# Mathematica Moravica, Vol. 9 (2005)

On Neutral Operations of (n,m)-groups
Mathematica Moravica, Vol. 9 (2005), 1–3.

Abstract. In this paper proposition on $\{1, n − m + 1\}$−neutral operations of $(n,m)$-groups is proved.
Keywords. $(n,m)$-group, $\{1, n − m + 1\}$−neutral operation of the $(n,m)$-groupoid.

Fixed Points for Nonself Mappings in Metric Spaces
Mathematica Moravica, Vol. 9 (2005), 5–11.

Abstract. In this paper we try to extend several well known fixed point theorems for nonself mappings in Banach spaces to mappings in metric spaces. To achieve this goal some additional requirements on convexity in metric spaces are needed. We introduce the notions of MP-convex and NMP-convex metric spaces and obtain several results on existence of fixed points for nonself non-expansive mappings in NMP-convex metric spaces. In particular, the notion of weakly inward mappings is generalized for mappings in metric spaces and the existence of fixed points is proved for mappings satisfying this condition.

Interlacing Theorem for the Laplacian Spectrum of a Graph
Mathematica Moravica, Vol. 9 (2005), 13–16.

Abstract. It is well known that the Interlacing theorem for the Laplacian spectrum of a finite graph and its induced sub graphs is not true in a general case. In this paper we completely describe all simple finite graphs for which this theorem is true. Besides, we prove a variant of the Interlacing theorem for Laplacian spectrum and induced sub graphs of a graph which is true in general case.
Keywords. Simple graphs, Laplacian spectrum.

On the p-reduced Energy of a Graph
Mathematica Moravica, Vol. 9 (2005), 17–20.

Abstract. Let $G$ be a simple connected graph of order $n$ and let $\lambda_{1}\geq\lambda_{2}\cdots\geq\lambda_{n}$ be the spectrum of $G$. Then the sum $S_{k}^{l}(G)=|\lambda_{k+1}|+|\lambda_{k+2}|+\cdots+|\lambda_{n-l}|$ is called $(k,l)$-reduced energy of $G$, where $k,l$ are two fixed nonnegative integers [2]. In this work, we make a generalization of the $(k,l)$-reduced energy as follows: for any fixed $p\in N$, the sum $S_{k}^{l}(G,p)=|\lambda_{k+1}|^{p}+|\lambda_{k+2}|^{p}+\cdots+|\lambda_{n-l}|^{p}$ is called the $p$-th $(k,l)$-reduced energy of the graph $G$. We also here introduce definitions of some other kinds of the $p$-reduced energies and we prove some properties of them.
Keywords. Simple graphs, Energy of the graph.

Fixed Point Theorem on FΛ-orbitally Complete Normed Spaces
Mathematica Moravica, Vol. 9 (2005), 21–24.

Abstract. Let $X$ be a normed space and $x_{0}\in X$. In this paper we prove the convergence of a convex sequence $x_{n}=\lambda x_{n-1}+(1-\lambda)f(x_{n-1})$, $\lambda\in(0,1)$ to the fixed point of the $f$, where $f: X \to X$ is the nonexpansive completely continuous operator, which satisfies some nonexpansive conditions.
Keywords. Convex sequence, fixed point, $f_{\lambda}$-orbitally complete space.

On m-Quasi-Irresolute Functions
Mathematica Moravica, Vol. 9 (2005), 25–41.

Abstract. In this paper we introduce a new notion of $m$-quasi irresolute functions as functions from a set satisfying some minimal conditions into a topological space. We obtain some characterizations and several properties of such functions. This function lead us to the formulation of a unified theory of $(\theta, s)$-continuity [26], $\alpha$-quasi irresolute [24], weakly $\theta$-irresolute [19], $\theta$-irresolute [27], $\beta$-quasi irresolute [23].
Keywords. $m$-structure, $(\theta,s)$-continuous, $\alpha$-quasi-irresolute, weakly $\theta$-irresolute, $\beta$-quasi-irresolute, $m$-compact, $S$-closed, $m$-quasi-closed graph.

Selection Theorems for Multivalued Generalized Contractions
Mathematica Moravica, Vol. 9 (2005), 43–52.

Abstract. The purpose of this paper is to report old and new selection results for multivalued operators. Main results of this paper concern with the existence of a Caristi type selection for some multivalued generalized contractions on metric spaces.
Keywords. Selection, Ciric type multivalued operator, Caristi mapping.

On a Method for Obtaining Iterative Formulas of Higher Order
Mathematica Moravica, Vol. 9 (2005), 53–58.

Abstract. In this paper a method for obtaining iterative formulas of higher order for finding roots of equations is obtained. These formulas include several already known results.
Keywords. Iteration formulas, approximate solutions of equations.

The Multiple Summation Formula and Polylogarithms
Mathematica Moravica, Vol. 9 (2005), 59–67.

Abstract. In this paper is given the formula: $F_{n}(x) = \sum_{k_{1}=1}^{\infty}\frac{x^{k_{1}}}{k_{1}}\sum_{k_{2}=1}^{k_{1}}\frac{x^{k_{2}}}{k_{2}}\cdots \sum_{k_{n}=1}^{k_{n-1}}\frac{x^{k_{n}}}{k_{n}} = \sum_{\sum_{j=1}^{n}j\cdot\alpha_{j}=n,\\ \alpha_{j}\geq 0} \frac{\prod_{k=1}^{n}\zeta_{k}^{\alpha_{k}}(x^{k})}{\prod_{k=1}^{n}k_{\alpha_{k}}\alpha_{k}!}$ $n\geq 1,\quad -1\leq x < 1$ with $\zeta_{k}(x) \equiv Li_{k}(x)\equiv \sum_{r=1}^{\infty}\frac{x^{r}}{r^{k}},\qquad (k\geq 0),$ and method by which it can be obtained.
Keywords. Multiple sums, multiple summation, polylogarithms.

Fréchet's Metric Spaces – 100th Next
Mathematica Moravica, Vol. 9 (2005), 69–75.

Abstract. In this survey I give fundamental historical facts on the metric spaces and some further consequences in history of mathematics. In the second part of this survey I give a historical oversee the fundamental facts on transversal spaces as a nature extension of Fréchet's, Kurepa's and Menger's spaces.

Mathematica Moravica, Vol. 9 (2005), 77–86.

Abstract. In this paper $(km, m)$-groups, $k\geq 3$, with one condition are described.
Keywords. $(n,m)$-group, $\{1,n-m+1\}$-neutral operation of the $(n,m)$-groupoid.
Abstract. In this article two characterization of $(n,m)$-groups for $n\geq 3m$ are proved (The case $m = 1$ is proved in [4]).
Keywords. $n$-group, $(n,m)$-groupoid, $(n,m)$-group, $\{1,n-m+1\}$-neutral operation of the $(n,m)$-groupoid.