Matemáticas
URI permanente para esta colecciónhttp://hdl.handle.net/11349/30143
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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.Ítem Aplicación y reconocimiento de herramientas informáticas de código sql, r y power bi en la optimización de la gestión del tratamiento de datos en la línea de negocios bienestarReyes Roa, Daniel Alberto; Masmela Caita, Luis AlejandroData analysis reveals phenomena and makes decisions based on large datasets, transforming our understanding of the world. In today's society, this practice is crucial for identifying problems and improving services, fostering innovation in line with the needs of the population. This project aims to present data management tools such as SQL, R, and Power BI, highlighting their role in optimizing data management. Their fundamentals and applications are explored through case studies, emphasizing best practices for efficient usage. Furthermore, it seeks to bridge the gap between university education and the professional career, enabling future mathematicians to contribute to the technological and scientific development of the country. This project is developed in R Markdown, combining text with Markdown formatting and R programming code in a single file, providing a dynamic environment for data exploration.Ítem Comportamiento caótico en sistemas dinámicos discretos unidimensionalesLozano Florez, Ludwig Steve; Bolaños Rivera, Yudy MarcelaThis monograph focuses on the analysis of one-dimensional discrete dynamical systems, exploring fundamental concepts such as stability, bifurcation and chaos. These systems exhibit a variety of behaviors under given initial conditions: they may stabilize in orbits, fail to converge, or display unpredictable and random behavior, typical characteristics of a chaotic system. The fundamental purpose of this work is to show in detail the behavior of orbits defined by the family of functions fμ(x) = μx(1 − x), whose dynamics varies with the parameter μ. The bifurcation phenomenon and its relation to changes in the behavior of the systems are analyzed. Finally, two definitions of chaos are provided: in the Li-Yorke sense and in the Devaney sense. The validity of the theoretical result is supported by a set of simulations.Ítem Un modelo de predicción para analizar el riesgo de crédito en MiBanco S.A.Romero Gutiérrez, Alisson Camila; Villarraga Poveda, Luis FernandoIn Colombia, the majority of commerce is sustained by microenterprises, which play a vital role in production and job creation. To ensure their optimal functioning, financial institutions have developed operational models based on offering financing and savings solutions through microcredits. Financial inclusion requires a precise analysis of credit risk and the calculation of indicators to determine the most suitable credit solution for both the microenterprise and the financial institution. This work proposes the use of microenterprise databases, for data security reasons, the models and results presented are based on simulated data. A characterization and prediction analysis of credit access is conducted, selecting significant variables and making curve adjustments using the least squares method, including regression lines, potential fitting, and data linearization, with the aim of finding optimal curves that adapt to data behavior. The conclusion presents the optimal models and a comparative analysis. To evaluate the models and validate the techniques used, the practice was carried out at MiBanco S.A.Ítem Metodología de cálculo de precios BMCEspitia Flórez, Juan Esteban; Másmela Caita, Luis Alejandro; Másmela Caita, Luis Alejandro [0000-0003-3882-4980]As part of the internship development at the Bolsa Mercantil de Colombia (BMC), this work aims to demonstrate a methodology that can be used to calculate national prices for certain products and allocate them to those lacking a sufficient sample. This is achieved by utilizing the Central Product Classification (CPC), provided by the National Administrative Department of Statistics (DANE), and a data source, in this case, the invoice records of the BMC. Additionally, two key tools are employed: Seasonal-Trend decomposition based on Loess (STL) and Dynamic Time Warping (DTW), providing an approximation of national prices in a dynamic economic environment like Colombia's.Ítem Autómatas ponderados sobre bimonoides fuertesArdila Burgos, Steven; Mejia Moreno, Carolina; Mejia Moreno, Carolina [0000-0002-2852-0644]Weighted automata make it possible to obtain information by assigning weights to transitions in the automata, through semantics. These weights can represent different types of information, such as the amount of resources necessary to execute a transition or the probability of its successful execution. The semantic interpretation may vary depending on the algebraic structure with which that is associated. Although the literature on weighted automata focuses mainly on semirings, the article mainly argues why the algebraic structure associated with an automaton must meet the minimum requirement of being a strong bimonoid, even if the half ring is perceived as more optimal. In addition, mathematical tools are presented, such as recognizable functions, which allow characterize automata through various languages and approach the study of bimonoids locally through finite subsets. The theory generated enables us to use mathematics and science computational in establishing relationships between the different semantics associated with auto- recognizable bushes. Also, it allows us to explore varied interpretations, recognizable functions, and local finite properties of bimonoids.Ítem Un modelo de predicción de llamadasDelgadillo Álvarez, Esteban; Villarraga Poveda, Luis FernandoThis internship was carried out in the Estrategia de Operaciones area of Refinancia S.A.S, an expert company in debt recovery. The area plays a fundamental role in decision-making with the aim of maximizing the collection from the assigned clients per month. Its main objective is to achieve the execution goals set by different partners while simultaneously reducing associated costs. To achieve this, a variety of strategies are employed, including the development of detailed management lists and the deployment of massive tools with personalized information, designed to allow clients to manage their debts autonomously.Ítem Modelo de clasificación en Machine Learning para productos del Fondo Nacional de Garantías S.A. - FNGDuque Carreño, Johan Camilo; Villarraga Poveda, Luis FernandoThis work details the implementation of a classification model in Machine Learning in order to analyze and evaluate guarantees in the scope of the National Guarantee Fund. Its main objective is to improve efficiency and precision in the detection of inconsistencies in guarantee claims by financial intermediaries, thus optimizing the process. The narrative is based on the author's experience during an internship at the National Guarantee Fund, specifically in areas such as the Vice Presidency of Operations and the Guarantees Subdirectorate, playing the role of technical analyst. The central focus of the work is to identify the financial intermediaries that tend to present the most inconsistencies and define the products and types of processes susceptible to notification. A database that includes all guarantees with some inconsistency at the time of the claim is used to illustrate the implementation of the classification model in Machine Learning. The code presented uses Python and libraries such as pandas, scikit-learn and openpyxl to load, process and build the model. classification. Mathematical and statistical knowledge is applied, such as linear algebra, probability and Boolean logic. In addition, machine learning is implemented using a RandomForestClassifier model, based on decision trees and graph theory, with optimization in model training. The theoretical framework focuses on the selection and preparation of data, the detailed exploration and appropriate choice of classes, until the implementation and evaluation of the model. The suitability of classification algorithms to optimize the prediction of inconsistencies and behaviors of the characteristics of the guarantees. This work contributes to the field of application of Machine Learning in financial entities, offering a specific and practical case study to improve processes in the context of the National Guarantee Fund. It is proposed to examine the implementation of classification models in a specific business context, evaluating their impact on decision making and improving operational efficiency.Ítem Implementación del modelo de regresión logística en el área comercial de W.C. InstalacionesGarcía Sierra, Johan Andrey; Villarraga Poveda, Luis FernandoLogistic regression represents a classification method used to anticipate the outcome of a variable categorized based on independent variables. By analyzing a data set composed of input variables and a binary outcome, it models the probability of occurrence of an event, commonly denoted as “yes” or “no.” Unlike linear regression, this approach accommodates the prediction of categorical outcomes and avoids assumptions about the normality of residuals. This statistical model was excutioned during an internship at Instalaciones Hidráulicas y Sanitarias W.C. S.A.S. where the development of tailored offers was developed at the request of potential clients in the commercial area. The company defines itself as an engineering entity specialized in providing services for the design and construction of hydraulic, sanitary, gas installations, etc. intended for residential, commercial, institutional and industrial buildings nationwide. Quarterly meetings were held in order to evaluate the performance of all areas through management indicators. The objective consisted on the data collection, cleansing and analysis in Python with the purpose of investigating its possible relationship with the current classification to offers. Data cleaning methods were used in the physical records, as well as in the Excel documents along with the creation of virtual tools for the registration for future entries. The process generated the presentation of a final report comprising the results obtained during the process.Ítem Una metodología para identificar operaciones alejadas en el mercado financieroDeschanel Ferreira, Steven; Villarraga Poveda, Luis FernandoA financial market is a place, physical or virtual, where financial assets are bought and sold. Financial assets are securities or accounting records that give the buyer the right to receive future income flows from the issuer or seller. These financial assets can be materialized through various financial instruments, such as bonds, stocks, and currencies. In order to ensure the reliability of the market, regulatory entities request that each participating agent monitor and track for the identification and prevention of potential manipulative or fraudulent behavior by their operators. One of the criteria for such monitoring includes the identification of transactions executed at rates (or prices) far from the best rates (or prices) of reference or average trading rates (or prices). This motivates the development of this project, which, based on mathematical tools, proposes a methodology to identify such operations.Ítem El modelo de Gompertz lineal con retardoPrieto Prada, Sergio Alejandro; Sanjuan Cuellar, Álvaro Arturo; Sanjuan Cuellar,Álvaro Arturo [0000-0002-0309-8299]The study of the Gompertz Model with Delay was carried out, analyzing it qualitatively by means of the Fourier transform method, with the objective of finding periodic solutions that can be interpreted in biological contexts, specifically in tumor growth.Ítem Un teorema de convergencia débil para puntos fijos comunes de dos transformaciones no lineales en espacios de HilbertBenavides Moreno, Juan Sebastián; Ramos Fernández, Julio Cesar; 0000-0002-7329-5263In this document, we thoroughly examine the article titled "A Weak Convergence Theorem for Common Fixed Points of Two Nonlinear Mappings in Hilbert Spaces," where, through the utilization of attractor properties and fixed points, an iteration scheme is analyzed that combines the simplified Baillon type and the Mann type to find a common fixed point of two commuting nonlinear transformations in Hilbert spaces.Ítem Categorías de modelos en espacios topológicosArismendy Tibaduiza, Jose Daniel; Giraldo Hernandez, Carlos AndresThe monograph delves into the concept of weak equivalences, highlighting their fundamental role in homotopy theory. Within this theoretical framework, morphisms with closure properties similar to isomorphism are examined; however, they allow for a more subtle interaction. Homotopical categories are investigated, such as topological spaces and chain complexes, along with some homotopic functors like homotopy groups and homology groups. Furthermore, the importance of model categories is emphasized, especially the H and Q model categories, which unify concepts such as fibrations, cofibrations, and weak equivalences in a coherent context. Emphasis is placed on the work of Quillen and his contributions to bifibrant replacements, which bring order and coherence to homotopy theory. This monograph explores connections between model categories in topological spaces and chain complexes, enriching the understanding of algebraic topology and homotopy theory.Ítem Espejos de Möbius y su reflejo en las matemáticasReyes Gómez, Diana Catherine; Ramos Fernández, Julio CésarWhat is the importance of the Möbius transforms in Möbius mirrors?, this is the main question of this monograph, but to be able to glimpse its answer it is first necessary to present the most important phenomena of optics, this with the intention of understanding precisely what effects light can present when it comes into contact with different media and materials, without forgetting that this work has a mathematical approach, so it seeks to make known the mathematics that supports these phenomena. Then a small introduction to the celestial body and the energy it emanates will be given, mainly to know where this theory is applied, and the last thing that is needed to understand, even superficially, the subject of Möbius mirrors, are the Möbius transforms. With this information, now we will be ready to face the Möbius mirrors, and their main characteristics, always from a mathematical point of view, which as presented in this monograph is the only way to give validity and certainty to the various theories and propositions present in each of the branches of scientific knowledge.Ítem Reservas matemáticas en ActuariaRodriguez Reyes, Jhonatan Styven; Villarraga Poveda, Luís FernandoThis thesis was conducted in the area of Financial Risk Management (FRM) at KPMG Advisory, Tax & Legal S.A.S, which focuses on providing audit and consulting support to various entities. FRM offers a wide range of services, including segmentation and classification models, recalculations and reasonableness assessments, financial projections, derivative valuations, among others. Furthermore, FRM specialists propose new methods to help clients optimize their processes and achieve more accurate results. We will focus on the calculation of three types of mathematical reserves for a client that has engaged KPMG. This will involve highlighting the differences between the client's calculations and those of KPMG, as well as automating these processes for future calculations within the company.Ítem Introducción a superficies de Riemann: fórmula de Riemann-HurwitzTorres Castillo, Jaider Daniel; Barajas Sicacha, Martín; Roa, Leonardo; Barajas Sicacha, Martín [0000-0001-9442-5015]; Roa Leguizamón, Leonardo [0000-0002-3102-7674]Riemann surfaces are complex manifolds of dimension 1. An analytical approach to the study of Riemann surfaces is motivated by the identification of these geometrical objects with the complex plane, in such a way that the classical results of complex analysis apply in this context. However, an algebraic approach becomes more prominent when we talk about describing and classifying these objects, and is primarily motivated by algebraic geometry. In this sense, we study Riemann surfaces in order to understand a first approach of this concept. We make a considerable number of classic examples explicit together with the respective results. We also study the algebraic properties of these objects, how they are related to each other, and some very important invariants.