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Mathematical proofs at a crossroad
 Theory Is Forever, Lectures Notes in Comput. Sci. 3113
, 2004
"... Abstract. For more than 2000 years, from Pythagoras and Euclid to Hilbert and Bourbaki, mathematical proofs were essentially based on axiomaticdeductive reasoning. In the last decades, the increasing length and complexity of many mathematical proofs led to the expansion of some empirical, experimen ..."
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Abstract. For more than 2000 years, from Pythagoras and Euclid to Hilbert and Bourbaki, mathematical proofs were essentially based on axiomaticdeductive reasoning. In the last decades, the increasing length and complexity of many mathematical proofs led to the expansion of some empirical, experimental, psychological and social aspects, yesterday only marginal, but now changing radically the very essence of proof. In this paper, we try to organize this evolution, to distinguish its different steps and aspects, and to evaluate its advantages and shortcomings. Axiomaticdeductive proofs are not a posteriori work, a luxury we can marginalize nor are computerassisted proofs bad mathematics. There is hope for integration! 1
WEIGHTED FOURIER IMAGE ANALYSIS AND MODELING
, 2008
"... A novel systematic framework of medical image analysis, weighted Fourier series (WFS) analysis is introduced. WFS is a combination of Fourier series and heat kernel smoothing. WFS effectively reduces the Gibbs phenomenon, improves the signal to noise ratio, and increases normality of the estimated ..."
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A novel systematic framework of medical image analysis, weighted Fourier series (WFS) analysis is introduced. WFS is a combination of Fourier series and heat kernel smoothing. WFS effectively reduces the Gibbs phenomenon, improves the signal to noise ratio, and increases normality of the estimated errors in the WFSbased generalized linear models. In estimating the parameters of WFS, the least squares estimation of WFS has been widely used but it is computationally inefficient. To address the computational inefficiency in the least squares estimation, much faster but less accurate iterative residual fitting (IRF) method has been proposed. The proposed adaptive iterative regression (AIR) technique inherits the computational efficiency of IRF and improves accuracy of IRF. AIR partitions the function space into a set of subspaces, and performs an extra orthogonalization procedure to reduce the bias of IRF estimation. A complimentary tool, the fast weighted