In (2), indefinite Kählerian manifolds have been examined from the point of view of holomorphic sectional curvature. Examples in (2) show that the analogue of Kulkarni's theorem (see (4), p. 173) for the holomorphic sectional curvature is false and the best possible result in the direction is:

Theorem 1 (known, (2)). Let M be a connected indefinite Kählerian manifold with complex dimension n ≥ 2. If the holomorphic sectional curvature of M is bounded above and bounded below, then M is an indefinite complex space form.

It is shown that if we know the Love's strain function due to the existence of any arbitrary axisymmetric singularity in the elastic whole space, then the corresponding Love's function due to the existence of the same singularity in the interior of one of two dissimilar semi-infinite solids separated by a dissimilar thick layer can be obtained by differentiation and integration of that for the elastic whole space. This discovery rests on the principle that to every singularity in the layered space there will correspond an infinite system of images. Remarkably, the number of images is finite when the phases have equal rigidities but different compressibilities. The analysis thus brings under one general theorem the solutions due to such influencing singularities as a normal point force, a normal point force doublet, an infinitesimal prismatic dislocation loop, a centre of dilatation and a dilatation doublet. From the theorem, we readily infer that the elastic field at large distances from any arbitrary axisymmetric influencing singularity becomes nearly that for two perfectly bonded semi-infinite solids without an interface layer.

Volume 98 (1985), 403–412

‘On the mobility of bodies in ℝn’

The proof of the following theorem which I gave in the above paper is incorrect. A correct proof is given here.

Theorem 2·1. Let ℬ be an intersection regular, finite collection of bodies, and ℱ a manoeuvre on ℬ. ℱ is strongly separating if and only if every component of S(ℬ, ℱ) is positive; and, if any S(i, j, p) (ℬ, ℱ) is negative, ℱ is colliding.

An action of a group, G, on a surface, F, consists of a homomorphism

ø: G → Homeo (F).

We will restrict our discussion to finite groups acting on closed, connected, orientable surfaces, with ø(g) orientation-preserving for all g ε G. In addition we will consider only effective (ø is injective) free actions. Free means that ø(g) is fixed-point-free for all g ε G, g ≠ 1. This paper addresses the classification of such actions.

Let U(1, n; ℂ) be the automorphism group of the Hermitian form

for . We can regard an element of U(1, n; ℂ) as a transformation acting on , where is the closure of the complex unit ball

The non-trivial elements of U(1, n; ℂ) fall into three conjugacy types, depending on the number and the location of their fixed points. Let g be a non-trivial element of U(1, n; ℂ). We call g elliptic if it has a fixed point in Bn and g parabolic if it has exactly one fixed point and this lies on the boundary ∂Bn. An element g will be called loxodromic if it has exactly two fixed points and they lie on ∂Bn.

Throughout, the word knot means a subspace of the 3-sphere S3 homeomorphic with the 1-sphere S1. Any knot can be expressed as a connected sum of a finite number of prime knots in a unique way (13), we consider the trivial knot a non-prime knot. (For higher dimensional knots, factorization and uniqueness have been studied in (1).) However given a knot it is difficult to determine if it is prime or not. We prove that totally knotted knots, see definition in §2, are prime in theorem 1, give a class of examples in theorem 2 and investigate how the last result can be applied to the conjecture that the family Y of unknotting number one knots are prime. (See problem 2 in (5).) At the end, prime tangles as defined by W. B. R. Lickerish are used to prove that in a certain family of knots, related somewhat to Y, there is just one non-prime knot: the square knot. The paper should be interpreted as being in the piecewise linear category. Standard definitions of 3-manifolds and knot theory may be found in (6) and (11) respectively.

In a recent paper (1) I have studied tangential properties of sets of infinite measure. This note represents a complementary study of sets and, in particular of arcs, of σ-finite linear measure. Suppose we have a linearly measurable set E of infinite measure that can be represented as the sum of sets of finite measure. Write En = En, 1 + En, 2, where En, 1 is the set of regular points of En and En, 2 that of irregular ones. The set F = ΣEn, 1 is a regular component of E and the set G = ΣEn, 2 an irregular one. A tangent toEn exists at almost all points of En, 1 and at almost no points of En, 2. Denote by Tn the set of all tangents to En and by T the sum-set ΣTn. T will be called a tangent-set to E. Lines of T are defined corresponding to almost all points of F and to the points of a subset of G of measure zero.

Let α1, …, αn be n ≥ 2 algebraic numbers such that log α1,…, log αn and 2πi are linearly independent over the field of rational numbers Q. It is well known (see (6), Ch. 1) that the Thue–Siegel–Roth theorem implies that, for each positive number δ, there are only finitely many integers b1,…, bn satisfying

where H denotes the maximum of the absolute values of b1, …, bn. However, such an argument cannot provide an explicit upper bound for the solutions of (1), because of the non-effective nature of the theorem of Thue–Siegel–Roth. An effective proof that (1) has only a finite number of solutions was given by Gelfond (6) in the case n = 2, and by Baker(1) for arbitrary n. The work of both these authors is based on arguments from the theory of transcendental numbers. Baker's effective proof of (1) has important applications to other problems in number theory; in particular, it provides an algorithm for solving a wide class of diophantine equations in two variables (2).

An account is given of experiments to determine the polarisation of downcoming waves of long wavelength, using the method devised by Ratcliffe and White. The results show that on the whole the polarisation is left-handed, although there is a tendency for right-handed polarisation in the region of 1500 metres.

We show that the proportion of polynomials of degree n over the finite field with q elements, which have a divisor of every degree below n, is given by c q n −1 + O(n −2). More generally, we give an asymptotic formula for the proportion of polynomials, whose set of degrees of divisors has no gaps of size greater than m. To that end, we first derive an improved estimate for the proportion of polynomials of degree n, all of whose non-constant divisors have degree greater than m. In the limit as q → ∞, these results coincide with corresponding estimates related to the cycle structure of permutations.

In his celebrated paper [7], Martinet showed the equivalence between the infinitesimal versality and the versality for [Ascr ]- and [Kscr ]-morphisms. By using this theorem, he obtained the following Theorem 1·1 which played one of the key roles for the classification of C∞ stable map-germs ([1, 7]).

In a recent paper Srivastav (2) considered the solution of certain two-dimensional mixed boundary-value problems in a wedge-shaped region. The problems were formulated as dual integral equations involving Mellin transforms and were reduced to the solution of a Fredholm integral equation of the second kind. In this paper it will be shown that a closed form solution to the problems treated in (2) may be obtained by elementary means.

We have called a linear partial differential operator P almost hypoelliptic in some of the variables if the distribution u is strongly regular in those variables in every open subset where this is true of Pu. In an earlier paper (l) we have characterised almost hypoelliptic operators with constant coefficients. Following techniques of Hörmander (2) and Yoshikawa(5) we obtain in this paper some a priori inequalities for almost hypoelliptic operators and derive results about the solvability of the adjoint operator P*. Our notation for various standard distribution spaces and multi-indices follows Hörmander (2).

In (3), the author defined the notion of a G-space. A G-space is a weaker notion than that of an H-space. The main purpose of this paper is to present various means of constructing G-spaces. As an application of some of the techniques of (3) and of this paper (though not an application of the concept of G-space) we shall prove the following theorem:

Theorem. Let G be a connected compact Lie group and let H be a connected subgroup of maximal rank. Then H3(G/H; Z) = 0. In fact, the Hurewicz homomorphism is trivial for odd dimensions.

We refer for general background to N. Bourbaki, Intégration, chapters iv and v. We consider a locally compact Hausdorff space R and denote the set of continuous functions with compact support by The Riesz-Markov theorem shows that there is a 1−1 correspondence between the set of regular Borel measures on R and the set of positive elements of the topological dual of . Let {μn}, μ be regular Borel probability measures on R. The sequence of measures {μn} is said to converge vaguely to μ if, for all . Thus the vague topology is that of simple convergence in . We shall call a μ-measurable set μ-quarrable if its boundary is a μ-null set.

A complete and satisfying account of the classification of pencils of binary cubics over an algebraically closed field was given by Newstead (2). Extending these results to the real case is not a matter of mere routine since new questions arise, for example the separation of roots of the cubics in a pencil (as well as their reality).

1. We suppose that f(t) is integrable L and periodic with period 2π and we denote the Fourier series of f(t) at the point t = x by

We denote the nth partial sum of the Fourier series by sn. We will prove here a general result (i.e Theorem 1) and some interesting results in connexion with associated series of a Fourier series of the type

We write

The object of the present paper is to prove the following theorems. (In all the following cases k is any constant > π if nothing else is mentioned.)

Let be the self-dual CCR algebra over a triplet (K, r, Γ), where K is assumed to be a Hilbert space, and let {τt}t ∈ R be a Bogoliubov automorphism group of induced by a strongly continuous one-parameter unitary group on K. In this paper, we introduce some continuity for linear functionals on and, under this continuity, we study existence, uniqueness, and non-uniqueness of KMS states for .

We extend to a wider class of rearrangement invariant spaces a result of S. Guerre concerning the extraction of almost symmetric sequences from a given weakly null sequence in Lp.