RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Sb.: Year: Volume: Issue: Page: Find

 Mat. Sb., 2000, Volume 191, Number 3, Pages 65–98 (Mi msb464)

Asymptotic behaviour of the partition function

V. Yu. Protasov

M. V. Lomonosov Moscow State University

Abstract: Given a pair of positive integers $m$ and $d$ such that $2\leqslant m\leqslant d$, for integer $n\geqslant 0$ the quantity $b_{m,d}(n)$, called the partition function is considered; this by definition is equal to the cardinality of the set
$$\{(a_0,a_1,…):n=\sum_ka_km^k, a_k\in\{0,…,d-1\}, k\geqslant 0\}.$$
The properties of $b_{m,d}(n)$ and its asymptotic behaviour as $n\to\infty$ are studied. A geometric approach to this problem is put forward. It is shown that
$$C_1n^{\lambda_1}\leqslant b_{m,d}(n)\leqslant C_2n^{\lambda_2},$$
for sufficiently large $n$, where $C_1$ and $C_2$ are positive constants depending on $m$ and $d$, and $\lambda_1=\varliminf\limits_{n\to\infty}\dfrac{\log b(n)}{\log n}$ and $\lambda_2=\varlimsup\limits_{n\to\infty}\dfrac{\log b(n)}{\log n}$ are characteristics of the exponential growth of the partition function. For some pair $(m,d)$ the exponents $\lambda_1$ and $\lambda_2$ 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 $C_1$ and $C_2$ are obtained.

DOI: https://doi.org/10.4213/sm464

Full text: PDF file (414 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2000, 191:3, 381–414

Bibliographic databases:

UDC: 511
MSC: Primary 11P81; Secondary 47A13

Citation: V. Yu. Protasov, “Asymptotic behaviour of the partition function”, Mat. Sb., 191:3 (2000), 65–98; Sb. Math., 191:3 (2000), 381–414

Citation in format AMSBIB
\Bibitem{Pro00} \by V.~Yu.~Protasov \paper Asymptotic behaviour of the~partition function \jour Mat. Sb. \yr 2000 \vol 191 \issue 3 \pages 65--98 \mathnet{http://mi.mathnet.ru/msb464} \crossref{https://doi.org/10.4213/sm464} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1773255} \zmath{https://zbmath.org/?q=an:1004.11056} \transl \jour Sb. Math. \yr 2000 \vol 191 \issue 3 \pages 381--414 \crossref{https://doi.org/10.1070/sm2000v191n03ABEH000464} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000088115700006} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-0034338840} 

• http://mi.mathnet.ru/eng/msb464
• https://doi.org/10.4213/sm464
• http://mi.mathnet.ru/eng/msb/v191/i3/p65

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. Yu. Protasov, “On the regularity of de Rham curves”, Izv. Math., 68:3 (2004), 567–606
2. V. Yu. Protasov, “On the Asymptotics of the Binary Partition Function”, Math. Notes, 76:1 (2004), 144–149
3. V. Yu. Protasov, “Piecewise smooth refinable functions”, St. Petersburg Math. J., 16:5 (2005), 821–835
4. Protasov V., “Applications of the joint spectral radius to some problems of functional analysis, probability and combinatorics”, 44th IEEE Conference on Decision and Control & European Control Conference, 2005, 3025–3030
5. V. Yu. Protasov, “Spectral factorization of 2-block Toeplitz matrices and refinement equations”, St. Petersburg Math. J., 18:4 (2007), 607–646
6. Blondel, VD, “On the complexity of computing the capacity of codes that avoid forbidden difference patterns”, IEEE Transactions on Information Theory, 52:11 (2006), 5122
7. Jungers, RM, “On the finiteness property for rational matrices”, Linear Algebra and Its Applications, 428:10 (2008), 2283
8. Jungers, RM, “Efficient algorithms for deciding the type of growth of products of integer matrices”, Linear Algebra and Its Applications, 428:10 (2008), 2296
9. Maesumi, M, “Optimal norms and the computation of joint spectral radius of matrices”, Linear Algebra and Its Applications, 428:10 (2008), 2324
10. Jungers R.M., Protasov V.Yu., Blondel V.D., “Computing the growth of the number of overlap-free words with spectra of matrices”, Latin 2008: Theoretical Informatics, Lecture Notes in Computer Science, 4957, 2008, 84–93
11. R. M. Jungers, V. Y. Protasov, “Counterexamples to the Complex Polytope Extremality Conjecture”, SIAM J Matrix Anal Appl, 31:2 (2009), 404
12. Jungers, RM, “Overlap-free words and spectra of matrices”, Theoretical Computer Science, 410:38–40 (2009), 3670
13. Yu. A. Alpin, “Bounds for Joint Spectral Radii of a Set of Nonnegative Matrices”, Math. Notes, 87:1 (2010), 12–14
14. Vladimir Y. Protasov, Raphaël M. Jungers, Vincent D. Blondel, “Joint Spectral Characteristics of Matrices: A Conic Programming Approach”, SIAM J Matrix Anal Appl, 31:4 (2010), 2146
15. Protasov V.Yu., “When do several linear operators share an invariant cone?”, Linear Algebra and Its Applications, 433:4 (2010), 781–789
16. Nicola Guglielmi, Vladimir Protasov, “Exact Computation of Joint Spectral Characteristics of Linear Operators”, Found Comput Math, 2012
17. Molteni G., “Representation of a 2-Power as Sum of K 2-Powers: the Asymptotic Behavior”, Int. J. Number Theory, 8:8 (2012), 1923–1963
18. Jun Liu, Mingqing Xiao, “Rank-one characterization of joint spectral radius of finite matrix family”, Linear Algebra and its Applications, 2013
19. V.Yu. Protasov, R.M. Jungers, “Lower and upper bounds for the largest Lyapunov exponent of matrices”, Linear Algebra and its Applications, 2013
20. Y. Nesterov, V. Y. Protasov, “Optimizing the Spectral Radius”, SIAM. J. Matrix Anal. & Appl, 34:3 (2013), 999
21. A.A.li Ahmadi, Raphaël.M.. Jungers, P.A.. Parrilo, Mardavij Roozbehani, “Joint Spectral Radius and Path-Complete Graph Lyapunov Functions”, SIAM J. Control Optim, 52:1 (2014), 687
22. J. Bochi, I. D. Morris, “Continuity properties of the lower spectral radius”, Proceedings of the London Mathematical Society, 2014
23. V.Y.. Protasov, Raphaël.M.. Jungers, “Resonance and marginal instability of switching systems”, Nonlinear Analysis: Hybrid Systems, 17 (2015), 81
24. Krenn D., Ralaivaosaona D., Wagner S., “Multi-Base Representations of Integers: Asymptotic Enumeration and Central Limit Theorems”, Appl. Anal. Discret. Math., 9:2 (2015), 285–312
25. Hare K.G., “Base-D Expansions With Digits 0 To Q-1”, Exp. Math., 24:3 (2015), 295–303
26. Czornik A., Niezabitowski M., “Stability of infinite-dimensional linear inclusions”, 2015 20th International Conference on Methods and Models in Automation and Robotics (MMAR ) (Miedzyzdroje, Poland), IEEE, 2015, 204–207
27. Czornik A., Jurgas P., Niezabitowski M., “Estimation of the Joint Spectral Radius”, Man–Machine Interactions 4, Advances in Intelligent Systems and Computing, 391, eds. Gruca A., Brachman A., Kozielski S., Czachorski T., Springer-Verlag Berlin, 2016, 401–410
28. Protasov V.Yu. Voynov A.S., “Matrix semigroups with constant spectral radius”, Linear Alg. Appl., 513 (2017), 376–408
29. Protasov V.Yu., “The Euler binary partition function and subdivision schemes”, Math. Comput., 86:305 (2017), 1499–1524
30. Guglielmi N., Laglia L., Protasov V., “Polytope Lyapunov Functions For Stable and For Stabilizable Lss”, Found. Comput. Math., 17:2 (2017), 567–623
31. V. Yu. Protasov, Ya. Wang, “Newman cyclotomic polynomials, refinable splines and the Euler binary partition function”, Sb. Math., 209:12 (2018), 1783–1802
•  Number of views: This page: 456 Full text: 88 References: 29 First page: 1