We write e(x) for e2πix and let ‖x‖ denote the distance of x from the nearest integer. The notation A ≪ B will mean |A| ≤ C|B| where C is a positive constant depending at most on an arbitrary positive number ε, and on an integer k. The letter p always denotes a prime number. The main results of the present paper are as follows.

Consider a slab which is made from a homogeneous and isotropic thermoelastic material and which occupies the region 0 ≤ x ≤ a, where x, y, z are the usual rectangular cartesian coordinates. Suppose that the slab undergoes a motion in which the displacement vector is parallel to the x-axis and the displacement and the temperature are functions of the coordinate x and the time t ( ≥ 0) only. Suppose too that the faces of the slab are clamped, that the face x = 0 is maintained at a constant temperature, and that heat is supplied to unit area of the face x = a at a prescribed rate h(t).

Reid (1974) derived “first approximations” to solutions of the Orr–Sommerfeld equation, which are uniformly valid in a full neighbourhood of a critical point. This paper shows that such approximations may be calculated to higher order, and makes a first step towards placing the theory on a rigorous basis by providing error bounds for the dominant-recessive approximations. These are obtained by generalizing methods discussed by Olver (1974) for second order linear ordinary differential equations.

Let K be a convex body (compact convex set with interior points) in d-dimensional euclidean space Ed, let D(K) denote its diameter, Δ(K) its minimal width, and

the number of lattice points (points of Ed with integer coordinates) in the interior of K. If G0(K) = 0, we call K lattice-point-free; in what follows, K will always be a lattice-point-free convex body.

Existence and uniqueness of classical solutions are established for the dissipative quasigeostrophic equations of geophysical fluid dynamics, using a priori estimates and a Schauder fixed point theorem. The flow is periodic in both horizontal directions and is bounded above and below by rigid flat surfaces. The Reynolds analogy of unit turbulent Prandtl number is assumed. Existence is proved for an arbitrary finite time, if it is further assumed that the surface temperatures vanish. Without this additional assumption existence is guaranteed only for a certain finite time, which is inversely proportional to the norms of the sources and initial conditions.

In this paper the authors obtain a series transformation of concentric circles into concentric rectangles.

We consider a body which occupies the open, bounded, regular region B, whose boundary is ∂B and whose closure is . We denote by da the element of surface area, by dυ the element of volume, and by n the outward unit normal. We suppose the behaviour of the body to be described by the equations of the quasi-static theory of homogeneous and isotropic thermoelasticity. These equations, which are obtained from the equations of the dynamical theory (see, for example, Carlson [1], Chadwick [2] or Boley and Weiner [3]) by omitting the inertial term pű from the right-hand side of the equation of motion (4), are: