Let Mm be a closed connected smooth manifold of dimension m, and set Pm = Mm – int Dm where Dm is a disc in M. In (4), Wall has the following exact sequence

where (M), (resp. (P)) is the Δ-set of diffeomorphisms of M (resp. P) as in(l), and [M/P] is the set of diffeomorphism classes of smooth manifolds obtained by glueing a disc to the boundary of P. In this paper we obtain some results on π0((M)) for particular M, and in the following sense: Using the techniques of (2), we can determine π0((P)), so the main portion of the paper is concerned with a discussion of the kernel of α. There is a map ω: π1((P)) → Γm+1, given by the obstruction to extending h∈π1((P)) to a concordance of the identity of M to itself, and it is clear that if this map is attached at the beginning of the sequence (*) we get exactness. We will obtain (for certain M) an alternative description of the image of α in terms of those homotopy spheres which can appear as the boundaries of thickenings.

In a letter dated 19 November 1988, addressed to Samuel Gitler, with copies to the three other authors of [2], J. F. Adams formulated the problem studied and solved in the present paper. In his letter, Adams provided the key ingredient for its solution and proved a first result. (See (1·2) and the second paragraph of the proof of (1·1)) We therefore wish to express our debt of gratitude to the late Professor Adams for his essential contribution, without which this paper could not have been written.

Solutions for the temperature distribution in a circular pipe have been given by various authors, notably Gretz, Nusselt, Goldstein; all these are cited in (1), section 266. I have already considered in one paper the temperature distribution in a circular pipe when viscous incompressible fluid is flowing through the pipe and rate of heat addition is an oscillatory function of time. In this paper expressions for the temperature distributions are derived when viscous incompressible fluid is flowing through the channel, the dissipation due to friction is neglected and the rate of heat addition is an exponential function of time. From the engineering point of view this situation has some importance.

It is well known that the operators mainly employed in quantum theory are hermitian; it is less well known amongst physicists that they are required, in addition, to be self-adjoint. This is essential for the validity of the result known in quantum theory as the representation theorem and in the mathematical theory as the resolution of the identity. The purpose of this paper is to show that the self-adjoint operators can be characterized by a condition which is nearer to having a physical significance than those given in the literature.

Using a criterion due to J. Tits [4] we shall show that it is easy to give a pair of conjugate matrices in GLn(ℂ) which freely generate a free group of rank two. The difficulty lies in producing two such matrices whose freeness does not follow directly from the known freeness of two 2 × 2 matrices. Finally we show that the well known problem as to whether the Burau representation of Artin's braid group B4 is faithful turns out to be arbitrarily close to being true in a large number of ways. Thanks are due to the referee for his helpful comments.

By a closed m-manifold we mean a C∞ compact connected m-dimensional manifold, without boundary. M will denote a closed m-manifold (m > 0), and En will denote Euclidean n-space. All embeddings will be C∞.

In this note we shall give two extensions of the results of (2) concerning the economic use of factory machines. Familiarity with (2) will be assumed and the same notation will be used. We shall show, first, that the number of decision elements required for the solution of the problem can be reduced considerably, and secondly, that the computer will also give the solutions to certain related problems.*

We consider a stochastically continuous process ω(t, α) with independent increments, whose sample functions are bounded in the unit interval 0 ≤ t ≤ 1 for almost all α. If ω(t, α) is a process with independent increments, the characteristic function of ω(t, α) is of the form exp {tψ(u)} where where F is a σ-finite measure with finite mass outside every neighbourhood of o, and a and σ are constants. There is no essential restriction in supposing ω(0, α) = 0.

The scattering amplitude for a potential expressible as a sum of n separable potentials is obtained as the solution of a set of n simultaneous linear equations. As n→∞ the Fredholm solution for the integral form of the Schrödinger equation is recovered.

We investigate the behaviour of families of meromorphic functions in the neighbourhood of points of non-normality and prove certain covering properties that complement Montel’s Theorem. In particular, we also obtain characterisations of non-normality in terms of such properties.

Let f(x, y) be a bounded real function on the unit square. Consider the Riemann integrals (if they exist)

In this paper, we consider the energy behaviour of Rayleigh surface waves in a homogeneous isotropic 3-dimensional half space with free boundary condition. It is known that Rayleigh waves with finite energy are asymptotically concentrated along the boundary. We clarify the domain where the energy of the Rayleigh waves concentrates and obtain the decay order of the energy distributed in the complement of the domain.

The view which the author has expressed regarding catalytic activity as due to frozen groups of atoms with strong specific external fields is further tested by a study of the stability of preparations of copper at the instant of formation, and during the subsequent life of the catalyst.

A considerable amount of experimental work has been done in recent years on the properties of the electrons which leave a metal surface when it is bombarded by an incident beam of electrons. Farnsworth, who first studied the energy distribution of the secondary beam systematically, showed that it is made up of two groups. The first has the same energy as the primary beam and may be regarded as consisting of reflected primary electrons; and the second has for the most part a very small energy, of the order of a few volts, and contains the true secondaries derived from the metal. The present writer in a preliminary note has described some measurements of the differential energy distribution by means of the magnetic spectrum method. Owing to an inadequate technique those experiments are considered to be valid only in the upper half of the energy range, where the form of the distribution has several features of particular interest when the mechanism of secondary emission and of electron reflection are considered. In this note, the author called attention to the peculiar depression in the curve immediately preceding the “reflected” peak. It was impossible then to insist on the reality of such an effect, owing to the probable presence of an instrumental cause (which was given there as the explanation). That it is real, however, has been demonstrated in a striking manner by Whiddington and Brown, who have succeeded in recording electrons as slow as 100 volts photographically on a specially sensitised film.

The stability and the stream lines of the symmetrical and unsymmetrical double rows in a channel have already been discussed by the author. It is the object of this paper to investigate the possibility of the existence of a double row whose axis is parallel to, and not necessarily coincident with, the axis of the channel. It will be seen that, if the system of vortices is to move forward as a whole, then the only possible types of double row are those that have been discussed previously, that is, the symmetrical and unsymmetrical systems, and in both of these cases the centre lines of the row and channel are coincident. It was shown previously that the symmetrical row is always unstable and that the unsymmetrical system is stable within a particular domain of stability.

Let χ(t) = a0tn – a1tn−1 + ⋯ + (−1)rartn−r be the chromatic polynomial of a graph, the characteristic polynomial of a matroid, or the characteristic polynomial of an arrangement of hyperplanes. For any integer k = 0, 1, …, r and real number x ⩾ k − r − 1, we obtain a linear bound of the coefficient sequence, that is

\begin{align*} {r+x\choose k}\leqslant \sum_{i=0}^{k}a_{i}{x\choose k-i}\leqslant {m+x\choose k}, \end{align*}

A k-3 cap in a three-dimensional Galois space, S3,q, is a set of k points, of which some 3, but no 4 are collinear. It is shown in (2) that for q ≥ 4.

1. This paper is concerned with certain properties of the solutions of differential equations of the type

where f, g have continuous partial derivatives up to the second order satisfying Lipschitz conditions in some bounded domain and are periodic in t with period 2π, ω is a positive constant and k is a small parameter. If f = 0, (1·1) is equivalent to

an equation representing the forced vibrations of a quasi-linear system of one degree of freedom.

For α a positive irrational, let 𝛢α be the subalgebra of continuous functions on the two-torus whose Fourier transform vanishes at (m, n) if m+αn < 0. These algebras were studied by Wermer and others, who proved properties such as maximality and characterised the Gelfand space. One of the major themes of current work in operator algebras is classification, but none of the properties which were investigated earlier distinguished between 𝛢α and αβ, if β is another positive irrational. We address this question. We also determine the automorphism group of 𝛢α.

1. In many problems in theoretical physics one encounters situations in which it is desirable to be able to expand orbitals centred at one particle about another centre, frequently, but not always, another particle.

There have long been available methods for permitting this to be effected for bound states but no such procedures are available for the continuum.

Typical examples of physical situations in which such procedures are desirable are positron scattering from atomic and molecular systems, electron scattering from molecules, and nucleon scattering from light nuclei, to mention only a few.

The author's interest arose from a desire to establish a general theory of slow electron scattering from molecules, and an account of such a theory will appear shortly (1).