The asymptotic properties of solutions of the non-linear eigenvalue problem, associated with the homogeneoud Dirichlet problem for

are investigated. Here f and g are smooth functions of position in a finite plane region with a smooth boundary. The results for the positive solution are well established, but knowledge of other branches of solutions is scarce. Here positive solutions are pieced together across lines partitioning the domain, and variational arguments are framed, as an effective means of locating the lines, so that the composite function is everywhere a solution of *. Heuristic arguments suggest strongly that there is a close relationship between the nodal lines of * and certain classes of weighted geodesic lines defined by the classical variational problem for the functional

which provides an effective basis for computation. Some results are proved but others remain conjectures. Analogous results are proved for the associated ordinary differential equation. The geometry of the solutions is surprisingly restricted when the coefficients are spatially variable. The arguments are extended to a class of reactive, diffusive systems. It is possible to predict the pattern of domains of different outcomes in terms of properties of the surface on which the reactions occur, without a knowledge of the chemical kinetics. The results appear to provide a basis for stringent testing of the postulated role of reactive-diffusive mechanisms in the formation of complex patterns in biological species.

The dual integral equations describing heat flow about a circular Heat Flux Sensor on the surface of a layered medium are derived and discussed, together with the extent to which the Heat Flux Sensor measures the heat flow which would occur in the absence of a Heat Flux Sensor. An asymptotic analysis provides new analytical results supporting those derived previously by numerical methods.

It is suggested that some properties of the general problem of a Heat Flux Sensor on the surface of a multiply-layered medium can be approximated by a lumped-parameter model depending on only four non-dimensional numbers: namely, two non-dimensional linear heat transfer coefficients, and essentially two non-dimensional thermal resistances. Some support for the lumped parameter model is provided.

Line distributions of Stokes flow singularities are used to model the flow around a slender body which is straddling a flat interface between two viscous fluids. Motion of the slender body parallel to the interface and normal to the interface is considered where the axis of symmetry of the slender body is always perpendicular to the undisturbed interface. Asymptotic approximations to the force distributions on the slender body are evaluated and the relative contributions of that part of the slender body in one fluid to the force distribution in the other fluid and of the interface interaction to the force distribution are examined. It is observed that a shielding region exists about the interface which is due to the interaction with that part of the slender body in the other fluid. Finally, for parallel motion, the first order interface deformation is calculated.

Exact wave-height solutions are presented for trapped waves over two new three-parameter depth topographies. Dispersive properties are calculated for both a semi-infinite and a truncated convex exponential profile, as well as for a semi-infinite concave profile. The analysis in all three cases is general in that both horizontal divergence and rotational effects are included. These solutions may be used for either high-frequency edge wave or low-frequency shelf wave studies by taking appropriate limits (f → 0 for edge wave and ε = f2L2/gH ≪ 1 for shelf waves).

Steady potential flow about a thin wing, flying in air above a dynamic water surface, is analysed in the asymptotic limit as the clearance-to-length ratio tends to zero. This leads to a non-linear integral equation for the one-dimensional pressure distribution beneath the wing, which is solved numerically. Results are compared with established “rigid-ground” and “hydrostatic” theories. Short waves lead to complications, including non-uniqueness, in some parameter ranges.

Sufficient conditions are obtained for the existence of a globally asymptotically stable strictly positive (componentwise) almost-periodic solution of a Lotka-Volterra system with almost periodic coefficients.

We solve a minimization problem in liver kinetics posed by Bass, et al., in this journal, (1984), pages 538–562. The problem is to choose the density functions for the location of two enzymes, in order to minimize the concentration of an intermediate form of a substance at the outlet of the liver. This form may be toxic to the rest of the body, but the second enzyme renders it harmless. It seems natural that the second enzyme should be downstream from the first. However, we can show that the minimum problem is sometimes solved by an overlap of the supports of the two density functions. Even more surprising is that, for certain forms of the kinetic functions and high levels of transformation of the first enzymatic reaction, some of the first enzyme should be located downstream from all the second enzyme. This suggests that the first reaction should be relatively slow.

The sums

are shown to approximate J0 (z); the error terms are series in higher order Bessel functions, leading with J2M (z). Similar sums approximate J1(z). These sums may be looked on as extensions of the Jacobi expansions for cos z and sin z in series of Bessel functions. They become numerically useful for M > |z|.

The main object of present paper is to obtain a finite summation of Srivastava's general triple hypergeometric series in terms of Kampé de Fériet's double hypergeometric series. A number of finite sums of Kampé de Fériet's double hypergeometric polynomials in terms of different kinds of single hypergeometric polynomials of higher order, are obtained. Some known results of Manocha and Sharma [9], [10], Munot [11], Pathan [12], Qureshi [15], Qureshi and Pathan [16] and Srivastava [26] are deduced as special cases. A result of Pathan [13, page 316 (1.2)] is also corrected here.