Functions and mappings aFrom 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 Map (mathematics) 168.1 Maps as functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168.2 Maps as morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168.3 Other uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.3.1 In logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.3.2 In graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.6 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 198.6.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.6.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.6.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Chapter 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 8Map (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.8.1 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. For instance, a map is a continuous function in topology, a linear transforma-tion in linear algebra, etc.Some authors, such as Serge Lang,[1] use function only to refer to maps in which the codomain is a set of numbers,i.e., a subset of the elds R or C, and the term mapping for more general functions.Sets of maps of special kinds are the subjects of many important theories: see for instance Lie group, mapping classgroup, permutation group.In the theory of dynamical systems, a map denotes an evolution function used to create discrete dynamical systems.See also Poincar map.A partial map is a partial function, and a total map is a total function. Related terms like domain, codomain, injective,continuous, etc. can be applied equally to maps and functions, with the same meaning. All these usages can be appliedto maps as general functions or as functions with special properties.In the communities surrounding programming languages that treat functions as rst class citizens, a map oftenrefers to the binary higher-order function that takes a function and a list [v0,v1,...,vn] as arguments and returns[(v0),(v1),...,(vn)], s.t. n 0.8.2 Maps as morphismsMain article: MorphismIn category theory, map is often used as a synonym for morphism or arrow, thus for something more general thana function.[2]168.3. OTHER USES 178.3 Other uses8.3.1 In logicIn formal logic, the termis sometimes used for a functional predicate, whereas a function is a model of such a predicatein set theory.8.3.2 In graph theoryAn example of a map in graph theory.In graph theory, a map is a drawing of a graph on a surface without overlapping edges (an embedding). If the surfaceis a plane then a map is a planar graph, similar to a political map.[3]8.4 See alsoBijection, injection and surjectionCategory theory18 CHAPTER 8. MAP (MATHEMATICS)Correspondence (mathematics)HomeomorphismHomomorphismList of chaotic mapsMapping class groupMorphismProjection (mathematics)Topology8.5 References[1] Lang, Serge (1971), Linear Algebra (2nd ed.), Addison-Wesley, p. 83[2] Simmons, H. (2011), An Introduction to Category Theory, Cambridge University Press, p. 2, ISBN 9781139503327.[3] Gross, Jonathan; Yellen, Jay (1998), Graph Theory and its applications, CRC Press, p. 294, ISBN 0-8493-3982-08.6. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 198.6 Text and image sources, contributors, and licenses8.6.1 Text A-equivalence Source: https://en.wikipedia.org/wiki/A-equivalence?oldid=591516732 Contributors: Michael Hardy, Oleg Alexandrov,Woohookitty, RussBot, Colonies Chris, JohnCD, Dharma6662000, David Eppstein, R'n'B, WereSpielChequers, Dawn Bard, JackSchmidt,KathrynLybarger, Simplix and Alpha Quadrant Algebraic function Source: https://en.wikipedia.org/wiki/Algebraic_function?oldid=673260293 Contributors: Michael Hardy, CharlesMatthews, Haukurth, Giftlite, Gene Ward Smith, Wmahan, Almit39, GrAfFiT, Photonique, Ksnow, Waldir, MarSch, Margosbot~enwiki,YurikBot, SmackBot, InverseHypercube, Bluebot, Bethling, Silly rabbit, Tsca.bot, BenWhitey, Krashlandon, Cronholm144, Mets501,Valoem, CBM, Dugwiki, AntiVandalBot, VictorAnyakin, Dekimasu, Johnbibby, MetsBot, Pomte, AlleborgoBot, Kai Su?, EmxBot,SieBot, VVVBot, JackSchmidt, ClueBot, PipepBot, JP.Martin-Flatin, DumZiBoT, Mattsem, Addbot, PV=nRT, Luckas-bot, AnomieBOT,Ciphers, Wisems, Isheden, GrouchoBot, Sawomir Biaa, EmausBot, WikitanvirBot, D.Lazard, ClueBot NG, The1337gamer, JochenBurghardt, Snowright, Elkevn and Anonymous: 46 Algebraic vector bundle Source:https://en.wikipedia.org/wiki/Algebraic_vector_bundle?oldid=644275002 Contributors:TakuyaMu-rata, Bearcat, David Eppstein, AnomieBOT, Xqbot and Alvin Seville Angle of parallelismSource: https://en.wikipedia.org/wiki/Angle_of_parallelism?oldid=656655746 Contributors: Michael Hardy, GeneWard Smith, Dissident, ArnoldReinhold, Rgdboer, FlaBot, Silly rabbit, Gracenotes, Majora4, CmdrObot, Epbr123, YK Times, Rocchini,LokiClock, Addbot, Vega2, Yobot, Unara, Maximilianklein, WillemienH and Anonymous: 7 Antihomomorphism Source: https://en.wikipedia.org/wiki/Antihomomorphism?oldid=655944002 Contributors: Zundark, Altenmann,Postdlf, Tobias Bergemann, Fropu, Waltpohl, Lumidek, Paul August, Aranel, Rgdboer, Cmdrjameson, Oleg Alexandrov, Woohookitty,Mathbot, Archelon, SmackBot, RDBury, Nbarth, SkyBoxx, LokiClock, Addbot, LaaknorBot, Yobot, Amirobot, Erik9bot, EmausBot,Daviddwd, Brad7777, Deltahedron and Anonymous: 11 Antilinear map Source: https://en.wikipedia.org/wiki/Antilinear_map?oldid=620118701 Contributors: Zundark, SimonP, Jimfbleak,Glenn, Phys, Fropu, Almit39, CtgPi, Rgdboer, Sin-man, Bo Jacoby, SmackBot, Maksim-e~enwiki, Mhss, Nbarth, Kjetil1001, Lesnail,Julian Mendez, RobHar, Lantonov, Miskimo, Synthebot, Addbot, , EdoBot and Anonymous: 7 Asanocontraction Source: https://en.wikipedia.org/wiki/Asano_contraction?oldid=667159679 Contributors: Michael Hardy, Yobot,John of Reading and Happysquirrel Map (mathematics) Source: https://en.wikipedia.org/wiki/Map_(mathematics)?oldid=656091129 Contributors: Patrick, Michael Hardy,, Glenn, AugPi, Charles Matthews, Dysprosia, McKay, Robbot, Romanm, Tobias Bergemann, Giftlite, Fropu, Curps, JorgeStol, Abdull, Rich Farmbrough, Zaslav, El C, Jumbuck, Dallashan~enwiki, MarkGallagher, Igny, Marudubshinki, RexNL, YurikBot,Taejo, Hede2000, Bota47, LarryLACa, Arthur Rubin, Benandorsqueaks, Luk, SmackBot, Adam majewski, Chuyelchulo~enwiki, JonAwbrey, Bjankuloski06en~enwiki, Newone, George100, Sdorrance, Arn7, Aitherios, Res2216restar, JAnDbot, David Eppstein, Der-Hexer, Pomte, Perel, Onkelringelhuth, LokiClock, Geometry guy, Kehrbykid, Reinderien, JP.Martin-Flatin, DragonBot, Allsaints23, Ne-penthes, Addbot, LaaknorBot, Yobot, 9258fahskh917fas, Goykhman212121, Erik9bot, Ibbn, Mastergreg82, ClueBot NG, Wcherowi,Luizpuodzius, ChrisGualtieri, YFdyh-bot, NitRav, Brirush, Ven. uvathanne sumana and Anonymous: 258.6.2 Images File:Angle_of_parallelism_half_plane_model.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/7c/Angle_of_parallelism_half_plane_model.svg License: CC BY 3.0 Contributors: Own work Original artist: Claudio Rocchini File:Commons-logo.svg Source: https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg License: ? 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Based on Image:Question book.png created by User:Equazcion Original artist:Tkgd2007 File:Y\char"005E\relax{}3-xy+1=0.png Source: https://upload.wikimedia.org/wikipedia/commons/0/06/Y%5E3-xy%2B1%3D0.pngLicense: CC-BY-SA-3.0 Contributors: http://en.wikipedia.org/wiki/File:Y%5E3-xy%2B1%3D0.png Original artist: Silly rabbit8.6.3 Content license Creative Commons Attribution-Share Alike 3.0