For isotropic incompressible hyperelastic materials the single function characterizing generalized shear deformations or as they are sometimes called anti-plane strain deformations must satisfy two distinct partial differential equations. Knowles [5] has recently given a necessary and sufficient condition for the strain–energy function of the material which if satisfied ensures that the two equations have consistent solutions. It is shown here for the general material not satisfying Knowles' criterion that the only possible consistent solution of the two partial differential equations are those which are already known to exist for all strain–energy functions. More general types of generalized shear deformations for such meterials are shown to exist only for special or restricted form ot the strain-energy function. In derving these results we also obtain an alternative derivation of Knowles' criterion.

Pontryagin's maximum principle is derived by elementary mathematical techniques. The conditions on the functions which enter are generally somewhat more stringent than in Pontryagin's derivation, but one (practically very awkward) condition of Pontryagin can be relaxed: continuity in the time variable can be replaced by a much weaker condition.

In this note we present a variable order continuation method for the solution of nonlinear equations when only a poor estimate of a solution is known. The method changes continuously from one which improves the global convergence characteristics to one which attains rapid convergence to a solution and proves to be more efficient than methods previously presented in [2].

A monoenergetic point-source solution of the steady-state cosmic-ray equation of transport for cosmic-rays in the interplanetary region in which monoenergetic particles are released isotropically and continuously from a fixed heliocentric position is derived by Lie Theory. A spherically-symmetric model of the propagation region is assumed incorporating anisotropic diffusion with a diffusion tensor symmetric about the radial direction, and the solar wind velocity is radial and of constant speed V. Because of the point release the solution is non-spherically-symmetric.

If a thin elastic circular plate B: ∣z∣ < 1 is clamped (simply supported, respectively) along its edge ∣z∣ = 1, its deflection at z ∈ B under a point load at ζ ∈ B, measured positively in the direction of the gravitational pull, is the biharmonic Green's function β(z, ζ) of the clamped plate (γ(z, ζ) of the simply supported plate, respectively). We ask: how do β(z, ζ) and γ(z, ζ) compare with the corresponding deflections β0(z, ζ) and γ0(z, ζ) of the punctured circular plate B0: 0 < ∣ z ∣ < 1 that is “clamped” or “simply supported”, respectively, also at the origin? We shall show that γ(z, ζ) is not affected by the puncturing, that is, γ(·, ζ) = γ0(·, ζ), whereas β(·, ζ) is:

on B0 × B0. Moreover, while β((·, ζ) is of constant sign, β0(·, ζ) is not. This gives a simple counterexmple to the conjecture of Hadamard [6] that the deflection of a clampled thin elastic plate be always of constant sign:

The biharmonic Gree's function of a clampled concentric circular annulus is not of constant sign if the radius of the inner boundary circle is sufficiently small.

Earlier counterexamples to Hadamard's conjecture were given by Duffin [2], Garabedian [4], Loewner [7], and Szegö [9]. Interest in the problem was recently revived by the invited address of Duffin [3] before the Annual Meeting of the American Mathematical Society in 1974. The drawback of the counterexample we will present is that, whereas the classical examples are all simply connected, ours is not. In the simplicity of the proof, however, there is no comparison.

The problem of determining a square integrable function from both its modulus and the modulus of its Fourier transform is studied. It is shown that for a large class of real functions the function is uniquely determined from this data. We also construct fundamental subsets of functions that are not uniquely determined. In quantum mechanical language, bound states are uniquely determined by their position and momentum distributions but, in general, scattering states are not.

Two problems involving the “best” solution X for a matrix equation V×V* = M are discussed, together with methods for their solution, and a generalization of one of the methods beyond matrix equations.

The irrotational flow of an incompressible, inviscid fluid over a spiliway is considered. The reciprocal ε of the Froude number is taken to be small and the method of matched asymptotic expansions is applied. The bed of the spillway is horizontal far upstream and makes an angle α with the horizontal far downstream. The inner expansion is valid upstream and over the spillway, but is invalid far downstream. The outer expansion which is valid downstream fails to satisfy the upstream conditions. Unknown constants in the outer expansion are determined by the matching and composite expansions obtained.

We consider a nonlinear singular perturbation problem on a semi-infinite interval, that is a generalization of the well-known Lagerstrom model equation intended to model low Reynolds number flow. By applying a Green's function method and the contraction mapping principle, we are able to obtain existence, uniqueness and asymptoticity results for this problem.

The effects on adsorption of the geometry of the solid may be studied through calculations based on a (distance)−ε (ε> 3) intermolecular potential. This paper establishes the result that the potential due to an infinitely long polygonal homogeneous solid prism, at position r in the plane of its right section, is – . Here ρi = ∣ r − ri ∣, where the ri are the position vectors of the n vertices of the polygon, and θij are the angles r − ri makes with the two sides of the polygon which meet at vertex ri. The g's are exact functions of θij. They are, in general, integrals of associated Legendre functions, but they are elementary for ε an even integer. A similar result holds for the potential within an infinitely long polygonal prismatic cavity. The analysis involves a systematic superposition schema and the concept of a supplementary potential with datum within the solid at infinity. The cases ε = 6 and ε = 4 are treated in detail and illustrative solutions given for the following configurations: semi-infinite laminae, deep rectangular cracks, square prisms, square prismatic cavities and regular n-gonal prismatic cavities.

The authors have noticed an oversight in Section 3.1 of this paper. To correct this error it is necessary to assume a uniform differentiability condition on G(x, h). This is required, for example, to imply δ on line 4 of p. 178 can be chosen independent of h. For brevity we note that this oversight has been corrected in [1], which is available from the authors. Also, the results of subsequent sections are unaffected.