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Anatoly Karatsuba
Anatoly Alexeyevich Karatsuba (his first name often spelled Anatolii) (russian: Анато́лий Алексе́евич Карацу́ба; Grozny, Soviet Union, 31 January 1937 – Moscow, Russia, 28 September 2008) was a Russian mathematician working in the field of analytic number theory, ''p''-adic numbers and Dirichlet series. For most of his student and professional life he was associated with the Faculty of Mechanics and Mathematics of Moscow State University, defending a D.Sc. there entitled "The method of trigonometric sums and intermediate value theorems" in 1966. He later held a position at the Steklov Institute of Mathematics of the Academy of Sciences. His textbook ''Foundations of Analytic Number Theory'' went to two editions, 1975 and 1983. The Karatsuba algorithm is the earliest known divide and conquer algorithm for multiplication and lives on as a special case of its direct generalization, the Toom–Cook algorithm. The main research works of Anatoly K ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grozny
Grozny ( rus, Грозный, p=ˈgroznɨj; ce, Соьлжа-ГӀала, translit=Sölƶa-Ġala), also spelled Groznyy, is the capital city of Chechnya, Russia. The city lies on the Sunzha River. According to the 2010 census, it had a population of 271,573 — up from 210,720 recorded in the 2002 census, but still only about two-thirds of 399,688 recorded in the 1989 census. It was previously known as (until 1870). Names In Russian, "Grozny" means "fearsome", "menacing", or "redoubtable", the same word as in Ivan Grozny (Ivan the Terrible). While the official name in Chechen is the same, informally the city is known as "" (""), which literally means "the city () on the Sunzha River ()". In 1996, during the First Chechen War, the Chechen separatists renamed the city Dzhokhar-Ghala ( ce, Джовхар-ГӀала, Dƶovxar-Ġala), literally Dzhokhar City, or Dzhokhar/Djohar for short, after Dzhokhar Dudayev, the first president of the Chechen Republic of Ichke ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karatsuba Algorithm
The Karatsuba algorithm is a fast multiplication algorithm. It was discovered by Anatoly Karatsuba in 1960 and published in 1962. Knuth D.E. (1969) ''The Art of Computer Programming. v.2.'' Addison-Wesley Publ.Co., 724 pp. It is a divide-and-conquer algorithm that reduces the multiplication of two ''n''-digit numbers to three multiplications of ''n''/2-digit numbers and, by repeating this reduction, to at most n^\approx n^ single-digit multiplications. It is therefore asymptotically faster than the traditional algorithm, which performs n^2 single-digit products. For example, the traditional algorithm requires (210)2 = 1,048,576 single-digit multiplications to multiply two 1024-digit numbers (''n'' = 1024 = 210), whereas the Karatsuba algorithm requires 310 = 59,049 thus being ~17.758 times faster. The Karatsuba algorithm was the first multiplication algorithm asymptotically faster than the quadratic "grade school" algorithm. The Toom–Cook algorithm (1963) is a fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prouhet–Tarry–Escott Problem
In mathematics, the Prouhet–Tarry–Escott problem asks for two disjoint multisets ''A'' and ''B'' of ''n'' integers each, whose first ''k'' power sum symmetric polynomials are all equal. That is, the two multisets should satisfy the equations :\sum_ a^i = \sum_ b^i for each integer ''i'' from 1 to a given ''k''. It has been shown that ''n'' must be strictly greater than ''k''. Solutions with k = n - 1 are called ''ideal solutions''. Ideal solutions are known for 3 \le n \le 10 and for n = 12. No ideal solution is known for n=11 or for n \ge 13. This problem was named after Eugène Prouhet, who studied it in the early 1850s, and Gaston Tarry and Edward B. Escott, who studied it in the early 1910s. The problem originates from letters of Christian Goldbach and Leonhard Euler (1750/1751). Examples Ideal solutions An ideal solution for ''n'' = 6 is given by the two sets and , because: : 01 + 51 + 61 + 161 + 171 + 221 = 11 + 21 + 101 + 121 + 201 + 211 : 02 + 5 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hua Luogeng
Hua Luogeng or Hua Loo-Keng (; 12 November 1910 – 12 June 1985) was a Chinese mathematician and politician famous for his important contributions to number theory and for his role as the leader of mathematics research and education in the People's Republic of China. He was largely responsible for identifying and nurturing the renowned mathematician Chen Jingrun who proved Chen's theorem, the best known result on the Goldbach conjecture. In addition, Hua's later work on mathematical optimization and operations research made an enormous impact on China's economy. He was elected a foreign associate of the US National Academy of Sciences in 1982. He was elected a member of the standing Committee of the first to sixth National people's Congress, Vice-Chairman of the sixth National Committee of the Chinese People's Political Consultative Conference (April 1985) and Vice-Chairman of the China Democratic League (1979). He joined the Communist Party of China in 1979. Hua did not rec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ivan Matveyevich Vinogradov
Ivan Matveevich Vinogradov ( rus, Ива́н Матве́евич Виногра́дов, p=ɪˈvan mɐtˈvʲejɪvʲɪtɕ vʲɪnɐˈɡradəf, a=Ru-Ivan_Matveyevich_Vinogradov.ogg; 14 September 1891 – 20 March 1983) was a Soviet mathematician, who was one of the creators of modern analytic number theory, and also a dominant figure in mathematics in the USSR. He was born in the Velikiye Luki district, Pskov Oblast. He graduated from the University of St. Petersburg, where in 1920 he became a Professor. From 1934 he was a Director of the Steklov Institute of Mathematics, a position he held for the rest of his life, except for the five-year period (1941–1946) when the institute was directed by Academician Sergei Sobolev. In 1941 he was awarded the Stalin Prize. In 1951 he became a foreign member of the Polish Academy of Sciences and Letters in Kraków. Mathematical contributions In analytic number theory, ''Vinogradov's method'' refers to his main problem-solving techni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moore Machine
In the theory of computation, a Moore machine is a finite-state machine whose current output values are determined only by its current state. This is in contrast to a Mealy machine, whose output values are determined both by its current state and by the values of its inputs. Like other finite state machines, in Moore machines, the input typicallinfluences the next state Thus the input may indirectly influence subsequent outputs, but not the current or immediate output. The Moore machine is named after Edward F. Moore, who presented the concept in a 1956 paper, “ Gedanken-experiments on Sequential Machines.” Formal definition A Moore machine can be defined as a 6-tuple (Q, q_0, \Sigma, O, \delta, G) consisting of the following: * A finite set of states Q * A start state (also called initial state) q_0 which is an element of Q * A finite set called the input alphabet \Sigma * A finite set called the output alphabet O * A transition function \delta : Q \times \Sigma \right ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
:Edward F
Edward is an English male given name. It is derived from the Anglo-Saxon name ''Ēadweard'', composed of the elements '' ēad'' "wealth, fortune; prosperous" and '' weard'' "guardian, protector”. History The name Edward was very popular in Anglo-Saxon England, but the rule of the Norman and Plantagenet dynasties had effectively ended its use amongst the upper classes. The popularity of the name was revived when Henry III named his firstborn son, the future Edward I, as part of his efforts to promote a cult around Edward the Confessor, for whom Henry had a deep admiration. Variant forms The name has been adopted in the Iberian peninsula since the 15th century, due to Edward, King of Portugal, whose mother was English. The Spanish/Portuguese forms of the name are Eduardo and Duarte. Other variant forms include French Édouard, Italian Edoardo and Odoardo, German, Dutch, Czech and Romanian Eduard and Scandinavian Edvard. Short forms include Ed, Eddy, Eddie, Ted, Teddy an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Soviet mathematician who contributed to the mathematics of probability theory, topology, intuitionistic logic, turbulence, classical mechanics, algorithmic information theory and computational complexity. Biography Early life Andrey Kolmogorov was born in Tambov, about 500 kilometers south-southeast of Moscow, in 1903. His unmarried mother, Maria Y. Kolmogorova, died giving birth to him. Andrey was raised by two of his aunts in Tunoshna (near Yaroslavl) at the estate of his grandfather, a well-to-do nobleman. Little is known about Andrey's father. He was supposedly named Nikolai Matveevich Kataev and had been an agronomist. Kataev had been exiled from St. Petersburg to the Yaroslavl province after his participation in the revolutionary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
FEE Method
In mathematics, the FEE method, or fast E-function evaluation method, is the method of fast summation of series of a special form. It was constructed in 1990 by Ekaterina Karatsuba and is so-named because it makes fast computations of the Siegel -functions possible, in particular of e^x. A class of functions, which are "similar to the exponential function," was given the name "E-functions" by Carl Ludwig Siegel. Among these functions are such special functions as the hypergeometric function, cylinder, spherical functions and so on. Using the FEE, it is possible to prove the following theorem: Theorem: Let y=f(x) be an elementary transcendental function, that is the exponential function, or a trigonometric function, or an elementary algebraic function, or their superposition, or their inverse, or a superposition of the inverses. Then : s_f(n) = O(M(n)\log^2n). \, Here s_f(n) is the complexity of computation (bit) of the function f(x) with accuracy up to n digits, M(n) is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Art Of Computer Programming
''The Art of Computer Programming'' (''TAOCP'') is a comprehensive monograph written by the computer scientist Donald Knuth presenting programming algorithms and their analysis. Volumes 1–5 are intended to represent the central core of computer programming for sequential machines. When Knuth began the project in 1962, he originally conceived of it as a single book with twelve chapters. The first three volumes of what was then expected to be a seven-volume set were published in 1968, 1969, and 1973. Work began in earnest on Volume 4 in 1973, but was suspended in 1977 for work on typesetting prompted by the second edition of Volume 2. Writing of the final copy of Volume 4A began in longhand in 2001, and the first online pre-fascicle, 2A, appeared later in 2001. The first published installment of Volume 4 appeared in paperback as Fascicle 2 in 2005. The hardback Volume 4A, combining Volume 4, Fascicles 0–4, was published in 2011. Volume 4, Fascicle 6 ("Satisfiability") was r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toom–Cook Multiplication
Toom–Cook, sometimes known as Toom-3, named after Andrei Toom, who introduced the new algorithm with its low complexity, and Stephen Cook, who cleaned the description of it, is a multiplication algorithm for large integers. Given two large integers, ''a'' and ''b'', Toom–Cook splits up ''a'' and ''b'' into ''k'' smaller parts each of length ''l'', and performs operations on the parts. As ''k'' grows, one may combine many of the multiplication sub-operations, thus reducing the overall computational complexity of the algorithm. The multiplication sub-operations can then be computed recursively using Toom–Cook multiplication again, and so on. Although the terms "Toom-3" and "Toom–Cook" are sometimes incorrectly used interchangeably, Toom-3 is only a single instance of the Toom–Cook algorithm, where ''k'' = 3. Toom-3 reduces 9 multiplications to 5, and runs in Θ(''n''log(5)/log(3)) ≈ Θ(''n''1.46). In general, Toom-''k'' runs in , where , ''ne'' is the time spent on sub-mu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
