|
Bell Series
In mathematics, the Bell series is a formal power series used to study properties of arithmetical functions. Bell series were introduced and developed by Eric Temple Bell. Given an arithmetic function f and a prime p, define the formal power series f_p(x), called the Bell series of f modulo p as: :f_p(x)=\sum_^\infty f(p^n)x^n. Two multiplicative functions can be shown to be identical if all of their Bell series are equal; this is sometimes called the ''uniqueness theorem'': given multiplicative functions f and g, one has f=g if and only if: :f_p(x)=g_p(x) for all primes p. Two series may be multiplied (sometimes called the ''multiplication theorem''): For any two arithmetic functions f and g, let h=f*g be their Dirichlet convolution. Then for every prime p, one has: :h_p(x)=f_p(x) g_p(x).\, In particular, this makes it trivial to find the Bell series of a Dirichlet inverse. If f is completely multiplicative, then formally: :f_p(x)=\frac. Examples The following is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Phi Function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In other words, it is the number of integers in the range for which the greatest common divisor is equal to 1. The integers of this form are sometimes referred to as totatives of . For example, the totatives of are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, since and . Therefore, . As another example, since for the only integer in the range from 1 to is 1 itself, and . Euler's totient function is a multiplicative function, meaning that if two numbers and are relatively prime, then . This function gives the order of the multiplicative group of integers modulo (the group of units of the ring \Z/n\Z). It is also used for defining the RS ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bell Numbers
In combinatorial mathematics, the Bell numbers count the possible partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan. In an example of Stigler's law of eponymy, they are named after Eric Temple Bell, who wrote about them in the 1930s. The Bell numbers are denoted B_n, where n is an integer greater than or equal to zero. Starting with B_0 = B_1 = 1, the first few Bell numbers are :1, 1, 2, 5, 15, 52, 203, 877, 4140, ... . The Bell number B_n counts the number of different ways to partition a set that has exactly n elements, or equivalently, the number of equivalence relations on it. B_n also counts the number of different rhyme schemes for n -line poems. As well as appearing in counting problems, these numbers have a different interpretation, as moments of probability distributions. In particular, B_n is the n -th moment of a Poisson distribution with mean 1. Counting Set partitions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mobius Function Of Order K
Moebius, Möbius or Mobius may refer to: People * August Ferdinand Möbius (1790–1868), German mathematician and astronomer * Theodor Möbius (1821–1890), German philologist * Karl Möbius (1825–1908), German zoologist and ecologist * Paul Julius Möbius (1853–1907), German neurologist * Dieter Moebius (1944–2015), German/Swiss musician * Mark Mobius (born 1936), emerging markets investments pioneer * Jean Giraud (1938–2012), French comics artist who used the pseudonym Mœbius Fictional characters * Mobius M. Mobius, a character in Marvel Comics * Mobius, also known as the Anti-Monitor, a supervillain in DC Comics Mathematics * Möbius energy, a particular knot energy * Möbius strip, an object with one surface and one edge * Möbius function, an important multiplicative function in number theory and combinatorics ** Möbius transform, transform involving the Möbius function ** Möbius inversion formula, in number theory * Möbius transformation, a particular ... [...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] |
Geometric Series
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each successive term can be obtained by multiplying the previous term by 1/2. In general, a geometric series is written as a + ar + ar^2 + ar^3 + ..., where a is the coefficient of each term and r is the common ratio between adjacent terms. The geometric series had an important role in the early development of calculus, is used throughout mathematics, and can serve as an introduction to frequently used mathematical tools such as the Taylor series, the complex Fourier series, and the matrix exponential. The name geometric series indicates each term is the geometric mean of its two neighboring terms, similar to how the name arithmetic series indicates each term is the arithmetic mean of its two neighboring terms. The sequence of geometric series ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
