Related papers: Solving the initial value problem for cellular aut…
A new iterative technique is presented for solving of initial value problem for certain classes of multidimensional linear and nonlinear partial differential equations. Proposed iterative scheme does not require any discretization,…
We consider the parity problem in one-dimensional, binary, circular cellular automata: if the initial configuration contains an odd number of 1s, the lattice should converge to all 1s; otherwise, it should converge to all 0s. It is easy to…
We consider random boolean cellular automata on the integer lattice, i.e., the cells are identified with the integers from 1 to $N$. The behaviour of the automaton is mainly determined by the support of the random variable that selects one…
Self-organizing complex systems can be modeled using cellular automaton models. However, the parametrization of these models is crucial and significantly determines the resulting structural pattern. In this research, we introduce and…
In this paper we use a Variational Quantum Algorithm to solve Initial Value Problems with the Implicit Crank-Nicolson and the Method of Lines (MoL) evolution schemes. The unknown functions use a spectral decomposition with the Fourier…
Determining properties of an arbitrary binary sequence is a challenging task if only local processing is allowed. Among these properties, the determination of the parity of 1s by distributed consensus has been a recurring endeavour in the…
In a recent paper [arXiv:1506.06649 [nlin.CG]], we presented an example of a 3-state cellular automaton which exhibits behaviour analogous to degenerate hyperbolicity often observed in finite-dimensional dynamical systems. We also…
The initial value problem is introduced after a thorough review of the essential geometry. The initial value equations are put into elliptic form using both conformal transformations and a treatment of the extrinsic curvature introduced…
We show that a behaviour analogous to degenerate hyperbolicity can occur in nearest-neighbour cellular automata (CA) with three states. We construct a 3-state rule by "lifting" elementary CA rule 140. Such "lifted" rule is equivalent to…
A Cellular Automata (CA) rule is presented that can generate "loop patterns" in a 2D grid under fixed boundary conditions. A loop is a cyclically closed path represented by one-cells enclosed by zero-cells. A loop pattern can contain…
We investigate number conserving cellular automata with up to five inputs and two states with the goal of comparing their dynamics with diffusion. For this purpose, we introduce the concept of decompression ratio describing expansion of…
We study initial value problems having dynamics ruled by discontinuous ordinary differential equations with the property of possessing a unique solution. We identify a precise class of such systems that we call solvable intitial value…
Given a finite set of local constraints, we seek a cellular automaton (i.e., a local and uniform algorithm) that self-stabilises on the configurations that satisfy these constraints. More precisely, starting from a finite perturbation of a…
We consider the problem of finding response curves for a class of binary two-dimensional cellular automata with $L$-shaped neighbourhood. We show that the dependence of the density of ones after an arbitrary number of iterations, on the…
In this paper, we propose a new approach for building cellular automata to solve real-world segmentation problems. We design and train a cellular automaton that can successfully segment high-resolution images. We consider a colony that…
In the computational-mechanics structural analysis of one-dimensional cellular automata the following automata-theoretic analogue of the \emph{change-point problem} from time series analysis arises: \emph{Given a string $\sigma$ and a…
Like many numerical methods, solvers for initial value problems (IVPs) on ordinary differential equations estimate an analytically intractable quantity, using the results of tractable computations as inputs. This structure is closely…
In the standard formulation of the occupancy problem one considers the distribution of r balls in n cells, with each ball assigned independently to a given cell with probability 1/n. Although closed form expressions can be given for the…
Motivated by an attempt to develop a method for solving initial value problems in a class of one dimensional periodic cellular automata (CA) associated with crystal bases and soliton equations, we consider a generalization of a simple…
If a cellular automaton (CA) is started with a single ON cell, how many cells will be ON after n generations? For certain "odd-rule" CAs, including Rule 150, Rule 614, and Fredkin's Replicator, the answer can be found by using the…