ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

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.

AMS Subject Classification (1991): 06B15; 06B10, 06B25

Keyword(s): Lattice, equivalence relation, embedding, coproduct, ideal, filter, upper continuous

Received October 1, 2007. (Registered under 6444/2009.)

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

AMS Subject Classification (1991): 06C10, 06D05

Keyword(s): Semimodular lattice, planar, distributive, modular

Received November 28, 2006, and in revised form February 10, 2007. (Registered under 6445/2009.)

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.

AMS Subject Classification (1991): 08A40

Received January 15, 2007, and in final form August 19, 2007. (Registered under 6446/2009.)

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

AMS Subject Classification (1991): 08A40

Keyword(s): clone, minimal clone, majority operation

Received February 22, 2007, and in revised form May 7, 2007. (Registered under 6447/2009.)

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.

AMS Subject Classification (1991): 08A40

Keyword(s): functional completeness, minimal clone, tournament

Received July 27, 2007, and in revised form October 16, 2007. (Registered under 6448/2009.)

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).

AMS Subject Classification (1991): 16S34, 16U60, 20C07, 20K10, 20K15, 20K20, 20K21

Keyword(s): p^{\omega +n}, totally projective groups, -projective groups, elongations, units, isomorphisms

Received November 25, 2005, and in revised form November 3, 2006. (Registered under 15/2005.)

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.

AMS Subject Classification (1991): 20K01

Keyword(s): Factorization, Simulated, Abelian Group

Received November 27, 2006, and in revised form July 6, 2007. (Registered under 6449/2009.)

Abstract. We derive necessary and sufficient conditions for the Birget--Rhodes prefix expansion [BR1] 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 [FG]. As a corollary, we obtain conditions for the prefix expansion to be inverse.

AMS Subject Classification (1991): 20M99

Keyword(s): prefix expansion, (weakly) left ample monoid, inverse monoid

Received October 31, 2006, and in final form March 15, 2007. (Registered under 6450/2009.)

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.

AMS Subject Classification (1991): 22E30, 22E27, 43A20

Keyword(s): nilpotent Lie group, Fourier inversion theorem, operator kernel, minimal ideals

Received November 21, 2005, and in revised form October 12, 2006. (Registered under 6451/2009.)

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.

AMS Subject Classification (1991): 34D05, 34D23; 37B25, 37B55

Received November 28, 2006, and in revised form August 25, 2007. (Registered under 6453/2009.)

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.

AMS Subject Classification (1991): 42A10, 42A14, 42A32; 26A15, 41A25

Received August 29, 2007, and in revised form September 25, 2007. (Registered under 6454/2009.)

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.

AMS Subject Classification (1991): 47A55, 47A75

Keyword(s): Spectral theory, compact resolvent, Jacobi matrices

Received November 25, 2006, and in revised from March 14, 2007. (Registered under 6455/2009.)

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 $[-r,\infty )$.

AMS Subject Classification (1991): 47A63, 47B15

Keyword(s): positive operator, chaotic order, Furuta inequality

Received September 14, 2006, and in revised form January 8, 2007. (Registered under 6456/2009.)

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.

AMS Subject Classification (1991): 47B35, 47L05, 47A15, 46E15, 47A60

Received August 15, 2006. (Registered under 6457/2009.)

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.

AMS Subject Classification (1991): 47B38, 47B35

Keyword(s): vector valued Hardy spaces and shift operators

Received October 19, 2005, and in final form May 4, 2007. (Registered under 6458/2009.)

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.

AMS Subject Classification (1991): 60F05, 60F15, 60F17; 60G50

Keyword(s): Almost sure central limit theorem, functional random-sum central limit theorem, logarithmic averages, summation methods, Wiener measure, empirical process

Received January 22, 2007, and in revised form October 18, 2007. (Registered under 6459/2009.)

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 $E\log |c(\varepsilon _0)|=0$.

AMS Subject Classification (1991): 60G10; 62M10, 91B84

Keyword(s): Augmented GARCH processes, Generalized autoregressive equations, Stochastic recurrence equations, Strictly stationary solutions

Received May 9, 2007. (Registered under 6460/2009.)

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 [8], 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.

AMS Subject Classification (1991): 60J65

Keyword(s): Brownian motion, Skorohod reflection, Bessel process

Received September 8, 2007. (Registered under 6461/2009.)

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.

AMS Subject Classification (1991): 60J80; 60J27, 60J85

Keyword(s): Integer--valued time series, INAR(1) model, nearly unstable model, Poisson approximation, Galton--Watson process

Received March 22, 2007, and in revised form June 29, 2007. (Registered under 6462/2009.)

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.

AMS Subject Classification (1991): 62G07, 60F17

Keyword(s): Long- and short-range dependence, randomization, kernel estimator, linear process, random- and fixed-design regression, smoothing dichotomy

Received January 22, 2007, and in final form October 25, 2007. (Registered under 6463/2009.)

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.

AMS Subject Classification (1991): 62F05

Received March 1, 2007, and in revised form June 29, 2007. (Registered under 6464/2009.)

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.

AMS Subject Classification (1991): 62N05, 62N01, 62N02; 60G44, 62E20

Keyword(s): Change-point detection, nonparametric regression, strong approximations, weighted approximations, weighted bootstrap, asymptotic distributions

Received May 14, 2007. (Registered under 6465/2009.)