By J. Aczél
Recently I taught brief classes on practical equations at numerous universities (Barcelona, Bern, Graz, Hamburg, Milan, Waterloo). My objective was once to introduce crucial equations and techniques of answer via real (not artifi cial) purposes which have been contemporary and with which I had anything to do. so much of them occurred to be on the topic of the social or behavioral sciences. All have been initially solutions to questions posed through experts within the respective utilized fields. the following I provide a a little bit prolonged model of those lectures, with newer effects and purposes integrated. As earlier wisdom simply the elemental proof of calculus and algebra are meant. components the place slightly extra (measure concept) is required and sketches of lengthier calcula tions are set in nice 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 numerous versions). A notice on numbering of statements and references: The numbering of Lemmata, Propositions, Theorems, Corollaries and (separately) formulae starts off anew in every one part. If quoted in one other part, the part quantity is further, 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 12 months, e.g. Daroczy-Losonczi [671. 1 1. An aggregation theorem for allocation difficulties. Cauchy equation for single-and multiplace services. extension theorems.
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Additional info for A Short Course on Functional Equations: Based Upon Recent Applications to the Social and Behavioral Sciences
Multiplicative, logarithmic functions. a(r)v(x) + v(r) = a(x)v(r) + v(x) 27 . L T Since u and thus v is not constant, a =1= equation (6) becomes aa(rx) - a = aa(r)a(x) - aa(r) ) o. +) , (13) (only one unknown function left). Functions satisfying (13) are called multiplicative. Also this equation can be reduced to the Cauchy equation or to (8). It is clear that we will do the latter by taking logarithms of both sides of (13). For this we have to establish that a is everywhere positive. Actually, a(x) 0 is a solution of (13) but we exclude this (since, by (12) and (7), it would cause u to be constant).
F = 0 of (14) so we The case where the dependent variable also has a ratio scale, that is, it transforms by a (homogeneous) linear transformation too (while the independent variables still undergo equal linear transformations), gives the functional equation u(rx) = R(r)u(x) (r EIR++, x EIR~+; u : 1R~+ ~ 1R++, 1R: 1R++ ~ 1R++) (18) (cf. (12)). This is simply the special case P(r) = 0 (r EIR++), u(x) >0 (x EIR~+) of (2). So from (8) L(r) =0 in (7), from (15) b = 0 (since that was the M(r) ¢:1 case) in (14) and from (17) R(r) 1 (since b = 0 in (16) would contradict u(x) > 0) and f is positive valued in (7) and (14) and b > 0 in (16).
Then we get (10). So the following holds. , Independent scales. Price Levels. Characters. ill Corollary 4. ) then in Theorem 1 u(xu···,x n ) = n :E Ci log Xi +b , i=1 or U(Xl"··'X n ) = a n n c. Xi' +b , i=l where a,b,cl, ... ,c n are arbitrary real constants. This too has been obtained by Luce (64] under the supposition that u is everywhere continuous. Our next case is where the independent variables all have interval scales with the same unz"t but with independent zeros, while the dependent variable also has an interval scale.
A Short Course on Functional Equations: Based Upon Recent Applications to the Social and Behavioral Sciences by J. Aczél