The purpose of this note is to extend the result which I established recently (see (3)) on the Diophantine equation

to some further equations of a similar kind. The following theorems will be proved.

An account is given of Eshelby's point-force method for solving elastic inclusion problems, and of his equations relating an in homogeneity to its equivalent inclusion. The introduction of complex variable formalism enables explicit solutions to be found in various two-dimensional cases. Strain energies are calculated. The equilibrium shape of an elliptic inclusion exhibits an interesting feature not previously expected. A fresh analysis of stress magnification effects is developed.

Assuming either the continuum hypothesis or Martin's axiom, we show that in the space βN – N, there are: (a) points which are not P-points, but which are also not limit points of any countable set, and (b) a countable set of points dense in itself such that each of the points is not a limit point of any countable discrete set. Our method is to construct such points in the Stone space of a measure algebra, and then embed that Stone space into βN – N. We also, by a similar use of measures, establish the independence of the existence of selective ultrafilters by showing that there are none in the random real model.

In this note we characterize those sequences kn such that the Hill estimator of the tail index based on the kn upper order statistics of a sample of size n from a Pareto-type distribution is strongly consistent.

A group G of homeomorphisms of a topological space X will be called discontinuous if

(1) the stabilizer of each point of X is finite, and

(2) each point x ∈ X; has a neighbourhood U such that any element of G not in the stabilizer of x maps U outside itself (i.e. if gx ≠ x then U ∩ gU is empty). The purpose of this note is to prove the following result.

Let denote the group of transformations of the upper-half complex plane U onto itself of the form

If, on U, we introduce the Riemannian metric ds = |dz| y−1 (z = x + iy), then U becomes a model of the hyperbolic plane and its group of isometries. The set of elements of type I, the orientation-preserving isometries form a subgroup of index two in , which we denote by .

Let k be a field of zero characteristic, and let F be a function field over k of genus g. We normalize each valuation v on F so that its order group consists of all rational integers, and for elements u1, …, un of F, not all zero, we define the (projective) height as

The sum formula on F shows that this is really a height on the projective space .

Let X be a compact polyhedron, Xn the topological product of n factors equal to X. The symmetric group Sn operates on Xn by permuting the factors, and hence if G is any subgroup of Sn we have an orbit space Xn/G obtained by identifying each point of Xn with its images under G. In particular Xn/Sn is the nth symmetric product of X, and if G is a cyclic subgroup of order n then Xn/G is the nth cyclic product of X.

This paper concerns steady plane periodic waves on the surface of an ideal liquid flowing above a horizontal bottom. The flow is irrotational. The volume flow rate is denoted by Q, the velocity potential by ø, the period in ø of the waves by 2L, and the maximum angle of inclination between the tangent to the surface and the horizontal by θm.

Krasovskii (12) established that, at each fixed Q and L, there exist wave solutions for each value of θm strictly between zero and ⅙π. We establish that, at each fixed Q and L, there exist wave solutions for each value of qc strictly between c and zero. Here qc is the flow speed at the crest, and

where g is the acceleration due to gravity. Krasovskii's set of solutions is included in the set that we obtain.

Generalizing the notion of left amenability for so-called F-algebras [12], we study the concept of ϕ-amenability of a Banach algebra A, where ϕ is a homomorphism from A to ℂ. We establish several characterizations of ϕ-amenability as well as some hereditary properties. In addition, some illuminating examples are given.

A form of the Hopf bifurcation theorem specially suited to systems in block-diagram form has been developed, which allows one to deal with high or infinite order linear elements solely in terms of their transfer functions. The results are proved by the method of harmonic balance, and, for general non-linear systems, lead to criteria for the existence and stability of bifurcated orbits generalizing those derived by various authors for systems of ordinary differential equations. In the particular case of control loops with a single non-linearity, a simple addition to the Nyquist diagram of the loop determines the amplitude and frequency of bifurcated orbits, and whether they occur when the equilibrium is stable or unstable. The analysis is independent of the central manifold theorem, and of Floquet theory.

An isotropic elastic material containing a random distribution of identical spherical particles of another elastic material is considered. The bonding between the spheres and the matrix is imperfect, so that slip may occur at interfaces when stress is applied to the medium. The shear stresses at the interface is assumed to be proportional to the amount of slip. The velocity and attenuation of the average harmonic elastic waves propagating through such a medium are calculated. The results are valid to the lowest order in frequency for wave lengths long compared with the radius of the sphere. The dynamic elastic moduli are obtained from these results and are compared with available results for welded contact. The variations in the P and S wave velocities for propagation across earthquake faults is discussed.

We apply the averaging theory of first, second and third order to the class of generalized polynomial Liénard differential equations. Our main result shows that for any n, m ≥ 1 there are differential equations of the form ẍ + f(x)ẋ + g(x) = 0, with f and g polynomials of degree n and m respectively, having at least [(n + m − 1)/2] limit cycles, where [·] denotes the integer part function.

In this paper we study the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker (GRW) spacetimes. In particular, we consider the following question: under what conditions must a compact spacelike hypersurface with constant higher order mean curvature in a spatially closed GRW spacetime be a spacelike slice? We prove that this happens, essentially, under the so called null convergence condition. Our approach is based on the use of the Newton transformations (and their associated differential operators) and the Minkowski formulae for spacelike hypersurfaces.

The theory of evolution equations in Hamiltonian form is developed by use of some differential complexes arising naturally in the formal theory of partial differential equations. The theory of integral invariants is extended to these infinite-dimensional systems, providing a natural generalization of the notion of a conservation law. A generalization of Noether's theorem is proved, giving a one-to-one correspondence between one-parameter (generalized) symmetries of a Hamiltonian system and absolute line integral invariants. Applications include a new solution to the inverse problem of the calculus of variations, an elementary proof and generalization of a theorem of Gel'fand and Dikiî on the equality of Lie and Poisson brackets for Hamiltonian systems, and a new hierarchy of conserved quantities for the Korteweg–de Vries equation.

The purpose of this paper is twofold. In order to state our first aim, let U denote the ‘infinite’ unitary group, and let BU be a classifying space for U. Then Bott (2),(3) has shown that we propose to investigate the ‘Postnikov system’ of BU.

The problem of statistical investigation is the description of a population, or Kollektiv, of which a sample has been observed. At best we can only state the probability that certain parameters of this population lie within assigned limits, i.e. specify their probability density. It has been shown, e.g. by von Mises(1), that this is only possible if we know the probability distribution of the parameter before the sample is taken. Bayes' theorem is based on the assumption that all values of the parameter in the neighbourhood of that observed are equally probable a priori. It is the purpose of this paper to examine what more reasonable assumption may be made, and how it will affect the estimate based on the observed sample.

Boundary effects in the classical bifurcation experiments of fluid mechanics have been the subject of much recent investigation, most notably for us the work of Benjamin(2) on bifurcating steady flows in the Taylor problem in a very short annulus. His paper broaches a fundamental conceptual issue: how does the number of cells in the primary flow, an integer, depend on the length L of the annulus, a real parameter? By the primary flow we mean the flow which develops at high Reynolds number R as the rotation speed is increased gradually from rest, L being held constant (say L = L*). The relevant part of Benjamin's data is presented in Fig. 0·1. For the very short annuli of his experiment the primary flow possessed either two of four cells only. (See Section 1 concerning the bias towards an even number of cells.) If L* > L2, L1 < L* < L2 or L* < L1 the primary flow has 4, 2 or 2 cells respectively; however when L1 < L* < L2 the evolution of the flow as R is increased is not smooth but suffers a jump as R passes the middle of the three intersections of the line {L = L*} with the cusped curve Γ in the figure. (It is now widely accepted that the development of cells in the flow occurs over a range of Reynolds numbers; this point is not the issue here.) The curve Γ divides the plane into two regions such that there are one or two stable, experimentally observable flows according as (R, L) lies in region I or region II, respectively. We may refer to the two stable states in region II as 2 cell or 4 cell, although their cellular structure is only partially developed at the moderate values of the Reynolds number occurring in the figure. One or the other of the two flows in region II loses stability across Γ, giving rise to the possibility of jumps when, as above, (R, L) crosses the boundary moving from region II to region I. In general if (R, L) is varied quasi-statically along a path which begins in region I, passes through region II, and re-enters region I, a jump will occur on re-entry if and only if the point H lies between the two points of Γ where the path crosses the boundary (between in the sense of distance along Γ).