Generalization of perfectly continuous,regular set-connected and clopen functions

Summary Noiri in 1984, Dontchev, Ganster and Reilly in recent years and Reilly and Vamanamurthy in 1983 introduced the notion of perfectly continuous, regular set-connected and clopen functions, respectively. The aim of this paper is to introduce the notion of a new class of functions which is called almost clopen functions including the classes of perfectly continuous, regular set-connected and clopen functions. Furthermore, properties of almost clopen functions are obtained and relationships among almost clopenness, perfect continuity, regular set-connectedness, clopenness and almost continuity are investigated.     

Fundamental groups of compact manifolds and symmetric geometry of noncompact type

We introduce the notion of geometrical engagement for actions of semisimple Lie groups and their lattices as a concept closely related to Zimmer's topological engagement condition. Our notion is a geometrical criterion in the sense that it makes use of Riemannian distances. However, it can be used together with the foliated harmonic map techniques introduced in [8] to establish foliated geometric superrigidity results for both actions and geometric objects. In particular, we improve the applications of the main theorem in [9] to consider nonpositively curved compact manifolds (not necessarily with strictly negative curvature). We also establish topological restrictions for Riemannian manifolds whose universal cover have a suitable symmetric de Rham factor (Theorem B), as well as geometric obstructions for nonpositively curved compact manifolds to have fundamental groups isomorphic to certain groups build out of cocompact lattices in higher rank simple Lie groups (Corollary 4.5). Received: October 22, 1997     

Approximation properties of modified Gamma operators

In this paper we present a survey of rates of pointwise approximation of modified Gamma operators Gn for locally bounded functions and absolutely continuous functions by using some inequalities and results of probability theory with the method of Bojanic-Cheng. In the paper a kind of locally bounded functions is introduced with different growth conditions in the fields of both ends of interval (0,+∞), and it is found out that the operators have different properties compared to the Gamma operators discussed in [X.M. Zeng, Approximation properties of Gamma operators, J. Math. Anal. Appl. 311 (2005) 389-401]. And we obtain two main theorems. Theorem 1 gives an estimate for locally bounded functions which subsumes the approximation of functions of bounded variation as a special case. Theorem 2 gives an estimate for absolutely continuous functions which is best possible in the asymptotical sense.     

Metric critical point theory: Potential Well Theorem and its applications

Recent results show evidence of the fact that the notion of critical point has a purely metric nature. In this paper, after giving the fundamental definitions of critical and regular points for continuous functions, we survey some applications of the nonsmooth critical point theory. Our basic tool is a Potential Well Theorem. Conferenza tenuta da A. Ioffe il 9 ottobre 1995     

Approaching Metric Domains

In analogy to the situation for continuous lattices which were introduced by Dana Scott as precisely the injective T₀ spaces via the (nowadays called) Scott topology, we study those metric spaces which correspond to injective T₀ approach spaces and characterise them as precisely the continuous lattices equipped with a unitary and associative [0,?∞?]-action. This result is achieved by a detailed analysis of the notion of cocompleteness for approach spaces.     

On <Emphasis Type="Italic">p</Emphasis>-convergent Operators on Banach Lattices

The notion of a p-convergent operator on a Banach space was originally introduced in 1993 by Castillo and Sánchez in the paper entitled “Dunford–Pettis-like properties of continuous vector function spaces”. In the present paper we consider the p-convergent operators on Banach lattices, prove some domination properties of the same and consider their applications (together with the notion of a weak p-convergent operator, which we introduce in the present paper) to a study of the Schur property of order p. Also, the notion of a disjoint p-convergent operator on Banach lattices is introduced, studied and its applications to a study of the positive Schur property of order p are considered.     

On Brøndsted–Rockafellar’s Theorem for convex lower semicontinuous epi-pointed functions in locally convex spaces

Mathematical Programming - In this work we give an extension of the Brøndsted–Rockafellar Theorem, and some of its important consequences, to proper convex lower-semicontinuous...     

Solutions to five problems on tensor products of lattices and related matters

The notion of a capped tensor product, introduced by G. Grätzer and the author, provides a convenient framework for the study of tensor products of lattices that makes it possible to extend many results from the finite case to the infinite case. In this paper, we answer several open questions about tensor products of lattices. Among the results that we obtain are the following:¶¶Theorem 2. Let A be a lattice with zero. If A ?B A \oplus B is a lattice for every lattice L with zero, then A is locally finite and A ?B A \oplus B is a capped tensor product for every lattice L with zero.¶¶Theorem 5. There exists an infinite, three-generated, 2-modular lattice K with zero such that K ?K K \oplus K is a capped tensor product.¶¶Here, 2-modularity is a weaker identity than modularity, introduced earlier by G. Grätzer and the author.     

Effective Borel measurability and reducibility of functions

The investigation of computational properties of discontinuous functions is an important concern in computable analysis. One method to deal with this subject is to consider effective variants of Borel measurable functions. We introduce such a notion of Borel computability for single‐valued as well as for multi‐valued functions by a direct effectivization of the classical definition. On Baire space the finite levels of the resulting hierarchy of functions can be characterized using a notion of reducibility for functions and corresponding complete functions. We use this classification and an effective version of a Selection Theorem of Bhattacharya‐Srivastava in order to prove a generalization of the Representation Theorem of Kreitz‐Weihrauch for Borel measurable functions on computable metric spaces: such functions are Borel measurable on a certain finite level, if and only if they admit a realizer on Baire space of the same quality. This Representation Theorem enables us to introduce a realizer reducibility for functions on metric spaces and we can extend the completeness result to this reducibility. Besides being very useful by itself, this reducibility leads to a new and effective proof of the Banach‐Hausdorff‐Lebesgue Theorem which connects Borel measurable functions with the Baire functions. Hence, for certain metric spaces the class of Borel computable functions on a certain level is exactly the class of functions which can be expressed as a limit of a pointwise convergent and computable sequence of functions of the next lower level. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some loose ends on unbounded order convergence

The notion of almost everywhere convergence has been generalized to vector lattices as unbounded order convergence, which proves to be a very useful tool in the theory of vector and Banach lattices. In this short note, we establish some new results on unbounded order convergence that tie up some loose ends. In particular, we show that every norm bounded positive increasing net in an order continuous Banach lattice is uo-Cauchy and that every uo-Cauchy net in an order continuous Banach lattice has a uo-limit in the universal completion.     

Construction of pseudorandom binary lattices by using the multiplicative inverse

In an earlier work Hubert and the authors of this paper introduced and studied the notion of pseudorandomness of binary lattices. Later in another paper the authors gave a construction for a large family of “good” binary lattices by using the quadratic characters of finite fields. Here, a further large family of “good” binary lattices is constructed by using finite fields and the notion of multiplicative inverse.     

The Cauchy formula with s-monogenic kernel and a functional calculus for noncommuting operators

The new notion of slice monogenic functions introduced in the paper [F. Colombo, I. Sabadini, D.C. Struppa, Slice monogenic functions, Israel J. Math. 171 (2009) 385-403] led us to define a new functional calculus for an n-tuple of not necessarily commuting operators, see [F. Colombo, I. Sabadini, D.C. Struppa, A new functional calculus for noncommuting operators, J. Funct. Anal. 254 (2008) 2255-2274]. In this paper we prove a Cauchy formula with slice monogenic kernel for the slice monogenic functions. This new Cauchy formula is the fundamental tool to prove that our functional calculus apply to a more general setting. Moreover, we deduce some fundamental properties of the functional calculus, for example: some algebraic properties, the Spectral Mapping Theorem and the Spectral Radius Theorem.     

相似剩余格及其对应逻辑系统的完备性

引入了相似剩余格的概念,讨论了剩余格上相似算子和等价算子的关系,并得到了真值剩余格和相似剩余格相互转化的方法.其次,研究了相似剩余格上的相似滤子,利用相似滤子刻画了可表示的相似剩余格.最后,引入了相似剩余格对应的逻辑系统,证明了其完备性定理,并得到了其成为半线性逻辑的条件.     

Fixräume regulärer operatoren und ein Korovkin-Satz

S. J. Bernau has introduced the notion of an exchange subspace of an L_p-space and has shown that the range of a contractive linear projection on an L_p-space (1 ? p < ∞, p ≠ 2) is an exchange subspace. In the present paper we define this notion for real Banach lattices with order continuous norm and prove among other things that fixed spaces of special regular operators on these spaces are exchange subspaces. As application we give a Korovkin theorem for sequences of contractions on real Banach lattices with an uniformly monotone norm.     

幺半环上模糊有限状态机的一些性质

状态机的很多性质在计算机等方面有着广泛的应用,因此对状态机的研究具有重要的意义.本文给出了幺半环上模糊有限状态机的概念,对状态之间的等价进行了定义,引入了同态的概念,得到同态定理和满同态分解定理,讨论了幺半环上模糊有限状态机在同态下的交换性质和连通性以及子状态机的可分离性.     

谓词转换器的拓扑语义

本文利用半拓扑空间的连续映射建立Dijkstra谓词转换器的拓扑语义。，引入D－半拓扑空间概念，用以建立相容谓词转换器的语义，引入SM-半拓扑空间概念，用以刻画连续谓词转换器的语义。针对不确定程序，本文引入了dI-domain上半稳定映射概念，给出了其等价刻画。     

Solvability of Implicit Complementarity Problems

In this paper, a new notion of exceptional family of elements (EFE) for a pair of functions involved in the implicit complementarity problem (ICP) is introduced. Based upon this notion and the Leray–Schauder Alternative, a general alternative is obtained which gives more general existence theorems for the implicit complementarity problem. Finally, via the techniques of continuous selections, these existence theorems are extended to the multi-valued implicit complementarity problems (MIPS).     

On S-Normal Spaces and Pre GS-Closed Functions

In this paper the notion of pre gs-closed functions is introduced and investigated. Presemiclosed functions are pre gs-closed but not conversely. It is shown that s-normality is preserved under continuous pre gs-closed surjections.     

On linear complexity of binary lattices,II

The linear complexity is an important and frequently used measure of unpredictably and pseudorandomness of binary sequences. In Part I of this paper, we extended this notion to two dimensions: we defined and studied the linear complexity of binary and bit lattices. In this paper, first we will estimate the linear complexity of a truly random bit (M,N)-lattice. Next we will extend the notion of k-error linear complexity to bit lattices. Finally, we will present another alternative definition of linear complexity of bit lattices.     

非对合剩余格的正规双极值模糊理想

本文对非对合剩余格的双极值模糊理想问题作进一步深入研究.引入了非对合剩余格的正规双极值模糊理想概念, 考察了其性质并获得了其若干等价刻画.同时, 给出了两类特殊的双极值模糊理想的定义, 分别称为极大双极值模糊理想和完全正规双极值模糊理想并讨论了它们的性质和相互关系.这些工作为进一步揭示非对合剩余格的结构特征拓展了研究思路.