Stochastic exponential growth is observed in a variety of contexts, including molecular autocatalysis, nuclear fission, population growth, inflation of the universe, viral social media posts, and financial markets. Yet literature on modeling the phenomenology of these stochastic dynamics has predominantly focused on one model, geometric Brownian motion (GBM), which can be described as the solut...

A short proof to a recent theorem of Giambruno and Mishchenko is given in this note. 1 The theorem The following theorem was recently proved by Giambruno and Mishchenko. Theorem 1. [1, Theorem 1] For every 0 < α < 1, there exist β > 1 and n0 ∈ N, such that for every partition λ of n > n0 with max{λ1, λ1} < αn f > β. The proof of Giambruno and Mishchenko is rather complicated and applies a cleve...

Both the computer drawing of the complement of the Mandelbrot-like set of a one-parameter dependent complex exponential family of maps and the computer drawing of the Julia sets of the maps of this family, grow with the maximal number of iterations we choose. Some graphic examples of this growth, which evoke the image of a garden, are shown here.

We say that Γ has uniform exponential growth if Ent Γ > 0. In [11], remarque 5.12, M. Gromov raised the question whether exponential growth always implies uniform exponential growth. The answer is negative, indeed, in [14] J.S. Wilson gave examples of finitely generated groups of exponential growth and non uniform exponential growth. Nevertheless, exponential growth implies uniform exponential ...

The Milnor-Wolf Theorem characterizes the finitely generated solvable groups which have exponential growth; a finitely generated solvable group has exponential growth iff it is not virtually nilpotent. Wolf showed that a finitely generated nilpotent by finite group has polynomial growth; then extended this by proving that polycyclic groups which are not virtually nilpotent have expontial growth...

The site frequency spectrum (SFS) and other genetic summary statistics are at the heart of many population genetic studies. Previous studies have shown that human populations have undergone a recent epoch of fast growth in effective population size. These studies assumed that growth is exponential, and the ensuing models leave an excess amount of extremely rare variants. This suggests that huma...