Màster universitari en Matemàtica Aplicada http://hdl.handle.net/2099.1/4636 2020-07-13T06:07:37Z Polar germs and singularity invariants http://hdl.handle.net/2099.1/25313 Polar germs and singularity invariants González Alonso, Víctor In this work we obtain an algorithm which explicitly recovers the cluster of singular points a germ of curve from the cluster of base points of the polar germs of the curve. To achieve this, we reinterpret the problem in terms of the recently developed theory of planar analytic morphisms. The result is a sort of local version of the known fact in projective geometry that the proper singular points of a plane projective algebraic curve are exactly the proper base points of its polar curves.. L'objectiu és revisar resultats clàssics de gèrmens polars de corba plana des de la nova perspectiva del desenvolupament recent de la teoria de morfismes analítics entre superfícies no singulars. Aquesta teoria ha estat desenvolupada des del punt de vista geomètric per Casas i des del punt de vista algebraic, a través de la teoria de valoracions, per Favre i Jonsson. S'estudiaran els dos punts de vista i s'intentaran aplicar a la teoria de polars. 2015-02-25T11:49:19Z González Alonso, Víctor In this work we obtain an algorithm which explicitly recovers the cluster of singular points a germ of curve from the cluster of base points of the polar germs of the curve. To achieve this, we reinterpret the problem in terms of the recently developed theory of planar analytic morphisms. The result is a sort of local version of the known fact in projective geometry that the proper singular points of a plane projective algebraic curve are exactly the proper base points of its polar curves.. L'objectiu és revisar resultats clàssics de gèrmens polars de corba plana des de la nova perspectiva del desenvolupament recent de la teoria de morfismes analítics entre superfícies no singulars. Aquesta teoria ha estat desenvolupada des del punt de vista geomètric per Casas i des del punt de vista algebraic, a través de la teoria de valoracions, per Favre i Jonsson. S'estudiaran els dos punts de vista i s'intentaran aplicar a la teoria de polars. The isoperimetric problem in Johnson graphs http://hdl.handle.net/2099.1/20544 The isoperimetric problem in Johnson graphs Gutiérrez, Víctor Diego It has been recently proved that the connectivity of distance regular graphs is the degree of the graph. We study the Johnson graphs $J(n,m)$, which are not only distance regular but distance transitive, with the aim to analyze deeper connectivity properties in this class. The vertex $k$-connectivity of a graph $G$ is the minimum number of vertices that have to be removed in order to separate the graph into two sets of at least $k$ vertices in each one. The isoperimetric function $\mu_G(k)$ of a graph $G$ is the minimum boundary among all subsets of vertices of fixed cardinality $k$. We give the value of the isoperimetric function of the Johnson graph $J(n,m)$ for values of $k$ of the form ${t\choose m}$, and provide lower and upper bounds for this function for a wide range of its parameter. The computation of the isoperimetric function is used to study the $k$-connectivity of the Johnson graphs as well. We will see that the $k$-connectivity grows very fast with $k$, providing much sensible information about the robustness of these graphs than just the ordinary connectivity. In order to study the isoperimetric function of Johnson graphs we use combinatorial and spectral tools. The combinatorial tools are based on compression techniques, which allow us to transform sets of vertices without increasing their boundary. In the compression process we will show that sets of vertices that induce Johnson subgraphs are optimal with respect to the isoperimetric problem. Upper bounds are obtained by displaying nested families of sets which interpolate optimal ones. The spectral tools are used to obtain lower bounds for the isoperimetric function. These tools allow us to display completely the isoperimetric function for Johnson graphs $J(n,3)$. . Distance regular graphs form a structured class of graphs which include well-known families, as the n-cubes. Isoperimetric inequalities are well understood for the cubes, but for the general class of distance regular class it has only been proved that the connectivity of these graphs equals the degree. As another test case, the project suggests to study the family of so-called Johnson graphs, which are not only distance regular but also distance transitive. Combinatorial and spectral techniques to analyze the problem are available. 2014-01-30T13:27:11Z Gutiérrez, Víctor Diego It has been recently proved that the connectivity of distance regular graphs is the degree of the graph. We study the Johnson graphs $J(n,m)$, which are not only distance regular but distance transitive, with the aim to analyze deeper connectivity properties in this class. The vertex $k$-connectivity of a graph $G$ is the minimum number of vertices that have to be removed in order to separate the graph into two sets of at least $k$ vertices in each one. The isoperimetric function $\mu_G(k)$ of a graph $G$ is the minimum boundary among all subsets of vertices of fixed cardinality $k$. We give the value of the isoperimetric function of the Johnson graph $J(n,m)$ for values of $k$ of the form ${t\choose m}$, and provide lower and upper bounds for this function for a wide range of its parameter. The computation of the isoperimetric function is used to study the $k$-connectivity of the Johnson graphs as well. We will see that the $k$-connectivity grows very fast with $k$, providing much sensible information about the robustness of these graphs than just the ordinary connectivity. In order to study the isoperimetric function of Johnson graphs we use combinatorial and spectral tools. The combinatorial tools are based on compression techniques, which allow us to transform sets of vertices without increasing their boundary. In the compression process we will show that sets of vertices that induce Johnson subgraphs are optimal with respect to the isoperimetric problem. Upper bounds are obtained by displaying nested families of sets which interpolate optimal ones. The spectral tools are used to obtain lower bounds for the isoperimetric function. These tools allow us to display completely the isoperimetric function for Johnson graphs $J(n,3)$. . Distance regular graphs form a structured class of graphs which include well-known families, as the n-cubes. Isoperimetric inequalities are well understood for the cubes, but for the general class of distance regular class it has only been proved that the connectivity of these graphs equals the degree. As another test case, the project suggests to study the family of so-called Johnson graphs, which are not only distance regular but also distance transitive. Combinatorial and spectral techniques to analyze the problem are available. Minimizers to reaction-diffusion PDEs, Sobolev inequalities, and monomial weights http://hdl.handle.net/2099.1/18308 Minimizers to reaction-diffusion PDEs, Sobolev inequalities, and monomial weights Ros Oton, Xavier Reaction-diffusion equations play a central role in PDE theory and its applications to other sciences. They also play an important role in geometric problems: the problem of prescribing a curvature on a manifold and parabolic flows on manifolds. The aim of this project is the study of the regularity of minimizers to reaction-diffusion equations in certain doimains with symmetries and, as a consequence of it, some new Sobolev inequalities with monomial weights.. Reaction-diffusion equations play a central role in PDE theory and its applications to other sciences. They also play an important role in geometric problems: the problem of prescribing a curvature on a manifold and parabolic flows on manifolds. The aim of this project is the study of the regularity of minimizers to reaction-diffusion equations in certain doimains with symmetries and, as a consequence of it, some new Sobolev inequalities with monomial weights Premi Évarist Galois 2012, atorgat per la Societat Catalana de Matemàtiques. 2013-05-28T13:24:15Z Ros Oton, Xavier Reaction-diffusion equations play a central role in PDE theory and its applications to other sciences. They also play an important role in geometric problems: the problem of prescribing a curvature on a manifold and parabolic flows on manifolds. The aim of this project is the study of the regularity of minimizers to reaction-diffusion equations in certain doimains with symmetries and, as a consequence of it, some new Sobolev inequalities with monomial weights.. Reaction-diffusion equations play a central role in PDE theory and its applications to other sciences. They also play an important role in geometric problems: the problem of prescribing a curvature on a manifold and parabolic flows on manifolds. The aim of this project is the study of the regularity of minimizers to reaction-diffusion equations in certain doimains with symmetries and, as a consequence of it, some new Sobolev inequalities with monomial weights Exploring network coding capabilities http://hdl.handle.net/2099.1/18307 Exploring network coding capabilities Ríos Robles, André In this MSc Thesis, the principles of Network Coding will be reviewed, ranging from the description of the model to the typical applications. Also, two solutions related to privacy and security issues will be analyzed by taking as starting point an encoding node performing independently Network Coding in multiple flows. An attacker with traffic analysis capabilities could infer whether a set of incoming and outgoing data packets belong to a given flow simply by analyzing the linear dependency between them. Recently, Wang et al. have proposed a simple deterministic mechanism known as ALNCode to solve this issue by mixing (intersecting) the multiple flows in order to hide the correlation between the incoming and outgoing packets in each flow. In this Thesis, we characterize this solution by using combinatorial tools over projective spaces that deliver an exact expression for the general intersection probability in a given dimension. In addition, we extend the analysis using the properties derived from the intersection graphs defined by the generated subspaces. Finally, using error correcting coding, we analyze the special case of erasure data packets in the network and how this affects the performance of Network Coding.. The purpose of this project is the study of the performance of network coding modelled on undirected graphs, including the effect of different graph topologies. It is also meant to present the state of the art of those techniques both from a theoretical and an applied point of view 2013-05-28T13:20:50Z Ríos Robles, André In this MSc Thesis, the principles of Network Coding will be reviewed, ranging from the description of the model to the typical applications. Also, two solutions related to privacy and security issues will be analyzed by taking as starting point an encoding node performing independently Network Coding in multiple flows. An attacker with traffic analysis capabilities could infer whether a set of incoming and outgoing data packets belong to a given flow simply by analyzing the linear dependency between them. Recently, Wang et al. have proposed a simple deterministic mechanism known as ALNCode to solve this issue by mixing (intersecting) the multiple flows in order to hide the correlation between the incoming and outgoing packets in each flow. In this Thesis, we characterize this solution by using combinatorial tools over projective spaces that deliver an exact expression for the general intersection probability in a given dimension. In addition, we extend the analysis using the properties derived from the intersection graphs defined by the generated subspaces. Finally, using error correcting coding, we analyze the special case of erasure data packets in the network and how this affects the performance of Network Coding.. The purpose of this project is the study of the performance of network coding modelled on undirected graphs, including the effect of different graph topologies. It is also meant to present the state of the art of those techniques both from a theoretical and an applied point of view Linealización por realimentación de una clase de manipulador con un número arbitrario de actuadores http://hdl.handle.net/2099.1/17486 Linealización por realimentación de una clase de manipulador con un número arbitrario de actuadores Reyes Jiménez, Àlex Este trabajo final de máster titulado "Linealización por reglamentación de una clase de manipulador con un número arbitrario de actuadores" estudia las salidas planas para un tipo particular de brazo robótico. El TFM que parte de un trabajo anterior en que se trataron brazos robóticos con 1 y 2 actuadores, amplia los resultados para cualquier número de actuadores. Durante el TFM se estudian en que caso los sistemas son linealizables, ya sea por reglamentación estática o bien por prolongaciones, dando condiciones suficientes y necesarias en cada caso; En qualsevol sistema robòtic (i mecànic en general) totalment actuat es pot planificar i/o fer seguiment de trajectòries arbitràries. Tanmateix, si el sistema és sub-actuat, no totes les trajectòries es poden assolir. D'altra banda, la sub-actuació pot ajudar a reduir el cost del robot. La propietat que permet planificar i fer seguiment de trajectòries és la platiutd diferencial. L'objectiu d'aquest projecte és estudiar l'esmentada propietat per a una classe de sistemes robòtics sub-actuats que tenen una específica distribució de la massa. Hi ha resultats coneguts per al cas que el sistema tingui un o dos actuadors. Es tracta, doncs, de generalitzar els resultats a un nombre arbitrari de controls. Les eines que s'utilitzaran en aquest projecte són rudiments d'equacions diferencials i càlculs amb parèntesis de Lie. 2013-03-04T12:35:31Z Reyes Jiménez, Àlex Este trabajo final de máster titulado "Linealización por reglamentación de una clase de manipulador con un número arbitrario de actuadores" estudia las salidas planas para un tipo particular de brazo robótico. El TFM que parte de un trabajo anterior en que se trataron brazos robóticos con 1 y 2 actuadores, amplia los resultados para cualquier número de actuadores. Durante el TFM se estudian en que caso los sistemas son linealizables, ya sea por reglamentación estática o bien por prolongaciones, dando condiciones suficientes y necesarias en cada caso En qualsevol sistema robòtic (i mecànic en general) totalment actuat es pot planificar i/o fer seguiment de trajectòries arbitràries. Tanmateix, si el sistema és sub-actuat, no totes les trajectòries es poden assolir. D'altra banda, la sub-actuació pot ajudar a reduir el cost del robot. La propietat que permet planificar i fer seguiment de trajectòries és la platiutd diferencial. L'objectiu d'aquest projecte és estudiar l'esmentada propietat per a una classe de sistemes robòtics sub-actuats que tenen una específica distribució de la massa. Hi ha resultats coneguts per al cas que el sistema tingui un o dos actuadors. Es tracta, doncs, de generalitzar els resultats a un nombre arbitrari de controls. Les eines que s'utilitzaran en aquest projecte són rudiments d'equacions diferencials i càlculs amb parèntesis de Lie. Past, present and challenges in Yang-Mills theory http://hdl.handle.net/2099.1/16489 Past, present and challenges in Yang-Mills theory Tejedor Sánchez, Jose Luís Aquest treball tracta sobre les teories gauge de la física teòrica, la seva evolució des de el primer treball de Weyl (1918) fins als nostres dies tenint en compte el treball fonamental elaborat el 1954 per Yang i Mills que extenia el concepte a les teories no abelianes. S'incideix en la seva aplicació al Model Standard de partícules elementals i es consideren alguns tòpics com els monòpols magnètics i la supersimetria. Per últim es posa de manifest la falta de demostració rigorosa d'una teoria de Yang Mills en 4 dimensions. . The aim of this project is a study of the Yang-Mills equations, and in particular its derivation from a variational principle, and a presentatioin of the phisical and mathematical problems that stand on the way of its quatization 2012-11-07T12:08:11Z Tejedor Sánchez, Jose Luís Aquest treball tracta sobre les teories gauge de la física teòrica, la seva evolució des de el primer treball de Weyl (1918) fins als nostres dies tenint en compte el treball fonamental elaborat el 1954 per Yang i Mills que extenia el concepte a les teories no abelianes. S'incideix en la seva aplicació al Model Standard de partícules elementals i es consideren alguns tòpics com els monòpols magnètics i la supersimetria. Per últim es posa de manifest la falta de demostració rigorosa d'una teoria de Yang Mills en 4 dimensions. . The aim of this project is a study of the Yang-Mills equations, and in particular its derivation from a variational principle, and a presentatioin of the phisical and mathematical problems that stand on the way of its quatization Methods for solving 1D Stefan problems with application to contact melting http://hdl.handle.net/2099.1/14984 Methods for solving 1D Stefan problems with application to contact melting De Decker, Michelle Contact melting is the process during which a phase change material is placed in contact with a substrate that is at a temperature above the phase change temperature. This leads to melting of the phase change material and a °owing liquid layer which is being squeezed out due to the weight of the solid pushing down upon it. This process arises in many engineering problems such as the production of construction materials and alloys. Other uses include thermal storage systems that rely on the storage of energy as latent heat, which is released upon melting. The mathematical modelling of contact melting involves two heat equations, one in the solid and one in the liquid phase, coupled with the Navier-Stokes equations for the °ow in the melt, a Stefan condition at the phase change interface and a force balance between the weight of the solid and the countering pressure in the melt. Consequently, in this thesis we will deal with the one dimensional heat equation and move on to the problem commonly known as the Stefan problem - or moving boundary problem. We will consider both one and two phase problems and eventually look at a contact melting problem. The focus of this dissertation is to develop a three dimensional model to describe a contact melting process and to develop and apply an approximation method with minimal error. 2012-04-02T10:39:48Z De Decker, Michelle Contact melting is the process during which a phase change material is placed in contact with a substrate that is at a temperature above the phase change temperature. This leads to melting of the phase change material and a °owing liquid layer which is being squeezed out due to the weight of the solid pushing down upon it. This process arises in many engineering problems such as the production of construction materials and alloys. Other uses include thermal storage systems that rely on the storage of energy as latent heat, which is released upon melting. The mathematical modelling of contact melting involves two heat equations, one in the solid and one in the liquid phase, coupled with the Navier-Stokes equations for the °ow in the melt, a Stefan condition at the phase change interface and a force balance between the weight of the solid and the countering pressure in the melt. Consequently, in this thesis we will deal with the one dimensional heat equation and move on to the problem commonly known as the Stefan problem - or moving boundary problem. We will consider both one and two phase problems and eventually look at a contact melting problem. The focus of this dissertation is to develop a three dimensional model to describe a contact melting process and to develop and apply an approximation method with minimal error. Smoothing and untangling of meshes on parameterized surfaces http://hdl.handle.net/2099.1/14565 Smoothing and untangling of meshes on parameterized surfaces Gargallo Peiró, Abel The aim of this work is to develop a simultaneous smoothing-untangling procedure for parametrized surfaces. To this end we will extend the definition of the quality metrics for planar meshes to parametrized surface meshes. Then, we will develop a minimization approach on the parametric space that will allow smoothing and untangling the surface mesh. To this end, first we will increase the robustness of the standard untangling techniques. Then, we will focus on the improvement of the computational efficiency of the developed approach. Finally, several examples will be presented in order to illustrate the capabilities of the proposed method.. L'objectiu d'aquest projecte és la millora de la qualitat de malles d'elements finits generades sobre superficies paramètriques. Aquestes malles podran estar formades per triangles o quadrilàters. Aquesta millora es realitzarà mitjançant la minimització numèrica d'una mesura de la qualitat de la malla definida en l'espai paramètric 2012-03-07T16:28:24Z Gargallo Peiró, Abel The aim of this work is to develop a simultaneous smoothing-untangling procedure for parametrized surfaces. To this end we will extend the definition of the quality metrics for planar meshes to parametrized surface meshes. Then, we will develop a minimization approach on the parametric space that will allow smoothing and untangling the surface mesh. To this end, first we will increase the robustness of the standard untangling techniques. Then, we will focus on the improvement of the computational efficiency of the developed approach. Finally, several examples will be presented in order to illustrate the capabilities of the proposed method.. L'objectiu d'aquest projecte és la millora de la qualitat de malles d'elements finits generades sobre superficies paramètriques. Aquestes malles podran estar formades per triangles o quadrilàters. Aquesta millora es realitzarà mitjançant la minimització numèrica d'una mesura de la qualitat de la malla definida en l'espai paramètric Mathematical methods of signal processing http://hdl.handle.net/2099.1/14560 Mathematical methods of signal processing Sayols Baixeras, Narcís The aim of this project is to present in a systematic way the more relevant mathematical methods of signal processing, and to explore how they are applied to speech and image precessing. After explaining the more common parts of a standard course in signal processing, we put special emphasis in two new tools that have played a significant role in signal processing in the past few years: pattern theory and wavelet theory. Finally, we use all these techniques to implement an algorithm that detects the wallpaper group of a plane mosaic taking an image of it as input and an algorithm that returns the phoneme sequence of a speech signal. The material in this memory can be grouped in two parts. The first part, consisting of the first six chapters, deals with the theoretical foundation of signal processing. It also includes materials related to plane symmetry groups. The second part, consisting of the last two chapters, is focussed on the applications. 2012-03-07T12:38:00Z Sayols Baixeras, Narcís The aim of this project is to present in a systematic way the more relevant mathematical methods of signal processing, and to explore how they are applied to speech and image precessing. After explaining the more common parts of a standard course in signal processing, we put special emphasis in two new tools that have played a significant role in signal processing in the past few years: pattern theory and wavelet theory. Finally, we use all these techniques to implement an algorithm that detects the wallpaper group of a plane mosaic taking an image of it as input and an algorithm that returns the phoneme sequence of a speech signal. The material in this memory can be grouped in two parts. The first part, consisting of the first six chapters, deals with the theoretical foundation of signal processing. It also includes materials related to plane symmetry groups. The second part, consisting of the last two chapters, is focussed on the applications. Transfer of energy in the nonlinear Schrödinger equationon http://hdl.handle.net/2099.1/14433 Transfer of energy in the nonlinear Schrödinger equationon Simon López, Adrià This work aims to use tools of Dynamical Systems to prove that the nonlinear Schr odinger equation presents a global instability. To do that, we rst reduce this PDE to a system of ODE's of dimension N which we call the Toy Model System. Consequently our new purpose is to study the existence of instability in a system of ODE's. The way of proving it will consist in taking invariant objects and showing that there exists a solution that ows near all of them. This strategy resembles Arnold Di usion. The contribution of this work is to use the so-called Silnikov coordinates to prove this result when N = 3. However, we detect a problem when we ow around one of this invariant objects (that, in suitable coordinates, can be seen as a saddle) that prevents us from completing the proof in the expected way.; L'objectiu d'aquest treball és demostrar una inestabilitat que presenta l'equació de Schrödinger no lineal, convenientment reduïda a un sistema finit d'EDO's, mitjançant tècniques pròpies dels Sistemes Dinàmics, com seria l'ús de les anomenades coordenades de Shilnikov.. Es tracta d'utilitzar tècniques avanzades de sistemes dinàmics, concretamet el mètode de Shilnikov, per tal de trobar, de manera constructiva, solucions de l'equació de Schrodinger nolineal "cubic defocusing" amb transferència considerable d'energia i, en particular, aplicar-ho al model de l'article: J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao. Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation. Invent. Math., 181(1):39¿113, 2010. 2012-02-28T12:48:43Z Simon López, Adrià This work aims to use tools of Dynamical Systems to prove that the nonlinear Schr odinger equation presents a global instability. To do that, we rst reduce this PDE to a system of ODE's of dimension N which we call the Toy Model System. Consequently our new purpose is to study the existence of instability in a system of ODE's. The way of proving it will consist in taking invariant objects and showing that there exists a solution that ows near all of them. This strategy resembles Arnold Di usion. The contribution of this work is to use the so-called Silnikov coordinates to prove this result when N = 3. However, we detect a problem when we ow around one of this invariant objects (that, in suitable coordinates, can be seen as a saddle) that prevents us from completing the proof in the expected way. L'objectiu d'aquest treball és demostrar una inestabilitat que presenta l'equació de Schrödinger no lineal, convenientment reduïda a un sistema finit d'EDO's, mitjançant tècniques pròpies dels Sistemes Dinàmics, com seria l'ús de les anomenades coordenades de Shilnikov.. Es tracta d'utilitzar tècniques avanzades de sistemes dinàmics, concretamet el mètode de Shilnikov, per tal de trobar, de manera constructiva, solucions de l'equació de Schrodinger nolineal "cubic defocusing" amb transferència considerable d'energia i, en particular, aplicar-ho al model de l'article: J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao. Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation. Invent. Math., 181(1):39¿113, 2010.