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Per a què serveix l’assignatura que estic cursant? Em servirà per la feina que exerciré en el futur? Tindré la possibilitat d’aplicar els nous coneixements adquirits? Gaudiré durant el meu procés de formació? Apropar les motivacions darrera la planificació de l’ensenyament a l’alumne ajuda a incloure’l en el procés d’aprenentatge i fer-lo entreveure les ramificacions que s’escauen. El serious game és un mètode pedagògic amb aspectes lúdics que, mitjançant un software, permet a l’alumne testejar els seus coneixements adquirits per resoldre casos específics en temps real. El serious game, acompanyat per un disseny dels continguts pedagògics, contextualitza en un entorn virtual d’entreteniment les aplicacions reals de la teoria ensenyada pel professor, i crea un vincle entre l’experiència de joc i les lliçons transmeses. Gràcies al recorregut pedagògic incorporat i els mecanismes d’avaluació, el joc guia l’estudiant en l’assimilació i la pràctica de l’objecte d’estudi, en un treball autònom fora de l’horari lectiu de l’assignatura o com a complement a les activitats d’aplicació pràctica a classe. Aquest treball presenta el serious game com a eina per respondre a les inquietuds dels alumnes i per que el professor pugui aprofitar-ne els aspectes d’aplicació pràctica, seguiment de l’alumne i motivacionals, assegurant l’assoliment de les competències previstes a l’assignatura. El projecte inclou una proposta de jocs per implementar en assignatures del Grau en Enginyeria en Tecnologies Industrials i un model d’implementació del serious game Mecagenius en l’assignatura de Sistemes de Fabricació a l’Escola Tècnica Superior d’Enginyeria Industrial de Barcelona. Es presenta un pla d’experimentació i validació de l’ús del joc com a complement a les sessions de laboratori de l’assignatura per oferir eines als estudiants per assolir un major coneixement pràctic dels sistemes de fabricació que s’hi ensenyen i obtenir una major implicació de l’alumne envers el contingut acadèmic en

The aim of this thesis is to create a CFD model of a fluidized bed to study the heat transfer having concentrated solar as a heat source. First, some background about concentrated solar energy, a review of its actual potentials and barriers and a comparison among other energy sources will be exposed. Then the fluidized bed concept and the recent studies about heat transfer will be presented and discussed. The core of the thesis will be based in creating a CFD model to predict the behavior and understand the interactions in a fluidized bed when is irradiated with concentrated solar. It has been decided to create a 2-D model of a rectangular fluidized bed with solid particles of SiC. First, the model has been tested and validated without the heat source and then the heat source has been added. Once the simple model has shown the desired results it has been upgraded including heat to the model. A crucial part of the study was how to model the heat source and how to implement it into the model. With the data obtained from current studies an approach of the heat source to a Gaussian distribution with its peak in the focal point has been made. Then this heat source has been introduced in the model affecting always the surface of the bed. Once the heat model has been set up two different cases have been studied: a fixed bed and a fluidized bed. In the fixed bed case, the temperature in the focal point increase very rapidly like in the current studies due to that a high amount of energy is concentrated in a very small region. On the fluidized bed case, the temperature of the bed fluctuates due to the high mixing and interaction that there is in the bed that improves the heat transfer between the solid particles.

On this Bachelor's Thesis we apply the method of layer potentials on two different contexts. On the first part of this work we will prove some important properties of the single and double layer potentials for the Laplacian on smooth domains and we will use them to prove uniqueness and existence of solution for the Dirichlet and Neumann problems (both exterior and interior). On the second part we use the method of the layer potentials to look for an approximate solution for the flow of air around a very thin wing and we use the solution found to compute the approximate force exerted over the wing.

The aim of this thesis is to give a method to produce families of complete Riemannian metrics with positive constant $\sigma_k$-curvature equal to $2^{-k}\binom{n}{k}$ on $M\setminus \Lambda^p$, where $M$ is a Riemannian manifold and $\Lambda^p$ is a smooth, closed, compact submanifold, whose dimension is $0 < p < \frac{n-2k}{2}$. To simplify the computations, we assume $k = 2$. However, we conjecture the method to work for $k > 2$. First, we give a geometrical motivation of the problem and then we establish a methodology to solve it using analysis tools. Most part of the analysis is concerned with the study of the mapping properties of the linearized operator around the approximate solutions.

En aquest treball, presentem succintament el problema de les paral·leles generat a partir del cinquè postulat d'Euclides i comentem alguns intents de demostrar-lo. Introduïm la geometria hiperbòlica, que és la que s'obté en substituir, en la geometria euclidiana, el postulat de les paral·leles per la seva negació. Analitzem tres articles de Beltrami. En el primer presenta el teorema conegut amb el seu nom, que caracteritza les superfícies que admeten un difeomorfisme (local) amb el pla que transformi geodèsiques en rectes. En el segon presenta un model pel pla hiperbòlic, el model projectiu. I en el darrer, descriu alguns models conformes pel pla hiperbòlic.

El present projecte es basa en l’elaboració d’un estudi del sistema de funcionament de l’Aula Lliure, un programa de l’ETSEIB que consisteix en que alumnes de cursos superiors de l’Escola ofereixen suport en la docència de diferents assignatures als alumnes de primer, rebent aquests últims el servei de forma gratuïta. L’objectiu d’aquest treball és, primerament, presentar l’Aula Lliure i el per què de la seva existència, donar evidència del seu bon funcionament i, finalment, analitzar diversos entorns semblants a l’ETSEIB per tal d’establir un document que indueixi unes primeres pautes del que seria la implantació del model en una altra Universitat. Es determina quina és l’estructura interna, així com el sistema d’organització i els seus processos bàsics de funcionament. S’estudia exhaustivament l’entorn social i econòmic, centrat en un enfocament acadèmic dels diferents agents universitaris. S’han examinat les dades recollides per l’Escola de les notes de l’assignatura Càlcul II obtingudes al Quadrimestre de Primavera del Curs 2014-2015, aplicant una anàlisi estadística en les qualificacions, per a poder veure quins són els resultats acadèmics segmentant els alumnes participants a l’Aula Lliure. L’elaboració del pressupost destinat a aquest programa durant l’any acadèmic 2014-2015 ha permès determinar quins serien els costos fixes que representaria implantar el programa en un altre Centre de semblants característiques, de les analitzades prèviament segons uns paràmetres qualificatius.

L objectiu d aquest treball de fi de grau és estudiar dues generalitzacions de l equació de la calor, la primera enlloc d un laplacià té un laplacià fraccionari i la segona un operador de la forma $Lu = J ∗ u − u$, i determinar certes hipòtesis sota les quals les seves solucions es comporten com les de l equació clàssica de la calor. Obtenim els dos operadors generalitzant el laplacià i estudiem algunes de les seves propietats, que aplicarem en els capítols principals del treball. El resultat més important parla del comportament asimptòtic de les solucions del model convolutiu, i afirma que la seva decaiguda és la mateixa que la de les solucions de l equació de difusió fraccionària. També estudiem la Γ-convergència dels funcionals que ens defineixen aquestes equacions per tal de demostrar que el seu límit coincideix en alguns casos.

This thesis is an introduction to algebraic K-theory, focusing on the study of rings, although we will give some geometric interpretations and some relations to topological K-theory. We will study the K_0 group. The first two chapters of the thesis include some basic notions of rings and modules, and category theory, especially regarding abelian categories. Afterwards, we present projective modules and their properties, that will be used to define the K_0 group in Chapter 4. This chapter will include basic definitions and the main theorems of the Grothendieck group of a ring. We will also show some relations with vector bundles and topological K-theory. The last chapter introduces the K_0 group of a category with exact sequences and the G_0 group. It includes the proofs of three important abstract theorems about K_0 and the Fundamental Theorem of G_0.

In this note we prove that an analytic symplectic action of a semisimple Lie algebra can be locally linearized in Darboux coordinates. This result yields simultaneous analytic linearization for Hamiltonian vector fields in a neighbourhood of a common zero. We also provide an example of smooth non-linearizable Hamiltonian action with semisimple linear part. The smooth analogue only holds if the semisimple Lie algebra is of compact type. An analytic equivariant b-Darboux theorem for b-Poisson manifolds and an analytic equivariant Weinstein splitting theorem for general Poisson manifolds are also obtained in the Poisson setting, Preprint

A dominating set S of a graph G is called locating-dominating, LD-set for short, if every vertex v not in S is uniquely determined by the set of neighbors of v belonging to S. Locating-dominating sets of minimum cardinality are called LD-codes and the cardinality of an LD-code is the location-domination number lambda(G). An LD-set S of a graph G is global if it is an LD-set of both G and its complement G'. The global location-domination number lambda g(G) is introduced as the minimum cardinality of a global LD-set of G. In this paper, some general relations between LD-codes and the location-domination number in a graph and its complement are presented first. Next, a number of basic properties involving the global location-domination number are showed. Finally, both parameters are studied in-depth for the family of block-cactus graphs., Postprint (published version)

We give a sharp lower bound for the self-intersection of a nef line bundle L on an irregular variety X in terms of its continuous global sections and the Albanese dimension of X, which we call the generalized Clifford-Severi inequality. We also extend the result to nef vector bundles and give a slope inequality for fibered irregular varieties. As a by-product we obtain a lower bound for the volume of irregular varieties; when X is of maximal Albanese dimension the bound is vol(X) >= 2n!chi(omega(X)) and it is sharp., Peer Reviewed, Postprint (published version)

We establish an integration by parts formula in bounded domains for the higher order fractional Laplacian (-Delta)(s) with s > 1. We also obtain the Pohozaev identity for this operator. Both identities involve local boundary terms, and they extend the identities obtained by the authors in the case s is an element of (0,1).; As an immediate consequence of these results, we obtain a unique continuation property for the eigenfunctions (-Delta)(s)phi = lambda phi in Omega, phi equivalent to 0 in R-n\Omega., Peer Reviewed, Postprint (published version)

We show that the conformally invariant fractional powers of the sub-Laplacian on the Heisenberg group are given in terms of the scattering operator for an extension problem to the Siegel upper halfspace. Remarkably, this extension problem is di erent from the one studied, among others, by Ca arelli and Silvestre., Peer Reviewed, Preprint

In the nineteenth century, at the Atacama Desert, a phenomenon of competition occurred between different industrial water desalination techniques, one of which had excellent results and a long life using solar energy. It is a significant fact that this technology was distributed and used in several places in the same desert, when its existence and / or results were diffused in London, New York, Madrid and Santiago de Chile. Nevertheless it seems no to be known in other parts of world before mid-twentieth century. We wish to know how the pendulum movement between Continuity and Innovation that affects the evolution of technology was taken over by the historical and cultural forces of the selection of technology. The history of water desalination in the nineteenth century in the Atacama Desert provides a relevant case study for the analyse of artifactual discard and intermittent duration of the objects. This study proposes an intense description of the process from the data obtained from a systematic revision of physical and electronic engineering archives and from magazines during the nineteenth and twentieth centuries, interviews with researchers / relevant at the field of solar desalination, and from a further examination of local archive files, in order to establish the history of solar desalination at Las Salinas, Sierra Gorda and Office Domeyko. These stations, built between 1872 and 1907 in the desert of Atacama, were the first solar desalination plants at industrial scale in the world. All this is expected to contribute to a better understanding of the overall process of evolution of technology and its impact on the problem of sustainability., En el siglo XIX en el desierto de Atacama ocurría un fenómeno de competencia entre diferentes técnicas industriales de desalación de agua, una de las cuales tuvo excelentes resultados y una larga duración utilizando energía solar. Es un hecho relevante que esta tecnología haya sido difundida y utilizada en otros puntos del mismo desierto, publicándose contemporáneamente su existencia y/o resultados en Londres, Nueva York, Madrid y Santiago de Chile, a pesar de lo cual su difusión en otros lugares del mundo tuvo lugar recién a mediados del siglo XX. Interesa conocer aquí cómo el proceso pendular de Continuidad e Innovación que afecta a la evolución de la tecnología era intervenido por las fuerzas histórico-culturales de la selección técnica. La historia de la desalación del agua en el siglo XIX en el desierto de Atacama aporta un caso ejemplar para el análisis de los fenómenos de descarte artefactual y la duración intermitente de los objetos. Este estudio propone una descripción densa del fenómeno a partir de la producción de datos obtenida de la revisión sistemática de archivos físicos y electrónicos de revistas de ingeniería del siglo XIX y del siglo XX, de entrevistas a investigadores/as relevantes del área de la desalación solar, además del examen de archivos locales, con lo cual es posible establecer el acaecer en la historia de las desaladoras solares de Las Salinas, Sierra Gorda y Oficina Domeyko que, construidas entre 1872 y 1907 en el desierto de Atacama, fueron las primeras desaladoras solares de escala industrial del mundo. Con esto se espera contribuir a la comprensión del proceso general de evolución de la tecnología y su incidencia en el problema de la sustentabilidad., Postprint (published version)

Consider the billiard map defined inside an analytic closed strictly convex curve Q. Given q>2 and 02 i 0

The main object of study of this thesis are invariant manifolds in the field of dynamical systems. We deal with two different and independent topics, namely, the study of exponentially small splitting of invariant manifolds in analytic unfoldings of the Hopf-zero singularity (in Part I) and the applications of dynamical systems in problems inspired by neuroscience (in Part II). In general, this thesis studies both theoretical and applied problems in dynamical systems, using analytical as well as computational tools. In Part I, we consider a certain class of generic unfoldings of the so-called Hopf-zero singularity. One can see that the truncation of the normal form at any finite order of such unfoldings possesses two saddle-focus critical points and, when the parameters lie on a certain curve, they are connected by a one- and a two-dimensional heteroclinic manifolds. However, considering the whole vector field, one expects these heteroclinic connections to be destroyed. This fact can lead to the birth of a homoclinic connection to one of the critical points, producing thus a Shilnikov bifurcation. For the case of $C^\infty$ unfoldings, this was proved by Broer and Vegter during the 80's, but for analytic unfoldings it has remained an open problem. Recently, under some assumptions on the size of the splitting of the heteroclinic connections, Dumortier, Ibáñez, Kokubu and Simó proved the existence of Shilnikov bifurcations in the analytic case. Our study concerns the splitting of the one- and two-dimensional heteroclinic connections. These cannot be detected in the truncation of the normal form at any order, and hence they are exponentially small with respect to one of the perturbation parameters. We give asymptotic formulas of these splittings and, in particular, we prove that under generic conditions the main assumptions made by Dumortier, Ibáñez, Kokubu and Simó hold. In Part II, we deal with tools to provide an accurate prediction of phase variations in an oscilla, El principal objecte d'estudi d'aquesta tesi són les varietats invariants en el camp dels sistemes dinàmics. Considerem dos temes diferents i independents, concretament l'estudi de l'escissió exponencialment petita de varietats invariants en desplegaments analí­tics de la singularitat Hopf-zero (a la Part I), i les aplicacions dels sistemes dinàmics en problemes inspirats per la neurociència (a la Part II). A la Part I, considerem una classe de desplegaments genèrics de l'anomenada singularitat Hopf-zero. Es pot veure que el truncament de la forma normal a qualsevol ordre finit d'aquests desplegaments té dos punts crítics de tipus sella-focus i, quan els paràmetres estan sobre una certa corba, estan connectats per dues varietats heteroclí­niques, una d'unidimensional i una de bidimensional. No obstant, si es considera tot el camp vectorial, s'espera que aquestes connexions heteroclí­niques desapareguin. Això pot causar el naixement d'una òrbita homoclí­nica en un dels dos punts crí­tics, produint així­ el que es coneix com una bifurcació de Shilnikov. En el cas de desplegaments $C^\infty$, això va ser provat per Broer i Vegter durant els anys 80, però el cas de desplegaments analí­tics ha quedat obert. Recentment, sota certes hipòtesis sobre la mida de l'escissió de les connexions heteroclí­niques, Dumortier, Ibáñez, Kokubu i Simó han provat l'existència de bifurcacions de Shilnikov en el cas analític. El nostre estudi consisteix en el càlcul del trencament de les connexions heteroclí­niques. Aquests trencaments no es poden detectar en la forma normal a cap ordre i, per tant, són exponencialment petits en un dels paràmetres de pertorbació. Donem fórmules asimptòtiques d'aquests trencaments i, en particular, provem que sota certes condicions genèriques les principals hipòtesis fetes per Dumortier, Ibàñez, Kokubu i Simó són vàlides. A la Part II, considerem eines per proporcionar una predicció acurada de la variació de fase en un oscil·lador subjecte a estímuls extern, Postprint (published version)

Most galaxies have a warped shape when they are seen from an edge-on point of view. In this work we apply dynamical system methods to find an explanation of this phenomenon that agrees with its abundance among galaxies, its persistence in time and the angular size of observed warps. Starting from a simple, but realistic, 3D galaxy model formed by a bar and a flat disc, we study the effect produced by a small misalignment between the angular momentum of the system and its angular velocity. To this end, a precession model is developed and considered, assuming that the bar behaves like a rigid body. In order to study the behaviour of the rigid body, we solve its Euler equations. We study the resulting solution in a precessing reference system, selected in such a way to make the angular momentum and angular velocity of the body time independent. After checking that the periodic orbits inside the bar keep being the skeleton of the inner system, even after inflicting a precession to the potential, we compute the invariant manifolds of the unstable periodic orbits departing from the equilibrium points at the ends of the bar to get evidence of their warped shapes. As is well known, from previous studies with 2D galaxy models, the invariant manifolds associated with these periodic orbits drive the arms and rings of barred galaxies and constitute the skeleton of these building blocks. Now, looking at them from an edge-on viewpoint, we find that these manifolds present warped shapes such as those recognized in observations with a close concordance in angles. In addition, test particle simulations have been performed to determine how the stars are affected by the applied precession, confirming in this way the theoretical results obtained. Once the behaviour of the precessing model is known, we develop the model with a more complex potential, including a spherical halo, in order to study the influence of each parameter that gives shape to the potential and to determine the effec, La mayoría de galaxias tienen forma alabeada cuando son vistas desde un punto de vista lateral. En este trabajo aplicamos métodos de sistemas dinámicos para encontrar una explicación de este fenómeno que concuerde con su abundancia entre galaxias, su persistencia en el tiempo y el tamaño de los ángulos de alabeo observados. Partiendo de un modelo de galaxia tridimensional sencillo, pero realista, formado por una barra y un disco delgado, estudiamos los efectos que produce un pequeño desalineamiento entre el momento angular del sistema y su velocidad angular. Con este fin, se desarrolla un modelo de precesión, asumiendo que la barra se comporta como un sólido rígido. Para estudiar el comportamiento del sólido rígido, resolvemos sus ecuaciones de Euler. Estudiamos la solución obtenida en un sistema de referencia de precesión, que hace que el momento y la velocidad angular del cuerpo sean constantes. Después de comprobar que las órbitas periódicas del interior de la barra siguen siendo el esqueleto del sistema, incluso después de aplicar una precesión al potencial, calculamos las variedades invariantes de las órbitas periódicas inestables que parten de los puntos de equilibrio en los extremos de la barra, obteniendo evidencias de sus formas alabeadas. Como es conocido, a partir de estudios previos con modelos bidimensionales de galaxias, las variedades invariantes asociadas con estas órbitas periódicas marcan la posición de los brazos y anillos de las galaxias barradas y constituyen el esqueleto de estos elementos. Ahora, observándolos desde un punto de vista lateral, hallamos que estas variedades presentan formas alabeadas análogas a las observadas, y con una gran concordancia de ángulos. Además, hemos realizado simulaciones de test de partículas para determinar como la precesión aplicada al potencial afecta a las estrellas, confirmando de esta manera los resultados teóricos obtenidos. Una vez se conoce el comportamiento del modelo de precesión, lo sofisticamos con un, Postprint (published version)

This thesis is focused on the study of dynamics and control of a multi-tethered satellite formation, where a multi-tethered formation is made up with several satellites (agents) connected by means of cables (tethers). The goal of the first part of the study is to evaluate the effect of tether mass on multi-tethered clusters. Due to the complexity of the formations analyzed, the stability of the formation is assessed through a numerical simulation. The behavior is evaluated in the ideal case of circular orbits, but also in non-ideal cases such as that of elliptical reference orbit or perturbed motion. For circular reference orbits, the dynamics of tethered satellite formation is studied, showing that tether mass affects formation dynamics for closed configurations featuring external tethers. This leads to significant instability effects affecting the position of deputies with respect to the parent body neglected by a more elementary modeling approach. When combined effect of orbit eccentricity and tether mass on tethered formations is analyzed, the most noticeable effect due to eccentricity is the increase in the variation of the local spin rate of the cluster between perigee and apogee passes of the reference elliptical orbit. This effect has consequences over the elongation of tethers, shape of tether oscillations and angular separation between adjacent tethers especially for open formations. When taking into account the J2 effect on massive tethered satellite formations, in the Earth¿facing scenario, the trajectory of the parent body presents oscillations of increasing amplitude in the direction perpendicular to the orbital plane. The second part of the study is focused on deriving a control law for position and attitude control of an Earth-facing double pyramid multi-tethered cluster. The control problem is decomposed in two levels: A first level to perform position and attitude coarse control of the formation as a whole, and a second level to achieve accurate po, La tesi es centra en l'estudi de la dinàmica i control de formacions de satèl·lits connectats per tethers. Aquestes formacions estan compostes per diversos satèl·lits (agents) connectats per cables (tethers). L'objectiu de la primera part de l'estudi, és l'avaluació de l'efecte de la massa a clústers connectats per múltiples tethers. Degut a la complexitat de les formacions analitzades, l'estabilitat de la formació s'analitza a través de simulacions. S'estudia el comportament pel cas ideal d'orbites circulars, així com en casos no ideals tals com orbites de referència el·liptiques, o moviment sota l'efecte de pertorbacions. La tesi analitza la dinàmica de les formacions per òrbites circulars, mostrant que l'efecte de la massa dels tethers afecta la dinàmica de formacions de geometria tancada (on el perímetre extern esta definit per tethers) amb un satèl·lit central. Aquest efecte dóna lloc a una clara inestabilitat que afecta la posició dels agents respecte a l'objecte central. Aquest efecte no és apreciable en models simplificats on s'ignora l'efecte de la massa al model. Quan es combina una òrbita de referència el·liptica amb un model que incorpora la massa dels tethers, l'efecte més notori és la variació de la velocitat de rotació local del clúster entre el pas per l'apogeu i perigeu de l'òrbita de referència. Aquest efecte té conseqüències sobre l'elongació dels tethers, la forma de les oscil·lacions, i la separació entre tethers adjacents (especialment en el cas de formacions obertes). Quan es té en compte l'efecte de la pertorbació J2, en el cas de formacions orientades envers la Terra, la trajectòria de l'objecte central presenta oscil·lacions d'amplitud creixent en la direcció perpendicular al pla orbital. La segona part de l'estudi es centra en la definició d'una llei de control per regular la posició i orientació d'un clúster amb geometria de doble piràmide orientat envers la Terra. El problema de control es descompon en dos nivells. Un primer nivell per u, Postprint (published version)

This master thesis is split into two separate main parts. Part I is a meta-analysis of randomized clinical trials (RCTs) and Part II is a simulation study. First, a meta-analysis was performed through a literature search of RCTs of the years 2010 and 2013. The goal was to confirm results of a preliminary work which analysed RCTs from the years 2004 and 2007. Second, different data sets of RCTs with repeated measurements and a continuous outcome were simulated with varying sample sizes, effect sizes, residual variances, random slope variances and correlation structures. The goal was to compare statistical methods that can be applied to longitudinal data.

We study the derivation, properties and solution methods of the classical and the hyperbolic heat transfer equations. Some results to approximate the hyperbolic model with the classical model for small values of the thermal relaxation time are given. Finally, two examples of application of the hyperbolic model are studied and discussed.

We describe prime ideals of height 2 minimally generated by three elements in a Gorenstein, Nagata local ring of Krull dimension 3 and multiplicity at most 3. This subject is related to a conjecture of Y. Shimoda and to a long-standing problem of J. Sally., Peer Reviewed, Postprint (author’s final draft)

Institut Hospital del Mar d'Investigacions Mèdiques, En este trabajo desarrollamos la teoría de los modelos gráficos mixtos de Markov (GMMs), y en concreto los modelos heterogéneos. Aplicamos esta teoría a la búsqueda de marcadores génicos que afectan a la varianza de la expresión de genes, y probamos nuestros resultados con una base de datos real.

ENGLISH: A construction of an asset is challenging work associated with huge responsibilities. Project manager’s main challenges in construction are generally Time-Cost-Quality dependent. In addition to these factors, Project Managers or construction Manager have to deal with numerous uncertainties in the process of construction. Considering its diverse nature and activity performed in construction. Here major role playing factor in success of a project is efficient scheduling of the construction project. While scheduling a project, the factors which are considered are mostly tangible, such as resources, labor and capital. My main objective in this research is to reduce the duration of the activity involved in construction, while considering qualitative factors like site organization, Labor Skills and quality of equipment used and reveal the effect of these tangible factors on duration of the project. Since these factors are difficult to be measured using classical mathematic theories. We have opted Fuzzy Logic Theory as our base, which can measure mathematically with use of Fuzzy Logic Tool Box in MATLAB®. In order to implement the proposed technique, various membership functions need to be estimated using judgement and guidance of experts. One of the main advantage of the proposed technique is that it can be easily implemented in existing computer programs and thus having a possibility to schedule a project efficiently.

Preprint

Zentrum Mathematik (Centre for Mathematics) Technische Universität München (Technical University of Munich)

The interest of taking technology to increasingly small scales has required great changes and advances in science. In particular, the well-known Fourier's law (very useful in large scales) has proved to be not accurate enough for describing heat transfer at smaller scales. The goal of this work is to introduce one of the alternatives to Fourier's law, the Cattaneo-Vernotte equation, and the resulting heat transfer model, the Hyperbolic Heat Equation. Numerical and analytical solutions will be provided and two case studies will be analysed, particularly studying the limit between the Parabolic and Hyperbolic Heat Equations at the nano and microscale.

In this article we derive the solution to the Schrödinger's equation in one dimension for a simple example, also the solution to the eigenvalue problem in a three dimensional ball, and then apply the results to deduce the eigenvalues of the Schrödinger's equation for the Hydrogen atom. This eigenvalues give rise to the energy emission lines of the Hydrogen. We also solve the initial-value problem of the Schrödigner's equation in $\R^n$ and the Nonlinear Schrödinger's equation locally, in $\R^3$. We explain the theoretical principles that underly the electronic orbitals of the atom, and deduce their approximated three dimensional aspect.

Aquest treball tracta sobre la història dels logaritmes des de la publicació de la Mirifici Logarithmorum Canonis Descriptio l any 1614 de John Napier (1550-1617) fins 20 anys més tard. Al llarg del treball es presenta l estudi del contingut de les obres principals sobre logaritmes de cinc autors: Napier, Henry Briggs (1561-1630), Edmund Gunter (1581-1626), Ezechiël De Decker (1603-1647) i Adriaen Vlacq (1600-1667). S hi analitzen les definicions, les propietats, les taules de logaritmes i les idees subjacents, tot elaborant taules que permetin la seva comparació. El treball inclou també uns Apèndixs amb reproduccions de les portades de les obres originals i de les taules de logaritmes que són calculades. L aportació de les taules comparatives permet adentrar-se a la matemàtica de principis del segle XVII i poder analitzar en català el contingut original de les primeres obres sobre logaritmes en llatí.

Phylogenetics is the study of the evolutionary relationships among different species through the analysis of sequences of biological data such as DNA. These relationships are usually represented using phylogenetic trees, which are tree diagrams that relate every species with its ancestor. Phylogenetic methods try to determine which tree best fits a given set of DNA sequences by using either distances among species or evolutionary models. In this line, the main goal of this project is to propose a new method for phylogenetic reconstruction, using a new coordinate system (Fourier coordinates), focused on 4-leaved trees under a certain evolutionary Markov model.

Phylogenetic reconstruction tries to recover the ancestral relationships among a group of contemporary species and represent them in a phylogenetic tree. To do it, it is useful to model evolution adopting a parametric statistic model. Using these models one is able to deduce polynomial relationships between the observed probabilities, known as phylogenetic invariants. Mathematicians have recently begun to be interested in the study of these polynomials and have developed techniques from algebraic geometry that have already been used in the study of phylogenetics. Nowadays there exist some phylogenetic reconstruction methods based in these phylogenetic invariants. In this project we study some theoretical results on stochasticity conditions of the parameters of the model and we analyze whether they give some new information to these reconstruction methods. We implement the conditions and analyze the results comparing them with the results provided by the reconstruction method Erik+2. Finally we propose a new reconstruction method based in the same ideas, with different implementation, and with very good results on simulated data.

En este trabajo se analiza la eficacia de una experiencia de innovación enel ámbito de la educación superior consistente en el uso de blogs educativos como herramienta didáctica. Dicha experiencia se ha llevado a cabo en los grupos de la asignatura de Álgebra lineal de primer curso de la titulación de grado impartida en la Escuela Superior de Ingenieros Industriales de Barcelona (ETSEIB) de la Universitat Politècnica de Catalunya (UPC). El resultado de la experiencia muestra que este recurso educativo acentúa el aprendizaje por encima de la enseñanza, potenciando el trabajo autónomo, la participación, la interacción y la colaboración de los estudiantes. Las características propias de los edublog hacen que con unos conocimientos básicos sobre tecnología digital, sean una herramienta fácil de utilizar en general y en particular para los estudiantes que llegan a la universidad ya que han nacido a partir del año 1990 y por tanto, son nativos digitales., Postprint (published version)

El uso de weblogs se va haciendo más habitual entre los docentes universitarios por lo que, se hace cada vez más necesario definir un buen sistema de evaluación. Los criterios de evaluación de un blog deben tener en cuenta la operatividad, la apariencia, la evidencia del trabajo realizado tanto en calidad como cantidad, los indicadores de reflexión y el nivel de participación y colaboración de los estudiantes. Teniendo en cuenta estos aspectos y para responder a esas necesidades hemos elegido la rúbrica como metodología para la evaluación de blogs. Para el tema específico de blogs utilizados en la asignatura álgebra lineal cursada en la ETSEIB de la Universidad Politécnica de Cataluña, se proponen dos rúbricas, una para la auto-evaluación y evaluación por pares, y la segunda para evaluar el trabajo del estudiante por parte del docente. Esta segunda rúbrica consiste en 3 sub-rúbricas, una para analizar la parte formal del blog del alumno (estructura, ortografía, nivel de idioma, entre otros) y las otras dos para evaluar los proyectos que los estudiantes deben realizar., Postprint (published version)

Presentació de l'obra matemàtica de A. Grothendieck., Postprint (published version)

We prove the existence and uniqueness of a traveling front and of its speed for the homogeneous heat equation in the half-plane with a Neumann boundary reaction term of unbalanced bistable type or of combustion type. We also establish the monotonicity of the front and, in the bistable case, its behavior at infinity. In contrast with the classical bistable interior reaction model, its behavior at the side of the invading state is of power type, while at the side of the invaded state its decay is exponential. These decay results rely on the construction of a family of explicit bistable traveling fronts. Our existence results are obtained via a variational method, while the uniqueness of the speed and of the front rely on a comparison principle and the sliding method., Peer Reviewed, Postprint (author’s final draft)

Cancer is a complex disease and thus is complicated to model. However, simple models that describe the main processes involved in tumoral dynamics, e.g., competition and mutation, can give us clues about cancer behavior, at least qualitatively, also allowing us to make predictions. Here, we analyze a simplified quasispecies mathematical model given by differential equations describing the time behavior of tumor cell populations with different levels of genomic instability. We find the equilibrium points, also characterizing their stability and bifurcations focusing on replication and mutation rates. We identify a transcritical bifurcation at increasing mutation rates of the tumor cells. Such a bifurcation involves a scenario with dominance of healthy cells and impairment of tumor populations. Finally, we characterize the transient times for this scenario, showing that a slight increase beyond the critical mutation rate may be enough to have a fast response towards the desired state (i.e., low tumor populations) by applying directed mutagenic therapies, Peer Reviewed, Postprint (author’s final draft)

In this article we develop tools to compute the Geometric Quantization of a symplectic manifold with respect to a regular In this article we develop tools to compute the Geometric Quantization of a symplectic manifold with respect to a regular Lagrangian foliation via sheaf cohomology and obtain important new applications in the case of real polarizations. The starting point is the definition of representation spaces due to Kostant. Besides the classical examples of Gelfand-Cetlin systems due to Guillemin and Sternberg [13] very few examples of explicit computations of real polarizations are known. The computation of Geometric Quantization in [13] is based on a theorem due to Śniatycki for fibrations [32] which identifies the representation space with the set of Bohr-Sommerfeld leaves determined by the integral action coordinates. In this article we check that the associated sheaf cohomology apparatus of Geometric Quantization satisfies Mayer-Vietoris and Künneth formulae. As a consequence, a new short proof of this classical result for fibrations due to Śniatycki is obtained. We also compute Geometric Quantization with respect to any generic regular Lagrangian foliation on a 2-torus and the case of the irrational flow. In the way, we recover some classical results in the computation of foliated cohomology of these polarizations., Peer Reviewed, Preprint

The spectral properties of special matrices have been widely studied, because of their applications. We focus on permutation matrices over a finite field and, more concretely, we compute the minimal annihilating polynomial, and a set of linearly independent eigenvectors from the decomposition in disjoint cycles of the permutation naturally associated to the matrix., Peer Reviewed, Postprint (published version)

Peer Reviewed, Postprint (published version)

This paper, which is the follow-up to part I, concerns the equation (-Delta)(s)v + G'(v) = 0 in R-n, with s is an element of (0, 1), where (-Delta)(s) stands for the fractional Laplacian-the infinitesimal generator of a Levy process.; When n = 1, we prove that there exists a layer solution of the equation (i.e., an increasing solution with limits +/- 1 at +/-infinity) if and only if the potential G has only two absolute minima in [-1, 1], located at +/- 1 and satisfying G'(-1) = G'(1) = 0. Under the additional hypotheses G ''(-1) > 0 and G ''(1) > 0, we also establish its uniqueness and asymptotic behavior at infinity. Furthermore, we provide with a concrete, almost explicit, example of layer solution.; For n >= 1, we prove some results related to the one-dimensional symmetry of certain solutions-in the spirit of a well-known conjecture of De Giorgi for the standard Laplacian., Peer Reviewed, Postprint (author’s final draft)

Motivated by phylogenetics, our aim is to obtain a system of low degree equations that define a phylogenetic variety on an open set containing the biologically meaningful points. In this paper we consider phylogenetic varieties defined via group-based models. For any finite abelian group G , we provide an explicit construction of codimX polynomial equations (phylogenetic invariants) of degree at most |G| that define the variety X on a Zariski open set U . The set U contains all biologically meaningful points when G is the group of the Kimura 3-parameter model. In particular, our main result confirms (Michalek, Toric varieties: phylogenetics and derived categories, PhD thesis, Conjecture 7.9, 2012) and, on the set U , Conjectures 29 and 30 of Sturmfels and Sullivant (J Comput Biol 12:204–228, 2005)., Peer Reviewed, Postprint (published version)

The purpose of this report is to study the dynamics of rigid blocks subjected to an earthquake ground motion. Such study could predict whether a rigid body overturns or remains standing. In particular, three cases are analysed in detail. Firstly, we deal with the one body case, for which it is studied the phase space when there is and when there is not a perturbation acting on the system. In particular, some numerical computations are carried out in order to find periodic orbits making use of the Poincaré impact map. Another special case of piecewise smooth systems is analyzed, a nonlinear impact oscillator. Finally, the one body case is extended to two blocks rocking. The purpose of this study is to look for some periodic orbits in order to know whether the motion of the blocks synchronizes.

In this article we prove an action-angle theorem for b-integrable systems on b-Poisson manifolds improving the action-angle theorem contained in [LMV11] for general Poisson manifolds in this setting. As an application, we prove a KAM-type theorem for b-Poisson manifolds., Postprint (published version)

Cotutela Universitat Politècnica de Catalunya i Université Mohammed V-Agdal, Information is such a valuable good of our time. Given that the transmission of information has always been subject to precision problems, knowing the obstacles existing between the transmitter and the receiver, eventual disruptions can happen anywhere in between, the physical means, channels involved with the exchange are never perfect and they are subject to errors that might result in loss of important data. Error correcting codes are a key element in the transmission and storage of digital information. In this thesis we study the possibility to redefine and improve properties of convolutional codes in terms of coding and decoding, with the help of the systems and control theory. For that matter, in chapter 1, we recall notions on coding theory, more specifically, on linear codes, both block and convolutional, redefining the convolutional codes as submodules of the F^n_{q} which is our main workspace. And we go through the prerequisites involved in the process of encoding and decoding, both for block and convolutional codes. And in order to approach them with tools of the systems theory, in chapter 2, we give the equivalence of the generating matrix in the form of a realization (A,B,C,D) of an input-output system. Then, we studied the concatenation because it has been proved to improve the transmission. In this work, we consider two big families of concatenation: serial concatenation, and parallel concatenation and two other models of concatenation called systematic serial concatenation and parallel interleaver concatenation. In chapter 3, we study control properties for each case. Nevertheless, we focus on the property of output-observability, and conditions to obtain it, particularly an easy iterative test is presented in order to discuss whether a code is output-observable. This test consists in calculating certain ranks of block matrices constructed from the matrices A, B, C, D. The output-observability property is very beneficial for the decoding as discusse, La información es un valioso bien de nuestro tiempo. Dado que la transmisión de la información siempre ha estado sujeta a problemas de precisión, conociendo los obstáculos existentes entre el transmisor y el receptor, las interrupciones eventuales pueden ocurrir en cualquier lugar en el medio, el medio físico, canal involucrado con el cambio nunca es perfecto y está sujeto a errores que podrán dar como resultado una pérdida de datos importantes. Dado que los códigos correctores de errores son un elemento clave en la transmisión y almacenamiento de información digital, por eso un más fácil y mejor uso abre interesantes oportunidades en la regulación de la transmisión de la información, el cual es una ventaja que ofrece la teoría de sistemas lineales y el álgebra lineal a la definición de los códigos de convolución. Esta es la razón por la que en esta tesis, seguimos esa perspectiva para estudiar la posibilidad de redefinir y mejorar las propiedades de los códigos de convolución en base a la codificación y descodificación, con la ayuda de los sistemas y la teoría de control. En este sentido, en el capítulo 1, recordamos nociones sobre la teoría de códigos, más específicamente, sobre los códigos lineales, tanto de bloques como de convolución, se redefinen los códigos convolucionales como submódulos de Fnq que es nuestro espacio principal de trabajo. Y damos un repaso a los requisitos previos necesarios en el proceso de codificación y descodificación, tanto para los códigos de bloque como los códigos convolucionales. Y con el fin de aproximarnos a los códigos convolucionales con las herramientas de la teoría de sistemas, en el capítulo 2, damos la equivalencia de la matriz generatriz en función de una realización (A;B;C;D) de un sistema de entrada-salida. A continuación, se estudia la concatenación porque es conocido que mejora la transmisión. En este trabajo, se consideran dos grandes familias de concatenación: la concatenación en serie, y la concatenación en paralelo, L'information est un bien de notre époque dont l'importance n'est plus à démontrer. Etant donné que la transmission de l'information a toujours été soumise à des problèmes de précisions, dûs aux obstacles existant entre le transmetteur and le récepteur, d'éventuelles perturbations peuvent arriver n'importe où, entre les canaux physiques, faisant partie du processus d'échange qui n'est jamais parfait et ils peuvent toujours être affectés par des erreurs créant d'importantes pertes d'information. Les codes correcteurs d'erreurs sont un élément clé dans la transmission et la conservation de l'information numérique. Etant donné que les codes correcteurs d'erreurs sont un élément clé dans la transmission et la conservation de l'information digitale, ainsi un meilleur et plus simple usage ouvre des opportunités plus intéressantes dans la régulation de la transmission de l'information, qui est l'avantage que la définition des codes convolutifs suivant la théorie des systèmes linéaires apporte, avec le matériel de l'algèbre linéaire. C'est pour cette raison que dans cette thèse, nous suivons cette perspective pour étudier la perspective d'étudier la possibilité de redéfinir et d'améliorer les propriétés des codes convolutifs en termes de codage et de décodage, grâce aux outils de la théorie des systèmes et de contrôle. A cet effet, dans le chapitre 1, nous rappelons des notions sur la théorie des codes linéaires, les codes en bloc ainsi que les codes convolutifs, redéfinissant les codes convolutifs comme des sous-modules de Fnq qui est notre principal espace de travail. Et c'est ainsi que nous invoquons tous les prérequis nécessaires pour le processus de codage et de décodage, pour ce qui est des codes en bloc, et des codes convolutifs. Et dans le but d'approcher ces derniers grâce aux outils de la théorie des systèmes, dans le chapitre 2, nous donnons l'équivalence de la matrice génératrice sous la forme d'une réalisation (A;B;C;D) d'un un système inputoutput. Ensuite, nou, Postprint (published version)

Determinar les característiques de bombeig d'un cor com a bomba volumétrica.Simular comportament cor sa, incloent-hi dinámica de vàlvules i l'evolució dels volums durant un cicle. Avaluar l'efecte de determinades cardiopaties comunicants sobre el bombeig. Ampliar simulació i fer-ne anàlisi.Introduir efectes no lineals i propietats reològiques reals de la sang., En l’actualitat, el nombre d’investigacions i de projectes realitzats amb la col·laboració de diverses disciplines de la ciència i la tecnologia és més elevat. Per tal de trobar equacions o lleis genèriques que permetin definir la realitat, cada vegada més es requereix un conjunt de coneixements de diversos àmbits. En el cas del present estudi, es pretén fer un anàlisi comparatiu de la capacitat de bombeig del cor en condicions normals i quan es presenten algunes cardiopaties congènites. Per assolir-ho, s’ha emprat els sistemes Bondgraph que permeten definir qualsevol realitat com un conjunt d’elements interconnectats entre si que dissipen energia, n’emmagatzemen, etc. Els resultats obtinguts obren les portes a assajar altres cardiopaties congènites o patologies cardiovasculars., Nowadays researches and projects that include all kind of science and technology disciplines are increasing in number. In order to find equations or general laws that can define reality, knowledge in several fields is increasingly required. In the case of this study, the aim is to make a comparative analysis of the pumping ability of the human heart between normal conditions and some congenital heart diseases. To do so, Bondgraph systems have been considered the best option because they allow to represent the reality as a set of interconnected elements which can dissipate energy, keep energy, etc. The obtained results lead to try other congenital heart diseases or some other cardiovascular diseases.

En l'article [CKSTT] els autors proven una inestabilitat global per a l'equació de Schrödinger cúbica desenfocant en el tor 2-dimensional. Per aconseguir-ho, detecten un sistema finit d'equacions diferencials ordinàries, l'anomenat Toy Model System, per al quan proven un resultat que típicament s'anomena de difusió: la connexió de N objectes invariants connectats entre ells mitjançant òrbites heteroclíniques. El que no esmenten és que la intersecció entre les varietats invariants de dos objectes consecutius no és transversal. Això fa descartar, pràcticament, que el mecanisme de difusió sigui regit per la coneguda difusió d'Arnold i, per tant, no hi ha una explicació geomètrica del motiu pel qual aquests objectes invariants poden ser connectats. L'objectiu principal de la tesi és proposar un mecanisme geomètric que justifiqui el fet que aquesta difusió és possible. En el Capítol 2 evidenciem que la intersecció entre les esmentades varietats no és transversal en el Toy Model System i mostrem, mitjançant exemples, que aquesta falta de transversalitat pot impedir la connexió en una cadena de transició. Un cop vist que la connexió pot no ser possible si la intersecció no és transversal, detectem el motiu per al qual sí que és possible en el cas del Toy Model System. La raó és l'alta dimensió del problema i que cada nova connexió pren lloc en un espai nou, en unes direccions que encara no s'han utilitzat. D'altra banda es troben exemples integrables per als quals aquesta difusió és possible, allunyant-nos del tot de la difusió d'Arnold, típica per a sistemes no integrables. En el mateix capítol es proposa un esquema de detecció per aquest tipus de difusió mitjançant el llenguatge dels h-sets i les relacions de cobriment, de manera que s'estableix una prova que justifica la connexió sempre i quan es provin abans certes relacions de cobriment. Mitjançant aquest esquema de detecció, en el Capítol 3 es prova la connexió en el cas del Toy Model System. Durant la prova, aconseg, In [CKSTT], the authors prove a global instability in the cubic defocusing nonlinear Schrödinger equation defined in the 2-dimensional torus. To obtain so, they detect a finite system of ordinary differential equations, the so-called Toy Model System, for which they prove, as it is typically called, a diffusion result: the connection of N invariant objects connected through heteroclinic orbits. What is not mentioned there is that the intersection between the invariant manifolds of two consecutive invariant objects is not transverse. With this fact we can almost discard that the mechanism for diffusion relies on the socalled Arnold diffusion so, then, there is no geometric explanation of why the invariant objects can be connected. The main goal of this thesis is to propose a geometric mechanism that justifies the fact that this diffusion is possible. In Chapter 2 we prove that, indeed, the intersection between the above mentioned invariant manifolds is not transverse in the Toy Model System and we show through examples that the lack of transversality can forbid the connection in a transition chain. Once we have seen that the connection could not be possible if the intersection is non-transverse, we detect the reason why it is possible in the Toy Model System. The main reason is the high dimension of the system and the fact that each new connection takes place in a new space, in some directions that have not been used before. In addition we present an integrable example for which the diffusion is feasible, totally discarding, then, the Arnold diffusion as a mechanism since it is typical for non integrable systems. In the same chapter, a scheme of detection for this new kind of diffusion is proposed through the language of h-sets and covering relations. A proof that justifies the connection if some covering relations hold is established. Through this scheme of detection, in Chapter 3 we prove the connection in the Toy Model System. We can also improve the original resu, Postprint (published version)

The aim of this thesis is to study the hyperinvariant and characteristic subspaces of a matrix, or equivalently, of an endomorphism of a finite dimensional vector space. We restrict ourselves to the case of matrices A with an splitting characteristic polynomial, leaving for future work the generalization for any characteristic polynomial. The subspaces A-hyperinvariant and A-characteristic are subclasses of A-invariant subspaces (those containing its image for A), a key concept in the theory of matrices. Specifically, the subspaces A-hiperinvariant are those that are also invariant for all matrices that commute with A, while the A characteristic are required that are only invariant for invertible matrices that commute with A. Both concepts first appeared in the mid-30s within the context of group theory. But it was not until the 70s that appears a characterization of the A-hiperinvariant subspaces and their lattice was described in the context of matrix theory. In 2009 appears an article of Astuti and Wimmer which shows that A-hyperinvariant and A-caracteristic subspaces are the same except in the field GF(2) . In this case, Shoda theorem gives necessary and sufficient conditions for the existence of characteristic non-hiperinvariant subspaces. But the description of these subspaces was an open problem which is solved in this thesis. Our first objective, therefore, is to analyze the behavior of the centralizer of a matrix (i.e., the set of matrices commuting with it), we will assume canonical form ( Jordan or Weyr). Specifically, we calculate the determinant of the matrices in the centralizer, which in particular allows to characterize the nonsingular. Furthermore, we determined the images of a given subspace respect to the set of all matrices of these centralizers, a result that will be key for further study of hiperinvariant subspaces. We begin this study, giving conditions for the existence of one-dimensional hiperinvariant subspaces. More generally, using the re, El objetivo principal de esta tesis es estudiar en profundidad los subespacios hiperinvariantes y característicos de una matriz, o equivalentemente, de un endomorfismo de un espacio vectorial de dimensión finita. Nos restringimos al caso de matrices A con polinomio característico totalmente descomponible, dejando para futuros trabajos la generalización para cualquier polinomio característico. Los subespacios A-hiperinvariantes y A-característicos son subclases de los subespacios A-invariantes (aquellos que contienen su imágen por A), concepto clave en la teoría de matrices. Concretamente, los subespacios A-hiperinvariantes son aquellos que también son invariantes para toda matriz que conmuta con A, mientras que a los A-característicos se les exige sólo que lo sean para las matrices inversibles que conmutan con A. Ambos conceptos aparecen por primera vez a mediados de los años 30 dentro del contexto de la teoría de grupos. Pero no es hasta los años 70 en que se da una caracterización de los subespacios A-hiperinvariantes y se describe su retículo dentro de la teoría de matrices. En el año 2009 aparece un artículo de Astuti y Wimmer donde se demuestra que en el caso de matrices A con polinomio característico totalmente descomponible, los subespacios característicos coinciden con los hiperinvariantes excepto si los coeficientes de A pertenecen a GF(2). En este casos, el Teorema de Shoda da condiciones necesarias y suficientes para la existencia de subespacios característicos no hiperinvariantes. Pero la descripción de este tipo de subespacios era un problema abierto que resolvemos en esta tesis. Nuestro primer objetivo, por tanto, será analizar el comportamiento del centralizador de una matriz, (esto es, el conjunto de matrices que conmutan con ella), que supondremos en forma canónica (de Jordan o de Weyr). Concretamente, calculamos el determinante de las matrices de dichos centralizadores, lo que en particular, permite caracterizar las no singulares. Por otra parte, d, Postprint (published version)

En aquest treball es valida un model que faci una predicció relativament ràpida i fiable de la direcció i velocitat del vent en un determinat territori per tal d’aplicar aquesta informació a l’estudi de viabilitat d’instal·lació d’un parc eòlic. Per dur-ho a terme, es recórrer a la utilització de la teoria de les Cadenes de Markov que, com s’explica, està basada en l’Àlgebra Lineal. Després d’una introducció general sobre l’energia eòlica i la teoria de les Cadenes de Markov, el treball es centra en realitzar una simulació de l’aplicació que s’està estudiant a Horta de Sant, per així poder arribar a deduccions generals del projecte. S’arriba a la conclusió que les Cadenes de Markov poden ser una eina molt útil per fer una predicció de la direcció i velocitat del vent de manera senzilla. Aquests models poden servir com a ajuda per les empreses eòliques a prendre decisions. A més, aquesta aplicació té un impacte ambiental positiu i necessita d’un pressupost relativament baix.

This works focus on computational aspects of the theory of singularities of plane algebraic curves. We show how to use the Puiseux factorization of a curve, computed through the Newton-Puiseux algorithm, to study the equisingularity type of a curve. We present a novel version of the Newton-Puiseux algorithm that can compute all the Puiseux factorization of any arbitrary polynomial, removing the restriction of reduced inputs. Next, we introduce the theory of infinitely near points and the concept of base points of an ideal. Finally, we develop a novel algorithm that, using our novel version of the Newton-Puiseux algorithm, computes the weighted cluster of base points of any two dimensional ideal from any set of generators.

The main goal of the thesis is to classify some particular Poisson structures, called b-Poisson, following results of Guillemin-Miranda-Pirés and Radko. To understand these results, this thesis contains some chapters of introduction to Poisson Geometry and Symplectic Geometry. There is also a new result extending the classification of Poisson structures to the non-orientable case. Moreover it is explained a way to obtain local classification of some Poisson structures that can lead to more original results.

This project covers the basics of Financial Portfolio Management theory through different stochastic optimization models. The concepts of uncertainty and stochastic models are introduced and proven to be more reliable than the deterministic models. The input data of the model now are random variables, and two types of stochastic models are analyzed in this project. The first type are the models that transform the random input into deterministic data before the model is run. Among them there are the classic Markowitz mean-variance (which is shown in a practical example for IBEX 35) and Black-Litterman models. The second type are the models that introduce the uncertainty of the input data explicitly in the models, which are prepared to handle with random variables. These models are advanced and complex to solve, but a detailed introduction to them is provided in this thesis: key features, general formulation and a multistage stochastic optimization modelling example with the same case of IBEX 35.