Minihypers and Linear Codes Meeting the Griesmer Bound: Improvements to Results of Hamada, Helleseth and Maekawa

This article improves results of Hamada, Helleseth and Maekawa on minihypers in projective spaces and linear codes meeting the Griesmer bound.In [10,12],it was shown that any -minihyper, with , where , is the disjoint union of points, lines,..., -dimensional subspaces. For q large, we improve on this result by increasing the upper bound on non-square, to non-square, square, , and (4) for square, p prime, p<3, to . In the case q non-square, the conclusion is the same as written above; the minihyper is the disjoint union of subspaces. When q is square however, the minihyper is either the disjoint union of subspaces, or the disjoint union of subspaces and one subgeometry . For the coding-theoretical problem, our results classify the corresponding codes meeting the Griesmer bound.     

Effects Connected with the Coincidence of Velocities in the Two-Velocity Dynamical System

The paper deals with the system

Complexity of a Noninterior Path-Following Method for the Linear Complementarity Problem

We study the complexity of a noninterior path-following method for the linear complementarity problem. The method is based on the Chen–Harker–Kanzow–Smale smoothing function. It is assumed that the matrix M is either a P-matrix or symmetric and positive definite. When M is a P-matrix, it is shown that the algorithm finds a solution satisfying the conditions Mx-y+q=0 and in at most

New Proof of the Semmes Inequality for the Derivative of the Rational Function

In the open disk of the complex plane, we consider the following spaces of functions: the Bloch space ; the Hardy--Sobolev space ; and the Hardy--Besov space . It is shown that if all the poles of the rational function R of degree n, , lie in the domain , then , where and depends only on . The second of these inequalities for the case of the half-plane was obtained by Semmes in 1984. The proof given by Semmes was based on the use of Hankel operators, while our proof uses the special integral representation of rational functions.     

The Spectrum of Lévy Constants for Quadratic Irrationalities and Class Numbers of Real Quadratic Fields

Let h(d) be the class number of the field and let be the Lévy constant. A connection between these constants is studied. It is proved that if d is large, then the value h(d) increases, roughly speaking, at the rate as grows. A similar result is obtained in the case where the value is close to , i.e., to the least possible value. In addition, it is shown that the interval contains no values of for prime p such that p 3 (mod 4). As a corollary, a new criterion for the equality h(d)=1 is obtained. Bibliography: 14 titles.     

Behavior of Automorphic L-Functions at the Center of the Critical Strip

Let be the Hecke eigenbasis of the space of -cusp forms of weight 2. Let p be a prime. Let be the Hecke L-series of form . The following statements are proved:

λ-Divergence of the Fourier Series of Continuous Functions of Several Variables

In this paper, we consider the behavior of rectangular partial sums of the Fourier series of continuous functions of several variables with respect to the trigonometric system. The Fourier series is called -convergent if the limit of rectangular partial sums over all indices for which for all j and k exists. In the space of arbitrary even dimension 2m we construct an example of a continuous function with an estimate of the modulus of continuity such that its Fourier series is -divergent everywhere for any .     

The inequality of Ky Fan and related results

In this survey paper, we present refinements, extensions, and variants of the inequality

On Invariance of Some Classes of Holomorphic Functions Under Integrodifferential Operators

The following classes of functions analytic in the unit disk are considered:

On the Uniqueness of the Recovery of Parameters of the Maxwell System from Dynamical Boundary Data

The paper deals with the problem of recovering the parameters (functions) and of the Maxwell dynamical system

Mahler measure and entropy for commuting automorphisms of compact groups

We compute the joint entropy ofd commuting automorphisms of a compact metrizable group. LetR _d = [u ₁ ^±1 ,...,[d ₁ ^±1 ] be the ring of Laurent polynomials ind commuting variables, andM be anR _d -module. Then the dual groupX _M ofM is compact, and multiplication onM by each of thed variables corresponds to an action _M of ^d by automorphisms ofX _M . Every action of ^d by automorphisms of a compact abelian group arises this way. IffR _d , our main formula shows that the topological entropy of is given by

A characterization of the periods of periodic points of 1-norm nonexpansive maps

In this paper the set of minimal periods of periodic points of 1-norm nonexpansive maps is studied. This set is denoted by R(n). The main goal is to present a characterization of R(n) by arithmetical and combinatorial constraints. More precisely, it is shown that , where denotes the set of periods of restricted admissible arrays on 2n symbols. The important point of this equality is that is determined by arithmetical and combinatorial constraints only, and that it can be computed in finite time. By using this equality the set R(n) is computed for . Furthermore it is shown that the largest element of R(n) satisfies:      

On Rounding Error Sums Related to the Circle Problem. ii

For a positive real parameter t, real numbers , , and , we consider sums , where is the rounding error function, i.e.\ . Generalizing and improving the main result of Part I of the paper we show that there exists an absolute constant such that for all , and all . Further, we give applications concerning the circle problem with linear, polynomial, and general weight.     

On the LIL for Self-Normalized Sums of IID Random Variables

Let be i.i.d. random variables and let, for each and . It is shown that a.s. whenever the sequence of self-normalized sums S _n /V _n is stochastically bounded, and that this limsup is a.s. positive if, in addition, X is in the Feller class. It is also shown that, for X in the Feller class, the sequence of self-normalized sums is stochastically bounded if and only if      

Class Numbers of Indefinite Binary Quadratic Forms and the Residual Indices of Integers Modulo p

Let h(d) be the class number of properly equivalent primitive binary quadratic forms ax²+bxy+cy² with discriminant d=b²-4ac. The behavior of h(5p²), where p runs over primes, is studied. It is easy to show that there are few discriminants of the form 5p² with large class numbers. In fact, one has the estimate

Problems on Extremal Decomposition of the Riemann Sphere

We apply a variant of the method of the extremal metric to some problems concerning extremal decompositions and related problems. Let be a system of distinct points on and let be the family of all systems of nonoverlapping simply connected domains on such that . Let

On the Growth of the Denominators of Convergents

Let log . We prove that there exist non-denumerably many pairwise not equivalent irrational numbers such that and where qn() denotes the denominator of the nth convergent of .     

The Number of Linear Extensions of the Boolean Lattice

Let L(Q ^t ) denote the number of linear extensions of the t-dimensional Boolean lattice Q ^t . We use the entropy method of Kahn to show that

Concentration phenomena in weakly coupled elliptic systems with critical growth*

In this paper we consider the weakly coupled elliptic system with critical growth

A New Critical Behavior for Nonlinear Wave Equations

We study the inhomogeneous semilinear wave equations on with initial values and ,where is a noncompact, complete manifold. We founda new critical behavior in the following sense. There exists ap* > 0. When 1 < p p*, the above problem hasno global solution for any nonnegative not identicallyzero and for any and ; when the problem has a global solution for some and some and . If , which is equipped with the Euclideanmetric, then . If we show that belongs to the blow upcase. Although homogeneous semilinear wave equations are known to exhibit acritical behavior for a long time, this seems to be the first result oninhomogeneous ones.