We consider a certain method of effectively calculating functions of the positive integers, related to the ‘transfinite recursion’ of Ackermann(1). It arises by use of well-ordered series, and one might hope that the functions so generated could be classified according to the ordinal of the series involved, but we shall see that the use of well-ordered series is not at all essential, and in any case no ordinal other than ω need be used. We investigate how any calculable function (that is, general recursive) can be derived by ordinal recursion (as we name our method), and prove a best possible result.

Suppose that X is any vector space on which it is possible to recognize a class of sets of such a nature that it is natural to call them ‘bounded’. (Precise conditions for such a class of sets are given in § 2.) Let L be any vector space of linear functionals on X which map each ‘bounded’ set into a bounded set. We say that a filter is boundedly-weakly convergent if it is convergent in the weak topology of the linear system [X, L] and contains a ‘bounded’ set. If M is a vector space of boundedly-weakly continuous linear functionals on X which includes L, we say that a subset S of M is limited if 〈 , f〉 converges uniformly for f ε S whenever is a boundedly-weakly convergent filter.

Let L1, …, Ln be n homogeneous linear forms in n variables u1, …, un, with non-zero determinant Δ. Suppose that L1, …, Lr have real coefficients, that Lr+1, …, Lr+s have complex coefficients, and that the form Lr+s+j is the complex conjugate of the form Lr+j for j = 1, …, s, where r + 2s = n. Let

for integral u1, …, un, not all zero. For any n numbers α1, …, αn of the same ‘type’ as the forms L1, …, Ln (that is, α1, …, αr real, αr+1, …, αr+s complex, αr+s+j = ᾱr+j), let

Some years ago I showed ((4), § 6, pp. 88–91) that the star domain K defined by the inequalities

has the minimum determinant Δ(K) = 2 and has an infinity of singular critical lattices. In this note I show that there is a unique irreducible star domain . That ís to say, there is just one star domain H contained in but different from K for which Δ(H) = Δ(K) = 2, and such that Δ(H′) < 2 for every star domain H′ contained in but different from H.

1·1. In this paper we shall be concerned with the equations

where K is a compact (completely continuous) linear operator in a Hilbert space , K is the adjoint of K, I is the identity operator, x and y are elements of ∥ x ∥ denotes the norm of x, and κ and σ are complex numbers.

The following ‘partial’ integral equation arises in the quantum theory of fields:

where f(x1x2) = f(x2x1), K(x1x2, y) = K(x2x1, y), and a symmetric solution of the equation is desired. The solution of (1) will be obtained as a special case of the solution of the following equation:

This note contains two counter-examples which have a bearing on work published in earlier papers (1, 2, 3) of the sequence. It has been shown that the Parseval formulae for Fourier cosine and sine transforms,

hold in a number of cases when two of the functions concerned are monotonic, and that there are corresponding results for Fourier series. In some of these theorems, the existence of one side of the relevant equation (in the ordinary finite sense) is postulated; others contain no such existence condition, and show that either both sides are finite and equal, or both sides are ‘ + ∞’. In the case of sine transforms, we always obtain theorems of the second type (which is clearly the stronger); but cosine transforms are more troublesome. The first of the examples which we shall give is designed to show that the cosine formula is not always true in the ‘finite or infinite’ sense when two of the functions are monotonic.

Experience has shown that the sections of an electronic digital computer which are easiest to maintain are those which have a simple logical structure. Not only can this structure be readily borne in mind by a maintenance engineer when looking for a fault, but it makes it possible to use fault-locating programmes and to test the equipment without the use of elaborate test gear. It is in the control section of electronic computers that the greatest degree of complexity generally arises. This is particularly so if the machine has a comprehensive order code designed to make it simple and fast in operation. In general, for each different order in the code some special equipment must be provided, and the more complicated the function of the order the more complex this equipment. In the past, fear of complicating unduly the control circuits of the machines has prevented the designers of electronic machines from providing such facilities as orders for floating-point operations, although experience with relay machines and with interpretive subroutines has shown how valuable such orders are.

In fundamental papers Bernstein (3) and Loève(8) have proved central limit theorems for wide classes of dependent variables. Their theorems are stated in terms of conditional distributions. In the case of dn-dependent variables (see § 3) they assume the existence, as the ‘conditioning’ variates vary, of finite upper bounds for certain conditional absolute moments higher than the second. More recently, Hoeffding and Robbins (7) have proved central limit theorems for m-dependent variables with finite third absolute moments, and Moran(10) has given a direct generalization of the Lindeberg-Lévy theorem for stationary discrete linear scalar processes.

Let pik (s, t) (i, k = 1, 2, …; s ≤ t) be the transition probabilities of a Markov process in a system with an enumerable set of states. The states are labelled by positive integers, and pik (s, t) is the conditional probability that the system be in state k at time t, given that it was in state i at an earlier time s. If certain regularity conditions are imposed on the pik, they can be shown to satisfy the well-known Kolmogorov equations§

The theoretical problem of the probable time of recurrence of a given state of a physical system which changes according to a specified random or stochastic process is in general not a very tractable one, but many situations are covered by the more particular methods and formulae given in this paper. Several of these are believed new, but others are also included for reference. We shall usually limit our attention to homogeneous processes, i.e. the transition probabilities do not depend explicitly on the time. Even so, a generally useful solution of the complete distributional problem of recurrence times would appear rather unlikely; for example, it would have to include as a special case the recurrence theory for trigonometrical or power series, problems of some complexity which have separately engaged the attention of mathematicians (see, for example, Kac (7)). Thus a manageable simplification of the complete distributional problem emerges only in particular but important classes of physical problems classifiable theoretically as Markoff processes or, somewhat more generally, as ‘renewal’ processes (cf. Feller (4)). In other cases, if we are content with mean values (which do not always exist), and are dealing with stationary processes with complete ergodic properties, such as a statistical mechanical system in equilibrium, the simple but powerful arguments originally used by Smoluchowski (11) are available. We first formulate the probability problem in symbolic terms, being more concerned in this paper with general structural relations than with special points of rigour, or specific applications.

In the serial test for sampling numbers (3, 4) an expression is used. It is in the form of a sum of squares and has previously been supposed to have asymptotically a χ2 (γ-variate) distribution. By evaluating it is shown that this supposition cannot be correct, since is not equal to the number of degrees of freedom. The evaluation is based on some results which can be used for testing a wide class of properties of random sequences. The definition of is generalized to deal with subsequences of length ν (instead of length 2) and the generalized , or is then decomposed into a linear form of simple variates defined in terms of discrete Fourier transforms. When the number, t, of digits in the sample space is prime these simple variates are shown to have asymptotically independent χ2 distributions with one degree of freedom each. The coefficients of the linear form are not 1 but they become 1 when the first and second differences with respect to ν are taken. Thus and have asymptotically χ2 distributions. The work is closely related to that of Bartlett (1) and Vajda (6).

A simple theory of gravitation is formulated in conformal Riemannian space-time. The metric is determined by a scalar function which satisfies a linear equation. A conclusion in favour of Einstein's general tensor theory is drawn from a discussion of the corrections to the Newtonian theory for purely gravitational phenomena. Finally the theory is applied to the cosmological problem and especially to the possibility of a steady-state universe. The velocity-distance law is shown to be compatible with a constant uniform distribution of matter without the need of artificial assumptions.

The diamagnetism of the conduction electrons gives rise to some of the most difficult problems in the theory of metals, the complete solution of which has not yet been found. Formally, the problem is equivalent to determining the density matrix

and the exact expression for ψ(r′, r, γ) for perfectly free electrons in a constant magnetic field H has recently been found by Sondheimer and Wilson(2). The extension of the theory to deal with quasi-bound electrons for all values of H seems to be out of the question, but an approximate partition function was given by Peierls (1), excluding terms of higher order than H2. In The theory of metals ((3), referred to as T.M.) I gave a more powerful and simpler method of dealing with the problem, based upon the properties of ψ(r′, r, γ), but since the solution was obtained as a power series in μ0Hγ, where it could at best determine only the normal diamagnetism.

The formulae for current and force which were employed in some of the initial work on magneto-hydrodynamic theory are not valid over the whole range of phenomena which the subject itself embraces. Generalized formulae, suitable for wider application, are derived by an extension of the magneto-ionic approach, and are applied to the simple case of plane-wave propagation in the direction of a primary magnetic field. The dispersion equation which results, as well as the basic current and force formulae, are reduced by approximations in a variety of limiting circumstances. Some earlier results are thereby regained and, in particular, the region of known validity of Alfvén's propagation relation is extended.

This work is consequent on a forthcoming joint paper by A. C. Stevenson and the present writer on the use of the torsion function in the solution of flexure problems. The treatment is general, but special attention is paid to doubly-connected cross-sections which can be mapped conformally on a circular annulus. The eccentric circles cross-section (4) is solved in order to illustrate the power of the method, and results are obtained for the cross-section bounded by confocal limacons. This is of interest, since it has been customary to regard these problems as largely determined by the cross-section of the cylinder; but these results show that the terms involving the elastic constants are now no longer negligible. The method is capable of giving useful information to aeronautical engineers, and the writer hopes to present in a later communication a treatment of an aerofoil-like profile.

In this paper a simple expression in finite terms is found for the small transverse displacement of a thin plane elastic plate due to a transverse force applied at an arbitrary point of the plate. The plate is clamped along the semi-infinite straight lines represented by AB, CD in Fig. 1, these lines being the only boundaries of the plate. The transverse displacement w at any point (x, y) of the plate is a biharmonic function of the variables (x, y) which vanishes together with its normal derivative at all points of the boundary. Clearly w is also a function of the coordinates (x0, y0) of the point of application of the force, and it is known ((5), p. 173) that it is a symmetrical function of the coordinate pairs (z, y) and (x0, y0); it is the Green's function associated with the differential equation and the boundary conditions.

The problem considered here is to determine the couple required to maintain the steady rotation about its axis of symmetry of an oblate spheroid which is placed in a uniform stream of viscous fluid moving parallel to this axis. The conditions of flow are assumed to be such as to permit the use of Oseen's approximation in the equations of motion. An expression in terms of spheroidal wave functions is derived for the transverse component of velocity in the fluid, and from this the expression for the couple is deduced. The disk rotating about the normal through its centre is considered as a special case, and some numerical results are given for certain values of a Reynolds number. A comparison is made, and agreement found, with results obtained by Jeffery in a previous paper which treated the case in which the mainstream was at rest.

The steady motion of a viscous fluid confined between two coaxial rotating disks is discussed both experimentally and theoretically. It is found experimentally that when the disks rotate in the same sense the main body of the fluid rotates as well, but if they rotate in opposite senses the main body of the fluid is almost at rest. An adequate theory is found to explain this.

A study is made of the actual trajectories of fluid particles in certain motions of classical hydrodynamics. When a solid body moves through an incompressible fluid, it induces a drift in the fluid, such that the final positions of the particles are further on than those from which they started. The drift-volume enclosed between the initial and final positions is equal to the volume corresponding to hydrodynamic mass, that is, the mass of fluid to be added to that of the solid in calculating its kinetic energy. This result is proved quite generally. The work involves integrals which are not absolutely convergent, and these are discussed in relation to the general mechanics of fluids. When the trajectories are considered of the fluid surrounding a rotating body, it is shown that the fluid particles slowly drift round the body, even though the motion is irrotational and without circulation. There seems to be in some respects a closer resemblance between the behaviour of the idealized hydrodynamic fluid and a real fluid than might be expected from the well-known discrepancies between them.