On the pillars of Functional Analysis

Funding for open access charge: Universidad de Granada/CBUA. This work is supported in part by Grupo de Investigacion FQ199 of the Junta de Andalucia (Spain) and by the IMAG-Maria de Maeztu Grant CEX2020-001105-M/AEI/10.13039/501100011033..Many authors consider that the main pillars of Functional Analysis are the Hahn–Banach Theorem, the Uniform Boundedness Principle and the Open Mapping Principle. The first one is derived from Zorn’s Lemma, while the latter two usually are obtained from Baire’s Category Theorem. In this paper we show that these three pillars should be either just two or at least eight, since the Uniform Boundedness Principle, the Open Mapping Principle and another five theorems are equivalent, as we show in a very elemental way. Since one can give an almost trivial proof of the Uniform Boundedness Principle that does not require the Baire’s theorem, we conclude that this is also the case for the other equivalent theorems that, in this way, are simultaneously proved in a simple, brief and concise way that sheds light on their nature.Universidad de Granada/CBUAJunta de Andalucia FQ199IMAG-Maria de Maeztu Grant CEX2020-001105-M/AEI/10.13039/50110001103

Analytic aspects of evolution algebras

We prove that every evolution algebra A is a normed algebra, for an l1-norm defined in terms of a fixed natural basis. We further show that a normed evolution algebra A is a Banach algebra if and only if A=A1⊕A0, where A1 is finite-dimensional and A0 is a zero-product algebra. In particular, every nondegenerate Banach evolution algebra must be finite-dimensional and the completion of a normed evolution algebra is therefore not, in general, an evolution algebra. We establish a sufficient condition for continuity of the evolution operator LB of A with respect to a natural basis B, and we show that LB need not be continuous. Moreover, if A is finite-dimensional and B={e1,…,en}, then LB is given by Le, where e=∑iei and La is the multiplication operator La(b)=ab, for b∈A. We establish necessary and sufficient conditions for convergence of (Lna(b))n, for all b∈A, in terms of the multiplicative spectrum σm(a) of a. Namely, (Lna(b))n converges, for all b∈A, if and only if σm(a)⊆Δ∪{1} and ν(1,a)≤1, where ν(1,a) denotes the index of 1 in the spectrum of La.The second author acknowledges funding from: the Distinguished Visitor Programme of the School of Mathematics and Statistics of University College Dublin, Project MTM2016-76327- C3-2-P of the Spanish Ministry of Economy, Industry and Competitiveness, Research Group FQM 199 of the Junta de Andalucía and European Union FEDER support

Classifying Evolution Algebras of Dimensions Two and Three

We classified evolution algebras of dimensions two and three. Evolution algebras of dimensions three were classified recently obtaining 116 non-isomorphic types of algebras. Herein, with a new approach, we classify these algebras into 14 non-isomorphic types of algebra, so that this new classification is easier to deal with.This work was partially supported by the Spanish Project MTM2016-76327-C3-2-P (AEI\FEDER, UE)

The multiplicative spectrum and the uniqueness of the complete norm topology

We define the spectrum of an element a in a non-associative algebra A according to a classical notion of invertibility (a is invertible if the multiplication operators La and Ra are bijective). Around this notion of spectrum, we develop a basic theoretical support for a non-associative spectral theory. Thus we prove some classical theorems of automatic continuity free of the requirement of associativity. In particular, we show the uniqueness of the complete norm topology of m-semisimple algebras, obtaining as a corollary of this result a well-known theorem of Barry E. Johnson (1967). The celebrated result of C.E. Rickart (1960) about the continuity of dense-range homomorphisms is also studied in the non-associative framework. Finally, because non-associative algebras are very suitable models in genetics, we provide here a hint of how to apply this approach in that context, by showing that every homomorphism from a complete normed algebra onto a particular type of evolution algebra is automatically continuous.Research supported by Junta de Andaluc´ıa grant FQM 0199

Pulse Processes in Networks and Evolution Algebras

In this paper, we merge two theories: that of pulse processes on weighted digraphs and that of evolution algebras. We enrich both of them. In fact, we obtain new results in the theory of pulse processes thanks to the new algebraic tool that we introduce in its framework, also extending the theory of evolution algebras, as well as its applications.This work was partially supported by the Spanish Project MTM2016-76327-C3-2-P (AEInFEDER, UE)

Comparing Textbooks of Calculus: The Case of the Derivative

Con la intención inicial de poner de manifiesto la dificultad en la selección de libros de referencia para recomendar en los primeros cursos de cálculo, se plantea una investigación en la que el concepto de derivada y los resultados esenciales del Cálculo Diferencial se comparan en dos textos de cálculo clásicos. Siguiendo una metodología de análisis de textos y utilizando la técnica del análisis de contenido, se establecen categorías de análisis que permiten concluir que, en muchos conceptos y resultados básicos relativos al cálculo diferencial, los significados que se manifiestan en los distintos textos pueden llegar a ser muy diferentes.With the initial aim of highlighting the difficulty in selecting reference books for undergraduate students of the first years of Calculus, we propose a research work to compare the notion of the derivative and many essential results of Differential Calculus in two classical texts. By following a rigorous methodology for analyzing texts and using the technique of content analysis, we establish categories of analysis to conclude that meanings of many concepts and basic results, in our case about Differential Calculus, can be very different in both texts.Este trabajo ha sido realizado parcialmente con ayuda del Proyecto “Conocimiento Didáctico del Profesor y Aprendizaje de Conceptos Matemáticos Escolares” (EDU2015-70565-P) del Plan Nacional de I+D+I (MICIN) y del Plan Andaluz de Investigación, Desarrollo e Innovación (Grupo FQM-193, Didáctica de la Matemática. Pensamiento Numérico)

Sobre el Teorema Principal de Wedderburn en álgebras de Banach complejas conmutativas

Memoria para optar al grado de Licenciada en Ciencias MatemáticasMemoria de licenciatura Univ. Granada, Departamento de Análisis Matemático. Leída el 11 de cotubre de 199

Continuidad de derivaciones aleatorias

María Victoria Velasco ColladoTesis: Univ. Granad

Sobre el Teorema Principal de Wedderburn en álgebras de Banach complejas conmutativas

Memoria para optar al grado de Licenciada en Ciencias MatemáticasMemoria de licenciatura Univ. Granada, Departamento de Análisis Matemático. Leída el 11 de cotubre de 199

Epsilon

Título, resumen y palabras clave en español e inglésResumen basado en el de la publicaciónLa Historia de las Matemáticas y el conocimiento de la génesis y evolución de los contenidos matemáticos desempeñan un papel fundamental en la formación de los profesores de matemáticas, como muchos autores vienen señalando desde hace ya más de un siglo. Se presenta una experiencia docente en relación con un Trabajo Fin de Máster realizado dentro del Programa de Máster Interuniversitario Matemáticas, que versa sobre la historia del Teorema de Rolle. Se muestra el perfil algebrista de Rolle y cómo expuso su Teorema en un contexto algebraico, totalmente alejado del Cálculo. Se indaga en el proceso mediante el cual el Teorema de Rolle muta del Álgebra al Análisis a lo largo de los siglos XVIII y XIX.ES