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.

There are, in projective space Σ of three dimensions, two famous quartic surfaces: W, the Weddle surface with six nodes ((13), p. 69, footnote) and K, the Kummer surface with sixteen ((s), p. 246, (7) passim). They are in birational correspondence and have the same non-singular model: the octavic base surface F of the net of quadrics in [5], which contain a given line λ and for which a given simplex S is self-polar ((5); (6)). One naturally takes S, with vertices X0, X1, X2, X3, X4, X5 as simplex of reference for homogeneous coordinates x0, x1, x2, x3, x4, x5; F is invariant under the harmonic inversions hj in the vertices Xj and opposite bounding primes xj = 0 of S. These six hj, mutually commutative and having identity for their product, generate an elementary abelian group of order 32. This representation of throws into prominence what may, in this context, be called its positive subgroup , of order 16, consisting of identity and the 15 products hjhk = hkhj; these are harmonic inversions in the edges XjXk and opposite bounding solids xj = xk = 0 of S. The coset of consists of the six hj and their ten products in threes, complementary products being the same (h0h1h2 ≡ h3h4h5) because of the product of all six hj being identity. These ten products are harmonic inversions in the ten pairs of opposite plane faces of S.

Let 0 < p < 1 be fixed and denote by G a random graph with point set , the set of natural numbers, such that each edge occurs with probability p, independently of all other edges. In other words the random variables eij, 1 ≤ i < j, defined by

are independent r.v.'s with P(eij = 1) = p, P(eij = 0) = 1 − p. Denote by Gn the subgraph of G spanned by the points 1, 2, …, n. These random graphs G, Gn will be investigated throughout the note. As in (1), denote by Kr a complete graph with r points and denote by kr(H) the number of Kr's in a graph H. A maximal complete subgraph is called a clique. In (1) one of us estimated the minimum of kr(H) provided H has n points and m edges. In this note we shall look at the random variables

the number of Kr's in Gn, and

the maximal size of a clique in Gn.

Two kinds of duality arising in studies of interaction models are discussed. The first kind, which has not previously been investigated, is related to algebraic properties of the coefficient ring. The second kind is the well-known geometric duality for planar graphs. The two dualities together lead to a perfectly symmetrical relationship for a general form of partition function.

This note is devoted to the construction of a 〈Q, EΦ〉-closed Fischer class which is not a formation. It provides a negative answer to the Questions 3–7 raised by Cossey in (1). All groups considered are finite and soluble.

Let be the class of groups possessing a subgroup of index n for each divisor n of the group order. McLain (7) initiated the formal investigation of and observed that every solvable group is a direct factor of an -group. However, subclasses of provide some interesting problems. Various subclasses of which satisfy other properties were studied by McLain; the upshot being that these classes approach supersolvability. This program was pursued by Humphreys (3) and Humphreys and Johnson (4), among others. In particular, , the largest quotient closed subclass of , was considered in (3) and (4). Humphreys (3) has shown an odd order -group is supersolvable, but provides some non-supersolvable groups. Our motivation is that if S ∈ QR0(S3), the formation generated by S3, and V is a faithful irreducible GF(2) [S]-module, the semidirect product VS ∈ .

If F is a free group on some fixed basis X, there is a mapping from F to the non-negative integers, given by sending an element of F to the length of the normal word in X±1 representing it. A similar mapping is obtained in the case of a free product of groups. Lyndon (3) considered mappings from an arbitrary group to the non-negative integers having certain properties in common with these mappings on free groups and free products.

If R is a Banach algebra and ø ∈ R′, the dual space, then we may define a bounded linear map by

We shall show that for suitable p the requirement that each be p-absolutely summing constrains R to be an operator algebra, or even, in certain cases, a uniform algebra. In this way we are able to give generalizations of results of Varopoulos (12) and Kaijser (4).

This paper will show that after localization at any given prime p, the infinite loop space structure on the space BSO is essentially unique. If the word ‘localization’ is replaced by ‘completion’, the result continues to hold; and both results continue to hold if the space BSO is replaced by the space BSU.

The work in this paper arises from observing a simple example of a potential in the plane, as the potential varies with changes in a ‘control’ parameter. To fix ideas, we first discuss the example in intuitive terms, and the rest of the paper is a detailed technical discussion. The example aids understanding of bifurcation theory and suggests interesting generalizations. We begin by describing the example, as follows.

Various elegant properties have been found for the waiting time distribution G for the queue GI/G/1 in statistical equilibrium, such as infinite divisibility ((1), p. 282) and that of having an exponential tail ((11), (2), p. 411, (1), p. 324). Here we derive another property which holds quite generally, provided the traffic intensity ρ < 1, and which is extremely simple, fitting in with the above results as well as yielding some useful properties in the form of upper and lower stochastic bounds for G which augment the bounds obtained by Kingman (5), (6), (8) and by Ross (10).

The W.K.B. solution for radio waves in a plane stratified ionosphere was obtained by Budden ((4), §§ 18.10 and 18.11) by a process using approximate diagonalization of the governing equations. Alternatively the solution may be obtained by specializing to the one-dimensional case the result of Bennett (1) which employs frequency scaling. Formally similar results were obtained by Ludwig (8) without explicitly introducing frequency scaling.

The analysis of the interactions between the electrons of different atoms is complicated by many effects. Unless the atoms are free and well separated, account has to be taken of covalency, overlap, exchange, and superexchange. But, although such effects are often important, it is sometimes useful to regard the electrons as being located on separate atoms. Two electrons, labelled 1 and 2, on respective atoms A and B, then interact principally through the Coulomb energy e2/r12. Neighbouring atoms L, M, N, …, which we suppose are spherically symmetric, make their presence felt by acting as polarizable spheres. The electric fields thereby produced modify the simple Coulombic expression for the energy of interaction of the two electrons. It is the purpose of this article to examine such modifications in detail.

The response of a supersonic laminar boundary layer is considered when fluid is blown obliquely into it from a slot in the boundary. The velocity and length scales associated with the slot blowing are chosen in such a way that the triple-deck theory of Stewartson (9) may be exploited. It is shown that, depending upon the blowing angle, there can be either a compressive or expansive interaction between the boundary layer and free stream upstream from the slot.

A uniform eigenstrain is prescribed within a spherical subregion of an isotropic linearly elastic half-space. Combining an application of potential theory with the stress-function approach of Papkovitch and Neuber, and starting with Eshelby's well-known solution for the homogeneous infinite solid, it is shown that the residual problem of potential to be solved is the determination of Boussinesq's first and second three-dimensional logarithmic potentials of volume distributions. Although explicit results are supplied only for the case of a spherical inclusion, the dependence of the solution on the infinite solid solution holds good for an arbitrarily shaped transforming inclusion. This can be established on the basis of the principle of superposition, considering that an arbitrary volume is essentially made up from an infinite spectrum of spherical regions.