The interleaved multichromatic number of a graph

For $k\ge 1$, we consider interleaved $k$-tuple colorings of the nodes of a graph, that is, assignments of $k$ distinct natural numbers to each node in such a way that nodes that are connected by an edge receive numbers that are strictly alternating between them with respect to the relation $<$. If it takes at least $\chi_{int}^k(G)$ distinct numbers to provide graph $G$ with such a coloring, then the interleaved multichromatic number of $G$ is $\chi_{int}^*(G)=\inf_{k\ge 1}\chi_{int}^k(G)/k$ and is known to be given by a function of the simple cycles of $G$ under acyclic orientations if $G$ is connected [1]. This paper contains a new proof of this result. Unlike the original proof, the new proof makes no assumptions on the connectedness of $G$, nor does it resort to the possible applications of interleaved $k$-tuple colorings and their properties

Probabilistic heuristics for disseminating information in networks

We study the problem of disseminating a piece of information through all the nodes of a network, given that it is known originally only to a single node. In the absence of any structural knowledge on the network other than the nodes' neighborhoods, this problem is traditionally solved by flooding all the network's edges. We analyze a recently introduced probabilistic algorithm for flooding and give an alternative probabilistic heuristic that can lead to some cost-effective improvements, like better trade-offs between the message and time complexities involved. We analyze the two algorithms both mathematically and by means of simulations, always within a random-graph framework and considering relevant node-degree distributions

Local heuristics and the emergence of spanning subgraphs in complex networks

We study the use of local heuristics to determine spanning subgraphs for use in the dissemination of information in complex networks. We introduce two different heuristics and analyze their behavior in giving rise to spanning subgraphs that perform well in terms of allowing every node of the network to be reached, of requiring relatively few messages and small node bandwidth for information dissemination, and also of stretching paths with respect to the underlying network only modestly. We contribute a detailed mathematical analysis of one of the heuristics and provide extensive simulation results on random graphs for both of them. These results indicate that, within certain limits, spanning subgraphs are indeed expected to emerge that perform well in respect to all requirements. We also discuss the spanning subgraphs' inherent resilience to failures and adaptability to topological changes

A novel evolutionary formulation of the maximum independent set problem

We introduce a novel evolutionary formulation of the problem of finding a maximum independent set of a graph. The new formulation is based on the relationship that exists between a graph's independence number and its acyclic orientations. It views such orientations as individuals and evolves them with the aid of evolutionary operators that are very heavily based on the structure of the graph and its acyclic orientations. The resulting heuristic has been tested on some of the Second DIMACS Implementation Challenge benchmark graphs, and has been found to be competitive when compared to several of the other heuristics that have also been tested on those graphs

Topics in Born-Infeld Electrodynamics

Classical version of Born-Infeld electrodynamics is recalled and its most important properties discussed. Then we analyze possible abelian and non-abelian generalizations of this theory, and show how certain soliton-like configurations can be obtained. The relationship with the Standard Model of electroweak interactions is also mentioned.Comment: (One new reference added). 15 pages, LaTeX. To be published in the Proceedings of XXXVII Karpacz Winter School edited in the Proceedings Series of American Mathematical Society, editors J. Lukierski and J. Rembielinsk

Lossy radial diffusion of relativistic Jovian electrons

The radial diffusion equation with synchrotron losses was solved by the Laplace transform method for near-equatorially mirroring relativistic electrons. The evolution of a power law distribution function was found and the characteristics of synchrotron burn-off are stated in terms of explicit parameters for an arbitrary diffusion coefficient. Emissivity from the radiation belts of Jupiter was studied. Asymptotic forms for the distribution in the strong synchrotron loss regime are provided