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Fast Wavelet Transform
The fast wavelet transform is a mathematical algorithm designed to turn a waveform or signal in the time domain into a sequence of coefficients based on an orthogonal basis of small finite waves, or wavelets. The transform can be easily extended to multidimensional signals, such as images, where the time domain is replaced with the space domain. This algorithm was introduced in 1989 by Stéphane Mallat. It has as theoretical foundation the device of a finitely generated, orthogonal multiresolution analysis (MRA). In the terms given there, one selects a sampling scale ''J'' with sampling rate of 2''J'' per unit interval, and projects the given signal ''f'' onto the space V_J; in theory by computing the scalar products :s^_n:=2^J \langle f(t),\varphi(2^J t-n) \rangle, where \varphi is the scaling function of the chosen wavelet transform; in practice by any suitable sampling procedure under the condition that the signal is highly oversampled, so : P_J x):=\sum_ s^_n\,\varphi(2^Jx ...
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Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ...
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Wavelets - DWT
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the number and direction of its pulses. Wavelets are imbued with specific properties that make them useful for signal processing. For example, a wavelet could be created to have a frequency of Middle C and a short duration of roughly one tenth of a second. If this wavelet were to be convolved with a signal created from the recording of a melody, then the resulting signal would be useful for determining when the Middle C note appeared in the song. Mathematically, a wavelet correlates with a signal if a portion of the signal is similar. Correlation is at the core of many practical wavelet applications. As a mathematical tool, wavelets can be used to extract information from many different kinds of data, including but not limited to aud ...
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Ronald Coifman
Ronald Raphael Coifman is the Sterling professor of Mathematics at Yale University. Coifman earned a doctorate from the University of Geneva in 1965, supervised by Jovan Karamata. Coifman is a member of the American Academy of Arts and Sciences, the Connecticut Academy of Science and Engineering, and the National Academy of Sciences. He is a recipient of the 1996 DARPA Sustained Excellence Award, the 1996 Connecticut Science Medal, the 1999 Pioneer Award of the International Society for Industrial and Applied Science, and the 1999 National Medal of Science. In 2013, he co-founded ThetaRay, a cyber security and big data analytics company. In 2018, he received the Rolf Schock Prize for Mathematics. References External links Scientific Data Has Become So Complex, We Have to Invent New Math to Deal With It Wired ''Wired'' (stylized as ''WIRED'') is a monthly American magazine, published in print and online editions, that focuses on how emerging technologies affect culture, ...
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Gregory Beylkin
Gregory Beylkin (born 16 March 1953) is an applied mathematician. Education and career He studied from 1970 to 1975 at the University of Leningrad, with Diploma in Mathematics in November 1975. From 1976 to 1979 he was a research scientist at the Research Institute of Ore Geophysics, Leningrad. From 1980 to 1982 he was a graduate student at New York University, where he received his PhD under the supervision of Peter Lax. From 1982 to 1983 Beylkin was an associate research scientist at the Courant Institute of Mathematical Sciences. From 1983 to 1991 he was a member of the professional staff of Schlumberger-Doll Research in Ridgefield, Connecticut. Since 1991 he has been a professor in the Department of Applied Mathematics at the University of Colorado Boulder. He was a visiting professor at Yale University, the University of Minnesota, and the Mittag-Leffler Institute and participated in 2012 and 2015 in the summer seminar on "Applied Harmonic Analysis and Sparse Approximation" at ...
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Barbara Burke Hubbard
Barbara Burke Hubbard (born 1948) is an American science journalist, mathematics popularizer, textbook author, and book publisher, known for her books on wavelet transforms and multivariable calculus. Life Burke Hubbard is the daughter of ''Los Angeles Times'' reporter Vincent J. Burke, and spent a year in high school living in Moscow when Burke was stationed there in 1964. She was an undergraduate at Harvard University, initially majoring in biology but switching to English, and graduating in 1969. She became a science writer for the Massachusetts Institute of Technology and a journalist for ''The Ithaca Journal'', and was the 1981 winner of the AAAS Westinghouse Science Journalism Award in the small newspaper category, for her articles on acid rain in ''The Ithaca Journal''. She married mathematician John H. Hubbard, with whom she has four children, and with her family has split her time between Ithaca, New York and Marseille, France, with shorter-term stays elsewhere. Books ...
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Cristina Pereyra
María Cristina Pereyra (born 1964) is a Venezuelan mathematician. She is a professor of mathematics and statistics at the University of New Mexico, and the author of several books on wavelets and harmonic analysis. Education and employment Pereyra was a member of the Venezuelan team for the 1981 and 1982 International Mathematical Olympiads. She earned a licenciado (the equivalent of a bachelor's degree) in mathematics in 1986 from the Central University of Venezuela. She went to Yale University for graduate studies, completing her Ph.D. there in 1993. Her dissertation, ''Sobolev Spaces On Lipschitz Curves: Paraproducts, Inverses And Some Related Operators'', was supervised by Peter Jones. After working for three years as an instructor at Princeton University, she joined the University of New Mexico faculty in 1996. Books Pereyra is the author or editor of: *''Lecture Notes on Dyadic Harmonic Analysis'' (Second Summer School in Analysis and Mathematical Physics, Cuernavaca, 2000 ...
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