On Λ-generalized closed sets in topological spaces

We introduce new classes of sets called Λ _g -closed sets and Λ _g -open sets in topological spaces. We also investigate several properties of such sets. It turns out that Λ _g -closed sets and Λ _g -open sets are weaker forms of closed sets and open sets, respectively and stronger forms of g-closed sets and g-open sets, respectively. Dedicated to Professor Maximilian Ganster on the occasion of his 50th birthday     

Point sets with low <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-discrepancy

In this paper we study the L _p -discrepancy of digitally shifted Hammersley point sets. While it is known that the (unshifted) Hammersley point set (which is also known as Roth net) with N points has L _p -discrepancy (p an integer) of order (log N)/N, we show that there always exists a shift such that the digitally shifted Hammersley point set has L _p -discrepancy (p an even integer) of order which is best possible by a result of W. Schmidt. Further we concentrate on the case p = 2. We give very tight lower and upper bounds for the L ₂-discrepancy of digitally shifted Hammersley point sets which show that the value of the L ₂-discrepancy of such a point set mostly depends on the number of zero coordinates of the shift and not so much on the position of these. This work is supported by the Austrian Research Fund (FWF), Project P17022-N12 and Project S8305.     

S-convexity revisited (fuzzy): long version

In this revisional article, we criticize (strongly) the use made by Medar et al., and those whose work they base themselves on, of the name ‘convexity’ in definitions which intend to relate to convex functions, or cones, or sets, but actually seem to be incompatible with the most basic consequences of having the name ‘convexity’ associated to them. We then believe to have fixed the ‘denominations’ associated with Medar’s (et al.) work, up to a point of having it all matching the existing literature in the field [which precedes their work (by long)]. We also expand his work scope by introducing s ₁-convexity concepts to his group of definitions, which encompasses only convex and its proper extension, s ₂-convex, so far. This article is a long version of our previous review of Medar’s work, published by FJMS (Pinheiro, M.R.: S-convexity revisited. FJMS, 26/3, 2007).     

I m -Quasi Upward Sets,with Their Best Approximations

The main purpose of this article is to introduce particular subsets of R ^I , which are not necessarily convex, and we call them I _m -quasi upward, or I _m -quasi downward. We show that these sets can be translated to downward or upward sets. We introduce the connection of these sets with downward and upward subsets of R ^I , and discuss the best approximation of these sets. Also we introduce embedded I _m -quasi upward and embedded I _m -quasi downward subsets of a normed space X.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-mixed intersection bodies

In this paper the author first introduce a new concept of L _p -dual mixed volumes of star bodies which extends the classical dual mixed volumes. Moreover, we extend the notions of L _p intersection body to L _p -mixed intersection body. Inequalities for L _p -dual mixed volumes of L _p -mixed intersection bodies are established and the results established here provide new estimates for these type of inequalities. This work was supported by the Natural Science Foundation of Zhejiang Province of China (Grant No. Y605065) and the Foundation of the Education Department of Zhejiang Province of China (Grant No. 20050392)     

Connections between stability and asymptotic stability of nonlinear switched systems

Here are considered nonlinear switched systems in which the switching occurs among a class of subsystems that are characterized by input–output properties stated in terms of L_p spaces of signals. The relationships between the L_p stability of each subsystem and the internal stability of the switched system are studied. In particular, conditions on the dwell time of the switching signals that guarantee the asymptotic stability of the overall system are provided. The connections among these conditions and the L_p input–output properties of the subsystems are investigated.     

Interval Valued Intuitionistic （S, T）-fuzzy Hv-submodules

On the basis of the concept of the interval valued intuitionistic fuzzy sets introduced by K. Atanassov, the notion of interval valued intuitionistic fuzzy Hv-submodules of an Hv-module with respect to a t-norm T and an s-norm S is given and the characteristic properties are described. The homomorphic image and the inverse image are investigated. In particular, the connections between interval valued intuitionistic （S, T）-fuzzy Hv-submodules and interval valued intuitionistic （S, T）-fuzzy submodules are discussed.     

Paired-Domination in <Emphasis Type="Italic">P</Emphasis><Subscript>5</Subscript>-Free Graphs

A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired-domination number of G, denoted by , is the minimum cardinality of a paired-dominating set of G. In [1], the authors gave tight bounds for paired-dominating sets of generalized claw-free graphs. Yet, the critical cases are not claws but subdivided stars. We here give a bound for graphs containing no induced P ₅, which seems to be the critical case.     

Some complete metrics on spaces of fuzzy subsets

The classes of the L_p,∞- and L_p-metrics play an important role to develop a probability theory in fuzzy sample spaces. All of these metrics are known to be separable, but not complete. The classes are closely related as for each L_p,∞-metric there exists some L_p-metric which induces the same topology. This paper deals with the completion of the L_p,∞- and L_p-metrics. We can also show that the relationship between the classes of L_p,∞- and L_p-metrics still holds for the obtained respective classes of their completions.     

The decomposition of Kv into K2×K5''''s

The decomposition of the complete graph Kv into Kr×Kc's, the products of Kr and Kc,is originated from the use of DNA library screening. In this paper, we consider the case where r=2 and c = 5, and show that such a decomposition exists if and only if v ≡ 1 (mod 25).     

Kp,q-factorization of complete bipartite graphs

Let K _m,nbe a complete bipartite graph with two partite sets having m and n vertices, respectively. A K _p,q-factorization of K _m,n is a set of edge-disjoint K _p,q-factors of K _m,n which partition the set of edges of K _m,n. When p = 1 and q is a prime number, Wang, in his paper “On K _1,k -factorizations of a complete bipartite graph” (Discrete Math, 1994, 126: 359—364), investigated the K _1,q -factorization of K _m,nand gave a sufficient condition for such a factorization to exist. In the paper “K _1,k -factorizations of complete bipartite graphs” (Discrete Math, 2002, 259: 301—306), Du and Wang extended Wang’s result to the case that q is any positive integer. In this paper, we give a sufficient condition for K _m,n to have a K _p,q-factorization. As a special case, it is shown that the Martin’s BAC conjecture is true when p : q = k : (k+ 1) for any positive integer k.     

Reliability Approximation for Markov Chain Imbeddable Systems

In the present article, a simple method is developed for approximating the reliability of Markov chain imbeddable systems. The approximating formula reduces the problem to the reliability assessment of smaller systems with structure similar to the original systems. Two specific reliability structures which have attracted considerable research interest recently (r-within-consecutive-k-out-of-n system and two dimensional r-within-k ₁ × k ₂-out-of-n ₁ × n ₂ system) are studied by the new approach and numerical calculations are carried out, which reveal the high quality of our approximations. Several possible extensions and generalizations are also presented in brief.     

<Emphasis Type="Italic">∂</Emphasis>-operations and <Emphasis Type="Italic">H</Emphasis><Subscript><Emphasis Type="Italic">V</Emphasis></Subscript>-fields

The hyperoperations, called theta-operations （δ）, are motivated from the usual property, which the derivative has on the derivation of a product of functions. Using any map on a set, one can define δ-operations. In this paper, we continue our study on the δ-operations on groupoids, rings, fields and vector spaces or on the corresponding hyperstructures. Using δ-operations one obtains, mainly, Hwstructures, which form the largest class of the hyperstructures. For representation theory of hyperstructures, by hypermatrices, one needs special Hv-rings or Hy-fields, so these hyperstructures can be used. Moreover, we study the relation of these δ-structures with other classes of hyperstructures, especially with the Hv-structures.     

态R_0代数

本文研究了R_0代数上有关态算子的问题.利用MV-代数上内态的引入方法引入了态算子,定义了态R_0代数,它是R_0代数的一般化.给出了一些非平凡态R_0代数的例子并讨论了态R_0代数的一些基本性质.在此基础上给出了态滤子和态局部R_0代数的概念,并利用态滤子刻画了态局部R_0代数.推广了局部R_0代数的相关理论.     

The L p -curvature images of convex bodies and L p -projection bodies

Associated with the L _p -curvature image defined by Lutwak, some inequalities for extended mixed p-affine surface areas of convex bodies and the support functions of L _p -projection bodies are established. As a natural extension of a result due to Lutwak, an L _p -type affine isoperimetric inequality, whose special cases are L _p -Busemann-Petty centroid inequality and L _p -affine projection inequality, respectively, is established. Some L _p -mixed volume inequalities involving L _p -projection bodies are also established.     

Characterisations of Chebyshev sets in c0

A subset MX of a normed linear space X is a Chebyshev set if, for every xX, the set of all nearest points from M to x is a singleton. We obtain a geometrical characterisation of approximatively compact Chebyshev sets in c₀. Also, given an approximatively compact Chebyshev set M in c₀ and a coordinate affine subspace Hc₀ of finite codimension, if M∩H≠, then M∩H is a Chebyshev set in H, where the norm on H is induced from c₀.     

A new multivariable <Subscript><Emphasis Type="BoldItalic">6</Emphasis></Subscript><Emphasis Type="BoldItalic">ψ</Emphasis><Subscript><Emphasis Type="BoldItalic">6</Emphasis></Subscript> summation formula

By multidimensional matrix inversion, combined with an A _r extension of Jackson’s ₈ φ ₇ summation formula by Milne, a new multivariable ₈ φ ₇ summation is derived. By a polynomial argument this ₈ φ ₇ summation is transformed to another multivariable ₈ φ ₇ summation which, by taking a suitable limit, is reduced to a new multivariable extension of the nonterminating ₆ φ ₅ summation. The latter is then extended, by analytic continuation, to a new multivariable extension of Bailey’s very-well-poised ₆ ψ ₆ summation formula. Partly supported by FWF Austrian Science Fund grants P17563-N13, and S9607 (the second is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”).     

F_2+vF_2上的循环码

本文研究了环F_2+vF_2上的循环码.利用标准形生成元集刻画了环F_2+vF_2上的循环码的代数结构,证明了环F_2+vF_2上的每一个非零的循环码均有唯一的标准形生成元集,进而得到了每一个循环码均是由一个多项式生成的.     

Lp-Lq DECAY ESTIMATES FOR HYPERBOLIC EQUATIONS WITH OSCILLATIONS IN COEFFICIENTS

This work is concerned with the proof of Lp -Lq decay estimates for solutions of the Cauchy problem for utt -λ2(t)b2(t)/Δu =0. The coefficient consists of an increasing smooth function λ and an oscillating smooth and bounded function b which are uniformly separated from zero. The authors‘ main interest is devoted to the critical case where one has an interesting interplay between the growing and the oscillating part.     

Weak Equality and Exact Sequences in H v -modules

The largest class of multivalued systems satisfying the module-like axioms is the H_v-module. H_v-modules first were introduced by Vougiouklis. In this paper we define weak equality between two subsets of an H_v-module and introduced the notion of exact sequences of H_v-modules. Also some results on the weak equality and exact sequences are given.