This paper gives a numerical method for estimating the Hausdorff-Besicovitch dimension where this differs from the fractal (or capacity or box-counting) dimension. The method has been implemented, and numerical results obtained for the set {1/n | n ∈ N} and the Cantor set. Comments about the practical use of the estimation algorithms are made.

Derivative-type ascent formulas are deduced for the kernels of certain half-space Dirichlet problems. These have the character of differentiation formulas for the Bessel functions but involve modifying variables after completing the differentiations. The Laplace equation and the equation of generalized axially-symmetric potential theory (GASPT) are considered in these. The methods employed also permit treating abstract versions of Dirichlet problems.

Gauss' Theorema Egregium produces a partial differential equation which relates the Gaussian curvature K to components of the metric tensor and its derivatives. Well-known partial differential equations (PDEs) such as the Schrödinger equation and the sine-Gordon equation can be derived from Gauss' equation for specific choices of K and coördinate systems. In this paper we consider a class of Bäcklund Transformations which corresponds to coördinate transformations on surfaces with a given Gaussian curvature. These Bäcklund Transformations lead to the construction of solutions to certain classes of non-linear second order PDEs of hyperbolic type by identifying these PDEs as the Gauss equation in some coördinate system. The possibility of solving the Cauchy Problem has also been explored for these classes of equations.

Reaction-diffusion systems are widely used to model the population densities of biological species competing for natural resources in their common habitat. It is often not too difficult to establish positive uniform upper bounds on solution components of such systems, but the task of establishing strictly positive uniform lower bounds (when they exist) can be quite troublesome. Two previously established criteria for the permanence (non-extinction and non-explosion) of solutions of general weakly-coupled competition-diffusion systems with diagonally convex reaction terms are used here as background to develop more easily verifiable and concrete conditions for permanence in various well-known competition diffusion models. These models include multi-component reaction-diffusion systems with (i) the by now classical Lotka-Volterra (logistic) reaction terms, (ii) higher order “logistic” interaction between the species, (iii) logistic-logarithmic reaction terms, (iv) Ayala-Gilpin-Ehrenfeld θ-interaction terms (which are used to model Drosophila competition), (v) logistic-exponential interaction between the species, (vi) Schoener-exploitation and (vii) modified Schoener-interference between the species. In (i) a known condition for permanence (for the ODE-system) is recovered, while in (ii)–(vii) new criteria for permanence are established.

We study models of economic equilibrium with fixed budgets and assuming superlinear connections between consumption and production. Extremal problems and the existence of equilibria are discussed for such models along with some related differential properties. Examples to illustrate the broad nature of the model are discussed.

Analytical, approximate and numerical methods are used to study the Neumann boundary value problem

− uxx + q2u = u2(1 + sin x), for 0 < x < π,

subject to ux(0) = 0, ux(π) = 0,

for q2 ∈ (0,∞). Asymptotic approximations to (1) are found for q2 small and q2 large. In the case where q2 is large u(x) ≈ 3qδ(x − π/2). When q2 = 0 we show that the only possible solution is u ≡ 0. However, there exist non-zero solutions for q2 > 0 as well as the trivial solution u ≡ 0. To O(q4) in the q2 small case u(x) = q2π(π + 2)−1, so that bifurcation occurs about the trivial solution branch u ≡ 0 at the first eigenvalue λ0 = 0 and in the direction of the first eigenfunction ξ0 = constant.

We obtain a bifurcation diagram for (1), which confirms that there exists a positive solution for q2 ∈ (0, 10). Symmetry-breaking bifurcations and blow-up behaviour occur on certain regions of the diagram. We show that all non-trival solutions to the problem must be positive.

The formal outer solution u = q2û appears to satisfy û = û2(1 + sin x), so that û ≡ 0 and û = (1 + sin x)−1 are possible limit solutions. However, in the non-trivial case ûx(0) = −1 and ûx(π) = 1; this means that û does not satisfy the boundary conditions required for a solution of (1). This behaviour usually implies that for q2 large a boundary layer exists near x = 0 (and one near x = π), which corrects the slope. However, we find no evidence for such a solution structure, and only find perturbations in the direction of a delta function about u ≡ 0. We show using the monotone convergence theorem for quadratic forms that the inverse of the operator on the left-hand side of (1) is strongly convergent as q2 → ∞. We show that strong convergence of the operator is sufficient to stop outer-layer behaviour occurring.

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 ∂1Bz, 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 ∂tBz, Ex and Ey on or above the conducting half-plane resulting from instantaneously stopping the current flow through the wires are derived.

The framework for accelerated spectral refinement for a simple eigenvalue developed in Part I of this paper is employed to treat the general case of a cluster of eigenvalues whose total algebraic multiplicity is finite. Numerical examples concerning the largest and the second largest multiple eigenvalues of an integral operator are given.

Multiple integrals in ten or twenty variables are often needed by atomic, molecular and nuclear physicists, because of the large number of degrees of freedom in the quantum systems with which they must deal. In statistics too there is often a need to evaluate integrals with many degrees of freedom. It is in mathematical finance, however, that the most striking examples are seen, with claims of integrals being evaluated during recent years with many hundreds of variables.

By means of piecewise continuous vector functions, which are analogues of the classical Lyapunov functions and via the comparison method, sufficient conditions are found for conditional, stability of the zero solution of a system of impulsive differential-difference equations.

We approximate a linear array of coupled harmonic oscillators as a symmetric circular array of identical masses and springs. The springs are taken to possess mass distributed along their lengths. We give a Lagrangian formulation of the problem of finding the natural frequencies of oscillation for the array. Damping terms are included by means of the Rayleigh dissipation function. A transformation to symmetry coordinates as determined by the group of rotations of the circle uncouples the equations of motion.

A problem of estimation of the critical Mach number for a class of carrying wing profiles with a fixed theoretical angle of attack is considered. The Chaplygin gas model is used to calculate the velocity field of the flow. The original problem is reduced to a special minimax problem. A solution is constructed for an extended class of flows including multivalent ones, hence M* is estimated from above. For a fixed interval [0, β0], β0 ≅ 3π/8, an estimate of M* is given from below.

The free surface flow of a finite depth fluid past a semi-infinite body is considered. The fluid is assumed to have constant vorticity throughout and the free surface is assumed to attach smoothly to the front face of the body. Numerical solutions are found using a boundary integral method in the physical plane and it is shown that solutions exist for all supercritical Froude numbers. The related problem of the cusp-like flow due to a submerged sink in a comer is also considered. Vorticity is included in the flow and it is shown that the behaviour of the solutions is qualitatively the same as that found in the problem described above.

This paper considers the solution of estimation problems based on the maximum likelihood principle when a fixed number of equality constraints are imposed on the parameters of the problem. Consistency and the asymptotic distribution of the parameter estimates are discussed as n → ∞, where n is the number of independent observations, and it is shown that a suitably scaled limiting multiplier vector is known. It is also shown that when this information is available then the good properties of Fisher's method of scoring for the unconstrained case extend to a class of augmented Lagrangian methods for the constrained case. This point is illustrated by means of an example involving the estimation of a mixture density.

Newton's method is applied to an operator that satisfies stronger conditions than those of Kantorovich. Convergence and error estimates are compared in the two situations. As an application, we obtain information on the existence and uniqueness of a solution for differential and integral equations.

Given the data (pi, ti, fi), i = 1,…,m, we consider the existence problem for the best least squares approximation of parameters for the 3-parametric exponential regression model. This problem does not always have a solution. In this paper it is shown that this problem has a solution provided that the data are strongly increasing at the ends.

Mixing rules model how the physical properties of a polymer, such as its relaxation modulus G(t), depend on the distribution w(m) of its molecular weights m. They are of practical importance because, among other things, they allow estimates of the molecular weight distribution (MWD) w(m) of a polymer to be determined from measurements of its physical properties including the relaxation modulus. The two most common mixing rules are “single” and “double” reptation. Various derivations for these rules have been published. In this paper, a conditional probability formulation is given which identifies that the fundamental essence of “double” reptation is the discrete binary nature of the “entanglements”, which are assumed to occur in the corresponding topological model of the underlying polymer dynamics. In addition, various methods for determining the MWD are reviewed, and the computation of linear functionals of the MWD motivated and briefly examined.

An initial value problem is considered for impulsive functional-differential equations. The impulses occur at fixed moments of time. Sufficient conditions are found for Lipschitz stability of the zero solution of these equations. An application in impulsive population dynamics is also discussed.