Step changes in current through either grounded or ungrounded wires lying on the surface of a uniformly conducting half-space produce image current sources within the surface the conductor. This image current is effectively the only source term for initial changes in ∂1 Bz, Ex and Ey. The general steady state electric and magnetic field components resulting from steady currents flowing through either grounded or ungrounded wires of finite length lying on the surface of a uniform half- space are derived. Then the operators mapping these steady fields into the early values of ∂t Bz, Ex and Ey on or above the conducting half-plane resulting from instantaneously stopping the current flow through the wires are derived.

The problem of principal component analysis of a symmetric matrix (finding a p-dimensional eigenspace associated with the largest p eigenvalues) can be viewed as a smooth optimization problem on a homogeneous space. A solution in terms of the limiting value of a continuous-time dynamical system is presented. A discretization of the dynamical system is proposed that exploits the geometry of the homogeneous space. The relationship between the proposed algorithm and classical methods are investigated.

This paper deals with the existence, uniqueness and qualitative properties of nonnegative and nontrivial solutions of a spatially heterogeneous Lotka-Volterra competition model with nonlinear diffusion. We give conditions in terms of the coefficients involved in the setting of the problem which assure the existence of nonnegative solutions as well as the uniqueness of a positive solution. In order to obtain these results we employ monotonicity methods, singular spectral theory and a fixed point index.

We consider in this article an evolutionary monotone follower problem in [0,1]. State processes under consideration are controlled diffusion processes , solutions of dyx(t) = g(yx(t), t)dt + σu(yx(t), t) dwt + dυt with yx(0) = x ∈[0, 1], where the control processes υt are increasing, positive, and adapted. The cost functional is of integral type, with certain explicit cost of control action including the cost of jumps. We shall present some analytic results of the value function, mainly its characterisation, by standard dynamic programming arguments.

We study the structure of solutions of an initial value problem arising in the study of steadily rotating spiral waves in the kinematic theory of excitable media. In particular, we prove that under certain conditions there is a unique global positive monotone increasing solution.

For Gauss–Turán quadrature formulae with an even weight function on the interval [−1, 1] and functions analytic in regions of the complex plane which contain in their interiors a circle of radius greater than I, the error term is investigated. In some particular cases we prove that the error decreases monotonically to zero. Also, for certain more general cases, we illustrate how to check numerically if this property holds. Some ℓ2-error estimates are considered.

Brockett has studied the isospectral flow Ḣ = [H, [H, N]], with [A, B] = AB ∔ BA, on spaces of real symmetric matrices. The flow diagonalises real symmetric matrices and can be used to solve linear programming problems with compact convex constraints. We show that the flow converges exponentially fast to the optimal solution of the programming problem and we give explicit estimates for the time needed by the flow to approach an ε-neighbourhood of the optimum. An interior point algorithm for the standard simplex is analysed in detail and a comparison is made with a continuous time version of Karmarkar algorithm.

If the terms of a series behave like n−k where k is an exactly known constant, a formula using two terms transforms the series into a series of terms like n−k −2 provided k ≠ 1. The multiple use of this transformation is demonstrated in summing three series.

Differentiation of the well-known addition theorem for Legendre polynomials produces results for sums over order m of products of various derivatives of associated Legendre functions. The same method is applied to the corresponding addition theorems for vector and tensor spherical harmonics. Results are also given for Chebyshev polynomials of the second kind, corresponding to ‘spin-weighted’ associated Legendre functions, as used in studies of distributions of rotations.

The main ideas of Hopf bifurcation theory and its relevance to the development of periodic motions of an autonomous system depending on a parameter are presented, and an algorithm for the computation of the orbits is described. It is then shown that a model system for the motion of a wheelset can be cast in the form amenable to Hopf bifurcation theory. Numerical results for the period and amplitudes of the lateral and yaw motions are obtained in terms of the forward speed of the wheelset, and the wheel-rail profile parameters.

It is found that the period of oscillation decreases while the lateral and yaw motion amplitudes increase as the forward speed increases, for any given rail and wheel profile. While the effect of wheel curvature on the lateral motions seems to be non-existent, its effect on the yaw motion amplitude and the period is to increase them very slightly as the wheel profile changes from a conical to a curved profile. On the other hand, the effect of rail curvature on the lateral amplitude, for instance, is significant; the larger the curvature the smaller the amplitude for a given forward speed.

A ratio game approach to the generalized fractional programming problem is presented and duality relations established. This approach suggests certain solution procedures for solving fractional programs involving several ratios in the objective function.

We analyse the performance of the additive observable proportional navigation guidance system, which is well-suited for low-cost homing missiles with bearings-only measurements. Closed-form solutions are derived for both manoeuvring and non-manoeuvring targets. Guidelines on how to select the navigation constants of the control law are presented. We show that the additive observable proportional navigation guidance system can cover a larger capture area than can a conventional proportional navigation system.

The generalized diffusion equation with a nonlinear source term which encompasses the Fisher, Newell-Whitehead and Fitzhugh-Nagumo equations as particular forms and appears in a wide variety of physical and engineering applications has been analysed for its generalized symmetries (isovectors) via the isovector approach. This yields a new and exact solution to the generalized diffusion equation. Further applications of group theoretic techniques on the travelling wave reductions of the Fisher, Newell-Whitehead and Fitzhugh-Nagumo equations result in integrability conditions and Lie vector fields for these equations. The Lie group of transformations obtained from the exponential vector fields reduces these equations in generalized form to a standard second-order differential equation of nonlinear type, which for particular cases become the Weierstrass and Jacobi elliptic equations. A particular solution to the generalized case yields the exact solutions that have been obtained through different techniques. The group-theoretic integrability relations of the Fisher and Newell-Whitehead equations have been cross-checked through Painlevé analysis, which yields a new solution to the Fisher equation in a complex-valued function form.

It is shown how to decompose a three-dimensional field periodic in two Cartesian coordinates into five parts, three of which are identically divergence-free and the other two orthogonal to all divergence-free fields. The three divergence-free parts coincide with the mean, poloidal and toroidal fields of Schmitt and Wahl; the present work, therefore, extends their decomposition from divergence-free fields to fields of arbitrary divergence. For the representation of known and unknown fields, each of the five subspaces is characterised by both a projection and a scalar representation. Use of Fourier components and wave coordinates reduces poloidal fields to the sum of two-dimensional poloidal fields, and toroidal fields to the sum of unidirectional toroidal fields.