In the May 1983 issue of Scientific American, A.T. Winfree published an article which gave an explanation of sudden cardiac arrest in terms of topology. His topological argument was modified in his 1987 book called “When time breaks down”. In this paper we fill what appears to be a gap in Winfree's topological arguments.

In this paper, we consider the existence of a family of periodic solutions of large amplitude when a pair of eigenvalues of the linear part of a first-order system of ordinary differential equations crosses the imaginary axis. We refer to this problem as a Hopf bifurcation problem at infinity. In our work, the nonlinearities may be discontinuous at the origin, and the proof of existence of periodic solutions is arrived at through the corresponding system of integral equations. The applicability of the result is demonstrated by the study of the dynamics of a train truck wheelset system.

The authors consider in this paper the inverse problem of finding a pair of functions (u, p) such that

where F, f, E, s, αi, βi, γi, gi, i = 1, 2, are given functions.

The existence and uniqueness of a smooth global solution pair (u, p) which depends continuously upon the data are demonstrated under certain assumptions on the data.

The controllable set of a controlled ordinary differential dynamic system to a given set is defined. Under certain reasonable conditions, the controllable set is characterised by a level set of the unique viscosity solution to some Hamilton-Jacobi-Bellman equation. The result is used to determine the asymptotic stable set of nonlinear autonomous differential equations.

Uncertainty principles like Heisenberg's assert an inequality obeyed by some measure of joint uncertainty associated with a function and its Fourier transform. The more groups under which that measure is invariant, the more that measure represents an intrinsic property of the underlying object represented by the given function. The Fourier transform is imbedded in a continuous group of operators, the fractional Fourier transforms, but the Heisenberg measure of overall spread turns out not to be invariant under that group. A new family is developed of measures that are invariant under the group of fractional Fourier transforms and that obey associated uncertainty principles. The first member corresponds to Heisenberg's measure but is generally smaller than his although equal to it in special cases.

This paper considers discrete multivariate processes with time-dependent rational spectral density matrices and gives a solution to the spectral factorisation problem. As a result, the corresponding state space representation for the process is obtained. The relationship between multivariate processes with time-dependent rational spectral density matrix functions and multivariate ARMA processes with time-dependent coefficients is discussed. Solutions for the prediction problem are given for the case when only finite data is available and the case when the whole history of the process is known.

A method is presented for the study of fully developed parallel flow of Newtonian viscous fluid in uniform straight ducts of very general cross-section. The method is based upon the concept of contour lines of constant velocity in a typical cross section of the duct, and uses the function which describes the contour lines as an independent variable to derive the integral momentum equation. The resulting ordinary integro-differential equation is, in principle, much easier to solve than the original momentum equation in partial differential equation form. Several illustrative examples of practical interest are included to explain the method of solution. Some of these solutions are compared with available solutions in the literature. All details are explained by graphs and tables. The method has several interesting features. The study has relevance to biomedical engineering research for blood and urinary tract flow.

The scattering and diffraction of harmonic SH waves by an arbitrarily shaped alluvial valley in a layered material is considered. The problem is solved in terms of boundary integral equations which yield a numerical solution.