Bivariate second--order linear partial differential equations and orthogonal polynomial solutions

In this paper we construct the main algebraic and differential properties and the weight functions of orthogonal polynomial solutions of bivariate second--order linear partial differential equations, which are admissible potentially self--adjoint and of hypergeometric type. General formulae for all these properties are obtained explicitly in terms of the polynomial coefficients of the partial differential equation, using vector matrix notation. Moreover, Rodrigues representations for the polynomial eigensolutions and for their partial derivatives of any order are given. Finally, as illustration, these results are applied to specific Appell and Koornwinder orthogonal polynomials, solutions of the same partial differential equation.Comment: 27 page

Linear partial divided-difference equation satisfied by multivariate orthogonal polynomials on quadratic lattices

In this paper, a fourth-order partial divided-difference equation on quadratic lattices with polynomial coefficients satisfied by bivariate Racah polynomials is presented. From this equation we obtain explicitly the matrix coefficients appearing in the three-term recurrence relations satisfied by any bivariate orthogonal polynomial solution of the equation. In particular, we provide explicit expressions for the matrices in the three-term recurrence relations satisfied by the bivariate Racah polynomials introduced by Tratnik. Moreover, we present the family of monic bivariate Racah polynomials defined from the three-term recurrence relations they satisfy, and we solve the connection problem between two different families of bivariate Racah polynomials. These results are then applied to other families of bivariate orthogonal polynomials, namely the bivariate Wilson, continuous dual Hahn and continuous Hahn, the latter two through limiting processes. The fourth-order partial divided-difference equations on quadratic lattices are shown to be of hypergeometric type in the sense that the divided-difference derivatives of solutions are themselves solution of the same type of divided-difference equations.Comment: 36 page

Antivivissecção na educação científica : uma proposta didática para a licenciatura em ciências biológicas

No contexto do presente trabalho, o termo vivissecção se refere a experimentos realizados com animais não humanos vivos no ensino visando observação, indução, ou constatação de fenômenos. O principal objetivo foi favorecer, através de uma estratégia didática, a construção de conceitos sobre antivivissecção pelos interlocutores da pesquisa. O trabalho foi desenvolvido durante um curso de formação ministrado a acadêmicos e professores da área de Licenciatura em Ciências Biológicas. O instrumento de coleta de dados consistiu em um questionário com questões abertas e fechadas. Os resultados chamam a atenção sobre a efetividade do planejamento e seleção adequada de recursos para a compreensão de conceitos sobre antivivissecção na Educação Científica. Também fornecem subsídios para se pensar a visão hegemônica utilitarista e especista no Ensino de Ciências

How is the biocompatibilty of dental biomaterials evaluated?

All biomaterials used in dentistry must be evaluated for biocompatibility using screening assays to protect patient health and safety. The purpose of this review is to explain the international biocompatibility guidelines, and to explain the structure of a test program. The test program requires the structured assessment of materials into four phases; general toxicity, local tissue irritation, pre-clinical, and clinical evaluation. Different types of screening assays are available, and it is important to understand the advantages and limitations of the various types of assays that are available, so that they can be selected for appropriateness and interpreted accurately. New scientific advances in terms of the chemical properties of dental materials, tissue engineering, stem cell, genetic transfer, biomaterial, and growth factor therapies are under development. These new therapies create improved opportunities to restore and regenerate oral tissues, but they can also present new hazards to patients. Prior to their clinical use, these new technologies must be proven to be safe, and not hazardous to human health. A structured biocompatibility assessment and advice on the selection of assays are outlined to evaluate these new therapies

¿Es la masa la medida de la inercia?

In this paper we analyse from a critical point of view the definition that has usually been given to inertia as «the resistance that a body presents to its movement being modified». We also reflect on the unwanted didactic implications that this definition -that grants bodies a «property of decision over their own movement»- can have on general physics courses

Fourth-order differential equations satisfied by the generalized co-recursive of all classical orthogonal polynomials. A study of their distribution of zeros

AbstractThe unique fourth-order differential equation satisfied by the generalized co-recursive of all classical orthogonal polynomials is given for any (but fixed) level of recursivity. Up to now, these differential equations were known only for each classical family separately and also for a specific recursivity level. Moreover, we use this unique fourth-order differential equation in order to study the distribution of zeros of these polynomials via their Newton sum rules (i.e., the sums of powers of their zeros) which are closely related with the moments of such distribution. Both results are obtained with the help of two programs built in Mathematica symbolic language

On a class of bivariate second-order linear partial difference equations and their monic orthogonal polynomial solutions

AbstractIn this paper we classify the bivariate second-order linear partial difference equations, which are admissible, potentially self-adjoint, and of hypergeometric type. Using vector matrix notation, explicit expressions for the coefficients of the three-term recurrence relations satisfied by monic orthogonal polynomial solutions are obtained in terms of the coefficients of the partial difference equation. Finally, we make a compilation of the examples existing in the literature belonging to the class analyzed in this paper, namely bivariate Charlier, Meixner, Kravchuk and Hahn orthogonal polynomials