Estudio e integración de una familia de ecuaciones en derivadas parciales de cuarto orden

En una memoria anterior sobre la aplicación de los funcionales abeloides a la resolución de las ecuaciones en derivadas parciales de cuarto orden con coeficientes constantes (1), se hizo la clasificación de las ecuaciones cuyo cono característico posee una generatriz tacnodal. Entre las ecuaciones de tipo totalmente hiperbólico se destacaron dos casos: según que el cono característico correspondiente sea de género O (primer caso), o bien sea de género 1 (segundo caso). En la citada memoria se estudia dicho primer caso, mientras que en ésta trataremos del segundo: con más precisión, se tratará de resolver las ecuaciones en derivadas parciales del tipo indicado cuyo cono característico además de ser de género 1, sea simétrico respecto al plano tangente al cono a lo largo de la generatriz tacnodal. El desarrollo de los cálculos es aquí muy penoso pero se puede evitar mediante consideraciones sintéticas sobre los resultados obtenidos en la menloria anteriormente citada.

Option valuation as an expectation in the complex domain: The Black-Scholes case

It is very well known that the first succesful valuation of a stock option was done by solving a deterministic partial differential equation (PDE) of the parabolic type with some complementary conditions specific for the option. In this approach, the randomness in the option value process is eliminated through a no-arbitrage argument. An alternative approach is to construct a replicating portfolio for the option. From this viewpoint the payoff function for the option is a random process which, under a new probabilistic measure, turns out to be of a special type, a martingale. Accordingly, the value of the replicating portfolio (equivalently, of the option) is calculated as an expectation, with respect to this new measure, of the discounted value of the payoff function. Since the expectation is, by definition, an integral, its calculation can be made simpler by resorting to powerful methods already available in the theory of analytic functions. In this paper we use precisely two of those techniques to find the well-known value of a European call

On the coincidence between the Shimomura's bargaining sets and the core

[cat] En l'article es dona una condició necessària per a que els conjunts de negociació definits per Shimomura (1997) i el nucli d'un joc cooperatiu amb utilitat transferible coincideixin. A tal efecte, s'introdueix el concepte de vectors de màxim pagament. La condició necessària consiteix a verificar que aquests vectors pertanyen al nucli del joc.

The Effect of a threshold proportional reinsurance strategy on ruin probabilities

[spa] En un modelo de Poisson compuesto, definimos una estrategia de reaseguro proporcional de umbral : se aplica un nivel de retención k1 siempre que las reservas sean inferiores a un determinado umbral b, y un nivel de retención k2 en caso contrario. Obtenemos la ecuación íntegro-diferencial para la función Gerber-Shiu, definida en Gerber-Shiu -1998- en este modelo, que nos permite obtener las expresiones de la probabilidad de ruina y de la transformada de Laplace del momento de ruina para distintas distribuciones de la cuantía individual de los siniestros. Finalmente presentamos algunos resultados numéricos.

Hamiltonian formalism for nonlocal Lagrangians

A Hamiltonian formalism is set up for nonlocal Lagrangian systems. The method is based on obtaining an equivalent singular first order Lagrangian, which is processed according to the standard Legendre transformation and then, the resulting Hamiltonian formalism is pulled back onto the phase space defined by the corresponding constraints. Finally, the standard results for local Lagrangians of any order are obtained as a particular case.

Line search multilevel optimization as computational methods for dense optical flow

We evaluate the performance of different optimization techniques developed in the context of optical flow computation with different variational models. In particular, based on truncated Newton methods (TN) that have been an effective approach for large-scale unconstrained optimization, we de- velop the use of efficient multilevel schemes for computing the optical flow. More precisely, we evaluate the performance of a standard unidirectional mul- tilevel algorithm - called multiresolution optimization (MR/OPT), to a bidrec- tional multilevel algorithm - called full multigrid optimization (FMG/OPT). The FMG/OPT algorithm treats the coarse grid correction as an optimiza- tion search direction and eventually scales it using a line search. Experimental results on different image sequences using four models of optical flow com- putation show that the FMG/OPT algorithm outperforms both the TN and MR/OPT algorithms in terms of the computational work and the quality of the optical flow estimation.

Neutralizing antibodies in Multiple Sclerosis a model-based analysis of Interferon beta signaling pathway in macrophages

Multiple Sclerosis is the most common non-traumatic cause of neurologicaldisability in young people. There is no cure yet, and until recently, few long-termtherapies existed. Interferon beta (IFNβ) was the first treatment, and remains the mostcommonly prescribed. One of the most significant problems of IFNβ therapy is theproduction of drug specific antibodies. Up to 45% of patients develop neutralizingantibodies (NAbs) to IFNβ products. The neutralizing antibody binds to the biologicalagent preventing its interaction with its receptor, inhibiting the biological action of theprotein, which abrogates the clinical efficacy of IFNβ treatment. Interferon-betamediates its response by binding to its high affinity cell surface receptor and initiatingthe JAK/STAT signalling cascade. In this project we have analyzed the IFNβ signalingpathway in macrophages when neutralizing antibodies are present. The response tothis pathway after IFNβ stimulation shows a transient oscillatory rhythm of STAT1phosphorylation, which varies as NAbs concentration increases. To improve ourunderstanding of that behavior, we extended an existing mathematical model based onnonlinear ordinary differential equations of JAK/STAT pathway by including IFN-NAbassociation and IFN-activation receptor. Combining our theoretical model withexperimental data we could study the role of neutralizing antibodies on the molecularresponse and determine its lifetime after cytokine stimulation.

Global periodicity conditions for maps and recurrences via Normal Forms

We face the problem of characterizing the periodic cases in parametric families of (real or complex) rational diffeomorphisms having a fixed point. Our approach relies on the Normal Form Theory, to obtain necessary conditions for the existence of a formal linearization of the map, and on the introduction of a suitable rational parametrization of the parameters of the family. Using these tools we can find a finite set of values p for which the map can be p-periodic, reducing the problem of finding the parameters for which the periodic cases appear to simple computations. We apply our results to several two and three dimensional classes of polynomial or rational maps. In particular we find the global periodic cases for several Lyness type recurrences

Integrability and non-integrability of periodic non-autonomous Lyness recurrences

This paper studies non-autonomous Lyness type recurrences of the form x_{n+2}=(a_n+x_n)/x_{n+1}, where a_n is a k-periodic sequence of positive numbers with prime period k. We show that for the cases k in {1,2,3,6} the behavior of the sequence x_n is simple(integrable) while for the remaining cases satisfying k not a multiple of 5 this behavior can be much more complicated(chaotic). The cases k multiple of 5 are studied separately.

Integrability and non-integrability of periodic non-autonomous Lyness recurrences (revised and enlarged version)

This paper studies non-autonomous Lyness type recurrences of the form xn+2 = (an+xn+1)=xn, where fang is a k-periodic sequence of positive numbers with primitive period k. We show that for the cases k 2 f1; 2; 3; 6g the behavior of the sequence fxng is simple (integrable) while for the remaining cases satisfying this behavior can be much more complicated (chaotic). We also show that the cases where k is a multiple of 5 present some di erent features.

Equacions diferencials i models

En tot cas, jo voldria que aquesta conferència fos això que he dit: una breu lliçó sobre la importància de les equacions diferencials. Parlaré d'elles des de el punt de vista del models, és a dir, dels fenòmens que modelitzeu. I intentaré explicar que malgrat el seu origen antic, totes elles segueixen presentant avui en dia problemes nous i interessants, tant des de el punt de vista teòric com pràctic.

Concepciones y creencias de los profesores universitarios de matemáticas acerca de la enseñanza de las ecuaciones diferenciales

La investigación que aquí presentamos es una aproximación a las concepciones y creencias de los profesores universitarios de matemáticas acerca de la enseñanza de las ecuaciones diferenciales en estudios científico-experimentales. A parte de los intentos por caracterizar a cada profesor en términos de sus concepciones y creencias, y de establecer el nivel de coherencia y consistencia de éstas, a partir de los resultados del análisis se explica la persistencia de la utilización de métodos tradicionales de enseñanza. Las diferencias y similitudes entre las concepciones y creencias de cada profesor, y el nivel de coherencia demostrado nos han permitido establecer tres grupos de profesores, a los que hemos denominado I, II y III.