Abstract

Propsrties of the dimension function dm are explored in the calss of normal spaces.

Keywords: Dimension dm

MSC: 54F45

Abstract

The boundary value problem $$ -y''+q(x)y=\lambda y+\int_0^{\pi}y\,d\sigma(x),\quad y(0)=y(\pi)=0, $$ is concerned, where $q\in C[0,\pi]$ and $\sigma$ is a function of bounded variation. It is proved that the system of eigenfunctions of the given problem is complete and minimal in $L^2(0,\pi)$, and also that functions of a certain class can be expanded into uniformly convergent series with respect to the mentioned system.

Keywords: Functional differential equation, boundary value problem, expansion theorem.

MSC: 34K10, 47E05

Abstract

We prove that a ccc linearly ordered topological space is metrizable if and only if it is divisible.

Keywords: Linearly ordered topological spaces

MSC: 54F05

Abstract

This article focuses on the issue of making a kind of preprocessor (called Automatic Modifier of Smalltalk Methods---AMSM) for the programming language Smalltalk. A general algorithm, based on the automatic modification of Smalltalk methods, is proposed. All modifications of related methods are realized in Smalltalk itself. The main parts of method-code and relevant classes are presented. Also, some characteristics (and possible) applications of AMSM are exposed.

Keywords: Smalltalk, automatic modification, method, environment

MSC: 68N15

Abstract

We prove an inequality that implies a 2-concave and $p$-convex Banach lattice is ``more'' uniformly convex than $L^p$.

Keywords: Banach space, uniform convexity

MSC: 46B20, 46B42

Abstract

A unified theory has been developed on the basis of the similarity in properties of perfect and allied types of functions. The theory intromites as a starting point a certain subset of $\Cal P(X)$, the power set of a nonvoid set $X$, and an operator $\alpha$ on $\Cal P(X)$; a second operator $\beta$ is also brought into action. This theory of $\beta$-perfect functions includes the theories of perfect, $\theta$-perfect and $\delta$-perfect functions and is seen to generate many new types of functions when different pairs of operators take the roles of the pair $(\alpha,\beta)$.

Keywords: $\beta$-perfect function, operator $\alpha$, $\beta$-set, $\gamma$-set, $\beta$-closed function, $\beta$-convergence and $\beta$-adherence of filter-bases.

MSC: 54C10

Abstract

En utilisant la fonctionelle $g$ définie par (1), on définit ainsi dit le $g$-angle dans un espace normé réel. On démontre que cet angle remplit les conditions de Menger (Définition~1) dans un espace strictement convexe. Le $g$-angle et l'angle de Wilson sont égaux seulement dans un espace préhilbertien. L'orthogonalité laquelle est engendrée par le $g$-angle et ainsi disant la $g$-orthogonalité ([3]) sont égaux seulement dans un espace préhilbertien. On donne encore quelques caractérisation nouvelles d'espace préhilbertien.

Keywords: $g$-angle

MSC: 46B20, 46C50

Abstract

In this note a notion of the support of a linear mapping from $C_b(T)$ into a locally convex space is introduced. Some of the properties are established.

Keywords: Locally convex space, support of a linear mapping.

MSC: 46A03, 46E10

Abstract

Counterexamples for two closed graph theorems from Köthe's monograph [5] are given.

Keywords: Locally convex spaces, closed graph theorem

MSC: 46A30

Abstract

Generalized recurrent Riemannian manifold is a Riemannian manifold whose curvature tensor satisfies the condition (1). In this paper we prove: If the associated $1$-form satisfies the condition (3), where $\gamma\ne1\ne2$, or, in the case $\gamam\ne\text{const}$, $\gamma_sA^s\ne0$, generalized recurrent Riemannian manifold reduces to a recurrent one.

Keywords: Generalized reccurent Riemannian manifold

MSC: 53C25

Abstract

In this paper we consider the expansion theorem in generalized eigenvectors of the operator $A=L+T$, where $L$ is a discrete, positive selfadjoint operator in a separable Hilbert space, and $T$ is a closed operator which is subordinated to $L$ in a certain sense.

Keywords: Eigenvector expansion, discrete operator, weak perturbations.

MSC: 47A70, 47A55

Abstract

This paper responds to the question of when are two topologies $\alpha$-equivalent by using some recently introduced classes of sets as well as the classes of regular open sets, nowhere dense sets and dense sets.

Keywords: $\alpha$-equivalen topologies

MSC: 54A05