Doctorado en Ciencias - Matemáticas

URI permanente para esta colecciónhttps://repositorio.unal.edu.co/handle/unal/82411

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Theory of associated and attached prime ideals over quantum algebras
(Universidad Nacional de Colombia - sede Bogotá, 2024) Higuera Rincón, Sebastián David; Reyes Villamil, Milton Armando; Higuera Rincón, Sebastián David [0009-0003-4520-2802]; Higuera Rincón, Sebastián David [Sebastian-Higuera-4]
En esta tesis estudiamos la propiedad nilpotente-reflexiva (RNP para abreviar) sobre álgebras cuánticas como anillos de polinomios torcidos y extensiones PBW torcidas. Investigamos la transferencia de esta propiedad entre anillos de coeficientes y anillos no conmutativos de tipo polinomial sobre estos anillos. Consideramos esta propiedad para localizaciones mediante elementos regulares de extensiones PBW torcidas. En relación con la propiedad RNP, estudiamos las nociones de anulador débil e ideal primo asociado nilpotente, y formulamos varios resultados que extienden o contribuyen a los correspondientes en la literatura. Adicionalmente, investigamos los ideales primos asociados de módulos inducidos sobre álgebras cuánticas y los caracterizamos. Nuestro trabajo es una contribución a una extensa investigación presentada por diferentes autores sobre las propiedades teóricas de los anillos no conmutativos y su descripción de los ideales primos asociados. Por otro lado, motivados por la investigación sobre los ideales primos adjuntos del módulo polinomial inverso sobre anillos de polinomios torcidos de tipo automorfismo, y considerando una familia muy importante de álgebras cuánticas conocidas como polinomios de Ore torcidos de orden superior, caracterizamos los ideales primos adjuntos de esos módulos sobre polinomios de Ore torcidos. También formulamos resultados sobre la dimensión uniforme y los primos asociados de módulos inducidos sobre algunas familias de álgebras cuánticas. Finalmente, estudiamos la dimensión couniforme para módulos polinomiales inversos sobre polinomios de Ore torcidos. (Texto tomado de la fuente)
On the noncommutative differential geometry of semi-graded Artin-Schelter regular algebras
(Universidad Nacional de Colombia, 2024) Rubiano Suárez, Andrés Alejandro; Reyes Villamil, Milton Armando; Rubiano Suárez, Andrés Alejandro [0000180012]; Rubiano Suárez, Andrés Alejandro [g4fvDBIAAAAJ&hl]; Rubiano Suárez, Andrés Alejandro [0009-0009-1633-8018]; Rubiano Suárez, Andrés Alejandro [Andres-Rubiano-11];
In this thesis we investigate the noncommutative differential geometry of families of semi-graded rings or semi-graded Artin-Schelter regular algebras. With this end, we present some definitions and preliminaries on these algebraic structures and on the smooth geometry of noncommutative algebras. We also present some differences between Artin-Schelter regular algebras and semi-graded Artin-Schelter regular algebras. Next, we concentrate our research on the smooth geometry of bi-quadratic algebras on three generators having PBW basis, double extension regular algebras of type (14641), diffusion algebras and skew Poincaré-Birkhoff-Witt extensions. Finally, we establish some related ideas with the aim of extending the Serre duality theorem for noncommutative projective schemes on graded algebras to the setting of semi-graded algebras. (Texto tomado de la fuente)
Métodos numéricos eficientes para ecuaciones diferenciales elípticas en medios porosos
(Universidad Nacional de Colombia, 2025) Cuervo Fernández, Omar Andrés; Galvis Arrieta, Juan Carlos; Galvis Arrieta, Juan Carlos [0000-0001-8904-1877]; Diseño y Análisis de Métodos Numéricos
Modeling flow and transport in porous media with uncertain properties is of critical importance in many scientific and engineering applications, including groundwater hydrology, petroleum reservoir simulation, and environmental risk assessment. The spatial heterogeneity of properties such as permeability is often modeled through stochastic processes defined by covariance functions. These formulations require the use of efficient numerical methods to simulate and quantify uncertainty. A brief review of modeling strategies and numerical approximations in this context is included in this thesis. The random coefficient field is represented using Karhunen Loève (KL) expansions and related spectral techniques, allowing for finite-dimensional parameterizations of infinite-dimensional random fields. This facilitates the application of both sampling-based and deterministic approximation methods in stochastic settings. This thesis investigates numerical methods based on the combination of finite element discretizations with random sampling techniques, including Monte Carlo and polynomial chaos expansions. The spatial discretization uses standard conforming and non-conforming methods, while uncertainty is treated using random inputs derived from the KL expansion. The main contributions of this thesis are as follows. First, the work focuses on simple design and efficient iterative solvers based on null space strategies and operator splitting that weakly depend on the coefficient realization dividing the computation into two stages, and offline stage that does not depend on the coefficient and an online stage that requires few operations per iterations. Second, a modified Monte Carlo method is developed and analyzed, in which the Galerkin matrices are approximated through sample-based numerical integration. This approach extends the standard Monte Carlo method and includes the study of cases where it provides more accurate and computationally efficient approximations of the quantities of interest. Numerical experiments are presented to illustrate the performance of the proposed methods. Initial theoretical estimates are provided for the approximation error, and their behavior is evaluated numerically across a range of model problems and solver configurations. Finally, this thesis concludes with a summary of the key findings from each chapter, and outlines several directions for future research, including the integration of the proposed solver strategies into multilevel Monte Carlo framework and adaptive sampling schemes. (Tomado de la fuente)
Morphisms between semi-graded rings and polynomial applications
(Universidad Nacional de Colombia, 2023) Ramírez Cubillos, María Camila; Reyes Villamil, Milton Armando
In this thesis, we study several kinds of morphisms between families of semi-graded rings with their corresponding polynomial applications. First, we present some ring-theoretical notions of these objects that are necessary throughout the thesis. With the aim of showing the generality of these rings in areas such as ring theory and noncommutative geometry, we include a non-exhaustive list of noncommutative algebras that are particular examples of these rings. Second, we develop a theory of cv-polynomials for iterated Ore extensions over arbitrary rings extending some results in the literature. We characterize these polynomials by using inner derivations of the coefficient ring, and also consider the problem of isomorphisms between these extensions. We illustrate our treatment with several noncommutative algebras. Third, for double Ore extensions introduced by Zhang and Zhang in the problem of classification of Artin-Schelter regular algebras of dimension four, we propose a theory of homomorphisms between them by introducing an adequate notion of cv-polynomial, and show that the computation of homomorphisms corresponding to these polynomials is non-trivial. Since there are no inclusions between the classes of all double Ore extensions of an algebra and of all length two iterated Ore extensions of the same algebra, we also present a comparison between theories of cv-polynomials between both families of algebras. We illustrate our results with Nakayama automorphisms of trimmed double Ore extensions. Finally, motivated by the research on maps between noncommutative projective spaces over $\mathbb{N}$-graded rings in the sense of Rosenberg and Van den Bergh, and the notion of schematicness introduced by Van Oystaeyen and Willaert to $\mathbb{N}$-graded rings with the aim of formulating a noncommutative scheme theory \`a la Grothendieck, we investigate maps in the setting of noncommutative projective spaces over schematic semi-graded rings, and extend different results from the category of schematic $\mathbb{N}$-graded rings to the category of schematic semi-graded rings.
On the cluster complex and theta functions for cluster Poisson varieties
(Universidad Nacional de Colombia, 2024) Melo López, Astrid Carolina; Nájera Chávez, Alfredo; Dorado Correa, Ivon Andrea; Melo Lopez, Astrid Carolina; https://orcid.org/0009-0006-9740-0386; https://www.researchgate.net/profile/Astrid-Carolina-Melo-Lopez
In this thesis, we establish an explicit description of the cluster complex, denoted as ∆^+_s(X), associated to a skew-symmetrizable cluster Poisson variety X and a seed s. Our approach involves a detailed description of the cones of ∆^+_s(X) and their facets using c-vectors. Specifically, each c-matrix determines a polyhedral cone of ∆^+_s(X), and its columns (c-vectors) determine the equations of the supporting hyperplanes of the respective cone. Additionally, we can describe the dimension of these cones through the implicit equalities of the system of inequalities derived from the associated c-matrix. Furthermore, we give formulas for the theta functions parametrized by the integral points of ∆^+s(X) using F -polynomials. In the special case where X is skew-symmetric and the quiver Q associated to s is acyclic, we describe the normal vectors of the supporting hyperplanes of the cones of ∆^+_s(X) using g-vectors of (non-necessarily rigid) objects in Kb(proj kQ). We use this to provide classes of examples where the ray generators of the maximal cones of ∆^+_s(X) can be expressed in terms of c-vectors. Simultaneously, we investigate a description of perfect matchings of snake graphs using routes (certain families of non-intersecting lattice paths) in connected acyclic-directed graphs. With this description, we define the poset of k-routes and use the well-known relation between perfect matchings and F-polynomials to give an alternative approach to describe F-polynomials for surface type cluster algebras, both recursively and non-recursively, using k-routes. Furthermore, we identify some open problems and propose strategies to address them in future research projects, ensuring the continuity and expansion of this work.
Algebro-geometric characterizations of commuting differential operators in semi-graded rings
(Universidad Nacional de Colombia, 2023) Niño Torres, Diego Arturo; Reyes Villamil, Milton Armando
In this thesis, we study algebro-geometric characterizations of commuting differential operators in families of semi-graded rings. First, we present some ring-theoretical notions of semi-graded rings that are necessary throughout the thesis. We include a non-exhaustive list of noncommutative rings that are particular examples of these rings. Second, to motivate the study of commuting differential operators beloging to noncommutative algebras, and hence to develop a possible Burchnall-Chaundy (BC) theory for them, we review algebraic and matrix results appearing in the literature on the theory of these operators in some families of semi-graded rings. Third, we introduce the notion of pseudo-multidegree function as a generalization of pseudo-degree function, and hence we establish a criterion to determine whether the centralizer of an element has finite dimension over a noncommutative ring having PBW basis. In this way, we formulate a BC theorem for rings having pseudo-multidegree functions. We illustrate our results with families of algebras appearing in ring theory and noncommutative geometry. Fourth, we develop a first approach to the BC theory for quadratic algebras having PBW bases defined by Golovashkin and Maksimov. We prove combinatorial properties on products of elements in these algebras, and then consider the notions of Sylvester matrix and resultant for quadratic algebras with the purpose of exploring common right factors. Then, by using the concept of determinant polynomial, we formulate the version of BC theory for these algebras. We present illustrative examples of the assertions about these algebras. Finally, we establish some bridging ideas with the aim of extending results on centralizers for graded rings to the setting of semi-graded rings.
Involutive and SAGBI bases for skew PBW extensions
(Universidad Nacional de Colombia, 2023) Suárez Gómez, Yésica Paola; Reyes, Armando
In this thesis, we study homological properties and SAGBI and Involutive bases of the noncommutative rings known as skew PBW extensions. First, we present some ring- theoretical notions of these extensions that are necessary throughout the thesis. With the aim of showing the generality of these objects in areas such as ring theory and noncommutative geometry, we include a non-exhaustive list of noncommutative algebras that are particular examples of these rings. Second, we characterize several homological properties of these ex- tensions. We provide a new and more general filtration to these extensions, and introduce the notion of σ-filtered skew PBW extension with the aim of studying its homological properties. We show that the homogenization of a σ-filtered skew PBW extension over a coefficient ring is a graded skew PBW extension over the homogenization of such a ring. By using this fact, we prove that if the homogenization of the coefficient ring is Auslander-regular, then the homogenization of the extension is a domain Noetherian, Artin-Schelter regular, Zariski and (ungraded) skew Calabi-Yau. Third, we present our proposal of SAGBI bases theory for skew PBW extensions over algebras. We consider the notion of reduction which is necessary in the characterization of these bases, and then establish an algorithm to find the normal form of an element. Then, we define what a SAGBI basis is, and formulate a criterion to determine when a subset of a skew PBW extension over a field is a SAGBI basis. In addition, we establish an algorithm to find a SAGBI basis from a subset contained in a subalgebra of a skew PBW extension. We illustrate our results with different examples of noncommutative algebras. We also investigate the problem of poly- nomial composition for SAGBI bases of subalgebras of skew PBW extensions. Finally, we present a theory of Involutive bases for skew PBW extensions over fields. We consider the notions of weak and strong Involutive bases, and then we define the involutive reduction process and involutive remainder that are necessary for the characterization of weak (strong) Involutive bases. Next, we introduce the notion of standard Involutive representation for elements of a subset of a skew PBW extension. Also, we give the definition of minimal Involutive basis and show the existence of a monic, involutively autoreduced, minimal Involutive basis. Finally, we establish different algorithms that compute involutive standard representations, principal involutive autoreduction, and an Involutive basis of a left ideal of a skew PBW extension. In this way, the existence of a finite Involutive basis for these ideals is proved by assuming that the involutive division is constructive Noetherian.
On the noncommutative geometry of semi-graded rings
(Universidad Nacional de Colombia, 2022) Chacón Capera, Andrés; Reyes Villamil, Milton Armando; Sac2
En esta tesis, establecemos diversas caracterizaciones topológicas del espectro no conmutativo de anillos semi-graduados al considerar la noción de topología débil de Zariski. Con este propósito, formulamos condiciones necesarias o suficientes para garantizar que familias de estos anillos definidos por endomorfismos y derivaciones sean anillos NI o anillos NJ. Presentamos resultados sobre la caracterización de diferentes tipos de elementos de anillos no conmmutativos tales como idempotentes, unidades, von Neumann regulares, π-regulares, y elementos limpios. También investigamos las nociones de anillo fuertemente armónico y de Gelfand sobre dichas familias de anillos semi-graduados. Nuestros resultados generalizan tratamientos desarrollados para anillos conmutativos, anillos de polinomios torcidos, y variadas familias de anillos N-graduados, y contribuyen a la investigación sobre estos temas que ha sido llevada a cabo parcialmente en la literatura. Por otra parte, investigamos la esquematicidad y el teorema de Serre-Artin-Zhang-Verevkin para anillos semi-graduados. Más exactamente, para los polinomios de Ore de orden superior generados por relaciones homogéneas y las extensiones torcidas de Poincaré-Birkhoff-Witt, formulamos condiciones necesarias o suficientes para garantizar la esquematicidad de estas familias de anillos. Desarrollamos una teoría de esquemas no conmutativa para anillos semi-graduados que no son necesariamente conexos y N-graduados. Con esta teoría, demostramos el teorema de Serre-Artin-Zhang-Verevkin para diversas familias de álgebras no N-graduadas que incluyen diferentes clases de anillos no conmutativos que surgen en la teoría de anillos y la geometría algebraica no conmutativa. Nuestro tratamiento contribuye a la investigación sobre este teorema desarrollada en la literatura. (Texto tomado de la fuente)
On generalized multiscale methods for flow in complex porous media and their applications
(2023-06-27) Contreras Hernandez, Luis Fernando; Fernando, Luis; Galvis Arrieta, Juan Carlos
In this document, the Generalized Multiscale Finite Element Method (GMsFEM) is studied, which deals with constructing multiscale spectral basis functions designed for high-contrast multiscale problems. The multiscale basis functions are built from the product of the eigenvectors, computed from a local spectral problem and a partition of unity over the study domain. The eigenvalues detect essential features of the solutions that are not captured by the initial multiscale basis functions. This document reviews the general convergence study where the error estimates are written in terms of the eigenvalues associated with the eigenvectors not used in the construction. Error analysis involves local and global norms that measure the convergence speed of the expansion of the solution in terms of local eigenvectors; this is achieved with a careful choice of the initial multiscale basis functions and the configuration of the eigenvalue problems. Two novel important numerical applications are presented: the first is the free-boundary dam problem posed on a heterogeneous high-contrast medium, where we introduce a fictitious time variable that motivates an adequate time discretization that can be understood as a fixed-point iteration. For the steady-state solution, we use the duality method to deal with the multivalued nonlinear terms involved; then, efficient approximations of pressure and saturation are calculated using the GMsFEM method. The second application is the solution of a parabolic equation. Here implementing time discretizations, such as finite differences or exponential integrators in the presence of a high contrast coefficient, it may not be practical in because each time iteration one needs the computation of matrix operators involving very large and extremely ill-conditioned sparse matrices. The GMsFEM is essential since it allows obtaining the solution of the problem more simply, allowing to combine the GMsFEM with the method of exponential integrators in time to get a good approximation of the final temporary solution. (Texto tomado de la fuente)
Conte de simetrías en objetos discretos
(Universidad Nacional de Colombia, 2023-01-31) Moreno Garzón, Andrés Ricardo; Ramírez, José Luis; Discremath: Matemáticas Discretas y Ciencias de la Computación
En este trabajo se estudia la ocurrencia en palabras, particiones de conjuntos y composiciones de enteros de las estadísticas pico simétrico, asociada a la ocurrencia del patrón 121, y pico asimétrico, asociada a la ocurrencia de los patrones 132 y 231, utilizando métodos combinatorios y analíticos como conteo directo, funciones generatrices, fórmulas recursivas y el método simbólico. En palabras, con base en las ideas y resultados obtenidos por Asakly en [1] se obtiene una nueva demostración por conteo para la cantidad de picos simétricos y asimétricos en palabras. Posteriormente se extienden estas ideas, métodos y resultados para el estudio de la ocurrencia de picos simétricos y asimétricos en particiones de conjuntos y composiciones donde no se tienen resultados previos asociados, obteniendo fórmulas cerradas para la cantidad de picos simétricos y asimétricos y extendiendo estos resultados a composiciones restringidas y composiciones palíndromas. (Texto tomado de la fuente)
Algebraic properties of weak quantum symmetries
(Universidad Nacional de Colombia, 2023-07-24) Calderón Mateus, Fabio Alejandro; Chelsea, Walton; Milton Armando, Reyes Villamil; Calderón, Fabio [0000-0003-1777-0805]
This thesis investigates the properties of weak bialgebras and weak Hopf algebras, their (co)representations, and applications in groupoids, path algebras, and Lie algebroids. The research employs algebraic and categorical techniques to explore the foundational properties of these structures, establishing connections between algebraic and categorical frameworks, and addressing open problems related to their actions on noncommutative graded algebras. By combining theoretical findings and practical examples, this work enhances our understanding of weak Hopf algebras as symmetry generators and their broader implications in various mathematical contexts. Our results contribute to the field of noncommutative algebra and Hopf algebras, paving the way for future research in these areas. (Texto tomado de la fuente)
Propiedades de algunos sistemas de polinomios ortogonales Sobolev en varias variables
(Universidad Nacional de Colombia, 2022-02-01) Salazar Morales, Omar; Dueñas Ruiz, Herbert Alonso; Grupo de Investigación en Polinomios Ortogonales y Aplicaciones
En este trabajo estudiamos algunas propiedades algebraicas y analíticas de los polinomios ortogonales en varias variables reales con respecto a un producto interno Sobolev continuo-discreto. Consideramos los polinomios Sobolev sobre diferentes dominios, a saber: un dominio producto; la bola unitaria; el simplex; y el cono. Nuestros principales resultados consisten en un método iterativo de construcción de los polinomios ortogonales con respecto al producto interno, propiedades que involucran su parte principal (continua), una fórmula de conexión, y algunos resultados sobre ecuaciones diferenciales parciales. Con el fin de ilustrar nuestras principales ideas, al final de este trabajo presentamos varios ejemplos numéricos en dos variables. Además, discutimos algunos problemas abiertos.
Effective computation of invariants of finite topological spaces
(Universidad Nacional de Colombia, 2021) Cuevas Rozo, Julián Leonardo; Lambán Pardo, Laureano; Romero Ibáñez, Ana; Sarria Zapata, Humberto
Hasta el momento, los métodos conocidos para el cálculo de invariantes de espacios topológicos finitos eran aplicables solamente a los posets de caras de complejos simpliciales o de CW-complejos regulares. En este trabajo hemos desarrollado versiones constructivas de algunos resultados teóricos de diferentes autores acerca de espacios finitos, produciendo en particular nuevos algoritmos para el cálculo explícito de algunos complejos de cadenas asociados a espacios finitos h-regulares y sus correspondientes generadores. Hasta donde sabemos, nuestro programa es el único software capaz de calcular grupos de homología de espacios topológicos finitos trabajando directamente sobre los posets. Hemos mejorado nuestros algoritmos sobre espacios finitos h-regulares mediante el uso de campos de vectores discretos, produciendo un nuevo algoritmo para construir dichos campos discretos definidos directamente sobre el poset, además de crear un proceso de h-regularización de espacios finitos permitiendo así ampliar la familia de espacios finitos h-regulares conocidos en la literatura. También hemos presentado una interfaz entre los sistemas de álgebra computacional SageMath y Kenzo. Nuestro trabajo ha permitido que ambos sistemas colaboren mutuamente en algunos cálculos que no pueden ser hechos de manera independiente por dichos programas. Más aún, hemos creado un módulo en SageMath implementando espacios topológicos finitos y algunos conceptos relacionados. Finalmente, hemos considerado algunas estrategias para estudiar diferentes alternativas para calcular campos de vectores discretos de mayor longitud sobre espacios finitos, haciendo uso de algunas técnicas de aprendizaje automático para obtener campos de vectores discretos de la mayor longitud posible. (Texto tomado de la fuente).
Procesos de conteo, sobredispersión y extensiones
(Universidad Nacional de Colombia, 2021-04-19) Cifuentes Amado, Maria Victoria; Cepeda-Cuervo, Edilberto
Se presenta una revisión detallada de modelos sobredispersos de regresión lineal y no lineal para datos de conteo, desde un enfoque Bayesiano. Se propone la función de distribución beta inclinada binomial, se estudian sus propiedades y se proponen los modelos de regresión lineal, donde se asignan estructuras de regresión a la media, parámetro de dispersión y parámetro de mixtura. Adicionalmente, se presentan las reparametrizaciones de las distribuciones beta binomial y binomial negativa, en términos de la media, y se establecen los modelos de regresión Bayesiana, proponiendo nuevas variables de trabajo para el parámetro de dispersión, que reducen la autocorrelación y mejoran la convergencia de las cadenas. Se define una extensión a modelos de regresión no lineal y se propone el algoritmo Bayesiano para este caso. Se definen nuevos modelos de regresión no lineal con exceso de ceros y se extiende la metodología Bayesiana propuesta en [21] para estos modelos: se desarrolla el algoritmo de Metropolis-Hastings y se proponen las variables de trabajo requeridas, a partir del método de aumento de datos de [74]. Se proponen modelos subdispersos no lineales doblemente generalizados para datos de conteo que presentan subdispersión con respecto a las distribuciones Poisson o binomial, y se definen funciones de cuasi-verosimilitud adecuadas, a partir del enfoque de [106], para el ajuste de los modelos Bayesianos. Finalmente, se proponen nuevos procesos de Poisson no homogéneos cíclicos y se definen modelos autoregresivos no lineales doblemente generalizados para series de conteo, basados en distribuciones de conteo subdispersas y se aplica el paradigma Bayesiano para la estimación. (Texto tomado de la fuente).
El problema de Cauchy asociado a una generalización de la ecuación Zakharov-Kuznetsov
(Universidad Nacional de Colombia, 2021-09-21) Rippe Espinosa, Miguel Angel; Rodríguez Blanco, Guillermo
En el presente trabajo, se tratan cuestiones tales como el buen planteamiento local en los espacios de Sobolev, espacios anisotrópicos con pesos y la existencia de ondas solitarias para el problema de valor inicial asociado a la ecuación: %En el presente trabajo, se estudia el buen planteamiento local en los espacios de Sobolev $H^s(\mathbb{R}^2)$ para $s>2$, del problema de valor inicial asociado a la ecuación: $$u_t-\partial_x\piz D_x^{1+\alpha}\pm D_y^{1+\beta}\pde u + u^pu_x=0,$$ donde $0\leq \alpha,\beta\leq1$ y $p\in\mathbb{Z}^+$, $x,y,t\in\Rn$. (Texto tomado de la fuente).
El problema de Cauchy asociado a una generalización de la ecuación Zakharov-Kuznetsov sobre el cilindro
(Universidad Nacional de Colombia, 2021-07) Albarracin Hernandez, Carolina; Rodríguez Blanco, Guillermo
In this work, we study questions related to the local well-posedness for the initial value problem associated to the partial differential equation, u_{t} − ∂_{x}(D_{x}^{α+1}u ± D_{y}^{β+1}u) + u^{p}u_{x} = 0, where 0 ≤ α, β ≤ 1 and p ∈ Z ^{+}, in the standard, anisotropic and weighted Sobolev spaces in R × T and T^{2}. For this purpose, we use parabolic regularization, localized Strichartz and energy estimates, together with a compactness argument, as well as, commutator estimates and remarkable properties of the Stein derivative. In addition, we show the existence of certain type of solitary wave in the cylinder.
Categorification of Some Integer Sequences and Its Applications
(Universidad Nacional de Colombia, 2020-07-30) Fernández Espinosa, Pedro Fernando; Moreno Cañadas, Agustín; TERENUFIA-UNAL
Categorification of real valued sequences, and in particular of integer sequences is a novel line of investigation in the theory of representation of algebras. In this theory introduced by Ringel and Fahr, numbers of a sequence are interpreted as invariants of objects of a given category. The categorification of the Fibonacci numbers via the structure of the Auslander-Reiten quiver of the 3-Kronecker quiver is an example of this kind of identifications. In this thesis, we follow the ideas of Ringel and Fahr to categorify several integer sequences but instead of using the 3-Kronecker quiver, we deal with a kind of algebras introduced recently by Green and Schroll called Brauer configuration algebras. Relationships between these algebras, some matrix problems and rational knots are used to interpret numbers in some integer sequences as invariants of indecomposable modules over path algebras of the 2-Kronecker quiver and the four subspace quiver. The results enable us to define the message of a Brauer Configuration and labeled Brauer configurations in order to give an interpretation of the number of perfect matchings of snake graphs, the number of homological ideals of some Nakayama algebras, and the number of k-paths linking two fixed points (associated to the Lindström problem) in a quiver as specializations of indecomposable modules over suitable Brauer configuration algebras. Actually, this setting can be also used to define the Gutman index of a tree (or the trace norm of a digraph, which is a fundamental notion in the topological index theory), magic squares, and different parameters of traffic flow models in terms of this kind of algebras. This research was partially supported by COLCIENCIAS convocatoria doctorados nacionales 785 de 2017.
Dynkin Functions and Its Applications
(Universidad Nacional de Colombia, 2020-10) Bravo Rios, Gabriel; Moreno Cañadas, Agustín; TERENUFIA-UNAL
Dynkin functions were introduced by Ringel as a tool to investigate combinatorial properties of hereditary artin algebras. According to Ringel, a Dynkin function consists of four sequences associated to An, Bn, Cn, Dn and five single values associated to the diagrams E6, E7, E8, F4 and G2. He also proposes to create an On-line Encyclopedia of Dynkin functions (OEDF) with the same purposes as the famous OEIS. Dynkin functions arise from the context of categorification of integer sequences, which according to Ringel and Fahr it means to consider suitable objects in a category instead of numbers of a given integer sequence. They gave a categorification of Fibonacci numbers by using the Gabriel's universal covering theory and the structure of the Auslander-Reiten quiver of the 3-Kronecker quiver. For instance, if Λ denotes a hereditary artin algebra associated to a Dynkin diagram ∆n then r(∆n) the number of indecomposable modules, a(∆n) the number of antichains in mod Λ, and tn(∆n) the number of tilting modules are Dynkin functions. In particular, we are focused on the way that some Dynkin functions act on Dynkin diagrams of type An. In this work, we follow the ideas of Ringel regarding Dynkin functions by investigating the number of sections in the Auslander-Reiten quiver of algebras of finite representation type. Dyck paths categories are introduced as a combinatorial model of the category of representations of quivers of Dynkin type An and it is shown an algebraic interpretation of frieze patterns as a direct sum of indecomposable objects of the category of Dyck paths. In particular, it is proved that there is a bijection between some Dyck paths and perfect matchings of some snake graphs. The approach allows us to give formulas for cluster variables in cluster algebras of Dynkin type An in terms of Dyck paths. At last but not least, it is introduced some Brauer configuration algebras such that the dimension of these algebras and its corresponding centers can be obtained via some combinatorial properties of the Catalan triangle. This research was partially supported by COLCIENCIAS convocatoria doctorados nacionales 785 de 2017.
Problema de Cauchy asociado a un ecuación del tipo KZKP con dispersión transversal fraccionaria
(2020-07-10) Morales Paredes, Jorge; Soriano Méndez, Félix Humberto; ECUACIONES DE EVOLUCIÓN
In this work it shall be studied the Cauchy problem for the following ZK-KP type equation u_{t}=u_{xxx}+HD_x^{\alpha}u_{yy}+uu_{x}, u(0)=\psi where 1\leq \alpha\leq 1, H denotes the Hilbert transform in the x variable and D_x is the \alpha^{th} fractional derivative defined via Fourier transform by D_x^{\alpha}f=|\xi|^{\alpha}\widehat{f}. It is showed the local well posedness in the ansisotropic Sobolev spaces H^{s_1,s_2} and examined ill-posedness properties for 1\leq \alpha < 0
The Auslander-Reiten Quiver of Equipped Posets of Finite Growth Representation Type, some Functorial Descriptions and Its Applications
(2020-05-18) Marín Gaviria, Isaías David; Moreno Cañadas, Agustín; Mendoza Hernández, Octavio; TERENUFIA-UNAL
The theory of representation of partially ordered sets or posets was introduced in the early 1970's as an effort to give an answer to the second Brauer-Thrall conjecture. Recall that one of the main goals of this theory is to give a complete description of the indecomposable objects of the category of representations of a given poset. Perhaps the most useful tool to obtain such classification are the algorithms of differentiation. For instance, Nazarova and Roiter introduced an algorithm known as the algorithm of differentiation with respect to a maximal point which allowed to Kleiner in 1972 to obtain a classification of posets of finite representation type. Soon afterwards between 1974 and 1977, this algorithm was used in 1981 by Nazarova and Zavadskij in order to give a criterion for the classification of posets of finite growth representation type. Actually, several years later, Zavadskij himself described the structure of the Auslander-Reiten quiver of this kind of posets, to do that, it was established that such an algorithm is in fact a categorical equivalence. Since the theory of representation of posets was developed in the 1980's and 1990's for posets with additional structures, for example, for posets with involution or for equipped posets by Bondarenko, Nazarova, Roiter, Zabarilo and Zavadskij among others. It was necessary to define a new class of algorithms to classify posets with these additional structures. In fact, Zavadskij introduced 17 algorithms. Algorithms, I-V (and some additional differentiations) were used by him and Bondarenko to classify posets with involution, whereas algorithms I, VII-XVII were used to classify equipped posets. In particular, algorithms I, VII, VIII and IX were used to classify equipped posets of finite growth representation type without paying attention to the behavior of the morphisms of the corresponding categories. In other words, it was obtained a classification of the objects without proving that the algorithms used to tackle the problems are in fact categorical equivalences, therefore, the main problem of the theory of the algorithms of differentiation consists of giving a detailed description of the behavior of the morphisms under these additive functors, such description allows to give a deep understanding of the Auslander-Reiten quiver of the corresponding categories. On the other hand, in the last few years has been noted a great interest in the application of the theory of representation of algebras in different fields of computer science, for example, in combinatorics, information security and topological data analysis. Ringel and Fahr, for instance, gave a categorification of Fibonacci numbers by using the 3-Kronecker quiver whereas representation of posets and the theory of posets have been used to analyze tactics of war and cyberwar. Besides, the theory of Auslander has been used to analyze big data via the homological persistent theory. In this research, it is proved that the algorithms of differentiation VIII-X induce categorical equivalences between some quotient categories, giving a description of the Auslander-Reiten quiver of some equipped posets by using the evolvent associated to these kind of posets. In this work, ideas arising from the theory of representation of equipped posets are used to give a categorification of Delannoy numbers. Actually, such numbers are interpreted as dimensions of some suitable equipped posets. We also interpret the algorithm of differentiation VII as a steganographic algorithm which allows to generate digital watermarks, such an algorithm can be also used to describe the behavior of some kind of informatics viruses, in fact, it is explained how this algorithm describe the infection-detection process when a computer network is affected for this type of malware. At last but not least, we recall that the theory of representation of equipped posets is a way to deal with the homogeneous biquadratic problem which is an open matrix problem, in this case, with respect to a pair of fields (F;G) with G a quadratic extension of the field F with respect to a polynomial of the form t^2 + q, q∈F. Actually, explicit solutions to this problem were given by Zavadskij who rediscovered in 2007 the Krawtchouk matrices introducing an interesting θ-transformation as well. In this research, such Krawtchouk matrices are used in order to give explicit solutions to non-linear systems of differential equations of the form X'(t)+AX^2 (t)=B, where X(t); X'(t); A and B are n×n square matrices. Tools arising from this solution are called in this work the Zavadskij calculus. This research was partially supported by COLCIENCIAS convocatoria doctorados nacionales 727 de 2015.

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