Generalized Differentiation of Parameter-Dependent Sets and Mappings

The paper concerns new aspects of generalized differentiation theory that plays a crucial role in many areas of modern variational analysis, optimization, and their applications. In contrast to the majority of previous developments, we focus here on generalized differentiation of parameter-dependent objects (sets, set-valued mappings, and nonsmooth functions), which naturally appear, e.g., in parametric optimization and related topics. The basic generalized differential constructions needed in this case are different for those known in parameter-independent settings, while they still enjoy comprehensive calculus rules developed in this paper

Metric Regularity of Mappings and Generalized Normals to Set Images

The primary goal of this paper is to study some notions of normals to nonconvex sets in finite-dimensional and infinite-dimensional spaces and their images under single-valued and set-valued mappings. The main motivation for our study comes from variational analysis and optimization, where the problems under consideration play a crucial role in many important aspects of generalized differential calculus and applications. Our major results provide precise equality formulas (sometimes just efficient upper estimates) allowing us to compute generalized normals in various senses to direct and inverse images of nonconvex sets under single-valued and set-valued mappings between Banach spaces. The main tools of our analysis revolve around variational principles and the fundamental concept of metric regularity properly modified in this paper

Room temperature 2D ferromagnetism in few-layered 1TT-CrTe2_{2}

Spin-related electronics using two dimensional (2D) van der Waals (vdW) materials as a platform are believed to hold great promise for revolutionizing the next generation spintronics. Although many emerging new phenomena have been unravelled in 2D electronic systems with spin long-range orderings, the scarcely reported room temperature magnetic vdW material has thus far hindered the related applications. Here, we show that intrinsic ferromagnetically aligned spin polarization can hold up to 316 K in a metallic phase of 1$T$-CrTe$_{2}$ in the few-layer limit. This room temperature 2D long range spin interaction may be beneficial from an itinerant enhancement. Spin transport measurements indicate an in-plane room temperature negative anisotropic magnetoresistance (AMR) in few-layered CrTe$_{2}$, but a sign change in the AMR at lower temperature, with -0.6$\%$ at 300 K and +5$\%$ at 10 K, respectively. This behavior may originate from the specific spin polarized band structure of CrTe$_{2}$. Our findings provide insights into magnetism in few-layered CrTe$_{2}$, suggesting potential for future room temperature spintronic applications of such 2D vdW magnets.Comment: 9 Pages, 4 Figure

Restrictive metric regularity and generalized differential calculus in Banach spaces

We consider nonlinear mappings f:X→Y between Banach spaces and study the notion of restrictive metric regularity of f around some point x¯, that is, metric regularity of f from X into the metric space E=f(X). Some sufficient as well as necessary and sufficient conditions for restrictive metric regularity are obtained, which particularly include an extension of the classical Lyusternik-Graves theorem in the case when f is strictly differentiable at x¯ but its strict derivative ∇f(x¯) is not surjective. We develop applications of the results obtained and some other techniques in variational analysis to generalized differential calculus involving normal cones to nonsmooth and nonconvex sets, coderivatives of set-valued mappings, as well as first-order and second-order subdifferentials of extended real-valued functions

Calculus of directional coderivatives and normal cones in Asplund spaces

We study the directional Mordukhovich normal cones to nonsmooth sets, coderivatives of set-valued mappings in Asplund spaces and establish extensive calculus results on these constructions under various operations of sets and mappings. We also develop calculus of the directional sequential normal compactness both in general Banach spaces and in Asplund spaces

Calculus of directional subdifferentials and coderivatives in Banach spaces

In this work we study the directional versions of Mordukhovich normal cones to nonsmooth sets, coderivatives of set-valued mappings, and subdifferentials of extended-real-valued functions in the framework of general Banach spaces. We establish some characterizations and basic properties of these constructions, and then develop calculus including sum rules and chain rules involving smooth functions. As an application, we also explore the upper estimates of the directional Mordukhovich subdifferentials and singular subdifferentials of marginal functions

Bornological Coderivative and Subdifferential Calculus in Smooth Banach Spaces

In this paper, we study bornological generalized differential properties of sets with nonsmooth boundaries, nonsmooth functions, and set-valued mappings in smooth Banach spaces. We establish a fuzzy intersection rule for bornological normal cones and develop fuzzy calculus for bornological generalized differential constructions as well as exact calculus for the limiting counterparts of these constructions

Potential Mechanisms of Triptolide against Diabetic Cardiomyopathy Based on Network Pharmacology Analysis and Molecular Docking

The incidence of heart failure was significantly increased in patients with diabetic cardiomyopathy (DCM). The therapeutic effect of triptolide on DCM has been reported, but the underlying mechanisms remain to be elucidated. This study is aimed at investigating the potential targets of triptolide as a therapeutic strategy for DCM using a network pharmacology approach. Triptolide and its targets were identified by the Traditional Chinese Medicine Systems Pharmacology database. DCM-associated protein targets were identified using the comparative toxicogenomics database and the GeneCards database. The networks of triptolide-target genes and DCM-associated target genes were created by Cytoscape. The common targets and enriched pathways were identified by the Gene Ontology (GO) and Kyoto Encyclopedia of Genes and Genomes (KEGG) pathway enrichment analyses. The gene-gene interaction network was analyzed by the GeneMANIA database. The drug-target-pathway network was constructed by Cytoscape. Six candidate protein targets were identified in both triptolide target network and DCM-associated network: STAT3, VEGFA, FOS, TNF, TP53, and TGFB1. The gene-gene interaction based on the targets of triptolide in DCM revealed the interaction of these targets. Additionally, five key targets that were linked to more than three genes were determined as crucial genes. The GO analysis identified 10 biological processes, 2 cellular components, and 10 molecular functions. The KEGG analysis identified 10 signaling pathways. The docking analysis showed that triptolide fits in the binding pockets of all six candidate targets. In conclusion, the present study explored the potential targets and signaling pathways of triptolide as a treatment for DCM. These results illustrate the mechanism of action of triptolide as an anti-DCM agent and contribute to a better understanding of triptolide as a transcriptional regulator of cytokine mRNA expression

Algorithms for the Shortest Path Improvement Problems under Unit Hamming Distance

In a shortest path improvement problem under unit Hamming distance (denoted by SPIUH), an edge weighted graph with a set of source-terminal pairs is given; we need to modify the lengths of edges by a minimum cost under unit Hamming distance such that the modified distances of the shortest paths are upper bounded by given values. The SPIUH problem on arborescent network is formulated as a 0-1 integer programming model. Some strongly polynomial time algorithms are designed for the problems on some special arborescent networks. Firstly, two greedy algorithms are proposed for problems on chain networks and special startree networks, respectively. Secondly, a strongly polynomial time algorithm is presented for the problem with a single source and constrained paths. Finally, a heuristic algorithm and its computational experiments are given for the SPIUH problem on general graphs