# Mathematica Moravica, Vol. 2 (1998)

Measure of Noncompactness on Uniform Spaces
Mathematica Moravica, Vol. 2 (1998), 1–8.
Abstract. In this paper we define measure of noncompactness on arbitrary uniform spaces, and give some their properties. We also give one characterization of complete uniform spaces.
Keywords. Measure of noncompactness, uniform space.

Implementation of Multi-Criterion Decision Making Models Recommendation
Mathematica Moravica, Vol. 2 (1998), 9–19.
Abstract. This paper gives some introductory notes on mul­ti-criterion decision making. It addresses to the known division into models of multi-purpose decision making and multi-attribution de­cision making or multi-criterion analysis, general notion of models, terminology, taxonomy of methods as well as significance of inter­active methods and the role of a decision maker in the procedure of solution making. The first out of a set of significant facets of implementation of multi-criterion decision making models is shown further: marginal solutions and their consequences, that is an exam­ination of a possibility of perfect solution of the said model.
Keywords. Multi-criterion optimization, multi-purpose decision making, multi-attribution optimization, multi-criterion analysis, mathematical model, methods, software.

Tauberian Theorems for Generalized Abelian Summability Methods (part of PhD thesis)
Mathematica Moravica, Vol. 2 (1998), 21–66.
Abstract. We introduce and study a significant generaliza­tion of Abel's summability method, and their corresponding limiting process. This leads to an analogue to Hardy-Littlewood Tauberian Theorem.The first section includes an introduction to some basic concepts of summability methods and a survey of classical and neoclassical results. In the second section a general summability method is designed and some related Tauberian theorems are established. In the third section higher order of Abel's summability methods are obtained as a special case of a general summability method and the general Littlewood theorem is proved for those summabil­ity methods. Finally we give Tauberian theorems corresponding to $(C, m)$-summability methods and present some further convergence theorems.
Keywords. General summability methods, General Tauberian theorem.

The Matuszewska Sequences
Mathematica Moravica, Vol. 2 (1998), 67–74.
Abstract. In this paper we prove a representation theorem for $\Delta RV$-sequences, which we call “Matuszewska sequences”, in the Bojanić-Seneta sense. We also find a connection between the class of $\Delta RV$-sequences and the functional Matuszewska class ERV, and the relations between the sequencial class $\Delta RV$ and the sequencial classes $RVS$ and $\ast RV$.
Keywords. Karamata's theory, sequencial class $\ast RV$, functional Matuszewska class, index funtion.

Theorem of Synthesis for Bisemilattice-valued Fuzzy Sets
Mathematica Moravica, Vol. 2 (1998), 75–83.
Abstract. Bisemilattice-valued fuzzy set ($B$-fuzzy set) is a mapping from a nonempty set to a bisemilattice. A $B$-fuzzy set has two families of level subsets, one for each ordering of the bisemilatÂ­tice. In this paper, necessary and sufficient conditions under which two families of subsets of a nonempty set are families of level subsets of a $B$-fuzzy set are given.
Keywords. Fuzzy sets, bisemilattices.

Application of an Expanded Min-Max Theorem and Tables of Decision Making for Solving Multi-criterion Conflict Situations
Mathematica Moravica, Vol. 2 (1998), 85–90.
Abstract. This paper proves that it is possible to define optimal strategy for multi-criterion conflict situation using Min-max theorem of Tasković in conjunction with tables of decision making.
Keywords. Multicriteria conflict situation, solving, John von Neumann's minimax principle, Tasković's minimax theorem, Minimax theory.

On the Location of the Zeros of Polynomials
Mathematica Moravica, Vol. 2 (1998), 91–96.
Abstract. In this paper we determine in the complex plane regions containing the zeros of the polynomial $P(z)=z^n+a_1z^{n-1}+a_2z^{n-2}+\cdots+a_{n-1}z+a_n,\qquad n\geq 3.$ We also obtain an expression which represents a upper bound for the moduli of the zeros of $P(z)$.
Keywords. Regions of the zeros, upper bound for the moduli of the zeros, the zeros in the halfplanes.

A Remark Concerning Zeros of One Class of Polynomials
Mathematica Moravica, Vol. 2 (1998), 97–108.
Abstract. In this paper, the distribution of zeros of a class of real polynomials is considered. In some cases the intervals, each one containing one zero, are determined with more accuracy.
Keywords. Distribution of the zeros, intervals of the zeros, zeros of a class of real polynomials.

Some Minimax Theorems on Ordered Sets
Mathematica Moravica, Vol. 2 (1998), 109–120.
Abstract. In this paper we continue the study and considerations of some minimax statements on ordered sets.
Keywords. Minimax theory, Partially ordered sets, Games theory, Saddle points, Transversal points, Borsuk-Ulam theorem, von Neumann's theory, Minimax Inequalities.

Schauder's 54th Problem in Scottish Book
Mathematica Moravica, Vol. 2 (1998), 121–132.
Abstract. The most famous of many problems in nonlinear analysis is Schauder's problem (Scottish Book, Problem 54) of the following form, that if $C$ is a nonempty convex compact subset of a linear topological space does every continuous mapping $f: C\to C$ has a fixed point?
The answer we give in this paper is yes. In this connection this paper proves and extends the Markoff-Kakutani theorem to arbitrary linear topological space as an immediate consequence of the preceding solution of Schauder's problem. During the last twenty years this old conjecture was intensively examined by many mathematians. For set in normed spaces this has been proved by Schauder and for sets in locally convex spaces by Tychonoff.
In this paper we prove that if $C$ is a nonempty convex compact subset of a linear topological space, then every continuous mapping $f: C\to C$ has a fixed point.
On the other hand, in this sense, we extend and connected former results of Brouwer, Schauder, Tychonoff, Markoff, Kakutani, Darbo, Sadovskij, Browder, Krasnoselskij, Ky Fan, Reinermann, Hukuhara, Mazur, Hahn, Ryll-Nardzewski, Day, Riedrich, Jahn, Eisenack-Fenske, Idzik, Kirk, Göhde, Granas, Dugundji, Klee and some others.
Keywords. Fixed point theorems, Brouwer's theorem, Schauder's theorem, Tychonof's theorem, Markoff's theorem, Kakutani's theorem, Sadovskij's theorem, Schauder's Conjecture (Scottish Book, Problem 54).

Transversal Spaces
Mathematica Moravica, Vol. 2 (1998), 133–142.
Abstract. In this paper we formulate a new structure of spaces which we call it transversal (upper or lower) spaces. We introduced the concept of transversal ordered (upper or lower) spaces as a natural extension of Fréchet's, Kurepa's and Menger's spaces.
Keywords. General ecart, distance, general distance, pseudometric, M-metric, Fréchet's and Kurepa's spaces, transversal spaces, Menger's spaces, bisection (upper or lower) functions.

New Geometric Fixed Point Theorems
Mathematica Moravica, Vol. 2 (1998), 143–148.
Abstract. In this paper it is proved the following main result that if $T$ is a self-map on a complete metric space $(X, \rho)$ and if there exists an upper semicontinuous bounded above function $G: X \to \mathbf{R}$ such that $\rho [x,Tx] \leq G(Tx)-G(x)$ for every $x \in X$, then $T$ has a fixed point in $X$. This paper presents and some other results of this type.
Keywords. Fixed point theorems, complete metric spaces, Caristi's theorem, Caristi-Kirk theorem, upper or lower semicontinuous functions.

On Topological n-Groups
Mathematica Moravica, Vol. 2 (1998), 149–159.
Abstract. Let $(Q,A)$ be an $n$-group, $^{-1}$ its inversing operation [:[13,16],1.3], $n \geq 2$, and let $Q$ be equipped with a topology $\mathcal{O}$. Then, in this paper, we say that $Q,A,\mathcal{O}$ is topological $n$-group iff: a) the $n$-ary operation $A$ is continuous in $\mathcal{O}$, and b) the $(n-1)$-ary operation $^{-1}$ is continuous in $\mathcal{O}$. The main result of the paper is the following proposition. Let $(Q,A)$ be an $n$-group, $n \geq 3$, and let $(Q,\{\cdot, \varphi,b\})$ be an arbitrary $nHG$-algebra associated to the $n$-group $(Q,A)$ [:[15],1.5]. Also, $Q$ is equipped with a topology $\mathcal{O}$. Then, $(Q,A,\mathcal{O})$ is a topological $n$-group iff the following statements hold: 1) $(Q,\cdot, \mathcal{O})$ is a topological group [:e.g. [7]], and 2) the unary operation $\varphi$ is continuous in $\mathcal{O}$.
Keywords. $n$-semigroups, $n$-quasigroups, $n$-groups, $\{1,n\}$-neutral operations on $n$-groupoids, inversing operation on $n$-group, $nHG$-algebras, topological groups.

Recurrence Sequences and Norlud-Bernoulli Polynomials
Mathematica Moravica, Vol. 2 (1998), 161–168.
Abstract. The purpose of this paper is to establish some identities containing Norlund-Bernoulli polynomials, which as one application, yield some results of Toscano [8], Kelisky [5] and Zhang and Guo [10] as special cases, as well as other identities involving Bernoulli-Euler and Fibonacci-Lucas or Pell and Pell-Lucas numbers.
Keywords. Recurrence sequence, Norlund-Bernoulli polynomial, Identity.

On $\{1,n\}$-neutral, Inversing and Skew Operations of $n$-Group
Mathematica Moravica, Vol. 2 (1998), 169–173.
Abstract. The skew operation in n-group $(n \geq 3)$ has been introduced in [1]. Using this operation, n-group has been described in [2] as a variety of type $\langle n,1\rangle$. Aim of this note is to connect the skew operation with neutral and inversing operations which have been introduced in [6] and [7].
Keywords. $n$-semigroups, $n$-quasigroups, $n$-groups, $\{1,n\}$-neutral operations on $n$-groupoids, inversing operation on $n$-group, skew operation on $n$-group.