Método dos elementos finitos baseado em polinómios de Hermite cúbicos, para resolução da equação de Black-Scholes não linear com opções europeias

Foi desenvolvido um algoritmo numérico para resolver uma equação diferencial parcial generalizada de Black-Scholes, que surge na precificação de opções europeias, considerando os custos de transação. O método Crank-Nicolson é usado para discretizar no tempo e o método de interpolação cúbica de Hermite para discretizar no espaço. A eficiência e precisão do método proposto são testadas numericamente e, os resultados confirmam o comportamento teórico das soluções, que também se encontra em boa concordância com a solução exata.A numerical algorithm for solving a generalized Black-Scholes partial differential equation, which arises in European option pricing considering transaction costs is developed. The Crank-Nicolson method is used to discretize in the temporal direction and the Hermite cubic interpolation method to discretize in the spatial direction. The efficiency and accuracy of the proposed method are tested numerically, and the results confirm the theoretical behaviour of the solutions, which is also found to be in good agreement with the exact solution

Construction of Concepts Images from Mathematical Models Obtained with the Tracker Software

Este trabalho tem por objetivo investigar o potencial de aliar a metodologia de Modelagem Matemática com o uso do programa Tracker com estudantes da graduação em Matemática-Licenciatura na reconstrução de modelos físicos, para construção de conceitos de função. Para isso foi realizado um estudo qualitativo de caráter exploratório, envolvendo atividades de campo. Nossos resultados parciais apontam para uma maior compreensão na construção dos modelos, já que os modelos matemáticos podem ser recriados a partir da correlação a modelos físicos já validados, também verificou-se que o programa Tracker possibilita uma excelente percepção visual de diferentes imagens de conceitos, que compreendemos favorecer a construção de definição de novos conceitos ou modificação e ampliação de conceitos já construídos no cognitivo do aluno.Este trabajo tiene como objetivo investigar el potencial de combinar la metodología de Modelado Matemático con el uso del programa Tracker con estudiantes de licenciatura en Matemáticas en la reconstrucción de modelos físicos para la construcción de conceptos funcionales. Para ello, se realizó un estudio exploratorio cualitativo, que involucró actividades de campo. Nuestros resultados parciales apuntan a una mayor comprensión en la construcción de los modelos, ya que los modelos matemáticos se pueden recrear a partir de la correlación a modelos físicos ya validados, también se verificó que el programa Tracker permite una excelente percepción visual de diferentes imágenes conceptuales, que entendemos favorecer la construcción de definición de nuevos conceptos o la modificación y expansión de conceptos ya construidos en el conocimiento cognitivo del alumno.This work aims to investigate the potential of combining the Mathematical Modeling methodology with the use of the software Tracker with undergraduate students in Mathematics in the reconstruction of physical models for the construction of function concepts. For this, a qualitative and exploratory study of bibliographic and experimental character was carried out, involving field activities. Our results point to a greater understanding in the construction of the models, since the mathematical models can be recreated from the correlation to physical models already validated. We also verified that the Tracker software enables an excellent visual perception of different concept images. &nbsp

Political, technical and pedagogical effects of the COVID-19 Pandemic in Mathematics Education: an overview of Brazil, Chile and Spain

En este artículo se discuten los problemas y las cuestiones que se plantean en Educación, en particular, en Educación Matemática a partir de la problematización de los efectos de la pandemia de COVID-19. Para ello, tres investigadores en Educación Matemática de Brasil, Chile y España se han reunido virtualmente. Con base en estas discusiones, el grupo proporcionó el contexto para cada uno de los tres países en términos de cómo han estado gestionando las acciones relacionadas al contexto de la pandemia y seleccionó tres historias vivenciadas, por ellos, en este contexto, las cuales las utilizaron como elementos materiales para una discusión conjunta de forma no localizada. Por medio de ese camino, como resultado, se muestra una visión general de los efectos de COVID-19 en diferentes países relacionados con la educación, que se clasificaron en tres dimensiones: técnica, política y pedagógica. Además de eso, se presentan cuestionamientos a la Educación Matemática producidos a partir de la problematización de los efectos de la pandemia.This article discusses the problems and questions that are raised in Education and specifically in Mathematics Education from the problematization of the effects of the pandemic of COVID-19. For this, it was brought together three researchers in Mathematics Education from Brazil, Chile and Spain, who started to meet virtually. Based on these discussions, the group provided the context for each of the three countries in terms of how they have been dealing with the pandemic and selected three tales they lived related to the pandemic, that were used as material elements for a joint discussion in a non localized way. With this path, as results, it is shown an overview of the effects of COVID-19 in different countries related to education, that were categorized in Technical, Political and Pedagogical dimensions. Besides that, questions are presented to Mathematics Education produced from the problematization of the effects of the pandemic.Este artigo discute os problemas e questões que surgem na Educação, em particular na Educação Matemática, a partir da problematização dos efeitos da pandemia COVID-19. Para isso, três pesquisadores em Educação Matemática do Brasil, Chile e Espanha se encontraram virtualmente. Com base nas discussões, o grupo forneceu o contexto para cada um dos três países em termos de como eles vêm gerenciando as ações relacionadas ao contexto da pandemia e selecionou três histórias que vivenciaram nesse contexto, que utilizaram como elementos materiais para uma discussão conjunta de forma não localizada. Por esse caminho, tem-se como resultado um panorama dos efeitos do COVID-19 na educação em diferentes países, os quais foram classificados em três dimensões: técnica, política e pedagógica. Além disso, são apresentadas questões à Educação Matemática produzidas a partir da problematização dos efeitos da pandemia

Peculiarities of smoothly undulating number

This notes presents results related to divisibility or multiplicity between two numbers in the class of integers called  smoothly undulating numbers of the type uz[n]. The main result is to characterize and display types of divisors of some types of numbers uz[n], and we show an algorithm to determine the greatest common divisor between two numbers uz[n].This note presents results related to divisibility or multiplicity between two numbers in the class of integers called smoothly undulating numbers of the type uz[n]. The main result is to characterize and display the types of divisors of some types of numbers uz[n], and we show an algorithm to determine the greatest common divisor between two numbers uz[n]

A new non-conformable derivative based on Tsallis’s q- exponential function

Neste artigo, uma nova derivada do tipo local é proposta e algumas propriedades básicas são estudadas. Esta nova derivada satisfaz algumas propriedades do cálculo de ordem inteira, por exemplo linearidade, regra do produto, regra do quociente e a regra da cadeia. Devido à função exponencial generalizada de Tsallis, podemos estender alguns dos resultados clássicos, a saber: teorema de Rolle, teorema do valor médio. Apresentamos a correspondente Q-integral a partir da qual surgem novos resultados. Especificamente, generalizamos a propriedade de inversão do teorema fundamental do cálculo e provamos um teorema associado à integração clássica por partes. Finalmente, apresentamos uma aplicação envolvendo equações diferenciais lineares por meio da Q-derivada.In this paper, a new derivative of local type is proposed and some basic properties are studied. This new derivative satisfies some properties of integer-order calculus, e.g. linearity, product rule, quotient rule and the chain rule. Because Tsallis' generalized exponential function, we can extend some of the classical results, namely: Rolle's theorem, the mean-value theorem. We present the corresponding Q-integral from which new results emerge. Specifically, we generalize the inversion property of the fundamental theorem of calculus and prove a theorem associated with the classical integration by parts. Finally, we present an application involving linear differential equations by means of Q derivative

On compact explicit formulas of the partial fraction decomposition and applications

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Modelo SIR para propagação da Covid-19 no Estado da Paraíba (Brasil)

Este trabalho visa aplicar o modelo compartimental do tipo SIR (Susceptível - Infectado - Removido) na evolução do Covid-19 no Estado da Paraíba e na cidade de Campina Grande - PB. Para tanto, os parâmetros do modelo foram considerados variáveis ao longo da evolução no tempo, dentro de um intervalo adequado. O sistema de equações diferenciais foi resolvido numericamente usando o método de Euler. Os parâmetros foram obtidos ajustando-se o modelo aos dados de infectados fornecidos pela Secretaria de Saúde do Estado da Paraíba. De acordo com os resultados obtidos, o modelo descreve bem a população infectada. Houve redução no número efetivo de reprodução no Estado da Paraíba e na Cidade de Campina Grande nos períodos analisados. Ressalta-se que, compreender a dinâmica de transmissão da infecção e avaliação da eficácia das medidas de controle, é crucial para avaliar o potencial de ocorrência de transmissão sustentada em novas áreas. O modelo também pode ser aplicado para descrever a dinâmica da epidemia em outras regiões e países.This work aims to apply the SIR-type compartmental model (Susceptible - Infected - Removed) in the evolution of Covid-19 in Paraíba's State and Campina Grande City. For that, the parameters of the model were considered to be variable during time evolution, within an appropriate range. The system of differential equations was solved numerically using the Euler method. The parameters were obtained by adjusting the model to the infected data provided by the Paraíba Health Department. According to the results obtained, the model describes the infected population well. There was a reduction in the effective reproduction number in Paraíba and the town of Campina Grande. It is noteworthy that understanding the dynamics of infection transmission and evaluating the effectiveness of control measures is crucial to assess the potential for sustained transmission to occur in new areas. The model can also be applied to describe epidemic dynamics in other regions and countries.

On the reachability tube of non-Newtonian first-order linear differential equations

A problem of practical interest is the determination of the reachability sets of ordinary differential equations with an external perturbation, or with a control. This problem can be extended to non-Newtonian spaces generated by continuous and injective functions α. This paper presents a method to determine the reachability tube of a family of non-Newtonian first-order linear differential equations with an external perturbation, or with a control, that belongs to a set of functions that are α-continuous and α-bounded. The reachability tube is determined explicitly in three non-Newtonian spaces that are associated with three α-generators. The results obtained are illustrated numerically.A problem of practical interest is the determination of the reachability sets of ordinary differential equations with an external perturbation, or with a control. This problem can be extended to non-Newtonian spaces generated by continuous and injective functions α. This paper presents the problem of determining the reachability tube of a family of non-Newtonian first-order linear differential equations with an external perturbation, or with a control, that belongs to a set of functions that are α-continuous and α-bounded. The reachability tube is determined explicitly in three non-Newtonian spaces that are associated with three α-generators. The results obtained are illustrated numerically

Simulation of the Onset turbulent flow around a Isothermal Complex Geometries: an analysis of thermofluid dynamic flow

In this work, in the area of Computational Fluid Dynamics (CFD), more specifically in the area of thermofluid dynamics for two-dimensional flows (2D), and also considering, the fluid-body interaction, allied to the phenomena of heat-transfer by mixed convection and the beginning of processes of the turbulent flow phenomenon in the fluid-body interaction, a study is proposed that demonstrates the efficiency in the analysis and simulation of these complex phenomena. We adopt an Eulerian approach for a fixed mesh, which is intended to represent the thermofluid dynamic movement, working together with a Lagrangian mesh, the latter being intended to discretize the immersed body. The strategy, in this work, allows approaching complex isothermal geometries, which present a certain aerodynamic degree on their surface, being popularly known as blunt body, where this, in turn, is immersed in an incompressible Newtonian fluid. One of the contributions of this work is the introduction of a simple but efficient method to calculate the Nusselt number. Regarding the process of validation and modeling of the physical phenomena of interest, that is, regarding the effectiveness of the methodology, called the Immersed Frontier, an implementation with low computational cost was carried out for the transfer of mixed convection heat, as well as for modeling the turbulence, namely, making use of the Spalart-Allmaras model, in the context of the URANS (Unsteady Reynolds Average Navier -Stokes) methodology. Numerical results showed good convergence with data available in the literature, which confirms the numerical precision and reliability of the adopted model

A methodology to obtain accurate potential energy Functions for diatomic systems: mathematical point of view

The mathematics used in physical chemistry has changed greatly in the past forty years and it will certainly continue to change more quickly. Theoretical chemists and physicists must have an acquaintance with abstract mathematics if they are to keep up with their field, as the mathematical language in which it is expressed changes. Thinking about it, in this article, we want to show some of the most important concepts of Mathematical Analysis involved in obtaining analytical functions to represent the potential energy interaction for diatomic systems. A basic guide for the construction of a potential based on Dunham's coefficients and an example of a new potential obtained from this methodology is also presented