ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. This is a continuation of the investigation on the partial elementary logic system introduced in [Wo97] to provide a basis for the construction of cylindric algebras describing properties of partial relations. We show the relationship between partial logic and classic elementary logic. We prove that classic logic is, in some general model-theoretic sense, interpretable in partial logic, but not vice versa. Moreover, we present --- via a theorem on the correspondence of models --- some close connections between models of partial and classic logics as well as between their theories.

AMS Subject Classification (1991): 03B60; 03C07, 03C52, 91A05, 91A80

Keyword(s): first order logic, cylindric algebras, partial relations, interpretability, two-person games, winning strategy, Boolean algebras of formulas, models, theories

Received February 22, 2010. (Registered under 13/2010.)

Abstract. A real-valued height function $f$ is defined on a closed rectangle $R$. A rectangle $S\subset R$ is an $f$-island if there exists an open set $G\subset R$ containing $S$ such that $f(x)< \inf_{S} f$ for every $x\in G\setminus S$. The set of all $f$-islands is called a {\it system of (rectangular) islands.} In this paper we prove that there exists a maximal system of islands of cardinality $\aleph_0$, and that the size of a maximal system of islands is either countable or continuum.

AMS Subject Classification (1991): 05A05, 54A25

Keyword(s): maximal systems of islands, countable, continuum, laminar system

Received September 7, 2009, and in final form March 3, 2010. (Registered under 161/2009.)

Abstract. In 1960, G. Grätzer and E. T. Schmidt proved that every finite distributive lattice can be represented as the congruence lattice of a sectionally complemented finite lattice $L$. For $u \leq v$ in $L$, they constructed a sectional complement, which is now called the {\it1960 sectional complement}. In 1999, G. Grätzer and E. T. Schmidt discovered a very simple way of constructing a sectional complement in the ideal lattice of a chopped lattice made up of two sectionally complemented finite lattices overlapping in only two elements---the Atom Lemma. The question was raised whether this simple process can be generalized to an algorithm that finds the 1960 sectional complement. In 2006, G. Grätzer and M. Roddy discovered such an algorithm---allowing a wide latitude how it is carried out. In this paper we prove that the wide latitude apparent in the algorithm is deceptive: whichever way the algorithm is carried out, it produces the same sectional complement. This solves, in fact, Problems 2 and 3 of the Grätzer--Roddy paper. Surprisingly, the unique sectional complement provided by the algorithm is the 1960 sectional complement, solving Problem 1 of the same paper.

AMS Subject Classification (1991): 06C15, 06B10

Keyword(s): sectionally complemented lattice, sectional complement, finite

Received September 26, 2009, and in revised form August 6, 2010. (Registered under 318/2009.)

Abstract. Let ${\cal K}$ denote the class of finite length semimodular lattices that have congruence-determining chain ideals. Assume that $L\in{\cal K}$ and $D$ is a $(0,1)$-sublattice of $\mathop{\rm Con} L$. We prove the existence of an $\overline L\in{\cal K}$ such that $L$ is a filter of $\overline L$ and the restriction mapping from $\mathop{\rm Con} \overline L$ to $\mathop{\rm Con} L$ is an isomorphism from $\mathop{\rm Con} \overline L$ onto $D$. The particular case $D=\{0,1\} $, not only for $L\in{\cal K}$, has intensively been studied by several papers, including [4], [5], [2] and [7].

AMS Subject Classification (1991): 06C10, 06B15

Keyword(s): lattice, semimodular, finite length, congruence lattice, embedding

Received May 11, 2009, and in final form March 17, 2011. (Registered under 66/2009.)

Abstract. The structure of the unit groups of the group algebra of the group ${C_3}^2 \rtimes C_2$ over any field of characteristic $3$ is established in terms of split extensions of cyclic groups.

AMS Subject Classification (1991): 16U60, 16S34, 20C05, 15A33

Keyword(s): unit group, group ring, group algebra, cyclic

Received October 3, 2009, and in revised form September 8, 2010. (Registered under 259/2009.)

Abstract. In a previous paper, we obtained conditions on a monoid $M$ for its prefix expansion to be either a left restriction monoid (in which case $M$ must be either `type-I' or `type-II') or a left ample monoid ($M$ is `type-Ia' or `type-IIa'). In the present paper, we demonstrate that there is some redundancy in these conditions. We therefore trim down the sets of conditions and show, by construction of suitable counterexamples, that the reduced sets of conditions are independent.

AMS Subject Classification (1991): 20M99

Keyword(s): monoid, unipotent, bipotent, type-I, type-II, type-Ia, type-IIa

Received September 1, 2009, and in final form January 14, 2010. (Registered under 99/2009.)

Abstract. In this paper we study the asymptotic properties of the polynomials orthogonal with respect to the modified Laguerre weight ${\prod_{k=1}^K(x-a_k)^{N_k}\over\prod _{j=1}^M(x-\eta_j)} x^\alpha e^{-x},$ where $N_k\in{\msbm N},$ $a_k,\eta_j< 0$ and $\eta_k\not=\eta_l$ for $k\not=l.$

AMS Subject Classification (1991): 33C45, 42C05

Keyword(s): orthogonal polynomials, modified Laguerre-type orthogonal polynomials, asymptotics

Received September 8, 2009, and in final form December 14, 2009. (Registered under 103/2009.)

Abstract. This paper considers the $n^{th}$-order boundary value problem consisting of the equation $$ -(\phi(u^{(n-1)}(x)))^{\prime }=f(x,u(x),\ldots,u^{(n-1)}(x)), x\in(0,1), $$ together with the boundary conditions $$\eqalign{ g_{i}(u,u^{\prime },\ldots,u^{(n-2) },u^{(i)}(0))&=0,\hbox{ }i=0,\ldots,n-3, \cr g_{n-2}(u,u^{\prime },\ldots,u^{(n-2)},u^{(n-2)}(0),u^{(n-1)}(0))&=0, \cr g_{n-1}(u,u^{\prime },\ldots,u^{(n-2)},u^{(n-2)}(1),u^{(n-1)}(1))&=0, }$$ where $\phi $ is an increasing homeomorphism such that $\phi(0)=0$, $n\geq2$ is an integer, $I:=[0,1]$, and $f\colon I\times{\msbm R}^n\rightarrow{\msbm R}$ is an $L^1$-Carathéodory function. Here, $g_{i}\colon(C(I))^{n-1}\times{\msbm R}\rightarrow{\msbm R}$, $i=0,\ldots,n-3$, and $g_{n-2}$, $g_{n-1}\colon( C(I))^{n-1}\times{\msbm R}^2\rightarrow{\msbm R}$ are continuous functions satisfying certain monotonicity assumptions. We present sufficient conditions on the nonlinearity and the boundary conditions to ensure the existence of solutions. Moreover, from the lower and upper solutions method, some information is given about the location of the solution and its qualitative properties. Due to the functional dependence in the boundary conditions, this work generalizes several results for higher order problems with many types of boundary conditions. The main results are illustrated with examples.

AMS Subject Classification (1991): 34B15, 34B10

Keyword(s): boundary value problems, increasing homeomorphism, functional boundary conditions, Nagumo condition, lower and upper solutions

Received October 23, 2009. (Registered under 5476/2009.)

Abstract. Recently, a generalization to the Pontryagin space setting of the notion of canonical (Hamiltonian) systems which involves a finite number of inner singularities has been given. The spectral theory of indefinite canonical systems was investigated with help of an operator model. This model consists of a Pontryagin space boundary triple and was constructed in an abstract way. Moreover, the construction of this operator model involves a procedure of splitting-and-pasting which is technical but at the present stage of development in general inevitable. In this paper we provide an isomorphic form of this operator model which acts in a finite dimensional extension of a function space naturally associated with the given indefinite canonical system. We give explicit formulae for the model operator and the boundary relation. Moreover, we show that under certain asymptotic hypotheses the procedure of splitting-and-pasting can be avoided by employing a limiting process. We restrict attention to the case of one singularity. This is the core of the theory, and by making this restriction we can significantly reduce the technical effort without losing sight of the essential ideas.

AMS Subject Classification (1991): 47E05, 46C20; 47B25, 34L05

Keyword(s): canonical system, Pontryagin space, boundary triple

Received September 4, 2009. (Registered under 102/2009.)

Abstract. Revisiting the results in [L1, L2], we consider the polynomial approximation on $(-1,1)$ with the weight $w(x)={\rm e}^{-(1-x^2)^{-\alpha }}$, $\alpha >0$. We introduce new moduli of smoothness, equivalent to suitable $K$-functionals, and we prove the Jackson theorem, also in its weaker form. Moreover, we state a new Bernstein inequality, which allows us to prove the Salem--Stechkin inequality. Finally, also the behaviour of the derivatives of the polynomials of best approximation is discussed.

AMS Subject Classification (1991): 41A10, 41A17, 41A25, 41A27

Keyword(s): Jackson theorems, Markoff--Bernstein inequalities, orthogonal polynomials, approximation by polynomials, one-sided approximation

Received November 10, 2009, and in revised form February 3, 2010. (Registered under 6175/2009.)

Abstract. The partial integrals of the $N$-fold Fourier integrals connected with elliptic polynomials (with a strictly convex level surface) are considered. It is proved that if $a+s>(N-1)/2$ and $ap=N$, then the Riesz means of the nonnegative order $s$ of the $N$-fold Fourier integrals of continuous finite functions from the Sobolev spaces $W_p^a(R^N)$ converge uniformly on every compact set, and if $a+s=(N-1)/2$, $ap=N$, then for any $x_0\in R^N$ there exists a continuous finite function from the Sobolev space $W_p^a(R^N)$ such that the corresponding Riesz means of the $N$-fold Fourier integrals diverge to infinity at $x_0$.

AMS Subject Classification (1991): 42B08, 42C14

Keyword(s): $N$-fold Fourier integrals, elliptic polynomials, continuous functions from the Sobolev spaces, uniformly convergence

Received May 28, 2009, and in revised form April 13, 2010. (Registered under 72/2009.)

Abstract. The spectral theory of a two-dimensional canonical (or `Hamiltonian') system is closely related with two notions, depending whether Weyl's limit circle or limit point case prevails. Namely, with its monodromy matrix or its Weyl coefficient, respectively. A Fourier transform exists which relates the differential operator induced by the canonical system to the operator of multiplication by the independent variable in a reproducing kernel space of entire $2$-vector valued functions or in a weighted $L^2$-space of scalar valued functions, respectively. Motivated from the study of canonical systems or Sturm--Liouville equations with a singular potential and from other developments in Pontryagin space theory, we have suggested a generalization of canonical systems to an indefinite setting which includes a finite number of inner singularities. We have constructed an operator model for such `indefinite canonical systems'. The present paper is devoted to the construction of the corresponding monodromy matrix or Weyl coefficient, respectively, and of the Fourier transform.

AMS Subject Classification (1991): 47E05, 46C20, 47B25, 46E22

Keyword(s): canonical system, Pontryagin space boundary triple, maximal chain of matrices, Weyl coefficient

Received August 4, 2009, and in revised form January 27, 2011. (Registered under 90/2009.)

Abstract. Let $X$ be an infinite dimensional complex Banach space, and let ${\cal L}(X)$ be the algebra of all bounded linear operators on $X$. In this paper, we prove that an additive surjective map $\phi\colon {\cal L}(X)\rightarrow{\cal L}(X)$ preserves the reduced minimum modulus if and only if either there exist bijective isometries $U\colon X\rightarrow X$ and $V\colon X\rightarrow X$ both linear or both conjugate linear such that $\phi(T)=UTV$ for all $T\in{\cal L}(X)$, or there exist bijective isometries $U\colon X^*\rightarrow X$ and $V\colon X\rightarrow X^*$ both linear or both conjugate linear such that $\phi(T)=UT^*V$ for all $T\in{\cal L}(X)$.

AMS Subject Classification (1991): 47B48, 47A10, 46H05

Keyword(s): reduced minimum modulus, isometry, additive preservers

Received August 25, 2009, and in final form April 21, 2010. (Registered under 96/2009.)

Abstract. If an operator $T$ defined on the intersection of a suitable couple of Banach spaces $X$ and $Y$ has an extension $T_X \in{\cal L}(X)$ and an extension $T_Y \in{\cal L}(Y)$, there arises the question whether the spectrum $\sigma(T_X)$ of $T_X$ is equal to that of $T_Y$. This paper is concerned with this question. As a typical result, in the case where $X \cap Y$ is dense in $X$ and in $Y$, it is proved that if $T$ has an extension that is bounded from $X+Y$ into $X \cap Y$, then $\sigma(T_X)=\sigma(T_Y)$. This result has an application to the $L^p$-spectral independence of bounded operators. On the other hand, in the case where $Y$ is continuously and not necessarily densely embedded into $X$, it is proved that if $X$ and $T \in{\cal L}(X,Y)$ satisfy certain conditions, then $\sigma(T_X)=\sigma(T_Y)$. This result gives an example that shows an operator in ${\cal L}(L^\infty({\msbm R}^N))$ and its part in ${\cal L}(C_0({\msbm R}^N))$ have the same spectrum.

AMS Subject Classification (1991): 47A10, 47A25, 47G10

Keyword(s): spectrum of bounded operator, $L^p$-spectral independence, product of operators, integral operator

Received November 1, 2009, and in revised form May 6, 2010. (Registered under 6079/2009.)

Abstract. We bound the difference in length of two curves in terms of their total curvatures and the Fréchet distance. The bound is independent of the dimension of the ambient Euclidean space, it improves upon a bound by Cohen-Steiner and Edelsbrunner, and it generalizes a result by Fáry and Chakerian.

AMS Subject Classification (1991): 14H50, 53A04, 26D15

Keyword(s): curves, Fréchet distance, inequalities, length, total curvature

Received September 6, 2010, and in revised form September 17, 2010. (Registered under 66/2010.)