THE {1}-AND {2}-INVERSES OF HOMOMORPHISMS OF r-MODULES

This note is a contribution to the application of generalized inverse of homomorphisms of modules in ring(module)theory.Using the{1}-and{2}-inverses of homomorphisms of modules,we characterize a class of rings and an important class of modules respectively.

Attractors of Dissipative Soliton Equation

In the paper ,we study longtime dynamic behavior of dissipative soliton equation existence of attractor ,geometrical structure of attractor dynamic behavior under the parametric perturbation of dissipative soliton equation.estimate of fractal dimension of attractor.

Dimensional Properties of Fractional Brownian Motion

Let B^α = {B^α（t）,t E R^N} be an （N,d）-fractional Brownian motion with Hurst index α∈ （0, 1）. By applying the strong local nondeterminism of B^α, we prove certain forms of uniform Hausdorff dimension results for the images of B^α when N 〉 αd. Our results extend those of Kaufman for one-dimensional Brownian motion.

Practicing the concept of “structuring” processing in the manufacture of polymer films

Polymer processing is a technology used to transfer raw materials into products with different shapes and functionalities and is a key step for polymer application.After years of development,the polymer processing task has changed from traditional processing,which mainly addresses the specific shapes of articles and focuses on the effect of processing on the structures and properties of polymers,to modern processing,which directly transforms a“designed structure”into commercial products via processing.It is the so-called“structuring”processing.Owing to the unique long-chain nature and slow topological relaxation,polymers are always driven and frozen into different nonequilibrium conformations,providing an effective way to design a given polymer material with desired structure and tunable performances via processing.Among the endless number of processing techniques,film casting is a prototypical pathway involving high supercooling or/and a strong flow field,based on which diverse thin polymer films have been successfully developed.In this review,taking isotactic polypropylene(i PP)film as an example,we highlight the strategy of“structuring”processing,in which we transform various crystalline structures of i PP into diverse commercial film products.

Generalized Smash Products

In this paper, we study the ring #(D,B) and obtain two very interesting results. First we prove in Theorem 3 that the category of rational left BU-modules is equivalent to both the category of #-rational left modules and the category of all (B,D)-Hopf modules D . Cai and Chen have proved this result in the case B = D = A. Secondly they have proved that if A has a nonzero left integral then A#A *rat is a dense subring of End k (A). We prove that #(A,A) is a dense subring of End k (Q), where Q is a certain subspace of #(A,A) under the condition that the antipode is bijective (see Theorem 18). This condition is weaker than the condition that A has a nonzero integral. It is well known the antipode is bijective in case A has a nonzero integral. Furthermore if A has nonzero left integral, Q can be chosen to be A (see Corollary 19) and #(A,A) is both left and right primitive. Thus A#A *rat ? #(A,A) ? End k (A). Moreover we prove that the left singular ideal of the ring #(A,A) is zero. A corollary of this is a criterion for A with nonzero left integral to be finite-dimensional, namely the ring #(A,A) has a finite uniform dimension.

The Uniform Dimension of Generalized Brownian Sheet

For the N-parameter d-dimensional Generalized Brownian Sheet (in short G. B. S.) satisfying some conditions, this paper proves that when 2N≤αd, with probability one, E∈B(R_+~N), there holds 2/βdim E+2N-2Nβ/α≤dim (E)≤2/αdim E. Especially when α=β=1, we have dim (E)=2 dim E. Noting that even if α=β=1, the G. B. S. is wider than the Brownian Sheet, thus we have extended the uniform dimension result of Mountford, T,S.