On Generic Well-posedness of Restricted Chebyshev Center Problems in Banach Spaces

Let B （resp. K, BC,KC） denote the set of all nonempty bounded （resp. compact, bounded convex, compact convex） closed subsets of the Banach space X, endowed with the Hausdorff metric, and let G be a nonempty relatively weakly compact closed subset of X. Let B° stand for the set of all F ∈B such that the problem （F, G） is well-posed. We proved that, if X is strictly convex and Kadec, the set KC ∩ B° is a dense Gδ-subset of KC ／ G. Furthermore, if X is a uniformly convex Banach space, we will prove more, namely that the set B ／B° （resp. K ／ B°, BC ／B°, KC ／ B°） is a-porous in B （resp. K,BC, KC）. Moreover, we prove that for most （in the sense of the Baire category） closed bounded subsets G of X, the set K ／ B° is dense and uncountable in K.     

Universal Similarity Factorization Equalities over Generalized Clifford Algebras

For any element a in a generalized 2^n-dimensional Clifford algebra Lln （F） over an arbitrary field F of characteristic not equal to two, it is shown that there exits a universal invertible matrix Pn over Lln（F） such that Pn^-1DnPn= φ（α）∈F^2n×2n, where φ（a） is a matrix representation of α over and Dα is a diagonal matrix consisting of a or its conjugate.     

Weak Hopf Algebras Corresponding to Borcherds-Cartan Matrices

Let y be a generalized Kac-Moody algebra with an integral Borcherds-Cartan matrix. In this paper, we define a d-type weak quantum generalized Kac-Moody algebra wUq^d（y）, which is a weak Hopf algebra. We also study the highest weight module over the weak quantum algebra wUdq^d（y） and weak A-forms of wUq^d（y）.     

The Existence of Maximal w^＊-closed Submodules in Nest Algebra Modules

Using the technique of annihilators, we characterize the w^＊-closure of the finite-partition submodule R（N, ～） of a weakly closed nest algebra module U completely; and by virtue of this characterization, we obtain a sufficient and necessary condition for U to have a maximal w^＊-closed submodule.     

Additive Mappings Preserving Rank-one Idempotents

A complete characterization of additive mappings φ ： B（X）→B（Y）, which map rank-one idempotents to themselves or to zero, is given.     

Weak Tensor Category and Related Generalized Hopf Algebras

There are at least two kinds of generalization of Hopf algebra, i.e. pre-Hopf algebra and weak Hopf algebra. Correspondingly, we have two kinds of generalized bialgebras, almost bialgebra and weak bialgebra. Let L = （L, ×, I, a, l, r） be a tensor category. By giving up I, l, r and keeping ×, a in L, the first author got so-called pre-tensor category L = （L, ×, a） and used it to characterize almost bialgebra and pre-Hopf algebra in Comm. in Algebra, 32（2）： 397-441 （2004）. Our aim in this paper is to generalize tensor category L = （L, ×, I, a, l, r） by weakening the natural isomorphisms l, r, i.e. exchanging the natural isomorphism ll^-1 = rr^-1 = id into regular natural transformations lll= l, rrr = r with some other conditions and get so-called weak tensor category so as to characterize weak bialgebra and weak Hopf algebra. The relations between these generalized （bialgebras） Hopf algebras and two kinds generalized tensor categories will be described by using of diagrams. Moreover, some related concepts and properties about weak tensor category will be discussed.     

On Maximal Abelian Self-adjoint Subalgebras of Factors of Type Ⅱ1

In this note, we show that if N is a proper subfactor of a factor M of type Ⅱ1 with finite Jones index, then there is a maximal abelian self-adjoint subalgebra （masa） A of N that is not a masa in ,M. Popa showed that there is a proper subfactor R0 of the hyperfinite type Ⅱ1 factor R such that each masa in R0 is also a masa in R. We shall give a detailed proof of Popa＇s result.     

On the Order of Magnitude of the Divisor Function

Let D be an increasing sequence of positive integers, and consider the divisor functions： d（n, D） =∑d｜n,d∈D,d≤√n1, d2（n,D）=∑[d,δ]｜n,d,δ∈D,[d,δ]≤√n1, where [d,δ]=1.c.m.（d,δ）. A probabilistic argument is introduced to evaluate the series ∑n=1^∞and（n,D） and ∑n=1^∞and2（n,D）.     

Jackson＇s Theorem on Bounded Symmetric Domains

Polynomial approximation is studied on bounded symmetric domain Ω in C^n for holomorphic function spaces X such as Bloch-type spaces, Bergman-type spaces, Hardy spaces, Ω algebra and Lipschitz space. We extend the classical Jackson＇s theorem to several complex variables：Eκ（f,X）≤ω（1/k,f,X）, where Eκ（f,X） is the deviation of the best approximation of f ∈X by polynomials of degree at most k with respect to the X-metric and ω（1/k,f,X） is the corresponding modulus of continuity.     

Lifting Unitary Elements with Application to Approximately Inner Automorphisms

Let A be a separable unital nuclear simple C＊-algebra with torsion K0 （A）, free K1 （A） and with the UCT. Let T ： A→M（K）/K be a unital homomorphism. We prove that every unitary element in the commutant of T（A） is an exponent, thus it is liftable. We also prove that each automorphism α on E with α ∈ Aut0（A） is approximately inner, where E is a unital essential extension of A by K and α is the automorphism on A induced by α.     

Hölder continuity of surfaces with bounded mean curvature at corners where Plateau boundaries meet free boundaries

Let $P={\rm \Gamma}\cap{\cal S}Let be the point of non-tangential intersection of a closed Jordan arc and an embedded, regular support surface . Let be a conformally parametrized solution of with partially free boundaries . It is proved, that is H?lder continuous up to with , whenever meets orthogonally along its free trace. This provides a regularity result for stationary minimal surfaces and is applicable also to surfaces of prescribed bounded mean curvature. Mathematics Subject Classification (2000) 53 A 10, 35 J 65, 35 R 35, 49 Q 05     

Minimum-Volume Enclosing Ellipsoids and Core Sets

We study the problem of computing a (1+ε)-approximation to the minimum-volume enclosing ellipsoid of a given point set . Based on a simple, initial volume approximation method, we propose a modification of the Khachiyan first-order algorithm. Our analysis leads to a slightly improved complexity bound of operations for . As a byproduct, our algorithm returns a core set with the property that the minimum-volume enclosing ellipsoid of provides a good approximation to that of . Furthermore, the size of depends on only the dimension d and ε, but not on the number of points n. In particular, our results imply that for .We thank the Associate Editor and an anonymous referee for handling our paper efficiently and for helpful comments and suggestions.This author was supported in part by NSF through Grant CCR-0098172.This author was supported in part by NSF through CAREER Grant DMI-0237415.     

The Density of Linear Symplectic Cocycles with Simple Lyapunov Spectrum in YIC（X, SL（2, R））

Let （X, G（X）, m） be a probability space with a-algebra G（X） and probability measure m. The set V in G is called P-admissible, provided that for any positive integer n and positive-measure set Vn∈ contained in V, there exists a Zn∈G such that Zn belong to Vn and 0 〈 m（Zn） 〈 1/n. Let T be an ergodic automorphism of （X, G） preserving m, and A belong to the space of linear measurable symplectic cocycles     

Homomorphisms between JC＊-algebras and Lie C＊-algebras

It is shown that every almost ＊-homomorphism h ： A→B of a unital JC＊-algebra A to a unital JC＊-algebra B is a ＊-homomorphism when h（rx） = rh（x） （r 〉 1） for all x∈A, and that every almost linear mapping h ： A→B is a ＊-homomorphism when h（2^nu o y） - h（2^nu） o h（y）, h（3^nu o y） - h（3^nu） o h（y） or h（q^nu o y） = h（q^nu） o h（y） for all unitaries u ∈A, all y ∈A, and n = 0, 1,.... Here the numbers 2, 3, q depend on the functional equations given in the almost linear mappings. We prove that every almost ＊-homomorphism h ： A→B of a unital Lie C＊-algebra A to a unital Lie C＊-algebra B is a ＊-homomorphism when h（rx） = rh（x） （r 〉 1） for all x ∈A.     

The Invariance Principle for the Total Length of the Nearest-Neighbor Graph

Let be the Poisson point process with intensity 1 in R^d and let be . We obtain a strong invariance principle for the total length of the nearest-neighbor graph on .     

Nonlinear Parabolic Equations with Singularities in Colombeau Vector Spaces

We consider nonlinear parabolic equations with nonlinear non-Lipschitz＇s term and singular initial data like Dirac measure, its derivatives and powers. We prove existence-uniqueness theorems in Colombeau vector space yC^1,W^2,2（[0,T）,R^n）,n ≤ 3. Due to high singularity in a case of parabolic equation with nonlinear conservative term we employ the regularized derivative for the conservative term, in order to obtain the global existence-uniqueness result in Colombeau vector space yC^1,L^2（[0,T）,R^n）,n≤ 3.     

On graded generalized local cohomology

Let be a homogeneous Noetherian ring with local base ring (R₀,m₀) and let M,N be two finitely generated graded R-modules. Let denote the i-th graded generalized local cohomology of N relative to M with support in . We study the vanishing, tameness and asymptotical stability of the homogeneous components of . Received: 22 March 2005; revised: 25 June 2005