MULTIPLE SOLUTIONS AND THEIR LIMITING BEHAVIOR OF COUPLED NONLINEAR SCHRDINGER SYSTEMS

We study the multiplicity of positive solutions and their limiting behavior as ε tends to zero for a class of coupled nonlinear Schrdinger system in R^N . We relate the number of positive solutions to the topology of the set of minimum points of the least energy function for ε suffciently small. Also, we verify that these solutions concentrate at a global minimum point of the least energy function.

NIH funding for disease categories related to neurodegenerative diseases

Introduction The National Institutes of Health （NIH）, a part of the U.S. Department of Health and Human Services is the nation＇s medical research agency-making important discoveries that improve health and save lives.Thanks in large part to NIH-funded medical research, Americans today are living longer and healthier. The NIH invests over $31.2＊ billion annually in medical research for the American people. More than 80% of the NIH＇s funding is awarded through almost 50 000 competitive grants to more than 325 000 researchers at over 3 000 universities, medical schools, and other research institutions in every state and around the world.

National Institutes of Health Funding for disease categories related to Neural Regeneration Research

The National Institutes of Health （NIH）, a part of the U.S. Department of Health and Human Services, is the nation＇s medical research agency-making important discoveries that improve health and save lives.Thanks in large part to NIH-funded medical research, Americans today are living longer and healthier. Life expectancy in the United States has jumped from 47 years in 1900 to 78 years as reported in 2009, and disability in people over age 65 has dropped dramatically in the past 3 decades. In recent years, nationwide rates of new diagnoses and deaths from all cancers combined have fallen significantly.

NIH Funding for Disease Categories related to Neural Regeneration Research

The National Institutes of Health （NIH）,a part of the U.S. Department of Health and Human Services,is the nation＇s medical research agency-making important discoveries that improve health and save lives.Thanks in large part to NIH-funded medical research,Americans today are living longer and healthier. Life expectancy in the United States has jumped from 47 years in 1900 to 78 years as reported in 2009,and disability in people over age 65 has dropped dramatically in the past 3 decades. In recent years,nationwide rates of new diagnoses and deaths from all cancers combined have fallen significantly.

Applications of balanced pairs

Let(X, Y) be a balanced pair in an abelian category. We first introduce the notion of cotorsion pairs relative to(X, Y), and then give some equivalent characterizations when a relative cotorsion pair is hereditary or perfect. We prove that if the X-resolution dimension of Y(resp. Y-coresolution dimension of X)is finite, then the bounded homotopy category of Y(resp. X) is contained in that of X(resp. Y). As a consequence, we get that the right X-singularity category coincides with the left Y-singularity category if the X-resolution dimension of Y and the Y-coresolution dimension of X are finite.

Resolving Subcategories of Triangulated Categories and Relative Homological Dimension

We introduce and study （pre）resolving subcategories of a triangulated category and the homological dimension relative to these subcategories. We apply the obtained properties to relative Gorenstein categories.

Relative Quotient Triangulated Categories

Let A be a finite dimensional algebra over a field k. We consider a subfunc- tor F of Ext1A（-, -）, which has enough projectives and injectives such that P（F） is of finite type, where P（F） denotes the set of F-projectives. One can get the relative derived category Db（A） of A-rood. For an F-self-orthogonal module TF, we discuss the relation between the relative quotient triangulated category Db（A）/Kb（addTF） and the relative stable category of the Frobenius category of TF-Cohen-Macaulay modules. In particular, for an F-Gorenstein algebra A and an F-tilting A-module TF, we get a triangle equiva- lence between DbF（A）/Kb（add TF） and the relative stable category of TF-Cohen-Macaulay modules. This gives the relative version of a result of Chen and Zhang.

Categorical resolutions of a class of derived categories

We clarify the relation between the subcategory D_(hf)~b(A) of homological finite objects in D^b(A)and the subcategory K^b(P) of perfect complexes in D^b(A), by giving two classes of abelian categories A with enough projective objects such that D_(hf)~b(A) = K^b(P), and finding an example such that D_(hf)~b(A)≠K^b(P). We realize the bounded derived category D^b(A) as a Verdier quotient of the relative derived category D_C^b(A), where C is an arbitrary resolving contravariantly finite subcategory of A. Using this relative derived categories, we get categorical resolutions of a class of bounded derived categories of module categories of infinite global dimension.We prove that if an Artin algebra A of infinite global dimension has a module T with inj.dimT <∞ such that ~⊥T is finite, then D^b(modA) admits a categorical resolution; and that for a CM(Cohen-Macaulay)-finite Gorenstein algebra, such a categorical resolution is weakly crepant.

Relative Derived Equivalences and Relative Homological Dimensions

Let A be a small abelian category.For a closed subbifunctor F of Ext_A^1（-,-）,Buan has generalized the construction of Verdier’s quotient category to get a relative derived category,where he localized with respect to F-acyclic complexes.In this paper,the homological properties of relative derived categories are discussed,and the relation with derived categories is given.For Artin algebras,using relative derived categories,we give a relative version on derived equivalences induced by F-tilting complexes.We discuss the relationships between relative homological dimensions and relative derived equivalences.