Functions and mappings abcFrom Wikipedia, the free encyclopediaContents1 A-equivalence 11.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Algebraic function 22.1 Algebraic functions in one variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.1 Introduction and overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 The role of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.3 Monodromy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Algebraic vector bundle 74 Angle of parallelism 84.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Antihomomorphism 115.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.2.1 Involutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Antilinear map 136.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Asano contraction 147.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14iii CONTENTS7.2 Location of zeroes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.3 Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Biholomorphism 168.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168.2 Riemann mapping theorem and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168.3 Alternative denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Bijection 199.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.2.1 Batting line-up of a baseball team . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.2.2 Seats and students of a classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.3 More mathematical examples and some non-examples . . . . . . . . . . . . . . . . . . . . . . . . 219.4 Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.5 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.6 Bijections and cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.7 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.8 Bijections and category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.9 Generalization to partial functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.10Contrast with . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249.14External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2410Bijection, injection and surjection 2510.1Injection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.2Surjection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2610.3Bijection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.4Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2710.5Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.5.1 Injective and surjective (bijective) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.5.2 Injective and non-surjective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.5.3 Non-injective and surjective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.5.4 Non-injective and non-surjective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2810.6Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2910.7Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2910.8History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2910.9See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29CONTENTS iii10.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2911Carleman matrix 3011.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3011.2Bell matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3111.3Jabotinsky matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3111.4Generalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3111.5Matrix properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3111.6Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3211.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3311.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312Carmichael function 3412.1Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3412.2Carmichaels theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3412.3Hierarchy of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.4Properties of the Carmichael function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.4.1 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.4.2 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.4.3 Primitive m-th roots of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.4.4 Exponential cycle length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.4.5 Average and typical value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3612.4.6 Lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3612.4.7 Small values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3612.4.8 Image of the function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3712.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3712.6Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3712.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3713Codomain 3813.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3913.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4013.3Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4013.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4014Constant function 4114.1Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4114.2Other properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4114.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4314.4External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4315Conway base 13 function 4415.1The Conway base 13 function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44iv CONTENTS15.1.1 Purpose. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4415.1.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4415.1.3 Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4415.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4515.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4516Correlation (projective geometry) 4616.1In two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4616.2In three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4616.3In higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4616.4Existence of correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4716.5Special types of correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4716.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4717Crystal Ball function 4817.1External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4918Map (mathematics) 5018.1Maps as functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5018.2Maps as morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5018.3Other uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5118.3.1 In logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5118.3.2 In graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5118.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5118.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5218.6Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 5318.6.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5318.6.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5418.6.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Chapter 1A-equivalenceIn mathematics,A -equivalence, sometimes called right-left equivalence, is an equivalence relation between mapgerms.Let M and N be two manifolds, and let f, g:(M, x) (N, y) be two smooth map germs.We say that f and gare A -equivalent if there exist dieomorphism germs : (M, x) (M, x) and : (N, y) (N, y) such that f= g .In other words, two map germs are A -equivalent if one can be taken onto the other by a dieomorphic change ofco-ordinates in the source (i.e.M ) and the target (i.e.N ).Let (Mx, Ny) denote the space of smooth map germs (M, x) (N, y). Let di(Mx) be the group of dieomorphismgerms(M, x) (M, x) and di(Ny) be the group of dieomorphism germs(N, y) (N, y). The groupG:=di(Mx) di(Ny) acts on(Mx, Ny) in the natural way: (, ) f =1 f . Under this ac-tion we see that the map germs f, g: (M, x) (N, y) are A -equivalent if, and only if, g lies in the orbit of f , i.e.g orbG(f) (or vice versa).A map germ is called stable if its orbit under the action of G := di(Mx) di(Ny) is open relative to the Whitneytopology. Since (Mx, Ny) is an innite dimensional space metric topology is no longer trivial. Whitney topologycompares the dierences in successive derivatives and gives a notion of proximity within the innite dimensionalspace. A base for the open sets of the topology in question is given by taking k -jets for every k and taking openneighbourhoods in the ordinary Euclidean sense. Open sets in the topology are then unions of these base sets.Consider the orbit of some map germorbG(f). The map germf is called simple if there are only nitely many otherorbits in a neighbourhood of each of its points. Vladimir Arnold has shown that the only simple singular map germs(Rn, 0) (R, 0) for 1 n 3 are the innite sequence Ak ( k N ), the innite sequence D4+k ( k N ), E6,E7, and E8.1.1 See alsoK-equivalence (contact equivalence)1.2 ReferencesM. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities. Graduate Texts in Mathematics,Springer.1Chapter 2Algebraic functionThis article is about algebraic functions in calculus, mathematical analysis, and abstract algebra. For functions inelementary algebra, see function (mathematics).In mathematics, an algebraic function is a function that can be dened as the root of a polynomial equation. Quiteoften algebraic functions can be expressed using a nite number of terms, involving only the algebraic operationsaddition, subtraction, multiplication, division, and raising to a fractional power:f(x) = 1/x, f(x) =x, f(x) =1 +x3x3/77x1/3are typical examples.However, some algebraic functions cannot be expressed by such nite expressions (as proven by Galois and NielsAbel), as it is for example the case of the function dened byf(x)5+f(x)4+x = 0In more precise terms, an algebraic function of degree n in one variable x is a functiony=f(x) that satises apolynomial equationan(x)yn+an1(x)yn1+ +a0(x) = 0where the coecients ai(x) are polynomial functions of x, with coecients belonging to a set S. Quite often, S= Q,and one then talks about function algebraic over Q", and the evaluation at a given rational value of such an algebraicfunction gives an algebraic number.Afunction which is not algebraic is called a transcendental function, as it is for example the case of exp(x), tan(x), ln(x), (x). A composition of transcendental functions can give an algebraic function:f(x) = cos(arcsin(x)) =1 x2.As an equation of degree n has n roots, a polynomial equation does not implicitly dene a single function, but nfunctions, sometimes also called branches. Consider for example the equation of the unit circle:y2+x2= 1. Thisdetermines y, except only up to an overall sign; accordingly, it has two branches:y= 1 x2.An algebraic function in m variables is similarly dened as a function y which solves a polynomial equation in m +1 variables:p(y, x1, x2, . . . , xm) = 0.It is normally assumed that p should be an irreducible polynomial. The existence of an algebraic function is thenguaranteed by the implicit function theorem.Formally, an algebraic function in m variables over the eld K is an element of the algebraic closure of the eld ofrational functions K(x1,...,xm).22.1. ALGEBRAIC FUNCTIONS IN ONE VARIABLE 32.1 Algebraic functions in one variable2.1.1 Introduction and overviewThe informal denition of an algebraic function provides a number of clues about the properties of algebraic func-tions.To gain an intuitive understanding, it may be helpful to regard algebraic functions as functions which can beformed by the usual algebraic operations: addition, multiplication, division, and taking an nth root. Of course, this issomething of an oversimplication; because of casus irreducibilis (and more generally the fundamental theorem ofGalois theory), algebraic functions need not be expressible by radicals.First, note that any polynomial functiony=p(x) is an algebraic function, since it is simply the solution y to theequationy p(x) = 0.More generally, any rational function y=p(x)q(x) is algebraic, being the solution toq(x)y p(x) = 0.Moreover, the nth root of any polynomial y=np(x) is an algebraic function, solving the equationynp(x) = 0.Surprisingly, the inverse function of an algebraic function is an algebraic function. For supposing that y is a solutiontoan(x)yn+ +a0(x) = 0,for each value of x, then x is also a solution of this equation for each value of y. Indeed, interchanging the roles of xand y and gathering terms,bm(y)xm+bm1(y)xm1+ +b0(y) = 0.Writing x as a function of y gives the inverse function, also an algebraic function.However, not every function has an inverse. For example, y = x2fails the horizontal line test: it fails to be one-to-one.The inverse is the algebraic function x = y . Another way to understand this, is that the set of branches of thepolynomial equation dening our algebraic function is the graph of an algebraic curve.2.1.2 The role of complex numbersFrom an algebraic perspective, complex numbers enter quite naturally into the study of algebraic functions. Firstof all, by the fundamental theorem of algebra, the complex numbers are an algebraically closed eld. Hence anypolynomial relation p(y, x) = 0 is guaranteed to have at least one solution (and in general a number of solutions notexceeding the degree of p in x) for y at each point x, provided we allow y to assume complex as well as real values.Thus, problems to do with the domain of an algebraic function can safely be minimized.Furthermore, even if one is ultimately interested in real algebraic functions, there may be no means to express thefunction in terms of addition, multiplication, division and taking nth roots without resorting to complex numbers (seecasus irreducibilis). For example, consider the algebraic function determined by the equationy3xy + 1 = 0.4 CHAPTER 2. ALGEBRAIC FUNCTIONA graph of three branches of the algebraic function y, where y3 xy + 1 = 0, over the domain 3/22/3< x < 50.Using the cubic formula, we gety= 2x3108 + 1281 12x3+3108 + 1281 12x36.For x 334, the square root is real and the cubic root is thus well dened, providing the unique real root. On theother hand, for x >334, the square root is not real, and one has to choose, for the square root, either non real-squareroot. Thus the cubic root has to be chosen among three non-real numbers. If the same choices are done in the twoterms of the formula, the three choices for the cubic root provide the three branches shown, in the accompanyingimage.It may be proven that there is no way to express this function in terms nth roots using real numbers only, even thoughthe resulting function is real-valued on the domain of the graph shown.On a more signicant theoretical level, using complex numbers allows one to use the powerful techniques of complexanalysis to discuss algebraic functions. In particular, the argument principle can be used to show that any algebraicfunction is in fact an analytic function, at least in the multiple-valued sense.Formally, let p(x, y) be a complex polynomial in the complex variables x and y. Suppose that x0 C is such that thepolynomial p(x0,y) of y has n distinct zeros. We shall show that the algebraic function is analytic in a neighborhoodof x0. Choose a system of n non-overlapping discs i containing each of these zeros. Then by the argument principle12i

ipy(x0, y)p(x0, y)dy= 1.By continuity, this also holds for all x in a neighborhood of x0. In particular, p(x,y) has only one root in i, given bythe residue theorem:fi(x) =12i

iypy(x, y)p(x, y)dywhich is an analytic function.2.2. HISTORY 52.1.3 MonodromyNote that the foregoing proof of analyticity derived an expression for a system of n dierent function elements fi(x),provided that x is not a critical point of p(x, y). Acritical point is a point where the number of distinct zeros is smallerthan the degree of p, and this occurs only where the highest degree term of p vanishes, and where the discriminantvanishes. Hence there are only nitely many such points c1, ..., cm.A close analysis of the properties of the function elements fi near the critical points can be used to show that themonodromy cover is ramied over the critical points (and possibly the point at innity). Thus the entire functionassociated to the fi has at worst algebraic poles and ordinary algebraic branchings over the critical points.Note that, away from the critical points, we havep(x, y) = an(x)(y f1(x))(y f2(x)) (y fn(x))since the fi are by denition the distinct zeros of p. The monodromy group acts by permuting the factors, and thusforms the monodromy representation of the Galois group of p. (The monodromy action on the universal coveringspace is related but dierent notion in the theory of Riemann surfaces.)2.2 HistoryThe ideas surrounding algebraic functions go back at least as far as Ren Descartes. The rst discussion of algebraicfunctions appears to have been in Edward Waring's 1794 An Essay on the Principles of Human Knowledge in whichhe writes:let a quantity denoting the ordinate, be an algebraic function of the abscissa x, by the common methodsof division and extraction of roots, reduce it into an innite series ascending or descending according tothe dimensions of x, and then nd the integral of each of the resulting terms.2.3 See alsoAlgebraic expressionAnalytic functionComplex functionElementary functionFunction (mathematics)Generalized functionList of special functions and eponymsList of types of functionsPolynomialRational functionSpecial functionsTranscendental function2.4 ReferencesAhlfors, Lars (1979). Complex Analysis. McGraw Hill.van der Waerden, B.L. (1931). Modern Algebra, Volume II. Springer.6 CHAPTER 2. ALGEBRAIC FUNCTION2.5 External linksDenition of Algebraic function in the Encyclopedia of MathWeisstein, Eric W., Algebraic Function, MathWorld.Algebraic Function at PlanetMath.org.Denition of Algebraic function in David J. Darling's Internet Encyclopedia of ScienceChapter 3Algebraic vector bundleIn mathematics, an algebraic vector bundle is a vector bundle for which all the transition maps are algebraic func-tions.Serre' theorem states that the category of algebraic vector bundles on an algebraic variety X is anti-equivalent to thecategory of locally free sheaves on X.All SU(2) -instantons over the sphere S4are algebraic vector bundles.7Chapter 4Angle of parallelismIf angle B is right and Aa and Bb are limiting parallel then the angle between Aa and AB is the is the angle of parallelismIn hyperbolic geometry, the angle of parallelism(a) , is the angle at one vertex of a right hyperbolic triangle thathas two asymptotic parallel sides. The angle depends on the segment length a between the right angle and the vertexof the angle of parallelism.Given a point o of a line, if we drop a perpendicular to the line from the point, then a is the distance along thisperpendicular segment, and or (a) is the least angle such that the line drawn through the point at that angle doesnot intersect the given line. Since two sides are asymptotic parallel,lima0(a) =12 and lima(a) = 0.There are ve equivalent expressions that relate (a)and a:sin (a) = sech a =1cosh a,cos (a) = tanh a,tan (a) = csch a =1sinh a,84.1. HISTORY 9tan(12(a)) = ea,(a) =12 gd(a),where sinh, cosh, tanh, sech and csch are hyperbolic functions and gd is the Gudermannian function.4.1 HistoryThe angle of parallelism was developed in 1840 in the German publication Geometrische Untersuchungen zurTheory der Parallellinien by Nicolai Lobachevsky.This publication became widely known in English after the Texas professor G. B. Halsted produced a translation in1891. (Geometrical Researches on the Theory of Parallels)The following passages dene this pivotal concept in hyperbolic geometry:The angle HAD between the parallel HA and the perpendicular AD is called the parallel angle (angle ofparallelism) which we will here designate by (p) for AD = p.[1]:13[2]4.2 DemonstrationThe angle of parallelism, , formulated as: (a) The angle between the x-axis and the line running from x, the center of Q, to y, they-intercept of Q, and (b) The angle from the tangent of Q at y to the y-axis.This diagram, with yellow ideal triangle, is similar to one found in a book by Smogorzhevsky.[3]In the Poincar half-plane model of the hyperbolic plane (see hyperbolic motions) one can establish the relation of to a with Euclidean geometry. Let Q be the semicircle with diameter on the x-axis that passes through the points(1,0) and (0,y), where y > 1. Since Q is tangent to the unit semicircle centered at the origin, the two semicirclesrepresent parallel hyperbolic lines. The y-axis crosses both semicircles, making a right angle with the unit semicircleand a variable angle with Q. The angle at the center of Q subtended by the radius to (0, y) is also because thetwo angles have sides that are perpendicular, left side to left side, and right side to right side. The semicircle Q hasits center at (x, 0), x < 0, so its radius is 1 x. Thus, the radius squared of Q isx2+y2= (1 x)2,hence10 CHAPTER 4. ANGLE OF PARALLELISMx =12(1 y2).The metric of the Poincar half-plane model of hyperbolic geometry parametrizes distance on the ray {(0, y) :y >0 } with natural logarithm. Let log y = a, so y = ea. Then the relation between and a can be deduced from thetriangle {(x, 0), (0, 0), (0, y)}, for example:tan =yx=2yy21=2eae2a1=1sinh a.4.3 References[1] Nicholaus Lobatschewsky (1840) G.B. Halsted translator (1891) Geometrical Researches on the Theory of Parallels, linkfrom Google Books[2] Bonola, Roberto (1955). Non-Euclidean geometry : a critical and historical study of its developments (Unabridged andunaltered republ. of the 1. English translation 1912. ed.). New York, NY: Dover. ISBN 0-486-60027-0.[3] A.S. Smogorzhevsky (1982) Lobachevskian Geometry, 12 Basic formulas of hyperbolic geometry, gure 37, page 60, MirPublishers, MoscowMarvin J. Greenberg (1974) Euclidean and Non-Euclidean Geometries, pp. 2113, W.H. Freeman &Company.Robin Hartshorne (1997) Companion to Euclid pp. 319, 325, American Mathematical Society, ISBN0821807978.Jeremy Gray (1989) Ideas of Space: Euclidean, Non-Euclidean, and Relativistic, 2nd edition, Clarendon Press,Oxford (See pages 113 to 118).Bla Kerkjrt (1966)LesFondementsdelaGomtry, Tome Deux, 97.6 Angle de paralllisme de lagomtry hyperbolique, pp. 411,2, Akademiai Kiado, Budapest.Chapter 5AntihomomorphismIn mathematics, an antihomomorphism is a type of function dened on sets with multiplication that reverses theorder of multiplication. An antiautomorphism is a bijective antihomomorphism, i.e., an antiisomorphism, from aset to itself. From being bijective it follows that it has an inverse, and that the inverse is also an antiautomorphism.5.1 DenitionInformally, an antihomomorphism is map that switches the order of multiplication.Formally, an antihomomorphism between X and Y is a homomorphism :X Yop , where Yop equals Y as a set,but has multiplication reversed: denoting the multiplication on Y as and the multiplication on Yop as , we havex y:= y x . The object Yop is called the opposite object to Y. (Respectively, opposite group, opposite algebra,opposite category etc.)This denition is equivalent to a homomorphism :Xop Y(reversing the operation before or after applying themap is equivalent). Formally, sending X to Xop and acting as the identity on maps is a functor (indeed, an involution).5.2 ExamplesIn group theory, an antihomomorphism is a map between two groups that reverses the order of multiplication. So if : X Y is a group antihomomorphism,(xy) = (y)(x)for all x, y in X.The map that sends x to x1 is an example of a group antiautomorphism. Another important example is the transposeoperation in linear algebra which takes rowvectors to column vectors. Any vector-matrix equation may be transposedto an equivalent equation where the order of the factors is reversed.With matrices, an example of an antiautomorphism is given by the transpose map. Since inversion and transposingboth give antiautomorphisms, their composition is an automorphism. This involution is often called the contragredientmap, and it provides an example of an outer automorphism of the general linear group GL(n,F) where F is a eld,except when |F|= 2 and n= 1 or 2 or |F| = 3 and n=1 (i.e., for the groups GL(1,2), GL(2,2), and GL(1,3))In ring theory, an antihomomorphism is a map between two rings that preserves addition, but reverses the order ofmultiplication. So : X Y is a ring antihomomorphism if and only if:(1) = 1(x+y) = (x)+(y)(xy) = (y)(x)1112 CHAPTER 5. ANTIHOMOMORPHISMfor all x, y in X.[1]For algebras over a eld K, must be a K-linear map of the underlying vector space. If the underlying eld has aninvolution, one can instead ask to be conjugate-linear, as in conjugate transpose, below.5.2.1 InvolutionsIt is frequently the case that antiautomorphisms are involutions, i.e. the square of the antiautomorphismis the identitymap; these are also called involutive antiautomorphisms.The map that sends x to its inverse x1 is an involutive antiautomorphism in any group.A ring with an involutive antiautomorphism is called a *-ring, and these form an important class of examples.5.3 PropertiesIf the target Y is commutative, then an antihomomorphism is the same thing as a homomorphism and an antiauto-morphism is the same thing as an automorphism.The composition of two antihomomorphisms is always a homomorphism, since reversing the order twice preservesorder. The composition of an antihomomorphism with a homomorphism gives another antihomomorphism.5.4 See alsoSemigroup with involution5.5 References[1] Jacobson, Nathan (1943). The Theory of Rings. Mathematical Surveys and Monographs2. American MathematicalSociety. p. 16. ISBN 0821815024.Weisstein, Eric W., Antihomomorphism, MathWorld.Chapter 6Antilinear mapIn mathematics, a mappingf : V Wfrom a complex vector space to another is said to beantilinear (orconjugate-linear or semilinear, though the latter term is more general) iff(ax +by) = af(x) +bf(y)for all a,b Cand all x,y V , where a andb are the complex conjugates of a and b respectively. The compositionof two antilinear maps is complex-linear.An antilinear map f: V W may be equivalently described in terms of the linear mapf: V W from V to thecomplex conjugate vector spaceW .Antilinear maps occur in quantummechanics in the study of time reversal and in spinor calculus, where it is customaryto replace the bars over the basis vectors and the components of geometric objects by dots put above the indices.6.1 ReferencesHorn and Johnson, Matrix Analysis, Cambridge University Press, 1985. ISBN 0-521-38632-2. (antilinearmaps are discussed in section 4.6).Budinich, P. and Trautman, A. The Spinorial Chessboard. Spinger-Verlag, 1988. ISBN 0-387-19078-3. (an-tilinear maps are discussed in section 3.3).6.2 See alsoLinear mapComplex conjugateSesquilinear formMatrix consimilarityTime reversal13Chapter 7Asano contractionIn complex analysis, a discipline in mathematics, and in statistical physics, the Asano contraction or AsanoRuellecontraction is a transformation on a separately ane multivariate polynomial. It was rst presented in 1970 by TaroAsano to prove the LeeYang theorem in the Heisenberg spin model case.This also yielded a simple proof of theLeeYang theorem in the Ising model. David Ruelle proved a general theorem relating the location of the roots of acontracted polynomial to that of the original. Asano contractions have also been used to study polynomials in graphtheory.7.1 DenitionLet (z1, z2, . . . , zn) be a polynomial which, when viewed as a function of only one of these variables is an anefunction. Such functions are called separately ane. For example, a + bz1 + cz2 + dz1z2 is the general form of aseparately ane function in two variables.Any separately ane function can be written in terms of any two of itsvariables as (zi, zj) = a +bzi +czj+dzizj . The Asano contraction (zi, zj) z sends to = a +dz .[1]7.2 Location of zeroesAsano contractions are often used in the context of theorems about the location of roots. Asano originally used thembecause they preserve the property of having no roots when all the variables have magnitude greater than 1.[2] Ruelleprovided a more general relationship which allowed the contractions to be used in more applications.[3] He showedthat if there are closed sets M1, M2, . . . , Mn not containing 0 such that cannot vanish unless zi Mi for someindex i , then=((zj, zk) z)() can only vanish if zi Mi for some index i =k, j or z MjMk whereMjMk= {ab; a Mj, b Mk}[4] Ruelle and others have used this theoremto relate the zeroes of the partitionfunction to zeroes of the partition function of its subsystems.7.3 UseAsano contractions can be used in statistical physics to gain information about a system from its subsystems. Forexample, suppose we have a system with a nite set of particles with magnetic spin either 1 or 1. For each site,we have a complex variable zx Then we can dene a separately ane polynomial P(z)=XcXzXwherezX=xX zx , cX=eU(X)andU(X) is the energy of the state where only the sites inX have positivespin. If all the variables are the same, this is the partition function. Now if =1 2 , then P(z) is obtainedfrom P(z1)P(z2) by contracting the variable attached to identical sites.[4] This is because the Asano contractionessentially eliminates all terms where the spins at a site are distinct in the P(z1) and P(z2) .Ruelle has also used Asano contractions to nd information about the location of roots of a generalization of matchingpolynomials which he calls graph-counting polynomials. He assigns a variable to each edge. For each vertex, he com-putes a symmetric polynomial in the variables corresponding to the edges incident on that vertex. The symmetricpolynomial contains the terms of degree equal to the allowed degree for that node. He then multiplies these sym-147.4. REFERENCES 15metric polynomials together and uses Asano contractions to only keep terms where the edge is present at both itsendpoints. By using the GraceWalshSzeg theorem and intersecting all the sets that can be obtained, Ruelle givessets containing the roots of several types of these symmetric polynomials. Since the graph-counting polynomial wasobtained from these by Asano contractions, most of the remaining work is computing products of these sets.[5]7.4 References[1] Lebowitz, Joel; Ruelle, David; Speer, Eugene (2012). Location of the LeeYang zeros and absence of phase transitionsin some Ising spin systems (PDF). Journal of Mathematical Physics 53. Retrieved 13 May 2015.[2] Asano, Taro (August 1970). Theorems on the Partition Functions of the Heisenberg Ferromagnets. Journal of thePhysical Society of Japan 29 (2): 350359.[3] Gruber, C.; Hintermann, A.; Merlini, D. (1977). Group Analysis of Classical Lattice Systems. Springer Berlin Heidelberg.p. 162. ISBN 978-3-540-37407-7. Retrieved 13 May 2015.[4] .Ruelle, David (1971). Extension of the LeeYang Circle Theorem (PDF). Physical Review Letters 26 (6):303304.Retrieved 13 May 2015.[5] Ruelle, David (1999). Zeros of Graph-Counting Polynomials (PDF). Communications in Mathematical Physics200:4356.Chapter 8BiholomorphismIn the mathematical theory of functions of one or more complex variables, and also in complex algebraic geometry, abiholomorphismor biholomorphic function is a bijective holomorphic function whose inverse is also holomorphic.8.1 Formal denitionFormally, a biholomorphic function is a function dened on an open subset U of the n -dimensional complex spaceCnwith values in Cnwhich is holomorphic and one-to-one, such that its image is an open set V in Cnand the inverse1: V U is also holomorphic. More generally, U and V can be complex manifolds. As in the case of functionsof a single complex variable, a sucient condition for a holomorphic map to be biholomorphic onto its image is thatthe map is injective, in which case the inverse is also holomorphic (e.g., see Gunning 1990, Theorem I.11).If there exists a biholomorphism :U V, we say that U and V are biholomorphically equivalent or that theyare biholomorphic.8.2 Riemann mapping theorem and generalizationsIf n = 1, every simply connected open set other than the whole complex plane is biholomorphic to the unit disc (thisis the Riemann mapping theorem). The situation is very dierent in higher dimensions. For example, open unit ballsand open unit polydiscs are not biholomorphically equivalent for n>1. In fact, there does not exist even a properholomorphic function from one to the other.8.3 Alternative denitionsIn the case of maps f : U C dened on an open subset U of the complex plane C, some authors (e.g., Freitag 2009,Denition IV.4.1) dene a conformal map to be an injective map with nonzero derivative i.e., f(z) 0 for every z inU. According to this denition, a map f : U C is conformal if and only if f: U f(U) is biholomorphic. Otherauthors (e.g., Conway 1978) dene a conformal map as one with nonzero derivative, without requiring that the mapbe injective. According to this weaker denition of conformality, a conformal map need not be biholomorphic eventhough it is locally biholomorphic. For example, if f: U U is dened by f(z) = z2with U = C{0}, then f isconformal on U, since its derivative f(z) = 2z 0, but it is not biholomorphic, since it is 2-1.8.4 ReferencesJohn B. Conway (1978). Functions of One Complex Variable. Springer-Verlag. ISBN 3-540-90328-3.John P. D'Angelo (1993). Several Complex Variables and the Geometry of Real Hypersurfaces. CRC Press.ISBN 0-8493-8272-6.168.4. REFERENCES 17Eberhard Freitag and Rolf Busam (2009). Complex Analysis. Springer-Verlag. ISBN 978-3-540-93982-5.Robert C. Gunning (1990). Introduction to Holomorphic Functions of Several Variables, Vol. II. Wadsworth.ISBN 0-534-13309-6.Steven G. Krantz (2002). Function Theory of Several Complex Variables. American Mathematical Society.ISBN 0-8218-2724-3.Thisarticleincorporatesmaterial frombiholomorphicallyequivalent onPlanetMath, whichislicensedundertheCreative Commons Attribution/Share-Alike License.18 CHAPTER 8. BIHOLOMORPHISM-1 1e-1 eeee-11 -1e0ReReImImez0 2/e-1The complex exponential function mapping biholomorphically a rectangle to a quarter-annulus.Chapter 9BijectionX 1234YDBCAA bijective function, f: X Y, where set X is {1, 2, 3, 4} and set Y is {A, B, C, D}. For example, f(1) = D.In mathematics, a bijection, bijective function or one-to-one correspondence is a function between the elementsof two sets, where every element of one set is paired with exactly one element of the other set, and every elementof the other set is paired with exactly one element of the rst set. There are no unpaired elements. In mathematical1920 CHAPTER 9. BIJECTIONterms, a bijective function f: X Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y.Abijection fromthe set X to the set Y has an inverse function fromY to X. If X and Y are nite sets, then the existenceof a bijection means they have the same number of elements. For innite sets the picture is more complicated, leadingto the concept of cardinal number, a way to distinguish the various sizes of innite sets.A bijective function from a set to itself is also called a permutation.Bijective functions are essential to many areas of mathematics including the denitions of isomorphism, homeomorphism,dieomorphism, permutation group, and projective map.9.1 DenitionFor more details on notation, see Function (mathematics) Notation.For a pairing between X and Y (where Y need not be dierent from X) to be a bijection, four properties must hold:1. each element of X must be paired with at least one element of Y,2. no element of X may be paired with more than one element of Y,3. each element of Y must be paired with at least one element of X, and4. no element of Y may be paired with more than one element of X.Satisfying properties (1) and (2) means that a bijection is a function with domain X. It is more common to seeproperties (1) and (2) written as a single statement: Every element of X is paired with exactly one element of Y.Functions which satisfy property (3) are said to be "onto Y " and are called surjections (or surjective functions).Functions which satisfy property (4) are said to be "one-to-one functions" and are called injections (orinjectivefunctions).[1] With this terminology, a bijection is a function which is both a surjection and an injection, or usingother words, a bijection is a function which is both one-to-one and onto.9.2 Examples9.2.1 Batting line-up of a baseball teamConsider the batting line-up of a baseball team (or any list of all the players of any sports team). The set X will be thenine players on the team and the set Y will be the nine positions in the batting order (1st, 2nd, 3rd, etc.) The pairingis given by which player is in what position in this order. Property (1) is satised since each player is somewhere inthe list. Property (2) is satised since no player bats in two (or more) positions in the order. Property (3) says thatfor each position in the order, there is some player batting in that position and property (4) states that two or moreplayers are never batting in the same position in the list.9.2.2 Seats and students of a classroomIn a classroom there are a certain number of seats. A bunch of students enter the room and the instructor asks themall to be seated. After a quick look around the room, the instructor declares that there is a bijection between the setof students and the set of seats, where each student is paired with the seat they are sitting in. What the instructorobserved in order to reach this conclusion was that:1. Every student was in a seat (there was no one standing),2. No student was in more than one seat,3. Every seat had someone sitting there (there were no empty seats), and4. No seat had more than one student in it.9.3. MORE MATHEMATICAL EXAMPLES AND SOME NON-EXAMPLES 21The instructor was able to conclude that there were just as many seats as there were students, without having to counteither set.9.3 More mathematical examples and some non-examplesFor any set X, the identity function 1X: X X, 1X(x) = x, is bijective.The function f: R R, f(x) = 2x + 1 is bijective, since for each y there is a unique x = (y 1)/2 such that f(x)= y. In more generality, any linear function over the reals, f:R R, f(x) = ax + b (where a is non-zero) is abijection. Each real number y is obtained from (paired with) the real number x = (y - b)/a.The function f: R (-/2, /2), given by f(x) = arctan(x) is bijective since each real number x is pairedwith exactly one angle y in the interval (-/2, /2) so that tan(y) = x (that is, y = arctan(x)). If the codomain(-/2, /2) was made larger to include an integer multiple of /2 then this function would no longer be onto(surjective) since there is no real number which could be paired with the multiple of /2 by this arctan function.The exponential function, g: R R, g(x) = ex, is not bijective: for instance, there is no x in R such that g(x) =1, showing that g is not onto (surjective). However if the codomain is restricted to the positive real numbersR+ (0, +) , then g becomes bijective; its inverse (see below) is the natural logarithm function ln.The function h:R R+, h(x) = x2is not bijective: for instance, h(1) = h(1) = 1, showing that h is not one-to-one (injective). However, if the domain is restricted to R+0[0, +) , then h becomes bijective; its inverseis the positive square root function.9.4 InversesAbijection f with domain X (functionally indicated by f: XY) also denes a relation starting in Y and going to X(by turning the arrows around). The process of turning the arrows around for an arbitrary function does not usuallyyield a function, but properties (3) and (4) of a bijection say that this inverse relation is a function with domain Y.Moreover, properties (1) and (2) then say that this inverse function is a surjection and an injection, that is, the inversefunction exists and is also a bijection. Functions that have inverse functions are said to be invertible.A function isinvertible if and only if it is a bijection.Stated in concise mathematical notation, a function f: X Y is bijective if and only if it satises the conditionfor every y in Y there is a unique x in X with y = f(x).Continuing with the baseball batting line-up example, the function that is being dened takes as input the name ofone of the players and outputs the position of that player in the batting order. Since this function is a bijection, it hasan inverse function which takes as input a position in the batting order and outputs the player who will be batting inthat position.9.5 CompositionThe compositiong f of two bijections f: X Y and g: Y Z is a bijection. The inverse ofg f is(g f)1=(f1) (g1) .Conversely, if the compositiong f of two functions is bijective, we can only say that f is injective and g is surjective.9.6 Bijections and cardinalityIf X and Y are nite sets, then there exists a bijection between the two sets X and Y if and only if X and Y havethe same number of elements. Indeed, in axiomatic set theory, this is taken as the denition of same number ofelements (equinumerosity), and generalising this denition to innite sets leads to the concept of cardinal number,a way to distinguish the various sizes of innite sets.22 CHAPTER 9. BIJECTIONX123YDBCAZPQRA bijection composed of an injection (left) and a surjection (right).9.7 PropertiesA function f: R R is bijective if and only if its graph meets every horizontal and vertical line exactly once.If X is a set, then the bijective functions from X to itself, together with the operation of functional composition(), form a group, the symmetric group of X, which is denoted variously by S(X), SX, or X! (X factorial).Bijections preserve cardinalities of sets: for a subset A of the domain with cardinality |A| and subset B of thecodomain with cardinality |B|, one has the following equalities:|f(A)| = |A| and |f1(B)| = |B|.If X and Y are nite sets with the same cardinality, and f: X Y, then the following are equivalent:1. f is a bijection.2. f is a surjection.3. f is an injection.For a nite set S, there is a bijection between the set of possible total orderings of the elements and the set ofbijections from S to S. That is to say, the number of permutations of elements of S is the same as the numberof total orderings of that setnamely, n!.9.8 Bijections and category theoryBijections are precisely the isomorphisms in the category Set of sets and set functions. However, the bijections are notalways the isomorphisms for more complex categories. For example, in the category Grp of groups, the morphismsmust be homomorphisms since they must preserve the group structure, so the isomorphisms are group isomorphismswhich are bijective homomorphisms.9.9. GENERALIZATION TO PARTIAL FUNCTIONS 239.9 Generalization to partial functionsThe notion of one-one correspondence generalizes to partial functions, where they are calledpartialbijections,although partial bijections are only required to be injective.The reason for this relaxation is that a (proper) partialfunction is already undened for a portion of its domain; thus there is no compelling reason to constrain its inverseto be a total function, i.e.dened everywhere on its domain.The set of all partial bijections on a given base set iscalled the symmetric inverse semigroup.[2]Another way of dening the same notion is to say that a partial bijection from A to B is any relation R (which turnsout to be a partial function) with the property that R is the graph of a bijection f:A B, where A is a subset of Aand likewise B B.[3]When the partial bijection is on the same set, it is sometimes called a one-to-one partial transformation.[4] Anexample is the Mbius transformation simply dened on the complex plane, rather than its completion to the extendedcomplex plane.[5]9.10 Contrast withThis list is incomplete; you can help by expanding it.Multivalued function9.11 See alsoInjective functionSurjective functionBijection, injection and surjectionSymmetric groupBijective numerationBijective proofCardinalityCategory theoryAxGrothendieck theorem9.12 Notes[1] There are names associated to properties (1) and (2) as well. A relation which satises property (1) is called a total relationand a relation satisfying (2) is a single valued relation.[2] Christopher Hollings (16 July 2014). Mathematics across the Iron Curtain: AHistory of the Algebraic Theory of Semigroups.American Mathematical Society. p. 251. ISBN 978-1-4704-1493-1.[3] Francis Borceux (1994). Handbook of Categorical Algebra: Volume 2, Categories and Structures. Cambridge UniversityPress. p. 289. ISBN 978-0-521-44179-7.[4] Pierre A. Grillet (1995). Semigroups: An Introduction to the Structure Theory. CRC Press. p. 228. ISBN 978-0-8247-9662-4.[5] John Meakin (2007). Groups and semigroups: connections and contrasts. In C.M. Campbell, M.R. Quick, E.F. Robert-son, G.C. Smith. Groups St Andrews 2005 Volume 2. Cambridge University Press. p. 367. ISBN 978-0-521-69470-4.preprint citing Lawson, M. V. (1998). The Mbius Inverse Monoid. Journal of Algebra 200 (2): 428. doi:10.1006/jabr.1997.7242.24 CHAPTER 9. BIJECTION9.13 ReferencesThis topic is a basic concept in set theory and can be found in any text which includes an introduction to set theory.Almost all texts that deal with an introduction to writing proofs will include a section on set theory, so the topic maybe found in any of these:Wolf (1998). Proof, Logic and Conjecture: A Mathematicians Toolbox. Freeman.Sundstrom (2003). Mathematical Reasoning: Writing and Proof. Prentice-Hall.Smith; Eggen; St.Andre (2006). A Transition to Advanced Mathematics (6th Ed.). Thomson (Brooks/Cole).Schumacher (1996). Chapter Zero: Fundamental Notions of Abstract Mathematics. Addison-Wesley.O'Leary (2003). The Structure of Proof: With Logic and Set Theory. Prentice-Hall.Morash. Bridge to Abstract Mathematics. Random House.Maddox (2002). Mathematical Thinking and Writing. Harcourt/ Academic Press.Lay (2001). Analysis with an introduction to proof. Prentice Hall.Gilbert; Vanstone (2005). An Introduction to Mathematical Thinking. Pearson Prentice-Hall.Fletcher; Patty. Foundations of Higher Mathematics. PWS-Kent.Iglewicz; Stoyle. An Introduction to Mathematical Reasoning. MacMillan.Devlin, Keith (2004). Sets, Functions, and Logic: An Introduction to Abstract Mathematics. Chapman & Hall/CRC Press.D'Angelo; West (2000). Mathematical Thinking: Problem Solving and Proofs. Prentice Hall.Cupillari. The Nuts and Bolts of Proofs. Wadsworth.Bond. Introduction to Abstract Mathematics. Brooks/Cole.Barnier; Feldman (2000). Introduction to Advanced Mathematics. Prentice Hall.Ash. A Primer of Abstract Mathematics. MAA.9.14 External linksHazewinkel, Michiel, ed. (2001), Bijection, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4Weisstein, Eric W., Bijection, MathWorld.Earliest Uses of Some of the Words of Mathematics: entry on Injection, Surjection and Bijection has the historyof Injection and related terms.Chapter 10Bijection, injection and surjectionIn mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in whicharguments (input expressions from the domain) and images (output expressions from the codomain) are related ormapped to each other.A function maps elements from its domain to elements in its codomain. Given a function f: A BThe function is injective (one-to-one) if every element of the codomain is mapped to by at most one elementof the domain. An injective function is an injection. Notationally:x, y A, f(x) = f(y) x = y.Or, equivalently (using logical transposition),x, y A, x = y f(x) = f(y).The function is surjective (onto) if every element of the codomain is mapped to by at least one element of thedomain. (That is, the image and the codomain of the function are equal.) A surjective function is a surjection.Notationally:y B, x A that such y= f(x).The function isbijective (one-to-oneandonto orone-to-onecorrespondence) if every element of thecodomain is mapped to by exactly one element of the domain. (That is, the function is both injective andsurjective.) A bijective function is a bijection.An injective function need not be surjective (not all elements of the codomain may be associated with arguments),and a surjective function need not be injective (some images may be associated with more than one argument). Thefour possible combinations of injective and surjective features are illustrated in the diagrams to the right.10.1 InjectionMain article: Injective functionFor more details on notation, see Function (mathematics) Notation.Afunction is injective (one-to-one) if every possible element of the codomain is mapped to by at most one argument.Equivalently, a function is injective if it maps distinct arguments to distinct images. An injective function is aninjection. The formal denition is the following.The function f: A B is injective i for all a, b A , we have f(a) = f(b) a = b.A function f : A B is injective if and only if A is empty or f is left-invertible; that is, there is a function g :f(A) A such that g o f = identity function on A. Here f(A) is the image of f.2526 CHAPTER 10. BIJECTION, INJECTION AND SURJECTIONX123YDBCAZPQRSInjective composition: the second function need not be injective.Since every function is surjective when its codomain is restricted to its image, every injection induces a bijectiononto its image. More precisely, every injection f : A B can be factored as a bijection followed by an inclusionas follows. Let fR : A f(A) be f with codomain restricted to its image, and let i : f(A) B be the inclusionmap from f(A) into B. Then f = i o fR. A dual factorisation is given for surjections below.The composition of two injections is again an injection, but if g o f is injective, then it can only be concludedthat f is injective. See the gure at right.Every embedding is injective.10.2 SurjectionMain article: Surjective functionA function is surjective (onto) if every possible image is mapped to by at least one argument. In other words, everyelement in the codomain has non-empty preimage. Equivalently, a function is surjective if its image is equal to itscodomain. A surjective function is a surjection. The formal denition is the following.The function f: A B is surjective i for all b B , there is a A such that f(a) = b.A function f : A B is surjective if and only if it is right-invertible, that is, if and only if there is a function g:B A such that f o g = identity function on B. (This statement is equivalent to the axiom of choice.)By collapsing all arguments mapping to a given xed image, every surjection induces a bijection dened on aquotient of its domain. More precisely, every surjection f : A B can be factored as a non-bijection followedby a bijection as follows. Let A/~ be the equivalence classes of A under the following equivalence relation: x ~y if and only if f(x) = f(y). Equivalently, A/~ is the set of all preimages under f. Let P(~) :A A/~ be theprojection map which sends each x in A to its equivalence class [x]~, and let fP : A/~ B be the well-denedfunction given by fP([x]~) = f(x). Then f = fP o P(~). A dual factorisation is given for injections above.The composition of two surjections is again a surjection, but if g o f is surjective, then it can only be concludedthat g is surjective. See the gure.10.3. BIJECTION 27X1234YDBCAZPQRSurjective composition: the rst function need not be surjective.10.3 BijectionMain article: Bijective functionA function is bijective if it is both injective and surjective. A bijective function is a bijection (one-to-one corre-spondence). A function is bijective if and only if every possible image is mapped to by exactly one argument. Thisequivalent condition is formally expressed as follow.The function f: A B is bijective i for all b B , there is a unique a A such that f(a) = b.A function f : A B is bijective if and only if it is invertible, that is, there is a function g: B A such thatg o f = identity function on A and f o g = identity function on B. This function maps each image to its uniquepreimage.The composition of two bijections is again a bijection, but if g o f is a bijection, then it can only be concludedthat f is injective and g is surjective.(See the gure at right and the remarks above regarding injections andsurjections.)The bijections from a set to itself form a group under composition, called the symmetric group.10.4 CardinalitySuppose you want to dene what it means for two sets to have the same number of elements. One way to do this isto say that two sets have the same number of elements if and only if all the elements of one set can be paired withthe elements of the other, in such a way that each element is paired with exactly one element. Accordingly, we candene two sets to have the same number of elements if there is a bijection between them. We say that the two setshave the same cardinality.Likewise, we can say that set A has fewer than or the same number of elements as set B if there is an injectionfrom A to B . We can also say that set A has fewer than the number of elements in set B if there is an injectionfrom A to B but not a bijection between A and B .28 CHAPTER 10. BIJECTION, INJECTION AND SURJECTIONX123YDBCAZPQRBijective composition: the rst function need not be surjective and the second function need not be injective.10.5 ExamplesIt is important to specify the domain and codomain of each function since by changing these, functions which wethink of as the same may have dierent jectivity.10.5.1 Injective and surjective (bijective)For every set A the identity function idA and thus specically R R : x x .R+ R+: x x2and thus also its inverse R+ R+: x x .The exponential function exp : R R+: x exand thus also its inverse the natural logarithm ln : R+R : x ln x10.5.2 Injective and non-surjectiveThe exponential function exp : R R : x ex10.5.3 Non-injective and surjectiveR R : x (x 1)x(x + 1) = x3xR [1, 1] : x sin(x)10.5.4 Non-injective and non-surjectiveR R : x sin(x)10.6. PROPERTIES 2910.6 PropertiesFor every function f, subset Aof the domain and subset Bof the codomain we have Af 1(f(A)) and f(f 1(B)) B. If f is injective we have A = f 1(f(A)) and if f is surjective we have f(f 1(B)) = B.For every function h : A C we can dene a surjection H : A h(A) : a h(a) and an injection I : h(A) C : a a. It follows that h = I H. This decomposition is unique up to isomorphism.10.7 Category theoryIn the category of sets, injections, surjections, and bijections correspond precisely to monomorphisms, epimorphisms,and isomorphisms, respectively.10.8 HistoryThis terminology was originally coined by the Bourbaki group.10.9 See alsoBijective functionHorizontal line testInjective moduleInjective functionPermutationSurjective function10.10 External linksEarliest Uses of Some of the Words of Mathematics: entry on Injection, Surjection and Bijection has the historyof Injection and related terms.Chapter 11Carleman matrixIn mathematics, a Carleman matrix is a matrix used to convert function composition into matrix multiplication.It is often used in iteration theory to nd the continuous iteration of functions which cannot be iterated by patternrecognition alone. Other uses of Carleman matrices occur in the theory of probability generating functions, andMarkov chains.11.1 DenitionThe Carleman matrix of a function f(x) is dened as:M[f]jk=1k![dkdxk(f(x))j]x=0,so as to satisfy the (Taylor series) equation:(f(x))j=k=0M[f]jkxk.For instance, the computation of f(x) byf(x) =k=0M[f]1,kxk.simply amounts to the dot-product of row 1 of M[f] with a column vector[1, x, x2, x3, ...].The entries of M[f] in the next row give the 2nd power of f(x) :f(x)2=k=0M[f]2,kxk,and also, in order to have the zero'th power off(x) inM[f] , we aadopt the row 0 containing zeros everywhereexcept the rst position, such thatf(x)0= 1 =k=0M[f]0,kxk= 1 +k=10 xk.Thus, the dot product of M[f] with the column vector[1, x, x2, ...]yields the column vector[1, f(x), f(x)2, ...]M[f] [1, x, x2, x3, ...]=[1, f(x), (f(x))2, (f(x))3, ...].3011.2. BELL MATRIX 3111.2 Bell matrixThe Bell matrix of a function f(x) is dened asB[f]jk=1j![djdxj (f(x))k]x=0,so as to satisfy the equation(f(x))k=j=0B[f]jkxj,so it is the transpose of the above Carleman matrix.11.3 Jabotinsky matrixEri Jabotinsky developed that concept of matrices 1947 for the purpose of representation of convolutions of polyno-mials. In an article Analytic Iteration (1963) he introduces the term representation matrix, and generalized thatconcept to two-way-innite matrices. In that article only functions of the type f(x)=a1x +k=2akxkare dis-cussed, but considered for positive *and* negative powers of the function. Several authors refer to the Bell matricesas Jabotinsky matrix since (D. Knuth 1992, W.D. Lang 2000), and possibly this shall grow to a more canonicalname.Analytic Iteration Author(s): Eri Jabotinsky Source: Transactions of the American Mathematical Society, Vol. 108,No. 3 (Sep., 1963), pp.457-477 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/1993593 Accessed: 19/03/2009 15:5711.4 GeneralizationA generalization of the Carleman matrix of a function can be dened around any point, such as:M[f]x0= Mx[x x0]M[f]Mx[x +x0]or M[f]x0= M[g] where g(x) = f(x +x0) x0 . This allows the matrix power to be related as:(M[f]x0)n= Mx[x x0]M[f]nMx[x +x0]11.5 Matrix propertiesThese matrices satisfy the fundamental relationships: M[f g] = M[f]M[g] , B[f g] = B[g]B[f] ,which makes the Carleman matrix M a (direct) representation of f(x) , and the Bell matrix B an anti-representationof f(x) . Here the term f g denotes the composition of functions f(g(x)) .Other properties include: M[fn] = M[f]n, wherefnis an iterated function and M[f1] = M[f]1, wheref1is the inverse function (if the Carleman matrix is invertible).32 CHAPTER 11. CARLEMAN MATRIX11.6 ExamplesThe Carleman matrix of a constant is:M[a] =1 0 0 a 0 0 a20 0 ............The Carleman matrix of the identity function is:Mx[x] =1 0 0 0 1 0 0 0 1 ............The Carleman matrix of a constant addition is:Mx[a +x] =1 0 0 a 1 0 a22a 1 ............The Carleman matrix of a constant multiple is:Mx[cx] =1 0 0 0 c 0 0 0 c2 ............The Carleman matrix of a linear function is:Mx[a +cx] =1 0 0 a c 0 a22ac c2 ............The Carleman matrix of a function f(x) =k=1fkxkis:M[f] =1 0 0 0 f1f2 0 0 f21 ............The Carleman matrix of a function f(x) =k=0fkxkis:M[f] =1 0 0 f0f1f2 f202f0f1f21+ 2f0f2 ............11.7. SEE ALSO 3311.7 See alsoBell polynomialsFunction compositionSchrders equation11.8 ReferencesR Aldrovandi, Special Matrices of Mathematical Physics: Stochastic, Circulant and Bell Matrices, World Sci-entic, 2001. (preview)R. Aldrovandi, L. P. Freitas, Continuous Iteration of Dynamical Maps, online preprint, 1997.P. Gralewicz, K. Kowalski, Continuous time evolution from iterated maps and Carleman linearization, onlinepreprint, 2000.K Kowalski and W-H Steeb, Nonlinear Dynamical Systems and Carleman Linearization, World Scientic,1991. (preview)D. Knuth, Convolution Polynomials arXiv online print, 1992Jabotinsky, Eri: Representation of Functions by Matrices. Application to Faber Polynomials in: Proceedingsof the American Mathematical Society, Vol. 4, No. 4 (Aug., 1953), pp. 546- 553 Stable jstor-URLChapter 12Carmichael functionIn number theory, the Carmichael function of a positive integer n, denoted (n) , is dened as the smallest positiveinteger m such thatam 1 (modn)for every integer a that is coprime to n. In more algebraic terms, it denes the exponent of the multiplicative groupof integers modulo n. The Carmichael function is also known as the reduced totient function or the least universalexponent function, and is sometimes also denoted (n) .The rst 36 values of (n) (sequence A002322 in OEIS) compared to Eulers totient function . (in bold if theyare dierent)12.1 Numerical example72= 49 1 (mod 8) because 7 and 8 are coprime (their greatest common divisor equals 1; they have no commonfactors) and the value of Carmichaels function at 8 is 2. Eulers totient function is 4 at 8 because there are 4 numberslesser than and coprime to 8 (1, 3, 5, and 7). Whilst it is true that 74 = 2401 1 (mod 8), as shown by Eulers theorem,raising 7 to the fourth power is unnecessary because the Carmichael function indicates that 7 squared equals 1 (mod8). Raising 7 to exponents greater than 2 only repeats the cycle 7, 1, 7, 1, ... . Because the same holds true for 3 and5, the Carmichael number is 2 rather than 4.[1]12.2 Carmichaels theoremFor a power of an odd prime, twice the power of an odd prime, and for 2 and 4, (n) is equal to the Euler totient(n); for powers of 2 greater than 4 it is equal to one half of the Euler totient:(n) ={(n) if n = 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 17, 19, 22, 23, 25, 26, 27, 29 . . .12(n) ifn = 8, 16, 32, 64, 128, 256 . . .Eulers function for prime powers is given by(pk) = pk1(p 1).By the fundamental theorem of arithmetic any n > 1 can be written in a unique wayn = pa11pa22. . . pa(n)(n)3412.3. HIERARCHY OF RESULTS 35where p1 < p2 < ... < p are primes and the ai > 0. (n = 1 corresponds to the empty product.)For general n, (n) is the least common multiple of of each of its prime power factors:(n) = lcm[(pa11), (pa22), . . . , (pa(n)(n))].Carmichaels theorem states that if a is coprime to n, thena(n) 1 (modn),where is the Carmichael function dened above. In other words, it asserts the correctness of the formulas. Thiscan be proven by considering any primitive root modulo n and the Chinese remainder theorem.12.3 Hierarchy of resultsSince (n) divides (n), Eulers totient function (the quotients are listed in A034380), the Carmichael functionis a stronger result than the classical Eulers theorem. Clearly Carmichaels theorem is related to Eulers theorem,because the exponent of a nite abelian group must divide the order of the group, by elementary group theory. Thetwo functions dier already in small cases: (15) = 4 while (15) = 8 (see A033949 for the associated n).Fermats little theorem is the special case of Eulers theorem in which n is a prime number p. Carmichaels theoremfor a prime p gives the same result, because the group in question is a cyclic group for which the order and exponentare both p 1.12.4 Properties of the Carmichael function12.4.1 Divisibilitya|b (a)|(b)12.4.2 CompositionFor all positive integers a and b it holds(lcm(a, b)) = lcm((a), (b))This is an immediate consequence of the recursive denition of the Carmichael function.12.4.3 Primitive m-th roots of unityLet a and n be coprime and let m be the smallest exponent with am 1(modn) , then it holdsm|(n)That is, the orders of primitive roots of unity in the ring of integers modulo n are divisors of (n) .12.4.4 Exponential cycle lengthFor a number n with maximum prime exponent of xmax under prime factorization, then for all a (including thosenot co-prime to n ) and all k xmax ,36 CHAPTER 12. CARMICHAEL FUNCTIONak ak+(n)(modn)In particular, for squarefree n ( xmax= 1 ), for all aa a(n)+1(modn)12.4.5 Average and typical valueFor any x > 16, and a constant B:1xnx(n) =xln xeB(1+o(1)) ln ln x/(ln ln ln x).[2][3]HereB= ep(1 1(p 1)2(p + 1)) 0.34537 .For all numbers N and all except o(N) positive integers n N:(n) = n/(ln n)ln ln ln n+A+o(1)where A is a constant,[3][4]A = 1 +plog p(p 1)2 0.2269688 .12.4.6 Lower boundsFor any suciently large number N and for any (ln ln N)3, there are at mostNe0.69(ln )1/3positive integers n N such that (n) ne.[5]For any sequence n1< n2< n3< of positive integers, any constant 0 < c < 1/ ln 2 , and any suciently largei:(ni) > (ln ni)c ln ln ln ni.[6][7]12.4.7 Small valuesFor a constant c and any suciently large positive A, there exists an integer n > A such that (n) < (ln A)c ln ln ln A.[7] Moreover, n is of the formn =(q1)|m and qprime isqfor some square-free integer m < (ln A)c ln ln ln A.[6]12.5. SEE ALSO 3712.4.8 Image of the functionThe set of values of the Carmichael function has counting functionx(log x)+o(1),where =1(1+loglog2)/(log2)=0.08607.[8]12.5 See alsoCarmichael number12.6 Notes[1] http://www25.brinkster.com/denshade/totient.html[2] Theorem 3 in Erds (1991)[3] Sndor & Crstici (2004) p.194[4] Theorem 2 in Erds (1991)[5] Theorem 5 in Friedlander (2001)[6] Theorem 1 in Erds 1991[7] Sndor & Crstici (2004) p.193[8] Ford, Kevin; Luca, Florian; Pomerance, Carl (27 August 2014). The image of Carmichaels -function. arXiv:1408.6506[math.NT].12.7 ReferencesErds, Paul; Pomerance, Carl; Schmutz, Eric (1991). Carmichaels lambda function. Acta Arithmetica 58:363385. ISSN 0065-1036. MR 1121092. Zbl 0734.11047.Friedlander, John B.; Pomerance, Carl; Shparlinski, Igor E. (2001). Period of the power generator andsmall values of the Carmichael function. Mathematics of Computation 70 (236): 15911605, 18031806.doi:10.1090/s0025-5718-00-01282-5. ISSN 0025-5718. MR 1836921. Zbl 1029.11043.Sndor, Jozsef; Crstici, Borislav (2004). Handbook of number theory II. Dordrecht: Kluwer Academic. pp.193195. ISBN 1-4020-2546-7. Zbl 1079.11001.Chapter 13CodomainXYf(x)f : X YxA function f from X to Y. The smaller oval inside Y is the image of f. Y is the codomain of f.In mathematics, the codomain or target set of a function is the set Y into which all of the output of the function isconstrained to fall. It is the set Y in the notation f: X Y. The codomain is also sometimes referred to as the rangebut that term is ambiguous as it may also refer to the image.The codomain is part of a function f if it is dened as described in 1954 by Nicolas Bourbaki,[1] namely a triple (X,Y, F), with F a functional subset[2] of the Cartesian product X Y and X is the set of rst components of the pairsin F (the domain).The set F is called the graph of the function.The set of all elements of the form f(x), where xranges over the elements of the domain X, is called the image of f. In general, the image of a function is a subset ofits codomain. Thus, it may not coincide with its codomain. Namely, a function that is not surjective has elements yin its codomain for which the equation f(x) = y does not have a solution.An alternative denition of function by Bourbaki [Bourbaki, op. cit., p. 77], namely as just a functional graph, doesnot include a codomain and is also widely used.[3] For example in set theory it is desirable to permit the domain of3813.1. EXAMPLES 39a function to be a proper class X, in which case there is formally no such thing as a triple (X, Y, F). With such adenition functions do not have a codomain, although some authors still use it informally after introducing a functionin the form f: X Y.[4][5][6][7][8]13.1 ExamplesFor a functionf : R Rdened byf : x x2, or equivalently f(x) =x2,the codomain of f is R , but f does not map to any negative number. Thus the image of f is the set R+0; i.e., theinterval [0, ).An alternative function g is dened thus:g : R R+0g : x x2.While f and g map a given x to the same number, they are not, in this view, the same function because they havedierent codomains. A third function h can be dened to demonstrate why:h: x x.The domain of h must be dened to be R+0:h: R+0RThe compositions are denotedh fh gOn inspection, h f is not useful. It is true, unless dened otherwise, that the image of f is not known; it is only knownthat it is a subset of R . For this reason, it is possible that h, when composed with f, might receive an argument forwhich no output is dened negative numbers are not elements of the domain of h, which is the square root function.Function composition therefore is a useful notion only when the codomain of the function on the right side of a com-position (not its image, which is a consequence of the function and could be unknown at the level of the composition)is the same as the domain of the function on the left side.The codomain aects whether a function is a surjection, in that the function is surjective if and only if its codomainequals its image. In the example, g is a surjection while f is not. The codomain does not aect whether a function isan injection.A second example of the dierence between codomain and image is demonstrated by the linear transformationsbetween two vector spaces in particular, all the linear transformations from R2to itself, which can be representedby the 22 matrices with real coecients. Each matrix represents a map with the domain R2and codomain R2.However, the image is uncertain. Some transformations may have image equal to the whole codomain (in this case40 CHAPTER 13. CODOMAINthe matrices with rank 2) but many do not, instead mapping into some smaller subspace (the matrices with rank 1 or0). Take for example the matrix T given byT=(1 01 0)which represents a linear transformation that maps the point (x, y) to (x, x). The point (2, 3) is not in the image ofT, but is still in the codomain since linear transformations from R2to R2are of explicit relevance. Just like all 22matrices, T represents a member of that set. Examining the dierences between the image and codomain can oftenbe useful for discovering properties of the function in question. For example, it can be concluded that T does nothave full rank since its image is smaller than the whole codomain.13.2 See alsoRange (mathematics)Domain of a functionSurjective functionInjective functionBijection13.3 Notes[1] N.Bourbaki (1954). Elements de Mathematique,Theorie des Ensembles. Hermann & cie. p. 76.[2] A set of pairs is functional i no two distinct pairs have the same rst component [Bourbaki, op. cit., p. 76][3] Forster 2003, pages 1011[4] Eccles 1997, quote 1, quote 2[5] Mac Lane 1998, page 8[6] Mac Lane, in Scott & Jech 1967, page 232[7] Sharma 2004, page 91[8] Stewart & Tall 1977, page 8913.4 ReferencesEccles, Peter J. (1997), An Introduction to Mathematical Reasoning: Numbers, Sets, and Functions, CambridgeUniversity Press, ISBN 978-0-521-59718-0Forster, Thomas (2003), Logic, Induction and Sets, Cambridge University Press, ISBN 978-0-521-53361-4Mac Lane, Saunders (1998), Categories for the working mathematician (2nd ed.), Springer, ISBN 978-0-387-98403-2Scott, Dana S.; Jech, Thomas J. (1967), Axiomatic set theory, Symposium in Pure Mathematics, AmericanMathematical Society, ISBN 978-0-8218-0245-8Sharma, A.K. (2004), Introduction To Set Theory, Discovery Publishing House, ISBN 978-81-7141-877-0Stewart, Ian; Tall, David Orme (1977), The foundations of mathematics, Oxford University Press, ISBN 978-0-19-853165-4Chapter 14Constant functionNot to be confused with function constant.In mathematics, a constant function is a function whose (output) value is the same for every input value.[1][2][3] Forexample, the function y(x) = 4 is a constant function because the value of y(x) is 4 regardless of the input value x(see image).14.1 Basic propertiesAs a real-valued function of a real-valued argument, a constant function has the general form y(x) = c or just y= c.Example: The function y(x) = 2 or just y= 2 is the specic constant function where the output valueis c=2 . The domain of this function is the set of all real numbers . The codomain of this functionis just {2}. The independent variable x does not appear on the right side of the function expression andso its value is vacuously substituted. Namely y(0)=2, y(2.7)=2, y()=2,.... No matter what value ofx is input, the output is 2.Real-world example: A store where every item is sold for the price of 1 euro.The graph of the constant function y= c is a horizontal line in the plane that passes through the point (0, c) .[4]In the context of a polynomial in one variable x, the non-zero constant function is a polynomial of degree 0 andits general form is f(x)=c , c =0 . This function has no intersection point with the x-axis, that is, it has no root(zero). On the other hand, the polynomialf(x) =0 is the identically zero function. It is the (trivial) constantfunction and every x is a root. Its graph is the x-axis in the plane.[5]A constant function is an even function, i.e. the graph of a constant function is symmetric with respect to the y-axis.In the context where it is dened, the derivative of a function is a measure of the rate of change of function valueswith respect to change in input values. Because a constant function does not change, its derivative is 0.[6] This isoften written: (c)= 0 . The converse is also true. Namely, if y'(x)=0 for all real numbers x, then y(x) is a constantfunction.[7]Example: Given the constant function y(x) = 2 . The derivative of y is the identically zero functiony(x) = (2)= 0 .14.2 Other propertiesFor functions between preordered sets, constant functions are both order-preserving and order-reversing; conversely,if f is both order-preserving and order-reversing, and if the domain of f is a lattice, then f must be constant.Every constant function whose domain and codomain are the same is idempotent.4142 CHAPTER 14. CONSTANT FUNCTIONConstant function y=4Every constant function between topological spaces is continuous.A constant function factors through the one-point set, the terminal object in the category of sets. This obser-vation is instrumental for F. William Lawvere's axiomatization of set theory, the Elementary Theory of theCategory of Sets (ETCS).[8]Every set X is isomorphic to the set of constant functions into it. For each element x and any set Y, there is aunique function x : Y X such that x(y) = x for all y Y. Conversely, if a function f: X Ysatisesf(y) = f(y) for all y, y Y, f is by denition a constant function.As a corollary, the one-point set is a generator in the category of sets.Every set X is canonically isomorphic to the function set X1, or hom set hom(1, X) in the categoryof sets, where 1 is the one-point set. Because of this, and the adjunction between cartesian productsand hom in the category of sets (so there is a canonical isomorphism between functions of two vari-ables and functions of one variable valued in functions of another (single) variable, hom(X Y, Z) =14.3. REFERENCES 43hom(X(hom(Y, Z)) ) the category of sets is a closed monoidal category with the cartesian product ofsets as tensor product and the one-point set as tensor unit. In the isomorphisms: 1 X =X =X 1: natural in X, the left and right unitors are the projections p1 and p2 the ordered pairs (, x)and (x, ) respectively to the element x , where is the unique point in the one-point set.A function on a connected set is locally constant if and only if it is constant.14.3 References[1] Tanton, James (2005). Encyclopedia of Mathematics. Facts on File, New York. p. 94. ISBN 0-8160-5124-0.[2] C.Clapham,J.Nicholson (2009). Oxford Concise Dictionary of Mathematics,Constant Function (PDF). Addison-Wesley. p. 175. Retrieved January 2014.[3] Weisstein, Eric (1999). CRC Concise Encyclopedia of Mathematics. CRC Press, London. p. 313. ISBN 0-8493-9640-9.[4] Dawkins, Paul (2007). College Algebra. Lamar University. p. 224. Retrieved January 2014.[5] Carter, John A.; Cuevas, Gilbert J.; Holliday, Berchie; Marks, Daniel; McClure, Melissa S.publisher=Glencoe/McGraw-Hill School Pub Co (2005). 1. Advanced Mathematical Concepts - Pre-calculus with Applications, Student Edition (1 ed.).p. 22. ISBN 978-0078682278.[6] Dawkins, Paul (2007). Derivative Proofs. Lamar University. Retrieved January 2014.[7] Zero Derivative implies Constant Function. Retrieved January 2014.[8] Leinster, Tom (27 Jun 2011). An informal introduction to topos theory. Retrieved 11 November 2014.Herrlich, Horst and Strecker, George E., Category Theory, Heldermann Verlag (2007).14.4 External linksWeisstein, Eric W., Constant Function, MathWorld.Constant function at PlanetMath.org.Chapter 15Conway base 13 functionThe Conway base 13 function is a function created by British mathematician John H. Conway as a counterexampleto the converse of the intermediate value theorem. In other words, even though Conways function f is not continuous,if f(a) < f(b) and an arbitrary value x is chosen such that f(a) < x < f(b), a point c lying between a and b can alwaysbe found such that f(c) = x. In fact, this function is even stronger than this: it takes on every real value in each intervalon the real line.15.1 The Conway base 13 function15.1.1 PurposeThe Conway base 13 function was created as part of a produce activity: in this case, the challenge was to producea simple-to-understand function which takes on every real value in every interval. It is thus discontinuous at everypoint.15.1.2 Denition(The following is Conways own notation.)The Conway base 13 function is a function f: R R dened as follows.If x R , write x as a tridecimal (a decimal in base 13) using the 13 underlined digit symbols 0, 1 , 2 , ..., 8 , 9 , + , , ; there should be no trailing recurring.There may be a leading sign, andsomewhere there will be a tridecimal point to distinguish the integer part from the fractional part; theseshould both be ignored in the sequel. (These digits can be thought of as having the values 0 to 12,respectively.)If from some point onwards, the tridecimal expansion ofx consists of an underlined signed ordinarydecimal number, r say, then dene f(x) = r , otherwise dene f(x) = 0 . For example,f(7+1 . 4+314159 . . .) = f(7+14+3141 . 59 . . .) = Note that the tridecimal point and earlier occurrences of + and are ignored, as there are later occurrences of non-decimal digits. (More precisely, to have the f(x) = r case, the trailing part must consist of either + or , followedby some nite number (possibly zero) of underlined decimal digits, followed by , followed by some number (possiblyinnitely many) of underlined decimal digits.Other possible cases can be permitted, but it makes no dierence tothe crucial properties of the function.)15.1.3 PropertiesThe functionfdened in this way satises the conclusion of the intermediate value theorem but is continuousnowhere. That is, on any closed interval [a, b] of the real line, f takes on every value betweenf(a) andf(b) .4415.2. REFERENCES 45More strongly, f takes as its value every real number somewhere within every open interval (a, b) .To prove this, let c (a, b) and r be any real number. Then c can have the tail end of its tridecimal representationmodied to ber (that is, r underlined, withr being written as a signed decimal), giving a new numberc . Byintroducing this modication suciently far along the tridecimal representation of c , the new number c will still liein the interval (a, b) and will satisfy f(c) = r .Thusf satises a property stronger than the conclusion of the intermediate value theorem. Moreover, iff werecontinuous at some point, f would be locally bounded at this point, which is not the case. Thus f is a spectacularcounterexample to the converse of the intermediate value theorem.15.2 ReferencesAgboola, Adebisi. Lecture. Math CS 120. University of California, Santa Barbara, 17 December 2005.15.3 See alsoDarboux functionChapter 16Correlation (projective geometry)This article is about correlation in projective geometry. For other uses, see correlation (disambiguation).In projective geometry, a correlation is a transformation of a d-dimensional projective space that transforms objectsof dimension k into objects of dimension d k 1, preserving incidence. Correlations are also called reciprocitiesor reciprocal transformations.16.1 In two dimensionsFor example, in the real projective plane points and lines are dual to each other. As expressed by Coxeter,A correlation is a point-to-line and a line-to-point transformation that preserves the relation of incidencein accordance with the principle of duality. Thus it transforms ranges into pencils, pencils into ranges,quadrangles into quadrilaterals, and so on.[1]Given a line m and P a point not on m, an elementary correlation is obtained as follows: for every Q on m form theline PQ. The inverse correlation starts with the pencil on P: for any line q in this pencil take the point m q. Thecomposition of two correlations that share the same pencil is a perspectivity.16.2 In three dimensionsIn a 3-dimensional projective space a correlation maps a point to a plane. As stated in one textbook:[2]If is such a correlation, every point P is transformed by it into a plane ' = P ; and conversely, everypoint P arises from a unique plane ' by the inverse transformation 1.Three-dimensional correlations also transform lines into lines, so they may be considered to be collineations of thetwo spaces.16.3 In higher dimensionsIn general n-dimensional projective space, a correlation takes a point to a hyperplane. This context was described byPaul Yale:A correlation of the projective space V* is an inclusion reversing permutation of the proper subspacesof V*.[3]4616.4. EXISTENCE OF CORRELATIONS 47He proves a theorem stating that a correlation interchanges joins and intersections, and for any subspace W*, thedimension of the image of W* under is (n 1) dim W* where n is the dimension of the vector space used toproduce the projective space.16.4 Existence of correlationsCorrelations can exist only if the space is self-dual. For dimensions 3 and higher, self-duality is easy to test: Acoordinatizing skeweld exists and self-duality fails if and only if the skeweld is not isomorphic to its opposite.16.5 Special types of correlationsIf a correlation is an involution (that is, two applications of the correlation equals the identity: 2(P) = P for allpoints P) then it is called a polarity.16.6 References[1] H. S. M. Coxeter (1974) Projective Geometry, second edition, page 57, University of Toronto Press ISBN 0-8020-2104-2[2] J. G. Semple and G. T. Kneebone (1952) Algebraic Projective Geometry, p 360, Clarendon Press[3] Paul B. Yale (1968, 1988. 2004) Geometry and Symmetry, chapter 6.9 Correlations and semi-bilinear forms, Dover Pub-lications ISBN 0-486-43835-XRobert J. Bumcroft (1969) Modern Projective Geometry, chapter 4.5 Correlations, page 90, Holt, Rinehart, andWinston .Robert A. Rosenbaum (1963) Introduction to Projective Geometry and Modern Algebra, page 198, Addison-Wesley.Chapter 17Crystal Ball functionExamples of the Crystal Ball function.The Crystal Ball function, named after the Crystal Ball Collaboration (hence the capitalized initial letters), is aprobability density function commonly used to model various lossy processes in high-energy physics.It consists ofa Gaussian core portion and a power-law low-end tail, below a certain threshold. The function itself and its rstderivative are both continuous.The Crystal Ball function is given by:f(x; , n, x, ) = N {exp((x x)222), forx x> A (B x x)n, forx x where4817.1. EXTERNAL LINKS 49A =(n||)n exp(||22)B=n|| ||N=1(C +D)C=n|| 1n 1 exp(||22)D =2(1 + erf(||2))N (Skwarnicki 1986) is a normalization factor and , n , x and are parameters which are tted with the data. erfis the error function.17.1 External linksJ. E. Gaiser, Appendix-F Charmonium Spectroscopy from Radiative Decays of the J/Psi and Psi-Prime, Ph.D.Thesis, SLAC-R-255 (1982). (This is a 205 page document in .pdf form the function is dened on p. 178.)M. J. Oreglia, A Study of the Reactions psi prime --> gamma gamma psi, Ph.D. Thesis, SLAC-R-236 (1980),Appendix D.T. Skwarnicki, A study of the radiative CASCADE transitions between the Upsilon-Prime and Upsilon reso-nances, Ph.D Thesis, DESY F31-86-02(1986), Appendix E.Chapter 18Map (mathematics)For other uses, see Map (disambiguation).In mathematics, the term mapping, usually shortened to map, refers to eitherA function, often with some sort of special structure, orA morphism in category theory, which generalizes the idea of a function.There are also a few, less common uses in logic and graph theory.18.1 Maps as functionsMain article: Function (mathematics)In many branches of mathematics, the term map is used to mean a function, sometimes with a specic property ofparticular importance to that branch. F