TFG - Facultad de Ciencias. Sección de Matemáticas
https://hdl.handle.net/10481/69595
2024-03-28T12:25:05ZLa ecuación logística discreta y sus aplicaciones a criptografía
https://hdl.handle.net/10481/84565
La ecuación logística discreta y sus aplicaciones a criptografía
Moya Díaz Santos, Nadia
The aim of this project is to analyse the relationship between two disciplines that at first sight
appear to be different: chaotic equations and cryptography. To do so, we are going to study how
chaotic equations can influence the design and security of cryptographic algorithms. Generally,
these algorithms are based on the generation of pseudo-random sequences that are difficult to
decrypt without the right key, but can be predictable. Thanks to the high randomness offered by
chaotic systems, we will see that they can provide a more solid basis in the cryptographic area.
Teorema de representación de Riesz para C ₀ ₀ (R)* y aplicaciones a EDPs
https://hdl.handle.net/10481/78408
Teorema de representación de Riesz para C ₀ ₀ (R)* y aplicaciones a EDPs
Galindo Pérez, Anabel
Starting from a partial derivative model that constitutes a conservation law, a mathematical reading of this fact is that the equations we work with preserve the L1 norm. This tells us that the solutions of the model, even before we know of its existence and/or uniqueness, will be bounded with this norm. This type of conservation tells us that our solutions live in a bounded space of the L1 space. However, this fact, a priori, does not give us a compactness or convergence with which we can work. In this project we will see that actually this boundedness in L1 can be used as a tool to establish a convergence of solutions, although not to a function of L1, but to a measure. In a second phase, it will be crucial to determine which equation verifies this limit measure. In order to have a global view of the steps we have followed to obtain this convergence, let us see a small summary of these results. While a bounded sequence of integrable functions does not have to converge, if we see this same sequence as a bounded sequence in the measure space, being the dual of a normed space (Riesz representation theorem), we can find a partial one that converges (Banach-Alouglu-Bourbaki theorem), taking into account that it will do so to a measure and with the weak-* topology. The general structure followed until we can arrive at the results that allow us to affirm the above-mentioned summary will be the following: in Capítulo 1 we will compile the results studied in the degree coming from diverse subjects, as well as introduce new concepts that will be necessary to us. As main source consulted, we highlight both the notes of the functional analysis course of the degree [PG10] and the books of Donald L. Cohn [Coh97] and Gerald B. Folland [Fol99]. In a second Chapter, we will illustrate all these results with an example extracted from [Nie03], thus addressing all the objectives foreseen in the end-of-degree proposal. In the first chapter, the most relevant results will be both the Banach-Alouglu- Bourbaki theorem and the Riesz representation theorem, redbeing able to emphasize also the isometric inclusion of the L1 space in the regular signed finite Borel measures, Mr, which we will present in this chapter since it is a new concept. The first of these theorems tells us that the closed balls in the dual of a normed space are compact, which will allow us to obtain some convergence on sequences seen within this dual. The second of these theorems, which gives the title to this memory, will allow us to identify the dual of the space of functions with the space of measures.
In order to arrive at both the inclusion of L1 in the measures and the Riesz theorem,
it will be crucial to define the integral of a function with respect to a measure. In
addition, through various examples in the project developed in different parts (as
appropriate), we will define and study a measure constructed from a function in
L1, which will be nothing more than the integral of the function with respect to the
Lebesgue measure as seen in Mathematical Analysis I. All these examples will aim at
identifying the space of integrable functions L1 with a subspace within the measures.
Another important new concept that we will introduce in this first chapter will
be that of a new topology, the weak-* topology, which has the advantage of having
more compact sets. This will be the topology that we will use in the Banach-Alouglu-
Bourbaki theorem to assert compactness of closed subsets of the dual.Therefore,
roughly speaking, every bounded sequence in the dual of a normed space will have
to have a partial that converges to the weak-* topology.
In Capítulo 2 we will start from a conservation law, namely mass, which will give
us an a priori estimate in L1 and we will see how the solutions behave when varying
a parameter of the equation. Since L1 is embedded in the measurements, as we have
already mentioned, the solutions will be bounded as measurements. Viewing the
measures as the dual of the space C00, being bounded they will have a convergent
subsuccession in the weak-* topology and to a measure.
This weak-* topology will come naturally when using it in the weak formulation
of the equations, since a measure coming from a function in L1 seen as an operator
on C00 will be nothing more than the integral of the product, as we will show in
Capítulo 1. Therefore, the difficulty of taking limits in the weak formulation will
reside in the nonlinear terms of the equation, which we will rewrite and prove the
bounding of the first moment, so that we can finally pass to the limit.; Cuando nos enfrentamos partiendo de un modelo en derivadas parciales que constituye
una ley de conservación, una lectura matemática de este hecho es que las
ecuaciones con las que trabajamos preservan la norma L1. Esto nos indica que las
soluciones del modelo, antes incluso de saber de su existencia y/o unicidad, van a
estar acotadas con esta norma. Este tipo de conservación nos indica que nuestras
soluciones viven en un espacio acotado de L1. Sin embargo, este hecho, a priori, no
nos da una compacidad o convergencia con la que podamos trabajar.
En esta memoria veremos que realmente esta acotación en L1 sí puede usarse
como herramienta para establecer una convergencia de soluciones, aunque no a una
función de L1, sino a una medida. En una segunda fase será crucial determinar qué
ecuación verifica esta medida límite. Para tener una visión global de los pasos que hemos
seguido para obtener esta convergencia, veamos un pequeño resumen de estos
resultados. Si consideramos una sucesión de funciones integrables acotada, esta no
tiene por qué converger. Aunque, viéndola como una sucesión acotada en el espacio
de medidas, al ser este el dual de un espacio normado (Teorema de representación
de Riesz), podemos encontrar una parcial convergente (Teorema de Banach-Alouglu-
Bourbaki), teniendo en cuenta que lo hará a una medida y con la topología débil-*.
La estructura general seguida hasta poder llegar a los resultados que nos permitan
afirmar el resumen antes comentado será la siguiente: en el Capítulo 1 recopilaremos
los resultados estudiados en el grado provenientes de diversas asignaturas, así como
introduciremos nuevos conceptos que nos serán necesarios más adelante. Como
principal fuente consultada destacamos tanto los apuntes de la asignatura de análisis
funcional del grado [PG10] como los libros de Donald L. Cohn [Coh97] y Gerald B.
Folland [Fol99]. En un segundo Capítulo, ilustraremos todos estos resultados con un
ejemplo extraído de [Nie03], quedando así abordados todos los objetivos previstos
en la propuesta de este proyecto.
En el primer Capítulo, los resultados más relevantes serán tanto el teorema de
Banach-Alouglu-Bourbaki como el teorema de representación de Riesz, pudiendo
destacar también la inclusión isométrica del espacio L1 en las medidas finitas signadas
regulares de Borel, Mr, las cuales presentaremos dentro de este al tratarse de
un nuevo concepto. El primero de estos teoremas nos dice que las bolas cerradas en
el dual de un espacio normado son compactas en cierta topología que definiremos
posteriormente, lo que nos permitirá obtener cierta convergencia sobre sucesiones vistas dentro de este dual. El segundo de ellos, el cual da título a esta memoria, nos
permitirá identificar el dual del espacio C00 con el espacio de las medidas Mr.
Para poder llegar tanto a la inclusión de L1 en las medidas como al teorema de
Riesz, será crucial definir la integral de una función respecto a una medida. Además,
a través de diversos ejemplos dentro de la memoria desarrollados en distintas partes
(según correspondan), definiremos y estudiaremos una medida construida a partir
de una función en L1, que no será más que la integral de la función respecto la
medida de Lebesgue como se vio en Análisis Matemático I. Todos estos ejemplos
tendrán como finalidad el poder identificar el espacio de las funciones integrables L1
con un subespacio dentro de las medidas.
Otro nuevo concepto importante que presentaremos en este primer capítulo será
el de una nueva topología, la topología débil-*, que posee la ventaja de tener más
conjuntos compactos. Esta será la topología que usaremos en el teorema de Banach-
Alouglu-Bourbaki para afirmar la compacidad de los subconjuntos cerrados del dual.
Por tanto, a grandes rasgos, toda sucesión acotada en el dual de un espacio normado
tendrá que tener una parcial que converja en la topología débil-*.
En el Capítulo 2 partiremos de una ley de conservación, concretamente de masa,
lo que nos dará una estimación, a priori, en L1. Con esto veremos cómo se comportan
las soluciones al variar un parámetro de la ecuación. Como L1 está embebido en las
medidas, hecho que ya habíamos comentado, las soluciones estarán acotadas como
medidas. Viendo las medidas como el dual del espacio C00, al estar acotadas, tendrán
una subsucesión convergente en la topología débil-* a una medida.
Esta topología débil-* aparecerá de manera natural al usarla en la formulación
débil de las ecuaciones, ya que una medida que provenga de una función en L1
vista como operador sobre C00, no será más que la integral del producto, esto será
probado en el Capítulo 1. Por tanto, la dificultad de tomar límites en la formulación
débil residirá en los términos no lineales de la ecuación. Para ello reescribiremos
y demostraremos la acotación del primer momento, lo que nos lleva finalmente a
poder pasar al límite.
Medidas de centralidad en grafos. Aplicación a ligas deportivas
https://hdl.handle.net/10481/76721
Medidas de centralidad en grafos. Aplicación a ligas deportivas
Gómez Romero, Gonzalo
Starting off with some basic concepts of graph theory and spectral theory, we will focus
our study in the application of mathematics to the problem of classification in competitive
leagues. Given that this problem is not completely solved, we want to offer a new view on
this subject, proving that we have really powerful tools to develop a satisfactory solution to
it with really interesting and desirable properties.
This study will show the great applicability of functional analysis, giving an abstraction of a
subject apparently not related to it, and obtaining new results helpful in the field of knowledge
that we are working on.
In the first Chapter, we give some basic definitions and results, already known, about graph
theory, so that we can define what centrality measures are, and giving the expressions of the
most commonly used in order to apply them in the next chapter.
The second Chapter is dedicated to the main matter of this study: the classification of competitive
leagues. It presents two main problems to solve. The first one is the problem of
classification of different leagues or the same league along different seasons attending to its
competitive balance which is a measure of concentration of wins, this is, how the victories
are distributed among the teams participating in the competition. The second problem that
is addressed in this chapter is the classification of the teams themselves in the league. This is
answering these questions: who is the winner of the league? Who is next in the classification?
Who is the last one? In other words, we are interested in giving a ranking method.
We will solve these two problems thanks to functional analysis. The first one will part from an
article written by Ávila-Cano and Triguero ([38]), and we will develop new results applying
norms and distances in the vector space Rn. This will grant us with plenty of tools that can
be used to give nice properties to the measures that we are looking for. We will define a new
generalization of the competitive balance measures, calling them competitive balance normed
measures and we will analyse and explain them in depth. The second one is a well known
problem with plenty of literature related to it (view [12] as an example), so we are going to
compile this literature and explain how we can apply graph theory to a competitive league,
in order to obtain centrality measures of the graphs that explain the situations given in these
leagues, so we can compare and conclude which centrality measure give us the best ranking
possible. We will see that this measure is the eigenvector centrality, so an important result
to apply this measure is the Perron-Frobenius Theorem, as it guarantees that this measure
always finds a ranking of the teams, this is, we can always find a solution to the expression
of the measure.
As the Perron-Frobenius becomes a fundamental result in our study, we dedicate the third
and final chapter of this assignment to, firstly, give an overview of the history of the theorem,
so we can understand in which context it appeared. Afterwards, we give the definition of the Perron’s Theorem, which can be considered as a first approximation of the result that
we are willing to prove. After giving all the pre-requisites that we need form the theory of
functional analysis, we finally give the demonstration of the Perron-Frobenius Theorem, and,
in order to provide the reader with a wider view on this subject, we also give an alternative
proof of the theorem, more based on a geometric approach to the problem.
In order to show the real importance of this result, we also enunciate the generalization of the
theorem to Banach spaces, and we give some of the many applications that this fundamental
result has in the field of scientific research.; El objetivo de este Trabajo Fin de Grado es aplicar técnicas y conocimientos propios del
Análisis Funcional y, en concreto, de la Teoría Espectral, a ciertos problemas de la vida real,
por lo que el enfoque de esta Memoria tiene un determinado componente interdisciplinar.
Como motivación e hilo conductor del trabajo, hemos elegido varios problemas del mundo
deportivo, que resultan extrapolables al contexto empresarial, dando una visión matemática
abstracta a cuestiones que ya han sido estudiadas desde otros campos como puede ser la
economía.
La propuesta de Trabajo Fin de Grado, que en un primer momento se formalizó tenía como
objetivo único el estudio de la centralidad del vector propio y, por tanto, del Teorema de
Perron-Frobenius. Buscando alguna aplicación vistosa que sirviese de motivación llegamos
al problema de la clasificación de equipos en una liga deportiva, en el que la centralidad
del vector propio había sido una herramienta propuesta en multitud de trabajos, como por
ejemplo [12], siendo referencia central en este enfoque el conocido artículo de J. P. Keener
[18] (que dispone de 349 citas) si bien el primero de todos ellos, que hace uso del Teorema
de Perrón Frobenius para clasificar equipos de fútbol, data de 1915 y se debe a E. Landau [21].
Cartografía y Geometría
https://hdl.handle.net/10481/76696
Cartografía y Geometría
García Olcina, Jose Manuel
This work deals with one of the oldest problems in geometry, the construction of maps of the
Earth. Although we know that does not have a perfectly spherical shape, we will simplify
the problem approximating it by a sphere of radius 1.
Estudio del Análisis Discriminante. Aplicación a datos reales
https://hdl.handle.net/10481/76620
Estudio del Análisis Discriminante. Aplicación a datos reales
Polonio Sánchez, María
The main aim of the study is to analyze Fisher’s Linear Discriminant Analysis and to later
apply real data using the R statistics software.
The LDA came about from a study made by Fisher in 1936 where, apart from the morphology
of flowers, he studied and evaluated a lineal function to establish the differences between
varieties of Iris (Setosa, Versicolor and Virgnica). Fisher did not verify all the hypotheses that
are currently considered when applying said technique, but established their foundations.
It consists in using a variable category that is a lineal combination of discriminating variables,
measured at intervals or through use of reason, to find existing differences between
the groups.
The LDA has two main aims. The first being to build discriminating functions, that allow us
to explain the belonging of an individual to a group, as well as establish the weight of each
variable in the discrimination. The second objective is to predict to which group it is most
probable the individual belongs, knowing only certain variables.
This classifying technique, included in multivariable dependency techniques ( those where
variables are divided into two groups: dependent variables and independent variables), is
applicable to many areas of knowledge. For example, in education, one tries to estimate
students’ academic performance based on educational and social factors. In medicine, it’s
used to diagnose illnesses and prescribe the most adequate treatment based on the characteristics
of the patient. Finally, one of the most remarkable uses is in the economic scope
for estimating cost effectiveness of a business based on variables such as income, debts and
the patrimony of said business. Furthermore, it deduces whether it would be beneficial for a
financial entity to approve a mortgage to its customers.