Fixed-point theorems in cones
Authorship
A.O.T.
Bachelor of Mathematics
A.O.T.
Bachelor of Mathematics
Defense date
07.04.2024 12:40
07.04.2024 12:40
Summary
This document comprehends several fixed point results on cones defined on Banach spaces and some of their applications to study the existence of solution to nonlinear differential equations. The first chapter establishes fundamental concepts such as Banach space, cone or Green's function, as weel as helpful results for next parts. Moreover, it is provided an axiomatically definition of fixed point index. Relaying on this last idea, in the second chapter we propose some fixed point theorems, highlighting the classic Krasnoselkii's Theorem. The last chapter includes different applications in which the previous results are used to study the existence and non-existence of one or multiple solutions to a differential problem in different regularity situations.
This document comprehends several fixed point results on cones defined on Banach spaces and some of their applications to study the existence of solution to nonlinear differential equations. The first chapter establishes fundamental concepts such as Banach space, cone or Green's function, as weel as helpful results for next parts. Moreover, it is provided an axiomatically definition of fixed point index. Relaying on this last idea, in the second chapter we propose some fixed point theorems, highlighting the classic Krasnoselkii's Theorem. The last chapter includes different applications in which the previous results are used to study the existence and non-existence of one or multiple solutions to a differential problem in different regularity situations.
Direction
LOPEZ SOMOZA, LUCIA (Tutorships)
LOPEZ SOMOZA, LUCIA (Tutorships)
Court
OTERO ESPINAR, MARIA VICTORIA (Chairman)
GONZALEZ DIAZ, JULIO (Secretary)
Jeremías López, Ana (Member)
OTERO ESPINAR, MARIA VICTORIA (Chairman)
GONZALEZ DIAZ, JULIO (Secretary)
Jeremías López, Ana (Member)
Regression models for zero-truncated, zero-inflated and hurdle count data
Authorship
M.S.M.
Bachelor of Mathematics
M.S.M.
Bachelor of Mathematics
Defense date
07.04.2024 17:10
07.04.2024 17:10
Summary
In this work we have studied regression models in which the response variable is a count that presents anomalies in the zero value, whether due to absence, defect or excess, that force us to consider different types of distributions other than the Poisson or the Negative Binomial distribution. We have studied the zero-truncated, zero-inflated and hurdle distributions, that provide us with a way to act in the event of those anomalies, and we have built regression models for each one of those distributions, illustrating their application with examples.
In this work we have studied regression models in which the response variable is a count that presents anomalies in the zero value, whether due to absence, defect or excess, that force us to consider different types of distributions other than the Poisson or the Negative Binomial distribution. We have studied the zero-truncated, zero-inflated and hurdle distributions, that provide us with a way to act in the event of those anomalies, and we have built regression models for each one of those distributions, illustrating their application with examples.
Direction
SANCHEZ SELLERO, CESAR ANDRES (Tutorships)
SANCHEZ SELLERO, CESAR ANDRES (Tutorships)
Court
OTERO ESPINAR, MARIA VICTORIA (Chairman)
GONZALEZ DIAZ, JULIO (Secretary)
Jeremías López, Ana (Member)
OTERO ESPINAR, MARIA VICTORIA (Chairman)
GONZALEZ DIAZ, JULIO (Secretary)
Jeremías López, Ana (Member)
Statistical analysis of statistical evidence of climate change
Authorship
A.S.S.
Bachelor of Mathematics
A.S.S.
Bachelor of Mathematics
Defense date
07.04.2024 17:50
07.04.2024 17:50
Summary
Nonparametric regression is a very interesting alternative to the usual linear regression models, since it provides a flexibility from which the user can benefit enormously in multiple situations. As with everything, nonparametric regression also has its disadvantages, and, as we shall see, the flexibility that one benefits from ends up paying for it. Throughout the paper we will work on the aforementioned nonparametric regression, focusing especially on the local linear estimator and its properties, as well as on the effect of the bandwidth parameter. In addition, we will discuss a tremendously useful tool connected to both the properties of the local linear estimator and the effect of the bandwidth parameter, called SiZer. To conclude the work, we will take data of great interest and topicality, such as those corresponding to climatological variables, and making use of everything seen throughout the study, we will try to carry out an analysis on them with the aim of supporting the existence of climate change, always bearing in mind that the study is still part of a final degree project and a rigorous study of climate change completely exceeds the objectives of the same.
Nonparametric regression is a very interesting alternative to the usual linear regression models, since it provides a flexibility from which the user can benefit enormously in multiple situations. As with everything, nonparametric regression also has its disadvantages, and, as we shall see, the flexibility that one benefits from ends up paying for it. Throughout the paper we will work on the aforementioned nonparametric regression, focusing especially on the local linear estimator and its properties, as well as on the effect of the bandwidth parameter. In addition, we will discuss a tremendously useful tool connected to both the properties of the local linear estimator and the effect of the bandwidth parameter, called SiZer. To conclude the work, we will take data of great interest and topicality, such as those corresponding to climatological variables, and making use of everything seen throughout the study, we will try to carry out an analysis on them with the aim of supporting the existence of climate change, always bearing in mind that the study is still part of a final degree project and a rigorous study of climate change completely exceeds the objectives of the same.
Direction
RODRIGUEZ CASAL, ALBERTO (Tutorships)
RODRIGUEZ CASAL, ALBERTO (Tutorships)
Court
OTERO ESPINAR, MARIA VICTORIA (Chairman)
GONZALEZ DIAZ, JULIO (Secretary)
Jeremías López, Ana (Member)
OTERO ESPINAR, MARIA VICTORIA (Chairman)
GONZALEZ DIAZ, JULIO (Secretary)
Jeremías López, Ana (Member)
Burnside's problem and its variations
Authorship
M.R.F.
Bachelor of Mathematics
M.R.F.
Bachelor of Mathematics
Defense date
07.04.2024 16:30
07.04.2024 16:30
Summary
In this work, we will deal in depth with the General Burnside Problem, proving it has a negative solution and giving examples where the answer is afirmative. First of all, we shall recall some basic notions of groups and start working with commuters and some of their properties. After this, we will prove that groups with exponents 2, 3 and 4, such as linear groups, are positive solutions to our problem. We will also talk about other variations of the Burnside Problem and possible formulations. Finally, we will construct three counterexamples to the problem after introducing ourselves to different mathematical concepts such as associative algebras or graph theory. These will provide us with the necessary knowledge to study the Golod-Shafarevich, Gupta-Sidki, and Grigorchuk groups properly.
In this work, we will deal in depth with the General Burnside Problem, proving it has a negative solution and giving examples where the answer is afirmative. First of all, we shall recall some basic notions of groups and start working with commuters and some of their properties. After this, we will prove that groups with exponents 2, 3 and 4, such as linear groups, are positive solutions to our problem. We will also talk about other variations of the Burnside Problem and possible formulations. Finally, we will construct three counterexamples to the problem after introducing ourselves to different mathematical concepts such as associative algebras or graph theory. These will provide us with the necessary knowledge to study the Golod-Shafarevich, Gupta-Sidki, and Grigorchuk groups properly.
Direction
LADRA GONZALEZ, MANUEL EULOGIO (Tutorships)
PAEZ GUILLAN, MARIA PILAR (Co-tutorships)
LADRA GONZALEZ, MANUEL EULOGIO (Tutorships)
PAEZ GUILLAN, MARIA PILAR (Co-tutorships)
Court
OTERO ESPINAR, MARIA VICTORIA (Chairman)
GONZALEZ DIAZ, JULIO (Secretary)
Jeremías López, Ana (Member)
OTERO ESPINAR, MARIA VICTORIA (Chairman)
GONZALEZ DIAZ, JULIO (Secretary)
Jeremías López, Ana (Member)
Gödel's incompleteness theorems
Authorship
A.C.V.
Bachelor of Mathematics
A.C.V.
Bachelor of Mathematics
Defense date
07.04.2024 09:30
07.04.2024 09:30
Summary
Gödel's incompleteness theorems, formulated by the Austrian logician Kurt Gödel in 1931, had a revolutionary impact on mathematics and philosophy. The first theorem establishes that in any sufficiently powerful logical system that includes the arithmetic of natural numbers, there will always be true statements that cannot be proven within that system. The second theorem denies the possibility of proving its own consistency. In this work, we will address the study of logical systems, which are formal frameworks where axioms can be expressed and theorems be proven. We will show completeness in certain of these systems in order to study ordinary arithmetic, with the ultimate goal of proving both incompleteness theorems.
Gödel's incompleteness theorems, formulated by the Austrian logician Kurt Gödel in 1931, had a revolutionary impact on mathematics and philosophy. The first theorem establishes that in any sufficiently powerful logical system that includes the arithmetic of natural numbers, there will always be true statements that cannot be proven within that system. The second theorem denies the possibility of proving its own consistency. In this work, we will address the study of logical systems, which are formal frameworks where axioms can be expressed and theorems be proven. We will show completeness in certain of these systems in order to study ordinary arithmetic, with the ultimate goal of proving both incompleteness theorems.
Direction
FERNANDEZ TOJO, FERNANDO ADRIAN (Tutorships)
FERNANDEZ TOJO, FERNANDO ADRIAN (Tutorships)
Court
DIAZ RAMOS, JOSE CARLOS (Chairman)
COSTOYA RAMOS, MARIA CRISTINA (Secretary)
Rodríguez López, Rosana (Member)
DIAZ RAMOS, JOSE CARLOS (Chairman)
COSTOYA RAMOS, MARIA CRISTINA (Secretary)
Rodríguez López, Rosana (Member)
Finite fields
Authorship
M.A.F.C.
Bachelor of Mathematics
M.A.F.C.
Bachelor of Mathematics
Defense date
07.18.2024 10:30
07.18.2024 10:30
Summary
In this work, the algebraic structure of finite field will be discussed, structure that has very particular characteristics and composition. It begins by briefly explaining the group, ring and field structures, to give rise to the first examples and properties of finite fields. After that, field extensions will be explained to get fully into the characterization of finite fields and to talk about the relation between their structure and their cardinal. In turn, the different ways to represent finite fields are described, some of them are more theoretical and other more practical. Subsequently, the work will talk about properties that will approximate finite fields theory to Galois theory, and will give information about the vector space structure that forms a finite field over its prime subfield. Finally, the work exposes the application from finite fields theory to coding theory, expressing its practical utility and giving as an example the reading code of the digital compact disc.
In this work, the algebraic structure of finite field will be discussed, structure that has very particular characteristics and composition. It begins by briefly explaining the group, ring and field structures, to give rise to the first examples and properties of finite fields. After that, field extensions will be explained to get fully into the characterization of finite fields and to talk about the relation between their structure and their cardinal. In turn, the different ways to represent finite fields are described, some of them are more theoretical and other more practical. Subsequently, the work will talk about properties that will approximate finite fields theory to Galois theory, and will give information about the vector space structure that forms a finite field over its prime subfield. Finally, the work exposes the application from finite fields theory to coding theory, expressing its practical utility and giving as an example the reading code of the digital compact disc.
Direction
ALONSO TARRIO, LEOVIGILDO (Tutorships)
ALONSO TARRIO, LEOVIGILDO (Tutorships)
Court
DIAZ RAMOS, JOSE CARLOS (Chairman)
COSTOYA RAMOS, MARIA CRISTINA (Secretary)
Rodríguez López, Rosana (Member)
DIAZ RAMOS, JOSE CARLOS (Chairman)
COSTOYA RAMOS, MARIA CRISTINA (Secretary)
Rodríguez López, Rosana (Member)
Carathéodory solutions for discontinuous ordinary differential equations
Authorship
L.I.M.
Bachelor of Mathematics
L.I.M.
Bachelor of Mathematics
Defense date
07.04.2024 10:30
07.04.2024 10:30
Summary
In this work, the basic theory of discontinuous ordinary differential equations is studied. First, a new notion of solution, the Carathéodory solution, must be defined. To achieve this, it is necessary to introduce certain characteristics of absolutely continuous functions. The two fundamental theorems presented prove the (local) existence and uniqueness of Carathéodory solutions. Considering the spatial dimension to be one, both extremal solutions and sub and supersolutions are also of great interest for the problems addressed. After presenting the conditions related to Carathéodory, new hypotheses are considered that suggest the possibility of assuming discontinuities not only with respect to the independent variable but also with respect to the dependent variable.
In this work, the basic theory of discontinuous ordinary differential equations is studied. First, a new notion of solution, the Carathéodory solution, must be defined. To achieve this, it is necessary to introduce certain characteristics of absolutely continuous functions. The two fundamental theorems presented prove the (local) existence and uniqueness of Carathéodory solutions. Considering the spatial dimension to be one, both extremal solutions and sub and supersolutions are also of great interest for the problems addressed. After presenting the conditions related to Carathéodory, new hypotheses are considered that suggest the possibility of assuming discontinuities not only with respect to the independent variable but also with respect to the dependent variable.
Direction
LOPEZ POUSO, RODRIGO (Tutorships)
LOPEZ POUSO, RODRIGO (Tutorships)
Court
DIAZ RAMOS, JOSE CARLOS (Chairman)
COSTOYA RAMOS, MARIA CRISTINA (Secretary)
Rodríguez López, Rosana (Member)
DIAZ RAMOS, JOSE CARLOS (Chairman)
COSTOYA RAMOS, MARIA CRISTINA (Secretary)
Rodríguez López, Rosana (Member)
Cohomological LS Category
Authorship
A.M.V.
Bachelor of Mathematics
A.M.V.
Bachelor of Mathematics
Defense date
07.04.2024 11:00
07.04.2024 11:00
Summary
The Lusternik-Schnirelmann category of a topological space is a well-known homotopy invariant, yet difficult to compute. The aim of this dissertation is to study a lower bound: the cohomological category. To achieve this, we will introduce simplicial complexes as well as simplicial and singular (co)homology. Additionally, we will explore the simplicial category and its corresponding cohomological version, which can be calculated using computational methods for any simplicial complex.
The Lusternik-Schnirelmann category of a topological space is a well-known homotopy invariant, yet difficult to compute. The aim of this dissertation is to study a lower bound: the cohomological category. To achieve this, we will introduce simplicial complexes as well as simplicial and singular (co)homology. Additionally, we will explore the simplicial category and its corresponding cohomological version, which can be calculated using computational methods for any simplicial complex.
Direction
Macías Virgós, Enrique (Tutorships)
MOSQUERA LOIS, DAVID (Co-tutorships)
Macías Virgós, Enrique (Tutorships)
MOSQUERA LOIS, DAVID (Co-tutorships)
Court
DIAZ RAMOS, JOSE CARLOS (Chairman)
COSTOYA RAMOS, MARIA CRISTINA (Secretary)
Rodríguez López, Rosana (Member)
DIAZ RAMOS, JOSE CARLOS (Chairman)
COSTOYA RAMOS, MARIA CRISTINA (Secretary)
Rodríguez López, Rosana (Member)
Introduction to Lie algebras through examples
Authorship
A.R.V.
Bachelor of Mathematics
A.R.V.
Bachelor of Mathematics
Defense date
07.04.2024 11:30
07.04.2024 11:30
Summary
Lie algebras are a type of non-associative algebras that are strongly linked to geometry. In this work we will focus on the algebraic part of them, introducing the basic notions of this structures. We will use different clasic algebras such as the special linear algebra, sl(n,F), or the Heisenberg algebra, heis(n,F), to show concepts such as the solvability, nilpotence or semisimplicity among others. We will also give a small classification of the low dimensional Lie algebras, as well as certain results that characterize certain types of algebras.
Lie algebras are a type of non-associative algebras that are strongly linked to geometry. In this work we will focus on the algebraic part of them, introducing the basic notions of this structures. We will use different clasic algebras such as the special linear algebra, sl(n,F), or the Heisenberg algebra, heis(n,F), to show concepts such as the solvability, nilpotence or semisimplicity among others. We will also give a small classification of the low dimensional Lie algebras, as well as certain results that characterize certain types of algebras.
Direction
LADRA GONZALEZ, MANUEL EULOGIO (Tutorships)
PAEZ GUILLAN, MARIA PILAR (Co-tutorships)
LADRA GONZALEZ, MANUEL EULOGIO (Tutorships)
PAEZ GUILLAN, MARIA PILAR (Co-tutorships)
Court
DIAZ RAMOS, JOSE CARLOS (Chairman)
COSTOYA RAMOS, MARIA CRISTINA (Secretary)
Rodríguez López, Rosana (Member)
DIAZ RAMOS, JOSE CARLOS (Chairman)
COSTOYA RAMOS, MARIA CRISTINA (Secretary)
Rodríguez López, Rosana (Member)
Global geometry of curves
Authorship
C.R.O.
Bachelor of Mathematics
C.R.O.
Bachelor of Mathematics
Defense date
07.04.2024 12:00
07.04.2024 12:00
Summary
The objective of this work is to contextualize, state and prove some of the most relevant theorems of the global theory of plane curves from a differential geometry perspective. Thus, we will address the study of Hopf's Umlaufsatz, Jordan's closed curve theorem, the isoperimetric inequality, Fenchel's theorem and the four-vertex theorem. After a short introduction to the basic concepts of differential geometry of plane curves, we will present the necessary tools for the study of each of the mentioned results, to finally provide a proof of each of them. Such proofs will be eminently geometric, although in several cases they will have an important topological and analytical component.
The objective of this work is to contextualize, state and prove some of the most relevant theorems of the global theory of plane curves from a differential geometry perspective. Thus, we will address the study of Hopf's Umlaufsatz, Jordan's closed curve theorem, the isoperimetric inequality, Fenchel's theorem and the four-vertex theorem. After a short introduction to the basic concepts of differential geometry of plane curves, we will present the necessary tools for the study of each of the mentioned results, to finally provide a proof of each of them. Such proofs will be eminently geometric, although in several cases they will have an important topological and analytical component.
Direction
DOMINGUEZ VAZQUEZ, MIGUEL (Tutorships)
DOMINGUEZ VAZQUEZ, MIGUEL (Tutorships)
Court
DIAZ RAMOS, JOSE CARLOS (Chairman)
COSTOYA RAMOS, MARIA CRISTINA (Secretary)
Rodríguez López, Rosana (Member)
DIAZ RAMOS, JOSE CARLOS (Chairman)
COSTOYA RAMOS, MARIA CRISTINA (Secretary)
Rodríguez López, Rosana (Member)
Introduction to stochastic optimization
Authorship
A.P.P.
Double bachelor degree in Mathematics and Physics
A.P.P.
Double bachelor degree in Mathematics and Physics
Defense date
07.17.2024 11:30
07.17.2024 11:30
Summary
In this work, we introduce stochastic optimization, which studies mathematical programming problems with uncertain data. In the first chapter, fundamental concepts of statistics, probability, and mathematical programming are presented, necessary for understanding and explaining the foundations of this topic. Next, we address two-stage stochastic problems, analyzing their main properties. This study is divided into two parts, considering the stochastic components of the problem, which can be either discrete or continuous. Finally, a solution method for these problems known as the L-Shaped Method is presented. Its algorithm is analyzed in detail, focusing on two of its fundamental components, \optimality cuts and feasibility cuts. Additionally, practical examples using the statistical software R are included to illustrate its resolution and application.
In this work, we introduce stochastic optimization, which studies mathematical programming problems with uncertain data. In the first chapter, fundamental concepts of statistics, probability, and mathematical programming are presented, necessary for understanding and explaining the foundations of this topic. Next, we address two-stage stochastic problems, analyzing their main properties. This study is divided into two parts, considering the stochastic components of the problem, which can be either discrete or continuous. Finally, a solution method for these problems known as the L-Shaped Method is presented. Its algorithm is analyzed in detail, focusing on two of its fundamental components, \optimality cuts and feasibility cuts. Additionally, practical examples using the statistical software R are included to illustrate its resolution and application.
Direction
CASARES DE CAL, MARIA ANGELES (Tutorships)
CASARES DE CAL, MARIA ANGELES (Tutorships)
Court
GONZALEZ MANTEIGA, WENCESLAO (Chairman)
PAEZ GUILLAN, MARIA PILAR (Secretary)
ALVAREZ DIOS, JOSE ANTONIO (Member)
GONZALEZ MANTEIGA, WENCESLAO (Chairman)
PAEZ GUILLAN, MARIA PILAR (Secretary)
ALVAREZ DIOS, JOSE ANTONIO (Member)
The Kalton-Peck space
Authorship
C.F.L.
Double bachelor degree in Mathematics and Physics
C.F.L.
Double bachelor degree in Mathematics and Physics
Defense date
07.16.2024 11:00
07.16.2024 11:00
Summary
This document will present the Kalton-Peck space, a solution of the Palais problem. This problem asks whether there exists a Banach space X, which is not Hilbert, such that it contains a subspace isomorphic to a Hilbert space H, such that X/H is also isomorphic to a Hilbert space. For this purpose, we will define twisted sums, which are the ideal setting for solving this problem. In addition, various concepts related to functional analysis will also be introduced, such as quasi-normed spaces, B-convexity or uniform convexity. These will be necessary to prove that the Kalton-Peck space is really a solution to this problem. Finally, some of the properties of this space will also be studied. In particular, it will be shown that it admits a Schauder basis and the form of its dual space.
This document will present the Kalton-Peck space, a solution of the Palais problem. This problem asks whether there exists a Banach space X, which is not Hilbert, such that it contains a subspace isomorphic to a Hilbert space H, such that X/H is also isomorphic to a Hilbert space. For this purpose, we will define twisted sums, which are the ideal setting for solving this problem. In addition, various concepts related to functional analysis will also be introduced, such as quasi-normed spaces, B-convexity or uniform convexity. These will be necessary to prove that the Kalton-Peck space is really a solution to this problem. Finally, some of the properties of this space will also be studied. In particular, it will be shown that it admits a Schauder basis and the form of its dual space.
Direction
LOSADA RODRIGUEZ, JORGE (Tutorships)
LOSADA RODRIGUEZ, JORGE (Tutorships)
Court
FEBRERO BANDE, MANUEL (Chairman)
BUEDO FERNANDEZ, SEBASTIAN (Secretary)
RODRIGUEZ GARCIA, JERONIMO (Member)
FEBRERO BANDE, MANUEL (Chairman)
BUEDO FERNANDEZ, SEBASTIAN (Secretary)
RODRIGUEZ GARCIA, JERONIMO (Member)
Mathematical methods of Artificial Intelligence
Authorship
V.O.Z.
Double bachelor degree in Mathematics and Physics
V.O.Z.
Double bachelor degree in Mathematics and Physics
Defense date
07.16.2024 12:30
07.16.2024 12:30
Summary
Artificial intelligence (AI) has been one of the major technological advancements of recent years. In this paper, we will see how mathematics play a crucial role in its development. We will begin by formally introducing artificial neurons and neural networks, starting from their analogy with biological neurons. The Universal Approximation Theorem and its subsequent generalization will be demonstrated, showing how a neural network can approximate any continuous function under minimally restrictive conditions regarding its architecture. Additionally, the backpropagation algorithm is introduced. Finally, we will see a few examples of AI applications in different fields.
Artificial intelligence (AI) has been one of the major technological advancements of recent years. In this paper, we will see how mathematics play a crucial role in its development. We will begin by formally introducing artificial neurons and neural networks, starting from their analogy with biological neurons. The Universal Approximation Theorem and its subsequent generalization will be demonstrated, showing how a neural network can approximate any continuous function under minimally restrictive conditions regarding its architecture. Additionally, the backpropagation algorithm is introduced. Finally, we will see a few examples of AI applications in different fields.
Direction
Nieto Roig, Juan José (Tutorships)
Nieto Roig, Juan José (Tutorships)
Court
FEBRERO BANDE, MANUEL (Chairman)
BUEDO FERNANDEZ, SEBASTIAN (Secretary)
RODRIGUEZ GARCIA, JERONIMO (Member)
FEBRERO BANDE, MANUEL (Chairman)
BUEDO FERNANDEZ, SEBASTIAN (Secretary)
RODRIGUEZ GARCIA, JERONIMO (Member)
Gödel's constructible universe
Authorship
P.S.F.
Double bachelor degree in Mathematics and Physics
P.S.F.
Double bachelor degree in Mathematics and Physics
Defense date
07.16.2024 18:30
07.16.2024 18:30
Summary
Most mathematical theories can be formalized within the ZFC system, which is a first-order logic theory. Gödel's Second Incompleteness Theorem prevents us from proving its consistency within ZFC itself, but it does not impose restrictions on relative consistency proofs. This means that, assuming a formal theory is consistent, we can prove the consistency of another. In our case, we will assume the consistency of a subset of ZFC axioms and proceed to prove the relative consistency of this theory with the remaining axioms. In fact, we will also demonstrate the relative consistency of ZFC with the Generalized Continuum Hypothesis. The fundamental tool in obtaining these results is model theory, which formalizes the intuitive concept of interpretation of a language. In this context, Gödel's constructible universe is a possible interpretation of ZFC set theory.
Most mathematical theories can be formalized within the ZFC system, which is a first-order logic theory. Gödel's Second Incompleteness Theorem prevents us from proving its consistency within ZFC itself, but it does not impose restrictions on relative consistency proofs. This means that, assuming a formal theory is consistent, we can prove the consistency of another. In our case, we will assume the consistency of a subset of ZFC axioms and proceed to prove the relative consistency of this theory with the remaining axioms. In fact, we will also demonstrate the relative consistency of ZFC with the Generalized Continuum Hypothesis. The fundamental tool in obtaining these results is model theory, which formalizes the intuitive concept of interpretation of a language. In this context, Gödel's constructible universe is a possible interpretation of ZFC set theory.
Direction
FERNANDEZ TOJO, FERNANDO ADRIAN (Tutorships)
FERNANDEZ TOJO, FERNANDO ADRIAN (Tutorships)
Court
FEBRERO BANDE, MANUEL (Chairman)
BUEDO FERNANDEZ, SEBASTIAN (Secretary)
RODRIGUEZ GARCIA, JERONIMO (Member)
FEBRERO BANDE, MANUEL (Chairman)
BUEDO FERNANDEZ, SEBASTIAN (Secretary)
RODRIGUEZ GARCIA, JERONIMO (Member)
Sum of squares and modular forms
Authorship
J.C.R.G.
Double bachelor degree in Mathematics and Physics
J.C.R.G.
Double bachelor degree in Mathematics and Physics
Defense date
09.11.2024 16:45
09.11.2024 16:45
Summary
The aim of this Final Degree Project is to answer some of the classic questions in Number Theory, such as which positive integers can be written as the sum of two squares or whether all positive integers are the sum of four squares. These questions are addressed from the point of view of Complex Analysis by means of the theory of modular forms. These are functions defined in the upper half-plane that admit different symmetries. Specifically, the Jacobi theta function will be used. First, definitions that are of interest for the comprehension of subsequent arguments are introduced, together with a series of results that are used when answering these questions. Then, the Jacobi theta function is defined, together with its properties that will later be used. Then, the proofs of the sum of two squares theorem, the sum of four squares and the sum of eight squares theorem according to Jacobi are presented, using the theory of modular forms and Jacobi theta function to prove the equivalence of the structural properties of the latter with other functions defined specifically to prove each theorem. Furthermore, an idea of the proof of the sum of three squares theorem, the treatment of which is more complex, is presented.
The aim of this Final Degree Project is to answer some of the classic questions in Number Theory, such as which positive integers can be written as the sum of two squares or whether all positive integers are the sum of four squares. These questions are addressed from the point of view of Complex Analysis by means of the theory of modular forms. These are functions defined in the upper half-plane that admit different symmetries. Specifically, the Jacobi theta function will be used. First, definitions that are of interest for the comprehension of subsequent arguments are introduced, together with a series of results that are used when answering these questions. Then, the Jacobi theta function is defined, together with its properties that will later be used. Then, the proofs of the sum of two squares theorem, the sum of four squares and the sum of eight squares theorem according to Jacobi are presented, using the theory of modular forms and Jacobi theta function to prove the equivalence of the structural properties of the latter with other functions defined specifically to prove each theorem. Furthermore, an idea of the proof of the sum of three squares theorem, the treatment of which is more complex, is presented.
Direction
Cao Labora, Daniel (Tutorships)
RIVERO SALGADO, OSCAR (Co-tutorships)
Cao Labora, Daniel (Tutorships)
RIVERO SALGADO, OSCAR (Co-tutorships)
Court
VIAÑO REY, JUAN MANUEL (Chairman)
Rodríguez López, Jorge (Secretary)
CARBALLES VAZQUEZ, JOSE MANUEL (Member)
VIAÑO REY, JUAN MANUEL (Chairman)
Rodríguez López, Jorge (Secretary)
CARBALLES VAZQUEZ, JOSE MANUEL (Member)
Notions of quantum cryptography
Authorship
R.A.R.
Double bachelor degree in Mathematics and Physics
R.A.R.
Double bachelor degree in Mathematics and Physics
Defense date
09.12.2024 16:00
09.12.2024 16:00
Summary
Quantum communication spawns contemporaneously to classical Information Theory, with a bigger potential for computations but a handicap when it comes to physical transmission. The present manuscript seeks to compare these two ways of sending information and to present the mathematical formalism behind quantum mechanics, as well as the most relevant protocols of this newly adapted cryptography and the first quantum error correcting codes.
Quantum communication spawns contemporaneously to classical Information Theory, with a bigger potential for computations but a handicap when it comes to physical transmission. The present manuscript seeks to compare these two ways of sending information and to present the mathematical formalism behind quantum mechanics, as well as the most relevant protocols of this newly adapted cryptography and the first quantum error correcting codes.
Direction
GAGO COUSO, FELIPE (Tutorships)
GAGO COUSO, FELIPE (Tutorships)
Court
OTERO ESPINAR, MARIA VICTORIA (Chairman)
GONZALEZ DIAZ, JULIO (Secretary)
Jeremías López, Ana (Member)
OTERO ESPINAR, MARIA VICTORIA (Chairman)
GONZALEZ DIAZ, JULIO (Secretary)
Jeremías López, Ana (Member)
Persistent homology in 3-manifolds
Authorship
P.T.M.
Double bachelor degree in Mathematics and Physics
P.T.M.
Double bachelor degree in Mathematics and Physics
Defense date
07.17.2024 12:15
07.17.2024 12:15
Summary
We begin with an introduction to topological data analysis, presenting the basic tools of this discipline: persistent homology, persistence diagrams and landscapes, as well as the stability theorem, among others. Specifically, we are interested in the application of persistent homology to finite samples of points on a manifold, using the construction of simplicial complexes on these samples, due to their potential to generate metric invariants. Next, we cover the fundamentals of hyperbolic geometry, studying some of the classical models of hyperbolic space, which is the universal cover of any hyperbolic manifold, focusing on the 3-dimensional case. The main objective of this section is to demonstrate Mostow's Rigidity Theorem in the compact case. To this end, we present some concepts and results used in the proof. As a consequence of this theorem, in the case of 3-dimensional hyperbolic manifolds, the metric is a topological invariant. Therefore, non-homeomorphic hyperbolic manifolds can be distinguished using metric invariants, such as those provided by persistent homology. Finally, we will put the previous techniques into practice by developing a program that samples random points on compact orientable 3-dimensional hyperbolic manifolds, calculates the corresponding persistence diagrams and landscapes, and compares the results obtained for any pair of given hyperbolic manifolds using hypothesis testing, with the aim of topologically distinguishing them with a certain degree of confidence.
We begin with an introduction to topological data analysis, presenting the basic tools of this discipline: persistent homology, persistence diagrams and landscapes, as well as the stability theorem, among others. Specifically, we are interested in the application of persistent homology to finite samples of points on a manifold, using the construction of simplicial complexes on these samples, due to their potential to generate metric invariants. Next, we cover the fundamentals of hyperbolic geometry, studying some of the classical models of hyperbolic space, which is the universal cover of any hyperbolic manifold, focusing on the 3-dimensional case. The main objective of this section is to demonstrate Mostow's Rigidity Theorem in the compact case. To this end, we present some concepts and results used in the proof. As a consequence of this theorem, in the case of 3-dimensional hyperbolic manifolds, the metric is a topological invariant. Therefore, non-homeomorphic hyperbolic manifolds can be distinguished using metric invariants, such as those provided by persistent homology. Finally, we will put the previous techniques into practice by developing a program that samples random points on compact orientable 3-dimensional hyperbolic manifolds, calculates the corresponding persistence diagrams and landscapes, and compares the results obtained for any pair of given hyperbolic manifolds using hypothesis testing, with the aim of topologically distinguishing them with a certain degree of confidence.
Direction
Álvarez López, Jesús Antonio (Tutorships)
Meniño Cotón, Carlos (Co-tutorships)
Álvarez López, Jesús Antonio (Tutorships)
Meniño Cotón, Carlos (Co-tutorships)
Court
TORRES LOPERA, JUAN FRANCISCO (Chairman)
CONDE AMBOAGE, MERCEDES (Secretary)
López Pouso, Óscar (Member)
TORRES LOPERA, JUAN FRANCISCO (Chairman)
CONDE AMBOAGE, MERCEDES (Secretary)
López Pouso, Óscar (Member)
Differential geometry of ruled surfaces and their application in architecture
Authorship
B.M.P.S.
Double bachelor degree in Mathematics and Physics
B.M.P.S.
Double bachelor degree in Mathematics and Physics
Defense date
07.16.2024 12:15
07.16.2024 12:15
Summary
A surface is called a ruled surface if through every point there is at least one straight line that lies on the surface. The aim of this project is to study this kind of surfaces and their properties in the context of differential geometry. In addition, we will consider a specific type of ruled surfaces which have null curvature, known as developable surfaces. Finally, we will examine their implementation in architecture, focusing on the analysis of the work done by architects Antoni Gaudí, Félix Candela, Santiago Calatrava and Frank Gehry.
A surface is called a ruled surface if through every point there is at least one straight line that lies on the surface. The aim of this project is to study this kind of surfaces and their properties in the context of differential geometry. In addition, we will consider a specific type of ruled surfaces which have null curvature, known as developable surfaces. Finally, we will examine their implementation in architecture, focusing on the analysis of the work done by architects Antoni Gaudí, Félix Candela, Santiago Calatrava and Frank Gehry.
Direction
Vázquez Abal, María Elena (Tutorships)
Vázquez Abal, María Elena (Tutorships)
Court
TORRES LOPERA, JUAN FRANCISCO (Chairman)
CONDE AMBOAGE, MERCEDES (Secretary)
López Pouso, Óscar (Member)
TORRES LOPERA, JUAN FRANCISCO (Chairman)
CONDE AMBOAGE, MERCEDES (Secretary)
López Pouso, Óscar (Member)
Prediction of financial returns
Authorship
M.C.L.
Bachelor of Mathematics
M.C.L.
Bachelor of Mathematics
Defense date
02.14.2024 10:00
02.14.2024 10:00
Summary
The goal of this project is predicting returns of financial assets series associated to IBEX35 for which they are going to use multivariate GARCH models. These models are focused in modeling conditinal variance that is the most important parameter for designing an investment wallet. To introduce GARCH models is necessary to know and manage autorregresive and moving average models (ARMA) because we are going to use similar tools for its building.
The goal of this project is predicting returns of financial assets series associated to IBEX35 for which they are going to use multivariate GARCH models. These models are focused in modeling conditinal variance that is the most important parameter for designing an investment wallet. To introduce GARCH models is necessary to know and manage autorregresive and moving average models (ARMA) because we are going to use similar tools for its building.
Direction
FEBRERO BANDE, MANUEL (Tutorships)
FEBRERO BANDE, MANUEL (Tutorships)
Court
TORRES LOPERA, JUAN FRANCISCO (Chairman)
CONDE AMBOAGE, MERCEDES (Secretary)
López Pouso, Óscar (Member)
TORRES LOPERA, JUAN FRANCISCO (Chairman)
CONDE AMBOAGE, MERCEDES (Secretary)
López Pouso, Óscar (Member)
Introduction to preconditioning in solving linear sistems
Authorship
M.E.F.
Bachelor of Mathematics
M.E.F.
Bachelor of Mathematics
Defense date
02.14.2024 10:45
02.14.2024 10:45
Summary
In this project we will try to give an approximation to the concept of preconditioning for solving systems of linear equations, focusing in the case of iterative methods with sparse matrices. We will begin by explaining the concept of solving a linear sistem of the form Au=b, one of the basic Matrix Numerical Analysis problems. We will see the different methods that exist to find a solution, distinguishing between direct and iterative methods. Next, we will define the concept of matrix condition number, to introduce the concept of precondition both for direct methods and for iterative methods. Then we will deduce the conjugate gradient method, an iterative method usually used to solve large linear systems whith positive definite symmetric matrix A. We will see the Cholesky factorization of a matrix to introduce later the incomplete Cholesky factorization, which allows us computing efficient preconditioners. We will also see how to apply the preconditioning to the conjugate gradient method to find the preconditioned conjugate gradient method. Finally, we will see some practices of applying the preconditioning and its effect in the number of iterations needed to reach the solution using the conjugate gradient method.
In this project we will try to give an approximation to the concept of preconditioning for solving systems of linear equations, focusing in the case of iterative methods with sparse matrices. We will begin by explaining the concept of solving a linear sistem of the form Au=b, one of the basic Matrix Numerical Analysis problems. We will see the different methods that exist to find a solution, distinguishing between direct and iterative methods. Next, we will define the concept of matrix condition number, to introduce the concept of precondition both for direct methods and for iterative methods. Then we will deduce the conjugate gradient method, an iterative method usually used to solve large linear systems whith positive definite symmetric matrix A. We will see the Cholesky factorization of a matrix to introduce later the incomplete Cholesky factorization, which allows us computing efficient preconditioners. We will also see how to apply the preconditioning to the conjugate gradient method to find the preconditioned conjugate gradient method. Finally, we will see some practices of applying the preconditioning and its effect in the number of iterations needed to reach the solution using the conjugate gradient method.
Direction
SALGADO RODRIGUEZ, MARIA DEL PILAR (Tutorships)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Tutorships)
Court
TORRES LOPERA, JUAN FRANCISCO (Chairman)
CONDE AMBOAGE, MERCEDES (Secretary)
López Pouso, Óscar (Member)
TORRES LOPERA, JUAN FRANCISCO (Chairman)
CONDE AMBOAGE, MERCEDES (Secretary)
López Pouso, Óscar (Member)
Regression on hierarchical data: multilevel models
Authorship
N.F.G.
Bachelor of Mathematics
N.F.G.
Bachelor of Mathematics
Defense date
02.14.2024 11:30
02.14.2024 11:30
Summary
In classic regression models, the observations of the variables are assumed to be independent, a fundamental hypothesis to derive the behavior of the parameter estimators. However, in practice it is not uncommon for data to present some type of dependence, for example, of a temporal or spatial nature, or just the practitioner knows that the data are not independent but it is not possible to establish a correlation function between the observations. This occurs, for example, when the observations are grouped in higher level units. In this work we will treat as an example the notes of different USC students, grouped by degrees and/or knowledge areas, so in the modeling process we could try to introduce some element that allows us to reflect on the source of extra variability that comes from the “group effect”. For this, we introduce (in addition to two classic models seen during the Mathematics Degree), new models not seen before, such as RANOVA and multilevel models with continuous responses, models with explanatory variable associated with the first or second level. Finally, we will obtain various predictions and conclusions from two studies that we will carry out.
In classic regression models, the observations of the variables are assumed to be independent, a fundamental hypothesis to derive the behavior of the parameter estimators. However, in practice it is not uncommon for data to present some type of dependence, for example, of a temporal or spatial nature, or just the practitioner knows that the data are not independent but it is not possible to establish a correlation function between the observations. This occurs, for example, when the observations are grouped in higher level units. In this work we will treat as an example the notes of different USC students, grouped by degrees and/or knowledge areas, so in the modeling process we could try to introduce some element that allows us to reflect on the source of extra variability that comes from the “group effect”. For this, we introduce (in addition to two classic models seen during the Mathematics Degree), new models not seen before, such as RANOVA and multilevel models with continuous responses, models with explanatory variable associated with the first or second level. Finally, we will obtain various predictions and conclusions from two studies that we will carry out.
Direction
CRUJEIRAS CASAIS, ROSA MARÍA (Tutorships)
CRUJEIRAS CASAIS, ROSA MARÍA (Tutorships)
Court
TORRES LOPERA, JUAN FRANCISCO (Chairman)
CONDE AMBOAGE, MERCEDES (Secretary)
López Pouso, Óscar (Member)
TORRES LOPERA, JUAN FRANCISCO (Chairman)
CONDE AMBOAGE, MERCEDES (Secretary)
López Pouso, Óscar (Member)
Introduction to Automatic Differentiation through Object-Oriented Programming
Authorship
V.F.P.P.
Bachelor of Mathematics
V.F.P.P.
Bachelor of Mathematics
Defense date
02.14.2024 12:15
02.14.2024 12:15
Summary
Automatic Differentiation consists on exact floating-point algorithms aiming to obtain different information of a function at the same time (derivative, gradient, hessian, ...). For an efficient implementation of this technique Object-Oriented Programming becomes very useful. The objective of this work consists on studying and utilizing Object-Oriented Programming in order to introduce into Automatic Differentiation. To this end, we will begin with simple cases such as calculating the value of a function and its derivative in a point. Next, we will get into more complicated cases such as calculating partial derivatives or high order derivatives by means of Taylor expansion coefficients.
Automatic Differentiation consists on exact floating-point algorithms aiming to obtain different information of a function at the same time (derivative, gradient, hessian, ...). For an efficient implementation of this technique Object-Oriented Programming becomes very useful. The objective of this work consists on studying and utilizing Object-Oriented Programming in order to introduce into Automatic Differentiation. To this end, we will begin with simple cases such as calculating the value of a function and its derivative in a point. Next, we will get into more complicated cases such as calculating partial derivatives or high order derivatives by means of Taylor expansion coefficients.
Direction
RODRIGUEZ GARCIA, JERONIMO (Tutorships)
PENA BRAGE, FRANCISCO JOSE (Co-tutorships)
RODRIGUEZ GARCIA, JERONIMO (Tutorships)
PENA BRAGE, FRANCISCO JOSE (Co-tutorships)
Court
TORRES LOPERA, JUAN FRANCISCO (Chairman)
CONDE AMBOAGE, MERCEDES (Secretary)
López Pouso, Óscar (Member)
TORRES LOPERA, JUAN FRANCISCO (Chairman)
CONDE AMBOAGE, MERCEDES (Secretary)
López Pouso, Óscar (Member)
Voronoi diagrams and application to locational optimization
Authorship
M.D.G.
Bachelor of Mathematics
M.D.G.
Bachelor of Mathematics
Defense date
02.15.2024 11:30
02.15.2024 11:30
Summary
This project begins with an historical introduction in order to provide context on Voronoi diagrams. In Chapter 1 basic knowledge on a variety of areas such as vector geometry, topology and graph theory is gathered. Furthermore, we specify mathematical notation that will be used on further chapters. Chapter 2 will describe Voronoi diagrams and give mathematical definitions of them and their dual, Delaunay tessellations, which derive naturally from the diagrams. Moreover, some properties of the diagrams are studied so as to describe two of their construction algorithms in Chapter 3. Lastly, Chapter 4 begins with a little introduction on mathematical programming with the purpose of studying how Voronoi diagrams can be applied to locational optimization problems. In Chapter 5 we conclude this project adding some observations and possible open fields in this area of study.
This project begins with an historical introduction in order to provide context on Voronoi diagrams. In Chapter 1 basic knowledge on a variety of areas such as vector geometry, topology and graph theory is gathered. Furthermore, we specify mathematical notation that will be used on further chapters. Chapter 2 will describe Voronoi diagrams and give mathematical definitions of them and their dual, Delaunay tessellations, which derive naturally from the diagrams. Moreover, some properties of the diagrams are studied so as to describe two of their construction algorithms in Chapter 3. Lastly, Chapter 4 begins with a little introduction on mathematical programming with the purpose of studying how Voronoi diagrams can be applied to locational optimization problems. In Chapter 5 we conclude this project adding some observations and possible open fields in this area of study.
Direction
CASAS MENDEZ, BALBINA VIRGINIA (Tutorships)
CASAS MENDEZ, BALBINA VIRGINIA (Tutorships)
Court
FEBRERO BANDE, MANUEL (Chairman)
BUEDO FERNANDEZ, SEBASTIAN (Secretary)
RODRIGUEZ GARCIA, JERONIMO (Member)
FEBRERO BANDE, MANUEL (Chairman)
BUEDO FERNANDEZ, SEBASTIAN (Secretary)
RODRIGUEZ GARCIA, JERONIMO (Member)
The optimal location problem
Authorship
A.D.F.
Bachelor of Mathematics
A.D.F.
Bachelor of Mathematics
Defense date
02.15.2024 12:00
02.15.2024 12:00
Summary
In this work we will address various location problems that have significant applications in diverse fields such as logistics, transportation, telecommunications, and many other areas. In particular, the focus will be on analyzing the Weber problem, which falls within the category of location problems and seeks to determine the location of a new point or facility in a way that minimizes the weighted sum of Euclidean distances to a given set of points. To tackle the Weber problem, we will delve into the Weiszfeld algorithm, a fundamental tool in the numerical resolution of such problems. Finally, a real-life example is presented, involving the optimal loca- tion calculation for a wine warehouse in Galicia, considering the restaurants to which it aims to supply wine and the companies that would provide resources.
In this work we will address various location problems that have significant applications in diverse fields such as logistics, transportation, telecommunications, and many other areas. In particular, the focus will be on analyzing the Weber problem, which falls within the category of location problems and seeks to determine the location of a new point or facility in a way that minimizes the weighted sum of Euclidean distances to a given set of points. To tackle the Weber problem, we will delve into the Weiszfeld algorithm, a fundamental tool in the numerical resolution of such problems. Finally, a real-life example is presented, involving the optimal loca- tion calculation for a wine warehouse in Galicia, considering the restaurants to which it aims to supply wine and the companies that would provide resources.
Direction
SAAVEDRA NIEVES, PAULA (Tutorships)
GINZO VILLAMAYOR, MARIA JOSE (Co-tutorships)
SAAVEDRA NIEVES, PAULA (Tutorships)
GINZO VILLAMAYOR, MARIA JOSE (Co-tutorships)
Court
FEBRERO BANDE, MANUEL (Chairman)
BUEDO FERNANDEZ, SEBASTIAN (Secretary)
RODRIGUEZ GARCIA, JERONIMO (Member)
FEBRERO BANDE, MANUEL (Chairman)
BUEDO FERNANDEZ, SEBASTIAN (Secretary)
RODRIGUEZ GARCIA, JERONIMO (Member)
Möbius transformations and the complex plane
Authorship
I.O.D.
Bachelor of Mathematics
I.O.D.
Bachelor of Mathematics
Defense date
02.15.2024 12:30
02.15.2024 12:30
Summary
In this document, we will study Möbius transformations, a type of holomorphic functions whose main characteristic is the preservation of angles. We will analyze in detail how these functions are derived from other simpler and well-known functions, and we will observe that they exhibit some properties of great interest, such as the transformation of circles into other circles or symmetry. Subsequently, we will explore how we can classify these transformations, and finally, we will examine some specific cases to conclude.
In this document, we will study Möbius transformations, a type of holomorphic functions whose main characteristic is the preservation of angles. We will analyze in detail how these functions are derived from other simpler and well-known functions, and we will observe that they exhibit some properties of great interest, such as the transformation of circles into other circles or symmetry. Subsequently, we will explore how we can classify these transformations, and finally, we will examine some specific cases to conclude.
Direction
Cao Labora, Daniel (Tutorships)
Cao Labora, Daniel (Tutorships)
Court
FEBRERO BANDE, MANUEL (Chairman)
BUEDO FERNANDEZ, SEBASTIAN (Secretary)
RODRIGUEZ GARCIA, JERONIMO (Member)
FEBRERO BANDE, MANUEL (Chairman)
BUEDO FERNANDEZ, SEBASTIAN (Secretary)
RODRIGUEZ GARCIA, JERONIMO (Member)
Discrete Calculus
Authorship
S.A.P.
Bachelor of Mathematics
S.A.P.
Bachelor of Mathematics
Defense date
02.14.2024 17:00
02.14.2024 17:00
Summary
In this paper we will be carrying out a brief study of linear difference equations, describing simultaneously their similarities and differences with differential equations. First of all, we will be introducing some basic concepts, such as the difference operator or what we define as linear difference equations, and then we will be developing them in order to obtain methods for solvingvthis kind of equations. During the second chapter we will be commencing by analysing first order equations. This will serve as a quick introduction to the general solving methods where we will be addressing about the definitions and the essential results about the resolution of linear difference equations; however, in a simple way, due to the fact that they are simplified equations. In addition, we will be approaching to the solution's qualitative study of linear difference equations and what is meant by asymptotic stability. In the third chapter we will be studying definitions, propositions and general methods to solve linear difference equations of the nth order with constant coeficients. In addition, we will be revolving around the concepts of linear dependence, Casoratian Matrix or resolution equation methods. To conclude, we will be analysing some practical examples of situations that can be modelled by linear difference equations. We will conclude this paper by epitomizing the main parallelisms between Differential and Discrete calculus in the context of linear difference and differential equations.
In this paper we will be carrying out a brief study of linear difference equations, describing simultaneously their similarities and differences with differential equations. First of all, we will be introducing some basic concepts, such as the difference operator or what we define as linear difference equations, and then we will be developing them in order to obtain methods for solvingvthis kind of equations. During the second chapter we will be commencing by analysing first order equations. This will serve as a quick introduction to the general solving methods where we will be addressing about the definitions and the essential results about the resolution of linear difference equations; however, in a simple way, due to the fact that they are simplified equations. In addition, we will be approaching to the solution's qualitative study of linear difference equations and what is meant by asymptotic stability. In the third chapter we will be studying definitions, propositions and general methods to solve linear difference equations of the nth order with constant coeficients. In addition, we will be revolving around the concepts of linear dependence, Casoratian Matrix or resolution equation methods. To conclude, we will be analysing some practical examples of situations that can be modelled by linear difference equations. We will conclude this paper by epitomizing the main parallelisms between Differential and Discrete calculus in the context of linear difference and differential equations.
Direction
CABADA FERNANDEZ, ALBERTO (Tutorships)
CABADA FERNANDEZ, ALBERTO (Tutorships)
Court
CABADA FERNANDEZ, ALBERTO (Student’s tutor)
CABADA FERNANDEZ, ALBERTO (Student’s tutor)
Foundations of a Connectivity Theory for Simplicial complexes
Authorship
A.X.M.G.
Bachelor of Mathematics
A.X.M.G.
Bachelor of Mathematics
Defense date
02.14.2024 12:00
02.14.2024 12:00
Summary
EL trabajo consistirá en la explicación con claridad de los resultados y conceptos presentados en el artículo: Foundations of a Connectivity Theory for Simplicial Complexes. En el que se expandirá el conccepto de homotopia a complejos simpliciales.Se introduirán antes algunos conceptos de topología y álgebra y después de presentar varios resultados sobre conexión en complejos simpliciales, se incluirá un teorema similar al de Seifert-Van Kampen para espacios topológicos.
EL trabajo consistirá en la explicación con claridad de los resultados y conceptos presentados en el artículo: Foundations of a Connectivity Theory for Simplicial Complexes. En el que se expandirá el conccepto de homotopia a complejos simpliciales.Se introduirán antes algunos conceptos de topología y álgebra y después de presentar varios resultados sobre conexión en complejos simpliciales, se incluirá un teorema similar al de Seifert-Van Kampen para espacios topológicos.
Direction
Gómez Tato, Antonio M. (Tutorships)
Gómez Tato, Antonio M. (Tutorships)
Court
Gómez Tato, Antonio M. (Student’s tutor)
Gómez Tato, Antonio M. (Student’s tutor)
The integral of Kurzweil-Stieltjes
Authorship
P.C.F.
Bachelor of Mathematics
P.C.F.
Bachelor of Mathematics
Defense date
02.15.2024 16:00
02.15.2024 16:00
Summary
In this work we will develop the theory corresponding to the Kurzweil-Stieltjes integral divided into three blocks. In the first chapter we present the integral and the elements that characterise it. In the second, we present the regulated functions and functions of bounded variation to conclude by presenting a result that assures its existence. Finally, in the chapter on properties we present results relating to classical integration such as integration by parts, the indefinite integral, the substitution theorem, absolute integrability and we end with the section with convergence theorems.
In this work we will develop the theory corresponding to the Kurzweil-Stieltjes integral divided into three blocks. In the first chapter we present the integral and the elements that characterise it. In the second, we present the regulated functions and functions of bounded variation to conclude by presenting a result that assures its existence. Finally, in the chapter on properties we present results relating to classical integration such as integration by parts, the indefinite integral, the substitution theorem, absolute integrability and we end with the section with convergence theorems.
Direction
FERNANDEZ FERNANDEZ, FRANCISCO JAVIER (Tutorships)
FERNANDEZ FERNANDEZ, FRANCISCO JAVIER (Tutorships)
Court
FERNANDEZ FERNANDEZ, FRANCISCO JAVIER (Student’s tutor)
FERNANDEZ FERNANDEZ, FRANCISCO JAVIER (Student’s tutor)
Teoría de Categorías
Authorship
M.M.M.M.
Bachelor of Mathematics
M.M.M.M.
Bachelor of Mathematics
Defense date
09.12.2024 09:00
09.12.2024 09:00
Summary
Category theory could be described as the multidimensional foundations of mathematics. It studies the different mathematical structures and the relationships between them, constructing bridges that connect and organize all kinds of areas of knowledge. In this sense, even though being quite formal and abstract, category theory constitutes an extremely powerful tool to solve very different problems. It enables one to translate given questions to an easier context or even write the solution on their own, as it is able to precisely describe some complex structures. The present work will serve as an introduction to this theory. It will explore the basic concepts, explain them thoroughly and provide some helpful examples.
Category theory could be described as the multidimensional foundations of mathematics. It studies the different mathematical structures and the relationships between them, constructing bridges that connect and organize all kinds of areas of knowledge. In this sense, even though being quite formal and abstract, category theory constitutes an extremely powerful tool to solve very different problems. It enables one to translate given questions to an easier context or even write the solution on their own, as it is able to precisely describe some complex structures. The present work will serve as an introduction to this theory. It will explore the basic concepts, explain them thoroughly and provide some helpful examples.
Direction
LADRA GONZALEZ, MANUEL EULOGIO (Tutorships)
TURDIBAEV , RUSTAM (Co-tutorships)
LADRA GONZALEZ, MANUEL EULOGIO (Tutorships)
TURDIBAEV , RUSTAM (Co-tutorships)
Court
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
Gröbner basis, Buchberger algorithm and applications
Authorship
B.Q.C.
Bachelor of Mathematics
B.Q.C.
Bachelor of Mathematics
Defense date
09.12.2024 09:40
09.12.2024 09:40
Summary
Gröbner basis are a fundamental concept for both computer algebra and algebraic geometry. Their importance is given due to their great applicability, since, for example, large complex systems of polynomial equations can be solved in a simpler way. In this paper, after providing the necessary context, both the Gröbner basis and their obtaining through the Buchberger algorithm will be studied. In addition, three applications in different disciplines will also be studied to show their importance, which are operations with ideals, integer programming and a recreational application to solving sudokus.
Gröbner basis are a fundamental concept for both computer algebra and algebraic geometry. Their importance is given due to their great applicability, since, for example, large complex systems of polynomial equations can be solved in a simpler way. In this paper, after providing the necessary context, both the Gröbner basis and their obtaining through the Buchberger algorithm will be studied. In addition, three applications in different disciplines will also be studied to show their importance, which are operations with ideals, integer programming and a recreational application to solving sudokus.
Direction
ALONSO TARRIO, LEOVIGILDO (Tutorships)
ALVITE PAZO, RAUL (Co-tutorships)
ALONSO TARRIO, LEOVIGILDO (Tutorships)
ALVITE PAZO, RAUL (Co-tutorships)
Court
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
A constructive approach to the inverse Galois problem
Authorship
D.Q.G.
Bachelor of Mathematics
D.Q.G.
Bachelor of Mathematics
Defense date
09.12.2024 10:20
09.12.2024 10:20
Summary
The inverse Galois problem poses the enigmatic question of whether for any finite group G, there is a polynomial whose Galois group over the field of rational numbers is G. This problem, first posed in the 19th century, remains unsolved, and this mathematical puzzle is part of the motivation of this work. Another incentive is to approach it from a constructive and computational point of view, finding polynomials whose Galois group is a given group.
The inverse Galois problem poses the enigmatic question of whether for any finite group G, there is a polynomial whose Galois group over the field of rational numbers is G. This problem, first posed in the 19th century, remains unsolved, and this mathematical puzzle is part of the motivation of this work. Another incentive is to approach it from a constructive and computational point of view, finding polynomials whose Galois group is a given group.
Direction
LADRA GONZALEZ, MANUEL EULOGIO (Tutorships)
LADRA GONZALEZ, MANUEL EULOGIO (Tutorships)
Court
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
Fréchet–Urysohn spaces
Authorship
V.F.G.
Bachelor of Mathematics
V.F.G.
Bachelor of Mathematics
Defense date
09.12.2024 10:00
09.12.2024 10:00
Summary
In this paper, we explore the use of sequences in general topology. We start with metric spaces and first countable spaces, and then continue with Fréchet-Urysohn spaces and sequential spaces. Each of these classes is contained within the next. We examine the relationships between the different spaces, characterizing one from another and observing how they behave under certain operations. The final aim is to characterize the Fréchet-Urysohn spaces as completely as possible.
In this paper, we explore the use of sequences in general topology. We start with metric spaces and first countable spaces, and then continue with Fréchet-Urysohn spaces and sequential spaces. Each of these classes is contained within the next. We examine the relationships between the different spaces, characterizing one from another and observing how they behave under certain operations. The final aim is to characterize the Fréchet-Urysohn spaces as completely as possible.
Direction
CARBALLES VAZQUEZ, JOSE MANUEL (Tutorships)
CARBALLES VAZQUEZ, JOSE MANUEL (Tutorships)
Court
CARBALLES VAZQUEZ, JOSE MANUEL (Student’s tutor)
CARBALLES VAZQUEZ, JOSE MANUEL (Student’s tutor)
Wavelets and applications
Authorship
S.C.C.
Bachelor of Mathematics
S.C.C.
Bachelor of Mathematics
Defense date
09.11.2024 09:30
09.11.2024 09:30
Summary
The aim of this paper is to study the famous Hilbert functional space defined by Lebesgue from the harmonic analysis approach. First, the Fourier transform is studied and its algebraic and analytic properties are explored. The Fourier inversion theorem, which allows us to recover a function from its transform by applying the inverse transform, and Plancherel's theorem, which makes feasible to extend the transform, initially defined in another functional space, to this Hilbert space, are demonstrated. Next, two phenomena connatural (and unwelcome) to Fourier's mathematical tools are presented and studied: the Gibbs phenomenon and the Heisenberg uncertainty principle. Finally, changing optics, wavelets are introduced to solve this problem. This last part focuses on characterising the wavelets that produce good approximations and on defining the wavelet transform following the roadmap provided by the Fourier transform, as well as briefly providing some examples and applications of wavelet theory.
The aim of this paper is to study the famous Hilbert functional space defined by Lebesgue from the harmonic analysis approach. First, the Fourier transform is studied and its algebraic and analytic properties are explored. The Fourier inversion theorem, which allows us to recover a function from its transform by applying the inverse transform, and Plancherel's theorem, which makes feasible to extend the transform, initially defined in another functional space, to this Hilbert space, are demonstrated. Next, two phenomena connatural (and unwelcome) to Fourier's mathematical tools are presented and studied: the Gibbs phenomenon and the Heisenberg uncertainty principle. Finally, changing optics, wavelets are introduced to solve this problem. This last part focuses on characterising the wavelets that produce good approximations and on defining the wavelet transform following the roadmap provided by the Fourier transform, as well as briefly providing some examples and applications of wavelet theory.
Direction
FERNANDEZ TOJO, FERNANDO ADRIAN (Tutorships)
FERNANDEZ TOJO, FERNANDO ADRIAN (Tutorships)
Court
VIAÑO REY, JUAN MANUEL (Chairman)
Rodríguez López, Jorge (Secretary)
CARBALLES VAZQUEZ, JOSE MANUEL (Member)
VIAÑO REY, JUAN MANUEL (Chairman)
Rodríguez López, Jorge (Secretary)
CARBALLES VAZQUEZ, JOSE MANUEL (Member)
Imre Lakatos. Proofs and refutations. The logic of mathematical discovery
Authorship
M.F.R.
Bachelor of Mathematics
M.F.R.
Bachelor of Mathematics
Defense date
09.11.2024 10:15
09.11.2024 10:15
Summary
The main objective of this Final Degree Project is to learn about and present the proof and refutation method proposed by the Hungarian mathematician and philosopher Imre Lakatos in his work Proofs and refutations: the logic of mathematical discovery. To do so, we will show what this method consists of in four different situations. First, we will analyse the discussion and proof of the Euler-Descartes Theorem on polyhedra. Then, we will talk about the discovery of the concept of uniform convergence and its relation with the continuity of the limit function of a sequence of continuous functions. We will then present some features of this method in the field of Measurement Theory and in relation to the role of boundedly varying functions in the Riemann-Stieltjes theory of integration.
The main objective of this Final Degree Project is to learn about and present the proof and refutation method proposed by the Hungarian mathematician and philosopher Imre Lakatos in his work Proofs and refutations: the logic of mathematical discovery. To do so, we will show what this method consists of in four different situations. First, we will analyse the discussion and proof of the Euler-Descartes Theorem on polyhedra. Then, we will talk about the discovery of the concept of uniform convergence and its relation with the continuity of the limit function of a sequence of continuous functions. We will then present some features of this method in the field of Measurement Theory and in relation to the role of boundedly varying functions in the Riemann-Stieltjes theory of integration.
Direction
LOSADA RODRIGUEZ, JORGE (Tutorships)
LOSADA RODRIGUEZ, JORGE (Tutorships)
Court
VIAÑO REY, JUAN MANUEL (Chairman)
Rodríguez López, Jorge (Secretary)
CARBALLES VAZQUEZ, JOSE MANUEL (Member)
VIAÑO REY, JUAN MANUEL (Chairman)
Rodríguez López, Jorge (Secretary)
CARBALLES VAZQUEZ, JOSE MANUEL (Member)
Identification of the mechanical behaviour of a beam and column composition
Authorship
H.M.H.
Bachelor of Mathematics
H.M.H.
Bachelor of Mathematics
Defense date
09.11.2024 16:00
09.11.2024 16:00
Summary
This paper consists in a case study included in the course of the visiting faculty at the USC Professor Vasant Matshagar. The aim of the study is to observe how is the mechanical behaviour of a portal frame composed of two columns and one beam. First of all, we are going to introduce the concepts of stress and strain, which are going to be useful, along with Hooke's law, to develop the 3D equations of elasticity. From the 3D model, and since the structure geometry and the applied loads have specific characteristics, a unidimensional model will be elaborated to simplify the calculation. We are going to use the Bernouilli-Euler's model and the bars model to develop it. To decide de boundary conditions, we consider that both columns are fixed to the ground and the portal frame is under the influence of gravity and strong wind gusts. Finally, we will make a stationary study of the model by a simulation using the COMSOL Multiphysics software. We are going to explain the steps of the simulation and contrast the results that we have been obtained.
This paper consists in a case study included in the course of the visiting faculty at the USC Professor Vasant Matshagar. The aim of the study is to observe how is the mechanical behaviour of a portal frame composed of two columns and one beam. First of all, we are going to introduce the concepts of stress and strain, which are going to be useful, along with Hooke's law, to develop the 3D equations of elasticity. From the 3D model, and since the structure geometry and the applied loads have specific characteristics, a unidimensional model will be elaborated to simplify the calculation. We are going to use the Bernouilli-Euler's model and the bars model to develop it. To decide de boundary conditions, we consider that both columns are fixed to the ground and the portal frame is under the influence of gravity and strong wind gusts. Finally, we will make a stationary study of the model by a simulation using the COMSOL Multiphysics software. We are going to explain the steps of the simulation and contrast the results that we have been obtained.
Direction
QUINTELA ESTEVEZ, PEREGRINA (Tutorships)
QUINTELA ESTEVEZ, PEREGRINA (Tutorships)
Court
VIAÑO REY, JUAN MANUEL (Chairman)
Rodríguez López, Jorge (Secretary)
CARBALLES VAZQUEZ, JOSE MANUEL (Member)
VIAÑO REY, JUAN MANUEL (Chairman)
Rodríguez López, Jorge (Secretary)
CARBALLES VAZQUEZ, JOSE MANUEL (Member)
Minimum cost flow problems: algorithms and applications
Authorship
S.F.R.
Bachelor of Mathematics
S.F.R.
Bachelor of Mathematics
Defense date
09.11.2024 13:00
09.11.2024 13:00
Summary
Minimum cost flow problems are optimization problems used to model and solve problems by minimizing costs and satisfying flow constraints. The it out-of-kilter algorithm is used in the resolution of minimum cost network flow problems and tries to optimize a flow through a network by meeting certain constraints. It works with dual feasibility, moving in primal and dual problems with the objective of reaching an optimal feasible solution and trying to achieve the complementary slack property. One application of these problems in which AMPL and the solver GUROBI have been used is the optimization of appointments to receive chemotherapy for cancer patients. We try to achieve an optimal way of appointing patients to receive treatment and make the waiting time as short as possible.
Minimum cost flow problems are optimization problems used to model and solve problems by minimizing costs and satisfying flow constraints. The it out-of-kilter algorithm is used in the resolution of minimum cost network flow problems and tries to optimize a flow through a network by meeting certain constraints. It works with dual feasibility, moving in primal and dual problems with the objective of reaching an optimal feasible solution and trying to achieve the complementary slack property. One application of these problems in which AMPL and the solver GUROBI have been used is the optimization of appointments to receive chemotherapy for cancer patients. We try to achieve an optimal way of appointing patients to receive treatment and make the waiting time as short as possible.
Direction
CASAS MENDEZ, BALBINA VIRGINIA (Tutorships)
DAVILA PENA, LAURA (Co-tutorships)
CASAS MENDEZ, BALBINA VIRGINIA (Tutorships)
DAVILA PENA, LAURA (Co-tutorships)
Court
CASAS MENDEZ, BALBINA VIRGINIA (Student’s tutor)
CASAS MENDEZ, BALBINA VIRGINIA (Student’s tutor)
Minimax theorem, optimization and learning
Authorship
N.O.G.
Bachelor of Mathematics
N.O.G.
Bachelor of Mathematics
Defense date
09.12.2024 12:20
09.12.2024 12:20
Summary
In this work, we will incorporate some historical background of game theory and John von Neumann’s trajectory. The essential focus will be on matrix games and their resolution. We will give a formal and direct proof of the minimax theorem. Subsequently, we will explore applications of optimization techniques to solve matrix games, both in their classical form using the duality theorem of linear programming, and in cases where the payoff function is vector-valued. An application to a real-world problem using optimization libraries in the R programming language will also be demonstrated. Finally, the learning method proposed by George W. Brown will be presented, whose convergence was demonstrated by Julia Robinson, and certain experiments illustrating it will be shown.
In this work, we will incorporate some historical background of game theory and John von Neumann’s trajectory. The essential focus will be on matrix games and their resolution. We will give a formal and direct proof of the minimax theorem. Subsequently, we will explore applications of optimization techniques to solve matrix games, both in their classical form using the duality theorem of linear programming, and in cases where the payoff function is vector-valued. An application to a real-world problem using optimization libraries in the R programming language will also be demonstrated. Finally, the learning method proposed by George W. Brown will be presented, whose convergence was demonstrated by Julia Robinson, and certain experiments illustrating it will be shown.
Direction
CASAS MENDEZ, BALBINA VIRGINIA (Tutorships)
CASAS MENDEZ, BALBINA VIRGINIA (Tutorships)
Court
CASAS MENDEZ, BALBINA VIRGINIA (Student’s tutor)
CASAS MENDEZ, BALBINA VIRGINIA (Student’s tutor)
Cooperative game values and cost allocation in condominiums.
Authorship
S.P.R.
Bachelor of Mathematics
S.P.R.
Bachelor of Mathematics
Defense date
09.11.2024 19:30
09.11.2024 19:30
Summary
In this paper we will adress the problem of the cost allocation in condo buildings using cooperative game theory, which studies the procedures to allocate the benefits or costs between the agents that are trying to cooperate on different situations in order to obtain the best result. We will focus on two different approaches: the elevator rule of Crettez and Deloche (2019), focussed on the French legislation, based on the idea that the owners shall pay according to the benefits obtained, and the egalitarian solutions of Alonso Meijide and others (2020), where they consider that the floors of the building naturally results on an Owen’s structure of a priori unions (1977). We will analyze its theorical foundations, its practical applications and the coaltional stability of the proposed methods. Eventually, we will present a practical example to show the implementation of these methods in real cases.
In this paper we will adress the problem of the cost allocation in condo buildings using cooperative game theory, which studies the procedures to allocate the benefits or costs between the agents that are trying to cooperate on different situations in order to obtain the best result. We will focus on two different approaches: the elevator rule of Crettez and Deloche (2019), focussed on the French legislation, based on the idea that the owners shall pay according to the benefits obtained, and the egalitarian solutions of Alonso Meijide and others (2020), where they consider that the floors of the building naturally results on an Owen’s structure of a priori unions (1977). We will analyze its theorical foundations, its practical applications and the coaltional stability of the proposed methods. Eventually, we will present a practical example to show the implementation of these methods in real cases.
Direction
CASAS MENDEZ, BALBINA VIRGINIA (Tutorships)
DAVILA PENA, LAURA (Co-tutorships)
CASAS MENDEZ, BALBINA VIRGINIA (Tutorships)
DAVILA PENA, LAURA (Co-tutorships)
Court
CASAS MENDEZ, BALBINA VIRGINIA (Student’s tutor)
CASAS MENDEZ, BALBINA VIRGINIA (Student’s tutor)
Anomaly detection in power lines
Authorship
D.P.V.
Double bachelor degree of Engeneering in Information Technology and Mathematics
D.P.V.
Double bachelor degree of Engeneering in Information Technology and Mathematics
Defense date
07.18.2024 13:00
07.18.2024 13:00
Summary
Power line inspection is a field in constant progress due to the importance of the entire electrical infrastructure in today's society. Most articles that address this topic focus on the monitoring of insulators, leaving aside a large number of components that also have relevance in the power grid. This paper explores the capability of versions of the RetinaNet, Single Shot Multibox Detector (SSD), You Only Look Once (YOLO) and Faster R-CNN models for the detection of a wider number of power grid elements. Finally, the performance of ResNet and EfficientNet classification models for the categorization of insulator defects is also studied.
Power line inspection is a field in constant progress due to the importance of the entire electrical infrastructure in today's society. Most articles that address this topic focus on the monitoring of insulators, leaving aside a large number of components that also have relevance in the power grid. This paper explores the capability of versions of the RetinaNet, Single Shot Multibox Detector (SSD), You Only Look Once (YOLO) and Faster R-CNN models for the detection of a wider number of power grid elements. Finally, the performance of ResNet and EfficientNet classification models for the categorization of insulator defects is also studied.
Direction
MUCIENTES MOLINA, MANUEL FELIPE (Tutorships)
Abado Bóveda, Silvia (Co-tutorships)
MUCIENTES MOLINA, MANUEL FELIPE (Tutorships)
Abado Bóveda, Silvia (Co-tutorships)
Court
VIDAL AGUIAR, JUAN CARLOS (Chairman)
DOSIL LAGO, RAQUEL (Secretary)
SAAVEDRA NIEVES, ALEJANDRO (Member)
VIDAL AGUIAR, JUAN CARLOS (Chairman)
DOSIL LAGO, RAQUEL (Secretary)
SAAVEDRA NIEVES, ALEJANDRO (Member)
Lifelong learning for edge computing applications in industry 4.0
Authorship
S.A.P.
Double bachelor degree of Engeneering in Information Technology and Mathematics
S.A.P.
Double bachelor degree of Engeneering in Information Technology and Mathematics
Defense date
07.18.2024 10:00
07.18.2024 10:00
Summary
In this work we develop an unsupervised time series encoder with continual learning, for its integration with a classification model. Specifically, we adapt an existing time series encoder to be trained with continual learning, allowing the generated encodings to be used as input for the classification model. In addition, in order to evaluate the quality of the encodings, we design a metrics and visualization methods. The experiments show that the performance when using the encoder with continual learning is comparable to the performance obtained when using offline learning, the former situation being more difficult for the model. The integration experiments show that when using the encodings as input for the classifier with continual learning, the performance obtained is satisfactory.
In this work we develop an unsupervised time series encoder with continual learning, for its integration with a classification model. Specifically, we adapt an existing time series encoder to be trained with continual learning, allowing the generated encodings to be used as input for the classification model. In addition, in order to evaluate the quality of the encodings, we design a metrics and visualization methods. The experiments show that the performance when using the encoder with continual learning is comparable to the performance obtained when using offline learning, the former situation being more difficult for the model. The integration experiments show that when using the encodings as input for the classifier with continual learning, the performance obtained is satisfactory.
Direction
MERA PEREZ, DAVID (Tutorships)
Fernández Castro, Bruno (Co-tutorships)
García Santaclara, Pablo (Co-tutorships)
MERA PEREZ, DAVID (Tutorships)
Fernández Castro, Bruno (Co-tutorships)
García Santaclara, Pablo (Co-tutorships)
Court
Cotos Yáñez, José Manuel (Chairman)
QUESADA BARRIUSO, PABLO (Secretary)
GAGO COUSO, FELIPE (Member)
Cotos Yáñez, José Manuel (Chairman)
QUESADA BARRIUSO, PABLO (Secretary)
GAGO COUSO, FELIPE (Member)
Review and analysis of quantum artificial intelligence models
Authorship
M.T.L.
Double bachelor degree of Engeneering in Information Technology and Mathematics
M.T.L.
Double bachelor degree of Engeneering in Information Technology and Mathematics
Defense date
07.18.2024 11:30
07.18.2024 11:30
Summary
Quantum computing is a promising field nowadays, aiming to a revolution in computer science by leveraging the advantages of quantum mechanics. Its combination with the field of artificial intelligence gives rise to the new field of Quantum Machine Learning. This thesis provides an introduction to its techniques and models. After a literature review, a specific model is selected: a classifier based on a novel technique known as data re-uploading. This model will be implemented using the standard Qiskit library and experimentation will be conducted to evaluate it.
Quantum computing is a promising field nowadays, aiming to a revolution in computer science by leveraging the advantages of quantum mechanics. Its combination with the field of artificial intelligence gives rise to the new field of Quantum Machine Learning. This thesis provides an introduction to its techniques and models. After a literature review, a specific model is selected: a classifier based on a novel technique known as data re-uploading. This model will be implemented using the standard Qiskit library and experimentation will be conducted to evaluate it.
Direction
BUGARIN DIZ, ALBERTO JOSE (Tutorships)
Fernández Pena, Anselmo Tomás (Co-tutorships)
BUGARIN DIZ, ALBERTO JOSE (Tutorships)
Fernández Pena, Anselmo Tomás (Co-tutorships)
Court
Cotos Yáñez, José Manuel (Chairman)
QUESADA BARRIUSO, PABLO (Secretary)
GAGO COUSO, FELIPE (Member)
Cotos Yáñez, José Manuel (Chairman)
QUESADA BARRIUSO, PABLO (Secretary)
GAGO COUSO, FELIPE (Member)
Automatic lip syncing with the Furhat social robot
Authorship
C.C.P.
Double bachelor degree of Engeneering in Information Technology and Mathematics
C.C.P.
Double bachelor degree of Engeneering in Information Technology and Mathematics
Defense date
07.19.2024 10:30
07.19.2024 10:30
Summary
The rise of artificial intelligence in social robotics and the proliferation of TTS (Text To Speech) systems mean that there can be an integration of both fields, as can be seen in the Furhat robot. This technology is limited to a few languages, recently it was successfully managed to integrate the Proxecto Nos’s TTS into this robot, which allowed Galician speech. However, lip syncing was not adjusted to the Galician language, resulting in an incomplete user experience. This work proposes an alternative lip synchronization for the social robot Furhat during its Galician speech. To achieve this, forced alignment will be used, through the Montreal Forced Aligner (MFA) tool. To align with MFA, we have to create a pronunciation dictionary and train acoustic models with this tool, as none of these resources are currently available for the Galician language.
The rise of artificial intelligence in social robotics and the proliferation of TTS (Text To Speech) systems mean that there can be an integration of both fields, as can be seen in the Furhat robot. This technology is limited to a few languages, recently it was successfully managed to integrate the Proxecto Nos’s TTS into this robot, which allowed Galician speech. However, lip syncing was not adjusted to the Galician language, resulting in an incomplete user experience. This work proposes an alternative lip synchronization for the social robot Furhat during its Galician speech. To achieve this, forced alignment will be used, through the Montreal Forced Aligner (MFA) tool. To align with MFA, we have to create a pronunciation dictionary and train acoustic models with this tool, as none of these resources are currently available for the Galician language.
Direction
CATALA BOLOS, ALEJANDRO (Tutorships)
BUGARIN DIZ, ALBERTO JOSE (Co-tutorships)
MAGARIÑOS IGLESIAS, MARIA DEL CARMEN (Co-tutorships)
CATALA BOLOS, ALEJANDRO (Tutorships)
BUGARIN DIZ, ALBERTO JOSE (Co-tutorships)
MAGARIÑOS IGLESIAS, MARIA DEL CARMEN (Co-tutorships)
Court
BARJA PEREZ, JAVIER (Chairman)
ORDOÑEZ IGLESIAS, ALVARO (Secretary)
MOSQUERA GONZALEZ, ANTONIO (Member)
BARJA PEREZ, JAVIER (Chairman)
ORDOÑEZ IGLESIAS, ALVARO (Secretary)
MOSQUERA GONZALEZ, ANTONIO (Member)
YourTurn!
Authorship
A.F.E.
Double bachelor degree of Engeneering in Information Technology and Mathematics
A.F.E.
Double bachelor degree of Engeneering in Information Technology and Mathematics
Defense date
07.19.2024 11:00
07.19.2024 11:00
Summary
The aim of this work is to develop a web application to allow a Magic game to be played in real time and remotely. The developed system will allow sharing the players' game table by using a video camera and will identify the cards, displaying their information and translating it in case the players speak a different language.
The aim of this work is to develop a web application to allow a Magic game to be played in real time and remotely. The developed system will allow sharing the players' game table by using a video camera and will identify the cards, displaying their information and translating it in case the players speak a different language.
Direction
TOBAR QUINTANAR, ALEJANDRO JOSE (Tutorships)
GARCIA LLORENS, LUIS VICENTE (Co-tutorships)
Ibán Sánchez, Armando (Co-tutorships)
TOBAR QUINTANAR, ALEJANDRO JOSE (Tutorships)
GARCIA LLORENS, LUIS VICENTE (Co-tutorships)
Ibán Sánchez, Armando (Co-tutorships)
Court
BARJA PEREZ, JAVIER (Chairman)
ORDOÑEZ IGLESIAS, ALVARO (Secretary)
MOSQUERA GONZALEZ, ANTONIO (Member)
BARJA PEREZ, JAVIER (Chairman)
ORDOÑEZ IGLESIAS, ALVARO (Secretary)
MOSQUERA GONZALEZ, ANTONIO (Member)
Effective adaptation of generative adversarial networks for processing multidimensional remote sensing images
Authorship
A.G.L.
Double bachelor degree of Engeneering in Information Technology and Mathematics
A.G.L.
Double bachelor degree of Engeneering in Information Technology and Mathematics
Defense date
07.19.2024 17:00
07.19.2024 17:00
Summary
In recent years, various deep learning classification models have been proposed, adding to the existing methods for classifying multispectral remote sensing images. Within this framework, in this work, a conditional generative adversarial network (GAN), based on the StyleGAN2 model architecture, was adapted to the problem of classifying multispectral remote sensing images. Subsequently, a study was conducted on the network's capability to both generate and classify high spatial resolution multispectral images corresponding to Galician rivers. Finally, the results achieved with this classification model were compared to other models used in multispectral remote sensing problems. It was observed that the conditional StyleGAN2 achieves results close to other classification schemes that do not use generated datasets, such as convolutional neural networks, but falls short of methods specifically designed to generate samples for classification problems with imbalanced classes, such as ResBaGAN.
In recent years, various deep learning classification models have been proposed, adding to the existing methods for classifying multispectral remote sensing images. Within this framework, in this work, a conditional generative adversarial network (GAN), based on the StyleGAN2 model architecture, was adapted to the problem of classifying multispectral remote sensing images. Subsequently, a study was conducted on the network's capability to both generate and classify high spatial resolution multispectral images corresponding to Galician rivers. Finally, the results achieved with this classification model were compared to other models used in multispectral remote sensing problems. It was observed that the conditional StyleGAN2 achieves results close to other classification schemes that do not use generated datasets, such as convolutional neural networks, but falls short of methods specifically designed to generate samples for classification problems with imbalanced classes, such as ResBaGAN.
Direction
Argüello Pedreira, Francisco Santiago (Tutorships)
Blanco Heras, Dora (Co-tutorships)
Argüello Pedreira, Francisco Santiago (Tutorships)
Blanco Heras, Dora (Co-tutorships)
Court
Fernández Pena, Anselmo Tomás (Chairman)
SACO LOPEZ, PEDRO JOSE (Secretary)
RODRIGUEZ PRESEDO, JESUS MARIA (Member)
Fernández Pena, Anselmo Tomás (Chairman)
SACO LOPEZ, PEDRO JOSE (Secretary)
RODRIGUEZ PRESEDO, JESUS MARIA (Member)
Survey and evaluation of homomorphic encryption algorithms
Authorship
H.V.R.
Double bachelor degree of Engeneering in Information Technology and Mathematics
H.V.R.
Double bachelor degree of Engeneering in Information Technology and Mathematics
Defense date
09.12.2024 10:30
09.12.2024 10:30
Summary
Homomorphic cryptography allows operations to be performed on encrypted data without having to decrypt it. This feature is very attractive because, in particular, it prevents third parties to whom calculations with that data are delegated from accessing it, opening up a wide range of applications. The problem of building a completely homomorphic cryptosystem, which allows arbitrary calculations with encrypted data, was considered the holy grail of cryptography for 30 years. During that time, countless attempts have been made to solve it, obtaining only partial solutions. In 2009, the work of Craig Gentry represented a revolution by introducing the bootstrapping technique, through which some cryptosystems could be made completely homomorphic. From this moment on, different families of cryptosystems with this property have emerged, all of them making use of Gentry's invention. The aim of this paper is to carry out a small comparative study between some of the most widely used completely homomorphic cryptosystems today, specifically BGV, BFV and TFHE, comparing the algorithms that compose them and measuring the execution times. Previously, homomorphic cryptography will be introduced theoretically, defining bootstrapping and making certain considerations regarding the security of these cryptosystems. In addition, a brief historical context and a brief review of the state of the art will be given, presenting the main families of completely homomorphic cryptosystems.
Homomorphic cryptography allows operations to be performed on encrypted data without having to decrypt it. This feature is very attractive because, in particular, it prevents third parties to whom calculations with that data are delegated from accessing it, opening up a wide range of applications. The problem of building a completely homomorphic cryptosystem, which allows arbitrary calculations with encrypted data, was considered the holy grail of cryptography for 30 years. During that time, countless attempts have been made to solve it, obtaining only partial solutions. In 2009, the work of Craig Gentry represented a revolution by introducing the bootstrapping technique, through which some cryptosystems could be made completely homomorphic. From this moment on, different families of cryptosystems with this property have emerged, all of them making use of Gentry's invention. The aim of this paper is to carry out a small comparative study between some of the most widely used completely homomorphic cryptosystems today, specifically BGV, BFV and TFHE, comparing the algorithms that compose them and measuring the execution times. Previously, homomorphic cryptography will be introduced theoretically, defining bootstrapping and making certain considerations regarding the security of these cryptosystems. In addition, a brief historical context and a brief review of the state of the art will be given, presenting the main families of completely homomorphic cryptosystems.
Direction
CARIÑENA AMIGO, MARIA PURIFICACION (Tutorships)
GAGO COUSO, FELIPE (Co-tutorships)
CARIÑENA AMIGO, MARIA PURIFICACION (Tutorships)
GAGO COUSO, FELIPE (Co-tutorships)
Court
Fernández Rivera, Francisco (Chairman)
SANTOS MATEOS, ROI (Secretary)
López Vilariño, David (Member)
Fernández Rivera, Francisco (Chairman)
SANTOS MATEOS, ROI (Secretary)
López Vilariño, David (Member)
Complex multiplication theory and Kronecker's Jugendtraum
Authorship
H.V.R.
Double bachelor degree of Engeneering in Information Technology and Mathematics
H.V.R.
Double bachelor degree of Engeneering in Information Technology and Mathematics
Defense date
09.12.2024 17:20
09.12.2024 17:20
Summary
Kronecker's Jugendtraum is related to the problem of finding all abelian extensions of a quadratic imaginary field, therefore generalising the Kronecker-Weber theorem. To study this issue, the main tool is a certain class of elliptic curves, with a specially rich arithmetic structure, called complex multiplication. In this work we will explore the connection between these two topics. We will start by introducing, on one side, the elliptic curves, defining over them a group structure, focusing on finite order points; and on the other, extensions of Q, emphasising those with an abelian Galois group, as well as those generated by roots of unity or elliptic curve points. Afterwards, we will present the concept of complex multiplication and, finally, with the help from group representations, all the previous theory will be used to study a particular case of Kronecker's Jugendtraum, looking at the abelian extensions of Q(i).
Kronecker's Jugendtraum is related to the problem of finding all abelian extensions of a quadratic imaginary field, therefore generalising the Kronecker-Weber theorem. To study this issue, the main tool is a certain class of elliptic curves, with a specially rich arithmetic structure, called complex multiplication. In this work we will explore the connection between these two topics. We will start by introducing, on one side, the elliptic curves, defining over them a group structure, focusing on finite order points; and on the other, extensions of Q, emphasising those with an abelian Galois group, as well as those generated by roots of unity or elliptic curve points. Afterwards, we will present the concept of complex multiplication and, finally, with the help from group representations, all the previous theory will be used to study a particular case of Kronecker's Jugendtraum, looking at the abelian extensions of Q(i).
Direction
GAGO COUSO, FELIPE (Tutorships)
RIVERO SALGADO, OSCAR (Co-tutorships)
GAGO COUSO, FELIPE (Tutorships)
RIVERO SALGADO, OSCAR (Co-tutorships)
Court
OTERO ESPINAR, MARIA VICTORIA (Chairman)
GONZALEZ DIAZ, JULIO (Secretary)
Jeremías López, Ana (Member)
OTERO ESPINAR, MARIA VICTORIA (Chairman)
GONZALEZ DIAZ, JULIO (Secretary)
Jeremías López, Ana (Member)
Reformulation in mixed integer nonlinear programming problems.
Authorship
S.A.P.
Double bachelor degree of Engeneering in Information Technology and Mathematics
S.A.P.
Double bachelor degree of Engeneering in Information Technology and Mathematics
Defense date
07.16.2024 16:00
07.16.2024 16:00
Summary
In this project we study a class of reformulations for polynomial programming problems. Initially, we present the theoretical foundation of these reformulations, which were originally developed for binary polynomial optimization problems. Then we extend these results to the broader context of polynomial optimization problems, including non-binary variables (integer and continuous). Finally, we evaluate the numeric results obtained after adapting and implementing these reformulations in a solver.
In this project we study a class of reformulations for polynomial programming problems. Initially, we present the theoretical foundation of these reformulations, which were originally developed for binary polynomial optimization problems. Then we extend these results to the broader context of polynomial optimization problems, including non-binary variables (integer and continuous). Finally, we evaluate the numeric results obtained after adapting and implementing these reformulations in a solver.
Direction
GONZALEZ DIAZ, JULIO (Tutorships)
Rodríguez Acevedo, Iria (Co-tutorships)
GONZALEZ DIAZ, JULIO (Tutorships)
Rodríguez Acevedo, Iria (Co-tutorships)
Court
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
Uniqueness criteria for first-order ODE systems
Authorship
D.A.F.
Bachelor of Mathematics
D.A.F.
Bachelor of Mathematics
Defense date
07.18.2024 09:30
07.18.2024 09:30
Summary
The theory of ordinary differential equations is one of the most important fields of mathematical analysis. Within this theory, the existence and uniqueness of solutions is one of the most studied issues by great mathematicians. The objective of this project is to study, from a theoretical point of view, the conditions needed to guarantee the uniqueness of solutions. We will start by introducing some basic concepts that will be necessary throughout the project. Next, various proofs for the Picard-Lipschitz Theorem will be presented, as it is one of the central results of this work. Afterwards, several generalizations of the previous result will be studied, reaching criteria that require weaker conditions than those in the Picard-Lipschitz Theorem, such as the Osgood, Nagumo and Montel-Tonelli criteria. Finally, results will be outlined that ensure the uniqueness of solutions where the hypothesis about the function hold with respect to the independent variable or with respect to an arbitrary vector of R2, providing alternative criteria those mentioned before.
The theory of ordinary differential equations is one of the most important fields of mathematical analysis. Within this theory, the existence and uniqueness of solutions is one of the most studied issues by great mathematicians. The objective of this project is to study, from a theoretical point of view, the conditions needed to guarantee the uniqueness of solutions. We will start by introducing some basic concepts that will be necessary throughout the project. Next, various proofs for the Picard-Lipschitz Theorem will be presented, as it is one of the central results of this work. Afterwards, several generalizations of the previous result will be studied, reaching criteria that require weaker conditions than those in the Picard-Lipschitz Theorem, such as the Osgood, Nagumo and Montel-Tonelli criteria. Finally, results will be outlined that ensure the uniqueness of solutions where the hypothesis about the function hold with respect to the independent variable or with respect to an arbitrary vector of R2, providing alternative criteria those mentioned before.
Direction
Rodríguez López, Jorge (Tutorships)
Rodríguez López, Jorge (Tutorships)
Court
DIAZ RAMOS, JOSE CARLOS (Chairman)
COSTOYA RAMOS, MARIA CRISTINA (Secretary)
Rodríguez López, Rosana (Member)
DIAZ RAMOS, JOSE CARLOS (Chairman)
COSTOYA RAMOS, MARIA CRISTINA (Secretary)
Rodríguez López, Rosana (Member)
Introduction to fractal sets
Authorship
A.B.A.
Bachelor of Mathematics
A.B.A.
Bachelor of Mathematics
Defense date
07.18.2024 10:00
07.18.2024 10:00
Summary
A fractal set is one that possesses a fractal dimension exceeding its topological dimension. Some exhibit self-similarity, being identical to the original in smaller scale details. We will explore what we understand by fractal dimension, providing examples of these and calculating it for some sets, as well as discussing the advantages and issues presented by each. Then, we will focus on fractals that exhibit self-similarity, which are of great practical interest. We will define iterative function systems and provide methods for representing these fractals, as well as ways to easily calculate their fractal dimension. Finally, we will explore some applications of these sets, primarily Brownian motion, but also applications in fractal antennas, image compression, and even in art.
A fractal set is one that possesses a fractal dimension exceeding its topological dimension. Some exhibit self-similarity, being identical to the original in smaller scale details. We will explore what we understand by fractal dimension, providing examples of these and calculating it for some sets, as well as discussing the advantages and issues presented by each. Then, we will focus on fractals that exhibit self-similarity, which are of great practical interest. We will define iterative function systems and provide methods for representing these fractals, as well as ways to easily calculate their fractal dimension. Finally, we will explore some applications of these sets, primarily Brownian motion, but also applications in fractal antennas, image compression, and even in art.
Direction
FERNANDEZ TOJO, FERNANDO ADRIAN (Tutorships)
FERNANDEZ TOJO, FERNANDO ADRIAN (Tutorships)
Court
DIAZ RAMOS, JOSE CARLOS (Chairman)
COSTOYA RAMOS, MARIA CRISTINA (Secretary)
Rodríguez López, Rosana (Member)
DIAZ RAMOS, JOSE CARLOS (Chairman)
COSTOYA RAMOS, MARIA CRISTINA (Secretary)
Rodríguez López, Rosana (Member)
Fourier series and solving partial differential equations in a higher dimension.
Authorship
T.G.R.
Bachelor of Mathematics
T.G.R.
Bachelor of Mathematics
Defense date
07.18.2024 11:00
07.18.2024 11:00
Summary
In this essay, we extend the study of the Fourier Series and Introduction to PDEs course on the use of Fourier series and the method of separation of variables for solving initial and boundary value problems with the heat, wave, and Laplace equations. The main objective was to explore more deeply the fundamental concepts and advanced techniques for the deduction and resolution of these mathematical models. We focused particularly on analytical and numerical methods that allow addressing problems in higher spatial dimensions, specifically two and three dimensions. For these dimensions, we highlighted the importance of boundary conditions and initial conditions in obtaining solutions. We employed the method of separation of variables and Fourier series as crucial tools to decompose and solve these equations in various practical scenarios. Finally, we provided concrete examples and graphical visualizations of the solutions obtained through specialized software, particularly Matlab and Maple. In this way, we aim to provide a more comprehensive and applied understanding of the techniques studied.
In this essay, we extend the study of the Fourier Series and Introduction to PDEs course on the use of Fourier series and the method of separation of variables for solving initial and boundary value problems with the heat, wave, and Laplace equations. The main objective was to explore more deeply the fundamental concepts and advanced techniques for the deduction and resolution of these mathematical models. We focused particularly on analytical and numerical methods that allow addressing problems in higher spatial dimensions, specifically two and three dimensions. For these dimensions, we highlighted the importance of boundary conditions and initial conditions in obtaining solutions. We employed the method of separation of variables and Fourier series as crucial tools to decompose and solve these equations in various practical scenarios. Finally, we provided concrete examples and graphical visualizations of the solutions obtained through specialized software, particularly Matlab and Maple. In this way, we aim to provide a more comprehensive and applied understanding of the techniques studied.
Direction
LOPEZ POUSO, RODRIGO (Tutorships)
LOPEZ POUSO, RODRIGO (Tutorships)
Court
DIAZ RAMOS, JOSE CARLOS (Chairman)
COSTOYA RAMOS, MARIA CRISTINA (Secretary)
Rodríguez López, Rosana (Member)
DIAZ RAMOS, JOSE CARLOS (Chairman)
COSTOYA RAMOS, MARIA CRISTINA (Secretary)
Rodríguez López, Rosana (Member)
Measure generation
Authorship
P.T.P.
Bachelor of Mathematics
P.T.P.
Bachelor of Mathematics
Defense date
07.18.2024 11:30
07.18.2024 11:30
Summary
In this work we will deal with general aspects of measure theory and then focus on the construction and properties of the Lebesgue-Stieltjes measure. We will begin by introducing an algebra of sets in order to construct on it the mentioned measure. Then, we will prove certain measure extension theorems and, working with the exterior measure, we will be able to construct the Lebesgue-Stieltjes measure space. Finally, we will study some interesting properties about its structure and form and we will treat the case of sets that are not measurable with respect to the Lebesgue-Stieltjes measure.
In this work we will deal with general aspects of measure theory and then focus on the construction and properties of the Lebesgue-Stieltjes measure. We will begin by introducing an algebra of sets in order to construct on it the mentioned measure. Then, we will prove certain measure extension theorems and, working with the exterior measure, we will be able to construct the Lebesgue-Stieltjes measure space. Finally, we will study some interesting properties about its structure and form and we will treat the case of sets that are not measurable with respect to the Lebesgue-Stieltjes measure.
Direction
FERNANDEZ FERNANDEZ, FRANCISCO JAVIER (Tutorships)
BUEDO FERNANDEZ, SEBASTIAN (Co-tutorships)
FERNANDEZ FERNANDEZ, FRANCISCO JAVIER (Tutorships)
BUEDO FERNANDEZ, SEBASTIAN (Co-tutorships)
Court
DIAZ RAMOS, JOSE CARLOS (Chairman)
COSTOYA RAMOS, MARIA CRISTINA (Secretary)
Rodríguez López, Rosana (Member)
DIAZ RAMOS, JOSE CARLOS (Chairman)
COSTOYA RAMOS, MARIA CRISTINA (Secretary)
Rodríguez López, Rosana (Member)
Fundamental Theorem of Hopf Modules
Authorship
B.A.G.
Bachelor of Mathematics
B.A.G.
Bachelor of Mathematics
Defense date
07.16.2024 09:00
07.16.2024 09:00
Summary
Hopf algebra theory first appeared in the 1940s through the work of the German topologist Heinz Hopf. Due to its numerous applications, mainly in the field of quantum physics, interest in this area of study has notably increased over the last few decades. The aim of this project is to provide a general overview of this theory, with the goal of proving a classic result: the Fundamental Theorem of Hopf Modules. Firstly, in order to define the concept of Hopf algebra, the notions of algebra, coalgebra, bialgebra and antipode are introduced, and some of their basic properties are studied. Subsequently, the notions of module over an algebra and comodule over a coalgebra are presented in order to establish the concept of Hopf module and prove the aforementioned theorem. Lastly, the importance of this result is showcased through some examples of application.
Hopf algebra theory first appeared in the 1940s through the work of the German topologist Heinz Hopf. Due to its numerous applications, mainly in the field of quantum physics, interest in this area of study has notably increased over the last few decades. The aim of this project is to provide a general overview of this theory, with the goal of proving a classic result: the Fundamental Theorem of Hopf Modules. Firstly, in order to define the concept of Hopf algebra, the notions of algebra, coalgebra, bialgebra and antipode are introduced, and some of their basic properties are studied. Subsequently, the notions of module over an algebra and comodule over a coalgebra are presented in order to establish the concept of Hopf module and prove the aforementioned theorem. Lastly, the importance of this result is showcased through some examples of application.
Direction
FERNANDEZ VILABOA, JOSE MANUEL (Tutorships)
RAMOS PEREZ, BRAIS (Co-tutorships)
FERNANDEZ VILABOA, JOSE MANUEL (Tutorships)
RAMOS PEREZ, BRAIS (Co-tutorships)
Court
GONZALEZ MANTEIGA, WENCESLAO (Chairman)
PAEZ GUILLAN, MARIA PILAR (Secretary)
ALVAREZ DIOS, JOSE ANTONIO (Member)
GONZALEZ MANTEIGA, WENCESLAO (Chairman)
PAEZ GUILLAN, MARIA PILAR (Secretary)
ALVAREZ DIOS, JOSE ANTONIO (Member)
Analysis of censored data
Authorship
A.C.P.
Bachelor of Mathematics
A.C.P.
Bachelor of Mathematics
Defense date
07.16.2024 10:00
07.16.2024 10:00
Summary
Censoring is a common occurrence in Survival Analysis and is linked to a partial loss of information. In this work, we illustrate how the traditional nonparametric estimators, such as the empirical cumulative distribution function and its conditional version, do not present good results in order to estimate the (conditional) cumulative distribution function of a right-censored random variable T. In the context of censored data, we introduce the Kaplan-Meier estimator and the Beran estimator, as estimators of the unconditional and conditional cumulative distribution function, respectively. For the Kaplan-Meier estimator, we also discuss its most relevant properties, which are essential to construct confidence intervals. To compare the behavior of the different estimators we perform several Montecarlo simulation studies using the statistical software R. Finally, we analyze a real data set that contains information about the survival time associated with a group of lung cancer patients by using the censored-adapted estimators.
Censoring is a common occurrence in Survival Analysis and is linked to a partial loss of information. In this work, we illustrate how the traditional nonparametric estimators, such as the empirical cumulative distribution function and its conditional version, do not present good results in order to estimate the (conditional) cumulative distribution function of a right-censored random variable T. In the context of censored data, we introduce the Kaplan-Meier estimator and the Beran estimator, as estimators of the unconditional and conditional cumulative distribution function, respectively. For the Kaplan-Meier estimator, we also discuss its most relevant properties, which are essential to construct confidence intervals. To compare the behavior of the different estimators we perform several Montecarlo simulation studies using the statistical software R. Finally, we analyze a real data set that contains information about the survival time associated with a group of lung cancer patients by using the censored-adapted estimators.
Direction
CONDE AMBOAGE, MERCEDES (Tutorships)
CONDE AMBOAGE, MERCEDES (Tutorships)
Court
GONZALEZ MANTEIGA, WENCESLAO (Chairman)
PAEZ GUILLAN, MARIA PILAR (Secretary)
ALVAREZ DIOS, JOSE ANTONIO (Member)
GONZALEZ MANTEIGA, WENCESLAO (Chairman)
PAEZ GUILLAN, MARIA PILAR (Secretary)
ALVAREZ DIOS, JOSE ANTONIO (Member)
Numerical solution of the heat equation
Authorship
L.G.V.
Bachelor of Mathematics
L.G.V.
Bachelor of Mathematics
Defense date
07.17.2024 09:00
07.17.2024 09:00
Summary
In this document we study the numerical solution of the heat equation, using finite difference methods and finite element methods. Specifically, we will implement (using MATLAB for this purpose) finite difference methods, for both one-dimensional and two-dimensional equations, giving several schemes and comparing them with different examples. With regard to the finite element method, we will only deal with the one-dimensional equation, using an implicit temporal scheme and treating the semi-discrete problem as an Sturm-Liouville problem. Finally, we analyze numerically the properties of stability, convergence and order of the different schemes and compare them using the solution of the introduced test examples.
In this document we study the numerical solution of the heat equation, using finite difference methods and finite element methods. Specifically, we will implement (using MATLAB for this purpose) finite difference methods, for both one-dimensional and two-dimensional equations, giving several schemes and comparing them with different examples. With regard to the finite element method, we will only deal with the one-dimensional equation, using an implicit temporal scheme and treating the semi-discrete problem as an Sturm-Liouville problem. Finally, we analyze numerically the properties of stability, convergence and order of the different schemes and compare them using the solution of the introduced test examples.
Direction
Ferrín González, José Luis (Tutorships)
Ferrín González, José Luis (Tutorships)
Court
GONZALEZ MANTEIGA, WENCESLAO (Chairman)
PAEZ GUILLAN, MARIA PILAR (Secretary)
ALVAREZ DIOS, JOSE ANTONIO (Member)
GONZALEZ MANTEIGA, WENCESLAO (Chairman)
PAEZ GUILLAN, MARIA PILAR (Secretary)
ALVAREZ DIOS, JOSE ANTONIO (Member)
The ROC curve and its application in Biomedicine
Authorship
M.N.S.
Bachelor of Mathematics
M.N.S.
Bachelor of Mathematics
Defense date
07.17.2024 10:00
07.17.2024 10:00
Summary
The ROC curve is a widely used statistical tool in the field of Biomedicine that allows for the evaluation of the effectiveness of a diagnostic test in distinguishing between sick and healthy patients. In other words, it studies the ability of a variable, known as the diagnostic variable, to detect the presence or absence of a disease. The discriminatory capacity of a test is subject to a value called the threshold, which allows for the classification of a patient based on whether the result for the variable of interest exceeds this value. In this work, we formalize the basic concepts for the definition of the ROC curve, along with the justification of its most relevant properties. We also present summary measures that enable the quantification of the classification efficiency of diagnostic tests and the comparison between them. Additionally, we briefly detail several methods for selecting the threshold. Once the theoretical or population version of the ROC curve is presented, we review different methods for its estimation, particularly examining both parametric and nonparametric methods. Finally, we illustrate the statistical concepts and techniques addressed in this work through two databases on the diagnosis of iron deficiency anaemia and heart failure. Throughout the work, we use free statistical software R in its version 4.4.1, and the code can be found in Appendix A.
The ROC curve is a widely used statistical tool in the field of Biomedicine that allows for the evaluation of the effectiveness of a diagnostic test in distinguishing between sick and healthy patients. In other words, it studies the ability of a variable, known as the diagnostic variable, to detect the presence or absence of a disease. The discriminatory capacity of a test is subject to a value called the threshold, which allows for the classification of a patient based on whether the result for the variable of interest exceeds this value. In this work, we formalize the basic concepts for the definition of the ROC curve, along with the justification of its most relevant properties. We also present summary measures that enable the quantification of the classification efficiency of diagnostic tests and the comparison between them. Additionally, we briefly detail several methods for selecting the threshold. Once the theoretical or population version of the ROC curve is presented, we review different methods for its estimation, particularly examining both parametric and nonparametric methods. Finally, we illustrate the statistical concepts and techniques addressed in this work through two databases on the diagnosis of iron deficiency anaemia and heart failure. Throughout the work, we use free statistical software R in its version 4.4.1, and the code can be found in Appendix A.
Direction
BORRAJO GARCIA, MARIA ISABEL (Tutorships)
BORRAJO GARCIA, MARIA ISABEL (Tutorships)
Court
GONZALEZ MANTEIGA, WENCESLAO (Chairman)
PAEZ GUILLAN, MARIA PILAR (Secretary)
ALVAREZ DIOS, JOSE ANTONIO (Member)
GONZALEZ MANTEIGA, WENCESLAO (Chairman)
PAEZ GUILLAN, MARIA PILAR (Secretary)
ALVAREZ DIOS, JOSE ANTONIO (Member)
Cubic spline interpolation
Authorship
A.B.G.
Bachelor of Mathematics
A.B.G.
Bachelor of Mathematics
Defense date
09.11.2024 15:30
09.11.2024 15:30
Summary
In this dissertation, we will introduce splines and prove how, in contrast to polynomial and piecewise interpolation, provide a superior tool for interpolation with regular functions. Once we present the intuitive idea, we will study the space of splines of degree m with respect to a partition of n+1 nodes, paying special attention to the basis of this space constituted by B-splines, which are of great importance in practical computations. Thereafter, we will solve the interpolation problem using cubic splines, as these are the most commonly used in practice. We will discuss the existence, uniqueness and the calculation algorithm, which we will implement in a MATLAB code to illustrate some examples. Finally, we will analyze the error between the interpolating spline and the function from which the interpolation nodes are derived, assuming certain regularity conditions in the latter.
In this dissertation, we will introduce splines and prove how, in contrast to polynomial and piecewise interpolation, provide a superior tool for interpolation with regular functions. Once we present the intuitive idea, we will study the space of splines of degree m with respect to a partition of n+1 nodes, paying special attention to the basis of this space constituted by B-splines, which are of great importance in practical computations. Thereafter, we will solve the interpolation problem using cubic splines, as these are the most commonly used in practice. We will discuss the existence, uniqueness and the calculation algorithm, which we will implement in a MATLAB code to illustrate some examples. Finally, we will analyze the error between the interpolating spline and the function from which the interpolation nodes are derived, assuming certain regularity conditions in the latter.
Direction
VIAÑO REY, JUAN MANUEL (Tutorships)
VIAÑO REY, JUAN MANUEL (Tutorships)
Court
GARCIA RODICIO, ANTONIO (Chairman)
ALVAREZ DIOS, JOSE ANTONIO (Secretary)
RODRIGUEZ CASAL, ALBERTO (Member)
GARCIA RODICIO, ANTONIO (Chairman)
ALVAREZ DIOS, JOSE ANTONIO (Secretary)
RODRIGUEZ CASAL, ALBERTO (Member)
An introduction to generalized additive models
Authorship
M.G.P.
Bachelor of Mathematics
M.G.P.
Bachelor of Mathematics
Defense date
09.11.2024 16:15
09.11.2024 16:15
Summary
Generalized additive models represent a very useful tool in data analysis due to their flexibility and ability to model non-linear relationships between variables. In this work, a review of linear and generalized linear regression models will be conducted, exposing their limitations and the need to employ more flexible methods, such as generalized additive models. These models introduce \textit{smooth} functions to model the relationships between the response variable and the explanatory variables. Their theoretical formulation will be presented, and the main estimation methods using splines will be examined. The introduced models, along with their limitations, will be illustrated through simulations. Finally, an application of the generalized additive model to a real database will be presented. This example will illustrate the advantages of this model in a real context, where the ability to adapt to non-linear patterns is essential for obtaining accurate and useful results.
Generalized additive models represent a very useful tool in data analysis due to their flexibility and ability to model non-linear relationships between variables. In this work, a review of linear and generalized linear regression models will be conducted, exposing their limitations and the need to employ more flexible methods, such as generalized additive models. These models introduce \textit{smooth} functions to model the relationships between the response variable and the explanatory variables. Their theoretical formulation will be presented, and the main estimation methods using splines will be examined. The introduced models, along with their limitations, will be illustrated through simulations. Finally, an application of the generalized additive model to a real database will be presented. This example will illustrate the advantages of this model in a real context, where the ability to adapt to non-linear patterns is essential for obtaining accurate and useful results.
Direction
CRUJEIRAS CASAIS, ROSA MARÍA (Tutorships)
VIDAL GARCIA, MARIA (Co-tutorships)
CRUJEIRAS CASAIS, ROSA MARÍA (Tutorships)
VIDAL GARCIA, MARIA (Co-tutorships)
Court
GARCIA RODICIO, ANTONIO (Chairman)
ALVAREZ DIOS, JOSE ANTONIO (Secretary)
RODRIGUEZ CASAL, ALBERTO (Member)
GARCIA RODICIO, ANTONIO (Chairman)
ALVAREZ DIOS, JOSE ANTONIO (Secretary)
RODRIGUEZ CASAL, ALBERTO (Member)
Mathematical modeling of the growth of cancerous tumors.
Authorship
C.L.G.
Bachelor of Mathematics
C.L.G.
Bachelor of Mathematics
Defense date
09.12.2024 11:00
09.12.2024 11:00
Summary
Comparative study of the best-known mathematical models to simulate tumor growth, such as the B. Gompertz model or the L. Bertalanffy model: brief description of the characteristics of a tumor, biological indicators that can be “mathematized”, important characteristics and properties of diferent models.
Comparative study of the best-known mathematical models to simulate tumor growth, such as the B. Gompertz model or the L. Bertalanffy model: brief description of the characteristics of a tumor, biological indicators that can be “mathematized”, important characteristics and properties of diferent models.
Direction
VIAÑO REY, JUAN MANUEL (Tutorships)
VIAÑO REY, JUAN MANUEL (Tutorships)
Court
GARCIA RODICIO, ANTONIO (Chairman)
ALVAREZ DIOS, JOSE ANTONIO (Secretary)
RODRIGUEZ CASAL, ALBERTO (Member)
GARCIA RODICIO, ANTONIO (Chairman)
ALVAREZ DIOS, JOSE ANTONIO (Secretary)
RODRIGUEZ CASAL, ALBERTO (Member)
Exploratory Statistical Analysis of Complex Data
Authorship
A.Q.D.
Bachelor of Mathematics
A.Q.D.
Bachelor of Mathematics
Defense date
09.12.2024 16:40
09.12.2024 16:40
Summary
This project aims to develop exploratory statistical techniques, focusing on the analysis of various statistical objects. The first chapter the basic concepts of OODA and its terminology are presented. In the second, various exploratory descriptive analysis techniques are explained, applied to complex data. Principal Component Analysis and its application to visualization is presented in the third chapter. The fourth and fifth chapters discuss supervised and unsupervised classification techniques, respectively. In the last chapter, a real data problem is introduced in which some of the techniques seen previously are applied.
This project aims to develop exploratory statistical techniques, focusing on the analysis of various statistical objects. The first chapter the basic concepts of OODA and its terminology are presented. In the second, various exploratory descriptive analysis techniques are explained, applied to complex data. Principal Component Analysis and its application to visualization is presented in the third chapter. The fourth and fifth chapters discuss supervised and unsupervised classification techniques, respectively. In the last chapter, a real data problem is introduced in which some of the techniques seen previously are applied.
Direction
GONZALEZ MANTEIGA, WENCESLAO (Tutorships)
GONZALEZ MANTEIGA, WENCESLAO (Tutorships)
Court
OTERO ESPINAR, MARIA VICTORIA (Chairman)
GONZALEZ DIAZ, JULIO (Secretary)
Jeremías López, Ana (Member)
OTERO ESPINAR, MARIA VICTORIA (Chairman)
GONZALEZ DIAZ, JULIO (Secretary)
Jeremías López, Ana (Member)
Biological dynamic systems optimization with algebraic differential equations
Authorship
C.C.P.
Double bachelor degree of Engeneering in Information Technology and Mathematics
C.C.P.
Double bachelor degree of Engeneering in Information Technology and Mathematics
Defense date
07.16.2024 16:40
07.16.2024 16:40
Summary
Mathematical modeling is essential for the understanding and handling of biological dynamic systems. Optimal control problems, which usually include algebraic differential equations as constrains, need to be used. For solving these kinds of problems, of infinite dimension, there are several different types of strategies. In this work, we will delve into direct methods, which perform a discretization of the optimal control problem, transforming it into a finite dimension nonlinear programming problem. In biology, most existing problems involve estimating system parameters from a series of experimental observations. For such a biological problem, a numerical study will be proposed for its solution, varying its configuration, and the results obtained will be commented in depth.
Mathematical modeling is essential for the understanding and handling of biological dynamic systems. Optimal control problems, which usually include algebraic differential equations as constrains, need to be used. For solving these kinds of problems, of infinite dimension, there are several different types of strategies. In this work, we will delve into direct methods, which perform a discretization of the optimal control problem, transforming it into a finite dimension nonlinear programming problem. In biology, most existing problems involve estimating system parameters from a series of experimental observations. For such a biological problem, a numerical study will be proposed for its solution, varying its configuration, and the results obtained will be commented in depth.
Direction
GONZALEZ RUEDA, ANGEL MANUEL (Tutorships)
GONZALEZ RUEDA, ANGEL MANUEL (Tutorships)
Court
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
Statistical learning for algorithm selection in optimisation problems
Authorship
A.F.E.
Double bachelor degree of Engeneering in Information Technology and Mathematics
A.F.E.
Double bachelor degree of Engeneering in Information Technology and Mathematics
Defense date
07.16.2024 18:00
07.16.2024 18:00
Summary
In this work, statistical learning techniques will be used to predict the best performing global optimiser for a non-linear mathematical programming problem. First of all, a method for solving integer linear programming problems and its adaptation to the nonlinear case, where new difficulties will arise, will be explained. Subsequently, the statistical learning problem and two techniques that allow to fit a model and create predictions will be presented: linear regression and single hidden layer neural networks. These techniques will allow learning to be performed on a set of problems and the results to be obtained, looking at the performance of the different optimisers.
In this work, statistical learning techniques will be used to predict the best performing global optimiser for a non-linear mathematical programming problem. First of all, a method for solving integer linear programming problems and its adaptation to the nonlinear case, where new difficulties will arise, will be explained. Subsequently, the statistical learning problem and two techniques that allow to fit a model and create predictions will be presented: linear regression and single hidden layer neural networks. These techniques will allow learning to be performed on a set of problems and the results to be obtained, looking at the performance of the different optimisers.
Direction
GONZALEZ DIAZ, JULIO (Tutorships)
GOMEZ CASARES, IGNACIO (Co-tutorships)
GONZALEZ DIAZ, JULIO (Tutorships)
GOMEZ CASARES, IGNACIO (Co-tutorships)
Court
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
De Rham cohomology
Authorship
C.L.A.
Double bachelor degree of Engeneering in Information Technology and Mathematics
C.L.A.
Double bachelor degree of Engeneering in Information Technology and Mathematics
Defense date
07.17.2024 10:10
07.17.2024 10:10
Summary
Differential forms constitute a highly relevant mathematical object in Differential Topology and Differential Geometry with important applications to Physics. The aim of this work is to study different uses of differential forms and to show how they allow for effectively addressing numerous problems. First, basic notions about fiber bundle theory will be given, which are fundamental to address the rest of the work. Following this, the concept of orientation of a manifold will be defined, and a brief introduction to semi-Riemannian manifolds will be presented. This will allow us to generalize basic concepts of vector calculus to differentiable manifolds. Next, integration on manifolds will be studied, and the generalized Stokes' Theorem will be proven. These concepts will also be addressed for the case when the manifold is non-orientable. Finally, de Rham cohomology and its main properties will be studied. In addition to all this, applications to electromagnetism of the studied concepts will be included throughout the work. Specifically, Maxwell's Equations will be formulated using differential forms, Stokes' Theorem will be used to give an integral version of the equations, and de Rham cohomology will be used to illustrate wormholes and magnetic monopoles, physical phenomena never detected experimentally.
Differential forms constitute a highly relevant mathematical object in Differential Topology and Differential Geometry with important applications to Physics. The aim of this work is to study different uses of differential forms and to show how they allow for effectively addressing numerous problems. First, basic notions about fiber bundle theory will be given, which are fundamental to address the rest of the work. Following this, the concept of orientation of a manifold will be defined, and a brief introduction to semi-Riemannian manifolds will be presented. This will allow us to generalize basic concepts of vector calculus to differentiable manifolds. Next, integration on manifolds will be studied, and the generalized Stokes' Theorem will be proven. These concepts will also be addressed for the case when the manifold is non-orientable. Finally, de Rham cohomology and its main properties will be studied. In addition to all this, applications to electromagnetism of the studied concepts will be included throughout the work. Specifically, Maxwell's Equations will be formulated using differential forms, Stokes' Theorem will be used to give an integral version of the equations, and de Rham cohomology will be used to illustrate wormholes and magnetic monopoles, physical phenomena never detected experimentally.
Direction
Álvarez López, Jesús Antonio (Tutorships)
Álvarez López, Jesús Antonio (Tutorships)
Court
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
Piecewise linear motion planning in robotics
Authorship
M.T.L.
Double bachelor degree of Engeneering in Information Technology and Mathematics
M.T.L.
Double bachelor degree of Engeneering in Information Technology and Mathematics
Defense date
07.17.2024 12:50
07.17.2024 12:50
Summary
The topological complexity is an invariant proposed by Michael Farber, arising in the context of motion planning in robotics. In this thesis, a discrete analogy proposed by Jesús González is developed: the simplicial complexity of a simplicial complex, which is equivalent to the topological complexity of its geometric realization but is computable using combinatorial methods.
The topological complexity is an invariant proposed by Michael Farber, arising in the context of motion planning in robotics. In this thesis, a discrete analogy proposed by Jesús González is developed: the simplicial complexity of a simplicial complex, which is equivalent to the topological complexity of its geometric realization but is computable using combinatorial methods.
Direction
Macías Virgós, Enrique (Tutorships)
MOSQUERA LOIS, DAVID (Co-tutorships)
Macías Virgós, Enrique (Tutorships)
MOSQUERA LOIS, DAVID (Co-tutorships)
Court
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
Methods of Classification and Ensemble of Classifiers in Supervised Learning
Authorship
A.G.L.
Double bachelor degree of Engeneering in Information Technology and Mathematics
A.G.L.
Double bachelor degree of Engeneering in Information Technology and Mathematics
Defense date
07.16.2024 12:30
07.16.2024 12:30
Summary
In this work, various ensemble techniques in supervised learning are analyzed, focusing on bagging, random forests, and AdaBoost. Initially, the fundamentals of statistical classification and supervised learning are explained. Subsequently, different strategies for combining classifier outputs when they consist of predictions and continuous values are examined. Finally, the ensemble methods are detailed, highlighting the characteristics that differentiate them.
In this work, various ensemble techniques in supervised learning are analyzed, focusing on bagging, random forests, and AdaBoost. Initially, the fundamentals of statistical classification and supervised learning are explained. Subsequently, different strategies for combining classifier outputs when they consist of predictions and continuous values are examined. Finally, the ensemble methods are detailed, highlighting the characteristics that differentiate them.
Direction
PATEIRO LOPEZ, BEATRIZ (Tutorships)
Rodríguez Acevedo, Iria (Co-tutorships)
PATEIRO LOPEZ, BEATRIZ (Tutorships)
Rodríguez Acevedo, Iria (Co-tutorships)
Court
GONZALEZ MANTEIGA, WENCESLAO (Chairman)
PAEZ GUILLAN, MARIA PILAR (Secretary)
ALVAREZ DIOS, JOSE ANTONIO (Member)
GONZALEZ MANTEIGA, WENCESLAO (Chairman)
PAEZ GUILLAN, MARIA PILAR (Secretary)
ALVAREZ DIOS, JOSE ANTONIO (Member)
Introduction to nonlinear optimization
Authorship
D.P.V.
Double bachelor degree of Engeneering in Information Technology and Mathematics
D.P.V.
Double bachelor degree of Engeneering in Information Technology and Mathematics
Defense date
07.17.2024 12:30
07.17.2024 12:30
Summary
This project offers a theorical approach of the fundamentals of nonlinear optimization. It begins with a review of the optimization-related content covered throughout the degree, aiming to establish the starting point. After introducing the necessary preliminaries for a correct formalization, the main points of the work are presented: optimality conditions and Lagrangian duality. Firstly, the Fritz John and Karush-Kuhn-Tucker optimality conditions for nonlinear problems are developed; and finally, the Lagrangian dual problem and the duality theorems are discussed. The relationships between the concepts presented and their generalizations or particular cases are addresed as the work progresses.
This project offers a theorical approach of the fundamentals of nonlinear optimization. It begins with a review of the optimization-related content covered throughout the degree, aiming to establish the starting point. After introducing the necessary preliminaries for a correct formalization, the main points of the work are presented: optimality conditions and Lagrangian duality. Firstly, the Fritz John and Karush-Kuhn-Tucker optimality conditions for nonlinear problems are developed; and finally, the Lagrangian dual problem and the duality theorems are discussed. The relationships between the concepts presented and their generalizations or particular cases are addresed as the work progresses.
Direction
GONZALEZ DIAZ, JULIO (Tutorships)
GONZALEZ DIAZ, JULIO (Tutorships)
Court
GONZALEZ MANTEIGA, WENCESLAO (Chairman)
PAEZ GUILLAN, MARIA PILAR (Secretary)
ALVAREZ DIOS, JOSE ANTONIO (Member)
GONZALEZ MANTEIGA, WENCESLAO (Chairman)
PAEZ GUILLAN, MARIA PILAR (Secretary)
ALVAREZ DIOS, JOSE ANTONIO (Member)
Existence of periodic solutions to the Mathieu equation
Authorship
E.P.V.
Double bachelor degree of Engeneering in Information Technology and Mathematics
E.P.V.
Double bachelor degree of Engeneering in Information Technology and Mathematics
Defense date
07.16.2024 17:45
07.16.2024 17:45
Summary
In this work, we will consider the dynamics of an electron beam guided by an axially symmetric periodic magnetic field. This type of mechanism is known as an electron gun or microwave valve and is a part of many scientific, industrial, and domestic electronic devices. Its dynamics can be modeled using the Mathieu equation, which is a second-order equation with singularities and can be treated as a particular case of the Hill equation. Thus, we will focus on modeling the phenomenon and describe some results that guarantee the existence of periodic solutions with constant sign.
In this work, we will consider the dynamics of an electron beam guided by an axially symmetric periodic magnetic field. This type of mechanism is known as an electron gun or microwave valve and is a part of many scientific, industrial, and domestic electronic devices. Its dynamics can be modeled using the Mathieu equation, which is a second-order equation with singularities and can be treated as a particular case of the Hill equation. Thus, we will focus on modeling the phenomenon and describe some results that guarantee the existence of periodic solutions with constant sign.
Direction
CABADA FERNANDEZ, ALBERTO (Tutorships)
CABADA FERNANDEZ, ALBERTO (Tutorships)
Court
FEBRERO BANDE, MANUEL (Chairman)
BUEDO FERNANDEZ, SEBASTIAN (Secretary)
RODRIGUEZ GARCIA, JERONIMO (Member)
FEBRERO BANDE, MANUEL (Chairman)
BUEDO FERNANDEZ, SEBASTIAN (Secretary)
RODRIGUEZ GARCIA, JERONIMO (Member)
Introduction to multiobjective programming
Authorship
J.R.F.
Bachelor of Mathematics
J.R.F.
Bachelor of Mathematics
Defense date
07.18.2024 09:00
07.18.2024 09:00
Summary
The goal of multiobjective optimization is to find the best feasible solutions to mathematical programming problems which have more than one objective function. In the first chapter, we introduce multiobjective programming from the perspective of a minimization problem. We also define our solutions based on this premise: those are Pareto optimal points or efficient points. Then, we characterize these points and justify when their existence is assured by certain results. In the end, we explain some concepts related to efficient points: weakly efficient points, strictly efficient points and properly efficient points. \par As we progress through our project, we describe different solving methods of multiobjective problems. Each approach is supported by a series of results to show its usefulness and the scope of problems it can address. Finally, we implement these methods in the R programming language.
The goal of multiobjective optimization is to find the best feasible solutions to mathematical programming problems which have more than one objective function. In the first chapter, we introduce multiobjective programming from the perspective of a minimization problem. We also define our solutions based on this premise: those are Pareto optimal points or efficient points. Then, we characterize these points and justify when their existence is assured by certain results. In the end, we explain some concepts related to efficient points: weakly efficient points, strictly efficient points and properly efficient points. \par As we progress through our project, we describe different solving methods of multiobjective problems. Each approach is supported by a series of results to show its usefulness and the scope of problems it can address. Finally, we implement these methods in the R programming language.
Direction
GONZALEZ RUEDA, ANGEL MANUEL (Tutorships)
GONZALEZ RODRIGUEZ, BRAIS (Co-tutorships)
GONZALEZ RUEDA, ANGEL MANUEL (Tutorships)
GONZALEZ RODRIGUEZ, BRAIS (Co-tutorships)
Court
GONZALEZ MANTEIGA, WENCESLAO (Chairman)
PAEZ GUILLAN, MARIA PILAR (Secretary)
ALVAREZ DIOS, JOSE ANTONIO (Member)
GONZALEZ MANTEIGA, WENCESLAO (Chairman)
PAEZ GUILLAN, MARIA PILAR (Secretary)
ALVAREZ DIOS, JOSE ANTONIO (Member)
Numerical resolution of blood flow problems in a conduit
Authorship
B.R.G.
Bachelor of Mathematics
B.R.G.
Bachelor of Mathematics
Defense date
07.18.2024 10:00
07.18.2024 10:00
Summary
A fully-implicit finite volume method for the simulation of one-dimensional blood flow is proposed. We first introduce the concept of hyperbolic systems of conservation laws, and the finite volume methods as a mean to find numerical solutions for this type of PDEs. Then, the 1D blood flow model is presented and split into three subsystems: the first one containing convective terms, the second one containing diffusive terms and the third one for the pressure variable. We then study a semi-implicit finite volume method that solves implicitly the last two stages and that discretizes explicitly the convective stage. Afterwards, we propose a novel scheme that discretizes the convective terms using an inexact Newton method combined with a BiCGSTAB algorithm. For the discretization of the flux term, we employ Rusanov or Ducros numerical flux functions. Finally, the new method is validated by comparing it with the semi-implicit scheme and exact solutions in a set of Riemann Problems in the context of blood flow simulation.
A fully-implicit finite volume method for the simulation of one-dimensional blood flow is proposed. We first introduce the concept of hyperbolic systems of conservation laws, and the finite volume methods as a mean to find numerical solutions for this type of PDEs. Then, the 1D blood flow model is presented and split into three subsystems: the first one containing convective terms, the second one containing diffusive terms and the third one for the pressure variable. We then study a semi-implicit finite volume method that solves implicitly the last two stages and that discretizes explicitly the convective stage. Afterwards, we propose a novel scheme that discretizes the convective terms using an inexact Newton method combined with a BiCGSTAB algorithm. For the discretization of the flux term, we employ Rusanov or Ducros numerical flux functions. Finally, the new method is validated by comparing it with the semi-implicit scheme and exact solutions in a set of Riemann Problems in the context of blood flow simulation.
Direction
VAZQUEZ CENDON, MARIA ELENA (Tutorships)
Busto Ulloa, Saray (Co-tutorships)
VAZQUEZ CENDON, MARIA ELENA (Tutorships)
Busto Ulloa, Saray (Co-tutorships)
Court
GONZALEZ MANTEIGA, WENCESLAO (Chairman)
PAEZ GUILLAN, MARIA PILAR (Secretary)
ALVAREZ DIOS, JOSE ANTONIO (Member)
GONZALEZ MANTEIGA, WENCESLAO (Chairman)
PAEZ GUILLAN, MARIA PILAR (Secretary)
ALVAREZ DIOS, JOSE ANTONIO (Member)
Statistical Learning Techniques for Classification and Regression: Study of the Adaboost Algorithm and Applications
Authorship
M.V.C.
Bachelor of Mathematics
M.V.C.
Bachelor of Mathematics
Defense date
07.18.2024 11:30
07.18.2024 11:30
Summary
Boosting algorithms are statistical learning techniques based on the use of combinations of weak classifiers with the aim of obtaining a more accurate final model. In this work, we will address one of the most widely used boosting methods: Adaboost or Adaptive Boosting. From a statistical point of view, the Adaboost algorithm seeks to minimize the error produced by weak classifiers in stages, modifying the weight assigned to each one based on its accuracy. The more complex the classification of an example, the more emphasis will be placed on it, and once the desired error is achieved in the training stage, the generalization to unlabeled data will proceed. In this work, an introduction to boosting methods will be presented, and the Adaboost method will be explored in depth, analyzing its theoretical foundations and evaluating its performance in practice compared to other statistical learning methods
Boosting algorithms are statistical learning techniques based on the use of combinations of weak classifiers with the aim of obtaining a more accurate final model. In this work, we will address one of the most widely used boosting methods: Adaboost or Adaptive Boosting. From a statistical point of view, the Adaboost algorithm seeks to minimize the error produced by weak classifiers in stages, modifying the weight assigned to each one based on its accuracy. The more complex the classification of an example, the more emphasis will be placed on it, and once the desired error is achieved in the training stage, the generalization to unlabeled data will proceed. In this work, an introduction to boosting methods will be presented, and the Adaboost method will be explored in depth, analyzing its theoretical foundations and evaluating its performance in practice compared to other statistical learning methods
Direction
PATEIRO LOPEZ, BEATRIZ (Tutorships)
GONZALEZ RODRIGUEZ, BRAIS (Co-tutorships)
PATEIRO LOPEZ, BEATRIZ (Tutorships)
GONZALEZ RODRIGUEZ, BRAIS (Co-tutorships)
Court
GONZALEZ MANTEIGA, WENCESLAO (Chairman)
PAEZ GUILLAN, MARIA PILAR (Secretary)
ALVAREZ DIOS, JOSE ANTONIO (Member)
GONZALEZ MANTEIGA, WENCESLAO (Chairman)
PAEZ GUILLAN, MARIA PILAR (Secretary)
ALVAREZ DIOS, JOSE ANTONIO (Member)
Autoregressive predictive models. Application to financial series
Authorship
M.B.V.
Bachelor of Mathematics
M.B.V.
Bachelor of Mathematics
Defense date
07.16.2024 10:00
07.16.2024 10:00
Summary
This paper consists on the analysis of autoregressive process applied to the financial sector. Thus, the project will be composed by a therorical framework, where different time series will be studied, and a practical section, where a full real example of a financial asset will be treated. With this purpose, we will start with a brief review of the linear regression model and some of its chracteristics. After this, we will formally introduce the concept of stochastic process and study linear autoregressive process in mean and variance, which are known as AR and ARCH models. Moreover, some non-linear process, parametric and non-parametric, that are commonly used in financial analysis will be treated. Finally, we will work with a practical case, where the chosen asset for its analysis is the value of Prosegur shares. Behavior of the mean and variance of the series will be studied and we will compare its volatility, as a risk measure, with other financial assets. For the calculations and analysis it will be used R software and all of the code used for this paper may be found on the appendix.
This paper consists on the analysis of autoregressive process applied to the financial sector. Thus, the project will be composed by a therorical framework, where different time series will be studied, and a practical section, where a full real example of a financial asset will be treated. With this purpose, we will start with a brief review of the linear regression model and some of its chracteristics. After this, we will formally introduce the concept of stochastic process and study linear autoregressive process in mean and variance, which are known as AR and ARCH models. Moreover, some non-linear process, parametric and non-parametric, that are commonly used in financial analysis will be treated. Finally, we will work with a practical case, where the chosen asset for its analysis is the value of Prosegur shares. Behavior of the mean and variance of the series will be studied and we will compare its volatility, as a risk measure, with other financial assets. For the calculations and analysis it will be used R software and all of the code used for this paper may be found on the appendix.
Direction
GONZALEZ MANTEIGA, WENCESLAO (Tutorships)
GONZALEZ MANTEIGA, WENCESLAO (Tutorships)
Court
TORRES LOPERA, JUAN FRANCISCO (Chairman)
CONDE AMBOAGE, MERCEDES (Secretary)
López Pouso, Óscar (Member)
TORRES LOPERA, JUAN FRANCISCO (Chairman)
CONDE AMBOAGE, MERCEDES (Secretary)
López Pouso, Óscar (Member)
Heteroskedasticity in regression models
Authorship
M.C.F.
Bachelor of Mathematics
M.C.F.
Bachelor of Mathematics
Defense date
07.16.2024 10:45
07.16.2024 10:45
Summary
In regression models, it is usual to suppose that the variance of the error is constant. However, this is not always true and causes the results obtained to be false. In this work, a review of the problem of heteroskedasticity in regression models will be made. Its causes and consequences will be studied, which can be very varied. Additionally, several methods to detect that the variance is not constant and procedures to address this problem will be explained. All this will be accompained by examples.
In regression models, it is usual to suppose that the variance of the error is constant. However, this is not always true and causes the results obtained to be false. In this work, a review of the problem of heteroskedasticity in regression models will be made. Its causes and consequences will be studied, which can be very varied. Additionally, several methods to detect that the variance is not constant and procedures to address this problem will be explained. All this will be accompained by examples.
Direction
SANCHEZ SELLERO, CESAR ANDRES (Tutorships)
SANCHEZ SELLERO, CESAR ANDRES (Tutorships)
Court
TORRES LOPERA, JUAN FRANCISCO (Chairman)
CONDE AMBOAGE, MERCEDES (Secretary)
López Pouso, Óscar (Member)
TORRES LOPERA, JUAN FRANCISCO (Chairman)
CONDE AMBOAGE, MERCEDES (Secretary)
López Pouso, Óscar (Member)
Voronoi diagrams
Authorship
P.D.M.C.
Bachelor of Mathematics
P.D.M.C.
Bachelor of Mathematics
Defense date
07.16.2024 11:30
07.16.2024 11:30
Summary
In this work we will study one of the most interesting constructions in computational geometry: the Voronoi diagrams. In the first chapter, the theoretical framework will be established, presenting the relevant definitions and results, as well as its duality relation with the Delaunay triangulation. Next, the two most commonly used algorithms for its construction, the “divide and conquer” algorithm and the “Fortune algorithm”, will be discussed. Finally, a couple of applications of Voronoi diagrams for solving classical proximity problems will be presented. The final appendix includes a Python implementation of “Fortune’s algorithm” and the applications of the last chapter.
In this work we will study one of the most interesting constructions in computational geometry: the Voronoi diagrams. In the first chapter, the theoretical framework will be established, presenting the relevant definitions and results, as well as its duality relation with the Delaunay triangulation. Next, the two most commonly used algorithms for its construction, the “divide and conquer” algorithm and the “Fortune algorithm”, will be discussed. Finally, a couple of applications of Voronoi diagrams for solving classical proximity problems will be presented. The final appendix includes a Python implementation of “Fortune’s algorithm” and the applications of the last chapter.
Direction
DIAZ RAMOS, JOSE CARLOS (Tutorships)
DIAZ RAMOS, JOSE CARLOS (Tutorships)
Court
TORRES LOPERA, JUAN FRANCISCO (Chairman)
CONDE AMBOAGE, MERCEDES (Secretary)
López Pouso, Óscar (Member)
TORRES LOPERA, JUAN FRANCISCO (Chairman)
CONDE AMBOAGE, MERCEDES (Secretary)
López Pouso, Óscar (Member)
Iterative methods for solving systems of nonlinear equations. Application to the solution of 1D boundary value problems.
Authorship
A.S.O.
Bachelor of Mathematics
A.S.O.
Bachelor of Mathematics
Defense date
07.17.2024 10:00
07.17.2024 10:00
Summary
Solving systems of nonlinear equations plays an important role in several fields of science and engineering. In this work, we study different iterative methods for solving systems of nonlinear equations. We study the fixed point method and Newton’s method in detail, analising the convergence under different hypotheses and we see the strengths and weaknesses of these methods. In addition, we describe some variants of Newton’s method, including discrete Newton’s method or Broyden’s method, which allow to overcome some of the weaknesses of Newton's method. Furthermore, we implement some of the methods in Matlab. Finally, Newton's method is applied to approximate the solution of a 1D boundary value problem.
Solving systems of nonlinear equations plays an important role in several fields of science and engineering. In this work, we study different iterative methods for solving systems of nonlinear equations. We study the fixed point method and Newton’s method in detail, analising the convergence under different hypotheses and we see the strengths and weaknesses of these methods. In addition, we describe some variants of Newton’s method, including discrete Newton’s method or Broyden’s method, which allow to overcome some of the weaknesses of Newton's method. Furthermore, we implement some of the methods in Matlab. Finally, Newton's method is applied to approximate the solution of a 1D boundary value problem.
Direction
SALGADO RODRIGUEZ, MARIA DEL PILAR (Tutorships)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Tutorships)
Court
TORRES LOPERA, JUAN FRANCISCO (Chairman)
CONDE AMBOAGE, MERCEDES (Secretary)
López Pouso, Óscar (Member)
TORRES LOPERA, JUAN FRANCISCO (Chairman)
CONDE AMBOAGE, MERCEDES (Secretary)
López Pouso, Óscar (Member)
Group actions in differential geometry
Authorship
A.S.M.V.
Bachelor of Mathematics
A.S.M.V.
Bachelor of Mathematics
Defense date
07.17.2024 10:45
07.17.2024 10:45
Summary
Lie groups have been a revolutionary tool in mathematics since their invention in the late 19th century. The primary objective of this work is to study the foundational theory of Lie group actions on differentiable manifolds and their relationship with homogeneous spaces, primarily from the perspective of Differential Geometry. To achieve this, a brief introduction to differentiable manifolds will be provided, followed by a thorough exposition of the necessary theory on Lie groups and Lie algebras. The core of the work will then be dedicated to the study of Lie group actions and their various types (such as proper actions), quotient manifolds, and orbits types, all illustrated with pertinent examples. Finally, a significant portion of the work will be devoted to homogeneous spaces, describing examples such as Grassmannians, and presenting several important results, notably the Homogeneous Space Construction Theorem and the Homogeneous Space Characterization Theorem.
Lie groups have been a revolutionary tool in mathematics since their invention in the late 19th century. The primary objective of this work is to study the foundational theory of Lie group actions on differentiable manifolds and their relationship with homogeneous spaces, primarily from the perspective of Differential Geometry. To achieve this, a brief introduction to differentiable manifolds will be provided, followed by a thorough exposition of the necessary theory on Lie groups and Lie algebras. The core of the work will then be dedicated to the study of Lie group actions and their various types (such as proper actions), quotient manifolds, and orbits types, all illustrated with pertinent examples. Finally, a significant portion of the work will be devoted to homogeneous spaces, describing examples such as Grassmannians, and presenting several important results, notably the Homogeneous Space Construction Theorem and the Homogeneous Space Characterization Theorem.
Direction
DOMINGUEZ VAZQUEZ, MIGUEL (Tutorships)
Otero Casal, Tomás (Co-tutorships)
DOMINGUEZ VAZQUEZ, MIGUEL (Tutorships)
Otero Casal, Tomás (Co-tutorships)
Court
TORRES LOPERA, JUAN FRANCISCO (Chairman)
CONDE AMBOAGE, MERCEDES (Secretary)
López Pouso, Óscar (Member)
TORRES LOPERA, JUAN FRANCISCO (Chairman)
CONDE AMBOAGE, MERCEDES (Secretary)
López Pouso, Óscar (Member)
The correlation coefficient. From Pearson's linear independence to the general independence of random variables.
Authorship
L.S.V.
Bachelor of Mathematics
L.S.V.
Bachelor of Mathematics
Defense date
07.17.2024 11:30
07.17.2024 11:30
Summary
This paper is a review of the most common dependence measures used to describe the relationships between random variables. In the first chapter we present some basic concepts related to probability theory that may be useful to introduce the concept of dependence in the following chapters. Already in the second chapter we will make a brief tour through the different notions of dependence and present a set of desirable properties for the global measures of association that we will discuss later. In the third chapter we will introduce the currently most recognized measure of dependence, Pearson's correlation coefficient, explaining its historical framework, properties and limitations. These limitations will lead us to present correlation coefficients based on the study of the ranks of the variables, which will provide a wider view of dependence with respect to Pearson's, such as Spearman's rho or Kendall's tau. In the fourth chapter we will study a nascent coefficient that arises again due to the deficiencies of the previous ones: the distance correlation coefficient. In addition, we will extend the study by giving some notions of dependence from the perspective of copula functions, making a preliminary tour of their definition and properties. Finally, in the final chapter we will interpret the above concepts based on a real dataset. To obtain the results we will use the R programming language and support our conclusions with graphs to facilitate the understanding of the study.
This paper is a review of the most common dependence measures used to describe the relationships between random variables. In the first chapter we present some basic concepts related to probability theory that may be useful to introduce the concept of dependence in the following chapters. Already in the second chapter we will make a brief tour through the different notions of dependence and present a set of desirable properties for the global measures of association that we will discuss later. In the third chapter we will introduce the currently most recognized measure of dependence, Pearson's correlation coefficient, explaining its historical framework, properties and limitations. These limitations will lead us to present correlation coefficients based on the study of the ranks of the variables, which will provide a wider view of dependence with respect to Pearson's, such as Spearman's rho or Kendall's tau. In the fourth chapter we will study a nascent coefficient that arises again due to the deficiencies of the previous ones: the distance correlation coefficient. In addition, we will extend the study by giving some notions of dependence from the perspective of copula functions, making a preliminary tour of their definition and properties. Finally, in the final chapter we will interpret the above concepts based on a real dataset. To obtain the results we will use the R programming language and support our conclusions with graphs to facilitate the understanding of the study.
Direction
GONZALEZ MANTEIGA, WENCESLAO (Tutorships)
GONZALEZ MANTEIGA, WENCESLAO (Tutorships)
Court
TORRES LOPERA, JUAN FRANCISCO (Chairman)
CONDE AMBOAGE, MERCEDES (Secretary)
López Pouso, Óscar (Member)
TORRES LOPERA, JUAN FRANCISCO (Chairman)
CONDE AMBOAGE, MERCEDES (Secretary)
López Pouso, Óscar (Member)
The problems of DEA and their applications
Authorship
L.E.S.
Bachelor of Mathematics
L.E.S.
Bachelor of Mathematics
Defense date
07.16.2024 17:20
07.16.2024 17:20
Summary
This study focuses on evaluating the efficiency of several airports in Spain, based on data collected during 2022. Using the Data Envelopment Analysis (DEA) technique, an analysis is performed after a detailed theoretical review of the methodology's fundamentals and applications. R software \cite{R}, an open-source tool, is used to process and analyze the data, thus calculating the efficiency of these entities within the air sector. This work not only contributes to the understanding of the competitive position of Spanish airports but also explores the potential for improvement in their operations, providing a valuable perspective for both researchers and professionals in the sector.
This study focuses on evaluating the efficiency of several airports in Spain, based on data collected during 2022. Using the Data Envelopment Analysis (DEA) technique, an analysis is performed after a detailed theoretical review of the methodology's fundamentals and applications. R software \cite{R}, an open-source tool, is used to process and analyze the data, thus calculating the efficiency of these entities within the air sector. This work not only contributes to the understanding of the competitive position of Spanish airports but also explores the potential for improvement in their operations, providing a valuable perspective for both researchers and professionals in the sector.
Direction
SAAVEDRA NIEVES, ALEJANDRO (Tutorships)
SAAVEDRA NIEVES, ALEJANDRO (Tutorships)
Court
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
Centrality measures in covert networks based on game theory
Authorship
A.F.P.
Bachelor of Mathematics
A.F.P.
Bachelor of Mathematics
Defense date
07.16.2024 18:40
07.16.2024 18:40
Summary
Numerous disciplines have converged in the study of social networks over the past few years. In such a globalized world, the communication between the elements that compose it and the role each one plays is key in analyzing its functioning. This work focuses on trying to identify the most important members of certain social networks that, for some reason, remain covert. Initially, classical centrality measures based on the structural arrangement of the group are proposed. To contrast this information with the social context of the network, the analysis will rely on an important branch of mathematics: cooperative game theory. Throughout the document, various influence measures based on the notions of this theory are constructed and compared. All these measures will be applied to the terrorist network responsible for the Paris and Brussels attacks of 2015 and 2016, respectively, with the aim of laying the groundwork for an analysis that can serve to allocate combat and surveillance resources in the most efficient way possible in the future.
Numerous disciplines have converged in the study of social networks over the past few years. In such a globalized world, the communication between the elements that compose it and the role each one plays is key in analyzing its functioning. This work focuses on trying to identify the most important members of certain social networks that, for some reason, remain covert. Initially, classical centrality measures based on the structural arrangement of the group are proposed. To contrast this information with the social context of the network, the analysis will rely on an important branch of mathematics: cooperative game theory. Throughout the document, various influence measures based on the notions of this theory are constructed and compared. All these measures will be applied to the terrorist network responsible for the Paris and Brussels attacks of 2015 and 2016, respectively, with the aim of laying the groundwork for an analysis that can serve to allocate combat and surveillance resources in the most efficient way possible in the future.
Direction
SAAVEDRA NIEVES, ALEJANDRO (Tutorships)
SAAVEDRA NIEVES, PAULA (Co-tutorships)
SAAVEDRA NIEVES, ALEJANDRO (Tutorships)
SAAVEDRA NIEVES, PAULA (Co-tutorships)
Court
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
Automorphism-Type Skew Polynomials in Coding Theory
Authorship
E.L.R.
Bachelor of Mathematics
E.L.R.
Bachelor of Mathematics
Defense date
07.17.2024 09:30
07.17.2024 09:30
Summary
The aim of this paper is to introduce sigma-cyclic codes, a class of linear codes constructed from a non-commutative ring of polynomials. These rings, commonly known as skew-polynomials, were introduced by Ore in 1933 as polynomial rings where the usual commutativity is modified by the effect of an endomorphism. We will start by reviewing fundamental concepts in Coding Theory, such as linear and cyclic codes. Following this, we will explore the structure of skew-polynomial rings, addressing key aspects like factorization and evaluation within these polynomials. Furthermore, we will conduct a comparative analysis with the properties of commutative rings. Finally, to conclude our paper, we will study the applications of automorphism-type skew-polynomials in coding theory. We will introduce the concept of sigma-cyclic codes and briefly justify their definition, highlighting their advantages over traditional cyclic codes.
The aim of this paper is to introduce sigma-cyclic codes, a class of linear codes constructed from a non-commutative ring of polynomials. These rings, commonly known as skew-polynomials, were introduced by Ore in 1933 as polynomial rings where the usual commutativity is modified by the effect of an endomorphism. We will start by reviewing fundamental concepts in Coding Theory, such as linear and cyclic codes. Following this, we will explore the structure of skew-polynomial rings, addressing key aspects like factorization and evaluation within these polynomials. Furthermore, we will conduct a comparative analysis with the properties of commutative rings. Finally, to conclude our paper, we will study the applications of automorphism-type skew-polynomials in coding theory. We will introduce the concept of sigma-cyclic codes and briefly justify their definition, highlighting their advantages over traditional cyclic codes.
Direction
GAGO COUSO, FELIPE (Tutorships)
PAEZ GUILLAN, MARIA PILAR (Co-tutorships)
GAGO COUSO, FELIPE (Tutorships)
PAEZ GUILLAN, MARIA PILAR (Co-tutorships)
Court
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
Topological groups
Authorship
M.M.F.
Bachelor of Mathematics
M.M.F.
Bachelor of Mathematics
Defense date
07.17.2024 10:50
07.17.2024 10:50
Summary
The main objective of this work is to serve as an introduction to the study of topological groups: topological spaces that also have a group structure in such a way that the multiplication and inversion maps, which are the group operations, are continuous. To this end, we begin by presenting the notion of a topological group, obtaining some of the first properties that follow from the definition, presenting the class of translations, which are important homeomorphisms of topological groups. Next, we move on to study topological groups locally, focusing on the neighborhoods of elements. We will address fundamental systems of neighborhoods, which serve to understand the fundamental role that the local analysis of the group’s identity element plays in understanding its topological structure. This will allow us to endow certain abstract groups with a topological group structure. We will apply these concepts to delve into the homomorphisms of topological groups, which will facilitate the study of homogeneous spaces and quotient space. Finally, we present the necessary concepts to understand G-homogeneous spaces, where G is a topological group, to conclude with the proof of the main result of the work: the characterization of G-homogeneous spaces. In order to make all these contents more accessible, throughout the work we try to illustrate the main theoretical results with non-trivial examples.
The main objective of this work is to serve as an introduction to the study of topological groups: topological spaces that also have a group structure in such a way that the multiplication and inversion maps, which are the group operations, are continuous. To this end, we begin by presenting the notion of a topological group, obtaining some of the first properties that follow from the definition, presenting the class of translations, which are important homeomorphisms of topological groups. Next, we move on to study topological groups locally, focusing on the neighborhoods of elements. We will address fundamental systems of neighborhoods, which serve to understand the fundamental role that the local analysis of the group’s identity element plays in understanding its topological structure. This will allow us to endow certain abstract groups with a topological group structure. We will apply these concepts to delve into the homomorphisms of topological groups, which will facilitate the study of homogeneous spaces and quotient space. Finally, we present the necessary concepts to understand G-homogeneous spaces, where G is a topological group, to conclude with the proof of the main result of the work: the characterization of G-homogeneous spaces. In order to make all these contents more accessible, throughout the work we try to illustrate the main theoretical results with non-trivial examples.
Direction
SANMARTIN LOPEZ, VICTOR (Tutorships)
LORENZO NAVEIRO, JUAN MANUEL (Co-tutorships)
SANMARTIN LOPEZ, VICTOR (Tutorships)
LORENZO NAVEIRO, JUAN MANUEL (Co-tutorships)
Court
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
Branched covering spaces
Authorship
C.S.S.
Bachelor of Mathematics
C.S.S.
Bachelor of Mathematics
Defense date
07.17.2024 11:30
07.17.2024 11:30
Summary
We present covering spaces and some group actions acting on the fibers of these spaces. Then, we introduce branched coverings, which are a generalization of the previous ones, and we prove the Riemann-Hurwitz Formula.
We present covering spaces and some group actions acting on the fibers of these spaces. Then, we introduce branched coverings, which are a generalization of the previous ones, and we prove the Riemann-Hurwitz Formula.
Direction
Álvarez López, Jesús Antonio (Tutorships)
MOSQUERA LOIS, DAVID (Co-tutorships)
Álvarez López, Jesús Antonio (Tutorships)
MOSQUERA LOIS, DAVID (Co-tutorships)
Court
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
Topological tools for understanding of complex systems
Authorship
J.S.M.
Bachelor of Mathematics
J.S.M.
Bachelor of Mathematics
Defense date
07.17.2024 12:10
07.17.2024 12:10
Summary
In this work, we introduce the concept of persistent homology and then explain how it can be a very useful tool in order to understand complex systems. Furthermore, we present different methods to build filtered simplicial complexes from a dataset that we want to study through topological techniques. Based on the research of M. Feng, we show particular applications of these tools on spatial systems involving social elements
In this work, we introduce the concept of persistent homology and then explain how it can be a very useful tool in order to understand complex systems. Furthermore, we present different methods to build filtered simplicial complexes from a dataset that we want to study through topological techniques. Based on the research of M. Feng, we show particular applications of these tools on spatial systems involving social elements
Direction
Gómez Tato, Antonio M. (Tutorships)
Gómez Tato, Antonio M. (Tutorships)
Court
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
GARCIA RIO, EDUARDO (Chairman)
RIVERO SALGADO, OSCAR (Secretary)
CASAS MENDEZ, BALBINA VIRGINIA (Member)
Tychonoff theorem and one of its applications
Authorship
M.C.P.
Bachelor of Mathematics
M.C.P.
Bachelor of Mathematics
Defense date
07.03.2024 10:00
07.03.2024 10:00
Summary
The Tychonoff theorem is a very important result in general topology that states that the product of compact spaces is always compact. In this memory, we prove the theorem for the case of an infinite product of spaces and then we deal with certain applications of it to the domain of functional analysis and differential equations, as it is the case, for example, of the Banach-Alaoglu theorem.
The Tychonoff theorem is a very important result in general topology that states that the product of compact spaces is always compact. In this memory, we prove the theorem for the case of an infinite product of spaces and then we deal with certain applications of it to the domain of functional analysis and differential equations, as it is the case, for example, of the Banach-Alaoglu theorem.
Direction
Álvarez López, Jesús Antonio (Tutorships)
MAJADAS MOURE, ALEJANDRO OMAR (Co-tutorships)
Álvarez López, Jesús Antonio (Tutorships)
MAJADAS MOURE, ALEJANDRO OMAR (Co-tutorships)
Court
VIAÑO REY, JUAN MANUEL (Chairman)
Rodríguez López, Jorge (Secretary)
CARBALLES VAZQUEZ, JOSE MANUEL (Member)
VIAÑO REY, JUAN MANUEL (Chairman)
Rodríguez López, Jorge (Secretary)
CARBALLES VAZQUEZ, JOSE MANUEL (Member)
Revisited Complex Analysis
Authorship
I.G.L.
Bachelor of Mathematics
I.G.L.
Bachelor of Mathematics
Defense date
07.03.2024 10:45
07.03.2024 10:45
Summary
In the analysis of complex functions, we often find results that we cannot easily understand and visualize. Through the representation of these results graphically we will understand these unknowns, we will visualize results and draw conclusions. At the end of the work we will be able to detect and classify many properties of the complex functions we have studied by looking only the representations.
In the analysis of complex functions, we often find results that we cannot easily understand and visualize. Through the representation of these results graphically we will understand these unknowns, we will visualize results and draw conclusions. At the end of the work we will be able to detect and classify many properties of the complex functions we have studied by looking only the representations.
Direction
TRINCHET SORIA, ROSA Mª (Tutorships)
TRINCHET SORIA, ROSA Mª (Tutorships)
Court
VIAÑO REY, JUAN MANUEL (Chairman)
Rodríguez López, Jorge (Secretary)
CARBALLES VAZQUEZ, JOSE MANUEL (Member)
VIAÑO REY, JUAN MANUEL (Chairman)
Rodríguez López, Jorge (Secretary)
CARBALLES VAZQUEZ, JOSE MANUEL (Member)
Vibrating strings and membranes: mathematics and sound.
Authorship
P.L.L.
Bachelor of Mathematics
P.L.L.
Bachelor of Mathematics
Defense date
07.03.2024 11:30
07.03.2024 11:30
Summary
This project addresses the solution of the wave equation in cases where its dimension is less than or equal to 3. In order to achieve this, the work begins with a derivation of the equation, followed by the solution of two categories of problems: those involving boundary conditions (Dirichlet or Neumann type) and the Cauchy's global problem. For the first one, the method of separation of variables will be used, while for the second one, we will present the method of spherical means and Hadamard's descent method. Additionally, throughout this document, various examples and graphical representations will be included to facilitate the understanding of the explained solutions.
This project addresses the solution of the wave equation in cases where its dimension is less than or equal to 3. In order to achieve this, the work begins with a derivation of the equation, followed by the solution of two categories of problems: those involving boundary conditions (Dirichlet or Neumann type) and the Cauchy's global problem. For the first one, the method of separation of variables will be used, while for the second one, we will present the method of spherical means and Hadamard's descent method. Additionally, throughout this document, various examples and graphical representations will be included to facilitate the understanding of the explained solutions.
Direction
LOPEZ POUSO, RODRIGO (Tutorships)
LOPEZ POUSO, RODRIGO (Tutorships)
Court
VIAÑO REY, JUAN MANUEL (Chairman)
Rodríguez López, Jorge (Secretary)
CARBALLES VAZQUEZ, JOSE MANUEL (Member)
VIAÑO REY, JUAN MANUEL (Chairman)
Rodríguez López, Jorge (Secretary)
CARBALLES VAZQUEZ, JOSE MANUEL (Member)
Numerical Simulation of the SEIR Model. Application to the COVID-19 Epidemic
Authorship
A.N.G.
Bachelor of Mathematics
A.N.G.
Bachelor of Mathematics
Defense date
07.03.2024 12:15
07.03.2024 12:15
Summary
This work addresses the problem of formulating accurate models that reflect the reality of the COVID-19 pandemic. Initially, a primary S-E-A-I-Q-R model is presented, which includes the imposition of quarantines and distinguishes between symptomatic and asymptomatic cases. Subsequently, a second, more precise model is introduced, which also considers delays in the isolation of infected individuals and a vaccination process. Two versions of this second model will be considered: one that is conservative in relation to the total population and another that is not. Finally, the numerical simulation of the results provided by the second model in its two versions will be carried out, for which the MATLAB platform will be an essential tool. The obtained results will be compared, and the stability and sensitivity of the solution to changes in the numerical values of certain parameters will be analyzed.
This work addresses the problem of formulating accurate models that reflect the reality of the COVID-19 pandemic. Initially, a primary S-E-A-I-Q-R model is presented, which includes the imposition of quarantines and distinguishes between symptomatic and asymptomatic cases. Subsequently, a second, more precise model is introduced, which also considers delays in the isolation of infected individuals and a vaccination process. Two versions of this second model will be considered: one that is conservative in relation to the total population and another that is not. Finally, the numerical simulation of the results provided by the second model in its two versions will be carried out, for which the MATLAB platform will be an essential tool. The obtained results will be compared, and the stability and sensitivity of the solution to changes in the numerical values of certain parameters will be analyzed.
Direction
QUINTELA ESTEVEZ, PEREGRINA (Tutorships)
QUINTELA ESTEVEZ, PEREGRINA (Tutorships)
Court
VIAÑO REY, JUAN MANUEL (Chairman)
Rodríguez López, Jorge (Secretary)
CARBALLES VAZQUEZ, JOSE MANUEL (Member)
VIAÑO REY, JUAN MANUEL (Chairman)
Rodríguez López, Jorge (Secretary)
CARBALLES VAZQUEZ, JOSE MANUEL (Member)
Quantum machine learning for variational problems
Authorship
R.A.R.
Double bachelor degree in Mathematics and Physics
R.A.R.
Double bachelor degree in Mathematics and Physics
Defense date
09.16.2024 17:00
09.16.2024 17:00
Summary
The Heisenberg-Ising hamiltonian or XXZ aims to model magnetism in materials, where the predominant effect is the interaction between spins. The most interesting components of this hamiltonian are its parameter of anisotropy Delta and its external magnetic field lambda, which modify the ground state energy profile and its quantum phase. We will present two variational algorithms to approximate the range of Delta between -1 and 1: the first of them is a type of VQE (Variational Quantum Eigensolver) with a novel approach in which the parameter of anisotropy is included within the variables to optimize; the second one will be a HVA (Hamiltonian Variational Ansatz), a method based on adiabatic quantum programming that will take into account the physics of the hamiltonian XXZ to reach its ground state.
The Heisenberg-Ising hamiltonian or XXZ aims to model magnetism in materials, where the predominant effect is the interaction between spins. The most interesting components of this hamiltonian are its parameter of anisotropy Delta and its external magnetic field lambda, which modify the ground state energy profile and its quantum phase. We will present two variational algorithms to approximate the range of Delta between -1 and 1: the first of them is a type of VQE (Variational Quantum Eigensolver) with a novel approach in which the parameter of anisotropy is included within the variables to optimize; the second one will be a HVA (Hamiltonian Variational Ansatz), a method based on adiabatic quantum programming that will take into account the physics of the hamiltonian XXZ to reach its ground state.
Direction
MAS SOLE, JAVIER (Tutorships)
Gómez Tato, Andrés (Co-tutorships)
MAS SOLE, JAVIER (Tutorships)
Gómez Tato, Andrés (Co-tutorships)
Court
MIGUEZ MACHO, GONZALO (Chairman)
González Fernández, Rosa María (Secretary)
BROCOS FERNANDEZ, MARIA DEL PILAR (Member)
MIGUEZ MACHO, GONZALO (Chairman)
González Fernández, Rosa María (Secretary)
BROCOS FERNANDEZ, MARIA DEL PILAR (Member)
Programming quantum computers through pulses
Authorship
C.F.L.
Double bachelor degree in Mathematics and Physics
C.F.L.
Double bachelor degree in Mathematics and Physics
Defense date
07.18.2024 09:30
07.18.2024 09:30
Summary
Quantum computing makes use of laws of quantum physics to solve numerical problems. Its minimum unit of information is the qubit, on which gates are applied to achieve the desired state. These gates are abstractions of underlying pulses that provoke the time evolution of the physical system that represents the qubits. In this paper we will present different methods to find the pulses corresponding to a given gate. On the one hand, we will use optimisation methods, which seek to minimise a cost function by modifying certain parameters related to the Hamiltonian of the system. On the other hand, we will use methods that rely on performing algebra on the Hamiltonians to obtain their results. We will also run a QAOA, a variational algorithm widely used in quantum computing, and compare the results of pulse computing versus gate-based quantum computing.
Quantum computing makes use of laws of quantum physics to solve numerical problems. Its minimum unit of information is the qubit, on which gates are applied to achieve the desired state. These gates are abstractions of underlying pulses that provoke the time evolution of the physical system that represents the qubits. In this paper we will present different methods to find the pulses corresponding to a given gate. On the one hand, we will use optimisation methods, which seek to minimise a cost function by modifying certain parameters related to the Hamiltonian of the system. On the other hand, we will use methods that rely on performing algebra on the Hamiltonians to obtain their results. We will also run a QAOA, a variational algorithm widely used in quantum computing, and compare the results of pulse computing versus gate-based quantum computing.
Direction
SANCHEZ DE SANTOS, JOSE MANUEL (Tutorships)
Mussa Juane, Mariamo (Co-tutorships)
SANCHEZ DE SANTOS, JOSE MANUEL (Tutorships)
Mussa Juane, Mariamo (Co-tutorships)
Court
REY LOSADA, CARLOS (Chairman)
ROMERO VIDAL, ANTONIO (Secretary)
DE LA FUENTE CARBALLO, RAUL (Member)
REY LOSADA, CARLOS (Chairman)
ROMERO VIDAL, ANTONIO (Secretary)
DE LA FUENTE CARBALLO, RAUL (Member)
Lienard's equation
Authorship
M.P.A.
Bachelor of Mathematics
M.P.A.
Bachelor of Mathematics
Defense date
07.03.2024 16:00
07.03.2024 16:00
Summary
Liénard’s equation generalizes the linear harmonic oscillator equation allows for the adequate modeling of certain planar differential equation systems where periodic motion exists. In this work, after providing some preliminary notions on the qualitative analysis of differential equations, we will study Liénard's Theorem, which guarantees the existence and uniqueness of a limit cycle for certain systems associated with Liénard’s equation. Additionally, we will explore in depth two examples, corresponding to the fields of electronics (the Van der Pol oscillator) and biology (the Zeeman heart model), which follow this structure. Finally, we will present two other theoretical resultsapplicable to other equations that further generalize the Liénard equation by introducing nonlinear coefficients that combine both the dependent variable and its derivative, along with their respective examples.
Liénard’s equation generalizes the linear harmonic oscillator equation allows for the adequate modeling of certain planar differential equation systems where periodic motion exists. In this work, after providing some preliminary notions on the qualitative analysis of differential equations, we will study Liénard's Theorem, which guarantees the existence and uniqueness of a limit cycle for certain systems associated with Liénard’s equation. Additionally, we will explore in depth two examples, corresponding to the fields of electronics (the Van der Pol oscillator) and biology (the Zeeman heart model), which follow this structure. Finally, we will present two other theoretical resultsapplicable to other equations that further generalize the Liénard equation by introducing nonlinear coefficients that combine both the dependent variable and its derivative, along with their respective examples.
Direction
Rodríguez López, Rosana (Tutorships)
Rodríguez López, Rosana (Tutorships)
Court
VIAÑO REY, JUAN MANUEL (Chairman)
Rodríguez López, Jorge (Secretary)
CARBALLES VAZQUEZ, JOSE MANUEL (Member)
VIAÑO REY, JUAN MANUEL (Chairman)
Rodríguez López, Jorge (Secretary)
CARBALLES VAZQUEZ, JOSE MANUEL (Member)
Introduction to Conservation Laws and Their Numerical Resolution
Authorship
A.P.D.V.R.
Bachelor of Mathematics
A.P.D.V.R.
Bachelor of Mathematics
Defense date
07.03.2024 16:45
07.03.2024 16:45
Summary
Mathematical modeling with conservation laws describes the behavior of a system by considering the conservation of some quantity, such as mass, energy, or momentum. These models are used in fields such as physics, engineering, and environmental science to understand and predict the behavior of complex systems, from fluid dynamics and traffic flow to chemical reactions and ecological interactions. In this work, we will first study the most important analytical properties of systems of conservation laws, including the concepts of classical solutions, weak solutions, entropy conditions, and the Riemann problem. Since it is generally not possible to obtain the exact solution of conservation laws, numerical methods will later be designed to numerically approximate the solutions of these systems, taking into account the characteristics and difficulties encountered. Special emphasis will be placed on the Godunov method, whose mathematical derivation is a direct consequence of the fundamental properties of conservation laws. Finally, MATLAB codes implementing the described methods will be provided.
Mathematical modeling with conservation laws describes the behavior of a system by considering the conservation of some quantity, such as mass, energy, or momentum. These models are used in fields such as physics, engineering, and environmental science to understand and predict the behavior of complex systems, from fluid dynamics and traffic flow to chemical reactions and ecological interactions. In this work, we will first study the most important analytical properties of systems of conservation laws, including the concepts of classical solutions, weak solutions, entropy conditions, and the Riemann problem. Since it is generally not possible to obtain the exact solution of conservation laws, numerical methods will later be designed to numerically approximate the solutions of these systems, taking into account the characteristics and difficulties encountered. Special emphasis will be placed on the Godunov method, whose mathematical derivation is a direct consequence of the fundamental properties of conservation laws. Finally, MATLAB codes implementing the described methods will be provided.
Direction
RODRIGUEZ GARCIA, JERONIMO (Tutorships)
RODRIGUEZ GARCIA, JERONIMO (Tutorships)
Court
VIAÑO REY, JUAN MANUEL (Chairman)
Rodríguez López, Jorge (Secretary)
CARBALLES VAZQUEZ, JOSE MANUEL (Member)
VIAÑO REY, JUAN MANUEL (Chairman)
Rodríguez López, Jorge (Secretary)
CARBALLES VAZQUEZ, JOSE MANUEL (Member)
The Riemann Zeta function
Authorship
I.R.P.
Bachelor of Mathematics
I.R.P.
Bachelor of Mathematics
Defense date
07.03.2024 17:30
07.03.2024 17:30
Summary
The main objective of this work is to study the Riemann Zeta function in depth and analyze the Riemann hypothesis. To this end, we begin by presenting some basic notions of complex analysis, which will be necessary throughout the work, along with a study of Euler’s Gamma function, which is highly related to the Riemann Zeta function. Subsequently, the definition of the Riemann Zeta function is introduced, as well as its fundamental properties and functional equation. Some specific values of the function, which are of interest, are also examined, with particular emphasis on its zeros, and some of its applications in other fields, such as quantum physics or linguistics, are shown. Lastly, the work focuses on the Riemann hypothesis. First, its historical context is examined, then the prime number theorem is demonstrated using the Riemann Zeta function, and finally, certain aspects associated with the hypothesis are presented. These aspects will be some equivalences or modifications of the hypothesis, its possible consequences, and the existing evidence about this hypothesis.
The main objective of this work is to study the Riemann Zeta function in depth and analyze the Riemann hypothesis. To this end, we begin by presenting some basic notions of complex analysis, which will be necessary throughout the work, along with a study of Euler’s Gamma function, which is highly related to the Riemann Zeta function. Subsequently, the definition of the Riemann Zeta function is introduced, as well as its fundamental properties and functional equation. Some specific values of the function, which are of interest, are also examined, with particular emphasis on its zeros, and some of its applications in other fields, such as quantum physics or linguistics, are shown. Lastly, the work focuses on the Riemann hypothesis. First, its historical context is examined, then the prime number theorem is demonstrated using the Riemann Zeta function, and finally, certain aspects associated with the hypothesis are presented. These aspects will be some equivalences or modifications of the hypothesis, its possible consequences, and the existing evidence about this hypothesis.
Direction
Cao Labora, Daniel (Tutorships)
Cao Labora, Daniel (Tutorships)
Court
VIAÑO REY, JUAN MANUEL (Chairman)
Rodríguez López, Jorge (Secretary)
CARBALLES VAZQUEZ, JOSE MANUEL (Member)
VIAÑO REY, JUAN MANUEL (Chairman)
Rodríguez López, Jorge (Secretary)
CARBALLES VAZQUEZ, JOSE MANUEL (Member)
Quadrature formulas
Authorship
L.R.S.
Bachelor of Mathematics
L.R.S.
Bachelor of Mathematics
Defense date
07.03.2024 18:15
07.03.2024 18:15
Summary
In this paper we will address the study of numerical methods for the approximate calculation of the definite integral through various quadrature formulas. First of all, we will talk about polynomial interpolatory quadrature formulas and we will offer a common glimpse to all of them. We will study in depth two important cases of the polynomial interpolatory quadrature formulas: the Newton-Cotes formulae and the Gaussian quadrature. Finally we will study composite quadrature formulae, which try to obtain better results. We will see how to obtain the coefficients of each respective formula and, for Gaussians, also the quadrature nodes. In each case we will talk about the quadrature error and give several results about convergence.
In this paper we will address the study of numerical methods for the approximate calculation of the definite integral through various quadrature formulas. First of all, we will talk about polynomial interpolatory quadrature formulas and we will offer a common glimpse to all of them. We will study in depth two important cases of the polynomial interpolatory quadrature formulas: the Newton-Cotes formulae and the Gaussian quadrature. Finally we will study composite quadrature formulae, which try to obtain better results. We will see how to obtain the coefficients of each respective formula and, for Gaussians, also the quadrature nodes. In each case we will talk about the quadrature error and give several results about convergence.
Direction
López Pouso, Óscar (Tutorships)
López Pouso, Óscar (Tutorships)
Court
VIAÑO REY, JUAN MANUEL (Chairman)
Rodríguez López, Jorge (Secretary)
CARBALLES VAZQUEZ, JOSE MANUEL (Member)
VIAÑO REY, JUAN MANUEL (Chairman)
Rodríguez López, Jorge (Secretary)
CARBALLES VAZQUEZ, JOSE MANUEL (Member)
Game Theory and Market Analysis
Authorship
I.C.Q.
Bachelor of Mathematics
I.C.Q.
Bachelor of Mathematics
Defense date
07.04.2024 10:00
07.04.2024 10:00
Summary
This project begins by giving a brief introduction to game theory, mentioning its different branches and some of its main fields of application. Chapters 1 and 2 are devoted to non-cooperative games in strategic form. In the first one we introduce some basic concepts and present several theorems on the existence and uniqueness of equilibria, and in the second one we describe two oligopoly models and calculate their equilibria under different circumstances. Chapters 3 and 4 focus on the study of cooperative or coalitional games. In Chapter 3 we introduce cooperative games with transferable utility, while in Chapter 4 we will discuss cooperative games without transferable utility, providing in both examples to understand their utility in markets and defining different types of stable allocations and solutions. Chapter 5 concludes with a series of final considerations and references to other results of interest.
This project begins by giving a brief introduction to game theory, mentioning its different branches and some of its main fields of application. Chapters 1 and 2 are devoted to non-cooperative games in strategic form. In the first one we introduce some basic concepts and present several theorems on the existence and uniqueness of equilibria, and in the second one we describe two oligopoly models and calculate their equilibria under different circumstances. Chapters 3 and 4 focus on the study of cooperative or coalitional games. In Chapter 3 we introduce cooperative games with transferable utility, while in Chapter 4 we will discuss cooperative games without transferable utility, providing in both examples to understand their utility in markets and defining different types of stable allocations and solutions. Chapter 5 concludes with a series of final considerations and references to other results of interest.
Direction
CASAS MENDEZ, BALBINA VIRGINIA (Tutorships)
CASAS MENDEZ, BALBINA VIRGINIA (Tutorships)
Court
OTERO ESPINAR, MARIA VICTORIA (Chairman)
GONZALEZ DIAZ, JULIO (Secretary)
Jeremías López, Ana (Member)
OTERO ESPINAR, MARIA VICTORIA (Chairman)
GONZALEZ DIAZ, JULIO (Secretary)
Jeremías López, Ana (Member)
Variable selection in regression models
Authorship
J.C.L.
Bachelor of Mathematics
J.C.L.
Bachelor of Mathematics
Defense date
07.04.2024 10:40
07.04.2024 10:40
Summary
Variable selection is a highly relevant topic, particularly in contexts where an abundance of potential covariates complicates model integration. Methods for selection can be categorized into those assuming a known model structure, typically linear, and those relying solely on the nature of involved variables (response and covariates). Classic approaches like Stepwise regression and modern techniques such as LASSO fall into the former category, applicable to linear models. The latter category includes methods like mRMR (minimum Redundancy Maximum Relevance) and distance correlation-based algorithms. This paper aims to introduce these techniques, analyzing their efficacy through simulations and real-world data examples.
Variable selection is a highly relevant topic, particularly in contexts where an abundance of potential covariates complicates model integration. Methods for selection can be categorized into those assuming a known model structure, typically linear, and those relying solely on the nature of involved variables (response and covariates). Classic approaches like Stepwise regression and modern techniques such as LASSO fall into the former category, applicable to linear models. The latter category includes methods like mRMR (minimum Redundancy Maximum Relevance) and distance correlation-based algorithms. This paper aims to introduce these techniques, analyzing their efficacy through simulations and real-world data examples.
Direction
FEBRERO BANDE, MANUEL (Tutorships)
FEBRERO BANDE, MANUEL (Tutorships)
Court
OTERO ESPINAR, MARIA VICTORIA (Chairman)
GONZALEZ DIAZ, JULIO (Secretary)
Jeremías López, Ana (Member)
OTERO ESPINAR, MARIA VICTORIA (Chairman)
GONZALEZ DIAZ, JULIO (Secretary)
Jeremías López, Ana (Member)
Optimization of problems with differential algebraic equations
Authorship
I.C.S.
Bachelor of Mathematics
I.C.S.
Bachelor of Mathematics
Defense date
07.04.2024 11:20
07.04.2024 11:20
Summary
The optimization of problems with differential constraints is described through the so-called optimal control problems, focus of this work. Therefore, after a presentation of the systems of differential algebraic equations, this kind of problems is introduced. Remarks and interpretations on its definition are provided, including also a brief theorical discussion, in which the maximum principle of Pontryagin is formulated, a highly important result on this field that means a first way of finding a solution. Next, the most frequently used direct methods for resolution are explained, which are based on a discretization of the original problem, leading to a nonlinear programming one which can be solved by using usual approaches such as sequential quadratic programming. Moreover, these methods are complemented with iterative algorithms of mesh refinement in order to improve the error of the discrete approximation obtained. Finally, some illustrative examples are solved employing an original code implemented in MATLAB.
The optimization of problems with differential constraints is described through the so-called optimal control problems, focus of this work. Therefore, after a presentation of the systems of differential algebraic equations, this kind of problems is introduced. Remarks and interpretations on its definition are provided, including also a brief theorical discussion, in which the maximum principle of Pontryagin is formulated, a highly important result on this field that means a first way of finding a solution. Next, the most frequently used direct methods for resolution are explained, which are based on a discretization of the original problem, leading to a nonlinear programming one which can be solved by using usual approaches such as sequential quadratic programming. Moreover, these methods are complemented with iterative algorithms of mesh refinement in order to improve the error of the discrete approximation obtained. Finally, some illustrative examples are solved employing an original code implemented in MATLAB.
Direction
GONZALEZ RUEDA, ANGEL MANUEL (Tutorships)
GONZALEZ RUEDA, ANGEL MANUEL (Tutorships)
Court
OTERO ESPINAR, MARIA VICTORIA (Chairman)
GONZALEZ DIAZ, JULIO (Secretary)
Jeremías López, Ana (Member)
OTERO ESPINAR, MARIA VICTORIA (Chairman)
GONZALEZ DIAZ, JULIO (Secretary)
Jeremías López, Ana (Member)
Fixed point theory and applications on differential equations
Authorship
L.M.F.P.
Bachelor of Mathematics
L.M.F.P.
Bachelor of Mathematics
Defense date
07.04.2024 12:00
07.04.2024 12:00
Summary
Fixed-point theory has proven to be a branch of mathematics with great potential for solving multiple problems in nonlinear analysis, such as proving the existence, uniqueness, or multiplicity of solutions to both integral and differential equations. The concepts of topological degree and fixed-point index are highly useful for proving various results that can be framed within this theory, such as Brouwer’s and Schauder’s fixed point theorems, Krasnoselskii’s theorem in cones, or the Legget-Willams’ theorem which establishes conditions for the existence of multiple fixed points of this kind. In this bachelor thesis, first, a detailed introduction to the concepts of topological degree and fixed-point index and their properties is presented. Then, results of the fixed-point theory that can be proven with the help of these concepts is shown. Finally, part of the developed theory is used to search for sufficient conditions for the existence of solution for a boundary value problem with Dirichlet-type conditions of a second-order differential equation.
Fixed-point theory has proven to be a branch of mathematics with great potential for solving multiple problems in nonlinear analysis, such as proving the existence, uniqueness, or multiplicity of solutions to both integral and differential equations. The concepts of topological degree and fixed-point index are highly useful for proving various results that can be framed within this theory, such as Brouwer’s and Schauder’s fixed point theorems, Krasnoselskii’s theorem in cones, or the Legget-Willams’ theorem which establishes conditions for the existence of multiple fixed points of this kind. In this bachelor thesis, first, a detailed introduction to the concepts of topological degree and fixed-point index and their properties is presented. Then, results of the fixed-point theory that can be proven with the help of these concepts is shown. Finally, part of the developed theory is used to search for sufficient conditions for the existence of solution for a boundary value problem with Dirichlet-type conditions of a second-order differential equation.
Direction
Rodríguez López, Jorge (Tutorships)
Rodríguez López, Jorge (Tutorships)
Court
OTERO ESPINAR, MARIA VICTORIA (Chairman)
GONZALEZ DIAZ, JULIO (Secretary)
Jeremías López, Ana (Member)
OTERO ESPINAR, MARIA VICTORIA (Chairman)
GONZALEZ DIAZ, JULIO (Secretary)
Jeremías López, Ana (Member)
Magnetic nanoparticles design for MPI
Authorship
B.M.P.S.
Double bachelor degree in Mathematics and Physics
B.M.P.S.
Double bachelor degree in Mathematics and Physics
Defense date
07.19.2024 09:30
07.19.2024 09:30
Summary
The aim of this work is to analyze the properties of magnetite nanoparticles and try to optimize their behavior in \magnetic particle imaging (MPI), a novel medical imaging technique that relies on the non-linearity of the magnetization curves of said nanoparticles and the existence of a saturation field in magnetic materials. We will examine how the particles magnetic anisotropy affects their magnetization and compare the results with the predictions of the Langevin theory of paramagnetism, which is often used in the study of MPI. To do this, we will simulate the evolution of magnetite particles magnetization in different scenarios using CESGA's computational resources. Firstly, we will assume that the particles only have one easy magnetization direction, in other words, they only have uniaxial anisotropy. Subsequently, we will consider particles closer to those existing in reality, which will present an intrinsic anisotropy due to their crystalline structure, cubic anisotropy in the case of magnetite, and will have an asymmetric shape, which will introduce a contribution of uniaxial anisotropy. In both cases we will initially assume that the easy axes corresponding to the uniaxial anisotropy are randomly distributed. However, since particles in MPI are in a viscous medium in which they can physically rotate with the applied magnetic field, we will also study how a specific orientation of these easy axes would affect the MPI signal.
The aim of this work is to analyze the properties of magnetite nanoparticles and try to optimize their behavior in \magnetic particle imaging (MPI), a novel medical imaging technique that relies on the non-linearity of the magnetization curves of said nanoparticles and the existence of a saturation field in magnetic materials. We will examine how the particles magnetic anisotropy affects their magnetization and compare the results with the predictions of the Langevin theory of paramagnetism, which is often used in the study of MPI. To do this, we will simulate the evolution of magnetite particles magnetization in different scenarios using CESGA's computational resources. Firstly, we will assume that the particles only have one easy magnetization direction, in other words, they only have uniaxial anisotropy. Subsequently, we will consider particles closer to those existing in reality, which will present an intrinsic anisotropy due to their crystalline structure, cubic anisotropy in the case of magnetite, and will have an asymmetric shape, which will introduce a contribution of uniaxial anisotropy. In both cases we will initially assume that the easy axes corresponding to the uniaxial anisotropy are randomly distributed. However, since particles in MPI are in a viscous medium in which they can physically rotate with the applied magnetic field, we will also study how a specific orientation of these easy axes would affect the MPI signal.
Direction
SERANTES ABALO, DAVID (Tutorships)
SERANTES ABALO, DAVID (Tutorships)
Court
REY LOSADA, CARLOS (Chairman)
ROMERO VIDAL, ANTONIO (Secretary)
DE LA FUENTE CARBALLO, RAUL (Member)
REY LOSADA, CARLOS (Chairman)
ROMERO VIDAL, ANTONIO (Secretary)
DE LA FUENTE CARBALLO, RAUL (Member)
Theoretical-computational study of ternary mixtures of ionic liquids with molecular solvents for electrochemical storage
Authorship
P.T.M.
Double bachelor degree in Mathematics and Physics
P.T.M.
Double bachelor degree in Mathematics and Physics
Defense date
07.19.2024 09:30
07.19.2024 09:30
Summary
In the present undergraduate thesis, molecular dynamics simulations will be performed on electrolytes based on ternary mixtures of ionic liquids (EAN), lithium salts (LiNO3), and molecular cosolvents (acetonitrile and water) of interest in electrochemical devices. After a review of the theoretical foundations of this discipline and familiarization with the software used in the simulations, the structural and dynamic properties of the aforementioned system will be analyzed for different solvent concentrations. These results will be used to compare the behavior of acetonitrile versus water, as well as to contrast them with the previously proposed theoretical hypotheses based on the structure of the molecules in the mixture.
In the present undergraduate thesis, molecular dynamics simulations will be performed on electrolytes based on ternary mixtures of ionic liquids (EAN), lithium salts (LiNO3), and molecular cosolvents (acetonitrile and water) of interest in electrochemical devices. After a review of the theoretical foundations of this discipline and familiarization with the software used in the simulations, the structural and dynamic properties of the aforementioned system will be analyzed for different solvent concentrations. These results will be used to compare the behavior of acetonitrile versus water, as well as to contrast them with the previously proposed theoretical hypotheses based on the structure of the molecules in the mixture.
Direction
Montes Campos, Hadrián (Tutorships)
MENDEZ MORALES, TRINIDAD (Co-tutorships)
Montes Campos, Hadrián (Tutorships)
MENDEZ MORALES, TRINIDAD (Co-tutorships)
Court
REY LOSADA, CARLOS (Chairman)
ROMERO VIDAL, ANTONIO (Secretary)
DE LA FUENTE CARBALLO, RAUL (Member)
REY LOSADA, CARLOS (Chairman)
ROMERO VIDAL, ANTONIO (Secretary)
DE LA FUENTE CARBALLO, RAUL (Member)
Optimization of Magnetic Nanoparticles for Cancer Treatment through Hyperthermia
Authorship
V.O.Z.
Double bachelor degree in Mathematics and Physics
V.O.Z.
Double bachelor degree in Mathematics and Physics
Defense date
07.18.2024 09:30
07.18.2024 09:30
Summary
The treatment of cancer through magnetic hyperthermia has generated great expectations in recent years. In this paper, we will review the theoretical framework and approaches that allow us to understand how the application of an alternating magnetic field to a system formed by magnetic nanoparticles leads to heat dissipation, which provokes cellular apoptosis. Taking Brezovich’s criterion (which considers safe field conditions for in vivo aplications) as a reference, a computational study will be conducted on the shapes and sizes that optimize the dissipated heat, first assuming a system of non-interacting nanoparticles and then a system with interaction. In both cases, a clear dependence on the particle ratio has been found, as well as higher performance for cubic particles with a side close to 20 nm. Furthermore, the inability of the uniaxial anisotropy approach to correctly model this system will be shown, in favor of a balance between this and cubic magnetocrystalline anisotropy. The simulations are based on solving the Landau-Lifshitz-Gilbert equation with the OOMMF software developed by NIST, thanks to the resources provided by CESGA.
The treatment of cancer through magnetic hyperthermia has generated great expectations in recent years. In this paper, we will review the theoretical framework and approaches that allow us to understand how the application of an alternating magnetic field to a system formed by magnetic nanoparticles leads to heat dissipation, which provokes cellular apoptosis. Taking Brezovich’s criterion (which considers safe field conditions for in vivo aplications) as a reference, a computational study will be conducted on the shapes and sizes that optimize the dissipated heat, first assuming a system of non-interacting nanoparticles and then a system with interaction. In both cases, a clear dependence on the particle ratio has been found, as well as higher performance for cubic particles with a side close to 20 nm. Furthermore, the inability of the uniaxial anisotropy approach to correctly model this system will be shown, in favor of a balance between this and cubic magnetocrystalline anisotropy. The simulations are based on solving the Landau-Lifshitz-Gilbert equation with the OOMMF software developed by NIST, thanks to the resources provided by CESGA.
Direction
SERANTES ABALO, DAVID (Tutorships)
SERANTES ABALO, DAVID (Tutorships)
Court
VAZQUEZ REGUEIRO, PABLO (Chairman)
ALEJO ALONSO, AARON JOSE (Secretary)
DEL PINO GONZALEZ DE LA HIGUERA, PABLO ALFONSO (Member)
VAZQUEZ REGUEIRO, PABLO (Chairman)
ALEJO ALONSO, AARON JOSE (Secretary)
DEL PINO GONZALEZ DE LA HIGUERA, PABLO ALFONSO (Member)
Electronic properties of graphene pentalayers
Authorship
P.S.F.
Double bachelor degree in Mathematics and Physics
P.S.F.
Double bachelor degree in Mathematics and Physics
Defense date
07.18.2024 09:30
07.18.2024 09:30
Summary
Electronic correlations are a type of interaction necessary to explain some exotic properties and states of matter, but they are not taken into account in band theory. Graphene multilayers have been a good material to study these states, and specifically, the pentalayer proves to be very interesting for this purpose. In this work, we will present the necessary tools for studying the bands of these materials, as well as the information derived from them, such as the density of states or the Fermi surfaces. We will focus on creating models for pentalayer graphene and attempt to associate the correlated states experimentally found in it with regions of high density of states in its bands. Models have been constructed with parameters for the Hamiltonian extracted from the literature, others by modifying these parameters, and even an attempt has been made to obtain an effective model for finding an optimal set of parameters. Although we ultimately did not find signals clearly related to the correlated states, a rather exhaustive study has been conducted that, if complemented, could yield more conclusive results.
Electronic correlations are a type of interaction necessary to explain some exotic properties and states of matter, but they are not taken into account in band theory. Graphene multilayers have been a good material to study these states, and specifically, the pentalayer proves to be very interesting for this purpose. In this work, we will present the necessary tools for studying the bands of these materials, as well as the information derived from them, such as the density of states or the Fermi surfaces. We will focus on creating models for pentalayer graphene and attempt to associate the correlated states experimentally found in it with regions of high density of states in its bands. Models have been constructed with parameters for the Hamiltonian extracted from the literature, others by modifying these parameters, and even an attempt has been made to obtain an effective model for finding an optimal set of parameters. Although we ultimately did not find signals clearly related to the correlated states, a rather exhaustive study has been conducted that, if complemented, could yield more conclusive results.
Direction
PARDO CASTRO, VICTOR (Tutorships)
Bascones Fernández de Velasco, Elena (Co-tutorships)
PARDO CASTRO, VICTOR (Tutorships)
Bascones Fernández de Velasco, Elena (Co-tutorships)
Court
VAZQUEZ REGUEIRO, PABLO (Chairman)
ALEJO ALONSO, AARON JOSE (Secretary)
DEL PINO GONZALEZ DE LA HIGUERA, PABLO ALFONSO (Member)
VAZQUEZ REGUEIRO, PABLO (Chairman)
ALEJO ALONSO, AARON JOSE (Secretary)
DEL PINO GONZALEZ DE LA HIGUERA, PABLO ALFONSO (Member)
Artificial Intelligence in a Meteorological Environment
Authorship
C.F.P.
Double bachelor degree in Mathematics and Physics
C.F.P.
Double bachelor degree in Mathematics and Physics
Defense date
07.18.2024 09:00
07.18.2024 09:00
Summary
In this work, real meteorological data were obtained from stations from Meteogalicia, and an Echo State Network model was programmed to try to adjust them. The objectives were the modeling and comprehension of the data, the detection and reproduction of extreme events, and, ultimately, to study the capacity of the model to predict future data. The analyses were performed, in order to be manageable and in accordance to the caracteristics of this work, only for a single station located in Ferrol, for data corresponding to summer between the years 2001 and 2023, both included. It was found that the model satisfactorily fitted the data, and that it was capable of finding extreme events with reliability. Training the model for this station, acceptable fits were obtained for stations corresponding to locations with a similar climate, but they were notably worse for places with different caracteristics. Regarding the prediction for future data, it was established that an acceptable prediction was obtained for a time equivalent to 12.5 % of the network training time, moment after which the model stopped providing useful data. No evidence of a significant evolution towards a greater climatic inestability was found. As future paths of study, there is the analysis for the rest of the seasons, as well as for different climates (it would be sensible to take a training station for each one of them). For longer predictions, the predicted values for the acceptable interval could be considered, then the model could be trained again including them, an so on.
In this work, real meteorological data were obtained from stations from Meteogalicia, and an Echo State Network model was programmed to try to adjust them. The objectives were the modeling and comprehension of the data, the detection and reproduction of extreme events, and, ultimately, to study the capacity of the model to predict future data. The analyses were performed, in order to be manageable and in accordance to the caracteristics of this work, only for a single station located in Ferrol, for data corresponding to summer between the years 2001 and 2023, both included. It was found that the model satisfactorily fitted the data, and that it was capable of finding extreme events with reliability. Training the model for this station, acceptable fits were obtained for stations corresponding to locations with a similar climate, but they were notably worse for places with different caracteristics. Regarding the prediction for future data, it was established that an acceptable prediction was obtained for a time equivalent to 12.5 % of the network training time, moment after which the model stopped providing useful data. No evidence of a significant evolution towards a greater climatic inestability was found. As future paths of study, there is the analysis for the rest of the seasons, as well as for different climates (it would be sensible to take a training station for each one of them). For longer predictions, the predicted values for the acceptable interval could be considered, then the model could be trained again including them, an so on.
Direction
Pérez Muñuzuri, Alberto (Tutorships)
García Selfa, David (Co-tutorships)
Pérez Muñuzuri, Alberto (Tutorships)
García Selfa, David (Co-tutorships)
Court
Varela Cabo, Luis Miguel (Chairman)
PARAJO VIEITO, JUAN JOSE (Secretary)
ARMESTO PEREZ, NESTOR (Member)
Varela Cabo, Luis Miguel (Chairman)
PARAJO VIEITO, JUAN JOSE (Secretary)
ARMESTO PEREZ, NESTOR (Member)
Relationship between structure and properties of nanomaterials for biomedic applications: Biosensing
Authorship
J.C.R.G.
Double bachelor degree in Mathematics and Physics
J.C.R.G.
Double bachelor degree in Mathematics and Physics
Defense date
07.18.2024 09:00
07.18.2024 09:00
Summary
The aim of this Final Degree Project is to review the state of the art of metal nanoparticles as biosensor systems. These can be synthesized by the breakdown of larger-scale structures (top-down approach) or by the assembly of individual atoms and molecules (bottom-up). Some of the possible ways to create nanoparticles are explained in a general way. The optical properties of metallic nanoparticles that give them interest in biosensing are presented, highlighting the surface plasmon resonance, which generates an electromagnetic field around the particle when irradiated by light. Enhanced Surface Raman Scattering is a spectroscopy technique that amplifies the field generated by the plasmon to allow the detection of low concentrations of analytes. This is because the plasmonics of the gold nanoparticles amplify the Raman signal of the analyte adsorbed on it and, as this signal is characteristic of each molecule, allows its identification. Another biosensing technique is colorimetry, which allows molecules to be identified by the change in the color of the solution in which they are found when they bond with gold nanoparticles and clump together. Finally, COMSOL Multiphysics is used to simulate the behavior of the electric field around gold nanoparticles of different morphologies, in addition to observing the amplification of the electric field in the gaps between several nanoparticles of the same geometry for different values of the gap.
The aim of this Final Degree Project is to review the state of the art of metal nanoparticles as biosensor systems. These can be synthesized by the breakdown of larger-scale structures (top-down approach) or by the assembly of individual atoms and molecules (bottom-up). Some of the possible ways to create nanoparticles are explained in a general way. The optical properties of metallic nanoparticles that give them interest in biosensing are presented, highlighting the surface plasmon resonance, which generates an electromagnetic field around the particle when irradiated by light. Enhanced Surface Raman Scattering is a spectroscopy technique that amplifies the field generated by the plasmon to allow the detection of low concentrations of analytes. This is because the plasmonics of the gold nanoparticles amplify the Raman signal of the analyte adsorbed on it and, as this signal is characteristic of each molecule, allows its identification. Another biosensing technique is colorimetry, which allows molecules to be identified by the change in the color of the solution in which they are found when they bond with gold nanoparticles and clump together. Finally, COMSOL Multiphysics is used to simulate the behavior of the electric field around gold nanoparticles of different morphologies, in addition to observing the amplification of the electric field in the gaps between several nanoparticles of the same geometry for different values of the gap.
Direction
TABOADA ANTELO, PABLO (Tutorships)
TABOADA ANTELO, PABLO (Tutorships)
Court
Varela Cabo, Luis Miguel (Chairman)
PARAJO VIEITO, JUAN JOSE (Secretary)
ARMESTO PEREZ, NESTOR (Member)
Varela Cabo, Luis Miguel (Chairman)
PARAJO VIEITO, JUAN JOSE (Secretary)
ARMESTO PEREZ, NESTOR (Member)
Mitigation Potential of Environmental Impact through the implementation of various production technologies in the European Union
Authorship
A.P.P.
Double bachelor degree in Mathematics and Physics
A.P.P.
Double bachelor degree in Mathematics and Physics
Defense date
07.19.2024 09:30
07.19.2024 09:30
Summary
This study examines the robustness of the static methodology for assessing the Mitigation Potential of Environmental Impact (IMPcc), a key environmental sustainability indicator. It analyzes the need to consider this indicator as dynamic when designing medium and long-term sustainable energy transition agendas, especially as the participation of renewable technologies increases. Using the OSeMOSYS simulation tool, which optimizes the overall costs of the energy system, a simplified reference system for Galicia from 2022 to 2050 is modeled. The methodology includes defining various transition scenarios, increasing in four steps (0.5, 1, 2 and 3 GW) the implemented technological capacity of three key technologies: solar photovoltaic, wind, and combined cycle. Annual emissions are analyzed, and the IMPcc is calculated to assess whether this indicator can remain constant and independent of the technology's share in the energy mix. The results indicate divergent behaviors among the technologies. Wind energy shows robustness, allowing the IMPcc to be considered constant. In contrast, solar photovoltaic presents a linearly increasing IMPcc, while the combined cycle does not show significant variations. This is due to the seasonal production of the technologies: wind and hydro generate more in winter, and solar photovoltaic in summer, with the combined cycle being more needed in summer to cover energy demand. Additionally, implementation and operating costs were analyzed, as well as the evolution of new implementation capacities and their contribution to production within the energy mix.
This study examines the robustness of the static methodology for assessing the Mitigation Potential of Environmental Impact (IMPcc), a key environmental sustainability indicator. It analyzes the need to consider this indicator as dynamic when designing medium and long-term sustainable energy transition agendas, especially as the participation of renewable technologies increases. Using the OSeMOSYS simulation tool, which optimizes the overall costs of the energy system, a simplified reference system for Galicia from 2022 to 2050 is modeled. The methodology includes defining various transition scenarios, increasing in four steps (0.5, 1, 2 and 3 GW) the implemented technological capacity of three key technologies: solar photovoltaic, wind, and combined cycle. Annual emissions are analyzed, and the IMPcc is calculated to assess whether this indicator can remain constant and independent of the technology's share in the energy mix. The results indicate divergent behaviors among the technologies. Wind energy shows robustness, allowing the IMPcc to be considered constant. In contrast, solar photovoltaic presents a linearly increasing IMPcc, while the combined cycle does not show significant variations. This is due to the seasonal production of the technologies: wind and hydro generate more in winter, and solar photovoltaic in summer, with the combined cycle being more needed in summer to cover energy demand. Additionally, implementation and operating costs were analyzed, as well as the evolution of new implementation capacities and their contribution to production within the energy mix.
Direction
LOPEZ AGUERA, Ma ANGELES (Tutorships)
LOPEZ AGUERA, Ma ANGELES (Tutorships)
Court
MORENO DE LAS CUEVAS, VICENTE (Chairman)
Liñeira del Río, José Manuel (Secretary)
TORRON CASAL, CAROLINA (Member)
MORENO DE LAS CUEVAS, VICENTE (Chairman)
Liñeira del Río, José Manuel (Secretary)
TORRON CASAL, CAROLINA (Member)
Neural Networks: Fundamentals and Application to Image Recognition
Authorship
C.F.P.
Double bachelor degree in Mathematics and Physics
C.F.P.
Double bachelor degree in Mathematics and Physics
Defense date
07.16.2024 11:30
07.16.2024 11:30
Summary
Neural networks constitute the topic of this project. As such, they are studied starting from their fundamentals, showing some of their main models, to finally end with the exploration of a specific model for dealing with images. Firstly, a general definition of neural networks is provided based upon their elements, and they are also considered from the point of view of graph theory. Afterwards, the original model for neural networks, the simple perceptron, is explored, as well as its natural extension, the multilayered perceptron. In both cases, their definition and theoretical results about their capacities and performance are provided. Finally, convolutional neural networks are studied, and a practical example is presented alongside a code for a problem dealing with image classification, for which the variation of different paramethers is studied according to their effect in the performance of the network.
Neural networks constitute the topic of this project. As such, they are studied starting from their fundamentals, showing some of their main models, to finally end with the exploration of a specific model for dealing with images. Firstly, a general definition of neural networks is provided based upon their elements, and they are also considered from the point of view of graph theory. Afterwards, the original model for neural networks, the simple perceptron, is explored, as well as its natural extension, the multilayered perceptron. In both cases, their definition and theoretical results about their capacities and performance are provided. Finally, convolutional neural networks are studied, and a practical example is presented alongside a code for a problem dealing with image classification, for which the variation of different paramethers is studied according to their effect in the performance of the network.
Direction
CRUJEIRAS CASAIS, ROSA MARÍA (Tutorships)
CRUJEIRAS CASAIS, ROSA MARÍA (Tutorships)
Court
GONZALEZ MANTEIGA, WENCESLAO (Chairman)
PAEZ GUILLAN, MARIA PILAR (Secretary)
ALVAREZ DIOS, JOSE ANTONIO (Member)
GONZALEZ MANTEIGA, WENCESLAO (Chairman)
PAEZ GUILLAN, MARIA PILAR (Secretary)
ALVAREZ DIOS, JOSE ANTONIO (Member)
Sequences of approximate solutions for ordinary differential equations
Authorship
P.V.G.
Bachelor of Mathematics
P.V.G.
Bachelor of Mathematics
Defense date
07.04.2024 12:30
07.04.2024 12:30
Summary
The main topic of this work are sequences of functions that converge to a solution of an ordinary differential equation. Some of the most relevant functional sequences in the history of mathematics are presented, following a chronological order, in Chapters 1-3, along with the motivation they were first introduced with: the proof of analyticity of solutions to ODE’s under hypothesis of analyticity (Cauchy’s Theorem and power series), and the proof of existence of solution (Peano’s Theorem and Euler polygonals) and uniqueness of solution (Picard-Lipschitz-Lindelöf’s Theorem and Picard’s Iteration Sequence) under different hypotheses. A more recent result, in which a sequence of solutions of perturbed problems (Walter’s sequence) is constructed in order to prove the existence of a maximal solution to an ODE under hypothesis of continuity, is presented in Chapter 4. Lastly, Chapter 5 deals with two numerical-symbolic methods, implemented in Matlab and SageMath, which allow to obtain the Taylor expansion of the solution to an initial-value problem, of degree as high as required, under hypothesis of analyticity.
The main topic of this work are sequences of functions that converge to a solution of an ordinary differential equation. Some of the most relevant functional sequences in the history of mathematics are presented, following a chronological order, in Chapters 1-3, along with the motivation they were first introduced with: the proof of analyticity of solutions to ODE’s under hypothesis of analyticity (Cauchy’s Theorem and power series), and the proof of existence of solution (Peano’s Theorem and Euler polygonals) and uniqueness of solution (Picard-Lipschitz-Lindelöf’s Theorem and Picard’s Iteration Sequence) under different hypotheses. A more recent result, in which a sequence of solutions of perturbed problems (Walter’s sequence) is constructed in order to prove the existence of a maximal solution to an ODE under hypothesis of continuity, is presented in Chapter 4. Lastly, Chapter 5 deals with two numerical-symbolic methods, implemented in Matlab and SageMath, which allow to obtain the Taylor expansion of the solution to an initial-value problem, of degree as high as required, under hypothesis of analyticity.
Direction
LOPEZ POUSO, RODRIGO (Tutorships)
LOPEZ POUSO, RODRIGO (Tutorships)
Court
DIAZ RAMOS, JOSE CARLOS (Chairman)
COSTOYA RAMOS, MARIA CRISTINA (Secretary)
Rodríguez López, Rosana (Member)
DIAZ RAMOS, JOSE CARLOS (Chairman)
COSTOYA RAMOS, MARIA CRISTINA (Secretary)
Rodríguez López, Rosana (Member)
An introduction to quantum computing
Authorship
Y.M.D.
Bachelor of Mathematics
Y.M.D.
Bachelor of Mathematics
Defense date
07.16.2024 11:45
07.16.2024 11:45
Summary
In this document, we will conduct a study of the foundations of quantum computing, such as qubits, a concept analogous to the bits used in classical computing, which act as the basic unit of information, their three-dimensional representation on a unit sphere, and the quantum entanglement effect that has no equivalent in the classical model and causes strong interactions between the qubits that compose it. This is the main reason why quantum computing has the ability to surpass traditional computing. Additionally, we will examine quantum circuits, mechanisms that allow transformations to be performed on the aforementioned information carried by the qubits, forming important algorithms whose application can both improve upon classical algorithms and complement them. Finally, we will review some of the main algorithms and provide a detailed study of Grover's search algorithm, including its implementation on a quantum simulator.
In this document, we will conduct a study of the foundations of quantum computing, such as qubits, a concept analogous to the bits used in classical computing, which act as the basic unit of information, their three-dimensional representation on a unit sphere, and the quantum entanglement effect that has no equivalent in the classical model and causes strong interactions between the qubits that compose it. This is the main reason why quantum computing has the ability to surpass traditional computing. Additionally, we will examine quantum circuits, mechanisms that allow transformations to be performed on the aforementioned information carried by the qubits, forming important algorithms whose application can both improve upon classical algorithms and complement them. Finally, we will review some of the main algorithms and provide a detailed study of Grover's search algorithm, including its implementation on a quantum simulator.
Direction
FERNANDEZ FERNANDEZ, FRANCISCO JAVIER (Tutorships)
FERNANDEZ FERNANDEZ, FRANCISCO JAVIER (Tutorships)
Court
FEBRERO BANDE, MANUEL (Chairman)
BUEDO FERNANDEZ, SEBASTIAN (Secretary)
RODRIGUEZ GARCIA, JERONIMO (Member)
FEBRERO BANDE, MANUEL (Chairman)
BUEDO FERNANDEZ, SEBASTIAN (Secretary)
RODRIGUEZ GARCIA, JERONIMO (Member)
Are rare functions really that rare?
Authorship
S.O.R.
Bachelor of Mathematics
S.O.R.
Bachelor of Mathematics
Defense date
07.16.2024 17:00
07.16.2024 17:00
Summary
In general terms, the aim of this work is to carry out a historical review of the construction and achievement of rigour in the field of Mathematical Analysis, focusing fundamentally on the 18th and 19th centuries, a primordial period for its development. In order to tackle this undertaking, we will introduce an unconventional concept, but one that will be of great use to us: the rare function. What do we mean by rare function? How do we measure such rarity? Are really that rare the functions that we are initially calling rare? Throughout the text we will try to answer all these questions, at the same time as we will study some particular cases and analyse their substantial importance in the course of history. With this aim, we will give priority to those examples of functions that have shown that some properties, until a certain moment considered pathological, were not in fact so, and we will evaluate how this fact led to the opening of new fronts of research that have ended up leading to the philosophy that is now in force.
In general terms, the aim of this work is to carry out a historical review of the construction and achievement of rigour in the field of Mathematical Analysis, focusing fundamentally on the 18th and 19th centuries, a primordial period for its development. In order to tackle this undertaking, we will introduce an unconventional concept, but one that will be of great use to us: the rare function. What do we mean by rare function? How do we measure such rarity? Are really that rare the functions that we are initially calling rare? Throughout the text we will try to answer all these questions, at the same time as we will study some particular cases and analyse their substantial importance in the course of history. With this aim, we will give priority to those examples of functions that have shown that some properties, until a certain moment considered pathological, were not in fact so, and we will evaluate how this fact led to the opening of new fronts of research that have ended up leading to the philosophy that is now in force.
Direction
TRINCHET SORIA, ROSA Mª (Tutorships)
TRINCHET SORIA, ROSA Mª (Tutorships)
Court
FEBRERO BANDE, MANUEL (Chairman)
BUEDO FERNANDEZ, SEBASTIAN (Secretary)
RODRIGUEZ GARCIA, JERONIMO (Member)
FEBRERO BANDE, MANUEL (Chairman)
BUEDO FERNANDEZ, SEBASTIAN (Secretary)
RODRIGUEZ GARCIA, JERONIMO (Member)
Calculus of variations and its applications
Authorship
H.P.R.
Bachelor of Mathematics
H.P.R.
Bachelor of Mathematics
Defense date
07.18.2024 11:30
07.18.2024 11:30
Summary
Along this work we will do an introduction to the most relevant concepts regarding the calculus of variations. We will begin with the classical examples that encouraged further development, which will lead us to the key result: the Euler-Lagrange equation. We will study a few specific cases and their resolution, as well as possible generalizations to higher-order derivatives and $n$-dimensional functions. Subsequently, we will also address real-life applications to problems that appear in physics or engineering.
Along this work we will do an introduction to the most relevant concepts regarding the calculus of variations. We will begin with the classical examples that encouraged further development, which will lead us to the key result: the Euler-Lagrange equation. We will study a few specific cases and their resolution, as well as possible generalizations to higher-order derivatives and $n$-dimensional functions. Subsequently, we will also address real-life applications to problems that appear in physics or engineering.
Direction
CABADA FERNANDEZ, ALBERTO (Tutorships)
CABADA FERNANDEZ, ALBERTO (Tutorships)
Court
FEBRERO BANDE, MANUEL (Chairman)
BUEDO FERNANDEZ, SEBASTIAN (Secretary)
RODRIGUEZ GARCIA, JERONIMO (Member)
FEBRERO BANDE, MANUEL (Chairman)
BUEDO FERNANDEZ, SEBASTIAN (Secretary)
RODRIGUEZ GARCIA, JERONIMO (Member)
Integral transforms
Authorship
S.R.M.
Bachelor of Mathematics
S.R.M.
Bachelor of Mathematics
Defense date
07.18.2024 12:15
07.18.2024 12:15
Summary
This work focuses on the study of Laplace and Fourier transforms, highlighting their importance in solving differential equations. Both transforms and their respective inverses are defined, and their main properties are analyzed, including their relationship with the convolution product. Practical examples are included to demonstrate their utility in solving both ordinary differential equations and partial differential equations, illustrating their applicability in real-world problems.
This work focuses on the study of Laplace and Fourier transforms, highlighting their importance in solving differential equations. Both transforms and their respective inverses are defined, and their main properties are analyzed, including their relationship with the convolution product. Practical examples are included to demonstrate their utility in solving both ordinary differential equations and partial differential equations, illustrating their applicability in real-world problems.
Direction
LOPEZ SOMOZA, LUCIA (Tutorships)
LOPEZ SOMOZA, LUCIA (Tutorships)
Court
FEBRERO BANDE, MANUEL (Chairman)
BUEDO FERNANDEZ, SEBASTIAN (Secretary)
RODRIGUEZ GARCIA, JERONIMO (Member)
FEBRERO BANDE, MANUEL (Chairman)
BUEDO FERNANDEZ, SEBASTIAN (Secretary)
RODRIGUEZ GARCIA, JERONIMO (Member)
Development of computer-based resources for the understanding of concepts and results of Mathematical Analysis
Authorship
M.R.I.
Bachelor of Mathematics
M.R.I.
Bachelor of Mathematics
Defense date
07.18.2024 13:00
07.18.2024 13:00
Summary
In this project, a review of fundamental concepts and results in mathematical analysis will be carried out. We will try to interpret their geometrical sense and elaborate codes with the help of the Maple program to facilitate their understanding. The idea lies in a deep understanding, beyond being able to carry out a demonstration that proves its veracity.
In this project, a review of fundamental concepts and results in mathematical analysis will be carried out. We will try to interpret their geometrical sense and elaborate codes with the help of the Maple program to facilitate their understanding. The idea lies in a deep understanding, beyond being able to carry out a demonstration that proves its veracity.
Direction
TRINCHET SORIA, ROSA Mª (Tutorships)
TRINCHET SORIA, ROSA Mª (Tutorships)
Court
FEBRERO BANDE, MANUEL (Chairman)
BUEDO FERNANDEZ, SEBASTIAN (Secretary)
RODRIGUEZ GARCIA, JERONIMO (Member)
FEBRERO BANDE, MANUEL (Chairman)
BUEDO FERNANDEZ, SEBASTIAN (Secretary)
RODRIGUEZ GARCIA, JERONIMO (Member)
Population models of Leslie-Gower type: analysis of their dynamics
Authorship
A.V.R.
Bachelor of Mathematics
A.V.R.
Bachelor of Mathematics
Defense date
07.17.2024 11:00
07.17.2024 11:00
Summary
The qualitative theory of differential equations is a discipline of Mathematics that describes the behavior of dynamic systems, which in general cannot be solved explicitly. This branch has applications in many areas, but we focus on the study of the Leslie-Gower predator-prey ecological model, which in some way extends the classical Lotka-Volterra model. Moreover, we analyze the influence of the Allee effect on this model.\\ In the first part of this project, we present the basic concepts of the qualitative theory of linear and nonlinear dynamic systems. Furthermore, we introduce new stability results and we explain the blow-up technique for degenerate singularities. Then, we contextualize and describe the Leslie-Gower model without the Allee efect, studying the local behavior of the singularities and carrying out a global analysis that allows us to analyze the long-term stability of the populations. Finally, we incorporate the Allee effect into the model and we study again the local and global dynamics of the model.
The qualitative theory of differential equations is a discipline of Mathematics that describes the behavior of dynamic systems, which in general cannot be solved explicitly. This branch has applications in many areas, but we focus on the study of the Leslie-Gower predator-prey ecological model, which in some way extends the classical Lotka-Volterra model. Moreover, we analyze the influence of the Allee effect on this model.\\ In the first part of this project, we present the basic concepts of the qualitative theory of linear and nonlinear dynamic systems. Furthermore, we introduce new stability results and we explain the blow-up technique for degenerate singularities. Then, we contextualize and describe the Leslie-Gower model without the Allee efect, studying the local behavior of the singularities and carrying out a global analysis that allows us to analyze the long-term stability of the populations. Finally, we incorporate the Allee effect into the model and we study again the local and global dynamics of the model.
Direction
Rodríguez López, Rosana (Tutorships)
BUEDO FERNANDEZ, SEBASTIAN (Co-tutorships)
Rodríguez López, Rosana (Tutorships)
BUEDO FERNANDEZ, SEBASTIAN (Co-tutorships)
Court
OTERO ESPINAR, MARIA VICTORIA (Chairman)
GONZALEZ DIAZ, JULIO (Secretary)
Jeremías López, Ana (Member)
OTERO ESPINAR, MARIA VICTORIA (Chairman)
GONZALEZ DIAZ, JULIO (Secretary)
Jeremías López, Ana (Member)
Lebesgue spaces
Authorship
T.B.R.
Bachelor of Mathematics
T.B.R.
Bachelor of Mathematics
Defense date
07.18.2024 12:00
07.18.2024 12:00
Summary
In this project work, our objective is to understand the differences between the spaces lp and Lp, p in [1, infty]. To achieve this goal, we will address fundamental notions about Banach spaces, as well as attempt to understand the concept of Schauder basis and all the results they entail. In particular, our main interest lies in seeing that spaces lp and Lp, p in [1, infty] are isomorphic to each other.
In this project work, our objective is to understand the differences between the spaces lp and Lp, p in [1, infty]. To achieve this goal, we will address fundamental notions about Banach spaces, as well as attempt to understand the concept of Schauder basis and all the results they entail. In particular, our main interest lies in seeing that spaces lp and Lp, p in [1, infty] are isomorphic to each other.
Direction
LOSADA RODRIGUEZ, JORGE (Tutorships)
LOSADA RODRIGUEZ, JORGE (Tutorships)
Court
LOSADA RODRIGUEZ, JORGE (Student’s tutor)
LOSADA RODRIGUEZ, JORGE (Student’s tutor)
Affine crystallographic groups
Authorship
A.D.A.
Bachelor of Mathematics
A.D.A.
Bachelor of Mathematics
Defense date
07.17.2024 11:00
07.17.2024 11:00
Summary
This work is focused on studying crystallographic groups. To define them, an introduction on topological groups, group actions and covering spaces is needed. These groups are characterized in Bieberbach theorems by using an algebraic and geometrical perspective, since they are intrinsically linked to Riemannian man ifolds. We will see how they allow us to prove the first part of 18th Hilbert’s problem. Next we will replace isometries with affine transformations so that affine crystallographic groups will emerge. Bieberbach’s theorems are not valid in this context, which has derived in the formulation of other generalizations, and finally in the conjectures by Milnor and Auslander.
This work is focused on studying crystallographic groups. To define them, an introduction on topological groups, group actions and covering spaces is needed. These groups are characterized in Bieberbach theorems by using an algebraic and geometrical perspective, since they are intrinsically linked to Riemannian man ifolds. We will see how they allow us to prove the first part of 18th Hilbert’s problem. Next we will replace isometries with affine transformations so that affine crystallographic groups will emerge. Bieberbach’s theorems are not valid in this context, which has derived in the formulation of other generalizations, and finally in the conjectures by Milnor and Auslander.
Direction
ALCALDE CUESTA, FERNANDO (Tutorships)
ALCALDE CUESTA, FERNANDO (Tutorships)
Court
ALCALDE CUESTA, FERNANDO (Student’s tutor)
ALCALDE CUESTA, FERNANDO (Student’s tutor)
A search for more efficient methods than the classical methods in the numerical solution of nonlinear equations.
Authorship
R.F.C.
Bachelor of Mathematics
R.F.C.
Bachelor of Mathematics
Defense date
07.16.2024 12:00
07.16.2024 12:00
Summary
The objective of this work is to find numerical methods for solving nonlinear equations that surpass the classical methods. First, the classical methods of bisection, secant, and Newton- Raphson are introduced, and some comparisons between them are performed. Next, more recent methods are considered, particularly Halley’s method and the hybrid methods of Dekker and Dekker-Brent, and their advantages over the previous methods are analyzed. Finally, comparisons between the different methods are carried out by testing them on different functions to study their convergence speed. The Matlab code used can be found in the appendix.
The objective of this work is to find numerical methods for solving nonlinear equations that surpass the classical methods. First, the classical methods of bisection, secant, and Newton- Raphson are introduced, and some comparisons between them are performed. Next, more recent methods are considered, particularly Halley’s method and the hybrid methods of Dekker and Dekker-Brent, and their advantages over the previous methods are analyzed. Finally, comparisons between the different methods are carried out by testing them on different functions to study their convergence speed. The Matlab code used can be found in the appendix.
Direction
RODRIGUEZ IGLESIAS, CARMEN (Tutorships)
RODRIGUEZ IGLESIAS, CARMEN (Tutorships)
Court
RODRIGUEZ IGLESIAS, CARMEN (Student’s tutor)
RODRIGUEZ IGLESIAS, CARMEN (Student’s tutor)
The integral transforms of Laplace and Fourier
Authorship
S.M.E.
Bachelor of Mathematics
S.M.E.
Bachelor of Mathematics
Defense date
07.16.2024 10:00
07.16.2024 10:00
Summary
In this work, we study Laplace and Fourier transforms. We define them, give examples, and explain their main properties. Additionally, we show how these mathematical tools can be used to solve integral equations as well as both ordinary and partial differential equations. Moreover, we illustrate their practical application through real-world problems, including the solution to the Abel mechanical problem and the heat equation.
In this work, we study Laplace and Fourier transforms. We define them, give examples, and explain their main properties. Additionally, we show how these mathematical tools can be used to solve integral equations as well as both ordinary and partial differential equations. Moreover, we illustrate their practical application through real-world problems, including the solution to the Abel mechanical problem and the heat equation.
Direction
Rodríguez López, Jorge (Tutorships)
Rodríguez López, Jorge (Tutorships)
Court
Rodríguez López, Jorge (Student’s tutor)
Rodríguez López, Jorge (Student’s tutor)
An introduction to function interpolation
Authorship
S.M.C.
Bachelor of Mathematics
S.M.C.
Bachelor of Mathematics
Defense date
07.16.2024 13:00
07.16.2024 13:00
Summary
In this paper three types of interpolation will be introduced. Firstly, the Lagrange polynomical interpolation. Secondly the rational interpolation and, lastly, the interpolation by spline functions. We will give results, methods and some code that will help us find the expression of the interpolating function, in all three types of interpolation. Meanwhile, we will compare some of the types of interpolation with others, to check which one is more efficient in each case, and we will show it with tables and graphs. We will also study the error made when interpolating a given function with the Lagrange interpolating polynomial.
In this paper three types of interpolation will be introduced. Firstly, the Lagrange polynomical interpolation. Secondly the rational interpolation and, lastly, the interpolation by spline functions. We will give results, methods and some code that will help us find the expression of the interpolating function, in all three types of interpolation. Meanwhile, we will compare some of the types of interpolation with others, to check which one is more efficient in each case, and we will show it with tables and graphs. We will also study the error made when interpolating a given function with the Lagrange interpolating polynomial.
Direction
RODRIGUEZ IGLESIAS, CARMEN (Tutorships)
RODRIGUEZ IGLESIAS, CARMEN (Tutorships)
Court
RODRIGUEZ IGLESIAS, CARMEN (Student’s tutor)
RODRIGUEZ IGLESIAS, CARMEN (Student’s tutor)
The Fast Fourier Transform. An application to the spectral analysis of periodic signals.
Authorship
G.M.D.C.
Bachelor of Mathematics
G.M.D.C.
Bachelor of Mathematics
Defense date
07.16.2024 09:00
07.16.2024 09:00
Summary
In the world of signal processing, the Fast Fourier Transform emerges as a fundamental tool for analyzing periodic signals. This algorithm computes the Discrete Fourier Transform, allowing a more efficient and faster spectral analysis. This work addresses the analysis and application of the Fast Fourier Transform. Starting with a historical review, it examines the fundamentals of Fourier Analysis and its evolution through the contributions of mathematicians such as D’Alembert, Euler, Bernoulli, and Gauss, who established the theoretical foundations of this discipline. The fundamentals of the Discrete Fourier Transform are described, along with its ability to convert discrete signals from the space or time domain to the frequency domain, facilitating precise analysis of their frequency components. The study also covers the fundamental properties of the Discrete Fourier Transform and the possible errors in its application, addressing important considerations for its effective use. The focus on the Fast Fourier Transform highlights its efficiency and speed, based on in the principle of ''divide and conquer''. A practical example is implemented in MATLAB, demonstrating its applicability to the spectral analysis of periodic signals.
In the world of signal processing, the Fast Fourier Transform emerges as a fundamental tool for analyzing periodic signals. This algorithm computes the Discrete Fourier Transform, allowing a more efficient and faster spectral analysis. This work addresses the analysis and application of the Fast Fourier Transform. Starting with a historical review, it examines the fundamentals of Fourier Analysis and its evolution through the contributions of mathematicians such as D’Alembert, Euler, Bernoulli, and Gauss, who established the theoretical foundations of this discipline. The fundamentals of the Discrete Fourier Transform are described, along with its ability to convert discrete signals from the space or time domain to the frequency domain, facilitating precise analysis of their frequency components. The study also covers the fundamental properties of the Discrete Fourier Transform and the possible errors in its application, addressing important considerations for its effective use. The focus on the Fast Fourier Transform highlights its efficiency and speed, based on in the principle of ''divide and conquer''. A practical example is implemented in MATLAB, demonstrating its applicability to the spectral analysis of periodic signals.
Direction
SALGADO RODRIGUEZ, MARIA DEL PILAR (Tutorships)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Tutorships)
Court
SALGADO RODRIGUEZ, MARIA DEL PILAR (Student’s tutor)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Student’s tutor)
Numerical Derivation Formulas
Authorship
A.S.M.
Bachelor of Mathematics
A.S.M.
Bachelor of Mathematics
Defense date
07.16.2024 16:00
07.16.2024 16:00
Summary
This research focuses on numerical derivation formulas, examining both the necessary results to derive them and the potential errors encountered during approximation. The concept of a numerical derivation formula will be defined, with an emphasis on polynomial interpolation-based numerical derivation formulas. Examples of these formulas will be provided, along with an examination of several particularly relevant cases. Additionaly, an in-deph investigation will be conducted, including Matlab examples, to analyze the errors associated with using these formulas for the differential calculation of a function. This investigation will explore the possible influence of these errors on the results and discuss potential expressions to represent them.
This research focuses on numerical derivation formulas, examining both the necessary results to derive them and the potential errors encountered during approximation. The concept of a numerical derivation formula will be defined, with an emphasis on polynomial interpolation-based numerical derivation formulas. Examples of these formulas will be provided, along with an examination of several particularly relevant cases. Additionaly, an in-deph investigation will be conducted, including Matlab examples, to analyze the errors associated with using these formulas for the differential calculation of a function. This investigation will explore the possible influence of these errors on the results and discuss potential expressions to represent them.
Direction
López Pouso, Óscar (Tutorships)
López Pouso, Óscar (Tutorships)
Court
López Pouso, Óscar (Student’s tutor)
López Pouso, Óscar (Student’s tutor)
Time Series Models
Authorship
M.S.C.
Bachelor of Mathematics
M.S.C.
Bachelor of Mathematics
Defense date
07.18.2024 10:00
07.18.2024 10:00
Summary
This work focuses on the analysis of time series, concentrating on autoregressive and moving average models. Firstly, the concept of time series will be introduced, along with some simple models and their respective properties. Then, it will delve deeper into the two aforementioned models, studying their characteristics and the future predictions they provide. All of this will be applied to different datasets to try to predict their future values and compare the predictions obtained by both models.
This work focuses on the analysis of time series, concentrating on autoregressive and moving average models. Firstly, the concept of time series will be introduced, along with some simple models and their respective properties. Then, it will delve deeper into the two aforementioned models, studying their characteristics and the future predictions they provide. All of this will be applied to different datasets to try to predict their future values and compare the predictions obtained by both models.
Direction
RODRIGUEZ CASAL, ALBERTO (Tutorships)
Bolón Rodríguez, Diego (Co-tutorships)
RODRIGUEZ CASAL, ALBERTO (Tutorships)
Bolón Rodríguez, Diego (Co-tutorships)
Court
Bolón Rodríguez, Diego (Student’s tutor)
RODRIGUEZ CASAL, ALBERTO (Student’s tutor)
Bolón Rodríguez, Diego (Student’s tutor)
RODRIGUEZ CASAL, ALBERTO (Student’s tutor)
Introduction to fractional order differential equations
Authorship
I.V.D.
Bachelor of Mathematics
I.V.D.
Bachelor of Mathematics
Defense date
07.16.2024 10:00
07.16.2024 10:00
Summary
This dissertation is dedicated to an introduction to calculus of fractional order, detailing the various definitions of fractional derivatives and their uses. On the other hand, results corresponding to differential equations of fractional order are introduced as well as some applications and their utility.
This dissertation is dedicated to an introduction to calculus of fractional order, detailing the various definitions of fractional derivatives and their uses. On the other hand, results corresponding to differential equations of fractional order are introduced as well as some applications and their utility.
Direction
Nieto Roig, Juan José (Tutorships)
Nieto Roig, Juan José (Tutorships)
Court
Nieto Roig, Juan José (Student’s tutor)
Nieto Roig, Juan José (Student’s tutor)
Fourier Transform and Its Applications
Authorship
M.D.C.
Bachelor of Mathematics
M.D.C.
Bachelor of Mathematics
Defense date
07.03.2024 12:00
07.03.2024 12:00
Summary
Partial Differential Equations (PDEs) are fundamental tools for modeling physical and natural phenomena that vary in space and time. These equations have a wide application in disciplines such as physics, engineering and applied mathematics. To address their resolution, the Fourier Transform emerges as an essential tool. This allows the problem to be transferred from the time or space domain to the frequency domain, which simplifies the resolution of the original equations. This work focuses on the concept of the Fourier Transform, exploring its most significant properties and results, and it analyzes how this tool plays a fundamental role in the resolution of PDEs.
Partial Differential Equations (PDEs) are fundamental tools for modeling physical and natural phenomena that vary in space and time. These equations have a wide application in disciplines such as physics, engineering and applied mathematics. To address their resolution, the Fourier Transform emerges as an essential tool. This allows the problem to be transferred from the time or space domain to the frequency domain, which simplifies the resolution of the original equations. This work focuses on the concept of the Fourier Transform, exploring its most significant properties and results, and it analyzes how this tool plays a fundamental role in the resolution of PDEs.
Direction
LOPEZ POUSO, RODRIGO (Tutorships)
LOPEZ POUSO, RODRIGO (Tutorships)
Court
LOPEZ POUSO, RODRIGO (Student’s tutor)
LOPEZ POUSO, RODRIGO (Student’s tutor)
Mathematical optimization in resource management
Authorship
T.P.T.
Bachelor of Mathematics
T.P.T.
Bachelor of Mathematics
Defense date
07.03.2024 09:30
07.03.2024 09:30
Summary
Nowadays, Mathematical Optimization has become a key tool for the efficient management of resources in various areas, such as advertising, laboratory research, and emergency management, including wildfires. Optimal resource allocation not only maximizes economic benefits but also reduces costs and the time needed to complete tasks. This degree thesis (TFG) focuses on the application of mathematical programming techniques, especially linear programming and integer programming, to solve practical problems related to resource distribution and allocation.
Nowadays, Mathematical Optimization has become a key tool for the efficient management of resources in various areas, such as advertising, laboratory research, and emergency management, including wildfires. Optimal resource allocation not only maximizes economic benefits but also reduces costs and the time needed to complete tasks. This degree thesis (TFG) focuses on the application of mathematical programming techniques, especially linear programming and integer programming, to solve practical problems related to resource distribution and allocation.
Direction
SAAVEDRA NIEVES, ALEJANDRO (Tutorships)
SAAVEDRA NIEVES, PAULA (Co-tutorships)
SAAVEDRA NIEVES, ALEJANDRO (Tutorships)
SAAVEDRA NIEVES, PAULA (Co-tutorships)
Court
SAAVEDRA NIEVES, ALEJANDRO (Student’s tutor)
SAAVEDRA NIEVES, ALEJANDRO (Student’s tutor)
Tonelli and Fubini Theorems in Measure Spaces
Authorship
M.S.P.
Bachelor of Mathematics
M.S.P.
Bachelor of Mathematics
Defense date
07.04.2024 18:00
07.04.2024 18:00
Summary
This work focuses on Tonelli and Fubini’s theorems within the context of measure spaces. First, it explains concepts related to measure spaces, how measures are generated, and the product measure, which will help in better understanding these theorems. Then, it analyzes the sections of sets and functions, along with some necessary definitions and lemmas to provide a detailed proof of Tonelli and Fubini’s theorems in different types of measure spaces. Finally, it discusses the specific example of these theorems in Lebesgue´s measure space, demonstrating their utility in this particular measure space.
This work focuses on Tonelli and Fubini’s theorems within the context of measure spaces. First, it explains concepts related to measure spaces, how measures are generated, and the product measure, which will help in better understanding these theorems. Then, it analyzes the sections of sets and functions, along with some necessary definitions and lemmas to provide a detailed proof of Tonelli and Fubini’s theorems in different types of measure spaces. Finally, it discusses the specific example of these theorems in Lebesgue´s measure space, demonstrating their utility in this particular measure space.
Direction
FERNANDEZ FERNANDEZ, FRANCISCO JAVIER (Tutorships)
FERNANDEZ FERNANDEZ, FRANCISCO JAVIER (Tutorships)
Court
FERNANDEZ FERNANDEZ, FRANCISCO JAVIER (Student’s tutor)
FERNANDEZ FERNANDEZ, FRANCISCO JAVIER (Student’s tutor)
Undergraduate dissertation
Authorship
L.G.A.
Bachelor of Mathematics
L.G.A.
Bachelor of Mathematics
Defense date
07.16.2024 10:00
07.16.2024 10:00
Summary
En este trabajo, se introduce al lector en los problemas de localización. Mediante una breve introducción histórica y una explicación de los conceptos básicos, que se emplean a lo largo del trabajo, se ubica al lector en el marco de la teoría de localización. A posteriori, se expone una formulación general para los problemas de localización, a partir de la cual, es posible escribir todos los problemas realizando simples modificaciones en ella. Se estudian las componentes básicas que forman un problema y se enumeran los diferentes tipos de problemas de localización existentes en función de sus objetivos. Se finaliza el trabajo centrándose en un problema concreto, el problema de la p-mediana, analizándolo y explicando cómo resolverlo mediante dos algoritmos diferentes.
En este trabajo, se introduce al lector en los problemas de localización. Mediante una breve introducción histórica y una explicación de los conceptos básicos, que se emplean a lo largo del trabajo, se ubica al lector en el marco de la teoría de localización. A posteriori, se expone una formulación general para los problemas de localización, a partir de la cual, es posible escribir todos los problemas realizando simples modificaciones en ella. Se estudian las componentes básicas que forman un problema y se enumeran los diferentes tipos de problemas de localización existentes en función de sus objetivos. Se finaliza el trabajo centrándose en un problema concreto, el problema de la p-mediana, analizándolo y explicando cómo resolverlo mediante dos algoritmos diferentes.
Direction
CASARES DE CAL, MARIA ANGELES (Tutorships)
CASARES DE CAL, MARIA ANGELES (Tutorships)
Court
CASARES DE CAL, MARIA ANGELES (Student’s tutor)
CASARES DE CAL, MARIA ANGELES (Student’s tutor)
Fourier series and partial differential equations in two and three dimensions
Authorship
P.L.D.
Bachelor of Mathematics
P.L.D.
Bachelor of Mathematics
Defense date
07.17.2024 10:30
07.17.2024 10:30
Summary
The main objective of this work is to extend concepts and ideas about Fourier series and their applications to partial differential equations to the case of higher dimensions. First, notions and results about Fourier series in one and more variables are introduced. The most relevant theorems on convergence, particularly uniform convergence, will be studied. The remaining chapters are dedicated to the heat, Laplace, and wave equations. These problems will be addressed from the formulation of their mathematical models to their resolution using the technique of separation of variables and the principle of superposition. Additionally, some results or phenomena that affect these equations will be included in this part.
The main objective of this work is to extend concepts and ideas about Fourier series and their applications to partial differential equations to the case of higher dimensions. First, notions and results about Fourier series in one and more variables are introduced. The most relevant theorems on convergence, particularly uniform convergence, will be studied. The remaining chapters are dedicated to the heat, Laplace, and wave equations. These problems will be addressed from the formulation of their mathematical models to their resolution using the technique of separation of variables and the principle of superposition. Additionally, some results or phenomena that affect these equations will be included in this part.
Direction
LOPEZ POUSO, RODRIGO (Tutorships)
LOPEZ POUSO, RODRIGO (Tutorships)
Court
LOPEZ POUSO, RODRIGO (Student’s tutor)
LOPEZ POUSO, RODRIGO (Student’s tutor)
An Introduction to abstract Measure Theory
Authorship
F.J.R.T.
Bachelor of Mathematics
F.J.R.T.
Bachelor of Mathematics
Defense date
07.16.2024 18:30
07.16.2024 18:30
Summary
Measure theory is an essential area of mathematical analysis, providing the foundation for integration and functional analysis. This study reviews the basic concepts and formally develops the central ideas, such as algebras and sigma-algebras, measures and their properties, and the integration of measurable functions. Additionally, it highlights the contributions of Henri Lebesgue, whose integral provides an extension and generalization of the Riemann integral, allowing for greater flexibility and applicability in mathematical analysis. The work is divided into two main parts: the first offers a historical and conceptual review of the evolution of the integral, where the concept of measure first appears, and the second focuses on the formalization and abstraction of the key concepts, providing a solid foundation for understanding and applying measure theory in various mathematical contexts.
Measure theory is an essential area of mathematical analysis, providing the foundation for integration and functional analysis. This study reviews the basic concepts and formally develops the central ideas, such as algebras and sigma-algebras, measures and their properties, and the integration of measurable functions. Additionally, it highlights the contributions of Henri Lebesgue, whose integral provides an extension and generalization of the Riemann integral, allowing for greater flexibility and applicability in mathematical analysis. The work is divided into two main parts: the first offers a historical and conceptual review of the evolution of the integral, where the concept of measure first appears, and the second focuses on the formalization and abstraction of the key concepts, providing a solid foundation for understanding and applying measure theory in various mathematical contexts.
Direction
TRINCHET SORIA, ROSA Mª (Tutorships)
TRINCHET SORIA, ROSA Mª (Tutorships)
Court
TRINCHET SORIA, ROSA Mª (Student’s tutor)
TRINCHET SORIA, ROSA Mª (Student’s tutor)
Epidemic models using ordinary differential equations
Authorship
S.E.C.
Bachelor of Mathematics
S.E.C.
Bachelor of Mathematics
Defense date
07.16.2024 10:30
07.16.2024 10:30
Summary
Compartmental models in epidemiology divide the population into Susceptible, Infectious and Recovered. It is an ordinary differential equations system and it supports the study of a pandemic's evolution using simple models. In this study we aim to explore the main properties and their application to some real situations.
Compartmental models in epidemiology divide the population into Susceptible, Infectious and Recovered. It is an ordinary differential equations system and it supports the study of a pandemic's evolution using simple models. In this study we aim to explore the main properties and their application to some real situations.
Direction
Nieto Roig, Juan José (Tutorships)
Nieto Roig, Juan José (Tutorships)
Court
Nieto Roig, Juan José (Student’s tutor)
Nieto Roig, Juan José (Student’s tutor)
Multinomial regression
Authorship
N.A.T.
Bachelor of Mathematics
N.A.T.
Bachelor of Mathematics
Defense date
07.16.2024 17:00
07.16.2024 17:00
Summary
Regression models are a widely used statistical tool in practice in order to establish the relationship between a dependent or response variable (commonly denoted by Y) and one or more explanatory or independent variables (commonly denoted by X). The most common scenario is when both the response variable and the explanatory variables are continuous, and the relationship between them is linear. Throughout this document, we will address an extension of the classic linear regession model in which we assume that the response variable is a categorical or qualitative variable. This kind of variable gives rise to multinomial regression models, which are a particular case of what are known as generalized linear models (GLM models). The estimation of the parameters associated with a multinomial model is the result of applying maximum likelihood procedures. In this document, we will see the estimation of these parameters as well as the application of inference techniques on them. Additionally, the different presented techniques will be illustrated using a real dataset.
Regression models are a widely used statistical tool in practice in order to establish the relationship between a dependent or response variable (commonly denoted by Y) and one or more explanatory or independent variables (commonly denoted by X). The most common scenario is when both the response variable and the explanatory variables are continuous, and the relationship between them is linear. Throughout this document, we will address an extension of the classic linear regession model in which we assume that the response variable is a categorical or qualitative variable. This kind of variable gives rise to multinomial regression models, which are a particular case of what are known as generalized linear models (GLM models). The estimation of the parameters associated with a multinomial model is the result of applying maximum likelihood procedures. In this document, we will see the estimation of these parameters as well as the application of inference techniques on them. Additionally, the different presented techniques will be illustrated using a real dataset.
Direction
CONDE AMBOAGE, MERCEDES (Tutorships)
CONDE AMBOAGE, MERCEDES (Tutorships)
Court
CONDE AMBOAGE, MERCEDES (Student’s tutor)
CONDE AMBOAGE, MERCEDES (Student’s tutor)
Semi-supervised learning for object detection
Authorship
C.L.A.
Double bachelor degree of Engeneering in Information Technology and Mathematics
C.L.A.
Double bachelor degree of Engeneering in Information Technology and Mathematics
Defense date
07.18.2024 16:30
07.18.2024 16:30
Summary
Object detection is one of the main challenges addressed in the field of Computer Vision. Traditional detectors require a large amount of labeled images for training. This presents a significant limitation, as image annotation is costly and data availability is often limited. In this context, semi-supervised learning for object detection emerges, addressing the scenario where there are few labeled images but a large number of unlabeled images. This work falls within this area and focuses on the study of Teacher-Student architectures. Specifically, the Unbiased Teacher v2 detector is studied. The aim of this work is to study different techniques that are beneficial in other learning contexts and analyze their applicability to this detector. Specifically, the following proposals will be analyzed: incorporation of modules typical of few-shot architectures (specifically the Gradient Decoupled Layer (GDL) and Prototypical Calibration Block (PCB) proposed in the DeFRCN detector); replacement of the fixed threshold strategy for filtering pseudo-labels with a flexible threshold strategy; and the use of a label assignment strategy based on Optimal Transport Assignment (OTA). The experiments conducted show that the use of GDL is beneficial for the detector's performance. Additionally, although the other proposals did not improve the detector, the experimentation revealed several challenges that arise when adapting these strategies to a detector with the characteristics of Unbiased Teacher v2. This information may be relevant for future analyses of the problem.
Object detection is one of the main challenges addressed in the field of Computer Vision. Traditional detectors require a large amount of labeled images for training. This presents a significant limitation, as image annotation is costly and data availability is often limited. In this context, semi-supervised learning for object detection emerges, addressing the scenario where there are few labeled images but a large number of unlabeled images. This work falls within this area and focuses on the study of Teacher-Student architectures. Specifically, the Unbiased Teacher v2 detector is studied. The aim of this work is to study different techniques that are beneficial in other learning contexts and analyze their applicability to this detector. Specifically, the following proposals will be analyzed: incorporation of modules typical of few-shot architectures (specifically the Gradient Decoupled Layer (GDL) and Prototypical Calibration Block (PCB) proposed in the DeFRCN detector); replacement of the fixed threshold strategy for filtering pseudo-labels with a flexible threshold strategy; and the use of a label assignment strategy based on Optimal Transport Assignment (OTA). The experiments conducted show that the use of GDL is beneficial for the detector's performance. Additionally, although the other proposals did not improve the detector, the experimentation revealed several challenges that arise when adapting these strategies to a detector with the characteristics of Unbiased Teacher v2. This information may be relevant for future analyses of the problem.
Direction
MUCIENTES MOLINA, MANUEL FELIPE (Tutorships)
CORES COSTA, DANIEL (Co-tutorships)
MUCIENTES MOLINA, MANUEL FELIPE (Tutorships)
CORES COSTA, DANIEL (Co-tutorships)
Court
VAZQUEZ CENDON, MARIA ELENA (Chairman)
CHAVES FRAGA, DAVID (Secretary)
SUAREZ GAREA, JORGE ALBERTO (Member)
VAZQUEZ CENDON, MARIA ELENA (Chairman)
CHAVES FRAGA, DAVID (Secretary)
SUAREZ GAREA, JORGE ALBERTO (Member)
Dental Diagnosis: Automatic Classification of Oral Panoramic X-ray Images Using Deep Learning Techniques
Authorship
E.P.V.
Double bachelor degree of Engeneering in Information Technology and Mathematics
E.P.V.
Double bachelor degree of Engeneering in Information Technology and Mathematics
Defense date
07.18.2024 12:30
07.18.2024 12:30
Summary
n the field of medical image analysis, the classification of dental X-rays is crucial for detecting pathologies, as it allows each image to be labeled with the presence or absence of a specific condition. Recently, convolutional neural networks (CNNs) have proven effective in this area. This thesis develops a model based on ResNet architectures to classify panoramic dental X-rays. Using a dataset provided by the OSRG from the University of Santiago de Compostela, which includes 8 classes of pathologies, data augmentation and class balancing techniques were applied to address class imbalance. Various ResNet architectures were explored, and hyperparameters such as batch size and image resolution were adjusted. This report details the experiments conducted, the results obtained, and the conclusions and potential extensions of the work.
n the field of medical image analysis, the classification of dental X-rays is crucial for detecting pathologies, as it allows each image to be labeled with the presence or absence of a specific condition. Recently, convolutional neural networks (CNNs) have proven effective in this area. This thesis develops a model based on ResNet architectures to classify panoramic dental X-rays. Using a dataset provided by the OSRG from the University of Santiago de Compostela, which includes 8 classes of pathologies, data augmentation and class balancing techniques were applied to address class imbalance. Various ResNet architectures were explored, and hyperparameters such as batch size and image resolution were adjusted. This report details the experiments conducted, the results obtained, and the conclusions and potential extensions of the work.
Direction
VILA BLANCO, NICOLAS (Tutorships)
CARREIRA NOUCHE, MARIA JOSE (Co-tutorships)
TOMAS CARMONA, INMACULADA (Co-tutorships)
VILA BLANCO, NICOLAS (Tutorships)
CARREIRA NOUCHE, MARIA JOSE (Co-tutorships)
TOMAS CARMONA, INMACULADA (Co-tutorships)
Court
VIDAL AGUIAR, JUAN CARLOS (Chairman)
DOSIL LAGO, RAQUEL (Secretary)
SAAVEDRA NIEVES, ALEJANDRO (Member)
VIDAL AGUIAR, JUAN CARLOS (Chairman)
DOSIL LAGO, RAQUEL (Secretary)
SAAVEDRA NIEVES, ALEJANDRO (Member)