Weak Rickart and dual weak Rickart objects in abelian categories

We introduce and investigate weak relative Rickart objects and dual weak relative Rickart objects in abelian categories. Several types of abelian categories are characterized in terms of (dual) weak relative Rickart properties. We relate our theory to the study of relative regular objects and (dual) relative Baer objects. We also give some applications to module and comodule categories.     

Free Modules over Cartesian Closed Topological Categories

The construction of free R-modules over a Cartesian closed topological category X is detailed (where R is a ring object in X), and it is shown that the insertion of generators is an embedding. This result extends the well-known construction of free groups, and more generally of free algebras over a Cartesian closed topological category. Mathematics Subject Classifications (2000) 18D15, 18D35, 18A40.     

Free Adjunction of Morphisms

We develop a general setting for the treatment of extensions of categories by means of freely adjoined morphisms. To this end, we study what we call composition graphs, i.e. large graphs with a partial binary operation on which we impose only rudimentary requirements. The quasicategory thus obtained contains the quasicategory of all categories as a full reflective subquasicategory; we characterize composition graphs for which this reflexion is of a particularly simple nature.This leads to the concept of semicategory; we apply semicategories to solve characterization problems concerning absolutely initial sources, absolute monosources and potential sections. For instance, we show that in any category, the absolutely initial sources are precisely the sources that contain a section.     

Generalized subdifferentials: a Baire categorical approach

We use Baire categorical arguments to construct pathological locally Lipschitz functions. The origins of this approach can be traced back to Banach and Mazurkiewicz (1931) who independently used similar categorical arguments to show that ``almost every continuous real-valued function defined on [0,1] is nowhere differentiable". As with the results of Banach and Mazurkiewicz, it appears that it is easier to show that almost every function possesses a certain property than to construct a single concrete example. Among the most striking results contained in this paper are: Almost every 1-Lipschitz function defined on a Banach space has a Clarke subdifferential mapping that is identically equal to the dual ball; if is a family of maximal cyclically monotone operators defined on a Banach space then there exists a real-valued locally Lipschitz function such that for each ; in a separable Banach space each non-empty weakcompact convex subset in the dual space is identically equal to the approximate subdifferential mapping of some Lipschitz function and for locally Lipschitz functions defined on separable spaces the notions of strong and weak integrability coincide.

     

Strongly Rickart objects in abelian categories: Applications to strongly regular and strongly Baer objects

We show how the theory of (dual) strongly relative Rickart objects may be employed in order to study strongly relative regular objects and (dual) strongly relative Baer objects in abelian categories. For each of them, we prove general properties, we analyze the behavior with respect to (co)products, and we study the transfer via functors. We also give applications to Grothendieck categories, (graded) module categories and comodule categories.     

Transfer of splitness with respect to a fully invariant short exact sequence in abelian categories

Abstract

We study the transfer via functors between abelian categories of the (dual) relative splitness of objects with respect to a fully invariant short exact sequence. We mainly consider fully faithful functors and adjoint pairs of functors. We deduce applications to Grothendieck categories, (graded) module categories and comodule categories.     

Strongly Rickart objects in abelian categories

We introduce and study (dual) strongly relative Rickart objects in abelian categories. We prove general properties, we analyze the behaviour with respect to (co)products, and we study the transfer via functors. We also give applications to Grothendieck categories, (graded) module categories and comodule categories. Our theory of (dual) strongly relative Rickart objects may be employed in order to study strongly relative regular objects and (dual) strongly relative Baer objects in abelian categories.     

On the category of hyper MV‐algebras

In this paper we study the category of hyper MV‐algebras and we prove that it has a terminal object and a coequalizer. We show that Jia's construction can be modified to provide a free hyper MV‐algebra by a set. We use this to show that in the category of hyper MV‐algebras the monomorphisms are exactly the one‐to‐one homomorphisms. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Free Monoids over Semigroups in a Monoidal Category: Construction and Applications

The adjunction of a unit to an algebraic structure with a given binary associative operation is discussed by interpreting such structures as semigroups and monoids respectively in a monoidal category. This approach then allows for results on the adjunction of counits to coalgebraic structures with a binary co-associative co-operation as well. Special attention is paid to situations where a given coalgebraic structure induces a “dual" algebraic one; here the compatibility of adjoining (co)units and dualization is examined. The extension of this process to starred algebraic structures and to monoid actions is discussed as well. Particular emphasis is given to examples from many areas of mathematics.     

广义控制系统在两种尺度意义下的稳定性

本文将两尺度稳定性理论引入了广义控制系统，并用类Ｌｉａｐｕｎｏｖ函数给出稳定和渐近稳定的充要条件。这些准则推广了关于广义控制系统和两尺度稳定性的已有结果，使我们能在一个统一的框架内研究广义控制系统稳定性。     

Statistical maps: A categorical approach

In probability theory, each random variable f can be viewed as channel through which the probability p of the original probability space is transported to the distribution p _f , a probability measure on the real Borel sets. In the realm of fuzzy probability theory, fuzzy probability measures (equivalently states) are transported via statistical maps (equivalently, fuzzy random variables, operational random variables, Markov kernels, observables). We deal with categorical aspects of the transportation of (fuzzy) probability measures on one measurable space into probability measures on another measurable spaces. A key role is played by D-posets (equivalently effect algebras) of fuzzy sets. Supported by VEGA 1/2002/06.     

Free boundary fluid flow in a trough: A numerical approach

This paper examines the slow flow of a viscous liquid in an open rectangular container, one side (the base) of which moves steadily along its own plane, thereby providing the driving force the liquid needs. Unlike the two vertical sides that are rigid and stationary, the top side is left open so that the upper part of the liquid is in contact with air and is being controlled by surface tension and gravity. A numerical procedure for obtaining solutions for the cases when the capillary numbers are small is provided and the curves of the free boundaries obtained here are presented for some flow parameters. The deviation of the shape of the free boundary is observed to be strongly dependent on the aspect ratio of the boundary (i.e., the ratio of the vertical to horizontal spread of the liquid) with its curvature changing sign in the interval [1, 1.5].     

Triple Brackets and Lax Morphism Categories

Various aspects of the traditional homotopy theory of topological spaces may be developed in an arbitrary 2-category C with zeros. In particular certain secondary composition operations called box brackets recently have been defined for C; these are similar to, but extend, the familiar Toda brackets in the topological case. In this paper we introduce further the notion of a suspension functor in C and explore the ramifications of relativizing the theory in terms of the associated lax morphism category of C, denoted mC. Four operations associated to a 3-box diagram are introduced and relations among them are clarified. The results and insights obtained, while by nature somewhat technical, yield effective and efficient techniques for computing many operations of Toda bracket type. We illustrate by recording some computations from the homotopy groups of spheres. Also the properties of a new operation, the 2-sided matrix Toda bracket, are explored.     

Assessing the default risk by means of a discrete‐time survival analysis approach

In this paper, the problem of company distress is assessed by means of a multi‐period model that exploits the potentialities of the survival analysis approach when both survival times and regressors are measured at discrete points in time. The discrete‐time hazards model can be used both as an empirical framework in the analysis of the causes of the deterioration process that leads to the default and as a tool for the prediction of the same event. Our results show that the prediction accuracy of the duration model is better than that provided by a single‐period logistic model. It is also shown that the predictive power of the discrete‐time survival analysis is enhanced when it is extended to allow for unobserved individual heterogeneity (frailty). Copyright © 2008 John Wiley & Sons, Ltd.     

Maximum Order of Triangle‐Free Graphs with a Given Rank

The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. We determine the maximum order of reduced triangle‐free graphs with a given rank and characterize all such graphs achieving the maximum order.     

A Taxonomy of Static Modules with Examples

The objective of this note is to study properties of static and adstatic modules that are linked to generated, cogenerated, presented, copresented, self-static, self-tilting, *-, *^s-, and n-star modules. Several numerical examples in various situations of ?_n-static modules are presented.     

Operator Characterizations,Rigidity and Constructions of (Ω,μ)-Frames

In this paper, first, we give some operator characterizations of (Ω,μ)-frames. We obtain that normalized tight (Ω,μ)-frames are precisely the (Ω,μ)-frames which are unitary equivalent to normalized tight (Ω,μ)-frames for some closed subspace ? of L²(Ω,μ) and (Ω,μ)-frames are precisely the (Ω,μ)-frames which are similar to normalized tight (Ω,μ)-frames for some closed subspace ? of L²(Ω,μ). We also characterize the alternate dual (Ω,μ)-frames through an operator equation. Then we establish some rigidity in the pairs of dual (super) (Ω,μ)-frames related with disjointness. Finally, we consider the constructions of (Ω,μ)-frames, including the constructions of new (Ω,μ)-frames or new pair of dual (Ω,μ)-frames from known ones and the constructions of the canonical dual of a (Ω,μ)-frame under certain conditions, which generalize the corresponding results on discrete frames.     

On ordered categories as a framework for fuzzification of algebraic and topological structures

Using the framework of ordered categories, the paper considers a generalization of the fuzzification machinery of algebraic structures introduced by Rosenfeld as well as provides a new approach to fuzzification of topological structures, which amounts to fuzzifying the underlying “set” of a structure in a suitably compatible way, leaving the structure itself crisp. The latter machinery allows the so-called “double fuzzification”, i.e., a fuzzification of something that is already fuzzified.     

两类锥广义伪不变凸性的刻画

研究了一类非光滑带约束的向量优化问题. 首先引入锥意义下的 FJ-伪不变凸I(II)型的概念; 然后将经典的Gordan择一定理推广到了带锥的情形,并在此基础上利用FJ向量驻点与(弱)有效解间的关系, 研究了锥FJ-伪不变凸I(II)型的等价刻画.