In a paper which appeared in this journal, Manocha and Sharma [6] obtained some results of Carlitz [4], Halim and Salain [5] and generalized a few of them by using fractional derivatives. The present paper is concerned with some erroneous results of this paper [6]. Many more sums of the product of hypergeometric polynomials are also obtained.

A class of partially structured nonlinear programming problems, containing the capacitated nonlinear minimum cost multicommodity flow problem, is considered. Such problems, although large, can often be solved efficiently and with minimal computational storage by gradient projection methods.

A review of Heckmann and Schücking's formulation of Newtonian cosmology is presented, which permits the discussion of models more general than those possessing both homogeneity and isotropy. In particular it is shown that all homogeneous cosmologies may be uniquely specified by the rate of shear tensor as an arbitrary function of time and specifying arbitrary initial values for expansion, rotation and density. Perturbations of these models are now discussed, with a view to their possible implications for galaxy formation. The Jeans criterion is shown to hold in all these models, even in the presence of viscosity; this generalizes a result of Bonnor which only applied to the isotropic case. Furthermore, Bonnor's analysis is considerably simplified in the present paper. Finally, a WKB-type of approximation procedure is described which appears to be successful in estimating the growth rate of unstable fluctuations.

The calculation of flows in pipe networks and in networks of mine shafts and the calculations of the currents in electrical circuits can be represented as variational problems. There are two approaches: the nodal method and the loop method. There is a variational representation for each of these. This paper describes the relationship between the two representations and in particular shows that the loop formulation is the Wolfe dual of the nodal formulation after the application of Legendre transformations to the variables and to the objective function.

A canonical form of the self-adjoint Matrix Riccati Differential Equation with constant coefficients is obtained in terms of extremal solutions of the self-adjoint Matrix Riccati Algebraic (steady-state) Equations. This form is exploited in order to obtain a convenient explicit solution of the transient problem. Estimates of the convergence rate to the steady state are derived.

We introduce a generalized form of the Hankel transform, and study some of its properties. A partial differential equation associated with the problem of transport of a heavy pollutant (dust) from the ground level sources within the framework of the diffusion theory is treated by this integral transform. The pollutant concentration is expressed in terms of a given flux of dust from the ground surface to the atmosphere. Some special cases are derived.

A nutrient-consumer model involving a distributed delay in material recycling and a discrete delay in growth response has been analysed. Various easily verifiable sets of sufficient conditions for global asymptotic stability of the positive equilibrium solution of the model equations have been obtained and the length of the delay in each case has been estimated.

In Section 3, since relation (6) is valid only for n ≥ 2r, the condition n ≥ r in relation (9) should be replaced by n ≥ 2r. When r ≤ n ≤ 2r − 1, relation (9) still holds but now relations (6) and (7) should be replaced, respectively, by the relations

and

the asterisk again denoting omission of the last column. The proof of (9) for r ≤ n ≤ 2r − 1 is exactly similar to its proof for n ≥ 2r.

A numerical method for calculation of the generalized Chakalov-Popoviciu quadrature formulae of Radau and Lobatto type, using the results given for the generalized Chakalov-Popoviciu quadrature formula, is given. Numerical results are included. As an application we discuss the problem of approximating a function f on the finite interval I = [a, b] by a spline function of degree m and variable defects dv, with n (variable) knots, matching as many of the initial moments of f as possible. An analytic formula for the coefficients in the generalized Chakalov-Popoviciu quadrature formula is given.

A three stage procedure for the analysis and least-cost design of looped water distribution networks is considered in this paper. The first stage detects spanning trees and identifies the true global optimum for the system. The second stage determines hydraulically feasible pipe flows for the network by the numerical solution of a set of non-linear simultaneous equations and shows that these solutions are contained within closed convex polygonal regions in the solution space bounded by singularities resulting from zero flows in individual pipes. Ideal pipe diameters, consistent with the pipe flows and the constant velocity constraint adopted to prevent the system degenerating into a branched network, are selected and costed. It is found that the most favourable optimum is in the vicinity of a vertex in the solution space corresponding to the minimum spanning tree. In the third stage, commercial pipes are specified and the design finalised. Upper bound formulae for the number of spanning trees and hydraulically feasible solutions in a network have also been proposed. The treatment of large networks by a heuristic procedure is described which is shown to result in significant economies compared with designs obtained by non-linear programming.

The diffusion equation is used to model and analyze sorption, a process used in the purification or separation of fluids. For the diffusion inside a spherical porous solid immersed in a limited-volume and well-stirred fluid, Ruthven [5], Crank [3] and, for the analogous flow of heat, Carslaw and Jaeger [2] give an eigenfunction expansion solution to the diffusion equation that provides accurate long-time solutions when only a few terms are used. However, to obtain the same accuracy for short-time solutions the number of eigenfunction terms required increases exponentially. An alternative error function solution of Carman and Haul [1] is accurate for sufficiently short times but not for long times. Although their solution is well quoted [3, 4, 6], Carman and Haul do not provide a derivation in their paper. This paper provides a full derivation of the short-time solution of Carman and Haul that uses only the first term of a negative exponential series in the Laplace domain. It is shown that the accuracy and range of the short-time result is improved by the inclusion of additional terms of the negative exponential series. An analysis of short-time and long-time resultsis presented, together with recommendations as to their use.

It is well known that the trapezoidal rule of quadrature is exact for linear functions on [0, 1], and easy to see that it is exact for functions of the form f = l+g where l is linear and g is odd about ½. Not so well known is an example of a continuous function for which the trapezoidal rule is exact but which does not have this form. In this paper we show that if the trapezoidal rule is exact for f then f has the form above provided it has either absolutely convergent Fourier series or continuous second derivative. We consider one-sided versions in which the approximate integrals are non-negative, and also characterize those sequences arising as the approximate integrals of a function with absolutely convergent Fourier series.

We show that a sequence of generalized eigenfunctions of a one-dimensional linear thermoelastic system with Dirichiet-Dirichlet boundary conditions forms a Riesz basis for the state Hilbert space. This develops a parallel result for the same system with Dirichlet-Neumann or Neumann-Dirichlet boundary conditions.

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.

A model for bushfire spread is proposed, in which radiative heat transfer, species consumption and flammable gas production are taken into account. It is shown that fire propagation in this model does not occur as a one-dimensional travelling wave, except for smouldering combustion of wet bushland. Numerical solutions for the evolution of a line fire are obtained using a diagonally implicit finite difference scheme, and the effects of firebreaks and uncleared combustible debris are studied. An energy theorem is presented for the case of a spreading two-dimensional fire, and numerical results are illustrated.

A train travels from one station to the next along a level track. The journey must be completed within a given time and it is desirable to minimise the energy required to drive the train. It has been shown with an appropriate formulation of the problem that an optimal strategy exists and that this strategy must satisfy a Pontryagin type criterion. In this paper the Pontryagin principle will be used to find the nature of the optimal strategy and this information will then be used to determine the precise optimal strategy.

The influence of thermal buoyancy on neutral wave modes in Poiseuille-Couette flow is considered. We examine the modifications to the asymptotic structure first described by Mureithi, Denier & Stott [16], who demonstrated that neutral wave modes in a strongly thermally stratified boundary layer are localized at the position where the streamwise velocity attains its maximum value. The present work demonstrates that such a flow structure also holds for Poiseuille-Couette flow but that a new flow structure emerges as the position of maximum velocity approaches the wall (and which occurs as the level of shear, present as a consequence of the Couette component of the flow, is increased). The limiting behaviour of these wave modes is discussed thereby allowing us to identify the parameter regime appropriate to the eventual restabilization of the flow at moderate levels of shear.

An integral inequality is established involving a probability density function on the real line and its first two derivatives. This generalizes an earlier result of Sato and Watari. If f denotes the probability density function concerned, the inequality we prove is that

under the conditions β > α 1 and 1/(β+1) < γ ≤ 1.