A Short Course on Functional Equations

1. Front Matter

   Pages i-vi

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2. Introduction

   • J. Aczél

   Pages 1-1

3. An aggregation theorem for allocation problems. Cauchy equation for single- and multiplace functions.Two extension theorems

   • J. Aczél

   Pages 2-21

4. Scale-invariant equal sacrifice in taxation. The linear-affine functional equation. Multiplicative and logarithmic functions

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   Pages 22-34

5. General forms of ‘laws of science’ without dimensional constants. The case of the same ratio scale for all variables. Generalized homogeneous functions. Inequality measures

   • J. Aczél

   Pages 35-51

6. General forms of ‘laws of sciences’ with (partially) independent ratio and interval scales. Multiplace multiplicative and logarithmic functions. Price levels.Endomorphisms of a real field. The exponential equation. Characters

   • J. Aczél

   Pages 52-70

7. Pexider’s equation and its extension. Quasi-extension of Cauchy’s equation. Determination of all generalized Hicks-neutral production functions

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   Pages 71-91

8. Determination of all Hicks-neutral production functions depending upon capital, labor and time (state of technology). The translation equation

   • J. Aczél

   Pages 92-105

9. The associativity equation. Synthesis of ratio judgements. The quasiarithmetic means. The Jensen equations. A conditional linear-affine equation. A characterization of root-mean-powers and of the geometric mean

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   Pages 106-132

10. Synthesis of measure judgements. Equations in a single variable. The Abel and Schröder equations. Iteration

    • J. Aczél

    Pages 133-153

11. Back Matter

    Pages 155-169

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Recently I taught short courses on functional equations at several universities (Barcelona, Bern, Graz, Hamburg, Milan, Waterloo). My aim was to introduce the most important equations and methods of solution through actual (not artifi­ cial) applications which were recent and with which I had something to do. Most of them happened to be related to the social or behavioral sciences. All were originally answers to questions posed by specialists in the respective applied fields. Here I give a somewhat extended version of these lectures, with more recent results and applications included. As previous knowledge just the basic facts of calculus and algebra are supposed. Parts where somewhat more (measure theory) is needed and sketches of lengthier calcula­ tions are set in fine print. I am grateful to Drs. J. Baker (Waterloo, Ont.), W. Forg-Rob (Innsbruck, Austria) and C. Wagner (Knoxville, Tenn.) for critical remarks and to Mrs. Brenda Law for care­ ful computer-typing of the manuscript (in several versions). A note on numbering of statements and references: The numbering of Lemmata, Propositions, Theorems, Corollaries and (separately) formulae starts anew in each section. If quoted in another section, the section number is added, e.g. (2.10) or Theorem 1.2. References are quoted by the last names of the authors and the last two digits of the year, e.g. Daroczy-Losonczi [671. 1 1. An aggregation theorem for allocation problems. Cauchy equation for single-and multiplace functions. Two extension theorems.

• functional equation
• inequality
• logarithm
• measure
• taxation

'... this book is not only an impressive example how mathematics can be applied to problems arising in the social and behavioral sciences, but also on outstanding course on functional equations. This course is short, as the title says, but leads astonishingly far, since the material is presented very economically.
I am convinced that every reader will enjoy this book greatly.'
Zeischrift für Operations Research, June 1989.

• Centre for Information Theory, University of Waterloo, Canada

  J. Aczél

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