|
Irrational Base Discrete Weighted Transform
In mathematics, the irrational base discrete weighted transform (IBDWT) is a variant of the fast Fourier transform using an irrational base; it was developed by Richard Crandall (Reed College), Barry Fagin (Dartmouth College) and Joshua Doenias ( NeXT Software) in the early 1990s using Mathematica. The IBDWT is used in the Great Internet Mersenne Prime Search's client Prime95 to perform FFT multiplication, as well as in other programs implementing Lucas-Lehmer test, such as CUDALucas and Glucas. References * Richard Crandall, Barry Fagin: ''Discrete weighted transforms and large-integer arithmetic'', Mathematics of Computation 62, 205, 305-324, January 1994PDF file * Richard Crandall Richard E. Crandall (December 29, 1947 – December 20, 2012) was an American physicist and computer scientist who made contributions to computational number theory. Background Richard Crandall was born in Ann Arbor, Michigan, and spent two years ...: ''Topics in Advanced Scientific Compu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fast Fourier Transform
A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa. The DFT is obtained by decomposing a sequence of values into components of different frequencies. This operation is useful in many fields, but computing it directly from the definition is often too slow to be practical. An FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse (mostly zero) factors. As a result, it manages to reduce the complexity of computing the DFT from O\left(N^2\right), which arises if one simply applies the definition of DFT, to O(N \log N), where N is the data size. The difference in speed can be enormous, especially for long data sets where ''N'' may be in the thousands or millions. In the presence of round-off error, many FFT algorithm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irrational Number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being '' incommensurable'', meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself. Among irrational numbers are the ratio of a circle's circumference to its diameter, Euler's number ''e'', the golden ratio ''φ'', and the square root of two. In fact, all square roots of natural numbers, other than of perfect squares, are irrational. Like all real numbers, irrational numbers can be expressed in positional notation, notably as a decimal number. In the cas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Crandall
Richard E. Crandall (December 29, 1947 – December 20, 2012) was an American physicist and computer scientist who made contributions to computational number theory. Background Richard Crandall was born in Ann Arbor, Michigan, and spent two years at Caltech before transferring to Reed College in Portland, Oregon, where he graduated in physics and wrote his undergraduate thesis on randomness. Career and accomplishments Crandall earned a Ph.D in theoretical physics from MIT. In 1978, he became a physics professor at Reed College, where he taught courses in experimental physics and computational physics for many years, ultimately becoming Vollum Professor of Science and director of the Center for Advanced Computation. He was also, at various times, Chief Scientist at NeXT, Inc., Chief Cryptographer and Distinguished Scientist at Apple, and head of Apple's Advanced Computation Group. He was a pioneer in experimental mathematics. He developed the irrational base discrete weighted trans ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reed College
Reed College is a private liberal arts college in Portland, Oregon. Founded in 1908, Reed is a residential college with a campus in the Eastmoreland neighborhood, with Tudor-Gothic style architecture, and a forested canyon nature preserve at its center. Referred to as one of "the most intellectual colleges in the country", Reed is known for its mandatory first-year humanities program, senior thesis, progressive politics, de-emphasis on grades, academic rigor, grade deflation, and unusually high proportion of graduates who go on to earn doctorates and other postgraduate degrees. The college has many prominent alumni, including over a hundred Fulbright Scholars, 67 Watson Fellows, and three Churchill Scholars; its 32 Rhodes Scholars are the second-highest count for a liberal arts college. Reed is ranked fourth in the United States for all postsecondary institutions for the percentage of its graduates who go on to earn a Ph.D., after Caltech, Harvey Mudd, and Swarthmore Colleg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barry Fagin
Barry may refer to: People and fictional characters * Barry (name), including lists of people with the given name, nickname or surname, as well as fictional characters with the given name * Dancing Barry, stage name of Barry Richards (born c. 1950), former dancer at National Basketball Association games Places Canada *Barry Lake, Quebec *Barry Islands, Nunavut United Kingdom * Barry, Angus, Scotland, a village ** Barry Mill, a watermill * Barry, Vale of Glamorgan, Wales, a town ** Barry Island, a seaside resort ** Barry Railway Company ** Barry railway station United States * Barry, Illinois, a city * Barry, Minnesota, a city * Barry, Texas, a city * Barry County, Michigan * Barry County, Missouri * Barry Township (other), in several states * Fort Barry, Marin County, California, a former US Army installation Elsewhere * Barry Island (Debenham Islands), Antarctica * Barry, New South Wales, Australia, a village * Barry, Hautes-Pyrénées, France, a commune Arts and e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dartmouth College
Dartmouth College (; ) is a private research university in Hanover, New Hampshire. Established in 1769 by Eleazar Wheelock, it is one of the nine colonial colleges chartered before the American Revolution. Although founded to educate Native Americans in Christian theology and the English way of life, the university primarily trained Congregationalist ministers during its early history before it gradually secularized, emerging at the turn of the 20th century from relative obscurity into national prominence. It is a member of the Ivy League. Following a liberal arts curriculum, Dartmouth provides undergraduate instruction in 40 academic departments and interdisciplinary programs, including 60 majors in the humanities, social sciences, natural sciences, and engineering, and enables students to design specialized concentrations or engage in dual degree programs. In addition to the undergraduate faculty of arts and sciences, Dartmouth has four professional and graduate schools: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joshua Doenias
Joshua () or Yehoshua ( ''Yəhōšuaʿ'', Tiberian: ''Yŏhōšuaʿ,'' lit. 'Yahweh is salvation') ''Yēšūaʿ''; syr, ܝܫܘܥ ܒܪ ܢܘܢ ''Yəšūʿ bar Nōn''; el, Ἰησοῦς, ar , يُوشَعُ ٱبْنُ نُونٍ '' Yūšaʿ ibn Nūn''; la, Iosue functioned as Moses' assistant in the books of Exodus and Numbers, and later succeeded Moses as leader of the Israelite tribes in the Hebrew Bible's Book of Joshua. His name was Hoshea ( ''Hōšēaʿ'', lit. 'Save') the son of Nun, of the tribe of Ephraim, but Moses called him "Yehoshua" (translated as "Joshua" in English),''Bible'' the name by which he is commonly known in English. According to the Bible, he was born in Egypt prior to the Exodus. The Hebrew Bible identifies Joshua as one of the twelve spies of Israel sent by Moses to explore the land of Canaan. In Numbers 13:1, and after the death of Moses, he led the Israelite tribes in the conquest of Canaan, and allocated lands to the tribes. According to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NeXT
Next may refer to: Arts and entertainment Film * ''Next'' (1990 film), an animated short about William Shakespeare * ''Next'' (2007 film), a sci-fi film starring Nicolas Cage * '' Next: A Primer on Urban Painting'', a 2005 documentary film Literature * ''Next'' (Crichton novel), a novel by Michael Crichton * ''Next'' (Hynes novel), a 2010 novel by James Hynes * ''Next'' (play), a play by Terrence McNally * '' Next: The Future Just Happened'', a 2001 non-fiction book by Michael Lewis Music Performers * Next (American band), an R&B trio * NEXT (Korean band), a South Korean rock band * Next (Chinese group), a boy group Albums * ''Next'' (ATB album), 2017 * ''Next'' (Journey album) or the title song, 1977 * ''Next'' (The Necks album) or the title instrumental, 1990 * ''Next'' (The Sensational Alex Harvey Band album) or the title song (see below), 1973 * ''Next'' (Sevendust album), 2005 * ''Next'' (Soulive album), 2002 * ''Next'' (Vanessa Williams album), 1997 * ''Next!' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematica
Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimization, plotting functions and various types of data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other programming languages. It was conceived by Stephen Wolfram, and is developed by Wolfram Research of Champaign, Illinois. The Wolfram Language is the programming language used in ''Mathematica''. Mathematica 1.0 was released on June 23, 1988 in Champaign, Illinois and Santa Clara, California. __TOC__ Notebook interface Wolfram Mathematica (called ''Mathematica'' by some of its users) is split into two parts: the kernel and the front end. The kernel interprets expressions (Wolfram Language code) and returns result expressions, which can then be displayed by the front end. The origin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Internet Mersenne Prime Search
The Great Internet Mersenne Prime Search (GIMPS) is a collaborative project of volunteers who use freely available software to search for Mersenne prime numbers. GIMPS was founded in 1996 by George Woltman, who also wrote the Prime95 client and its Linux port MPrime. Scott Kurowski wrote the back end PrimeNet server to demonstrate volunteer computing software by Entropia, a company he founded in 1997. GIMPS is registered as Mersenne Research, Inc. with Kurowski as Executive Vice President and board director. GIMPS is said to be one of the first large scale volunteer computing projects over the Internet for research purposes. , the project has found a total of seventeen Mersenne primes, fifteen of which were the largest known prime number at their respective times of discovery. The largest known prime is 282,589,933 − 1 (or M82,589,933 for short) and was discovered on December 7, 2018, by Patrick Laroche. On December 4, 2020, the project passed a major milestone afte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime95
Prime95, also distributed as the command-line utility mprime for FreeBSD and Linux, is a freeware application written by George Woltman. It is the official client of the Great Internet Mersenne Prime Search (GIMPS), a volunteer computing project dedicated to searching for Mersenne primes. It is also used in overclocking to test for system stability. Although most of its source code is available, Prime95 is not free and open-source software because its end-user license agreement states that if the software is used to find a prime qualifying for a bounty offered by the Electronic Frontier Foundation, then that bounty will be claimed and distributed by GIMPS. Finding Mersenne primes by volunteer computing Prime95 tests numbers for primality using the Fermat primality test (referred to internally as PRP, or "probable prime"). For much of its history, it used the Lucas–Lehmer primality test, but the availability of Lucas–Lehmer assignments was deprecated in April 2021 to increas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
