Asymptotic behaviour of the partition function
- M.V. Lomonosov Moscow State University, Moscow (Russian Federation)
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.
- OSTI ID:
- 21202924
- Journal Information:
- Sbornik. Mathematics, Vol. 191, Issue 3; Other Information: DOI: 10.1070/SM2000v191n03ABEH000464; Country of input: International Atomic Energy Agency (IAEA); ISSN 1064-5616
- Country of Publication:
- United States
- Language:
- English
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