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Title: Asymptotic behaviour of the partition function

Abstract

Given a pair of positive integers m and d such that 2{<=}m{<=}d, for integer n{>=}0 the quantity b{sub m,d}(n), called the partition function is considered; this by definition is equal to the cardinality of the set. The properties of b{sub m,d}(n) and its asymptotic behaviour as n{yields}{infinity} are studied. A geometric approach to this problem is put forward. It is shown that C{sub 1}n{sup {lambda}{sub 1}}{<=}b{sub m,d}(n){<=}C{sub 2}n{sup {lambda}{sub 2}}, for sufficiently large n, where C{sub 1} and C{sub 2} are positive constants depending on m and d. For some pair (m,d) the exponents {lambda}{sub 1} and {lambda}{sub 2} are calculated as the logarithms of certain algebraic numbers; for other pairs the problem is reduced to finding the joint spectral radius of a suitable collection of finite-dimensional linear operators. Estimates of the growth exponents and the constants C{sub 1} and C{sub 2} are obtained.

Authors:
 [1]
  1. M.V. Lomonosov Moscow State University, Moscow (Russian Federation)
Publication Date:
OSTI Identifier:
21202924
Resource Type:
Journal Article
Journal Name:
Sbornik. Mathematics
Additional Journal Information:
Journal Volume: 191; Journal Issue: 3; Other Information: DOI: 10.1070/SM2000v191n03ABEH000464; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1064-5616
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ASYMPTOTIC SOLUTIONS; MATHEMATICAL LOGIC; PARTITION FUNCTIONS

Citation Formats

Protasov, V Yu. Asymptotic behaviour of the partition function. United States: N. p., 2000. Web. doi:10.1070/SM2000V191N03ABEH000464; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
Protasov, V Yu. Asymptotic behaviour of the partition function. United States. doi:10.1070/SM2000V191N03ABEH000464; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
Protasov, V Yu. Sun . "Asymptotic behaviour of the partition function". United States. doi:10.1070/SM2000V191N03ABEH000464; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
@article{osti_21202924,
title = {Asymptotic behaviour of the partition function},
author = {Protasov, V Yu},
abstractNote = {Given a pair of positive integers m and d such that 2{<=}m{<=}d, for integer n{>=}0 the quantity b{sub m,d}(n), called the partition function is considered; this by definition is equal to the cardinality of the set. The properties of b{sub m,d}(n) and its asymptotic behaviour as n{yields}{infinity} are studied. A geometric approach to this problem is put forward. It is shown that C{sub 1}n{sup {lambda}{sub 1}}{<=}b{sub m,d}(n){<=}C{sub 2}n{sup {lambda}{sub 2}}, for sufficiently large n, where C{sub 1} and C{sub 2} are positive constants depending on m and d. For some pair (m,d) the exponents {lambda}{sub 1} and {lambda}{sub 2} are calculated as the logarithms of certain algebraic numbers; for other pairs the problem is reduced to finding the joint spectral radius of a suitable collection of finite-dimensional linear operators. Estimates of the growth exponents and the constants C{sub 1} and C{sub 2} are obtained.},
doi = {10.1070/SM2000V191N03ABEH000464; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)},
journal = {Sbornik. Mathematics},
issn = {1064-5616},
number = 3,
volume = 191,
place = {United States},
year = {2000},
month = {4}
}