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*SIAM* *REVIEW*

"... applications, he has encouraged many young mathematicians in the United States to choose secular rather than monastic mathematics. ..."

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applications, he has encouraged many young mathematicians in the United States to choose secular rather than monastic mathematics.

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Posted With Permission of *SIAM* BOOK *REVIEW* *SIAM* *REVIEW*

"... Wavelet theory is an attempt to address the pervasive problem of describing the frequency content of a function locally in time. The wavelet approach is to analyze a function using an appropriate family of dilates and translates of one or more wavelets. Although this term is relatively new, wavelet- ..."

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Wavelet theory is an attempt to address the pervasive problem of describing the frequency content of a function locally in time. The wavelet approach is to analyze a function using an appropriate family of dilates and translates of one or more wavelets. Although this term is relatively new, wavelet-like techniques have been independently invented over the past 30 years in harmonic analysis, quantum mechanics, signal processing, etc. Some of wavelet theory has therefore been the rediscovery of those ideas, but the introduction of a single logical framework has allowed new insights and the formulation of new problems not possible before. Moreover, the applicability of wavelets to such diverse fields has led to an unusual—and highly productive—feedback between mathematicians, physicists, and engineers. This book is a greatly expanded and updated version of ten lectures given by Daubechies at a CBMS conference at the University of Lowell in June, 1990. It is an introductory-level mathematics text, intended to introduce mathematicians and other scientists to wavelets. In this respect it is superficially similar to Chui’s book [C1]; however, whereas Chui is exclusively mathematical and concentrates on only a few aspects of wavelet theory, Daubechies effectively surveys most of the current theory and specifically discusses the links with other areas, especially signal processing.

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method. *SIAM* *Review*, 21:339-360, 1979.

"... [2] P. A. Businger. Numerically stable deflation of Hessenberg and symmetric tridiagonal matrices. BIT, 11:262-270, 1971. [3] J. Cuppen. A divide and conquer method for the symmetric tridiagonal eigenproblem. Numer. Math., 36:177-195, 1981. ..."

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[2] P. A. Businger. Numerically stable deflation of Hessenberg and symmetric tridiagonal matrices. BIT, 11:262-270, 1971. [3] J. Cuppen. A divide and conquer method for the symmetric tridiagonal eigenproblem. Numer. Math., 36:177-195, 1981.

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*SIAM* *Review* is currently published by Society for Industrial and Applied Mathematics.

, 2008

"... Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at ..."

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Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at

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Key words: *SIAM* *Review* · citations · paper length · referee delay 2010 MSC: 00A99 (General: Miscellaneous topics) Background

, 2000

"... SIAM Review is examined for referee delay, citations, and paper length after the reorganization of the journal in 1999. A single, very-highly cited arti-cle was responsible for all the increase to the impact factor during the past decade; the reorganization did not improve the journal overall. Some ..."

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*SIAM*

*Review*is examined for referee delay, citations, and paper length after the reorganization of the journal in 1999. A single, very-highly cited arti-cle was responsible for all the increase to the impact factor during the past decade; the reorganization did not improve the journal overall. Some

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Newman. Power-law distributions in empirical data. *SIAM* *review*, 51(4):661–

, 2009

"... The poweRlaw package provides code to fit heavy tailed distributions, including discrete and continuous power-law distributions. The fitting procedure follows the method detailed in Clauset et al. 1. The parameter values are obtained by maximising the likelihood. The cut-off value, xmin, is estimate ..."

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The poweRlaw package provides code to fit heavy tailed distributions, including discrete and continuous power-law distributions. The fitting procedure follows the method detailed in Clauset et al. 1. The parameter values are obtained by maximising the likelihood. The cut-off value, xmin, is estimated by minimising the Kolmogorov-Smirnoff statistic.

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*SIAM* *Review* 41(1):102–112, 1999 ILL-CONDITIONED MATRICES ARE COMPONENTWISE NEAR TO SINGULARITY

"... Abstract. For a square matrix normed to 1, the normwise distance to singularity is well known to be equal to the reciprocal of the condition number. In this paper we give an elementary and self-contained proof for the fact that an ill-conditioned matrix is also not far from a singular matrix in a co ..."

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Abstract. For a square matrix normed to 1, the normwise distance to singularity is well known to be equal to the reciprocal of the condition number. In this paper we give an elementary and self-contained proof for the fact that an ill-conditioned matrix is also not far from a singular matrix in a componentwise sense. This is shown to be true for any weighting of the componentwise distance. In words: Ill-conditioned means for matrix inversion nearly ill-posed also in the componentwise sense. 1. Introduction. The

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[*Review* published in *SIAM* *Review*, Vol. 53, Issue 1, pp. 210–211.] Introduction to the Mathematics of Sub- division Surfaces. By Lars-Erik Andersson and

"... xxiv+356 pp., hardcover. ISBN 978-0-898716- ..."

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*SIAM* *REVIEW* c ○ 2000 Society for Industrial and Applied Mathematics Vol. 42, No. 1, pp. 43–67 Fractional Splines and Wavelets ∗

"... Abstract. We extend Schoenberg’s family of polynomial splines with uniform knots to all fractional degrees α>−1. These splines, which involve linear combinations of the one-sided power functions xα + =max(0,x)α, are α-Hölder continuous for α>0. We construct the corresponding B-splines by takin ..."

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Abstract. We extend Schoenberg’s family of polynomial splines with uniform knots to all fractional degrees α>−1. These splines, which involve linear combinations of the one-sided power functions xα + =max(0,x)α, are α-Hölder continuous for α>0. We construct the corresponding B-splines by taking fractional finite differences and provide an explicit characterization in both time and frequency domains. We show that these functions satisfy most of the properties of the traditional B-splines, including the convolution property, and a generalized fractional differentiation rule that involves finite differences only. We characterize the decay of the B-splines that are not compactly supported for nonintegral α’s. Their most astonishing feature (in reference to the Strang–Fix theory) is that they have a fractional order of approximation α + 1 while they reproduce the polynomials of degree ⌈α⌉. Forα> − 1, they satisfy all the requirements for a multiresolution analysis of 2 L2 (Riesz bounds, two-scale relation) and may therefore be used to build new families of wavelet bases with a continuously varying order parameter. Our construction also yields symmetrized fractional B-splines which provide the connection with Duchon’s general theory of radial (m, s)-splines (including thin-plate splines). In particular, we show that the symmetric version of our splines can be obtained as the solution of a variational problem involving the norm of a fractional derivative.