ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. Given a class of first order structures ${\cal K}$ and an algebra $A$ of the type of ${\cal K}$, we define the set of $A$-{\it structures in ${\cal K}$}, in symbols ${\cal K}_A$, as the set of members of ${\cal K}$ whose underlying algebra is $A$. In this paper we pose the following problem: When the closure of ${\cal K}$ under the common operations (taking substructures, direct products,$\ldots $) can be expressed in terms of local properties concerning the sets ${\cal K}_A$ and global properties relating ${\cal K}_A$ and ${\cal K}_B$ whenever there is an algebra homomorphism from $A$ into $B$. We obtain some results in this direction which hold, in particular, for classes of structures axiomatized by equality-free sentences. The fundamental result says that the classes ${\cal K}$ for which ${\cal K}_A$ is an algebraic closure system, for all algebras $A$, are precisely the (equality-free) strict universal Horn classes.

AMS Subject Classification (1991): 03C52, 03C30

Keyword(s): Set of structures on an algebra, filter extension, local and global properties, closure condition

Received December 23, 1997 and in final form August 31, 1998. (Registered under 2669/2009.)

Abstract. Suppose $\cal V$ is a pointed variety of algebras of a type $\tau $, that is, $\cal V$ has a nullary operation $0$ such that each $A$ in $\cal V$ has $\{0_A\} $ as its smallest subalgebra. Each $n$-ary term $p(x_1,x_2,\ldots,x_n)$ of $\cal V$ determines a corresponding $n$-tuple of unary terms $\langle p_1,p_2,\ldots,p_n\rangle $, where $$p_j(x)=p(0,0,\ldots,0,x,0,\ldots,0)$$ equals $p$ evaluated with $x$ as $j$-th argument and $0$ elsewhere. If each $n$-ary term $p$ is uniquely determined by $\langle p_1,p_2,\ldots,p_n\rangle $ up to equivalence in $\cal V$, then $\cal V$ is said to have {\it linear terms}. The variety $R$-Mod of modules over a ring $R$ with unit has linear terms, since each $n$-ary term is equivalent to some linear combination $\Sigma_{j=1}^nr_jx_j$ which is determined by an $n$-tuple of coefficients in $R$. More generally, the variety of semimodules over a semiring $S$ with unit has linear terms. Pointed unary varieties also have linear terms. The properties of varieties with linear terms are studied.

AMS Subject Classification (1991): 08B99

Received April 15, 1997. (Registered under 2670/2009.)

Abstract. We study the polynomials $p_n(x)$ where $p_n(k)$ counts the number of integer-coordinate lattice points $(x_1,\ldots,x_n)$ with $\sum_{i=1}^n|x_i|\le k $. Using the fact that the polynomials $i^nn!p_n(-1/2-ix/2)$ are the classical Meixner polynomials of the second kind, we are able to prove finiteness results on the number of solutions of the diophantine equation $p_n(x)=p_m(y)$.

AMS Subject Classification (1991): 11D41, 26C10, 33C25, 05A15

Received October 20, 1997 and in revised form October 7, 1998. (Registered under 2671/2009.)

Abstract. A ring $R$ is said to be quasi directly finite if for any $a,b \in R$ such that $a+b = ab$ we have $ab = ba$; otherwise $R$ is quasi directly infinite. In order to study the latter rings, we introduce a ring ${\cal C}$ of $N \times N$ matrices over $Z$ generated by two elements, where $N$ is the set of nonnegative integers and $Z$ is the ring of integers. We characterize the ring ${\cal C}$ in several ways including the fact that ${\cal C}$ is isomorphic to the augmentation ideal of a semigroup ring. The main result shows that a ring $R$ is quasi directly infinite if and only if it contains a certain homomorphic image of the ring ${\cal C}$. Several ramifications of this result provide further characterizations of such rings as well as their relationship with directly infinite rings.

AMS Subject Classification (1991): 16P99, 15A36, 20M25

Received November 21, 1997 and in revised form October 12, 1998. (Registered under 2672/2009.)

Abstract. A mapping $\alpha $ on a poset $P$ is {\it decreasing} if $p \alpha\leq p$ for $p \in P$. For a class of finite posets [lattices] $\cal U$ let ${\rm DEnd} \cal U$ stand for the class of the semigroups which can be faithfully represented by decreasing order [lattice] endomorphisms of posets [lattices] from $\cal U$. We consider the classes ${\rm DEnd} \cal U$ for various $\cal U$ in comparison with the two classes ${\rm DEnd} {\cal C}$ and ${\rm DEnd} {\cal P}$ studied earlier, where ${\cal C}$ [${\cal P}$] stands for the class of all finite chains [posets].

AMS Subject Classification (1991): 20M20, 20M30, 06A07

Received April 7, 1998 and in revised form September 28, 1998. (Registered under 2673/2009.)

Abstract. We generalize a theorem pertaining to the factorization of inequalities of infinite series. The generalization means that we replace the function $x^p$, playing a crucial role in the antecedent theorem, by a more general function $\varphi(x)$.

AMS Subject Classification (1991): 26D15, 40D09

Received November 25, 1998 and in revised form December 9, 1998. (Registered under 2674/2009.)

Abstract. In this paper the summability of conjugate Laplace series of functions or measures is investigated. Criteria for the Abel summability, the Cesàro summability (pointwise and in $L^p$-norm) and the pointwise convergence are given.

AMS Subject Classification (1991): 33C55, 40G05, 40G10

Received January 12, 1998. (Registered under 2675/2009.)

Abstract. A comparison result is presented for linear impulsive delay differential equations (IDDE), and a new monotone iterative technique is obtained for nonlinear IDDE, which improve the results [3,4] in the literature.

AMS Subject Classification (1991): 34K15, 34K25, 34C10

Keyword(s): Impulsive delay differential equation, monotone iterative technique, comparison theorem

Received October 13, 1997 and in revised form January 13, 1999. (Registered under 2676/2009.)

Abstract. The regularity property of a norm-sequence $\rho(n)=\|T^n\| (n\in{\msbm N})$ ensures that the operator $T$ can be intertwined with an isometry $V$, which relation can be exploited to obtain a lot of information for $T$ itself, as it was shown in [1] and [2]. In [3] general sufficient conditions of regularity were provided. In the present note a necessary and sufficient condition of regularity is given. Applying this criterion a non-regular norm-sequence $\rho $ of positive radius is exhibited, settling the question, posed in [1] and [3], in the negative.

AMS Subject Classification (1991): 40A99, 47A99

Received September 30, 1998. (Registered under 2677/2009.)

Abstract. We extend known Tauberian results concerning the power series method of summability to results concerning the more general summability method $D_{\lambda,a}$ based on the Dirichlet series $\sum a_n e^{- \lambda_nx}.$

AMS Subject Classification (1991): 40G10, 40E05

Keyword(s): Tauberian, Dirichlet series methods

Received November 11, 1998 and in revised form December 21, 1998. (Registered under 2678/2009.)

Abstract. Properties of reproducing kernel function, Bessel's inequality, Parseval's identity in terms of an orthogonal basis of the space of all polynomials in $d$, where $d\in N\setminus\{1\} $, variables are studied.

AMS Subject Classification (1991): 42C05, 33C50

Keyword(s): multivariate orthogonal polynomials

Received February 26, 1998 and in revised form September 21, 1998. (Registered under 2679/2009.)

Abstract. A new constant $C(X)$ for a Banach space $X$ is introduced and it is proved that $X$ has the weak Banach--Saks property whenever $C(X)< 2$. Morover, it is shown that the Kottman constant $D(X)< 2$ implies that $X$ has the Banach--Saks property whenever $X$ is a Köthe sequence space. Nakano sequence spaces with the Banach--Saks property are charaterized. It is also proved that Cesàro sequence space $\mathop{\rm ces} _p$ has Banach--Saks type $p$.

AMS Subject Classification (1991): 46B20, 46E30

Keyword(s): Köthe Sequence Space, Cesàro Sequence Space, Packing Constant, Banach--Saks Property, Nakano Sequence Space, Weak Banach--Saks Property

Received January 26, 1998 and in revised form September 29, 1998. (Registered under 2680/2009.)

Abstract. The purpose of this paper is to obtain some new estimates of the norms of functions in Lorentz spaces $L_{q,r}$ ($q\ge2,r>0$), which are analogous to the estimates given by V.I. Kolyada [3] in the case of Lebesgue spaces. We shall also get estimates that generalize a well-known theorem of Marcinkiewicz and Zygmund [6]. The second half of our paper is devoted to obtain conjugate estimates for the ones mentioned above.

AMS Subject Classification (1991): 46E35, 26A15

Received May 1, 1998 and in revised form October 2, 1998. (Registered under 2681/2009.)

Abstract. Let $A$ and $B$ be bounded operators defined on a Hilbert space $H$ with a kernel condition $\ker A \subset\ker B$. We define a quotient $B/A$ to be a mapping $Au \to Bu, u \in H$. It is known that the family of all quotients contains all closed operators. In this paper we investigate some properties of the smallest and the largest positive selfadjoint extensions of a given positive symmetric quotient $B/A$, that is, $A^{\ast }B=B^{\ast }A \ge0$ (with respect to an order introduced below).

AMS Subject Classification (1991): 47A05, 47A99

Received March 26, 1998 and in final form December 15, 1998. (Registered under 2682/2009.)

Abstract. This paper deals with approximation---in the Banach algebra of all bounded linear operators on an infinite-dimensional (possibly nonseparable) real or complex Hilbert space---by the semi-$\alpha $-Fredholm operators. These are the operators which are either left or right invertible modulo the closed two-sided ideal of the algebra above which is associated to an infinite cardinal number $\alpha $, less than or equal to the Hilbert dimension of the space. The boundaries of all semi-$\alpha $-Fredholm components, as well as the boundaries of their closures, are characterized in terms of the approximate nullities of their elements and of the respective adjoints. The distances from a bounded linear operator $T$ to several sets related to the semi-$\alpha $-Fredholm operators are also computed. In particular, formulas for the distances to the semi-$\alpha $-Fredholm components, to their boundaries and to the boundaries of their closures are given. For each distance, a formula in terms of the weighted reduced minimi moduli of $T$ and a formula in terms of the minima of the weighted spectra of the modulus of $T$ are given.

AMS Subject Classification (1991): 47A58, 47A53

Keyword(s): \alpha, semi-Fredholm and semi--Fredholm operators, nonseparable Hilbert spaces, \alpha, semi--Fredholm components and boundaries, distance formulas, \alpha, reduced minimum modulus of weight

Received April 15, 1998 and in revised form November 26, 1998. (Registered under 2683/2009.)

Abstract. Various bounds for perturbation of eigenvalues of normal matrices are shown to remain true for compact perturbations of normal operators, if ``extended'' eigenvalue sets are considered in the sense introduced by T. Kato.

AMS Subject Classification (1991): 47A75, 15A42; 47A55, 47B10

Keyword(s): Spectral variation, Eigenvalue perturbation

Received March 2, 1995 and in revised form July 10, 1998. (Registered under 2684/2009.)

Abstract. We prove that the essentially normal composition operators with closed range on the space $H^2$ must be normal. We also give some other partial answers to the following general question. Is every noncompact essentially normal composition operator on $ H^2$ normal?

AMS Subject Classification (1991): 47B38, 47B15

Received December 10, 1997 and in final form August 26, 1998. (Registered under 2685/2009.)

Abstract. Our goal is twofold: (i) We give a unified treatment of the results initiated by Hardy in 1928 when he proved that the space $L^p({\msbm T})$ for any $1\le p< \infty $, is invariant under the $(C, 1)$-transform of the Fourier coefficients. (ii) We prove new results on the harmonic Cesàro operator ${\cal C}$ and the harmonic Copson operator ${\cal C}^*$ applied to functions defined on either the half real line ${\msbm R}_+$, or the whole real line ${\msbm R}$, or the torus ${\msbm T}$. Among others, we prove that the harmonic Copson operator $C^*$ is bounded on BMO, as well as from the subspace of the even functions in the real Hardy space $H^1$ into $L^1$.

AMS Subject Classification (1991): 47B48; 42A16, 42A38

Keyword(s): harmonic Cesàro operator, harmonic Copson operator, adjoint operator, inverse operator, spectrum of an operator, $L^p$-spaces, BMO, $H^1$-space, real Hardy space, Fourier transform, Fourier coefficient

Received January 14, 1998 and in revised form December 9, 1998. (Registered under 2686/2009.)

Abstract. Under Kato's or Tanabe's conditions, we prove that the solution of the Cauchy Problem for temporally inhomogeneous evolution systems depending analytically on a parameter $z$ varying in some domain $\Omega $, is itself analytic in $z$, $z\in\Omega $. This generalizes a result of [K] from the temporally homogeneous case to the temporally inhomogeneous case.

AMS Subject Classification (1991): 47D03, 47D05, 47D10

Received December 15, 1997 and in revised form February 27, 1998. (Registered under 2687/2009.)

Abstract. In this paper we consider composite multifunctions, i. e. multifunctions which are of the form $F=S\circ G$ with $S$ and $G$ both multifunctions which are u.s.c. and have compact and convex values in a Banach space. The resulting composition need not be convex-valued. Nevertheless, we show that a degree function can be defined for such compositions so as to satisfy the three basic properties (normalization, additivity and homotopy invariance). Also we show that under certain conditions this degree is independent of the way the composition is defined and we examine the relation between the boundary conditions and the degree. We also derive a product formula for condensing multifunctions and define a degree function for semicondensing multivalued vector fields. Finally we examine the fixed point index for weakly inward multifunctions.

AMS Subject Classification (1991): 47H11, 55M25, 47H04

Received December 30, 1996. (Registered under 2688/2009.)

Abstract. Consider a lower semicontinuous nonempty closed convex set-valued map $Q$ from a compact Hausdorff topological space ${\cal T}$ to ${\msbm R}^r$. To $Q$, there correspond a subset ${\msbm Q}$ of $C({\cal T},{\msbm R}^r)$ and a functional $q$ on ${\cal T}\times{\msbm R}^r$. Expressions for the tangent cone of ${\msbm Q}$ is given in terms of the corresponding concepts for $Q(t)$. The image space of each of the maps $Q \mapstochar\rightarrow {\msbm Q}$ and $Q \mapstochar\rightarrow q$ is completely described for this case and for the case when $Q(t)$ is open or has a nonempty interior for all $t$ in ${\cal T}$.

AMS Subject Classification (1991): 54C60, 54C65; 47H04, 58C06

Keyword(s): Set-valued maps, lower semicontinuous, C, C({\cal T}, closed and open-convex sets in, {\msbm R}^r), support functional, tangent and normal cones

Received March 10, 1998 and in revised form December 17, 1998. (Registered under 2689/2009.)

Abstract. Coincidence theorems for expansive mappings are shown. As applications, the existence of common solutions for a class of functional equations arising in dynamic programming is discussed. The results presented in this paper generalize, improve and unify some recent results.

AMS Subject Classification (1991): 54H25, 47H10

Received May 29, 1997 and in revised form July 22, 1998. (Registered under 2690/2009.)

Abstract. We give necessary and sufficient conditions for a pair of mappings to possess a common fixed point, which extend properly the results of Chang [3], Fisher [4], Jungck [5], Khan and Fisher [8], Rhoades, Tiwary and Singh [10] and Sessa, Mukherjee and Som [11]. As applications, the existence and uniqueness of common solutions for a class of functional equations in dynamic programming are discussed.

AMS Subject Classification (1991): 54H25, 47H10

Received October 16, 1997 and in revised form July 22, 1998. (Registered under 2691/2009.)

Abstract. Let $X(t) (t \in{\msbm R}_+)$ be a stable process in a random scenery. The Hausdorff dimension of certain level sets is determined and the existence of the local time of $X(t)$ is proved.

AMS Subject Classification (1991): 60G17, 60G18

Received August 4, 1998 and in revised form January 4, 1999. (Registered under 2692/2009.)