Showing papers by "Michael Drmota published in 2020"
TL;DR: In this article, the authors consider planar maps adjusted with a regular critical Boltzmann distribution and show that the expected number of pattern occurrences of a given map is asymptotically linear when the number n of edges goes to infinity.
10 citations
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TL;DR: In this article, it was shown that the asymptotic estimate for the coefficients of the solutions of (so-called) positive catalytic equations has a universal and universal behavior, and the central limit theorem for parameters that can be encoded by catalytic functional equations, even when they are not positive.
Abstract: Catalytic equations appear in several combinatorial applications, most notably in the numeration of lattice path and in the enumeration of planar maps. The main purpose of this paper is to show that the asymptotic estimate for the coefficients of the solutions of (so-called) positive catalytic equations has a universal asymptotic behavior. In particular, this provides a rationale why the number of maps of size $n$ in various planar map classes grows asymptotically like $c\cdot n^{-5/2} \gamma^n$, for suitable positive constants $c$ and $\gamma$. Essentially we have to distinguish between linear catalytic equations (where the subexponential growth is $n^{-3/2}$) and non-linear catalytic equations (where we have $n^{-5/2}$ as in planar maps). Furthermore we provide a quite general central limit theorem for parameters that can be encoded by catalytic functional equations, even when they are not positive.
8 citations
DOI•
01 Jan 2020
TL;DR: It is shown that X_n/n → c in probability (for some explicit c>0) and for so-called subcritial subclasses of planar maps like outerplanar maps the authors obtain a central limit theorem, too.
Abstract: The main goal of this paper is to determine the asymptotic behavior of the number X_n of cut-vertices in random planar maps with n edges It is shown that X_n/n → c in probability (for some explicit c>0) For so-called subcritial subclasses of planar maps like outerplanar maps we obtain a central limit theorem, too
7 citations
TL;DR: In this article, the structure of equilibria of a coagulation-fragmentation-death model of silicosis is analyzed and exact multiplicity results in the particular case of piecewise constant coefficients, results on existence and non-existence of equilibrium in the general case, as well as precise asymptotics for the infinite series that arise in the case of power law coefficients.
Abstract: We analyse the structure of equilibria of a coagulation–fragmentation–death model of silicosis. We present exact multiplicity results in the particular case of piecewise constant coefficients, results on existence and non-existence of equilibria in the general case, as well as precise asymptotics for the infinite series that arise in the case of power law coefficients.
6 citations
TL;DR: In this article, it was shown that for sequences of integers with digit properties in two coprime bases, the von Mangoldt and Mobius functions can be approximated with Fourier analysis, Diophantine approximation, and combinatorial arguments.
Abstract: We estimate the sums ∑n≤xΛ(n)f(n)g(n)exp(2iπϑn) and ∑n≤xμ(n)f(n)g(n)exp(2iπϑn), where Λ denotes the von Mangoldt function (and μ the Mobius function) whenever q1 and q2 are two coprime bases, f (resp., g) is a strongly q1-multiplicative (resp., strongly q2-multiplicative) function of modulus 1, and ϑ is a real number. The goal of this work is to introduce a new approach to study these sums involving simultaneously two different bases combining Fourier analysis, Diophantine approximation, and combinatorial arguments. We deduce from these estimates a prime number theorem (and Mobius orthogonality) for sequences of integers with digit properties in two coprime bases.
6 citations
TL;DR: The goal of this paper is to obtain quantitative results on the number and on the size of maximal independent sets and maximal matchings in several block-stable graph classes that satisfy a proper sub-criticality condition.
Abstract: The goal of this paper is to obtain quantitative results on the number and on the size of maximal independent sets and maximal matchings in several block-stable graph classes that satisfy a proper sub-criticality condition. In particular we cover trees, cacti graphs and series-parallel graphs. The proof methods are based on a generating function approach and a proper singularity analysis of solutions of implicit systems of functional equations in several variables. As a byproduct, this method extends previous results of Meir and Moon for trees [Meir, Moon: On maximal independent sets of nodes in trees, Journal of Graph Theory 1988].
5 citations
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TL;DR: In this paper, it was shown that the logarithmic densities of any automatic sequence along squares and primes are known to exist and are computable, and for these subsequences a criterion to decide whether the densities exist.
Abstract: In this paper we develop a method to transfer density results for primitive automatic sequences to logarithmic-density results for general automatic sequences. As an application we show that the logarithmic densities of any automatic sequence along squares $(n^2)_{n\geq 0}$ and primes $(p_n)_{n\geq 1}$ exist and are computable. Furthermore, we give for these subsequences a criterion to decide whether the densities exist, in which case they are also computable.
In particular in the prime case these densities are all rational. We also deduce from a recent result of the third author and Lemanczyk that all subshifts generated by automatic sequences are orthogonal to any bounded multiplicative aperiodic function.
3 citations
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TL;DR: In this article, it was shown that strongly $b$-multiplicative functions of modulus $1$ along squares are asymptotically orthogonal to the Mobius function.
Abstract: We prove that strongly $b$-multiplicative functions of modulus $1$ along squares are asymptotically orthogonal to the Mobius function. This provides examples of sequences having maximal entropy and satisfying this property.
1 citations
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TL;DR: This paper focuses on universal compression for memoryless sources, and presents precise analysis for the maximal minimax and the average minimax for constrained distributions, and shows that the sequential algorithm based on modified Krichevsky-Trofimov estimator is asymptotically optimal up to $O(1)$ for both maximal and average redundancies.
Abstract: Sequential probability assignment and universal compression go hand in hand. We propose sequential probability assignment for non-binary (and large alphabet) sequences with empirical distributions whose parameters are known to be bounded within a limited interval. Sequential probability assignment algorithms are essential in many applications that require fast and accurate estimation of the maximizing sequence probability. These applications include learning, regression, channel estimation and decoding, prediction, and universal compression. On the other hand, constrained distributions introduce interesting theoretical twists that must be overcome in order to present efficient sequential algorithms. Here, we focus on universal compression for memoryless sources, and present precise analysis for the maximal minimax and the average minimax for constrained distributions. We show that our sequential algorithm based on modified Krichevsky-Trofimov (KT) estimator is asymptotically optimal up to $O(1)$ for both maximal and average redundancies. This paper follows and addresses the challenge presented in \cite{stw08} that suggested "results for the binary case lay the foundation to studying larger alphabets".
1 citations
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TL;DR: In this paper, a sufficient and necessary condition for the existence of a distribution function in the Zeckendorf expansion is provided. But this condition is not applicable to the case of the Fibonacci sequence.
Abstract: In 1972 Delange observed in analogy of the classical Erd\H os-Wintner theorem that $q$-additive functions $f(n)$ has a distribution function if and only if the two series $\sum f(d q^j)$, $\sum f(d q^j)^2$ converge. The purpose of this paper is to provide quantitative versions of this theorem as well as generalizations to other kinds of digital expansions. In addition to the $q$-ary and Cantor case we focus on the Zeckendorf expansion that is based on the Fibonacci sequence, where we provide a sufficient and necessary condition for the existence of a distribution function, namely that the two series $\sum f(F_j)$, $\sum f(F_j)^2$ converge (previously only a sufficient condition was known).