Related papers: Multivariate growth and cogrowth
Fixing a subgroup $\Gamma$ in a group $G$, the commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $\Delta$ of $G$ with $[\Gamma: \Gamma \cap \Delta][\Delta : \Gamma \cap \Delta] = n$. For pairs…
The growth rate of the large-scale structure of the universe has been advocated as the observable par excellence for testing gravity on cosmological scales. By considering linear-order deviations from General Relativity, we show that…
We investigate the relationship between geometric, analytic and probabilistic indices for quotients of the Cayley graph of the free group ${\rm Cay}(F_n)$ endowed with variable edge lengths, by an arbitrary subgroup $G$ of $F_n$. Our main…
We study the matching problem of regular tree languages, that is, "$\exists \sigma:\sigma(L)\subseteq R$?" where $L,R$ are regular tree languages over the union of finite ranked alphabets $\Sigma$ and $\mathcal{X}$ where $\mathcal{X}$ is an…
The $\Gamma$ growth model is an effective parameterization employed across various scientific disciplines and scales to depict growth. It has been demonstrated that the cosmic star formation rate density (CSFRD) can also be described…
In this paper, we investigate the convergence of language models (LMs) trained under different random seeds, measuring convergence as the expected per-token Kullback--Leibler (KL) divergence across seeds. By comparing LM convergence as a…
We study the growth rate of the number of maximal arithmetic subgroups of bounded covolumes in a semi-simple Lie group using an extension of the method due to Borel and Prasad.
Alternative cosmological models have been proposed to alleviate the tensions reported in the concordance cosmological model, or to explain the current accelerated phase of the universe. One way to distinguish between General Relativity and…
This work is concerned with regular languages defined over large alphabets, either infinite or just too large to be expressed enumeratively. We define a generic model where transitions are labeled by elements of a finite partition of the…
The growth rate of matter perturbation and the expansion rate of the Universe can be used to distinguish modified gravity and dark energy models in explaining cosmic acceleration. We explore here the inclusion of spatial curvature into the…
Evolution, the engine behind the survival and growth of life on Earth, operates through the population-based process of reproduction. Inspired by this principle, this paper formally defines a newly emerging problem -- the population-based…
In this paper, we study saddle point (SP) problems, focusing on convex-concave optimization involving functions that satisfy either two-sided quadratic functional growth (QFG) or two-sided quadratic gradient growth (QGG)--novel conditions…
Evolutionary algorithms excel in solving complex optimization problems, especially those with multiple objectives. However, their stochastic nature can sometimes hinder rapid convergence to the global optima, particularly in scenarios…
The ghost-free theory of massive gravity with two dynamical metrics has been shown to produce viable cosmological expansion, where the late-time acceleration of the Universe is due to the finite range of the gravitational interaction rather…
We investigate subalgebras in free Lie algebras, the main tool being relative growth and cogrowth functions. Our study reveals drastic differences in the behavior of proper finitely generated subalgebras and nonzero subideals. For instance,…
We analyze a large-scale snapshot of del.icio.us and investigate how the number of different tags in the system grows as a function of a suitably defined notion of time. We study the temporal evolution of the global vocabulary size, i.e.…
This thesis explores how concepts of formal language theory can be used to study left-orderable groups. It analyses the languages formed by their positive cones and demonstrates how the abstract families of languages (AFLs) in the Chomsky…
We study the LEF growth function of a finitely generated LEF group $\Gamma$, which measures the orders of finite groups admitting local embeddings of balls in a word metric on $\Gamma$. We prove that any sufficiently smooth increasing…
Suppose $X$ and $Y$ are finite complexes, with $Y$ simply connected. Gromov conjectured that the number of mapping classes in $[X,Y]$ which can be realized by $L$-Lipschitz maps grows asymptotically as $L^\alpha$, where $\alpha$ is an…
Let $G$ be a free group of rank $n$ and $H\subset G$ its subgroup of finite index. Then $H$ is also a free group and the rank $m$ of $H$ is determined by Schreier's formula $m-1=(n-1)\cdot|G:H|.$ Any subalgebra of a free Lie algebra is also…