ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. We prove that there exists a lattice whose congruence lattice is not isomorphic to the congruence lattice of any lattice with $m$-permutable congruences. Our proof also extends to a wider class of algebras with $m$-permutable congruences. In order to do this we use and further develop the method invented by F. Wehrung for solving Dilworth's congruence lattice problem. To minimize the cardinality of our construction, we use the free trees combinatorial principle of P. Růžička.

AMS Subject Classification (1991): 06B10, 08A30, 06A12

Keyword(s): algebraic lattice, variety, congruence

Received March 6, 2007, and in revised form October 2, 2007. (Registered under 6000/2009.)

Abstract. We show that in a finite semimodular lattice, the ordering of join-irreducible congruences is done in a special type of sublattice, we call a tight $S_7$.

AMS Subject Classification (1991): 06C10, 06B10

Keyword(s): Semimodular lattice, planar, congruence, prime interval

Received April 21, 2007, and in final form January 31, 2008. (Registered under 6001/2009.)

Abstract. It has been recently shown [3] that the lattice effect algebras can be treated as a subvariety of the variety of so-called basic algebras. The main goal of this paper is to describe the variety of basic algebras generated by the horizontal sum of two 3-element MV-chains. It is shown that this variety is modulo the variety of distributive lattice effect algebras characterized by a unique simple identity.

AMS Subject Classification (1991): 08A30, 08B10

Keyword(s): Lattice effect algebra, basic algebra, antitone involution, variety

Received April 26, 2007, and in revised form November 22, 2007. (Registered under 6002/2009.)

Abstract. We present an {\it effective method} for the construction of a nonnegative realization of a real coefficient scalar transfer function having a single dominant (positive) pole and complex poles within the spectral disc, {\it all of arbitrary orders}. The nonnegativity of the impulse response is not assumed, but the nonnegativity of the coefficients of the dominant terms in the partial fraction decomposition of the transfer function. If a coefficient in this decomposition is sufficiently large, then a {\it general realization algorithm} is applicable with {\it a priori estimation of the dimension} of the obtained nonnegative realization. {\it An example shows the practical application} of the realization process.

AMS Subject Classification (1991): 15A48, 15A60, 93B15

Keyword(s): invariant cone, nonnegative realization of a primitive SISO transfer function, a priori estimation of the dimension, multiple complex poles, realization algorithm

Received December 22, 2006, and in revised form December 19, 2007. (Registered under 6003/2009.)

Abstract. We show that there are uncountably many varieties of semirings between the variety generated by ${\bf Rel}(2)$, the semiring of binary relations on a two-element set, and the variety generated by its particular subsemiring ${\bf\Sigma }_7$ (which is proved to be nonfinitely based in an earlier paper, as a first instance of such kind). An analogous result is obtained for semigroup varieties. As a consequence, we are able to exhibit a pseudorecursive variety of additively idempotent semirings.

AMS Subject Classification (1991): 16Y60, 20M07, 08B05

Received February 12, 2007, and in revised form June 20, 2007. (Registered under 6004/2009.)

Abstract. In this paper we prove a generalization of the classical Hermite--Hadamard inequality for convex (concave) functions extending it to $n$ different nodes, without using any further restrictions on the function. We use the concept of iterated integrals of $f$ to effect our purpose. Moreover we apply the main result to the function family $f(u) = u^\alpha(\alpha\ge 0)$ and useful identities for sums and products are given.

AMS Subject Classification (1991): 26A51

Received September 12, 2005, and in final form February 1, 2008. (Registered under 6005/2009.)

Abstract. We characterize those positive functions on a Boolean algebra $A$ which can be represented as the variation of a quasi-measure on $A$ with values in an Abelian normed group $G$. We also show that if there exists such a representation, then there is one in which $G$ is an $F^*$-lattice.

AMS Subject Classification (1991): 28B10, 28B05, 28A12, 28A60

Received March 8, 2007, and in revised form January 31, 2008. (Registered under 6006/2009.)

Abstract. This paper deals with the Tricomi problem for the Lavrent'ev-Bitsadze equation for the special case in which the elliptic part of the domain is a semi-disk. The connections amongst values of solutions of the Tricomi problem for the Lavrent'ev--Bitsadze equation on the characteristics in the hyperbolic part of the mixed domain are investigated. Inversion formulas for some integral operators are obtained. It is shown that Gellerstedt problems can be reduced to the Tricomi problem. Henceforth, two theorems and one corollary are obtained for the problem under consideration.

AMS Subject Classification (1991): 35M

Keyword(s): the Tricomi problem, the Lavrent'ev-Bitsadze equation, the Gellerstedt problems

Received April 4, 2007, and in revised form October 24, 2007. (Registered under 6007/2009.)

Abstract. In this paper we recall recent results that are direct consequences of the fact that $(w_{\infty }(\lambda),w_{\infty}(\lambda)) $ is a Banach algebra. Then we define the set $W_{\tau }=D_{\tau }w_{\infty }$ and characterize the sets $W_{\tau}(A) $ where $A$ is either of the operators $\Delta $, $\Sigma $, $\Delta(\lambda) $, or $C(\lambda)$. Afterwards we consider the sets $[A_{1},A_{2}] _{W_{\tau }}$ and give conditions for these sets to be in the form $W_{\tau }$. Finally we apply the previous results to obtain characterizations of matrix transformations in sets that generalize the sets of lacunary sequences such as $(N_{\theta }^{\infty }(\Delta ),N_{\xi }^{\infty }(\Delta ^{+h}))$, $(N_{\theta }^{\infty }(D_{1/\tau }C^+(\lambda )),N_{\xi }^{\infty })$ and $(N_{\theta }^{\infty }(\Delta(\mu )),N_{\xi }^{\infty })$.

AMS Subject Classification (1991): 40C05, 40J05, 46A15

Received April 21, 2007, and in revised form May 9, 2007. (Registered under 6008/2009.)

Abstract. De la Vallée Poussin means are used to prove Jackson--Favard type estimates for weighted algebraic polynomials with Jacobi and Laguerre-like weights.

AMS Subject Classification (1991): 41A10, 42C10, 33C45

Keyword(s): De la Vallée Poussin mean, Jackson-type inequality, weighted polynomial approximation, Laguerre and Jacobi polynomials

Received March 29, 2007, and in revised form April 28, 2008. (Registered under 6009/2009.)

Abstract. We consider Steklov operators in spaces of continuous functions on the real line and on a bounded interval. We study the connections of these operators with some second-order degenerate parabolic problems establishing a general Voronovskaja type formula.

AMS Subject Classification (1991): 41A36, 35K65

Keyword(s): Steklov operators, Representation of semigroups, Evolution problems

Received February 8, 2007, and in revised form May 4, 2007. (Registered under 6010/2009.)

Abstract. We consider the double Fourier series of functions $f\colon{\msbm T}^2\rightarrow{\msbm C}$, where ${\msbm T}^2$ is the two-dimensional torus. We prove sufficient conditions on the convergence of the double series whose terms are the $\beta $-th power of the absolute value of the Fourier coefficients of the function $f$ in question. These conditions are given in terms of moduli of continuity, of bounded variation in the sense of Vitali or Hardy and Krause, and of the mixed partial derivate in case $f$ is an absolutely continuous function. Our results extend the classical theorems of O. Szász and A. Zygmund from single to double Fourier series.

AMS Subject Classification (1991): 42A20, 42B99

Keyword(s): double Fourier series, absolute convergence, multiplicative moduli of continuity, multiplicative Lipschitz class, functions of bounded variation in the sense of Vitali and of Hardy and Krause, absolutely continuous functions of two variables, double versions of the theorems of O. Szász and A. Zygmund

Received December 11, 2006. (Registered under 6011/2009.)

Abstract. Strong summability results are proved for $d$-dimensional Fourier transforms of $f\in L_p(\log L)^{d-1}({\msbm R}^d)$ and $f\in L_p({\msbm R}^d)$.

AMS Subject Classification (1991): 42B08, 42A24, 42A38

Keyword(s): Fourier transforms, strong summability, \theta, -summability, strong maximal function, Lebesgue points

Received March 29, 2007, and in revised form May 21, 2007. (Registered under 6012/2009.)

Abstract. Let $A,B,$ and $C$ be unital $C^{\ast }$-algebras with $B$ injective. Let $C$ be a subalgebra of $A$ and $B$ with $I_{C}=I_{A}$ and $I_{C}=I_{B}$, let $M$ be a complex subspace of $A$ with $c_{1}Mc_{2}\subseteq M$ for all $c_{1},c_{2}\in C$, and let $L\colon M\rightarrow B $ be a $w_{\rho }$ completely bounded $C$-bihomomorphism. Then there exists a $C$-bihomomorphism extension $ \widetilde{L}\colon A\rightarrow B$ of $L$ with $\|\widetilde{L}\|_{W_{\rho cb}}=\|L\|_{W_{\rho cb}}$ $(0< \varrho\leq 2)$. Let $A_{i}$ be a unital $C^{\ast }$-algebra and $L_{i}\colon A_{i}\rightarrow B(H_{i})$ be a completely bounded map $(i=1,2)$. We provide $w_{\varrho }$ norms on $A_{1}\otimes_{\min }A_{2}$ and inequalities involving $\|L_{1}\otimes_{\min }L_{2}\|_{w_{\varrho }cb}$.

AMS Subject Classification (1991): 46L05, 46L10

Received September 25, 2006, and in revised form November 8, 2007. (Registered under 6013/2009.)

Abstract. In this note we first make a modest improvement of an inequality of Brown--Lomonosov--Simonic on certain transitive operator algebras. Then we use a result of Foias--Pasnicu--Voiculescu to provide similar inequalities for algebras generated by a quasitriangular operator.

AMS Subject Classification (1991): 47A15

Keyword(s): Invariant subspace, hyperinvariant subspace, quasitriangular, transitive operator

Received January 3, 2008, and in revised form February 11, 2008. (Registered under 6014/2009.)

Abstract. Let $f\colon X\rightarrow X$ be a continuous map on a Hausdorff topological space $X$ without isolated points. We show that if the orbit of a point $x\in X$ under $f$ is dense in $X$ while the orbit of $x$ under $f^N, N>1,$ is not, then the space $X$ decomposes into a family of sets relative to which the behaviour of $f$ is simple to describe. This decomposition solves a problem that P. S. Bourdon posed in 1996 ([3]). A slight variant of our result also provides a new argument for the celebrated theorem of S. Ansari [1]: If $T$ is a hypercyclic operator on a topological vector space $X$ then $T$ and $T^N$ have the same sets of hypercyclic vectors ($N\geq1$).

AMS Subject Classification (1991): 47A16; 37B05

Keyword(s): Dense orbit, Hypercyclic operator, Powers of hypercyclic operators, Ansari's theorem

Received September 21, 2007, and in revised form October 17, 2007. (Registered under 6015/2009.)

Abstract. We discuss the two closely related, but different concepts of weak and almost weak stability for the powers of a contraction on a separable Hilbert space. Extending Halmos' and Rohlin's theorems in ergodic theory as a model, we show that the set of all weakly stable contractions is of first category while the set of all almost weakly stable contractions is of second category and is residual. Analogous statements for unitary and isometric operators are also proved.

AMS Subject Classification (1991): 47A35, 37A25

Keyword(s): Power bounded operators, Hilbert space, stability, weak and strong mixing

Received December 15, 2006, and in final form March 28, 2008. (Registered under 6016/2009.)

Abstract. Let $T$ be a bounded linear operator on a complex Hilbert space $H$. In this paper, we prove: (i) $T$ has property ($\beta $) if and only if $\widetilde{T}_{p,r}=|T|^pU|T|^r$ $(p+r=1)$ has property ($\beta $). (ii) If $T$ belongs to Class $wF(p,r,q)$ operators, and $\lambda $ is an isolated point of the spectrum of $T$, $E$ the Riesz idempotent, with respect to $\lambda $, of $T$, then $\mathop{\rm Ker} (T -\lambda )=EH$ if $\lambda\not=0$. (iii) Weyl's theorem and a-Browder's theorem hold for Class $wF(p,r,q)$ operators. (iv) The spectral mapping theorem holds for the Weyl spectrum of $T$ and for the essential approximate point spectrum of $T$.

AMS Subject Classification (1991): 47A10, 47B20

Keyword(s): wF(p, class, r, operators, q), single valued extension property, \beta, Bishop's property(), Weyl's theorem, a-Browder's theorem, Browder's theorem

Received October 19, 2006, and in revised form June 12, 2007. (Registered under 6017/2009.)

Abstract. The paper deals with the norm calculations of the composition operator on Fock space over ${\msbm C}$. If $0< p< \infty $ and $C_{\varphi }\colon F^p\rightarrow F^p$ is the composition operator defined by $C_{\varphi }f=f\circ\varphi $, then it has been shown that the composition operator is bounded if and only if $\varphi(z)=az+b$, where either $|a|< 1$ and $b\in{\msbm C}$ or $|a|=1$ and $b=0$. Further, when $p=2$, it was proved that $\|C_{\varphi }\|_{2}=e^{{|b|^2\over4(1-|a|^2)}}$. In this paper we prove that for $p>2$ and $|a|< 1$ the norm of the composition operator is $\|C_{\varphi }\|_{p}=e^{|b|^2\over2p(1-|a|^2)}$.

AMS Subject Classification (1991): 47B33

Keyword(s): Composition operator, Fock space

Received March 1, 2007, and in revised form September 8, 2007. (Registered under 6018/2009.)

Abstract. In 1976, S. Stahl formulated the conjecture on the multichromatic number of the Kneser graphs: For any positive integers $m$ and $n$ with $m\geq2n$, $\chi_{nq+r}(KG_{m,n})=mq+m-2n+2r$, where $0\leq q$ and $0< r \leq n$. It is well kown that $\chi_{nq}(KG_{m,n})=mq$, moreover, Stahl's conjecture is equivalent to the claim that $\chi_{nq+1}(KG_{m,n})\geq mq+m-2n+2$. In $[4]$ Stahl proved that the gap between $\chi_{nq}$ and $\chi_{nq+1}$ is arbitrarily large if $n$ is fixed and $m$ is large enough. We shall prove here that $\chi_{nq+1}(KG_{m,n})\geq mq+3$ for any positive integers $m,n$ and $q$.

AMS Subject Classification (1991): 05C15

Received January 9, 2007, and in revised form April 11, 2007. (Registered under 1/2007.)

Abstract. The notion of $H$-convexity is a generalized convexity notion with many metrical and combinatorial applications (e.g., in distance geometry, combinatorial geometry, Minkowski geometry, and abstract convexity), and $H$-convex sets are simply defined with the help of a finite or infinite system $H$ of unit vectors in Euclidean $n$-space. In [BM2], [BM3], and [BM4] we investigated non-onesided, so-called $M$-complete systems of unit vectors and some of their applications in combinatorial geometry. In particular, we established a condition under which the vector (or Minkowski) sum of any two $H$-convex sets is again $H$-convex, and conditions for $H$-separability of $H$-convex sets. In both cases the notion of $M$-completeness, defined for the vector systems $H$, plays the key role. Here we study properties of {\it maximal} non-onesided, $M$-complete vector systems $\overline H$ and $\hat H$ in the unit sphere ${\msbm S}^{n-1}$, which means that any non-onesided, $M$-complete vector system containing them coincides with ${\msbm S}^{n-1}$. On the other hand, we prove for closed systems, which are symmetric with respect to the origin, that the systems $\overline H$ and $\hat H$ are also {\it universal}, i.e., under some natural condition every non-onesided, $M$-complete vector system distinct from ${\msbm S}^{n-1}$ is contained in $\overline H$ or in $\hat H$. Some examples illustrate the obtained results.

AMS Subject Classification (1991): 32F17, 32F99, 52A01, 52A20, 52A30

Keyword(s): direct decomposition, direct vector sum, generalized convexity notion, H, -convexity, M, -complete vector system, Minkowski addition, positive linear combination, universality, vector sum

Received May 23, 2007, and in revised form October 1, 2007. (Registered under 6/2007.)

Abstract. Let $X_t$ be a Lévy process and $V_t=\sigma ^2 t+ \sum_{0< s\le t} (\Delta X_s)^2, t>0$, its quadratic variation process, where $\Delta X_t=X_t-X_{t-} $ denotes the jump process of $X$. When $X$ is symmetric, we show that the self-normalized process $Y_t:=X_{t}/\sqrt{V_t}$ converges in distribution as $t\downarrow0$ to an a.s. finite, nondegenerate random variable, if and only if (i) $X_t$ is in the domain of attraction of a nondegenerate stable random variable $S_\alpha $; that is, if and only if, for some nonstochastic function $b(t)>0$, $X_t/b(t)\mathop{\longrightarrow}^{\mathrm D} S_\alpha $, as $t\downarrow0$; or else, (ii) the tail of the Lévy measure of $X$ is slowly varying at 0. This is proved as an application of criteria we set out for the joint convergence of $X_t$ and $V_t$, after norming (and centering, in the case of $X$), to infinitely divisible, and, in particular, to stable, limit rvs, as $t\downarrow0$, either continuously, or through a subsequence.

AMS Subject Classification (1991): 60F05, 60F15, 60G51, 62E20, 62G30

Keyword(s): Lévy process, self-normalized, small-time behavior, domain of attraction

Received July 4, 2007, and in revised form February 11, 2008. (Registered under 6019/2009.)

Abstract. We study some properties of the local time of the asymmetric Bernoulli walk on the line. These properties are very similar to the corresponding ones of the simple symmetric random walks in higher ($d\geq3$) dimension, which we established in the recent years. The goal of this paper is to highlight these similarities.

AMS Subject Classification (1991): 60G50; 60F15, 60J55

Keyword(s): transient random walk, local time, occupation time, strong theorems

Received September 21, 2007, and in revised form January 7, 2008. (Registered under 6020/2009.)

Abstract. The Riemann zeta process is a stochastic process $\{Z(\sigma ), \sigma >1\} $ with independent increments and marginal distributions whose characteristic functions are proportional to the Riemann zeta function along vertical lines ${\msbm R}e s = \sigma $. We establish functional limit theorems for the zeta process and other related processes as arguments $\sigma $ approach the pole at $s=1$ of the zeta function (from above).

AMS Subject Classification (1991): 60G51, 60F17, 11N37

Keyword(s): Erdős--Kac theorem, functional limit theorem, geometric process, Riemann zeta function, zeta process

Received August 14, 2007, and in revised form March 13, 2008. (Registered under 6021/2009.)

Abstract. In this paper, we consider the kernel density estimator constructed from the product-limit estimator of an unknown continuous distribution when the data are subjected to random left truncation and right censorship. We obtained a law of the iterated logarithm for the exact pointwise convergence rate of the kernel density estimator.

AMS Subject Classification (1991): 62G05, 62G07, 62G20, 60F15

Keyword(s): Law of the iterated logarithm, strong approximation, counting process, martingales, stochastic integrals, product-limit estimator

Received June 8, 2007, and in final form April 15, 2008. (Registered under 6022/2009.)

Abstract. We study the asymptotic behavior of large deviation probabilities for a general class of tail index estimators. This new class consists of the generalized version of the weighted least-squares estimators proposed by Viharos [9] and also contains the class of kernel estimators obtained by Csörgő et al. [3]. Based on the large deviation probabilities, a comparison of the members of this class can be made. The Hill estimator turns out to have optimal rate of convergence within a subclass of estimators.

AMS Subject Classification (1991): 62G32, 60F10

Received August 14, 2007, and in revised form November 20, 2007. (Registered under 6023/2009.)

Abstract. In this paper a possible definition of the third order LRD is given in terms of the bispectrum and third order cumulants. It is shown that three particular second order LRD processes fulfil the requirements of the third order LRD. Some conjectures are closing the paper.

AMS Subject Classification (1991): 62M10, 37M10; 62M15, 91B70

Received June 8, 2007, and in final form April 25, 2008. (Registered under 6024/2009.)