arXiv subjects

Akane Kawaharada

Publications and source records attributed to Akane Kawaharada.

6 recordsLinked to original sources

arXiv2026

Sufficiency of Unit Coefficients for Binary Orbits in Uniformly Weighted Linear Cellular Automata

This paper investigates the classification of spatio-temporal patterns generated by linear cellular automata with uniform weights (LCA-UW) over the ring ${\mathbb Z} / n {\mathbb Z}$. While these systems are governed by the state size $n$ and a transition coefficient $c$, their combined influence produces a vast array of patterns that are difficult to organize through exhaustive observation. We introduce a binary projection operator $\mathcal{B}$ to focus on the fundamental structural evolution (infinite binary orbits) of these automata. Our main result demonstrates a fundamental reduction principle. For any coefficient $c$ that shares prime factors with $n$, the generated infinite binary orbit eventually coincides with the orbit of an LCA-UW with some reduced state size and a unit coefficient $c=1$. We prove that for a fixed $n$, there exist exactly $2^m - 1$ distinct types of binary orbits, where $m$ is the number of distinct prime factors of $n$. This theorem effectively collapses the two-dimensional parameter space $(n, c)$ into a one-dimensional search over $n$, providing a streamlined framework for the topological and fractal classification of LCA-UW dynamics.

math.DS↗

arXiv2023

Why it is sufficient to consider only the case where the seed of linear cellular automata is $1$

When using a cellular automaton (CA) as a fractal generator, consider orbits from the single site seed, an initial configuration that gives only a single cell a positive value. In the case of a two-state CA, since the possible states of each cell are $0$ or $1$, the "seed" in the single site seed is uniquely determined to be the state $1$. However, for a CA with three or more states, there are multiple candidates for the seed. For example, for a $3$-state CA, the possible states of each cell are $0$, $1$, and $2$, so the candidates for the seed are $1$ and $2$. For a $4$-state CA, the possible states of each cell are $0$, $1$, $2$, and $3$, so the candidates for the seed are $1$, $2$, and $3$. Thus, as the number of possible states of a CA increases, the number of seed candidates also increases. In this paper, we prove that for linear CAs it is sufficient to consider only the orbit from the single site seed with the seed $1$.

math.DS↗

arXiv2022

$D$-dimensional cellular automata provide Salem's singular function $L_α$ with $α=1/(2D+1)$ and $1/(2^D+1)$

Salem's singular function is strictly increasing, continuous, and has a derivative equal to zero almost everywhere in $[0,1]$; it is also known as de Rham's singular function or Lebesgue's singular function. The parameter of Salem's singular function $L_α$ is $α\in (0, 1)$ and $α\neq 1/2$. Our previous studies have shown that for some cases of which the limit set of spatio-temporal pattern of a cellular automaton (CA) is fractal, Salem's singular function with $α= 1/3$, $1/4$, or $1/5$ is given by projecting the pattern onto the time axis. However, it remained unclear whether there exists a CA that gives Salem's singular function with a parameter $α$ equal to the multiplicative inverse of an integer greater than $5$. In this paper, we construct CAs giving Salem's singular function with $α= 1/(2D+1)$ and $α= 1/(2^D+1)$ for each dimension $D \geq 1$. This implies that there exist CAs that give Salem's function with a parameter $α$ equal to the multiplicative inverse of any integer greater than or equal to $3$. We also present the results of numerical experiments showing that for $D \leq 5$, the functions given by $D$-dimensional linear symmetric $2$-state radius-$1$ CAs other than the above two types cannot be Salem's function with $α= 1/M$ for $M \in {\mathbb Z}_{\geq 3}$. In addition to the square lattice, the triangular and hexagonal lattices can be considered as regular lattices in the two-dimensional plane, and we also discuss functions obtained from CAs on these lattices.

math-ph↗

arXiv2022

Cellular automata that generate symmetrical patterns give singular functions

In this paper, we mainly study linear one-dimensional and two-dimensional elementary cellular automata that generate symmetrical spatio-temporal patterns. For spatio-temporal patterns of cellular automata from the single site seed, we normalize the number of nonzero states of the patterns, take the limits, and give one-variable functions for the limit sets. We can obtain a one-variable function for each limit set and show that the resulting functions are singular functions, which are non-constant, are continuous everywhere, and have a zero derivative almost everywhere. We show that for Rule 90, a one-dimensional elementary cellular automaton (CA), and a two-dimensional elementary CA, the resulting functions are Salem's singular functions. We also discuss two nonlinear elementary CAs, Rule 22, and Rule 126. Although their spatio-temporal patterns are different from that of Rule 90, their resulting functions from the number of nonzero states equal the function of Rule 90.

nlin.CG↗

arXiv2021

Discontinuous Riemann integrable functions emerging from cellular automata

This paper presents discontinuous Riemann integrable functions on the unit interval $[0, 1]$ derived from the dynamics of two-dimensional elementary cellular automata. Based on the self-similarities of their orbits, we write down the numbers of nonzero states in the spatial and spatio-temporal patterns and obtain discontinuous Riemann integrable functions by normalizing the values. We calculate the integrals of the two obtained functions over $[0, 1]$ and demonstrate the relationship between them.

math.DS↗

arXiv2020

Singular function emerging from one-dimensional elementary cellular automaton Rule 150

In this paper, we give a singular function on a unit interval derived from the dynamic of the one-dimensional elementary cellular automaton Rule 150. We describe properties of the resulting function, that is strictly increasing, uniformly continuous, and differentiable almost everywhere, and we show that it is not differentiable at dyadic rational points. We also give functional equations that the function satisfies, and show that the function is the only solution of the functional ones.

math.DS↗