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Observable system responses and functional equations

The following general problem arises in many fields, ranging from engineering applications like process control and signal processing to the physical, biological, and social sciences: we wish to construct a mathematical model describing the relationship between different sets of variables, subject to certain reasonable constraints on input-output behavior. In the case of linear systems, observable behavior (specifically, the principle of superposition) strongly constrains the structure of compatible models (e.g., dynamic models are necessarily characterized by a convolution representation), a result that follows via the Cauchy functional equation if we impose even weak regularity conditions (e.g., continuity at any one point). In contrast, relaxing the principle of superposition to a requirement of homogeneity leads to a much larger class of compatible models that have important connections with nonlinear signal processing (for example, the median filter and many of its most popular variants satisfy homogeneity but not superposition). An analog of the equation F(x)=t^r F(x) (t>0) of positive homogeneity of degree r leads to the Cobb-Douglas production functions important in economics and to the general form of scientific laws that are invariant under ratio conversions (e.g., conversion from kilograms to pounds). In all these cases, as in many others, the key to establishing relations between observable forms of behavior and the structure of those models compatible with that behavior lies in identifying and solving functional equations, in particular the power equation p(xy)=p(x)p(y) and other Cauchy-type and related equations. Consistent aggregation of production functions leads to the generalized bisymmetry equation F[G(x,y),H(u,v)] = K[L(x,u),M(y,v)], further generalized to m x n variables. (Strangely, Cobb-Douglas production functions are not consistent with aggregation by addition). For the original bisymmetry equation F[F(x,y),F(u,v)]=F[F(x,u),F(y,v)], generalized to n x n variables, the general continuous strictly monotonic solution is essentially of the form F(x_1,...,x_n) = phi^{-1}[sum_{k=1}^n b_kphi(x_k)]. The question when such functions are homogeneous is answered with aid of Cauchy's power equation and helps to determine all homogeneous homomorphic nonlinear digital filter structures.

Type of Seminar:
Public Seminar
Prof. Janos Aczel and Ronald K. Pearson
Faculty of Mathematics University of Waterloo Waterloo, Ontario N2L 3G1 Canada
Jun 06, 2000   15:30

ETH Zentrum, ETZ E6, Gloriastrasse 35, 8006 Zürich
Contact Person:

Ronald K. Pearson
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Biographical Sketch:
János Aczél has been with the University of Waterloo(Canada) since 1965 as Professor and (since 1969) Distinguished Professor of Mathematics (and directed its Centre for Information Theory and Qualitative Economics); since 1993 Distinguished Professor Emeritus. Previous academic positions were at the University of Cologne (Germany), the Kossuth University of Debrecen, the Technical University of Miskolc and the University of Szeged (Hungary). He had shorter time visiting appointments at 20 universities and two research institutes in North America, Europe, Australia, Asia, and Africa. His Ph.D. is in mathematical analysis from the University of Budapest. He is Fellow of the Royal Society of Canada (Academy of Science) and Foreign Member of the Hungarian Academy of Sciences. He received the J. R. Cajal Medal from the Council of Scientific Research (Madrid) and honorary degrees from the University of Karlsruhe (in Economics), the University of Graz, the University of Miskolc (in Engineering), and the Silesian University, Katowice (in Mathematics). His research, which is both theoretical and applied, includes functional equations and applications, in rough chronological order to nomography, webs, geometrical objects, probability theory, information measures, index numbers, group decision making, aggregation, production functions, laws of science, theory of measurement and utility theory. He is the author or co-author of 8 books, editor of two books, author or co-author of over 250 scientific papers, and editor of several journals, including Aequationes Mathematicae, which he founded.