant with | s | > 1. It is showed that for any solution f (z) of the Schr?der equation f (sz) = R(f (z)), Julia directions of f (z) are also Borel directions of f (z). Received: 2 May 2005; revised: 22 December 2005     

Integral means and Dirichlet integral for analytic functions

For normalized analytic functions f in the unit disk, the estimate of the integral means is important in certain problems in fluid dynamics, especially when the functions are non‐vanishing in the punctured unit disk . We consider the problem of finding the extremal function f which maximizes the integral means for f belong to certain classes of analytic functions related to sufficient conditions of univalence. In addition, for certain subclasses of the class of normalized univalent and analytic functions, we solve the extremal problem for the Yamashita functional where denotes the area of the image of under . The first problem was originally discussed by Gromova and Vasil'ev in 2002 while the second by Yamashita in 1990.     

Integral representations of harmonic functions in half spaces

In this paper, using a modified Poisson kernel in an upper half-space, we prove that a harmonic function u(z) in a upper half space with its positive part u⁺(x)=max{u(x),0} satisfying a slowly growing condition can be represented by its integral in the boundary of the upper half space, the integral representation is unique up to the addition of a harmonic polynomial, vanishing in the boundary of the upper half space and that its negative part u⁻(x)=max{−u(x),0} can be dominated by a similar slowly growing condition, this improves some classical result about harmonic functions in the upper half space.     

Kernels and Choquet capacities

Summary M. Brelot showed that the capacity corresponding to a function-kernel is a Choquet capacity, provided that the kernel satisfies the principle of equilibrium, the weak domination principle and the adjoint kernel satisfies the weak principle of equilibrium. This result is not applicable for a series of important kernels in potential theory (e.g. the fundamental solution of the heat equation, or the Kolmogorov equation), since the above principles no longer hold in this situation. New principles for function kernels guaranteeing that the capacity is a Choquet capacity are introduced and applied in the framework of balayage spaces. In particular, polar and adjoint polar sets are shown to coincide in this context.     

Defect of a nodal hypersurface

In this note we give formulas for the Hodge numbers of a nodal hypersurface in a smooth complex projective fourfold. Received: 10 May 2000 / Revised version: 7 November 2000     

Robustness of the controllability for the heat equation under the influence of multiple impulses and delays

For many control systems in real life, impulses and delays are intrinsic properties that do not modify their behavior. Thus, we conjecture that under certain conditions the abrupt changes and delays as perturbations of a system, that could model a real situation, do not modify properties such as controllability. In this regard, we prove the approximate controllability of the semilinear heat equation under the influence of multiple impulses and delays, this is done by using new techniques, avoiding fixed point theorems, employed by A.E. Bashirov et al.     

Bases d'ouverts réguliers et formule de Poisson pour l'équation de la chaleur

Summary Nous donnons une formule de Poisson pour certains polyèdres. Nous résolvons ainsi le problème de Dirichlet pour l'équation de la chaleur dans ces domaines.

     

A survey of pseudo (v,k, λ)-designs

This work is an attempt to give a complete survey of all known results about pseudo (v, k, )-designs. In doing this, the author hopes to bring more attention to his conjecture given in Section 6; an affirmative answer to this conjecture would settle completely the existence and construction problem for a pseudo (v, k, )-design in terms of the existence of an appropriate (v, k, )-design.     

A maximum principle for node voltages in finitely structured,transfinite, electrical networks

Transfinite electrical networks have unique finite-powered voltage-current regimes given in terms of branch voltages and branch currents, but they do not in general possess unique node voltages. However, if their structures are sufficiently restricted, those node voltages will exist and will satisfy a maximum principle much like that which holds for ordinary infinite electrical networks. The structure that is imposed in order to establish these results generalized the idea of local-finiteness. Other properties that do not hold in general for transfinite networks but do hold under the imposed structure are Kirchhoff's current laws for nodes of any ranks and the permissibility of connecting pure voltage sources to such nodes. This work lays the foundation for a theory of transfinite random walks, which will be the subject of a subsequent work.This work was supported by the National Science Foundation under the grants DMS-9200738 and MIP-9200748.     

Some remarks concerning kissing numbers,blocking numbers and covering numbers

This article shows an inequality concerning blocking numbers and Hadwiger's covering numbers and presents a strange phenomenon concerning kissing numbers and blocking numbers. As a simple corollary, we can improve the known upper bounds for Hadwiger's covering numbers ford-dimensional centrally symmetric convex bodies to 3 ^d –1.