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Works presented

Review and comparison of dimensionality reduction techniques.
Authorship
A.A.M.
Bachelor of Mathematics
Defense date
07.16.2025 11:45
Summary
Dimensionality reduction techniques are fundamental procedures in statistics for simplifying datasets while losing the least amount of information as possible. The objective of this work is to thoroughly review some of these methods, with a particular focus on one of the most widely used techniques: Principal Component Analysis. Additionally, we will discuss other more recent nonlinear techniques, which have been gaining popularity in recent years. Finally, to emphasize the practical importance of these techniques, we will present application examples with real datasets to explore the challenges of interpretation and processing they entail.
Direction
PATEIRO LOPEZ, BEATRIZ (Tutorships)
Court
CRUJEIRAS CASAIS, ROSA MARÍA (Chairman)
PENA BRAGE, FRANCISCO JOSE (Secretary)
DOMINGUEZ VAZQUEZ, MIGUEL (Member)
The Foundations of Mathematics and Set Theory: A Review
Authorship
L.A.C.
Bachelor of Mathematics
Defense date
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.
Direction
ALONSO TARRIO, LEOVIGILDO (Tutorships)
Court
ALONSO TARRIO, LEOVIGILDO (Student’s tutor)
Spectral graph theory
Authorship
I.A.A.
Bachelor of Mathematics
Defense date
07.15.2025 19:00
Summary
Given a graph, it is possible to associate a matrix that encodes the connections between its vertices, known as the adjacency matrix; another particularly relevant matrix is the so-called Laplacian matrix. The study of the eigenvalues of these matrices provides important information about the structure of the graph. The aim of this work is to formulate some results in spectral graph theory and to study a relevant theorem in the area, such as Kirchhoff’s theorem, which allows one to compute the number of spanning trees of a graph from the determinant of a submatrix of the Laplacian.
Direction
RIVERO SALGADO, OSCAR (Tutorships)
Court
RIVERO SALGADO, OSCAR (Student’s tutor)
Curves in Lorentz-Minkowski space
Authorship
F.A.D.
Bachelor of Mathematics
Defense date
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.
Direction
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)
Diophantine Equations in the Mathematical Olympiads
Authorship
M.A.R.
Bachelor of Mathematics
Defense date
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.
Direction
GAGO COUSO, FELIPE (Tutorships)
RIVERO SALGADO, OSCAR (Co-tutorships)
Court
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
Defense date
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.
Direction
DOMINGUEZ VAZQUEZ, MIGUEL (Tutorships)
Court
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
Defense date
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.
Direction
CRUJEIRAS CASAIS, ROSA MARÍA (Tutorships)
Court
Majadas Soto, José Javier (Chairman)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Secretary)
CASARES DE CAL, MARIA ANGELES (Member)
Goodness-of-fit tests.
Authorship
E.A.A.
Bachelor of Mathematics
Defense date
07.15.2025 10:00
Summary
The development of this project focuses on the theoretical and practical study of the chi-squared test, which is used to test certain hypotheses on a data sample. Firstly, the historical context of the test is presented, including its origin and evolution. In the theoretical study, the test is formalized by defining its corresponding test statistic and presenting the main results related to it, including those concerning the asymptotic convergence of the statistic. The case in which the null hypothesis is simple is distinguished from the case in which it belongs to a parametric family. In the practical part, simulations of the test are carried out in its different cases using the R software, with the aim of analyzing and characterizing its actual behavior. The calibration of the test (cases in which the null hypothesis holds) and its power (cases in which the null hypothesis does not hold) are studied. Certain practical recommendations mentioned during the degree are also tested.
Direction
RODRIGUEZ CASAL, ALBERTO (Tutorships)
Bolón Rodríguez, Diego (Co-tutorships)
Court
Bolón Rodríguez, Diego (Student’s tutor)
RODRIGUEZ CASAL, ALBERTO (Student’s tutor)
The Riemann mapping theorem
Authorship
P.A.G.
Double bachelor degree in Mathematics and Physics
Defense date
07.16.2025 16:40
Summary
This document addresses the Riemann mapping theorem, an essential result in complex analysis that establishes the existence of conformal mappings between simply connected sets and the unit disk. To this end, the necessary theoretical framework is introduced, beginning with a brief introduction to the theorem, along with some fundamental definitions and results on holomorphic functions. Later, the focus shifts to Möbius transformations, a key tool in this work that, together with Schwarz’s lemma, plays a fundamental role in characterizing the biholomorphic automorphisms of the unit disk. This theoretical development concludes with the study of certain properties of the space of holomorphic functions, thereby providing a foundation for giving a rigorous proof of the theorem. Finally, the importance of the theorem is highlighted by presenting some relevant applications in other scientific fields, such as fluid mechanics in physics, and an algorithm is provided to approximately compute the mapping described by the theorem.
Direction
CAO LABORA, DANIEL (Tutorships)
Court
LOPEZ POUSO, RODRIGO (Chairman)
PEON NIETO, ANA (Secretary)
SEOANE MARTINEZ, MARIA LUISA (Member)
Theoretical-computational study of criticality in Ising-type models
Authorship
P.A.G.
Double bachelor degree in Mathematics and Physics
Defense date
07.17.2025 09:30
Summary
This document addresses the Ising model, widely used in the study of phase transitions, from both theoretical and computational perspectives. It briefly introduces concepts related to critical phenomena, followed by a description of the model and some well-known theoretical developments and results in one- and two-dimensional lattices with nearest-neighbor interactions. A computational study is then carried out, introducing the Metropolis and Wolff algorithms, and demonstrating their ergodicity and compliance with the detailed balance equation. These two algorithms are compared, showing that the Wolff algorithm achieves faster convergence near the critical point. Using a custom-developed code, phase transitions are characterized in systems ranging from one to four dimensions with nearest-neighbor interactions, and the model is implemented on complex small-world networks, both one- and two-dimensional.
Direction
MENDEZ MORALES, TRINIDAD (Tutorships)
Montes Campos, Hadrián (Co-tutorships)
Court
ZAS ARREGUI, ENRIQUE (Chairman)
GARCIA FEAL, XABIER (Secretary)
FONDADO FONDADO, ALFONSO (Member)
Mathematical modelling of optimal dosage in drug administration
Authorship
I.A.O.
Bachelor of Mathematics
Defense date
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.
Direction
QUINTELA ESTEVEZ, PEREGRINA (Tutorships)
Court
CRUJEIRAS CASAIS, ROSA MARÍA (Chairman)
PENA BRAGE, FRANCISCO JOSE (Secretary)
DOMINGUEZ VAZQUEZ, MIGUEL (Member)
Clifford algebras and spin groups
Authorship
D.A.D.A.
Double bachelor degree in Mathematics and Physics
Defense date
07.16.2025 10:00
Summary
The aim of this work is to be a starting point for the study of Spin groups through the formalism of Clifford algebras. In order to do so, we first define and construct these algebras through tensor algebras and we study their main properties as superalgebras, their definition in terms of generators and relations and their connection with exterior algebra. Afterwards, we perform a first classification of the low-dimensional Clifford real algebras by using the Z_2-graded tensor product before proceeding to the complete classification of said algebras ---in the real and complex case---supported by several isomorphisms proved in this work, together with the Bott periodicity theorem. Afterwards, we pass on to defining the Spin group as a subgroup of the units of the Clifford algebra and, after studying the latter’s actions on the whole algebra, we check using this same action that the Spin groups are a double cover of the special orthogonal group SO(n). We also study some additional properties of these groups and classify some of the low dimensional cases.
Direction
DIAZ RAMOS, JOSE CARLOS (Tutorships)
Lorenzo Naveiro, Juan Manel (Co-tutorships)
Court
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
Analysis of the B+ to mumutaunu decay from a theoretical and experimental point of view
Authorship
D.A.D.A.
Double bachelor degree in Mathematics and Physics
Defense date
07.17.2025 09:30
Summary
This work aims to be a preliminary study, both from a theoretical and an experimental point of view, of the decay B+ to mumutaunu in the context of the LHCb experiment. First, a theoretical development based on bibliographic sources is carried out, performing a similar calculation with the goal of understanding the underlying mechanisms and adapting them to the case of interest; the ultimate objective of this analysis is to obtain an initial reference numerical value for the branching ratio (BR) of this decay. Next, we design an analysis aiming to detect a signal peak for this decay or to set an upper limit on the BR; to this end, we address the problem of missing information due to the undetectable neutrino by introducing the corrected mass. We present the data selection methods, the elimination of combinatorial background using machine learning tools, and a study of the ability to distinguish a signal peak through simulations and fits, along with statistical tools such as Wilks’ theorem. Finally, we present the progress achieved in this initial analysis and conclude with the work to be developed for a future extension of this study.
Direction
CID VIDAL, XABIER (Tutorships)
Fernández Gómez, Miguel (Co-tutorships)
Court
ZAS ARREGUI, ENRIQUE (Chairman)
GARCIA FEAL, XABIER (Secretary)
FONDADO FONDADO, ALFONSO (Member)
Analysis and application of Dijkstra's algorithm in route optimisation: in searth of the shortest path
Authorship
A.B.V.
Double bachelor degree in Mathematics and Physics
Defense date
07.15.2025 09:10
Summary
In this project, we start off from basic concepts regarding flow problems in networks and then looked at more specific ones that would help us achieve our main goal, which is to explain different algorithms for solving the shortest path problem. In the first chapter we will introduce concepts from graph theory and explain important notions for the following chapters, such as the definition of node, arc or cost. In addition, we will introduce minimun cost flow problems joint with their mathematical formulations. Finally, we will analyze a very important property when solving this type of problem: unimodularity, along with some properties that make solving them easier. In the second chapter we will focus on the shortest path problem, describing that there are different types of algorithms for solving it with particular emphasis on Dijkstra's algorithm. We will also provide some implentations of this and other algorithms that can help solve the problem more efficiently. Additionally, the explanations will be accompanied by examples to make the algorithms easier to understand. Finally, the last chapter presents a practical application of shortest path computation within the Spanish railway network. It includes a detailed description of the Python implementation, the data used, and a graphical representation of the computed routes.
Direction
GONZALEZ RUEDA, ANGEL MANUEL (Tutorships)
Court
Majadas Soto, José Javier (Chairman)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Secretary)
CASARES DE CAL, MARIA ANGELES (Member)
Baroclinic instability and its sensitivity to vertical temperature and wind profiles in atmospheric layers
Authorship
A.B.V.
Double bachelor degree in Mathematics and Physics
Defense date
07.17.2025 09:30
Summary
This project aims to study baroclinic instability in the atmosphere through theoretical analysis and numerical simulations. In the theoretical section, the key factors influencing this phenomenon will be explained, such as vertical wind shear, thermal gradients, and atmospheric stratification, as well as the role of planetary rotation and the Coriolis force. Next, numerical simulations will be carried out to analyze the impact of variations in temperature and wind profiles on the growth of baroclinic disturbances. The results will allow for an assessment of how these atmospheric conditions affect the evolution of baroclinic waves, comparing the simulations with theoretical predictions. Finally, the agreement between theory and simulations will be discussed, highlighting the conditions that favor the development of baroclinic instability and its potential applications in weather forecasting.
Direction
MIGUEZ MACHO, GONZALO (Tutorships)
CRESPO OTERO, ALFREDO (Co-tutorships)
Court
ZAS ARREGUI, ENRIQUE (Chairman)
GARCIA FEAL, XABIER (Secretary)
FONDADO FONDADO, ALFONSO (Member)
Application of ODEs to biological models
Authorship
C.B.M.
Bachelor of Mathematics
Defense date
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.
Direction
Rodríguez López, Jorge (Tutorships)
Court
Rodríguez López, Jorge (Student’s tutor)
Brouwer fixed point theorem
Authorship
A.B.M.
Bachelor of Mathematics
Defense date
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.
Direction
Rodríguez López, Jorge (Tutorships)
Court
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
A review of scheduling problems and applications
Authorship
S.B.G.
Bachelor of Mathematics
Defense date
07.15.2025 09:55
Summary
Scheduling problems involve the efficient allocation of limited resources to a set of tasks, considering specific time constraints and optimality criteria. In the first chapter, notation used across the paper will be detailed. The second and third chapters review different variants of the problem, presenting in each case the corresponding algorithm to find the optimal solution, along with the discussion of certain associated theoretical properties. The fourth chapter includes the implementation of these algorithms in R code. Finally, in the fifth chapter, conclusions will be drawn, highlighting the relevance of these problems both in everyday live as well as professional contexts.
Direction
GONZALEZ RUEDA, ANGEL MANUEL (Tutorships)
Court
Majadas Soto, José Javier (Chairman)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Secretary)
CASARES DE CAL, MARIA ANGELES (Member)
Symmetries and integrating factors in the solution of first-order ordinary differential equations.
Authorship
A.C.M.
Bachelor of Mathematics
Defense date
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.
Direction
BUEDO FERNANDEZ, SEBASTIAN (Tutorships)
SANMARTIN LOPEZ, VICTOR (Co-tutorships)
Court
BUEDO FERNANDEZ, SEBASTIAN (Student’s tutor)
SANMARTIN LOPEZ, VICTOR (Student’s tutor)
Population dynamics
Authorship
A.C.P.
Bachelor of Mathematics
Defense date
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.
Direction
Diz Pita, Érika (Tutorships)
Court
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
Defense date
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.
Direction
COSTOYA RAMOS, MARIA CRISTINA (Tutorships)
Court
COSTOYA RAMOS, MARIA CRISTINA (Student’s tutor)
Estimation of level sets to study the Velutina wasp
Authorship
J.C.P.
Bachelor of Mathematics
Defense date
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.
Direction
SAAVEDRA NIEVES, PAULA (Tutorships)
ALONSO PENA, MARIA (Co-tutorships)
Court
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
Defense date
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.
Direction
ALONSO TARRIO, LEOVIGILDO (Tutorships)
Court
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
Comparison of search technologies in evaluation benchmarks for the study of misinformation in the context of health-related queries
Authorship
X.C.A.
Double Bachelor's Degree in Informatics Engineering and Mathematics
Defense date
07.18.2025 11:00
Summary
The presence of misinformation in web search results related to health is a concerning issue both socially and scientifically, as it can negatively influence users' decision-making and lead to serious health consequences. This phenomenon, which gained visibility because of the COVID-19 pandemic, is one of the current research areas in Information Retrieval (IR), around which this bachelor thesis is structured. The main goal of the project is to explore how to distinguish relevant and accurate documents from harmful ones, given a specific search intent. With this goal in mind, a study structured along three research lines is proposed. First, a systematic analysis is conducted on the performance of state-of-the-art search systems with respect to this task. Second, a new technique is designed, implemented, and evaluated, based on Large Language Models (LLMs), for generating alternative versions of user queries in such a way that the variants promote the retrieval of relevant and accurate documents, while discouraging the retrieval of harmful ones. Lastly, a study is presented on predicting the presence of health misinformation in search results, for which techniques from related fields are tested, and a new LLM-based predictor, specifically tailored for this task, is designed, implemented, and evaluated. The findings of this work support the potential of LLMs in the field of IR, as they manage to improve the effectiveness of state-of-the-art search systems. Moreover, the project addresses the literature gap with regard to misinformation prediction in queries, while also showing the superior capability of LLMs for this task compared to more general techniques. Parts of the contributions from this project have been accepted for publication at the ACM SIGIR 2025 conference.
Direction
Losada Carril, David Enrique (Tutorships)
FERNANDEZ PICHEL, MARCOS (Co-tutorships)
Court
MOSQUERA GONZALEZ, ANTONIO (Chairman)
TOBAR QUINTANAR, ALEJANDRO JOSE (Secretary)
QUESADA BARRIUSO, PABLO (Member)
Quantum Computing. Principles and Applications
Authorship
A.C.S.
Double bachelor degree in Mathematics and Physics
Defense date
07.16.2025 17:20
Summary
This report studies the foundations of Quantum Computing in a fully mathematical manner, abstracting from the underlying real physical systems. Firstly, we study the fundamentals of Quantum Physics, exploring concepts and properties of Hilbert spaces over the field of complex numbers. Next, we define the concepts of qubit and p-qubit, along with the quantum logic gates that operate on them. Finally, we develop some important algorithms with specific applications that show the relevance of this kind of logic. The aim of this document is to serve as an introduction, based on concepts covered by the Mathematics undergraduate program, to the world of Quantum Computing, without requiring any previous knowledge of Physics, in doing so, it seeks to present easier access to understanding or developing quantum algorithms.
Direction
FERNANDEZ FERNANDEZ, FRANCISCO JAVIER (Tutorships)
Court
LOPEZ POUSO, RODRIGO (Chairman)
PEON NIETO, ANA (Secretary)
SEOANE MARTINEZ, MARIA LUISA (Member)
Evaluation of the Effective Temperature through Quantum Simulation
Authorship
A.C.S.
Double bachelor degree in Mathematics and Physics
Defense date
07.16.2025 17:00
Summary
An important line of research in the field of quantum computing is focused on noise detection and mitigation. In this report, we will focus on methods for measuring the effective temperature, a quantity used for the estimation of the residual population of the excited state of a qubit, due to thermal fluctuations in the device. First, we study the physical systems of transmon superconducting qubits and their dispersive readout process, which are the ones available in QMIO. Based on them, we develop a stochastic simulation on which the measurement methods can be validated. Finally, we study a method for measuring the effective temperature based on e-f Rabi oscillations, validating its hypotheses and obtaining results through simulation.
Direction
MAS SOLE, JAVIER (Tutorships)
Gómez Tato, Andrés (Co-tutorships)
Court
MIRA PEREZ, JORGE (Chairman)
CID VIDAL, XABIER (Secretary)
MOSQUEIRA REY, JESUS MANUEL (Member)
Optimization and equity
Authorship
N.C.A.
Bachelor of Mathematics
Defense date
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.
Direction
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)
High-precision quadrature
Authorship
J.C.S.
Bachelor of Mathematics
Defense date
07.16.2025 17:30
Summary
In this work, we address the study of numerical methods for the approximate calculation of definite integrals through quadrature formulas, fundamental techniques when the exact value of the integral is unknown. Specifically, we focus on polynomial interpolatory formulas, which approximate the true value by integrating an interpolation polynomial. We examine in depth three main quadrature methods. First of all, the Gauss formulas, known for their high accuracy using a few nodes. Then, the Romberg method, which combines the composite trapezoidal rule with Richardson extrapolation. Finally, we study the use of endpoint corrections for the composite trapezoidal and Simpson’s rules. For each method, we explore its theoretical formulation, error behavior, and the conditions required to achieve accuracy. Additionally, we provide examples and tables illustrating the variation of the approximation error for each method.
Direction
López Pouso, Óscar (Tutorships)
BARRAL RODIÑO, PATRICIA (Co-tutorships)
Court
CRUJEIRAS CASAIS, ROSA MARÍA (Chairman)
PENA BRAGE, FRANCISCO JOSE (Secretary)
DOMINGUEZ VAZQUEZ, MIGUEL (Member)
Sudoku: An Application of Operations Research
Authorship
L.D.G.
Bachelor of Mathematics
Defense date
07.15.2025 10:40
Summary
This work focuses on the mathematical and computational study of Sudoku solving, starting from its connection with Latin squares. Its existence and enumeration are analyzed, laying the groundwork for a solid theoretical framework to understand the underlying structure of Sudoku. The game is then described in a historical and formal context, examining its rules and properties, the total number of possible boards, as well as the problem of the minimum number of clues required to guarantee a unique solution. The most commonly used manual solving techniques, both basic and advanced, are also described. The main part of this work explores several solution methodologies using mathematical programming, including linear programming, backtracking algorithms, evolutionary methods (such as genetic algorithms), and simulated annealing, along with graph-based models. Furthermore, the analysis is extended to some Sudoku variants. Finally, a comparative evaluation of the computational efficiency of all proposed methodologies is presented, based on implementations in the R programming language.
Direction
SAAVEDRA NIEVES, ALEJANDRO (Tutorships)
DAVILA PENA, LAURA (Co-tutorships)
Court
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
Defense date
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.
Direction
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)
The fast Fourier transform
Authorship
P.D.V.
Double Bachelor's Degree in Informatics Engineering and Mathematics
Defense date
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.
Direction
LOPEZ SOMOZA, LUCIA (Tutorships)
Court
QUINTELA ESTEVEZ, PEREGRINA (Chairman)
TRINCHET SORIA, ROSA Mª (Secretary)
DIAZ RAMOS, JOSE CARLOS (Member)
Study and optimization of Octree models for neighbourhood searches in 3D point clouds
Authorship
P.D.V.
Double Bachelor's Degree in Informatics Engineering and Mathematics
Defense date
07.18.2025 12:30
Summary
This work presents an approach for efficient neighbour search in 3D point clouds, particularly those acquired via LiDAR technology. The proposed method involves a spatial reordering of the point cloud based on Space-Filling Curves (SFCs), coupled with efficient implementations of Octree-based search algorithms. The study explores how Morton and Hilbert SFCs can optimize the organization of three-dimensional data, improving the performance of neighbourhood queries and the construction of spatial structures over the cloud. Several Octree variants are proposed and evaluated, analyzing their impact on the efficiency and scalability of the proposed method. Experimental results demonstrate that SFC reordering, along with specialized Octree-based search algorithms, significantly enhances spatial data access, providing a robust solution for applications requiring fast processing of large 3D point cloud datasets. Part of this work was presented at the XXXV Jornadas de Paralelismo organized by SARTECO.
Direction
Fernández Rivera, Francisco (Tutorships)
YERMO GARCIA, MIGUEL (Co-tutorships)
Court
Pardo López, Xosé Manuel (Chairman)
SUAREZ GAREA, JORGE ALBERTO (Secretary)
CARIÑENA AMIGO, MARIA PURIFICACION (Member)
Statistical methods in bioinformatics
Authorship
C.D.R.
Bachelor of Mathematics
Defense date
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.
Direction
CASARES DE CAL, MARIA ANGELES (Tutorships)
Court
CRUJEIRAS CASAIS, ROSA MARÍA (Chairman)
PENA BRAGE, FRANCISCO JOSE (Secretary)
DOMINGUEZ VAZQUEZ, MIGUEL (Member)
Crossed modules of groups
Authorship
M.E.L.
Bachelor of Mathematics
Defense date
07.16.2025 18:00
Summary
The mathematical concept of a crossed module was introduced by J. H. C. Whitehead in 1949 with the aim of modeling spaces of homotopy 2-type. Crossed modules exist for a wide variety of algebraic structures; in this work, we will consider exclusively the group structure. In this context, a crossed module of groups is a group homomorphism \mu: M \to N together with an action of the group N on the group M by automorphisms, such that two fundamental conditions are satisfied: the equivariance of \mu and the Peiffer identity. Crossed modules have different algebraic properties and are categorically equivalent to structures such as Cat1 -grupos, strict 2-groups, and strict categorical groups. Describing these properties and equivalences will be the main objective of this work.
Direction
LADRA GONZALEZ, MANUEL EULOGIO (Tutorships)
RAMOS PEREZ, BRAIS (Co-tutorships)
Court
LOPEZ POUSO, RODRIGO (Chairman)
PEON NIETO, ANA (Secretary)
SEOANE MARTINEZ, MARIA LUISA (Member)
Abstract Measure and Integration: Unveiling the Radon Nikodym Theorem
Authorship
F.E.L.
Bachelor of Mathematics
Defense date
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.
Direction
TRINCHET SORIA, ROSA Mª (Tutorships)
Court
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
Automatic Differentiation through computational graphs
Authorship
L.E.G.
Bachelor of Mathematics
Defense date
07.17.2025 10:00
Summary
Automatic Differentiation is a differentiation technique that combines the ideas of symbolic and numerical differentiation with the goal of accurately evaluating derivatives of functions at given points. Two main approaches are typically used: forward mode and reverse mode. Although both will be discussed, the primary focus of this work is on the reverse mode, due to its advantages in scenarios involving functions with many input variables. The study explores the benefits of reverse mode for gradient computation, its extension to multiple independent variables and higher-order derivatives, and the construction of the computational graph required for its implementation in MATLAB. Finally, practical applications are examined, particularly in the context of gradient-based optimization processes, such as those used in training neural networks.
Direction
PENA BRAGE, FRANCISCO JOSE (Tutorships)
RODRIGUEZ GARCIA, JERONIMO (Co-tutorships)
Court
LOPEZ POUSO, RODRIGO (Chairman)
PEON NIETO, ANA (Secretary)
SEOANE MARTINEZ, MARIA LUISA (Member)
Introduction to Markov Chains and its Applications to Luck Games.
Authorship
A.F.B.
Bachelor of Mathematics
Defense date
07.16.2025 12:30
Summary
Markov chains are processes in which the probability of future events does not depend on the past, but only on the current situation. These particular chains are very useful in several fields such as biology, chemistry, physics and computer science. In this project we will analyze Markov chains and apply the studied concepts to games based on luck, like Blackjack or Snakes and Ladders. These games, besides depending on chance, are often influenced by the players’ strategies; therefore, understanding the theory behind them can help the players making better decisions.
Direction
AMEIJEIRAS ALONSO, JOSE (Tutorships)
Bolón Rodríguez, Diego (Co-tutorships)
Court
CRUJEIRAS CASAIS, ROSA MARÍA (Chairman)
PENA BRAGE, FRANCISCO JOSE (Secretary)
DOMINGUEZ VAZQUEZ, MIGUEL (Member)
Exploring OEDs
Authorship
D.F.C.
Bachelor of Mathematics
Defense date
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.
Direction
López Pouso, Óscar (Tutorships)
Court
QUINTELA ESTEVEZ, PEREGRINA (Chairman)
TRINCHET SORIA, ROSA Mª (Secretary)
DIAZ RAMOS, JOSE CARLOS (Member)
Metaheuristics of the TSP: A didactic and computational tour.
Authorship
E.F.D.S.
Double Bachelor's Degree in Informatics Engineering and Mathematics
Defense date
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.
Direction
CASAS MENDEZ, BALBINA VIRGINIA (Tutorships)
Court
RODRIGUEZ CASAL, ALBERTO (Chairman)
ALONSO TARRIO, LEOVIGILDO (Secretary)
SALGADO SECO, MODESTO RAMON (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
Defense date
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.
Direction
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)
Computational aspects of goodness-of-fit tests for linear regression models
Authorship
G.F.F.
Bachelor of Mathematics
Defense date
07.16.2025 13:15
Summary
This work will present an introduction to the multiple linear regression model and to para- metric regression models, as well as to the classical assumptions typically imposed on them. One of these assumptions is the linearity, respectively the parametric form, of the model; which will lead us to focus on the goodness-of-fit test introduced by Stute (1997), which is highly useful in order to check if a certain model follows some specified parametric form. Due to the complexity of estimating the distribution of the proposed test statistic, we will present a bootstrap approxi- mation, more specifically the wild bootstrap about the residuals of the considered model in order to carry out the calibration of the test in practice. Later, we will implement this test in R and we will carry out a simulation study with the aim of assessing the good performance of the test empirically; verifying that it respects the nominal level under the null hypothesis and it shows a good power, that is, the test will reject the null hypothesis when we consider models under the alternative hypothesis. Finally, we will present a real data application which will allow us to illustrate the utility of the presented procedure in practice.
Direction
CONDE AMBOAGE, MERCEDES (Tutorships)
Court
CRUJEIRAS CASAIS, ROSA MARÍA (Chairman)
PENA BRAGE, FRANCISCO JOSE (Secretary)
DOMINGUEZ VAZQUEZ, MIGUEL (Member)
Game theory and logistics in the fishing sector
Authorship
U.F.G.
Bachelor of Mathematics
Defense date
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.
Direction
CASAS MENDEZ, BALBINA VIRGINIA (Tutorships)
Court
RODRIGUEZ CASAL, ALBERTO (Chairman)
ALONSO TARRIO, LEOVIGILDO (Secretary)
SALGADO SECO, MODESTO RAMON (Member)
Chebotarev's density theorem
Authorship
G.F.L.
Bachelor of Mathematics
Defense date
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.
Direction
RIVERO SALGADO, OSCAR (Tutorships)
Court
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
Defense date
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.
Direction
Ferrín González, José Luis (Tutorships)
Court
Ferrín González, José Luis (Student’s tutor)
Memetic algorithms for the MC-TTRP
Authorship
N.F.O.
Double Bachelor's Degree in Informatics Engineering and Mathematics
Defense date
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.
Direction
CASAS MENDEZ, BALBINA VIRGINIA (Tutorships)
Court
Majadas Soto, José Javier (Chairman)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Secretary)
CASARES DE CAL, MARIA ANGELES (Member)
Reconstruction of Phylogenetic trees using Quantum Computing
Authorship
N.F.O.
Double Bachelor's Degree in Informatics Engineering and Mathematics
Defense date
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.
Direction
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)
Introduction to the Black-Scholes model
Authorship
M.F.P.
Bachelor of Mathematics
Defense date
07.16.2025 16:00
Summary
Published in 1973, the Black-Scholes model represented a significant breakthrough in the theory of financial option pricing, as it provided an explicit solution for the theoretical price of European options. The objective of this paper is to present an introduction to this model. The document begins by explaining basic financial concepts and then addresses the necessary mathematical foundations on which the model is based, particularly highlighting the geometric Brownian motion and Itô’s Lemma. Subsequently, with the support of these tools, the Black-Scholes differential equation is formally derived, followed by the presentation of the explicit formula for European options. A practical case is also included to illustrate the usefulness of this model in real-world situations. Finally, the paper presents a series of factors that limit the model’s applicability in today’s financial markets, and discusses possible extensions and modifications designed to adapt the model to these realities.
Direction
AMEIJEIRAS ALONSO, JOSE (Tutorships)
GINZO VILLAMAYOR, MARIA JOSE (Co-tutorships)
Court
CRUJEIRAS CASAIS, ROSA MARÍA (Chairman)
PENA BRAGE, FRANCISCO JOSE (Secretary)
DOMINGUEZ VAZQUEZ, MIGUEL (Member)
Stadistical analysis of frequency tables
Authorship
X.F.S.
Bachelor of Mathematics
Defense date
07.16.2025 16:45
Summary
In this paper, a review on statistical methods for the analysis of frequency tables is presented. First, the probability distributions for discrete variables most relevant to frequency tables are reviewed: Binomial, Multinomial , Poisson and Hypergeometric. Next, a review of fundamental concepts related to Statistical Inference and its applications to contingency tables is carried out. Moreover bidimensional contingency tables are studied, starting with $2\times2$ tables and continuing with general $I\times J$ tables. The chi-squared test, the likelihood ratio test and Fisher's exact test will be reviewed. Not only are ilustrative examples presented, but also very visual graphical methods such as barplots, fourfold plots, sieve diagrams and mosaic plots. The paper ends with the study of multidimensional contingency tables, which involves the study of partial and marginal tables, culminating in the Mantel-Haenszel test and Breslow-Day-Tarone test. Odds-ratios are revelaed to be a very useful tool for the analysis of association forms in a contingency table. The paper if accompanied of examples, as well as \verb*|R| code for the implementation of the considered statistical methods.
Direction
SANCHEZ SELLERO, CESAR ANDRES (Tutorships)
Court
CRUJEIRAS CASAIS, ROSA MARÍA (Chairman)
PENA BRAGE, FRANCISCO JOSE (Secretary)
DOMINGUEZ VAZQUEZ, MIGUEL (Member)
Comparison of classical and machine learning methodologies in time series analysis
Authorship
A.F.M.
Bachelor of Mathematics
Defense date
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.
Direction
PATEIRO LOPEZ, BEATRIZ (Tutorships)
Court
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
Defense date
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.
Direction
CRUJEIRAS CASAIS, ROSA MARÍA (Tutorships)
Court
Majadas Soto, José Javier (Chairman)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Secretary)
CASARES DE CAL, MARIA ANGELES (Member)
Analysis and Comparison of Concept Drift Detection and Adaptation Techniques
Authorship
F.F.M.
Double Bachelor's Degree in Informatics Engineering and Mathematics
Defense date
07.17.2025 10:00
Summary
This work compares different techniques for the detection and adaptation of Concept Drift, a phenomenon that arises in dynamic and non-stationary environments where the statistical relationships between the model's variables change over time, affecting its performance in some cases. The main objective is to evaluate various drift detectors present in the Literature, along with the analysis of several adaptation techniques following detection. The study is carried out in an experimental setting using artificial datasets that simulate different types of drift, applied to a classification model. Classical algorithms from the RiverML library are analyzed, in scenarios with and without prior knowledge of the environment. In the first case, a hyperparameter optimization is applied using a novel method based on Random Search. Regarding KSWIN, one of the evaluated algorithms, a modification developed as a complementary part of the Bachelor's Thesis in Mathematics is incorporated. This modification introduces statistical techniques such as multiple hypothesis testing and the Benjamini Hochberg correction to improve the detection process, as well as a system for identifying the type of drift through non-parametric inference, which is considered innovative in the Literature. The results highlight the strengths and limitations of both the detectors and the adaptation strategies analyzed. While some algorithms, such as HDDMW, show good overall performance, the choice of the most appropriate detector largely depends on the use case and the type of drift present. Likewise, adaptation based on mini batches offers solid performance compared to periodic retraining. In addition, the proposed modification of KSWIN outperforms other detectors in terms of balancing false positives and false negatives during the detection process, and establishes a solid foundation for the method of drift type identification.
Direction
MERA PEREZ, DAVID (Tutorships)
Court
LADRA GONZALEZ, MANUEL EULOGIO (Chairman)
LOPEZ FANDIÑO, JAVIER (Secretary)
VIDAL AGUIAR, JUAN CARLOS (Member)
Special functions in solving partial differential equations
Authorship
C.F.S.
Double bachelor degree in Mathematics and Physics
Defense date
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.
Direction
LOPEZ POUSO, RODRIGO (Tutorships)
Court
QUINTELA ESTEVEZ, PEREGRINA (Chairman)
TRINCHET SORIA, ROSA Mª (Secretary)
DIAZ RAMOS, JOSE CARLOS (Member)
Undergraduate dissertation
Authorship
C.F.S.
Double bachelor degree in Mathematics and Physics
Defense date
07.17.2025 10:00
Summary
The van der Waals material 1T CrSe2 consists of strongly covalently bonded layers in plane which are held together out of plane by weak van der Waals forces. These systems exhibit an intermediate behavior between localized electrons with well defined magnetic moments at each atomic site and itinerant electrons delocalized across conduction bands without a net moment assignable to each atom To investigate its magnetic ordering and electronic instabilities we performed ab initio calculations using the WIEN2k code employing the FP LAPW plus local orbitals method and the PBE generalized gradient approximation. First the rigid 1 por 1 monolayer was characterized its ground state is ferromagnetic and its density of states DOS is dominated at the Fermi level by Cr t2g orbitals indicating electronic instabilities To capture the charge density waves CDWs associated with Peierls modes we constructed 2 por 2 and sqrt3 por sqrt3 supercells. Relaxation of the 2 por 2 cell reveals Cr tetramers and a partial band gap characteristic of a unidirectional Peierls mechanism. The sqrt3 por sqrt3 reconstruction simultaneously trimerizes all three t2g orbitals opening pseudogaps above and below the Fermi level and leaving only a single residual flat band Because conventional DFT tends to underestimate the local Coulomb interaction we applied the LDA U correction to the Cr d orbitals. For moderate values of U the pseudogap opening is enhanced and the ferromagnetic CDW phase is further stabilized whereas an excessive U induces subband reordering that reintroduces DOS peaks at EF
Direction
PARDO CASTRO, VICTOR (Tutorships)
Court
MIRA PEREZ, JORGE (Chairman)
CID VIDAL, XABIER (Secretary)
MOSQUEIRA REY, JESUS MANUEL (Member)
Statistical Modeling of Sports Data
Authorship
A.G.A.
Double bachelor degree in Mathematics and Physics
Defense date
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.
Direction
RODRIGUEZ CASAL, ALBERTO (Tutorships)
Court
CRUJEIRAS CASAIS, ROSA MARÍA (Chairman)
PENA BRAGE, FRANCISCO JOSE (Secretary)
DOMINGUEZ VAZQUEZ, MIGUEL (Member)
Automatic Segmentation of Preclinical Magnetic Resonance Imaging
Authorship
A.G.A.
Double bachelor degree in Mathematics and Physics
Defense date
07.16.2025 09:00
Summary
The main objective of this work is to develop an automated system for the detection and segmentation of glioblastomas, an aggressive type of brain tumor, in animal models (mice and rats) using preclinical magnetic resonance imaging (MRI). For this purpose, supervised learning techniques are employed to automatically segment the glioblastoma and accurately calculate its volume. Chapter 3 presents the U-Net model, a convolutional neural network specialized in medical segmentation tasks. This model is trained using MRI images of mice with manual segmentations (ground truth), with the aim of automatically locating the glioblastoma in new images. The impact of different hyperparameter configurations is explored, and the models performance is evaluated using specific segmentation metrics. Chapter 4 focuses on the development of predictive models to estimate tumor volume in mice, based on the same images used in the previous chapter. The process includes the extraction of radiomic features, their comparison with those previously obtained by researcher Sara Ortega, the selection of the most relevant variables, and the training of regression models. Again, various combinations of hyperparameters are analyzed to study their influence on prediction quality. In Chapter 5, the previous procedure is replicated, this time using rat images. In addition, an analysis is conducted on the impact of increasing the sample size on the performance of the predictive models, training the algorithms with different amounts of data. Finally, the possibility of building a model to predict animal survival from the images was considered. However, a preliminary analysis revealed that the available data were insufficient to obtain reliable predictions, so this possibility was proposed as future research.
Direction
IGLESIAS REY, RAMON (Tutorships)
Court
Pérez Muñuzuri, Vicente (Chairman)
GALLAS TORREIRA, ABRAHAM ANTONIO (Secretary)
RODRIGUEZ GONZALEZ, JUAN ANTONIO (Member)
Shallow water equations: analytic and numeric solutions
Authorship
M.G.C.
Bachelor of Mathematics
Defense date
07.17.2025 10:40
Summary
Shallow water equations are of notable relevance in hydraulics and environmental sciences. Having access to a simple mathematical model capable of describing reality with precision comes with substantial benefits. Following this motivation, this work is targeted at introducing the shallow water equations and together with their analytical and numeric solutions. We begin studying the mathematical properties of the corresponding system of hyperbolic conservation laws, so that we can next obtain some particular solutions and finally approach the design and validation of numerical methods for solving the equations. The Matlab code developed is based in finite volume schemes of first order and allows the approximate solution of the classic Riemann problem.
Direction
VAZQUEZ CENDON, MARIA ELENA (Tutorships)
BUSTO ULLOA, SARAY (Co-tutorships)
Court
LOPEZ POUSO, RODRIGO (Chairman)
PEON NIETO, ANA (Secretary)
SEOANE MARTINEZ, MARIA LUISA (Member)
Introduction to Minimal Surfaces
Authorship
J.G.G.
Bachelor of Mathematics
Defense date
07.16.2025 10:45
Summary
The investigation of the theory of minimal surfaces, which still remains fully in force, began in the 18th century with valuable contributions from illustrious mathematicians such as L. Euler or J. Lagrange. We begin this work by collecting the first approaches and definitions, sometimes from different perspectives -such as physic, geometric or analytic- of minimal surfaces. One of the most significant advances in this subject occurred between 1861 and 1864 with the incorporation of Complex Analysis to the study of minimal surfaces, culminating in what we know today as the Weierstrass-Enneper Representation, which we also discuss in this text. Finally, we move on to study one of the main tools in the context of surface theory: the Maximum Principle. This result allows us to compare and distinguish surfaces through the analysis of their respective mean curvatures.
Direction
SANMARTIN LOPEZ, VICTOR (Tutorships)
Court
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
What to do with a sample and a computer? Applying the bootstrap methodology to the calculation of confidence intervals
Authorship
C.G.G.
Bachelor of Mathematics
Defense date
07.16.2025 09:15
Summary
In Statistics, one of the main objectives is to estimate the value of a parameter that characterizes a population. Likewise, it is of interest to study the properties of such estimates, such as the uncertainty associated with the estimator. This is the aim of the jackknife method, from which the bootstrap method emerged as an improved version. Uniform bootstrap, which is the simplest version, is useful in situations where the population distribution is completely unknown and only the sample is available. However, when certain properties of the underlying distribution are known, other variants can be used that yield better results than the uniform one. This work presents the different bootstrap procedures, as well as the associated algorithms, in combination with the Monte Carlo method. Currently, the applications of the bootstrap are numerous. One of the most relevant is the construction of confidence intervals for various parameters. In this work, we compare, based on their coverage error, the method based on the asymptotic Normal distribution and three bootstrap variants: the basic percentile method, the percentil-t, and the symmetrized percentilt. These three procedures, whose construction is based on the pivotal method, differ on the definition of the pivot used. Notably, the absolute studentized pivot used in the symmetrized percentil-t provides more accurate confidence intervals.
Direction
BORRAJO GARCIA, MARIA ISABEL (Tutorships)
Court
CRUJEIRAS CASAIS, ROSA MARÍA (Chairman)
PENA BRAGE, FRANCISCO JOSE (Secretary)
DOMINGUEZ VAZQUEZ, MIGUEL (Member)
The Functional Linear Regression Model
Authorship
L.G.R.
Bachelor of Mathematics
Defense date
07.16.2025 10:00
Summary
Functional data analysis is the branch of statistics concerned with the study of functions as probabilistic objects, rather than scalars or vectors, as is the case in the traditional statistical framework. The aim of this work is to provide an introduction to the statistics of functional data, establishing its theoretical foundations with regression as a centric point. Two functional linear regression models are formulated, depending on whether the response variable is scalar or functional, and a range of estimation methods as well as significance testing procedures applicable to each model are presented.
Direction
GONZALEZ MANTEIGA, WENCESLAO (Tutorships)
Court
CRUJEIRAS CASAIS, ROSA MARÍA (Chairman)
PENA BRAGE, FRANCISCO JOSE (Secretary)
DOMINGUEZ VAZQUEZ, MIGUEL (Member)
Bankruptcy problems
Authorship
H.G.S.
Bachelor of Mathematics
Defense date
07.15.2025 11:25
Summary
The motivation stems from a concrete and highly significant problem that had to be resolved years ago and left the world paralyzed: the COVID crisis. The initial distribution of masks was crucial in limiting the virus’s spread, forcing different states to face an allocation problem of a scarce resource across their territories. This work examines the bankruptcy problem, a classic model in game theory that can also be approached axiomatically, representing situations in which a group of agents claims more than what is available from a limited resource. This type of problem arises in numerous social and economic contexts where it is necessary to distribute a scarce good among different interested parties, such as geopolitics, business economics, or public asset management. The main objective of this work is to conduct a bibliographic review of the different proposed solutions for these situations, studying their mathematical properties, conceptual justifications and potential applications, whether using existing game theory models or developing specific ones. Various rules are analyzed, such as proportional division, the Talmud solution, the Shapley value, priority rules, and geometric rules, among others, comparing their behavior in terms of fairness, consistency, and efficiency, all applied to the practical case.
Direction
SAAVEDRA NIEVES, ALEJANDRO (Tutorships)
Court
Majadas Soto, José Javier (Chairman)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Secretary)
CASARES DE CAL, MARIA ANGELES (Member)
L functions of elliptic curves and modular forms
Authorship
J.G.C.
Double bachelor degree in Mathematics and Physics
Defense date
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.
Direction
RIVERO SALGADO, OSCAR (Tutorships)
Court
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
Shortcuts to adiabaticity in simple quantum systems
Authorship
J.G.C.
Double bachelor degree in Mathematics and Physics
Defense date
07.16.2025 09:00
Summary
Given a quantum system in which we want to modify some control parameter without changes in its energy level, the adiabatic theorem allows us to do it. In exchange, the speed at which this change is made must be sufficiently small. This leads to a greater vulnerability of the system to effects such as noise or decoherence that result in the lost of its quantum properties. In order to solve this difficulty, the methods known as shortcuts to adiabaticity are designed. They allow us to accelerate the system preparation time without having to depend on the adiabatic theorem. In particular, in this work we will focus on the method known as Counterdiabatic Driving (CD) based on the introduction of an additional term in the hamiltonian of the system that avoids transitions between instantaneous eigenstates. First, we will motivate and give the precise statement of the adiabatic theorem. Then, we will introduce the CD formalism and apply it to several simple systems. Finally, a brief introduction to others shortcuts to adiabaticity and their limits will be given.
Direction
Vázquez Ramallo, Alfonso (Tutorships)
Court
Pérez Muñuzuri, Vicente (Chairman)
GALLAS TORREIRA, ABRAHAM ANTONIO (Secretary)
RODRIGUEZ GONZALEZ, JUAN ANTONIO (Member)
Generating functions in the calculation of power indices.
Authorship
C.G.F.
Bachelor of Mathematics
Defense date
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.
Direction
SAAVEDRA NIEVES, ALEJANDRO (Tutorships)
DAVILA PENA, LAURA (Co-tutorships)
Court
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
Defense date
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.
Direction
BUEDO FERNANDEZ, SEBASTIAN (Tutorships)
LOIS PRADOS, CRISTINA (Co-tutorships)
Court
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
Defense date
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.
Direction
AMEIJEIRAS ALONSO, JOSE (Tutorships)
Court
Majadas Soto, José Javier (Chairman)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Secretary)
CASARES DE CAL, MARIA ANGELES (Member)
Robust vision-language models for small objects
Authorship
N.G.S.D.V.
Double Bachelor's Degree in Informatics Engineering and Mathematics
Defense date
07.17.2025 11:00
Summary
This work focuses on the optimization of vision-language models (VLMs) applied to visual question answering (VQA) tasks on videos containing small objects, a scenario that presents significant computational challenges due to the large number of irrelevant visual tokens generated. To address this issue, a method is proposed based on the integration of a detector that identifies relevant visual regions in the video frames, allowing the filtering of background-associated visual tokens generated by the vision module (ViT) before being processed by the language model (LLM). Experimental results show that this filtering effectively eliminates a large proportion of visual tokens, leading to a notable reduction in the computational complexity of the language model and, consequently, a decrease in the overall system complexity, without compromising performance. Furthermore, an improvement in the LLM’s execution time is observed, contributing to greater efficiency in textual processing. However, the overall inference time is still influenced by the ViT, which remains the main bottleneck due to high-resolution image processing, as well as by the additional computational cost introduced by the detector. This work validates the use of filtering techniques as an effective strategy to improve the efficiency of VLMs and opens new lines of research aimed at optimizing visual processing and exploring lighter-weight detectors.
Direction
MUCIENTES MOLINA, MANUEL FELIPE (Tutorships)
CORES COSTA, DANIEL (Co-tutorships)
Court
LADRA GONZALEZ, MANUEL EULOGIO (Chairman)
LOPEZ FANDIÑO, JAVIER (Secretary)
VIDAL AGUIAR, JUAN CARLOS (Member)
Combinatorial Optimization and Heuristic Algorithms
Authorship
D.G.G.
Bachelor of Mathematics
Defense date
07.15.2025 12:10
Summary
The work will primarily consist of an in-depth analysis of combinatorial optimization problems, particularly three of them: the network flow problem, the knapsack problem, and the traveling salesman problem. The latter will include a practical simulation applied to real-world scenarios, allowing for an analysis of the efficiency of its main heuristics.
Direction
CASAS MENDEZ, BALBINA VIRGINIA (Tutorships)
Court
Majadas Soto, José Javier (Chairman)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Secretary)
CASARES DE CAL, MARIA ANGELES (Member)
A review of finite difference algorithms in parabolic models
Authorship
Y.G.G.
Bachelor of Mathematics
Defense date
07.15.2025 10:30
Summary
In this work, we will present different algorithms using the finite difference method to solve parabolic partial differential equations (PDEs). These numerical methods will be oriented towards simulating an application in bio-heat transfer. We will begin with the formalization of the parabolic model, followed by a focus on a specific case of significant relevance in bioengineering: the Pennes equation. The finite difference method will be introduced as a discretization tool, along with formulations in alternative coordinate systems. Next, four classical algorithms will be developed and analyzed: the explicit method, the implicit method, the Crank-Nicolson method, and the method of lines. For each of these, we will provide a detailed formulation, numerical analysis, MATLAB implementation, and validation. Finally, the model will be applied to a real case of breast tissue with cyst, validating the obtained results and discussing their practical usefulness.
Direction
QUINTELA ESTEVEZ, PEREGRINA (Tutorships)
Court
QUINTELA ESTEVEZ, PEREGRINA (Student’s tutor)
Mathematical models for the regulation of cell volume
Authorship
J.A.G.B.
Bachelor of Mathematics
Defense date
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.
Direction
CABADA FERNANDEZ, ALBERTO (Tutorships)
Court
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
Defense date
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.
Direction
VIAÑO REY, JUAN MANUEL (Tutorships)
Court
QUINTELA ESTEVEZ, PEREGRINA (Chairman)
TRINCHET SORIA, ROSA Mª (Secretary)
DIAZ RAMOS, JOSE CARLOS (Member)
Spatial data modeling
Authorship
N.H.C.
Bachelor of Mathematics
Defense date
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.
Direction
RODRIGUEZ CASAL, ALBERTO (Tutorships)
Court
Majadas Soto, José Javier (Chairman)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Secretary)
CASARES DE CAL, MARIA ANGELES (Member)
Nilpotent centers at polynomial systems.
Authorship
A.I.V.
Bachelor of Mathematics
Defense date
07.17.2025 11:20
Summary
In this bachelor's final thesis, conducted within the field of Mathematical Analysis, we tackle the typification of global centers at bidimensional polynomial dynamic systems, which is an open problem. After gathering a collection of fundamental results about dynamic systems theory, local and global centers are introduced by a classification attending to their linearization. Owing to the problem’s complexity, we confine the study to find nilpotent centers at systems with quintic homogeneous polynomials and under certain symmetries. Hamiltonian systems were considered too. Studying global centers requires a analysis into orbit behaviour near the infinity of the euclidean plane. To face it, we draw on the Poincaré compactification. This technique projects the vector field defined in R2 onto a sphere, where points at infinity are identified with its equator line. Finally, in order to understand the local structure of some singularities, we use blow up transformations that expand the singular point along a whole straight line, making them easier to manage.
Direction
OTERO ESPINAR, MARIA VICTORIA (Tutorships)
Diz Pita, Érika (Co-tutorships)
Court
LOPEZ POUSO, RODRIGO (Chairman)
PEON NIETO, ANA (Secretary)
SEOANE MARTINEZ, MARIA LUISA (Member)
The Newton and discretized Newton methods for nonlinear systems of equations.
Authorship
A.J.P.
Bachelor of Mathematics
Defense date
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.
Direction
MUÑOZ SOLA, RAFAEL (Tutorships)
Court
MUÑOZ SOLA, RAFAEL (Student’s tutor)
Introduction to time series models
Authorship
I.L.C.
Double Bachelor's Degree in Informatics Engineering and Mathematics
Defense date
07.17.2025 09:10
Summary
The study of time series is a fundamental tool in the analysis of data that evolve over time, allowing us to model and predict phenomena in fields such as economics, meteorology, engineering or social sciences. This Final Degree Thesis deals with the analysis of time series from a classical statistical perspective, considering that the observed series are realisations of stochastic processes. Autoregressive (AR), moving average (MA), mixed (ARMA), integrated (ARIMA) and seasonal (SARIMA) models are presented in detail, explaining their mathematical formulation, parameter estimation methods and associated forecasting techniques. A methodology for time series analysis is also proposed, which will be used in the analysis of two real time series. In addition, a simulation study is included to illustrate and evaluate the estimation and forecasting processes of the models.
Direction
CRUJEIRAS CASAIS, ROSA MARÍA (Tutorships)
Court
Majadas Soto, José Javier (Chairman)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Secretary)
CASARES DE CAL, MARIA ANGELES (Member)
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
Defense date
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.
Direction
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)
Permutation groups in the classification of idempotent evolution algebras
Authorship
A.L.P.
Bachelor of Mathematics
Defense date
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.
Direction
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)
Theory of lower and upper solutions applied to ordinary differential equations.
Authorship
A.L.R.
Bachelor of Mathematics
Defense date
07.15.2025 16:00
Summary
The method of lower and upper solutions has been studied since the last decade of the 19th century due to its effectiveness in the analysis of nonlinear differential equations. This approach allows the establishment of the existence of solutions without the need to solve the problem explicitly. In this work, we first present the application of the lower and upper solution method to second-order ordinary differential equations with periodic conditions. Various existence results are proved, which also ensure that the solution or solutions are bounded by the lower and the upper solution. Next, we describe the monotone method, a constructive technique based on Green’s functions and the appropriate choice of a pair of lower and upper solutions. Unlike the previous case, this method is presented to ordinary differential equations of arbitrary order and under general boundary conditions, not necessarily periodic. The procedure generates monotone sequences that converge to the extremal solutions of the problem.
Direction
CABADA FERNANDEZ, ALBERTO (Tutorships)
Court
LOPEZ POUSO, RODRIGO (Chairman)
PEON NIETO, ANA (Secretary)
SEOANE MARTINEZ, MARIA LUISA (Member)
Mathematical Methods of Artificial Inteligence
Authorship
P.L.P.
Double Bachelor's Degree in Informatics Engineering and Mathematics
Defense date
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.
Direction
Nieto Roig, Juan José (Tutorships)
Court
Nieto Roig, Juan José (Student’s tutor)
Exploitation of Large Language Models for Automatic Annotation of Credibility
Authorship
P.L.P.
Double Bachelor's Degree in Informatics Engineering and Mathematics
Defense date
07.17.2025 11:30
Summary
This thesis focuses on the application of state-of-the-art language models, such as GPT-4 and LlaMa 3, to the labeling of health-related documents in the context of the TREC project. The main objective is to evaluate the possibility of replacing annotations made by human experts with labels generated by LLMs. The field of Information Retrieval requires large labeled datasets, which are created by expert human annotators, a process that is costly in terms of both time and money. If it can be proven that these human annotations can be replaced by automatically generated ones, this would represent a major advance in the generation of high-quality datasets. In this work, we will label the same web documents that were labeled by humans; this will allow us to analyze discrepancies between human labels and those generated by the models. We also studied the effect that the instructions given to the language model have on the accuracy of the labeling. We based our methodology on a publication by Microsoft researchers, in which the relevance of each document is labeled. The results obtained in thomas2024 were very satisfactory and were implemented in Bing due to their improvement in time, cost, and quality compared to crowdsourced labelers. Our results represent an advance over this previous publication, as we carry out labeling of more complex features such as medical correctness and credibility. The results obtained in our work were in some cases very similar to those of Paul Thomas et al, so we consider them positive enough to replace human labels.
Direction
Losada Carril, David Enrique (Tutorships)
FERNANDEZ PICHEL, MARCOS (Co-tutorships)
Court
LADRA GONZALEZ, MANUEL EULOGIO (Chairman)
LOPEZ FANDIÑO, JAVIER (Secretary)
VIDAL AGUIAR, JUAN CARLOS (Member)
Determination of spatial dependence using variograms.
Authorship
C.L.L.
Bachelor of Mathematics
Defense date
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
Direction
FEBRERO BANDE, MANUEL (Tutorships)
Court
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
Defense date
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.
Direction
RIVERO SALGADO, OSCAR (Tutorships)
Court
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
The problem of comparing populations: what to do when the t- test can not be applied
Authorship
A.L.P.
Bachelor of Mathematics
Defense date
07.16.2025 10:45
Summary
Comparing populations is a topic within Statistical Inference which allows detecting differences among several groups. This document starts with an introduction to this issue, followed by a chapter dedicated to presenting basic statistical concepts which are needed to correctly understand the full document. Among these concepts, hypothesis testing stands out. After that, the problem of comparing population is carefully explained in order to compare means, variances and distribution functions. On the one hand, we will work with two populations or with more than two populations and, on the other hand, parametric methods are distinguished from non-parametric methods. The first methods assume normality, with the t-test being part of this group, whereas the second methods do not make any kind of assumption about the target populations. Two tests will be added in order to check the normality hypothesis: the Lilliefors test and the Shapiro-Wilk test. Other tests to check for variance equality will be included as well, such as the Levene’s test. Finally, the theoretical tools explained previously will be used to analyze a dataset on forest fires in Galicia in 2006 using software R.
Direction
BORRAJO GARCIA, MARIA ISABEL (Tutorships)
Court
CRUJEIRAS CASAIS, ROSA MARÍA (Chairman)
PENA BRAGE, FRANCISCO JOSE (Secretary)
DOMINGUEZ VAZQUEZ, MIGUEL (Member)
(Finitely) universal categories
Authorship
D.L.P.
Bachelor of Mathematics
Defense date
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.
Direction
COSTOYA RAMOS, MARIA CRISTINA (Tutorships)
Court
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
Sylow Theory and low order groups
Authorship
I.M.H.
Bachelor of Mathematics
Defense date
07.15.2025 12:00
Summary
The objective of this work is to study finite groups of order less than or equal to 30, grouping them into different categories according to their structure in prime factors, fundamentally using techniques such as Sylow Theory. We will also introduce new tools to complete our study such as the semi-direct product. Normally, the analysis of the different groups will be done for the general case and then for the specific orders, but, if the task is complicated in the general case, the study will focus only on the specific case, of which an in-depth study will be attempted.
Direction
FERNANDEZ RODRIGUEZ, ROSA Mª (Tutorships)
COSTOYA RAMOS, MARIA CRISTINA (Co-tutorships)
Court
COSTOYA RAMOS, MARIA CRISTINA (Student’s tutor)
FERNANDEZ RODRIGUEZ, ROSA Mª (Student’s tutor)
Quantum algorithms
Authorship
Y.M.R.
Bachelor of Mathematics
Defense date
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.
Direction
FERNANDEZ TOJO, FERNANDO ADRIAN (Tutorships)
Court
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
Women mathematicians in history
Authorship
T.M.S.
Bachelor of Mathematics
Defense date
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.
Direction
Diz Pita, Érika (Tutorships)
DAVILA PENA, LAURA (Co-tutorships)
Court
QUINTELA ESTEVEZ, PEREGRINA (Chairman)
TRINCHET SORIA, ROSA Mª (Secretary)
DIAZ RAMOS, JOSE CARLOS (Member)
Undergraduate dissertation
Authorship
D.M.M.
Bachelor of Mathematics
Defense date
07.15.2025 16:40
Summary
A research project focused on algebraic curves is the first step toward entering the realm of varieties. The geometric perspective from which they are studied serves as an important motivation for certain foundations and theorems in commutative algebra. Using the knowledge acquired in introductory courses on ring theory and number theory, one can draft the foundations of algebraic geometry. After introducing the notions of affine and projective varieties, it is common to present the concept of a singular point. The main idea to study this points is to find which rings provide local information about the curves and, furthermore, whether it is possible to build examples of simple curves that preserve the properties of more complex ones.
Direction
ALONSO TARRIO, LEOVIGILDO (Tutorships)
ALVITE PAZO, RAUL (Co-tutorships)
Court
LOPEZ POUSO, RODRIGO (Chairman)
PEON NIETO, ANA (Secretary)
SEOANE MARTINEZ, MARIA LUISA (Member)
Functional Analysis by G.D. Lugovaia and S.N. Sherstniov. Solved exercises.
Authorship
M.O.P.
Bachelor of Mathematics
Defense date
07.15.2025 17:20
Summary
In this Final Degree Project, we will present the basic concepts of the theory of C*-algebras taught by Galina Dmitrievna Lugovaia and Anatoli Nikolaevich Sherstniov, as well as solve the exercises they propose. We will begin by introducing Banach algebras, studying their algebraic and topological properties, and culminating with the Guelfand representation, which relates them to spaces of continuous functions. Next, we will study the particular case of C*-algebras, of great interest in the field of functional analysis. We will analyze their structure and most important concepts: positive elements, approximate identities, morphisms, states...
Direction
LOSADA RODRIGUEZ, JORGE (Tutorships)
Court
LOPEZ POUSO, RODRIGO (Chairman)
PEON NIETO, ANA (Secretary)
SEOANE MARTINEZ, MARIA LUISA (Member)
Model Specification in Survival Analysis
Authorship
A.A.P.B.
Bachelor of Mathematics
Defense date
07.17.2025 09:55
Summary
This project goes over some specification contrasts based on the distribution function and the density function for both complete data samples and samples with random right censoring. Specification contrasts are explained, along with their elements. A nonparametric estimator for the distribution function is proposed in case of a complete sample, its properties are studied and contrasts are layed out using different distances. Nonparametric estimation of the density function for complete samples is discussed, as well as some contrasts for it. Survival analysis is presented, focusing on some of the most important functions for studying lifetime data and frequently used parametric families; censoring, and especially random right censoring, are also introduced. A nonparametric estimator for the distribution function under random right censoring is proposed and parametric estimation is discussed; contrasts for complete samples are adapted to this scenario. A nonparametric estimator for the density function under random right censoring is given, together with a contrast based on it. A nonparametric continuous estimator for the distribution function is studied along with a contrast using it.
Direction
GONZALEZ MANTEIGA, WENCESLAO (Tutorships)
VIDAL GARCIA, MARIA (Co-tutorships)
Court
Majadas Soto, José Javier (Chairman)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Secretary)
CASARES DE CAL, MARIA ANGELES (Member)
Dynamical Systems
Authorship
M.P.Q.
Bachelor of Mathematics
Defense date
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.
Direction
LOPEZ SOMOZA, LUCIA (Tutorships)
Court
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
Classification trees and optimisation
Authorship
I.Q.R.
Double Bachelor's Degree in Informatics Engineering and Mathematics
Defense date
07.17.2025 10:40
Summary
This paper studies classification trees, a technique widely used in machine learning due to its simplicity, predictive capacity and ease of interpretation. In particular, different strategies for their construction are analysed, with special attention to methods based on mathematical optimisation. Three representative approaches are considered: Random Forests (RF), as a heuristic model based on tree assembly; Optimal Classification Trees (OCT), which poses the problem as a mixed integer linear optimisation; and Optimal Randomized Classification Trees (ORCT), which uses a continuous formulation that improves scalability while maintaining interpretability. The paper begins with a review of the fundamentals of statistical classification and decision tree-based methods. This is followed by a detailed description of optimisation models that allow the construction of optimal trees. Finally, a comparative empirical study is performed using five datasets of varying complexity, evaluating each model for accuracy, training time, interpretability and practical feasibility. The results show that RF offers high performance at low computational cost, while ORCT strikes a balance between accuracy and scalability. In contrast, OCT, while theoretically attractive, has computational limitations that restrict its use to smaller scale problems.
Direction
GONZALEZ DIAZ, JULIO (Tutorships)
Court
Majadas Soto, José Javier (Chairman)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Secretary)
CASARES DE CAL, MARIA ANGELES (Member)
Automatic Text Analysis Technologies for Personality Trait Estimation
Authorship
I.Q.R.
Double Bachelor's Degree in Informatics Engineering and Mathematics
Defense date
07.17.2025 10:15
Summary
This work is framed within the field of Text-based Personality Computing (TPC), which seeks to estimate personality traits from texts written by users using natural language processing (NLP) techniques. Traditionally, personality traits are measured with questionnaires, but these methods have limitations such as the subjectivity of the answers or the difficulty of applying them on a large scale. Thanks to advances in PLN, it is now possible to analyse texts and predict certain traits without the need for surveys. In this paper, we have used a Reddit dataset with information about the personality traits of its users and applied modern techniques to compare their texts with the items of the NEO-FFI questionnaire. Through this process, Big-5 scores were estimated, the results were evaluated and MBTI traits were derived from these results. The proposed approach offers a simple, scalable and interpretable alternative for automatic personality analysis.
Direction
Losada Carril, David Enrique (Tutorships)
FERNANDEZ PICHEL, MARCOS (Co-tutorships)
Court
CATALA BOLOS, ALEJANDRO (Chairman)
Triñanes Fernández, Joaquín Ángel (Secretary)
GONZALEZ DIAZ, JULIO (Member)
Topology of viral evolution.
Authorship
L.M.Q.T.
Bachelor of Mathematics
Defense date
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.
Direction
Gómez Tato, Antonio M. (Tutorships)
Court
RODRIGUEZ CASAL, ALBERTO (Chairman)
ALONSO TARRIO, LEOVIGILDO (Secretary)
SALGADO SECO, MODESTO RAMON (Member)
Spherical Regression
Authorship
R.R.B.
Bachelor of Mathematics
Defense date
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.
Direction
ALONSO PENA, MARIA (Tutorships)
Court
ALONSO PENA, MARIA (Student’s tutor)
Quantum Computing Algorithm for Optimization
Authorship
P.R.P.
Bachelor of Mathematics
Defense date
07.15.2025 12:00
Summary
In the field of quantum computing, there is considerable interest in leveraging quantum properties to solve optimization problems, due to their potential advantage over classical methods. This thesis introduces the fundamentals of quantum computing, covering both basic elements (the qubit and the p-qubit) as well as quantum gates and the measurement process. It then explores how optimization problems can be modeled using Hamiltonians, with particular focus on the QUBO and Ising models. The work delves into adiabatic quantum computing (adiabatic theorem) and presents the Quantum Approximate Optimization Algorithm (QAOA). It also details the hybrid classical-quantum architecture of the Variational Quantum Eigensolver (VQE), its variational optimization techniques, and practical applications. Finally, the performance of classical versus quantum computing is compared through complexity classes, the notion of quantum advantage is discussed, and future lines of mathematical research in this area are explored.
Direction
PENA BRAGE, FRANCISCO JOSE (Tutorships)
Court
PENA BRAGE, FRANCISCO JOSE (Student’s tutor)
Optimal control of discrete systems and ordinary differential equations
Authorship
D.R.C.
Bachelor of Mathematics
Defense date
07.15.2025 18:00
Summary
In this work we will study optimal control problems without restrictions, both in the discrete and continuous cases. The first thing that will be sought in these problems will be to provide methods that allow to calculate the gradient of the cost functional, $\tilde J$, in the discrete case. These will be particularized in the case where the problem is evolutionary, taking advantage of this structure to reduce the computational cost. In the continuous case, problems where the state is given by an EDO will be considered. The way in which these problems will be approached will consist of: discretizing the problem so that it falls within the framework of the previous sections, or calculating the directional derivatives of the cost functional, using methods that will be provided in this work, thus allowing the gradient to be calculated by approximating the control space by a finite-dimensional space. The methods of the discrete case will be used to solve parameter estimation problems, while those of the continuous case will solve a problem related to the motion of a cart-pole system.
Direction
RODRIGUEZ GARCIA, JERONIMO (Tutorships)
Court
LOPEZ POUSO, RODRIGO (Chairman)
PEON NIETO, ANA (Secretary)
SEOANE MARTINEZ, MARIA LUISA (Member)
Topologycal data analysis and dynamical systems.
Authorship
R.R.L.D.L.V.
Bachelor of Mathematics
Defense date
07.16.2025 11:30
Summary
In this work, we will explore the main concepts of topological data analysis (TDA): we will study various homology theories and examine the idea of persistence. Additionally, we will introduce dynamical systems and study entropy in depth. Finally, we will analyze Shub’s conjecture, which relates the entropy of the system to the spectral radius between homology groups, and we will seek a version of the result using persistent homology
Direction
Álvarez López, Jesús Antonio (Tutorships)
Meniño Cotón, Carlos (Co-tutorships)
Court
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
Schauder basis in Banach Spaces
Authorship
I.R.R.
Bachelor of Mathematics
Defense date
07.16.2025 16:00
Summary
We will study the basic concepts of Schauder basis, mainly related to the Functional Analysis. We will see some of their properties, types and utilities in characterization of spaces attributes, focusing on reflexivity and the internal structure of spaces, giving special treat to the spaces of sequences like $c_0$ and $\ell_p$, laying emphasis on $\ell_1$ among latter. Furthermore, we will give some notes related to existence and uniqueness of Schauder basis and try to show how this basis are a natural extension of Hamel basis of finite spaces as well as a generalization of Hilbert basis to more generic Banach spaces.
Direction
LOSADA RODRIGUEZ, JORGE (Tutorships)
Court
LOPEZ POUSO, RODRIGO (Chairman)
PEON NIETO, ANA (Secretary)
SEOANE MARTINEZ, MARIA LUISA (Member)
Manifolds with a warped product structure
Authorship
A.R.S.
Double bachelor degree in Mathematics and Physics
Defense date
07.16.2025 12:15
Summary
Pseudo-Riemannian manifolds with a local product structure have played a fundamental role in addressing a wide range of geometric and physical problems. In particular, such manifolds pro- vide one of the most commonly used geometric frameworks for constructing Einstein Riemannian metrics and constitute the underlying structure of most relativistic spacetimes. The aim of this work is to conduct a systematic study of manifolds with a local warped product structure, focusing on their curvature properties.
Direction
GARCIA RIO, EDUARDO (Tutorships)
CAEIRO OLIVEIRA, SANDRO (Co-tutorships)
Court
GARCIA RODICIO, ANTONIO (Chairman)
CAO LABORA, DANIEL (Secretary)
Gómez Tato, Antonio M. (Member)
Propagation of Ultrahigh-energy cosmic rays in the Universe
Authorship
A.R.S.
Double bachelor degree in Mathematics and Physics
Defense date
07.17.2025 09:30
Summary
Ultrahigh-energy cosmic rays (UHECRs) are the most energetic particles detected to date in the Universe, reaching energies around the exaelectronvolt (EeV). Their study is fundamental to explore physics beyond the Standard Model, analyze the structure of the Galactic magnetic field, and address relevant questions in the field of cosmology. Currently, there are multiple lines of research focused on understanding these events, and observatories such as the Pierre Auger Observatory and the Telescope Array are comitted to detecting this type of extragalactic particles. This work studies the propagation of these cosmic rays within the Galaxy, aiming to analyze how the Galactic magnetic field influences such propagation and how the sources from which they originate could be identified. To this end, the astrophysical simulation framework CRPropa is used, which allows simulations to be performed through its Python interface. In this way, simulations of the backtracking of UHECRs detected by the Pierre Auger Observatory are carried out, obtaining both their arrival direction at the Galactic boundary and the deflection experienced during their trajectory. The study focuses on performing simulations by varying the components of the magnetic field used within the JF12 model [1, 2], considering both the presence of small-scale random turbulence and its absence, as well as the type of particle being propagated.
Direction
ALVAREZ MUÑIZ, JAIME (Tutorships)
Court
ACOSTA PLAZA, EVA MARIA (Chairman)
VIEITES DIAZ, MARIA (Secretary)
Wu , Bin (Member)
Differentiation on normed spaces and minimization of functionals
Authorship
A.R.T.
Bachelor of Mathematics
Defense date
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).
Direction
Rodríguez López, Jorge (Tutorships)
Court
Rodríguez López, Jorge (Student’s tutor)
Time series analysis for maize crops in Galicia
Authorship
M.S.R.
Bachelor of Mathematics
Defense date
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.
Direction
SAAVEDRA NIEVES, PAULA (Tutorships)
Court
SAAVEDRA NIEVES, PAULA (Student’s tutor)
Regression and Classification with Random Forests
Authorship
A.S.G.
Bachelor of Mathematics
Defense date
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.
Direction
FEBRERO BANDE, MANUEL (Tutorships)
Court
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
Defense date
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.
Direction
FERNANDEZ FERNANDEZ, FRANCISCO JAVIER (Tutorships)
Court
FERNANDEZ FERNANDEZ, FRANCISCO JAVIER (Student’s tutor)
Complex dynamics and the Mandelbrot set
Authorship
D.S.C.
Bachelor of Mathematics
Defense date
07.17.2025 12:00
Summary
In the 1960s, thanks to the rise of computers, which made it possible to produce graphical representations, and the work of the Polish mathematician Benoît Mandelbrot, a special kind of sets became popular: fractals. The aim of this work is to rigurously define the Mandelbrot set, one of the many fractals studied by him. To achieve this, we will explore in detail the dinamycs of the quadratic family.
Direction
BUEDO FERNANDEZ, SEBASTIAN (Tutorships)
CAO LABORA, DANIEL (Co-tutorships)
Court
LOPEZ POUSO, RODRIGO (Chairman)
PEON NIETO, ANA (Secretary)
SEOANE MARTINEZ, MARIA LUISA (Member)
Statistic models for dertermining the thickness of the Greenland ice sheet
Authorship
V.S.S.P.
Bachelor of Mathematics
Defense date
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.
Direction
FEBRERO BANDE, MANUEL (Tutorships)
Court
CABADA FERNANDEZ, ALBERTO (Chairman)
BORRAJO GARCIA, MARIA ISABEL (Secretary)
MUÑOZ SOLA, RAFAEL (Member)
Solving systems of linear equations with non-Hermitian complex matrices
Authorship
A.S.M.
Bachelor of Mathematics
Defense date
07.16.2025 12:00
Summary
The aim of this dissertation is to present an iterative technique, the Conjugate Orthogonal Conjugate Gradient (COCG) method, designed to solve linear systems Ax = b whose coefficient matrix A is complex, symmetric in the real sense, and non-Hermitian. Because COCG belongs to the family of Krylov subspace methods, we first introduce these methods for real matrices and then show how several of them can be extended to the complex case. The exposition begins with the Conjugate Gradient (CG) and Biconjugate Gradient (BCG) algorithms, and culminates with a detailed presentation of the COCG scheme. Algorithms for implementing the different methods will be described, and some practical examples using matrices obtained from Matrix Market will be presented.
Direction
SALGADO RODRIGUEZ, MARIA DEL PILAR (Tutorships)
Court
SALGADO RODRIGUEZ, MARIA DEL PILAR (Student’s tutor)
Introduction to reinforcement learning
Authorship
R.T.L.
Double Bachelor's Degree in Informatics Engineering and Mathematics
Defense date
07.17.2025 11:25
Summary
This work presents an introduction to reinforcement learning from a mathematical perspective, with special emphasis on its connection to dynamic programming. Although reinforcement learning is an independent field within artificial intelligence, many of its core ideas such as sequential decision-making, utility functions, or optimal policies originate from dynamic programming. For this reason, the study begins by addressing dynamic programming as the conceptual and formal foundation upon which reinforcement learning is built. Once the section on dynamic programming is concluded, the most important aspects of reinforcement learning are introduced. Understanding Markov decision processes will be essential for grasping the theory. The work also discusses topics derived from this paradigm, such as the exploration exploitation dilemma, and examines various solution algorithms, including Monte Carlo methods and temporal-difference learning. Finally, the study explores how reinforcement learning can be a powerful tool in the field of mathematical optimization. To this end, several classical optimization problems are analyzed, showing how they can be reformulated within the reinforcement learning framework. This allows the application of RL algorithms to complex problems that are not easily solved by traditional methods.
Direction
GONZALEZ DIAZ, JULIO (Tutorships)
Court
Majadas Soto, José Javier (Chairman)
SALGADO RODRIGUEZ, MARIA DEL PILAR (Secretary)
CASARES DE CAL, MARIA ANGELES (Member)
Mobile application for musical ear training learning
Authorship
R.T.L.
Double Bachelor's Degree in Informatics Engineering and Mathematics
Defense date
07.17.2025 12:30
Summary
This work will describe the process of creating a mobile application for learning musical ear training. It will begin with a preliminary study where the potential of the idea will be explored. We will identify who might be interested in the application and why. A business model will also be proposed, as mobile applications are relatively easy to introduce into the market. The requirements that will define the application's functionality will be specified. A chapter will be devoted to software engineering, explaining the lifecycle model and describing the planning process. Use cases for the previously defined functional requirements will also be included. The design section will cover different aspects such as the graphical interface, system architecture, and software design. Finally, the testing plan will be presented and the success or failure of the project will be evaluated.
Direction
TOBAR QUINTANAR, ALEJANDRO JOSE (Tutorships)
Court
Argüello Pedreira, Francisco Santiago (Chairman)
PENA BRAGE, FRANCISCO JOSE (Secretary)
Carreira Nouche, María José (Member)
Statistics for forensic genetics.
Authorship
A.V.C.
Bachelor of Mathematics
Defense date
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.
Direction
CASARES DE CAL, MARIA ANGELES (Tutorships)
Court
CRUJEIRAS CASAIS, ROSA MARÍA (Chairman)
PENA BRAGE, FRANCISCO JOSE (Secretary)
DOMINGUEZ VAZQUEZ, MIGUEL (Member)
From Galois theory to class field theory
Authorship
L.V.M.
Bachelor of Mathematics
Defense date
07.17.2025 12:30
Summary
The aim of this thesis is to justify the introduction of Class Field Theory starting from Galois Theory and Algebraic Number Theory, using the initial motivation provided by Artin and other mathematicians from the early 20th century. The work will begin with a review of some results from Galois theory and algebraic number theory that predate the development of class field theory, such as the Kronecker--Weber theorem. Then, based on the results of Weber, Hilbert, and Artin, the foundational results of Class Field Theory will be formulated. The rest of the work will be devoted to exploring their implications, with a particular focus on the connections with Galois Theory.
Direction
RIVERO SALGADO, OSCAR (Tutorships)
Court
RIVERO SALGADO, OSCAR (Student’s tutor)
Ordinary Differential Equations with Applications to Economics
Authorship
C.V.F.
Bachelor of Mathematics
Defense date
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.
Direction
Rodríguez López, Rosana (Tutorships)
Court
Rodríguez López, Rosana (Student’s tutor)
Conformal maps: an introduction
Authorship
Y.X.X.C.
Bachelor of Mathematics
Defense date
07.17.2025 17:00
Summary
In this paper we make an introductory study of conformal map in the context of complex analysis, with special emphasis on Möbius transformations. Since these applications arise from the theory of holomorphic functions, we devote the first part of the paper to review the basic tools of complex analysis. Then, we introduce the general concept of conformal map and show that every injective holomorphic function is a conformal map, illustrating it with examples. On this basis, the paper focuses on Möbius transformations, the only conformal automorphism of the extended complex plane. We present its matrix representation, study its invariants (points, circles, cross-ratio and symmetry) and classify according to the number of fixed points (elliptic, hyperbolic, parabolic, loxodromic). Finally, we develop in Maple a set of procedures to facilitate the intuitive understanding of these concepts.
Direction
TRINCHET SORIA, ROSA Mª (Tutorships)
Court
TRINCHET SORIA, ROSA Mª (Student’s tutor)