A full family of multimodal maps on the circle

We exhibit a family of trigonometric polynomials inducing a family of 2m-multimodal maps on the circle which contains all relevant dynamical behavior.

Coincidence Wecken homotopies versus Wecken homotopies relative to a fixed homotopy in one of the maps

We study the 1-parameter Wecken problem versus the restricted Wecken problem, for coincidence free pairs of maps between surfaces. For this we use properties of the function space between two surfaces and of the pure braid group on two strings of a surface. When the target surface is either the 2-sphere or the torus it is known that the two problems are the same. We classify most pairs of homotopy classes of maps according to the answer of the two problems are either the same or different when the target is either projective space or the Klein bottle. Some partial results are given for surfaces of negative Euler characteristic. (C) 2010 Elsevier B.V. All rights reserved.

Neural networks and statistical analysis for classification of corneal videokeratography maps based on Zernike coefficients: a quantitative comparison

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.

The Tausk controversy on the foundations of quantum mechanics: Physics, philosophy, and politics

In 1966 the Brazilian physicist Klaus Tausk (b. 1927) circulated a preprint from the International Centre for Theoretical Physics in Trieste, Italy, criticizing Adriana Daneri, Angelo Loinger, and Giovanni Maria Prosperi`s theory of 1962 on the measurement problem in quantum mechanics. A heated controversy ensued between two opposing camps within the orthodox interpretation of quantum theory, represented by Leon Rosenfeld and Eugene P. Wigner. The controversy went well beyond the strictly scientific issues, however, reflecting philosophical and political commitments within the context of the Cold War, the relationship between science in developed and Third World countries, the importance of social skills, and personal idiosyncrasies.