A recurrence relation for the restricted partition functionp(n,m) which denotes the number of partitions of a positive integern such that the product of all the summands in each partition ism is given. The functionp(n,m) is tabulated forn andm in the range 1,2,..., 100.

Summary

A new stability functional is introduced for analyzing the stability and consistency of linear multistep methods. Using it and the general theory of [1] we prove that a linear multistep method of design orderq≧p≧1 which satisfies the weak stability root condition, applied to the differential equationy′ (t)=f (t, y (t)) wheref is Lipschitz continuous in its second argument, will exhibit actual convergence of ordero(h^p−1) ify has a (p−1)th derivativey^(p−1) that is a Riemann integral and ordero(h^p) ify^(p−1) is the integral of a function of bounded variation. This result applies for a functiony taking on values in any real vector space, finite or infinite dimensional.

A terminating algorithm is developed for the problem of finding the point of smallest Euclidean norm in the convex hull of a given finite point set in Euclideann-space, or equivalently for finding an “optimal” hyperplane separating a given point from a given finite point set. Its efficiency and accuracy are investigated, and its extension to the separation of two sets and other convex programming problems described.

In the models discussed in sections 1 – 12, the idea has been that the experimentally recorded electrical activity of neurones, whether fluctuations in membrane potential, spike discharges or some measure of the activity of neural aggregates, is stochastic, and the problem has been to obtain models which generate a stochastic process with properties similar to the observed samples of activity. These models may be abstract, or may contain parameters and variables which correspond to biophysically measurable quantities, and so the models might give insights into possible mechanisms generating the stochastic activity. In this chapter I will consider a different kind of problem: does the stochastic nature of neural activity have any functional implications?