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URI permanente para esta colecciónhttp://hdl.handle.net/11349/30143
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Ítem Solución del problema de Basilea generalizado utilizando el teorema de Mittag-Leffler(Universidad Distrital Francisco José de Caldas) Sandoval Salazar, Andrés Felipe; Sanjuán Cuéllar, Álvaro Arturo; Sanjuán Cuéllar, Álvaro Arturo [0000-0002-0309-8299]In this work, the generalized Basel problem is solved using the Mittag-Leffler theorem from complex analysis. A detailed study of the theory, the partial fraction decomposition of the function cot(πz), and its Laurent series expansion is carried out. As a result, expressions are obtained that allow the series composing the problem to be determined in terms of Bernoulli numbers.Ítem El algoritmo de Dijkstra desde el álgebra tropical(Universidad Distrital Francisco José de Caldas) Ramos Narváez, Daniel; Cifuentes Vargas, VerónicaThis work presents a formulation and algorithmic implementation of the shortest path problem in directed graphs using the tropical algebra framework, starting with a theoretical description of tropical semirings and their application to optimization problems, followed by the construction of a symbolic algorithm capable of operating on tropical graph representations; the methodology is validated through its application to a road network in the city of Bogotá, represented as a graph derived from real geographic data, and through symbolic evaluations and optimal path visualizations, it is demonstrated that the tropical approach is not only mathematically sound but also useful and efficient from a computational perspective; the work concludes that tropical algebra enables new ways to model, analyze, and solve classical graph theory problems.Ítem Sobre mutaciones de Quivers en arreglos de pseudolíneas(Universidad Distrital Francisco José de Caldas) Romero Sepulveda , Daniel Andrés; Cifuentes Vargas , Verónica; Cifuentes Vargas, Verónica [0009-0000-9741-8902]This text explores how pseudoline arrangements, along with a series of local moves on these arrangements, called braid moves or triangle flips, correspond exactly to the operations known as quiver mutations. The concept of quiver mutation originates from the work of Sergey Fomin and Andrei Zelevinsky, titled Introduction to Cluster Algebras, where quivers are introduced and their connection to cluster algebras is developed through the notion of quiver mutations.Ítem Estudio de la ecuación logística integrando respuestas funcionales de tipo Holling(Universidad Distrital Francisco José de Caldas) Ávila Fuquene, Angie Dayanna; Trejos Ángel, Deccy Yaneth; Trejos Ángel, Deccy Yaneth [0000-0001-7586-9091]This work presents a qualitative study of the logistic equation when integrated with Holling-type functional responses I, II, and III. The dynamic properties of the resulting systems are analyzed, including the existence and stability of equilibrium points. For each case, the ecological implications of the model are discussed, representing predator-prey interactions and environmental carrying capacity. The analysis is complemented by visualizations using phase diagrams, which provide a graphical interpretation of the qualitative solutions of the studied models. This approach reveals the dynamic richness that can emerge even from mathematically simple models, highlighting the relevance of functional responses in ecological modeling. Furthermore, the qualitative analysis of differential equations is emphasized as an essential tool for understanding nonlinear systems, allowing the identification of global behaviors, regions of growth or extinction, and the comparison of the impact of different predation forms. This type of study offers a solid foundation for applications in theoretical ecology, conservation, and natural resource management.Ítem Modelo predictivo de desembolsos de libranzas basado en datos operativos del Banco de Bogotá(Universidad Distrital Francisco José de caldas) Hernández Riveros, Camilo Andrés; Másmela Caita , Luis AlejandroThis paper develops a series of SARIMA models aimed at early alert detection and predicting disbursements of drafts corresponding to the main agreements of Banco de Bogotá. This initiative seeks to strengthen decision-making in the Commercial Boost area by anticipating atypical behavior or significant drops in disbursement flows. To this end, a process of data cleansing and temporal analysis is carried out, taking into account the seasonality and behavioral patterns of each agreement. The models constructed allow for the generation of adjusted forecasts that constitute a key input for commercial planning and the optimization of future operational strategies.Ítem Topologías generalizadas y sus aplicaciones(Universidad Distrital Francisco José de Caldas) Castañeda Castañeda, Julián Alexander; Giraldo Hernández, Carlos AndrésThis work presents a study on generalized topological spaces, an extension of the theory of classical topological spaces introduced by Ákos Császár in the 20th century. These generalized spaces allow for a more flexible structure through a family of subsets known as generalized topology, which are utilized to analyze key properties such as compactness, connectedness, and separation conditions. In this context, category theory is introduced as a formal framework for relating these spaces through continuous morphisms and commutative diagrams, exploring concepts such as products, equalizers, and pullbacks within the category of generalized topological spaces. Additionally, applications of this structure are presented in areas such as graph theory and databases, highlighting its potential for the organization and classification of complex structures. The methodology includes formalizing definitions, proving theorems, and providing illustrative examples that emphasize the properties of generalized topological spaces and their applications in both pure mathematics and data analysis.Ítem Análisis y predicción de deserciones en campaña de INTOUCH CX(Universidad Distrital Francisco José de Caldas) Ospina Mogollón, Elver Yamid; Trejos Ángel , Deccy Yaneth; Trejos Ángel Deccy Yaneth [0000-0001-7586-9091]This project aims to analyze the desertion of employees in the company. HE seeks to characterize those who remain longer and shorter in the organization. Using visual tools such as Excel and R, decision trees are developed to identify these patterns and develop a profile of the ideal candidate. This profile will be presented to the recruitment area with the purpose of reducing the dropout rate in the company. Likewise, a prediction model was developed using the tool Google Colab, in order to identify employees who could leave the company company in the short, medium and long term.Ítem Análisis predictivo sobre la efectividad de canales de comunicación en clientes con mora alta(Universidad Distrital Francisco José de Caldas) Bocanegra Arias , Kevin Daniel; Trejos Ángel , Deccy Yaneth; Trejos Ángel, Deccy Yaneth [0000-0001-7586-9091]This report aims to analyze the messaging history of collection communications carried out by the company Refinancia, with the purpose of developing predictive models such as logistic regression, neural networks, and decision trees. These models will enable the identification and prioritization of the most suitable messaging channel for each client, based on their specific characteristics. The evaluated channels include advisor phone calls, SMS messages, and HTML format emails.Ítem La regla del trapecio para funciones complejas(Universidad Distrital Francisco José de Caldas) Gómez Moreno, Rotmmel Nicolás; Ramos Férnandez, Julio CésarThis monograph studies the application of the trapezoidal rule in the integration of complex functions over curves in the complex plane. Two fundamental approaches are presented: one based on the decomposition of the complex function into its real and imaginary parts, and another that directly uses the values of the function over a partition of the curve. Both methods allow the approximation of complex integrals through finite sums and provide a practical way to tackle calculations where analytical solutions are not possible. Additionally, the approximation error and its behavior are analyzed, showing that it decreases with the square of the number of partitions, provided the function involved is sufficiently smooth. The methods are illustrated with examples over closed contours, and the numerically obtained results are compared with exact values, demonstrating the effectiveness of the method even in complex settings. The theoretical foundation is supported by complex analysis and classical numerical analysis, which allows for a rigorous justification of all formulas used.Ítem Fibraciones(Universidad Distrital Francisco José de Caldas) Lugo Mazo, Hollman Stiven; Galvis Quiroga, Robinson Duván; Giraldo Hernández, Carlos AndrésEn esta monografía se abordan los conceptos homotópicos iniciales empleados en topología algebraica, utilizados en el proceso de clasificación de los espacios topológicos. Comenzando con los conceptos de H-grupo y H-cogrupo, los cuales representan una base en la algebrización de objetos geométricos. Posteriormente se definen los grupos de homotopía de orden superior, los cuales son invariantes algebraicos asociados a los espacios. Finalizamos con los objetos de estudio de esta monografía, los espacios fibrados, los cuales permiten relacionar los grupos de homotopía de diferentes espacios.Ítem Análisis de aplicaciones de mejor aproximación en espacios de Hilbert(Universidad Distrital Francisco José de Caldas) Vargas Ruiz, Aura Yorleny; Guerrero Guio, John Alejandro; Ramos Fernández, Julio CésarThis work explores the Best Approximation Mappings (BAM) in Hilbert spaces, starting from basic notions and progressing to more recent definitions. The main reference is the article Best Approximation Mappings in Hilbert Spaces by Bauschke, Ouyang, and Wang (2020), which introduces a general characterization of this type of mapping. Throughout the document, examples and results are included to illustrate how these mappings enable a broader approach to approximation problems.Ítem Curvas asintóticas suavemente encajadas en R4(Universidad Distrital Francisco Jose de Caldas) Olarte Rojas, Daniel Felipe; Barajas Sichacá, Martín; Barajas Sichacá, Martín [0000-0001-9442-5015]We will conduct a local study of curves in R4, using concepts such as curves parameterized by arc length, and in doing so, we will construct the Frenet tetrahedron of a curve using the Gram-Schmidt orthonormalization process, which will facilitate the local study of curves. We will then define a smoothly embedding of surface in R4 with codimension 2. After defining the surfaces, we will study the concept of surface curvature to understand its relationship with respect to the second fundamental form. Then, using the classical definition of an asymptotic curve of a surface, we will demonstrate that the characterization is preserved with respect to normal curvature. We will also conduct a study of an asymptotic curve to show that the classical definition of an asymptotic curve is consistent with the characterization of asymptotic curves using ellipse curvature. Finally, elliptical, hyperbolic, and parabolic points are studied to see if they retain the properties found on a 3-dimensional surface.Ítem Dualidad de curvas, ecuaciones diferenciales binarias y la transformada de Legendre(Universidad Distrital Francisco José de Caldas) Orduz Baez, Camilo; Barajas Sichacá, Martín; Barajas Sichacá, Martín [0000-0001-9442-5015]We consider binary differential equations of the form a(x, y)(dy)^2 + 2b(x, y)dxdy + c(x, y)(dx)^2 = 0 where a,b,c are smooth functions that vanish at (0,0). In particular, we consider points where (b(x, y))^2 − a(x, y)c(x, y))≥ 0. Under those conditions there exists a classification up to diffeomorphism with the jet space of the normal forms of the equations given by: 1. Lemon: y(dy)^2 + 2xdxdy − y(dx)^2 = 0 2. Star: y(dy)^2 − 2xdxdy − y(dx)^2 = 0 3. Monstar: y(dy)^2 + 1/2 xdxdy − y(dx)^2 = 0 Applying the Legendre transform to the models, we obtain their integral curves due to duality with the corresponding curves of the Legendre transform. The purpose of this work is to show the construction of the classification and the duality of those models under the Legendre transformation.Ítem Incrustaciones contextualizadas de palabras con ELMO(Universidad Distrital Francisco José de Caldas) Segura González, David Stiven; Másmela Caita, Luis Alejandro; Másmela Caita, Luis Alejandro [0000-0003-3882-4980]; García Barreto, Germán Alberto (Catalogador)Natural language processing (NLP) is an essential and evolving field within machine learning, with applications such as machine translation, chatbots, sentiment analysis and plagiarism detection. Machine learning models for NLP seek efficient representations of words using different encodings, most notably word embeddings, which provide a simplified vector representation. However, these traditional models often omit the context of words. In this sense, ELMo (Embeddings from Language Models), a model that considers the context to generate dynamic vector representations, has emerged. ELMo employs a bidirectional language model (biLM), based on neural networks such as CNN , LSTM , and High-Way Network , allowing to capture context and solve polysemy problems. Introduced in 2018 by researchers at the Allen NLP Institute and the University of Washington, ELMo represents a significant advance in the field.Ítem Análisis comparativo de factores demográficos entre clientes caídos y no caídos para predecir KTPS en refinancia(Universidad Distrital Francisco José de Caldas) Pineda López, Julián David; Masmela Caita, Luis Alejandro; Masmela Caita, Luis Alejandro [0000-0003-3882-4980]; García Barreto, Germán Alberto (Catalogador)This document presents a comprehensive analysis of the demographic variables influencing the payment capacity (KTP) of Refinancia's clients, utilizing an advanced neural network model alongside a logistic regression model. Key determinants such as payment amount, credit score, and city segmentation are identified and evaluated, demonstrating their critical impact on predicting payment behavior. Additionally, surprising findings are highlighted, such as the negative influence of certain delinquency ranges and specific occupations, providing a detailed perspective to optimize collection strategies and client segmentation.Ítem Volúmenes de politopos matroidales(Universidad Distrital Francisco José de Caldas) León Ciprián, Bayron Ignacio; Tamayo López, Sergio Andrés; Bravo Ríos, Gabriel; Bravo Ríos, Gabriel [0000-0003-1386-6658]In this work, the volume of matroid polytopes is studied through the properties of generalized permutohedra. Initially, a detailed contextualization of polytopes is presented through convex sets. Next, an algebraic structure for polytopes is provided via the Minkowski sum and the convex hull of two or more polytopes. Subsequently, the definition of a matroid is approached from the perspective of independent sets and its equivalence in terms of its bases, which is highly relevant as it allows the rank of a matroid to be defined. Some properties of matroids are mentioned, and the beta invariant of a matroid is described—a key tool for defining the volume of a matroid polytope, which arises when defining a polytope over a set of vectors associated with the base of a matroid. Likewise, generalized permutohedra are discussed, starting from the usual permutohedron. Additionally, the concept of a mixed volume over a convex body is described. Throughout the work, various examples and graphs are presented to visually illustrate the concepts and results discussed in the paper and found in the bibliography.Ítem Sobre operadores autoadjuntos definidos en superficies de R^n(Universidad Distrital Francisco José de Caldas) Rodríguez Pinilla, Juan Camilo; Barajas Sichacá, Martín; Barajas Sichacá, Martín [000-0001-9442-5015]This work focuses on the study of geometric properties of surfaces in Rn through self-adjoint operators, more specifically in R4 of codimension 2. It explores how codimension affects geometric properties and the operators involved. The analysis will include a comparison of Weingarten shape operators, with the aim of identifying patterns and relationships that can contribute to a better understanding of differential geometry in higher-dimensional spaces. The results obtained are expected to provide new theoretical perspectives and potential applications in related fields.Ítem Grupo fundamental de espacios topológicos no euclidianos(Universidad Distrital Francisco José de Caldas) Rodríguez Olaya, Alejandro; Giraldo Hernández, Carlos Andrés; Giraldo Hernández, Carlos Andrés [0009-0009-6528-1310]The study of topology is important because this field allows problems from other areas of mathematics to be solved more effectively and simply. In the study of topological spaces, the problem of classifying and identifying homeomorphic spaces arises. To address this, the theory of topological invariants is developed; however, this tool is not sufficient in some cases, so work is done to associate a group to a topological space, which is called the fundamental group. However, this study is usually developed for topological spaces with Euclidean topologies. Our objective is to study the fundamental group of certain topological spaces with non-Euclidean topologies. To do this, we will review some topics that are fundamental to the study of the fundamental group, such as connected components and path-connected components of a space, until we reach the point of knowing how to compute the fundamental group, thus gaining insight into the nature of the fundamental group of some of these spaces.Ítem Detección de depresión mayor por medio de la lógica difusa(Universidad Distrital Francisco José de Caldas) Ardila Ortiz, Laura Daniela; Giraldo Hernández, Carlos Andrés; Ardila Ortiz, Laura Daniela [0009-0009-0810-0894]The early detection and analysis of psychological disorders is a topic of great interest, addressed in the fields of artificial intelligence, psychology, and medicine. Neuro-fuzzy models have proven to be an effective tool for tackling this problem by combining fuzzy logic with neural network theory. This work proposes a new neuro-fuzzy model designed to assist in the diagnosis of major depressive disorder in individuals, with the potential to be extended to other psychological disorders such as anxiety, obsessive-compulsive disorder, among others.Ítem Aritmética tropical y programación dinámica(Universidad Distrital Francisco José de Caldas) Trujillo Henao , Jefersson Giusep; Cifuentes Vargas , VerónicaThis monograph paper examines the application of the Floyd-Warshall algorithm in the context of tropical arithmetic, adjusting its structure to address optimization problems in weighted graphs using tropical sum and product. Through the construction of a tropical adjacency matrix and its boosting, the minimum weights representing the optimal alignment costs are determined. In the first phase, the tropical Floyd-Warshall algorithm is used to solve optimization problems in dynamic programming, particularly in path minimization, which is applied to the computation of optimal routes between nodes, highlighting its potential in various planning areas. Next, a biological sequence alignment problem is modeled, representing it in an alignment graph, where the lowest weight paths that reflect the optimal alignment are identified, validating the effectiveness of the tropical model in sequence-related problems. This approach provides a robust framework for addressing combinatorial optimization problems.