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

This article is cited in 31 scientific papers (total in 31 papers)

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

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

Bibliographic databases:

UDC: 511
MSC: Primary 11P81; Secondary 47A13
Received: 23.06.1999

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

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    This publication is cited in the following articles:
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    2. V. Yu. Protasov, “On the Asymptotics of the Binary Partition Function”, Math. Notes, 76:1 (2004), 144–149  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. V. Yu. Protasov, “Piecewise smooth refinable functions”, St. Petersburg Math. J., 16:5 (2005), 821–835  mathnet  crossref  mathscinet  zmath
    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  crossref  mathscinet  isi  scopus  scopus  scopus
    5. V. Yu. Protasov, “Spectral factorization of 2-block Toeplitz matrices and refinement equations”, St. Petersburg Math. J., 18:4 (2007), 607–646  mathnet  crossref  mathscinet  zmath  elib
    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  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    7. Jungers, RM, “On the finiteness property for rational matrices”, Linear Algebra and Its Applications, 428:10 (2008), 2283  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    9. Maesumi, M, “Optimal norms and the computation of joint spectral radius of matrices”, Linear Algebra and Its Applications, 428:10 (2008), 2324  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    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  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
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    13. Yu. A. Alpin, “Bounds for Joint Spectral Radii of a Set of Nonnegative Matrices”, Math. Notes, 87:1 (2010), 12–14  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
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    16. Nicola Guglielmi, Vladimir Protasov, “Exact Computation of Joint Spectral Characteristics of Linear Operators”, Found Comput Math, 2012  crossref  mathscinet  isi  scopus  scopus  scopus
    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  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    18. Jun Liu, Mingqing Xiao, “Rank-one characterization of joint spectral radius of finite matrix family”, Linear Algebra and its Applications, 2013  crossref  mathscinet  isi  scopus  scopus  scopus
    19. V.Yu. Protasov, R.M. Jungers, “Lower and upper bounds for the largest Lyapunov exponent of matrices”, Linear Algebra and its Applications, 2013  crossref  mathscinet  isi  scopus  scopus  scopus
    20. Y. Nesterov, V. Y. Protasov, “Optimizing the Spectral Radius”, SIAM. J. Matrix Anal. & Appl, 34:3 (2013), 999  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    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  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    22. J. Bochi, I. D. Morris, “Continuity properties of the lower spectral radius”, Proceedings of the London Mathematical Society, 2014  crossref  scopus  scopus
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    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  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    25. Hare K.G., “Base-D Expansions With Digits 0 To Q-1”, Exp. Math., 24:3 (2015), 295–303  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    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  crossref  isi  scopus  scopus  scopus
    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  crossref  isi  scopus
    28. Protasov V.Yu. Voynov A.S., “Matrix semigroups with constant spectral radius”, Linear Alg. Appl., 513 (2017), 376–408  crossref  mathscinet  zmath  isi  scopus
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    30. Guglielmi N., Laglia L., Protasov V., “Polytope Lyapunov Functions For Stable and For Stabilizable Lss”, Found. Comput. Math., 17:2 (2017), 567–623  crossref  mathscinet  zmath  isi  elib  scopus
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