By J. Aczél

ISBN-10: 902772377X

ISBN-13: 9789027723772

ISBN-10: 9400937490

ISBN-13: 9789400937499

Recently I taught brief classes on useful equations at a number of universities (Barcelona, Bern, Graz, Hamburg, Milan, Waterloo). My target used to be to introduce an important equations and techniques of answer via real (not artifi­ cial) functions which have been contemporary and with which I had whatever to do. such a lot of them occurred to be regarding the social or behavioral sciences. All have been initially solutions to questions posed via experts within the respective utilized fields. right here I provide a just a little prolonged model of those lectures, with newer effects and purposes incorporated. As prior wisdom simply the fundamental proof of calculus and algebra are intended. components the place a little bit extra (measure conception) is required and sketches of lengthier calcula­ tions are set in superb print. i'm thankful to Drs. J. Baker (Waterloo, Ont.), W. Forg-Rob (Innsbruck, Austria) and C. Wagner (Knoxville, Tenn.) for serious comments and to Mrs. Brenda legislation for care­ ful computer-typing of the manuscript (in a number of versions). A observe on numbering of statements and references: The numbering of Lemmata, Propositions, Theorems, Corollaries and (separately) formulae begins anew in each one part. If quoted in one other part, the part quantity is additional, e.g. (2.10) or Theorem 1.2. References are quoted by means of the final names of the authors and the final digits of the yr, e.g. Daroczy-Losonczi [671. 1 1. An aggregation theorem for allocation difficulties. Cauchy equation for single-and multiplace capabilities. extension theorems.

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Extra resources for A Short Course on Functional Equations: Based Upon Recent Applications to the Social and Behavioral Sciences

Sample text

9) So v(x) = <1>( log x) is the general solution of (8), where dJ is an arbitrary solution of (9) and. by (7), u (x) = 0>( log x) + b. where b is an arbitrary constant. 5

For this purpose, first put x = 1 into (5): u(r) = + f3(r) o:(r)u(l) and, by subtracting this from (5), get the functional equation v(rx) = o:(r)v(x) + v(r) (6) for the function v defined by v(x) = u(x) - u(l) (x EIR-,-+) . The equation (6) contains only two unknown functions. solving (6) we ~istinguish two cases. (7) In Section 2 26 First, if o:{r) = 1 for all positive r, then (6) goes over into v{rx) = v{r) + v{x) (x,r E lR++) (8) Functions satisfying (8) are called logarithmic. 23}. Indeed, writing we get

So we have the following. Corollary 2. The general positive, continuous and nonconstant utility functions generating scale invariant relative sacrifices, that is, the general posz"iive, continuous and nonconstant solutions of (15) or (16) are given by Section 2 30 (17) and by (18) where a =1= 0, c =1= 0, B trary real constants. > 0, but otherwise a, B, c are arbi- Of course, further restrictions on the utility functions are reasonable, for instance that u be increasing. Then in (10) and (17) a > 0, in (14) and (18) either a > 0, c > or a < 0, c < 0.

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A Short Course on Functional Equations: Based Upon Recent Applications to the Social and Behavioral Sciences by J. Aczél

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