SEMINARIS / SEMINARS
Curs 2011/2012
- 26 d'octubre de 2011
David Juher
L'arbre mínim per a un període d'entropia zero donat
The minimum tree for a given zero-entropy period
In this talk, we prove a formula which computes, for any given natural number n, the minimum number of endopoints of a tree so that there exists a zero-entropy continuous map defined on it having a period n orbit.
- 23 de novembre (1a part) i 14 de desembre (2a
part) de 2011
Joan Saldaña
Les funcions generatrius de probabilitat i epidèmies en xarxes
Una de les tècniques més utilitzades per a l'anàlisi de procesos en xarxa són les funcions generatrius de probabilitat (FGP). Una de les aplicacions més recents d'aquesta tècnica és l'estudi d'epidèmies en xarxes on neixen i moren nodes (individus). En la xerrada es vol fer una presentació de com s'utilitzen les FGP en uns models senzills d'epidèmies en xarxes i discutir la seva possible aplicació per resoldre problemes oberts en xarxes dinàmiques.
Probability generating functions and epidemics in networks
One of the techniques used to analyze processes on networks are the so-called probability generating functions (PGF). One of the recent applications of this technique is the study of epidemics in networks where nodes (individuals) are created (newborns)and deleted (deaths). In the talk we present on how PGF are used in the analysis of simple epidemic models on networks and discuss their application to solve open problems about epidemic models defined on dynamic networks.
- 1 de febrer de 2012
Jordi Ripoll
Graph Spectra with applications to network models
In this talk we will review some results about Spectra for networks, i.e. basically the set of eigenvalues of the adjacency matrix of the network. We will apply these results to the computation of the basic reproduction number R0 for epidemic network models through the next generation matrix.
- 7 de febrer de 2012
Óscar Angulo
Estudio numérico y analítico de modelos asociados a la hematopoiesis
La hematopoiesis es el proceso por el que se producen y regulan las distintas poblaciones de células sanguíneas. Está basado en una sucesión de mecanismos complejos de diferenciación de las células madre. Estas diferentes diferenciaciones, que ocurren en la médula ósea, están principalmente reguladas por la población total de las células hematopoieticas. Este proceso exhibe a menudo anormalidades in la producción de células sanguíneas, que causan las denominadas enfermedades hematológicas. Una enfermedad hematológica severa es la Leucemia Myelogeneous Crónica (CML), un cáncer de los glóbulos blancos. En algunos casos, ésta exhibe oscilaciones periódicas en las contabilidades de todas las células sanguíneas.
Presentamos un modelo de la dinámica de la hematopoiesis. Analizamos la estabilidad asintótica de los equilibrios y numéricamente ilustramos nuestros resultados y obtenemos la dinámica del modelo que relacionamos con las observaciones de la CML periódica.
- 29 de febrer de 2012
Xavier Jarque (UB)
Sierpinski Julia sets for quadratic rational maps
Sierpinski (1882-1969) discovered, among many other topologically interesting sets, the Sierpisnki carpet. The set is obtained recursively in the following manner. Start with a square of length 1. Remove a central square of length 1/3. The figure can now be naturally divided in 8 squares of length 1/3. Remove the central square of length 1/9 from each one of them. Do the same procedure up to the limit. The set we get is called the Sierpinski carpet. Why is so important? It was proved by Sierpinski that it is a universal plane continuum in the sense that it contains a homeomorphic copy of any planar, one-dimensional, compact and connected set. .
Later, in 1953, Whyburn showed that any planar set that is compact, connected, locally connected, nowhere dense (does not contain open sets), and has the property that any two complementary domains are bounded by disjoint simple closes curves is homeomorphic to the Sierpinski carpet. Sets with this property are known as Sierpinski curves.
The connection of all these previous topological results has a strong relation with complex dynamics because the Julia set of a rational map is always compact and nowhere dense (unless is the whole sphere). Sufficient (and necessary) conditions to get the connectivity and local connectivity are known. So, a natural question is to find out when you can guarantee that the Julia set of a rational map is a Sierpinski curve.
I wil present some examples given in the literature and some new results we have found in the family of quadratic rational maps.
- 28 de març de 2012
Víctor Mañosa (UPC)
Birational maps on elliptic curves: blending dynamics and algebraic geometry
Birational planar maps possessing a rational first integral, preserve a foliation of the plane given by algebraic curves. We will review some results which state that the most typical situation is that this algebraic foliation will be given either by conics and straight lines or by elliptic curves. In the last case we will see some nice results showing that the group structure of the elliptic foliation characterizes the dynamics of any birational map preserving it. All these results are classical and well-known in algebraic geometry, see [2] and [3]. This will be the main core of the talk, and it is aimed to be expository and addressed to a general audience.
To exemplify the above stuff we will see how it works on the Lyness map F(x,y)=(y,(a+y)/x). This map preserves an algebraic foliation given by curves which are, generically, elliptic. We will see how on each of these elliptic curves the map is an affine action in terms of the group structure of the curve. In fact, we will see that the Lyness’ one is a universal family of elliptic curves.
Finally we will review the group structure of rational elliptic curves and its relation with the existence of rational periodic orbits; we will do a brief digression on numerical experiments; and will give a negative answer to a conjecture of Zeeman (and an open problem of Bastien an Rogalski) about the existence of rational 9-periodic orbits of the Lyness map, see [1,4] and [5].
[1] G. Bastien, M. Rogalski, Global behavior of the solutions of Lyness' difference equation u_{n+2}u_n=u_{n+1}+a, J. Difference Equations and Appl. 10 (2004), 977-1003.
[2] J. Duistermaat. Discrete Integrable Systems. QRT Maps and Elliptic Surfaces. Springer-Verlag, 2010.
[3] D. Jogia. J.A.G. Roberts, F. Vivaldi, An algebraic geometric approach to integrable maps of the plane. Journal of Physics A, 39 (2006), 1133--1149.
[4] A. Gasull, V. Mañosa, X. Xarles. Rational Periodic Sequences for the Lyness Equation. Discrete and Continuous Dynamical Systems -series A. 32 (2012), 587-604.
[5] E.C. Zeeman. Geometric unfolding of a difference equation, Preprint Hertford College, Oxford (1996). Unpublished.
- 25 d'abril de 2012
Pau Roldan (UPC)
Difusió al llarg de resonàncies al problema restringit de tres cossos
És ben conegut pels astrònoms que, al Cinturó d'Asteroides situat entre les òrbites de Mart i Júpiter, la distribució dels asteroides presenta els nomenats "forats de Kirkwood" (http://en.wikipedia.org/wiki/Kirkwood_gap). Aquests forats coincideixen exactament amb les ressonàncies orbitals entre Asteroide i Júpiter.
Nosaltres demostrem l'existència d'òrbites de difusió per a l'Asteroide que mostren un canvi dràstic en la seva excentricitat, mentre que el seu semieix major es manté casi constant. Per tant, el nostre mecanisme dóna una justificació rigorosa dels forats de Kirkwood. Aquest treball és una col·laboració amb el Jacques Féjoz (Université Paris-Dauphine and Observatoire de Paris), el Marcel Guàrdia (University of Maryland at College Park), i el Vadim Kaloshin (University of Maryland at College Park). Per a més informació, veure el "preprint" http://arxiv.org/abs/1109.2892
- 20 de juny de 2012
M.C. Ricardo Meraz Sánchez (Universidad Nacional Autónoma de México)
Estado de la pesquería de camarón café, Farfantepenaeus californiensis (Holmes, 1900) en el suroeste del golfo de California evaluando el error en el proceso de evaluación.
El camarón café es la especie de camarón mas abundante en el Pacifico mexicano. En el suroeste del golfo de california existe la flota camaronera más grande de México, con más de 700 barcos que capturan anualmente cerca de 7,000 toneladas. El estado de la pesquería de camarón café fue evaluado utilizando datos comerciales de captura y esfuerzo del suroeste del golfo de California de 1995-2011. Un modelo dinámico de biomasa de Schaefer y técnicas de re-muestreos fueron utilizadas para analizar la captura por unidad de esfuerzo de la pesquería. Dos hipótesis en el proceso de evaluación fueron examinada: Error de observación y de proceso. El estimador de error de observación tuvo un mejor ajuste a los datos que el estimador de error de proceso. Este resultado puede deberse a que en actualmente es imposible discriminar el esfuerzo de pesca para las diferentes especies de camarón, cambios en el poder de pesca por la mejora de las embarcaciones y el sub-reporte que pudiera haber de las capturas. La pesquería de camarón café en el suroeste del golfo de california presento síntomas de sobrexplotación. Sin embargo, las capturas reportadas en los últimos años pareciera que el recurso presenta una rápida recuperación de la población.
Curs 2010/2011
- 18 octubre 2010
Marta Pellicer
Modelització de la fase inflamatòria en la cicatrització de ferides
Modelling the inflamatory phase in wound healing
Wound healing is an extremely complicated process and still not fully understood, moreover when diabetis mellitus is present. The first phase of this process, the inflammatory phase, is where there exists a major difference between diabetic and nondiabetic wound healing. This work (in progress) is related with the modeling and analysis of the dynamics of some of the main agents involved in this first phase. We propose a reaction-difusion system as a model, that aims at generalizing the previous existing approach of J.Sherratt and H.Waugh, where an ODE system was proposed as a first approach for this situation.
Joint work with N. Cónsul (U. Politècnica de Catalunya) and S.M. Oliva (U. de Sao Paulo, Brazil).
- 15 de novembre de 2010
Maria Aguareles
Estructura de solucions amb defectes topològics al pla
Structure of topological defect solutions in the plane
Una de les tècniques més utilitzades per a l'anàlisi de procesos en xarxa són les funcions generatrius de probabilitat (FGP). Una de les aplicacions més recents d'aquesta tècnica és l'estudi d'epidèmies en xarxes on neixen i moren nodes (individus). En la xerrada es vol fer una presentació de com s'utilitzen les FGP en uns models senzills d'epidèmies en xarxes i discutir la seva possible aplicació per resoldre problemes oberts en xarxes dinàmiques. In this talk we will deal with some non-linear partial differential equations of the form -Laplace(u) = F(u) that have solutions with a non-vanishing degree or winding number. These type of solutions arise in many physical contexts like for instance in superconductivity or nematic liquid crystals where they are usually known as vortices or topological defects. In particular we will focus on solutions in the whole plane with a single zero, in which case the partial differential equation reduces to an ordinary differential equation by means of a radial symmetry of the form u=f(r)e^{in\phi}, being n the degree of the solution. We will henceforth work with this ordinary differential equation on f(r) and we will prove existence of these solutions using a rather tricky fixed point theorem while uniqueness will be proved using a so-called sliding method that we will briefly explain. We will also show that with our derivation one also gets monotonicity and some other properties at the origin and at infinity, such as for instance the fact that the derivatives of the solution at the origin become exponentially small as the degree increases.
Joint work with I. Baldomà (U. Politècnica de Catalunya).
- 20 de desembre de 2010
Esther Barrabés
Un cas límit pel problema de l'anell planetari de Maxwell
A limit cas of the "Ring Problem"
We study the dynamics of an extremely idealized model of a planetary ring. In particular, we study the motion of an infinitesimal particle moving under the gravitational influence of a large central body and a regular n-gon of smaller bodies as n tends to infinity. Our goal is to gain insight into the structure of thin, isolated rings.
Joint work with J.M. Cors (U. Politècnica de Catalunya) and G.R.Hall (Boston University)
- 22 de febrer de 2011
Albert Avinyó
Obtenció de funcions que equalitzen una densitat donada mitjançant el mètode del transport de la calor
Heat transport method to construct density-equalizing maps
The problem of producing a transformation of a domain into itself such that its jacobian determinant is given at each point is an interesting geometrical problem with some applications in others areas of science and technology. When this transformation is applied to geographical maps in order to make each country or region to have an area proportional to a given quantity, like the number of inhabitants, the gross domestic product or others, the output is usually called a Geographical Cartogram.
In this talk we analyze mathematically the heat transport method to produce these transformations, but in some particular cases where the initial data has line or angle discontinuities in the plane. In this situation, the conclusion reinforces the conjecture that the algorithm is always well-posed, in accordance with the experience of the extensive numerical uses of it in the last years.
Joint work with J. Solà-Morales and M. València of Universitat Politècnica de Catalunya
- 22 de març de 2011
David Juher
Maximitzant entropia de cicles en arbres
Maximizing entropy of cycles on trees
In this talk we give a partial characterization of the periodic tree patterns of maximum entropy. More precisely, let n be a natural number and let K be the maximum of the topological entropies of all n-periodic tree patterns. We prove that each n-periodic pattern with entropy K is irreducible and simplicial, and that all such patterns are maximodal -in the sense that its monotone representative has no local homeomorphisms at the points of the invariant set.
Joint work with Ll. Alsedà i F. Mañosas (U. Autònoma de Barcelona)
- 12 d'abril de 2011
Laura Garcia
Evitant collisions entre satèl.lits al voltant de la Terra
Avoiding collisions between satellites around the Earth
Whether talking about formations of spacecraft around Earth, or one spacecraft which can collide with an obstacle (such as space junk), we must find a rapid methodology to identify collisions and find a new trajectory which spend as little fuel as possible and avoids collisions between spacecraft. The methodology presented is a combination of a methodology based on finite elements (which gives better results in terms of fuel consumption) and an analytical methodology (which is much faster in terms of computing time).
Joint work with Luke Sauter and Phil Palmer from Surrey Space Centre
- 17 de maig de 2011
Sara Costa
Coexistència d'un conjunt innombrable de conjunts atractors en el cilindre
Coexistence of uncountable many attracting sets on the cylinder
We work
with two-dimensional skew-products defined on the cylinder from two
functions f and p sucht that f is a continuous degree-one
circle
map, and p satisfies some properties of concavity and
monotonicity. We prove that when f has no periodic
points
(rotation interval is a singleton), there exists finitely many
attracting sets. In contrast, when the rotation interval of any of the
lifting of f is non-degenerate, we prove the existence of
uncountably many attracting sets, each of them related with an
irrational rotation number.
Joint work wiht Ll. Alsedà from UAB.
Joint work wiht Ll. Alsedà from UAB.