Sum of a random number of independent random variables

Presentació feta a classe2021/20222n quadrimestr

The Galton-Watson process

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Gaussian random vectors

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Random walks

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Generating and characteristic functions

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Sequences of random variables

Presentació feta a classe2021/20222n quadrimestr

Distance-layer structure of the De Bruijn and Kautz digraphs: analysis and application to deflection routing

This is the peer reviewed version of the following article: Fàbrega, J.; Martí, J.; Muñoz, X. Distance-layer structure of the De Bruijn and Kautz digraphs: analysis and application to deflection routing. "Networks", 29 Juliol 2023, which has been published in final form at https://onlinelibrary.wiley.com/doi/10.1002/net.22177. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions. This article may not be enhanced, enriched or otherwise transformed into a derivative work, without express permission from Wiley or by statutory rights under applicable legislation. Copyright notices must not be removed, obscured or modified. The article must be linked to Wiley’s version of record on Wiley Online Library and any embedding, framing or otherwise making available the article or pages thereof by third parties from platforms, services and websites other than Wiley Online Library must be prohibited.In this article, we present a detailed study of the reach distance-layer structure of the De Bruijn and Kautz digraphs, and we apply our analysis to the performance evaluation of deflection routing in De Bruijn and Kautz networks. Concerning the distance-layer structure, we provide explicit polynomial expressions, in terms of the degree of the digraph, for the cardinalities of some relevant sets of this structure. Regarding the application to defection routing, and as a consequence of our polynomial description of the distance-layer structure, we formulate explicit expressions, in terms of the degree of the digraph, for some probabilities of interest in the analysis of this type of routing. De Bruijn and Kautz digraphs are fundamental examples of digraphs on alphabet and iterated line digraphs. If the topology of the network under consideration corresponds to a digraph of this type, we can perform, in principle, a similar vertex layer description.Partially supported by the Ministerio de Ciencia e Innovación/Agencia Estatal de Investigación, Spain, and the European Regional Development Fund under project PGC2018-095471-B-I00; and by AGAUR from the Catalan Government under project 2017SGR-1087.Peer ReviewedPostprint (author's final draft

Equivalent characterizations of the spectra of graphs and applications to measures of distance-regularity

The spectrum of a graph usually provides a lot of information about its combinatorial structure. Moreover,from the spectrum, the so-called predistance polynomials can be defined, as a generalization, for any graph, of the distancepolynomials of a distance-regular graph. Going further, the preintersection numbers generalize the intersection numbers ofa distance-regular graph. This paper describes, for any graph, the closed relationships between its spectrum, predistancepolynomials, and preintersection numbers. Then, some applications to derive combinatorial properties of the given graph,most of them related to some fundamental characterizations of distance-regularity, are presented. In particular, the so-called‘spectral excess theorem’ is revisited. This result states that a connected regular graph is distance-regular if and only if itsspectral excess, which is a value computed from the spectrum, equals the average excess, that is, the mean of the numbers ofvertices at maximum distance from every vertex.Peer ReviewedPostprint (author's final draft

Enhancing dynamic student learning by teamwork in innovative projects at an Erasmus Mundus master subject adapted to the EHEA (EEES)

The EHEA (EEES in Spain) is creating common academic degree standards comparable and compatible between the European countries. The definition of the new Credit unit, the ECTS, focused on the hours of study devoted by the students, is also promoting the renovation of the educational process at the higher studies of Graduate Degree and Master. We describe the experience of innovation of the teaching practice of the “Fibres & Telecommunications” subject of the Master PhotonicsBCN (part of the Erasmus Mundus Master EUROPHOTONICS), by introducing team9 working techniques for promoting: the development of cooperative learning, interpersonal skills, and positive interdependence among the students, not forgetting the individual initiative and responsibility. The teamwork is oriented to small innovation projects (inspired in state9of9the9art research challenges faced by the European FP7 EURO9FOS Network9 of 9Excellence) promoting interactive and dynamic student learning process. Each team member is spontaneously encouraged to learn as much as possible on both: the area of his task inside the common project, and also the influence of his task in the over all results of the project. This is implemented by the use of commercial software for professional systems deployment (VPIphotonics TM ), which allows each team to check the real viability of their innovative solutionPostprint (published version

Further topics in connectivity

Continuing the study of connectivity, initiated in §4.1 of the Handbook, we survey here some (sufficient) conditions under which a graph or digraph has a given connectivity or edge-connectivity. First, we describe results concerning maximal (vertex- or edge-) connectivity. Next, we deal with conditions for having (usually lower) bounds for the connectivity parameters. Finally, some other general connectivity measures, such as one instance of the so-called “conditional connectivity,” are considered. For unexplained terminology concerning connectivity, see §4.1.Peer ReviewedPostprint (published version