Search

Let $\mathcal A$ be a infinite dimensional C*-algebra and $1

When optimization theorists consider optimization problems in infinite dimensional spaces, they need to deal with closed convex subsets(usually cones) which mostly have empty interior. These subsets often prevent optimization theorists from applying powerful techniques to study these optimization problems. In this paper, by nonsupport point, we present generating spaces which are relative to a Banach space and a nonsupport point of its convex closed subset. Then for optimization problems in infinite dimensional spaces, in some general cases, we replace original spaces by generating spaces while containing solutions. Thus this method enable us to apply powerful classical techniques to optimization problems in very general class of infinite dimensional spaces. Based on functional analysis, from classical Banach spaces to separable Banach spaces, from Banach lattice to latticization, we give characterizations of generating spaces and conclude that they are actually linearly isometric to $L_\infty$($\ell _\infty$) or their closed subspaces. Thus continuous linear functional involved in these techniques could be chosen from $L_\infty^*$($\ell_\infty^*$). After that, applications in Penalty principle, Lagrange duality and scalarization function are further studied by this method.

Assume that $X$ is a complex separable infinite dimensional Banach space and $\mathcal{B}(X)$ denotes the Banach algebra of all bounded linear operators from $X$ to itself. In 1970, P.R. Halmos raised ten open problems in Hilbert spaces. The third one is the following: If an intransitive operator $T$ has an inverse, is its inverse also intransitive? This question is closely related to the invariant subspace problem. Ever since Enflo's celebrated counterexample on $\ell_1$ answered the invariant subspace problem in negative, the Banach space setting of the third question of Halmos has become more interesting. In this paper, we give an affirmative answer to this problem under certain spectral conditions. As an application, we show that for an invertible operator $T$ with Dunford's Property ($C$), if $T^{-1}$ is intransitive and there exists a connected component $\Omega$ of $int\sigma(T^{-1})^\land$ which is off the origin such that $\Omega\cap\rho_F(T^{-1})\neq \emptyset$, then $T$ is also intransitive. In the end of the paper, we show that a sufficient and necessary condition for that there exists a bounded linear operator without non-trivial invariant subspaces on the infinite dimensional space $L_1(\Omega,\sum,\mu)$ (resp., $C(K)$, the space of bounded continuous functions on a complete metric space $K$) is that $(\Omega,\sum,\mu)$ is $\sigma$-finite (resp., $K$ is compact)., Comment: 30pages

The purpose of this paper is devoted to studying representation of measures of non generalized compactness, in particular, measures of noncompactness, of non-weak compactness, and of non-super weak compactness, etc, 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 $\mathbb T$ from the super space $\mathscr B$ of all 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, regular measure, homogeneous measure, sublinear measure of non generalized compactness $\mu$ on $X$, there is a convex function $\digamma$ on the cone $V=\mathbb T(\mathscr B)$ which is Lipschitzian on each bounded set of $V$ such that \[\digamma(\mathbb T(B))=\mu(B),\;\;\forall\;B\in\mathscr B.\] 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 Goebel, Rzymowski, and Bana\'{s}.

A long-standing question in the theory of measures of noncompactness is that for the Kuratowski measure of noncompactness $\alpha$ defined on a metric space $M$, and for every bounded subset $B\subset M$, is there a countable subset $B_0\subset B$ such that $\alpha(B_0)=\alpha(B)$? In this paper, we give an affirmative answer to the question above. It is done by showing that for each nonempty set $B$ of a Banach space, there is a countable subset $B_0\subset B$ so that $B$ is strongly finitely representable in $B_0$, and that there is a free ultrafilter $\mathcal U$ so that $B$ is affinely isometric to a subset of the ultrapower $[{\rm co}(B_0)]_\mathcal U$ of ${\rm co}(B_0)$.

High-throughput sequencing can detect tens of thousands of genes in parallel, providing opportunities for improving the diagnostic accuracy of multiple diseases including sepsis, which is an aggressive inflammatory response to infection that can cause organ failure and death. Early screening of sepsis is essential in clinic, but no effective diagnostic biomarkers are available yet. Here, we present a novel method, Recurrent Logistic Regression, to identify diagnostic biomarkers for sepsis from the blood transcriptome data. A panel including five immune-related genes, LRRN3, IL2RB, FCER1A, TLR5, and S100A12, are determined as diagnostic biomarkers (LIFTS) for sepsis. LIFTS discriminates patients with sepsis from normal controls in high accuracy (AUROC = 0.9959 on average; IC = [0.9722-1.0]) on nine validation cohorts across three independent platforms, which outperforms existing markers. Our analysis determined an accurate prediction model and reproducible transcriptome biomarkers that can lay a foundation for clinical diagnostic tests and biological mechanistic studies.

In this paper, we first show that for a Banach space $X$ there is a fully order reversing mapping $T$ from ${\rm conv}(X)$ (the cone of all 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 $T:{\rm conv}(X)\rightarrow {\rm conv}(X)$ there exist a linear isomorphism $U:X\rightarrow X^*$, $x_0^*, \;\varphi_0\in X^*$, $\alpha>0$ and $r_0\in\mathbb R$ so that \begin{equation}\nonumber (Tf)(x)=\alpha(\mathcal Ff)(Ux+x^*_0)+\langle\varphi_0,x\rangle+r_0,\;\;\forall x\in X, \end{equation} where $\mathcal F: {\rm conv}(X)\rightarrow {\rm conv}(X^*)$ 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 $S:{\rm semn}(X)\rightarrow {\rm semn}(X)$ there is a linear isomorphism $U:X\rightarrow X$ so that \begin{equation}\nonumber (Sf)(x)=f(Ux),\;\;\forall f\in{\rm semn}(X),\;x\in X, \end{equation} where ${\rm semn}(X)$ is the cone of all lower semicontinuous seminorms on $X$.