222 research outputs found
El conjunt d'escapament en dinàmica holomorfa
No full textTreballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2022, Director: Xavier Jarque i Ribera[en] In this report we will study discrete dynamical systems that come of the iteration of holomorphic functions in the complex plane. In fact, we will focus on polynomial and transcendental entire functions. We will study the points that under iteration have orbits tending to infinity. This set is called the escaping set and it’s been studied at length in this paper
Dimensió de Hausdorff i sistemes dinàmics
No full textTreballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2018, Director: Xavier Jarque i Ribera i Núria Fagella Rabionet[en] In this final degree project it is studied the different types of dimension, like the fractal one, Box-Counting and Hausdorff’s. We will focus on finding and comparing these dimensions through several objects, among them, fractal sets. Then, we will study Julia’s set, its shapes and at the end we will connect those with the different definitions of dimensions
The war of attrition
No full textTreballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2023, Director: Xavier Jarque i Ribera[en] This paper presents a negotiation model known as the War of Attrition, in which two players compete for a good that loses value until they reach an agreement on distribution (what part of this good each will keep). The mathematical tool we use to study this model is game theory, and more specifically, the theory of non-cooperative games with incomplete information. The resolution of the model, that is, the characterization of the equilibria, will involve the study of special solutions of differential equations in the plane
The war of attrition
No full textTreballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2023, Director: Xavier Jarque i Ribera[en] This paper presents a negotiation model known as the War of Attrition, in which two players compete for a good that loses value until they reach an agreement on distribution (what part of this good each will keep). The mathematical tool we use to study this model is game theory, and more specifically, the theory of non-cooperative games with incomplete information. The resolution of the model, that is, the characterization of the equilibria, will involve the study of special solutions of differential equations in the plane
Dimensió de Hausdorff i sistemes dinàmics
No full textTreballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2018, Director: Xavier Jarque i Ribera i Núria Fagella Rabionet[en] In this final degree project it is studied the different types of dimension, like the fractal one, Box-Counting and Hausdorff’s. We will focus on finding and comparing these dimensions through several objects, among them, fractal sets. Then, we will study Julia’s set, its shapes and at the end we will connect those with the different definitions of dimensions
Costly Voting Models
No full textTreballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2024, Director: Oriol Tejada i Xavier Jarque i RiberaWe review different game-theoretical models of elections where voters incur voting costs. In those models, we focus on the equilibrium equations and see how these change with different assumptions on the fundamentals of the model. We provide additional proofs and further detail some existing ones as well as analyze some interesting concepts such as self-defeating polls, handicaps and false-consensus. All of the models focus on the concept of pivotal voter. By looking into these models, we aim to deepen understanding of voting dynamics and their implications
Estudi de camps vectorials polinomials mitjançant la compactificació de Poincaré
No full textTreballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2023, Director: Xavier Jarque i Ribera[en] The study of solutions escaping to infinity is an important tool in order to understand the global portrait of a dynamical system in . The Poincaré compactification, which is a method to extend analytically a vector field in a compact manifold (in fact, to an sphere), is one of the most used methods to study the dynamic of vector fields in a neighbourhood of infinity. The main purpose of the project is to follow the path of its construction not only in the euclidean plane (in where it is most used), but also in the -dimensional euclidean space, to end up applying it in the study of several dynamical systems
Existència i unicitat de solucions d'equacions diferencials ordinàries i amb retard
No full textTreballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2024, Director: Xavier Jarque i RiberaAquest és un treball sobre teoremes d’existència i unicitat de solucions de diversos tipus d’equacions diferencials. Comencem per les equacions diferencials ordinàries, demostrant els teoremes de Picard i de Peano. Tot seguit, introduïm les equacions diferencials amb retard i estudiem els casos de retard discret i retard proporcional. Finalment, estudiem alguns models on apareixen equacions diferencials amb retard.This is a project about existence and uniqueness of solutions theorems for various types of differential equations. We start with ordinary differential equations, proving Picard’s and Peano’s Theorems. Next, we introduce delay differential equations and study the cases of discrete delay and proportional delay. Finally, we study some models where delay differential equations are present
Condicions suficients per equilibri de Nash i models de diferenciació espacial
No full textTreballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2020, Director: Xavier Jarque i Ribera[en] The Cournot competition model created in 1838 is the predecesor of the application of game theory to find equilibrium conditions for the competition. This work starts with the concepts of game theory developed during the twentieth century to be able to study spatial competition models of Hotelling’s type and several dimensions.
An equilibrium existence theorem for n-dimensional models with any m firms will be presented. This theorem was proposed by Caplin and Nalebuff in 1991 and we will detail the formal proof to show the relation between the assumptions made in the model and the final result.
These results will be used to find an equilibrium in a specific model of spatial competition in two dimensions
El mètode de Newton com a sistema dinàmic
No full textTreballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2022, Director: Xavier Jarque i Ribera[en] Newton’s method, first introduced in 1669, is one of the most well-known root-finding algorithm. At the end of the nineteenth century, it emerged the idea of study the algorithm as a dynamical system in the complex plane, with the aim of understand the behavior of the method in a global way. The main goal of this thesis is to understand Newton’s method applied to polynomials as a rational function, studying properties and convergence
results, as well as, showing an algorithm to find all roots of a complex polynomial through Newton’s method. We are also going to study the iterative system applied to a special case of entire function and we will give a possible innovation to find all roots of a complex plynomial with Newton’s method through this type of functions
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