Generating and Adding Flows on Locally Complete Metric Spaces

As a generalization of a vector field on a manifold, the notion of an arc field on a locally complete metric space was introduced in \cite{BC}. In that paper, the authors proved an analogue of the Cauchy-Lipschitz Theorem i.e they showed the existence and uniqueness of solution curves for a time independent arc field. In this paper, we extend the result to the time dependent case, namely we show the existence and uniqueness of solution curves for a time dependent arc field. We also introduce the notion of the sum of two time dependent arc fields and show existence and uniqueness of solution curves for this sum.Comment: 29 pages,6 figure

Heavy Viable Trajectories of Controlled Systems

We define and study the concept of heavy viable trajectories of a controlled system with feedbacks. Viable trajectories are trajectories satisfying at each instant given constraints on the state. The controls regulating viable trajectories evolve according a set-valued feedback map. Heavy viable trajectories are the ones which are associated to the controls in the feedback map whose velocity has at each instant the minimal norm. We construct the differential equation governing the evolution of the controls associated to heavy viable trajectories and we prove their existence

An additive subfamily of enlargements of a maximally monotone operator

We introduce a subfamily of additive enlargements of a maximally monotone operator. Our definition is inspired by the early work of Simon Fitzpatrick. These enlargements constitute a subfamily of the family of enlargements introduced by Svaiter. When the operator under consideration is the subdifferential of a convex lower semicontinuous proper function, we prove that some members of the subfamily are smaller than the classical $\epsilon$-subdifferential enlargement widely used in convex analysis. We also recover the epsilon-subdifferential within the subfamily. Since they are all additive, the enlargements in our subfamily can be seen as structurally closer to the $\epsilon$-subdifferential enlargement

The Regulation of MS-KIF18A Expression and Cross Talk with Estrogen Receptor

This study provides a novel view on the interactions between the MS-KIF18A, a kinesin protein, and estrogen receptor alpha (ERα) which were studied in vivo and in vitro. Additionally, the regulation of MS-KIF18A expression by estrogen was investigated at the gene and protein levels. An association between recombinant proteins; ERα and MS-KIF18A was demonstrated in vitro in a pull down assay. Such interactions were proven also for endogenous proteins in MBA-15 cells were detected prominently in the cytoplasm and are up-regulated by estrogen. Additionally, an association between these proteins and the transcription factor NF-κB was identified. MS-KIF18A mRNA expression was measured in vivo in relation to age and estrogen level in mice and rats models. A decrease in MS-KIF18A mRNA level was measured in old and in OVX-estrogen depleted rats as compared to young animals. The low MS-KIF18A mRNA expression in OVX rats was restored by estrogen treatment. We studied the regulation of MS-KIF18A transcription by estrogen using the luciferase reporter gene and chromatin immuno-percipitation (ChIP) assays. The luciferase reporter gene assay demonstrated an increase in MS-KIF18A promoter activity in response to 10−8 M estrogen and 10−7M ICI-182,780. Complimentary, the ChIP assay quantified the binding of ERα and pcJun to the MS-KIF18A promoter that was enhanced in cells treated by estrogen and ICI-182,780. In addition, cells treated by estrogen expressed higher levels of MS-KIF18A mRNA and protein and the protein turnover in MBA-15 cells was accelerated. Presented data demonstrated that ERα is a defined cargo of MS-KIF18A and added novel insight on the role of estrogen in regulation of MS-KIF18A expression both in vivo and in vitro

About intrinsic transversality of pairs of sets

The article continues the study of the ‘regular’ arrangement of a collection of sets near a point in their intersection. Such regular intersection or, in other words, transversality properties are crucial for the validity of qualification conditions in optimization as well as subdifferential, normal cone and coderivative calculus, and convergence analysis of computational algorithms. One of the main motivations for the development of the transversality theory of collections of sets comes from the convergence analysis of alternating projections for solving feasibility problems. This article targets infinite dimensional extensions of the intrinsic transversality property introduced recently by Drusvyatskiy, Ioffe and Lewis as a sufficient condition for local linear convergence of alternating projections. Several characterizations of this property are established involving new limiting objects defined for pairs of sets. Special attention is given to the convex case