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### 2013/2014 Seminar Series

Seminar Series 2013/2014

#### Thursday 10th April 2014, 12:00 am

(Seminari de Matemàtica Aplicada, Ed. P4)

Miguel Camelo (Dpt. Arquitectura i Tecnologia de Computadors, Universitat de Girona)

Geometric routing on word-metric spaces

Abstract:

Geometric routing is a strategy to send messages among nodes in a topology by using geometric information of their position in some metric space. For single-path routing, a greedy forwarding technique is used. Greedy forwarding brings the messages closer to the destination in each step using local information only. Each node forwards the message to the neighbor that minimizes the distance to the destination in each step (greedy). Some examples of well-known metric spaces are the Euclidean space, the hyperbolic space and a normed vector space. Cayley graphs, a kind of graphs that arise from algebraic groups, has been studied and proposed as a general kind of graphs to represent the underlying interconnection network of data centers. Due to these graphs arise from group theory, it is possible to use algebraic tools to simplify their study for many applications. These graphs are vertex symmetric, which makes possible to implement the same routing scheme in each node of the network, have hierarchical structure, which allows recursive construction, and have high connectivity, which facilitates fault tolerance. Additionally, these graphs have a metric called the word metric. This metric measures the length of the shortest path in the Cayley graph between two elements of the group. Combining the fact that any Cayley graph is a word metric space, we can use its metric to perform geometric routing on them. The vertices in the graph are labeled with the elements of the group and the distance between vertices can be computed efficiently by using a Shortlex Automatic Structure. This structure, which is a set of automatons that encode the global geometry of the graph, can compute the shortest path between any pair of vertices by reducing their labels to their shortlex equivalent word. Our work is focused in the design of a geometric routing scheme for this kind of graphs and to extend its applicability on general graphs by using graph embedding techniques.

#### Friday 7th March 2014, 10:00 am

(Seminari de Matemàtica Aplicada, Ed. P4)

Esther Barrabés (Universitat de Girona)

Central configurations of twisted rings in the plane

Abstract:

Considerem el problema newtonià de N cossos al pla. Una configuració no és res més que una disposició concreta dels N cossos. I una configuració central (CC) és una disposició dels N cossos tal que la força que actua sobre cada cos (o l'acceleració) apunta cap al centre de masses del conjunt i és proporcional a la distància del cos al centre de masses (i amb la mateixa constant de proporcionalitat per a tots). Per exemple, per a N=2, tota configuració és central; per a N qualsevol i totes les masses iguals, l'N-àgon regular és una configuració central. A partir de les configuracions centrals, es poden construir solucions homogràfiques del problema de N cossos, en particular homotètiques (només escalament), o rígides (d'equilibri relatiu, només amb rotacions). Per exemple, a partir d'una CC col.lineal de 3 cossos i afegint una rotació rígida, es pot obtenir una solució del problema Terra+Lluna+satèl.lit amb el satèl.lit sempre darrera la Lluna (en permanent eclipse). Nosaltres prenem k grups de n cossos (N=kn), de manera que tots els cossos d'un mateix grup tenen la mateixa massa (però cossos de grups diferents poden tenir masses diferents) i estan disposats en els vèrtexs d'un n-àgon regular. Deduïm les equacions de CC en general i passem a estudiar en detall com son les CC en el cas de k=2 grups i n arbitrari. Sorprenentment obtenim que els casos n=2,3 i 4 tenen particularitats que els diferencien entre si, mentre que per a n major o igual a 5, essencialment s'obtenen els mateixos resultats. Per al cas k=3 només hem explorat el problema numèricament donada la complexitat de les equacions.

#### Thursday 14th November 2013, 12:00 am

(Seminari de Matemàtica Aplicada, Ed. P4)

Alejandro Luque (Universitat de Barcelona)

Motion of charged particles in ABC magnetic fields

Abstract:

The analysis of the motion of a charged particle under the action of ABC magnetic fields can be interpreted as a toy model for the motion of plasma charged particles in a tokamak. In this talk we describe some dynamical aspects of this problem by using several tools of dynamical systems. We will try to give an overview of different kinds of phenomena, but we will focus on the existence of quasi-periodic solutions. This is a joint work with Daniel Peralta-Salas.

#### Thursday 10th October 2013, 12:00 am

(Seminari de Matemàtica Aplicada, Ed. P4)

Winfried Just (Ohio University, USA)

Discrete approximations of continuous models

Abstract:

Real-world systems can be usually be modeled in several different frameworks such as systems of differential equations, stochastic processes, or discrete dynamical systems like Boolean networks. If an ODE system D and a Boolean system B supposedly model the same natural system, we would want to know under which conditions the two systems are guaranteed to be consistent, that is, to make qualitatively equivalent predictions. This talk will review some results that have been obtained along these lines by the presenter and his collaborators in [1] and [2] and outline some open problems for future research on this topic.

[1] D. Terman, S. Ahn, X. Wang, and W. Just; Reducing neuronal networks to discrete dynamics, Physica D 237 (2008) 324--338.
[2] W. Just, M. Korb, B. Elbert, and T. Young; Two classes of ODE models with switch-like behavior, Physica D, In press.