Abstract

A projective plane $\Cal P_m$ is a Ber's subplane of a finite projective plane $\Cal P_n$ if every point and line of $\Cal P_n\setminus\Cal P_m$ is incident to some line and some point, respectively, of $\Cal P_n$. It is known that the order of the plane $\Cal P_n$ and its Ber's subplane $\Cal P_m$ satisfy the equation $n=m^2$. In this article we prove some properties of finite projective planes $\Cal P_n$ having disjoint Ber's subplanes covering it.

Keywords: Projective plane, Ber's subplane.

MSC: 51E15

Abstract

Sharp results concerning coefficients, distortion theorem and the radius of convexity for the class $P_p^*(\alpha,\beta,\xi)$ are determined. Furthermore it is shown that the class $P_p^*(\alpha,\beta,\xi)$ is closed under convex linear combinations. The extreme points of $P_p^*(\a,\b,\xi)$ are also determined.

Keywords: $p$-valent function, distortion theorem, radius of convexity.

MSC: 30C45

Abstract

Let $X$ be a Banach space over a topological semifield and $T_1$, $T_2\:X\to X$ two maps satisfying the condition (1). Then $T_1$ and $T_2$ have a common fixed point.

Keywords: Banach space over a topological semifield, common fixed point, Cauchy sequence, sequentially complete metric space.

MSC: 54H25, 47H10

Abstract

When the finiteness condition in the definition of precompactness is replaced by countability, a new notion in the class of quasi-uniform spaces is introduced, and we call it {\it pre-Lindelöfness}. Some properties of pre-Lindelöf spaces, in particular in the subclass of quasi-pseudo-metric spaces, are investigated.

Keywords: Quasi-pseudo-metric, quasi-uniform, (hereditarily) Lindelöf, (hereditarily) precompact, (hereditarily) pre-Lindelöf, separable.

MSC: 54E15, 54E99, 54D20

Abstract

We define locally (on $C\subset R^2$) subadditive functions $f$, $f\:C\to R$, by $$ f(x_1+x_2,y_1+y_2)\leq f(x_1,y_1)+f(x_2,y_2),\quad(x_1,y_1),(x_2,y_2)\in C, $$ where $C$ is some cone in $R^2$. The purpose of the paper is to find explicit form of such functions.

Keywords: (Locally) subadditive function.

MSC: 26B25

Abstract

The paper gives a special construction of those words (binary sequences) of length $n$ over alphabet $\{0,1\}$ in which the subword $0\undersetbrace{k-1}\to{11\dots11}0$ is forbidden for some natural number~$k$. This number of words is counted in two different ways, which gives some new combinatorial identities.

Keywords: Subword.

MSC: 05A15

Abstract

A multicomponent alternating direction finite difference scheme for solving the wave equation with several variables is considered. Its stability and convergence are investigated in the case when the solution of the initial-boundary value problem belongs to a Sobolev space.

Keywords: Finite difference scheme, wave equation.

MSC: 65M15, 65Y05

Abstract

In this paper a new approach to resolving the parts explosion problem [1] will be presented. The parts explosion problem is well known as a problem that is beyond the capabilities of classical relational algebra. This problem can not be solved in a natural way in an SQL system by means of recursive calls, because SQL does not support recursion in a classical sense. Suggested solution is recursive and it is based on a different concept of a result of an SQL query embedded in a programming language.

Keywords: Explosion problem, LISP/SQL.

MSC: 68N15