Download A Short Course on Functional Equations: Based Upon Recent by J. Aczél PDF

By J. Aczél

Recently I taught brief classes on useful equations at numerous universities (Barcelona, Bern, Graz, Hamburg, Milan, Waterloo). My goal used to be to introduce an important equations and strategies of resolution via genuine (not artifi­ cial) functions which have been fresh and with which I had whatever to do. such a lot of them occurred to be on the topic of the social or behavioral sciences. All have been initially solutions to questions posed by means of experts within the respective utilized fields. right here I supply a a little bit prolonged model of those lectures, with more moderen effects and functions integrated. As prior wisdom simply the fundamental proof of calculus and algebra are intended. components the place a bit of extra (measure idea) is required and sketches of lengthier calcula­ tions are set in advantageous print. i'm thankful to Drs. J. Baker (Waterloo, Ont.), W. Forg-Rob (Innsbruck, Austria) and C. Wagner (Knoxville, Tenn.) for severe feedback and to Mrs. Brenda legislations for care­ ful computer-typing of the manuscript (in numerous versions). A observe 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 extra, e.g. (2.10) or Theorem 1.2. References are quoted via 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 capabilities. extension theorems.

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

Example text

R(xn+s)-s) For the function u defined by (33) for y>s (that is, Yi >s, i=l, ... ,n), equation (32) is transformed (with y = x+s) into Section 3 48 (y u(ry) = u(y) > s) (34) . The additional restriction r > 1 can be lifted (as long as r E 1R++) since (34) can be written as 1 u(z)=u(-z) (r>l,z>s), r so (34) holds also for r u(ry) < 1. So we have u(y) (y = > s,r (35) EIR-"-+) which is the particular case R(r) = 1 of (18) or (25), again on another domain. But, substituting r = 1/11 y II, we have again u(y) = g( II ~ il ) for y =1= 0 and u(O) = b (arbitrary) , which indeed satisfies (35) with arbitrary g and b.

3. General forms of 'laws of science' without dimensional constants. The case of the same ratio scale for all variables. Generalized homogeneous functions. Inequality measures. In the previous section changes of scales played an important role. This role IS quite fundamental in· the natural, behavioral and social sciences. Dimensional analysis (for instance in physics) is based on the recognition that the laws (formulae) of science should be equations, both sides of which have the same (physical) dimensions - if there are no dimensional constants.

39 R(r) = 1, P{r) = L{r) (8) (r ElR++) . M(r) , (9) where AI is a multiplicative function, that is, a solution of M(rs) = M{r)M{s) (r,s ElR++) . ) In view of (9) and with the notation w{x) = v{x) +a (11) , equation (3) goes over into w(rx) = M(r)w{x) (r ElR++, x ElR~+). (12) Similarly as above, this yields w for instance in the form X2 xn = M(xdf(-, ... , - ) . Xl (13) Xl where this time f(t 2 , ... ,t n ) = w(1,t 2 , ... ,t n ). With (11) and (4) we have Section 3 40 (14) Conversely, every u of the form (14), with arbitrary M satisfying (10), arbitrary function 1 and arbitrary constant b, satisfies (2): X2 xn u(rx) = A1(r )M(xd/(-, ...

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