ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

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.

AMS Subject Classification (1991): 08A55, 08A60, 05C20, 05C90, 08A30

Received July 15, 1998, and in final form January 21, 1999. (Registered under 2694/2009.)

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.

AMS Subject Classification (1991): 08B05

Keyword(s): finite algebra, clone, finitely based

Received December 10, 1998. (Registered under 2695/2009.)

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.

AMS Subject Classification (1991): 11A63, 11K55, 11M41, 11R04

Received October 28, 1998, and in revised form April 12, 1999. (Registered under 2696/2009.)

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.

AMS Subject Classification (1991): 26D15, 26A51

Keyword(s): Quasiarithmetic means, Power means, Separation, Hölder and Minkowski-type inequalities

Received October 29, 1998. (Registered under 2697/2009.)

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

AMS Subject Classification (1991): 30E20, 47B38, 46E15

Received October 5, 1998, and in revised form March 18, 1999. (Registered under 2698/2009.)

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.

AMS Subject Classification (1991): 34C10, 34C15

Received May 7, 1998. (Registered under 2699/2009.)

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.

AMS Subject Classification (1991): 34C25

Keyword(s): Nonsmooth critical point theory, locally Lipschitz functional, generalized subdifferential, saddle point theorem, eigenvalue, nonresonance condition, Palais-Smale condition, Poincaré-Wirtinger inequality

Received September 9, 1997, and in revised form January 25, 1999. (Registered under 2700/2009.)

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.

AMS Subject Classification (1991): 35A20, 35A22, 35C10, 35C20

Received January 22, 1999, and in revised form April 21, 1999. (Registered under 2701/2009.)

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.

AMS Subject Classification (1991): 35B27, 73B27

Received November 17, 1998, and in revised form April 12, 1999. (Registered under 2702/2009.)

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.

AMS Subject Classification (1991): 41A55, 65D32

Received May 21, 1998, and in revised form February 25, 1999. (Registered under 2703/2009.)

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.

AMS Subject Classification (1991): 42A16, 42A24, 42A38

Received February 3, 1999, and in revised form March 9, 1999. (Registered under 2704/2009.)

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].

AMS Subject Classification (1991): 42B08, 42B15, 42C10, 46B45, 46E30

Received February 15, 1999. (Registered under 2705/2009.)

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.

AMS Subject Classification (1991): 42C10; 43A75

Received February 3, 1999, and in revised form May 21, 1999. (Registered under 2706/2009.)

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.

AMS Subject Classification (1991): 44A05, 40E05

Keyword(s): Integral transform, continuity theorems, Abelian theorems, Tauberian theorems, dominated variation, regular variation

Received January 29, 1999. (Registered under 2707/2009.)

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.

AMS Subject Classification (1991): 46F12

Received May 21, 1998, and in revised form November 27, 1998. (Registered under 2708/2009.)

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.

AMS Subject Classification (1991): 47A13, 47B20, 47A53

Keyword(s): Taylor-Browder spectrum, joint Weyl spectrum, dominant operator, M-hyponormal operator

Received February 9, 1999. (Registered under 2709/2009.)

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.

AMS Subject Classification (1991): 47A57, 47A45

Received April 13, 1999. (Registered under 2710/2009.)

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]).

AMS Subject Classification (1991): 47B35, 15A60

Received April 29, 1998, and in revised form January 29, 1999. (Registered under 2711/2009.)

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.

AMS Subject Classification (1991): 47D03, 47A67

Received February 5, 1999. (Registered under 2712/2009.)

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 $B(H)$.

AMS Subject Classification (1991): 47D50, 47B49, 46L40

Keyword(s): Reflexivity, automorphisms, ideals of operators, Wigner's unitary-antiunitary theorem

Received January 15, 1999, and in revised form March 26, 1999. (Registered under 2713/2009.)

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.

AMS Subject Classification (1991): 51E21

Received May 26, 1998, and in revised form March 11, 1999. (Registered under 2714/2009.)

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.

AMS Subject Classification (1991): 60G10, 60G60

Received February 18, 1999, and in final form August 5, 1999. (Registered under 2715/2009.)