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Geometric grid classes of permutations have proven to be key in investigations of classical permutation pattern classes. By considering the representation of gridded permutations as words in a trace monoid, we prove that every geometric…
We give a general asymptotic formula for the growth rate of the number of indecomposable summands in the tensor powers of representations of finite groups, over a field of arbitrary characteristic. In characteristic zero we obtain…
The group of automorphisms of a symbolic dynamical system is countable, but often very large. For example, for a mixing subshift of finite type, the automorphism group contains isomorphic copies of the free group on two generators and the…
We study growth rates for strongly continuous semigroups. We prove that a growth rate for the resolvent on imaginary lines implies a corresponding growth rate for the semigroup if either the underlying space is a Hilbert space, or the…
For a finite alphabet $\mathcal{A}$ and shift $X\subseteq\mathcal{A}^{\mathbb{Z}}$ whose factor complexity function grows at most linearly, we study the algebraic properties of the automorphism group ${\rm Aut}(X)$. For such systems, we…
In this paper we investigate iterative roots of strictly monotone upper semi-continuous multifunctions having finitely many jumps. Known results are concerning roots of order 2 for multifunctions of exact one jump. For the general…
The purpose of this note is threefold. First we state a few conjectures that allow us to rigorously derive a theory which is asymptotic in N (the number of agents) that describes transients in large arrays of (identical) linear damped…
Let $\mathscr{S}$ be the class of all structures whose growth rate on orbits of subsets of size $n$ is not faster than $\frac{2^n}{p(n)}$ for any polynomial $p$. In this article we give a complete classification of all structures in…
Intersection growth concerns the asymptotic behavior of the index of the intersection of all subgroups of a group that have index at most n. In this note we show that the intersection growth of some groups may not be a nicely behaved…
Given an $\mathbb{N}$-weighted tree automaton, we give a decision procedure for exponential vs polynomial growth (with respect to the input size) in quadratic time, and an algorithm that computes the exact polynomial degree of growth in…
It is observed that the conjugacy growth series of the infinite fini-tary symmetric group with respect to the generating set of transpositions is the generating series of the partition function. Other conjugacy growth series are computed,…
We study the entropy rate of pattern sequences of stochastic processes, and its relationship to the entropy rate of the original process. We give a complete characterization of this relationship for i.i.d. processes over arbitrary…
We study the growth rate of a sequence which measures the uniform norm of the differential under the iterates of maps. On symplectically hyperbolic manifolds, we show that this sequence has at least linear growth for every non-identical…
Asymptotic quadratic growth rates of saddle connections and families of periodic cylinders on translation tori with n marked points are studied. For any marking the existence of limits of the quadratic growth rate is shown using elementary…
We introduce a quantitative characterization of subgroup alternatives modeled on the Tits alternative in terms of group laws and investigate when this property is preserved under extensions. We develop a framework that lets us expand the…
In this paper, the author (1) compares subnormal closures of finite sets in free groups; (2) proves that the exponential growth rate (e.g.r.), i.e., the limit of the n-th roots of g(n), where g(n) is the growth function of a subgroup H with…
In this article we continue the study of automorphism groups of constant length substitution shifts and also their topological factors. We show that up to conjugacy, all roots of the identity map are letter exchanging maps, and all other…
For a subshift over a finite alphabet, a measure of the complexity of the system is obtained by counting the number of nonempty cylinder sets of length $n$. When this complexity grows exponentially, the automorphism group has been shown to…
The synchronization problem is investigated for the class of locally strongly transitive automata introduced in a previous work of the authors. Some extensions of this problem related to the notions of stable set and word of minimal rank of…
Large-scale integration of inverter-based resources into power grids worldwide is challenging their stability and security. This paper takes a closer look at synchronous condensers as a solution to mitigate stability challenges caused by…