Time-evolution of the Rule 150 cellular automaton activity from a Fibonacci iteration

可逆基本二阶元胞自动机115的奇特案例

The Curious Case of Reversible Elementary Second Order Cellular Automaton 115

证明了可逆基本二阶元胞自动机规则115在有限初始配置下是周期的，并研究了具有有趣周期函数的有限配置族。

Sumsets of Wythoff Sequences, Fibonacci Representation, and Beyond

Let $α= (1+\sqrt{5})/2$ and define the lower and upper Wythoff sequences by $a_i = \lfloor i α\rfloor$, $b_i = \lfloor i α^2 \rfloor$ for $i \geq 1$. In a recent interesting paper, Kawsumarng et al. proved a number of results about numbers representable as sums of the form $a_i + a_j$, $b_i + b_j$, $a_i + b_j$, and so forth. In this paper I show how to derive all of their results, using one simple idea and existing free software called Walnut. The key idea is that for each of their sumsets, there is a relatively small automaton accepting the Fibonacci representation of the numbers represented.…

The GraftalLace Cellular Automaton

We introduce our GraftalLace Cellular Automaton in short GLCA which is a new one-dimensional cellular automaton on the regular square lattice. It makes a monochromatic infinite directed graph otherwise an octal number triangle or number trapezoid by partly influences the states of the neighbour cells with bit operations. We show new ways to make symmetric fractals like Sierpinski triangle and Pascal triangle modulo 3 and unknown complex patterns. We find reversible rules and show possibilities to represent and extend our automaton in different ways. 2D version of GLCA can be represented as a 3…

Generalized energies and integrable D^{(1)}_n cellular automaton

We introduce generalized energies for a class of U_q(D^{(1)}_n) crystals by using the piecewise linear functions that are building blocks of the combinatorial R. They include the conventional energy in the theory of affine crystals as a special case. It is shown that the generalized energies count the particles and anti-particles in a quadrant of the two dimensional lattice generated by time evolutions of an integrable D^{(1)}_n cellular automaton. Explicit formulas are conjectured for some of them in the form of ultradiscrete tau functions.

Fermionic partition functions for a periodic soliton cellular automaton

Fermionic formulas in combinatorial Bethe ansatz consist of sums of products of q-binomial coefficients. There exist refinements without a sum that are known to yield partition functions of box-ball systems with a prescribed soliton content. In this paper, such a refined fermionic formula is extended to the periodic box-ball system and a q-analogue of the Bethe root counting formula for XXZ chain at $Δ=\infty$.