Extending Sets of Idempotents to Ring Extensions

In this paper the idea of an intrinsic extension of a ring, first proposed by Faith and Utumi, is generalized and studied in its own right. For these types of ring extensions, it is shown that, with relatively mild conditions on the base ring, R, a complete set of primitive idempotents (a complete set of left triangulating idempotents, a complete set of centrally primitive idempotents) can be constructed for an intrinsic extension, T, from a corresponding set in the base ring R. Examples and applications are given for rings that occur in functional analysis and group ring theory.     

On Primitive Idempotents of the Strong Endomorphism Monoid of a Graph

51.IntroductionandPreIiminariesThemonoidofendomorphismsofagraph,inparticular,thatofstrongendomorphismsofagraph,hasbeentheobjectofresearchesinthetheoryofsemigroupsforquitesometime(cf.Llj-Lloj).LlJandL2jcanserveasasurvey.Theaimoftheseresearchesistocon-tributetothealgebraicanalysisofgraphs.Thesubjectyieldssomeinterestsincetheresultsoftheseresearchesmayopenvastpossibilitiesforapplicationsofsemigrouptheorytographtheo-ry.InL1]andL3j,ithasbeenprovedthatforfinitegraphG,sEnd(G),themonoidofstrongen…     

The set of idempotents in the weakly almost periodic compactification of the integers is not closed

This paper answers negatively the question of whether the sets of idempotents in the weakly almost periodic compactifications of and are closed.

     

The Covering Relation of Idempotents in the Strong Endomorphism Monoid of a Graph

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L-拓扑空间的单点超F紧化

给出L-拓扑空间的单点超F紧化的一种具体作法,以及局部超F紧性的定义,并证明了:(1)局部超F紧性是L-好推广;(2)一个L-拓扑空间是局部超F紧T2空间当且仅当其单点超F紧化空间是超F紧T2空间;(3)单点超F紧化在同胚意义下是唯一的。     

Geometry of the copositive and completely positive cones

The copositive cone, and its dual the completely positive cone, have useful applications in optimisation, however telling if a general matrix is in the copositive cone is a co-NP-complete problem. In this paper we analyse some of the geometry of these cones. We discuss a way of representing all the maximal faces of the copositive cone along with a simple equation for the dimension of each one. In doing this we show that the copositive cone has faces which are isomorphic to positive semidefinite cones. We also look at some maximal faces of the completely positive cone and find their dimensions. Additionally we consider extreme rays of the copositive and completely positive cones and show that every extreme ray of the completely positive cone is also an exposed ray, but the copositive cone has extreme rays which are not exposed rays.     

最小Q-过程中断时Martin流出边界与Ray-Knight紧化的关系

在最小Q-过程中断条件下证明了Martin流出边界与Ray-Knight紧化之间存在一定的映射关系.并举例说明如果最小Q-过程中断且无限流出,则Ray-Knight紧化可能比Martin流出边界要多出一些点.因此在Markov链的构造论中不能只考虑Martin边界.     

Joins and Meets of Symmetric Idempotents

We show that the poset of symmetric idempotents of a (non-commutative) unital ring with involution in the obvious order need not admit finite joins and meets.      

Spectral Radius Algebras of Idempotents

We show that the spectral radius algebras of certain quadratic operators possess nontrivial invariant subspaces. Additionally, such algebras properly contain the operator’s commutant, so that the invariant subspaces are in some sense beyond hyperinvariant. The spectral radius algebras of idempotents are completely described and, as a consequence, it is shown that every intransitive collection of operators must be contained in a norm-closed proper spectral radius algebra.      

Cone dominance and efficiency in DEA

This paper incorporates cones on virtual multipliers of inputs and outputs into DEA analysis. Cone DEA models are developed to generalize the dual of the BCC models as well as congestion models. Input-output data and/or numbers of DMUs for BCC models are inadequate to capture many aspects where judgments, expert opinions, and other external information should be taken into analysis. Cone DEA models, on the other hand, offer improved definitions of efficiency over general cone and polyhedral cone structures. The relationships between cone models and BCC models as well as those between cone models and congestion models are discussed in the development. Two numerical examples are provided to illustrate our findings.     

Some notes on the paper “Cone metric spaces and fixed point theorems of contractive mappings”

Huang and Zhang reviewed cone metric spaces in 2007 [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468-1476]. We shall prove that there are no normal cones with normal constant M<1 and for each k>1 there are cones with normal constant M>k. Also, by providing non-normal cones and omitting the assumption of normality in some results of [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468-1476], we obtain generalizations of the results.     

Cone Characterizations of Approximate Solutions in Real Vector Optimization

Approximate nondominated solutions of real vector optimization problems are characterized using the concept of translated cones. Relationships between these solutions and Pareto nondominated points are examined, and the problem of optimizing over the set of approximate solutions is addressed. This research was supported by the National Science Foundation, Grant DMS-0425768, and by the Automotive Research Center, a US Army TACOM Center of Excellence for Modeling and Simulation of Ground Vehicles at the University of Michigan     

On the Cone of Bounded Lower Semicontinuous Functions

We prove that the cone of bounded lower semicontinuous functions defined on a Tychonoff space X is algebraically and structurally isomorphic and isometric to a convex cone contained in the cone of all bounded lower semicontinuous functions defined on the Stone-Cech compactification βX if and only if the space X is normal. We apply this theorem to the study of relationship between a class of multivalued maps and sublinear operators. Using these results, we obtain new proofs of theorems about continuous selections.__________Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 886–902.Original Russian Text Copyright ©2005 by Yu. E. Linke.     

Flow Compactifications of Nondiscrete Monoids, Idempotents and Hindman's Theorem

We describe the extension of the multiplication on a not-necessarily-discrete topological monoid to its flow compactification. We offer two applications. The first is a nondiscrete version of Hindman's Theorem, and the second is a characterization of the projective minimal and elementary flows in terms of idempotents of the flow compactification of the monoid.     

On Solidness of Polar Cones

We investigate the properties of cones whose polars are solid in different polar topologies. By a standard duality argument, we obtain a number of necessary and sufficient conditions for closed convex cones to be solid in various locally convex spaces. From this, we can deduce easily the extensions of previous related results. Furthermore, we construct a class of closed convex cones in some Banach spaces, which are not solid but whose polars satisfy the angle property. This solves the Han conjecture in the negative.     

Coercivity and Strong Semismoothness of the Penalized Fischer-Burmeister Function for the Symmetric Cone Complementarity Problem

For the nonlinear complementarity problem (NCP), Chen et al. (Math. Program., 88:211–216, 2000) proposed a penalized Fischer-Burmeister (FB) function that has most desirable properties among complementarity functions (C-functions). Motivated by their work, the authors showed (Kum and Lim in Penalized Complementarity Functions on Symmetric Cones, submitted, 2009) that this function naturally extends to a C-function for the symmetric cone complementarity problem (SCCP). In this note, we show that the main coercivity property of this function for NCP also extends to the SCCP. The proof uses a new trace inequality on Euclidean Jordan algebras. We also show that the penalized FB function is strongly semismooth in the case of a semidefinite cone and a second-order cone. This work was supported by the Korea Research Foundation Grant KRF-2008-314-C00039.     

幂等矩阵线性组合的可逆性

设T1,T2,T3是三个不同的两两相互可交换的n×n非零的三次幂等矩阵,并且c1,c2,c3是非零数.本文主要给出了线性组合c1T1 c2T2 c3T3可逆性的刻画.