Uniform growth rate

In an evolutionary system in which the rules of mutation are local in nature, the number of possible outcomes after $m$ mutations is an exponential function of $m$ but with a rate that depends only on the set of rules and not the size of the original object. We apply this principle to find a uniform upper bound for the growth rate of certain groups including the mapping class group. We also find a uniform upper bound for the growth rate of the number of homotopy classes of triangulations of an oriented surface that can be obtained from a given triangulation using $m$ diagonal flips.Comment: 13 pages, 5 figures, minor revisions, final version appears in Proc. Amer. Math. So

Growth rate for beta-expansions

Let $\beta>1$ and let m>\be be an integer. Each x\in I_\be:=[0,\frac{m-1}{\beta-1}] can be represented in the form $$x=\sum_{k=1}^\infty \epsilon_k\beta^{-k},$$ where $\epsilon_k\in\{0,1,...,m-1\}$ for all $k$ (a $\beta$-expansion of $x$). It is known that a.e. $x\in I_\beta$ has a continuum of distinct $\beta$-expansions. In this paper we prove that if $\beta$ is a Pisot number, then for a.e. $x$ this continuum has one and the same growth rate. We also link this rate to the Lebesgue-generic local dimension for the Bernoulli convolution parametrized by $\beta$. When $\beta<\frac{1+\sqrt5}2$, we show that the set of $\beta$-expansions grows exponentially for every internal $x$.Comment: 21 pages, 2 figure

On growth rate and contact homology

It is a conjecture of Colin and Honda that the number of Reeb periodic orbits of universally tight contact structures on hyperbolic manifolds grows exponentially with the period, and they speculate further that the growth rate of contact homology is polynomial on non-hyperbolic geometries. Along the line of the conjecture, for manifolds with a hyperbolic component that fibers on the circle, we prove that there are infinitely many non-isomorphic contact structures for which the number of Reeb periodic orbits of any non-degenerate Reeb vector field grows exponentially. Our result hinges on the exponential growth of contact homology which we derive as well. We also compute contact homology in some non-hyperbolic cases that exhibit polynomial growth, namely those of universally tight contact structures non-transverse to the fibers on a circle bundle

Mach Stem Height and Growth Rate Predictions

A new, more accurate prediction of Mach stem height in steady flow is presented. In addition, starting with a regular reflection in the dual-solution domain, the growth rate of the Mach stem from the time it is first formed till it reaches its steady-state height is presented. Comparisons between theory, experiments, and computations are presented for the Mach stem height. The theory for the Mach stem growth rate in both two and three dimensions is compared to computational results. The Mach stem growth theory provides an explanation for why, once formed, a Mach stem is relatively persistent

The Λ\LambdaCDM growth rate of structure revisited

We re-examine the growth index of the concordance $\Lambda$ cosmology in the light of the latest 6dF and {\em WiggleZ} data. In particular, we investigate five different models for the growth index $\gamma$, by comparing their cosmological evolution using observational data of the growth rate of structure formation at different redshifts. Performing a joint likelihood analysis of the recent supernovae type Ia data, the Cosmic Microwave Background shift parameter, Baryonic Acoustic Oscillations and the growth rate data, we determine the free parameters of the $\gamma(z)$ parametrizations and we statistically quantify their ability to represent the observations. We find that the addition of the 6dF and {\em WiggleZ} growth data in the likelihood analysis improves significantly the statistical results. As an example, considering a constant growth index we find $\Omega_{m0}=0.273\pm 0.011$ and $\gamma=0.586^{+0.079}_{-0.074}$.Comment: 8 pages, 5 figures, Accepted for publication by International J. of Modern Physics D (IJMPD). arXiv admin note: substantial text overlap with arXiv:1203.672