Asymptotic densities from the modified Montroll-Weiss equation for coupled CTRWs

AbstractLet x = $\begin{array}{} \displaystyle \{x_n\}_{n=1}^{\infty} \end{array}$ be a sequence of positive numbers, and 𝓙x be the collection of all subsets A ⊆ ℕ such that $\begin{array}{} \displaystyle \sum_{k\in A} \end{array}$xk < +∞. The aim of this article is to study how large the summable subsequence could be. We define the upper density of summable subsequences of x as the supremum of the upper asymptotic densities over 𝓙x, SUD in brief, and we denote it by D*(x). Similarly, the lower density of summable subsequences of x is defined as the supremum of the lower asymptotic densities over 𝓙x, SLD in brief, and we denote it by D*(x). We study the properties of SUD and SLD, and also give some examples. One of our main results is that the SUD of a non-increasing sequence of positive numbers tending to zero is either 0 or 1. Furthermore, we obtain that for a non-increasing sequence, D*(x) = 1 if and only if $\begin{array}{} \displaystyle \liminf_{k\to\infty}nx_n=0, \end{array}$ which is an analogue of Cauchy condensation test. In particular, we prove that the SUD of the sequence of the reciprocals of all prime numbers is 1 and its SLD is 0. Moreover, we apply the results in this topic to improve some results for distributionally chaotic linear operators.

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A Two-Dimensional Cellular Automaton Crystal with Irrational Density

New Constructions in Cellular Automata ◽

10.1093/oso/9780195137170.003.0007 ◽

2003 ◽

Author(s):

David Griffeath ◽

Dean Hickerson

Keyword(s):

Cellular Automaton ◽

Asymptotic Density ◽

Coarse Grain ◽

Constant Density ◽

Mathematical Framework ◽

Two Dimensional ◽

Empty Cell ◽

Asymptotic Shape ◽

Conway’S Game Of Life ◽

Asymptotic Densities

We solve a problem posed recently by Gravner and Griffeath [4]: to find a finite seed A0 of 1s for a simple {0, l}-valued cellular automaton growth model on Z2 such that the occupied crystal An after n updates spreads with a two-dimensional asymptotic shape and a provably irrational density. Our solution exhibits an initial A0 of 2,392 cells for Conway’s Game Of Life from which An cover nT with asymptotic density (3 - √5/90, where T is the triangle with vertices (0,0), (-1/4,-1/4), and (1/6,0). In “Cellular Automaton Growth on Z2: Theorems, Examples, and Problems” [4], Gravner and Griffeath recently presented a mathematical framework for the study of Cellular Automata (CA) crystal growth and asymptotic shape, focusing on two-dimensional dynamics. For simplicity, at any discrete time n, each lattice site is assumed to be either empty (0) or occupied (1). Occupied sites after n updates grows linearly in each dimension, attaining an asymptotic density p within a limit shape L: . . . n-1 A → p • 1L • (1) . . . This phenomenology is developed rigorously in Gravner and Griffeath [4] for Threshold Growth, a class of monotone solidification automata (in which case p = 1), and for various nonmonotone CA which evolve recursively. The coarse-grain crystal geometry of models which do not fill the lattice completely is captured by their asymptotic density, as precisely formulated in Gravner and Griffeath [4]. It may happen that a varying “hydrodynamic” profile p(x) emerges over the normalized support L of the crystal. The most common scenario, however, would appear to be eq. (1), with some constant density p throughout L. All the asymptotic densities identified by Gravner and Griffeath are rational, corresponding to growth which is either exactly periodic in space and time, or nearly so. For instance, it is shown that Exactly 1 Solidification, in which an empty cell permanently joins the crystal if exactly one of its eight nearest (Moore) neighbors is occupied, fills the plane with density 4/9 starting from a singleton.

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