Instituto de Matemática

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Now showing 1 - 16 of 16
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    A look at generalized degenerate Bernoulli and Euler matrices
    (2023-06-16) Juan Hernández
    In this paper, we consider the generalized degenerate Bernoulli/Euler polynomial matrices and study some algebraic properties for them. In particular, we focus our attention on some matrix-inversion formulae involving these matrices. Furthermore, we provide analytic properties for the so-called generalized degenerate Pascal matrix of the first kind, and some factorizations for the generalized degenerate Euler polynomial matrix.
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    Behind Jarratt’s Steps: Is Jarratt’s Scheme the best version of itself?
    (2023-06-15) Elaine Segura
    In this paper, we analyze the stability of the family of iterative methods designed by Jarratt using complex dynamics tools. This allows us to conclude whether the scheme known as Jarratt’s method is the most stable among all the elements of the family. We deduce that classical Jarratt’s scheme is not the only stable element of the family. We also obtain information about the members of the class with chaotical behavior. Some numerical results are presented for confirming the convergence and stability results.
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    Combined matrices and conditioning
    (2021-08-12) Máximo Santana
    In this work, we study a lower bound of the condition number of a matrix by its combined matrix. In particular, we construct a special combined matrix in such a way that the sums of its columns are lower bounds of the condition number of the matrix. Cases for special matrices as unitary matrices are considered.
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    Combined matrix of diagonally equipotent matrices
    (2023-09-12) Máximo Santana
    Let (𝐴)=𝐴∘𝐴−𝑇 be the combined matrix of an invertible matrix 𝐴, where ∘ means the Hadamard product of matrices. In this work, we study the combined matrix of a nonsingular matrix, which is an 𝐻 -matrix whose comparison matrix is singular. In particular, we focus on (𝐴) when 𝐴 is diagonally equipotent, and we study whether(𝐴) is an 𝐻 -matrix and to which class it belongs. Moreover, we give some properties on the diagonal dominance of these matrices and on their comparison matrices.
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    Convergence and stability of a new parametric class of iterative processes for nonlinear systems
    (2023-03-16) Antmel Rodríguez
    In this manuscript, we carry out a study on the generalization of a known family of multipoint scalar iterative processes for approximating the solutions of nonlinear systems. The convergence analysis of the proposed class under various smooth conditions is provided. We also study the stability of this family, analyzing the fixed and critical points of the rational operator resulting from applying the family on low-degree polynomials, as well as the basins of attraction and the orbits (periodic or not) that these points produce. This dynamical study also allows us to observe which members of the family are more stable and which have chaotic behavior. Graphical analyses of dynamical planes, parameter line and bifurcation planes are also studied. Numerical tests are performed on different nonlinear systems for checking the theoretical results and to compare the proposed schemes with other known ones.
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    Darboux transformations for orthogonal differential systems and differential Galois theory
    (2023-03-31) Primitivo Acosta
    Darboux developed an ingenious algebraic mechanism to construct infinite chains of “integrable’’ second-order differential equations as well as their solutions. After a surprisingly long time, Darboux’s results were rediscovered and applied in many frameworks, for instance in quantum mechanics (where they provide useful tools for supersymmetric quantum mechanics), in soliton theory, Lax pairs and many other fields involving hierarchies of equations. In this paper, we propose a method which allows us to generalize the Darboux transformations algorithmically for tensor product constructions on linear differential equations or systems. We obtain explicit Darboux transformations for third-order orthogonal systems (so(3,CK) systems) as well as a framework to extend Darboux transformations to any symmetric power of SL(2,C) -systems. We introduce SUSY toy models for these tensor products, giving as an illustration the analysis of some shape invariant potentials. All results in this paper have been implemented and tested in the computer algebra system Maple.
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    Desarrollo profesional del docente universitario de matemática en la República Dominicana
    (2022-10-28) Alicia Virginia Martín S.; Carmen Evarista Matías
    Objetivo: El artículo expone una síntesis del modelo teórico aportado para dirigir el proceso de desarrollo profesional de los docentes universitarios que imparten Matemática en República Dominicana. Métodos: Se empleó la revisión documental, los estudios lógicos e históricos y la modelación. Resultado: Se conformó un modelo teórico como marco de referencia para la dirección del desarrollo profesional del docente universitario que imparte Matemática y una estrategia pedagógica estructurada en tres fases que posibilita la concreción en la práctica de la propuesta teórica. Transformación, ISSN: 2077-2955, RNPS: 098, enero-abril 2023, 19 (1), 195-214 Universidad de Camagüey “Ignacio Agramonte Loynaz” 196 Conclusión: El modelo teórico y la estrategia pedagógica que viabiliza la dirección del desarrollo profesional, permiten garantizar transformaciones pertinentes y de calidad, acorde con las exigencias de la sociedad.
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    Design and dynamical behavior of a fourth order family of iterative methods for solving nonlinear equations
    (2024-02-28) Arleen Ledesma Collado
    In this paper, a new fourth-order family of iterative schemes for solving nonlinear equations has been proposed. This class is parameter-dependent and its numerical performance depends on the value of this free parameter. For studying the stability of this class, the rational function resulting from applying the iterative expression to a low degree polynomial was analyzed. The dynamics of this rational function allowed us to better understand the performance of the iterative methods of the class. In addition, the critical points have been calculated and the parameter spaces and dynamical planes have been presented, in order to determine the regions with stable and unstable behavior. Finally, some parameter values within and outside the stability region were chosen. The performance of these methods in the numerical section have confirmed not only the theoretical order of convergence, but also their stability. Therefore, the robustness and wideness of the attraction basins have been deduced from these numerical tests, as well as comparisons with other existing methods of the same order of convergence.
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    Differential Properties of Jacobi-Sobolev Polynomials and Electrostatic Interpretation
    (2023-08-06) Juan Toribio-Milane; Javier Quintero-Roba; Héctor Pijeira-Cabrera
    We study the sequence of monic polynomials {𝑆𝑛}𝑛⩾0, ortogonal with respect to the Jacobi-Sobolev inner product ⟨ 𝑓, 𝑔 ⟩𝑠 = ∫_{-1}^{1} 𝑓(𝑥)𝑔(𝑥)𝑑𝜇𝛼,𝛽(𝑥) + ∑_{𝑗=1}^{𝑁} ∑_{𝑘=0}^{𝑑𝑗} 𝑑𝑗𝜆𝑗,𝑘 𝑓(𝑘)(𝑐𝑗)𝑔(𝑘)(𝑐𝑗), where 𝑁, 𝑑𝑗 ∈ ℤ^+, 𝜆𝑗,𝑘 ⩾ 0, 𝑑𝜇𝛼,𝛽(𝑥) = (1−𝑥)^𝛼(1+𝑥)^𝛽𝑑𝑥, 𝛼,𝛽 > −1, and 𝑐𝑗∈𝑅∖(−1,1). A connection formula that relates the Sobolev polynomials 𝑆𝑛 with the Jacobi polynomials is provided, as well as the ladder differential operators for the sequence {𝑆𝑛}⩾0 and a second-order differential equation with a polynomial coefficient that they satisfied. We give sufficient conditions under which the zeros of a wide class of Jacobi-Sobolev polynomials can be interpreted as the solution of an electrostatic equilibrium problem of n unit charges moving in the presence of a logarithmic potential. Several examples are presented to illustrate this interpretation.
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    Hessenberg – Sobolev matrices and Favard type theorem
    (2022-12-01) Ignacio Pérez Yzquierdo
    We study the relation between certain non-degenerate lower Hessenberg infinite matrices G and the existence of sequences of orthogonal polynomials with respect to Sobolev inner products. In other words, we extend the well-known Favard theorem for Sobolev orthogonality. We characterize the structure of the matrix G and the associated matrix of formal moments MG in terms of certain matrix operators.
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    Las metas académicas en función del género y la nacionalidad y su relación con el rendimiento en estudiantes de secundaria, modalidad en artes
    (2023-05-11) Lamec Antonio Fabián
    El objetivo del presente trabajo fue analizar las perspectivas de los alumnos respecto a sus metas académicas en relación con el sexo, la nacionalidad y el rendimiento en estudiantes de secundaria del Centro Educativo, en Artes, Julio Alberto Hernández, Santiago, República Dominicana. La muestra seleccionada fue de 187 alumnos, 61.5% eran mujeres y 38.5% hombres, además se consideró la nacionalidad, con 89.8% de jóvenes dominicanos y 10.2% haitianos, la investigación tiene un enfoque cuantitativo y un diseño transversal no experimental. La prueba t no mostro diferencias significativas para la selección de los alumnos por sexo (p = 0.07) ni por nacionalidad (p = 0.304). Se utilizo el cuestionario llamado escala de Metas Académicas (CME) que cuenta con 20 ítems con respuesta tipo Likert, con un nivel de fiabilidad de 0.806 al aplicar el alfa de Cronbach. Los estudiantes respondieron con promedios muy bajos en la MRS, indicando el poco interés por buscar el reconocimiento de padres, amigos y docentes, además, se encontró ML significativamente más elevadas en las mujeres (p = 0.025), sin embargo los nacionales haitianos tienen ML (p = 0.023) y MRS (p = 0.038) significativamente más altas que los jóvenes dominicanos, en el caso del rendimiento académico en lengua española los jóvenes haitianos presentan rendimientos significativamente más altos (p = 0.001). Finalmente se ofrecen conclusiones que serán de ayuda para futuros trabajos en esta línea, para verificar el efecto de los factores ML, MA y MRS en otras situaciones académicas.
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    Liouvillian solutions for second order linear differential equations with Laurent plynomial coefficient
    (2023-05-09) Primitivo Acosta
    This paper is devoted to a complete parametric study of Liouvillian solutions of the general trace-free second order differential equation with a Laurent polynomial coefficient. This family of equations, for fixed orders at 0 and ∞ of the Laurent polynomial, is seen as an affine algebraic variety. We prove that the set of Picard-Vessiot integrable differential equations in the family is an enumerable union of algebraic subvarieties. We compute explicitly the algebraic equations of its components. We give some applications to well known subfamilies, such as the doubly confluent and biconfluent Heun equations, and to the theory of algebraically solvable potentials of Shrödinger equations. Also, as an auxiliary tool, we improve a previously known criterium for a second order linear differential equations to admit a polynomial solution.
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    Los problemas de optimización en el cálculo diferencial de una variable
    (2022-08-18) Neel Báez; Ramón Blanco Sánchez; Wendy Heredia Soriano
    Objetivo. Este artículo tiene como fin proponer orientaciones didácticas para el trabajo de los estudiantes con los problemas de optimización, con el propósito de influir en su destreza con el fin de solucionarlos para lo cual se realiza un análisis de las posibles causas que limitan el desempeño de los estudiantes. Métodos. Fueron empleados tanto métodos teóricos como prácticos: los primeros para la caracterización del estado del arte y la construcción del marco teórico; de los de naturaleza práctica se empleó el análisis documental para conocer las causas de los errores en los instrumentos de evaluación aplicados a los estudiantes, así como su correspondencia con lo descrito en la bibliografía especializada. Resultados. Se esclarecen como causas que limitan la solución de problemas de optimización a las dificultades en el uso del lenguaje matemático, el trabajo con las funciones y la transferencia Transformación, ISSN: 2077-2955, RNPS: 2098, mayo-agosto 2022, 18 (2), 317-335 Universidad de Camagüey “Ignacio Agramonte Loynaz” 318 de registros semióticos. De igual forma, se proponen procedimientos para la solución de problemas de optimización. Conclusiones. La modelación del problema, el buen uso y comprensión del lenguaje matemático como un recurso necesario, la utilización de los asistentes matemáticos como medio para inducir y comprobar resultados son procedimientos recomendables para favorecer la solución de problemas de optimización.
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    Mixed-type hypergeometric Bernoulli-Gegenbauer polynomials
    (2023-09-15) Dionicio Peralta
    In this paper, we consider a novel family of the mixed-type hypergeometric Bernoulli– Gegenbauer polynomials. This family represents a fascinating fusion between two distinct categories of special functions: hypergeometric Bernoulli polynomials and Gegenbauer polynomials. We focus our attention on some algebraic and differential properties of this class of polynomials, including its explicit expressions, derivative formulas, matrix representations, matrix-inversion formulas, and other relations connecting it with the hypergeometric Bernoulli polynomials. Furthermore, we show that unlike the hypergeometric Bernoulli polynomials and Gegenbauer polynomials, the mixed-type hypergeometric Bernoulli– Gegenbauer polynomials do not fulfill either Hanh or Appell conditions.
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    Sequentially ordered Sobolevinner product and Laguerre – Sobolev polynomials
    (2023-04-20) Juan Hernández
    We study the sequence of polynomials {𝑆𝑛}𝑛≥0{𝑆𝑛}𝑛≥0 that are orthogonal with respect to the general discrete Sobolev-type inner product ⟨𝑓, 𝑔⟩{𝗌} = ∫ 𝑓(𝑥)𝑔(𝑥)𝑑𝜇(𝑥) + ∑ 𝑁𝑗=1 ∑ 𝑑𝑗𝑘=0 𝜆𝑗,𝑘𝑓(𝑘)(𝑐𝑗)𝑔(𝑘)(𝑐𝑗), ⟨𝑓, 𝑔⟩{𝑠} = ∫ 𝑓(𝑥)𝑔(𝑥)𝑑𝜇(𝑥) + ∑ 𝑗=1 𝑁 ∑ 𝑘=0 𝑑𝑗𝜆𝑗,𝑘𝑓(𝑘)(𝑐𝑗)𝑔(𝑘) (𝑐𝑗) where 𝜇𝜇 is a finite Borel measure whose support supp(𝜇) is an infinite set of the real line, 𝜆𝑗,𝑘≥0, and the mass points 𝑐𝑖, 𝑖=1,…,𝑁 are real values outside the interior of the convex hull of supp(𝜇)supp𝜇 (𝑐𝑖∈R\𝐂h(supp(𝜇))∘)𝑐𝑖∈𝑅\𝐶h(supp(𝜇))∘). Under some restriction of order in the discrete part of ⟨·,·⟩𝗌⟨·,·⟩𝑠, we prove that 𝑆𝑛𝑆𝑛 has at least 𝑛−𝑑∗𝑛−𝑑* zeros on 𝐂h(supp(𝜇))∘𝐶h(supp𝜇)∘, being 𝑑∗𝑑* the number of terms in the discrete part of ⟨·,·⟩𝗌⟨·,·⟩𝑠. Finally, we obtain the outer relative asymptotic for {𝑆𝑛}{𝑆𝑛} in the case that the measure 𝜇𝜇 is the classical Laguerre measure, and for each mass point, only one order derivative appears in the discrete part of ⟨·,·⟩𝗌⟨·,·⟩𝑠.
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    What the teacher must master to direct the learning process
    (2022-09-16) Neel Báez
    Determining what the Mathematics teacher must master is a fundamental aspect to specify what the correct preparation of said teacher should be, which is a widely debated topic, on which there are various proposals for action, in each of which can be find positive aspects, despite the different approaches found in the specialized literature, at least there is consensus that such preparation must be composed of mathematical knowledge and didactic knowledge, which is supported by research, experiences and proposals aimed at achieving a good preparation of these professionals; several authors have specified the notion of Pedagogical Content Knowledge for the teaching of Mathematics, understanding as such the mastery of the mathematical content to be explained together with the appropriate didactic procedures to explain said content, other authors refer to the Specific Knowledge of the Content, as the “mathematical knowledge to teach”. The objective of this work is to scientifically argue the domain of epistemological and ontological characteristics of Mathematics, which can allow the teacher to establish the appropriate link between mathematical knowledge and didactic knowledge.