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

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].

A spatio-temporal point process model for particle growth

A spatio-temporal model of particle or star growth is defined, whereby new unit masses arrive sequentially in discrete time. These unit masses are referred to as candidate stars, which tend to arrive in mass-dense regions and then either form a new star or are absorbed by some neighbouring star of high mass. We analyse the system as time increases, and derive the asymptotic growth rate of the number of stars as well as the size of a randomly chosen star. We also prove that the size-biased mass distribution converges to a Poisson–Dirichlet distribution. This is achieved by embedding our model into a continuous-time Markov process, so that new stars arrive according to a marked Poisson process, with locations as marks, whereas existing stars grow as independent Yule processes. Our approach can be interpreted as a Hoppe-type urn scheme with a spatial structure. We discuss its relevance for and connection to models of population genetics, particle aggregation, image segmentation, epidemic spread, and random graphs with preferential attachment.

Minkowski Dimension and Explicit Tube Formulas for p-Adic Fractal Strings

The theory of complex dimensions describes the oscillations in the geometry (spectra and dynamics) of fractal strings. Such geometric oscillations can be seen most clearly in the explicit volume formula for the tubular neighborhoods of a p-adic fractal string L p , expressed in terms of the underlying complex dimensions. The general fractal tube formula obtained in this paper is illustrated by several examples, including the nonarchimedean Cantor and Euler strings. Moreover, we show that the Minkowski dimension of a p-adic fractal string coincides with the abscissa of convergence of the geometric zeta function associated with the string, as well as with the asymptotic growth rate of the corresponding geometric counting function. The proof of this new result can be applied to both real and p-adic fractal strings and hence, yields a unifying explanation of a key result in the theory of complex dimensions for fractal strings, even in the archimedean (or real) case.

The genealogical decomposition of a matrix population model with applications to the aggregation of stages

Matrix projection models are a central tool in many areas of population biology. In most applications, one starts from the projection matrix to quantify the asymptotic growth rate of the population (the dominant eigenvalue), the stable stage distribution, and the reproductive values (the dominant right and left eigenvectors, respectively). Any primitive projection matrix also has an associated ergodic Markov chain that contains information about the genealogy of the population. In this paper, we show that these facts can be used to specify any matrix population model as a triple consisting of the ergodic Markov matrix, the dominant eigenvalue and one of the corresponding eigenvectors. This decomposition of the projection matrix separates properties associated with lineages from those associated with individuals. It also clarifies the relationships between many quantities commonly used to describe such models, including the relationship between eigenvalue sensitivities and elasticities. We illustrate the utility of such a decomposition by introducing a new method for aggregating classes in a matrix population model to produce a simpler model with a smaller number of classes. Unlike the standard method, our method has the advantage of preserving reproductive values and elasticities. It also has conceptually satisfying properties such as commuting with changes of units.

The critical dimension for -measures

The critical dimension of an ergodic non-singular dynamical system is the asymptotic growth rate of sums of consecutive Radon–Nikodým derivatives. This has been shown to equal the average coordinate entropy for product odometers when the size of individual factors is bounded. We extend this result to $G$-measures with an asymptotic bound on the size of individual factors. Furthermore, unlike von Neumann–Krieger type, the critical dimension is an invariant property on the class of ergodic $G$-measures.

Bursting water balloons

The impact and rupture of a water-filled latex balloon on a flat, rigid surface is investigated using high-speed photography. Three distinct stages of the flow are observed, for which physical explanations are given. As the balloon lands and deforms, waves are formed on the balloon’s surface for which the restoring force is tension in the latex. These waves are shown to closely obey linear potential theory for constant surface tension. Should the balloon rupture, a crack forms, from which the membrane retracts. Spray is simultaneously ejected from the water’s surface, a consequence of a shear instability in the wake behind the retracting membrane. At later times, a larger-scale growth of the interfacial amplitude is observed, for which the generation mechanism is momentum in the water due to the preburst waves. However, it is argued that this is also a manifestation of the same mechanism that drives Richtmyer–Meshkov instability (RMI). Further, it is shown experimentally that this growth of the interface may also occur when there is no density difference across the balloon, a situation that does not arise for the standard RMI. An analytical model is then derived to predict the interfacial growth for such an interface, and is shown to predict the asymptotic growth rate of the interface accurately.

Effect of mutualist partner identity on plant demography

Mutualisms play a central role in the origin and maintenance of biodiversity. Because many mutualisms have strong demographic effects, interspecific variation in partner quality could have important consequences for population dynamics. Nevertheless, few studies have quantified how a mutualist partner influences population growth rates, and still fewer have compared the demographic impacts of multiple partner species. We used integral projection models parameterized with three years of census data to compare the demographic effects of two ant species – Crematogaster laevis and Pheidole minutula – on populations of the Amazonian ant-plant Maieta guianensis. Estimated population growth rates were positive (i.e., λ>1) for all ant-plant combinations. However, populations with only Pheidole minutula had the highest asymptotic growth rate (λ=1.23), followed by those colonized by Crematogaster laevis (λ=1.16), and in which the partner ant alternated between C. laevis and P. minutula at least once during our study (λ=1.15). Our results indicate that the short-term superiority of a mutualist partner – in this system P. minutula is a better defender of plants against herbivores than C. laevis – can have long-term demographic consequences. Furthermore, the demographic effects of switching among alternative partners appear to be context-dependent, with no benefits to plants hosting C. laevis but a major cost of switching to plants hosting P. minutula. Our results underscore the importance of expanding the study of mutualisms beyond the study of pair-wise interactions to consider the demographic costs and benefits of interacting with different, and multiple, potential partners.

Effect of mutualist partner identity on plant demography

Mutualisms play a central role in the origin and maintenance of biodiversity. Because many mutualisms have strong demographic effects, interspecific variation in partner quality could have important consequences for population dynamics. Nevertheless, few studies have quantified how a mutualist partner influences population growth rates, and still fewer have compared the demographic impacts of multiple partner species. We used integral projection models parameterized with three years of census data to compare the demographic effects of two ant species – Crematogaster laevis and Pheidole minutula – on populations of the Amazonian ant-plant Maieta guianensis. Estimated population growth rates were positive (i.e., λ>1) for all ant-plant combinations. However, populations with only Pheidole minutula had the highest asymptotic growth rate (λ=1.23), followed by those colonized by Crematogaster laevis (λ=1.16), and in which the partner ant alternated between C. laevis and P. minutula at least once during our study (λ=1.15). Our results indicate that the short-term superiority of a mutualist partner – in this system P. minutula is a better defender of plants against herbivores than C. laevis – can have long-term demographic consequences. Furthermore, the demographic effects of switching among alternative partners appear to be context-dependent, with no benefits to plants hosting C. laevis but a major cost of switching to plants hosting P. minutula. Our results underscore the importance of expanding the study of mutualisms beyond the study of pair-wise interactions to consider the demographic costs and benefits of interacting with different, and multiple, potential partners.