The differential equation

which is of importance in boundary-layer theory, has recently been discussed in detail for all non-negative values of the parameter λ(1). I wish to point out here that when λ = − 1 the equation can be solved completely in terms of known functions.

Given a1, . . ., ar ∈ ℚ \ {0, ±1}, the Schinzel–Wójcik problem is to determine whether there exist infinitely many primes p for which the order modulo p of each a1, . . ., ar coincides. We prove on the GRH that the primes with this property have a density and in the special case when each ai is a power of a fixed rational number, we show unconditionally that such a density is non zero. Finally, in the case when all the ai's are prime, we express the density it terms of an infinite product.

It is found that the correlation functions of quantum stochastic processes are very similar to the corresponding functions for classical stochastic processes. It is shown also that if the state for the process satisfies the KMS boundary condition then this is reflected in the analyticity of the continuation of the correlation.

The Gliding of a Plate on a Stream of Finite Depth*

Page 597. The line after equation (44) should read

Proc. Cambridge Phil. Soc. 31 (1935), 589–603.

Owing to its physical and chemical properties being greatly different from those of any of the liquids which have hitherto been used in the Wilson cloud chamber, mercury has been used in the experiments described in this paper and the condensation phenomena of its vapour at different temperatures observed. Before constructing the apparatus it was considered necessary to get from theoretical considerations some idea about the magnitude of the critical supersaturation for mercury vapour in equilibrium with a drop carrying unit charge. Assuming that J. J. Thomson's formula.

where s is the supersaturation of mercury vapour in equilibrium with a drop of mercury of radius a, charge e, density σ and surface tension T, the value of which is assumed here to be independent of the radius of the drop, K the specific inductive capacity of the dielectric surrounding the drop, and R the gas constant for one gramme of weight, all at temperature θ, can be applied to the present problem, this critical supersaturation sm is given by the formula

Integral transformations analogous to the Nörlund means have been introduced and investigated by Kuttner, Knopp and Vanderburg(6), (5), (4). It is known that with any regular Nörlund mean (N, p) there is associated a function

regular for |z| < 1, and if we have two Nörlund means (N, p) and (N, r), where (N, pr is regular, while the function is regular for |z| ≤ 1 and different) from zero at z = 1, then q(z) = r(z)p(z) belongs to a regular Nörlund mean (N, q). Concerning Nörlund means Peyerimhoff(7) and Miesner (3) have recently obtained the relation between the convergence fields of the Nörlund means (N, p) and (N, r) on the one hand and the convergence field of the Nörlund mean (N, q) on the other hand.

Let O2 denote the group of isometrics of the plane R2 fixing the origin and let PL2 denote the space of piecewise-linear (PL) homeomorphisms of R2 fixing the origin. Akiba has proved that PL2 is homotopy equivalent to O2(1). The purpose of this note is to point out that this result can be proved by the methods of the author in (4). A complete proof of this result using these methods appeared in the author's Ph.D. thesis(5).

Let γ be an irreducible variety of dimension d, and let In denote an involution of order n on Γ. The image of In is a variety C, also irreducible and of dimension d, and Γ is said to be a multiple of C. Among the possible multiples of a given variety C there is a class of particular interest; this arises when the involution In possesses the following properties:

(1) it is unramified, that is, in every set of the involution the n points are all distinct;

(2) it is generated by an Abelian group G of order n of birational transformations of Γ into itself. Given any point P of Γ the n transformations of G transform P into n points P1 = P, P2, …, Pn which constitute a set of In. The two conditions together imply that the function field κ of Γ is an unramified Abelian Galois extension of the function field k of C, and this shows that the study of the particular class of multiple varieties—which we may call normal multiple varieties when it is necessary to give a name to them—is analogous to the study of a well-known branch of the theory of numbers.

Let a cyclic group of odd prime order p act on a ℤ(p)-Poincaré duality space X. We prove a relation between the Witt classes associated to the [ ]p-cohomology rings of the fixed point set of this action and of X. This is applied to show a similar result for actions of finite p-groups on ℤ(p)-homology manifolds.

The results of the present paper were developed some years ago for quite trivial purposes. It seems possible, however, to judge by a recent publication, that they may sooner or later have scientific utility. They have also a certain slight intrinsic interest.

KMS states of a twisted convolution algebra of Schwartz functions on a vector group are classified and related to KMS states of twisted L1-algebras for certain subquotients. The KMS states for the subquotient algebras are also related to Fock states of vector groups. In the particular case of the subquotient Tn × ℤn of ℚ2n this links the Fock space construction of the theta functions with their appearance in KMS states of loop groups and in the Kac character formula.

It is well known that a regular neighbourhood of a polyhedron in a piecewise linear manifold may be regarded as a simplicial mapping cylinder. The aim of this paper is to show that if the polyhedron is a locally unknotted submanifold of the interior then the class of maps giving rise to such regular neighbourhoods has a simple characterization. At the same time, it is possible to answer the question: Given a simplicial map f defined on a combinatorial manifold, when is the image of f also a combinatorial manifold? Marshall Cohen has answered this question when the image is required to be isomorphic to the domain; the methods used here are those developed in (1), to which the reader is referred for definitions and notation.

The expectation, variance and covariance for different states of a k-state Markov chain have been given by Patankar (6), Whittle (7), Good (3) and Bhat (l). Patankar's results involve the k latent roots of the determinantal equation. As it is not easy to determine the latent roots when k > 2, the actual asymptotic values of variances and covariances cannot be readily evaluated. Whittle gives exact probability distributions for the transitions, but the moments have been obtained after some gross approximations. Good (3) and Bhat(l) have given the first two moments and product moment for the frequency of different states. By using certain methods developed by Iyer and Kapur(4), the first four cumulants and product cumulants for the transition numbers of a two-state Markov chain were calculated and presented in an earlier publication(5).

Since Thom first introduced the notion of the ‘dual’ of a Steenrod square, in (12), it has become apparent that calculation with such duals in the cohomology of, say, a simplicial complex X will often yield information about the impossibility of embedding X in Sn, for certain values of n. For example, the celebrated theorem that cannot be embedded in can easily be proved in this way. In this paper, we seek to generalize this method to any pair of extraordinary cohomology theories h* and k*, and natural stable cohomology operation θ: h* → k*. We show in section 3 that a simplicial embeddingf: X → Sn gives rise via the Alexander duality isomorphism to a homology operation

which is independent of n, the particular embedding f, and even the particular triangulations of X and Sn. If h* and k* are multiplicative cohomology theories, there are Kronecker products

if h0(S0) = k0(S0) = G, a field, and the Kronecker products make h*, h* and k*, k* into dual vector spaces over G, then can be turned into a cohomology operation c(θ): k*(X)→h*(X), by using this duality. This is certainly true if h* = k* = H*(;Zp), p prime, and in this case we have the original situation considered by Thom, who showed, for example, that

The theory of surfaces and of manifolds of higher dimension is greatly handicapped by the lack of exact information concerning the extension of the theorem of Riemann-Roch for curves. Allied to this is the difficulty of finding an expression for the arithmetic genus Pa of primals in terms of their defining elements. This paper seeks to examine certain particular applications of the theorem.

The classical configuration connected with the names of Miquel and Clifford associates with a set of n straight lines in a plane a point called their Miquel point when n is even, and a circle, their Clifford circle, when n is odd. When the number of lines n in a set is even, the omission of these, one at a time, gives n subsets whose associated Clifford circles are concurrent at the Miquel point of the original set; when n is odd, the omission of the lines one at a time gives n subsets whose associated Miquel points are concyclic on the Clifford circle of the original set. To start the chain, it is only necessary to define the Miquel point of two lines as their common point.

The method of Part I(3) is applied to problems involving boundary conditions on an equatorial plane.

The topological theory of covering spaces may be used to prove results in group theory, for instance, the Kuros-Reidemeister-Schreier theorem (1). It seems likely that such methods can be applied to prove the Freiheitsatz (4) and the identity theorem (3), and also perhaps Lyndon's conjecture, that the normal closure in a free group F of a, single element r is freely generated by conjugates of r. However, although these problems may easily be stated in topological terms, no such proof is at present known. In this paper we prove a related result.

The purpose of this paper is to investigate the asymptotic behaviour as r → 1− of the integrals

and f is an analytic function on the unit disk Δ which has non-tangential limits at almost every point on ∂Δ. The paper is divided into three parts. In the first part we consider the case where λ ≠ 1/k, in the second the somewhat more delicate case when λ = 1/k and in the third part we concentrate on some problems related to the case λ = k = 1.