ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. In 1970, R. Freese proved that the variety ${\bf M}^4$ generated by modular lattices of width at most $4$ has a finite basis. As an application, he obtained a complete description of all subdirectly irreducible members of this variety. We obtain an intuitive description of how congruences generated by a prime interval spread in a modular lattice of width at most $4$, and apply the result to reprove Freese's description of subdirectly irreducible lattices of width at most $4$.

AMS Subject Classification (1991): 06C05; 06B20

Keyword(s): lattice, modular, width, subdirectly irreducible, snake, weakly atomic

Received July 11, 2006, and in revised form November 24, 2006. (Registered under 5951/2009.)

Abstract. We show, under a weak assumption on the term $p$, that a variety of general algebras satisfies the congruence identity $p(\alpha_1, \ldots, \alpha_n) \subseteq q(\alpha_1, \ldots, \alpha_n )$ if and only if it satisfies the tolerance identity $p(\Theta_1, \ldots, \Theta_n) \subseteq q(\Theta_1, \ldots, \Theta_n)$, provided we restrict ourselves to tolerances representable as $R \circ R^-$. Varieties in which every tolerance is representable include all congruence permutable varieties and all varieties of lattices. For arbitrary tolerances, the congruence identity $p(\alpha_1, \ldots, \alpha_n) \subseteq q(\alpha_1, \ldots, \alpha_n )$ is equivalent to the identity $p(\Theta_1 \circ\Theta _1, \ldots, \Theta_n \circ\Theta _n) \subseteq q(\Theta_1 \circ\Theta _1, \ldots, \Theta_n \circ\Theta _n)$. See Theorems 3, 4 and 5. Our arguments essentially deal with labeled graphs, rather than terms; we try to clarify the role of graphs in the study of Mal'cev conditions (see especially Proposition 21 and Theorem 22).

AMS Subject Classification (1991): 08A30, 08B05

Keyword(s): Congruence, tolerance identity, Mal'cev condition, labeled graph, regular term, representable tolerance

Received November 13, 2006, and in revised form February 17, 2007. (Registered under 5952/2009.)

Abstract. Using the computational algebra system GAP and the package LAGUNA, we checked that all 2-groups of order not greater than 32 are determined by normalized unit groups of their modular group algebras over the field of two elements.

AMS Subject Classification (1991): 16S34, 20C05

Keyword(s): modular group algebras

Received July 10, 2006. (Registered under 5953/2009.)

Abstract. Let ${\rm A}(G)$ denote the automorphism group of a group $G$. A polynomial automorphism of $G$ is an automorphism of the form $x\mapsto(v_{1}^{-1}x^{\epsilon_{1}}v_{1})\ldots(v_{m}^{-1}x^{\epsilon_{m}}v_{m})$. We prove that if $G$ is nilpotent (resp. metabelian), then so is the subgroup of ${\rm A}(G)$ generated by all polynomial automorphisms.

AMS Subject Classification (1991): 20F28, 20F16, 20F18

Keyword(s): polynomial automorphism, metabelian group, nilpotent group, IA-automorphism

Received February 14, 2006, and in revised form June 9, 2006. (Registered under 5954/2009.)

Abstract. In this paper we prove that every $E$-unitary cover of an orthodox semigroup with a regular band of idempotents arises from a close embedding into an almost factorizable orthodox semigroup. Since every orthodox semigroup has an $E$-unitary cover, this shows that every orthodox semigroup with a regular band of idempotents is closely embeddable into an almost factorizable orthodox semigroup.

AMS Subject Classification (1991): 20M19, 20M10

Received December 8, 2006, and in revised form January 29, 2007. (Registered under 5955/2009.)

Abstract. Under various finiteness conditions, sets on which a given commutative semigroup $S$ acts have subdirect decompositions whose factors are readily constructed in terms of $S$.

AMS Subject Classification (1991): 20M20; 20M14.

Keyword(s): \pi, Key words and phrases:-regular semigroup, finitely generated commutative semigroup, Ponizovsky factor, act, bijective act, cancellative act, nilpotent act, elementary act, subelementary act, subdirect product

Received March 7, 2005, and in revised form October 1, 2006. (Registered under 5956/2009.)

Abstract. We improve and unify two theorems proved in a previous paper by the second author. The idea of the improvement is due to an assertion of the first author made in his referee's report of this paper.

AMS Subject Classification (1991): 26A15, 42A10

Received October 24, 2006. (Registered under 5957/2009.)

Abstract. In this note, some embedding relations among many important functional classes are considered. Results of Leindler [7] are extended and improved.

AMS Subject Classification (1991): 26A15, 42A10

Received September 14, 2006, and in revised form September 25, 2006. (Registered under 5958/2009.)

Abstract. Three theorems proved in [9] for sequences of rest bounded variation are extended in such a way that in the new theorems the Fourier cosine coefficients belong to the new class of sequences of mean rest bounded variation.

AMS Subject Classification (1991): 26A15, 42A16

Received September 2, 2006. (Registered under 5959/2009.)

Abstract. We introduce the $\omega $-tangential maximal function and the $\omega $-Littlewood--Paley function associated with Beurling's generalized distributions, which are extensions of Schwartz's distributions in terms of the weight function $\omega $. We compare the $\nu $-weighted $L^p$-norms of these maximal functions, and relations among several kinds of maximal functions. Also, we study the extension of Beurling's distributions on $R^n$ to $R^{n+1}_+$ by means of convolutions. As an application of these results we establish estimates between some of the above $\omega $-maximal functions associated to the extension.

AMS Subject Classification (1991): 42B25

Keyword(s): \omega, -tangential-maximal functions, \omega, -Littlewood--Paley functions

Received May 17, 2006, and in final form February 26, 2007. (Registered under 5960/2009.)

Abstract. We define spaces $H_{p}^+(w),$ $H_{p}^-(w)$ for weight functions $w$ with singularities in finitely many points and show that the dimension of $H_{p}^+(w)\cap H_{p}^-(w)$ is finite and depends on the number of singularities. We find the codimension in $L^p(w)$ of the subspaces generated by $H_{p}^+(w)\cup H_{p}^-(w).$ Necessary and sufficient conditions on the weight function $w$ are found so that the natural projection from $L^p(w)$ onto $H_{p}^+(w)$ exists for $1< p< \infty.$ It is also shown that no natural projection from $L^1(w)$ onto $H_{1}^+(w)$ may exist for any weight function $w $ under consideration.

AMS Subject Classification (1991): 42C15, 42C30

Keyword(s): H^p, weightedspaces, projection operator

Received August 21, 2003, and in revised form September 20, 2006. (Registered under 5961/2009.)

Abstract. We consider the space $E=E(\Omega,\|. \|)$ as the commutative C*-algebra ${\cal C}_0(\Omega )$ equipped with a norm $\|. \|$ having the monotonicity property $\|f \|\ge\|g \|$ if $| f | \ge | g | $. We show there exists a finest partition $\Pi $ of the underlying space $\Omega $ along with a function $m\colon\Omega \to{\msbm R}_+$ with the following properties: $\sup_{S\in\Pi } \# S < \infty $, $0<\inf m\le\sup m < \infty $ and each $E$-Hermitian operator $A$ can be written in the matrix form $Af(\omega ) = \sum_{\eta\in S} a^{(S)}_{\omega\eta } f(\eta )$, $\omega\in S \in\Pi _E$ with some system $[ a^{(S)} : S \in\Pi ]$ of matrices $a^{(S)} = [ a^{(S)}_{\omega\eta } ]_{\omega,\eta\in S}$ indexed with the elements of $\Omega $ and we have $\{f| _{S} : \|f \|\le1\} = \{\varphi\in {\cal C}(S): \sum_{\omega\in S} | \varphi(\omega ) | ^2 \le1\} $ for any partition member $S\in\Pi $. Hence, generalizing the Banach--Stone theorem, we obtain matrix descriptions for surjective isometries $E(\Omega,\|. \|) \to E(\widetilde{\Omega },\|. \|^\sim )$. We apply this result to show that unlike in the classical case of spectral norms, the linear isometric equivalence of the spaces $E(\Omega,\|. \|)$ and $E(\widetilde{\Omega },\|. \|^\sim )$ does not imply the existence of a positive surjective linear isometry in general, disproving a conjecture on Sunada type theorems for generalized Reinhardt domains.

AMS Subject Classification (1991): 46E15, 46B42, 28C05

Keyword(s): Banach lattice, Reinhardt domain, C*-algebra

Received November 25, 2006. (Registered under 5962/2009.)

Abstract. A certain class of higher rank Exel--Laca algebras, satisfying a Cuntz--Krieger type uniqueness theorem, is introduced.

AMS Subject Classification (1991): 46L05, 46L55

Received July 24, 2006, and in final form November 25, 2006. (Registered under 5963/2009.)

Abstract. Assume that $T$ is a bounded linear operator on a Banach space $X$. Let $W$ be a closed $T$-invariant subspace of $X.$ In this paper, the relationships between the spectral and Fredholm properties of $T$ and those of the pair of operators $T_{W}$ and $\widehat{T_{W}}$ are studied ($T_{W}$ is the restriction of $T$ to $W$ and $\widehat{T_{W}}$ is the operator determined by $T$ on $X/W).$ These results are applied to operators with infinite ascent or infinite strong descent.

AMS Subject Classification (1991): 47A05, 47A10, 47A53

Received September 21, 2005. (Registered under 5964/2009.)

Abstract. A localized version of the single-extension property is studied, for a bounded linear operator $T$ acting on a Banach space, at the points $\lambda\in {\msbm C}$ such that $\lambda I-T$ is quasi-Fredholm. This property is also studied at the points $\lambda\in {\msbm C}$ which are not limit points of the approximate point spectrum and the surjectivity spectrum. As a consequence, we improve a classical Putnam result about the non-isolated boundary points of the spectrum. From the characterizations of this property we shall also deduce several results on cluster points of some distinguished parts of the spectrum.

AMS Subject Classification (1991): 47A10, 47A11; 47A53, 47A55

Keyword(s): Localized SVEP, quasi-Fredholm operators, B, -Fredholm operators

Received July 7, 2006, and in final form January 15, 2007. (Registered under 5965/2009.)

Abstract. In paper [5] B. Sz.-Nagy proved his famous theorem about contractions in complex Hilbert spaces. In this paper we generalize his theorem to quaternion Hilbert spaces.

AMS Subject Classification (1991): 47A10

Keyword(s): Quaternion Hilbert spaces, operator, spectrum, contractions

Received July 11, 2006, and in revised form October 4, 2006. (Registered under 5966/2009.)

Abstract. We introduce a topology in the set of all semiclosed operators in a Hilbert space and investigate the topological structure by using the method of quotients of bounded operators. Under our topology, it is shown that the set of all closed operators is open in the set of all semiclosed operators, and that the topology restricted to the set of all closed operators is strictly stronger than the topology induced from the gap metric.

AMS Subject Classification (1991): 47A65

Keyword(s): semiclosed operators, quotients of bounded operators, semiclosed subspaces

Received April 4, 2005, and in revised form December 11, 2006. (Registered under 5967/2009.)

Abstract. Consider a pencil of rays in the Euclidean or Hyperbolic plane. The question may arise whether a polygon with equal angles can be constructed in such a way that the vertices are located on the given set of rays. We will discuss the solutions for triangles and quadrilaterals where the conditions are exactly given.

AMS Subject Classification (1991): 51M20

Received May 16, 2006, and in revised form January 16, 2007. (Registered under 8/2006.)

Abstract. Merging asymptotic expansions are established for the distribution functions of suitably centered and normed cumulative winnings in a full sequence of generalized St.$ $Petersburg games. These expansions are given in terms of suitably chosen members from the classes of subsequential semistable infinitely divisible asymptotic distribution functions and certain derivatives of these functions, where the classes themselves are determined by the two parameters of the game. Depending upon the most interesting cases of the tail parameter, which include the classical St.$ $Petersburg game, the expansions yield best possible rates of uniform merge with the selected semistable distribution functions.

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

Received January 18, 2007, and in final form May 16, 2007. (Registered under 5968/2009.)

Abstract. A collector samples with replacement a set of $n\ge2$ distinct coupons until he has, for the first time, all the coupons with only $m_n\in\{0,1,\ldots,n-1\} $ missing. If $m_n\to\infty $ and $(n-m_n)/\sqrt{n}\to\infty $ as $n\to\infty $, then the asymptotic distribution of the standardized random number of necessary draws is normal. With a Fourier-analytic method, we give a bound for the rates of convergence in these central limit theorems.

AMS Subject Classification (1991): 60F05

Received November 10, 2006. (Registered under 5969/2009.)

Abstract. We deduce a partial version of the KMT (1975) inequality for coupling the uniform empirical process with a sequence of Brownian bridges via the construction used by Csörgő and Révész (CsR) (1978) for their similar coupling of the uniform quantile process with another sequence of Brownian bridges. These constructions are pivoted on the KMT (1975, 1976) inequalities for approximating partial sums by a Wiener process (Brownian motion).

AMS Subject Classification (1991): 60F17, 60F15, 60G50, 62G30

Keyword(s): Empirical and quantile processes, Brownian bridge approximations, Hungarian construction

Received January 8, 2007, and in revised form March 6, 2007. (Registered under 5970/2009.)

Abstract. For a sequence of real-valued i.i.d. mean $0$ random variables $\{X, X_{n}; n \geq1 \} $ with partial sums $S_{n} = \sum_{i=1}^n X_{i}$, $n \geq1$, conditions are provided for $\{X, X_{n}; n \geq1 \} $ to enjoy one-sided iterated logarithm type behavior of the form $0 < \limsup_{n \rightarrow\infty } S_{n}/\sqrt{nh(n)} < \infty $ almost surely where $h(\cdot )$ is a positive, nondecreasing function which is slowly varying at infinity. New results are obtained as special cases and some open problems are posed.

AMS Subject Classification (1991): 60F15, 60G50

Keyword(s): Sums of i.i.d. random variables, law of the iterated logarithm, one-sided iterated logarithm type behavior, almost surely, slowly varying function

Received January 4, 2007, and in revised form May 10, 2007. (Registered under 5971/2009.)

Abstract. Data truncation is a problem in scientific investigations. So far, statistical models and inferences are mostly based on the assumption that the survival and truncation times are independent, which can be unrealistic in applications. In a nonparametric setting, we discuss identifiability of the conditional and unconditional survival and hazard functions when the survival times are subject to dependent truncation, namely, the survival time is dependent on the truncation time. Nonparametric kernel estimators of these unknowns are proposed. Usefulness of the nonparametric estimators is demonstrated through their theoretical properties, an application and a simulation study.

AMS Subject Classification (1991): 62N02, 62F12

Keyword(s): Conditional distribution, dependent truncation, hazard rate, identifiability, kernel, nonparametric estimation, survival function, truncation

Received January 15, 2007. (Registered under 5972/2009.)