The Foundations of Mathematics and Set Theory: A Review
Authorship
L.A.C.
Bachelor of Mathematics
L.A.C.
Bachelor of Mathematics
Defense date
02.13.2025 16:30
02.13.2025 16:30
Summary
The objective of this work is to present, in a clear and simple way, the evolution of mathematics with a focus on logic from its beginnings in Ancient Greece to the 19th and 20th centuries. The necessary basic concepts will be defined and the development of Set Theory, which encompasses fundamental concepts such as the Axiom of Choice, the Continuum Hypothesis and the Zermelo-Fraenkel Axiomatization, will be studied. In addition, the relationship between these and their impact on modern mathematics will be analyzed.
The objective of this work is to present, in a clear and simple way, the evolution of mathematics with a focus on logic from its beginnings in Ancient Greece to the 19th and 20th centuries. The necessary basic concepts will be defined and the development of Set Theory, which encompasses fundamental concepts such as the Axiom of Choice, the Continuum Hypothesis and the Zermelo-Fraenkel Axiomatization, will be studied. In addition, the relationship between these and their impact on modern mathematics will be analyzed.
Direction
ALONSO TARRIO, LEOVIGILDO (Tutorships)
ALONSO TARRIO, LEOVIGILDO (Tutorships)
Court
ALONSO TARRIO, LEOVIGILDO (Student’s tutor)
ALONSO TARRIO, LEOVIGILDO (Student’s tutor)
Curves in Lorentz-Minkowski space
Authorship
F.A.D.
Bachelor of Mathematics
F.A.D.
Bachelor of Mathematics
Defense date
07.02.2025 10:00
07.02.2025 10:00
Summary
The objective of this paper is to analyze the basic results of the local theory of curves in threedimensional Lorentz-Minkowski space. The different causality of the curves considered implies differences in the construction of the associated curvature functions. Once the corresponding versions of the Frenet trihedron, as well as the curvature and the torsion of the curve, have been obtained, we seek to obtain a Fundamental Theorem that guarantees the existence and the uniqueness of curves with prefixed curvature and torsion. Unlike the euclidean situation, while the existence results have a natural correspondence, uniqueness is far from being fulfilled unless additional conditions on the curve’s causality are imposed.
The objective of this paper is to analyze the basic results of the local theory of curves in threedimensional Lorentz-Minkowski space. The different causality of the curves considered implies differences in the construction of the associated curvature functions. Once the corresponding versions of the Frenet trihedron, as well as the curvature and the torsion of the curve, have been obtained, we seek to obtain a Fundamental Theorem that guarantees the existence and the uniqueness of curves with prefixed curvature and torsion. Unlike the euclidean situation, while the existence results have a natural correspondence, uniqueness is far from being fulfilled unless additional conditions on the curve’s causality are imposed.
Direction
GARCIA RIO, EDUARDO (Tutorships)
Vázquez Abal, María Elena (Co-tutorships)
GARCIA RIO, EDUARDO (Tutorships)
Vázquez Abal, María Elena (Co-tutorships)
Court
QUINTELA ESTEVEZ, PEREGRINA (Chairman)
TRINCHET SORIA, ROSA Mª (Secretary)
DIAZ RAMOS, JOSE CARLOS (Member)
QUINTELA ESTEVEZ, PEREGRINA (Chairman)
TRINCHET SORIA, ROSA Mª (Secretary)
DIAZ RAMOS, JOSE CARLOS (Member)
Diophantine Equations in the Mathematical Olympiads
Authorship
M.A.R.
Bachelor of Mathematics
M.A.R.
Bachelor of Mathematics
Defense date
02.12.2025 19:45
02.12.2025 19:45
Summary
The main goal of this work is to explore and analyze several methods to solve the diophantine equations that appear in mathematical olympiads. It tries to understand how these equations, that require integer solutions, are used in competition problems and the way in which the theoretical concepts translate into technics useful to solve them. In this way, the work is divided in three chapters. The first one, about the history of these problems. The second one, about different types of diophantine equations and their resolution. And finaly, a sellection of problems that can be found in local, national and international mathematical olympiads.
The main goal of this work is to explore and analyze several methods to solve the diophantine equations that appear in mathematical olympiads. It tries to understand how these equations, that require integer solutions, are used in competition problems and the way in which the theoretical concepts translate into technics useful to solve them. In this way, the work is divided in three chapters. The first one, about the history of these problems. The second one, about different types of diophantine equations and their resolution. And finaly, a sellection of problems that can be found in local, national and international mathematical olympiads.
Direction
GAGO COUSO, FELIPE (Tutorships)
RIVERO SALGADO, OSCAR (Co-tutorships)
GAGO COUSO, FELIPE (Tutorships)
RIVERO SALGADO, OSCAR (Co-tutorships)
Court
RODRIGUEZ CASAL, ALBERTO (Chairman)
ALONSO TARRIO, LEOVIGILDO (Secretary)
SALGADO SECO, MODESTO RAMON (Member)
RODRIGUEZ CASAL, ALBERTO (Chairman)
ALONSO TARRIO, LEOVIGILDO (Secretary)
SALGADO SECO, MODESTO RAMON (Member)
Solition symetry for elliptical problems with an overdetermined boundary .
Authorship
I.A.V.
Bachelor of Mathematics
I.A.V.
Bachelor of Mathematics
Defense date
02.13.2025 12:00
02.13.2025 12:00
Summary
In this project we will consider some of the fundamental aspects of the research papers that conform the beginning of the research on elliptic partial differential equations with overdetermined boundary conditions, that is where both Dirichlet and Neumann boundary conditions are present. In the first place we will analyze Serrin's Theorem, along with its proof, which is based on finding symmetries using the moving plane method along with the maximum principles. We will also study Weinberger's alternative and more concise proof in which more classical analytical methods are applied. Lastly we will also propose a series of examples in the field of physics in which elliptic partial differential equations where overdetermined boundary conditions appear with the goal of to show the usefulness and importance of the research done in this area.
In this project we will consider some of the fundamental aspects of the research papers that conform the beginning of the research on elliptic partial differential equations with overdetermined boundary conditions, that is where both Dirichlet and Neumann boundary conditions are present. In the first place we will analyze Serrin's Theorem, along with its proof, which is based on finding symmetries using the moving plane method along with the maximum principles. We will also study Weinberger's alternative and more concise proof in which more classical analytical methods are applied. Lastly we will also propose a series of examples in the field of physics in which elliptic partial differential equations where overdetermined boundary conditions appear with the goal of to show the usefulness and importance of the research done in this area.
Direction
DOMINGUEZ VAZQUEZ, MIGUEL (Tutorships)
DOMINGUEZ VAZQUEZ, MIGUEL (Tutorships)
Court
RODRIGUEZ CASAL, ALBERTO (Chairman)
ALONSO TARRIO, LEOVIGILDO (Secretary)
SALGADO SECO, MODESTO RAMON (Member)
RODRIGUEZ CASAL, ALBERTO (Chairman)
ALONSO TARRIO, LEOVIGILDO (Secretary)
SALGADO SECO, MODESTO RAMON (Member)
An introduction to circular data
Authorship
S.A.L.
Bachelor of Mathematics
S.A.L.
Bachelor of Mathematics
Defense date
07.03.2025 09:15
07.03.2025 09:15
Summary
Circular data is data that can be identified as points or vectors within the unit circle. In this bachelor thesis we will consider classic statistical tools designed for this kind of data, as well as introduce the most important distribution models and inference procedures, including tests (about uniformity, or goodness-of-fit) and estimates of the parameters. This circular theory will be exemplified using simulated and real data in R software. This work is structured in three differentiated chapters. In the first one, we will dive in the definitions of descriptive statistics for circular data (measures of location, dispersion and graphical representation); in the second one, we will study how to construct circular distributions and expound the most significant ones; and in the last one, we will show basic inference instruments and model fitting for a single sample.
Circular data is data that can be identified as points or vectors within the unit circle. In this bachelor thesis we will consider classic statistical tools designed for this kind of data, as well as introduce the most important distribution models and inference procedures, including tests (about uniformity, or goodness-of-fit) and estimates of the parameters. This circular theory will be exemplified using simulated and real data in R software. This work is structured in three differentiated chapters. In the first one, we will dive in the definitions of descriptive statistics for circular data (measures of location, dispersion and graphical representation); in the second one, we will study how to construct circular distributions and expound the most significant ones; and in the last one, we will show basic inference instruments and model fitting for a single sample.
Direction
CRUJEIRAS CASAIS, ROSA MARÍA (Tutorships)
CRUJEIRAS CASAIS, ROSA MARÍA (Tutorships)
Court
Majadas Soto, José Javier (Chairman)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Secretary)
CASARES DE CAL, MARIA ANGELES (Member)
Majadas Soto, José Javier (Chairman)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Secretary)
CASARES DE CAL, MARIA ANGELES (Member)
Mathematical modelling of optimal dosage in drug administration
Authorship
I.A.O.
Bachelor of Mathematics
I.A.O.
Bachelor of Mathematics
Defense date
07.02.2025 10:30
07.02.2025 10:30
Summary
This work develops a pharmacokinetic/pharmacodynamic (PK/PD) model to optimize drug administration in chemotherapy treatments. The aim is to find a dosing strategy that minimizes tumor volume while keeping the total amount of drug administered constant. A differential equation-based model is considered, using a modified version of the Gompertz model, which is then implemented in MATLAB. Here, numerical results are compared with exact solutions and results from the literature are reproduced. Subsequently, an optimization problem with clinical constraints, linked to the described model, is presented. The theoretical results indicate that the optimal solution is to administer a greater number of doses evenly, provided that the imposed constraints are satisfied. The reproduced literature results match the model's predictions, validating its implementation. It is concluded that there are more effective treatments strategies than those normally used, and the importance of moving towards more realistic clinical applications is emphasized, highlighting the potential of mathematical tools in personalized therapeutic planning.
This work develops a pharmacokinetic/pharmacodynamic (PK/PD) model to optimize drug administration in chemotherapy treatments. The aim is to find a dosing strategy that minimizes tumor volume while keeping the total amount of drug administered constant. A differential equation-based model is considered, using a modified version of the Gompertz model, which is then implemented in MATLAB. Here, numerical results are compared with exact solutions and results from the literature are reproduced. Subsequently, an optimization problem with clinical constraints, linked to the described model, is presented. The theoretical results indicate that the optimal solution is to administer a greater number of doses evenly, provided that the imposed constraints are satisfied. The reproduced literature results match the model's predictions, validating its implementation. It is concluded that there are more effective treatments strategies than those normally used, and the importance of moving towards more realistic clinical applications is emphasized, highlighting the potential of mathematical tools in personalized therapeutic planning.
Direction
QUINTELA ESTEVEZ, PEREGRINA (Tutorships)
QUINTELA ESTEVEZ, PEREGRINA (Tutorships)
Court
CRUJEIRAS CASAIS, ROSA MARÍA (Chairman)
PENA BRAGE, FRANCISCO JOSE (Secretary)
DOMINGUEZ VAZQUEZ, MIGUEL (Member)
CRUJEIRAS CASAIS, ROSA MARÍA (Chairman)
PENA BRAGE, FRANCISCO JOSE (Secretary)
DOMINGUEZ VAZQUEZ, MIGUEL (Member)
Application of ODEs to biological models
Authorship
C.B.M.
Bachelor of Mathematics
C.B.M.
Bachelor of Mathematics
Defense date
02.13.2025 12:30
02.13.2025 12:30
Summary
Ordinary differential equations (ODEs) are a fundamental tool for modeling dynamic processes in various disciplines. This work focuses on the application of ODEs to biological models, particularly those related to the spread of infectious diseases. The study provides a detailed analysis of the SIR model and its extensions, such as the SEIR and SIRS models, aiming to understand epidemiological dynamics and the stability of equilibrium states. Furthermore, a specific case study on the spread of HIV in Cuba is presented using a nonlinear extension of the SIR model. The analysis includes both analytical and numerical solutions, as well as an evaluation of the impact of different control and eradication strategies. The results highlight the importance of the basic reproduction number R0 and interventions such as vaccination in mitigating disease transmission.
Ordinary differential equations (ODEs) are a fundamental tool for modeling dynamic processes in various disciplines. This work focuses on the application of ODEs to biological models, particularly those related to the spread of infectious diseases. The study provides a detailed analysis of the SIR model and its extensions, such as the SEIR and SIRS models, aiming to understand epidemiological dynamics and the stability of equilibrium states. Furthermore, a specific case study on the spread of HIV in Cuba is presented using a nonlinear extension of the SIR model. The analysis includes both analytical and numerical solutions, as well as an evaluation of the impact of different control and eradication strategies. The results highlight the importance of the basic reproduction number R0 and interventions such as vaccination in mitigating disease transmission.
Direction
Rodríguez López, Jorge (Tutorships)
Rodríguez López, Jorge (Tutorships)
Court
Rodríguez López, Jorge (Student’s tutor)
Rodríguez López, Jorge (Student’s tutor)
Brouwer fixed point theorem
Authorship
A.B.M.
Bachelor of Mathematics
A.B.M.
Bachelor of Mathematics
Defense date
07.02.2025 10:00
07.02.2025 10:00
Summary
This work studies the Brouwer fixed point theorem and its impact on various areas of mathematics. It presents the classical formulation along with several equivalent historical results and formulations, with particular emphasis on the no-retraction theorem. Furthermore, it includes a construction of the Brouwer degree, used in a direct proof of the theorem. The second part explores applications to differential equations, such as the shooting method and the existence of periodic solutions. Finally, two generalizations of the Brouwer fixed point theorem are addressed: the Kakutani and Schauder fixed point theorems, along with their respective applications in game theory and the resolution of differential equations.
This work studies the Brouwer fixed point theorem and its impact on various areas of mathematics. It presents the classical formulation along with several equivalent historical results and formulations, with particular emphasis on the no-retraction theorem. Furthermore, it includes a construction of the Brouwer degree, used in a direct proof of the theorem. The second part explores applications to differential equations, such as the shooting method and the existence of periodic solutions. Finally, two generalizations of the Brouwer fixed point theorem are addressed: the Kakutani and Schauder fixed point theorems, along with their respective applications in game theory and the resolution of differential equations.
Direction
Rodríguez López, Jorge (Tutorships)
Rodríguez López, Jorge (Tutorships)
Court
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
Symmetries and integrating factors in the solution of first-order ordinary differential equations.
Authorship
A.C.M.
Bachelor of Mathematics
A.C.M.
Bachelor of Mathematics
Defense date
02.12.2025 10:00
02.12.2025 10:00
Summary
It is well known that there is no general rule for solving first-order ordinary differential equations(ODEs), but rather a variety of methods, many of which can be expressed in the language of integrating factors. Unfortunately, there is no technique that allows for the explicit determination of integrating factors for an arbitrary differential equation. However, the Norwegian mathematician Sophus Lie (1842-1899) developed, based on the symmetries of differential equations, a unified procedure for their determination. The aim of this work is to study symmetries and integrating factors as a method of solving first-order ordinary differential equations.
It is well known that there is no general rule for solving first-order ordinary differential equations(ODEs), but rather a variety of methods, many of which can be expressed in the language of integrating factors. Unfortunately, there is no technique that allows for the explicit determination of integrating factors for an arbitrary differential equation. However, the Norwegian mathematician Sophus Lie (1842-1899) developed, based on the symmetries of differential equations, a unified procedure for their determination. The aim of this work is to study symmetries and integrating factors as a method of solving first-order ordinary differential equations.
Direction
BUEDO FERNANDEZ, SEBASTIAN (Tutorships)
SANMARTIN LOPEZ, VICTOR (Co-tutorships)
BUEDO FERNANDEZ, SEBASTIAN (Tutorships)
SANMARTIN LOPEZ, VICTOR (Co-tutorships)
Court
BUEDO FERNANDEZ, SEBASTIAN (Student’s tutor)
SANMARTIN LOPEZ, VICTOR (Student’s tutor)
BUEDO FERNANDEZ, SEBASTIAN (Student’s tutor)
SANMARTIN LOPEZ, VICTOR (Student’s tutor)
Population dynamics
Authorship
A.C.P.
Bachelor of Mathematics
A.C.P.
Bachelor of Mathematics
Defense date
07.02.2025 10:45
07.02.2025 10:45
Summary
In this paper, differential equation systems, difference equations and matrix models applied to population evolution will be studied. Beginning with the study of a single species, and continuing with the analysis of the joint evolution of multiple species coexisting in an environment. This part will be structured according to the classic interspecific interactions: competition, mutualism or predation, with special emphasis on the latter and on the adaptations that the corresponding models under go depending on the different biological characteristics.
In this paper, differential equation systems, difference equations and matrix models applied to population evolution will be studied. Beginning with the study of a single species, and continuing with the analysis of the joint evolution of multiple species coexisting in an environment. This part will be structured according to the classic interspecific interactions: competition, mutualism or predation, with special emphasis on the latter and on the adaptations that the corresponding models under go depending on the different biological characteristics.
Direction
Diz Pita, Érika (Tutorships)
Diz Pita, Érika (Tutorships)
Court
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
Evolution algebras and associated graphs
Authorship
D.C.G.
Bachelor of Mathematics
D.C.G.
Bachelor of Mathematics
Defense date
07.02.2025 12:00
07.02.2025 12:00
Summary
Evolution algebras are commutative but generally non-associative algebras, introduced in 2006 to model genetic inheritance situations that do not follow Mendelian laws. Since then, several authors, most notably J.P. Tian, have developed the theory of these algebras, examining their properties and the connections they exhibit with other areas of study, such as non-associative algebras, graph theory, and biology. The aim of this work is to explore the interactions between evolution algebras, biology, and graph theory, first by studying the properties of these algebras, and then by delving into the associated directed graphs and practical biological scenarios.
Evolution algebras are commutative but generally non-associative algebras, introduced in 2006 to model genetic inheritance situations that do not follow Mendelian laws. Since then, several authors, most notably J.P. Tian, have developed the theory of these algebras, examining their properties and the connections they exhibit with other areas of study, such as non-associative algebras, graph theory, and biology. The aim of this work is to explore the interactions between evolution algebras, biology, and graph theory, first by studying the properties of these algebras, and then by delving into the associated directed graphs and practical biological scenarios.
Direction
COSTOYA RAMOS, MARIA CRISTINA (Tutorships)
COSTOYA RAMOS, MARIA CRISTINA (Tutorships)
Court
COSTOYA RAMOS, MARIA CRISTINA (Student’s tutor)
COSTOYA RAMOS, MARIA CRISTINA (Student’s tutor)
Estimation of level sets to study the Velutina wasp
Authorship
J.C.P.
Bachelor of Mathematics
J.C.P.
Bachelor of Mathematics
Defense date
07.02.2025 10:00
07.02.2025 10:00
Summary
The Asian wasp or Vespa velutina nigrithorax, has become one of the most problematic invasive species in the Galician community due to its great ecological impact. The main objective of this work will be the study of the spatial distribution of the species, making use of the nests sighted throughout the Galician territory, from its beginnings in 2014 until 2024. Throughout the work, several advanced statistical techniques will be used, such as the non-parametric kernel density estimation, from which the high density regions are obtained, and data weighting strategies. The results obtained show the presence of a clear bias in the database; by using unweighted data, rural areas are underrepresented due to the lack of possible observers. However, conclusions change completely after weighting, obtaining an analysis more in line with reality. In conclusion, the methodology used in this work offers conclusions adjusted to the real situation, facilitating the work of public entities when planning and managing pest control, being able to adopt the methods of this study to different fields and situations.
The Asian wasp or Vespa velutina nigrithorax, has become one of the most problematic invasive species in the Galician community due to its great ecological impact. The main objective of this work will be the study of the spatial distribution of the species, making use of the nests sighted throughout the Galician territory, from its beginnings in 2014 until 2024. Throughout the work, several advanced statistical techniques will be used, such as the non-parametric kernel density estimation, from which the high density regions are obtained, and data weighting strategies. The results obtained show the presence of a clear bias in the database; by using unweighted data, rural areas are underrepresented due to the lack of possible observers. However, conclusions change completely after weighting, obtaining an analysis more in line with reality. In conclusion, the methodology used in this work offers conclusions adjusted to the real situation, facilitating the work of public entities when planning and managing pest control, being able to adopt the methods of this study to different fields and situations.
Direction
SAAVEDRA NIEVES, PAULA (Tutorships)
ALONSO PENA, MARIA (Co-tutorships)
SAAVEDRA NIEVES, PAULA (Tutorships)
ALONSO PENA, MARIA (Co-tutorships)
Court
ALONSO PENA, MARIA (Student’s tutor)
SAAVEDRA NIEVES, PAULA (Student’s tutor)
ALONSO PENA, MARIA (Student’s tutor)
SAAVEDRA NIEVES, PAULA (Student’s tutor)
Elliptic curves and applications in cryptography
Authorship
X.C.A.
Double Bachelor's Degree in Informatics Engineering and Mathematics
X.C.A.
Double Bachelor's Degree in Informatics Engineering and Mathematics
Defense date
07.02.2025 11:30
07.02.2025 11:30
Summary
The aim of this work is to provide a thorough study of elliptic curves, a particular case of algebraic curves that has occupied a prominent position in several branches of mathematics, such as algebraic geometry and number theory, and that has found significant applications in modern cryptography. In order to provide a detailed analysis on both the theoretical aspects and their applications, this bachelor thesis begins by introducing key concepts and results from algebraic geometry that serve as the fundamental framework for the subsequent development. Next, the formal definition of an elliptic curve is presented, as well as its classification based on the j-invariant. After discussing one of its most notable properties, that being its group structure, the focus shifts to the theoretical properties of elliptic curves over finite fields, which are the main object of interest in the final part, where their practical application in cryptography is explored.
The aim of this work is to provide a thorough study of elliptic curves, a particular case of algebraic curves that has occupied a prominent position in several branches of mathematics, such as algebraic geometry and number theory, and that has found significant applications in modern cryptography. In order to provide a detailed analysis on both the theoretical aspects and their applications, this bachelor thesis begins by introducing key concepts and results from algebraic geometry that serve as the fundamental framework for the subsequent development. Next, the formal definition of an elliptic curve is presented, as well as its classification based on the j-invariant. After discussing one of its most notable properties, that being its group structure, the focus shifts to the theoretical properties of elliptic curves over finite fields, which are the main object of interest in the final part, where their practical application in cryptography is explored.
Direction
ALONSO TARRIO, LEOVIGILDO (Tutorships)
ALONSO TARRIO, LEOVIGILDO (Tutorships)
Court
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
Optimization and equity
Authorship
N.C.A.
Bachelor of Mathematics
N.C.A.
Bachelor of Mathematics
Defense date
07.03.2025 09:55
07.03.2025 09:55
Summary
In this work, we analyze conditions of equity in optimization problems through the study of three problems. First, we address the optimization of vehicle routing optimization in the context of humanitarian aid, which requires an equitable approach due to the nature of the situation. Second, we examine conditions of equity related to locating a group of facilities, offering different formulations of the problem. Finally, we study equity in the allocation of water resources, for which we analyze a real case.
In this work, we analyze conditions of equity in optimization problems through the study of three problems. First, we address the optimization of vehicle routing optimization in the context of humanitarian aid, which requires an equitable approach due to the nature of the situation. Second, we examine conditions of equity related to locating a group of facilities, offering different formulations of the problem. Finally, we study equity in the allocation of water resources, for which we analyze a real case.
Direction
CASAS MENDEZ, BALBINA VIRGINIA (Tutorships)
DAVILA PENA, LAURA (Co-tutorships)
CASAS MENDEZ, BALBINA VIRGINIA (Tutorships)
DAVILA PENA, LAURA (Co-tutorships)
Court
Majadas Soto, José Javier (Chairman)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Secretary)
CASARES DE CAL, MARIA ANGELES (Member)
Majadas Soto, José Javier (Chairman)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Secretary)
CASARES DE CAL, MARIA ANGELES (Member)
Surfaces in Lorentz Minkowski space
Authorship
M.D.G.
Bachelor of Mathematics
M.D.G.
Bachelor of Mathematics
Defense date
07.02.2025 10:30
07.02.2025 10:30
Summary
Model spaces for surfaces with non negative constant Gauss curvature correspond to the plane and the sphere. However, Hilbert’s Theorem establishes the impossibility of the existence of complete regular surfaces with constant negative Gauss curvature in three dimensional Euclidean space. Therefore, we employ Lorentzian geometry and, in particular, the three dimensional Minkowski metric to construct models of hyperbolic geometry. We analyze the hyperboloid and Poincaré disk models, with special attention to the behavior of their geodesics.
Model spaces for surfaces with non negative constant Gauss curvature correspond to the plane and the sphere. However, Hilbert’s Theorem establishes the impossibility of the existence of complete regular surfaces with constant negative Gauss curvature in three dimensional Euclidean space. Therefore, we employ Lorentzian geometry and, in particular, the three dimensional Minkowski metric to construct models of hyperbolic geometry. We analyze the hyperboloid and Poincaré disk models, with special attention to the behavior of their geodesics.
Direction
GARCIA RIO, EDUARDO (Tutorships)
Vázquez Abal, María Elena (Co-tutorships)
GARCIA RIO, EDUARDO (Tutorships)
Vázquez Abal, María Elena (Co-tutorships)
Court
QUINTELA ESTEVEZ, PEREGRINA (Chairman)
TRINCHET SORIA, ROSA Mª (Secretary)
DIAZ RAMOS, JOSE CARLOS (Member)
QUINTELA ESTEVEZ, PEREGRINA (Chairman)
TRINCHET SORIA, ROSA Mª (Secretary)
DIAZ RAMOS, JOSE CARLOS (Member)
The fast Fourier transform
Authorship
P.D.V.
Double Bachelor's Degree in Informatics Engineering and Mathematics
P.D.V.
Double Bachelor's Degree in Informatics Engineering and Mathematics
Defense date
07.02.2025 11:00
07.02.2025 11:00
Summary
Since their rediscovery in the fifties, the class of algorithms known as Fast Fourier Transforms (FFTs) have been fundamental in numerous fields within mathematics, science and engineering. It comes as no surprise that the Cooley-Tukey algorithm (commonly known as the FFT) is widely recognized as one of the most important algorithms of the 20th century. On this project, we aim to provide an structured and well-grounded approach to the development of FFTs. We begin with the mathematical foundations of Lp spaces and the continuous Fourier Transform, that provide a new way to look into functions via their spectrum of frequencies. Later, we introduce the Discrete Fourier Transform (DFT) as a numerical tool to enable Fourier Methods. Computing the DFT for large input sizes is only doable because of FFT algorithms. Finally, we present a brief overview of two major application domains: digital signal processing and data compression. In particular, we review digital audio filters and examine the role of FFTs in JPEG image compression.
Since their rediscovery in the fifties, the class of algorithms known as Fast Fourier Transforms (FFTs) have been fundamental in numerous fields within mathematics, science and engineering. It comes as no surprise that the Cooley-Tukey algorithm (commonly known as the FFT) is widely recognized as one of the most important algorithms of the 20th century. On this project, we aim to provide an structured and well-grounded approach to the development of FFTs. We begin with the mathematical foundations of Lp spaces and the continuous Fourier Transform, that provide a new way to look into functions via their spectrum of frequencies. Later, we introduce the Discrete Fourier Transform (DFT) as a numerical tool to enable Fourier Methods. Computing the DFT for large input sizes is only doable because of FFT algorithms. Finally, we present a brief overview of two major application domains: digital signal processing and data compression. In particular, we review digital audio filters and examine the role of FFTs in JPEG image compression.
Direction
LOPEZ SOMOZA, LUCIA (Tutorships)
LOPEZ SOMOZA, LUCIA (Tutorships)
Court
QUINTELA ESTEVEZ, PEREGRINA (Chairman)
TRINCHET SORIA, ROSA Mª (Secretary)
DIAZ RAMOS, JOSE CARLOS (Member)
QUINTELA ESTEVEZ, PEREGRINA (Chairman)
TRINCHET SORIA, ROSA Mª (Secretary)
DIAZ RAMOS, JOSE CARLOS (Member)
Statistical methods in bioinformatics
Authorship
C.D.R.
Bachelor of Mathematics
C.D.R.
Bachelor of Mathematics
Defense date
07.02.2025 11:15
07.02.2025 11:15
Summary
The current paper focuses on the study of DNA sequences, employing statistical methods as key tools. Diverse issues related to the assembly process associated with genome sequencing will be studied. Previously, concepts of probability and random variables will be reviewed, as well as an introduction to stochastic processes, specifically, Poisson processes, which are necessary for modelling the sequencing process. Lastly, a practical case study, related to the bacterial genome will be presented, by accessing genetic databases and using specialized software for sequence assembly.
The current paper focuses on the study of DNA sequences, employing statistical methods as key tools. Diverse issues related to the assembly process associated with genome sequencing will be studied. Previously, concepts of probability and random variables will be reviewed, as well as an introduction to stochastic processes, specifically, Poisson processes, which are necessary for modelling the sequencing process. Lastly, a practical case study, related to the bacterial genome will be presented, by accessing genetic databases and using specialized software for sequence assembly.
Direction
CASARES DE CAL, MARIA ANGELES (Tutorships)
CASARES DE CAL, MARIA ANGELES (Tutorships)
Court
CRUJEIRAS CASAIS, ROSA MARÍA (Chairman)
PENA BRAGE, FRANCISCO JOSE (Secretary)
DOMINGUEZ VAZQUEZ, MIGUEL (Member)
CRUJEIRAS CASAIS, ROSA MARÍA (Chairman)
PENA BRAGE, FRANCISCO JOSE (Secretary)
DOMINGUEZ VAZQUEZ, MIGUEL (Member)
Abstract Measure and Integration: Unveiling the Radon Nikodym Theorem
Authorship
F.E.L.
Bachelor of Mathematics
F.E.L.
Bachelor of Mathematics
Defense date
07.02.2025 12:15
07.02.2025 12:15
Summary
This dissertation presents a comprehensive and detailed study of the concept of measure, starting with positive measures and gradually extending the analysis to more general cases, such as real and complex measures. Special attention is given to one of the fundamental results in Measure Theory: the Radon Nikodym Theorem, which, along with the Lebesgue Decomposition Theorem, is an essential tool for understanding the structure and behavior of measures. To support the development of these topics, the necessary background in Measure Theory and Functional Analysis is introduced, including key definitions, auxiliary propositions, and intermediate results that enable a rigorous formulation and proof of the theorems. Additionally, a historical review is included, highlighting the contributions of Henri Lebesgue, Johann Radon, and Otton Nikodym to the development of the theorem that bears their names. The study is completed with a detailed academic example that illustrates the application of the Radon Nikodym Theorem in a concrete setting. Overall, this work aims to provide a solid and accessible understanding of one of the most relevant theorems in contemporary mathematical analysis.
This dissertation presents a comprehensive and detailed study of the concept of measure, starting with positive measures and gradually extending the analysis to more general cases, such as real and complex measures. Special attention is given to one of the fundamental results in Measure Theory: the Radon Nikodym Theorem, which, along with the Lebesgue Decomposition Theorem, is an essential tool for understanding the structure and behavior of measures. To support the development of these topics, the necessary background in Measure Theory and Functional Analysis is introduced, including key definitions, auxiliary propositions, and intermediate results that enable a rigorous formulation and proof of the theorems. Additionally, a historical review is included, highlighting the contributions of Henri Lebesgue, Johann Radon, and Otton Nikodym to the development of the theorem that bears their names. The study is completed with a detailed academic example that illustrates the application of the Radon Nikodym Theorem in a concrete setting. Overall, this work aims to provide a solid and accessible understanding of one of the most relevant theorems in contemporary mathematical analysis.
Direction
TRINCHET SORIA, ROSA Mª (Tutorships)
TRINCHET SORIA, ROSA Mª (Tutorships)
Court
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
Exploring OEDs
Authorship
D.F.C.
Bachelor of Mathematics
D.F.C.
Bachelor of Mathematics
Defense date
07.02.2025 11:30
07.02.2025 11:30
Summary
Some problems modeled by ordinary differential equations will be studied. Although the approach will be numerical, access to the solutions will be achieved using existing software, without requiring significant programming work. The student may choose some problems from the following list, taken from what will be the main reference for this work: Caffeine elimination from the bloodstream. Classical pursuit problems. The parachutist problem. Beam theory and spaghetti strength. Eigenstates of the Schrödinger equation. Adjoints and optimization. Moon, sun, and tides. Nonlinear pendulum. SIR model for epidemics. Designed non-uniqueness. Metastability, radioactivity, and quantum tunneling. Chaos in a food web. Linearized Lorenz trajectories. Transition to turbulence in a pipe. Sending a spacecraft to a destination. Arrhenius chemical reaction. Band gaps and forbidden frequencies. Why is it hotter in New York than in San Francisco? Jacobi sine function. Solitons and the KdV equation.
Some problems modeled by ordinary differential equations will be studied. Although the approach will be numerical, access to the solutions will be achieved using existing software, without requiring significant programming work. The student may choose some problems from the following list, taken from what will be the main reference for this work: Caffeine elimination from the bloodstream. Classical pursuit problems. The parachutist problem. Beam theory and spaghetti strength. Eigenstates of the Schrödinger equation. Adjoints and optimization. Moon, sun, and tides. Nonlinear pendulum. SIR model for epidemics. Designed non-uniqueness. Metastability, radioactivity, and quantum tunneling. Chaos in a food web. Linearized Lorenz trajectories. Transition to turbulence in a pipe. Sending a spacecraft to a destination. Arrhenius chemical reaction. Band gaps and forbidden frequencies. Why is it hotter in New York than in San Francisco? Jacobi sine function. Solitons and the KdV equation.
Direction
López Pouso, Óscar (Tutorships)
López Pouso, Óscar (Tutorships)
Court
QUINTELA ESTEVEZ, PEREGRINA (Chairman)
TRINCHET SORIA, ROSA Mª (Secretary)
DIAZ RAMOS, JOSE CARLOS (Member)
QUINTELA ESTEVEZ, PEREGRINA (Chairman)
TRINCHET SORIA, ROSA Mª (Secretary)
DIAZ RAMOS, JOSE CARLOS (Member)
Study and application of AWS Rekognition for automatic recognition of clothing labels in user images.
Authorship
E.F.D.S.
Double Bachelor's Degree in Informatics Engineering and Mathematics
E.F.D.S.
Double Bachelor's Degree in Informatics Engineering and Mathematics
Defense date
02.20.2025 10:00
02.20.2025 10:00
Summary
Nowadays there are multiple tools to perform image classification processes, such as convolutional neural networks and transformers. However, the Zara brand continues to perform labeling manually, resulting in a set of inaccurate labels. For this reason, this study explores the implementation of automated methods to improve the results obtained manually. The purpose of this research is to evaluate and analyze the effectiveness of the AWS Rekognition Custom Labels service for labeling garments. The adopted strategy aims to identify the limits of the service for the referred task through a feasibility analysis of the source dataset. The project development starts with a preliminary analysis of the dataset to determine its suitability for model training. Subsequently, an examination of the service constraints is performed, considering five main variables: the total number of images, the interrelationship between labels, the type of label, the number of images available for each label, and the influence of each label on the others. To achieve this, several resources will be used such as the service itself, an initial dataset and a REST API developed for this project. The main findings include the low relevance of the total number of images, as well as the limitations associated with the type of tag and the importance of the tags not being overly related.
Nowadays there are multiple tools to perform image classification processes, such as convolutional neural networks and transformers. However, the Zara brand continues to perform labeling manually, resulting in a set of inaccurate labels. For this reason, this study explores the implementation of automated methods to improve the results obtained manually. The purpose of this research is to evaluate and analyze the effectiveness of the AWS Rekognition Custom Labels service for labeling garments. The adopted strategy aims to identify the limits of the service for the referred task through a feasibility analysis of the source dataset. The project development starts with a preliminary analysis of the dataset to determine its suitability for model training. Subsequently, an examination of the service constraints is performed, considering five main variables: the total number of images, the interrelationship between labels, the type of label, the number of images available for each label, and the influence of each label on the others. To achieve this, several resources will be used such as the service itself, an initial dataset and a REST API developed for this project. The main findings include the low relevance of the total number of images, as well as the limitations associated with the type of tag and the importance of the tags not being overly related.
Direction
Carreira Nouche, María José (Tutorships)
Rodríguez Díez, Helio (Co-tutorships)
Carreira Nouche, María José (Tutorships)
Rodríguez Díez, Helio (Co-tutorships)
Court
ARIAS RODRIGUEZ, JUAN ENRIQUE (Chairman)
Querentes Hermida, Raquel Esther (Secretary)
PIÑEIRO POMAR, CESAR ALFREDO (Member)
ARIAS RODRIGUEZ, JUAN ENRIQUE (Chairman)
Querentes Hermida, Raquel Esther (Secretary)
PIÑEIRO POMAR, CESAR ALFREDO (Member)
Metaheuristics of the TSP: A didactic and computational tour.
Authorship
E.F.D.S.
Double Bachelor's Degree in Informatics Engineering and Mathematics
E.F.D.S.
Double Bachelor's Degree in Informatics Engineering and Mathematics
Defense date
02.13.2025 12:45
02.13.2025 12:45
Summary
During the history of computing, routing problems have attracted great interest due to their multiple applications in different fields, such as planning and logistics. This study focuses on the traveling salesman problem or TSP. Specifically, on the techniques to solve it in an approximate way in polynomial time, the metaheuristics. The main objective of this study is to provide a guide to understand four of the most important ones, both theoretically and computationally. For this purpose, a literature review was performed, finding relevant information of them and synthesizing it. They are: tabu search, simulated annealing, genetic algorithm and ant colony optimization. For the computational part, R implementations of all metaheuristics were performed and evaluated with different instances of the TSPLIB library. As a result, it was obtained that there is no metaheuristic better than the rest in all aspects. Tabu search and ant colony optimization obtain very promising results in terms of distance to optimal cost, however, they are temporarily more expensive than the other two. Simulated annealing obtains somewhat worse results than the previous ones, but in a very fast way. Finally, the genetic algorithm obtains very bad results in a relatively acceptable time. In conclusion, this work serves as a guide to people who want to understand these concepts.
During the history of computing, routing problems have attracted great interest due to their multiple applications in different fields, such as planning and logistics. This study focuses on the traveling salesman problem or TSP. Specifically, on the techniques to solve it in an approximate way in polynomial time, the metaheuristics. The main objective of this study is to provide a guide to understand four of the most important ones, both theoretically and computationally. For this purpose, a literature review was performed, finding relevant information of them and synthesizing it. They are: tabu search, simulated annealing, genetic algorithm and ant colony optimization. For the computational part, R implementations of all metaheuristics were performed and evaluated with different instances of the TSPLIB library. As a result, it was obtained that there is no metaheuristic better than the rest in all aspects. Tabu search and ant colony optimization obtain very promising results in terms of distance to optimal cost, however, they are temporarily more expensive than the other two. Simulated annealing obtains somewhat worse results than the previous ones, but in a very fast way. Finally, the genetic algorithm obtains very bad results in a relatively acceptable time. In conclusion, this work serves as a guide to people who want to understand these concepts.
Direction
CASAS MENDEZ, BALBINA VIRGINIA (Tutorships)
CASAS MENDEZ, BALBINA VIRGINIA (Tutorships)
Court
RODRIGUEZ CASAL, ALBERTO (Chairman)
ALONSO TARRIO, LEOVIGILDO (Secretary)
SALGADO SECO, MODESTO RAMON (Member)
RODRIGUEZ CASAL, ALBERTO (Chairman)
ALONSO TARRIO, LEOVIGILDO (Secretary)
SALGADO SECO, MODESTO RAMON (Member)
Game theory and logistics in the fishing sector
Authorship
U.F.G.
Bachelor of Mathematics
U.F.G.
Bachelor of Mathematics
Defense date
02.13.2025 13:30
02.13.2025 13:30
Summary
Game theory is a mathematical discipline that studies decision problems that involve various agents. We differenciate between cooperative and non cooperative games, which are distinguished by the existence or not of mechanisms for establish binding agreements. Two basic concepts are the Shapley value and the subgame perfect equilibrium, taken from cooperative games with transferable utility and games in extensive form. In this work we will use the tools mentioned above to understand and explain a recent investigation about fish aggregating devices. This leads to a possible increase of fishing firms profits, besides a beneficial contribution for the enviroment in terms of fuel reduction and CO2 emissions. Along with theoretical considerations, the aim is also to show an empirical analysis of this problem.
Game theory is a mathematical discipline that studies decision problems that involve various agents. We differenciate between cooperative and non cooperative games, which are distinguished by the existence or not of mechanisms for establish binding agreements. Two basic concepts are the Shapley value and the subgame perfect equilibrium, taken from cooperative games with transferable utility and games in extensive form. In this work we will use the tools mentioned above to understand and explain a recent investigation about fish aggregating devices. This leads to a possible increase of fishing firms profits, besides a beneficial contribution for the enviroment in terms of fuel reduction and CO2 emissions. Along with theoretical considerations, the aim is also to show an empirical analysis of this problem.
Direction
CASAS MENDEZ, BALBINA VIRGINIA (Tutorships)
CASAS MENDEZ, BALBINA VIRGINIA (Tutorships)
Court
RODRIGUEZ CASAL, ALBERTO (Chairman)
ALONSO TARRIO, LEOVIGILDO (Secretary)
SALGADO SECO, MODESTO RAMON (Member)
RODRIGUEZ CASAL, ALBERTO (Chairman)
ALONSO TARRIO, LEOVIGILDO (Secretary)
SALGADO SECO, MODESTO RAMON (Member)
Chebotarev's density theorem
Authorship
G.F.L.
Bachelor of Mathematics
G.F.L.
Bachelor of Mathematics
Defense date
07.02.2025 17:00
07.02.2025 17:00
Summary
The aim of this Bachelor’s Thesis is to study Chebotarev’s density theorem, exploring some of its applications, especially the factorization of polynomials modulo p. In the first part, we will present an initial approach to the connection between Galois theory and the factorization of polynomials modulo p, examining its relationship with other results such as the law of quadratic reciprocity. We will then explain the role played by Chebotarev’s density theorem and discuss further applications.
The aim of this Bachelor’s Thesis is to study Chebotarev’s density theorem, exploring some of its applications, especially the factorization of polynomials modulo p. In the first part, we will present an initial approach to the connection between Galois theory and the factorization of polynomials modulo p, examining its relationship with other results such as the law of quadratic reciprocity. We will then explain the role played by Chebotarev’s density theorem and discuss further applications.
Direction
RIVERO SALGADO, OSCAR (Tutorships)
RIVERO SALGADO, OSCAR (Tutorships)
Court
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
Numerical Methods in Ansys Fluent
Authorship
N.F.M.
Bachelor of Mathematics
N.F.M.
Bachelor of Mathematics
Defense date
07.03.2025 11:00
07.03.2025 11:00
Summary
In this publication, we will study some of the most commonly used numerical methods for solving differential equations. We will verify their theoretical properties by solving a physical initial value problem. Finally, we will conclude with a comparison between the available methods and those used by a commercial software for fluid modeling, such as Ansys Fluent.
In this publication, we will study some of the most commonly used numerical methods for solving differential equations. We will verify their theoretical properties by solving a physical initial value problem. Finally, we will conclude with a comparison between the available methods and those used by a commercial software for fluid modeling, such as Ansys Fluent.
Direction
Ferrín González, José Luis (Tutorships)
Ferrín González, José Luis (Tutorships)
Court
Ferrín González, José Luis (Student’s tutor)
Ferrín González, José Luis (Student’s tutor)
Reconstruction of Phylogenetic trees using Quantum Computing
Authorship
N.F.O.
Double Bachelor's Degree in Informatics Engineering and Mathematics
N.F.O.
Double Bachelor's Degree in Informatics Engineering and Mathematics
Defense date
02.20.2025 10:30
02.20.2025 10:30
Summary
Quantum computing is a field of computer science that uses principles of quantum physics to solve problems more efficiently than classical computing, especially in areas such as optimization. Bioinformatics, on the other hand, is a field that combines elements of biology and computer science to analyze large biological data sets. A prominent example of this discipline is genomics, which includes the generation of phylogenetic trees, key tools for understanding the biological evolution of species. The reconstruction of these trees represents a computational problem that is very difficult to solve due to its complexity. This work explores whether quantum computing can offer effective solutions to address this problem. In this context, the performance of quantum computation and quantum optimization algorithms has been studied, with emphasis on Quantum Annealing and the Quantum Approximate Optimization Algorithm (QAOA). Based on these approaches, a quantum algorithm capable of reconstructing phylogenies by cutting graphs has been developed. The proposed algorithm was implemented and tested on currently available quantum hardware, obtaining satisfactory results that demonstrate its potential to solve complex problems in the area of bioinformatics.
Quantum computing is a field of computer science that uses principles of quantum physics to solve problems more efficiently than classical computing, especially in areas such as optimization. Bioinformatics, on the other hand, is a field that combines elements of biology and computer science to analyze large biological data sets. A prominent example of this discipline is genomics, which includes the generation of phylogenetic trees, key tools for understanding the biological evolution of species. The reconstruction of these trees represents a computational problem that is very difficult to solve due to its complexity. This work explores whether quantum computing can offer effective solutions to address this problem. In this context, the performance of quantum computation and quantum optimization algorithms has been studied, with emphasis on Quantum Annealing and the Quantum Approximate Optimization Algorithm (QAOA). Based on these approaches, a quantum algorithm capable of reconstructing phylogenies by cutting graphs has been developed. The proposed algorithm was implemented and tested on currently available quantum hardware, obtaining satisfactory results that demonstrate its potential to solve complex problems in the area of bioinformatics.
Direction
Fernández Pena, Anselmo Tomás (Tutorships)
PICHEL CAMPOS, JUAN CARLOS (Co-tutorships)
Fernández Pena, Anselmo Tomás (Tutorships)
PICHEL CAMPOS, JUAN CARLOS (Co-tutorships)
Court
ARIAS RODRIGUEZ, JUAN ENRIQUE (Chairman)
Querentes Hermida, Raquel Esther (Secretary)
PIÑEIRO POMAR, CESAR ALFREDO (Member)
ARIAS RODRIGUEZ, JUAN ENRIQUE (Chairman)
Querentes Hermida, Raquel Esther (Secretary)
PIÑEIRO POMAR, CESAR ALFREDO (Member)
Memetic algorithms for the MC-TTRP
Authorship
N.F.O.
Double Bachelor's Degree in Informatics Engineering and Mathematics
N.F.O.
Double Bachelor's Degree in Informatics Engineering and Mathematics
Defense date
07.03.2025 10:40
07.03.2025 10:40
Summary
A metaheuristic is a high-level search procedure designed to guide subordinate heuristics in order to efficiently explore solution spaces in complex optimization problems, especially those where exact methods are computationally infeasible. These techniques do not guarantee to find the optimal solution, but seek to obtain good quality solutions in reasonable times, which makes them especially useful in real environments.Within this framework, evolutionary algorithms, which are inspired by principles of biological evolution to explore complex search spaces, stand out. Among them, genetic and memetic algorithms are particularly relevant. Genetic algorithms employ mechanisms such as selection, crossover and mutation to generate new solutions, while memetic algorithms combine this global exploration with local improvement strategies to further optimize each solution. These methods have been successfully applied in solving a variety of complex problems, including routing problems. These consist of finding the optimal set of paths that a fleet of vehicles must take to serve a set of customers. A generalization of routing problems is the multicompartment truck and trailer routing problem (MC-TTRP). This problem considers two types of compartmentalized vehicles, trucks and trailers that must be towed, and two types of customers with different service constraints and requiring multiple types of cargo, resulting in the existence of multiple types of routes to optimize distribution. In this work we have explored genetic and memetic algorithms, studying how the operators used work and how to obtain a memetic algorithm that allows us to solve a complex problem. Routing problems have also been studied, with a greater emphasis on the MC-TTRP, offering a linear and mixed integer programming model that enables us to model the problem in a mathematical way. Using this knowledge, a C++ algorithm has been implemented to obtain the optimal routes for any instance of the MC-TTRP.
A metaheuristic is a high-level search procedure designed to guide subordinate heuristics in order to efficiently explore solution spaces in complex optimization problems, especially those where exact methods are computationally infeasible. These techniques do not guarantee to find the optimal solution, but seek to obtain good quality solutions in reasonable times, which makes them especially useful in real environments.Within this framework, evolutionary algorithms, which are inspired by principles of biological evolution to explore complex search spaces, stand out. Among them, genetic and memetic algorithms are particularly relevant. Genetic algorithms employ mechanisms such as selection, crossover and mutation to generate new solutions, while memetic algorithms combine this global exploration with local improvement strategies to further optimize each solution. These methods have been successfully applied in solving a variety of complex problems, including routing problems. These consist of finding the optimal set of paths that a fleet of vehicles must take to serve a set of customers. A generalization of routing problems is the multicompartment truck and trailer routing problem (MC-TTRP). This problem considers two types of compartmentalized vehicles, trucks and trailers that must be towed, and two types of customers with different service constraints and requiring multiple types of cargo, resulting in the existence of multiple types of routes to optimize distribution. In this work we have explored genetic and memetic algorithms, studying how the operators used work and how to obtain a memetic algorithm that allows us to solve a complex problem. Routing problems have also been studied, with a greater emphasis on the MC-TTRP, offering a linear and mixed integer programming model that enables us to model the problem in a mathematical way. Using this knowledge, a C++ algorithm has been implemented to obtain the optimal routes for any instance of the MC-TTRP.
Direction
CASAS MENDEZ, BALBINA VIRGINIA (Tutorships)
CASAS MENDEZ, BALBINA VIRGINIA (Tutorships)
Court
Majadas Soto, José Javier (Chairman)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Secretary)
CASARES DE CAL, MARIA ANGELES (Member)
Majadas Soto, José Javier (Chairman)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Secretary)
CASARES DE CAL, MARIA ANGELES (Member)
Comparison of classical and machine learning methodologies in time series analysis
Authorship
A.F.M.
Bachelor of Mathematics
A.F.M.
Bachelor of Mathematics
Defense date
07.02.2025 12:00
07.02.2025 12:00
Summary
When taking into consideration a set of data, one can find independent observations or observations that present some kind of spacial or temporal dependence, as we see in the case of time series. By taking this dependence into account, the statistical theory of time series analysis naturally appears, as we will be discussing it over the next pages. The objective of this piece of work is the description and comparison of the different models and methodologies about time series analysis. From this comparison made on the base of the following terms: accuracy, simpleness, interpretability and computational efficiency, I have reached the conclusion that the most appropriate models vary depending on each case.
When taking into consideration a set of data, one can find independent observations or observations that present some kind of spacial or temporal dependence, as we see in the case of time series. By taking this dependence into account, the statistical theory of time series analysis naturally appears, as we will be discussing it over the next pages. The objective of this piece of work is the description and comparison of the different models and methodologies about time series analysis. From this comparison made on the base of the following terms: accuracy, simpleness, interpretability and computational efficiency, I have reached the conclusion that the most appropriate models vary depending on each case.
Direction
PATEIRO LOPEZ, BEATRIZ (Tutorships)
PATEIRO LOPEZ, BEATRIZ (Tutorships)
Court
CRUJEIRAS CASAIS, ROSA MARÍA (Chairman)
PENA BRAGE, FRANCISCO JOSE (Secretary)
DOMINGUEZ VAZQUEZ, MIGUEL (Member)
CRUJEIRAS CASAIS, ROSA MARÍA (Chairman)
PENA BRAGE, FRANCISCO JOSE (Secretary)
DOMINGUEZ VAZQUEZ, MIGUEL (Member)
Mathematical Aspects of Concept Drift
Authorship
F.F.M.
Double Bachelor's Degree in Informatics Engineering and Mathematics
F.F.M.
Double Bachelor's Degree in Informatics Engineering and Mathematics
Defense date
07.03.2025 11:25
07.03.2025 11:25
Summary
This work addresses the phenomenon of Concept Drift, which arises in dynamic and nonstationary environments where the statistical relationships between model variables change over time, thus affecting the performance of machine learning algorithms. The main objective is to develop a modification of the KSWIN algorithm, part of the RiverML library, which is based on the Kolmogorov-Smirnov test. The proposed modification incorporates multiple hypothesis tests and the Benjamini-Hochberg correction in order to enhance the statistical robustness of the test and reduce the false positive rate. Several configurations of the detector are proposed, targeting both the monitoring of data drawn from continuous distributions and the evaluation of performance metrics. For the latter approach, a mechanism is introduced to identify the type of drift, using non-parametric inference techniques. For the first case, a testing environment with artificially generated data is designed. In the second, the work integrates a comparative study developed in a Bachelor’s Thesis in Computer Engineering, focused on the empirical evaluation of several drift detection algorithms from the literature. The experiments show a significant reduction in the false positive rate without compromising test power, improving the effectiveness of both the original algorithm and other classical detectors. Furthermore, the ability to identify the type of drift adds practical value to one of the proposed configurations.
This work addresses the phenomenon of Concept Drift, which arises in dynamic and nonstationary environments where the statistical relationships between model variables change over time, thus affecting the performance of machine learning algorithms. The main objective is to develop a modification of the KSWIN algorithm, part of the RiverML library, which is based on the Kolmogorov-Smirnov test. The proposed modification incorporates multiple hypothesis tests and the Benjamini-Hochberg correction in order to enhance the statistical robustness of the test and reduce the false positive rate. Several configurations of the detector are proposed, targeting both the monitoring of data drawn from continuous distributions and the evaluation of performance metrics. For the latter approach, a mechanism is introduced to identify the type of drift, using non-parametric inference techniques. For the first case, a testing environment with artificially generated data is designed. In the second, the work integrates a comparative study developed in a Bachelor’s Thesis in Computer Engineering, focused on the empirical evaluation of several drift detection algorithms from the literature. The experiments show a significant reduction in the false positive rate without compromising test power, improving the effectiveness of both the original algorithm and other classical detectors. Furthermore, the ability to identify the type of drift adds practical value to one of the proposed configurations.
Direction
CRUJEIRAS CASAIS, ROSA MARÍA (Tutorships)
CRUJEIRAS CASAIS, ROSA MARÍA (Tutorships)
Court
Majadas Soto, José Javier (Chairman)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Secretary)
CASARES DE CAL, MARIA ANGELES (Member)
Majadas Soto, José Javier (Chairman)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Secretary)
CASARES DE CAL, MARIA ANGELES (Member)
Special functions in solving partial differential equations
Authorship
C.F.S.
Double bachelor degree in Mathematics and Physics
C.F.S.
Double bachelor degree in Mathematics and Physics
Defense date
07.03.2025 10:00
07.03.2025 10:00
Summary
The solution by separation of variables of, for example, the wave equation in a circular spatial domain leads us to Bessel functions as the fundamental functions for obtaining series solutions. This final-year project is devoted to studying Bessel functions, along with other special functions, and demonstrating their applications in solving partial differential equations (PDEs) in circular or cylindrical spatial domains.
The solution by separation of variables of, for example, the wave equation in a circular spatial domain leads us to Bessel functions as the fundamental functions for obtaining series solutions. This final-year project is devoted to studying Bessel functions, along with other special functions, and demonstrating their applications in solving partial differential equations (PDEs) in circular or cylindrical spatial domains.
Direction
LOPEZ POUSO, RODRIGO (Tutorships)
LOPEZ POUSO, RODRIGO (Tutorships)
Court
QUINTELA ESTEVEZ, PEREGRINA (Chairman)
TRINCHET SORIA, ROSA Mª (Secretary)
DIAZ RAMOS, JOSE CARLOS (Member)
QUINTELA ESTEVEZ, PEREGRINA (Chairman)
TRINCHET SORIA, ROSA Mª (Secretary)
DIAZ RAMOS, JOSE CARLOS (Member)
Statistical Modeling of Sports Data
Authorship
A.G.A.
Double bachelor degree in Mathematics and Physics
A.G.A.
Double bachelor degree in Mathematics and Physics
Defense date
07.02.2025 12:45
07.02.2025 12:45
Summary
Throughout this work, an application of the supervised learning model Random Forest to sports data is presented. Specifically, data associated with NBA teams from recent seasons. In the first chapter, a brief introduction to supervised learning algorithms is provided, with a particular emphasis on the bias-variance tradeoff, a fundamental problem in this type of model. Next, a systematic description of decision trees is given. These are among the simplest supervised learning models but serve as essential components in more complex models such as Random Forest. In Chapter 3, the Random Forest model is introduced as defined by Leo Breiman in 2001. Additionally, key results related to its relative error reduction and variance are presented. Finally, in the last chapter, the Random Forest model is applied to advanced statistics of NBA teams. Both a classification case and a regression case will be analyzed. In each scenario, the dependence of the models on their hyperparameters will be studied, and the results will be compared with other commonly used models for this type of problem.
Throughout this work, an application of the supervised learning model Random Forest to sports data is presented. Specifically, data associated with NBA teams from recent seasons. In the first chapter, a brief introduction to supervised learning algorithms is provided, with a particular emphasis on the bias-variance tradeoff, a fundamental problem in this type of model. Next, a systematic description of decision trees is given. These are among the simplest supervised learning models but serve as essential components in more complex models such as Random Forest. In Chapter 3, the Random Forest model is introduced as defined by Leo Breiman in 2001. Additionally, key results related to its relative error reduction and variance are presented. Finally, in the last chapter, the Random Forest model is applied to advanced statistics of NBA teams. Both a classification case and a regression case will be analyzed. In each scenario, the dependence of the models on their hyperparameters will be studied, and the results will be compared with other commonly used models for this type of problem.
Direction
RODRIGUEZ CASAL, ALBERTO (Tutorships)
RODRIGUEZ CASAL, ALBERTO (Tutorships)
Court
CRUJEIRAS CASAIS, ROSA MARÍA (Chairman)
PENA BRAGE, FRANCISCO JOSE (Secretary)
DOMINGUEZ VAZQUEZ, MIGUEL (Member)
CRUJEIRAS CASAIS, ROSA MARÍA (Chairman)
PENA BRAGE, FRANCISCO JOSE (Secretary)
DOMINGUEZ VAZQUEZ, MIGUEL (Member)
L functions of elliptic curves and modular forms
Authorship
J.G.C.
Double bachelor degree in Mathematics and Physics
J.G.C.
Double bachelor degree in Mathematics and Physics
Defense date
07.02.2025 17:45
07.02.2025 17:45
Summary
L functions are functions defined in the complex plane that allow us to obtain arithmetic information from analytic properties such as the location of their zeros, poles or the fulfillment of a certain functional equation. Moreover, they allow us to connect objects of different nature like elliptic curves, of geometric nature, and modular forms, of analytic nature, through the modularity theorem that establishes a correspondence between them through its associated L functions. In this work, we will focus on the study of L functions associated to generalizations of modular forms, the so-called automorphic forms, and Galois representations. In particular, we will begin by introducing Galois representations and their connections with elliptic curves and modular forms. Then, we will study automorphic forms and representations in the case of GL2 where Tate's thesis techniques to establish functional equations of its L functions will be introduced. In the next two chapters these concepts will be generalised to the general case of an arbitrary reductive algebraic group. All this will be studied placing it within the Langlands program that generalises the connection between elliptic curves and modular forms to a more general context.
L functions are functions defined in the complex plane that allow us to obtain arithmetic information from analytic properties such as the location of their zeros, poles or the fulfillment of a certain functional equation. Moreover, they allow us to connect objects of different nature like elliptic curves, of geometric nature, and modular forms, of analytic nature, through the modularity theorem that establishes a correspondence between them through its associated L functions. In this work, we will focus on the study of L functions associated to generalizations of modular forms, the so-called automorphic forms, and Galois representations. In particular, we will begin by introducing Galois representations and their connections with elliptic curves and modular forms. Then, we will study automorphic forms and representations in the case of GL2 where Tate's thesis techniques to establish functional equations of its L functions will be introduced. In the next two chapters these concepts will be generalised to the general case of an arbitrary reductive algebraic group. All this will be studied placing it within the Langlands program that generalises the connection between elliptic curves and modular forms to a more general context.
Direction
RIVERO SALGADO, OSCAR (Tutorships)
RIVERO SALGADO, OSCAR (Tutorships)
Court
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
Generating functions in the calculation of power indices.
Authorship
C.G.F.
Bachelor of Mathematics
C.G.F.
Bachelor of Mathematics
Defense date
02.12.2025 12:30
02.12.2025 12:30
Summary
Within the field of game theory, weighted majority games play a fundamental role in the analysis of voting processes in parliaments and committees. This work introduces this class of games, focusing on the study of power indices, a solution concept that assigns a measure of influence or power to the players involved in the voting process. Among the power indices available in the literature, we will consider five: Shapley-Shubik, Banzhaf, Johnston, Colomer-Martínez, and Johnston-Colomer-Martínez. Their mathematical properties will be examined, practical applications will be provided, and their computational cost will be assessed. To facilitate the computation of these five indices, we will develop methods based on generating functions, which are combinatorial tools that allow us to derive, through polynomials, the necessary components for their calculation. Furthermore, we will model a new scenario in which players can form alliances, leading to what are known as games with coalition structure. For these games, we will introduce two additional power indices: Owen and Banzhaf-Owen, along with computation methods based on generating functions. Finally, these concepts will be applied to a practical case: the analysis of the Spanish Parliament. We will examine changes in the distribution of power among political parties between the general elections held in November 2019 and July 2023, as well as the consequences of members of parliament switching between parliamentary groups during the XV Legislature. The \textit{powerindexR} library within the statistical software R will be used to compute the power indices in these scenarios.
Within the field of game theory, weighted majority games play a fundamental role in the analysis of voting processes in parliaments and committees. This work introduces this class of games, focusing on the study of power indices, a solution concept that assigns a measure of influence or power to the players involved in the voting process. Among the power indices available in the literature, we will consider five: Shapley-Shubik, Banzhaf, Johnston, Colomer-Martínez, and Johnston-Colomer-Martínez. Their mathematical properties will be examined, practical applications will be provided, and their computational cost will be assessed. To facilitate the computation of these five indices, we will develop methods based on generating functions, which are combinatorial tools that allow us to derive, through polynomials, the necessary components for their calculation. Furthermore, we will model a new scenario in which players can form alliances, leading to what are known as games with coalition structure. For these games, we will introduce two additional power indices: Owen and Banzhaf-Owen, along with computation methods based on generating functions. Finally, these concepts will be applied to a practical case: the analysis of the Spanish Parliament. We will examine changes in the distribution of power among political parties between the general elections held in November 2019 and July 2023, as well as the consequences of members of parliament switching between parliamentary groups during the XV Legislature. The \textit{powerindexR} library within the statistical software R will be used to compute the power indices in these scenarios.
Direction
SAAVEDRA NIEVES, ALEJANDRO (Tutorships)
DAVILA PENA, LAURA (Co-tutorships)
SAAVEDRA NIEVES, ALEJANDRO (Tutorships)
DAVILA PENA, LAURA (Co-tutorships)
Court
CABADA FERNANDEZ, ALBERTO (Chairman)
BORRAJO GARCIA, MARIA ISABEL (Secretary)
MUÑOZ SOLA, RAFAEL (Member)
CABADA FERNANDEZ, ALBERTO (Chairman)
BORRAJO GARCIA, MARIA ISABEL (Secretary)
MUÑOZ SOLA, RAFAEL (Member)
Introduction to bifurcations in ordinary differential equations
Authorship
A.G.L.
Bachelor of Mathematics
A.G.L.
Bachelor of Mathematics
Defense date
02.12.2025 13:15
02.12.2025 13:15
Summary
The study of the qualitative behaviour of differential equations seeks to obtain properties of the solutions without the need to know them explicitly. This approach acquires special relevance when parameters are incorporated into the equation, since small variations in them can lead to very significant changes, having effects on the number of singular points, their stability or the appearance of oscillatory solutions. This is the idea behind the theory of bifurcations, which will be explored in depth by means of the most typical examples in one and two dimensions: the tangent, transcritical, pitchfork and Hopf bifurcations. For each of them, the qualitative behaviour of a type equation will be explored, followed by a generic study in which the conditions that characterise it will be obtained.
The study of the qualitative behaviour of differential equations seeks to obtain properties of the solutions without the need to know them explicitly. This approach acquires special relevance when parameters are incorporated into the equation, since small variations in them can lead to very significant changes, having effects on the number of singular points, their stability or the appearance of oscillatory solutions. This is the idea behind the theory of bifurcations, which will be explored in depth by means of the most typical examples in one and two dimensions: the tangent, transcritical, pitchfork and Hopf bifurcations. For each of them, the qualitative behaviour of a type equation will be explored, followed by a generic study in which the conditions that characterise it will be obtained.
Direction
BUEDO FERNANDEZ, SEBASTIAN (Tutorships)
LOIS PRADOS, CRISTINA (Co-tutorships)
BUEDO FERNANDEZ, SEBASTIAN (Tutorships)
LOIS PRADOS, CRISTINA (Co-tutorships)
Court
CABADA FERNANDEZ, ALBERTO (Chairman)
BORRAJO GARCIA, MARIA ISABEL (Secretary)
MUÑOZ SOLA, RAFAEL (Member)
CABADA FERNANDEZ, ALBERTO (Chairman)
BORRAJO GARCIA, MARIA ISABEL (Secretary)
MUÑOZ SOLA, RAFAEL (Member)
Model-based clustering
Authorship
N.G.S.D.V.
Double Bachelor's Degree in Informatics Engineering and Mathematics
N.G.S.D.V.
Double Bachelor's Degree in Informatics Engineering and Mathematics
Defense date
07.03.2025 12:10
07.03.2025 12:10
Summary
Clustering is an unsupervised statistical technique that aims to automatically identify homogeneous groups of observations within a dataset. Its usefulness has been consolidated across various disciplines, particularly in the current context of massive data generation, thanks to its ability to identify groups in complex and high-dimensional data. Although heuristic methods such as k-means or hierarchical techniques have traditionally been used, these approaches present limitations, such as the lack of a solid theoretical foundation or the difficulty in determining the optimal number of groups. In contrast, model-based clustering (MBC) offers a statistically grounded alternative by modeling the data as a finite mixture of probability distributions. This approach allows for rigorous inferences, the selection of appropriate models, justifiable determination of the number of groups, and the evaluation of uncertainty in the assignment of observations. This work presents the theoretical foundations of model-based clustering, with a focus on Gaussian mixture models, which are the most widely used, as well as the EM algorithm for parameter estimation and model selection criteria, including the choice of the number of clusters. Additionally, practical examples are presented using the mclust package in R.
Clustering is an unsupervised statistical technique that aims to automatically identify homogeneous groups of observations within a dataset. Its usefulness has been consolidated across various disciplines, particularly in the current context of massive data generation, thanks to its ability to identify groups in complex and high-dimensional data. Although heuristic methods such as k-means or hierarchical techniques have traditionally been used, these approaches present limitations, such as the lack of a solid theoretical foundation or the difficulty in determining the optimal number of groups. In contrast, model-based clustering (MBC) offers a statistically grounded alternative by modeling the data as a finite mixture of probability distributions. This approach allows for rigorous inferences, the selection of appropriate models, justifiable determination of the number of groups, and the evaluation of uncertainty in the assignment of observations. This work presents the theoretical foundations of model-based clustering, with a focus on Gaussian mixture models, which are the most widely used, as well as the EM algorithm for parameter estimation and model selection criteria, including the choice of the number of clusters. Additionally, practical examples are presented using the mclust package in R.
Direction
AMEIJEIRAS ALONSO, JOSE (Tutorships)
AMEIJEIRAS ALONSO, JOSE (Tutorships)
Court
Majadas Soto, José Javier (Chairman)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Secretary)
CASARES DE CAL, MARIA ANGELES (Member)
Majadas Soto, José Javier (Chairman)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Secretary)
CASARES DE CAL, MARIA ANGELES (Member)
Mathematical models for the regulation of cell volume
Authorship
J.A.G.B.
Bachelor of Mathematics
J.A.G.B.
Bachelor of Mathematics
Defense date
07.03.2025 10:30
07.03.2025 10:30
Summary
In this work, we study a model of differential equations that reflects the variation of cell volume caused by different biological factors. First, we will review basic concepts related to Ordinary Differential Equations and then introduce the concept of Brouwer degree and some of its main properties. We will also examine how the model fits different types of cells and how its parameters vary. In addition, we will study the stability of our model with the help of specific cases. Finally, we will explore the possibility that the model admits non-trivial T-periodic solutions through various results, which will be illustrated with some examples.
In this work, we study a model of differential equations that reflects the variation of cell volume caused by different biological factors. First, we will review basic concepts related to Ordinary Differential Equations and then introduce the concept of Brouwer degree and some of its main properties. We will also examine how the model fits different types of cells and how its parameters vary. In addition, we will study the stability of our model with the help of specific cases. Finally, we will explore the possibility that the model admits non-trivial T-periodic solutions through various results, which will be illustrated with some examples.
Direction
CABADA FERNANDEZ, ALBERTO (Tutorships)
CABADA FERNANDEZ, ALBERTO (Tutorships)
Court
QUINTELA ESTEVEZ, PEREGRINA (Chairman)
TRINCHET SORIA, ROSA Mª (Secretary)
DIAZ RAMOS, JOSE CARLOS (Member)
QUINTELA ESTEVEZ, PEREGRINA (Chairman)
TRINCHET SORIA, ROSA Mª (Secretary)
DIAZ RAMOS, JOSE CARLOS (Member)
Numerical solution for polinomial ecuations
Authorship
A.H.T.
Bachelor of Mathematics
A.H.T.
Bachelor of Mathematics
Defense date
07.03.2025 11:00
07.03.2025 11:00
Summary
This work addresses the numerical solution of polynomial equations using computational methods. Its main objective is to describe and apply techniques to bound, locate, and approximate the roots of polynomials with real and complex coefficients. We develop both the theory and practice of classical methods such as Horner’s scheme for efficient evaluation of polynomials and their derivatives; root bounding techniques (Lehmer Schur for complex roots; Laguerre Thibault, Newton, and Sturm for real roots); and approximation algorithms (Newton, Bernoulli, Bairstow, and Graeffe Lobachevsky) all of this last four implemented in MATLAB. We show that the choice of method depends critically on the polynomial’s nature: root multiplicity, separation between roots, and the presence of complex conjugates. Our numerical experiments reveal, among other findings, that Newton’s method with deflation is robust for simple roots; Bairstow’s method is optimal for finding complex conjugate pairs without resorting to complex arithmetic; and Bernoulli’s method converges rapidly for a single dominant root, but is sensitive to the initial guess. The thesis includes verified MATLAB code for every root approximation algorithm, validated against test polynomials. In conclusion, we emphasize the importance of combining analytical techniques (root separation and bounding) with numerical algorithms to ensure both accuracy and reliability.
This work addresses the numerical solution of polynomial equations using computational methods. Its main objective is to describe and apply techniques to bound, locate, and approximate the roots of polynomials with real and complex coefficients. We develop both the theory and practice of classical methods such as Horner’s scheme for efficient evaluation of polynomials and their derivatives; root bounding techniques (Lehmer Schur for complex roots; Laguerre Thibault, Newton, and Sturm for real roots); and approximation algorithms (Newton, Bernoulli, Bairstow, and Graeffe Lobachevsky) all of this last four implemented in MATLAB. We show that the choice of method depends critically on the polynomial’s nature: root multiplicity, separation between roots, and the presence of complex conjugates. Our numerical experiments reveal, among other findings, that Newton’s method with deflation is robust for simple roots; Bairstow’s method is optimal for finding complex conjugate pairs without resorting to complex arithmetic; and Bernoulli’s method converges rapidly for a single dominant root, but is sensitive to the initial guess. The thesis includes verified MATLAB code for every root approximation algorithm, validated against test polynomials. In conclusion, we emphasize the importance of combining analytical techniques (root separation and bounding) with numerical algorithms to ensure both accuracy and reliability.
Direction
VIAÑO REY, JUAN MANUEL (Tutorships)
VIAÑO REY, JUAN MANUEL (Tutorships)
Court
QUINTELA ESTEVEZ, PEREGRINA (Chairman)
TRINCHET SORIA, ROSA Mª (Secretary)
DIAZ RAMOS, JOSE CARLOS (Member)
QUINTELA ESTEVEZ, PEREGRINA (Chairman)
TRINCHET SORIA, ROSA Mª (Secretary)
DIAZ RAMOS, JOSE CARLOS (Member)
Spatial data modeling
Authorship
N.H.C.
Bachelor of Mathematics
N.H.C.
Bachelor of Mathematics
Defense date
07.03.2025 12:55
07.03.2025 12:55
Summary
Spatial data represent geographic locations whose analysis enables the detection of potential spatial patterns and structures. This Bachelor's Thesis presents an introduction to non-parametric bivariate density estimation, with a focus on the use of the kernel estimator in order to obtain smooth representations of the spatial distribution of the data. The different methods will be illustrated through the implementation of R code, applied to real data on leukemia cases and controls recorded in the northwest of England. The main objective is to identify significant spatial clusters that may contribute to the understanding of the observed epidemiological patterns.
Spatial data represent geographic locations whose analysis enables the detection of potential spatial patterns and structures. This Bachelor's Thesis presents an introduction to non-parametric bivariate density estimation, with a focus on the use of the kernel estimator in order to obtain smooth representations of the spatial distribution of the data. The different methods will be illustrated through the implementation of R code, applied to real data on leukemia cases and controls recorded in the northwest of England. The main objective is to identify significant spatial clusters that may contribute to the understanding of the observed epidemiological patterns.
Direction
RODRIGUEZ CASAL, ALBERTO (Tutorships)
RODRIGUEZ CASAL, ALBERTO (Tutorships)
Court
Majadas Soto, José Javier (Chairman)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Secretary)
CASARES DE CAL, MARIA ANGELES (Member)
Majadas Soto, José Javier (Chairman)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Secretary)
CASARES DE CAL, MARIA ANGELES (Member)
The Newton and discretized Newton methods for nonlinear systems of equations.
Authorship
A.J.P.
Bachelor of Mathematics
A.J.P.
Bachelor of Mathematics
Defense date
07.02.2025 11:00
07.02.2025 11:00
Summary
This work studies Newton's method applied to systems of nonlinear equations, as well as its discretized variant, in which derivatives are approximated using finite differences. The exposition begins with the one-dimensional case to aid understanding, and then generalizes to the multidimensional setting. The theoretical part includes a detailed analysis of the local convergence of both methods. Finally, both methods are implemented in MATLAB, and their properties are illustrated through three numerical examples.
This work studies Newton's method applied to systems of nonlinear equations, as well as its discretized variant, in which derivatives are approximated using finite differences. The exposition begins with the one-dimensional case to aid understanding, and then generalizes to the multidimensional setting. The theoretical part includes a detailed analysis of the local convergence of both methods. Finally, both methods are implemented in MATLAB, and their properties are illustrated through three numerical examples.
Direction
MUÑOZ SOLA, RAFAEL (Tutorships)
MUÑOZ SOLA, RAFAEL (Tutorships)
Court
MUÑOZ SOLA, RAFAEL (Student’s tutor)
MUÑOZ SOLA, RAFAEL (Student’s tutor)
Efficient semantic segmentation of land cover images using an encoder-decoder architecture
Authorship
I.L.C.
Double Bachelor's Degree in Informatics Engineering and Mathematics
I.L.C.
Double Bachelor's Degree in Informatics Engineering and Mathematics
Defense date
02.20.2025 11:30
02.20.2025 11:30
Summary
In the area of remote sensing, there is great interest in collecting land cover information to identify and classify the different types of surfaces present on the ground, such as vegetated areas, water bodies, urban soils, grasslands, forests or agricultural areas, among others. On the other hand, semantic image segmentation allows assigning a label to each pixel of the image, classifying them into different categories or specific classes, which facilitates the interpretation and analysis of satellite or aerial images. The use of deep learning techniques has proven to be effective in the field of computer vision, specifically in semantic segmentation tasks. However, these models are very computationally expensive, and often require the use of specialised hardware and optimisation techniques to improve the efficiency and feasibility of training and inference. In this Bechelor Thesis, the aim is to test different models with encoder-decoder architecture, trying to improve the efficiency and feasibility of training even with large amounts of data. From the existing parallelism techniques for multiGPU training, data parallelism will be used, selecting a PyTorch module that implements it in an efficient way. In addition, using 16-bit mixed floating-point precision reduces memory usage and makes better use of the GPU hardware, performing training in half the time without affecting the quality of the segmentation.
In the area of remote sensing, there is great interest in collecting land cover information to identify and classify the different types of surfaces present on the ground, such as vegetated areas, water bodies, urban soils, grasslands, forests or agricultural areas, among others. On the other hand, semantic image segmentation allows assigning a label to each pixel of the image, classifying them into different categories or specific classes, which facilitates the interpretation and analysis of satellite or aerial images. The use of deep learning techniques has proven to be effective in the field of computer vision, specifically in semantic segmentation tasks. However, these models are very computationally expensive, and often require the use of specialised hardware and optimisation techniques to improve the efficiency and feasibility of training and inference. In this Bechelor Thesis, the aim is to test different models with encoder-decoder architecture, trying to improve the efficiency and feasibility of training even with large amounts of data. From the existing parallelism techniques for multiGPU training, data parallelism will be used, selecting a PyTorch module that implements it in an efficient way. In addition, using 16-bit mixed floating-point precision reduces memory usage and makes better use of the GPU hardware, performing training in half the time without affecting the quality of the segmentation.
Direction
Argüello Pedreira, Francisco Santiago (Tutorships)
Blanco Heras, Dora (Co-tutorships)
Argüello Pedreira, Francisco Santiago (Tutorships)
Blanco Heras, Dora (Co-tutorships)
Court
ARIAS RODRIGUEZ, JUAN ENRIQUE (Chairman)
Querentes Hermida, Raquel Esther (Secretary)
PIÑEIRO POMAR, CESAR ALFREDO (Member)
ARIAS RODRIGUEZ, JUAN ENRIQUE (Chairman)
Querentes Hermida, Raquel Esther (Secretary)
PIÑEIRO POMAR, CESAR ALFREDO (Member)
Permutation groups in the classification of idempotent evolution algebras
Authorship
A.L.P.
Bachelor of Mathematics
A.L.P.
Bachelor of Mathematics
Defense date
07.02.2025 18:30
07.02.2025 18:30
Summary
An evolution algebra over a field is an algebra endowed with a basis such that the product of any pair of distinct basis elements is always zero. Finite-dimensional idempotent evolution algebras have the property that their automorphism group is finite and admits a representation via permutations. In the context of the group realization problem, the natural question arises whether every permutation representation of a finite group can be realized through a finite dimensional idempotent evolution algebra. This work introduces the necessary theory to understand the problem and discusses the main results found in the literature.
An evolution algebra over a field is an algebra endowed with a basis such that the product of any pair of distinct basis elements is always zero. Finite-dimensional idempotent evolution algebras have the property that their automorphism group is finite and admits a representation via permutations. In the context of the group realization problem, the natural question arises whether every permutation representation of a finite group can be realized through a finite dimensional idempotent evolution algebra. This work introduces the necessary theory to understand the problem and discusses the main results found in the literature.
Direction
COSTOYA RAMOS, MARIA CRISTINA (Tutorships)
FERNANDEZ RODRIGUEZ, ROSA Mª (Co-tutorships)
COSTOYA RAMOS, MARIA CRISTINA (Tutorships)
FERNANDEZ RODRIGUEZ, ROSA Mª (Co-tutorships)
Court
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
Mathematical Methods of Artificial Inteligence
Authorship
P.L.P.
Double Bachelor's Degree in Informatics Engineering and Mathematics
P.L.P.
Double Bachelor's Degree in Informatics Engineering and Mathematics
Defense date
07.03.2025 10:00
07.03.2025 10:00
Summary
This thesis explores the mathematical foundations of artificial intelligence, focusing on neural networks and their research lines. It begins with a detailed analysis of neural networks, covering foundational concepts such as architecture and training, and also research topics like expressivity, optimization, generalization, and explainability. The Vapnik-Chervonenkis (VC) dimension is introduced as a theoretical framework to quantify the capacity of models, offering insights into their generalization ability and limitations. To address the curse of dimensionality, the thesis discusses dimensionality reduction techniques, including principal component analysis (PCA) and linear discriminant analysis (LDA), showcasing their role in improving model efficiency without sacrificing performance. Finally, the mathematical capabilities of large language models like GPT are evaluated. Leveraging examples from reasoning and problem-solving tasks, this work investigates how these models process and generate mathematically rigorous outputs.
This thesis explores the mathematical foundations of artificial intelligence, focusing on neural networks and their research lines. It begins with a detailed analysis of neural networks, covering foundational concepts such as architecture and training, and also research topics like expressivity, optimization, generalization, and explainability. The Vapnik-Chervonenkis (VC) dimension is introduced as a theoretical framework to quantify the capacity of models, offering insights into their generalization ability and limitations. To address the curse of dimensionality, the thesis discusses dimensionality reduction techniques, including principal component analysis (PCA) and linear discriminant analysis (LDA), showcasing their role in improving model efficiency without sacrificing performance. Finally, the mathematical capabilities of large language models like GPT are evaluated. Leveraging examples from reasoning and problem-solving tasks, this work investigates how these models process and generate mathematically rigorous outputs.
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)
Determination of spatial dependence using variograms.
Authorship
C.L.L.
Bachelor of Mathematics
C.L.L.
Bachelor of Mathematics
Defense date
02.12.2025 16:30
02.12.2025 16:30
Summary
This work provides an introduction to geostatistics, focusing particularly on the concept of the variogram, a structure that quantifies spatial dependence, and the Kriging spatial interpolation method. To this end, the theoretical foundations of spatial dependence are presented as the basis for the development of the variogram, including both its experimental and theoretical conception, as well as the different existing models and the reasons why it may fail to properly model spatial dependence. Next, the theory behind the Kriging interpolation method is introduced, along with its different variants: ordinary, universal, and multivariate Kriging. Finally, a practical case is presented to illustrate the usefulness of these concepts, aiming to model the interpolation of the pollutants SO2, PM10 and NOx in the Galician territory using the R libraries gstat and sm
This work provides an introduction to geostatistics, focusing particularly on the concept of the variogram, a structure that quantifies spatial dependence, and the Kriging spatial interpolation method. To this end, the theoretical foundations of spatial dependence are presented as the basis for the development of the variogram, including both its experimental and theoretical conception, as well as the different existing models and the reasons why it may fail to properly model spatial dependence. Next, the theory behind the Kriging interpolation method is introduced, along with its different variants: ordinary, universal, and multivariate Kriging. Finally, a practical case is presented to illustrate the usefulness of these concepts, aiming to model the interpolation of the pollutants SO2, PM10 and NOx in the Galician territory using the R libraries gstat and sm
Direction
FEBRERO BANDE, MANUEL (Tutorships)
FEBRERO BANDE, MANUEL (Tutorships)
Court
CABADA FERNANDEZ, ALBERTO (Chairman)
BORRAJO GARCIA, MARIA ISABEL (Secretary)
MUÑOZ SOLA, RAFAEL (Member)
CABADA FERNANDEZ, ALBERTO (Chairman)
BORRAJO GARCIA, MARIA ISABEL (Secretary)
MUÑOZ SOLA, RAFAEL (Member)
A Journey Through Fermat's Last Theorem
Authorship
L.L.R.
Bachelor of Mathematics
L.L.R.
Bachelor of Mathematics
Defense date
07.03.2025 10:00
07.03.2025 10:00
Summary
This work presents an approach to some of the ideas that were developed during the exploration and eventual proof of Fermat’s Last Theorem. Starting with the cases n=3 and n=4, we will explore certain aspects related to the arithmetic of number fields. The central part of the report focuses on the study of the proof of Fermat’s Theorem for regular primes. Finally, we offer a brief overview of some of the ideas developed in the 20th century around the concept of modularity, which ultimately led to Andrew Wiles’ proof of the result in the 1990s.
This work presents an approach to some of the ideas that were developed during the exploration and eventual proof of Fermat’s Last Theorem. Starting with the cases n=3 and n=4, we will explore certain aspects related to the arithmetic of number fields. The central part of the report focuses on the study of the proof of Fermat’s Theorem for regular primes. Finally, we offer a brief overview of some of the ideas developed in the 20th century around the concept of modularity, which ultimately led to Andrew Wiles’ proof of the result in the 1990s.
Direction
RIVERO SALGADO, OSCAR (Tutorships)
RIVERO SALGADO, OSCAR (Tutorships)
Court
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
(Finitely) universal categories
Authorship
D.L.P.
Bachelor of Mathematics
D.L.P.
Bachelor of Mathematics
Defense date
07.03.2025 10:45
07.03.2025 10:45
Summary
One of the classical problems that has driven significant advances in algebra is the Inverse Galois Problem, proposed by Hilbert in 1892. Inspired by this problem and following a similar logical framework, the group realization problem emerged in the early 20th century, posing a seemingly simple question: given a category C and a group G, does there exist an object in C whose automorphism group is isomorphic to G? When this holds for all (finite) groups, the category is said to be (finitely) universal. One of the earliest breakthroughs in this area is due to R. Frucht, who in 1939 proved that the category of finite simple graphs is finitely universal. Since then, the problem has been studied in many categories and remains an active area of research in Algebra. The aim of this paper is to introduce the group realization problem, present the most relevant tools for its study and then apply these techniques to address, for the first time in the literature, the finite universality of the category of fusion rings - algebraic structures that naturally arise both in algebra and in certain theoretical physics contexts within the current research framework.
One of the classical problems that has driven significant advances in algebra is the Inverse Galois Problem, proposed by Hilbert in 1892. Inspired by this problem and following a similar logical framework, the group realization problem emerged in the early 20th century, posing a seemingly simple question: given a category C and a group G, does there exist an object in C whose automorphism group is isomorphic to G? When this holds for all (finite) groups, the category is said to be (finitely) universal. One of the earliest breakthroughs in this area is due to R. Frucht, who in 1939 proved that the category of finite simple graphs is finitely universal. Since then, the problem has been studied in many categories and remains an active area of research in Algebra. The aim of this paper is to introduce the group realization problem, present the most relevant tools for its study and then apply these techniques to address, for the first time in the literature, the finite universality of the category of fusion rings - algebraic structures that naturally arise both in algebra and in certain theoretical physics contexts within the current research framework.
Direction
COSTOYA RAMOS, MARIA CRISTINA (Tutorships)
COSTOYA RAMOS, MARIA CRISTINA (Tutorships)
Court
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
Quantum algorithms
Authorship
Y.M.R.
Bachelor of Mathematics
Y.M.R.
Bachelor of Mathematics
Defense date
07.03.2025 11:30
07.03.2025 11:30
Summary
In this Bachelor’s Final Project, two of the most important quantum algorithms are presented, Shor’s factorization algorithm and Grover’s search algorithm. Previously, the bases of this type of computation are detailed, from a purely mathematical perspective. More specifically, the concepts of p-qubit and the operations that can be performed between them through the use of logic gates are developed. Also, subroutines such as the calculation of the quantum Fourier transform and the quantum phase algorithm are presented. Both of them are indispensible in many quantum algorithms. Lastly, an in-depth study of Shor’s and Grover’s algorithm is carried out, accompanied by geometric interpretations and examples to facilitate their understanding.
In this Bachelor’s Final Project, two of the most important quantum algorithms are presented, Shor’s factorization algorithm and Grover’s search algorithm. Previously, the bases of this type of computation are detailed, from a purely mathematical perspective. More specifically, the concepts of p-qubit and the operations that can be performed between them through the use of logic gates are developed. Also, subroutines such as the calculation of the quantum Fourier transform and the quantum phase algorithm are presented. Both of them are indispensible in many quantum algorithms. Lastly, an in-depth study of Shor’s and Grover’s algorithm is carried out, accompanied by geometric interpretations and examples to facilitate their understanding.
Direction
FERNANDEZ TOJO, FERNANDO ADRIAN (Tutorships)
FERNANDEZ TOJO, FERNANDO ADRIAN (Tutorships)
Court
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
Women mathematicians in history
Authorship
T.M.S.
Bachelor of Mathematics
T.M.S.
Bachelor of Mathematics
Defense date
07.03.2025 11:30
07.03.2025 11:30
Summary
In this Bachelor's Thesis, we will study the presence of female figures in mathematics throughout history, as well as their contributions to the different fields of this science, with a special focus on the work of five women from recent centuries. To achieve this objective, the project will begin with an introduction to the history of mathematics, highlighting its most relevant aspects while also providing an overview of the sociopolitical context of those different eras. This will be followed by an enumeration of the reasons behind the silencing of women in this history, many of which will later be reflected in the biographies that make up the chapters of this work. In each chapter, a brief biography of the selected women will be presented, followed by an in-depth study of their work and the importance of their contributions in the development of science. Among the works analyzed are Observaciones de pasos por dos verticales, the first astronomy PhD Thesis written by a Spanish woman; the Cauchy Kovalevskaya Theorem, of great utility in the context of partial differential equations; the definition of a Noetherian ring, fundamental in commutative algebra; and the branch and bound method, essential in the field of operations research.
In this Bachelor's Thesis, we will study the presence of female figures in mathematics throughout history, as well as their contributions to the different fields of this science, with a special focus on the work of five women from recent centuries. To achieve this objective, the project will begin with an introduction to the history of mathematics, highlighting its most relevant aspects while also providing an overview of the sociopolitical context of those different eras. This will be followed by an enumeration of the reasons behind the silencing of women in this history, many of which will later be reflected in the biographies that make up the chapters of this work. In each chapter, a brief biography of the selected women will be presented, followed by an in-depth study of their work and the importance of their contributions in the development of science. Among the works analyzed are Observaciones de pasos por dos verticales, the first astronomy PhD Thesis written by a Spanish woman; the Cauchy Kovalevskaya Theorem, of great utility in the context of partial differential equations; the definition of a Noetherian ring, fundamental in commutative algebra; and the branch and bound method, essential in the field of operations research.
Direction
Diz Pita, Érika (Tutorships)
DAVILA PENA, LAURA (Co-tutorships)
Diz Pita, Érika (Tutorships)
DAVILA PENA, LAURA (Co-tutorships)
Court
QUINTELA ESTEVEZ, PEREGRINA (Chairman)
TRINCHET SORIA, ROSA Mª (Secretary)
DIAZ RAMOS, JOSE CARLOS (Member)
QUINTELA ESTEVEZ, PEREGRINA (Chairman)
TRINCHET SORIA, ROSA Mª (Secretary)
DIAZ RAMOS, JOSE CARLOS (Member)
Dynamical Systems
Authorship
M.P.Q.
Bachelor of Mathematics
M.P.Q.
Bachelor of Mathematics
Defense date
07.03.2025 12:15
07.03.2025 12:15
Summary
A dynamical system is just a system of equations that varies over time: if it goes through real numbers, we will talk about continuous dynamical systems; when it takes integer values, they will be called discrete dynamical systems. The objective of this work is to make a theoretical introduction to dynamical systems in order to study the two existing particular cases later on. We will see how continuous dynamical systems can be considered equivalent to ordinary differential equations thanks to the Global Existence Theorem. Moreover, we will study the different types of attracting or repelling sets that define their phase portraits. On the other hand, we will approach discrete dynamical systems in a more graphic way, talking about periodic and hyperbolic points. Then, we will focus on a specific example (the quadratic family) and, to finish, we will introduce the concept of chaos.
A dynamical system is just a system of equations that varies over time: if it goes through real numbers, we will talk about continuous dynamical systems; when it takes integer values, they will be called discrete dynamical systems. The objective of this work is to make a theoretical introduction to dynamical systems in order to study the two existing particular cases later on. We will see how continuous dynamical systems can be considered equivalent to ordinary differential equations thanks to the Global Existence Theorem. Moreover, we will study the different types of attracting or repelling sets that define their phase portraits. On the other hand, we will approach discrete dynamical systems in a more graphic way, talking about periodic and hyperbolic points. Then, we will focus on a specific example (the quadratic family) and, to finish, we will introduce the concept of chaos.
Direction
LOPEZ SOMOZA, LUCIA (Tutorships)
LOPEZ SOMOZA, LUCIA (Tutorships)
Court
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
Topology of viral evolution.
Authorship
L.M.Q.T.
Bachelor of Mathematics
L.M.Q.T.
Bachelor of Mathematics
Defense date
02.12.2025 19:00
02.12.2025 19:00
Summary
In the last decades, several topological tools for data analysis in different areas have been developed. The present work aims to explain simplicial homology and persistent homology, and their application in biology as a method to study and predict the viral evolution, not very well known nor controlled. Specifically, we will focus on the flu virus (Influenza A) and the Human Immunodeficiency Virus (HIV), both for their prevalence and mortality rate in humans, as well as the disposition of its data and the suitability of the explained topological methods for its study.
In the last decades, several topological tools for data analysis in different areas have been developed. The present work aims to explain simplicial homology and persistent homology, and their application in biology as a method to study and predict the viral evolution, not very well known nor controlled. Specifically, we will focus on the flu virus (Influenza A) and the Human Immunodeficiency Virus (HIV), both for their prevalence and mortality rate in humans, as well as the disposition of its data and the suitability of the explained topological methods for its study.
Direction
Gómez Tato, Antonio M. (Tutorships)
Gómez Tato, Antonio M. (Tutorships)
Court
RODRIGUEZ CASAL, ALBERTO (Chairman)
ALONSO TARRIO, LEOVIGILDO (Secretary)
SALGADO SECO, MODESTO RAMON (Member)
RODRIGUEZ CASAL, ALBERTO (Chairman)
ALONSO TARRIO, LEOVIGILDO (Secretary)
SALGADO SECO, MODESTO RAMON (Member)
Spherical Regression
Authorship
R.R.B.
Bachelor of Mathematics
R.R.B.
Bachelor of Mathematics
Defense date
07.03.2025 12:00
07.03.2025 12:00
Summary
This project studies a regression model for spherical variables, in other words, those that are defined on the surface of a sphere. It begins with an introduction to the basic and necessary concepts and results of simple linear regression, multiple and nonlinear regression; and then begin to work with spherical data, incorporating the fundamental definitions. For this purpose, we will use graphical representations. We will also incorporate the most important distributions, with the von Mises-Fisher distribution being of particular interest, as it will be the one we will use in subsequent chapters. Once all the prior knowledge has been presented, we will put it into practice by conducting a simulation study using R. In this study, we introduce the rotation model for spherical regression, explaining some of its main properties and interpreting the results obtained. Finally, we'll discuss and explain the potential limitations of this project (exclusive use of simulated data, references to proofs, etc). We'll also discuss how this work could be extended, for example, by switching to a larger dimension or even by mentioning other well-known distribution models.
This project studies a regression model for spherical variables, in other words, those that are defined on the surface of a sphere. It begins with an introduction to the basic and necessary concepts and results of simple linear regression, multiple and nonlinear regression; and then begin to work with spherical data, incorporating the fundamental definitions. For this purpose, we will use graphical representations. We will also incorporate the most important distributions, with the von Mises-Fisher distribution being of particular interest, as it will be the one we will use in subsequent chapters. Once all the prior knowledge has been presented, we will put it into practice by conducting a simulation study using R. In this study, we introduce the rotation model for spherical regression, explaining some of its main properties and interpreting the results obtained. Finally, we'll discuss and explain the potential limitations of this project (exclusive use of simulated data, references to proofs, etc). We'll also discuss how this work could be extended, for example, by switching to a larger dimension or even by mentioning other well-known distribution models.
Direction
ALONSO PENA, MARIA (Tutorships)
ALONSO PENA, MARIA (Tutorships)
Court
ALONSO PENA, MARIA (Student’s tutor)
ALONSO PENA, MARIA (Student’s tutor)
Differentiation on normed spaces and minimization of functionals
Authorship
A.R.T.
Bachelor of Mathematics
A.R.T.
Bachelor of Mathematics
Defense date
07.03.2025 10:00
07.03.2025 10:00
Summary
The aim of this work is to expand the knowledge regarding differential calculus, encompasing a more generalized environment than the one considered in the bachelor degree. In the first part, we will establish the concept of Fréchet’s derivative, comparing it with that of Gâteau’s derivative, in order to study, from then on, numerous results based on both. We will use these ideas to work with the second-order and higher derivatives, which will allow us, in turn, to analyze the necessary and sufficient conditions for the existence of function extrema. In the second part, we will contextualize the calculus of variations and break down the process that will allow us to obtain the Euler-Lagrange equation, in order to apply it later to specific problems, such as the revolution surface with minimun area or the curve of fastest descent (or brachistochrone curve).
The aim of this work is to expand the knowledge regarding differential calculus, encompasing a more generalized environment than the one considered in the bachelor degree. In the first part, we will establish the concept of Fréchet’s derivative, comparing it with that of Gâteau’s derivative, in order to study, from then on, numerous results based on both. We will use these ideas to work with the second-order and higher derivatives, which will allow us, in turn, to analyze the necessary and sufficient conditions for the existence of function extrema. In the second part, we will contextualize the calculus of variations and break down the process that will allow us to obtain the Euler-Lagrange equation, in order to apply it later to specific problems, such as the revolution surface with minimun area or the curve of fastest descent (or brachistochrone curve).
Direction
Rodríguez López, Jorge (Tutorships)
Rodríguez López, Jorge (Tutorships)
Court
Rodríguez López, Jorge (Student’s tutor)
Rodríguez López, Jorge (Student’s tutor)
Time series analysis for maize crops in Galicia
Authorship
M.S.R.
Bachelor of Mathematics
M.S.R.
Bachelor of Mathematics
Defense date
07.02.2025 11:30
07.02.2025 11:30
Summary
Forage maize production in Galicia is key to the region’s primary sector, being one of the crops that employs the most workers due to its importance in cattle feed. In this study, we use real data on forage maize yield and various climatic variables provided by the Agricultural Research Center of Mabegondo to analyze their relationship through multiple regression models. In parallel, we examine the evolution of average temperature and precipitation in Galicia using time-series modeling techniques, with the aim of fitting statistically validated ARIMA models with time series. These models allow us to generate climate forecasts and interpret how changes in meteorological conditions may influence agricultural productivity in the medium and long term, within a context shaped by climate change.
Forage maize production in Galicia is key to the region’s primary sector, being one of the crops that employs the most workers due to its importance in cattle feed. In this study, we use real data on forage maize yield and various climatic variables provided by the Agricultural Research Center of Mabegondo to analyze their relationship through multiple regression models. In parallel, we examine the evolution of average temperature and precipitation in Galicia using time-series modeling techniques, with the aim of fitting statistically validated ARIMA models with time series. These models allow us to generate climate forecasts and interpret how changes in meteorological conditions may influence agricultural productivity in the medium and long term, within a context shaped by climate change.
Direction
SAAVEDRA NIEVES, PAULA (Tutorships)
SAAVEDRA NIEVES, PAULA (Tutorships)
Court
SAAVEDRA NIEVES, PAULA (Student’s tutor)
SAAVEDRA NIEVES, PAULA (Student’s tutor)
Regression and Classification with Random Forests
Authorship
A.S.G.
Bachelor of Mathematics
A.S.G.
Bachelor of Mathematics
Defense date
07.02.2025 13:30
07.02.2025 13:30
Summary
The Random Forests method is a machine learning technique, used for both regression and classification. It is based on the construction and combination of multiple decision trees, thus obtaining an improvement in the accuracy of predictions and reducing the risk of overfitting. This algorithm employs a method known as Bagging, based on generating multiple random subsets of data and training each tree on each of them. In addition, at each node, a subset is randomly selected, providing the model with diversity. Finally, in the case of classification, the final prediction is obtained through majority voting among the trees, whereas in regression, the final prediction is computed as the average of the individual tree predictions. Random Forests is a robust and versatile model that reduces the impact of outliers and allows us to work effectively with high-dimensional data.
The Random Forests method is a machine learning technique, used for both regression and classification. It is based on the construction and combination of multiple decision trees, thus obtaining an improvement in the accuracy of predictions and reducing the risk of overfitting. This algorithm employs a method known as Bagging, based on generating multiple random subsets of data and training each tree on each of them. In addition, at each node, a subset is randomly selected, providing the model with diversity. Finally, in the case of classification, the final prediction is obtained through majority voting among the trees, whereas in regression, the final prediction is computed as the average of the individual tree predictions. Random Forests is a robust and versatile model that reduces the impact of outliers and allows us to work effectively with high-dimensional data.
Direction
FEBRERO BANDE, MANUEL (Tutorships)
FEBRERO BANDE, MANUEL (Tutorships)
Court
CRUJEIRAS CASAIS, ROSA MARÍA (Chairman)
PENA BRAGE, FRANCISCO JOSE (Secretary)
DOMINGUEZ VAZQUEZ, MIGUEL (Member)
CRUJEIRAS CASAIS, ROSA MARÍA (Chairman)
PENA BRAGE, FRANCISCO JOSE (Secretary)
DOMINGUEZ VAZQUEZ, MIGUEL (Member)
Fundamental Integral Calculos Theorem for Lebesgue's integral
Authorship
J.S.S.
Bachelor of Mathematics
J.S.S.
Bachelor of Mathematics
Defense date
07.02.2025 11:00
07.02.2025 11:00
Summary
This final degree proyect will consist on enunciate and proof the Fundamental Integral Calculos Theorem for Lebesgue's integral, what will finally be done at chapter 7. Before that, in order to reach that point with the necessary theoretical basis, there will be estudied properties, definition, and variated results about absolute contiuos functions (chapter 6); also bounded variation functions (chapter 3) and, particullary, their differentiation (chapter 5), for what we will use as a tool the Dini Derivatives (chapter 4). It will be also necessary for advancing on these concepts' comprehension certain results such as Vitali's Covering Theorem, which will be explained in chapter 2; and of course, an important basis of Measuring Theory, explained on the first chapter, the preliminaries, for not leaving us until the end of the proyect.
This final degree proyect will consist on enunciate and proof the Fundamental Integral Calculos Theorem for Lebesgue's integral, what will finally be done at chapter 7. Before that, in order to reach that point with the necessary theoretical basis, there will be estudied properties, definition, and variated results about absolute contiuos functions (chapter 6); also bounded variation functions (chapter 3) and, particullary, their differentiation (chapter 5), for what we will use as a tool the Dini Derivatives (chapter 4). It will be also necessary for advancing on these concepts' comprehension certain results such as Vitali's Covering Theorem, which will be explained in chapter 2; and of course, an important basis of Measuring Theory, explained on the first chapter, the preliminaries, for not leaving us until the end of the proyect.
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)
Statistic models for dertermining the thickness of the Greenland ice sheet
Authorship
V.S.S.P.
Bachelor of Mathematics
V.S.S.P.
Bachelor of Mathematics
Defense date
02.12.2025 17:15
02.12.2025 17:15
Summary
Over the course of this dissertation, we present and explore the use of Kriging models in fitting a solution to the geostatistical problem of estimating the total size of the Greenland ice sheet, both in volume and extention. In addition, we evaluate the prediction performance of these and other models, comparing their precision in relation to their respective complexity.
Over the course of this dissertation, we present and explore the use of Kriging models in fitting a solution to the geostatistical problem of estimating the total size of the Greenland ice sheet, both in volume and extention. In addition, we evaluate the prediction performance of these and other models, comparing their precision in relation to their respective complexity.
Direction
FEBRERO BANDE, MANUEL (Tutorships)
FEBRERO BANDE, MANUEL (Tutorships)
Court
CABADA FERNANDEZ, ALBERTO (Chairman)
BORRAJO GARCIA, MARIA ISABEL (Secretary)
MUÑOZ SOLA, RAFAEL (Member)
CABADA FERNANDEZ, ALBERTO (Chairman)
BORRAJO GARCIA, MARIA ISABEL (Secretary)
MUÑOZ SOLA, RAFAEL (Member)
Statistics for forensic genetics.
Authorship
A.V.C.
Bachelor of Mathematics
A.V.C.
Bachelor of Mathematics
Defense date
07.02.2025 14:15
07.02.2025 14:15
Summary
Kinship analysis is a key area within forensic genetics. This paper explores the mathematical foundations necessary to address such cases, with a particular focus on the standard trio and standard duo scenarios. These cases are presented in detail, following an introduction to the essential genetic concepts required to understand the terminology. The work also develops the probabilistic and statistical notions needed to model these situations, enabling their resolution through various estimation techniques and hypothesis testing approaches. This rigorous mathematical framework supports objective and consistent interpretation of genetic results, ensuring their scientific validity. Additionally, the R software packages Familias and paramlink are introduced as practical tools for implementing the analytical methods discussed, with illustrative examples provided throughout. The overall aim is to emphasize the critical role of a strong mathematical foundation in the application of forensic genetics and other scientific disciplines.
Kinship analysis is a key area within forensic genetics. This paper explores the mathematical foundations necessary to address such cases, with a particular focus on the standard trio and standard duo scenarios. These cases are presented in detail, following an introduction to the essential genetic concepts required to understand the terminology. The work also develops the probabilistic and statistical notions needed to model these situations, enabling their resolution through various estimation techniques and hypothesis testing approaches. This rigorous mathematical framework supports objective and consistent interpretation of genetic results, ensuring their scientific validity. Additionally, the R software packages Familias and paramlink are introduced as practical tools for implementing the analytical methods discussed, with illustrative examples provided throughout. The overall aim is to emphasize the critical role of a strong mathematical foundation in the application of forensic genetics and other scientific disciplines.
Direction
CASARES DE CAL, MARIA ANGELES (Tutorships)
CASARES DE CAL, MARIA ANGELES (Tutorships)
Court
CRUJEIRAS CASAIS, ROSA MARÍA (Chairman)
PENA BRAGE, FRANCISCO JOSE (Secretary)
DOMINGUEZ VAZQUEZ, MIGUEL (Member)
CRUJEIRAS CASAIS, ROSA MARÍA (Chairman)
PENA BRAGE, FRANCISCO JOSE (Secretary)
DOMINGUEZ VAZQUEZ, MIGUEL (Member)
Ordinary Differential Equations with Applications to Economics
Authorship
C.V.F.
Bachelor of Mathematics
C.V.F.
Bachelor of Mathematics
Defense date
02.13.2025 13:00
02.13.2025 13:00
Summary
This Bachelor's Thesis focuses on the analysis of certain ordinary differential equations applied to the study of economic models. Throughout this work, five key models are addressed: the Phillips curve, the Harrod-Domar model, the Solow-Swan model, the Goodwin model, and the dynamic Leontief model, which allow for the description of fundamental economic phenomena, from the relationship between unemployment and wages to the interaction among productive sectors. Each model has been contextualized, solved, and analyzed in detail, highlighting both its contributions and its limitations, with the aim of better understanding its usefulness and exploring possible improvements for its application in modern economies.
This Bachelor's Thesis focuses on the analysis of certain ordinary differential equations applied to the study of economic models. Throughout this work, five key models are addressed: the Phillips curve, the Harrod-Domar model, the Solow-Swan model, the Goodwin model, and the dynamic Leontief model, which allow for the description of fundamental economic phenomena, from the relationship between unemployment and wages to the interaction among productive sectors. Each model has been contextualized, solved, and analyzed in detail, highlighting both its contributions and its limitations, with the aim of better understanding its usefulness and exploring possible improvements for its application in modern economies.
Direction
Rodríguez López, Rosana (Tutorships)
Rodríguez López, Rosana (Tutorships)
Court
Rodríguez López, Rosana (Student’s tutor)
Rodríguez López, Rosana (Student’s tutor)