
ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

429429
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Friedrich Wehrung

429443

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.)
George Grätzer,
Edward Knapp

445462

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 3element 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.)
Christopher Hollings

519545

Abstract. We derive necessary and sufficient conditions for the BirgetRhodes 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.)
Jean Ludwig,
Carine MolitorBraun,
Laurent Scuto

547591

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.)
Vilmos Totik,
Péter P. Varjú

593611
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Jitsuro Sugie,
Masakazu Onitsuka

613636

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, $\phi _p(x') = x'^{p2}x'$ 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.)
Ferenc Móricz,
József Németh

637647

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.)
Petru A. Cojuhari,
Jan Janas

649667

Abstract. General criteria of discreteness of the spectrum for operators associated to positive and unbounded Jacobi matrices are established. These criteria unify wellknown 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.)
Jiangtao Yuan,
Zongsheng Gao

669681

Abstract. This paper is to discuss the Furuta inequality and the Furutatype 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.)
Edward Azoff,
Ruben A. MartinezAvendano,
James Solazzo

683728

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 ``semiinfinite'' dimensional settings.
AMS Subject Classification
(1991): 47B35, 47L05, 47A15, 46E15, 47A60
Received August 15, 2006. (Registered under 6457/2009.)
Ronald G. Douglas,
Takahiko Nakazi,
Michio Seto

729744

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 twodimensional 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.)
Zdzisł{a}w Rychlik,
Wojciech Skublewski,
Tomasz Walczyński

745765

Abstract. In this paper we present functional randomsum 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 randomsum 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 randomsum 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.)
Alexander Aue,
István Berkes,
Lajos Horváth

767779

Abstract. We provide a characterization of strictly stationary solutions to the stochastic recurrence equation $z_k=c(\varepsilon _{k1})z_{k1}+g(\varepsilon _{k1})$ with Borelmeasurable 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.)
Bálint Tóth,
Bálint Vető

781788

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 Skorohodreflection. In the recent paper [8], Jon Warren proves inter alia that $Z(t):= X^+(t)Y^(t)$ is a threedimensional Besselprocess. 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.)
László Györfi,
Márton Ispány,
Gyula Pap,
Katalin Varga

789815

Abstract. An inhomogeneous firstorder integervalued 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):
Integervalued time series,
INAR(1) model,
nearly unstable model,
Poisson approximation,
GaltonWatson process
Received March 22, 2007, and in revised form June 29, 2007. (Registered under 6462/2009.)
Artur Bryk,
Jan Mielniczuk

817838

Abstract. We consider a fixeddesign regression model with longrange dependent errors which form a moving average process. Taking into account different behavior of regression estimators in such a model and in a randomdesign 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 shortrange dependence,
randomization,
kernel estimator,
linear process,
random and fixeddesign regression,
smoothing dichotomy
Received January 22, 2007, and in final form October 25, 2007. (Registered under 6463/2009.)
Erika Fülöp,
Gyula Pap

839864

Abstract. We consider a discrete time HeathJarrowMorton 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 noarbitrage 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 changepoint occurs in a nonparametric regression model. Since many quantities are unspecified, the results are not distributionfree. A weighted bootstrap approach is used to approximate the limiting distributions.
AMS Subject Classification
(1991): 62N05, 62N01, 62N02; 60G44, 62E20
Keyword(s):
Changepoint detection,
nonparametric regression,
strong approximations,
weighted approximations,
weighted bootstrap,
asymptotic distributions
Received May 14, 2007. (Registered under 6465/2009.)

883895
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