Volume 49 , issue 2 ( 1997 ) | back |

$H$-projecting in $n$-dimensional Euclidean space $E^n$ | 77$-$80 |

**Abstract**

Several types of projecting in $n$-dimensional Euclidean spaces are known. In this article we define a new type of projecting of the $n$-dimensional Euclidean space onto its fixed plane. We shall prove some properties of this projecting. It will be shown that so defined projecting is a central projecting with an $(n-3)$-dimensional subspace as a center.

**Keywords:** $H$-projecting.

**MSC:** 51N15

Corrections to ``Some Generalizations of $T_D$-Spaces'' and ``A Generalization of Normal Spaces'' | 81$-$84 |

**Abstract**

Some corrections to the papers ``Some Generalizations of $T_D$-Spaces (Mat. Vesnik {\bf34} (1982), 221--230)'' and ``A Generalization of Normal Spaces (ibid\. {\bf35} (1983), 1--10)'' are given.

**Keywords:** Semi-$T_D$-spaces, semi-$T_1$-spaces, $s$-normal spaces.

**MSC:** 54D15

On certain class of rational functions whose derivatives have positive real parts | 85$-$88 |

**Abstract**

For a certain class of rational functions we give a sufficient condition such that their derivatives have positive real parts.

**Keywords:** Univalent function.

**MSC:** 30C45

Boundary value problem with shift for two simply connected regions | 89$-$92 |

**Abstract**

In this paper a boundary value problem for two analytic functions in different simply connected regions is considered. If index corresponeded to the problem is not greater than zero, then the solavability of that problem is shown without any further conditions and its solution is obtained in an explicit form~(4). However, if the index is greater than zero, then the problem is solvable and its solution is given by~(5) if real conditions of solvability~(6) hold.

**Keywords:** Boundary value problem.

**MSC:** 30E25

On weighted parametric information measure | 93$-$97 |

**Abstract**

Shannon's measure of information plays a very important role for measuring diversity in plants and animals. But this measure does not deal with growth models other than exponential. Thus we need a parametric model which is suitable to all types of distributions. In this communication, one such measure depending upon two real parameters has been developed under a set of certain axioms and its particular cases have been studied.

**Keywords:** Shannon's measure, parametric information measure.

**MSC:** 62B10; 94A17

Application of interpolation theory to the analysis of the convergence rate for finite difference schemes of parabolic type | 99$-$107 |

**Abstract**

In this paper we show how the theory of interpolation of function spaces can be used to establish convergence rate estimates for finite difference schemes. As a model problem we consider the first initial-boundary value problem for the heat equation with variable coefficients. We assume that the solution of the problem and the coefficients of the equation belong to the corresponding Sobolev spaces. Using interpolation theory we construct a fractional-order convergence rate estimate which is consistent with the smoothness of the data.

**Keywords:** Initial-boundary Value
Problems, Finite Differences, Interpolation of Function Spaces, Sobolev
Spaces, Convergence Rate Estimates.

**MSC:** 65M15; 46B70

Common fixed point theorem for compatible mappings and compatible mappings of type (A) | 109$-$114 |

**Abstract**

The purpose of this paper is to prove some theorems which generalize a theorem of M. Telči, K. Tas and B. Fisher on compatible mappings and compatible mappings of type~(A).

**Keywords:** Compatible mappings, compatible mappings of type (A), common fixed
points.

**MSC:** 54H25

A time-dependent biochemical system | 115$-$122 |

**Abstract**

Following Selgrade [1] we consider a model of a positive feedback biochemical system in a periodic environment. Results of existence, uniqueness and stability for solutions of such systems are presented.

**Keywords:** Positive feedback, biochemical system, periodic environment.

**MSC:** 92C45

Elementary solution of Vecua equation with analytic coefficients on $Z$, $\overline{Z}$ | 123$-$128 |

**Abstract**

An explicit solution of Vecua equation with analytic coefficients on $z$, $\bar z$ is given in the form of a series which depends on the coefficients $A$, $B$, $F$ of the equation, and an analytic function $\Phi(z)$ in the role of an arbitrary integration constant.

**Keywords:** Vecua equation.

**MSC:** 34A20; 30D05

Some estimates of the remainder in the expressions for the eigenvalue asymptotics of some singular integral operators | 129$-$137 |

**Abstract**

Some estimates are given for the remainder in the expressions for eigenvalue asymptotics of operators of the type of Riesz' potential and logarithmic potential. As a corollary an estimate is obtained of the spectral asymptotics of an operator playing a role in the thin airplane wing theory. The method is elementary and it is based on the properties of the singular values of compact operators.

**Keywords:** Operator with Riesz' potential, operator with logarithimic
potential, eigenvalue asymptotics.

**MSC:** 47B10