Acta Sci. Math. (Szeged)
65(1999), 439--460

Abstract. The main algebraic result of this paper contains a characterization of the weak subalgebra lattice of a unary partial algebra of a given finite unary type. This lattice must satisfy the conditions from [Bar] and moreover, one combinatorial condition concerning its atoms and join--irreducible elements. In a subsequent part [Pió2] we solve this problem for infinite unary types. Recall that in [Pió1] we reduced our algebraic problem for finite unary types to the following question: let $\bf G$ be a graph (which may have infinite sets of vertices and edges) and let $n$ be a natural number; when can $\bf G$ be directed ({\rm i.e.} when can its edges be directed) in such a way that at most $n$ directed edges start from each vertex? In the present paper we first solve this graph problem and hence we easily obtain the solution of our algebraic problem for finite unary types.

Acta Sci. Math. (Szeged)
65(1999), 461--467

Abstract. There exists an infinite ascending chain of finitely generated clones on a nine-element set for which the corresponding algebras are alternately finitely based and inherently nonfinitely based.

Acta Sci. Math. (Szeged)
65(1999), 469--492

Abstract. It is easy to prove that, given two integers $d\ge2$, $k\ge1$ and a sequence $\omega\in \{0,\ldots,d-1\} ^k$, the frequency of occurrence of $\omega $ in the expansions of the first $N$ integers with respect to the base $d$ tends, when $N\to\infty $, to a limit which only depends on $k$. This paper gives a suitable method to generalize this result to certain algebraic bases, and extends to it the Champernowne's construction.

Acta Sci. Math. (Szeged)
65(1999), 493--503

Abstract. In this paper Hölder and Minkowski type inequalities are considered for quasiarithmetic means. The situation when these inequalities are consequences of the classical Hölder and Minkowski inequalities for power means is characterized. The proofs involve some recent results of the author on the separation of quasiarithmetic means by power means.

Acta Sci. Math. (Szeged)
65(1999), 505--513

Abstract. We show that the Cesàro operator is bounded on the space of Cauchy transforms.

Acta Sci. Math. (Szeged)
65(1999), 515--527

Abstract. We establish some oscillation theorems for the nonlinear differential equation $$ [p(t)g(x)x']'+q(t)f(x)=r(t) $$ where $q,r\colon[t_0,+\infty )\to{\msbm R}$ are continuous functions and $p\colon[t_0,+\infty )\to(0,+\infty )$, $g\colon{\msbm R}\to(0,+\infty )$, $f\colon{\msbm R}\to{\msbm R}$ are continuously differentiable functions.

Acta Sci. Math. (Szeged)
65(1999), 529--542

Abstract. In this paper we study a quasilinear second order ordinary differential equation with periodic boundary conditions and a discontinuous vector field. We pass to a multivalued problem by filling in the gaps at the discontinuity points. Then, for the multivalued problem we prove the existence of a solution using techniques from the nonsmooth critical point theory.

Acta Sci. Math. (Szeged)
65(1999), 543--551

Abstract. We study summability of formal power series in a variable $t$, whose coefficients are functions of another variable $z$ and are assumed to satisfy certain differential recursion formulas. Such series arise naturally as formal solutions of certain partial differential equations. The results obtained here generalize earlier work for the heat equation by Lutz, Miyake and Schäfke, resp. W. Balser, as well as classical results concerning convergence of formal power series solutions of partial differential equations.

Acta Sci. Math. (Szeged)
65(1999), 553--566

Abstract. In this paper we study the effective behavior of composites with a nonlinear material behavior and a special type of microstructure. The microstructure we study is obtained by reiteration, i.e., introducing several local scales. We derive bounds on the effective shear properties and prove that the bounds are sharp when the number of scales tends to infinity.

Acta Sci. Math. (Szeged)
65(1999), 567--575

Abstract. Let $d\alpha $ be a measure on $[a,b]$ and let $p_k\ge1$, $k=1,2,\ldots,n$, be arbitrary fixed real numbers. The existence, uniqueness, and characterizations of a solution of the extremal problem $$\min_{a< x_1\le\ldots \le x_n\le b}\int_a^b\prod_{k=1}^n|x-x_k|^{p_k}d\alpha(x)$$ are given.

Acta Sci. Math. (Szeged)
65(1999), 577--584

Abstract. We consider the Riemann means of trigonometric series. Results proved: (i) A necessary and sufficient condition for a trigonometric series to be the Fourier series of a function in the periodic real Hardy space $H^1(T)$ is the boundedness of its Riemann means $S_h$ in $H^1$-norm. (ii) A necessary and sufficient condition for a trigonometric series to be the Fourier series of a function in the periodic ${\rm BMO}$ space is the boundedness of its Riemann means in the ${\rm BMO}$-norm.

Acta Sci. Math. (Szeged)
65(1999), 585--595

Abstract. In the paper [5] we found effective conditions for multipliers for double Walsh series with respect to homogeneous Banach spaces, using the notions of complementary spaces and summability factors. The aim of this paper is to find relations between several classes of multipliers for double Walsh series, using the above notions. We use all the notations and definitions of paper [5].

Acta Sci. Math. (Szeged)
65(1999), 597--610

Abstract. We obtain several results about sets of uniqueness for classes of Vilenkin systems which show that, in general, the more rapidly the parameters $p_k$ grow, the thinner the corresponding sets of uniqueness must be.

Acta Sci. Math. (Szeged)
65(1999), 611--633

Abstract. We consider continuity theorems for multiplicative integral transforms $\hat{f}$ of functions $f\colon(0, \infty ) \to{\msbm R},$ i.e., we relate the limit behaviours $f_n \to f$ with $\hat{f_n} \to\hat {f}.$ These continuity theorems based on sequential compactness arguments lead to Abelian-- and ratio--Tauberian theorems for these integral transforms.

Acta Sci. Math. (Szeged)
65(1999), 635--649

Abstract. In this paper we introduce new classes of Boehmians involving the Hankel convolution. ${\cal S}$-asymptotic behaviours of Zemanian distributions and Boehmians are analyzed.

Acta Sci. Math. (Szeged)
65(1999), 651--655

Abstract. In this note we give a spectral characterization of the Taylor--Browder spectrum for a doubly commuting $n$-tuple of some dominant operators properly containing $M$-hyponormal operators. This extends earlier results obtained by M. Chō.

Acta Sci. Math. (Szeged)
65(1999), 657--686

Abstract. This paper presents a refined and constructive version of the commutant lifting theorem (see [Sz.-NF3]) which includes the Treil--Volberg generalization of that theorem (see [TV]). This theory is used to solve a new variant of the Sarason interpolation problem.

Acta Sci. Math. (Szeged)
65(1999), 687--700

Abstract. The goal of this paper is to find the norm of special Gramians. As an illustrative example we determine the norm of the Hilbert matrix of infinite size using a method which is different from the usual one based on the Nehari extension problem (see [4]).

Acta Sci. Math. (Szeged)
65(1999), 701--726

Abstract. Extending former investigations on single operators with regular norm-sequences, representations of discrete abelian semigroups are studied. It is shown that if the representation exhibits ``regular'' norm-behaviour, then it can be related to an isometric representation, with useful containment properties between the corresponding spectra. As an application, a well-known stability theorem on bounded representations is extended to this general situation.

Acta Sci. Math. (Szeged)
65(1999), 727--736

Abstract. We present the following reflexivity-like result concerning the automorphism group of the $C^*$-algebra $B(H)$, $H$ being a separable Hilbert space. Let $\phi\colon B(H)\to B(H)$ be a multiplicative map (no linearity or continuity is assumed) which can be approximated at every point by automorphisms of $B(H)$ (these automorphisms may, of course, depend on the point) in the operator norm. Then $\phi $ is an automorphism of the algebra $B(H)$.

Acta Sci. Math. (Szeged)
65(1999), 737--748

Abstract. A new construction method for blocking sets in ${\rm PG}(2,q^2)$ is presented. We prove that the class of linear blocking sets introduced recently by Lunardon, Polito, and Polverino is closed under our construction. Various examples of blocking sets with 0,1,2 Rédei lines are also obtained by our method.

Acta Sci. Math. (Szeged)
65(1999), 749--765

Abstract. Recently there has been some development of limit theory for ``nonisotropic" random fields, under mixing conditions in which the mixing rates are (in a nontrivial way) allowed to differ in the different coordinate directions. Here a class of strictly stationary random fields is constructed in order to ``separate'' various possible mixing rates for certain mixing conditions on such nonisotropic random fields.

Acta Sci. Math. (Szeged)
65(1999), 767--784