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On a general theorem of set theory leading to the Gibbs, Bose-Einstein, and Pareto distributions as well as to the Zipf-Mandelbrot law for the stock market. (English. Russian original) Zbl 1112.91036

Math. Notes 78, No. 6, 807-813 (2005); translation from Mat. Zametki 78, No. 6, 870-877 (2005).
Summary: The notion of density of a finite set is introduced. We prove a general theorem of set theory which refines the Gibbs, Bose-Einstein, and Pareto distributions as well as the Zipf law.

MSC:

91G99 Actuarial science and mathematical finance
91B80 Applications of statistical and quantum mechanics to economics (econophysics)
91B24 Microeconomic theory (price theory and economic markets)
82B99 Equilibrium statistical mechanics
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[1] W. Hurewicz and H. Wallman, Dimension Theory, Princeton Mathematical Series, vol. 4. Princeton University Press, Princeton, NJ, 1941; Russian translation: Moscow, 1948. · JFM 67.1092.03
[2] V. P. Maslov, ”Nonlinear averages in economics,” Mat. Zametki [Math. Notes], 78 (2005), no. 3, 377–395. · Zbl 1153.91429
[3] V. P. Maslov, ”The law of large deviations in number theory. Computable functions of several arguments and decoding,” Dokl. Ross. Akad. Nauk [Russian Acad. Sci. Dokl. Math.], 404 (2005), no. 6, 731–736. · Zbl 1121.03050
[4] G. V. Koval’ and V. P. Maslov, ”On estimates for a large partition function,” (to appear). · Zbl 1108.82026
[5] B. Mandelbrot, Structure formelle des textes et communication, Word, vol. 10. no.1, New York, 1954. · Zbl 0059.13802
[6] V. P. Maslov, ”The principle of increasing complexity of portfolio formation on the stock exchange,” Dokl. Ross. Akad. Nauk [Russian Acad. Sci. Dokl. Math.], 404 (2005), no. 4, 446–450. · Zbl 1201.91244
[7] V. P. Maslov, ”A refinement of the Zipf law for frequency dictionaries and stock exchanges,” Dokl. Ross. Akad. Nauk [Russian Acad. Sci. Dokl. Math.], 405 (2005), no. 5.
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