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Erdős–Kac Theorem
In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ''ω''(''n'') is the number of distinct prime factors of ''n'', then, loosely speaking, the probability distribution of : \frac is the standard normal distribution. (\omega(n) is sequence A001221 in the OEIS.) This is an extension of the Hardy–Ramanujan theorem, which states that the normal order of ''ω''(''n'') is log log ''n'' with a typical error of size \sqrt. Precise statement For any fixed ''a'' < ''b'', : where is the normal (or "Gaussian") distribution, defined as : More generally, if ''f''(''n'') is a strongly |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations ( Diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic obj ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Function
In number theory, an additive function is an arithmetic function ''f''(''n'') of the positive integer variable ''n'' such that whenever ''a'' and ''b'' are coprime, the function applied to the product ''ab'' is the sum of the values of the function applied to ''a'' and ''b'':Erdös, P., and M. Kac. On the Gaussian Law of Errors in the Theory of Additive Functions. Proc Natl Acad Sci USA. 1939 April; 25(4): 206–207online/ref> f(a b) = f(a) + f(b). Completely additive An additive function ''f''(''n'') is said to be completely additive if f(a b) = f(a) + f(b) holds ''for all'' positive integers ''a'' and ''b'', even when they are not coprime. Totally additive is also used in this sense by analogy with totally multiplicative functions. If ''f'' is a completely additive function then ''f''(1) = 0. Every completely additive function is additive, but not vice versa. Examples Examples of arithmetic functions which are completely additive are: * The restriction of the logarit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Journal Of Mathematics
The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United States, established in 1878 at the Johns Hopkins University by James Joseph Sylvester, an English-born mathematician who also served as the journal's editor-in-chief from its inception through early 1884. Initially W. E. Story was associate editor in charge; he was replaced by Thomas Craig in 1880. For volume 7 Simon Newcomb became chief editor with Craig managing until 1894. Then with volume 16 it was "Edited by Thomas Craig with the Co-operation of Simon Newcomb" until 1898. Other notable mathematicians who have served as editors or editorial associates of the journal include Frank Morley, Oscar Zariski, Lars Ahlfors, Hermann Weyl, Wei-Liang Chow, S. S. Chern, André Weil, Harish-Chandra, Jean Dieudonné, Henri Cartan, Stephen Smale, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pál Turán
Pál Turán (; 18 August 1910 – 26 September 1976) also known as Paul Turán, was a Hungarian mathematician who worked primarily in extremal combinatorics. He had a long collaboration with fellow Hungarian mathematician Paul Erdős, lasting 46 years and resulting in 28 joint papers. Life and education Turán was born into a Jewish family in Budapest on 18 August 1910.At the same period of time, Turán and Erdős were famous answerers in the journal '' KöMaL''. He received a teaching degree at the University of Budapest in 1933 and the PhD degree under Lipót Fejér in 1935 at Eötvös Loránd University. As a Jew, he fell victim to numerus clausus, and could not get a university job for several years. He was sent to labour service at various times from 1940-44. He is said to have been recognized and perhaps protected by a fascist guard, who, as a mathematics student, had admired Turán's work. Turán became associate professor at the University of Budapest in 1945 and ful ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alfréd Rényi
Alfréd Rényi (20 March 1921 – 1 February 1970) was a Hungarian mathematician known for his work in probability theory, though he also made contributions in combinatorics, graph theory, and number theory. Life Rényi was born in Budapest to Artúr Rényi and Borbála Alexander; his father was a mechanical engineer, while his mother was the daughter of philosopher and literary critic Bernhard Alexander; his uncle was Franz Alexander, a Hungarian-American psychoanalyst and physician. He was prevented from enrolling in university in 1939 due to the anti-Jewish laws then in force, but enrolled at the University of Budapest in 1940 and finished his studies in 1944. At this point, he was drafted to forced labour service, from which he escaped. He then completed his PhD in 1947 at the University of Szeged, under the advisement of Frigyes Riesz. He married Katalin Schulhof (who used Kató Rényi as her married name), herself a mathematician, in 1946; their daughter Zsuzsanna was ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
EKT Plot
EKT Gdynia - one of the most popular Polish shanty-bands, created by the guitarist and vocalist Jan Wydra in 1986. Ireneusz Wójcicki joined the group in 1987. From the first group the drummer, Jacek Fimiak, has also been playing. During the past 20 years of the existence EKT Gdynia has won many distinctions in sea shanties festivals in Poland and abroad (f.e. in Canada), and hits like ''24 February'' ( pol: ''24 lutego''), ''The Beskid Pub'' ( pol: ''Bar w Beskidzie'') or ''People don't sell your dreams'' ( pol: ''Ludzie, nie sprzedawajcie swych marzeń'') have contributed to the sea shanty achievements. Lineup * Jan Wydra - vocal, acoustic guitar * Ireneusz "Messalina" Wójcicki - vocal, tambourine *Jacek Fimiak - drums, vocal *Kamil Szewczyk - bass guitar, *Krzysztof Kowalewski - electric guitar An electric guitar is a guitar that requires external amplification in order to be heard at typical performance volumes, unlike a standard acoustic guitar (however combina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sieve Theory
Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. The prototypical example of a sifted set is the set of prime numbers up to some prescribed limit ''X''. Correspondingly, the prototypical example of a sieve is the sieve of Eratosthenes, or the more general Legendre sieve. The direct attack on prime numbers using these methods soon reaches apparently insuperable obstacles, in the way of the accumulation of error terms. In one of the major strands of number theory in the twentieth century, ways were found of avoiding some of the difficulties of a frontal attack with a naive idea of what sieving should be. One successful approach is to approximate a specific sifted set of numbers (e.g. the set of prime numbers) by another, simpler set (e.g. the set of almost prime numbers), which is typically somewhat larger than the original set, and easier to analyze. More sophisticated sieves als ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Limit Theorem
In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselves are not normally distributed. The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions. This theorem has seen many changes during the formal development of probability theory. Previous versions of the theorem date back to 1811, but in its modern general form, this fundamental result in probability theory was precisely stated as late as 1920, thereby serving as a bridge between classical and modern probability theory. If X_1, X_2, \dots, X_n, \dots are random samples drawn from a population with overall mean \mu and finite variance and if \bar_n is the sample mean of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lindeberg Condition
In probability theory, Lindeberg's condition is a sufficient condition (and under certain conditions also a necessary condition) for the central limit theorem (CLT) to hold for a sequence of independent random variables. Unlike the classical CLT, which requires that the random variables in question have finite variance and be both independent and identically distributed, Lindeberg's CLT only requires that they have finite variance, satisfy Lindeberg's condition, and be independent. It is named after the Finnish mathematician Jarl Waldemar Lindeberg. Statement Let (\Omega, \mathcal, \mathbb) be a probability space, and X_k : \Omega \to \mathbb,\,\, k \in \mathbb, be ''independent'' random variables defined on that space. Assume the expected values \mathbb\, _k= \mu_k and variances \mathrm\, _k= \sigma_k^2 exist and are finite. Also let s_n^2 := \sum_^n \sigma_k^2 . If this sequence of independent random variables X_k satisfies Lindeberg's condition: : \lim_ \frac\sum_^n \mathbb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Order Of An Arithmetic Function
In number theory, a normal order of an arithmetic function is some simpler or better-understood function which "usually" takes the same or closely approximate values. Let ''f'' be a function on the natural numbers. We say that ''g'' is a normal order of ''f'' if for every ''ε'' > 0, the inequalities : (1-\varepsilon) g(n) \le f(n) \le (1+\varepsilon) g(n) hold for ''almost all'' ''n'': that is, if the proportion of ''n'' ≤ ''x'' for which this does not hold tends to 0 as ''x'' tends to infinity. It is conventional to assume that the approximating function ''g'' is continuous and monotone. Examples * The Hardy–Ramanujan theorem: the normal order of ω(''n''), the number of distinct prime factors of ''n'', is log(log(''n'')); * The normal order of Ω(''n''), the number of prime factors of ''n'' counted with multiplicity, is log(log(''n'')); * The normal order of log(''d''(''n'')), where ''d''(''n'') is the number of divisors of ''n'', is log(2)& ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Erdős
Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory. Much of his work centered around discrete mathematics, cracking many previously unsolved problems in the field. He championed and contributed to Ramsey theory, which studies the conditions in which order necessarily appears. Overall, his work leaned towards solving previously open problems, rather than developing or exploring new areas of mathematics. Erdős published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed. He firmly believed mathematics to be a social activity, living an itinerant lifestyle with the sole purpose of writing mathematical papers with other mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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