MULTIPLE SOLUTIONS FOR ASYMPTOTICALLY LINEAR ELLIPTIC EQUATIONS INVOLVING NATURAL GROWTH TERM

In this article, under some conditions on the behaviors of the perturbed function f（x, s） or its primitive F（x,s） =∫so f（x,t）dt near infinity and near zero, a class of asymptotically linear elliptic equations involving natural growth term is studied. By computing the critical group, the existence of three nontrivial solutions is proved.

TOWARD A THEORY OF WEAK I SEQUENCES

The author introduces a notion of weakIsequences and characterizes such sequences by means of homological methods.This notion extends the notion of weakMsequences and thus extends the notions of generalized Cohen Macaulay modules and Buchsbaum modules to more general cases.

Cellular Chain Complex of Small Covers with Integer Coefficients and Its Application

Let Pn be a simple n-polytope with a Z2-characteristic function λ. And h is a Morse function over Pn. Then the small cover Mn（λ） corresponding to the pair （Pn, λ） has a cell structure given by h. From this cell structure we can derive a cellular chain complex of Mn（λ） with integer coefficients. In this paper, firstly, we discuss the highest dimensional boundary morphism ？n of this cellular chain complex and get that ？n=0 or 2 by a natural way. And then, from the well-known result that the submanifold corresponding to （F, λF） is naturally a small cover with dimension k, where F is any k-face of Pn and λF is the restriction of λ on F, we get that ？k=0 or ±2 for 0 ≤ k 〈 n. Finally, by using the definition of inherited characteristic function which is the restriction of λ on the faces of Pn, we get a way to calculate the homology groups of Mn（λ）. Applying our result to a 3-small cover we have that the homology groups of any 3-small cover is torsion-free or has only 2-torsion.

Overexpression of enhancer of zests homolog 2 in lymphoma

Objective This article aimed to review the biological characteristics of enhancer of zests homolog 2 （EZH2）, and the transcriptional repression mechanism of action of EZH2 in tumors, particularly in the progression of lymphoma. Data sources The data cited in this review were mainly obtained from the articles listed in PubMed and HighWare that were published from March 2004 to April 2012. The search terms were ＂enhancer of zests homolog 2＂, ＂polycomb group＂, and ＂lymphoma＂. Study selection Articles regarding the mechanism of EZH2 in post-transcriptional modification, functions of polycomb group proteins, and the roles of EZH2 in lymphoma were selected. Results EZH2 acts as oncogene and involved in many kinds of tumors. Moreover, it plays an important role in tumorigenesis and lymphomagenesis by promoting the proliferation and aggressiveness of neoplastic cells, facilitating malignant tumor cell diffusion, and mediating transcriptional silencing. Conclusion EZH2 mediated transcriptional repression through its methyltransferase activity at the chromatin level has certain influence on lymphoma, and there might exist a therapeutic window for the development of new agents and identification of novel diagnostic markers based on EZH2.

On Proper and Exact Relative Homological Dimensions

In Enochs'relative homological dimension theory occur the(co)resolvent and(co)proper dimensions,which are defined by proper and coproper resolutions constructed by precovers and preenvelopes,respectively.Recently,some authors have been interested in relative homological dimensions defined by just exact sequences.In this paper,we contribute to the investigation of these relative homological dimensions.First we study the relation between these two kinds of relative homological dimensions and establish some transfer results under adjoint pairs.Then relative global dimensions are studied,which lead to nice characterizations of some properties of particular cases of self-orthogonal subcategories.At the end of this paper,relative derived functors are studied and generalizations of some known results of balance for relative homology are established.