# Mathematica Moravica, Vol. 7 (2003)

Weak Asymptotic Equivalence Relation and Inverse Functions in the Class $OR$
Mathematica Moravica, Vol. 7 (2003), 1–6.

Abstract. If $f(x)$ is a continuous, strictly increasing and unbounded function defined on an interval $[a,+\infty)$, $(a>0)$, in this paper we shall prove that $f^{-1}(x)$, $(x\geq a)$ belongs to the Karamata class $OR$ of all $\mathcal{O}$-regularly varying functions, if and only if for every function $g(x)$, $(x\geq a)$, which satisfies $f(x) \asymp g(x)$ as $x\to\infty$, we have $f^{-1}(x) \asymp g^{-1}(x)$ as $x\to +\infty$. Here, $\asymp$ is the weak asymptotic equivalence relation. We shall also prove some variants of the previous theorem, in which, except the weak, we also deal with the strong asymptotic equivalence relation.
Keywords. Regular variation, asymptotic equivalence, inversion.

The Numerical Function of a $\ast$-Regularly Varying Sequence
Mathematica Moravica, Vol. 7 (2003), 7–10.

Abstract. In this paper, we impose some conditions under which there is a close relation between the asymptotic behaviour of a $\ast$-regularly varying sequence and the asymptotic behaviour of its numerical function $\delta_{c}(x)$, $x>0$.
Keywords. Regularly varying sequence, numerical function.

Mathematica Moravica, Vol. 7 (2003), 11–14.

Abstract. This paper is to present two fixed points theorems for nonexpansive mappings, defined on general convex topological spaces. This theorems, for mappings defined on metric spaces are given by Y. Kijma and W. Takahashi ([3]).
Keywords. Fixed point, admissible sets, general convex topological spaces, general convex structure.

A Note on Nearly Paracompactness
Mathematica Moravica, Vol. 7 (2003), 15–21.

Abstract. The purpose of the present paper is to study some properties of nearly paracompactness, $\alpha$-Hausdorff subset, almost closed mappings and closed graphs.
Keywords. Compact, $\alpha$-Hausdorff, $\alpha$-nearly paracompact, $\alpha$-paracompact, $\alpha$-regular, almost closed mappings, closed graph, equivalence relation.

Infinitely Distributive Elements in Posets
Mathematica Moravica, Vol. 7 (2003), 23–32.

Abstract. Infinitely distributive and codistributive elements in posets are studied. It is proved that an element $a$ in a poset $P$ has these properties if and only if the image of a has the corresponding properties in the Dedekind MacNeille completion of $P$. An application of the order theoretical results to a poset of weak congruences is presented.
Keywords. Infinite distributive, elements, infinite codistributive elements, congruence, $\omega$-stable, complete congruence.

Some Types of Relative Paracompactness
Mathematica Moravica, Vol. 7 (2003), 33–42.

Abstract. This paper is a continuation, of the study of relative topological properties. We use a characterization of paracom-pactness via a certain selection principle to introduce five types of relative paracompactness provide examples showing that none of them coincide with each other and establish some results concerning finite unions of subspaces which are relatively paracompact in one or another of the defined senses.
Keywords. Locally finite family, point finite family, open cover, relative paracompactness, $\mathsf{S}(\mathcal{O}, \mathcal{O})_{lf}$, $\mathsf{S}(\mathcal{O},\mathcal{O})_{pƒ}$.

New Recurrent Formulae of $P(n)$ and $\tau(n)$ Functions
Mathematica Moravica, Vol. 7 (2003), 43–49.

Abstract. In this paper a new recurrent formulae of partition function $P(n)$ and Ramanujan's tau function $\tau (n)$ are given.
Keywords. Partition function, Ramanujan's tau function.

A Method for Revocation in Group Signature Schemes
Mathematica Moravica, Vol. 7 (2003), 51–59.

Abstract. A group signature scheme allows any group member to sign on behalf of the group in an anonymous and unlinkable fashion. In the event of a dispute, a designated trusted entity can reveal the identity of the signer. In this paper we propose a revocation method for group signatures based on the group signature scheme from [12]. This method requires no time periods and offers constant length signatures.
Keywords. Group signature scheme, revocation of group members, Okamoto-Shiraishi assumption.

On Some Fixed Point Theorems for Mappings Satisfying a New Type of Implicit Relation
Mathematica Moravica, Vol. 7 (2003), 61–66.

Abstract. In this paper we introduce a new class of functions $F: R_{+}^{6}\to R$ such that the fulfillment of the inequality of type (3) for $x, у\in X$, ensures the existence and the uniqueness of a fixed point.
Keywords. Fixed point, complete metric space, compact metric space, implicit relation.

Inequalities for Wallis' Product
Mathematica Moravica, Vol. 7 (2003), 67–72.

Abstract. Some of inequalities for Wallis' products, stronger than inequality of Kazarinoff, are given in this paper.
Keywords. Wallis' product; inequality of Cesaro-Buchner; inequality of Kazarinoff.

An Extension of Baire's Category Theorem to Relator Spaces
Mathematica Moravica, Vol. 7 (2003), 73–89.

Abstract. As a particular case of a more general result, we show that if $\mathcal{R}$ is a topological, topologically filtered, topologically regular relator on $X$ such that $\mathcal{R}$ is either topologically relatively locally sequentially compact, or uniformly countable and properly sequentially convergence-adherence complete, then $\mathcal{R}$ is a Baire relator.
If $X$ is a nonvoid set, then by a relator $\mathcal{R}$ on $X$ we mean a nonvoid family of binary relations on $X$. The relator $\mathcal{R}$ is called a Baire relator if the fat subsets of the relator space $X(\mathcal{R})$ are not meager. A subset $A$ of $X(\mathcal{R})$ is called fat if $\mathrm{int}_{\mathcal{R}}(A)\neq\emptyset.$ While, the set $A$ is called meager if it is a countable union of rare (nowhere dense) sets.
Keywords. Generalized uniformities, Baire's category theorem.

Transversal Intervally Spaces
Mathematica Moravica, Vol. 7 (2003), 91–106.

Abstract. In this paper we formulate a new structure of spaces which we call it transversal (upper or lower) intervally spaces. We introduce this concept as a natural extension of transversal probabilistic and Menger's spaces. Transversal intervally spaces are a new concept of spaces in the fixed point theory and further a new way in the nonlinear analysis. In this sense, we introduce notions of the intervally contractions on upper and lower transversal intervally spaces and prove some fixed point statements.
Keywords. General ecart, distance, Fréchet's spaces, Kurepa's spaces, Monger's spaces, transversal spaces, transversal probabilistic spaces, transversal intervally spaces, transverse,bisection functions, fixed points, intervally contractions, probabilistic contractions.

Fundamental Facts on Translational $\mathcal{O}$-Regularly Varying Functions
Mathematica Moravica, Vol. 7 (2003), 107–152.

Abstract. In this paper we introduce three new classes of functions under names translational slowly varying, translational regularly varying and translational $\mathcal{O}$-regularly varying functions. All classes have important applications in the study of asymptotic processes. In this sense, Uniform Convergence Theorem, Characterization Theorem and Representation Theorem are the main results of this paper for all cite classes of functions. This results are closely connected with the Karamata's theory of regularly varying functions. Also, in this paper we introduce three classes of sequences under names translational slowly varying, translational regularly varying and translational $\mathcal{O}$-regularly varying sequences. All three classes have important applications in the study of asymptotic processes. The results are of relevance in connection with limit statements in various branches of probability theory and ergodic theory.
Keywords. Translational slowly varying function, translational regularly varying function, translational $\mathcal{O}$-regularly varying function, uniform convergence, characterization, representation, slowly varying function, regularly varying function, er­godic theory, $\mathcal{O}$-regularly varying functions, Karamata's theory, translational regularly varying sequences, embedding sequences in functions, exponential representations of sequences and functions, Karamata's theory of sequences, translational $\mathcal{O}$-regularly varying sequences.

Survey on Transversal Normed Spaces
Mathematica Moravica, Vol. 7 (2003), 153–174.

Abstract. In this paper we formulate a new structure of spaces which we call it transversal (upper or lower) normed spaces. Combination of algebraic and transversal structures opens up the possibility of studying linear transformation of one transversal normed space into another. This concept of spaces is a natural extension of Banach spaces. Most of our work in this paper centers around forms of three fundamental theorems relating to bounded linear transformations: form of the Hahn-Banach theorem, form of the open mapping theorem and form of the Banach-Steinhaus theorem.
Keywords. Transversal spaces, transversal normed spaces (upper or lower), upper normed spaces, lower normed spaces, form of the Hahn-Banach theorem, form of the open mapping theorem, form of the Banach-Steinhaus theorem, form of the uniform boundedness theorem.

Fixed Points on Transversal Edges Spaces
Mathematica Moravica, Vol. 7 (2003), 175–186.

Abstract. In this paper we formulate a new structure of spaces which we call it edges (upper or lower) transversal spaces. Also, in this sense, we describe a class of conditions sufficient for the existence of a fixed point on edges (upper or lower) transversal spaces.
Keywords. Transversal spaces, Edges (upper or lower) transversal spaces, Fréchet's spaces, fixed point theorems, diametral $\varphi$-contraction, upper (or lower) edges contraction, edges (upper or lower) continuous, method of successive approximations.

On $nB$-Algebras
Mathematica Moravica, Vol. 7 (2003), 187–191.

Abstract. In the present paper: 1) we define $nB$-algebra $(Q;B,\mathbf{e})$ of the type $\langle n,n-2\rangle$, so that (among others) for $n=2$ $(Q;B,\mathbf{e})$ is a $B$-algebra; and 2) $nB$-algebra $(Q;B,\mathbf{e})$ is described as an $n$-group $(Q;A)$.
Keywords. $B$-algebra, $nB$-algebra, $n$-group.

A Comment on Near-$P$-Polyagroups (Polyagroups)
Mathematica Moravica, Vol. 7 (2003), 193–197.

Abstract. In this article one proposition on $\{1,n\}$-neutral operation ín near-$P$-polyagroups (polyagroups) is proved.
Keywords. $n$-groupoid, $n$-quasigroup, $n$-group, polyagroup, $NP$-polyagroup.
Some Remarks on Near-$P$-Polyagroups and Polyagroups
Abstract. In this paper the Hosszu-Gluskin Theorem for near-$P$-polyagroups (polyagroups) is proved.
Keywords. $n$-groupoid, $n$-quasigroup, $n$-group, polyagroup, near-$P$-polyagroup.