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Mat. Zametki, 2008, Volume 84, Issue 1, Pages 69–98 (Mi mz5195)  

This article is cited in 16 scientific papers (total in 16 papers)

On the Distribution of Integer Random Variables Satisfying Two Linear Relations

V. P. Maslova, V. E. Nazaikinskiib

a M. V. Lomonosov Moscow State University
b A. Ishlinsky Institite for Problems in Mechanics, Russian Academy of Sciences

Abstract: We consider the multiplicative (in the sense of Vershik) probability measure corresponding to an arbitrary real dimension $d$ on the set of all collections $\{N_j\}$ of integer nonnegative numbers $N_j$, $j=l_0,l_0+1,…$, satisfying the conditions
$$ \sum_{j=l_0}^\infty jN_{j}\le M, \qquad \sum_{j=l_0}^\infty N_j=N, $$
where $l_0,M,N$ are natural numbers. If $M,N\to\infty$ and the rates of growth of these parameters satisfy a certain relation depending on $d$, and $l_0$ depends on them in a special way (for $d\ge2$ we can take $l_0=1$), then, in the limit, the “majority” of collections (with respect to the measure indicated above) concentrates near the limit distribution described by the Bose–Einstein formulas. We study the probabilities of the deviations of the sums $\sum_{j=l}^{\infty} N_j$ from the corresponding cumulative integrals for the limit distribution. In an earlier paper (see [6]), we studied the case $d=3$.

Keywords: Bose–Einstein distribution, multiplicative measure, cumulative distribution, cumulative integral, Bose particles

DOI: https://doi.org/10.4213/mzm5195

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English version:
Mathematical Notes, 2008, 84:1, 73–99

Bibliographic databases:

UDC: 519.2+531.19
Received: 14.06.2008

Citation: V. P. Maslov, V. E. Nazaikinskii, “On the Distribution of Integer Random Variables Satisfying Two Linear Relations”, Mat. Zametki, 84:1 (2008), 69–98; Math. Notes, 84:1 (2008), 73–99

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. P. Maslov, “Thermodynamics of Nanostructures”, Math. Notes, 84:4 (2008), 592–595  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. V. P. Maslov, “Transition to the Condensate State for Classical Gases and Clusterization”, Math. Notes, 84:6 (2008), 785–813  mathnet  crossref  crossref  mathscinet  isi
    3. V. P. Maslov, “New distribution formulas for classical gas, clusters, and phase transitions”, Theoret. and Math. Phys., 157:2 (2008), 1577–1594  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    4. Maslov V. P., “New Look at Thermodynamics of Gas and at Clusterization”, Russ. J. Math. Phys., 15:4 (2008), 493–510  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    5. V. P. Maslov, “On the new distribution generalizing the Gibbs, Bose–Einstein, and Pareto distributions”, Math. Notes, 85:5-6 (2009), 613–622  crossref  mathscinet  zmath  isi  elib  scopus
    6. Maslov V. P., “Threshold levels in economics and time series”, Math. Notes, 85:3-4 (2009), 305–321  crossref  zmath  isi  elib  scopus
    7. Maslov V. P., “Theorems on the debt crisis and the occurrence of inflation”, Math. Notes, 85:1-2 (2009), 146–150  crossref  zmath  isi  elib  scopus
    8. Maslov V. P., “On the boundedness law for the number of words in an overabundant dictionary”, Math. Notes, 85:1-2 (2009), 296–301  crossref  zmath  isi  elib  scopus
    9. Maslov V. P., “Theory of chaos and its application to the crisis of debts and the origin of inflation”, Russ. J. Math. Phys., 16:1 (2009), 103–120  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    10. Maslov V., “Dequantization, Statistical Mechanics and Econophysics”, Tropical and Idempotent Mathematics, Contemporary Mathematics, 495, ed. Litvinov G. Sergeev S., Amer Mathematical Soc, 2009, 239–279  crossref  mathscinet  zmath  isi
    11. Maslov V.P., “New global distributions in number theory and their applications”, J. Fixed Point Theory Appl., 8:1 (2010), 81–111  crossref  mathscinet  zmath  isi  scopus
    12. V. P. Maslov, “Mathematical Solution of the Gibbs Paradox”, Math. Notes, 89:2 (2011), 266–276  mathnet  crossref  crossref  mathscinet  isi
    13. Maslov V.P., “Mixture of New Ideal Gases and the Solution of the Gibbs and Einstein Paradoxes”, Russ. J. Math. Phys., 18:1 (2011), 83–101  crossref  mathscinet  zmath  isi  elib  scopus
    14. Maslov V.P., “The Relationship Between the Van-der-Waals Model and the Undistinguishing Statistics of Objectively Distinguishable Objects. the New Parastatistics”, Russ. J. Math. Phys., 21:1 (2014), 99–111  crossref  mathscinet  zmath  isi  elib  scopus
    15. Maslov V.P., “Large negative numbers in number theory, thermodynamics, information theory, and human thermodynamics”, Russ. J. Math. Phys., 23:4 (2016), 510–528  crossref  mathscinet  zmath  isi  scopus
    16. Maslov V.P., “Negative energy, debts, and disinformation from the viewpoint of analytic number theory”, Russ. J. Math. Phys., 23:3 (2016), 355–368  crossref  mathscinet  zmath  isi  elib  scopus
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