A local description of space and time in which translations are included in the group of gauge transformations is studied using the formalism of fibre bundles. It is shown that the flat Minkowski space–time may be obtained from a non-flat connection in a de Sitter structured fibre bundle by choosing at least two different cross-sections. The interaction terms in the covariant derivative of a Dirac wave function that correspond to translations may be interpreted as the mass term of the Dirac equation, and then the two cross-sections (gauges) correspond to the description of a fermion and antifermion respectively.

Increase in dimensionality of the signal space for a fixed bandwidth leads to exponential growth in the number of different signals which must be encoded. In this paper we determine the best subspace of orthogonal functions which can be used to minimise the worst ratio of peak power to RMS power. A mathematical formulation of this problem has been made and it has been found that the Fourier basis satisfies the required constraints of optimality in terms of form factor (peak/RMS ratio).

Sufficient conditions for the controllability of nonlinear neutral Volterra integrodifferential systems with implicit derivative are established. The results are a generalisation of the previous results, through the notions of condensing map and measure of noncompactness of a set.

Necessary and sufficient conditions for optimality in the control of linear differential systems ẋ = Ax + Bu with Stieltjes boundary conditions , where ν is an r × n matrix valued measure of bounded variation, are obtained, Feedback-like control is given in the case of quadratic performance.

The resolution of many problems in probability depends on being able to provide sufficiently good upper or lower bounds to certain moments of distributions. A striking example from the literature of a result that can offer such bounds was given by Pó1ya over sixty years ago as the following theorem (see [7, Vol. II, p. 144] and [7, Vol. I, p. 94]).

By noting that it is possible to interchange the roles of the solution vector x and the vector of Lagrange multipliers λ in the restricted least squares problem we are able to give a very efficient implementation of Clark's subset selection algorithm. Numerical results are presented for several selection heuristics.

The oscillatory squeeze film problem is solved for the simple fluid in the sense of Noll. It is shown that dynamic properties of polymeric liquids can be measured on the plastometer in the oscillatory mode. This should be useful to food and other technologists who have to deal with awkward, highly viscous materials.

A new version of Jensen's inequality is established for probability distributions on the non-negative real numbers which are characterized by moments higher than the first. We deduce some new sharp bounds for Laplace-Stieltjes transforms of such distribution functions.

In the absence of surface tension, the problem of determining a travelling surface pressure distribution that displaces a portion of the free surface in a prescribed manner has been solved by several authors, and this “planing-surface” problem is reasonably well understood. The effect of inclusion of surface tension is to change, in a dramatic way, the singularity in the integral equation that describes the problem. It is now necessary in general to allow for isolated impulsive pressure, as well as a smooth distribution over the wetted length. Such pressure points generate jump discontinuities in free-surface slope. Numerical results are obtained here for a class of problems in which there is a single impulse located at the leading edge of the planing surface and detachment with continuous slope at the trailing edge. These results do not appear to approach the classical results in the limit as the surface tension approaches zero, a paradox that is resolved in Part II, which follows.

A simple eigenvalue and a corresponding wavefunction of a Schrödinger operator is initially approximated by the Galerkin method and by the iterated Galerkin method of Sloan. The initial approximation is iteratively refined by employing three schemes: the Rayleigh-Schrödinger scheme, the fixed point scheme and a modification of the fixed point scheme. Under suitable conditions, convergence of these schemes is established by considering error bounds. Numerical results indicate that the modified fixed point scheme along with Sloan's method performs better than the others.

Sufficient conditions are obtained for the existence of a globally attracting positive periodic solution of the mutualism model

where ri, Ki, αi ∈ C(R, R+) and αi > Ki, i = 1, 2, τi, σi ∈ C(R, R+), i = 1, 2, and ri, Ki, αi, τi, σi (i = 1, 2) and functions of period ω > 0.

In this paper we propose an easy-to-implement algorithm for solving general nonlinear optimization problems with nonlinear equality constraints. A nonmonotonic trust region strategy is suggested which does not require the merit function to reduce its value in every iteration. In order to deal with large problems, a reduced Hessian is used to replace a full Hessian matrix. To avoid solving quadratic trust region subproblems exactly which usually takes substantial computation, we only require an approximate solution which requires less computation. The calculation of correction steps, necessary from a theoretical view point to overcome the Maratos effect but which often brings in negative results in practice, is avoided in most cases by setting a criterion to judge its necessity. Global convergence and a local superlinear rate are then proved. This algorithm has a good performance.

A quasi-Newton method (QNM) in infinite-dimensional spaces for identifying parameters involved in distributed parameter systems is presented in this paper. Next, the linear convergence of a sequence generated by the QNM algorithm is also proved. We apply the QNM algorithm to an identification problem for a nonlinear parabolic partial differential equation to illustrate the efficiency of the QNM algorithm.

Assuming a travelling oscillating pressure source model, this paper sets out to investigate the observation of surface gravity waves generated by a cyclone moving with constant speed v. It is shown that when the source frequency is near the critical resonant value g/4ν, large amplitude waves may be generated. There is some agreement with observations of waves from cyclone Pam of February, 1974.

A uniform approximation to the description of a linear oscillator's slow resonant transition is calculated. If the time scale of the transition is ɛ−1, the approximation contains explicitly the 0(1) and 0(ɛ½) terms, and fixes a uniform 0(ɛ) error bound.

We answer a few questions raised by S. Fitzpatrick concerning the representation of maximal monotone operators by convex functions. We also examine some other questions concerning this representation and other ones which have recently emerged.

An improved model of water cresting towards horizontal wells is presented, using a hodograph solution to the lateral edge drive model with a constant potential boundary at a finite outer radius. In this model, the water crest tends to a horizontal asymptote far from the well, correcting previous approaches which led to the unphysical result of a crest which never levels off but, rather, tends to a parabolic curve. The hodograph solution yields the shape of the free water-oil interface. It also yields integral representations for the lengths of boundary segments, and these have enabled the derivation of an explicit expression for the critical rate in terms of the outer radius. The critical rates calculated using the improved model do not differ significantly from those calculated using previous approaches. The main advantage of the model, therefore, is not a correction to the quantitative predictions of the critical rate, but the removal of physical inconsistencies in the underlying theory.

A hierarchy of bilinear Lotka-Volterra equations with a unified structure is proposed. The bilinear Bäcklund transformation for this hierarchy and the corresponding canonical Lax pair are obtained. Furthermore, the nonlinear superposition formula is proved rigorously.