Matemáticas
URI permanente para esta colecciónhttp://hdl.handle.net/11349/30143
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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.Ítem Desarrollo e implementación de un modelo estadístico para la predicción de las ventas mensuales de las primas de seguros del estado(Universidad Distrital Francisco José de Caldas) Garzón Marín , Karla Niccole; Másmela Caita , Luis AlejandroThis study presents the implementation of a statistical model to forecast the monthly sales of insurance premiums at the Antiguo Country branch of Seguros del Estado. Using historical data from 2014 onward, the Box-Jenkins methodology was applied, employing the ARIMA model to identify temporal patterns and dependencies. A rigorous statistical analysis was conducted, including stationarity tests such as the Dickey-Fuller test, as well as autocorrelation and partial autocorrelation functions (ACF and PACF) to determine the model’s optimal parameters. Additionally, diagnostic tests such as Box-Ljung and Jarque-Bera were used to validate the model’s accuracy and reliability. The results confirm the effectiveness of the proposed model in anticipating future demand, enabling the company to optimize resource allocation and enhance strategic decision-making. Given its accuracy and robustness, this model can be replicated in other branches.Ítem Desatando nudos y teoremas : grupo fundamental y teorema de Seifert Van Kampen(Universidad Distrital Francisco José de Caldas) Arciniegas Barreto, Yoseth Stiven; Giraldo Hernández, Carlos AndrésThis work presents a detailed study of the construction of the fundamental group, starting with the definition of homotopy and the implications of bringing this concept into an algebraic context. Subsequently, the construction of free groups and amalgamated free products is addressed, with the aim of presenting the Seifert–van Kampen theorem and exploring one of its applications in knot theory. Using this theorem, it is shown that the knot group can be considered in R^3 or S^3, as both are isomorphic.Ítem Fibrados Vectoriales: el fundamento detrás de la geometría diferencial(Universidad Distrital Francisco José de Caldas) Rodríguez Cruz, Diego Andrés; Barajas Sichacá, Martín; Barajas Sichacá Martín [0000-0001-9442-5015]The following work has the purpose of making an intuitive and easy to understand theoretical introduction about vector bundles, using as a resource differential geometry and the relationship between these two areas of mathematics. Establishing in principle the basic concepts of a bundle, the object that precedes the vector bundle and from where the latter arises. In addition, the theory of tangent fibers and vector fields on smooth varieties is constructed, connecting the terms by means of a proposition, which constructs a tangent fiber as a vector fiber and establishing examples to visualize this result. The approach that the paper seeks to develop is by means of graphs, figures, illustrations and correspondences, so that it is interactive and visually appealing to the reader. Bundle theory is a recent area in mathematics and generates several important results for physics, topology and algebraic geometry.Ítem Aproximación de las soluciones de ecuaciones diferenciales de primer orden por medio de métodos numéricosMisnaza Leguizamo, Alan Steven; López León, Camilo Andrés; Sanjuán Cúellar, Álvaro Arturo; Sanjuán Cúellar, Álvaro Arturo [0000-0002-0309-8299]This work explores the effectiveness of two numerical methods: Euler and Runge-Kutta, used to solve initial value problems. First, we analytically solve the differential equations and then apply both numerical methods to approximate their solutions. By comparing the approximations with the analytical solutions, we find that the Runge-Kutta method is more effective for the second initial value problem due to its lower error. Furthermore, we deduce that the smaller the interval size used for the approximation, the more reliable it will be compared to the analytical solution of the initial value problem. Subsequently, we applied the same procedure to a "simple" third initial value problem. The results showed that the previous assertion did not hold, revealing a high sensitivity to initial conditions that led to chaotic behavior. This finding aligns with chaos theory, indicating that small variations can trigger dramatically different outcomes. Finally, this study demonstrates that numerical methods are not always effective for all initial value problems, particularly the Runge-Kutta method for solutions that exhibit discontinuities, singularities, or high oscillation, leading to chaotic behaviors. This study underscores the importance of understanding the limitations of numerical methods in predicting solutions for initial value problems and paves the way for future work in improving these techniques and understanding chaos in dynamic systems.Ítem Optimización estática para funciones de tipo Cobb-DouglasGarcía Sánchez, Maritza Lizeth; Villarraga Poveda, Luis FernandoThe general objective of this monograph is to study the static optimization theory applied to Cobb-Douglas type functions. Specifically, an analysis of the theory and how it is interpreted considering a generalized Cobb-Douglas type function is carried out. Unconstrained and equality constrained optimization problems are examined, highlighting an example that models a Cobb-Douglas function under a budget equality constraint. As a complement, an interactive web page “https://optimizacion-cobb-douglas-maritza-garcias-projects.vercel.app/” is developed. This tool implements a software designed to solve optimization problems in Cobb-Douglas type functions with 1 to 5 independent variables and allows visualizing both the mathematical and economic interpretation of the concepts of static optimization in Cobb-Douglas type functions, contributing to a more effective and significant application. The technologies used in the development of the page and the explanation of their respective functionality can be found in the annexes.Ítem Espacios recubridores: un enfoque gráfico de la teoría(Universidad Distrital Francisco José de Caldas) López Martínez, Mike Hiarol; Salas Lizarazo, Deiby Esteban; Barajas Sichacá, Martín; Barajas Sichacá, Martín [0000-0001-9442-5015]This work addresses fundamental concepts of algebraic topology, focusing on group actions on topological spaces. It explores structures such as quotient spaces, orbits, and covering spaces, using examples like the action of the integer group on the real line and the antipodal action on the sphere, leading to the construction of spaces such as the circle, the torus, and the projective plane. Additionally, it examines the relationship between properly discontinuous groups and covering spaces, as well as the importance of covering transformations and the fundamental group in understanding the algebraic structure of spaces.Ítem Aplicación de las matemáticas a la sostenibilidadRomero Garnica, Julián Yesid; Villarraga Poveda, Luis FernandoThis work aims to identify the areas with the greatest susceptibility to drought in Colombia, together with the clients and offices of the Banco de las Microfinanzas Bancamía S.A., to develop strategies that promote the growth of the placement of microcredits and insurance adapted to the needs of customers in those regions. As a case study, the municipality of La Unión, located in the department of Antioquia, is examined to determine the level of drought that it has experienced and that which has been predicted for the year 2024. This is done by calculating the Drought Index. Standardized Precipitation (SPI), which is obtained by fitting the monthly precipitation data recorded in a specific period to a gamma distribution. Subsequently, the fitted data are normalized and the drought level is classified according to the estimated value obtained. The next step is to use programming and geolocation tools, such as Python and ArcGIS Pro, to create climate vulnerability maps for all of Colombia, thus identifying areas susceptible to drought together with Bancamía clients and offices.Ítem Estudio del ausentismo en el Banco de Bogotá: análisis estadístico, descriptivo y temporal con un enfoque en salud mentalCárdenas Martínez, Harim Sofía; Trejos Angel, Deccy Yaneth; Trejos Angel, Deccy Yaneth [0000-0001-7586-9091]This project focuses on the analysis of absenteeism, examining in detail its causes, its impact on the organization and the data needed to develop effective strategies for its mitigation, with a special emphasis on mental health. This analysis allows managers to better anticipate and manage the absence factor within the bank, building and improving a healthier and more productive work environment; which is crucial for those seeking to implement evidence-based strategies to improve staff retention and satisfaction, reduce absenteeism costs and improve employee health, thus optimizing work dynamics and organizational performance.Ítem Explorando la política: una visión con complejos simpliciales.Leguízamo Guzmán, Angelly Vanessa; Bermúdez Bohórquez, Nicol Daniela; Giraldo Hernández, Carlos AndrésSimplicial complexes, as a particular example of a topological space, offer a different perspective for understanding the dynamics within political and social structures. These complexes are mathematical constructions that represent sets of elements called simplices, which can be used to model relationships between actors such as political parties, legislators, or voters. By applying the theory of simplicial complexes, contemporary political dynamics can be understood and represented, thus providing a mathematical alternative for their representation. In this work, we represent a political structure composed of five electoral candidates for the 2023 Mayor of Bogotá, evaluating them in four areas of interest (Gender, Transportation, Health, and Education) through natural language processing focused on text analysis and mining. Using this approach, we construct simplicial complexes through the Vietoris-Rips method, generating a mathematical representation that allows us to analyze the relationships between government plans and the potential for collaboration among agents within these political structures. In doing so, we propose a method for constructing simplicial complexes that complements existing methodologies, such that the constructed complexes can explain the interactions between different political structures.Ítem Arboles de decisiones (Modelo clasificación de accidente)Peñuela Cardenas, Sergio; Villarraga Poveda, Luis FernandoWith the aim of improving Seguros Bolívar's understanding of work-related accidents linked to traffic incidents and thus establishing recovery measures for the claims authorized by the ARL, we have developed and validated predictive models. These models are based on information obtained from a representative sample of claims. Through the analysis of data from 2023, which includes the number of accidents classified as work-related traffic incidents, our project seeks to provide Seguros Bolívar with crucial information to effectively identify and categorize accidents reported to the ARL. This will enable the company to make informed and proactive decisions.Ítem Análisis de frecuencia en tarificacion de Seguros BolívarDiaz Silva, Michael Sebastian; Villarraga Poveda, Luis FernandoThis thesis was done while doing an internship at Seguros Bolívar, a leading entity in the insurance sector that is part of the business conglomerate called Grupo Bolívar. During my time at Seguros Bolívar, I had the opportunity to immerse myself in the area of property insurance, where I was able to acquire a deep knowledge of the functioning and needs of the non-life insurance sector, with a particular focus on home insurance. The main objective of this thesis is to develop a predictive model for home insurance. In a business context increasingly oriented towards data analysis and evidence-based decision making, the ability to foresee events or trends has become essential. This model will be designed to predict the claim count for a given policy, taking into account a variety of factors such as the type of housing, the cost of the property, among others. By more accurately predicting claims, the insurer can optimize its pricing processes, thus offering more personalized and competitive coverage to its policyholders. In addition, this model contributes to proactive risk management, allowing the company to anticipate possible adverse events and take preventive measures to mitigate their impact. In short, this work not only represents an academic effort, but also a tangible contribution to the insurance sector, facilitating informed decision-making and improving the quality of the services offered by Seguros Bolívar.Ítem Homología persistente(Universidad Distrital Francisco José de Caldas) González López, Juliana; Giraldo Hernández, Carlos AndrésThe basic theory of persistent homology is presented as a tool that provides geometric information for a set of points. Through the concepts of filtration and persistence, examples are given that demonstrate the topological shape of different data clouds. Using the GUDHI library in Python, persistent homology calculation is performed, showcasing computational challenges that limit effective analysis on large datasets.Ítem Elementos básicos de la teoría analítica y cualitativa de las ecuaciones diferenciales ordinariasHernández Suarique, Juan Sebastián; Niño Mateus, Eduar Fernando; Bolaños Rivera, Yudy MarcelaIn general, it is well known that solving systems of differential equations can be a very complex task, and sometimes it is impossible to obtain explicit solutions. Even when this is possible, studying such solutions can be very complicated. Because of this, it is very important to have different tools for study from both quantitative and qualitative perspectives. This work conducts a study of the fundamental concepts of the theory of linear ordinary differential equation systems (LODES) using two approaches: quantitative and qualitative. It aims to serve as an integral and essential guide to understanding and applying these concepts in different fields of study. We begin the study with theorems that allow us to guarantee the existence, uniqueness, and method for calculating the solutions of linear equation systems, which are fundamental in the theory of ODEs, as they establish conditions under which it is possible to find a unique and valid solution for a given system. We also address Jordan forms, which will allow us to find the solution of a system of linear differential equations with constant coefficients, using the eigenvalues and eigenvectors of the system's coefficient matrix. Finally, we present a complete classification of phase diagrams of linear differential equation systems in the plane.Ítem Desarrollo e implementación de un modelo estadístico para la predicción de la demanda de medicamentos en Droguerías Colsubsidio(Universidad Distrital Francisco José de Caldas) Moreno Pérez , Yazmin Andrea; Másmela Caita, Luis Alejandro; Másmela Caita, Luis Alejandro [0000-0003-3882-4980]The project carried out during the internship was based on the development and implementation of an ARIMA model for the monthly forecast of the demand for medicines in Colsubsidio drugstores. This initiative arises as a response to the problem of inventory levels, since having an excess or shortage of medicines has a negative impact on the company's ability to provide an optimal service to its customers. This imbalance becomes a critical factor that directly affects the overall success of the organization, therefore, the implementation of this model seeks to mitigate the imbalances in inventory levels and ensure a more efficient management that allows to meet the needs of customers in a timely and accurate manner, and thus also contribute to a good competitiveness of the company in the market.Ítem Manejo de bases de datos por medio del análisis topológico de clústeres(Universidad Distrital Francisco José de Caldas) Rodríguez Rincón, Yeimi Daniela; Montaño Rodríguez, Gennifer; Giraldo Hernandez, CarlosThe connection between general topology and the data clustering method, known as clustering, is studied, which aims to group a set of data according to their similarity properties. Initially, statistical techniques are used to construct a dendrogram that classifies the data. This dendrogram is complemented with topological theory, which allows determining the number of groups into which the data should be separated. These groups generate a basis for a topology on the data set, from which additional information will be extracted to understand their behavior, using the topological concepts of interior, closure, boundary, and exterior.Ítem Teorema de Picard(Universidad Distrital Francisco José de Caldas) Aponte Ávila, Néstor Heli; Sanjuán Cuéllar, Álvaro ArturoIn this paper we study the proof for Picard's Theorem presented in Complex Analysis (Ahlfors, 1979). To this end, we make a study of the lambda modular function, some of its properties, and the Monodromy Theorem, a result coming from concepts related to the analytic continuation of functions.