Acta Sci. Math. (Szeged)
73(2008), 429--429

The Béla Szőkefalvi-Nagy Medal for the year 2007 has been awarded to Hari Bercovici, Indiana University, Bloomington, U. S. A..

Acta Sci. Math. (Szeged)
73(2007), 429--443

Abstract. We prove that the lattice $\mathop{\rm Eq} \Omega $ of all equivalence relations on an infinite set $\Omega $ contains, as a $0,1$-sublattice, the $0$-coproduct of two copies of itself, thus answering a question by G. M. Bergman. Hence, by using methods initiated by de Bruijn and further developed by Bergman, we obtain that $\mathop{\rm Eq} \Omega $ also contains, as a sublattice, the coproduct of $2^{\mathop{\rm card}\Omega }$ copies of itself.

Acta Sci. Math. (Szeged)
73(2007), 445--462

Abstract. We construct all planar semimodular lattices in three simple steps from the direct product of two chains.

Acta Sci. Math. (Szeged)
73(2007), 463--470

Abstract. We determine the primitive positive clones $F$ on finite sets $A$ with at least three elements for which $(A;F)$ is simple and idempotent, and the primitive positive clones $F$ having all constant operations for which $(A;F)$ either generates a congruence distributive variety or is a simple algebra that is not strongly abelian.

Acta Sci. Math. (Szeged)
73(2007), 471--486

Abstract. We characterize minimal clones generated by a majority function containing at most seven ternary operations.

Acta Sci. Math. (Szeged)
73(2007), 487--495

Abstract. We investigate minimal clones of operations on finite sets such that the corresponding algebra is functionally complete. We find that among the 24 isomorphism types of minimal clones over the 3-element set, determined by Béla Csákány, five are functionally complete. One of these minimal clones is generated by a binary operation corresponding to a tournament. We prove that a finite groupoid determined by a tournament is functionally complete, provided it is simple. Since almost all tournaments give rise to simple groupoids, this shows that there are a large number of functionally complete minimal clones.

Acta Sci. Math. (Szeged)
73(2007), 497--510

Abstract. Suppose $G$ is an abelian $p$-group and $F$ is a field. The Structure, Direct Factor and Isomorphism Theorems are proved for the group algebra $FG$ in the class of all restricted direct products of totally projective groups with $p^{\omega +n}$-projective groups ($n\in{\msbm N}_0={\msbm N}\cup\{0\} $) under some additional conditions on $F$. Specifically, we establish the following: Let $G$ be a separate $p^{\omega +1}$-totally projective $p$-group and ${\msbm F}_p$ the finite $p$-element field. Then $G$ is a direct factor of $V({\msbm F}_pG)$ such that $V({\msbm F}_pG)/G$ is separate $p^{\omega +1}$-totally projective and ${\msbm F}_pG$ as an ${\msbm F}_p$-algebra determines $G$ up to isomorphism. The results obtained strengthen assertions due to W. May (Proc. Amer. Math. Soc., 1979 and 1988) as well as due to the author (Compt. rend. Acad. bulg. Sci., 2001) and (Acta Math. Vietnamica, 2004).

Acta Sci. Math. (Szeged)
73(2007), 511--517

Abstract. It has been shown that in a factorization of an abelian group involving only simulated factors one factor must be periodic. This has been generalised in certain groups to allow one factor to satisfy weaker conditions but with stronger conditions on the simulated factors. In this paper we prove such results for all finite abelian groups and also remove the extra restriction on the simulated factors.

Acta Sci. Math. (Szeged)
73(2007), 519--545

Abstract. We derive necessary and sufficient conditions for the Birget--Rhodes prefix expansion [br1984] of a monoid to be (weakly) left ample, thereby proving analogues of the results already obtained for the related Szendrei expansion by Fountain, Gomes and Gould [fgg] and Fountain and Gomes [fg1990]. As a corollary, we obtain conditions for the prefix expansion to be inverse.

Acta Sci. Math. (Szeged)
73(2007), 547--591

Abstract. We generalize some aspects of the classical Fourier inversion theorem to the class of connected, simply connected, nilpotent Lie groups. In this setting, the generalized Fourier transform is the operator valued map $f \mapstochar\rightarrow (\pi_l(f))_{l \in{\eufm g}^*/Ad^*G}$. These operators are characterized by operator kernels. We construct a retract to the generalized Fourier transform which maps into the Schwartz space ${\cal S}(G)$, by limiting ourselves to a suitable set of families of operator kernels. This is done via variable Lie structures.

Acta Sci. Math. (Szeged)
73(2007), 593--611

Acta Sci. Math. (Szeged)
73(2007), 613--636

Abstract. The equation considered in this paper is $$ \left(a(t)\phi_p(x')\right )' + b(t)\phi_p(x') + c(t)\phi_p(x) = 0. $$ Here, $\phi_p(x') = |x'|^{p-2}x'$ with $p > 1$. The coefficients $a(t)$, $b(t)$ and $c(t)$ are not assumed to be positive. The main purpose is to present sharp conditions for the global asymptotic stability of the zero solution of a system equivalent to this differential equation. Sufficient conditions are also given for the zero solution to be globally attractive. Our results are new even in the linear case ($p = 2$). Some suitable examples are included to illustrate the main theorem. Global phase portraits are also attached for a deeper understanding. Finally, certain changes of variable are used to broaden the application of our results.

Acta Sci. Math. (Szeged)
73(2007), 637--647

Abstract. We study the relation between the generalized Zygmund classes of functions defined by Leindler [2] and the class of functions possessing a given rate of the strong approximation by their Fourier series. Furthermore, we present sufficient conditions in terms of Fourier coefficients to ensure a given rate of this strong approximation; and these conditions are also necessary in the special case when the Fourier coefficients are all nonnegative.

Acta Sci. Math. (Szeged)
73(2007), 649--667

Abstract. General criteria of discreteness of the spectrum for operators associated to positive and unbounded Jacobi matrices are established. These criteria unify well-known previous results on compactness of the resolvent of the above matrices.

Acta Sci. Math. (Szeged)
73(2007), 669--681

Abstract. This paper is to discuss the Furuta inequality and the Furuta-type operator function $f_{r,s}(p)=(A^{{r}/{2}}B^pA^{{r}/{2}})^{(s+r)/(p+r)}$ under $\log A\geq\log B$. Firstly, we provide a direct proof of the best possibility of the Furuta inequality with negative powers by using Tanahashi's two invertible operators. Secondly, it is well known that $f_{r,s}(p)$ is decreasing for $p\geq\max \{s,0\} $ when $r>0$ and $s>-r$. We show that, for each $r>0$ and $s>-r$, the monotone interval $[\max\{s,0\},\infty )$ is the unique one for the function $f_{r,s}(p)$ under chaotic order in the interval $[-r,\infty )$.

Acta Sci. Math. (Szeged)
73(2007), 683--728

Abstract. The space ${\cal H}ank$ of Hankel operators acting on the Hardy space ${\bf H}^2$ is a module over ${\bf H}^\infty $. There is a natural correspondence between weak* closed submodules of ${\cal H}ank$ and individual inner functions, and we apply work of V. Kapustin on Jordan models to characterize which submodules are reflexive in terms of the canonical factorization of these functions. We also prove that reflexivity of any weak* closed subspace of ${\cal H}ank$ is equivalent to reflexivity of the largest ${\bf H}^\infty $ module it contains. Analogous results are obtained in the finite dimensional and ``semi-infinite'' dimensional settings.

Acta Sci. Math. (Szeged)
73(2007), 729--744

Abstract. In this paper, we study closed invariant subspaces under the action of a unilateral shift and a truncated shift in the Hardy space that takes values in a two-dimensional Hilbert space. We deal with characteristic functions, unitary equivalence and $C^{\ast }$-algebras on these spaces.

Acta Sci. Math. (Szeged)
73(2007), 745--765

Abstract. In this paper we present functional random-sum central limit theorems in $L^2[0,1]$ with almost sure convergence for independent nonidentically distributed random variables. We consider the case where the summation random indices and partial sums are independent. In the past decade several authors have investigated the almost sure functional central limit theorems and related logarithmic limit theorems for partial sums of independent random variables. We extend this theory to almost sure versions of the functional random-sum central limit theorems in $L^2[0,1]$. The almost sure random functional limit theorem for the empirical process in $L^2[0,1]$ is presented, too.

Acta Sci. Math. (Szeged)
73(2007), 767--779

Abstract. We provide a characterization of strictly stationary solutions to the stochastic recurrence equation $z_k=c(\varepsilon_{k-1})z_{k-1}+g(\varepsilon_{k-1})$ with Borel-measurable functions $c$ and $g$, and independent, identically distributed random variables $\{\varepsilon_k\} $. Strictly stationary solutions that are functions of the past, respectively, of the future exist if and only if the expected value $E\log |c(\varepsilon_0)|$ is negative, respectively, positive. The main result of the paper is to show that there is no solution that is a function of the past or the future if $E\log |c(\varepsilon_0)|=0$.

Acta Sci. Math. (Szeged)
73(2007), 781--788

Abstract. Let $B(t)$, $X(t)$ and $Y(t)$ be independent standard 1d Brownian motions. Define $X^+(t)$ and $Y^-(t)$ as the trajectories of the processes $X(t)$ and $Y(t)$ pushed upwards and, respectively, downwards by $B(t)$, according to Skorohod-reflection. In the recent paper [warren], Jon Warren proves inter alia that $Z(t):= X^+(t)-Y^-(t)$ is a three-dimensional Bessel-process. In this note, we present an alternative, elementary proof of this fact.

Acta Sci. Math. (Szeged)
73(2007), 789--815

Abstract. An inhomogeneous first-order integer-valued autoregressive (INAR(1)) process is investigated, where the autoregressive type coefficient converges to one and the immigration mean tends to zero at appropriate speeds. It is shown that the process converges weakly to a Poisson or a compound Poisson distribution.

Acta Sci. Math. (Szeged)
73(2007), 817--838

Abstract. We consider a fixed-design regression model with long-range dependent errors which form a moving average process. Taking into account different behavior of regression estimators in such a model and in a random-design regression model discussed in Csörgő and Mielniczuk [5], we introduce an artificial randomization of grid points at which observations are taken in order to diminish the impact of strong dependence of errors. The resulting estimator is shown to exhibit smoothing dichotomy with the variance in both cases tending to $0$ more quickly than in the fixed design case. Moreover, we establish a uniform convergence rate of the regression function estimators which also reflects the dichotomous behaviour of the regression estimator. Simulation results indicate significant improvement for moderate sample sizes when randomization is employed.

Acta Sci. Math. (Szeged)
73(2007), 839--864

Abstract. We consider a discrete time Heath--Jarrow--Morton type forward interest rate model, where the interest rate curves are driven by a geometric spatial autoregression field. Local asymptotic normality is proved for stable and unstable no-arbitrage models containing a simple stochastic discounting factor. Based on these results, asymptotically optimal tests are constructed for testing the autoregression parameter.

Acta Sci. Math. (Szeged)
73(2007), 865--882

Abstract. Asymptotic results are obtained for weighted empirical processes appropriate for testing whether a change-point occurs in a nonparametric regression model. Since many quantities are unspecified, the results are not distribution-free. A weighted bootstrap approach is used to approximate the limiting distributions.

Acta Sci. Math. (Szeged)
73(2007), 883--895