1. Introduction. 1.1 In (5), (6) and (7) I have catalogued the classes of involutions of the orthogonal, symplectic and unitary groups over perfect fields of characteristic 2. So, too, were obtained the involution classes of the well-known relatives of these groups. For each group the relevant classes have simple and explicit descriptions in terms of the geometry of its setting, and an overall pattern can be seen.

Introduction. One of the most satisfying theorems in the theory of convex sets states that a closed connected subset of a topological linear space which is locally convex is convex. This was first established in En by Tietze and was later extended by other authors (see (3)) to a topological linear space. A generalization of Tietze's theorem which appears in (2) shows if S is a closed subset of a topological linear space such that the set Q of points of local non-convexity of S is of cardinality n < ∞ and S ~ Q is connected, then S is the union of n + 1 or fewer convex sets. (The case n = 0 is Tietze's theorem.)

Introduction. This paper gives some embedding and unknotting theorems for bounded manifolds in the PL and differential categories. The theorems are of a similar nature to the embedding and unknotting theorems of Irwin and Zeeman (9, 18), except that the boundaries are allowed to move and the appropriate connectivity conditions become conditions on the relative homotopy groups of the manifolds modulo the boundary, or some suitable part of the boundary.

1. Introduction. It is known (6) that the connectivity conditions on the manifolds in Irwin's embedding theorem (7) can be replaced by a connectivity condition on the map. This has led to the suggestion that in Hudson's analogue of Irwin's theorem for bounded manifolds (5), (6), the connectivity conditions on the manifolds modulo boundary can be replaced by the corresponding connectivity condition on the map modulo boundary. The main purpose of the present note is to show that this is in fact false, as is the formal analogue of Irwin's and Hudson's theorems for manifold triads. Its other purpose is to announce a related embedding theorem for manifolds with boundary which appeared in (3).

1. Introduction. We are concerned with measures µ, in the measure algebra M(G) of a locally compact Abelian group G, which have independent (mutually singular) powers. In (6), Williamson showed that the spectrum, σ(µ) of a Hermitian independent power measure µ satisfying ∥µ∥n=∥µn∥ for positive integer n, contains an infinity of points on the real axis. He conjectured that, in fact, σ(µ) is the disc {λ:|λ|≤∥µ∥}. Taylor (5), has recently proved that, in the case G = R, any positive continuous independent power µ has σ(µ) = {λ:|λ|≤∥µ∥}. His methods depend on his deep and beautiful theory of critical points. Here we verify Williamson's conjecture, give an elementary proof of Taylor's result, and give a simple characterization of the class of LCA groups for which the natural generalization is valid.

1. Introduction. Let A be a subset of a Hausdorff topological linear space. By a convex series of elements of A we mean a series of the form where an∈A and λn ≥ 0 for each n, and . We say that A is:

(i) CS-closed if it contains the sum of every convergent convex series of its elements;

(ii) CS-compact if every convex series of its elements converges to a point of A (this bold terminology is chosen because sets satisfying this condition turn out to have properties analogous to those of compact sets).

In the contour integral

the functions g( ) and f( ) are analytic functions of their arguments, and N is a large positive parameter. When N tends to ∞, asymptotic expansions can usually be found by the method of steepest descents, which shows that the principal contributions arise from the saddle-points, i.e. the values of z at which ∂f/∂z = 0. The position of the saddle-points varies with the l parameters, denoted by α = (α1,α2,…,αl), and it may happen that two or more saddle-points coincide at a certain value of α. The asymptotic expansions given by the ordinary method of steepest descents are then non-uniform. The case of a single parameter and two nearly coincident saddle- points was studied earlier and leads to Airy functions. Here we are concerned with the case of l parameters and of m saddle-points which are nearly coincident when α is near 0. It is then known that f(z,α) can be transformed locally into a polynomial of degree m + 1, and accordingly we here restrict f(z,α) to be such a polynomial. The form of the resulting asymptotic expansion had already been conjectured (correctly as we now see), and involves certain special functions of m–1 variables. To establish the validity of the expansion we must show that the remainder after any number of terms is of smaller magnitude than the last term retained in the expansion. In other words, we must establish an inequality between integrals which is to hold for sufficiently small α and sufficiently large N, and which involves 1 parameters and m nearly coincident saddle-points.

1. Introduction. Bernoulli trials. Consider a sequence of Bernoulli trials. Let p, assumed to satisfy 0<p < 1, be the probability of success at any given trial and let q = 1–p. If Nn is the number of successes in the first n trials, it is well known that Nn/n→p almost surely as n→∞ so that for every ∈> 0,

as n→∞, and it is clearly of great interest to know quantitatively how this probability depends upon n and ∈.

It is well known that in an even dimensional Riemannian space there always exists a particular scalar density which is such that its associated Euler–Lagrange tensor vanishes identically. In this note we show that each of these scalar densities can be expressed as the ordinary divergence of a contravariant vector density. Furthermore, there exists an entire class of contravariant vector densities which enjoy this property. These results are discussed with particular reference to the general theory of relativity.

This paper concludes the investigation into the short-wave asymptotic motion due to a cylinder heaving on water of finite constant depth, and we now present a rigorous method for an arbitrary smooth cylinder which intersects the free surface normally. The aim is to justify those assumptions made in a previous paper concerning the asymptotic values on the cylinder of the two related auxiliary potentials–only one of which need be considered–which were introduced to enable an evaluation of the descriptive wave-making and virtual-mass coefficients. This is done by constructing an integral equation of the second kind whose kernel is small and deducing the leading term in the iteration solution. The method is an extension of the method for infinite depth which is treated first and it is found that there is a simple relationship between the kernels. The argument depends to some extent on results established in another previous paper–which is generalized herein–on the rigorous method for the special case of a half-immersed circular cylinder.

A non-linear Klein–Gordon equation is used to discuss the theory of slowly varying, weakly non-linear wave trains. An averaged variational principle is used to obtain transport equations for the slow variations which incorporate the leading order modulation and non-linear terms. Linearized transport equations are used to discuss instabilities.

Wave-front singularities for the displacement functions, associated with the radiation of linear elastic waves from a point source embedded in a finitely strained two-dimensional elastic solid, are examined in detail. It is found that generally the singularities are of order d–½ where d measures distance away from the front. However, in certain exceptional cases singularities of order d–-n where n = ¼, ⅔, ¾, may be encountered.

In the asymptotic solution of the title equation for simple boundary data on the rectangle 0 ≤ x ≤ l, 0 ≤ y ≤ 1 the bottom corners are not passive, as has previously been supposed, but in fact generate the parabolic side layers at x = o, l. The top corners, though passive, lead to unconventional elliptic quarter-plane problems. Implications for the MHD duct problem are indicated.

1. Introduction. In a paper on the many nucleon problem (l), which we shall henceforth call I, the determination of the irreducible representations of the four-dimensional unitary group were found from a decomposition of its infinitesimal ring U04 The method of decomposition made use of the four primitive four-component idempotents (projection operators) of the Dirac ring each of which, as was recognized long ago by Eddington (2), can be identified with a possible charge-spin state of a Dirac particle. Some experimental justification for the representations was also provided, and it is the purpose of this paper to apply the same tools to the many electron problem. In particular, matrices will be derived for the spin multiplets of a system of r electrons, and it will be shown how the model can account for the atomic shell structure and orbital angular momentum.