Enumeration of Mapping Classes for the Torus

We enumerate periodic, reducible, and Anosov elements of the mapping class group of the torus, and determine their growth function. Then we prove that almost all elements of the mapping class group of the torus are Anosov.     

Milnor invariants for spatial graphs

Link-homotopy has been an active area of research for knot theorists since its introduction by Milnor in the 1950s. We introduce a new equivalence relation on spatial graphs called component-homotopy, which reduces to link-homotopy in the classical case. Unlike previous attempts at generalizing link-homotopy to spatial graphs, our new relation allows analogues of some standard link-homotopy results and invariants.In particular we can define a type of Milnor group for a spatial graph under component-homotopy, and this group determines whether or not the spatial graph is splittable. More surprisingly, we will also show that whether the spatial graph is splittable up to component-homotopy depends only on the link-homotopy class of the links contained within it. Numerical invariants of the relation will also be produced.     

Smash Products for Secondary Homotopy Groups

We construct a smash product operation on secondary homotopy groups yielding the structure of a lax symmetric monoidal functor. Applications on cup-one products, Toda brackets and Whitehead products are considered. The second author was partially supported by the Spanish Ministry of Education and Science under MEC-FEDER grants MTM2004-01865 and MTM2004-03629, the postdoctoral fellowship EX2004-0616, and a Juan de la Cierva research contract.     

几类非Abel群之增广商群的结构

记ZG为有限群G的整群环,△n(G)为增广理想△(G)的n次幂,Qn(G)=△"(G)/△n 1(G)为G的增广商群.本文考虑了二面体群D2tk(k 奇)和m次对称群Sm,证明了Qn(D2tk)为秩不超过2t 1的基本2-群以及Qn(Sm)≌Z2.     

2-Cocycles of original deformative Schr?dinger-Virasoro algebras

Both original and twisted Schr?dinger-Virasoro algebras, and also their deformations were introduced and investigated in a series of papers by Henkel, Roger and Unterberger. In the present paper we aim at determining the 2-cocycles of original deformative Schr?dinger-Virasoro algebras. This work was supported by the National Natural Science Foundation of China (Grant Nos. 10471091, 10671027), Foundation of Shanghai Education Committee (Grant No. 06FZ029) and “One Hundred Talents Program” from University of Science and Technology of China     

A duality theorem for generalized local cohomology

We prove a duality theorem for graded algebras over a field that implies several known duality results: graded local duality, versions of Serre duality for local cohomology and of Suzuki duality for generalized local cohomology, and Herzog-Rahimi bigraded duality.

     

A Criterion for the Second Real Continuous Bounded Cohomology of a Locally Compact Group to be Finite-Dimensional

We present a brief review of the theory of quasi-characters and quasi-representations and prove a necessary and sufficient condition that the second real continuous bounded cohomology of a locally compact group to be finite-dimensional. This criterion is established by using the properties of continuous pseudocharacters on a locally compact group.     

On Representations Associated with Completely n-Positive Linear Maps on Pro-C^＊-Algebras

It is shown that an n × n matrix of continuous linear maps from a pro-C^＊-algebra A to L（H）, which verifies the condition of complete positivity, is of the form [V^＊TijФ（·）V]^n i,where Ф is a representation of A on a Hilbert space K, V is a bounded linear operator from H to K, and j=1,[Tij]^n i,j=1 is a positive element in the C^＊-algebra of all n×n matrices over the commutant of Ф（A） in L（K）. This generalizes a result of C. Y.Suen in Proc. Amer. Math. Soc., 112（3）, 1991, 709-712. Also, a covariant version of this construction is given.     

A continuity theorem for Stinespring's dilation

We show a continuity theorem for Stinespring's dilation: two completely positive maps between arbitrary C^∗-algebras are close in cb-norm if and only if we can find corresponding dilations that are close in operator norm. The proof establishes the equivalence of the cb-norm distance and the Bures distance for completely positive maps. We briefly discuss applications to quantum information theory.