Functions and mappings alFrom 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.10 Contrast with . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249.14 External 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4918Derivative 5018.1Dierentiation and derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5118.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5218.1.2 Rigorous denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5318.1.3 Denition over the hyperreals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5318.1.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5418.1.5 Continuity and dierentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5418.1.6 The derivative as a function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5418.1.7 Higher derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5618.1.8 Inection point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5718.2Notation (details) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5718.2.1 Leibnizs notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5718.2.2 Lagranges notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5818.2.3 Newtons notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5818.2.4 Eulers notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5918.3Rules of computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5918.3.1 Rules for basic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5918.3.2 Rules for combined functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6018.3.3 Computation example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6118.4Derivatives in higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6118.4.1 Derivatives of vector valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6118.4.2 Partial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6118.4.3 Directional derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6318.4.4 Total derivative, total dierential and Jacobian matrix . . . . . . . . . . . . . . . . . . . . 6418.5Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66CONTENTS v18.6History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6618.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6718.8Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6718.9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6818.9.1 Print . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6818.9.2 Online books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6818.9.3 Web pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6819Dieology 6919.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6919.2Smooth manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7019.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7019.4External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7020Dierential coecient 7121Discontinuous linear map 7221.1A linear map from a nite-dimensional space is always continuous . . . . . . . . . . . . . . . . . . 7221.2A concrete example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7321.3A nonconstructive example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7321.4General existence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7421.5Role of the axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7421.6Closed operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7421.7Impact for dual spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7521.8Beyond normed spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7521.9Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7521.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7522Domain of a function 7722.1Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7822.2Natural domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7822.3Domain of a partial function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7822.4Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7822.5Real and complex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7822.6More examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7922.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7922.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7923Eective domain 8023.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8024Elasticity of a function 8124.1Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8124.2Estimating point elasticities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82vi CONTENTS24.3Semi-elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8224.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8224.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8325Embedding 8425.1Topology and geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8425.1.1 General topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8425.1.2 Dierential topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8425.1.3 Riemannian geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8525.2Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8525.2.1 Field theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8525.2.2 Universal algebra and model theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8525.3Order theory and domain theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8625.4Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8625.4.1 Normed spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8625.5Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8625.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8725.7Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8725.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8725.9External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8826Empty function 8926.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8927Equiareal map 9027.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9028Function (mathematics) 9128.1Introduction and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9328.2Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9428.3Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9528.4Specifying a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9628.4.1 Graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9628.4.2 Formulas and algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9628.4.3 Computability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9728.5Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9728.5.1 Image and preimage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9728.5.2 Injective and surjective functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9828.5.3 Function composition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9828.5.4 Identity function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10028.5.5 Restrictions and extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10028.5.6 Inverse function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10028.6Types of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100CONTENTS vii28.6.1 Real-valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10128.6.2 Further types of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10128.7Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10128.7.1 Currying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10228.8Variants and generalizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10228.8.1 Alternative denition of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10228.8.2 Partial and multi-valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10228.8.3 Functions with multiple inputs and outputs . . . . . . . . . . . . . . . . . . . . . . . . . . 10328.8.4 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10428.9History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10428.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10428.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10428.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10528.13Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10528.14External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10629Function application 10729.1Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10729.2As an operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10729.3Other instances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10829.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10830Function composition 10930.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10930.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11030.3Composition monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11030.4Functional powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11030.5Alternative notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11330.6Composition operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11330.7In programming languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11330.8Multivariate functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11330.9Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11430.10Typography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11430.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11430.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11530.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11530.14External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11531Functional decomposition 11631.1Basic mathematical denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11631.1.1 Example: Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11631.1.2 Example: Decomposition of continuous functions . . . . . . . . . . . . . . . . . . . . . . 117viii CONTENTS31.2Motivation for decomposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11731.3Philosophical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11731.3.1 Reductionist tradition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11731.3.2 Characteristics of hierarchy and modularity . . . . . . . . . . . . . . . . . . . . . . . . . 11831.3.3 Inevitability of hierarchy and modularity . . . . . . . . . . . . . . . . . . . . . . . . . . . 11831.4Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11831.4.1 Knowledge representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11931.4.2 Database theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11931.4.3 Machine learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11931.4.4 Software architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11931.4.5 Signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12031.4.6 Systems engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12031.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12031.6Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12131.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12132Generalized Ozaki cost function 12432.1Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12432.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12433Geometric transformation 12533.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12633.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12633.3Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12634Glide reection 12734.1Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12734.2Glide reection in cellular automata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12834.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12934.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12934.5External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12935Graph of a function 13035.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13235.1.1 Functions of one variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13235.1.2 Functions of two variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13335.1.3 Normal to a graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13335.2Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13435.3Tools for plotting function graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13435.3.1 Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13435.3.2 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13435.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13435.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134CONTENTS ix35.6External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13436High-dimensional model representation 13536.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13536.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13537History of the function concept 13637.1Functions before the 17th century. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13637.2The notion of function in analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13637.2.1 Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13637.2.2 Fourier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13737.2.3 Cauchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13737.2.4 Lobachevsky and Dirichlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13737.2.5 Dedekind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13837.2.6 Hardy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13837.3The logicians function prior to 1850 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13937.4The logicians function 18501950 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13937.4.1 George Booles The Laws of Thought 1854; John Venns Symbolic Logic 1881 . . . . . . . . 13937.4.2 Freges Begrisschrift 1879 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14037.4.3 Peano 1889 The Principles of Arithmetic 1889 . . . . . . . . . . . . . . . . . . . . . . . . 14137.4.4 Bertrand Russells The Principles of Mathematics 1903 . . . . . . . . . . . . . . . . . . . . 14137.4.5 Evolution of Russells notion of function 19081913 . . . . . . . . . . . . . . . . . . . . 14137.5The formalists function": David Hilberts axiomatization of mathematics (19041927) . . . . . . . 14237.6Development of the set-theoretic denition of function . . . . . . . . . . . . . . . . . . . . . . . 14337.6.1 Russells paradox 1902 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14337.6.2 Zermelos set theory (1908) modied by Skolem (1922) . . . . . . . . . . . . . . . . . . . 14337.6.3 The WienerHausdorKuratowski ordered pair denition 19141921 . . . . . . . . . . 14437.6.4 Schnnkels notion of function as a many-one correspondence 1924 . . . . . . . . . . 14437.6.5 Von Neumanns set theory 1925 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14537.6.6 Bourbaki 1939 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14537.7Since 1950 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14537.7.1 Notion of function in contemporary set theory . . . . . . . . . . . . . . . . . . . . . . . 14537.7.2 Relational form of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14637.8Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14637.9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15037.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15237.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15238Homeomorphism 15338.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15338.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15338.2.1 Non-examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154x CONTENTS38.3Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15538.4Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15538.5Informal discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15638.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15638.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15638.8External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15639Homography (computer vision) 15739.13D plane to plane equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15739.2Mathematical denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15939.3Ane homography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15939.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15939.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16039.6External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16040Homomorphic secret sharing 16140.1Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16140.2Example: decentralized voting protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16140.2.1 Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16240.2.2 Vulnerabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16340.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16340.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16341Horizontal translation 16441.1Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16441.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16442HubbardStratonovich transformation 16542.1Calculation of resulting eld theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16542.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16543Hypercomplex analysis 16643.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16643.2External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16744Identity function 16844.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16944.2Algebraic property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16944.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16944.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16944.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16945Inclusion map 17045.1Applications of inclusion maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171CONTENTS xi45.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17145.3Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17145.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17146Injective function 17246.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17346.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17446.3Injections can be undone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17746.4Injections may be made invertible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17746.5Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17746.6Proving that functions are injective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17846.7See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17846.8Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17946.9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17946.10External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17947Integral 18047.1History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18147.1.1 Pre-calculus integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18147.1.2 Newton and Leibniz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18247.1.3 Formalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18247.1.4 Historical notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18247.2Terminology and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18247.3Interpretations of the integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18347.4Formal denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18647.4.1 Riemann integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18647.4.2 Lebesgue integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18747.4.3 Other integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18847.5Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18947.5.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18947.5.2 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19047.5.3 Conventions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19147.6Fundamental theorem of calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19247.6.1 Statements of theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19247.7Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19347.7.1 Improper integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19347.7.2 Multiple integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19547.7.3 Line integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19747.7.4 Surface integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19747.7.5 Integrals of dierential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19847.7.6 Summations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20047.8Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200xii CONTENTS47.8.1 Analytical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20047.8.2 Symbolic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20147.8.3 Numerical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20147.8.4 Mechanical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20347.8.5 Geometrical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20447.9Some important denite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20447.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20447.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20547.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20547.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20647.13.1 Online books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20648Inversion transformation 20748.1Early use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20748.2Transformation on coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20748.3Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20848.4Physical evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20848.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20849Involution (mathematics) 20949.1General properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20949.2Involution throughout the elds of mathematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21049.2.1 Pre-calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21049.2.2 Euclidean geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21049.2.3 Projective geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21049.2.4 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21049.2.5 Quaternion algebra, groups, semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 21149.2.6 Ring theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21149.2.7 Group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21149.2.8 Mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21249.2.9 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21249.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21249.4Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21249.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21250Isometry 21450.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21450.2Formal denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21450.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21550.4Linear isometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21650.5Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21650.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216CONTENTS xiii50.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21750.8Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21751Iterated function 21851.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21851.2Abelian property and Iteration sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21851.3Fixed points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21951.4Limiting behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21951.5Fractional iterates and ows, and negative iterates . . . . . . . . . . . . . . . . . . . . . . . . . . . 21951.6Some formulas for fractional iteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22051.6.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22051.6.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22051.6.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22151.7Conjugacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22151.8Markov chains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22151.9Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22251.10Means of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22251.11In computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22251.12Denitions in terms of iterated functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22251.13Lies data transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22251.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22351.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22352Jnsson function 22452.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22453K-equivalence 22553.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22553.2KV-equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22553.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22553.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22654KolmogorovArnold representation theorem 22754.1Original references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22754.2Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22754.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22755Laver function 22955.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22955.2Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22955.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22956Left and right derivative 23056.1Derivatives arising from one-sided limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230xiv CONTENTS56.1.1 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23156.2Dierential operators acting to the left or the right . . . . . . . . . . . . . . . . . . . . . . . . . . 23156.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23256.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23257Limit of a function 23357.1History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23357.2Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23357.3Functions of a single variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23457.3.1 Existence and one-sided limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23457.3.2 More general subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23557.3.3 Deleted versus non-deleted limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23657.3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23657.4Functions on metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23757.5Functions on topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23857.6Limits involving innity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23857.6.1 Limits at innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23857.6.2 Innite limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23957.6.3 Alternative notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24057.6.4 Limits at innity for rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24157.7Functions of more than one variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24257.8Sequential limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24257.9Other characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24257.9.1 In terms of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24257.9.2 In non-standard calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24257.9.3 In terms of nearness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24357.10Relationship to continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24357.11Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24357.11.1 Chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24457.11.2 Limits of special interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24457.11.3 L'Hpitals rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24557.11.4 Summations and integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24657.12See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24657.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24658Linear map 24858.1Denition and rst consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24858.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24958.3Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24958.4Examples of linear transformation matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25158.5Forming new linear maps from given ones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25258.6Endomorphisms and automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252CONTENTS xv58.7Kernel, image and the ranknullity theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25258.8Cokernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25358.8.1 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25458.9Algebraic classications of linear transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 25458.10Change of basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25558.11Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25558.12Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25558.13See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25558.14Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25658.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25659List of limits 25759.1Limits for general functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25759.2Limits of general functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25759.3Notable special limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25859.4Simple functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25859.5Logarithmic and exponential functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25859.6Trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25959.7Near innities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25960Locally nite operator 26060.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26061Logit 26161.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26161.2History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26261.3Uses and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26261.4Comparison with probit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26261.5See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26361.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26461.7Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26462Lorentz transformation 26562.1History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26562.2Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26662.3Frames in standard conguration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26662.3.1 Boost in the x-direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26662.3.2 Boost in the y or z directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26862.3.3 Boost in any direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26962.3.4 Composition of two boosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27162.4Visualizing the transformations in Minkowski space . . . . . . . . . . . . . . . . . . . . . . . . . 27262.4.1 Rapidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27462.5Transformation of other physical quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275xvi CONTENTS62.6Special relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27562.6.1 Transformation of the electromagnetic eld . . . . . . . . . . . . . . . . . . . . . . . . . 27662.6.2 The correspondence principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27662.7Spacetime interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27662.8See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27862.9References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27862.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27962.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27963Map (mathematics) 28063.1Maps as functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28063.2Maps as morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28063.3Other uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28163.3.1 In logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28163.3.2 In graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28163.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28163.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28263.6Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 28363.6.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28363.6.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29063.6.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295Chapter 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) Les Fondements de la Gomtry, 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 Vto 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 inXhave 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 Vin 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