We study the deformation of a long slender drop of viscosity ζμ suspended in another liquid of viscosity μ. Interfacial tension causes the drop to become spherical when there is no fluid motion. When the flow is weak the drop is slightly perturbed, and this case was studied by Taylor (7). Computing the flow around an exact sphere, he used the resulting imbalance in the normal stresses to predict the perturbed drop shape. When the drop is in viscid or slightly viscous (ζ ≪ 1), and when the flow is stronger, the drop becomes long and slender. Previous slender-body analyses (Taylor (8) Buckmaster (2, 3), Acrivos & Lo(1) and Hinch & Acrivos(5)) predict pointed ends, but break down in the neighbourhood of these ends. Here we adopt an approach similar to that of Taylor (7). The zero Reynolds number flow around a spindle-shaped drop with pointed ends is computed exactly. Interfacial tension does not quite balance the hydrodynamic stress, and the resulting imbalance in the normal stresses is used to predict a more accurate representation of the drop shape.

Earlier study by one of the authors on the convergence of non-homogeneous stochastic chains is continued here. Several results more general than those considered earlier relating convergence of such chains to ergodicity of certain smaller chains are presented.

Let (M, g) be an arbitrary space-time of dimension ≥ 2 and let d = d(g): M × M → ℝ ∪ {∞} (where d(p, q) = 0 for q∉J+(p)) denote the Lorentzian distance function of (M, g). Also let C(M, g) denote the space of Lorentzian metrics for M globally con-formal to g. Here g1 is said to be globally conformal to g if there exists a smooth function Ω: M → (0, ∞) such that g1 = Ωg.

When a system of differential equations is derivable from a variational principle, an outstanding consequence is the connection which ensues between symmetries and conservation laws. The idea behind the present work is to use this consequence as a new characterization of the variational principle itself. For general systems of equations there is a rather weak form of the connection, as will be demonstrated; but it has highly arbitrary and non-invariant features, and, for instance, there is not a formula associating a particular infinitesimal invariance with a particular constant of the motion. In terms of the modern technical concept which will be employed, the transformation is not natural. However, if the system is derivable from a variational principle, there follows a preferred transformation; and it has certain invariance properties which accord with the concept of naturality.

It was first pointed out by Sommerfeld, around the turn of the century, that certain multi-valued solutions of the wave equation in ℝ3 can be used to deal with the problem of scattering by a wedge, or reflection in a corner. The older literature on this subject is extensive; see ((4), chapter 5) for references up to 1958. The object of this Note is to give an explicit and elementary construction of a forward fundamental solution of the wave equation, of this type, in ℝn+1, where n ≥ 2; for n = 2 this includes Sommer-feld's original result.

The b-completion of a smooth manifold with linear connection was introduced by Schmidt (6). He pointed out that, in a spacetime, an inextensible timelike curve of finite proper length and bounded acceleration has an endpoint on the b-boundary. However, it is not clear in what sense boundedness was meant. Geroch(1) suggested that any acceptable method of completing a spacetime should have this property with boundedness interpreted through the Lorentz norm. This result for the b-completion was given by Rosso(4), based on his earlier paper (3), which, however, contains some arguments that we do not accept. We give another proof of the result and improve it by weakening the condition of boundedness of the acceleration.

Bona and Smith (6) have suggested that the coupled system of equations

has the same formal justification as other Boussinesq-type models for the two-way propagation of one-dimensional water waves of small but finite amplitude in a channel with a flat bottom. The variables u and η represent the velocity and elevation of the free surface, respectively. Using the energy invariant

they show that for a restricted, but nevertheless physically relevant, class of initial data, the system (1·1) has solutions which exist for all time, and that in such circumstances the wave height is bounded solely in terms of the initial data.

The author is grateful to Les Lander for pointing out an error in the stability section of (1). In fact Theorems 5 and 7 are incorrect. Recently Arkeryd proved a stability theorem for the infinite-dimensional case in the context of the imperfect bifurcation theory of Golubitsky and Schaeffer(3). In his result finitely many derivatives are controlled, the number depending on the codimension of the singularity unfolded. In this note we shall present a stability theorem involving the determinacy of the singularity. The context is the parameter-free potential case, that is, catastrophe theory. The proof is without recourse to the finite-dimensional results, and the theorem concludes an account of a part of singularity theory in Banach spaces, in which the author has tried to use as little as possible of the finite-dimensional theory (1, 2).

The solution set for the (sourceless) Yang-Mills equations on a spacetime with compact Cauchy surface is a smooth manifold (i.e. the equations are linearization stable) except at solutions that are symmetric. At such symmetric solutions, the structure is described by a homogeneous quadratic form. The degeneracy space for this form is tangent to a manifold of symmetric solutions. Symmetry breaking occurs for perturbations in the nondegenerate directions of the quadratic form. The terms ‘symmetry’ and ‘stability’ in the present work are compared to these terms as used elsewhere in the literature.

In the spirit of generalized coordinates in Lagrangian mechanics, the finite deformation of a material element is considered to be specified by arbitrary geometric variables, and the associated generalized stresses are generated by work conjugacy. The comparative quantitative influences of such coordinate choices are investigated for the main state variables in anisotropic thermoelasticity, namely the moduli, compliances, thermal coefficients of expansion, pressure coefficients, and specific heats. Transformation formulae are obtained to connect the respective values of the state variables for different choices. Special attention is given to the identification of coordinate-invariant magnitudes and relationships. The entire analysis is carried farther for coordinates in a class of tensor measures of finite strain; such measures are used widely in current theories of the mechanical response of solid materials, but apparently not yet in regard to thermal phenomena.