to the complex plane which is called an Lfunction . In the classical cases, already, one knows that useful information is contained in the values and behaviour of the Lfunction at points where the series representation does not converge. The general term Lfunction here includes many known types ... that one would wish to see generalized location of zeros and poles functional equation Lfunction functional equation Lfunction , with respect to some vertical line Re s constant interesting values ... for p adic Lfunction p adic Lfunction s, which describe certain Galois module s. The statistics of the zero ... be regarded as complementary to it Langlands work relates largely to Artin Lfunction Artin Lfunction s, which, like Hecke Lfunction Hecke s L functions , were defined several decades earlier, and to L ... research programs. See also Generalized Riemann hypothesis Dirichlet Lfunction Modularity theorem Artin conjecture L functions Artin conjecture Special values of L functions Shimizu Lfunction References ... mathematical world a breakthrough third degree transcendental Lfunction revealed, Physorg.com , March ... Lfunction SpringerEOM title Lfunction id L function&oldid 19281 last Lavrik first A.F. DEFAULTSORT LFunction Category Zeta and L functions de L Funktion es Funci n L fr Fonction L it Funzione ...The theory of L functions has become a very substantial, and still largely conjectural , part of contemporary analytic number theory . In it, broad generalisations of the Riemann zeta function and the Dirichlet LfunctionL series for a Dirichlet character are constructed, and their general properties, in most cases still out of reach of proof, are set out in a systematic way. L functions We should distinguish at the outset between the L series , an infinite series representation for example the Dirichlet series for the Riemann zeta function , and the Lfunction, the function in the complex plane that is its analytic continuation . The general constructions start with an L series, defined first ... more details
In mathematics, a Hecke Lfunction may refer to an Lfunction of a modular form an Lfunction of a Hecke character . Mathematical disambiguation ... more details
In algebraic number theory , an equivariant Artin Lfunction is a function associated to a finite Galois extension of global field s created by packaging together the various Artin Lfunction s associated with the extension. Each extension has many traditional Artin L functions associated with it, corresponding to the group character character s of Representation mathematics representations of the Galois group. By contrast, each extension has a unique corresponding equivariant Lfunction. Equivariant L functions have become increasingly important as a wide range of conjectures and theorems in number theory have been developed around them. Among these are the Brumer&ndash Stark conjecture , the Coates Sinnott conjecture , and a recently developed equivariant Iwasawa conjecture equivariant version of the Iwasawa main conjecture main conjecture in Iwasawa theory . unreferenced date January 2011 Category Field theory Category Algebraic number theory Category Zeta and L functions numtheory stub ... more details
In mathematics , a Dirichlet L series is a function of the form math L s, chi sum n 1 infty frac chi .... By analytic continuation , this function can be extended to a meromorphic function on the whole complex plane , and is then called a Dirichlet Lfunction and also denoted L s , . These functions are named ... s 1 similar to that of the Riemann zeta function are known to exist for all Dirichlet L functions. Just as the Riemann zeta function is conjectured to obey the Riemann hypothesis , so the Dirichlet L ... character is completely multiplicative , its Lfunction can also be written as an Euler product in the half plane of absolute convergence math L s, chi prod p left 1 chi p p s right 1 text for text ... math Lambda s, chi left frac pi k right s a 2 Gamma left frac s a 2 right L s, chi , math where denotes the Gamma function and the symbol a is given by math a begin cases 0 & mbox if chi 1 1, 1 & mbox if chi 1 1, end cases math one has the functional equation Lfunction functional equation math ... zeta function The Dirichlet L functions may be written as a linear combination of the Hurwitz zeta function Hurwitz zeta function at rational values. Fixing an integer k 1, the Dirichlet L functions ... modulo k . Then we can write its Dirichlet Lfunction as math L s, chi sum n 1 infty frac chi ... Generalized Riemann hypothesis Lfunction Modularity theorem Artin conjecture L functions Artin ... that also bears his name. In the course of the proof, Dirichlet shows that Nowrap L s , is non zero at s 1. Moreover, if is principal, then the corresponding Dirichlet Lfunction has a simple pole at s 1. Zeros of the Dirichlet L functions If is a primitive character with &minus 1 1, then the only zeros of L s , with Re s 0 are at the negative even integers. If is a primitive character with &minus 1 &minus 1, then the only zeros of L s , with Re s 0 are at the negative odd integers ... m k and m 1, 2, ..., k . This means that the Hurwitz zeta function for rational q has analytic properties ... more details
In mathematics, the term standard Lfunction refers to a particular type of automorphic Lfunction described by Robert P. Langlands . ref Armand Borel, Automorphic L functions , Proc. Symp. Pure Math 33, American Mathematical Society, 1979, A. Borel and W. Casselman, editors. ref Here, standard refers to the finite dimensional representation r being the standard representation of the Langlands dual L group as a matrix group. Relations to other L functions Standard L functions are thought to be the most general type of Lfunction . Conjecturally, they include all examples of L functions, and in particular are expected to coincide with the Selberg class . Furthermore, all L functions over arbitrary number field s are widely thought to be instances of standard L functions for the general linear group GL n over the rational numbers Q. This makes them a useful testing ground for statements about L functions, since it sometimes affords structure from the theory of automorphic form s. Analytic properties These L functions were proven to always be entire by Roger Godement and Herv Jacquet , ref Roger Godement and Herve Jacquet, Zeta functions of simple algebras , Springer Lecture Notes in Mathematics, volume 260, 1972. ref with the sole exception of Riemann zeta function Riemann function , which arises for n 1. Another proof was later given by Freydoon Shahidi using the Langlands Shahidi method see ref Stephen Gelbart and Freydoon Shahidi, Analytic Properties of Automorphic L functions , Academic Press, New York, 1988. ref for a useful broader discussion . References reflist Category Zeta and L functions ... more details
In mathematics , motivic L functions are a generalization of Hasse Weil Lfunction Hasse Weil L functions to general motive algebraic geometry motives over global field s. The local L factor at a finite ... realization of the motive. It is conjectured that, like other LfunctionL functions , that each motivic Lfunction can be analytic continuation analytically continued to a meromorphic function on the entire complex plane and satisfies a functional equation relating the LfunctionL s , M of a motive M to nowrap L 1 &minus s , M sup sup , where M sup sup is the dual of the motive M . ref Another common normalization of the L functions consists in shifting the one used here so that the functional ... of the motive. ref Examples Basic examples include Artin Lfunction Artin L functions and Hasse Weil L functions. It is also known harv Scholl 1990 , for example, that a motive can be attached to a newform i.e. a primitive cusp form , hence their L functions are motivic. Conjectures Several conjectures exist concerning motivic L functions. It is believed that motivic L functions should all arise as automorphic Lfunction automorphic L functions , ref harvnb Langlands 1980 ref and hence should be part of the Selberg class . There are also conjectures concerning the values of these L functions at integers generalizing those known for the Riemann zeta function , such as Deligne s conjecture L functions Deligne s conjecture on special values of L functions , the Beilinson conjecture , and the Bloch Kato conjecture L functions Bloch Kato conjecture on special values of L functions . Notes reflist ... fontions L et p riodes d int grales contribution url http www.ams.org online bks pspum332 pspum332 ptIV 8.pdf title Automorphic Forms, Representations, and L Functions editor last Borel editor first ... first Robert P. author link Robert Langlands contribution L functions and automorphic representations ...?id SDPP 1969 1970 11 2 A4 0 L functions footer Category Zeta and L functions Category Algebraic geometry ... more details
Unreferenced date December 2009 In mathematics , the Lfunction s of number theory are expected to have several characteristic properties, one of which is that they satisfy certain functional equation s . There is an elaborate theory of what these equations should be, much of which is still conjectural. For example, the Riemann zeta function has a functional equation relating its value at the complex number s with its value at 1 &minus s . In every case this relates to some value s that is only ... number real field . There is a similar equation for the Dirichlet Lfunction s, but this time relating ... character , sup sup its complex conjugate, the Lfunction multiplied by a gamma factor, and ... formula Lfunction approximate functional equation DEFAULTSORT Functional Equation LFunction Category Zeta and L functions Category Functional equations fr quation fonctionnelle fonction L ja pt Equa o funcional fun o L ... in the line . Therefore use of the functional equation is basic, in order to study the zeta function in the whole complex plane . The functional equation in question for the Riemann zeta function ... the gamma function . This is now read as an extra factor in the Euler product for the zeta function ... zeta function of a number field K , with an appropriate gamma factor that depends only on the embeddings ... is a Gauss sum formed from . This equation has the same function on both sides if and only if is a real ... sums, the value is always 1, so no such simple zero can exist the function is even about the point ... Hecke character s, for which his proof based on theta function s also worked. These characters and their associated L functions are now understood to be strictly related to complex multiplication ... zeta function s, arising at a fundamental level for the analogue of Poincar duality in tale cohomology . The Euler products of the Hasse Weil zeta function for an algebraic variety V over a number ... more details
In mathematics , an Artin Lfunction is a type of Dirichlet series associated to a linear representation .... Cf. Hasse Weil Lfunction for a similar situation. ref The Artin Lfunction math L rho,s math ... Lfunction Dirichlet Lfunction s when K is the rational number field, and as Hecke Lfunction Hecke Lfunction s in general . Novelty comes in with abelian group non abelian G and their representations ... function splits into a product of Artin L functions, for each irreducible representation of G . For example ... representation of degree 2, an Artin Lfunction for such a representation occurs, squared, in the factorisation ... zeta function for the trivial representation and an Lfunction of Dirichlet s type for the signature representation. Functional equation Artin L functions satisfy a functional equation Lfunction functional equation . The functionL s , is related in its values to L 1 &minus s , , where denotes ... the same Lfunction on each side. It is, algebraically speaking, the case when is a real representation ... The Artin conjecture on Artin L functions states that the Artin LfunctionL , s of a non trivial ... for Dirichlet Lfunction s. More generally Artin showed that the Artin conjecture is true ... representation if the Galois representation is irreducible, such that the Artin Lfunction of the Galois representation is the same as the automorphic Lfunction of the automorphic representation .... See also Equivariant Lfunction Notes Reflist group note References Cite journal first E. last Artin ... DEFAULTSORT Artin LFunction Category Zeta and L functions Category Class field theory es Funci n L ... non abelian class field theory is to incorporate the complex analytic nature of Artin L functions ... G math is the Galois group of the finite extension math L K math of number fields, the Artin math L math function math L rho,s math is defined by an Euler product . For each prime ideal math mathfrak P math in math L math , there is an Euler factor, which is easiest to define in the case where math ... more details
DISPLAYTITLE p adic Lfunction In mathematics , a p adic zeta function , or more generally a p adic Lfunction , is a function analogous to the Riemann zeta function , or more general LfunctionL functions , but whose domain of a function domain and codomain target are p adic where p is a prime number ... adic number p adic numbers Q sub p sub or its algebraic closure . The source of a p adic Lfunction ... Leopoldt gave the first construction of a p adic Lfunction harv Kubota Leopoldt 1964 &mdash is via the p adic interpolation of special values of L functions special values of L functions . For example, Kubota Leopoldt used Kummer s congruence s for Bernoulli number s to construct a p adic Lfunction, the p adic Riemann zeta function sub p sub s , whose values at negative odd integers are those of the Riemann zeta function at negative odd integers up to an explicit correction factor . p adic L functions arising in this fashion are typically referred to as analytic p adic L functions . The other major source of p adic L functions&mdash first discovered by Kenkichi Iwasawa &mdash is from the arithmetic ... of cyclotomic fields or even more general towers. A p adic Lfunction arising in this way is typically called an arithmetic p adic Lfunction as it encodes arithmetic data of the Galois module involved ... that the Kubota Leopoldt p adic Lfunction and an arithmetic analogue constructed by Iwasawa ... The Dirichlet Lfunction is given by the analytic continuation of math L s, chi sum n frac chi n n s prod p text prime frac 1 1 chi p p s math The Dirichlet Lfunction at negative integers is given ... p adic LfunctionL sub p sub s , interpolates the Dirichlet Lfunction with the Euler factor at p removed. More precisely, L sub p sub s ,&chi is the unique continuous function of the p ... integers n divisible by p &minus 1. The right hand side is just the usual Dirichlet Lfunction, except ... p adic L functions are constructed or expected , the statement that they agree is called the main ... more details
In mathematics, an automorphic Lfunction is a functionL s , , r of a complex variable s , associated ... are 1. General linear groups harvtxt Godement Jacquet 1972 constructed the automorphic L functions for general linear groups with r the standard representation so called standard Lfunction s and verified ... representation r of the Langlands dual group sup L sup G of G , generalizing the Dirichlet L series ... 1979 and harvtxt Arthur Gelbart 1991 gave surveys of automorphic L functions. Properties Automorphic L functions should have the following properties which have been proved in some cases but are still conjectural in other cases . The LfunctionL s , , r should be a product over the places v of F of local L functions. L s , , r L s , sub v sub , r sub v sub Here the automorphic representation sub v sub is a tensor product of the representations sub v sub of local groups. The Lfunction is expected to have an analytic continuation as a meromorphic function of all complex s , and satisfy a functional equation L s , , r s , , r L 1 s , , r sup &or sup where the factor s , , r is a product ... in Tate s thesis . The Langlands functoriality conjectures imply that all automorphic L functions are equal to L functions of general linear groups, so this would prove the analytic continuation and functional ... editor1 last Coates editor1 first John editor2 last Taylor editor2 first M. J. title L functions ... CBO9780511526053.003 mr 1110389 year 1991 volume 153 chapter Lectures on automorphic L functions pages ..., representations and L functions Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977 ... year 1979 volume XXXIII chapter Automorphic L functions pages 27 61 Citation last1 Cogdell first1 James W. last2 Kim first2 Henry H. last3 Murty first3 Maruti Ram title Lectures on automorphic L functions ... Stephen title Explicit constructions of automorphic L functions publisher Springer Verlag location ... year 1971 Category Automorphic forms Category Zeta and L functions Category Langlands program ... more details
In mathematics, a Shimizu Lfunction , introduced by harvs txt authorlink Hideo Shimizu year 1963 last Shimizu is a Dirichlet series associated to a totally real number field totally real algebraic number field . harvs txt last1 Atiyah first1 Michael Francis author1 link Michael Atiyah last2 Donnelly first2 H. last3 Singer first3 I. M. author3 link Isadore Singer title Eta invariants, signature defects of cusps, and values of L functions url http dx.doi.org 10.2307 2006957 doi 10.2307 2006957 id MR 707164 year 1983 journal Annals of Mathematics Annals of Mathematics. Second Series issn 0003 486X volume 118 issue 1 pages 131 177 defined the signature defect of the boundary of a manifold as the eta invariant , the value as s 0 of their eta function, and used this to show that Hirzebruch s signature defect of a cusp of a Hilbert modular surface can be expressed in terms of the value at s 0 or 1 of a Shimizu Lfunction. Definition Suppose that K is a totally real algebraic number field, M is a lattice in the field, and V is a subgroup of maximal rank of the group of totally positive units preserving the lattice. The Shimazu L series is given by math L M,V,s sum mu in M 0 V frac operatorname sign N mu N mu s math References Citation last1 Atiyah first1 Michael Francis author1 link Michael Atiyah last2 Donnelly first2 H. last3 Singer first3 I. M. title Geometry and analysis of Shimizu L functions url http www.jstor.org stable 12685 id MR 674920 year 1982 journal Proceedings of the National Academy of Sciences Proceedings of the National Academy of Sciences of the United States of America issn 0027 8424 volume 79 issue 18 pages 5751 Citation last1 Atiyah first1 Michael Francis author1 link Michael Atiyah last2 Donnelly first2 H. last3 Singer first3 I. M. title Eta invariants, signature defects of cusps, and values of L functions url http dx.doi.org 10.2307 2006957 doi 10.2307 ... 486X volume 77 pages 33 71 Category zeta and L functions ... more details
wiktionary functionFunction may refer to Diatonic function , a term in music theory Function E 40 song , a 2012 song by American rapper E 40 featuring YG rapper YG , iAmSu & Problem Function biology , explaining why a feature survived selection Function computer science , or subroutine, a portion of code within a larger program, performs a specific task Function engineering , related to the selected property of a system Function language , in linguistics, a way of achieving an aim using language Function mathematics , an abstract entity that associates an input to a corresponding output according to some rule Function model , a structured representation of the functions, activities or processes Function object , or functor or functionoid, a concept of object oriented programming Function Drinks , a beverage company based in Redondo Beach, California. An organised event such as a party or meeting See also Functionalism disambiguation Function hall Functional disambiguation Functionality in polymer chemistry see Structural unit Functor disambiguation bg bs Funkcija vor ca Funci desambiguaci cs Funkce da Funktion de Funktion et Funktsioon es Funci n eo Funkcio eu Funtzio argipena fr Fonction ko id Fungsi it Funzione lt Funkcija lmo Funziun nl Functie ja no Funksjon nn Funksjon pl Funkcja ujednoznacznienie pt Fun o desambigua o ro Func ie dezambiguizare ru simple Function sk Funkcia sl Funkcija razlo itev sr sh Funkcija razvrstavanje sv Funktion olika betydelser th uk zh ... more details
In mathematics, S function may refer to sigmoid function Schur polynomials In physics, it may refer to Action physics action functional mathdab Short pages monitor This long comment was added to the page to prevent it from being listed on Special Shortpages. It and the accompanying monitoring template were generated via Template Long comment. Please do not remove the monitor template without removing the comment as well. ... more details
Image VEST Core4 LowLevel.png thumbnail 320px right VEST 4 T function followed by a transposition layer In cryptography , a T function is a bijection bijective mapping that updates every bit of the state computer science state in a way that can be described as math x i x i f x 0, cdots, x i 1 math , or in simple words an update function in which each bit of the state is updated by a linear combination of the same bit and a function of a subset of its less significant bits. If every single less significant bit is included in the update of every bit in the state, such a T function is called triangular . Thanks to their bijectivity no collisions, therefore no entropy loss regardless of the used Boolean function s and regardless of the selection of inputs as long as they all come from one side of the output bit , T functions are now widely used in cryptography to construct block cipher s, stream cipher s, PRNG s and cryptographic hash function hash functions . T functions were first proposed in 2002 by Alexander Klimov A. Klimov and Adi Shamir A. Shamir in their paper A New Class of Invertible Mappings . Ciphers such as TSC 1 , TSC 3 , TSC 4 , ABC stream cipher ABC , Mir 1 and VEST are built with different types of T functions. Because arithmetic operation s such as addition , subtraction and multiplication are also T functions triangular T functions , software efficient word based T functions can be constructed by combining bitwise logic with arithmetic operations. Another important property of T functions based on arithmetic operations is predictability of their period mathematics period , which is highly attractive to cryptographers. Although triangular T functions are naturally vulnerable to guess and determine attacks, well chosen bitwise transposition mathematics transposition ... bit. Subsequent transposition of the output bits and iteration of the T function also do not affect ... and losing the T function bias of depending only on the less significant bits of the state. References ... more details
In mathematics , a zeta function is usually a function mathematics function analogous to the original example the Riemann zeta function math zeta s sum n 1 infty frac 1 n s . math Zeta functions include Airy zeta function , related to the zeros of the Airy function Arithmetic zeta function Artin Mazur zeta function Artin Mazur zeta function of a dynamical system Barnes zeta function Beurling zeta function of Beurling generalized primes Dedekind zeta function Dedekind zeta function of a number field Real analytic Eisenstein series Epstein zeta function Epstein zeta function of a quadratic form. Goss zeta function of a function field Hasse Weil zeta function Hasse Weil zeta function of a variety Hurwitz zeta function Hurwitz zeta function A generalization of the Riemann zeta function Ihara zeta function Ihara zeta function of a graph Igusa zeta function Igusa zeta function Jacobi zeta function This is related to elliptic functions and is not analogous to the Riemann zeta function. Lfunction , a twisted zeta function. Lefschetz zeta function Lefschetz zeta function of a morphism Lerch zeta function Lerch zeta function A generalization of the Riemann zeta function Local zeta function of a characteristic p variety Matsumoto zeta function Minakshisundaram Pleijel zeta function of a Laplacian Motivic zeta function of a motive Mordell Tornheim zeta function of several variables Multiple zeta function p adic zeta function of a p adic number Prime zeta function Like the Riemann zeta function, but only summed over primes. Riemann zeta function The archetypal example. Selberg zeta function Selberg zeta function of a Riemann surface Shimizu Lfunction Shintani zeta function Weierstrass zeta function This is related to elliptic functions and is not analogous to the Riemann zeta function. Witten zeta function of a Lie group Zeta function operator Zeta function of an operator See also Artin conjecture L functions Artin conjecture Birch and Swinnerton Dyer conjecture Riemann hypothesis ... more details
In microeconomics , the expenditure function describes the minimum amount of money an individual needs to achieve some level of utility, given a utility function and prices. Formally, if there is a utility function math u math that describes preferences over L commodities, the expenditure function math e p, u textbf R L times textbf R rightarrow textbf R math says what amount of money is needed to achieve a utility math u math if prices are set by math p math . This function is defined by math e p, u min x in geq u p cdot x math where math geq u x in textbf R L u x geq u math is the set of all bundles that give utility at least as good as math u math . See also Expenditure minimization problem Hicksian demand function Utility maximization problem References Andreu Mas Colell, Michael D. Whinston and Jerry R. Green Microeconomic Theory , 2007, ISBN 0195102681, pp. 59 60, The Expenditure Function DEFAULTSORT Expenditure Function Category Consumer theory az X rc funksiyas de Ausgabenfunktion vi H m chi ti u ... more details
The Patterson function is used to solve the phase problem in X ray crystallography . It was introduced in 1935 by Arthur Lindo Patterson 1902 1966 while he was a visiting researcher in the laboratory of Bertram Eugene Warren 1902 1991 at MIT. ref A. L. Patterson Z. Krist . A90 , 517 1935 . ref The Patterson function is defined as math P u,v,w sum limits h k l left F h k l right 2 e 2 pi i hu kv lw . math It is essentially the Fourier transform of the intensities rather than the structure factor s. The Patterson function is also equivalent to the electron density convolution convolved with its inverse math P vec u rho vec r rho vec r . math Furthermore, a Patterson map of N points will have N N&minus 1 peaks, excluding the central peak and any overlap. The peaks in the Patterson function are the interatomic distance vectors weighted by the product of the number of electrons in the atoms concerned. Because for each vector between atoms i and j there is an oppositely oriented vector of the same length between atoms j and i , the Patterson function always has centrosymmetry . One dimensional example Consider the series of Dirac delta function delta function s given by math f x delta x 3 delta x 2 delta x 5 3 delta x 8 5 delta x 10 . , math Then the Patterson function is math P u 5 delta u 10 18 delta u 8 9 delta u 6 6 delta u 5 6 delta u 3 18 delta u 2 45 delta u , math math 18 delta u 2 6 delta u 3 6 delta u 5 9 delta u 6 18 delta u 8 5 delta u 10 . , math References reflist DEFAULTSORT Patterson Function Category Crystallography de Patterson Methode fr M thode de Patterson he ja sr Patersonova funkcija ... more details
multiple issues expert November 2008 unreferenced November 2008 In mathematical field of geometric group theory , a length function is a function that assigns a number to each element of a group. Definition A length function L G &rarr R sup sup on a group mathematics group G is a function satisfying math begin align L e & 0, L g 1 & L g L g 1 g 2 & leq L g 1 L g 2 , quad forall g 1, g 2 in G. end align math Compare with the axioms for a Metric mathematics metric and a filtered algebra . Word metric main Word metric An important example of a length is the word metric given a presentation of a group by generators and relations, the length of an element is the length of the shortest word expressing it. Coxeter group s including the symmetric group have combinatorial important length functions, using the simple reflections as generators thus each simple reflection has length 1 . A longest element of a Coxeter group is both important and unique up to conjugation up to different choice of simple reflections . Properties A group with a length function does not form a filtered group , meaning that the sublevel set s math S i g mid ell g leq i math do not form subgroups in general. However, the group ring group algebra of a group with a length functions forms a filtered algebra the axiom math ell gh leq ell g ell h math corresponds to the filtration axiom. PlanetMath attribution id 4365 title Length function Category Group theory Category Geometric group theory ... more details
math vartheta 3 q math is the Jacobi theta function . Conjectures div style float right clear right Image Liouville.svg thumb none Summatory Liouville functionL n up to n 10 sup 4 sup . The readily visible oscillations are due to the first non trivial zero of the Riemann zeta function. Image Liouville big.svg thumb none Summatory Liouville functionL n up to n 10 sup 7 sup ... graph of the summatory Liouville functionL n up to n 2 × 10 sup ... title Liouville Function springer author A.F. Lavrik title Liouville function id L l059620 Reflist ...The Liouville function , denoted by n and named after Joseph Liouville , is an important function mathematics function in number theory . If n is a positive integer , then n is defined as math lambda n 1 Omega n , , math where Big Omega function &Omega n is the number of prime number prime divisor factors of n , counted with multiplicity OEIS A008836 . is multiplicative function completely multiplicative since n is additive function additive . The number one has no prime factors, so 1 0 and therefore 1 1. The Liouville function satisfies the Identity mathematics identity math sum d ... function s Dirichlet inverse is the absolute value of the Mobius function . Series The Dirichlet series for the Liouville function gives the Riemann zeta function as math frac zeta 2s zeta s sum n 1 infty frac lambda n n s . math The Lambert series for the Liouville function is math sum ... Liouville function M n up to n 10 sup 3 sup div The P lya conjecture is a conjecture made by George P lya in 1919. Defining math L n sum k 1 n lambda k , math the conjecture states that math L n leq 0 math for n 1. This turned out to be false. The smallest counter example is n 906150257, found by Minoru Tanaka in 1980. It has since been shown that L n ..., Sign Changes in Sums of the Liouville Function , Mathematics of Computation 77 2008 , no. 263 ... more details
is the period of the function then math L 2 pi k math Generalizations Antiperiodic functions One ... , a periodic function is a function mathematics function that repeats its values in regular ... s, wave s, and other phenomena that exhibit Frequency periodicity . Any function which is not periodic is called aperiodic . Image Periodic function illustration.svg thumb right 300px An illustration of a periodic function with period math P. math Definition A function f is said to be periodic ... a least positive ref For some functions, like a constant function or the indicator function of the rational number s, a least positive period may not exist the infimum of possible positve P being zero . ref constant P with this property, it is called the prime period . A function with period ... . Geometrically, a periodic function can be defined as a function whose graph exhibits translational symmetry . Specifically, a function f is periodic with period P if the graph of f is invariant ... tessellation s of the plane. A function that is not periodic is called aperiodic . Examples Image Sine.svg thumb right 350px A graph of the sine function, showing two complete periods For example, the sine function is periodic with period 2 &pi , since math sin x 2 pi sin x , math for all values of x . This function repeats on intervals of length 2 &pi see the graph to the right . Everyday ... as periodic functions, all with the same period. For a function on the real number s or on the integer s, that means that the entire Graph of a function graph can be formed from copies of one particular portion, repeated at regular intervals. A simple example of a periodic function is the function ... f 1.5 f 2.5 ... 0.5. The graph of the function f is the sawtooth wave . Image Sine cosine plot.svg 300px ... function s sine and cosine are common periodic functions, with period 2 see the figure on the right . The subject of Fourier series investigates the idea that an arbitrary periodic function ... more details
Image Bump2D illustration.png right thumb 250px An illustration of a bump function in two variables. In mathematics , a bump function is a function math f Bbb R n to Bbb R math on a Euclidean space math Bbb R n math which is both smooth function smooth in the sense of having continuous derivatives of all ... distributions . Examples Image Mollifier illustration.png right thumb 280px The function &Psi x . The function ... 1 0 & mbox otherwise end cases math is an example of a bump function in one dimension. It is clear from the construction that this function has compact support. The proof of smoothness follows along the same lines as for the related function discussed in the Non analytic smooth function article. This function can be interpreted as the Gaussian function math e y 2 math scaled to fit into the unit .... math A simple example of a bump function in math n math variables is obtained by taking the product of math n math copies of the above bump function in one variable, so math Phi x 1, x 2, dots, x n ... and math U math is an open set containing math K, math there exists a bump function math phi math ... small neighborhood of math K, math this amounts to being able to construct a function that is 1 on math ... in math U, math so math K subset V o subset V subset U. math The indicator function characteristic function math chi V math of math V math will be equal to 1 on math V math and math 0 math outside of math ... This function is not smooth however. The key idea is to smooth math chi V math a bit, by taking the convolution of math chi V math with a mollifier . The latter is just a bump function with a very small ... function math Phi math from the previous section and performing appropriate scalings. Properties and uses While bump functions are smooth, they cannot be analytic function analytic unless they vanish ... as mollifier s, as smooth cutoff function s, and to form smooth partitions of unity . They are the most ... operations. For instance, the sum, product, or convolution of two bump functions is again a bump function ... more details
A Chapman function describes the integration of atmospheric absorption electromagnetic radiation absorption along a slant path on a spherical earth, relative to the vertical case. It applies for any quantity with a concentration exponential decay decreasing exponentially with increasing altitude . To a first approximation, valid at small zenith angle s, the Chapman function for absorption optics optical absorption is equal to math sec z , math where z is the zenith angle and sec denotes the secant function . The Chapman function is named after Sydney Chapman astronomer Sydney Chapman . See also Atmospheric physics Ionosphere References Chapman, S., Absorption and dissociative or ionising effects of monochromatic radiation in an atmosphere on a rotating earth , Proc. Phys. Soc., London, 43 , 1047 1055, 1931 Smith III, F. L. and C. Smith, Numerical evaluation of Chapman s grazing incidence integral ch X, , J. Geophys. Res., 77 , 3592 3597, 1972 External links http scienceworld.wolfram.com chemistry ChapmanFunction.html Chapman function at Science World http www mpl.sri.com software chapman chapman.html David L. Huestis at SRI Category Radio frequency propagation Category Special functions Physics stub ... more details
In mathematics , an ru R R function , or Rvachev function , is a real valued function whose sign does not change if none of the signs of its arguments does not change, that is, if its sign is determined solely by the signs of its arguments. ref V.L. Rvachev, On the analytical description of some geometric objects , Reports of Ukrainian Academy of Sciences , vol. 153 , no. 4, 1963, pp. 765 767 in Russian ref ref V. Shapiro, Semi analytic geometry with R Functions, Acta Numerica, Cambridge University Press, 2007, 16 239 303 ref Interpreting positive values as true and negative values as false , an R function is transformed into a companion Boolean function the two functions are called friends . For instance, the R function x , y min x , y is one possible friend of the logical conjunction AND . R functions are used in computer graphics and geometric modeling in the context of implicit surface s and the function representation . They also appear in certain boundary value problem s, and are also popular in certain artificial intelligence applications, where they are used in pattern recognition . R functions were first proposed by ru , Vladimir Logvinovich Rvachev ref http users.kpi.kharkov.ua apm all rva75en.htm 75 years to Vladimir L. Rvachev 75th anniversary biographical tribute ref lang ru in 1963, though the name, R functions , was given later on by Ekaterina L. Rvacheva Yushchenko, in memory of their father, Logvin Fedorovich Rvachev, who perished in Stalin s concentration camps. See also Function representation Notes references References http sal cnc.me.wisc.edu Research meshless R functions R functions.html Meshfree Modeling and Analysis, R Functions University of Wisconsin http docs.lib.purdue.edu dissertations AAI3263546 Pattern Recognition Methods Based on Rvachev Functions Purdue University http hyperfun.org wiki doku.php?id frep main Shape Modeling and Computer Graphics with Real Functions ... more details
In mathematical optimization , the perturbation function is any function mathematics function which relates to primal and dual problem s. The name comes from the fact that any such function defines a perturbation of the initial problem. In many cases this takes the form of shifting the constraints. ref ... function is called the perturbation function, and the perturbation function is called the bifunction ... . Then given the function math f X to mathbb R cup infty math , we can define the primal problem by math inf x in X f x . , math If there are constraint conditions, these can be built in to the function ... function convex analysis indicator function . Then math F X times Y to mathbb R cup infty math is a perturbation function if and only if math F x,0 f x math . ref name BWG ref name Zalinescu cite book ... Verlag Berlin GmbH isbn 978 3 8325 2503 3 ref For any choice of perturbation function F weak duality ... For instance, if F is proper convex function proper , jointly convex function convex , lower semi ... pairs. Given a primal problem minimize f x and a related perturbation function F x,y then the Lagrangian math L X times Y to mathbb R cup infty math is the negative conjugate of F with respect to y i.e. the concave conjugate . That is the Lagrangian is defined by math L x,y inf y in Y left F x,y ... F 0,y sup y in Y inf x in X L x,y leq inf x in X sup y in Y L x,y inf x in X F x,0 . math If the primal ... function is math F x,y f x I mathbb R d y g x math . Thus the connection to Lagrangian duality can be seen, as L can be trivially seen to be math L x,y begin cases f x y g x & text if y in mathbb ... operator math T Y to X math , and assume the primal objective function math f x math including the constraints by way of the indicator function can be written as math f x J x,Tx math such that math J X times Y to mathbb R cup infty math . Then the perturbation function is given by math F x,y J x,Tx y math . In particular if the primal objective is math f x g Tx math then the perturbation function ... more details
In mathematics, omega function or function may refer to Pearson Cunningham function Lambert W function Wright Omega function Mathematical disambiguation ... more details
Encyclopedia
top
