The main theme of this paper is that under wide conditions the product of a large number of square non-negative matrices is close to a positive matrix of rank 1. (A non-negative matrix is one whose elements are non-negative real numbers.)

The model is closely connected with a model by Lamb and Scully (10). Atoms described as two-level systems, initially in an incoherent superposition of the two levels, interact successively during a time T with an electromagnetic field of which only one mode is taken into consideration. In the limit as infinitely many atoms have interacted, it is shown that the field either approaches a thermal distribution or is excited to arbitrarily high Photon numbers according to whether or not the lower level of the atoms is initially more probable than the upper level. It is also shown that in any case the correlations between pure Photon number states converge to 0. If the atoms are initially in the upper level it is proved that the Photon number grows roughly as the square root of the number of atoms that have interacted. Throughout the discussion number-theoretical properties of T play a disturbing role. The last mentioned result in fact depends on a sharp (but arbitrary) value for T and is therefore disqualified for physical interpretation.

The aim of the present note is to point to a strange fact, through an example, in connexion with the asymptotic distribution of sums of a random number of independent and identically distributed (i.i.d.) random variables. We shall formulate our conclusion after the introduction of some notations.

A previous application of the theory, to the nonlinear boundary-value problem for steady flows of a viscous fluid in a bounded domain, is first retraced in order to verify a general theorem concerning the indices of multiple solutions. Then, in §5, the bearing of the theorem on questions of the bifurcation of steady flows is discussed, and the conclusion is drawn that a transcritical form of bifurcation is virtually universal in practice. In §6 it is proved that a flow represented by a solution with index i = – 1 is necessarily unstable, and hence it appears that lack of uniqueness generally implies the existence of an unstable steady flow.

Global existence, uniqueness and regularity of solutions and continuous dependence of solutions on varied initial data are established for the initial-value problem for the coupled system of equations

This system has the same formal justification as a model for the two-way propagation of (one-dimensional) long waves of small but finite amplitude in an open channel of water of constant depth as other versions of the Boussinesq equations. A feature of the analysis is that bounds on the wave amplitude η are obtained which are valid for all time.

That causality implies the Lorentz group was first proved by Zeeman (1). Later, Barruchi (2) and Teppati(3) proved a generalised version of the theorem by using completely different arguments. In this paper, we give a very simple and elementary geometric proof of the same theorem.

An incompressible magnetohydrodynamic flow, bounded below by a semi-infinite solid wall insulator with a plane surface, is weakly perturbed by two-dimensional (i.e. cylindrical) sources immersed within the fluid. Certain results of Chee-Seng(1) are employed to establish an exact analytic solution for arbitrary source functions. Both direct emission and reflected disturbance fields are hyperliptic, i.e. partly hyperbolic and partly elliptic. The reflected hyperbolic mode is propagated as a Riemann invariant of a modified incident field and its Hilbert transform. This propagation occurs along reflected characteristics whose (acute) inclination with the interface is generally unequal to that of the incident characteristics, but which are, instead, parallel to those non-incident characteristic rays emanating away from the interface.

We consider in this note two problems which arise naturally in Smale's paper (3).

First of all we show that it is easy to extend Smale's results on the dynamics of LCR circuits with non-linear elements to circuits containing triode valves. It should be noted that transistors and other n-ports (1) can also be easily accommodated within this theory with suitable differentiability assumptions. We then further extend the theory to include elements whose voltage depends explicitly on time.

The initial value problem for the equation

has been studied recently as a model for long waves in nonlinear dispersive systems. Benjamin, Bona and Mahony (2) introduced this equation as an alternative to the KdV equation of Korteweg-de Vries. Hence, it is referred to as the BBM equation. They studied solutions u(x, t) of the BBM equation for t ≥ 0 and x∈(− ∞, ∞), satisfying u(x, 0) = g(x). Bona and Bryant(1) carried through the study of the BBM equation for t ≥ 0 and x ∈ [0, ∞), satisfying u(x, 0) = g(x) and u(0, t) = h(t). The aim of this paper is to study the equation

where At and Bt are mappings defined on subsets of Banach spaces, especially when At is a second order elliptic operator and B is a differential operator of lower order, defined on an unbounded subset Ω of .

The paper continues earlier work on the generalization in n-dimensional space of the differential equations governing the propagation of electromagnetic waves in inhomogeneous anisotropic ionospheric plasma. Here there are investigated certain properties of the equations and their solutions that are independent of the dimension n of the space in which propagation takes place. The isotropic case is discussed, together with a special anisotropic model for which the various planes of circular polarization are geometrically related to each other.

A line source whose strength varies sinusoidally with time and also with the co-ordinate measured along its length is situated parallel to the shoreline of a beach of angle ¼π0. Both long-and short-wave solutions are found. It is shown that for certain positions of the source, long waves are not radiated to infinity, while in the short-wave regime, the solutions take the form of edge-waves, with resonances occurring at certain wavenumbers. Computations of the free-surface contours are presented for a range of wavenumbers.