# Δελτίο της Ελληνικής Μαθηματικής Εταιρίας: Τεύχος 43

Στοιχεία Τεύχους
Περιοδικό Δελτίο της Ελληνικής Μαθηματικής Εταιρίας 43 2000 2000 - 0072-7466 - Αγγλική-Γαλλική-Γερμανική
A Generalized Treatment of the Rounding Problem. Rounding errors of Linear Functions
In 1967 F. Mosteller, C. Youtz and D. Zahn in {|it ``The distribution of sums of rounded percentages''} (Demography 4, 850--858) investigated the frequence that may have the rounded percentages in order to fail to add 100 % and mostly the distributions of sums of rounded percentages in some specific cases. Later on, in 1979, P. Diaconis and D. Freedman assessed the probability that a table of rounded percentages adds to 100% by giving a mathematical treatment of this phenomenon in the case that the table is drawn from a multinomial distribution or from a mixture of multinomial distributions. M. L. Balinski and S. T. Rachev in 1992 continued their work by introducing the so-called {it stationary rules} and by considering vectors and matrices under varying assumptions regarding the probabilistic structure of the data that is to be rounded. Following the abovementioned approaches the author in {it ``Probabilistic approaches to the rounding problem''} (Serdica Bulgaricae, Mathimaticae Publicationes, 1993) examined the rounding problem under a variety of possible rounding rules and more specifically studied the vector (p,h) of rouding in the particular case where \$p\$ is uniformly distributed on the Simplex S_n. The aim of the present paper is to study the vector problem by modyfing the objective of rounding originally considered. In particular the author gives new results concerning with the vector problem in the case that the components of p are independent and identically distributed random variables as well as with the rounding errors of linear functions of such variables.
Studying graph - stability to apprehend relative hardness of constructive-non-constructive approximation
It is known that polynomial approximation theory, developed as the main tool for investigating {|bf NP}-complete problems, can also motivate numerous questions addressed to the heart of complexity theory. The aim of the present paper is to describe a general thought process for the study of the relative hardness between determining solutions and computing (approximately or exactly) optimal solution-values of combinatorial optimization problems. In the particular case of the maximum independent set, this thought process leads us to define classes of independent set problems, the approximability of which is particularly interesting.
Multi-Point Boundary Value Problems with Nonlinear Boundary Conditions
The aim of the present paper is to give existence results for some multi-point value problems with nonlinear boundary conditions (BVP for short) regarding second order ordinary differential equations. In particular the author by using the Nonlinear Alternative of the well-known Topological Transversality and a variation of the well-known Leray-Schauder Alternative in a different way without using the a-priori bounds on all relevant solutions of the related one parameter family of problems produces a {|it ``forbidden value''} of the norm of solutions, that is, a value which the norm never takes on. (The original result is given in theorem 3.1). Moreover many examples may be derived from the general form of the existence results based on specific BVP problems.
Functional calculus and spectral theory of locally C*-sums of locally m-convex C*-algebras
The aim of the present paper is to describe some consequences of functional calculus for locally C*-algebras, which are further applied to locally C*-sums of locally m-convex C*-algebras. In particular, the structure space of such an algebra is involved which yields an example of a locally C*-sum. This is used next in providing a sort of continuity of the spectrum and of the seminorms defining the topology of the algebras appeared in the above locally C*-sum.
A LIP Immersion of Lipschitz Manifolds Modelled on Some Banach Spaces
The aim of the present paper is to study the analogous problem given by J. Luukkainen and J. Vaisala for second countable, paracompact Lipschitz n- manifolds, n|ge 1 and by I. Colojoara for H-Lipschitz (H being a separable Hilbert space) second countable, paracompact manifolds in the case of B-Lipschitz manifolds, where B is a Banach space. In particular the paper describes author's attempt to give a thorough answer to the following problem: If X is a B-Lipschitz manifold then what are the conditions that B and X have in order to exist a map f: X rightarrow B having the following properties: item a. f is injective |item b. f and f^{-1}: f(X) |rightarrow X are locally Lipschitz |item c. f(X) is closed.''
m-Point Boundary Value Problems
The aim of the present paper is to study the problem of existence of a solution for a specific multi-point boundary value problem for second order ordinary differential equations. In particular the author gives the suitable conditions for the existence of a solution for the above boundary value problem by using Leray-Schauder Continuation theorem.
On the size of integer solutions of elliptic equations, II
Let K be a number field of degree d and let a and b be comprime algebraic integers in K. The aim of the present paper is to refine A. Pinter's bound for the size of the integer solutions of elliptic equations in the rational case given in {On the magnitude of integer points on elliptic curves} (Bull. Austral. Math. Soc. 52 (1995), 195-199). In particular in this paper the author with the essential contribution of D. Poulakis presents an upper estimate for the size of the integer solutions of the specific elliptic equation y^2=x^3+ax+b depending only on the prime factors of the discriminant of the polynomial f(X)=X^3+aX+b and not on its height in the important case that a and b are coprime algebraic integers in arbitrary K. (The original result is given in section 2, Theorem 1).
Rank one elements in Banach algebras
The aim of the present paper is to compare the two definitions of rank one elements of A banach algebra A given by P. Nylen and L. Rodman (based on the first property that rank one operators must satisfy) and by J. Erdos, S. Giotopoulos and N. Katseli (based on the second property that rank one operators must satisfy). According to author's approach the previous definitions are equivalent in the case that A is a semi-prime complex Banach algebra. (The original result is given in Theorem 2.1).
A Generalization of Pachpatte Difference Inequalities
It is known that a large number of research works dealing with discrete inequalities and their applications have appeared during the last three decades. Furthermore recurrent inequalities involving sequences of real numbers considered as discrete analogues of the Gronwall-Bellman inequality or its variants have been used in the analysis of the finite difference equations such as those of infinity type given by B. G. Pachpatte in {On certain new finite difference inequalities} (Indian J. Pure Appl. Math. 24(1993), 373-384). The aim of the present paper is to study the structure of three analogues of the Gronwall-Bellman inequalities of infinity type given by B. G. Pachpatte by introducing a general form for such inequalities. According to author's approach it is given two generalized inequalities in the case of one dimensional functions defined on Z to R_+. (The original results given in section 2, Theorem 1 and 2 generalize the corresponding Theorem 1 in the abovementioned B. G. Pachpatte's paper) and a generalized inequality for functions of two-variables defined on Z|times Z into P_+. (The original result given in section 3, Theorem 1 generalize the analogous Theorem 4 in the B. G. Pachpatte's paper). The previous generalizations are supported by using the mean value theorem instead of the elementary calculations that are used in B. G. Pachpatte's paper.