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Omega Function (other)
In mathematics, omega function refers to a function using the Greek letter omega, written ω or Ω. \Omega (big omega) may refer to: * The lower bound in Big O notation, f \in \Omega (g)\,\!, meaning that the function f\,\! dominates g\,\! in some limit * The prime omega function \Omega(n)\,\!, giving the total number of prime factors of n\,\!, counting them with their multiplicity. * The Lambert W function \Omega(x)\,\!, the inverse of y = x\cdot e^ \,\!, also denoted W(x)\,\!. * Absolute Infinity \omega (omega) may refer to: * The Wright Omega Function \omega(x)\,\!, related to the Lambert W Function * The Pearson–Cunningham function \omega_(x) * The prime omega function In number theory, the prime omega functions \omega(n) and \Omega(n) count the number of prime factors of a natural number n. Thereby \omega(n) (little omega) counts each ''distinct'' prime factor, whereas the related function \Omega(n) (big omega) ... \omega(n)\,\!, giving the number of distinct prime fac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Omega
Omega (; capital: Ω, lowercase: ω; Ancient Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the twenty-fourth and final letter in the Greek alphabet. In the Greek numeric system/ isopsephy ( gematria), it has a value of 800. The word literally means "great O" (''ō mega'', mega meaning "great"), as opposed to omicron, which means "little O" (''o mikron'', micron meaning "little"). In phonetic terms, the Ancient Greek Ω represented a long open-mid back rounded vowel , comparable to the "aw" of the English word ''raw'' in dialects without the cot–caught merger, in contrast to omicron which represented the close-mid back rounded vowel , and the digraph ''ου'' which represented the long close-mid back rounded vowel . In Modern Greek, both omega and omicron represent the mid back rounded vowel or . The letter omega is transliterated into a Latin-script alphabet as ''ō'' or simply ''o''. As the final letter in the Greek alphabet, omega ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big O Notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for '' Ordnung'', meaning the order of approximation. In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; a famous example of such a difference is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth rates: d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Omega Function
In number theory, the prime omega functions \omega(n) and \Omega(n) count the number of prime factors of a natural number n. Thereby \omega(n) (little omega) counts each ''distinct'' prime factor, whereas the related function \Omega(n) (big omega) counts the ''total'' number of prime factors of n, honoring their multiplicity (see arithmetic function). That is, if we have a prime factorization of n of the form n = p_1^ p_2^ \cdots p_k^ for distinct primes p_i (1 \leq i \leq k), then the respective prime omega functions are given by \omega(n) = k and \Omega(n) = \alpha_1 + \alpha_2 + \cdots + \alpha_k. These prime factor counting functions have many important number theoretic relations. Properties and relations The function \omega(n) is additive and \Omega(n) is completely additive. \omega(n)=\sum_ 1 If p divides n at least once we count it only once, e.g. \omega(12)=\omega(2^2 3)=2. \Omega(n) =\sum_ 1 =\sum_\alpha If p divides n \alpha \geq 1 times then we count the exponent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lambert W Function
In mathematics, the Lambert function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse relation of the function , where is any complex number and is the exponential function. For each integer there is one branch, denoted by , which is a complex-valued function of one complex argument. is known as the principal branch. These functions have the following property: if and are any complex numbers, then :w e^ = z holds if and only if :w=W_k(z) \ \ \text k. When dealing with real numbers only, the two branches and suffice: for real numbers and the equation :y e^ = x can be solved for only if ; we get if and the two values and if . The Lambert relation cannot be expressed in terms of elementary functions. It is useful in combinatorics, for instance, in the enumeration of trees. It can be used to solve various equations involving exponentials (e.g. the maxima of the Planck, Bose–Einstein, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolute Infinite
The Absolute Infinite (''symbol'': Ω) is an extension of the idea of infinity proposed by mathematician Georg Cantor. It can be thought of as a number that is bigger than any other conceivable or inconceivable quantity, either finite or transfinite. Cantor linked the Absolute Infinite with God, Cited as ''Cantor 1883b'' by Jané; with biography by Adolf Fraenkel; reprinted Hildesheim: Georg Olms, 1962, and Berlin: Springer-Verlag, 1980, . Original article. and believed that it had various mathematical properties, including the reflection principle: every property of the Absolute Infinite is also held by some smaller object.''Infinity: New Research and Frontiers'' by Michael Heller and W. Hugh Woodin (2011)p. 11 Cantor's view Cantor said: Cantor also mentioned the idea in his letters to Richard Dedekind (text in square brackets not present in original): The Burali-Forti paradox The idea that the collection of all ordinal numbers cannot logically exist seems paradoxical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pearson–Cunningham Function
In statistics, the Cunningham function or Pearson–Cunningham function ω''m'',''n''(''x'') is a generalisation of a special function introduced by and studied in the form here by . It can be defined in terms of the confluent hypergeometric function ''U'', by :\displaystyle \omega_(x) = \fracU(m/2-n,1+m,x). The function was studied by Cunningham in the context of a multivariate generalisation of the Edgeworth expansion for approximating a probability density function based on its (joint) moments. In a more general context, the function is related to the solution of the constant-coefficient diffusion equation The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's law ..., in one or more dimensions. The function ω''m'',''n''(''x'') is a solution of the differential equation for ''X'': ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
