The purpose of this thesis is to define equivalence classes of Cauchy sequences of rational numbers and the operations of taking a sum and a product and then to show that this system is an uncountable, ordered, complete field. In so doing, a mathematical system is obtained which is isomorphic to the real number system.

Submitted by Jean Medeiros (jeanletras@uepb.edu.br) on 2017-09-12T12:42:53Z No. of bitstreams: 1 PDF - José Elias da Silva.pdf: 9535482 bytes, checksum: 4f018a51c1e15736072db257c4b86319 (MD5) / Approved for entry into archive by Secta BC (secta.csu.bc@uepb.edu.br) on 2017-10-26T16:05:13Z (GMT) No. of bitstreams: 1 PDF - José Elias da Silva.pdf: 9535482 bytes, checksum: 4f018a51c1e15736072db257c4b86319 (MD5) / Made available in DSpace on 2017-10-26T16:05:13Z (GMT). No. of bitstreams: 1 PDF - José Elias da Silva.pdf: 9535482 bytes, checksum: 4f018a51c1e15736072db257c4b86319 (MD5) Previous issue date: 2016-10-28 / In this study we work two constructions of the real numbers system. The construction the system of real numbers by cuts or straight sections in the set of rational numbers, advanced by Dedekind, and the construction of the real number as equivalence class of fundamental sequences of rational numbers, idea introducel by Cantor. Related to this approach, we dedicate a Chapter to show density of the rational num- bers and irrational numbers in the set of real numbers. Later, in a more synthesized form than the above constructions,we present other ap- proachs which the fundamental idea of real numbers is more clear. Finally we use method axiomatic in order to show the uniqueness of the real numbers system, thus, we conclude that there is a complete and orderly body which is unique up to isomorphism . This unique body is named the real numbers body. / Neste trabalho serão estudadas duas construções do sistema dos números reais. A construção do sistema dos números reais por cortes na reta ou secções no conjunto dos números racionais, avançada por Dedekind, e a construção do número real como classe de equivalência de sucessões fundamentais de números racionais, ideia protagonizada por Cantor. Relacionado com este tema, um capítulo deste trabalho será dedicado à aplicação da densidade dos números racionais e irracionais. Posteriormente, e de uma forma mais sintetizada do que nas anteriores, são apresentadas outras construções, procurando tornar mais claro a ideia fundamental subjacente ao conceito de número real. Por fim, utiliza-se o método axiomático com o intuito de mostrar a unicidade do sistema dos números reais, isto é, concluir finalmente que existe um corpo completo e ordenado, e apenas um a menos de um isomorfismo, do conjunto dos números reais.

[EN] Fuzzy mathematics has constituted a wide field of research, since L. A. Zadeh introduced in 1965 the concept of fuzzy set. In particular, the problem of constructing a satisfactory theory of fuzzy metric spaces has been investigated by several authors. In 1994, George and Veeramani introduced and studied a notion of fuzzy metric space that constituted a modification of the one given by Kramosil and Michalek. Several authors have contributed to the study of this kind of fuzzy metrics, from the mathematical point of view and for their applications. In this thesis we have contributed to develop the study of these fuzzy metrics, from the mathematical point of view, and we approached the problem of measuring perceptual colour-difference between samples of colour using one of these fuzzy metrics. The contributions of the study carried out in this thesis is summarized as follows: \begin{enumerate} \item[(i)] We have made a detailed study of the fuzzy metric space $(X,M,\cdot)$ where $M$ is given on $X=[0,\infty[$ by $M(x,y,t)=\frac{\min\{x,y\}+t}{\max\{x,y\}+t}$ and others related to it. As a consequence we have introduced five questions in fuzzy metrics related to continuity, extension, contractivity and completion. \item[(ii)] We have answered an open question constructing a fuzzy metric space $(X,M,\ast)$ in which the assignment $f(t)=\lim_n M(a_n,b_n,t)$, where $\{a_n\}$ and $\{b_n\}$ are $M$-Cauchy sequences in $X$, is not a continuous function on $t$. The response to this question has allowed us to characterize the class of completable strong fuzzy metric spaces. \item[(iii)] We have introduced and studied a stronger concept than convergence of sequences in fuzzy metric spaces, which we call $s$-convergence. In our study, we have gotten a characterization of those spaces in which every convergent sequence is $s$-convergent and we have given a classification of fuzzy metrics attending to the behaviour of the fuzzy metric with respect to the different types of convergence. \item[(iv)] We have studied, in the context of fuzzy metric spaces, when certain families of open balls centered at a point are local bases for this point. \item[(v)] We have answered two open questions related to standard convergence, a stronger concept than convergence of sequences in fuzzy metric spaces, introduced in a natural way attending to the concept of standard Cauchy sequence (introduced in \cite{adomain}). These responses have led us to establish conditions under which Cauchyness and convergence should be considered \textit{compatible}. \item[(vi)] As a practical application, we have shown that a certain fuzzy metric is useful for measuring perceptual colour-differences between colour samples. \end{enumerate} / [ES] La matemática fuzzy ha constituido un amplio campo en la investigación, desde que en 1965 L. A. Zadeh introdujo el concepto de conjunto fuzzy. En particular, la construcción de una teoría satisfactoria de espacios métricos fuzzy ha sido un problema investigado por muchos autores. En 1994, George y Veeramani introdujeron y estudiaron una noción de espacio métrico fuzzy que constituía una modificación de la anteriormente dada por Kramosil y Michalek. Muchos autores han contribuido al estudio de este tipo de métricas fuzzy, desde el punto de vista matemático y de sus aplicaciones. En esta tesis hemos contribuido al desarrollo del estudio de estas métricas fuzzy, desde el punto de vista matemático, y hemos abordado el problema de la medida de la diferencia perceptual de color utilizando una de estas métricas. Las contribuciones que aportamos en esta tesis a dicho estudio, se resumen a continuación: \begin{enumerate} \item[(i)] Hemos hecho un estudio detallado del espacio métrico fuzzy $(X,M,\cdot)$ donde $M$ está dada sobre $[0,\infty[$ por la expresión $M(x,y,t)=\frac{\min\{x,y\}+t}{\max\{x,y\}+t}$ y de otros espacios métricos fuzzy relacionados con el. Como consecuencia de este estudio hemos introducido cinco cuestiones en la teoría de las métricas fuzzy relacionadas con continuidad, extensión, contractividad y completación. \item[(ii)] Hemos respondido a una cuestión abierta construyendo un espacio métrico fuzzy $(X,M,\ast)$ en el cual la asignación $f(t)=\lim_n M(a_n,b_n,t)$, donde $\{a_n\}$ y $\{b_n\}$ son sucesiones $M$-Cauchy, no es una función continua sobre $t$. La respuesta a esta cuestión nos ha permitido caracterizar la clase de los espacios métricos fuzzy strong completables. \item[(iii)] Hemos introducido y estudiado un concepto más fuerte que el de convergencia de sucesiones en espacios métricos fuzzy, al que hemos llamado $s$-convergencia. En nuestro estudio hemos conseguido una caracterización de aquellos espacios métricos fuzzy en los cuales toda sucesión convergente es $s$-convergente y hemos dado una clasificación de los espacios métricos fuzzy atendiendo a su comportamiento con respecto a los diferentes tipos de convergencia que se da en él. \item[(iv)] Hemos estudiado, en el contexto de los espacios métricos fuzzy, cuando ciertas familias de bolas abiertas centradas en un punto son base local de este punto. \item[(v)] Hemos respondido a dos cuestiones abiertas relacionadas con la convergencia standard, un concepto más fuerte que el de convergencia de sucesiones en espacios métricos fuzzy, introducido de forma natural a partir del concepto de sucesión de Cauchy standard (introducido en \cite{adomain}). Estas respuestas nos han llevado a establecer unas condiciones bajo las cuales un concepto relacionado con el concepto de sucesión de Cauchy y un concepto relacionado con el de convergencia deberían satisfacer para ser consideradas \textsl{compatibles}. \item[(vi)] Como aplicación práctica, hemos mostrado que una cierta métrica fuzzy es útil para medir diferencia perceptual de color entre muestras de color. \end{enumerate} / [CAT] La matemàtica fuzzy ha constituït un ampli camp en la investigació, des que el 1965 L. A. Zadeh va introduir el concepte de conjunt fuzzy. En particular, la construcció d'una teoria satisfactòria d'espais mètrics fuzzy ha estat un problema investigat per molts autors. El 1994, George i Veeramani introduiren i estudiaren una noció d'espai mètric fuzzy que constituïa una modificació de la donada per Kramosil i Michalek anteriorment. Molts autors han contribuït a l'estudi d'aquest tipus de mètriques fuzzy, des del punt de vista matemàtic i de les seves aplicacions. En aquesta tesi hem contribuït al desenvolupament de l'estudi d'aquestes mètriques fuzzy, des del punt de vista matemàtic, i hem abordat el problema de la mesura de la diferència perceptiva de color utilitzant aquestes mètriques. Les contribucions que aportem en aquesta tesi a tal estudi es resumeixen a continuació: \begin{enumerate} \item[(i)] Hem fet un estudi detallat de l'espai mètric fuzzy $(X,M,\cdot)$ on $M$ està donada sobre $[0,\infty[$ per l'expressió $M(x,y,t)=\frac{\min\{x,y\}+t}{\max\{x,y\}+t}$ i d'altres espais mètrics fuzzy relacionats amb ell. Com a conseqüència d'aquest estudi hem introduït cinc qüestions en la teoria de les mètriques fuzzy relacionades amb continuïtat, extensió, contractividad i completació. \item[(ii)] Hem respost a una qüestió oberta construint un espai mètric fuzzy $ (X, M, \ast) $ en el qual l'assignació $ f (t) = \lim_n M (a_n, b_n, t) $, on $ \{a_n\} $ i $ \{b_n \} $ són successions $ M $-Cauchy, no és una funció contínua sobre $ t $. La resposta a aquesta qüestió ens ha permès caracteritzar la classe dels espais mètrics fuzzy strong completables. \item[(iii)] Hem introduït i estudiat un concepte més fort que el de convergència de successions en espais mètrics fuzzy, al qual hem anomenat $ s $-Convergència. En el nostre estudi hem aconseguit una caracterització d'aquells espais mètrics fuzzy en els quals tota successió convergent és $ s $-convergente i hem donat una classificació dels espais mètrics fuzzy atenent al seu comportament respecte als diferents tipus de convergència que es dóna en ell. \item[(iv)] Hem estudiat, en el context dels espais mètrics fuzzy, quan certes famílies de boles obertes centrades en un punt són base local d'aquest punt. \item[(v)] Hem respost a dues qüestions obertes relacionades amb la convergència estàndard, un concepte més fort que el de convergència de successions en espais mètrics fuzzy, introduït de forma natural a partir del concepte de successió de Cauchy estàndard (introduït en \cite{adomain}). Aquestes respostes ens han portat a establir unes condicions sota les quals un concepte relacionat amb el concepte de successió de Cauchy i un concepte relacionat amb el de convergència haurien de satisfer per a ser considerats \textsl{compatibles}. \item[(vi)] Com a aplicació pràctica, hem mostrat que una certa mètrica fuzzy és útil per mesurar la diferència perceptiva de color entre mostres de color. \end{enumerate} / Miñana Prats, JJ. (2015). Fuzzy metric spaces and applications to perceptual colour-differences [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/50612 / TESIS

The purpose of the thesis is to study completeness of abstract spaces. In particular, we study completeness in partial metric spaces, partial metric type spaces, dislocated metric spaces, dislocated metric type spaces and symmetric spaces that are generalizations of metric spaces. It is well known that complete metric spaces have a wide range of applications. For instance, the classical Banach contraction principle is phrased in the context of complete metric spaces. Analogously, the Banach's xed point theorem and xed point results for Lipschitzian maps are discussed in this context, namely in, partial metric spaces and metric type spaces. Finally, xed point results are presented for symmetric spaces. / Geography / Ph. D. (Mathematics)

The notion of a fuzzy metric space due to George and Veeramani has many advantages in analysis since many notions and results from classical metric space theory can be extended and generalized to the setting of fuzzy metric spaces, for instance: the notion of completeness, completion of spaces as well as extension of maps. The layout of the dissertation is as follows: Chapter 1 provide the necessary background in the context of metric spaces, while chapter 2 presents some concepts and results from classical metric spaces in the setting of fuzzy metric spaces. In chapter 3 we continue with the study of fuzzy metric spaces, among others we show that: the product of two complete fuzzy metric spaces is also a complete fuzzy metric space. Our main contribution is in chapter 4. We introduce the concept of a standard fuzzy pseudo metric space and present some results on fuzzy metric identification. Furthermore, we discuss some properties of t-nonexpansive maps. / Mathematical Sciences / M. Sc. (Mathematics)

The notion of a fuzzy metric space due to George and Veeramani has many advantages in analysis since many notions and results from classical metric space theory can be extended and generalized to the setting of fuzzy metric spaces, for instance: the notion of completeness, completion of spaces as well as extension of maps. The layout of the dissertation is as follows: Chapter 1 provide the necessary background in the context of metric spaces, while chapter 2 presents some concepts and results from classical metric spaces in the setting of fuzzy metric spaces. In chapter 3 we continue with the study of fuzzy metric spaces, among others we show that: the product of two complete fuzzy metric spaces is also a complete fuzzy metric space. Our main contribution is in chapter 4. We introduce the concept of a standard fuzzy pseudo metric space and present some results on fuzzy metric identification. Furthermore, we discuss some properties of t-nonexpansive maps. / Mathematical Sciences / M. Sc. (Mathematics)

The purpose of the thesis is to study completeness of abstract spaces. In particular, we study completeness in partial metric spaces, partial metric type spaces, dislocated metric spaces, dislocated metric type spaces and symmetric spaces that are generalizations of metric spaces. It is well known that complete metric spaces have a wide range of applications. For instance, the classical Banach contraction principle is phrased in the context of complete metric spaces. Analogously, the Banach's xed point theorem and xed point results for Lipschitzian maps are discussed in this context, namely in, partial metric spaces and metric type spaces. Finally, xed point results are presented for symmetric spaces / Mathematical Sciences / Ph. D. (Mathematics)