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This paper reports on patterns exhibiting self-replication with spontaneous, inheritable mutations and exponential genetic drift in Neural Cellular Automata. Despite the models not being explicitly trained for mutation or inheritability,…
The transport and chemical reactions of solutes are modelled as a cellular automaton in which molecules of different species perform a random walk on a regular lattice and react according to a local probabilistic rule. The model describes…
In 2010 Menon and Srinivasan published a conjecture for the statistical structure of solutions $\rho$ to scalar conservation laws with certain Markov initial conditions, proposing a kinetic equation that should suffice to describe…
In this paper we present two interesting properties of stochastic cellular automata that can be helpful in analyzing the dynamical behavior of such automata. The first property allows for calculating cell-wise probability distributions over…
A deautonomized version of the two-dimensional Toda lattice equation is presented. Its ultra-discrete analogue and soliton solutions are also discussed.
We define a new version of $\mathbb A^1$-homology, called cellular $\mathbb A^1$-homology, for smooth schemes over a field that admit an increasing filtration by open subschemes with cohomologically trivial closed strata. We provide several…
A very natural construction of integrable extensions of soliton systems is presented. The extension is made on the level of evolution equations by a modification of the algebra of dynamical fields. The paper is motivated by recent works of…
A folding process is applied to fused a^(1)_r defects to construct defects for the non-simply laced affine Toda field theories of c^(1)_n, d^(2)_n and a^(2)_2n at the classical level. Support for the hypothesis that these defects are…
Acoustic solitons can be obtained by considering the propagation of large amplitude sound waves across a set of Helmholtz resonators. The model proposed by Sugimoto and his coauthors has been validated experimentally in previous works. Here…
We define and study a few properties of a class of random automata networks. While regular finite one-dimensional cellular automata are defined on periodic lattices, these automata networks, called randomized cellular automata, are defined…
Recently Prosen and Mej\'ia-Monasterio (J. Phys. A: Math. Theor. 49 (2016) 185003) obtained exact nonequilibrium steady states of an integrable and reversible cellular automaton driven by some stochastic boundary conditions. In this paper,…
A delay analogue of the box and ball system (BBS) is presented. This new soliton cellular automaton is constructed by the ultra-discretization of the delay discrete Lotka-Volterra equation, which is an integrable delay analogue of the…
For some non-linear field theories which allow for soliton solutions, submodels with infinitely many conservation laws can be defined. Here we investigate the symmetries of the submodels, where in some cases we find a symmetry enhancement…
The basis for most of the ideas mentioned in this paper is the theory of cellular automata. A cellular automata contains a regular grid of cells, with each cell having a pre-defined set of finite states. The initial state is determined at…
A new class of automata networks is defined. Their evolution rules are determined by a probability measure p on the set of all integers Z and an indicator function I_A on the interval [0,1]. It is shown that any cellular automaton rule can…
In this paper, we study a class of cellular automata (CA) called stable cellular automata (SCA) that preserve stability by reflection, modulo-recurrent, and richness. After applying these automata to Sturmian words, we determine some of…
Gauge invariant conservation laws for the linear and angular momenta are studied in a certain 2+1 dimensional first order dynamical model of vortices in superconductivity. In analogy with fluid vortices it is possible to express the linear…
A family of reversible deterministic cellular automata, including the rules 54 and 201 of [Bobenko et al., Commun. Math. Phys. 158, 127 (1993)] as well as their kinetically constrained quantum (unitary) or stochastic deformations, is shown…
We present Mass-Conserving Evolution (MaCE), a general method for implementing mass conservation in Cellular Automata (CA). MaCE is a simple evolution rule that can be easily 'attached' to existing CAs to make them mass-conserving, which…
We Propose A Novel Automaton Model which uses Arithmetic Operations as the Evolving Rules, each cell has the states of the Natural Numbers k = (N), a radius of r = 1/2 and operates on an arbitrary input size. The Automaton reads an…