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De Bruijn Function
In analytic number theory, the Dickman function or Dickman–de Bruijn function ''ρ'' is a special function used to estimate the proportion of smooth numbers up to a given bound. It was first studied by actuary Karl Dickman, who defined it in his only mathematical publication, which is not easily available, and later studied by the Dutch mathematician Nicolaas Govert de Bruijn. Definition The Dickman–de Bruijn function \rho(u) is a continuous function that satisfies the delay differential equation :u\rho'(u) + \rho(u-1) = 0\, with initial conditions \rho(u) = 1 for 0 ≤ ''u'' ≤ 1. Properties Dickman proved that, when a is fixed, we have :\Psi(x, x^)\sim x\rho(a)\, where \Psi(x,y) is the number of ''y''-smooth (or ''y''-friable In materials science, friability ( ), the condition of being friable, describes the tendency of a solid substance to break into smaller pieces under stress or contact, especially by rubbing. The opposite of friable i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pollard P-1
Pollard may refer to: Places in the United States * Pollard, Alabama, a town * Pollard, Arkansas, a city * Pollard, Kansas, an unincorporated community People * Pollard (surname), a list of people with the surname * Pollard Hopewell (between 1786 and 1789 – 1813), midshipman in the United States Navy * Charles Pollard Olivier (1884–1975), American astronomer * James Pollard Espy (1785–1860), American meteorologist * Ngoia Pollard Napaltjarri (c. 1948–2022), Australian indigenous (Warlpiri people) artist * Thomas Pollard Sampson (1875–1961), Australian architect Flora and fauna *Pollard, a tree affected by pollarding (cropping of the upper branches) *Pollard, a deer which has cast its antlers *Pollard or polled livestock, hornless livestock of normally-horned species *Pollard, the European chub (''Squalius cephalus''), a freshwater fish Mathematics *Several algorithms created by British mathematician John Pollard: ** Pollard's kangaroo algorithm ** Pollard's ''p'' &m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramanujan Journal
''The Ramanujan Journal'' is a peer-reviewed scientific journal covering all areas of mathematics, especially those influenced by the Indian mathematician Srinivasa Ramanujan. The journal was established in 1997 and is published by Springer Science+Business Media. According to the ''Journal Citation Reports'', the journal has a 2021 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... of 0.804. References External links * {{DEFAULTSORT:Ramanujan Journal, The English-language journals Mathematics journals Springer Science+Business Media academic journals Academic journals established in 1997 9 times per year journals Srinivasa Ramanujan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson-Dirichlet Distribution
In probability theory, Poisson-Dirichlet distributions are probability distributions on the set of nonnegative, non-increasing sequences with sum 1, depending on two parameters \alpha \in [0,1) and \theta \in (-\alpha, \infty). It can be defined as follows. One considers independent random variables (Y_n)_ such that Y_n follows the beta distribution of parameters 1-\alpha and \theta+n \alpha. Then, the Poisson-Dirichlet distribution PD(\alpha, \theta) of parameters \alpha and \theta is the law of the random decreasing sequence containing Y_1 and the products Y_n \prod_^(1-Y_k). This definition is due to Jim Pitman and Marc Yor. It generalizes Kingman's law, which corresponds to the particular case \alpha = 0. Number theory Patrick Billingsley has proven the following result: if n is a uniform random integer in \, if k \geq 1 is a fixed integer, and if p_1 \geq p_2 \geq \dots \geq p_k are the k largest prime divisors of n (with p_j arbitrarily defined if n has le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rough Number
A ''k''-rough number, as defined by Finch in 2001 and 2003, is a positive integer whose prime factors are all greater than or equal to ''k''. ''k''-roughness has alternately been defined as requiring all prime factors to strictly exceed ''k''.p. 130, Naccache and Shparlinski 2009. Examples (after Finch) #Every odd positive integer is 3-rough. #Every positive integer that is congruent to 1 or 5 mod 6 is 5-rough. #Every positive integer is 2-rough, since all its prime factors, being prime numbers, exceed 1. See also * Buchstab function, used to count rough numbers * Smooth number Notes References * Finch's definition from Number Theory Archives* "Divisibility, Smoothness and Cryptographic Applications", D. Naccache and I. E. Shparlinski, pp. 115–173 in ''Algebraic Aspects of Digital Communications'', eds. Tanush Shaska and Engjell Hasimaj, IOS Press, 2009, . The On-Line Encyclopedia of Integer Sequences The On-Line Encyclopedia of Integer Sequences (OEIS) is an online ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Of Computation
''Mathematics of Computation'' is a bimonthly mathematics journal focused on computational mathematics. It was established in 1943 as ''Mathematical Tables and Other Aids to Computation'', obtaining its current name in 1960. Articles older than five years are available electronically free of charge. Abstracting and indexing The journal is abstracted and indexed in Mathematical Reviews, Zentralblatt MATH, Science Citation Index, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. According to the '' Journal Citation Reports'', the journal has a 2020 impact factor of 2.417. References External links * Delayed open access journals English-language journals Mathematics journals Academic journals established in 1943 American Mathematical Society academic journals Bimonthly journals {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trapezoidal Rule
In calculus, the trapezoidal rule (or trapezium rule in British English) is a technique for numerical integration, i.e., approximating the definite integral: \int_a^b f(x) \, dx. The trapezoidal rule works by approximating the region under the graph of the function f(x) as a trapezoid and calculating its area. It follows that \int_^ f(x) \, dx \approx (b-a) \cdot \tfrac(f(a)+f(b)). The integral can be even better approximated by Partition of an interval, partitioning the integration interval, applying the trapezoidal rule to each subinterval, and summing the results. In practice, this "chained" (or "composite") trapezoidal rule is usually what is meant by "integrating with the trapezoidal rule". Let \ be a partition of [a,b] such that a=x_0 < x_1 < \cdots < x_ < x_N = b and be the length of the -th subinterval (that is, ), then |
Polylogarithm
In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function of order and argument . Only for special values of does the polylogarithm reduce to an elementary function such as the natural logarithm or a rational function. In quantum statistics, the polylogarithm function appears as the closed form of integrals of the Fermi–Dirac distribution and the Bose–Einstein distribution, and is also known as the Fermi–Dirac integral or the Bose–Einstein integral. In quantum electrodynamics, polylogarithms of positive integer order arise in the calculation of processes represented by higher-order Feynman diagrams. The polylogarithm function is equivalent to the Hurwitz zeta function — either function can be expressed in terms of the other — and both functions are special cases of the Lerch transcendent. Polylogarithms should not be confused with polylogarithmic functions, nor with the offset logarithmic integra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if for every x_0 in its domain, its Taylor series about x_0 converges to the function in some neighborhood of x_0 . This is stronger than merely being infinitely differentiable at x_0 , and therefore having a well-defined Taylor series; the Fabius function provides an example of a function that is infinitely differentiable but not analytic. Definitions Formally, a function f is ''real analytic'' on an open set D in the real line if for any x_0\in D one can write f(x) = \sum_^\infty a_ \left( x-x_0 \right)^ = a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 + \cdots in which the coefficients a_0, a_1, \dots a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Integral
In mathematics, the exponential integral Ei is a special function on the complex plane. It is defined as one particular definite integral of the ratio between an exponential function and its argument. Definitions For real non-zero values of ''x'', the exponential integral Ei(''x'') is defined as : \operatorname(x) = -\int_^\infty \fract\,dt = \int_^x \fract\,dt. The Risch algorithm shows that Ei is not an elementary function. The definition above can be used for positive values of ''x'', but the integral has to be understood in terms of the Cauchy principal value due to the singularity of the integrand at zero. For complex values of the argument, the definition becomes ambiguous due to branch points at 0 and Instead of Ei, the following notation is used, :E_1(z) = \int_z^\infty \frac\, dt,\qquad, (z), 0. Properties Several properties of the exponential integral below, in certain cases, allow one to avoid its explicit evaluation through the definition ab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golomb–Dickman Constant
In mathematics, the Golomb–Dickman constant, named after Solomon W. Golomb and Karl Dickman, is a mathematical constant, which arises in the theory of random permutations and in number theory. Its value is :\lambda = 0.62432 99885 43550 87099 29363 83100 83724\dots It is not known whether this constant is rational or irrational. Its simple continued fraction is given by ; 1, 1, 1, 1, 1, 22, 1, 2, 3, 1, .../math>, which appears to have an unusually large number of 1s. Definitions Let ''a''''n'' be the average — taken over all permutations of a set of size ''n'' — of the length of the longest cycle in each permutation. Then the Golomb–Dickman constant is : \lambda = \lim_ \frac. In the language of probability theory, \lambda n is asymptotically the expected length of the longest cycle in a uniformly distributed random permutation of a set of size ''n''. In number theory, the Golomb–Dickman constant appears in connection with the average size of the l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
