Webs, iteration groups, and equivalent changes in probabilities
Authors:
János Aczél, Günter Rote and Jens Schwaiger
Journal:
Quart. Appl. Math. 54 (1996), 475-499
MSC:
Primary 39B12; Secondary 39B22
DOI:
https://doi.org/10.1090/qam/1402406
MathSciNet review:
MR1402406
Full-text PDF Free Access
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Abstract: Yew-Kwang Ng [12] listed several “reasonable properties” for equivalent changes of probabilities and other proportions. He produced a family of functions satisfying all properties and asked whether there exist essentially different ones. We show that this is the case, by constructing uncountably many families of functions satisfying all properties. We show also that there are no other solutions. Our method establishes connections with webs (nets) and iteration groups. This may be of interest both in itself and for applications.
J. Aczél, Quasigroups, nets, and nomograms, Adv. Math. 1, 383–450 (1965)
J. Aczél, Funktionskomposition, Iterationsgruppen und Gewebe, Arch. Math. (Basel) 17, 469–475 (1966)
G. Blanton, Smoothness in disjoint groups of real functions under composition, Aequationes Math. 35, 1–16 (1988)
G. Blanton and J. A. Baker, Iteration groups generated by $C^{n}$ functions, Arch. Math. (Brno) 18, 121–128 (1982)
W. Blaschke and G. Bol, Geometrie der Gewebe, Springer, Berlin, 1938
R. H. Bruck, A Survey of Binary Systems, Springer, Berlin, Heidelberg, New York, 1971
R. L. Devaney, Chaos, Fractals, and Dynamics: Computer Experiments in Mathematics, Addison-Wesley, Menlo Park, California, 1990
O. Hölder, Die Axiome der Quantität und die Lehre vom Maβ, Berichte über die Verhandlungen der königlich sächsischen Gesellschaft der Wissenschaften zu Leipzig, mathematisch-physische Classe 53, 1–64 (1901)
D. H. Krantz, R. D. Luce, P. Suppes, and A. Tversky, Foundations of Measurement, vol. 1: Additive and Polynomial Representations, Academic Press, New York, London, 1971
M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, PWN-Uniwersytet Ślski, Warszawa, Kraków, Katowice, 1985
M. A. McKiernan, On iteration groups and related functional equations, Aequationes Math. 31, 37–46 (1986)
Yew-Kwang Ng, Equivalent changes/differences for percentages and probabilities, Quart. Appl. Math. 49, 289–301 (1991)
K. S. Sibirski, Introduction to Topological Dynamics, Noordhoff, Leyden, 1975
M. C. Zdun, Continuous and Differentiable Iteration Semigroups, Uniwersytet Ślski, Katowice, 1979
J. Aczél, Quasigroups, nets, and nomograms, Adv. Math. 1, 383–450 (1965)
J. Aczél, Funktionskomposition, Iterationsgruppen und Gewebe, Arch. Math. (Basel) 17, 469–475 (1966)
G. Blanton, Smoothness in disjoint groups of real functions under composition, Aequationes Math. 35, 1–16 (1988)
G. Blanton and J. A. Baker, Iteration groups generated by $C^{n}$ functions, Arch. Math. (Brno) 18, 121–128 (1982)
W. Blaschke and G. Bol, Geometrie der Gewebe, Springer, Berlin, 1938
R. H. Bruck, A Survey of Binary Systems, Springer, Berlin, Heidelberg, New York, 1971
R. L. Devaney, Chaos, Fractals, and Dynamics: Computer Experiments in Mathematics, Addison-Wesley, Menlo Park, California, 1990
O. Hölder, Die Axiome der Quantität und die Lehre vom Maβ, Berichte über die Verhandlungen der königlich sächsischen Gesellschaft der Wissenschaften zu Leipzig, mathematisch-physische Classe 53, 1–64 (1901)
D. H. Krantz, R. D. Luce, P. Suppes, and A. Tversky, Foundations of Measurement, vol. 1: Additive and Polynomial Representations, Academic Press, New York, London, 1971
M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, PWN-Uniwersytet Ślski, Warszawa, Kraków, Katowice, 1985
M. A. McKiernan, On iteration groups and related functional equations, Aequationes Math. 31, 37–46 (1986)
Yew-Kwang Ng, Equivalent changes/differences for percentages and probabilities, Quart. Appl. Math. 49, 289–301 (1991)
K. S. Sibirski, Introduction to Topological Dynamics, Noordhoff, Leyden, 1975
M. C. Zdun, Continuous and Differentiable Iteration Semigroups, Uniwersytet Ślski, Katowice, 1979
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Article copyright:
© Copyright 1996
American Mathematical Society
