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}

A Topological Approach to Fuzzy Sets

This thesis presents a topological approach to studying fuzzy setsby means of modifier operators. Modifier operators are mathematical models, e.g., for hedges, and we present briefly different approaches to studying modifier operators. We are interested in compositional modifier operators, modifiers for short, and these modifiers depend on binary relations. We show that if a modifier depends on a reflexive and transitive binary relation on U, then there exists a unique topology on U such that this modifier is the closure operator in that topology. Also, if U is finite then there exists a lattice isomorphism between the class of all reflexive and transitive relations and the class of all topologies on U. We define topological similarity relation "≈" between L-fuzzy sets in an universe U, and show that the class LU/ ≈ is isomorphic with the class of all topologies on U, if U is finite and L is suitable. We consider finite bitopological spaces as approximation spaces, and we show that lower and upper approximations can be computed by means of α-level sets also in the case of equivalence relations. This means that approximations in the sense of Rough Set Theory can be computed by means of α-level sets. Finally, we present and application to data analysis: we study an approach to detecting dependencies of attributes in data base-like systems, called information systems.

Indirect likelihood inference (revised)

Standard indirect Inference (II) estimators take a given finite-dimensional statistic, Z_{n} , and then estimate the parameters by matching the sample statistic with the model-implied population moment. We here propose a novel estimation method that utilizes all available information contained in the distribution of Z_{n} , not just its first moment. This is done by computing the likelihood of Z_{n}, and then estimating the parameters by either maximizing the likelihood or computing the posterior mean for a given prior of the parameters. These are referred to as the maximum indirect likelihood (MIL) and Bayesian Indirect Likelihood (BIL) estimators, respectively. We show that the IL estimators are first-order equivalent to the corresponding moment-based II estimator that employs the optimal weighting matrix. However, due to higher-order features of Z_{n} , the IL estimators are higher order efficient relative to the standard II estimator. The likelihood of Z_{n} will in general be unknown and so simulated versions of IL estimators are developed. Monte Carlo results for a structural auction model and a DSGE model show that the proposed estimators indeed have attractive finite sample properties.

Quasiconformal mappings and inequalities involving special functions

This PhD thesis in Mathematics belongs to the field of Geometric Function Theory. The thesis consists of four original papers. The topic studied deals with quasiconformal mappings and their distortion theory in Euclidean n-dimensional spaces. This theory has its roots in the pioneering papers of F. W. Gehring and J. Väisälä published in the early 1960’s and it has been studied by many mathematicians thereafter. In the first paper we refine the known bounds for the so-called Mori constant and also estimate the distortion in the hyperbolic metric. The second paper deals with radial functions which are simple examples of quasiconformal mappings. These radial functions lead us to the study of the so-called p-angular distance which has been studied recently e.g. by L. Maligranda and S. Dragomir. In the third paper we study a class of functions of a real variable studied by P. Lindqvist in an influential paper. This leads one to study parametrized analogues of classical trigonometric and hyperbolic functions which for the parameter value p = 2 coincide with the classical functions. Gaussian hypergeometric functions have an important role in the study of these special functions. Several new inequalities and identities involving p-analogues of these functions are also given. In the fourth paper we study the generalized complete elliptic integrals, modular functions and some related functions. We find the upper and lower bounds of these functions, and those bounds are given in a simple form. This theory has a long history which goes back two centuries and includes names such as A. M. Legendre, C. Jacobi, C. F. Gauss. Modular functions also occur in the study of quasiconformal mappings. Conformal invariants, such as the modulus of a curve family, are often applied in quasiconformal mapping theory. The invariants can be sometimes expressed in terms of special conformal mappings. This fact explains why special functions often occur in this theory.

Essais en économetrie et économie de l'éducation

Cette thèse est organisée en trois chapitres. Les deux premiers s'intéressent à l'évaluation, par des méthodes d'estimations, de l'effet causal ou de l'effet d'un traitement, dans un environnement riche en données. Le dernier chapitre se rapporte à l'économie de l'éducation. Plus précisément dans ce chapitre j'évalue l'effet de la spécialisation au secondaire sur le choix de filière à l'université et la performance. Dans le premier chapitre, j'étudie l'estimation efficace d'un paramètre de dimension finie dans un modèle linéaire où le nombre d'instruments peut être très grand ou infini. L'utilisation d'un grand nombre de conditions de moments améliore l'efficacité asymptotique des estimateurs par variables instrumentales, mais accroit le biais. Je propose une version régularisée de l'estimateur LIML basée sur trois méthodes de régularisations différentes, Tikhonov, Landweber Fridman, et composantes principales, qui réduisent le biais. Le deuxième chapitre étend les travaux précédents, en permettant la présence d'un grand nombre d'instruments faibles. Le problème des instruments faibles est la consequence d'un très faible paramètre de concentration. Afin d'augmenter la taille du paramètre de concentration, je propose d'augmenter le nombre d'instruments. Je montre par la suite que les estimateurs 2SLS et LIML régularisés sont convergents et asymptotiquement normaux. Le troisième chapitre de cette thèse analyse l'effet de la spécialisation au secondaire sur le choix de filière à l'université. En utilisant des données américaines, j'évalue la relation entre la performance à l'université et les différents types de cours suivis pendant les études secondaires. Les résultats suggèrent que les étudiants choisissent les filières dans lesquelles ils ont acquis plus de compétences au secondaire. Cependant, on a une relation en U entre la diversification et la performance à l'université, suggérant une tension entre la spécialisation et la diversification. Le compromis sous-jacent est évalué par l'estimation d'un modèle structurel de l'acquisition du capital humain au secondaire et de choix de filière. Des analyses contrefactuelles impliquent qu'un cours de plus en matière quantitative augmente les inscriptions dans les filières scientifiques et technologiques de 4 points de pourcentage.

Good-visibility computation using graphics hardware

Aquesta tesi tracta del disseny, implementació i discussió d'algoritmes per resoldre problemes de visibilitat i bona-visibilitat utilitzant el hardware gràfic de l'ordinador. Concretament, s'obté una discretització dels mapes de multi-visibilitat i bona-visibilitat a partir d'un conjunt d'objectes de visió i un conjunt d'obstacles. Aquests algoritmes són útils tant per fer càlculs en dues dimensions com en tres dimensions. Fins i tot ens permeten calcular-los sobre terrenys.

Space partitioning for scalable K-means

K-Means is a popular clustering algorithm which adopts an iterative refinement procedure to determine data partitions and to compute their associated centres of mass, called centroids. The straightforward implementation of the algorithm is often referred to as `brute force' since it computes a proximity measure from each data point to each centroid at every iteration of the K-Means process. Efficient implementations of the K-Means algorithm have been predominantly based on multi-dimensional binary search trees (KD-Trees). A combination of an efficient data structure and geometrical constraints allow to reduce the number of distance computations required at each iteration. In this work we present a general space partitioning approach for improving the efficiency and the scalability of the K-Means algorithm. We propose to adopt approximate hierarchical clustering methods to generate binary space partitioning trees in contrast to KD-Trees. In the experimental analysis, we have tested the performance of the proposed Binary Space Partitioning K-Means (BSP-KM) when a divisive clustering algorithm is used. We have carried out extensive experimental tests to compare the proposed approach to the one based on KD-Trees (KD-KM) in a wide range of the parameters space. BSP-KM is more scalable than KDKM, while keeping the deterministic nature of the `brute force' algorithm. In particular, the proposed space partitioning approach has shown to overcome the well-known limitation of KD-Trees in high-dimensional spaces and can also be adopted to improve the efficiency of other algorithms in which KD-Trees have been used.

Linear-wavelet networks

This paper proposes a nonlinear regression structure comprising a wavelet network and a linear term. The introduction of the linear term is aimed at providing a more parsimonious interpolation in high-dimensional spaces when the modelling samples are sparse. A constructive procedure for building such structures, termed linear-wavelet networks, is described. For illustration, the proposed procedure is employed in the framework of dynamic system identification. In an example involving a simulated fermentation process, it is shown that a linear-wavelet network yields a smaller approximation error when compared with a wavelet network with the same number of regressors. The proposed technique is also applied to the identification of a pressure plant from experimental data. In this case, the results show that the introduction of wavelets considerably improves the prediction ability of a linear model. Standard errors on the estimated model coefficients are also calculated to assess the numerical conditioning of the identification process.

HiPP: A Novel Hierarchical Point Placement Strategy and its Application to the Exploration of Document Collections

Point placement strategies aim at mapping data points represented in higher dimensions to bi-dimensional spaces and are frequently used to visualize relationships amongst data instances. They have been valuable tools for analysis and exploration of data sets of various kinds. Many conventional techniques, however, do not behave well when the number of dimensions is high, such as in the case of documents collections. Later approaches handle that shortcoming, but may cause too much clutter to allow flexible exploration to take place. In this work we present a novel hierarchical point placement technique that is capable of dealing with these problems. While good grouping and separation of data with high similarity is maintained without increasing computation cost, its hierarchical structure lends itself both to exploration in various levels of detail and to handling data in subsets, improving analysis capability and also allowing manipulation of larger data sets.

Two-Phase Mapping for Projecting Massive Data Sets

Most multidimensional projection techniques rely on distance (dissimilarity) information between data instances to embed high-dimensional data into a visual space. When data are endowed with Cartesian coordinates, an extra computational effort is necessary to compute the needed distances, making multidimensional projection prohibitive in applications dealing with interactivity and massive data. The novel multidimensional projection technique proposed in this work, called Part-Linear Multidimensional Projection (PLMP), has been tailored to handle multivariate data represented in Cartesian high-dimensional spaces, requiring only distance information between pairs of representative samples. This characteristic renders PLMP faster than previous methods when processing large data sets while still being competitive in terms of precision. Moreover, knowing the range of variation for data instances in the high-dimensional space, we can make PLMP a truly streaming data projection technique, a trait absent in previous methods.

Coherent state quantization of paragrassmann algebras

By using a coherent state quantization of paragrassmann variables, operators are constructed in finite Hilbert spaces. We thus obtain in a straightforward way a matrix representation of the paragrassmann algebra. This algebra of finite matrices realizes a deformed Weyl-Heisenberg algebra. The study of mean values in coherent states of some of these operators leads to interesting conclusions.

Irreducible morphisms of categories of complexes

We study irreducible morphisms of complexes. In particular, we show that the irreducible morphisms having one (finite) irreducible submorphism fall into three canonical forms and we give necessary and sufficient conditions for a given morphism of that type to be irreducible. Our characterization of the above mentioned type of irreducible morphisms of complexes characterizes also some class of irreducible morphisms of the derived category D(-)(A) for A a finite dimensional k-algebra, where k is a field. (C) 2009 Published by Elsevier Inc.

Lie algebra methods for the applications to the statistical theory of turbulence

Approximate Lie symmetries of the Navier-Stokes equations are used for the applications to scaling phenomenon arising in turbulence. In particular, we show that the Lie symmetries of the Euler equations are inherited by the Navier-Stokes equations in the form of approximate symmetries that allows to involve the Reynolds number dependence into scaling laws. Moreover, the optimal systems of all finite-dimensional Lie subalgebras of the approximate symmetry transformations of the Navier-Stokes are constructed. We show how the scaling groups obtained can be used to introduce the Reynolds number dependence into scaling laws explicitly for stationary parallel turbulent shear flows. This is demonstrated in the framework of a new approach to derive scaling laws based on symmetry analysis [11]-[13].

Hyperbolicity of semigroup algebras

Let A be a finite-dimensional Q-algebra and Gamma subset of A a Z-order. We classify those A with the property that Z(2) negated right arrow U(Gamma) and refer to this as the hyperbolic property. We apply this in case A = K S is a semigroup algebra, with K = Q or K = Q(root-d). A complete classification is given when KS is semi-simple and also when S is a non-semi-simple semigroup. (c) 2008 Elsevier Inc. All rights reserved.

Tilt-critical algebras of tame type

Let A be a finite dimensional k-algebra over an algebraically closed field. Assume A=kQ/I where Q is a quiver without oriented cycles. We say that A is tilt-critical if it is not tilted but every proper convex subcategory of A is tilted. We describe the tilt-critical algebras which are strongly simply connected and tame.