Asymptotic Growth

2012 ◽  
pp. 377-387
Author(s):  
Jeff Edmonds
Keyword(s):  
Asymptotic Growth

Resolution of T. Ward's Question and the Israel–Finch Conjecture: Precise Analysis of an Integer Sequence Arising in Dynamics

2014 ◽  
Vol 24 (1) ◽  
pp. 195-215
Author(s):  
JEFFREY GAITHER ◽  
GUY LOUCHARD ◽  
STEPHAN WAGNER ◽  
MARK DANIEL WARD
Keyword(s):  
Weighted Average ◽  
Analytic Number Theory ◽  
Combinatorial Structure ◽  
Asymptotic Growth ◽  
First Order ◽  
Integer Sequence ◽  
Precise Analysis ◽  
Formula Group ◽  
Fourth Author ◽  
Image Position

We analyse the first-order asymptotic growth of \[ a_{n}=\int_{0}^{1}\prod_{j=1}^{n}4\sin^{2}(\pi jx)\, dx. \] The integer an appears as the main term in a weighted average of the number of orbits in a particular quasihyperbolic automorphism of a 2n-torus, which has applications to ergodic and analytic number theory. The combinatorial structure of an is also of interest, as the ‘signed’ number of ways in which 0 can be represented as the sum of ϵjj for −n ≤ j ≤ n (with j ≠ 0), with ϵj ∈ {0, 1}. Our result answers a question of Thomas Ward (no relation to the fourth author) and confirms a conjecture of Robert Israel and Steven Finch.

Asymptotic growth bounds for the Vlasov-Poisson system in a half space

2015 ◽  
Vol 9 ◽  
pp. 6791-6803
Author(s):  
Athraa Neamah Albukhuttar ◽  
Xianwen Zhang
Keyword(s):  
Half Space ◽  
Poisson System ◽  
Asymptotic Growth

The Ample Cone for a K3 Surface

2011 ◽  
Vol 63 (3) ◽  
pp. 481-499 ◽  
Author(s):  
Arthur Baragar
Keyword(s):  
Hausdorff Dimension ◽  
Number Field ◽  
K3 Surface ◽  
Picard Number ◽  
Asymptotic Growth ◽  
Group Of Automorphisms ◽  
Line Parallel ◽  
Ample Cone

Abstract In this paper, we give several pictorial fractal representations of the ample or K¨ahler cone for surfaces in a certain class of K3 surfaces. The class includes surfaces described by smooth (2, 2, 2) forms in ℙ1 × ℙ1 × ℙ1 defined over a sufficiently large number field K that have a line parallel to one of the axes and have Picard number four. We relate the Hausdorff dimension of this fractal to the asymptotic growth of orbits of curves under the action of the surface's group of automorphisms. We experimentally estimate the Hausdorff dimension of the fractal to be 1.296 ± .010.

Growth in Baumslag–Solitar groups II: The Bass–Serre tree

2020 ◽  
Vol 30 (02) ◽  
pp. 339-378
Author(s):  
Jared Adams ◽  
Eric M. Freden
Keyword(s):  
Growth Rate ◽  
Minimal Length ◽  
Geometric Object ◽  
Asymptotic Growth ◽  
Growth Series ◽  
Context Sensitive ◽  
Counting Algorithms ◽  
Self Similar ◽  
Context Free ◽  
Asymptotic Growth Rate

Denote the Baumslag–Solitar family of groups as [Formula: see text]). When [Formula: see text] we study the Bass–Serre tree [Formula: see text] for [Formula: see text] as a geometric object. We suggest that the irregularity of [Formula: see text] is the principal obstruction for computing the growth series for the group. In the particular case [Formula: see text] we exhibit a set [Formula: see text] of normal form words having minimal length for [Formula: see text] and use it to derive various counting algorithms. The language [Formula: see text] is context-sensitive but not context-free. The tree [Formula: see text] has a self-similar structure and contains infinitely many cone types. All cones have the same asymptotic growth rate as [Formula: see text] itself. We derive bounds for this growth rate, the lower bound also being a bound on the growth rate of [Formula: see text].

Bisexual Galton–Watson branching processes with superadditive mating functions

1986 ◽  
Vol 23 (03) ◽  
pp. 585-600 ◽  
Author(s):  
D. J. Daley ◽  
David M. Hull ◽  
James M. Taylor
Keyword(s):  
Growth Rate ◽  
Branching Process ◽  
Critical Case ◽  
Branching Processes ◽  
Upper And Lower Bounds ◽  
Simple Criterion ◽  
Asymptotic Growth ◽  
Extinction Probabilities ◽  
Mating Function ◽  
Asymptotic Growth Rate

For a bisexual Galton–Watson branching process with superadditive mating function there is a simple criterion for determining whether or not the process becomes extinct with probability 1, namely, that the asymptotic growth rate r should not exceed 1. When extinction is not certain (equivalently, r > 1), simple upper and lower bounds are established for the extinction probabilities. An example suggests that in the critical case that r = 1, some condition like superadditivity is essential for ultimate extinction to be certain. Some illustrative numerical comparisons of particular mating functions are made using a Poisson offspring distribution.

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