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Dickman 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) integers below ''x''. Ramaswami later gave a rigorous proof that for fixed ''a'', \Psi(x,x^) was asymptotic to x \rho(a), with the error bound :\Psi(x,x^)=x\rho(a)+O(x/\log x) in big O notation. Applications Th ... [...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 (born c. 1948), Australian indigenous (Warlpiri people) artist * Thomas Pollard Sampson (1875–1961), Australian architect Flora and fauna *Pollard, an animal or a tree which has been polled (had its antlers or horns, or branches removed): **Pollard, a deer which has cast its antlers **Pollard or polled livestock, hornless livestock of normally-horned species **Pollard, a tree affected by pollarding, a method for shaping trees, cropping the branches above head-height * Pollard, the European chub (''Squalius cephalus''), a freshwater ... [...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 scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... of 0.804. References External links * {{DEFAULTSORT:Ramanujan Journal, The English-language journals Mathematics journals Springer Science+Business Media academic journals Publications established in 1997 9 times per year journals Srinivasa Ramanujan ... [...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 (OEIS) lists ''p''-rough numbers for small ''p'': * 2-rough numbers: A00 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Buchstab Function
The Buchstab function (or Buchstab's function) is the unique continuous function \omega: \R_\rightarrow \R_ defined by the delay differential equation :\omega(u)=\frac 1 u, \qquad\qquad\qquad 1\le u\le 2, : (u\omega(u))=\omega(u-1), \qquad u\ge 2. In the second equation, the derivative at ''u'' = 2 should be taken as ''u'' approaches 2 from the right. It is named after Alexander Buchstab, who wrote about it in 1937. Asymptotics The Buchstab function approaches e^ \approx 0.561 rapidly as u\to\infty, where \gamma is the Euler–Mascheroni constant. In fact, :, \omega(u)-e^, \le \frac, \qquad u\ge 1, where ''ρ'' is the Dickman function. Also, \omega(u)-e^ oscillates in a regular way, alternating between extrema and zeroes; the extrema alternate between positive maxima and negative minima. The interval between consecutive extrema approaches 1 as ''u'' approaches infinity, as does the interval between consecutive zeroes.p. 131, Cheer and Goldston 1990. Applications The ... [...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 The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as ... of 2.417. References External links * Delayed open access journals English-language journals Mathematics journals Publications ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trapezoidal Rule
In calculus, the trapezoidal rule (also known as the trapezoid rule or trapezium rule; see Trapezoid for more information on terminology) is a technique for 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 trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sums, and is sometimes defined this way. The integral can be even better approximated by 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 ,b/math> such that a=x_0 < x_1 < \cdots < x_ < x_N = b and be the length of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 integral which h ... [...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 its Taylor series about ''x''0 converges to the function in some neighborhood for every ''x''0 in its domain. 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 + a_3 (x-x_0)^3 + \cdots in which the coefficients a_0, a_1, \dots are real numbers and the series is convergent to f(x) for x in a neighborhood of x_0. Alternatively, a real analytic function is an infinitely differentiable function such that the Taylor series at any point x_0 in its domain ... [...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 abov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golomb–Dickman Constant
In mathematics, the Golomb–Dickman constant 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. Definitions Let ''a''''n'' be the average — taken over all permutations of a set of size ''n'' — of the length of the longest cyclic permutation, 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 value, expected length of the longest cycle in a discrete uniform distribution, 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 largest prime factor of an integer. More precisely, :\lambda = \lim_ \frac1n \sum_^n \frac, where P_1(k) is the largest prime factor of ''k'' . So if ''k'' is a ''d'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal De Théorie Des Nombres De Bordeaux
A journal, from the Old French ''journal'' (meaning "daily"), may refer to: *Bullet journal, a method of personal organization *Diary, a record of what happened over the course of a day or other period *Daybook, also known as a general journal, a daily record of financial transactions *Logbook, a record of events important to the operation of a vehicle, facility, or otherwise *Record (other) *Transaction log, a chronological record of data processing *Travel journal In publishing, ''journal'' can refer to various periodicals or serials: *Academic journal, an academic or scholarly periodical **Scientific journal, an academic journal focusing on science **Medical journal, an academic journal focusing on medicine **Law review, a professional journal focusing on legal interpretation *Magazine, non-academic or scholarly periodicals in general **Trade magazine, a magazine of interest to those of a particular profession or trade **Literary magazine, a magazine devoted to literat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
