Matemáticas
URI permanente para esta colecciónhttp://hdl.handle.net/11349/30143
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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]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]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.Ítem Comparación de métricas de similitud en el método de imputación de datos k-vecinos más cercanos(Universidad Distrital Francisco José de Caldas) Niño Traslaviña, Gisel Fernanda; Másmela Caita, Luis AlejandroThe treatment of missing data is a common problem in data analysis, and data imputation is a widely used technique to address this issue. However, the choice of the appropriate imputation method can significantly influence the analysis results. Therefore, it is crucial to investigate and compare different imputation methods to understand their performance and effectiveness in various situations. In this context, this project focuses on the k-nearest neighbors data imputation methodology. It proposes to compare variations of this method using different similarity metrics such as Chebyshev, Canberra, Manhattan, Euclidean, and cosine similarity to evaluate its performance in estimating means from incomplete datasets.Ítem Espacios Lineales Tropicales(Universidad Distrital Francisco José de Caldas) Gómez Quintero, Andrés; Garay Salas, Cristhian; Cifuentes Vargas, VerónicaIn this work, tropical linear spaces were studied, particularly their combinatorial structure. These structures can be used to prove purely combinatorial results in matroid theory.Ítem Análisis y aplicaciones del método de Glove en el procesamiento del lenguaje natural: fundamentos e Implementación(Universidad Distrital Francisco José de Caldas) Jiménez Aguirre, Edwin Fernando; Másmela Caita, Luis AlejandroThis work covers the GloVe (Global Vectors) model, developed by Stanford University in 2014 in his article "GloVe: Global Vectors for Word Representation"[7]. The methodology will be Qualitative-Quantitative type and the type of research aims to be explanatory. It is proposed establish a contextualization of the GloVe method, which allows a thorough understanding of the mathematical-theoretical foundation behind this model, exemplify it manually and with a simulation in Python that reveals its applicability in the world of "Natural Processing Language” and that serves as a basis for future research in the Deep Learning topic.Ítem Una nota sobre la desigualdad de simpson mediante integrales generalizadas ponderadas(Universidad Distrital Francisco José de Caldas) Palacios Prieto , Hordenell; Ramos Fernández , Julio Cesar; García Barreto, Germán Alberto (Catalogador)This paper studies the article “A note about Simpson’s Inequality via weighted generalized integrals”, which was written by Juan Eduardo Nápoles Valdés and Florencia Rabossi [12]. Simpson’s inequality is explored from a novel perspective, using generalized weighted integrals to obtain new results and generalizations of previously known inequalities. The paper also highlights the importance of convex functions and their role in the development of these inequalities.Ítem Clasificación de formas cuadráticas con un enfoque en el lema morse y el lema de separación(Universidad Distrital Francisco José de Caldas) Alonso Herrea, Dairon Manuel; Velasquez Riaño, David Fernando; Barajas Sichacá, Martín; Barajas Sichacá, Martín [ 0000-0001-9442-5015 ]Quadratic forms are fundamental mathematical objects that have intrigued mathematicians for centuries. These second-degree polynomial functions have found their application in a wide range of disciplines, from physics and engineering to economics and computer science. Their study has not only revealed properties inherent to algebraic structures, but has also allowed practical problems to be tackled with elegance and precision. In this monograph, we explore in depth quadratic forms and their classification through two powerful tools: Morse's Lemma and the Separation Lemma. These lemmas, although seemingly abstract, provide insightful insight into the nature of quadratic forms and how they behave in different contexts. The monograph begins with a comprehensive review of quadratic forms, from their basic definitions to their more advanced properties. Then, we dive into the fascinating world of Morse's Lemma, a result that establishes a deep connection between topology and the theory of quadratic forms. We will explore how this lemma allows us to understand the structure of manifolds defined by quadratic forms and how we can use it to effectively classify them. We then delve into the Separation Lemma, another crucial result that sheds light on the relationship between quadratic forms and sets in Euclidean space. This lemma provides us with tools to distinguish between different quadratic forms and understand their behavior in terms of convex sets. Throughout this monograph, we not only focus on the abstract theory, but also explore concrete applications and illustrative examples that demonstrate the relevance and versatility of these concepts.Ítem Análisis y predicción de deserciones en campaña de INTOUCH CX(Universidad Distrital Francisco José de Caldas) Ospina Mogollón, Elver Yamid; Trejos Ángel, Deccy Yanteh; Trejos Ángel; Deccy Yanteh [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 Extracción de insights clave para la toma de decisiones a partir de comentarios negativos de detractores del Banco BBVA(Universidad Distrital Francisco José de Caldas) Jerez Hernández , Wilson Eduardo; Másmela Caita, Luis AlejandroThis work presents the development of a sentiment analysis model based on Machine Learning techniques, with the objective of classifying and obtaining insights from negative comments issued by detractors of Banco BBVA. The main purpose is to identify areas of improvement in the bank's services to increase customer satisfaction and reduce the number of detractors. To achieve this, natural language processing (NLP) is used on textual data obtained from social networks and internal surveys.Ítem Desarrollo e implementación de sistemas de recomendación(Universidad Distrital Francisco José de Caldas) Aldana Contreras, Laura Camila; Masmela Caita, Luis AlejandroThis project was developed in collaboration with Compensar, one of the leading welfare and compensation fund entities in Colombia, with the aim of optimizing product selection for events through a content-based recommendation system. To achieve this, advanced natural language processing (NLP) techniques were implemented, utilizing the TF-IDF (Term Frequency-Inverse Document Frequency) model to extract and analyze key product attributes, such as category, name, and suggested event. Based on keyword analysis and textual similarities, the system identifies the most relevant products for each event. Additionally, a Multinomial Naive Bayes model was incorporated for product categorization, enabling a more efficient and precise organization of items. This model, trained with previously labeled data, improves the alignment between products and different event types, facilitating decision-making in planning and logistics. The primary objective of the system is to provide personalized and accurate recommendations, enhancing the user experience and optimizing event management at Compensar. The system was validated using key metrics, such as classification accuracy and user feedback, ensuring its effectiveness and scalability across different data scenarios.Ítem Aplicación de tiempos de viaje como semimétrica en k-means para la agrupación y optimización de rutasLasprilla Acevedo, Julián Felipe; Masmela Caita, Luis AlejandroThis work explores the adaptation of the K-means algorithm to use travel times as a semimetric, applied to the analysis of georeferenced data in an urban context, specifically in the city of Bogotá. By incorporating travel times obtained via the Google Directions API, the model allows the formation of clusters that reflect not only geographical proximity but also traffic conditions, making the clustering process more realistic in complex urban environments. To evaluate the quality of the clustering, validation indices such as the Silhouette index and the Davies-Bouldin index were employed, which indicated optimal configurations for the clusters. This approach contributes to logistical planning in the city and opens up opportunities for future research. Potential extensions include the use of georeferenced time series to capture traffic patterns throughout the day and the inclusion of different modes of transportation, such as public transit or walking. It is also proposed to investigate variations in the travel time semimetric, which could facilitate the use of advanced techniques such as spectral clustering, further enhancing the analysis of spatial data in urban applications.Ítem Comparación e Interpretación de algoritmos de machine learning para la predicción de conversión de clientes potenciales en el Banco de BogotáHerrera Wilches, Juan Camilo; Villarraga Poveda, Luis FernandoThis work focuses on the implementation of a predictive model for Banco de Bogotá, aimed at identifying potential customers with a high probability of becoming active clients. A database containing historical information about the bank's users was utilized, allowing for exploratory analysis and the selection of relevant features. Initially, algorithms such as Random Forest, Support Vector Machines (SVM), and logistic regression were applied; however, the results were not satisfactory. To address this, an ensemble model based on SVM was developed, which handles overfitting using Tomek Links, imputes data with K-Nearest Neighbors (KNN), and selects the most important features through logistic regression. As a result, a more robust final model was achieved, enhancing its predictive capacity, as confirmed by sensitivity and specificity metrics.Ítem Implementación de un modelo de Backtesting para la evaluación del IBNR(Universidad Distrital Francisco José de Caldas) Parra Contreras, María Camila; Trejos Angel, Deccy Yaneth; Trejos Angel, Deccy Yaneth [0000-0001-7586-9091]This report will focus on the description and analysis of the internship carried out at the insurance company La Previsora S.A, specifically in the Actuarial Management. During the internship period, a Backtesting model was studied and applied with the objective of evaluating the effectiveness of the calculations used for the Incurred But Not Reported IBNR reserve in most of the company's technical branches. This model is based on a mean square error of prediction (MSEP), which is used to define a confidence interval that allows determining whether the calculations made for the reserve are adequate or not.Ítem Modelo double exponential smoothing para pronosticar saldos diarios de algunos productos en BBVACortes Acosta, Michael Steven; Villarraga Poveda , Luis FernandoIn the financial context, accurate balance forecasting is crucial for strategic and operational decision-making. This thesis focuses on the application of the Double Exponential Smoothing (DES) model as an effective tool for predicting daily balances for selected products at BBVA. The project was developed in several stages. First, a construction and adjustment process was carried out based on historical daily balance data, identifying the structure to be used in our forecasts. Based on this construction, the double exponential smoothing model was selected due to its ability to handle data with linear trends and its computational simplicity. Finally, the model's performance is observed and confidence intervals are constructed by combining the bootstrap and standard normal distributions.