Simulation of function Y = f(X) by means of random choices of increments deltaX and deltaY (Method of Expectancies)

Abstract

In many mathematical models of growth phenomena, an elemental area ¢A is
added to an object growing in the plane; in doing so, the perimeter of the object changes
with the area. If ¢A is an elemental area (a square of sides equal to unity), it turns out
that the changes of perimeter, ¢P; may have only …ve possible values: ¢P = 4; 2; 0;¡2; -4;
depending upon the place where ¢A is added to the cluster. Thus, the function relating
the area and the perimeter, A = f(P), may be predicted if the probabilities of the di¤erent
changes of perimeter are known (or measured). During the aggregation of the n-th particle,
the area and the perimeter will be
An+1 = An + ¢A and Pn+1 = Pn + ¢P
respectively. We will herein present the method used with success in growth phenomena but
in a more general fashion. We assume that we try to generate any function, Y = f(X), by
means of any (…nite) number of increments ¢X and ¢Y chosen at random from a given set
of possibilities for each of them. Thus, the purpose of this paper is the study of the algorithm
Yn+1 = Yn + ¢Y and Xn+1 = Xn + ¢X
at the n-th step of the growth of the function.