Judith R. Goodstein: Einstein’s Italian Mathematicians – Ricci,Levi-Civita,and the Birth of General Relativity

Mathematische Semesterberichte -     

A note on Hadwiger’s conjecture for W5-free graphs with independence number two

The Hadwiger number of a graph $G$, denoted $h\left(G\right)$, is the largest integer $t$ such that $G$ contains $K_t$ as a minor. A famous conjecture due to Hadwiger in 1943 states that for every graph $G$, $h\left(G\right)\ge \chi \left(G\right)$, where $\chi \left(G\right)$ denotes the chromatic number of $G$. Let $\alpha \left(G\right)$ denote the independence number of $G$. A graph is $H$-free if it does not contain the graph $H$ as an induced subgraph. In 2003, Plummer, Stiebitz and Toft proved that $h\left(G\right)\ge \chi \left(G\right)$ for all $H$-free graphs $G$ with $\alpha \left(G\right)\le 2$, where $H$ is any graph on four vertices with $\alpha \left(H\right)\le 2$, $H=C_5$, or $H$ is a particular graph on seven vertices. In 2010, Kriesell subsequently generalized the statement to include all forbidden subgraphs $H$ on five vertices with $\alpha \left(H\right)\le 2$. In this note, we prove that $h\left(G\right)\ge \chi \left(G\right)$ for all $W_5$-free graphs $G$ with $\alpha \left(G\right)\le 2$, where $W_5$ denotes the wheel on six vertices.     

Macrophage Immunomodulation Through New Polymers that Recapitulate Functional Effects of Itaconate as a Power House of Innate Immunity

Itaconate (ITA) is an emerging powerhouse of innate immunity with therapeutic potential that is limited in its ability to be administered in a soluble form. A library of polyester materials that incorporate ITA into polymer backbones resulting in materials with inherent immunoregulatory behavior is developed. Harnessing hydrolytic degradation release from polyester backbones, ITA polymers result in the mechanism specific immunoregulatory properties on macrophage polarization in vitro. In a functional assay, the polymer-released ITA inhibits bacterial growth on acetate. Translation to an in vivo model of biomaterial associated inflammation, intraperitoneal injection of ITA polymers demonstrate a rapid resolution of inflammation in comparison to a control polymer silicone, demonstrating the value of sustained biomimetic presentation of ITA.     

Compactness and convergence rates in the combinatorial integral approximation decomposition

The combinatorial integral approximation decomposition splits the optimization of a discrete-valued control into two steps: solving a continuous relaxation of the discrete control problem, and computing a discrete-valued approximation of the relaxed control. Different algorithms exist for the second step to construct piecewise constant discrete-valued approximants that are defined on given decompositions of the domain. It is known that the resulting discrete controls can be constructed such that they converge to a relaxed control in the \(\hbox {weak}^*\) topology of \(L^\infty \) if the grid constant of this decomposition is driven to zero. We exploit this insight to formulate a general approximation result for optimization problems, which feature discrete and distributed optimization variables, and which are governed by a compact control-to-state operator. We analyze the topology induced by the grid refinements and prove convergence rates of the control vectors for two problem classes. We use a reconstruction problem from signal processing to demonstrate both the applicability of the method outside the scope of differential equations, the predominant case in the literature, and the effectiveness of the approach.

     

Performance of PKI-based security mechanisms in mobile ad hoc networks

Security for ad hoc network environments has received a lot of attention as of today. Previous work has mainly been focussing on secure routing, fairness issues, and malicious node detection. However, the issue of introducing and conserving trust relationships has received considerably less attention. In this article, we present a scalable method for the use of public key certificates and their revocation in mobile ad hoc networks (MANETs). With the LKN-ad hoc security framework (LKN-ASF) a certificate management protocol has been introduced, bringing PKI technology to MANETs. In addition a performance analysis of two different revocation approaches for MANETs will be presented.     

Sur les surfaces lisses de ℙ4

Sans résuméEspace projectif sur un corps algébriquement clos de caractéristique zéro