# Mathematica Moravica, Vol. 1 (1997)

## Abstracts only

Dragoslav Mitrinović (Scientific Work and Facts)
Mathematica Moravica, Vol. 1 (1997), 1–6.

A Fixed Point Theorem for Upper Semicontinuous Multifunctions on Compact Menger Spaces
Mathematica Moravica, Vol. 1 (1997), 7–10.
Abstract. In this paper we prove a fixed point theorem for upper semicontinuous multifunction defined on compact Menger spaces. As its application we prove one theorem of Iohvidov-Fans type.
Keywords. Upper semicontinuous multifunctions, Iohvidov-Fans type theorem, fixed point, Menger space, $H$-Space.

Semilaticces of Left Completely Archimedean Semigroups
Mathematica Moravica, Vol. 1 (1997), 11–16.
Abstract. In this paper we introduce the notion of a left completely Archimedean semigroup, which is a generalization of the notion of a completely Archimedean semigroup. We give certain characterizations of semilattices of left completely Archimedean semigroups and some new results concerning semilattices of completely Archimedean semigroups.
Keywords. Archimedean semigroups, Munn's lemma, Semilattices, Characterization of semilattices of left completely Archimedean semigroups.

Decision Making on the Choice of Supplier in Monopolism and Blockade Conditions
Mathematica Moravica, Vol. 1 (1997), 17–26.
Abstract. An approach has been developed to help in decision making in conditions of monopolism and constricted possibilities for making out of alternatives. A hypothetical situation, modeled in such a way to be as complex as possible for the decision maker was analyzed.
Keywords. Decision making, multicriteria ranking, blocade condition.

Characterizations of Some Classes of Functions Related to Karamata Theory
Mathematica Moravica, Vol. 1 (1997), 27–32.
Abstract. In this paper we prove some characterizations of several classes of functions related to Karamata theory in terms of the corresponding index functions.
Keywords. Karamata theory, index function, regularly varying functions, $\mathcal{O}$-regularly varying functions, Class of Matuszewska functions

A Statement of Tauber Type for Power Series
Mathematica Moravica, Vol. 1 (1997), 33–34.
Abstract. In this paper we give a generalization of a statement of Hardy and Littlewood from 1914.
Keywords. Tauber's theory, Slowly varying functions, power series, Karamata's theory, Hardy-Littlewood theorem.

Geometric Condition for Local Finiteness of a Lattice of Convex Sets
Mathematica Moravica, Vol. 1 (1997), 35–40.
Abstract. Convex sets of states and the corresponding normal functionals defined on the Hilbert space containing them play a central role in the foundations of General Quantum Mechanics (see [4, 1]). The design of quantum logics for reasoning about General Quantum Mechanics leads naturally to the theory of orthomodular lattices [4, 1, 2], for example, as lattices of closed convex sets in a (complex) Hilbert space. The quantum logics developed for such a purpose can be much easier to work in if the corresponding lattice of convex sets is finite, or at least locally finite (in the classical or intuitionistic case, this holds, because truth assignments are made in a boolean algebra a distributive lattice). We give elementary examples which establish the following facts: the lattice of convex subsets of a Hilbert space is not locally finite, it is not modular (hence not distributive), and locally finite lattices of closed convex sets in any Hilbert space have very restrictive geometric arrangements of their members.
Keywords. Distributive lattice, modular lattice, locally finite lattice, freely generated lattice, convex sets.

$\{1\}$-inverses of Square Matrices and Rational Canonical Form
Mathematica Moravica, Vol. 1 (1997), 41–49.
Abstract. In this paper we solve the first Penrose's equation $AXA = A$ for square real matrices A using the rational canonical form of matrices. The idea is to find ${1}$-inverse $X$ of $A$ using similarity $X = TZT^{-1}$, where $Z$ is $\{1\}$-inverse of $B$ and $A = TBT^{-1}$ is the rational canonical representation of $A$.
Keywords. Penrose's equation, $\{1\}$-inverses of square matrices, rational canonical representation, representation of $\{1\}$-inverses.

Solving Systems of Linear Equations by Means of Mathematical Spectra
Mathematica Moravica, Vol. 1 (1997), 51–58.
Abstract. This paper is a continuation of papers [3, 11]. In [11] we describe an interpreter applicable on mathematical spectra. In [3] we describe applications of the interpreter in computation of determinants of real matrices and exact computation of determinants of integer matrices, using the methods presented in [8] and [9]. In this paper we investigate application of the interpreter in solving a system of linear equations. In the direct step during the solving of a given system of linear equations, we use several functions introduced in [3], together with the functions described in [11]. In the direct step we use more effective of two methods, introduced in [9]. For the inverse step we introduce a new type of mathematical spectra, called the appended spectra, and define the corresponding function ($append) for its implementation. Keywords. Mathematical spectra, interpreter, Linear equations, Means of mathematical spectra, Appended spectra, Corresponding function. Non-isomorphic Affine Finites$\langle Bb,E\rangle$-nets Mathematica Moravica, Vol. 1 (1997), 59–63. Download PDF file: (54kB) | Abstract and keywords Abstract. It is known that for each$n\in \mathbb{N}$there exist affine finites$\langle Nn,E\rangle$-nets$(A_{n-1}(n,q),\|)$with parameters$(q,q^{n-1}+q^{n-2}+\cdots+q+1,q^{n-2})$, where$q$is prime power. In the paper we prove that for each$n\in\mathbb{N}$,$n>2$, and any prime power$q$there exist non-isomorphic affine finites$\langle Nn,E\rangle$-nets with equal parameters$(q,q^{n-1}+q^{n-2}+\cdots+q+1,q^{n-2})$. Keywords. Affine finites, Non-isomorphic affine nets, Affine block Two Fixed Point Theorems in Normed Spaces Mathematica Moravica, Vol. 1 (1997), 65–68. Download PDF file: (51kB) | Abstract and keywords Abstract. In this paper we consider the convergence of sequences defined by$x_{n+1}=\lambda x_{n}+(1-\lambda)f(x_{n})$, for$\lambda\in(0,1)$, to a fixed point of mapping$f:X\to X$where$X$is a$f_{\lambda}$-orbitally completed subset of normed linear space. Keywords. Fixed point, Krasnoselskij's theorem, Mann's iteration, Fixed points in normed spaces,$f_{\lambda}$-orbitally completeness. Addenda to the Monograph "Recent Advances in Geometric Inequalities" Mathematica Moravica, Vol. 1 (1997), 69–72. Download PDF file: (43kB) | Abstract and keywords Abstract. In this paper we prove some inequalities related to the elements of a triangle. These results are refinements or generalizations of the results from [4] and some are of a new nature. This paper is a continuation of [3]. We shall follow the terminology of [4]. Keywords. Cauchy's inequality, Geometric inequalities, Jensen's inequality, Bajer's identity. Multicriteria Approach to Solving Real Conflict Situations Mathematica Moravica, Vol. 1 (1997), 73–83. Download PDF file: (44kB) | Abstract and keywords Abstract. This paper describes a way of solving problems regarding decisionmaking in real multicriteria conflict situations. The problem has been solved first and then it was found that the theoretical background can be based upon a quite new theorem of M. Tasković convering nonlinear functional analysis which is in fact an extension of well known John von Neuman's Minimax theorem. Keywords. Multicriteria conflict situation solving, John von Neuman's minimax principle, Tasković's minimax theorem, Minimax theory. Representations of$\{3\}$,$\{4\}$,$\{1,3\}$,$\{1,4\}$Inverses Mathematica Moravica, Vol. 1 (1997), 85–92. Download PDF file: (61kB) | Abstract and keywords Abstract. In this paper we obtain a general solution of the Penrose's equation (3), solution of the equation (4) and a general solution of the system of Penrose's equations (1), (3) and (1), (4). We also introduce a determinantal representation of the classes of$\{3\}$,$\{4\}$,$\{1,3\}$and$\{1,4\}$-inverses of complex matrices. Keywords. Penrose's equation, System of Penrose's, Characterization of inverses, Determinantal representation. Some Inequalities for Bisectors and Other Elements of Triangle Mathematica Moravica, Vol. 1 (1997), 93–100. Download PDF file: (47kB) | Abstract and keywords Abstract. In this paper are given the proof of the inequlity$\sum\frac{w_{a}}{m_{a}}>1$and some its improvements, and the proof of inequality$\sum\frac{h_{a}}{w_{a}}>1$. Keywords. Geometrical Inequalities, Inequalities for bisectors and medians of triangle, Inequalities of triangle. On Some Inequalities with Factorials Mathematica Moravica, Vol. 1 (1997), 101–104. Download PDF file: (44kB) | Abstract and keywords Abstract. In this note some inequalities for finite sums and products with factorials are given. Keywords. Analytic inequalities, Inequalities with factorials, Kurepa's left factorials, Convex inequalities. A Directly Extension of Caristi Fixed Point Theorem Mathematica Moravica, Vol. 1 (1997), 105–108. Download PDF file: (56kB) | Abstract and keywords Abstract. In this paper it is proved that if$T$is a self-map on a complete metric space$(X,\rho)$and if there exist a lower semicontinuous function$G:\to \mathbb{R}_{+}^{0}$and an arbitrary fixed integer$k\geq 0$such that $\rho[x,Tx]\leq G(x)-G(Tx)+\cdots +G(T^{2k}x)-G(T^{2k+1}x)$ and$G(T^{2i+1}x)\leq G(T^{2i}x)$for$i=0,1,\dots,k$and for every$x\in X$, then$T$has a fixed point$\xi$in$X$. Keywords. Fixed point theorems, complete metric space, Caristi's theorem, Caristi-Kirk theorem. Extensions of Hardy-Littlewood-Pólya and Karamata Majorization Principles Mathematica Moravica, Vol. 1 (1997), 109–126. Download PDF file: (73kB) | Abstract and keywords Abstract. The following main result is proved: Let$J\subset\mathbb{R}$be an open interval and let$x_{i},y_{i}\in J$,$(i=1,\ldots,n)$, be real numbers such that fulfilling $x_{1}\geq\cdots\geq x_{n},\qquad y_{1}\geq\cdots\geq y_{n}.$ Then, a necessary and sufficient condition in order that $\sum_{i=1}^{n}f(x_{i})\geq 2\sum_{i=1}^{n}f(y_{i})-n\max\bigl\{f(a),f(b),g(f(a),f(b))\bigr\}$ holds for every general convex function$f: J\to \mathbb{R}$which is in contact with function$g: f^{2}(J)\to\mathbb{R}$and for arbitrary$a,b\in J$, ($a\geq x_{i}\geq b$for$i=1,\ldots,n\$), is that $\sum_{i=1}^{k}y_{i}\leq\sum_{i=1}^{k}x_{i}\quad (k=1,\ldots,n-1),\qquad \sum_{i=1}^{n}y_{i}=\sum_{i=1}^{n}x_{i}.$
Keywords. General convex functions, Convex functions, Inequalities, Majorizatioin principle, Hardy-Littlewood-Pólya majorization principle, Karamata theorem, Characterization of majorization.

General Convex Topological Spaces and Fixed Points
Mathematica Moravica, Vol. 1 (1997), 127–134.
Abstract. In this paper we shall consider general convexity which is described in an abstract from topological spaces. Also, we formulate a fixed point theorem for general nonexpansive mappings in the general convex topological spaces. This result extends previous results of M. Brodskij, D. Milman, W. Kirk, K. Goebel, W. Takashi, F. Browder, D. Göhde and some others.
Keywords. General convex structures, general convex topological spaces, fixed point theorem, Browder-Göhde-Kirk theorem, normal structure, Smulian property, general nonexapnsive mappings