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- ItemDifferential Properties of Jacobi-Sobolev Polynomials and Electrostatic Interpretation(2023-08-06)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.
- ItemWhat the teacher must master to direct the learning process(2022-09-16)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.
- ItemSequentially ordered Sobolevinner product and Laguerre – Sobolev polynomials(2023-04-20)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.
- ItemMixed-type hypergeometric Bernoulli-Gegenbauer polynomials(2023-09-15)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.
- ItemLos problemas de optimización en el cálculo diferencial de una variable(2022-08-18)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.