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Möbius Inversion Formula
In mathematics, the classic Möbius inversion formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors. It was introduced into number theory in 1832 by August Ferdinand Möbius. A large generalization of this formula applies to summation over an arbitrary locally finite partially ordered set, with Möbius' classical formula applying to the set of the natural numbers ordered by divisibility: see incidence algebra. Statement of the formula The classic version states that if and are arithmetic functions satisfying : g(n)=\sum_f(d)\quad\textn\ge 1 then :f(n)=\sum_\mu(d)g\left(\frac\right)\quad\textn\ge 1 where is the Möbius function and the sums extend over all positive divisors of (indicated by d \mid n in the above formulae). In effect, the original can be determined given by using the inversion formula. The two sequences are said to be Möbius transforms of each other. The formula is also correct if and are fun ... [...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 points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Function
Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. That is, when is the identity function, the equality is true for all values of to which can be applied. Definition Formally, if is a set, the identity function on is defined to be a function with as its domain and codomain, satisfying In other words, the function value in the codomain is always the same as the input element in the domain . The identity function on is clearly an injective function as well as a surjective function, so it is bijective. The identity function on is often denoted by . In set theory, where a function is defined as a particular kind of binary relation, the identity function is given by the identity relation, or ''diagonal'' of . Algebraic properties If is any function, then we h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philip Hall
Philip Hall FRS (11 April 1904 – 30 December 1982), was an English mathematician. His major work was on group theory, notably on finite groups and solvable groups. Biography He was educated first at Christ's Hospital, where he won the Thompson Gold Medal for mathematics, and later at King's College, Cambridge. He was elected a Fellow of the Royal Society in 1951 and awarded its Sylvester Medal in 1961. He was President of the London Mathematical Society in 1955–1957, and awarded its Berwick Prize in 1958 and De Morgan Medal in 1965. Publications * * * See also * Abstract clone * Commutator collecting process * Isoclinism of groups * Regular p-group * Three subgroups lemma * Hall algebra, and Hall polynomials * Hall subgroup * Hall–Higman theorem * Hall–Littlewood polynomial * Hall's universal group * Hall's marriage theorem * Hall word * Hall–Witt identity * Irwin–Hall distribution * Zappa–Szép product In mathematics, especially group theory, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Louis Weisner
Louis Weisner (born 1899–1988) was an American-Canadian mathematician at the University of New Brunswick who introduced Weisner's method. He graduated in 1923 from Columbia University with a Ph.D. in mathematics. His thesis ''Groups whose maximal cyclic subgroups are independent'' was supervised by Frank Nelson Cole. As a postdoc, Weisner was an instructor at the University of Rochester The University of Rochester (U of R, UR, or U of Rochester) is a private university, private research university in Rochester, New York. The university grants Undergraduate education, undergraduate and graduate degrees, including Doctorate, do .... At Hunter College he was appointed an instructor in 1927 and was successively promoted to assistant professor and associate professor. When he was an associate professor in 1954, the Board of Higher Education of the City of New York charged him with "neglect of duty" and "conduct unbecoming a member of the staff" because of his alleged involveme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iverson's Convention
In mathematics, the Iverson bracket, named after Kenneth E. Iverson, is a notation that generalises the Kronecker delta, which is the Iverson bracket of the statement . It maps any statement to a function of the free variables in that statement. This function is defined to take the value 1 for the values of the variables for which the statement is true, and takes the value 0 otherwise. It is generally denoted by putting the statement inside square brackets: = \begin 1 & \text P \text \\ 0 & \text \end In other words, the Iverson bracket of a statement is the indicator function of the set of values for which the statement is true. The Iverson bracket allows using capital-sigma notation without summation index. That is, for any property P(k) of the integer k, \sum_kf(k)\, (k)= \sum_f(k). By convention, f(k) does not need to be defined for the values of for which the Iverson bracket equals ; that is, a summand f(k) textbf/math> must evaluate to 0 regardless of whether f(k) is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Product
In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard Euler. This series and its continuation to the entire complex plane would later become known as the Riemann zeta function. Definition In general, if is a bounded multiplicative function, then the Dirichlet series :\sum_ \frac\, is equal to :\prod_ P(p, s) \quad \text \operatorname(s) >1 . where the product is taken over prime numbers , and is the sum :\sum_^\infty \frac = 1 + \frac + \frac + \frac + \cdots In fact, if we consider these as formal generating functions, the existence of such a ''formal'' Euler product expansion is a necessary and sufficient condition that be multiplicative: this says exactly that is the product of the whenever factors as the product of the powers of distinct primes . An important special case is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Zeta Function
In mathematics, the prime zeta function is an analogue of the Riemann zeta function, studied by . It is defined as the following infinite series, which converges for \Re(s) > 1: :P(s)=\sum_ \frac=\frac+\frac+\frac+\frac+\frac+\cdots. Properties The Euler product for the Riemann zeta function ''ζ''(''s'') implies that : \log\zeta(s)=\sum_ \frac n which by Möbius inversion gives :P(s)=\sum_ \mu(n)\frac n When ''s'' goes to 1, we have P(s)\sim \log\zeta(s)\sim\log\left(\frac \right). This is used in the definition of Dirichlet density. This gives the continuation of ''P''(''s'') to \Re(s) > 0, with an infinite number of logarithmic singularities at points ''s'' where ''ns'' is a pole (only ''ns'' = 1 when ''n'' is a squarefree number greater than or equal to 1), or zero of the Riemann zeta function ''ζ''(.). The line \Re(s) = 0 is a natural boundary as the singularities cluster near all points of this line. If one defines a sequence :a_n=\prod_ \frac=\prod_ \f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduced Fraction
An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and −1, when negative numbers are considered). In other words, a fraction is irreducible if and only if ''a'' and ''b'' are coprime, that is, if ''a'' and ''b'' have a greatest common divisor of 1. In higher mathematics, "irreducible fraction" may also refer to rational fractions such that the numerator and the denominator are coprime polynomials. Every positive rational number can be represented as an irreducible fraction in exactly one way.. An equivalent definition is sometimes useful: if ''a'' and ''b'' are integers, then the fraction is irreducible if and only if there is no other equal fraction such that or , where means the absolute value of ''a''. (Two fractions and are ''equal'' or ''equivalent'' if and only if ''ad'' = ''bc''.) For example, , , and are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Inverse
In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet. Definition If f , g : \mathbb\to\mathbb are two arithmetic functions from the positive integers to the complex numbers, the ''Dirichlet convolution'' is a new arithmetic function defined by: : (f*g)(n) \ =\ \sum_ f(d)\,g\!\left(\frac\right) \ =\ \sum_\!f(a)\,g(b) where the sum extends over all positive divisors ''d'' of ''n'', or equivalently over all distinct pairs of positive integers whose product is ''n''. This product occurs naturally in the study of Dirichlet series such as the Riemann zeta function. It describes the multiplication of two Dirichlet series in terms of their coefficients: :\left(\sum_\frac\right) \left(\sum_\frac\right) \ = \ \left(\sum_\frac\right). Properties The set of arithmetic functions forms a commutative ring, the , under pointwise addition, where is d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interval (mathematics)
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other examples of intervals are the set of numbers such that , the set of all real numbers \R, the set of nonnegative real numbers, the set of positive real numbers, the empty set, and any singleton (set of one element). Real intervals play an important role in the theory of integration, because they are the simplest sets whose "length" (or "measure" or "size") is easy to define. The concept of measure can then be extended to more complicated sets of real numbers, leading to the Borel measure and eventually to the Lebesgue measure. Intervals are central to interval arithmetic, a general numerical computing technique that automatically provides guaranteed enclosures for arbitrary formulas, even in the presence of uncertainties, mathematical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'Codomain. ''Encyclopedia of Mathematics''/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number a+bi, is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with rea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
