Facultad de Ciencias Matemáticas y Naturales
URI permanente para esta comunidadhttp://hdl.handle.net/11349/30142
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Ítem La alternativa de FredholmMolano Cabrera, Santiago Alejandro; Lesmes Acosta, Milton del CastilloIn this monograph we present the Fredholm alternative from its bases in linear algebra to integral operators. With the key idea of establishing that finding solutions of Fredholm second-rate integral equations can be reduced only to find solutions of linear algebraic systems related to them according to their kernel.Ítem Formas cuadráticas de Carcajes y teorema de GabrielAguirre Escobar, Yeimy Paola; Mateus Moreno, Edwin Leonardo; Cifuentes Vargas, Verónica; Fernández Espinosa, Pedro FernandoIn this monograph we will reconstruct the proof of Gabriel's theorem describing the number of indecomposable representations of Dynkin type quivers. For this purpose, tools such as: quadratic forms, the group of reflections and the Coxeter transform will be used. In addition, using the tools described above, some Auslander-Reiten shells for An and Dn type shells are constructed.Ítem Perturbaciones singulares sobre una clase de ecuaciones diferencialesRamírez Parada, David Leonardo; Sanjuán Cuéllar, Álvaro ArturoIn this paper we deal with a particular equation of a class of partial differential equations. We focus on the continuity of the solution which is given by the method of characteristic curves. More specifically, the generalized Burgers equation. This is a differential equation well known for mathematical analysis and for its applications. It is commonly used in traffic flow and fluid motion models. In finite time there is a singularity also called blow-up that generates discontinuities even if the initial conditions are continuous. We describe the continuous solution given by the singularity. And we employ the singular perturbation technique to find a correction to the position at the discontinuity.Ítem Sobre operador escalera y ecuación diferencial asociada a los polinomios clásicosQuiroga Vargas, Eyder David; Mora Valbuena, Luis OriolA theoretical synthesis is made in some areas of the mathematical views during the undergraduate to have a conceptual base of tools necessary for the monograph development. Following this, a compendium of the general theory of orthogonal polynomials is presented, the relationship of three terms is demonstrated, the Christoffel-Darboux formula is demonstrated, the Kernel of polynomials are presented, and the families of orthogonal polynomials of Hermite and Laguerre are presented in detail. Finally, the definition of the Fredholm determinant is presented, the integral operator defined by the subtraction of two operators is presented, the differential recurrence relationship satisfied by the orthogonal polynomial systems with respect to a weight function under certain integration criteria is demonstrated, deduce two operators called creation and annihilation, the relationship between these two operators, the application of these operators in the Hermite orthogonal polynomials and deduce the second order differential equation satisfied by the orthogonal polynomial systems.Ítem Sobre la distribución de Rayleigh generalizada y transmutadaVelásquez Bohórquez, Fredy Alexander; Villarraga Poveda, Luis FernandoThe present work is intended to extend the generalized Rayleigh distribution using the quadratic range transmutation map studied by Shaw and Buckley in “The alchemy of probability distributions: beyond gramcharlier expansions, and a skew-kurtotic-normal distribution from a rank transmutation map ”Where a model comes from which the main properties and characteristics will be studied to subsequently perform statistical inference, show some applications with real and simulated data. Finally, the main conclusions are presented.Ítem Reconstrucción demostración teorema de no encaje afín de GromovCalderón Díaz, Alirio; Julio Arrieta, Carlos AntonioIn this memoir, we present a proof of Gromov's affine non-squeezing theorem. We start by recalling in Chapter 2 a few basic concepts of linear algebra, as well as, in Chapter 3, the definition of symplectic vector space, isotropic and co-isotropic subspace, Lagrangian subspace and symplectic subspace. We then proceed to Chapter 4, where we recall the definition of the symplectic group and give a proof of Gromov's affine non-squeezing theorem.Ítem Operadores en espacios de sucesiones de LorentzRivera Sarmiento, María Alejandra; Ramos Fernández, Julio CésarThis work consists in the study of the Lorentz sequence spaces l(p,q), which is a generalization of lp spaces, for this reason a characterization of the counting measure and the decreasing rearrangement that equip this space is done to be able to describe the space and thus be able to define a semi-norm in it, also is establishes and demonstrates certain results that allow to know how the multiplication operator defined on that space behaves.Ítem Estructura de redes neuronales (MLP) y su aplicación como aproximador universalSosa Jerez, Lexly Vanessa; Zamora Alvarado, Laura Camila; Másmela Caita, Luis AlejandroThe present work is oriented to the study of the theoretical support of artificial neural networks (MPL - Multilayer Perceptron), compared to their classification capacity. For this purpose, the concepts that allow their mathematical understanding are formally defined, in addition to the implementation of said notions in the elaboration of a code that shows, through particular examples, the classification capacity of neural networks based on the theory developed here. Additionally, the ability of neural networks to approximate functions is evidenced, through an example applied to the developed code, this result is possible thanks to the Universal Approximation Theorem of Functions.Ítem Acerca de la conjetura de Goldbach para polinomios enterosVeloza Bernal, Sergio Daniel; Cifuentes Vargas, VerónicaIn this chapter boarded the Conjecture of Golbach in a context of Rings and Fields, whose object is show how does it work, the idea is see the irreducibility of a Polynomial of the same manner which is tackled the concept of number prime in the numbers integers, the end is deduct the possibility of get two polynomials irreducibility's give us as sume one polynomial anyone selected.Ítem ParacompacidadCruz Guerrero, María Angélica; Julio Arrieta, Carlos AntonioThe compact spaces are not enough by themselves to give a measure to the given spaces, the generalization of these, the paracompact spaces allow to provide the desired metric to work on the space, it is therefore necessary to know how to share and the most important properties of the paracompact space, for example, that the product of the paracompact spaces is not paracompact, but that by adding the hypothesis that at least one of the spaces is compact, the product will finally be paracompact.Ítem Visión categórica del concepto de grupoLeón Benítez, Bertha Giselle; Ochoa Castillo, Carlos OrlandoThis project is done with the aim of studying and analyzing the concept of group in a categorical environment. This concept encompasses the concept of group seen as a category, the category of groups that includes the class of these structures and the group structure present in a category. The motivation for this work is dedicated to some of the various connections between groups and other structures. . It focuses on three different aspects of the relationship between categories and groups: groups in a category, the category of groups, and finally groups as categories.Ítem Algunos elementos de las dinámicas tipo LorenzHernández López, Omar Eduardo; Ochoa Castillo, Carlos OrlandoIn the mid-20th century, with the power to model behavior of climate Edward Lorenz designed a model from of differential equations, which have a behavior chaotic thus giving rise to the well-known Lorenz butterfly (or attractor strange Lorenz), with the introduction of this system originated a whole new theory about chaos, making this system a object of much study not only in mathematics but in different fields of science such as biology, physics or economics and thus presenting over time different investigations with Lorenz-like models. In order to delve into this topic, the objective of this work is the basic theory that is needed to understand the behavior of the Lorenz system as well as a mayor system dimension that is derived from this, going through the topics like stability, bifurcation, the relationship between the exponents of Lyapunov and the behavior of a system.Ítem Una propuesta de modelo de estimación de pérdidas con base en el triángulo run-off y el método lda: una comparación frente al método Chain-LadderPlazas Rojas, Gerson; Másmela Caita, Luis AlejandroThe estimation of losses due to accidents is important because it provides the amount destined to meet the obligations to the insured, this amount is known as a reserve for loss. However there are several methods in this regard, basically each of these seek to make a forecast of the reserve derived from the claims that have occurred but not reported (IBNR). That is why it is vital to study, detail and propose robust methods to meet the objective. This is the description of the main components of one of the most commonly used methodologies in the Colombian insurance market, the Chain Ladder method. Which contemplates in its calculations a high degree of knowledge of the business. This method bases its calculation on the historical behavior of the central statistics of the probability distribution of the development factors, understood by this, as the growth rate of the losses. An alternative method based on the LDA model (Loss Distribution Approach) most frequently used to estimate the value in operational risk is proposed. This method allows to use and integrate the historical probabilistic behavior of the frequency of the losses and the severity. The possibilities of the Monte Carlo simulation are used to study different scenarios of the projections generated based on the LDA proposal. It is found that the proposed method allows not only to know a static estimate of the loss, but also to identify different possible values associated with different measures of probability of the occurrence of these events. In this case, the objectives of this work are to identify and study the most common methods for estimating the reserve of casualties, in addition to studying the LDA method and applying it to the calculation of the reserve estimate. Last, but not least, is a comparison of the proposed LDA method with the Chain Ladder method. The work is divided into four parts: the first one, the two parts that are necessary for the development of the work, such as safe business concepts, the methods for calculating the reserve and the LDA method. In the second, there is a description of the data taken for the analysis and application of the Chain Ladder method and the proposed LDA method for the calculation of the reserve. In the third part, there are the results of the study, comparing the methods already described. Finally, the conclusions are presented.Ítem Constantes óptimas para el operador de hardy-littlewood actuando sobre grafos finitosPUENTES SOLER, ANGIE YURANI; RAMOS FERNÁNDEZ, JULIO CÉSARThis monograph consists of the study of functions and spaces of sequences on finite graphs, and with the help of the counting measure, it is possible to describe and observe the behavior of Hardy-Littlewood's maximal operator on finite graphs; This establishes and demonstrates certain results, which are the main basis for giving dimensions for some specific finite graphs, and asymptotic equivalences for other finite graphs.Ítem Métricas Riemannianas en k Superficies de R^n, Un Acercamiento a Grupos de HeisenbergRodríguez Quevedo, Luisa Paulina; Julio Arrieta, Carlos AntonioThe following work seeks to study geometry in Heisenberg groups. A Heisenberg group of dimension three, noted as H3 is a subgroup of the group GL (3; R) denoting the group of three-order and invertible square matrices. Since all Lie algebra of dimension n can be represented within the square matrix algebra of order n × n, we will see that H3 is a closed subgroup of GL (3; R) and with the properties of Lie groups will be provided with a Metric invariant to the left that in turn will allow to deduce the fundamental forms, geodesics and normal functions in hypersurfaces. At the end a brief introduction to minimal surfaces in H3 is given with some examples.Ítem Operadores topológico y retículosPáez Cañón, Katherinne Rocío; Ochoa Castillo, Carlos OrlandoThis work tries to show a connection between algebra and topology, based on fundamental lattice theory and Kuratowski's operators, verifying that operators interior, closure and complement denoted by {i, a, c}, they can generate a Boolean algebra of 16 elements.Ítem Aplicativo en Knime - Cuentas maestras pagadorasVenegas Chaparro, Stefania; Villarraga Poveda, Luis FernandoThis thesis presents a workflow carried out on the Knime data mining platform, which relates to an application that allows consolidating all the report files of the information of the Master Paying Accounts that each banking entity supervised by the Financial Superintendence of Colombia makes before the Ministry of Finance and Public Credit on a monthly basis within the first twenty (20) days of each month. In the application, it is possible to identify the territorial entities that have incurred in alleged risk events within the framework of Decree 028 of 2008 and complementary regulations regarding the reporting and operation conditions of Master Accounts, and consequently, of Master Paying Accounts, in order to ensure the non-suspension of transfers of resources from the General System of Participations, thus ensuring the proper management of public resources and a high coverage of the product.Ítem Aplicaciones de las formas diferenciales en el análisis complejoPedrozo Quiceno, Jorge Camilo; Orduz Chávez, Juan Esteban; Sanjuán, Arturo; Sanjuán, Arturo [0000-0002-0309-8299]In this work we provide a proof of the Cauchy’s Theorem for analytic functions using the theory of 1-forms over R2. For this we are making a study of 1-forms and his properties. Whith this application of the vectorial analysis to complex analysis, we present an unconventional proof for the Cauchy’s Theorem.Ítem Existencia, Unicidad y Regularidad de las Soluciones a la Ecuación de Onda a través del Teorema de Hille-YosidaTiria Bulla, Julián Felipe; Sanjuán Cuellar, Alvaro ArturoDifferential equations have been a topic of discussion that emerged in the eighteenth century As a question to the behavior of the string of a violin and how to model this Phenomenon mathematically. Some related questions were being studied From the times of Pythagoras from the arhones that the rope generated according to its Length and arrangement of music in front of another set of strings. They were great mathematicians Such as Euler, D'Alembert and Daniel Bernoulli who addressed this problem, For the time, generated great questions by the methods of solution presented. This time the existence, uniqueness and regularity of the wave equation Supported in the spaces of Sobolev and the theorem of Hille-Yosida.Ítem Introducción a las álgebras de configuración de BrauerRodríguez López, Eder Alejandro; Suárez Guzmán, Javier Steven; Cifuentes Vargas, VerónicaIn this paper we develop a brief introduction to a new class of finite dimensional and wild nature algebras, called Brauer configuration algebras. of finite dimension and wild nature, called Brauer configuration algebras. For the construction of these algebras algebras we take as a starting point a combinatorial object consisting mainly of a set of vertices and polygons. a set of vertices and polygons which are called Brauer configurations. In this paper the whole path from the combinatorial object to the presentation of the algebraic structure is presented. algebraic structure, we also study some interesting properties of these algebras among which we highlight the formula for calculating the dimension of the algebraic structure. the formula to calculate the dimension of these algebras and the dimension of their center in terms of combinatorial objects. in terms of combinatorial objects. For this purpose, particular examples are developed to illustrate the basic concepts illustrate the basic concepts and are related to recent research.