The growth rate of trajectories of a quadratic differential

A sufficient condition for the existence of holomorphic quadratic differential on a non-compact simply-connected Riemann surface with prescribed horizontal and vertical trees is obtained. In particular, for any pair of complete ℝ-trees of finite vertices with (n + 2) infinite edges, there exists a polynomial quadratic differential on ℂ of degree n such that the associated vertical and horizontal trees are isometric to the given pair.

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

International Journal of Algebra and Computation ◽

10.1142/s0218196720500022 ◽

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].

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Bisexual Galton–Watson branching processes with superadditive mating functions

Journal of Applied Probability ◽

10.1017/s0021900200111763 ◽

1986 ◽

Vol 23 (03) ◽

pp. 585-600 ◽

Cited By ~ 1

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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Decomposition of integrable holomorphic quadratic differential on Riemann surface of infinite type

Science China Mathematics ◽

10.1007/s11425-010-4055-y ◽

2010 ◽

Vol 53 (8) ◽

pp. 2039-2044 ◽

Cited By ~ 1

Author(s):

Tao Cheng

Keyword(s):

Riemann Surface ◽

Quadratic Differential ◽

Infinite Type ◽

Holomorphic Quadratic Differential

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Bounds for the Asymptotic Growth Rate of an Age-Dependent Branching Process

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700007114 ◽

1969 ◽

Vol 10 (1-2) ◽

pp. 231-235 ◽

Cited By ~ 3

Author(s):

P. J. Brockwell

Keyword(s):

Growth Rate ◽

Population Size ◽

Branching Process ◽

Expected Number ◽

Lattice Distribution ◽

Asymptotic Growth ◽

Age Dependent ◽

The Mean ◽

Image Position ◽

Asymptotic Growth Rate

Let M(t) denote the mean population size at time t (conditional on a single ancestor of age zero at time zero) of a branching process in which the distribution of the lifetime T of an individual is given by Pr {T≦t} =G(t), and in which each individual gives rise (at death) to an expected number A of offspring (1λ A λ ∞). expected number A of offspring (1 < A ∞). Then it is well-known (Harris [1], p. 143) that, provided G(O+)-G(O-) 0 and G is not a lattice distribution, M(t) is given asymptotically by where c is the unique positive value of p satisfying the equation .

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Bisexual Galton–Watson branching processes with superadditive mating functions

Journal of Applied Probability ◽

10.2307/3213999 ◽

1986 ◽

Vol 23 (3) ◽

pp. 585-600 ◽

Cited By ~ 45

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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SINGULARITIES OF QUADRATIC DIFFERENTIALS AND EXTREMAL TEICHMÜLLER MAPPINGS DEFINED BY DEHN TWISTS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788709000032 ◽

2009 ◽

Vol 87 (2) ◽

pp. 275-288 ◽

Cited By ~ 3

Author(s):

C. ZHANG

Keyword(s):

Riemann Surface ◽

Finite Type ◽

Quadratic Differential ◽

Quadratic Differentials ◽

Closed Geodesics ◽

Dehn Twists ◽

Holomorphic Quadratic Differential ◽

Simple Closed Geodesics ◽

Disk Component ◽

Teichmüller Mapping

AbstractLet S be a Riemann surface of finite type. Let ω be a pseudo-Anosov map of S that is obtained from Dehn twists along two families {A,B} of simple closed geodesics that fill S. Then ω can be realized as an extremal Teichmüller mapping on a surface of the same type (also denoted by S). Let ϕ be the corresponding holomorphic quadratic differential on S. We show that under certain conditions all possible nonpuncture zeros of ϕ stay away from all closures of once punctured disk components of S∖{A,B}, and the closure of each disk component of S∖{A,B} contains at most one zero of ϕ. As a consequence, we show that the number of distinct zeros and poles of ϕ is less than or equal to the number of components of S∖{A,B}.

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Some Local Asymptotic Laws for the Cauchy Process on the Line

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/2007/81934 ◽

2007 ◽

Vol 2007 ◽

pp. 1-9 ◽

Cited By ~ 1

Author(s):

A. Chukwuemeka Okoroafor

Keyword(s):

Growth Rate ◽

Sojourn Time ◽

Escape Rate ◽

Time Process ◽

Rate Process ◽

Monotone Functions ◽

Asymptotic Growth ◽

Asymptotic Size ◽

Asymptotic Growth Rate ◽

Cauchy Process

This paper investigates the lim inf behavior of the sojourn time process and the escape rate process associated with the Cauchy process on the line. The monotone functions associated with the lower asymptotic growth rate of the sojourn time are characterized and the asymptotic size of the large values of the escape rate process is developed.

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