ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. The equivalences ${\cal D}, {\cal L}$ and ${\cal R},$ defined initially on semigroups by J. A. Green, are used to study both noncommutative lattices and their congruence lattices, with particular attention given to the effects of assuming that some or all of these equivalences are congruences. Several specialized classes of noncommutative lattices are considered, including some that are simple algebras. Occurrences of distribution in noncommutative lattices as well as their congruence lattices are also considered.

AMS Subject Classification (1991): 06F05, 08A30, 20M10

Keyword(s): noncommutative lattices, congruences, Green's equivalences

Received February 7, 2001, and in revised form January 26, 2002. (Registered under 2854/2009.)

Abstract. This paper contributes to the investigation of general relation algebras in connection with first order definable operations: e.g. Boolean systems of relations with projections (BSP) are algebras of relations closed with respect to set-theoretical operations definable by first order formulas without equality. As in the case of relational clones and Krasner algebras, BSP are Galois closed sets with respect to a Galois connection -- the strong invariance -- between operations (here unary operations) and relations. They can internally be described also as extensions of Krasner algebras. Variations of the first order formulas under considerations lead to several Galois connections the Galois closed elements of which are also completely characterized. In a unified setting instead of unary functions we use multifunctions as objects corresponding to relations w.r.t. the Galois connection.

AMS Subject Classification (1991): 08A02, 03G99, 06A15

Keyword(s): relation algebra, Galois connection, strongly invariant relation, first order logic without equality, multifunction

Received April 3, 1998, and in revised form March 30, 2001. (Registered under 2855/2009.)

Abstract. We present some new families of collapsing monoids. These monoids form large intervals in the submonoid lattices of the full transformation semigroups. Some of these intervals have cardinalities $\ge2^{2^{cn}}$ where $n$ is the size of the base set.

AMS Subject Classification (1991): 08A40

Received March 28, 2001, and in revised form July 25, 2001. (Registered under 2856/2009.)

Abstract. We show that there are many natural algebraic constructions under which dualisability is not always preserved. In particular, we exhibit two dualisable unary algebras whose product is not dualisable.

AMS Subject Classification (1991): 08A60, 08C15, 18A40

Keyword(s): Natural duality, dualisability, unary algebra

Received December 8, 2000, and in final form March 16, 2001. (Registered under 2857/2009.)

Abstract. We use some commutator theory together with a recent result by K. Kearnes and A. Szendrei in order to provide a non-trivial implication between two congruence identities strictly weaker than modularity.

AMS Subject Classification (1991): 08B99, 06B20, 08A30, 08B10

Keyword(s): Congruence (lattice) identities, congruence modularity, weak difference term, commutator of congruences

Received April 29, 1999, and in revised form April 23, 2000. (Registered under 2858/2009.)

Abstract. This is a complete characterization of all possible simultaneous relations between the clones of uniformly continuous maps of two metric spaces and the respective clones of their continuous maps, in terms of the equality, isomorphism and elementary equivalence of their initial clone segments. In conjunction with earlier results, the apparatus introduced here gives a full characterization of the equality, isomorphism and elementary equivalence of clone segments for two topological spaces and their various lower and upper modifications, and a similar characterization of the segments of centralizer clones for two algebras with at least three non-nullary operations and their respective reducts.

AMS Subject Classification (1991): 54C05, 08C05

Keyword(s): clone, clone segment, finitary algebraic theory, subtheory, functors preserving finite products, algebras and their reducts, categories of universal algebras, categories of uniform or topological spaces, topological modifications, equality of clone segments, isomorphism of clone segments, elementary equivalence of clone segments

Received May 9, 2001, and in revised form April 7, 2002. (Registered under 2859/2009.)

Abstract. Trinomials which define canonical number systems are characterized in terms of their coefficients.

AMS Subject Classification (1991): 11R04, 11R16, 11R21, 12D99

Received February 12, 2001, and in revised form November 8, 2001. (Registered under 2860/2009.)

Abstract. Using the characterization of groups with abelian Sylow 2-subgroups, we deduce some splitting criteria. The main result is: if $G$ is a group with abelian Sylow 2-subgroups without non-trivial solvable factor groups and without non-trivial solvable normal subgroups, then any extension of $G$ splits over $G$. Also, we give new proofs of some known theorems about splitting over normal subgroups with abelian Sylow subgroups.

AMS Subject Classification (1991): 20D40, 20F17

Received March 25, 2001, and in revised form April 12, 2002. (Registered under 2861/2009.)

Abstract. It is proved that the class of quasi-monotonic sequences with the additional assumption $\Sigma c_n/n < \infty $ is not comparable to the class of $\delta $-quasi-monotonic sequences with the assumption $\Sigma n^\gamma\delta _n < \infty $, $\gamma >0$; furthermore none of them is comparable to the class of sequences of rest bounded variation.

AMS Subject Classification (1991): 26D15, 40-99, 42A20

Keyword(s): Inequalities, embedding relations, sums, \delta, -quasi monotone sequences, R^+_0 BV, -sequences, sine and cosine series

Received January 30, 2001. (Registered under 2862/2009.)

Abstract. Consider ${\bf T}=\{z \in {\bf C}:|z|=1\}$, the unit circle with the usual normalized arc-length measure ${\cal L}$. We give a simple sufficient condition (a Divergence Criterion), with a completely self-contained and elementary proof, for the divergence of ergodic averages along subsequences in ${\bf N}$. As an application, we give a very elementary argument of the following result. Let $(n_k)_1^\infty$ be any increasing sequence in ${\bf N}$ with strictly increasing gaps, i.e., $n_{k+1}-n_{k}>n_{k}-n_{k-1}, k\geq 2$. Let $0<\rho<1$ be given. Then there exists an ergodic rotation $\tau \colon {\bf T}\to {\bf T}$ such that for any given $\epsilon >0$, there are infinitely many $f \in L^\infty({\bf T})$ satisfying $$ {\cal L}\Big( \big\{z \in {\bf T}: \overline{\lim}{1 \over l}\sum_{k=1}^{l}f \circ \tau^{10^{n_k}}(z)- \underline{\lim}{1 \over l}\sum_{k=1}^lf \circ\tau^{10^{n_k}}(z) \geq \rho \big\}\Big)\geq 1 -\epsilon.$$

AMS Subject Classification (1991): 28D99, 60F99

Received March 27, 2001. (Registered under 2863/2009.)

Abstract. The equation $$x''+a^2(t)x=0, a(t):=a_k\ \hbox{ if }\ t_{k-1} \le t< t_k, \ \hbox{ for }\ k=1,2,\ldots $$ is considered, where the sequence $\{a_k\} ^\infty_{k=1}$ $(a_k>0, k=1,2,\ldots )$ is given, and $t_{k+1}-t_k$, $k=1,2,\ldots $ are totally independent random variables uniformly distributed on interval $[0,1]$. The probability of events $\gamma =0$, $\Gamma =0$, and $\Gamma >0$ are studied, where $$\gamma :=\liminf_{t\to\infty }\left(x^2(t)+{(x'(t))^2\over a(t)}\right ),\qquad \Gamma :=\limsup_{t\to\infty }\left(x^2(t)+{(x'(t))^2\over a(t)}\right ).$$

AMS Subject Classification (1991): 34D20, 34F05

Received April 3, 2002, and in revised form October 21, 2002. (Registered under 2864/2009.)

Abstract. In this paper we prove some abstract minimax principles for nonsmooth locally Lipschitz energy functionals and then we use those abstract results to study semilinear and quasilinear hemivariational inequalities at resonance. We permit the possibility of strong resonance at $\pm\infty $ and using a variational approach, based on the nonsmooth critical point theory of Chang, we prove the existence of nontrivial solutions and multiple solutions for semilinear and quasilinear hemivariational inequalities at resonance.

AMS Subject Classification (1991): 35J20, 35J85, 35R70

Keyword(s): hemivariational inequalities, strong resonance, locally Lipschitz functional, subdifferential, nonsmooth Cerami condition, critical point, minimax principle, nonsmooth Saddle Point Theorem, Ekeland variational principle, Rayleigh quotient, principal eigenvalue, p-Laplacian

Received March 3, 2000, and in revised form April 18, 2002. (Registered under 2865/2009.)

Abstract. We show that four successive enlargements of the Sidon--Telyakovskii's class ${\cal ST}$, introduced as new integrability and $L^1$-convergence classes, are identical. For even trigonometric series, they coincide with the wellknown even classes ${\cal F}_p$, $p>1$, introduced by Fomin in 1978. For general trigonometric series, they coincide with a Fomin-type integrability class introduced by F. Móricz in 1991. It is somewhat surprising that several `different' enlargements of ${\cal ST}$ should yield only equivalent and indeed more complicated descriptions of the Fomin's and the Fomin-type classes. We also prove that the Fomin-type classes for general series, due to F. Móricz, are subclasses of $(dv^2)'$, one of the largest known integrability and $L^1$-convergence classes, and discuss other relationships between the known integrability classes. Furthermore, we show that the Fomin-type theorems for general series can be directly deduced from the original Fomin's results for even, i.e. cosine series.

AMS Subject Classification (1991): 42A16, 42A20

Received June 5, 2000, and in final form November 6, 2001. (Registered under 2866/2009.)

Abstract. A generalized uniqueness problem for Rademacher series has been posed and solved.

AMS Subject Classification (1991): 43A70, 43A75, 42C10

Keyword(s): Rademacher function, Uniqueness

Received May 8, 2001, and in revised form August 1, 2001. (Registered under 2867/2009.)

Abstract. Conditions for pointwise Fourier inversion using Cesàro means of a given order are established on rank one compact symmetric spaces.

AMS Subject Classification (1991): 43A85

Received April 23, 2001. (Registered under 2868/2009.)

Abstract. A bounded linear operator $ T, $ respectively an $n$-tuple $ T $ of commuting bounded operators, on a complex Banach space $ {\cal X} $ is strongly harmonic if it is contained in a unital commutative strongly harmonic closed subalgebra $ {\cal A} \subset B({\cal X}). $ Every strongly harmonic operator is decomposable in the sense of Foiaş and every strongly harmonic $n$-tuple is decomposable in the sense of Frunză. On the other hand, it is proven that the class of strongly harmonic operators is quite large and that operators in this class have very nice properties. If an elementary operator is determined by two strongly harmonic $ n$-tuples, then it is strongly harmonic, and its local spectra are in a simple connection with the analytic local spectra of $2n$-tuple of the coefficients.

AMS Subject Classification (1991): 47B40, 47B47, 47B48

Received February 27, 2001, and in revised form April 23, 2001. (Registered under 2869/2009.)

Abstract. Even though there are no asymptotic distributions in the usual sense, we show that the distribution functions of the suitably centered and normed cumulative winnings in a full sequence of generalized St.Petersburg games merge together uniformly with completely specified semistable infinitely divisible distribution functions at certain fast rates, depending upon the tail parameter of the game.

AMS Subject Classification (1991): 60F05, 60E07, 60G50

Received February 12, 2002, and in final form June 25, 2002. (Registered under 2870/2009.)

Abstract. Let $X=\{X(t), t\geq0, {\msbm P}^x, x \in G \} $ be the Brownian motion on the Sierpiński gasket $G$. We prove that there exist two positive constants $c$ and $C$ such that for every $x \in G$, ${\msbm P}^x$-a.s. for all $t \in[0, \infty )$, we have $ ct \leq\varphi-m({\rm Gr}(X[0,t]))\leq Ct$, where ${ \rm Gr}X([0,t])=\{(s, X(s)): 0 \leq s \leq t \} $ is the graph set of $X$, $$\varphi(s)=s^{1+ \log3/\log2 - \log3/\log5}(\log\log {1}/{s})^{ \log3/\log5}, s \in(0, {1}/{8}],$$ and $\varphi $-$m$ denotes Hausdorff $\varphi $-measure.

AMS Subject Classification (1991): 60G17, 60J60, 28A78

Keyword(s): Brownian motion on the Sierpiński gasket, Hausdorff measure, graph

Received April 23, 2001, and in revised form October 24, 2001. (Registered under 2871/2009.)

Abstract. We survey eleven papers by György Pollák published from 1973 to 1989 and devoted to various aspects of the theory of semigroup varieties: hereditarily finitely based varieties, permutation identities, covers in varietal lattices.

AMS Subject Classification (1991): 20M07, 08B05, 08B15

Received February 18, 2002. (Registered under 2873/2009.)

Abstract. We prove that if $\{x,y\},\{u,v\} $ are two sets of generating pairs for a free group $F$ satisfying the equation $ [x,y^n] = [u,v^m]$ then $n = m$. Further if $n = m \ge2$ then $y$ is conjugate in $F$ to $v^{\pm1}$. This theorem rose out of a question concerning Schottky groups. The method of proof is used to consider certain related equations in free groups and generalizations to genus one Fuchsian groups.

AMS Subject Classification (1991): 20E05

Keyword(s): Free Groups, Equations, Test Elements, Scottky Groups

Received January 19, 2001, and in revised form July 12, 2001. (Registered under 2874/2009.)

Abstract. A periodic-basis function (PBF) is a function of the form $$ \phi(u)=\sum_{k\in{\msbm Z}}\widehat{\phi }(k) e^{iku}, u\in{\msbm R}, $$ where the sequence of Fourier coefficients $\{\widehat{\phi }(k) : k\in{\msbm Z}\} $ satisfies the following conditions: $$ \widehat{\phi }(k)=\widehat{\phi }(-k), k\in{\msbm Z}, \hbox{ and } \sum_{k\in{\msbm Z}}|\widehat{\phi }(k)|< \infty. $$ A PBF $\phi $ is said to be strictly positive definite if every Fourier coefficient of $\phi $ is positive. It is known that if $\phi $ is strictly positive definite, then given any continuous $2\pi $-periodic function $f$ and any triangular array $\{\theta_{j,\mu } : 1\le j\le\mu, \mu\in {\msbm N}\} $ of distinct points in $[-\pi,\pi )$, there exists a unique PBF interpolant $ I(\theta ):= \sum_{j=1}^\mu a_j\phi(\theta -\theta_{j,\mu })$, $a_j\in{\msbm R}$, such that $ I(\theta_{k,\mu })=f(\theta_{k,\mu })$, $1\le k\le\mu $. This paper studies the uniform convergence of these PBF interpolants to the approximand $f$. Even though there is a rather well-developed theory which supplies various results of this nature, it also has the shortcoming that if $\phi $ is very smooth, then the class of functions $f$ which can be simultaneously approximated and interpolated by PBF interpolants is highly restricted. The primary objective of this paper is to suggest an oversampling strategy to overcome this problem. Specifically, it is shown that by increasing the dimension of the underlying space of approximants/interpolants judiciously, one can construct PBF interpolants (based on very smooth $\phi $) that converge to approximands which are only assumed to be continuous. The main tool in the analysis is a periodic version of a result of Szabados on algebraic polynomials, the proof of which relies on the trigonometric version of a fundamental theorem due to Erdős.

AMS Subject Classification (1991): 41A05, 41A30, 42A08, 42A12

Received November 21, 2000, and in revised form July 17, 2001. (Registered under 2875/2009.)

Abstract. We prove Bernstein type inequalities for algebraic polynomials on the interval $I:=[-1,1]$ and for trigonometric polynomials on {\msbm R} when the roots of the polynomials are outside of a certain domain of the complex plane. The cases of real vs. complex coefficients are handled separately. In case of trigonometric polynomials with real coefficients and root restriction, the $L_p$-situation will also be considered. In most cases, the sharpness of the estimates will be shown.

AMS Subject Classification (1991): 41A17

Received February 26, 2001, and in revised form August 8, 2001. (Registered under 2876/2009.)

Abstract. A bounded linear operator in a Banach space is called Koliha--Drazin invertible (generalized Drazin invertible) if ${0}$ is not an accumulation point of its spectrum. In this paper the main result is the stability of the Koliha--Drazin invertible operators with finite nullity under commuting Riesz operator perturbations. We also generalize some recent results of Castro, Koliha and Wei, and characterize the perturbation of the Koliha--Drazin invertible operators with essentialy equal eigenprojections at zero.

AMS Subject Classification (1991): 47A05, 47A53, 15A09

Keyword(s): generalized Drazin inverse, perturbation, Riesz operator

Received January 2, 2001, and in revised form March 26, 2001. (Registered under 2877/2009.)

Abstract. We give a sufficient condition involving local spectra for an operator on a separable Banach space to be hypercyclic. Similar conditions are given for supercyclicity. These spectral conditions allow us to characterize the hyponormal operators with hypercyclic adjoints and those with supercyclic adjoints.

AMS Subject Classification (1991): 47A10, 47A11, 47A16, 47B20, 47B40

Keyword(s): Hypercyclic, supercyclic, hyponormal, (\beta ), (\delta ), propertiesand

Received February 7, 2001, and in final form October 2, 2001. (Registered under 2878/2009.)