On the Decomposition Theorems for C∗-Algebras

Elliott dimension drop interval algebra is an important class among all C^?-algebras in the classification theory. Especially, they are building stones of AHD algebra and the latter contains all AH algebras with the ideal property of no dimension growth.In this paper, the authors will show two decomposition theorems related to the Elliott dimension drop interval algebra. Their results are key steps in classifying all AH algebras with the ideal property of no dimension growth.     

Going-down functors, the Künneth formula, and the Baum-Connes conjecture

We study the connection between the Baum-Connes conjecture for a locally compact group G with coeefficient A and the Künneth formula for the K-theory of tensor products by the corresponding crossed product . The main tool for this is obtained by an application of a general reduction procedure which allows us to analyze certain functors connected to the topological K-theory of a group in terms of their restrictions to compact subgroups. We also discuss several other interesting applications of this method, including a general extension result for the Baum-Connes conjecture.     

Representable K-Theory for σ-C*-algebras

We define a version of K-theory on the category of -C ^*-algebras (countable inverse limits of C ^*-algebras). Our theory is homotopy invariant, has long exact sequences and a Milnor sequence, and satisfies Bott periodicity. On C ^*-algebras it gives the ordinary K-theory, and on the space of continuous functions on a countable direct limit X of compact Hausdorff spaces, it gives the representable K-theory of X. (We do not claim that our theory is in general a representable functor.) We also define an equivariant version, and discuss several related groups.Partially supported by a National Science Foundation Postdoctoral Fellowship.     

Constructions and Dévissage in Hermitian K-Theory

We first give the good definition of the Hermitian K-theory of an exact category with duality. Then we prove a Dévissage theorem which allows us to compare the Hermitian K-theory of a number field with the Hermitian K-theory of its integers.     

K-Theory of Semi-local Rings with Finite Coefficients and étale Cohomology

Let A be a commutative semi-local ring containing 1/2. We construct natural isomor-phisms

A K¨unneth Formula for Finite Sets*

In this paper, the authors define the homology of sets, which comes from and contains the ideas of path homology and embedded homology. Moreover, A K¨unneth formula for sets associated to the homology of sets is given.     

On the classification of certain real rank zero C*-algebras

In this paper,we give a classification of real rank zero C^*-algebras that can be expressed as inductive limits of a sequence of a subclass of Elliott-Thomsen algebras C.     

AF C*-代数中的Lie理想

本文描述了AF C*-代数中闭Lie理想,证明了如果AF C*-代数A中的线性流形L 是A的闭Lie理想,则存在A的闭结合理想I和A的典型masa D中的闭子代数EI使得[A,I](?)L(?)I EI,并且A中每一个这种形式的闭子空间都是A的闭Lie理想.     

Hilbert C*模及其有界模映射

设E为C^*代数A上可数生成的Hilbert模,B(E)为E上有界模映射全体,则B(E)保距同构于K(E)的左乘子,其中K(E)为E上“紧”模映射全体。当A为无限维本原C^*代数且E为自对偶模,则E为代数有限生成。     

Hilbert C*模上的Gabor酉系统

本文引入了Hilbert C*-模上的Gabor酉系统的概念,研究了它的基本性质,证明了膨胀定理,并给出某个酉系统的两个正规框架向量不相交的等价条件.     

A Characterization of the Standard Tori in C2 as Compact Lagrangian ξ-Submanifolds*

In this paper, the authors give a characterization theorem for the standard tori S¹(a) × S¹(b), a, b > 0, as the compact Lagrangian ξ-submanifolds in the two-dimensional complex Euclidean space C² , and obtain the best version of a former rigidity theorem for compact Lagrangian ξ-submanifold in C² . Furthermore, their argument in this paper also proves a new rigidity theorem which is a direct generalization of a rigidity theorem by Li and Wang for Lagrangian self-shrinkers in C².     

Convergence of Gradient Algorithms for Nonconvex C1+α Cost Functions*

This paper is concerned with convergence of stochastic gradient algorithms with momentum terms in the nonconvex setting. A class of stochastic momentum methods, including stochastic gradient descent, heavy ball and Nesterov’s accelerated gradient,is analyzed in a general framework under mild assumptions. Based on the convergence result of expected gradients, the authors prove the almost sure convergence by a detailed discussion of the effects of momentum and the number of upcrossings. It is worth noting that there are not additional restrictions imposed on the objective function and stepsize.Another improvement over previous results is that the existing Lipschitz condition of the gradient is relaxed into the condition of H¨older continuity. As a byproduct, the authors apply a localization procedure to extend the results to stochastic stepsizes.     

C1,α-partial Regularity of Nonlinear Parabolic Systems

We prove C^{1,α}-partial regularity of weak solution of nonlinear parabolic systems u^i_t - D_αA^α_i(x, t, u, Du) = B_i(x, t, u, Du), \quad i=1,…, N under the main assumption that A^α_i and B_i, satisfy the natural growth condition.     

Balanced path decomposition of λ K n,n and λ K n,n *

Let P _k denote a path with k edges and λ K _n,n denote the λ-fold complete bipartite graph with both parts of size n. In this paper, we obtain the necessary and sufficient conditions for λ K _n,n to have a balanced P _k -decomposition. We also obtain the directed version of this result.     

C*代数l上的左模的半双线性型的稳定性

本文分别研究了 C^*-代数l上的单位左模和具有实阶零性质的CC^*代数l上的单位左模的半双线性型的 Hyers-Ulam 稳定性.     

Extremal K¨ahler Metrics of Toric Manifolds*

This paper is a survey of some recent developments concerning extremal K¨ahler metrics on Toric Manifolds.     

Real structures of Teichmüller spaces, Dehn twists, and moduli spaces of real curves

An orientation reversing involution of a topological compact genus surface induces an antiholomorphic involution of the Teichmüller space of genus g Riemann surfaces. Two such involutions and are conjugate in the mapping class group if and only if the corresponding orientation reversing involutions and of are conjugate in the automorphism group of . This is equivalent to saying that the quotient surfaces and are homeomorphic. Hence the Teichmüller space has distinct antiholomorphic involutions, which are also called real structures of ([7]). This result is a simple fact that follows from Royden's theorem ([4]) stating that the the mapping class group is the full group of holomorphic automorphisms of the Teichmüller space (). Let and be two real structures that are not conjugate in the mapping class group. In this paper we construct a real analytic diffeomorphism such that This mapping d is a product of full and half Dehn–twists around certain simple closed curves on the surface . This has applications to the moduli spaces of real algebraic curves. A compact Riemann surface admitting an antiholomorphic involution is a real algebraic curve of the topological type . All fixed–points of the real structure of the Teichmüller space , are real curves of the above topological type and every real curve of that topological type is represented by an element of the fixed–point set of . The fixed–point set is the Teichmüller space of real algebraic curves of the corresponding topological type. Given two different real structures and , let d the the real analytic mapping satisfying (1). It follows that d maps onto and is an explicit real analytic diffeomorphism between these Teichmüller spaces. Received 8 December 1997; accepted 12 August 1998     

Surfaces with pg?=?q?=?1, K2?=?8 and nonbirational bicanonical map

We prove that if the bicanonical map of a minimal surface of general type S with p _g  = q = 1 and K_S²=8{K_{S}^2=8} is nonbirational, then it is a double cover onto a rational surface. An application of this theorem is the complete classification of minimal surfaces of general type with p_g=q=1, K_S²=8{p_{g}=q=1, K_{S}^2=8} and nonbirational bicanonical map.