Related papers: Solving the initial value problem for cellular aut…
Cellular automata (CA) are fully discrete alternatives to partial differential equations (PDE). For PDEs, one often considers the Cauchy problem, or initial value problem: find the solution of the PDE satisfying a given initial condition.…
We consider the problem of computing a response curve for binary cellular automata -- that is, the curve describing the dependence of the density of ones after many iterations of the rule on the initial density of ones. We demonstrate how…
We propose dynamical systems defined on algebra of lattices, which we call `lattice equations'. We give exact general solutions of initial value problems for a class of lattice equations, and evaluate the complexity of the solutions.…
In order to solve an initial value problem by the variational iteration method, a sequence of functions is produced which converges to the solution under some suitable conditions. In the nonlinear case, after a few iterations the terms of…
We study initial value problem for a system consisting of an integer order and distributed-order fractional differential equation describing forced oscillations of a body attached to a free end of a light viscoelastic rod. Explicit form of…
We consider initial value problems of nonlinear dynamical systems, which include physical parameters. A quantity of interest depending on the solution is observed. A discretisation yields the trajectories of the quantity of interest in many…
This paper presents a classification of Cellular Automata rules based on its properties at the nth iteration. Elaborate computer program has been designed to get the nth iteration for arbitrary 1-D or 2-D CA rules. Studies indicate that the…
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…
In this article, we discuss a novel approach to solving number sequence problems, in which sequences of numbers following unstated rules are given, and missing terms are to be inferred. We develop a methodology of decomposing test sequences…
This paper introduces, up to the author's knowledge, for the first time the generalized initial value problem. In this problem, given an ordinary differential equation defined in some set, the initial conditions are mapped to a subset of…
We consider the problem of finding the density of 1's in a configuration obtained by $n$ iterations of a given cellular automaton (CA) rule, starting from disordered initial condition. While this problems is intractable in full generality…
We discuss example of an elementary cellular automaton for which the density of ones decays toward its limiting value as a power of the number of iterations $n$. Using the fact that this rule conserves the number of blocks 10 and that…
We establish essentially optimal bounds on the complexity of initial-value problems in the randomized and quantum settings. For this purpose we define a sequence of new algorithms whose error/cost properties improve from step to step. These…
We search for regularities in the sequences of numbers of preimages for elementary cellular automata. For 46 out of 88 "minimal" rules, we find recognizable patterns, usually in the form of second order recurrence equations with constant…
We formulate the initial value problem for causal variational principles in the continuous setting on a compact metric space. The existence and uniqueness of solutions is analyzed. The results are illustrated by simple examples.
For the string baryon model "triangle" the initial-boundary value problem is stated and solved in general. This problem implies defining a classical motion of the system on the base of given initial position and initial velocities of string…
We study the initial value problem for actions which contain non-trivial functions of integrals of local functions of the dynamical variable. In contrast to many other non-local actions, the classical solution set of these systems is at…
Initial value problems -- a system of ordinary differential equations and corresponding initial conditions -- can be used to describe many physical phenomena including those arise in classical mechanics. We have developed a novel approach…
We consider the problem of computing a response curve for binary cellular automata -- that is, the curve describing the dependence of the density of ones after many iterations of the rule on the initial density of ones. We demonstrate how…
Given a (finite) string of zeros and ones, we report a way to determine if the number of ones is less than, greater than, or equal to a prescribed number by applying two sets of cellular automaton rules in succession. Thus, we solve the…