非紧性测度的零化度和完全性

Banach空间的非紧性测度(measure of noncompactness,MNC)μ,按照它的零点集kerμ构成的超空间划分成三大类:(1)完全MNC(kerμ恰好是X的所有非空相对紧集构成的超空间K,它蕴涵了在MNC的应用中极为重要的广义Cantor交的性质);(2)不完全MNC,也称为MNC(kerμ是K的一个非空子集,并满足广义Cantor交性质);(3)广义MNC(K是kerμ的一个非空子集,它包括了非弱紧性测度等广义非紧性测度).不要求广义Cantor交性质的MNC称为准MNC,而验证这个性质是否成立往往在技术上又是极为困难的.长期以来,人们不知道这条“额外”的性质是否独立于MNC定义中的其他条件.另一个长期令人困惑的基本问题是,一个MNC是否能够控制一个完全MNC?本文主要研究这两个问题.其结果是通过建立Banach空间准MNC和广义MNC的表示定理,证明广义Cantor交假设是独立的,同时给出MNC能够控制一个完全MNC的特征.

Representation of measures of noncompactness and its applications related to an initial value problem in Banach spaces

This paper is devoted to studying the representation of measures of non-generalized compactness,in particular,measures of noncompactness,of non-weak compactness and of non-super weak compactness,defined on Banach spaces and its applications.With the aid of a three-time order-preserving embedding theorem,we show that for every Banach space X,there exist a Banach function space C(K)for some compact Hausdorff space K and an order-preserving affine mapping T from the super space B of all the nonempty bounded subsets of X endowed with the Hausdorff metric to the positive cone C(K)^(+) of C(K),such that for every convex measure,in particular,the regular measure,the homogeneous measure and the sublinear measure of non-generalized compactnessμon X,there is a convex function F on the cone V=T(B)which is Lipschitzian on each bounded set of V such that F(T(B))=μ(B),■B∈B.As its applications,we show a class of basic integral inequalities related to an initial value problem in Banach spaces,and prove a solvability result of the initial value problem,which is an extension of some classical results due to Bana′s and Goebel(1980),Goebel and Rzymowski(1970)and Rzymowski(1971).

Banach空间的保度量(等距)映射

Banach空间的保度量(等距)映射研究自从Mazur-Ulam定理(1932)(Banach空间之间的保度量满射一定是仿射映射)开始,已经进行了80多年.本文主要对Banach空间上的保度量映射、或者等距映射及其推广形式—扰动等距和粗等距的研究历史进行回顾,同时也包含一些新的结果.本文重点放在近十年来的研究进展,以及目前该领域所关注的问题介绍.

Every Banach space admits a homogenous measure of non-compactness not equivalent to the Hausdorff measure

In this paper, we show that every infinite dimensional Banach space admits a homogenous measure of non-compactness not equivalent to the Hausdorff measure. Therefore, it resolves a long-standing question.

Erratum to: Every Banach space admits a homogenous measure of non-compactness not equivalent to the Hausdorff measure

[1, Theorem 4.4] states that every infinite dimensional Banach space admits a homogenous measure of noncompactness not equivalent to the Hausdorff measure. Howevere, there is a gap in the proof. In fact,we found that [1, Lemma 4.3] is not true. In this erratum, we give a corrected proof of [1, Theorem 4.4].

On order-preserving and order-reversing mappings defined on cones of convex functions

In this paper,we first show that for a Banach space X,there is a fully order-reversing mapping T from conv(X)(the cone of all the extended real-valued lower semicontinuous proper convex functions defined on X)onto itself if and only if X is reflexive and linearly isomorphic to its dual X^(*).Then we further prove the following generalized Artstein-Avidan-Milman representation theorem:For every fully order-reversing mapping T:conv(X)→conv(X),there exist a linear isomorphism U:X→X^(*),x_(0)^(*),φ_(0)∈X^(*),α>0 and r_0∈R so that(Tf)(x)=α(Ff)(Ux+x_(0)^(*))+<φ_(0),x>+r_(0),■x∈X where T:conv(X)→conv(X^(*))is the Fenchel transform.Hence,these resolve two open questions.We also show several representation theorems of fully order-preserving mappings defined on certain cones of convex functions.For example,for every fully order-preserving mapping S:semn(X)→semn(X),there is a linear isomorphism U:X→X so that(Sf)(x)=f(Ux),■f∈semn(X),x∈X where semn(X)is the cone of all the lower semicontinuous seminorms on X.

GNL-Scorer: a generalized model for predicting CRISPR on-target activity by machine learning and featurization

CRISPR/Cas9 is an adaptive immunity system in bacteria and most archaea(Koonin and Makarova,2009;Horvath and Barrangou,2010).The CRISPR/Cas9 gene editing system is comprised of two key components,a small guide RNA(gRNA)and a Cas9 endonuclease(Deltcheva etal.,2011;Jineketal.,2012).