Angel F. Tenorio

Universidad Pablo de Olavide | UPO · Department of Economics, Quantitative Methods and Economic History

Even if it has been less than a decade and a half since Tian introduced his concept of evolution algebras to represent algebraically non-Mendelian rules in Genetics, their study is becoming increasingly widespread mainly due to their applications to many scientific disciplines. In order to facilitate further research on the topic, this paper deals...

In this paper, we focus on the link between evolution algebras and (pseudo)digraphs. We study some theoretical properties about this association and determine the properties of the (pseudo)digraphs associated with each type of evolution algebras. We also analyze the isomorphism classes for each configuration associated with these algebras providing...

En los últimos años, como docentes que imparten asignaturas de Matemá-ticas en primer curso de los distintos grados de la Facultad de Ciencias Empre-sariales de la Universidad Pablo de Olavide, de Sevilla, hemos observado serias carencias en el alumnado que inicia sus estudios no solo en las competencias matemáticas básicas, sino también en el prop...

As it is well-known there exist matrix representations of any given finite-dimensional complex Lie algebra. More concretely, such representations can be obtained by means of an isomorphic matrix Lie algebra consisting of upper-triangular square matrices. However, there is no general information about the minimal order for the matrices involved in s...

This paper studies the link between isomorphic digraphs and isomorphic Leibniz algebras, determining in detail this fact when using (psuedo)digraphs of 2 and 3 vertices associated with Leibniz algebras according to their isomorphism classes. Moreover, we give the complete list with all the combinatorial structures of 3 vertices associated with Leib...

Es ampliamente conocida la relevancia que tiene el uso de las nuevas tecnologías de la información y comunicación en la vida cotidiana como ciudadanos y ciudadanas profesionales y competentes. En ese sentido, la Universidad Pablo de Olavide ha llevado a cabo una fuerte apuesta. Desde hace ya varios años, se fomenta institucionalmente el desarrollo...

In this paper, we introduce an algorithmic procedure that computes abelian subalgebras and ideals of a given finite-dimensional Malcev algebra. All the computations are performed by using the non-zero brackets in the law of the algebra as input. Additionally, the algorithm also computes the α and β invariants of these algebras, and as a supporting...

In this paper, the maximal abelian dimension is algorithmically and computationally studied for the Lie algebra hn, of n×n upper-triangular matrices. More concretely, we define an algorithm to compute abelian subalgebras of hn besides programming its implementation with the symbolic computation package MAPLE. The algorithm returns a maxi-mal abelia...

In this paper, we present an algorithmic method to describe and classify filiform Lie algebras by considering the maximal dimension of abelian ideals and studying the relation between them and some known invariants of filiform Lie algebras. To do so, we have proved some new results about structure coefficients in the general law of filiform Lie alg...

We compare two teaching methodologies for explaining two topics concerning Input-output analysis, autonomous sets and fundamental products, in the context of bachelor degrees in Economics and Business Administration. In addition, we expound the traditional methodology based on studying the technical coefficients matrix and the subsets of rows comin...

This paper deals with the problem of locating the minimal number of parking areas being necessary for dangerous goods in the European Road Transport Network. To obtain a near-optimal solution for this problem, we introduce the design of a new graph-based algorithm to locate parking areas in such a way that drivers can obey the regulations related t...

This paper exemplifies the potential of GeoGebra as didactic resource for teaching Mathematics not only in High School but even in University. To be more precise, our main goal consists in putting forward the usefulness of GeoGebra as working tool so that our students manipulate several numerical (both recursive and iterative) methods to solve nonl...

Abelian subalgebras play an important role in the study of Lie algebras and their properties and structures. In this paper, the historical evolution of this concept is shown, analyzing the current status for the research on this topic. So, the main results obtained from previous years are indicated and commented here. Additionally, a list of some r...

This paper is the second in a series giving the foundations for associating Leibniz algebras with combinatorial structures of dimension 2. On this occasion, we characterize Leibniz algebras of dimension greater than 3 associated with (psuedo)digraphs and, consequently, we also study the converse, proving some properties about (pseudo)digraphs assoc...

The main goal of this paper is to analyze an economy through its symmetric input-output table by using Graph Theory. After providing a definition of the characteristic digraph for the economy, the authors give some properties and algorithms to characterize and compute the fundamental products and, later, the autonomous sets of the studied economy....

Mostramos la evolución histórica de la Teoría de Juegos desde los primeros atisbos en el s. XVIII hasta su formulación formal y estudio en profundidad en relación a cuestiones de índole económico, político y social. Nos centraremos tanto en los inicios como en el punto álgido de esta teoría: la década de 1950. Para mostrar la relevancia alcanzada p...

En este trabajo se realiza un estudio estadístico descriptivo de los resultados que el alumnado en el Grado de Ingeniería Informática de la Universidad Pablo de Olavide obtuvieron al resolver las cuestiones que les fueron planteadas en un aprueba inicial, al comienzo del curso académico, donde se planteaban cuestiones para obtener información sobre...

In this work, we design an algorithmic method to associate combinatorial structures with finite-dimensional Malcev algebras. In addition to its theoretical study, we have performed the implementation of procedures to construct the digraph associated with a given Malcev algebra (if its associated combinatorial structure is a digraph) and, conversely...

In this paper, we show an algorithmic procedure to compute abelian subalgebras and ideals of a given finite-dimensional Leibniz algebra, starting from the non-zero brackets in its law. In order to implement this method, the symbolic computation package MAPLE 12 is used. Moreover, we also show a brief computational study considering both the computi...

In this paper, we introduce an algorithmic process to associate Leibniz algebras with combinatorial structures. More concretely, we have designed an algorithm to automatize this method and to obtain the restrictions over the structure coefficients for the law of the Leibniz algebra and so determine its associated combinatorial structure. This algor...

A well-known result on Lie Theory states that every finite-dimensional complex solvable Lie algebra can be represented as a matrix Lie algebra, with upper-triangular square matrices as elements. However, this result does not specify which is the minimal order of the matrices involved in such representations. Hence, the main goal of this paper is to...

Given a finite-dimensional Leibniz algebra with certain basis, we show how to associate such a Lie algebra with a combinatorial structure of dimension 2. In some particular cases, this structure can be reduced to a digraph or a pseudodigraph. In this paper, we study some theoretical properties about this association and we determine the type of Lei...

In this paper, we study how two important ideals of a given Lie algebra \(\mathfrak {g}\) (namely, the center \(Z(\mathfrak {g})\) and the derived Lie algebra \(\mathcal {D}(\mathfrak {g})\) ) can be translated into the language of Graph Theory. In this way, we obtain some criteria and characterizations of these ideals using Graph Theory.

This paper shows how to associate the Lie algebra gn (n ∈ N), of n×n strictly upper-triangular matrices, with a specific 2-dimensional triangular configuration. The properties of those configu-rations are analyzed here by using Graph Theory and, finally, applied to obtain some results about Representation Theory for nilpotent Lie algebras.

The problem of Lie algebras’ classification, in their different varieties, has been dealt with by theory researchers since the early 20 th century. This problem has an intrinsically infinite nature since it can be inferred from the results obtained that there are features specific to each field and dimension. Despite the hundreds of attempts publis...

This paper deals with the use of differential equations and finite difference methods for solving several problems in the field of Economics and Business Administration. Economics usually needs to study the evolution of the values which are taken by a given variable in different moments. If the time variable works in a continuous way, its evolution...

This paper deals with several operations on graphs and combinatorial structures linking them with their associated Lie algebras. More concretely, our main goal is to obtain some criteria to determine when there exists a Lie algebra associated with a combinatorial structure arising from those operations. Additionally, we show an algorithmic method f...

This paper is devoted to the study of the maximal dimension of abelian Lie subalgebras of a particular family of finite-dimensional Lie algebras: namely, Lie algebra hn, of n × n upper-triangular matrices. Although its maximal dimension of abelian Lie subalgebras can be computed by using different techniques and methods (some being purely theoretic...

In this paper, we compute minimal faithful representations of filiform Lie algebras by means of strictly upper-triangular matrices. To obtain such representations, we use nilpotent Lie algebras n, of n × n strictly upper-triangular matrices, because any given (filiform) nilpotent Lie algebra admits a Lie-algebra isomorphism with a subalgebra of n f...

In this paper, we revise some ideas on Graph Theory and develop them to explain and determine three usual concepts of Input–Output Analysis: fundamental products, autonomous sets, and indecomposable matrices. Firstly, we define a digraph associated with the given economic system (i.e., with its technical coefficients matrix). Secondly, we consider...

This paper shows a characterization of digraphs of three vertices associated with Lie algebras, as well as determining the list of isomorphism classes for Lie algebras associated with these digraphs. Additionally, we introduce and implement two algorithmic procedures related to this study: the first is devoted to draw, if exists, the digraph associ...

This paper deals with the maximal abelian dimension of a Lie algebra, that is, the maximal value for the dimensions of its abelian Lie subalgebras. Indeed, we compute the maximal abelian dimension for every nilpotent Lie algebra of dimension less than 7 and for the Heisenberg algebra Hk, with k ∈ N. In this way, an algorithmic procedure is introduc...

In this paper, we show an algorithmic procedure to compute abelian subalgebras and ideals of finite-dimensional Lie algebras, starting from the non-zero brackets in its law. In order to implement this method, we use the symbolic computation package MAPLE 12. Moreover, we also give a brief computational study considering both the computing time and...

This paper studies the law of the Lie algebras \frak{h}_n associated with a particular type of Lie groups: the Lie groups H_n formed by all the n × n upper-triangular matrices without zeros in their main diagonal. Indeed, these laws are obtained by means of a computational algorithm which we have constructed and particularly implemented by using th...

This paper describes an algorithm to compute the law of the Lie algebra associated with the Lie group G n , formed of all the n×n upper-unitriangular matrices. The goal of this paper is to show the algorithm that computes the law of and its implementation using the symbolic computation package MAPLE. In addition, the complexity of the algorithm is...

This paper studies the law of the Lie algebras hn associated with a particular type of Lie groups: the Lie groups Hn formed by all the n × n upper-triangular matrices without zeros in their main diagonal. Indeed, these laws are obtained by means of a computational algorithm which we have constructed and particularly implemented by using the symboli...

This work shows how to associate the Lie algebra hnhn, of upper triangular matrices, with a specific combinatorial structure of dimension 22, for n∈Nn∈N. The properties of this structure are analyzed and characterized. Additionally, the results obtained here are applied to obtain faithful representations of solvable Lie algebras.

In this paper, we study the structure and properties of those n-dimensional Lie algebras associated with either summed structures of complete graphs or some families of digraphs, having into consideration that all these combinatorial structures are made up of n vertices. Our main goal is to obtain criteria determining when a Lie algebra is associat...

El presente artículo muestra cómo evaluar al alumnado en la asignatura Métodos Matemáticos para la Ingeniería de la Universidad Pablo de Olavide, mediante una evaluación continua de tipo portafolio en la que el alumnado irá realizando una serie de actividades por tema que serán las que se emplearán para su evaluación final y que realizarán bajo la...

En Isla Mágica se puede intuir cómo se vivía y se pensaba en el siglo XVI y, si nos fijamos con detenimiento, podemos intuir como eran las costumbres de la época o a qué se dedicaba la gente. En este capítulo vamos a intentar pasar un rato divertido averiguando qué Matemáticas conocían entonces y aprendiendo más cosas de Geometría, Cálculo, Probabi...

In this paper, we study the families of n-dimensional Lie algebras associated with a combinatorial structure made up of n vertices and with its edges forming a complete simple, undirected graph. Moreover, some properties are characterized for these structures using Lie theory, giving some examples and representations. Furthermore, we also study the...

This paper shows a new computational method to obtain filiform Lie algebras, which is based on the relation between some known invariants of these algebras and the maximal dimension of their abelian ideals. Using this relation, the law of each of these algebras can be completely determined and characterized by means of the triple consisting of its...

In this paper, we compute minimal faithful unitriangular matrix representations of filiform Lie algebras. To do it, we use the nilpotent Lie algebra, \mathfrakgn\mathfrak{g}_n, formed of n ×n strictly upper-triangular matrices. More concretely, we search the lowest natural number n such that the Lie algebra \mathfrak gn\mathfrak g_n contains a...

The main goal of this paper is to compute the maximal abelian dimension of each solvable non-decomposable Lie algebra of dimension less than 7. To do it, we apply an algorithmic method which goes ruling out non-valid maximal abelian dimensions until obtaining its exact value. Based on Mubarakzyanov and Turkowsky's classical classifications of solva...

This paper shows an overview about the concept and the philosophy of WebQuest. Besides, most of WebQuest are studied between those which can be found in Internet dealing with Statistics and Probability in Spanish. To do it, these are analyzed in-depth by indicating the aspects which are favorable and those which can be improved.

Entre las capacidades que nuestros alumnos deben desarrollar, se encuentran la búsqueda y uso de la información. Para este propósito, presentamos una actividad muy útil y aplicable en distintos campos de la investigación, como es la WebQuest. Mostraremos un ejemplo de WebQuest sobre Mujeres en la Historia de las Matemáticas, donde los alumnos podrá...

Infinity is an important and fascinating subject. In fact, the paradoxical twists and turns of infinity have baffled many great thinkers throughout the history. However, there does not exist one single �infinity� concept, instead it is the name of a notion that depends on the context in which you are using it. Most people have sort of an intuitive...

Este artículo trata el concepto y la utilización de las WebQuest por parte del profesorado de Matemáticas en los diferentes niveles educativos, haciendo énfasis tanto en Educación Secundaria Obligatoria como en Bachillerato. Más concretamente, se comentan algunas de las escasas WebQuest existentes en Internet y versando sobre la Historia de las Mat...

It is well-known that there exists a close link between Lie Theory and Relativity Theory. Indeed, the set of all symmetries of the metric in our four-dimensional spacetime is a Lie group. In this paper we try to study this link in depth, by dealing with three particular types of Lie algebras: h(n) algebras, g(n) algebras and Heisenberg algebras. Ou...

Abstract—The set of all abelian subalgebras is computationally obtained for any given finite-dimensional Lie algebra, starting from the nonzero brackets in its law. More concretely, an algorithm is described and implemented,to compute,a basis for each non- trivial abelian subalgebra with the help of the symbolic computation package MAPLE. Finally,...

This paper deals with the maximal abelian dimension of the Lie algebra hn, of n×n upper-triangular matrices. Regarding this, we obtain an algorithm which computes abelian subalgebras of hn as well as its implementation (and a computational study) by using the symbolic computation package MAPLE, where the order n of the matrices in hn is the unique...

In this paper, the maximal abelian dimension is computationally obtained for an arbitrary finite-dimensional Lie algebra, defined by its nonzero brackets. More concretely, we describe and implement an algorithm which computes such a dimension by running it in the symbolic computation package MAPLE. Finally, we also show a computational study relate...

This paper shows an algorithm which computes the law of the Lie algebra associated with the complex Lie group of n × n upper-triangular matrices with exponential elements in their main diagonal. For its implementation two procedures are used, respectively, to define a basis of the Lie algebra and the nonzero brackets in its law with respect to that...

This paper is devoted to show and explain some applications of Lie Theory to solve some problems in Economics and Mathematical Finance. So we put forward and discuss mathematical aspects and approaches for several economic problems which have been previously considered in the literature. Besides we also show our advances on this topic, mentioning s...

En este artículo se presenta una visión global de lo que se entiende por Web- Quest (WQ en lo sucesivo) y se muestra su posible utilización por el profesor de Matemáticas de Secundaria y de Bachillerato para motivar al alumnado en sus clases y despertar en él el interés y el gusto por la asignatura. Se comentan algunas de las escasas WQ existentes...

Resumen Durante los últimos cursos académicos, son varias las sesiones realizadas en el Aula Abierta de Mayores de la Universidad Pablo de Olavide que estaban dedicadas a tratar algunos contenidos matemáticos con una perspectiva atractiva y útil para el alumnado (cuya edad supera los 55 años). A este respecto, se buscaron temáticas que estuviesen p...

Themain goal of this paper is to compute aminimal matrix representation for each non-isomorphic nilpotent Lie algebra of dimension less than 6. Indeed, for each of these algebras, we search the natural number n ∈ N \ {1} such that the linear algebra �n, formed by all the n × n complex strictly upper-triangular matrices, contains a representation of...

In this paper some results on the structure of finite-dimensional Lie algebras are obtained by means of the concept of maximal abelian dimension. More concretely, a sufficient condition is given for the solvability in finite-dimensional Lie algebras by using maximal abelian dimensions. Besides, a necessary condition for the nilpotency is also state...

This paper deals with the computation of abelian subalgebras of the solvable Lie algebra hn, of n × n upper-triangular matrices. Firstly, we construct an algorithm to find abelian Lie subalgebras in a given Lie algebra hn. This algorithm allows us to compute an abelian subalgebra up to a certain dimension. Such a dimension is proved to be equal to...

La ponencia que aquí se presenta persigue un triple objetivo. Por una parte, mostrarle al lector en general, y al profesorado de Secundaria y Bachillerato en particular, el concepto de WebQuest (WQ) y sus principales aplicaciones a la enseñanza como herramienta sencilla y cómoda de utilizar por el alumnado de estos niveles en el estudio de cualquie...

En este artículo se presenta una visión global de lo que se entiende por WebQuest y se comentan varias de las existentes en Internet relativas al bloque de contenidos de Geometría y que están en castellano. Para ello, se realiza un estudio a fondo de las mismas, indicando aquellos aspectos que pueden considerarse favorables y los que podrían mejora...

En este artículo, los autores pretenden mostrar y explicar cómo la Teoría de Lie se puede aplicar a la resolución de algunos problemas relativos a la Economía y a las Finanzas. Concretamente, se realiza un análisis de dos de esos problemas y se discuten tanto sus aspectos matemáticos como el acercamiento hecho desde la Teoría de Lie para su resoluc...

We compute the largest dimension of the Abelian Lie subalgebras contained in the Lie algebra \mathfrakgn \mathfrak{g}_n of n×n strictly upper triangular matrices, where n ∈ ℕ \ {1}. We do this by proving a conjecture, which we previously advanced, about this dimension. We introduce an algorithm and use it first to study the two simplest particul...

A diferencia de una mera exposici�on de datos puramente biogr�a�cos, este art��culo pretende dar a conocer a cualquier persona en general, y a los matem�aticos y cient���cos en particular, aquellas an�ecdotas menos conocidas, leyendas m�as o menos ajustadas a la realidad, correspondencia enviada y recibida, y sobre todo, opiniones de otras personas...

En la presente comunicación explicamos algunas de las herramientas de la Geometría Diferencial y, en concreto, de la Teoría de Lie con las que se trabaja actualmente en Economía. Se indican las condiciones que se exigen a las funciones de producción y la definición de un tipo de progreso técnico denominado de tipo Lie, consistente en exigir las tre...

Entre las capacidades que nuestros alumnos deben desarrollar, se encuentran la búsqueda y uso de la información. Para este propósito, podemos emplear las TIC y, en particular, Internet. Vamos a presentar una actividad muy útil en el Aula de Matemáticas dedicada a la investigación: las WebQuest. Expondremos su estructura, indicando algunas herramien...

Cuadernillo con diversas actividades didácticas de Matemáticas destinado al alumnado y profesorado de primaria y secundaria que realice una visita escolar al parque temático Isla Mágica, siendo la primera vez que se realiza una publicación en este sentido.

According to Ado and Cartan Theorems, every Lie algebra of finite dimension can be represented as a Lie subalgebra of the Lie algebra associated with the general linear group of matrices. We show in this paper a method to obtain the simply connected Lie group associated with a nilpotent Lie algebra, by using unipotent matrices. Two cases are distin...

El Espacio Europeo de Educación Superior persigue implantar un sistema educativo que facilite la movilidad de los estudiantes y permita una homologación automática de los títulos expedidos por las universidades europeas. Con este fin, se introduce el Sistema Europeo de Transferencia de Créditos que define un nuevo concepto de “crédito”, que cuantif...

Con la entrada del Sistema Europeo de Transferencia de Créditos (ECTS) y la instauración del Espacio Europeo de Educación Superior, las asignaturas de una titulación no pueden limitarse a tratar unos contenidos y unos procedimientos, sino que han de capacitar a los alumnos para la obtención de una serie de competencias. Entre ellas se encuentran la...

In this paper we explain some tools of Differential Geometry. In detail we deal with Lie Theory, which is currently being investigated in Economics. Firstly we indicate the conditions demanded to production functions in such studies, and the definition of a particular type of technical progress: the Lie type. For this type, the three properties of...

In this paper we explain some tools of Differential Geometry. In detail we deal with Lie Theory, which is currently being investigated in Economics. Firstly we indicate the conditions demanded to production functions in such studies, and the definition of a particular type of technical progress: the Lie type. For this type, the three properties of...