Bruce Calvert

Another K-winners-take-all analog neural network

An analog Hopfield type neural network is given, that identifies the K largest components of a list of d of N real numbers. The neurons are identical, with a tanh characteristic, and the weight matrix is symmetric and fully filled. The list to be processed is a summand of the input currents of the neurons, and the network is started from zero. We provide easily computable restrictions on the parameters. The main emphasis here is on the magnitude of the neuronal gain. A complete mathematical analysis is given. The trajectories are shown to eventually have positive components precisely in the positions given by the K largest elements in the input list.

Performance analysis for a K-winners-take-all analog neural network: basic theory

In a previous work, the authors proposed an analog Hopfield-type neural network that identified the K largest components of a list of real numbers. In this work, we identify computable restrictions on the parameters, in order that the network can repeatedly process lists, one after the other, at a given rate. A complete mathematical analysis gives analytical bounds for the time required in terms of circuit parameters, the length of the lists, and the relative separation of list elements. This allows practical setting of circuit parameters for required clocking times. The emphasis is on high gain functioning of each neuron. Numerical investigations show the accuracy of the theoretical predictions, and study the influence of various parameters on performance.

The Downs-Thomson Effect in a Markov Process

Suppose customers pass through a network of two queues in parallel. A statedependent routing policy gives individuals their quickest journey. The Downs-Thomson effect is any increase in the long-run expected journey time caused by an increase in the service rates. This effect may occur.

Operating points in infinite nonlinear networks approximated by finite networks

Given a nonlinear infinite resistive network, an operating point can be determined by approximating the network by finite networks obtained by shorting together various infinite sets of nodes, and then taking a limit of the nodal potential functions of the finite networks. Initially, by taking a completion of the node set of the infinite network under a metric given by the resistances, limit points are obtained that represent generalized ends, which we call “terminals,” of the infinite network. These terminals can be shorted together to obtain a generalized kind of node, a special case of a 1-node. An operating point will involve Kirchhoff’s current law holding at 1-nodes, and so the flow of current into these terminals is studied. We give existence and bounds for an operating point that also has a nodal potential function, which is continuous at the 1-nodes. The existence is derived from the said approximations.

Unicursal resistive networks

We give some existence results for a resistive network in which the components are neither voltage nor current controlled; that is, they are merely unicursal. In fact we allow coupling. Degree arguments give existence and bounds. We study several ways of avoiding the requirement of eventual passivity. No-gain and passive multiterminal elements are included. The results are extended to infinite networks.

Local and global extrema for functions of several variables

Let p : R 2 → R be a polynomial with a local minimum at its only critical point. This must give a global minimum if the degree of p is 1980 Mathematics subject classification (Amer. Math. Soc.) : 26 B 99, 26 C 99.

Infinite nonlinear resistive networks, after Minty

This paper gives the existence of and bounds for the branch voltages and currents in a given infinite network, which on each branch are related by a maximal monotone resistance function. We give a simpler proof than that of Minty, with smaller bounds, in the finite network case.

Nonexpansive projections onto two-dimensional subspaces of Banach spaces

We show that if a three dimensional normed space X has two linearly independent smooth points e and f such that every two-dimensional subspace containing e or f is the range of a nonexpansive projection then X is isometrically isomorphic to l p (3) for some p , 1 c 0 and l p , 1 p ≤ ∞, and a characterisation of real Hilbert spaces.

Neural networks with an infinite number of cells

Abstract We give existence and uniqueness results for the equations describing the dynamics of some neural networks for which there are infinitely many cells.

Global asymptotic stability for a linear system in a saturated mode

We study an ordinary differential equation which models the behaviour of a class of neural networks, which are similar to the Hopfield networks, and which can give convergence in finite time. For application to optimisation problems, and other applications, we are concerned with global asymptotic stability. This paper gives new results on this topic.