The radiation from 198Au (half-value period 2·76 ± 0·02 day) have been studied, by absorption and coincidence methods, in a wide variety of experimental conditions. For the most part the results confirm those obtained by other investigators, but, in addition, a fairly intense γ-radiation of about 65 KeV. energy has been detected. This radiation has been distinguished from the weaker fluorescent K X-radiation of mercury, emitted as the result of internal conversion of one of the γ-rays of 198Au, and it has been shown that it is not the K X-radiation of platinum. Disintegration of 198Au by K-electron capture, if it occurs at all, cannot take place in more than about 15% of all disintegrations.

In (1) Dvoretsky proved, using very ingenious methods, that every centrally symmetric convex body of sufficiently high dimension contains a central k-dimensional section which is almost spherical. Here we shall extend this result (Corollary to Theorem 2) to k-dimensional sections through an arbitrary interior point of any convex body.

An example illustrates (a) that the use of indefinite inner products can illuminate Probabilistic Wiener–Hopf Theory in symmetrizable cases, and (b) that half-winding probabilities should be thought of as branching measures for Ray processes. The paper's wide variety of techniques could make it instructive for research students.

In recent years many papers have dealt with the zero-dimensional multiple point sets of smooth immersions of closed manifolds in Euclidean spaces (see, for example, [2, 6, 7, 8, 9, 10, 13, 14, 15, 21]). In the present paper we deal with the case when the double points form a two-dimensional surface and consider the following question: Which surfaces can occur as double point surfaces of self-transverse immersions of closed n-manifolds in ℝ2n−2? We prove the following

This paper proves the existence of solutions for the discrete coagulation–fragmentation equation over all times, even when source and efflux terms are present. The hypotheses required cover most physical applications. Roughly speaking, the hypotheses ensure a finite flux of mass through the system. The techniques used, which extend those in I of this series, may apply to other infinite systems of differential equations.

Basis functions are derived and tabulated for the double-valued representations of the triclinic, monoclinic, orthorhombic, trigonal, tetragonal and hexagonal crystallographic point groups using the method of projection operators.

The time-dependent solutions of an infinite set of differential-difference equations arising from queueing theory and models of ‘living’ polymer are expressed in terms of modified Bessel functions. Explicit solutions are available for constant values of a parameter describing the arrival rate or monomer concentration; for time-dependent parameter a formal solution is obtained in terms of a function which satisfies a Volterra type integral equation of the second kind. These results are used as the basis of a numerical method of solving the infinite set of differential equations when the time-dependent parameter itself satisfies a differential equation.

It is a classical result (5) that the algebraic surfaces which admit a continuous group of birational self-transformations, or automorphisms, and which are neither rational nor scrollar, belong to one or other of two families: (i) the elliptic surfaces; these possess an elliptic group of ∞1 automorphisms; (ii) the hyperelliptic surfaces of rank 1 (which, for brevity, we shall call hyperelliptic); these possess a completely transitive permutable group of ∞2 automorphisms. The chief properties of the two classes of surface, in so far as they are required in the present work, are described in § 1.

1. The problem to be discussed in the present paper arises when the finite resolving time of a recording apparatus is taken into account. Events are divided into two classes, recorded and unrecorded. Any recorded event is followed by a dead interval of a certain length of time, during which any other event which occurs will be unrecorded. A typical example is an α-particle counter; a recorded particle causes the chamber to ionize, and no other particle can be recorded until the chamber has deionized. In the case when the dead interval is of constant length τ, if the observation time t lies between (n − 1)τ and nτ, the number of recorded events must be 0, 1, 2, …, n − 1 or n. We shall assume that the probability of an event occurring in the interval [t, t + dt] is λdt, λ being constant; this is the case of most practical interest. The probability distribution of recorded events for dead intervals of constant length has been determined by Ruark and Devol (2). The explicit expression of this distribution is fairly complicated, and it is therefore difficult to manipulate. The method employed in the present paper leads naturally to the Laplace transform of the distribution with respect to the time t, and this is relatively simple. The method can easily be generalized to deal with the case of dead intervals which are not all equal, but follow a probability distribution u(τ) dτ. Finally the Laplace transform is very convenient for determining the asymptotic behaviour of the distribution of recorded events for large times of observation.

Lagrangian coordinates are used in conjunction with the self-similarity hypothesis in order to examine the problem of the vertical entry at constant speed of a wedge and also of a cone into an incompressible fluid initially at rest. Certain known properties of the free surface are recovered in a very direct and simple manner and new exact results concerning the inclination of the free surface and the angle of contact with the rigid surface are obtained.

Let SsTt denote a space of s + t dimensions with the property that the square of the distance between any two points

is of the form

where s of the ε's are equal to + 1 and t of them to — 1.

The aim of this article is to discuss a paradox of importance in the quantum theory of measurement. The paradox was first propounded by Einstein, Podolsky and Rosen in 1935(1), and was discussed subsequently by Bohr(2), Furry(3), Schrödinger(4, 5), as well as in texts (6, 7). We begin by formulating the paradox.

By means of purely qualitative arguments which do not depend on any particular model, the general scheme of stable nuclei and the isotopic breadth of nuclei with odd charge number are explained.

The breadth of the isobaric region can be obtained if the numerical values of certain energies are known. Though these can be estimated only very roughly, the values for the breadth of the isobaric region obtained in this way are in good agreement with the experimental values. The increase in the breadth of the isobaric region from light nuclei to heavier nuclei can be explained, but no plausible explanation has been found for the fact that the breadth decreases again for the heaviest nuclei.

It is clear from § 1 that we shall have quite a large number of theorems to consider, as we may distribute the conditions of monotony and existence in a good many different ways. It is therefore convenient to collect together the various results for series and for transforms in tabular form, for reference and comparison; this is done at the end of this section.

The transform table is very simple. The theorems contained in it have been discussed at length in M.F.(I); they may be summarized by saying that (1) holds in all the cases considered, except perhaps in the consine case of section [B] where the truth is not known. The series results are apparently more complicated as well as more numerous, since there are extra conditions to be imposed in [A, 1], [B, 1], [C′, 1] and [C, 2]. However, these are certainly satisfied if we suppose f and g to be positive whenever they are given to be monotonic (a condition which automatically holds in the transform theorems, since in these the monotonic functions tend to zero at infinity). If we confine our attention to this case, the series results are what we should expect from the transform ones, except that the difficulty in section [B] for transforms is not reproduced for series in [B, 1] or [B′, 1]; it is reproduced in [B, 2].

Throughout this paper we deal only with complex and semi-simple algebras. Let B be such an algebra. We denote the socle of B as SB. B is a modular annihilator algebra if B/SB is a radical algebra, i.e. if every element of B is quasi-regular modulo the socle of B; see (1) or (12). Now assume that B is a modular annihilator algebra and a Banach algebra. Then any semi-simple closed subalgebra of B is a modular annihilator algebra by ((4), Cor. to Theorem 4·2,). It is not true, however, that any semi-simple subalgebra A of B is a modular annihilator algebra, even when A is a Banach algebra in some norm. We give a simple example to illustrate this. Let A be the algebra of all complex functions f, continuous on the closed unit disk D in the complex plane, analytic in the interior of D, and such that f(0) = 0. A is a Banach algebra in the usual sup norm over D. Now consider the norm on A defined by

Let B be the completion of A in this norm. A has an involution * defined by and also ‖ff*‖ = ‖f‖2 for all f ∈ A. Therefore B is a B*-algebra. It is not difficult to verify that the only non-zero multiplicative linear functionals on A which are continuous with respect to the norm ‖·‖, are the point evaluations at 1/n, n = 1, 2 …. It follows that every non-zero multiplicative linear functional on B is an extension of one of these point evaluations to B. Thus B can be identified with the algebra of all complex sequences which converge to zero.

1. The object of this paper is to discuss conditions of validity of the Parseval formulae for Fourier integrals:

where the transforms are defined by ordinary convergence; we shall not be concerned with the more elegant theory in which they are given by convergence in mean.

For any integer N ≽ 3 we will construct a 3-manifold which can be described as + 1 surgery on N distinct knots in S3. We will also give examples of 3-manifolds which are N-fold cyclic branched covers of S3 over 2 distinct knots. Brakes (2) discovered the first examples of 3-manifolds with multiple knot surgery descriptions. Our construction is much different and follows directly from the construction used by Lickerish (6) to describe a manifold which has 2 distinct knot surgery descriptions. Giller (5) has given examples of 3-manifolds which arise as cyclic branched covers over distinct knots in S3. Qur construction is similar, but the knots are much easier to distinguish, being iterated torus knots.

A family ℳ of closed circular discs in the plane is called a saturated family or a saturated system of circles if (i) the infimum r of the radii of the discs in ℳ is positive and (ii) every closed disc of radius r in the plane intersects at least one disc in ℳ. For a saturated family ℳ, we denote by S the point-set union of the interiors of the members of ℳ and by S(l) the part of S inside the circular disc of radius l centred at the origin. We define the lower density ρℳ of the saturated family ℳ as

where V(S(l)) denotes the Lebesgue measure of the set S(l).

The problem of the torsion of beams of ⊥- and L-cross-sections has received attention from very few authors despite its important technical applications. The first mathematical solution in this connection was obtained by F. Kötter in 1908 for an L-section both of whose arms are infinite. He attacked the problem by the use of the known solution of the rectangle and by application of the scheme of conformal transformation. Kötter's method, however, does not lend itself readily to the solution of the problem involving more than one re-entrant angle. The first solution for the torsion of a beam whose cross-section is a rectilinear polygon of n sides was published in 1921 by E. Trefftz who also applied his method to an infinite L-section. Recently I. S. Sokolnikoff has suggested a more general method depending upon the fundamental theorem of potential theory that a harmonic function is uniquely determined by the values assigned along the boundary of the region within which the harmonic function is sought, the boundary condition and the region being subject to certain well-known assumptions of continuity, connectivity, etc. As an illustration of his method he has given an approximate solution for a ⊥-section whose flange and web are both infinite.

1. The results. Let l be an odd prime, r ≥ 1, and let

be the quaternion group of order 4lr, as given by generators and relations. Throughout N is a tamely ramified normal number field with Galois group Gal (N/Q) = H (a ‘quaternion field’), and its ring of integers. We are interested in the structure of as a module over the integral group ring ZH. Deriving, first, certain classgroup invariants for locally free ZH-modules, we shall then determine those for the module in terms of the arithmetic invariants of N/Q. When 1 ≡ – 1 (mod 4), this yields again a Galois module interpretation of Artin root numbers quite analogous to that in (2). On the other hand for l ≡ 1 (mode 4), we shall get a weak ‘normal integral basis theorem’. The original impetus for this work came from computations of J. Queyrut, who – in different language – obtained these results in the case l = 3, r = 1 (cf. (7)). The tools, we are using, come from the general theory developed in recent years with such concrete applications in mind, and it is perhaps of interest to see how the various ‘strands’, on root numbers (cf. (3), (4)), on locally free modules (cf. (5)), and on Galois module structure (cf. (6)) are here pulled together. For technical reasons, we shall impose on N the slight further restriction, that l be non-ramified, although our results would remain true without this. Both the statements and the proofs of the theorem depend on ideas contained in (5) and (6). The reader who is prepared to take them for granted should, however, be able to read the present paper independently of those papers.