Algebraic Description of Shape Invariance Revisited

We revisit the algebraic description of shape invariance method in one-dimensional quantum mechanics. In this note we focus on four particular examples: the Kepler problem in flat space, the Kepler problem in spherical space, the Kepler problem in hyperbolic space, and the Rosen-Morse potential problem. Following the prescription given by Gangopadhyaya et al., we first introduce certain nonlinear algebraic systems. We then show that, if the model parameters are appropriately quantized, the bound-state problems can be solved solely by means of representation theory.Comment: 12 pages, 8 eepic figures; minor correction

Temporal evolution of a circular membrane subject to various boundary conditions

In this paper we illustrate a standard problem of mathematical physics, by exploring the potential of using a computer algebra system in a classroom experiment. Although many textbooks describe the problem and solve its particular examples to variable extent, to the student is always left the task of imagining, after a few tedious calculations, the temporal evolution of the system. The aim of this paper is, using algebraic computation, to give a complete route to the problem of an asymmetrically perturbed circular membrane in viscous media with the usual boundary conditions, including the computer animation of the results

Redes neurais e análise estatística para classificação de mapas topográficos da córnea baseados em coeficientes de Zernike: uma comparação quantitativa

PURPOSE: The main goal of this study was to develop and compare two different techniques for classification of specific types of corneal shapes when Zernike coefficients are used as inputs. A feed-forward artificial Neural Network (NN) and discriminant analysis (DA) techniques were used. METHODS: The inputs both for the NN and DA were the first 15 standard Zernike coefficients for 80 previously classified corneal elevation data files from an Eyesys System 2000 Videokeratograph (VK), installed at the Departamento de Oftalmologia of the Escola Paulista de Medicina, São Paulo. The NN had 5 output neurons which were associated with 5 typical corneal shapes: keratoconus, with-the-rule astigmatism, against-the-rule astigmatism, "regular" or "normal" shape and post-PRK. RESULTS: The NN and DA responses were statistically analyzed in terms of precision ([true positive+true negative]/total number of cases). Mean overall results for all cases for the NN and DA techniques were, respectively, 94% and 84.8%. CONCLUSION: Although we used a relatively small database, results obtained in the present study indicate that Zernike polynomials as descriptors of corneal shape may be a reliable parameter as input data for diagnostic automation of VK maps, using either NN or DA.OBJETIVOS: Nosso principal objetivo neste trabalho foi de desenvolver e comparar duas técnicas diferentes para classificação de superfícies corneanas. Uma rede neural artificial alimentada adiante e análise descriminante foram as técnicas de classificação comparadas neste trabalho. MÉTODOS: As entradas para ambos os métodos de classificação foram os primeiros 15 coeficientes de Zernike para 80 córneas mensuradas anteriormente em um topógrafo Eyesys instalado no Departamento de Oftalmologia da Escola Paulista de Medicina - UNIFESP. A rede neural tem 5 saídas que foram associados aos cinco casos típicos contidos na base de dados: ceratocone, astigmatismo a favor da regra, astigmatismo contra a regra, formato "regular" ou "normal" e pós-PRK. RESULTADOS: Os resultados de ambos os métodos foram estatisticamente analisados em termos de precisão. Os resultados gerais para ambos os métodos de redes neurais e análise discriminante foram 94% e 84,8%, respectivamente. CONCLUSÃO: Embora tenha-se utilizado uma base de dados relativamente pequena, os resultados obtidos aqui indicam que os polinômios de Zernike podem ser um parâmetro de entrada para classificação de diferentes formatos de córnea, tanto para uso com redes neurais ou análise discriminante