On generalized neighbourhood systems

It is shown in the paper [1] that every generalized topology can be generated by a generalized neighbourhood system. Following the paper [3], we discuss some questions related to this construction. Research supported by Hungarian Foundation for Scientific Research, grant No. T 49786.     

Weak MV-algebras

In a recent paper [CHAJDA, I.—KüHR, J.: A non-associative generalization of MV-algebras, Math. Slovaca 57, (2007), 301–312], authors introduced and studied a non-associative generalization of MV-algebras called NMV-algebras. In contrast to MV-algebras, sections (i.e. principal filters) in NMV-algebras which are proper (i.e. are not MV-algebras), do not admit a structure of an NMV-algebra with respect to the operations defined in a natural way. The aim of the paper is to present a new class of algebras generalizing MV-algebras but sharing the above property. The financial support by the grant of Czech Government MSM 6198959214 is gratefully acknowledged.     

Applications of k-covers

We investigate a role of k-covers in selection principles theory. Some results about relationships between Ramsey theory and game theory and selection principles that involve k-covers are also given.     

A characterization of the algebra <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">β</Emphasis></Subscript>(Ω)

We study some properties of algebras of continuous functions on a locally compact space, these algebras being equipped with the topology defined by a family of multiplication operators (β-uniform algebras). We prove an analog of a theorem due to Sheinberg for β-uniform algebras [see Uspekhi Mat. Nauk, 32:5 (197) (1977), 203–204].     

A unified theory of R-continuity

We introduce and investigate R-M-continuous functions defined between sets satisfying some minimal conditions. The functions enable us to formulate a unified theory of modifications of R-continuity [22]: R-irresoluteness [6], R-preirresoluteness [7].      

Criterions for detecting the existence of the exponential dichotomies in the asymptotic behavior of the solutions of variational equations

We prove that the admissibility of any pair of vector-valued Schäffer function spaces (satisfying a very general technical condition) implies the existence of a “no past” exponential dichotomy for an exponentially bounded, strongly continuous cocycle (over a semiflow). Roughly speaking the class of Schäffer function spaces consists in all function spaces which are invariant under the right-shift and therefore our approach addresses most of the possible pairs of admissible spaces. Complete characterizations for the exponential dichotomy of cocycles are also obtained. Moreover, we involve a concept of a “no past” exponential dichotomy for cocycles weaker than the classical concept defined by Sacker and Sell (1994) in [23]. Our definition of exponential dichotomy follows partially the definition given by Chow and Leiva (1996) in [4] in the sense that we allow the unstable subspace to have infinite dimension. The main difference is that we do not assume a priori that the cocycle is invertible on the unstable space (actually we do not even assume that the unstable space is invariant under the cocycle). Thus we generalize some known results due to O. Perron (1930) [14], J. Daleckij and M. Krein (1974) [7], J.L. Massera and J.J. Schäffer (1966) [11], N. van Minh, F. Räbiger and R. Schnaubelt (1998) [26].     

A unified theory of<Emphasis Type="Italic">θ</Emphasis>-continuity for functions

We introduce a new notion ofθ-M-continuous functions as functions from a set satisfying some minimal conditions into a set satisfying some minimal conditions. We obtain some characterizations and several properties of such functions. The functions enable us to formulate a unified theory ofθ-continuity [15],θ-semi-continuity [4], quasi-irresoluteness [9],θ-preirresoluteness [29], and weakβ-irresoluteness [30].     

Amply regular graphs and block designs

We study the amply regular diameter d graphs Γ such that for some vertex a the set of vertices at distance d from a is the set of points of a 2-design whose set of blocks consists of the intersections of the neighborhoods of points with the set of vertices at distance d-1 from a. We prove that the subgraph induced by the set of points is a clique, a coclique, or a strongly regular diameter 2 graph. For diameter 3 graphs we establish that this construction is a 2-design for each vertex a if and only if the graph is distance-regular and for each vertex a the subgraph Γ₃(a) is a clique, a coclique, or a strongly regular graph. We obtain the list of admissible parameters for designs and diameter 3 graphs under the assumption that the subgraph induced by the set of points is a Seidel graph. We show that some of the parameters found cannot correspond to distance-regular graphs.     

On subadditive functions and ψ-additive mappings

In [4], assuming among others subadditivity and submultiplicavity of a function ψ: [0, ∞)→[0, ∞), the authors proved a Hyers-Ulam type stability theorem for “ψ-additive” mappings of a normed space into a normed space. In this note we show that the assumed conditions of the function ψ imply that ψ=0 and, consequently, every “ψ-additive” mapping must be additive     

On the topology of spherically symmetric space-times

Spherically symmetric space-times have attained considerable attention ever since the early beginnings of the theory of general relativity. In fact, they have appeared already in the papers of K. Schwarzschild [12] and W. De Sitter [5] which were published in 1916 and 1917 respectively soon after Einstein's epoch-making work [7] in 1915. The present survey is concerned mainly with recent results pertainig to the toplogy of spherically symmetric space-times. Definition. By space-time a connected time-oriented 4-dimensional Lorentz manifold is meant. If (M,<,>) is a space-time, and Φ: SO(3)×M→M an isometric action such that the maximal dimension of its orbits is equal to 2, then the action Φ is said to be spherical and the space-time is said to be spherically symmetric [8]; [11]. Likewise, isometric actions Ψ: O(3)×M→M are also considered ([10], p. 365; [4]) which will be called quasi-spherical if the maximal dimension of its orbits is 2 and then the space-time is said to be quasi-spherically symmetric here. Each quasi-spherical action yields a spherical one by restricting it to the action of SO(3); the converse of this statement will be considered elsewhere. The main results concerning spherically symmetric space-times are generally either of local character or pertaining to topologically restricted simple situations [14], and earlier results of global character are scarce [1], [4], [6], [13]. A report on recent results concerning the global geometry of spherically symmetric space-times [16] is presented below.     

Generalization of a theorem of Luxemburg and Zaanen

We give a generalization of a result of Luxemburg and Zaanen [6] and we deduce some interesting consequences.     

β级星像函数类和凸像函数类的扩展问题

本文利用文[2,3]的引理和算子L(a,c)f(z)的一些性质.结合Hadamard乘积,研究了算子L(a,c)f(z),获得了L(a,c)f(z)∈S*(β)和L(a,c)f(z)∈K(β)的充分条件,推广了文[2,3]的相关结论.     

A unified theory of weak continuity for functions

In this paper we introduce a new notion of weaklyM-continuous functions as functions from a set satisfying some minimal conditions into a set satisfying some minimal conditions. We obtain some characterizations and several properties of such functions. This function leads to the formulation of a unified theory of weak continuity [27], almosts-continuity [43],p(θ)-continuity [10] andp-continuity [59].     

Totally balanced combinatorial optimization games

Combinatorial optimization games deal with cooperative games for which the value of every subset of players is obtained by solving a combinatorial optimization problem on the resources collectively owned by this subset. A solution of the game is in the core if no subset of players is able to gain advantage by breaking away from this collective decision of all players. The game is totally balanced if and only if the core is non-empty for every induced subgame of it.?We study the total balancedness of several combinatorial optimization games in this paper. For a class of the partition game [5], we have a complete characterization for the total balancedness. For the packing and covering games [3], we completely clarify the relationship between the related primal/dual linear programs for the corresponding games to be totally balanced. Our work opens up the question of fully characterizing the combinatorial structures of totally balanced packing and covering games, for which we present some interesting examples: the totally balanced matching, vertex cover, and minimum coloring games. Received: November 5, 1998 / Accepted: September 8, 1999?Published online February 23, 2000     

On elongations of totally projective groups and <Emphasis Type="Italic">α</Emphasis>-<Emphasis Type="Italic">Σ</Emphasis>-groups

We prove that a special α-elongation of a totally projective group is an α-Σ-group if and only if it is a totally projective group. This parallels our recent result in [5].     

A note on W-I-continuous functions

In [1], Aç?kgöz et al. introduced and investigated the notions of w-I-continuous and w*-I-continuous functions in ideal topological spaces. In this paper, we investigate their relationships with continuous and θ-continuous functions.     

Strong convergence theorems for countable families of asymptotically relatively nonexpansive mappings with applications

The purpose of this paper is by using the hybrid iterative method to prove some strong convergence theorems for approximating a common element of the set of solutions to a system of generalized mixed equilibrium problems and the set of common fixed points for two countable families of closed and asymptotically relatively nonexpansive mappings in Banach space. The results presented in the paper improve and extend the corresponding results of Su et al. [Y.F. Su, H.K. Xu, X. Zhang, Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications, Nonlinear Anal. 73 (2010) 3890-3906], Li and Su [H.Y. Li, Y.F. Su, Strong convergence theorems by a new hybrid for equilibrium problems and variational inequality problems, Nonlinear Anal. 72 (2) (2010) 847-855], Chang et al. [S.S. Chang, H.W. Joseph Lee, Chi Kin Chan, A new hybrid method for solving a generalized equilibrium problem solving a variational inequality problem and obtaining common fixed points in Banach spaces with applications, Nonlinear Anal. TMA 73 (2010) 2260-2270], Kang et al. [J. Kang, Y. Su, X. Zhang, Hybrid algorithm for fixed points of weak relatively nonexpansive mappings and applications, Nonlinear Anal. HS 4 (4) (2010) 755-765], Matsushita and Takahashi [S. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in Banach spaces, J. Approx. Theory 134 (2005) 257-266], Tan et al. [J.F. Tan, S.S. Chang, M. Liu, J.I. Liu, Strong convergence theorems of a hybrid projection algorithm for a family of quasi-?-asymptotically nonexpansive mappings, Opuscula Math. 30 (3) (2010) 341-348], Takahashia and Zembayashi [W. Takahashi, K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal. 70 (2009) 45-57] and Wattanawitoon and Kumam [K. Wattanawitoon, P. Kumam, Strong convergence theorems by a new hybrid projection algorithm for fixed point problem and equilibrium problems of two relatively quasi-nonexpansive mappings, Nonlinear Anal. Hybrid Systems 3 (2009) 11-20] and others.     

On the sequential order continuity of the <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">K</Emphasis>)-space

As shown in [1], for each compact Hausdorff space K without isolated points, there exists a compact Hausdorff P′-space X but not an F-space such that C(K) is isometrically Riesz isomorphic to a Riesz subspace of C(X). The proof is technical and depends heavily on some representation theorems. In this paper we give a simple and direct proof without any assumptions on isolated points. Some generalizations of these results are mentioned.