The notion of basic topological binary relations is introduced, and corresponding to each such relation we define -regularity, -connectedness, and a separation axiom . The family of separation axioms so obtained is shown to include all the standard axioms T0, T1, …, T5, while familiar properties such as hyperconnectedness and ultraconnectedness are among the class of connectedness conditions. The regularity conditions include R0, R1, and regularity itself.

1. Introduction. A Markov Renewal Process (MRP) with m(<∞) states is one which records at each time t, the number of times a system visits each of the m states up to time t, if the system moves from state to state according to a Markov chain with transition probability matrix P0 = [pij] and if the time required for each successive move is a random variable whose distribution function (d.f.) depends on the two states between which the move is made. Thus, if the system moves from state i to state j, the holding time in the state i has Fij(x) as its d.f. (i, j = 1,2, …, m).

§ 1. The general quaternary cubic

can be expressed in various interesting canonical forms involving suitably chosen linear forms Xi, Yi, Ai. Thus, referred to the pentahedron. X1X2X3X4X5 of Sylvester the cubic becomes the sum of five cubes

with its Hessian in the form

where the coefficients ai may if necessary be taken as equal to unity and the five linear forms Xi each contain four independent parameters, making a total of twenty parameters which is the number of coefficients aijk in the given cubic C3

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.

In this paper we investigate the relationship between genus and localization of finitely generated (torsion-free abelian)-by-finite groups and genus and localization in integral representation theory. In particular when such a group has finite commutator subgroup we can determine the genus exactly.

The purpose of this note is to indicate the relation between (a) the problem of evaporation from a finite area of rectangular shape into a turbulent air stream and of the boundary layer on a flat plate of finite width, and (b) the problem of the boundary layer on a semi-infinite plate and evaporation from an infinite quadrant.

1. Introduction. This note is a sequel to (1), the notation of which will be used without further explanation. In (2), certain deductions from the theorems proved in (1) were made in the case where K(x, t) is derived from a function K(x), a.c. in [0, 1], by the equation

The main aim of this paper is to give some non-trivial results that exhibit the difference and similarity between Kähler and symplectic manifolds. To be precise, it is known that simply connected symplectic manifolds of dimension greater than 8, in general, do not satisfy the formality satisfied by all Kähler manifolds. In this paper we show that such non-formality of simply connected symplectic manifolds occurs even in dimension 8. We do this by some complicated but explicit construction of a simply connected non-formal symplectic manifold of dimension 8. In this construction we essentially use a variation of the construction of a simply connected symplectic manifold by Gompf. As a consequence, we can give infinitely many simply connected non-formal symplectic manifolds of any even dimension no less than 8.

Secondly, we show that every compact symplectic manifold admitting a semi-free Hamiltonian circle action with only isolated fixed points must satisfy the strong Lefschetz property satisfied by all Kähler manifolds. This result shows that the strong Lefschetz property for the symplectic manifold admitting Hamiltonian circle actions is closely related to their fixed point set, as expected.

Let G be a locally compact topological group with left Haar measure, m; let M(G) denote the bounded regular Borel measures on G and let Lp(G) denote the equivalence classes of pth power integrable functions on G with respect to the left Haar measure.

In Part I of this series it was shown that in a metal with overlapping energy bands, one of which is nearly full and the other nearly empty, there exists a critical temperature below which spontaneous magnetization will be present, Tc being given approximately by the equations

where kθ is the exchange energy term postulated by Stoner, b is the ratio of excess vacancies in the d-band to the maximum possible number of parallel spins, and ζd0 is the equivalent energy depth of the vacant levels in the d-band.

1. Let F be an irreducible surface in [4] with apparent triple points, t in number; let P be an arbitrary point in the space and p1,…,pt the trisecant lines which can be drawn through P. Then, if F lies on at least ∞1 cubic primals, any one of these which passes through P will contain p1,…,pt in consequence and the linear system will be compounded of the congruence of trisecant lines of F. By a well-known theorem it follows that the grade of the system is zero. The free surface of intersection Φ of two cubic primals will be ruled, having t generators, each a trisecant of F, passing through every point on it.

1. A few months ago, in the course of teaching an elementary class. I had occasion to discuss problems of the familiar type, ‘if α, β, γ are the roots of the equation x3 + qx + r = 0, form the equation whose roots are α2 + βγ, β2 + γα, γ2 + αβ'. After I had explained the device of replacing βγ by αβγ/α = −r/α, so that the roots of the new equation are the same rational function of the separate roots of the original equation, I was asked ‘whether such a transformation was always possible’. On investigation the answer to the question proved to be a surprising one.

We shall verify Miller's conjectured characterization of parts in measure algebras and obtain also a simpler proof of the main result of (2). The basic pattern of argument follows Miller while the chief novelty resides in the approximation method using Wiener–Ditkin sets which is introduced in section 2.

In this paper we consider Dehn surgery and essential annuli whose two boundary components are in distinct components of the boundary of a 3-manifold.

Let Nl be an orientable 3-manifold with boundary, Kl a knot in Nl, and N2 the 3-manifold obtained by performing γ-Dehn surgery Kl. In detail, let Vl be a regular neighbourhood Kl, X = Nl − int Vl the exterior of Kl, T the toral component ∂Vl of ∂X, and γ a slope on T. Then we obtain the 3-manifold N2 by attaching a solid torus V2 to X so that γ bounds a disc in V2. Let K2 be the core of V2. Let π be the slope of a meridian loop of Kl, and Δ the distance between the slopes π and γ, i.e. the minimal number of intersection points of the two slopes on T. Suppose for i = 1 and 2 that Ni contains a proper annulus Ai such that the two components of ∂Ai are essential loops on distinct incompressible components of ∂Ni. Then note that Ai is essential, i.e. incompressible and ∂-incompressible in Ni.

1. The relation

in which we suppose each of a, b, c, a2 − c2 to be different from zero, leads, from a value θ, to two values of φ, which we may denote by θ1 and θ−1. Each of these, put in place of θ, leads, beside the value θ of φ, to another value of φ, say, respectively, θ2 and θ−2. If θ2 or θ−2 be put for θ, the same relation leads to two values of φ, say, θ1 and θ3 or θ−1 and θ−3, respectively. And so on. It may happen that θn is the same as θ, in which case also θ−n is the same as θ; this we may express by saying that the relation is closed, or that there is closure, after n links. It is the object of the present note to express in reduced form, in terms of a, b, c, the condition that this may be so. Evidently, if n = pq, the condition of closure after n links is satisfied when the condition for closure after p links (or also after q links) is satisfied. But there is a condition for closure after n links which is not satisfied for any less number of links; this is the condition which we call the reduced or proper condition for closure after n links, and it is this which we seek to express.

Let Σan be a given infinite series with the sequence of partial sums {Sn}. Let the sequence be defined by

where is given by

The importance of experimental investigation of the intensity of scattering of X-rays by crystals has long been realized, since it affords valuable information regarding the charge distribution in the atoms of the crystal. Recently Debye has pointed out the importance of similar investigations of the scattering by gases. Here diffraction effects may be observed due to regularities in the structure of the gas molecules as distinct from crystal diffraction, where the regularities are due to the orderly arrangement of crystal atoms. This provides a means of obtaining important information as to the structure of molecules, and Debye and his collaborators have obtained interesting experimental results in this field. Debye has also introduced an approximate theory of the results obtained, but owing to the great complexity of the molecules considered it is at present impossible to proceed any further in the theoretical discussion. However, the case of molecular hydrogen may be worked out for short X-rays, since the charge distribution in the ground state of the molecule is fairly well known. It is the purpose of this paper to calculate the intensities of short X-rays scattered from hydrogen gas. The case of other diatomic molecules will not be very dissimilar.

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.

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.