Modelos dinámicos en finanzas

El problema de la modelización dinámica enfinanzas tiene mucho que ver con el tipo deproblema que se pretende estudiar. Es preciso teneren cuenta el subyacente así como las magnitudesque se pretende estimar para elegir el modeloadecuado.-

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 (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.

Optimized back-focal-plane interferometry directly measures forces of optically trapped particles

Back-focal-plane interferometry is used to measure displacements of optically trapped samples with very high spatial and temporal resolution. However, the technique is closely related to a method that measures the rate of change in light momentum. It has long been known that displacements of the interference pattern at the back focal plane may be used to track the optical force directly, provided that a considerable fraction of the light is effectively monitored. Nonetheless, the practical application of this idea has been limited to counter-propagating, low-aperture beams where the accurate momentum measurements are possible. Here, we experimentally show that the connection can be extended to single-beam optical traps. In particular, we show that, in a gradient trap, the calibration product κ·β (where κ is the trap stiffness and 1/β is the position sensitivity) corresponds to the factor that converts detector signals into momentum changes; this factor is uniquely determined by three construction features of the detection instrument and does not depend, therefore, on the specific conditions of the experiment. Then, we find that force measurements obtained from back-focal-plane displacements are in practice not restricted to a linear relationship with position and hence they can be extended outside that regime. Finally, and more importantly, we show that these properties are still recognizable even when the system is not fully optimized for light collection. These results should enable a more general use of back-focal-plane interferometry whenever the ultimate goal is the measurement of the forces exerted by an optical trap.

Optimized back-focal-plane interferometry directly measures forces of optically trapped particles

Back-focal-plane interferometry is used to measure displacements of optically trapped samples with very high spatial and temporal resolution. However, the technique is closely related to a method that measures the rate of change in light momentum. It has long been known that displacements of the interference pattern at the back focal plane may be used to track the optical force directly, provided that a considerable fraction of the light is effectively monitored. Nonetheless, the practical application of this idea has been limited to counter-propagating, low-aperture beams where the accurate momentum measurements are possible. Here, we experimentally show that the connection can be extended to single-beam optical traps. In particular, we show that, in a gradient trap, the calibration product κ·β (where κ is the trap stiffness and 1/β is the position sensitivity) corresponds to the factor that converts detector signals into momentum changes; this factor is uniquely determined by three construction features of the detection instrument and does not depend, therefore, on the specific conditions of the experiment. Then, we find that force measurements obtained from back-focal-plane displacements are in practice not restricted to a linear relationship with position and hence they can be extended outside that regime. Finally, and more importantly, we show that these properties are still recognizable even when the system is not fully optimized for light collection. These results should enable a more general use of back-focal-plane interferometry whenever the ultimate goal is the measurement of the forces exerted by an optical trap.

Notas del primer seminario de integrabilidad de la Universitat Politècnica de Catalunya

Estas notas corresponden a las exposiciones presentadas en el \emph{Primer Seminario de Integrabilidad}, dentro de lo que se denomina \emph{Aula de Sistemas Din\'amicos}. Durante este evento se realizaron seis conferencias, todas presentadas por miembros del grupo de Sistemas Din\'amicos de la UPC. El programa desarrollado fue el siguiente:\\\begin{center}AULA DE SISTEMAS DIN\'AMICOS\end{center}\begin{center}\texttt{http://www.ma1.upc.es/recerca/seminaris/aulasd-cat.html}\end{center}\begin{center}SEMINARIO DE INTEGRABILIDAD\end{center}\begin{center}Martes 29 y Mi\'ercoles 30 de marzo de 2005\\Facultad de Matem\'aticas y Estad\'{\i}stica, UPC\\Aula: Seminario 1\end{center}\bigskip\begin{center}PROGRAMA Y RES\'UMENES\end{center}{\bf Martes 29 de marzo}\begin{itemize}\item15:30. Juan J. Morales-Ruiz. \emph{El problema de laintegrabilidad en Sistemas Din\'amicos}\medskip {\bf Resumen.} En esta presentaci\'on se pretende dar unaidea de conjunto, pero sin entrar en detalles, sobre las diversasnociones de integrabilidad, asociadas a nombres de matem\'aticostan ilustres como Liouville, Galois-Picard-Vessiot, Lie, Darboux,Kowalevskaya, Painlev\'e, Poincar\'e, Kolchin, Lax, etc. Adem\'astambi\'en mencionaremos la revoluci\'on que supuso en los a\~nossesenta del siglo pasado el descubrimiento de Gardner, Green,Kruskal y Miura sobre un nuevo m\'etodo para resolver en algunoscasos determinadas ecuaciones en derivadas parciales. \medskip\item16:00. David G\'omez-Ullate. \emph{Superintegrabilidad, pares deLax y modelos de $N-$cuerpos en el plano}\medskip{\bf Resumen.} Introduciremos algunas t\'ecnicas cl\'asicas paraconstruir modelos de N-cuerpos integrables, como los pares de Laxo la din\'amica de los ceros de un polinomio. Revisaremos lanoci\'on de integrabilidad Liouville y superintegrabilidad, ydiscutiremos un nuevo m\'etodo debido a F. Calogero para contruirmodelos de N-cuerpos en el plano con muchas \'orbitasperi\'odicas. La exposici\'on se acompa\~nar\'a de animaciones delmovimiento de los cuerpos, y se plantear\'an algunos problemasabiertos.\medskip\item17:00. Pausa\medskip\item17:30. Yuri Fedorov. \emph{An\'alisis de Kovalevskaya--Painlev\'ey Sistemas Algebraicamente Integrables}\medskip{\bf Resumen.} Muchos sistemas integrables poseen una propiedadremarcable: todas sus soluciones son funciones meromorfas deltiempo como una variable compleja. Tal comportamiento, que serefiere como propiedad de Kovalevskaya-Painleve (KP) y que se usafrecuentemente como una ensayo de integrabilidad, no es accidentaly tiene unas ra\'{\i}ces geom\'etricas profundas. En esta charladescribiremos una clase de tales sistemas (conocidos como lossistemas algebraicamente integrables) y subrayaremos suspropiedades geom\'etricas principales que permiten predecir laestructura de las soluciones complejas y adem\'as encontrarlasexpl\'{\i}citamente. Eso lo ilustraremos con algunos sistemas dela mec\'anica cl\'asica. Tambi\'en mencionaremos unasgeneralizaciones \'utiles de la noci\'on de integrabilidadalgebraica y de la propiedad KP.\end{itemize}\medskip{\bf Mi\'ercoles 30 de marzo}\begin{itemize}\item 15:30. Rafael Ram\'{\i}rez-Ros. \emph{El m\'etodo de Poincar\'e}\medskip{\bf Resumen.} Dado un sistema Hamiltoniano aut\'onomo cercano acompletamente integrable Poincar\'e prob\'o que, en general, noexiste ninguna integral primera adicional uniforme en elpar\'ametro de perturbaci\'on salvo el propio Hamiltoniano.Esbozaremos las ideas principales del m\'etodo de prueba ycomentaremos algunas extensiones y generalizaciones.\newpage\item16:30. Chara Pantazi. \emph{El M\'etodo de Darboux}\medskip{\bf Resumen.} Darboux, en 1878, present\'o su m\'etodo paraconstruir integrales primeras de campos vectoriales polinomialesutilizando sus curvas invariantes algebraicas. En estaexposici\'on presentaremos algunas extensiones del m\'etodocl\'asico de Darboux y tambi\'en algunas aplicaciones.\medskip\item17:30. Pausa\medskip\item18:00. Juan J. Morales-Ruiz. \emph{M\'etodos recientes paradetectar la no integrabilidad}\medskip{\bf Resumen.} En 1982 Ziglin utiliza la estructura de laecuaci\'on en variaciones de Poincar\'e (sobre una curva integralparticular) como una herramienta fundamental para detectar la nointegrabilidad de un sistema Hamiltoniano. En esta charla sepretende dar una idea de esta aproximaci\'on a la nointegrabilidad, junto con t\'ecnicas m\'as recientes queinvolucran la teor\'{\i}a de Galois de ecuaciones diferencialeslineales, haciendo \'enfasis en los ejemplos m\'as que en lateor\'{\i}a general. Ilustraremos estos m\'etodos con resultadossobre la no integrabilidad de algunos problemas de $N$ cuerpos enMec\'anica Celeste.\end{itemize}

N=1 SQCD-like theories with N_f massive flavors from AdS/CFT and beta functions

We study new supergravity solutions related to large-N c N=1 supersymmetric gauge field theories with a large number N f of massive flavors. We use a recently proposed framework based on configurations with N c color D5 branes and a distribution of N f flavor D5 branes, governed by a function N f S(r). Although the system admits many solutions, under plausible physical assumptions the relevant solution is uniquely determined for each value of x ≡ N f /N c . In the IR region, the solution smoothly approaches the deformed Maldacena-Núñez solution. In the UV region it approaches a linear dilaton solution. For x < 2 the gauge coupling β g function computed holographically is negative definite, in the UV approaching the NSVZ β function with anomalous dimension γ 0 = −1/2 (approaching − 3/(32π 2)(2N c  − N f )g 3)), and with β g  → −∞ in the IR. For x = 2, β g has a UV fixed point at strong coupling, suggesting the existence of an IR fixed point at a lower value of the coupling. We argue that the solutions with x > 2 describe a"Seiberg dual" picture where N f  − 2N c flips sign.

On the computation of reducible invariant tori on a parallel computer

We present an algorithm for the computation of reducible invariant tori of discrete dynamical systems that is suitable for tori of dimensions larger than 1. It is based on a quadratically convergent scheme that approximates, at the same time, the Fourier series of the torus, its Floquet transformation, and its Floquet matrix. The Floquet matrix describes the linearization of the dynamics around the torus and, hence, its linear stability. The algorithm presents a high degree of parallelism, and the computational effort grows linearly with the number of Fourier modes needed to represent the solution. For these reasons it is a very good option to compute quasi-periodic solutions with several basic frequencies. The paper includes some examples (flows) to show the efficiency of the method in a parallel computer. In these flows we compute invariant tori of dimensions up to 5, by taking suitable sections.

Differential methylation of TCF7L2 promoter in peripheral blood DNA in newly diagnosed, drug-naïve patients with type 2 diabetes

TCF7L2 is the susceptibility gene for Type 2 diabetes (T2D) with the largest effect on disease risk that has been discovered to date. However, the mechanisms by which TCF7L2 contributes to the disease remain largely elusive. In addition, epigenetic mechanisms, such as changes in DNA methylation patterns, might have a role in the pathophysiology of T2D. This study aimed to investigate the differences in terms of DNA methylation profile of TCF7L2 promoter gene between type 2 diabetic patients and age- and Body Mass Index (BMI)- matched controls. We included 93 type 2 diabetic patients that were recently diagnosed for T2D and exclusively on diet (without any pharmacological treatment). DNA was extracted from whole blood and DNA methylation was assessed using the Sequenom EpiTYPER system. Type 2 diabetic patients were more insulin resistant than their matched controls (mean HOMA IR 2.6 vs 1.8 in controls, P<0.001) and had a poorer beta-cell function (mean HOMA B 75.7 vs. 113.6 in controls, P<0.001). Results showed that 59% of the CpGs analyzed in TCF7L2 promoter had significant differences between type 2 diabetic patients and matched controls. In addition, fasting glucose, HOMA-B, HOMA-IR, total cholesterol and LDL-cholesterol correlated with methylation in specific CpG sites of TCF7L2 promoter. After adjustment by age, BMI, gender, physical inactivity, waist circumference, smoking status and diabetes status uniquely fasting glucose, total cholesterol and LDL-cholesterol remained significant. Taken together, newly diagnosed, drug-naïve type 2 diabetic patients display specific epigenetic changes at the TCF7L2 promoter as compared to age- and BMI-matched controls. Methylation in TCF7L2 promoter is further correlated with fasting glucose in peripheral blood DNA, which sheds new light on the role of epigenetic regulation of TCF7L2 in T2D.

Genotypic comparison of Pantoea agglomerans plant and clinical strains

Pantoea agglomerans strains are among the most promising biocontrol agents for avariety of bacterial and fungal plant diseases, particularly fire blight of apple and pear. However, commercial registration of P. agglomerans biocontrol products is hampered because this species is currently listed as a biosafety level 2 (BL2) organism due to clinical reports as an opportunistichuman pathogen. This study compares plant-origin and clinical strains in a search for discriminating genotypic/phenotypic markers using multi-locus phylogenetic analysis and fluorescent amplified fragment length polymorphisms (fAFLP) fingerprinting.Results: Majority of the clinical isolates from culture collections were found to be improperly designated as P. agglomerans after sequence analysis. The frequent taxonomic rearrangements underwent by the Enterobacter agglomerans/Erwinia herbicola complex may be a major problem in assessing clinical associations within P. agglomerans. In the P. agglomerans sensu stricto (in the stricter sense) group, there was no discrete clustering of clinical/biocontrol strains and no marker was identified that was uniquely associated to clinical strains. A putative biocontrol-specific fAFLP marker was identified only in biocontrol strains. The partial ORF located in this band corresponded to an ABC transporter that was found in all P. agglomerans strains. Conclusion: Taxonomic mischaracterization was identified as a major problem with P.agglomerans, and current techniques removed a majority of clinical strains from this species. Although clear discrimination between P. agglomerans plant and clinical strains was not obtained with phylogenetic analysis, a single marker characteristic of biocontrol strains was identified whichmay be of use in strain biosafety determinations. In addition, the lack of Koch's postulate fulfilment, rare retention of clinical strains for subsequent confirmation, and the polymicrobial nature of P. agglomerans clinical reports should be considered in biosafety assessment of beneficial strains in this species

Matemáticas y juego = Mathematics and games

In this article we try to look at the learning of mathematics through games in the first years of schooling. The use of game resources in the class should not be carried out in a uniquely intuitive way but rather in a manner that contains some preliminary reflections such as, what do we understand by games? Why use games as a resource in the Mathematics classroom? And what does its use imply?