Functional Equations in Several Variables: With Applications to Mathematics, Information Theory and to the Natural and Social Sciences (Encyclopedia of Mathematics and its Applications, Vol. 31)

Deals with modern theory of functional equations in several variables and their applications to mathematics, information theory, and the natural, behavioral, and social sciences. The authors emphasize applications, although not at the expense of theory, and have kept the prerequisites to a minimum; the reader should be familiar with calculus and some simple structures of algebra and have a basic knowledge of Lebesque integration. For the applications the authors have included references and explained the results used. The book is designed so that the chapters may be read almost independently of each other, enabling a selection of material to be chosen for introductory and advanced courses. Each chapter concludes with exercises and further results, 400 in all, which extend and test the material presented in the text. The history of functional equations is well documented in a final chapter which is complemented by an encyclopedic bibliography of over 1600 items. This volume will be of interest to professionals and graduate students in pure and applied mathematics. Cover......Page 1 ENCYCLOPAEDIA OF MATHEMATICS AND ITS APPLICATIONS......Page 2 Title......Page 4 Copyright......Page 5 CONTENTS......Page 6 Dedication......Page 9 PREFACE......Page 10 FURTHER INFORMATION......Page 15 1 Axiomatic motivation of vector addition......Page 16 Exercises and further results......Page 23 2.1 General considerations, extensions, and regular solutions......Page 26 2.2 General solutions......Page 33 Exercises and further results......Page 37 3 Three further Cauchy equations. An application to information theory......Page 40 Exercises and further results......Page 46 4.1 Multiplace and vector functions......Page 49 4.2 A matrix functional equation and a characterization of densities in the theory of geometric objects......Page 53 4.3 Pexider equations......Page 57 4.4 Cauchy-type equations on semigroups......Page 61 Exercises and further results......Page 63 5.1 Cauchy's equation and the exponential equation for complex functions......Page 67 5.2 Endomorphisms of the real and complex fields......Page 72 5.3 Bohr groups......Page 75 5.4 Recursive entropies......Page 81 Exercises and further results......Page 84 6 Conditional Cauchy equations. An application to geometry and a characterization of the Heaviside function......Page 88 Exercises and further results......Page 97 7.1 Extensions and quasi-extensions......Page 99 7.2 Extensions almost everywhere and integral transforms......Page 107 7.3 Consensus allocations......Page 114 Exercises and further results......Page 116 8 D'Alembert's functional equation. An application to noneuclidean mechanics......Page 118 Exercises and further results......Page 126 9.1 Equations containing images of sets and chronogeometry......Page 129 9.2 Sets on which bounded additive functions are continuous......Page 136 Exercises and further results......Page 142 10.1 Functional equations and extreme points......Page 144 10.2 Totally monotonic functions and extreme rays......Page 150 103 A characterization of strictly convex normed spaces......Page 153 10.4 Isometries in real normed spaces......Page 157 10.5 A topology on the set of all solutions of a functional equation: the Bohr group......Page 163 10.6 Valuations on the fields of rational and of real numbers......Page 171 Exercises and further results......Page 176 11.1 Quadratic functional: a characterization of inner product spaces......Page 180 11.2 Triangles in normed spaces: a second characterization of inner product spaces......Page 193 11.3 Orthogonal additivity......Page 200 11.4 An application to gas dynamics......Page 206 Exercises and further results......Page 209 12 Some related equations and systems of equations. Applications to combinatorics and Markov processes......Page 216 Exercises and further results......Page 222 13 Equations for trigonometric and similar functions......Page 224 Exercises and further results......Page 240 14 A class of equations generalizing d'Alembert and Cauchy Pexider-type equations......Page 243 Exercises and further results......Page 252 15 A further generalization of Pexider's equation. A uniqueness theorem. An application to mean values.......Page 255 Exercises and further results......Page 265 16.1 Expansions of the Cauchy equation from curves......Page 269 16.2 Cylindrical conditions......Page 276 16.3 Additive number theoretical functions and related equations......Page 280 16.4 An application to mean codeword lengths......Page 282 16.5 Totally additive number theoretical functions and their generalizations......Page 289 16.6 Further equations for number theoretical functions......Page 296 Exercises and further results......Page 298 17 Mean values, mediality and self-distributivity......Page 302 Exercises and further results......Page 311 18 Generalized mediality. Connection to webs and nomograms......Page 313 Exercises and further results......Page 322 19 Further composite equations. An application to averaging theory......Page 324 19.1 One-parameter subgroups of affine groups......Page 326 19.2 Another example of determining one-parameter subgroups......Page 334 19.3 Two more composite equations......Page 341 19.4 Reynolds and averaging operators......Page 345 19.5 Interpolating and extension operators......Page 349 19.6 Derivation operators......Page 352 Exercises and further results......Page 355 20 Homogeneity and some generalizations. Application to economics......Page 360 Exercises and further results......Page 367 21.1 Definition of linear and quadratic functions by functional equations in the Middle Ages and application of an implied characterization by Galileo......Page 370 21.2 The functional equations of the logarithm and of the exponential function......Page 375 21.3 Some functional equations in the works of Euler......Page 377 21.4 Functional equations arising from physics......Page 378 21.5 The binomial theorem and Cauchy's equations......Page 380 21.6 Cauchy equations after Cauchy......Page 386 21.7 Further equations......Page 388 21.8 Recent developments......Page 392 Notation and symbols......Page 394 Hints to selected 'exercises and further results'......Page 397 BIBLIOGRAPHY......Page 403 AUTHOR INDEX......Page 464 SUBJECT INDEX......Page 473 This Treatise Deals With Modern Theory Of Functional Equations In Several Variables And Their Applications To Mathematics, Information Theory, And The Natural, Behavioural And Social Sciences. The Authors Have Chosen To Emphasize Applications, Though Not At The Expense Of Theory, So They Have Kept The Prerequisites To A Minimum; The Reader Need Be Familiar Only With Calculus And Elementary Algebra, And Have A Basic Knowledge Of Lebesgue Integration. Where, For Certain Applications, More Advanced Topics Are Needed, The Authors Have Included References And Explained The Results Used. Moreover, The Book Has Been Designed So That The Chapters Can Be Read Almost Independently Of Each Other, Enabling A Selection Of Material To Be Chosen For Introductory And Advanced Courses. At The End Of Each Chapter Are Included Exercises And Further Results, Some 400 In All, Which Extend The Material Presented In The Text And Also Test It. The History Of Functional Equations Is Well Documented In A Final Chapter Which Is Complemented By An Encyclopedic Bibliography Running To Over 1600 Items. J. Aczel, J. Dhombres. Title From Publisher's Bibliographic System (viewed On 01 Jun 2016). Mode Of Access: World Wide Web.